| {"id": "1.1/i", "chapter": "1", "split": "chapter_1", "problem_number": "1.1", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "In a monopolized industry, the demand function has a constant elasticity: $q = D(p) = p^{-\\epsilon}$ where $\\epsilon > 1$ is the elasticity of demand. Marginal cost is constant and equal to $c$. (i) Under a social planner (or a competitive industry) outcome with price equal to marginal cost, what is the resulting total welfare $W^c$ as a function of $c$ and $\\epsilon$? Symbols (for final answer): - `c`: constant marginal cost - `epsilon`: elasticity of demand (epsilon > 1)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "c**(1-epsilon)/(epsilon-1)", "reference_answer_sympy": "c**(1-epsilon)/(epsilon-1)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "c**(1-epsilon)/(epsilon-1)", "reference_reasoning": "In the social planner/competitive outcome with constant marginal cost c, efficiency implies p=c. Then q^c = D(c) = c^{-epsilon}. Inverse demand is p(q)=q^(-1/epsilon). Total welfare equals total surplus: W^c = \\int_0^{q^c} p(q) dq - c q^c. Compute \\int_0^{q^c} q^(-1/epsilon) dq = [epsilon/(epsilon-1) * q^((epsilon-1)/epsilon)]_0^{q^c} = epsilon/(epsilon-1) * (q^c)^((epsilon-1)/epsilon) = epsilon/(epsilon-1) * c^{-(epsilon-1)}. Subtract cost c q^c = c * c^{-epsilon} = c^(1-epsilon). Thus W^c = (epsilon/(epsilon-1) - 1) c^(1-epsilon) = c^(1-epsilon)/(epsilon-1).", "allowed_symbols": ["c", "epsilon"], "symbol_definitions": {"c": "constant marginal cost", "epsilon": "elasticity of demand (epsilon > 1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.1/ii", "chapter": "1", "split": "chapter_1", "problem_number": "1.1", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "In a monopolized industry, the demand function has a constant elasticity: $q = D(p) = p^{-\\epsilon}$ where $\\epsilon > 1$ is the elasticity of demand. Marginal cost is constant and equal to $c$. (ii) Compute the welfare loss, $WL$, under monopoly, defined as $WL \\equiv W^c - W^m$. Symbols (for final answer): - `c`: constant marginal cost - `epsilon`: elasticity of demand (ε), with epsilon > 1", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "c**(1-epsilon)/(epsilon-1)*(1-((2*epsilon-1)/(epsilon-1))*(epsilon/(epsilon-1))**(-epsilon))", "reference_answer_sympy": "c**(1-epsilon)/(epsilon-1)*(1-((2*epsilon-1)/(epsilon-1))*(epsilon/(epsilon-1))**(-epsilon))", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "c**(1-epsilon)/(epsilon-1)*(1-((2*epsilon-1)/(epsilon-1))*(epsilon/(epsilon-1))**(-epsilon))", "reference_reasoning": "Under perfect competition, p^c=c and q^c=c^{-\\varepsilon}. Under monopoly, p^m=\\frac{\\varepsilon}{\\varepsilon-1}c and q^m=(p^m)^{-\\varepsilon}. Total surplus at price p is W(p)=\\int_p^{\\infty} x^{-\\varepsilon} dx + (p-c)p^{-\\varepsilon} = \\frac{p^{1-\\varepsilon}}{\\varepsilon-1} + (p-c)p^{-\\varepsilon}. Hence W^c=W(c)=\\frac{c^{1-\\varepsilon}}{\\varepsilon-1}. Also W^m=W(p^m)=\\frac{(p^m)^{1-\\varepsilon}}{\\varepsilon-1}+(p^m-c)(p^m)^{-\\varepsilon}. Substituting p^m=\\frac{\\varepsilon}{\\varepsilon-1}c yields W^m=c^{1-\\varepsilon}\\left(\\frac{\\varepsilon}{\\varepsilon-1}\\right)^{-\\varepsilon}\\left(\\frac{2\\varepsilon-1}{\\varepsilon-1}\\right). Therefore WL=W^c-W^m=\\frac{c^{1-\\varepsilon}}{\\varepsilon-1}\\left[1-\\left(\\frac{2\\varepsilon-1}{\\varepsilon-1}\\right)\\left(\\frac{\\varepsilon}{\\varepsilon-1}\\right)^{-\\varepsilon}\\right].", "allowed_symbols": ["c", "epsilon"], "symbol_definitions": {"c": "constant marginal cost", "epsilon": "elasticity of demand (ε), with epsilon > 1"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.1/iii#1", "chapter": "1", "split": "chapter_1", "problem_number": "1.1", "sub_id": "iii", "pair_index": 1, "pair_count": 3, "question": "In a monopolized industry, the demand function has a constant elasticity: $q = D(p) = p^{-\\epsilon}$ where $\\epsilon > 1$ is the elasticity of demand. Marginal cost is constant and equal to $c$. (iii-a) Compute the relative dead-weight loss ratio $WL/W^c$ as a function of $\\epsilon$ (it should not depend on $c$). Symbols (for final answer): - `epsilon`: elasticity of demand parameter (epsilon>1)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1 - ((2*epsilon - 1)/(epsilon - 1))*((epsilon - 1)/epsilon)**epsilon", "reference_answer_sympy": "1 - ((2*epsilon - 1)/(epsilon - 1))*((epsilon - 1)/epsilon)**epsilon", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1 - ((2*epsilon - 1)/(epsilon - 1))*((epsilon - 1)/epsilon)**epsilon", "reference_reasoning": "Inverse demand is p(q)=q^(-1/epsilon). Competitive outcome: p=c so q_c=c^(-epsilon) and W_c=∫_0^{q_c}(p(q)-c)dq = c^(1-epsilon)/(epsilon-1). Monopoly: maximize π(q)=q^(1-1/epsilon)-cq. FOC gives (1-1/epsilon)q^(-1/epsilon)=c, so p_m=epsilon/(epsilon-1)*c and q_m=c^(-epsilon)*((epsilon-1)/epsilon)^epsilon. Monopoly total surplus W_m=∫_0^{q_m}(p(q)-c)dq = epsilon/(epsilon-1)*(q_m)^(1-1/epsilon)-c q_m. Using (q_m)^(-1/epsilon)=p_m=epsilon/(epsilon-1)c implies (q_m)^(1-1/epsilon)=q_m*epsilon/(epsilon-1)c, hence W_m=c q_m*(epsilon^2/(epsilon-1)^2 - 1)=c q_m*(2epsilon-1)/(epsilon-1)^2. Substitute q_m to get W_m=c^(1-epsilon)*((epsilon-1)/epsilon)^epsilon*(2epsilon-1)/(epsilon-1)^2. Therefore W_m/W_c = ((2epsilon-1)/(epsilon-1))*((epsilon-1)/epsilon)^epsilon, and WL/W_c = 1 - W_m/W_c.", "allowed_symbols": ["epsilon"], "symbol_definitions": {"epsilon": "elasticity of demand parameter (epsilon>1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.1/iii#2", "chapter": "1", "split": "chapter_1", "problem_number": "1.1", "sub_id": "iii", "pair_index": 2, "pair_count": 3, "question": "In a monopolized industry, the demand function has a constant elasticity: $q = D(p) = p^{ -\\varepsilon}$ where $\\varepsilon > 1$ is the elasticity of demand. Marginal cost is constant and equal to $c$. (iii-b) True or False: The deadweight loss $WL(\\varepsilon)$ is monotone in $\\varepsilon$. (You may use that $WL(\\varepsilon)=\\dfrac{c^{1-\\varepsilon}}{\\varepsilon-1}\\big(1-K(\\varepsilon)\\big)$ where $K(\\varepsilon)=\\dfrac{2\\varepsilon-1}{\\varepsilon-1}\\big(\\dfrac{\\varepsilon-1}{\\varepsilon}\\big)^{\\varepsilon}\\in(0,1)$.)", "question_type": "judge", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "Using the provided expression, as \\varepsilon \\downarrow 1 we have W^c= c^{1-\\varepsilon}/(\\varepsilon-1) \\to +\\infty while K(\\varepsilon)\\to 1, and a first-order expansion implies WL(\\varepsilon)\\to +\\infty. As \\varepsilon\\to\\infty, W^c\\to 0 (market size shrinks with \\varepsilon for typical parameterizations), and since 0\\le WL\\le W^c, it follows WL(\\varepsilon)\\to 0. A positive continuous function that diverges to +\\infty near \\varepsilon=1 but tends to 0 as \\varepsilon\\to\\infty cannot be monotone. Therefore the statement that WL is monotone is false.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.1/iii#3", "chapter": "1", "split": "chapter_1", "problem_number": "1.1", "sub_id": "iii", "pair_index": 3, "pair_count": 3, "question": "In a monopolized industry, the demand function has a constant elasticity: $q = D(p) = p^{-\\epsilon}$ where $\\epsilon > 1$ is the elasticity of demand. Marginal cost is constant and equal to $c$. (iii-c) Compute the fraction $\\Pi^m/W^c$ of potential consumer surplus that can be captured by the monopolist. Symbols (for final answer): - `epsilon`: elasticity of demand (epsilon > 1)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "((epsilon-1)/epsilon)**epsilon", "reference_answer_sympy": "((epsilon-1)/epsilon)**epsilon", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "((epsilon-1)/epsilon)**epsilon", "reference_reasoning": "Competitive surplus is W^c = c^(1-epsilon)/(epsilon-1). Monopoly profit is Pi^m = (p^m-c)q^m = (c/(epsilon-1)) q^m = c^(1-epsilon)/(epsilon-1) * ((epsilon-1)/epsilon)^epsilon. Dividing gives Pi^m/W^c = ((epsilon-1)/epsilon)^epsilon = (1-1/epsilon)^epsilon.", "allowed_symbols": ["epsilon"], "symbol_definitions": {"epsilon": "elasticity of demand (epsilon > 1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.3/i#1", "chapter": "1", "split": "chapter_1", "problem_number": "1.3", "sub_id": "i", "pair_index": 1, "pair_count": 3, "question": "A monopolist's marginal cost of supplying a good to consumers is $\\bar{c} = c+ t$ (where $t$ is a unit commodity tax). Let $p_m(\\bar{c})$ denote the corresponding monopoly price. (i) For the demand function $p = q^{-1/\\epsilon}$, compute $\\dfrac{dp^m}{d\\bar{c}}$. Symbols (for final answer): - `epsilon`: demand elasticity parameter ε (assume ε>1)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "epsilon/(epsilon-1)", "reference_answer_sympy": "epsilon/(epsilon-1)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "epsilon/(epsilon-1)", "reference_reasoning": "The inverse demand function is \\[ p(q) = q^{-1/\\epsilon}, \\quad \\epsilon > 1. \\] The monopolist's profit is \\[ \\pi(q) = p(q)q - \\bar{c}q = q^{1 - \\frac{1}{\\epsilon}} - \\bar{c}q. \\] The first-order condition is \\[ \\frac{d\\pi}{dq} = \\left(1 - \\frac{1}{\\epsilon}\\right) q^{-1/\\epsilon} - \\bar{c} = 0. \\] Thus, \\[ \\left(1 - \\frac{1}{\\epsilon}\\right) q^{-1/\\epsilon} = \\bar{c}. \\] Since \\(p = q^{-1/\\epsilon}\\), the monopoly price is \\[ p^m = \\frac{\\bar{c}}{1 - \\frac{1}{\\epsilon}} = \\frac{\\epsilon}{\\epsilon - 1}\\bar{c}. \\] Differentiating with respect to \\(\\bar{c}\\), we obtain \\[ \\frac{dp^m}{d\\bar{c}} = \\frac{\\epsilon}{\\epsilon - 1}. \\] \\[ \\boxed{\\frac{dp^m}{d\\bar{c}} = \\frac{\\epsilon}{\\epsilon - 1}}. \\]", "allowed_symbols": ["epsilon"], "symbol_definitions": {"epsilon": "demand elasticity parameter ε (assume ε>1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.3/i#2", "chapter": "1", "split": "chapter_1", "problem_number": "1.3", "sub_id": "i", "pair_index": 2, "pair_count": 3, "question": "A monopolist's marginal cost of supplying a good to consumers is $\\bar{c} = c+ t$ (where $t$ is a unit commodity tax). Let $p_m(\\bar{c})$ denote the corresponding monopoly price. (i) Compute $\\frac{dp_m}{d\\bar{c}}$ for the following demand function: $p = \\alpha - \\beta q^{\\delta}$. Symbols (for final answer): - `delta`: demand curvature parameter in p = alpha - beta*q^delta", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/(delta+1)", "reference_answer_sympy": "1/(delta+1)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/(delta+1)", "reference_reasoning": "For constant marginal cost \\(\\bar c\\), the monopoly FOC is \\(p(q)+q p'(q)=\\bar c\\). Differentiate implicitly w.r.t. \\(\\bar c\\): \\((2p'(q)+q p''(q))\\,dq^m/d\\bar c=1\\). Then \\(dp^m/d\\bar c=p'(q^m)\\,dq^m/d\\bar c=\\frac{p'(q^m)}{2p'(q^m)+q^m p''(q^m)}\\). For \\(p=\\alpha-\\beta q^\\delta\\), \\(p'(q)=-\\beta\\delta q^{\\delta-1}\\) and \\(p''(q)=-\\beta\\delta(\\delta-1)q^{\\delta-2}\\). Substituting and simplifying yields \\(dp^m/d\\bar c=1/(\\delta+1)\\).", "allowed_symbols": ["delta"], "symbol_definitions": {"delta": "demand curvature parameter in p = alpha - beta*q^delta"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.3/i#3", "chapter": "1", "split": "chapter_1", "problem_number": "1.3", "sub_id": "i", "pair_index": 3, "pair_count": 3, "question": "A monopolist's marginal cost of supplying a good to consumers is $\\bar{c} = c+ t$ (where $t$ is a unit commodity tax). Let $p_m(\\bar{c})$ denote the corresponding monopoly price. (i) Compute $dp^m/d\\bar{c}$ for the following demand function: $p = a - b\\ln q$. Symbols (for final answer): - dp_m_d_cbar: the derivative $\\frac{dp^m}{d\\bar c}$", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1", "reference_answer_sympy": "1", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1", "reference_reasoning": "For a monopolist with constant marginal cost cbar, the FOC is p(q)+q p'(q)=cbar. Differentiate implicitly w.r.t. cbar: (2p'(q)+q p''(q)) dq_m/d_cbar = 1, so dp_m/d_cbar = p'(q_m) dq_m/d_cbar = p'(q_m)/(2p'(q_m)+q_m p''(q_m)). For p=a-b ln q: p'(q)=-b/q and p''(q)=b/q^2. Then 2p'(q)+q p''(q)=2(-b/q)+q*(b/q^2)=(-2b/q)+(b/q)=(-b/q). Hence dp_m/d_cbar = (-b/q)/(-b/q)=1.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.3/ii", "chapter": "1", "split": "chapter_1", "problem_number": "1.3", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "A monopolist's marginal cost of supplying a good to consumers is $\\bar{c} = c+ t$ (where $t$ is a unit commodity tax). Let $p_m(\\bar{c})$ denote the corresponding monopoly price. (ii) Sumner (1981) uses an ingenious approach to estimate the elasticity of demand—and thus the degree of monopoly power—in the American cigarette industry. He notes that in the United States, commodity taxes—and therefore the generalized cost $\\bar{c}$—vary across states. Although data on $c$ are hard to obtain, data on $t$ are readily available. Sumner uses varying levels of taxation across states to estimate the elasticity of demand. Bulow and Pfleiderer (1983) argue that the method has limited applicability. True or False: Cross-state variation in the unit tax $t$ (hence in $\\bar c$) is sufficient, without additional functional-form restrictions on demand, to uniquely identify the demand elasticity $\\varepsilon$ from the observed relationship between monopoly price $p_m$ and $t$.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "Under monopoly pricing, the FOC implies a markup condition linking price, marginal cost, and the elasticity at the chosen price. Sumner’s approach uses tax-induced cost variation to infer elasticity from how price responds to cost. However, Bulow–Pfleiderer show that with general (non-constant-elasticity) demand, the pass-through of cost into price depends not only on the level of elasticity but also on how elasticity changes with price (demand curvature). Therefore the mapping from observed price changes to a unique elasticity is not identified without strong functional-form restrictions (e.g., constant elasticity). Additional concerns include potential correlation of taxes with other state-level demand shifters. Hence tax variation alone is not sufficient for unique identification in general.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.5/main", "chapter": "1", "split": "chapter_1", "problem_number": "1.5", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider a multiproduct firm with monopoly power over two goods, indexed by $i=1,2$. The firm produces goods at cost $C(q_1,q_2)$ and faces demand functions $q_i = D_i(p_1,p_2)$. Assume that the two goods are perfect complements, so that demand depends only on the total price: \\[ D_1(p_1,p_2) = D_2(p_1,p_2) = D(P), \\quad \\text{where } P = p_1 + p_2. \\] Starting from the multiproduct monopoly first-order condition \\[ D_i + p_i \\frac{\\partial D_i}{\\partial p_i} + \\sum_{j \\neq i} p_j \\frac{\\partial D_j}{\\partial p_i} = \\sum_j \\frac{\\partial C}{\\partial q_j} \\frac{\\partial D_j}{\\partial p_i}, \\quad \\text{for } i=1,2, \\] Compute that the pricing problem which can be expressed in terms of the total price $P$, and that the firm's problem reduces to that of a single-product monopolist. In particular, express the resulting condition using $D_P = D(P)$, $Dprime_P = D'(P)$, and the system marginal cost $MC_s$. Symbols (for final answer): - `D_P`: demand level D(P) - `Dprime_P`: derivative D'(P) - `MC_s`: system marginal cost MC_s(D(P)) = ∂C/∂q1(D(P),D(P)) + ∂C/∂q2(D(P),D(P)) - `P`: total price P = p1 + p2", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "D_P + (P - MC_s)*Dprime_P", "reference_answer_sympy": "D_P + (P - MC_s)*Dprime_P", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "D_P + (P - MC_s)*Dprime_P", "reference_reasoning": "With perfect complements, q1=q2=D(P) where P=p1+p2. Also, for either i, \\partial D_1/\\partial p_i=\\partial D_2/\\partial p_i=D'(P). Substituting into the multiproduct FOC yields D(P)+p1 D'(P)+p2 D'(P)=(\\partial C/\\partial q1+\\partial C/\\partial q2)D'(P). Using P=p1+p2 and defining MC_s(D(P))=(\\partial C/\\partial q1+\\partial C/\\partial q2) evaluated at (D(P),D(P)), rearrange to D(P)+(P-MC_s(D(P)))D'(P)=0, which is the standard single-product monopoly FOC for total price P and demand D(P).", "allowed_symbols": ["D_P", "Dprime_P", "MC_s", "P"], "symbol_definitions": {"D_P": "demand level D(P)", "Dprime_P": "derivative D'(P)", "MC_s": "system marginal cost MC_s(D(P)) = ∂C/∂q1(D(P),D(P)) + ∂C/∂q2(D(P),D(P))", "P": "total price P = p1 + p2"}, "review_status": "confirmed_by_expert"} | |
| {"id": "1.6/i", "chapter": "1", "split": "chapter_1", "problem_number": "1.6", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "A power plant (or a hotel, or an airline) faces two types of demand: off-peak $(q_1 = D_1(p_1))$ and peak $(q_2 = D_2(p_2))$, where $D_1(p) = \\lambda D_2(p)$ with $\\lambda < 1$. (For simplicity, the demands are independent.) The marginal cost of production is $c$ (as long as capacity is not satiated). The marginal cost of investing one unit of capacity is $\\gamma$. The same capacity serves peak and off-peak demands. (i) True or False: If off-peak demand is small relative to peak demand in the sense that at the optimum the off-peak unconstrained monopoly quantity does not exceed capacity (so off-peak capacity does not bind while peak capacity binds), then the monopolist sets off-peak output by $MR_1=c$ and chooses capacity so that the peak price satisfies $p_2=c+\\gamma$ (equivalently, peak marginal revenue equals $c+\\gamma$ under the binding capacity constraint).", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "With capacity K, profit is p1 q1 + p2 q2 - c(q1+q2) - gamma K subject to q1<=D1(p1), q2<=D2(p2), and q1<=K, q2<=K. If off-peak demand is small so that capacity does not bind off-peak but binds in peak (D1(p1)<=K < D2(p2)), then off-peak is unconstrained and the standard monopoly condition applies: MR1=c. In peak, output is constrained at q2=K, so increasing K by dK increases peak sales by dK while not affecting off-peak sales. The marginal benefit of capacity is (p2-c)*dq2/dK = (p2-c). Optimality requires p2-c=gamma, hence p2=c+gamma; under the binding capacity constraint the firm sets price so demand equals K, so this corresponds to peak marginal revenue equaling c+gamma. Therefore the statement is true under the stated 'small off-peak demand' condition.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.6/ii", "chapter": "1", "split": "chapter_1", "problem_number": "1.6", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "A power plant (or a hotel, or an airline) faces two types of demand: off-peak $(q_1 = D_1(p_1))$ and peak $(q_2 = D_2(p_2))$, where $D_1(p) = \\lambda D_2(p)$ with $\\lambda < 1$. (For simplicity, the demands are independent.) The marginal cost of production is $c$ (as long as capacity is not satiated). The marginal cost of investing one unit of capacity is $\\gamma$. The same capacity serves peak and off-peak demands. (ii) Consider the case where off-peak demand is sufficiently large so that capacity is binding in both periods at the optimum. Assume constant-elasticity demand with elasticity \\(\\epsilon > 1\\). Compute the optimal peak price \\(p_2\\) in the interior solution. Symbols (for final answer): - `c`: marginal cost of production - `epsilon`: constant elasticity parameter (epsilon>1) - `gamma`: marginal cost of investing one unit of capacity - `lambda`: relative off-peak demand scale (lambda<1)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(2*c+gamma)/((1-1/epsilon)*(1+lambda**(1/epsilon)))", "reference_answer_sympy": "(2*c+gamma)/((1-1/epsilon)*(1+lambda**(1/epsilon)))", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(2*c+gamma)/((1-1/epsilon)*(1+lambda**(1/epsilon)))", "reference_reasoning": "When off-peak demand is not small, capacity binds in both periods, so K=q1=q2 and D1(p1)=D2(p2)=K. With constant-elasticity demands D2(p)=A p^{-epsilon} and D1(p)=lambda A p^{-epsilon}, inverse demands satisfy p1(K)=lambda^(1/epsilon) p2(K). Profit is pi(K)=(p1(K)-c)K+(p2(K)-c)K-gamma K. The FOC for an interior optimum is MR1(K)+MR2(K)=2c+gamma. For isoelastic demand, MR_i(K)=p_i(K)*(1-1/epsilon). Hence (1-1/epsilon)(p1+p2)=2c+gamma. Using p1=lambda^(1/epsilon)p2 gives (1-1/epsilon)(1+lambda^(1/epsilon))p2=2c+gamma, so p2=(2c+gamma)/((1-1/epsilon)(1+lambda^(1/epsilon))).", "allowed_symbols": ["c", "epsilon", "gamma", "lambda"], "symbol_definitions": {"c": "marginal cost of production", "epsilon": "constant elasticity parameter (epsilon>1)", "gamma": "marginal cost of investing one unit of capacity", "lambda": "relative off-peak demand scale (lambda<1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.7/i", "chapter": "1", "split": "chapter_1", "problem_number": "1.7", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "The monopoly producer of a single good has a constant unit cost $c(w(t))$ at time $t$, where $w(t)$ is the firm's \"experience\" at that date. (Assume $c > 0$, $c' < 0$, and $\\lim_{t \\to \\infty} c(t) > 0$.) Time is continuous and runs from zero to infinity. Experience accumulates with production: $dw(t)/dt = q(t)$, where $q(t)$ is production at time $t$. (Those who have done the empirical work have assumed, as we do, that production exhibits constant instantaneous returns to scale and that the appropriate measure of experience is cumulative output.) Let $R(q)$ denote the revenue function as a function of quantity (supposing demand is invariant). Assume $R' > 0$ and $R'' < 0$. Let $r$ denote the interest rate. The monopolist's objective function is \\[ \\int_0^{\\infty} [R(q(t)) - c(w(t))q(t)]e^{-rt}dt. \\] (i) Show that at each instant the monopolist sets marginal revenue equal to the average (discounted) unit cost in the future: \\[ A(t) = \\int_t^{\\infty} c\\_w\\_s\\, r e^{-r(s-t)}\\,ds. \\] Hint: Consider the current cost and the future savings from changing $q(t)$ slightly. Symbols (for final answer): - `c_w_s`: unit cost evaluated at experience w(s), treated as a symbol for integration purposes - `r`: interest rate - `s`: future time (integration variable) - `t`: current time", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "r*Integral(c_w_s*exp(-r*(s-t)), (s, t, oo))", "reference_answer_sympy": "r*Integral(c_w_s*exp(-r*(s-t)), (s, t, oo))", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "r*Integral(c_w_s*exp(-r*(s-t)), (s, t, oo))", "reference_reasoning": "Define A(t) as the discounted average of future unit costs. A marginal increase in q(t) raises current profit by R'(q(t))dq - c(w(t))dq and increases experience thereafter, lowering future unit costs. The first-order condition can be written as R'(q(t)) = A(t), where A(t) equals the discounted average future unit cost. Using q(s)=w_dot(s) and integration by parts yields A(t)=r\\int_t^\\infty c(w(s))e^{-r(s-t)}ds.", "allowed_symbols": ["c_w_s", "r", "s", "t"], "symbol_definitions": {"c_w_s": "unit cost evaluated at experience w(s), treated as a symbol for integration purposes", "r": "interest rate", "s": "future time (integration variable)", "t": "current time"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.7/ii", "chapter": "1", "split": "chapter_1", "problem_number": "1.7", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "The monopoly producer of a single good has a constant unit cost $c(w(t))$ at time $t$, where $w(t)$ is the firm's \"experience\" at that date. (Assume $c > 0$, $c' < 0$, and $\\lim_{t \\to \\infty} c(t) > 0$.) Time is continuous and runs from zero to infinity. Experience accumulates with production: $dw(t)/dt = q(t)$, where $q(t)$ is production at time $t$. (Those who have done the empirical work have assumed, as we do, that production exhibits constant instantaneous returns to scale and that the appropriate measure of experience is cumulative output.) Let $R(q)$ denote the revenue function as a function of quantity (supposing demand is invariant). Assume $R' > 0$ and $R'' < 0$. Let $r$ denote the interest rate. The monopolist's objective function is \\[ \\int_0^{\\infty} [R(q(t)) - c(w(t))q(t)]e^{-rt}dt. \\] (ii) Judge whether the following statement is true or false: Along an interior optimal path with $r>0$, output is weakly decreasing over time (i.e., $\\dot q(t)\\le 0$ for all $t$).", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "Set up the current-value Hamiltonian H = R(q) - c(w)q + λ q with state equation \\dot w = q. The interior FOC is R'(q) - c(w) + λ = 0, i.e. R'(q)=c(w)-λ. The current-value costate equation is \\dot λ = rλ - ∂H/∂w = rλ + c'(w)q. Since c'(w)<0 and q≥0, we have \\dot λ - rλ = c'(w)q ≤ 0. With the transversality condition, this implies λ(t)≥0. Differentiate the FOC over time: R''(q)\\dot q = d/dt(c(w)-λ)=c'(w)\\dot w-\\dot λ=c'(w)q-\\dot λ. Substitute \\dot λ = rλ + c'(w)q to get R''(q)\\dot q = -rλ. Because R''(q)<0, r>0, and λ≥0, it follows that \\dot q ≥ 0 (strict if λ>0). Therefore output is weakly increasing over time.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.8/ii", "chapter": "1", "split": "chapter_1", "problem_number": "1.8", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "Both the monopolist and the consumers are infinite-lived. The unit production cost is 0. The consumers' valuations, $v$, are uniformly distributed on $[0, 1/(1 - \\delta)]$ (which amounts to saying that the per-period valuation is uniformly distributed on $[0, 1]$). A consumer with valuation $v$ has utility $\\delta^t(v - p_t)$ if he buys at date $t$ at price $p_t$, where $\\delta$ is the discount factor. The monopolist's intertemporal profit is \\[ \\sum_{t=1}^{\\infty} \\delta^t p_t q_t, \\] where $q_t$ is the quantity sold (the number of consumers who buy) at time $t$. Look for a linear and stationary equilibrium: When facing at some date a price $p$, any consumers with valuations exceeding $w(p) = \\lambda p$ buy and consumers with lower valuations do not buy, where $\\lambda > 1$. Conversely, if at some date consumers with valuations exceeding $v$ have bought and the others have not, the monopolist charges $p(v) = \\mu v$, where $\\mu < 1$. (ii) Using the relationship between $\\lambda$ and $\\mu$ implied by the monopolist's optimization problem, compute the value of \\[ 1 - 2\\lambda \\mu + \\delta \\lambda^2 \\mu^2. \\] Symbols (for final answer): - `delta`: discount factor - `lambda`: consumer cutoff multiplier in w(p)=lambda*p - `mu`: monopolist pricing multiplier in p(v)=mu*v", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "From the monopolist's FOC with respect to p_1: v - 2*lambda*p_1 + delta*lambda*p_2 = 0. Using the linear pricing rules p_1 = mu*v and p_2 = mu*(lambda*p_1) = lambda*mu^2*v, divide by v (v>0) to obtain 1 - 2*lambda*mu + delta*lambda^2*mu^2 = 0.", "allowed_symbols": ["delta", "lambda", "mu"], "symbol_definitions": {"delta": "discount factor", "lambda": "consumer cutoff multiplier in w(p)=lambda*p", "mu": "monopolist pricing multiplier in p(v)=mu*v"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.8/iii", "chapter": "1", "split": "chapter_1", "problem_number": "1.8", "sub_id": "iii", "pair_index": 1, "pair_count": 1, "question": "Both the monopolist and the consumers are infinite-lived. The unit production cost is 0. The consumers' valuations, $v$, are uniformly distributed on $[0, 1/(1 - \\delta)]$ (which amounts to saying that the per-period valuation is uniformly distributed on $[0, 1]$). A consumer with valuation $v$ has utility $\\delta^t(v - p_t)$ if he buys at date $t$ at price $p_t$, where $\\delta$ is the discount factor. The monopolist's intertemporal profit is \\[ \\sum_{t=1}^{\\infty} \\delta^t p_t q_t, \\] where $q_t$ is the quantity sold (the number of consumers who buy) at time $t$. Look for a linear and stationary equilibrium: When facing at some date a price $p$, any consumers with valuations exceeding $w(p) = \\lambda p$ buy and consumers with lower valuations do not buy, where $\\lambda > 1$. Conversely, if at some date consumers with valuations exceeding $v$ have bought and the others have not, the monopolist charges $p(v) = \\mu v$, where $\\mu < 1$. (iii) Compute $\\lambda-1-\\delta\\lambda(1-\\mu)$ by using consumer $w(p)$'s indifference equation. Symbols (for final answer): - `delta`: discount factor - `lambda`: linear cutoff parameter in w(p)=lambda*p - `mu`: linear pricing parameter in p(v)=mu*v", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "Consumer with valuation w(p)=lambda*p is indifferent between buying now at price p and waiting one period, in which case the monopolist charges p(w)=mu*w=mu*lambda*p. Indifference: lambda*p - p = delta*(lambda*p - mu*lambda*p). Dividing by p>0 gives lambda-1 = delta*lambda*(1-mu). Therefore lambda-1-delta*lambda*(1-mu)=0.", "allowed_symbols": ["delta", "lambda", "mu"], "symbol_definitions": {"delta": "discount factor", "lambda": "linear cutoff parameter in w(p)=lambda*p", "mu": "linear pricing parameter in p(v)=mu*v"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.8/iv", "chapter": "1", "split": "chapter_1", "problem_number": "1.8", "sub_id": "iv", "pair_index": 1, "pair_count": 1, "question": "Both the monopolist and the consumers are infinite-lived. The unit production cost is 0. The consumers' valuations, $v$, are uniformly distributed on $[0, 1/(1 - \\delta)]$ (which amounts to saying that the per-period valuation is uniformly distributed on $[0, 1]$). A consumer with valuation $v$ has utility $\\delta^t(v - p_t)$ if he buys at date $t$ at price $p_t$, where $\\delta$ is the discount factor. The monopolist's intertemporal profit is \\[ \\sum_{t=1}^{\\infty} \\delta^t p_t q_t, \\] where $q_t$ is the quantity sold (the number of consumers who buy) at time $t$. Look for a linear and stationary equilibrium: When facing at some date a price $p$, any consumers with valuations exceeding $w(p) = \\lambda p$ buy and consumers with lower valuations do not buy, where $\\lambda > 1$. Conversely, if at some date consumers with valuations exceeding $v$ have bought and the others have not, the monopolist charges $p(v) = \\mu v$, where $\\mu < 1$. (iv) Compute that when $\\delta$ tends to $1$, the monopolist's profit tends to what? Symbols (for final answer): - `delta`: discount factor, with limit taken as delta -> 1 from below", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "(iv) Solving for $\\lambda$ and $\\mu$ gives \\[ \\lambda \\mu = (1 - \\sqrt{1 - \\delta})/\\delta, \\] \\[ \\lambda = (\\sqrt{1 - \\delta})^{-1}, \\] \\[ \\mu = (\\sqrt{1 - \\delta} - (1 - \\delta))/\\delta. \\] Notice that $\\lim_{\\delta \\to 1} \\mu = 0.$", "allowed_symbols": ["delta"], "symbol_definitions": {"delta": "discount factor, with limit taken as delta -> 1 from below"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.9/i", "chapter": "1", "split": "chapter_1", "problem_number": "1.9", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Both the monopolist and the consumers are infinite-lived. The unit production cost is 0. The consumers' valuations, $v$, are uniformly distributed on $[0, 1/(1 - \\delta)]$ (which amounts to saying that the per-period valuation is uniformly distributed on $[0, 1]$). A consumer with valuation $v$ has utility $\\delta^t(v - p_t)$ if he buys at date $t$ at price $p_t$, where $\\delta$ is the discount factor. The monopolist's intertemporal profit is \\[ \\sum_{t=1}^{\\infty} \\delta^t p_t q_t, \\] where $q_t$ is the quantity sold (the number of consumers who buy) at time $t$. Look for a linear and stationary equilibrium: When facing at some date a price $p$, any consumers with valuations exceeding $w(p) = \\lambda p$ buy and consumers with lower valuations do not buy, where $\\lambda > 1$. Conversely, if at some date consumers with valuations exceeding $v$ have bought and the others have not, the monopolist charges $p(v) = \\mu v$, where $\\mu < 1$. In contrast with the previous conditions, assume that the monopolist commits himself to a sequence of prices \\[ (p_1, p_2, p_3, \\ldots). \\] (i) True/False: In an optimal committed price policy, the monopolist must sometimes use a strictly increasing price sequence, so he cannot restrict attention to non-increasing price sequences $p_1 \\ge p_2 \\ge p_3 \\ge \\cdots$.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "If a committed price sequence has an increase at some date t relative to an earlier date s<t (i.e., p_t > p_s), then any consumer who would buy at t at price p_t would strictly prefer to buy earlier at s at the lower price p_s (same good, earlier consumption, and discounting). Hence demand at t is zero whenever p_t exceeds the minimum of earlier prices. Therefore replacing each p_t by min{p_1,...,p_t} weakly increases discounted profit: it does not reduce earlier sales and can only weakly increase sales at t while not raising price. Iterating yields a non-increasing sequence without lowering profit, so an optimal policy can be taken non-increasing.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.9/iii", "chapter": "1", "split": "chapter_1", "problem_number": "1.9", "sub_id": "iii", "pair_index": 1, "pair_count": 1, "question": "Both the monopolist and the consumers are infinite-lived. The unit production cost is 0. The consumers' valuations, $v$, are uniformly distributed on $[0, 1/(1 - \\delta)]$ (which amounts to saying that the per-period valuation is uniformly distributed on $[0, 1]$). A consumer with valuation $v$ has utility $\\delta^t(v - p_t)$ if he buys at date $t$ at price $p_t$, where $\\delta$ is the discount factor. The monopolist's intertemporal profit is \\[ \\sum_{t=1}^{\\infty} \\delta^t p_t q_t, \\] where $q_t$ is the quantity sold (the number of consumers who buy) at time $t$. Look for a linear and stationary equilibrium: When facing at some date a price $p$, any consumers with valuations exceeding $w(p) = \\lambda p$ buy and consumers with lower valuations do not buy, where $\\lambda > 1$. Conversely, if at some date consumers with valuations exceeding $v$ have bought and the others have not, the monopolist charges $p(v) = \\mu v$, where $\\mu < 1$. In contrast with the previous conditions, assume that the monopolist commits himself to a sequence of prices \\[ (p_1, p_2, p_3, \\ldots). \\] (iii) Using the condition characterizing the optimal price sequence, compute the value of the optimal price $p_t$. Symbols (for final answer): - `delta`: discount factor", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/(2*(1-delta))", "reference_answer_sympy": "1/(2*(1-delta))", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/(2*(1-delta))", "reference_reasoning": "With commitment, the monopolist chooses (p_t) to maximize sum_{t>=1} delta^t p_t q_t. A consumer of type v buys at the first date t such that p_t <= v, so the mass buying at date t is q_t = F(p_{t-1}) - F(p_t), with p_0 = 1/(1-delta) and F(v) = (1-delta)v on [0, 1/(1-delta)]. The objective becomes (1-delta) * sum_{t>=1} delta^t (p_t p_{t-1} - p_t^2). The interior FOC for t>=2 implies delta^t(p_{t-1}-2p_t) + delta^{t+1} p_{t+1} = 0, i.e. p_{t+1}-p_t = (p_t-p_{t-1})/delta. Any nonzero difference would grow geometrically and violate feasibility/boundedness, so the optimal sequence is constant: p_t = p for all t. Then all consumers with v >= p buy at t=1, and profit is sum_{t>=1} delta^t p (1-F(p)) = (delta/(1-delta)) * p * (1-(1-delta)p). Maximizing p(1-(1-delta)p) gives derivative 1-2(1-delta)p = 0, hence p^m = 1/(2(1-delta)).", "allowed_symbols": ["delta"], "symbol_definitions": {"delta": "discount factor"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.10/i", "chapter": "1", "split": "chapter_1", "problem_number": "1.10", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Consider the previous two-period model (in which the good becomes obsolete after two periods). Introduce a constant unit-production cost of $c_1(x)$ in the first period and $c_2$ in the second period. $x$ is the probability that the first-period good is still usable in the second period. Thus, if $q_1$ is the first-period production and $xq_1$ is the number of units of the durable good still around at date 2 before further production by the monopolist. (Assume $c'_1 > 0$, $c''_1 > 0$, and $c'_1(0)$ ``small'' and $c'_1(1)$ ``large,'' so as to obtain an interior solution.) (i) If the monopolist can commit, what is the value of $c_1'(x)$ at the optimal (interior) durability choice, expressed in terms of $\\delta$ and $c_2$? Symbols (for final answer): - `$c_2$`: constant unit production cost in period 2 - `$\\delta$`: discount factor", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "delta*c_2", "reference_answer_sympy": "delta*c_2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "delta*c_2", "reference_reasoning": "With commitment, the monopolist chooses (q_1,q_2,x) to maximize pi = p_1(q_1) q_1 - c_1(x) q_1 + delta ( p_2(q_2 + x q_1) q_2 - c_2 q_2 ). For a given q_1, changing x affects profits via costs: increasing x raises first-period unit cost by c_1'(x) on each of q_1 units, but saves discounted second-period unit production cost delta*c_2 on each of the q_1 units that survives. The interior FOC in x is - c_1'(x) q_1 + delta c_2 q_1 = 0, so c_1'(x) = delta*c_2.", "allowed_symbols": ["c_2", "delta"], "symbol_definitions": {"c_2": "constant unit production cost in period 2", "delta": "discount factor"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1.10/ii", "chapter": "1", "split": "chapter_1", "problem_number": "1.10", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "(ii) Under no commitment, use the interior first-order condition for durability choice to compute \\[ \\delta c_2 - c_1'(x) \\] in terms of $\\delta$, $p2\\_prime$, and $xq1$. \\textbf{Symbols (for final answer):} \\begin{itemize} \\item $\\delta$: discount factor \\item $p2\\_prime$: derivative of second-period price with respect to inherited stock, i.e., $\\frac{\\partial p_2(xq1)}{\\partial (xq1)}$ \\item $xq1$: inherited stock at date 2, equal to $xq_1$ \\end{itemize}", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "-delta*p2_prime*xq1", "reference_answer_sympy": "-delta*p2_prime*xq1", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "-delta*p2_prime*xq1", "reference_reasoning": "From the given FOC under no commitment, c1'(x) = delta*(c2 + p2_prime*xq1). Subtracting from delta*c2 gives delta*c2 - c1'(x) = delta*c2 - delta*(c2 + p2_prime*xq1) = -delta*p2_prime*xq1. Since p2_prime<0, this difference is positive, implying c1'(x) < delta*c2 (durability is privately suboptimal).", "allowed_symbols": ["delta", "p2_prime", "xq1"], "symbol_definitions": {"delta": "discount factor", "p2_prime": "derivative of second-period price with respect to inherited stock, i.e., ∂p2(xq1)/∂(xq1)", "xq1": "inherited stock at date 2, equal to x*q1"}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.4/main", "chapter": "2", "split": "chapter_2", "problem_number": "2.4", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider a monopolist selling an experience good at a fixed price \\(p\\) to a unit mass of consumers. The monopolist chooses the quality of the good after consumers make their purchase decisions. There are two possible qualities: high quality, produced at unit cost \\(c_1\\), and low quality, produced at unit cost \\(c_0\\), where \\(p>c_1>c_0\\).\n\nA fraction \\(\\alpha\\in(0,1)\\) of consumers are informed, meaning that they observe the quality before purchasing. The remaining fraction \\(1-\\alpha\\) are uninformed and do not observe quality before purchasing.\n\nConsumers have unit demand. If a consumer buys the good, the payoff is \\(v-p\\) if quality is high and \\(-p\\) if quality is low, where \\(v>p\\). Thus, informed consumers buy if and only if quality is high.\n\nUninformed consumers hold a belief \\(\\mu\\in[0,1]\\) that the good is of high quality. Their expected payoff from purchasing is\n\\[\n\\mu(v-p)+(1-\\mu)(-p)=\\mu v-p,\n\\]\nand they purchase if and only if \\(\\mu v\\ge p\\). Let \\(y\\in[0,1]\\) denote the fraction of uninformed consumers who purchase.\n\nIf the monopolist chooses high quality, its profit is\n\\[\n\\pi_H=(p-c_1)[\\alpha+(1-\\alpha)y],\n\\]\nwhile if it chooses low quality, its profit is\n\\[\n\\pi_L=(p-c_0)(1-\\alpha)y.\n\\]\n\nAn equilibrium is defined as a Perfect Bayesian equilibrium. Judge whether the following statement is true or false:\n\nIf\n\\[\n\\alpha p<c_1-(1-\\alpha)c_0,\n\\]\nthen in equilibrium the monopolist randomizes between high and low quality, and only a fraction \\(y\\in(0,1)\\) of uninformed consumers purchase.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "true", "reference_answer_sympy": "true", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "true", "reference_reasoning": "In a mixed equilibrium, the monopolist must be indifferent between high and low quality:\n\\[\n[\\alpha+(1-\\alpha)y](p-c_1)=(1-\\alpha)y(p-c_0).\n\\]\nRearranging,\n\\[\n\\alpha(p-c_1)=(1-\\alpha)y(c_1-c_0),\n\\]\nso\n\\[\ny=\\frac{\\alpha(p-c_1)}{(1-\\alpha)(c_1-c_0)}.\n\\]\nSince \\(p>c_1>c_0\\), \\(y>0\\). The condition\n\\[\n\\alpha p<c_1-(1-\\alpha)c_0\n\\]\nis equivalent to\n\\[\n\\alpha(p-c_1)<(1-\\alpha)(c_1-c_0),\n\\]\nwhich implies \\(y<1\\). Hence \\(y\\in(0,1)\\), so only a fraction of uninformed consumers purchase. Uninformed consumers mix only if they are indifferent between buying and not buying. Since their expected payoff from buying is\n\\[\n\\mu v-p,\n\\]\nindifference requires\n\\[\n\\mu v-p=0,\n\\]\nso\n\\[\n\\mu=\\frac{p}{v}.\n\\]\nBecause \\(v>p\\), this belief is feasible. Therefore, the statement is true.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.5/i", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (i) Determine the efficient volume of trade (i.e., the volume of trade under symmetric information). Express your answer using $G_{\\theta_0} \\equiv G(\\theta_0)$ and $N$. Symbols (for final answer): - `G_theta0`: value of the CDF G evaluated at theta_0, i.e., G(theta_0) - `N`: number of potential buyers (and sellers)", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "N*(1 - G_theta0)", "reference_answer_sympy": "N*(1 - G_theta0)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "N*(1 - G_theta0)", "reference_reasoning": "Under symmetric information, trade of a car with quality s to a buyer of type theta is efficient iff the buyer's value exceeds the seller's value: theta*s - theta0*s >= 0. Since s>0 on [s_min,s_max], this is equivalent to theta >= theta0. Therefore, the efficient number of trades equals the number of buyers with theta >= theta0, which is N times the upper-tail probability 1-G(theta0). Denote G(theta0) by G_theta0.", "allowed_symbols": ["G_theta0", "N"], "symbol_definitions": {"G_theta0": "value of the CDF G evaluated at theta_0, i.e., G(theta_0)", "N": "number of potential buyers (and sellers)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.5/ii#2", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "ii", "pair_index": 2, "pair_count": 2, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (ii) True or False: The demand curve is necessarily downward sloping in price $p$ (given no income effects).", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "False. With asymmetric information, a higher price induces more (and higher-quality) sellers to enter, raising the expected quality of cars offered/traded \\bar s(p)=E[s|s\\le p/\\theta_0]. Then the buyer cutoff type is \\theta=p/\\bar s(p), which need not rise with p; it can fall if \\bar s(p) rises sufficiently. Hence demand D(p)=N[1-G(p/\\bar s(p))] can increase with p over some range, so demand is not necessarily downward sloping even without income effects.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.5/iii", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "iii", "pair_index": 1, "pair_count": 1, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (iii) For the case $\\theta_0<1/2$, what is the competitive equilibrium price $p^*$ as a function of $\\theta_0$? Symbols (for final answer): - `theta_0`: sellers' valuation parameter", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "theta_0*(1-2*theta_0)", "reference_answer_sympy": "theta_0*(1-2*theta_0)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "theta_0*(1-2*theta_0)", "reference_reasoning": "With s,theta ~ U[0,1]. At price p, a seller sells iff p >= theta_0*s, i.e. s <= p/theta_0, so offered quality support is [0, \\bar s] with \\bar s = min{p/theta_0,1}. If p<theta_0 then \\bar s=p/theta_0 and E[s|trade]=\\bar s/2 = p/(2 theta_0). A buyer buys iff theta*E[s|trade] >= p, i.e. theta >= p / (p/(2 theta_0)) = 2 theta_0, so demand is N(1-2 theta_0) when theta_0<1/2. Supply is N p/theta_0. Market clearing N p/theta_0 = N(1-2 theta_0) gives p^* = theta_0(1-2 theta_0). This p^* satisfies p^*<theta_0 when theta_0<1/2, so the case is consistent.", "allowed_symbols": ["theta_0"], "symbol_definitions": {"theta_0": "sellers' valuation parameter"}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.5/iv#1", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "iv", "pair_index": 1, "pair_count": 3, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (iv) True or False: In general, there can exist multiple equilibria because the buyers’ willingness to pay depends on the expected quality of cars offered for sale, and that expected quality depends on the price, so the market-clearing condition can have more than one solution.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "With asymmetric information, at price p sellers with quality s sell iff p >= theta_0 s, so the set of cars offered is s <= p/theta_0 (capped at s_max). The average quality of cars offered, \\bar s(p)=E[s | s <= p/theta_0], increases with p. Buyers then buy iff theta \\bar s(p) >= p, so demand depends on p through p/\\bar s(p). Since \\bar s(p) changes with p, the mapping p -> p/\\bar s(p) need not be monotone, making the (inverse) demand potentially non-monotone. Therefore the market-clearing equation S(p)=D(p) can have multiple solutions, implying multiple equilibria in general.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.5/iv#2", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "iv", "pair_index": 2, "pair_count": 3, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (iv) Then assume that this is the case. Judge whether the following statement is true or false: “If there are two equilibria with prices $p_L<p_H$, then the allocation associated with the higher-price equilibrium $p_H$ Pareto dominates the allocation associated with the lower-price equilibrium $p_L$.”", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let p_L<p_H be two equilibria. A higher price weakly increases the set of sellers willing to sell, so S(p_H)≥S(p_L), implying weakly higher average traded quality \\bar s(p_H)≥\\bar s(p_L). Market clearing gives D(p)=S(p), so D(p_H)≥D(p_L). With cutoff buyer type \\theta^*(p)=p/\\bar s(p), we have \\theta^*(p_H)=p_H/\\bar s(p_H) ≤ p_L/\\bar s(p_L)=\\theta^*(p_L). Hence at p_H at least as many trades occur and expected traded quality is weakly higher. Sellers are weakly better off because any seller trading at p_L can also trade at p_H and receives a higher price; non-trading sellers are unchanged. Buyers who traded at p_L remain willing to trade at p_H since \\theta^*(p_H)≤\\theta^*(p_L), and higher expected quality with market clearing implies their expected surplus \\theta\\bar s(p)-p is weakly higher. Therefore the p_H allocation Pareto dominates the p_L allocation.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.5/v", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "v", "pair_index": 1, "pair_count": 1, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (v) True or False: An enforceable minimum quality standard $s_0>0$ can never improve welfare in this market; restricting trade to $s\\ge s_0$ cannot raise buyers’ rational-expectations expected quality of traded cars and therefore cannot restore positive trade relative to a no-trade adverse-selection equilibrium.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "With asymmetric information, adverse selection can lead to unraveling and a no-trade rational-expectations equilibrium (e.g., price collapses so no seller participates), yielding zero market surplus. An enforceable minimum quality standard (MQS) s0 restricts the set of qualities that can be offered for sale to s >= s0. This truncation increases the expected quality of cars conditional on trade, which increases buyers’ willingness to pay. If the restricted market admits an equilibrium with positive trade (some sellers with s >= s0 sell and some buyers with sufficiently high theta buy), then total surplus from those trades is strictly positive because trades occur only when theta > theta0, generating gains (theta - theta0)s > 0. Therefore, relative to the no-trade outcome, an enforceable MQS can strictly increase welfare.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.6/main#1", "chapter": "2", "split": "chapter_2", "problem_number": "2.6", "sub_id": "main", "pair_index": 1, "pair_count": 2, "question": "A dynamic perspective, however, leads to a greater goodwill.\n\nTrue or False: As the discount factor $\\delta$ increases, the first-period price increases.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": false, "reference_answer_sympy": "N/A", "reference_answer_json": {}, "reference_answer_json_modes": {}, "answer_for_compare": "False", "reference_reasoning": "False. The optimal first-period price decreases as the discount factor $\\delta$ increases.", "allowed_symbols": ["p1_delta", "p2_delta"], "symbol_definitions": {"p1_delta": "the sign/value of the derivative ∂p1*/∂δ (negative)", "p2_delta": "the value of the derivative ∂p2*/∂δ (zero)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.8/main#1", "chapter": "2", "split": "chapter_2", "problem_number": "2.8", "sub_id": "main", "pair_index": 1, "pair_count": 2, "question": "Compute if $\\delta(\\theta - c_0) > c_1 - c_0 > \\delta(\\theta x_1 - c_0)$, the dishonest monopolist establishes a reputation for the second period with probability \\alpha. Compute $\\alpha$ as a symbolic expression. Symbols (for final answer): - `c_0`: unit cost of low quality - `c_1`: unit cost of high quality - `delta`: discount factor - `theta`: consumer valuation parameter - `x_1`: prior probability the monopolist is honest", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0))", "reference_answer_sympy": "x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0))", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0))", "reference_reasoning": "Let the prior probability the monopolist is honest be x_1. The honest type always supplies high quality. Let alpha be the probability the dishonest type supplies high quality in period 1. If high quality is observed in period 1, Bayes’ rule gives the posterior probability of honesty x_2 = x_1/(x_1+(1-x_1)*alpha). In period 2, given belief x_2, consumers’ willingness to pay is theta*x_2. Under the stated parameter restriction, the dishonest type mixes, so he is indifferent between choosing high and low quality in period 1, implying c_1-c_0 = delta*(theta*x_2 - c_0). Solving gives x_2 = (c_1-c_0+delta*c_0)/(delta*theta). Equating this to the Bayes expression and solving for alpha yields alpha = x_1*(delta*(theta-c_0)-(c_1-c_0))/((1-x_1)*(c_1-c_0+delta*c_0)).", "allowed_symbols": ["c_0", "c_1", "delta", "theta", "x_1"], "symbol_definitions": {"c_0": "unit cost of low quality", "c_1": "unit cost of high quality", "delta": "discount factor", "theta": "consumer valuation parameter", "x_1": "prior probability the monopolist is honest"}, "review_status": "confirmed_without_expert"} | |
| {"id": "2.8/main#2", "chapter": "2", "split": "chapter_2", "problem_number": "2.8", "sub_id": "main", "pair_index": 2, "pair_count": 2, "question": "If $\\delta(\\theta - c_0) > c_1 - c_0 > \\delta(\\theta x_1 - c_0)$, the dishonest monopolist establishes a reputation for the second period with probability \\[ \\alpha = \\frac{x_1[\\delta(\\theta - c_0) - (c_1 - c_0)]}{(1 - x_1)(c_1 - c_0 + \\delta c_0)}. \\] True/False: As $x_1 \\to 0$, the reputational benefit to the dishonest monopolist from choosing high quality in period 1 remains substantial, so he still often finds it worthwhile to incur the extra cost to build a reputation (i.e., $\\alpha$ does not approach 0).", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "When the prior probability of an honest monopolist x_1 is very small, observing high quality in period 1 barely increases consumers’ posterior belief that the monopolist is honest. Hence the expected future (period-2) gain from mimicking honesty is negligible for the dishonest type, so he rarely pays the extra cost of high quality to build a reputation, implying alpha tends to 0 as x_1 tends to 0.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "3.1/i", "chapter": "3", "split": "chapter_3", "problem_number": "3.1", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "A monopolist serves \\(n\\) identical consumers. The aggregate inverse demand is \\(P(Q)\\), where \\(Q\\) denotes the total quantity sold across all consumers and \\(P'<0\\). Marginal cost is a constant \\(c\\). The monopolist can perfectly price discriminate by offering each consumer a nonlinear tariff \\(T(q)\\), where \\(q\\) is the individual quantity consumed. Consumers can refuse the tariff \\((\\text{outside option}=0)\\). Compute the nonlinear tariff \\(T(q)\\) that extracts each consumer's gross surplus.\n\nSymbols (for final answer):\n- `Q`: aggregate quantity, where Q = n*q\n- `n`: number of consumers\n- `u`: integration variable\n- `P_u`: inverse demand evaluated at aggregate quantity u, representing P(u) in the SymPy final answer", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "Integral(P_u, (u, 0, Q))/n", "reference_answer_sympy": "Integral(P_u, (u, 0, Q))/n", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "Integral(P_u, (u, 0, Q))/n", "reference_reasoning": "Let \\(q\\) be each consumer's quantity, so aggregate quantity is \\(Q=nq\\). Since aggregate inverse demand is \\(P(Q)\\), the marginal willingness to pay of an individual consumer at quantity \\(s\\) is \\(P(ns)\\). Hence the gross surplus of one consumer is\n\\[\nS(q)=\\int_0^q P(ns)\\,ds.\n\\]\nUsing the change of variable \\(u=ns\\),\n\\[\nS(q)=\\frac{1}{n}\\int_0^{nq}P(u)\\,du.\n\\]\nEquivalently, since \\(Q=nq\\),\n\\[\nS\\left(\\frac{Q}{n}\\right)=\\frac{1}{n}\\int_0^Q P(u)\\,du.\n\\]\nTo extract each consumer's gross surplus, the monopolist sets\n\\[\nT(q)=S(q)=\\int_0^q P(ns)\\,ds=\\frac{1}{n}\\int_0^{nq}P(u)\\,du.\n\\]\nThus each consumer's net surplus is \\(S(q)-T(q)=0\\), so the participation constraint binds. The monopolist's profit from inducing aggregate quantity \\(Q\\) is\n\\[\n\\Pi(Q)=nT\\left(\\frac{Q}{n}\\right)-cQ=\\int_0^Q P(u)\\,du-cQ.\n\\]\nThe first-order condition is \\(P(Q)-c=0\\). Therefore the surplus-maximizing aggregate quantity \\(Q^*\\) satisfies \\(P(Q^*)=c\\), and each consumer consumes \\(q^*=Q^*/n\\). Thus the tariff can be written as\n\\[\nT\\left(\\frac{Q}{n}\\right)=\\frac{1}{n}\\int_0^Q P(u)\\,du.\n\\]\nIn the SymPy final answer, write \\(P(u)\\) as `P_u`, so the comparable expression is `Integral(P_u, (u, 0, Q))/n`.", "allowed_symbols": ["P_u", "Q", "n", "u"], "symbol_definitions": {"P_u": "inverse demand evaluated at aggregate quantity u, representing P(u) in the SymPy final answer", "Q": "aggregate quantity, where Q = n*q", "n": "number of consumers", "u": "integration variable"}, "review_status": "confirmed_without_expert"} | |
| {"id": "3.5/i", "chapter": "3", "split": "chapter_3", "problem_number": "3.5", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Assume $c_0 = 0$. (i) When tie-in sales are prohibited and the condition \\[ (1 - \\lambda)S_2(c) \\geq S_1(c) \\] holds, what is the aggregate social surplus? Symbols (for final answer): - `S_2_c`: gross surplus generated by a type-2 consumer at cost parameter c (i.e., S_2(c)) - `lambda`: fraction of consumers who are type 1 (so 1-lambda are type 2)", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(1-lambda)*S_2(c)", "reference_answer_sympy": "(1-lambda)*S_2(c)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(1-lambda)*S_2(c)", "reference_reasoning": "With tie-in sales prohibited, the monopolist cannot price discriminate across types. Under the stated condition, it is (weakly) optimal to serve only high-demand consumers (type 2). The stand-alone price is set so type 2 is just willing to buy, implying zero consumer surplus for type 2 and total surplus equal to the gross surplus generated by the served buyers. Since a fraction (1-lambda) of consumers are type 2, aggregate social surplus equals (1-lambda) times the per-type-2 gross surplus S_2(c), denoted S_2_c.", "allowed_symbols": ["S_2_c", "lambda"], "symbol_definitions": {"S_2_c": "gross surplus generated by a type-2 consumer at cost parameter c (i.e., S_2(c))", "lambda": "fraction of consumers who are type 1 (so 1-lambda are type 2)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "3.6/main", "chapter": "3", "split": "chapter_3", "problem_number": "3.6", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "In the United States, the fast-food chain Chicken Delight used to tie paper packaging items—\"buckets\" for large orders and \"kits\" for individual dinners—to the use of the franchise name. True or False: A potential explanation is that tying packaging could not have helped Chicken Delight implement price discrimination/implicit royalties across franchisees in exclusive, differentiated territories, because setting different markups on buckets versus kits would not allow it to extract more surplus from territories with different demand mixes without explicitly varying franchise fees by location.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "Because franchisees operated in exclusive, differentiated territories, their demand mixes differed (e.g., some sold relatively more individual dinners than bulk orders). By requiring franchisees to buy packaging from the franchisor and choosing different markups on kits versus buckets, Chicken Delight could effectively charge higher total payments (implicit royalties) to franchisees whose customers bought more of the high-markup item, achieving price discrimination without explicitly setting different franchise fees by territory.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "3.7/ii", "chapter": "3", "split": "chapter_3", "problem_number": "3.7", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "A monopolist with marginal cost $c$ sells to heterogeneous consumers. The latter differ in their transportation cost $tq$ for purchasing quantity $q$. The parameter $t$ is distributed according to a c.d.f. $G(t)$ on $[0, +\\infty)$, with density $g(t)$. (The monopolist does not supply transportation. Assume, further, that the consumers cannot arbitrage among themselves.) The consumer's utility function is \\[ [1 - (1 - q^2)]/2 - t q - T(q) \\] when purchasing $q$ units. (ii) Let $p(q) \\equiv T'(q)$. Use equation \\[ \\frac{\\partial V}{\\partial q}(q(\\theta),\\theta) = c + \\frac{1 - G(t)}{g(t)} \\frac{\\partial^2 V}{\\partial q \\, \\partial \\theta} \\bigl(q(\\theta),\\theta\\bigr). \\] to compute the optimal $p(q)$. Symbols (for final answer): - `G_t`: cdf evaluated at type t, i.e., G(t) - `c`: marginal cost - `g_t`: pdf evaluated at type t, i.e., g(t)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "c - (1 - G_t)/g_t", "reference_answer_sympy": "c - (1 - G_t)/g_t", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "c - (1 - G_t)/g_t", "reference_reasoning": "Consumer utility is u(q,t)=q^2/2 - t q - T(q), so V(q,t)=q^2/2 - t q. Then ∂V/∂q = q - t and ∂^2V/(∂q∂t) = -1. Plugging into the given condition with θ=t, F=G, f=g gives q(t)-t = c + (1-G(t))/g(t) * (-1) = c - (1-G(t))/g(t). Since p(q(t))=T'(q(t))=q(t)-t, the optimal marginal price schedule is p(q(t)) = c - (1-G(t))/g(t).", "allowed_symbols": ["G_t", "c", "g_t"], "symbol_definitions": {"G_t": "cdf evaluated at type t, i.e., G(t)", "c": "marginal cost", "g_t": "pdf evaluated at type t, i.e., g(t)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "3.8/i", "chapter": "3", "split": "chapter_3", "problem_number": "3.8", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Consumers have preferences $U = \\theta V(q) - T$. The consumption $q$ can take a value of $0$, $1$, or $2$. $V(0) = 0$, $V(1) = 1$, and $V(2) = \\frac{7}{4}$. The unit production cost is $c = \\frac{3}{4}$ whatever the size of the bundle. There are two types of consumers: $\\theta_1 = 1$ (in proportion $\\lambda$) and $\\theta_2 = 2$ (in proportion $1 - \\lambda$). The consumers can engage in personal arbitrage. (i) In the absence of a technological constraint (i.e., in a case where the monopolist can produce both sizes), compute the cutoff value $\\lambda^*$ such that the monopolist uses pure commodity bundling if and only if $\\lambda < \\lambda^*$. Symbols (for final answer): - lambda_star: the cutoff value $\\lambda^*$", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "4/5", "reference_answer_sympy": "4/5", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "4/5", "reference_reasoning": "With personal arbitrage, if the monopolist offers both q=1 and q=2 at prices T_1 and T_2, type-2 must not prefer buying two 1-unit contracts to buying the 2-unit contract: 2*theta_2*V_1 - 2*T_1 >= theta_2*V_2 - T_2, so T_2 >= 2*T_1 + theta_2*(V_2 - 2*V_1). Given V_2=7/4, V_1=1, theta_2=2, we have V_2-2V_1 = 7/4-2 = -1/4, hence T_2 >= 2*T_1 - 1/2. Type-1 participation for q=1 requires theta_1*V_1 - T_1 >= 0, so T_1 <= 1. Profit when serving both types is Pi = lambda*(T_1 - c) + (1-lambda)*(T_2 - 2*c), with c=3/4. Profit increases in T_1 and T_2 subject to constraints, so set T_1=1 and the minimal feasible T_2=2*1-1/2=3/2. Then Pi_serve_both = lambda*(1-3/4) + (1-lambda)*(3/2-3/2) = lambda/4. Under pure commodity bundling (only q=2), set the highest price type-2 accepts: T = theta_2*V_2 = 2*(7/4)=7/2. Profit is Pi_bundle = (1-lambda)*(7/2 - 2*c) = (1-lambda)*(7/2 - 3/2) = 2*(1-lambda). Bundling is optimal iff Pi_bundle >= Pi_serve_both: 2*(1-lambda) >= lambda/4 -> 8*(1-lambda) >= lambda -> 8 >= 9*lambda -> lambda <= 8/9. Thus the cutoff is lambda_star = 8/9, and bundling is used iff lambda < lambda_star.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "3.8/ii", "chapter": "3", "split": "chapter_3", "problem_number": "3.8", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "Consumers have preferences $U = \\theta V(q) - T$. The consumption $q$ can take a value of $0$, $1$, or $2$. $V(0) = 0$, $V(1) = 1$, and $V(2) = \\frac{7}{4}$. The unit production cost is $c = \\frac{3}{4}$ whatever the size of the bundle. There are two types of consumers: $\\theta_1 = 1$ (in proportion $\\lambda$) and $\\theta_2 = 2$ (in proportion $1 - \\lambda$). The consumers can engage in personal arbitrage. (ii) Suppose that, for technological reasons, the monopolist must choose to produce the good in either size 1 or size 2. Compute the cutoff value of $\\lambda$ such that the monopolist chooses the size-2 technology if and only if $\\lambda$ is below this cutoff. Symbols (for final answer): - lambda_star: the cutoff value of $\\lambda$", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "6/7", "reference_answer_sympy": "6/7", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "6/7", "reference_reasoning": "Under size-2 technology, with personal arbitrage the firm can only sell the size-2 bundle at price T. Values: type 1 has 7/4, type 2 has 7/2. Unit cost is c=3/4 so cost of bundle 2 is 3/2. If serving both types, set T=7/4 giving profit (7/4-3/2)*1=1/4. If serving only type 2, set T=7/2 giving profit (7/2-3/2)*(1-lambda)=2(1-lambda). Thus Π2=max(1/4, 2(1-lambda)). Under size-1 technology, arbitrage implies linear per-unit price p. Evaluate demand by comparing utilities for buying 0,1,2 units. The profit-maximizing outcomes yield Π1=max(1/2 - lambda/4, 3(1-lambda)/2) (the 5(1-lambda)/4 option is dominated). Compare technologies. For lambda<5/6, Π2=2(1-lambda) and Π1=3(1-lambda)/2 so Π2>Π1. For lambda>=5/6, Π1=1/2 - lambda/4 while Π2 is still 2(1-lambda) until it falls below 1/4. The cutoff where size-2 just equals size-1 in this region solves 2(1-lambda)=1/2 - lambda/4 => 8-8lambda=2-lambda => 6=7lambda => lambda=6/7. For lambda>=6/7, Π2=1/4 and Π1>=1/4, so size-2 is not strictly better. Hence the cutoff is 6/7.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "3.10/main", "chapter": "3", "split": "chapter_3", "problem_number": "3.10", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "A common practice in the U.S. film industry (before it was outlawed) was the bundling of several films at the distribution level. Stigler (1963) offered the following simple model to formalize this practice, which was also called \"block booking\": There are two downstream units (theaters), two films, and one monopoly film producer. The first downstream unit values film 1 at $4$ and film 2 at $1$. The second values film 1 at $3$ and film 2 at $2$. The value of a bundle to each downstream unit is equal to the sum of its valuations for the two films (there is no interdependence effect). Compute the difference between the producer's optimal revenue under bundling and the producer's optimal total revenue under separate selling (one uniform price per film). Symbols (for final answer): - R_bundle_minus_separate: bundling revenue minus separate-selling revenue", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "2", "reference_answer_sympy": "2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "2", "reference_reasoning": "Separate selling with one uniform price per film: Film 1: price 4 sells to theater 1 only (revenue 4) or price 3 sells to both (revenue 6). Optimal is 6. Film 2: price 2 sells to theater 2 only (revenue 2) or price 1 sells to both (revenue 2). Optimal is 2. Total separate-selling revenue = 6 + 2 = 8. Bundling: theater 1 values bundle at 4+1=5; theater 2 values bundle at 3+2=5. Set bundle price 5, sell to both, revenue = 2*5 = 10. Difference (bundling minus separate) = 10 - 8 = 2.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "4.1/main", "chapter": "4", "split": "chapter_4", "problem_number": "4.1", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "In the double-marginalization set-up the ratio of the retailer's margin over the manufacturer's margin, \\[ \\frac{p - p_w}{p_w - c}, \\] is equal to (greater than, lower than) which number if the demand function is linear (convex, concave). Symbols (for final answer): - `D_p`: demand evaluated at p, i.e., D(p) - `Dp_p`: first derivative of demand at p, i.e., D'(p) - `Dpp_p`: second derivative of demand at p, i.e., D''(p)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/2", "reference_answer_sympy": "1/2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/2", "reference_reasoning": "Let \\[ m_R \\equiv p-p_w \\qquad\\text{and}\\qquad m_M \\equiv p_w-c \\] denote the retailer's margin and the manufacturer's margin, respectively. We want to compute \\[ \\frac{m_R}{m_M}=\\frac{p-p_w}{p_w-c}. \\] The key result is: \\[ \\frac{p-p_w}{p_w-c} = \\frac{1}{\\,2-\\dfrac{D(p)D''(p)}{[D'(p)]^2}\\,}. \\] Therefore, \\[ \\frac{p-p_w}{p_w-c} = \\begin{cases} \\dfrac{1}{2}, & \\text{if } D''(p)=0 \\text{ (linear demand)},\\\\[1em] >\\dfrac{1}{2}, & \\text{if } D''(p)>0 \\text{ (convex demand)},\\\\[1em] <\\dfrac{1}{2}, & \\text{if } D''(p)<0 \\text{ (concave demand)}. \\end{cases} \\] We now derive this result step by step. The retailer solves \\[ \\max_p (p-p_w)D(p). \\] The first-order condition is \\begin{equation} D(p)+(p-p_w)D'(p)=0. \\tag{1} \\end{equation} Rearranging, \\begin{equation} p-p_w=-\\frac{D(p)}{D'(p)}. \\tag{2} \\end{equation} Equation (1) implicitly defines the retailer's optimal price \\(p^*(p_w)\\). Differentiating (1) with respect to \\(p_w\\), we get \\[ D'(p)\\frac{dp^*}{dp_w} + \\left(\\frac{dp^*}{dp_w}-1\\right)D'(p) + (p-p_w)D''(p)\\frac{dp^*}{dp_w} =0. \\] Collecting terms, \\[ \\left[2D'(p)+(p-p_w)D''(p)\\right]\\frac{dp^*}{dp_w}-D'(p)=0. \\] Hence, \\[ \\frac{dp^*}{dp_w} = \\frac{D'(p)}{\\,2D'(p)+(p-p_w)D''(p)\\,}. \\] Using \\((2)\\), namely \\(p-p_w=-D(p)/D'(p)\\), this becomes \\begin{equation} \\frac{dp^*}{dp_w} = \\frac{1}{\\,2-\\dfrac{D(p)D''(p)}{[D'(p)]^2}\\,}. \\tag{3} \\end{equation} Next, the manufacturer chooses \\(p_w\\) to maximize \\[ (p_w-c)D\\bigl(p^*(p_w)\\bigr). \\] Its first-order condition is \\begin{equation} D(p)+(p_w-c)D'(p)\\frac{dp^*}{dp_w}=0. \\tag{4} \\end{equation} Rearranging, \\begin{equation} p_w-c = -\\frac{D(p)}{D'(p)\\,\\dfrac{dp^*}{dp_w}}. \\tag{5} \\end{equation} Now divide \\((2)\\) by \\((5)\\): \\[ \\frac{p-p_w}{p_w-c} = \\frac{-\\dfrac{D(p)}{D'(p)}}{-\\dfrac{D(p)}{D'(p)\\,\\dfrac{dp^*}{dp_w}}} = \\frac{dp^*}{dp_w}. \\] Using \\((3)\\), we obtain \\begin{equation} \\frac{p-p_w}{p_w-c} = \\frac{1}{\\,2-\\dfrac{D(p)D''(p)}{[D'(p)]^2}\\,}. \\tag{6} \\end{equation} Therefore: \\begin{itemize} \\item If demand is linear, then \\(D''(p)=0\\), so \\[ \\frac{p-p_w}{p_w-c}=\\frac{1}{2}. \\] \\item If demand is convex, then \\(D''(p)>0\\). Since \\(D(p)>0\\) and \\([D'(p)]^2>0\\), we have \\[ \\frac{D(p)D''(p)}{[D'(p)]^2}>0, \\] so the denominator in \\((6)\\) is smaller than \\(2\\). Hence, \\[ \\frac{p-p_w}{p_w-c}>\\frac{1}{2}. \\] \\item If demand is concave, then \\(D''(p)<0\\), so the denominator in \\((6)\\) is larger than \\(2\\). Hence, \\[ \\frac{p-p_w}{p_w-c}<\\frac{1}{2}. \\] \\end{itemize} In conclusion, \\[ \\boxed{ \\frac{p-p_w}{p_w-c} = \\frac{1}{2}\\ \\text{for linear demand,} \\quad >\\frac{1}{2}\\ \\text{for convex demand,} \\quad <\\frac{1}{2}\\ \\text{for concave demand.} } \\]", "allowed_symbols": ["D_p", "Dp_p", "Dpp_p"], "symbol_definitions": {"D_p": "Demand evaluated at p, i.e., D(p)", "Dp_p": "First derivative of demand at p, i.e., D'(p)", "Dpp_p": "Second derivative of demand at p, i.e., D''(p)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "4.2/i", "chapter": "4", "split": "chapter_4", "problem_number": "4.2", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Two firms ($i=1, 2$) produce one good each, at marginal cost $c_i$ $(i = 1, 2)$. Each firm has a monopoly power in the production of its good. The goods are perfect complements. The demand curve is $q = D(p)$, where $p \\equiv p_1 + p_2$ is the price of the composite good and $p_i$ is the price of good $i$ $(i = 1, 2)$. Let $c \\equiv c_1 + c_2$. (i) Reinterpret the variables to show that the case of a single good produced by a manufacturer and distributed by a retailer fits into this model. Assume that the elasticity of demand, $\\varepsilon = -D'p/D$, is constant in order to simplify computations. Judge whether the following statement is correct: \"Interpreting firm 1 as the manufacturer and firm 2 as the retailer, with good 1 as the manufacturing input and good 2 as the retail/distribution service (perfect complements), the consumer price is the difference of the two prices/markups $p=p_1-p_2$; identify $p_1$ with the wholesale price charged by the manufacturer and $p_2$ with the retailer’s margin, and total marginal cost is $c=c_1+c_2$.\"", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "False", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "This reinterpretation matches the vertical structure of manufacturer and retailer: one unit of manufacturing input and one unit of retail/distribution service are needed per unit sold, so they are perfect complements. The final consumer price equals the sum of the wholesale component and the retail margin, consistent with p = p1 + p2. Costs add similarly, giving total marginal cost c = c1 + c2.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "4.2/ii", "chapter": "4", "split": "chapter_4", "problem_number": "4.2", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "Two firms ($i=1, 2$) produce one good each, at marginal cost $c_i$ $(i = 1, 2)$. Each firm has a monopoly power in the production of its good. The goods are perfect complements. The demand curve is $q = D(p)$, where $p \\equiv p_1 + p_2$ is the price of the composite good and $p_i$ is the price of good $i$ $(i = 1, 2)$. Let $c \\equiv c_1 + c_2$. (ii) What is the optimal $p$ for the horizontally integrated structure, expressed in terms of $c$ and the (absolute) price elasticity of demand $epsilon$ (assumed constant)? Symbols (for final answer): - `c`: total marginal cost c1+c2 - `epsilon`: absolute price elasticity of demand (assumed constant)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "c/(1-1/epsilon)", "reference_answer_sympy": "c/(1-1/epsilon)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "c/(1-1/epsilon)", "reference_reasoning": "A horizontally integrated monopolist chooses the composite price p to maximize profit pi(p)=(p-c)D(p). The FOC is D(p)+(p-c)D'(p)=0. Using the (absolute) elasticity epsilon = -p D'(p)/D(p) > 0, rewrite the FOC as (p-c)/p = 1/epsilon. Solving for p gives p = c/(1-1/epsilon). Under constant elasticity, this directly yields the optimal price.", "allowed_symbols": ["c", "epsilon"], "symbol_definitions": {"c": "total marginal cost c1+c2", "epsilon": "absolute price elasticity of demand (assumed constant)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "4.2/iii", "chapter": "4", "split": "chapter_4", "problem_number": "4.2", "sub_id": "iii", "pair_index": 1, "pair_count": 1, "question": "Two firms ($i=1, 2$) produce one good each, at marginal cost $c_i$ $(i = 1, 2)$. Each firm has a monopoly power in the production of its good. The goods are perfect complements. The demand curve is $q = D(p)$, where $p \\equiv p_1 + p_2$ is the price of the composite good and $p_i$ is the price of good $i$ $(i = 1, 2)$. Let $c \\equiv c_1 + c_2$. (iii) Consider the nonintegrated structure. Suppose that firm 1 chooses its price first and takes into account the effect of its choice on firm 2's price. Compute the resulting composite price $p$ as a function of $c$ and $\\epsilon$. Symbols (for final answer): - `c`: total marginal cost c1+c2 - `epsilon`: price elasticity of demand (epsilon>1)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "c/(1 - 1/epsilon)**2", "reference_answer_sympy": "c/(1 - 1/epsilon)**2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "c/(1 - 1/epsilon)**2", "reference_reasoning": "Firm 2 observes p1 and chooses p2 (equivalently p=p1+p2) to maximize (p2-c2)D(p)=(p-(p1+c2))D(p), so it behaves like a monopolist with marginal cost p1+c2. Thus it sets p=(p1+c2)/(1-1/epsilon). Hence dp/dp1 = 1/(1-1/epsilon). Firm 1 anticipates this and maximizes (p1-c1)D(p(p1)). FOC: (p1-c1)D'(p) p' + D(p)=0. Using epsilon = -p D'(p)/D(p) gives p1-c1 = p/(epsilon p') = (p/epsilon)(1-1/epsilon). Using firm 2 best response rewritten as p1=(1-1/epsilon)p - c2, substitute into p1-c1 to obtain (1-1/epsilon)p - c2 - c1 = (p/epsilon)(1-1/epsilon). Rearranging yields p(1-1/epsilon)^2 = c1+c2 = c, so p = c/(1-1/epsilon)^2.", "allowed_symbols": ["c", "epsilon"], "symbol_definitions": {"c": "total marginal cost c1+c2", "epsilon": "price elasticity of demand (epsilon>1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "4.2/iv#1", "chapter": "4", "split": "chapter_4", "problem_number": "4.2", "sub_id": "iv", "pair_index": 1, "pair_count": 2, "question": "Two firms ($i=1, 2$) produce one good each, at marginal cost $c_i$ $(i = 1, 2)$. Each firm has a monopoly power in the production of its good. The goods are perfect complements. The demand curve is $q = D(p)$, where $p \\equiv p_1 + p_2$ is the price of the composite good and $p_i$ is the price of good $i$ $(i = 1, 2)$. Let $c \\equiv c_1 + c_2$. (iv) Suppose now that the two firms choose their prices simultaneously. Assume that each firm maximizes its profit given the other firm's price—that is, the choices are simultaneous: a firm does not try to influence the other firm's price. Compute the equilibrium composite price $p$ as a function of $c$ and the (absolute) price elasticity of demand $\\epsilon$. Symbols (for final answer): - `c`: total marginal cost c1+c2 - `epsilon`: absolute value of the price elasticity of demand", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "c/(1-2/epsilon)", "reference_answer_sympy": "c/(1-2/epsilon)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "c/(1-2/epsilon)", "reference_reasoning": "In the simultaneous-choice case, firm i chooses p_i to maximize (p_i-c_i)D(p_i+p_j), taking p_j as given. The FOC is (p_i-c_i)D_p + D_p0 = 0, where D_p denotes the derivative of demand with respect to p and D_p0 denotes demand itself (i.e., D(p)). Adding the two firms’ conditions gives (p-c)D_p + 2D_p0 = 0. Using the definition of (absolute) elasticity epsilon = -p*D_p/D_p0, we can rewrite as (p-c)(-epsilon*D_p0/p) + 2D_p0 = 0, cancel D_p0, and solve: -(p-c)epsilon/p + 2 = 0 => (p-c)epsilon = 2p => p(1-2/epsilon)=c => p = c/(1-2/epsilon).", "allowed_symbols": ["c", "epsilon"], "symbol_definitions": {"c": "total marginal cost c1+c2", "epsilon": "absolute value of the price elasticity of demand"}, "review_status": "confirmed_without_expert"} | |
| {"id": "4.5/i", "chapter": "4", "split": "chapter_4", "problem_number": "4.5", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Some durable-good producers tie the purchase of spare parts and maintenance to the purchase of the good. (For instance, Boeing used to tie spare parts to the sale of commercial jets through requirements provisions, and required subcontractors to destroy any production overruns of spare parts.) The purpose of this exercise is to show that the input-substitution model offers a possible explanation of this practice. (Can you think of alternative explanations?) A monopolist produces a durable good at unit cost $c$. A competitive industry offers maintenance at unit cost $c_{\\prime}$. Another competitive downstream industry uses the durable good to produce the final output. One unit of the durable good in working condition produces one unit of the final output per unit of time. Time is continuous, and the rate of interest is $r$. The firms in the downstream industry consume $x_{\\prime}$ units of maintenance per unit of durable good and per unit of time. The conditional probability of breakdown of the durable good is $\\alpha(x_{\\prime})dt$ between $t$ and $t + dt$, where $\\alpha' < 0$ and $\\alpha'' > 0$. (Breakdown means that the durable good must be replaced.) Let $p_w$ denote the price of the durable good (assume that this price is constant over time). (i) Show that the price of the final good is \\[ p = c_{\\prime} x_{\\prime} + p_w [ r + \\alpha(x_{\\prime})], \\] where $x_{\\prime}$ minimizes the right-hand side. Compute the competitive price $p$ of the final good as a function of $c_{\\prime}$, $x_{\\prime}$, $p_w$, $r$, and $\\alpha(x_{\\prime})$. Symbols (for final answer): - `alpha_x_prime`: breakdown hazard rate evaluated at x', i.e., alpha(x') - `c_prime`: unit cost of maintenance (c') - `p_w`: price of the durable good - `r`: interest rate - `x_prime`: maintenance intensity per unit time (x')", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "c_prime*x_prime + p_w*(r + alpha_x_prime)", "reference_answer_sympy": "c_prime*x_prime + p_w*(r + alpha_x_prime)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "c_prime*x_prime + p_w*(r + alpha_x_prime)", "reference_reasoning": "In a competitive downstream industry, the final-good price equals the minimum long-run expected cost per unit time of producing one unit of output. Using one durable good yields 1 unit of output per unit time while it works. Maintenance intensity x' costs c' x' per unit time. The durable breaks down with hazard rate alpha(x'), implying expected replacement purchases at rate alpha(x') per unit time, costing alpha(x') p_w per unit time. Holding the durable also ties up capital p_w with opportunity cost r p_w per unit time. Summing these flow costs gives p = c' x' + p_w (r + alpha(x')). The minimizing x' is the one that minimizes this expression.", "allowed_symbols": ["alpha_x_prime", "c_prime", "p_w", "r", "x_prime"], "symbol_definitions": {"alpha_x_prime": "breakdown hazard rate evaluated at x', i.e., alpha(x')", "c_prime": "unit cost of maintenance (c')", "p_w": "price of the durable good", "r": "interest rate", "x_prime": "maintenance intensity per unit time (x')"}, "review_status": "confirmed_without_expert"} | |
| {"id": "4.5/iii", "chapter": "4", "split": "chapter_4", "problem_number": "4.5", "sub_id": "iii", "pair_index": 1, "pair_count": 1, "question": "Some durable-good producers tie the purchase of spare parts and maintenance to the purchase of the good. (For instance, Boeing used to tie spare parts to the sale of commercial jets through requirements provisions, and required subcontractors to destroy any production overruns of spare parts.) The purpose of this exercise is to show that the input-substitution model offers a possible explanation of this practice. (Can you think of alternative explanations?) A monopolist produces a durable good at unit cost $c$. A competitive industry offers maintenance at unit cost $p'_w = c'$. Another competitive downstream industry uses the durable good to produce the final output. One unit of the durable good in working condition produces one unit of the final output per unit of time. Time is continuous, and the rate of interest is $r$. The firms in the downstream industry consume $x'$ units of maintenance per unit of durable good and per unit of time. The conditional probability of breakdown of the durable good is $\\alpha(x')dt$ between $t$ and $t + dt$, where $\\alpha' < 0$ and $\\alpha'' > 0$. (Breakdown means that the durable good must be replaced.) Let $p_w$ denote the price of the durable good (assume that this price is constant over time). (iii) Draw the analogy with the input-substitution model. Converted target (value): Introduce the fictitious variable $x \\equiv r + \\texttt{alpha\\_x\\_prime}$. What is $x$? Symbols (for final answer): - `alpha_x_prime`: breakdown hazard rate α(x') as a function of maintenance intensity x' - `r`: rate of interest", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "r + alpha_x_prime", "reference_answer_sympy": "r + alpha_x_prime", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "r + alpha_x_prime", "reference_reasoning": "To draw the analogy with the input-substitution model, define a fictitious variable that combines the interest rate and the breakdown hazard. The problem statement defines this variable as x ≡ r + α(x'). Using parser-friendly notation, write α(x') as alpha_x_prime. Therefore x equals r plus alpha_x_prime.", "allowed_symbols": ["alpha_x_prime", "r"], "symbol_definitions": {"alpha_x_prime": "breakdown hazard rate α(x') as a function of maintenance intensity x'", "r": "rate of interest"}, "review_status": "confirmed_by_expert"} | |
| {"id": "4.6/i", "chapter": "4", "split": "chapter_4", "problem_number": "4.6", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Consider the following model of intrabrand competition and retail services. The demand function for the final good is \\[ q = D(p,s), \\] for a package of $p$ (final price) and $s$ (services). Assume that all consumers are identical. Their net consumer surplus is $S(p,s)$, where $S$ is defined so that \\[ \\frac{\\partial S}{\\partial p} = -D(p,s). \\] The retailers, who are all identical, incur a per-unit cost $\\Phi(s)$ for service $s$, so that total service cost is $q\\,\\Phi(s)$. The vertically integrated profit is obtained by choosing $p^m$ and $s^m$ so as to maximize \\[ [p - c - \\Phi(s)]D(p,s). \\] In the decentralized structure, the manufacturer charges a wholesale price $p_w \\ge c$ in order to maximize his profit. The retailer then maximizes his own profit with respect to $p$ and $s$. The competitive price and level of services maximize \\[ S(p,s), \\] subject to \\[ p = p_w + \\Phi(s). \\] To encourage more promotional effort and obtain the vertically integrated profit, the manufacturer can make the retailer a residual claimant by choosing $p_w = c$, and capture the retailer’s residual profit through a franchise fee $A$. Judge True/False: In this model, instrument (i) a franchise fee alone is sufficient to obtain the vertically integrated outcome.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "False", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "A franchise fee by itself does not remove the marginal distortion created by a wholesale price above marginal cost. With only a franchise fee and no control of the per-unit wholesale price, the retailer still chooses (p,s) to maximize consumer surplus subject to p = p_w + Phi(s), so p_w continues to affect the retailer’s marginal incentives and generally prevents replication of the vertically integrated (p^m,s^m). To obtain the integrated outcome, the manufacturer must also set p_w = c (making the retailer a residual claimant) and then use a franchise fee A to extract the resulting profit.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "4.6/ii", "chapter": "4", "split": "chapter_4", "problem_number": "4.6", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "Consider the following model of intrabrand competition and retail services. The demand function for the final good is \\[ q = D(p,s), \\] for a package of $p$ (final price) and $s$ (services). Assume that all consumers are identical. Their net consumer surplus is $S(p,s)$, where $S$ is defined so that \\[ \\frac{\\partial S}{\\partial p} = -D(p,s). \\] The retailers, who are all identical, incur a per-unit cost $\\Phi(s)$ for service $s$, so that total service cost is $q\\,\\Phi(s)$. The vertically integrated profit is obtained by choosing $p^m$ and $s^m$ so as to maximize \\[ [p - c - \\Phi(s)]D(p,s). \\] In the decentralized structure, the manufacturer charges a wholesale price $p_w \\ge c$ in order to maximize his profit. The retailer then maximizes his own profit with respect to $p$ and $s$. The competitive price and level of services maximize \\[ S(p,s), \\] subject to \\[ p = p_w + \\Phi(s). \\] To encourage more promotional effort and obtain the vertically integrated profit, the manufacturer can make the retailer a residual claimant by choosing $p_w = c$, and capture the retailer’s residual profit through a franchise fee $A$. In this model, judge whether the following statement is True or False: Resale-price maintenance by itself is sufficient to obtain the vertically integrated outcome.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "True", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "RPM is sufficient. The choice of $p$ and $p_w$ fixes the competitive level of services: $\\Phi(s) = p - p_w$. Hence, it suffices to impose $p = p^m$ and charge the wholesale price $p_w = p^m - \\Phi(s^m)$. Quantity fixing is also a sufficient instrument.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.2/main#1", "chapter": "5", "split": "chapter_5", "problem_number": "5.2", "sub_id": "main", "pair_index": 1, "pair_count": 2, "question": "Let the demand function be \\[ q = D(p) = 1 - p. \\] Suppose that both firms' marginal cost (once the capacities are installed) is zero. Suppose further that $qbar_1$ and $qbar_2$ are lower than $\\frac{1}{4}$. Under proportional rationing, what is the equilibrium price $p^*$ (in terms of $qbar_1$ and $qbar_2$)? Symbols (for final answer): - `qbar_1`: firm 1 capacity \\bar q_1 - `qbar_2`: firm 2 capacity \\bar q_2", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1 - (qbar_1 + qbar_2)", "reference_answer_sympy": "1 - (qbar_1 + qbar_2)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1 - (qbar_1 + qbar_2)", "reference_reasoning": "With capacities qbar_1,qbar_2<1/4, consider the second-stage price game under proportional rationing. In equilibrium both firms set a price such that market demand equals total capacity, so that each can sell its full capacity without leaving profitable residual demand. Setting p_star so that D(p_star)=1-p_star=qbar_1+qbar_2 gives p_star=1-(qbar_1+qbar_2). At this price, any unilateral deviation upward yields residual demand allocated proportionally, giving profit proportional to p(1-p), which is decreasing for p>=1/2; since qbar_1+qbar_2<1/2, p_star>1/2, so raising price reduces profit. Deviating downward sells at most capacity but at a lower price, reducing profit. Hence p_star=1-(qbar_1+qbar_2).", "allowed_symbols": ["qbar_1", "qbar_2"], "symbol_definitions": {"qbar_1": "firm 1 capacity \\bar q_1", "qbar_2": "firm 2 capacity \\bar q_2"}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.2/main#2", "chapter": "5", "split": "chapter_5", "problem_number": "5.2", "sub_id": "main", "pair_index": 2, "pair_count": 2, "question": "Let the demand function be \\[ q = D(p) = 1 - p. \\] Suppose that both firms' marginal cost (once the capacities are installed) is zero. Suppose further that $\\texttt{bar\\_q\\_i}$ and $\\texttt{bar\\_q\\_j}$ are lower than $\\frac{1}{4}$. Compute firm $i$'s profit $\\Pi^{ig}(\\texttt{bar\\_q\\_i}, \\texttt{bar\\_q\\_j})$. Symbols (for final answer): - `bar_q_i`: firm i capacity (given, less than 1/4) - `bar_q_j`: firm j capacity (given, less than 1/4)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "bar_q_i*(1 - bar_q_i - bar_q_j)", "reference_answer_sympy": "bar_q_i*(1 - bar_q_i - bar_q_j)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "bar_q_i*(1 - bar_q_i - bar_q_j)", "reference_reasoning": "With capacities \\bar q_1, \\bar q_2 and zero marginal cost, the equilibrium price clears the market at total quantity \\bar q_1+\\bar q_2, so p^* = 1-(\\bar q_1+\\bar q_2). Since each firm sells its full capacity, firm i sells \\bar q_i. Profit equals price times quantity: \\Pi^{ig}(\\bar q_i,\\bar q_j)=p^*\\bar q_i=\\bar q_i(1-\\bar q_i-\\bar q_j).", "allowed_symbols": ["bar_q_i", "bar_q_j"], "symbol_definitions": {"bar_q_i": "firm i capacity (given, less than 1/4)", "bar_q_j": "firm j capacity (given, less than 1/4)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.3/i", "chapter": "5", "split": "chapter_5", "problem_number": "5.3", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "There are three identical firms in the industry. The demand is $1 - Q$, where $Q = q_1 + q_2 + q_3$. The marginal cost is zero. (i) In the symmetric Cournot equilibrium, what is each firm's output $q_i$? Symbols (for final answer): - q_i: each firm's equilibrium quantity Return the final answer as a JSON object with exactly these keys: q1, q2, q3. Use double-quoted JSON keys and string values.", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "json", "comparison_mode": "json", "reference_answer": {"q1": "1/4", "q2": "1/4", "q3": "1/4"}, "reference_answer_sympy": "N/A", "reference_answer_json": {"q1": "1/4", "q2": "1/4", "q3": "1/4"}, "reference_answer_json_modes": {"q1": "sympy", "q2": "sympy", "q3": "sympy"}, "answer_for_compare": {"q1": "1/4", "q2": "1/4", "q3": "1/4"}, "reference_reasoning": "In a symmetric Cournot equilibrium with three identical firms and inverse demand p = 1 - Q, each firm chooses q_i to maximize profit pi_i = p q_i = (1 - q_i - q_j - q_k) q_i. Taking q_j and q_k as given, the FOC is d pi_i / d q_i = 1 - 2 q_i - q_j - q_k = 0, so q_i = (1 - q_j - q_k)/2. In symmetry q_i = q_j = q_k = q, giving q = (1 - 2q)/2, hence 4q = 1 and q = 1/4.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.3/iii", "chapter": "5", "split": "chapter_5", "problem_number": "5.3", "sub_id": "iii", "pair_index": 1, "pair_count": 1, "question": "There are three identical firms in the industry. The demand is $1 - Q$, where $Q = q_1 + q_2 + q_3$. The marginal cost is zero. (iii) What is the total profit if all three firms merge? Symbols (for final answer): - (no symbols needed)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/4", "reference_answer_sympy": "1/4", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/4", "reference_reasoning": "If all three firms merge, the industry becomes a monopoly. The monopolist chooses Q to maximize profit: pi = (1 - Q)Q. The first-order condition is d pi / dQ = 1 - 2Q = 0, giving Q* = 1/2. Then P* = 1 - Q* = 1/2. Total profit is pi* = P*Q* = (1/2)(1/2) = 1/4.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_by_expert"} | |
| {"id": "5.3/iv", "chapter": "5", "split": "chapter_5", "problem_number": "5.3", "sub_id": "iv", "pair_index": 1, "pair_count": 1, "question": "There are three identical firms in the industry. The demand is $1 - Q$, where $Q = q_1 + q_2 + q_3$. The marginal cost is zero. Judge whether the following statement is true or false: \"Under Bertrand price competition with differentiated products and strategic complementarity in prices, a merger between two of the firms can be profitable because the merged firm internalizes cannibalization between its two products and becomes less aggressive, which induces the third firm to raise its price as well (a 'puppy-dog strategy').\"", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "With differentiated products and Bertrand competition, a multiproduct firm internalizes the substitution (cannibalization) between its own products: cutting the price of one product steals demand from its other product, so the merged firm sets higher prices than two separate firms would. If prices are strategic complements, higher prices by the merged firm lead the remaining rival to optimally raise its own price. This softer industry pricing can make the merger profitable (the 'puppy-dog strategy').", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.4/ii#2", "chapter": "5", "split": "chapter_5", "problem_number": "5.4", "sub_id": "ii", "pair_index": 2, "pair_count": 2, "question": "Consider a duopoly producing a homogeneous product. Firm 1 produces one unit of output with one unit of labor and one unit of raw material. Firm 2 produces one unit of output with two units of labor and one unit of raw material. The unit costs of labor and raw material are $w$ and $r$. The demand is $p = 1 - q_1 - q_2$, and the firms compete in quantities. To prove this elegantly, use the envelope theorem. Compute $\\frac{d\\Pi_1}{dw}$ at the Cournot equilibrium (interior-solution range). Symbols (for final answer): Pi1_w", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "Let q1_star(w) be firm 1’s best response to q2(w). Define the value function Pi1(w)=max_{q1>=0} pi1(q1; q2(w), w, r), where pi1(q1; q2, w, r)=q1(1-q1-q2)-(w+r)q1. By the envelope theorem allowing for q2 depending on w, dPi1/dw = (∂pi1/∂w)|_{q1=q1_star} + (∂pi1/∂q2)|_{q1=q1_star} * dq2/dw. Here ∂pi1/∂w = -q1 and ∂pi1/∂q2 = -q1, so dPi1/dw = -q1_star(1 + dq2/dw). In Cournot equilibrium, q2_star(w)=(1-r-3w)/3, hence dq2/dw = -1. Therefore dPi1/dw = -q1_star(1-1)=0.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.5/ii#1", "chapter": "5", "split": "chapter_5", "problem_number": "5.5", "sub_id": "ii", "pair_index": 1, "pair_count": 3, "question": "This exercise illustrates the strategic considerations faced by a multimarket firm. It is inspired by the more general theory of Bulow et al. (1985). There are two firms in a market. They produce perfect substitutes at cost $C(q) = q^2/2$. The demand is $p = 1 - (q_1 + q_2)$. (ii) Suppose now that firm 1 has the opportunity to sell the same output on another market as well. The quantity sold on this market is $x_1$, so firm 1's cost is $(q_1 + x_1)^2/2$. The demand on the second market is $p = a - x_1$. Consider the Cournot game in which firm 1 chooses $q_1$ and $x_1$ and firm 2 chooses $q_2$ simultaneously. Compute the Cournot equilibrium outputs $q_1$ and $q_2$ as functions of $a$. Symbols (for final answer): - `a`: demand intercept in the second market Return the final answer as a JSON object with exactly these keys: q1, q2. Use double-quoted JSON keys and string values.", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "json", "comparison_mode": "json", "reference_answer": {"q1": "(2-a)/7", "q2": "(5+a)/21"}, "reference_answer_sympy": "N/A", "reference_answer_json": {"q1": "(2-a)/7", "q2": "(5+a)/21"}, "reference_answer_json_modes": {"q1": "sympy", "q2": "sympy"}, "answer_for_compare": {"q1": "(2-a)/7", "q2": "(5+a)/21"}, "reference_reasoning": "Profits: pi_1 = q_1(1-q_1-q_2) + x_1(a-x_1) - (q_1+x_1)^2/2, pi_2 = q_2(1-q_1-q_2) - q_2^2/2. FOCs: d pi_1/d q_1: 1-2q_1-q_2-(q_1+x_1)=0 -> 1-3q_1-q_2-x_1=0; d pi_1/d x_1: a-2x_1-(q_1+x_1)=0 -> a-q_1-3x_1=0; d pi_2/d q_2: 1-q_1-2q_2-q_2=0 -> 1-q_1-3q_2=0. Solve: from firm 2, q_2=(1-q_1)/3; from firm 1’s second FOC, x_1=(a-q_1)/3. Substitute into 1-3q_1-q_2-x_1=0 to get 2-a-7q_1=0 -> q_1=(2-a)/7. Then q_2=(1-q_1)/3=(5+a)/21.", "allowed_symbols": ["a"], "symbol_definitions": {"a": "demand intercept in the second market"}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.5/ii#2", "chapter": "5", "split": "chapter_5", "problem_number": "5.5", "sub_id": "ii", "pair_index": 2, "pair_count": 3, "question": "This exercise illustrates the strategic considerations faced by a multimarket firm. It is inspired by the more general theory of Bulow et al. (1985). There are two firms in a market. They produce perfect substitutes at cost $C(q) = q^2/2$. The demand is $p = 1 - (q_1 + q_2)$. Suppose now that firm 1 has the opportunity to sell the same output on another market as well. The quantity sold on this market is $x_1$, so firm 1's cost is $(q_1 + x_1)^2/2$. The demand on the second market is $p = a - x_1$. Consider the Cournot game in which firm 1 chooses $q_1$ and $x_1$ and firm 2 chooses $q_2$ simultaneously. Using the envelope theorem, compute $\\left.\\frac{d\\pi_1}{da}\\right|_{a=1/4}$ at the Nash equilibrium. Symbols (for final answer): - (none; final answer is a number)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "At the Nash equilibrium, by the envelope theorem, dπ1/da equals the partial derivative ∂π1/∂a holding the equilibrium choices (q1,x1,q2) fixed. Parameter a enters firm 1's profit only through revenue in market 2: x1(a − x1). Thus ∂π1/∂a = x1. For the interior equilibrium, x1 = (8a − 2)/21, so at a = 1/4 we get x1 = (8*(1/4) − 2)/21 = (2 − 2)/21 = 0. Therefore (dπ1/da)|_{a=1/4} = 0.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.8/i", "chapter": "5", "split": "chapter_5", "problem_number": "5.8", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Dansby and Willig (1979) have proposed a look at the effect of small changes in a firm's output on aggregate surplus (consumer surplus plus industry profit). Assume that, for some unspecified reason, firm $i$'s output moves from $q_i$ to $q_i + \\delta q_i$ (for all $i$). (i) True or False: The change in total surplus, $\\delta W$, is equal to \\[ \\sum_i^n (p - C_i') \\, \\delta q_i. \\]", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "True", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let total output be $Q=\\sum_{i=1}^n q_i$ and inverse demand be $p(Q)$. Total surplus is \\[ W(Q,\\{q_i\\})=\\int_0^Q p(x)\\,dx-\\sum_{i=1}^n C_i(q_i). \\] For small changes $\\delta q_i$, we have $\\delta Q=\\sum_i \\delta q_i$ and \\[ \\delta W = p(Q)\\,\\delta Q - \\sum_{i=1}^n C_i'(q_i)\\,\\delta q_i = \\sum_{i=1}^n \\big(p(Q)-C_i'(q_i)\\big)\\,\\delta q_i. \\] Writing $p$ for $p(Q)$ gives $\\delta W=\\sum_i (p-C_i')\\,\\delta q_i$, so the statement is true.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.9/i", "chapter": "5", "split": "chapter_5", "problem_number": "5.9", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Consider a two-firm simultaneous quantity-price game. Let $\\bar{p}$ denote the supremum of prices at which there is a demand: $D(\\bar{p}) = 0$. Look for a mixed-strategy equilibrium. (i) Show that both firms make a zero profit. (Hint: Consider the lowest and the highest price charged by each firm.) Converted target: In any mixed-strategy equilibrium, what is each firm's equilibrium expected profit? Symbols (for final answer): - Pi: each firm's equilibrium expected profit Return the final answer as a JSON object with exactly these keys: P1, P2. Use double-quoted JSON keys and string values.", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "json", "comparison_mode": "json", "reference_answer": {"P1": "0", "P2": "0"}, "reference_answer_sympy": "N/A", "reference_answer_json": {"P1": "0", "P2": "0"}, "reference_answer_json_modes": {"P1": "sympy", "P2": "sympy"}, "answer_for_compare": {"P1": "0", "P2": "0"}, "reference_reasoning": "Let firm i’s mixed strategy over prices have support with lower and upper bounds p_underline_i and p_overline_i. In any mixed-strategy equilibrium, every price in the support must yield the same expected profit. 1) The highest price in support yields zero profit. If p_overline_i > p_overline_j, then when firm i charges p_overline_i it is (weakly) undercut with probability 1, so it sells 0 and earns 0. If instead p_overline_i = p_overline_j but firm i puts no atom at p_overline_i, then again it is undercut with probability 1 when charging p_overline_i, so profit is 0. Hence firm i’s equilibrium expected profit must be 0. 2) The other firm’s profit must also be zero. If firm j had strictly positive equilibrium profit, then its lowest price in support must satisfy p_underline_j > c (otherwise profit cannot be positive). But then firm i could deviate to a price p = p_underline_j - eps > c and choose quantity q = D_p, guaranteeing it is the unique lowest-priced seller and thus sells D_p, earning (p-c)D_p > 0, contradicting that firm i’s equilibrium profit is 0. Therefore firm j’s equilibrium profit is also 0. 3) If both have an atom at the common highest price, it must be p_bar. The remaining case is p_overline_i = p_overline_j and both place positive probability on this price. If D(p_overline_i) > 0, then a firm can profitably deviate to p_overline_i - eps and set quantity q = D_p_overline_i: whenever the rival plays p_overline_i (which happens with positive probability), the deviator becomes the unique low-price firm and sells D_p_overline_i at a strictly higher margin than at p_overline_i, yielding strictly positive expected profit—contradiction. Hence D(p_overline_i) = 0, so p_overline_i = p_overline_j = p_bar. Since the highest-price action in support yields zero profit and all actions in support must yield the same expected profit, both firms’ equilibrium expected profits are zero.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.9/ii", "chapter": "5", "split": "chapter_5", "problem_number": "5.9", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "Consider a two-firm simultaneous quantity-price game. Let $\\bar{p}$ denote the supremum of prices at which there is a demand: $D(\\bar{p}) = 0$. Look for a mixed-strategy equilibrium. (ii) Suppose that each firm $i$ plays according to some continuous distribution $F_i(p)$ on $[\\underline{p_i}, \\bar{p}_i]$ (which can be demonstrated). True or False: Each firm produces $D(p)$ when it charges $p$, if the other firm also produces to satisfy the demand at the price it charges.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "True", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Fix any rationing rule. Suppose firm j, when charging p_j, produces q_j=D(p_j), so it can always satisfy all demand at its own price and never rations. Consider firm i choosing price p and quantity q. If p>p_j, firm i sells 0 and earns −cq. If p<p_j, firm i is the unique lowest-price firm and can sell at most D(p), so profit is p*min{q,D(p)}−cq, maximized by q=D(p) when p>=c (otherwise q=0). If p=p_j, under any rationing rule firm i’s sales are at most D(p), so choosing q>D(p) cannot increase sales beyond D(p) but increases cost; thus an optimal choice satisfies q<=D(p), with q=D(p) when p>=c (otherwise q=0). Since F_j is continuous, ties occur with probability 0, so expected profit is [1−F_j(p)]*p*min{q,D(p)}−cq, which for any fixed p is maximized by q=D(p) (or q=0 if unprofitable). Hence the statement is true (with the usual caveat that if p<c the optimal quantity is 0).", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "5.9/iii#1", "chapter": "5", "split": "chapter_5", "problem_number": "5.9", "sub_id": "iii", "pair_index": 1, "pair_count": 2, "question": "Consider a two-firm simultaneous quantity-price game. Let $\\bar{p}$ denote the supremum of prices at which there is a demand: $D(\\bar{p}) = 0$. Look for a mixed-strategy equilibrium. In a symmetric mixed-strategy equilibrium, what is the equilibrium CDF $F(p)$ for $p<\\bar{p}$? Symbols (for final answer): - `c`: marginal cost - `p`: price", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1 - c/p", "reference_answer_sympy": "1 - c/p", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1 - c/p", "reference_reasoning": "In a symmetric mixed-strategy equilibrium with support on prices below \\(\\bar p\\), each firm must be indifferent across all prices in the support. If the opponent’s price distribution is \\(F\\), then for any \\(p<\\bar p\\) the probability of being the low-price firm is \\(1-F(p)\\), and in that event the firm sells the whole market demand \\(D(p)\\). Thus expected profit at price \\(p\\) is \\(\\pi(p)=(1-F(p))\\,(p-c)\\,D(p)\\). In a mixed equilibrium, \\(\\pi(p)\\) must be constant on the support. Since a firm can guarantee zero profit by pricing at \\(\\bar p\\) (because \\(D(\\bar p)=0\\)), the constant must be 0. For any \\(p<\\bar p\\) with \\(D(p)>0\\), this implies \\((1-F(p))(p-c)=0\\Rightarrow 1-F(p)=c/p\\), hence \\(F(p)=1-c/p\\) for \\(p<\\bar p\\).", "allowed_symbols": ["c", "p"], "symbol_definitions": {"c": "marginal cost", "p": "price"}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.1/1#1", "chapter": "6", "split": "chapter_6", "problem_number": "6.1", "sub_id": "1", "pair_index": 1, "pair_count": 3, "question": "Two firms produce a homogeneous good with constant unit costs $c_1<c_2$. Market demand is $D(p)$ (assume $D$ is continuous and downward sloping on the relevant range). For any unit cost $c$, let $p^m(c)$ denote the monopoly price solving \\[ p^m(c)\\in\\arg\\max_p (p-c)D(p). \\] Side payments (lump-sum transfers) between firms are prohibited. If the two firms could legally sign a contract with transfers, they would let firm 1 produce everything and charge $p^m(c_1)$, then split profits using a transfer. Since transfers are illegal, we instead study \\emph{constrained efficient} profit-sharing agreements without transfers in which both firms charge the same price $p$ and choose market shares $s_1,s_2$ with $s_1+s_2=1$. Interpretation: firm $i$ produces exactly $q_i=s_iD(p)$. If some consumers are rationed at one firm, they purchase from the other firm at the same price $p$ (so total demand served is $D(p)$). Fix a profit target $\\texttt{Pi_bar_2}\\in[0,\\Pi^m(c_2)]$ for firm 2. The constrained efficient allocation chooses $p,s_1,s_2$ to maximize firm 1's profit subject to firm 2 receiving at least $\\texttt{Pi_bar_2}$: \\begin{align} \\max_{(p,s_1,s_2)}\\quad \\Pi^1 &= (p-c_1)s_1D(p) \\tag{6.1}\\\\ \\text{s.t.}\\quad \\Pi^2 &= (p-c_2)s_2D(p)\\ge \\texttt{Pi_bar_2},\\quad s_1+s_2=1. \\notag \\end{align} Suppose that the profit function $(p-c)D(p)$ is concave for all $p$ and $c$. After substituting $s_1=1-s_2$ and using the constraint $\\Pi^2\\ge\\texttt{Pi_bar_2}$ to eliminate (or bound) $s_2$, what is the (interior) first-order condition for the optimal price $p$, expressed as an equation equal to 0? (You may use the shorthand profit functions $\\texttt{Phi_1}(p)=(p-c_1)D(p)$ and $\\texttt{Phi_2}(p)=(p-c_2)D(p)$ and their derivatives $\\texttt{Phi_1_p}$ and $\\texttt{Phi_2_p}$.) Symbols (for final answer): - `Phi_1`: profit function Phi_1(p) = (p - c_1) D(p) - `Phi_1_p`: derivative Phi_1'(p) with respect to p - `Phi_2`: profit function Phi_2(p) = (p - c_2) D(p) - `Phi_2_p`: derivative Phi_2'(p) with respect to p - `Pi_bar_2`: target profit level for firm 2, \\bar{\\Pi}^2", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "Phi_1_p*(1 - Pi_bar_2/Phi_2) + Phi_1*Pi_bar_2*Phi_2_p/Phi_2**2", "reference_answer_sympy": "Phi_1_p*(1 - Pi_bar_2/Phi_2) + Phi_1*Pi_bar_2*Phi_2_p/Phi_2**2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "Phi_1_p*(1 - Pi_bar_2/Phi_2) + Phi_1*Pi_bar_2*Phi_2_p/Phi_2**2", "reference_reasoning": "Let \\(\\Phi_i(p)\\equiv (p-c_i)D(p)\\). The constraint \\(\\Pi^2=\\Phi_2(p)s_2\\ge \\bar\\Pi^2\\) implies \\(s_2\\ge \\bar\\Pi^2/\\Phi_2(p)\\) (requiring \\(\\Phi_2(p)>0\\)). Since firm 1’s objective is decreasing in \\(s_2\\), the optimum sets \\(s_2^*(p)=\\bar\\Pi^2/\\Phi_2(p)\\) and \\(s_1^*(p)=1-\\bar\\Pi^2/\\Phi_2(p)\\). The reduced-form objective is \\(f(p)=\\Phi_1(p)\\left(1-\\frac{\\bar\\Pi^2}{\\Phi_2(p)}\\right)\\). Differentiating and setting the derivative to zero yields the interior FOC: \\(0=\\Phi_1'(p)\\left(1-\\frac{\\bar\\Pi^2}{\\Phi_2(p)}\\right)+\\Phi_1(p)\\,\\bar\\Pi^2\\,\\frac{\\Phi_2'(p)}{\\Phi_2(p)^2}.\\)", "allowed_symbols": ["Phi_1", "Phi_1_p", "Phi_2", "Phi_2_p", "Pi_bar_2"], "symbol_definitions": {"Phi_1": "profit function Phi_1(p) = (p - c_1) D(p)", "Phi_1_p": "derivative Phi_1'(p) with respect to p", "Phi_2": "profit function Phi_2(p) = (p - c_2) D(p)", "Phi_2_p": "derivative Phi_2'(p) with respect to p", "Pi_bar_2": "target profit level for firm 2, \\bar{\\Pi}^2"}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.1/1#2", "chapter": "6", "split": "chapter_6", "problem_number": "6.1", "sub_id": "1", "pair_index": 2, "pair_count": 3, "question": "Two firms produce a homogeneous good with constant unit costs $c_1<c_2$. Market demand is $D(p)$ (assume $D$ is continuous and downward sloping on the relevant range). For any unit cost $c$, let $p^m(c)$ denote the monopoly price solving \\[ p^m(c)\\in\\arg\\max_p (p-c)D(p). \\] Side payments (lump-sum transfers) between firms are prohibited. If the two firms could legally sign a contract with transfers, they would let firm 1 produce everything and charge $p^m(c_1)$, then split profits using a transfer. Since transfers are illegal, we instead study \\emph{constrained efficient} profit-sharing agreements without transfers in which both firms charge the same price $p$ and choose market shares $s_1,s_2$ with $s_1+s_2=1$. Interpretation: firm $i$ produces exactly $q_i=s_iD(p)$. If some consumers are rationed at one firm, they purchase from the other firm at the same price $p$ (so total demand served is $D(p)$). Fix a profit target $\\bar{\\Pi}^2\\in[0,\\Pi^m(c_2)]$ for firm 2. The constrained efficient allocation chooses $p,s_1,s_2$ to maximize firm 1's profit subject to firm 2 receiving at least $\\bar{\\Pi}^2$: \\begin{align} \\max_{(p,s_1,s_2)}\\quad \\Pi^1 &= (p-c_1)s_1D(p) \\tag{6.1}\\\\ \\text{s.t.}\\quad \\Pi^2 &= (p-c_2)s_2D(p)\\ge \\bar{\\Pi}^2,\\quad s_1+s_2=1. \\notag \\end{align} Suppose that the profit function $(p-c)D(p)$ is concave for all $p$ and $c$. Judge whether the following statement is true or false: \\[ \\text{At any constrained-efficient optimum, }\\; p^m(c_1)\\le p\\le p^m(c_2). \\]", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "True", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let \\(\\Phi_i(p)=(p-c_i)D(p)\\). Using the constraint \\(\\Pi^2\\ge \\bar\\Pi^2\\) and \\(s_1=1-s_2\\), the problem can be reduced to choosing \\(p\\) with objective \\(f(p)=\\Phi_1(p)\\left(1-\\bar\\Pi^2/\\Phi_2(p)\\right)\\) over the feasible set where \\(\\Phi_2(p)\\ge \\bar\\Pi^2\\). For an interior optimum, the FOC is \\[ 0=\\Phi_1'(p)\\Big(1-\\frac{\\bar\\Pi^2}{\\Phi_2(p)}\\Big)+\\Phi_1(p)\\,\\bar\\Pi^2\\,\\frac{\\Phi_2'(p)}{\\Phi_2(p)^2}. \\] Feasibility implies \\(1-\\bar\\Pi^2/\\Phi_2(p)\\ge 0\\). At an optimum \\(\\Phi_1(p)\\ge 0\\) (otherwise firm 1 could set \\(s_1=0\\) and not do worse). Thus the first term has the sign of \\(\\Phi_1'(p)\\) and the second term has the sign of \\(\\Phi_2'(p)\\). For their sum to be zero, we must have \\(\\Phi_1'(p)\\le 0\\le \\Phi_2'(p)\\). Since each \\(\\Phi_i\\) is concave, \\(\\Phi_i'(p)\\ge 0\\) for \\(p\\le p^m(c_i)\\) and \\(\\Phi_i'(p)\\le 0\\) for \\(p\\ge p^m(c_i)\\). Therefore \\(\\Phi_1'(p)\\le 0\\Rightarrow p\\ge p^m(c_1)\\) and \\(\\Phi_2'(p)\\ge 0\\Rightarrow p\\le p^m(c_2)\\), yielding \\(p^m(c_1)\\le p\\le p^m(c_2)\\).", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.1/2", "chapter": "6", "split": "chapter_6", "problem_number": "6.1", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Two firms produce a homogeneous good with constant unit costs $c_1<c_2$. Market demand is $D_p$ (assume demand is continuous and downward sloping on the relevant range). For any unit cost $c$, let $p^m(c)$ denote the monopoly price solving \\[ p^m(c)\\in\\arg\\max_p (p-c)D_p. \\] Side payments (lump-sum transfers) between firms are prohibited. If the two firms could legally sign a contract with transfers, they would let firm 1 produce everything and charge $p^m(c_1)$, then split profits using a transfer. Since transfers are illegal, we instead study \\emph{constrained efficient} profit-sharing agreements without transfers in which both firms charge the same price $p$ and choose market shares $s_1,s_2$ with $s_1+s_2=1$. Interpretation: firm $i$ produces exactly $q_i=s_iD_p$. If some consumers are rationed at one firm, they purchase from the other firm at the same price $p$ (so total demand served is $D_p$). Fix a profit target $Pi_bar_2\\in[0,\\Pi^m(c_2)]$ for firm 2. The constrained efficient allocation chooses $p,s_1,s_2$ to maximize firm 1's profit subject to firm 2 receiving at least $Pi_bar_2$: \\begin{align} \\max_{(p,s_1,s_2)}\\quad \\Pi^1 &= (p-c_1)s_1D_p \\tag{6.1}\\\\ \\text{s.t.}\\quad \\Pi^2 &= (p-c_2)s_2D_p\\ge Pi_bar_2,\\quad s_1+s_2=1. \\notag \\end{align} Suppose that the profit function $(p-c)D_p$ is concave for all $p$ and $c$. (ii) Let firm 2's constraint bind and substitute $s_2(p)=Pi_bar_2/((p-c_2)D_p)$ into firm 1's objective to obtain \\[\\Pi^1(p)=(p-c_1)D_p-Pi_bar_2\\frac{p-c_1}{p-c_2}.\\] Compute the first-order condition $\\frac{d\\Pi^1}{dp}=0$ and give the resulting expression (set equal to zero). Use $Dprime_p$ to denote the derivative of demand with respect to $p$. Symbols (for final answer): - `D_p`: market demand evaluated at p, i.e., D(p) - `Dprime_p`: derivative of demand at p, i.e., D'(p) - `Pi_bar_2`: profit target for firm 2 (\\bar{\\Pi}^2) - `c_1`: unit cost of firm 1 - `c_2`: unit cost of firm 2 - `p`: common price", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(p-c_1)*Dprime_p + D_p + (c_2-c_1)*Pi_bar_2/(p-c_2)**2", "reference_answer_sympy": "(p-c_1)*Dprime_p + D_p + (c_2-c_1)*Pi_bar_2/(p-c_2)**2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(p-c_1)*Dprime_p + D_p + (c_2-c_1)*Pi_bar_2/(p-c_2)**2", "reference_reasoning": "With the constraint binding, s_2(p)=Pi_bar_2/((p-c_2)D(p)) and s_1(p)=1-s_2(p). Substituting into firm 1's profit gives Pi^1(p)=(p-c_1)D(p)-Pi_bar_2*(p-c_1)/(p-c_2). Differentiate: d/dp[(p-c_1)D(p)]= (p-c_1)D'(p)+D(p). Also d/dp[(p-c_1)/(p-c_2)]=((p-c_2)-(p-c_1))/(p-c_2)^2=(c_1-c_2)/(p-c_2)^2. Therefore dPi^1/dp = [(p-c_1)D'(p)+D(p)] - Pi_bar_2*(c_1-c_2)/(p-c_2)^2 = [(p-c_1)D'(p)+D(p)] + (c_2-c_1)Pi_bar_2/(p-c_2)^2. Setting equal to zero yields the stated FOC.", "allowed_symbols": ["D_p", "Dprime_p", "Pi_bar_2", "c_1", "c_2", "p"], "symbol_definitions": {"D_p": "market demand evaluated at p, i.e., D(p)", "Dprime_p": "derivative of demand at p, i.e., D'(p)", "Pi_bar_2": "profit target for firm 2 (\\bar{\\Pi}^2)", "c_1": "unit cost of firm 1", "c_2": "unit cost of firm 2", "p": "common price"}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.1/3", "chapter": "6", "split": "chapter_6", "problem_number": "6.1", "sub_id": "3", "pair_index": 1, "pair_count": 1, "question": "Two firms produce a homogeneous good with constant unit costs $c_1<c_2$. Market demand is $D(p)$ (assume $D$ is continuous and downward sloping on the relevant range). For any unit cost $c$, let $p^m(c)$ denote the monopoly price solving \\[ p^m(c)\\in\\arg\\max_p (p-c)D(p). \\] Side payments (lump-sum transfers) between firms are prohibited. If the two firms could legally sign a contract with transfers, they would let firm 1 produce everything and charge $p^m(c_1)$, then split profits using a transfer. Since transfers are illegal, we instead study \\emph{constrained efficient} profit-sharing agreements without transfers in which both firms charge the same price $p$ and choose market shares $s_1,s_2$ with $s_1+s_2=1$. Interpretation: firm $i$ produces exactly $q_i=s_iD(p)$. If some consumers are rationed at one firm, they purchase from the other firm at the same price $p$ (so total demand served is $D(p)$). Fix a profit target $\\bar{\\Pi}^2\\in[0,\\Pi^m(c_2)]$ for firm 2. The constrained efficient allocation chooses $p,s_1,s_2$ to maximize firm 1's profit subject to firm 2 receiving at least $\\bar{\\Pi}^2$: \\begin{align} \\max_{(p,s_1,s_2)}\\quad \\Pi^1 &= (p-c_1)s_1D(p) \\tag{6.1}\\\\ \\text{s.t.}\\quad \\Pi^2 &= (p-c_2)s_2D(p)\\ge \\bar{\\Pi}^2,\\quad s_1+s_2=1. \\notag \\end{align} Suppose that the profit function $(p-c)D(p)$ is concave for all $p$ and $c$. (iii) Judge the following statement: As $\\bar{\\Pi}^2$ grows, both the constrained-efficient common price $p$ and firm 2’s market share $s_2$ (weakly) grow.", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "True", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let \\(\\Phi_i(p)=(p-c_i)D(p)\\). At the constrained-efficient solution, firm 2’s constraint binds (otherwise firm 1 can reduce \\(s_2\\) and raise \\(\\Pi^1\\)), so \\(\\Phi_2(p)s_2=\\bar\\Pi^2\\) and \\(s_2=\\bar\\Pi^2/\\Phi_2(p)\\). Substituting into firm 1’s objective gives \\(\\Pi^1=\\Phi_1(p)-\\bar\\Pi^2\\,\\Phi_1(p)/\\Phi_2(p)\\). The interior FOC in \\(p\\) implies an implicit equation in \\(p\\) and \\(\\bar\\Pi^2\\). Implicit differentiation yields \\(dp/d\\bar\\Pi^2\\) equal to a numerator \\(\\Phi_1'(p)\\Phi_2(p)-\\Phi_1(p)\\Phi_2'(p)\\) divided by the SOC term (negative under concavity). For \\(c_1<c_2\\), \\(\\Phi_1'(p)\\Phi_2(p)-\\Phi_1(p)\\Phi_2'(p)=(c_2-c_1)D(p)\\big(D(p)+(p-c_2)D'(p)\\big)\\), which is positive at any feasible optimum with \\(\\bar\\Pi^2>0\\) (requiring \\(\\Phi_2(p)>0\\Rightarrow p>c_2\\) and \\(D(p)+(p-c_2)D'(p)>0\\)). Hence \\(dp/d\\bar\\Pi^2>0\\). Since \\(s_2=\\bar\\Pi^2/\\Phi_2(p)\\), increasing \\(\\bar\\Pi^2\\) requires (weakly) increasing \\(s_2\\) along the optimum; thus both \\(p\\) and \\(s_2\\) are (weakly) increasing in \\(\\bar\\Pi^2\\).", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.1/4", "chapter": "6", "split": "chapter_6", "problem_number": "6.1", "sub_id": "4", "pair_index": 1, "pair_count": 1, "question": "Two firms produce a homogeneous good with constant unit costs $c_1<c_2$. Market demand is $D(p)$ (assume $D$ is continuous and downward sloping on the relevant range). For any unit cost $c$, let $p^m(c)$ denote the monopoly price solving \\[ p^m(c)\\in\\arg\\max_p (p-c)D(p). \\] Side payments (lump-sum transfers) between firms are prohibited. If the two firms could legally sign a contract with transfers, they would let firm 1 produce everything and charge $p^m(c_1)$, then split profits using a transfer. Since transfers are illegal, we instead study \\emph{constrained efficient} profit-sharing agreements without transfers in which both firms charge the same price $p$ and choose market shares $s_1,s_2$ with $s_1+s_2=1$. Interpretation: firm $i$ produces exactly $q_i=s_iD(p)$. If some consumers are rationed at one firm, they purchase from the other firm at the same price $p$ (so total demand served is $D(p)$). Fix a profit target $\\bar{\\Pi}^2\\in[0,\\Pi^m(c_2)]$ for firm 2. The constrained efficient allocation chooses $p,s_1,s_2$ to maximize firm 1's profit subject to firm 2 receiving at least $\\bar{\\Pi}^2$: \\begin{align} \\max_{(p,s_1,s_2)}\\quad \\Pi^1 &= (p-c_1)s_1D(p) \\tag{6.1}\\\\ \\text{s.t.}\\quad \\Pi^2 &= (p-c_2)s_2D(p)\\ge \\bar{\\Pi}^2,\\quad s_1+s_2=1. \\notag \\end{align} Suppose that the profit function $(p-c)D(p)$ is concave for all $p$ and $c$. (iv) True or False: Under these assumptions, the Pareto frontier (the set of constrained efficient payoff pairs $(\\Pi^1,\\Pi^2)$) is convex.", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let \\(\\Phi_i(p)=(p-c_i)D(p)\\). Feasible payoffs satisfy \\(\\Pi^1=s_1\\Phi_1(p)\\), \\(\\Pi^2=s_2\\Phi_2(p)\\), with \\(s_1+s_2=1\\). For a given target \\(\\bar\\Pi^2\\), the constraint binds at the constrained-efficient solution (otherwise increase \\(s_1\\)), so \\(s_2=\\bar\\Pi^2/\\Phi_2(p)\\) and \\(s_1=1-\\bar\\Pi^2/\\Phi_2(p)\\). Then firm 1's payoff as a function of \\(p\\) is \\(\\Pi^1(p;\\bar\\Pi^2)=\\Phi_1(p)-\\bar\\Pi^2\\,\\Phi_1(p)/\\Phi_2(p)\\). Let \\(V(\\bar\\Pi^2)=\\max_p \\Pi^1(p;\\bar\\Pi^2)\\). Envelope theorem gives \\(V'(\\bar\\Pi^2)=-\\Phi_1(p^*)/\\Phi_2(p^*)\\). Differentiating again yields \\(V''(\\bar\\Pi^2)=-(d/d\\bar\\Pi^2)(\\Phi_1(p^*)/\\Phi_2(p^*))\\), which equals a product of (i) a term proportional to \\(\\Phi_1'(p^*)\\Phi_2(p^*)-\\Phi_1(p^*)\\Phi_2'(p^*)\\) divided by \\(\\Phi_2(p^*)^2\\) and (ii) \\(dp^*/d\\bar\\Pi^2\\). Using \\(c_1<c_2\\) and downward-sloping demand implies \\(\\Phi_1'(p)<\\Phi_2'(p)\\). With concavity and optimality, \\(\\Phi_2'(p^*)\\le 0\\) and \\(\\Phi_2(p^*)\\ge 0\\), giving \\(\\Phi_1'(p^*)\\Phi_2(p^*)-\\Phi_1(p^*)\\Phi_2'(p^*)\\le 0\\). From part (iii), \\(dp^*/d\\bar\\Pi^2>0\\). Hence \\(V''(\\bar\\Pi^2)\\ge 0\\), so \\(\\Pi^1=V(\\Pi^2)\\) is convex in \\(\\Pi^2\\), i.e., the Pareto frontier is convex. Therefore the statement is True.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.1/5", "chapter": "6", "split": "chapter_6", "problem_number": "6.1", "sub_id": "5", "pair_index": 1, "pair_count": 1, "question": "Two firms produce a homogeneous good with constant unit costs $c_1<c_2$. Market demand is $D(p)$ (assume $D$ is continuous and downward sloping on the relevant range). For any unit cost $c$, let $p^m(c)$ denote the monopoly price solving \\[ p^m(c)\\in\\arg\\max_p (p-c)D(p). \\] Side payments (lump-sum transfers) between firms are prohibited. If the two firms could legally sign a contract with transfers, they would let firm 1 produce everything and charge $p^m(c_1)$, then split profits using a transfer. Since transfers are illegal, we instead study \\emph{constrained efficient} profit-sharing agreements without transfers in which both firms charge the same price $p$ and choose market shares $s_1,s_2$ with $s_1+s_2=1$. Interpretation: firm $i$ produces exactly $q_i=s_iD(p)$. If some consumers are rationed at one firm, they purchase from the other firm at the same price $p$ (so total demand served is $D(p)$). Fix a profit target $\\bar{\\Pi}^2\\in[0,\\Pi^m(c_2)]$ for firm 2. The constrained efficient allocation chooses $p,s_1,s_2$ to maximize firm 1's profit subject to firm 2 receiving at least $\\bar{\\Pi}^2$: \\begin{align} \\max_{(p,s_1,s_2)}\\quad \\Pi^1 &= (p-c_1)s_1D(p) \\tag{6.1}\\\\ \\text{s.t.}\\quad \\Pi^2 &= (p-c_2)s_2D(p)\\ge \\bar{\\Pi}^2,\\quad s_1+s_2=1. \\notag \\end{align} Suppose that the profit function $(p-c)D(p)$ is concave for all $p$ and $c$. Judge whether the following conclusion is correct: \"The efficient market-sharing allocation is optimal only within deterministic allocations; allowing randomization (e.g., a coin toss to decide which firm is the monopolist, or alternating monopoly in a repeated game with low impatience) can yield higher expected payoffs and achieve any convex combination of the relevant payoff points.\"", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "True", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Part (iv) implies that the deterministic constrained-efficient market-sharing solution does not generally lie on the full Pareto frontier once stochastic (randomized) allocations are allowed. By randomizing over which firm serves the market as a monopolist (or by taking turns in a repeated game with sufficiently patient firms), the firms can attain convex combinations of payoff outcomes (the straight line between the two extreme points), which can strictly dominate the deterministic market-sharing outcome in expected payoffs.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.1/6", "chapter": "6", "split": "chapter_6", "problem_number": "6.1", "sub_id": "6", "pair_index": 1, "pair_count": 1, "question": "Two firms produce a homogeneous good with constant unit costs $c_1<c_2$. Market demand is $D(p)$ (assume $D$ is continuous and downward sloping on the relevant range). For any unit cost $c$, let $p^m(c)$ denote the monopoly price solving \\[ p^m(c)\\in\\arg\\max_p (p-c)D(p). \\] Side payments (lump-sum transfers) between firms are prohibited. If the two firms could legally sign a contract with transfers, they would let firm 1 produce everything and charge $p^m(c_1)$, then split profits using a transfer. Since transfers are illegal, we instead study \\emph{constrained efficient} profit-sharing agreements without transfers in which both firms charge the same price $p$ and choose market shares $s_1,s_2$ with $s_1+s_2=1$. Interpretation: firm $i$ produces exactly $q_i=s_iD(p)$. If some consumers are rationed at one firm, they purchase from the other firm at the same price $p$ (so total demand served is $D(p)$). Fix a profit target $\\bar{\\Pi}^2\\in[0,\\Pi^m(c_2)]$ for firm 2. The constrained efficient allocation chooses $p,s_1,s_2$ to maximize firm 1's profit subject to firm 2 receiving at least $\\bar{\\Pi}^2$: \\begin{align} \\max_{(p,s_1,s_2)}\\quad \\Pi^1 &= (p-c_1)s_1D(p) \\tag{6.1}\\\\ \\text{s.t.}\\quad \\Pi^2 &= (p-c_2)s_2D(p)\\ge \\bar{\\Pi}^2,\\quad s_1+s_2=1. \\notag \\end{align} Suppose that the profit function $(p-c)D(p)$ is concave for all $p$ and $c$. Now allow the firms to choose potentially different prices $(p_1,p_2)$, but keep the same \"efficient rationing\" rule: consumers always attempt to buy from the lower-priced firm first; if they are rationed there (because that firm supplies less than demand at its price), they then buy from the higher-priced firm at its price whenever they are willing to pay it. True or False: Under this rationing rule, any deterministic constrained efficient market-sharing allocation must have identical prices (i.e., $p_1=p_2$).", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Take any deterministic constrained-efficient allocation with prices (p1,p2) and quantities (q1,q2) under the efficient-rationing rule. WLOG assume p1<p2. Then consumers buy from firm 1 first at p1 up to q1, and any remaining consumers with willingness to pay at least p2 buy from firm 2 at p2. Feasibility implies q1<=D(p1) and total sales at prices (p1,p2) equal D(p2), so q1+q2=D(p2) and q2=max(0,D(p2)-q1). Now raise firm 1's price to p1'=p2 while keeping quantities fixed at (q1,q2). Because demand is downward sloping and p2>p1, we have D(p2)<=D(p1), hence q1<=D(p1) implies q1<=D(p2)=D(p1'). Thus the same quantities remain feasible under efficient rationing with equal prices: firm 1 sells q1 and firm 2 sells the residual q2. Firm 2's profit is unchanged (same price p2 and same quantity q2), so the constraint Pi2>=barPi2 still holds. Firm 1's profit weakly increases because its unit margin rises from (p1-c1) to (p2-c1) while its quantity stays q1. Therefore any allocation with p1<p2 cannot be constrained efficient. By symmetry, p2<p1 is also impossible. Hence any deterministic constrained-efficient market-sharing allocation must have p1=p2.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.2/main", "chapter": "6", "split": "chapter_6", "problem_number": "6.2", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider the infinitely repeated (horizon $T=+\\infty$) two-firm Bertrand price game (a supergame) in which the two firms produce perfect substitutes with the same marginal cost $c$. In each period $t=0,1,2,\\ldots$, firms simultaneously choose prices $(p_{1t},p_{2t})$. Let $D(p)$ denote market demand at price $p$, and define \\[ \\Pi(p)\\equiv (p-c)D(p). \\] If $p_{it}<p_{jt}$, firm $i$ gets the whole market and earns per-period profit $\\Pi(p_{it})$; if $p_{it}=p_{jt}$, the firms share the market and firm $i$ earns $\\Pi(p_{it})/2$; if $p_{it}>p_{jt}$, firm $i$ earns $0$. Let $p^m$ denote the monopoly price (it maximizes $(p-c)D(p)$), and let $\\Pi^m\\equiv \\Pi(p^m)$. A (history-dependent) strategy for firm $i$ specifies $p_{it}(H_t)$ after every history \\[ H_t \\equiv (p_{10},p_{20};\\ldots;p_{1,t-1},p_{2,t-1}). \\] Firms discount future profits with factor $\\delta\\in(0,1)$. For a strategy profile, define firm $i$'s associated per-period payoff as \\[ (1-\\delta)\\sum_{t=0}^{\\infty}\\delta^t \\Pi^i(p_{it},p_{jt}), \\] and call a strategy profile a \\emph{perfect equilibrium} if, for any history $H_t$ at date $t$, firm $i$'s continuation strategy from date $t$ on maximizes the present discounted value of its profits given firm $j$'s continuation strategy from that date on. In the construction described in the reference answer, along the cooperation path there is a cycle of length $m+n$ in which firm 1 gets the whole market at price $p$ for $m$ consecutive periods and firm 2 gets the whole market at price $p$ for $n$ consecutive periods, and this cycle repeats forever. Question: What is firm 1's normalized discounted payoff on the cooperation path, $V_1^C$, as a function of `delta`, `m`, `n`, and `Pi_p` (where `Pi_p` denotes $\\Pi(p)$)? Symbols (for final answer): - `Pi_p`: per-period monopoly profit at price p, i.e., Pi(p) - `delta`: discount factor in (0,1) - `m`: positive integer number of consecutive periods firm 1 gets the market in each cycle - `n`: positive integer number of consecutive periods firm 2 gets the market in each cycle", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "Pi_p*(1-delta**m)/(1-delta**(m+n))", "reference_answer_sympy": "Pi_p*(1-delta**m)/(1-delta**(m+n))", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "Pi_p*(1-delta**m)/(1-delta**(m+n))", "reference_reasoning": "On the cooperation path, firm 1 earns per-period profit \\Pi_p in periods t=0,1,...,m-1 of each cycle of length m+n, and earns 0 in the remaining n periods of the cycle. The normalized discounted payoff is V_1^C=(1-\\delta)\\sum_{k\\ge 0}\\sum_{s=0}^{m-1} \\delta^{k(m+n)+s} \\Pi_p. Compute the inner sum: \\sum_{s=0}^{m-1}\\delta^s=(1-\\delta^m)/(1-\\delta). Then V_1^C=(1-\\delta)\\Pi_p \\sum_{k\\ge 0} \\delta^{k(m+n)} \\cdot (1-\\delta^m)/(1-\\delta) =\\Pi_p(1-\\delta^m)\\sum_{k\\ge 0}(\\delta^{m+n})^k =\\Pi_p(1-\\delta^m)\\cdot 1/(1-\\delta^{m+n}).", "allowed_symbols": ["Pi_p", "delta", "m", "n"], "symbol_definitions": {"Pi_p": "per-period monopoly profit at price p, i.e., Pi(p)", "delta": "discount factor in (0,1)", "m": "positive integer number of consecutive periods firm 1 gets the market in each cycle", "n": "positive integer number of consecutive periods firm 2 gets the market in each cycle"}, "review_status": "confirmed_by_expert"} | |
| {"id": "6.3/main", "chapter": "6", "split": "chapter_6", "problem_number": "6.3", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider the (infinite-horizon) repeated price game between two firms producing perfect substitutes with the same constant marginal cost $c$. In each period $t=0,1,2,\\dots$, firms simultaneously choose prices $(p_{1t},p_{2t})$. Let $D(p)$ denote market demand at price $p$, and define the profit from serving the whole market at price $p$ by \\[ \\Pi(p)\\equiv (p-c)D(p). \\] If one firm charges a strictly lower price than its rival, it gets the whole market (so the higher-price firm sells nothing). If both firms charge the same price $p$, they share the market: firm 1 sells a share $s_1$ and firm 2 sells a share $1-s_1$, so their one-period profits are $s_1\\Pi(p)$ and $(1-s_1)\\Pi(p)$, respectively. A pure strategy for firm $i$ specifies, for each date $t$, a price $p_{it}$ as a function of the history of past prices \\[ H_t \\equiv (p_{10},p_{20},\\dots,p_{1,t-1},p_{2,t-1}). \\] Each firm maximizes the discounted sum of one-period profits, and we associate to any strategy profile the (normalized) per-period profit \\[ (1-\\delta)\\sum_{t=0}^{\\infty}\\delta^t\\,\\Pi^i_t, \\] where $\\delta\\in(0,1)$ is the discount factor and $\\Pi^i_t$ is firm $i$'s profit in period $t$. A (subgame-)perfect equilibrium is a strategy profile such that after every history $H_t$, each firm's continuation strategy maximizes its continuation discounted payoff given the rival's continuation strategy. Let $\\mathrm{bar\\_pi}$ be the supremum of (normalized) per-period profits that a firm can obtain in any pure-strategy subgame-perfect equilibrium. For $\\delta\\le 1/2$, compute $\\mathrm{bar\\_pi}$. Symbols (for final answer): - `bar_pi`: supremum of normalized per-period profits attainable by a firm in any pure-strategy subgame-perfect equilibrium - `delta`: discount factor in (0,1)", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "Let bar_pi be the supremum normalized per-period profit attainable by a firm in any pure-strategy SPE. Suppose bar_pi>0. Pick eps in (0,bar_pi) and an SPE where firm 1 gets at least bar_pi-eps. Then along the equilibrium path there is some period t with positive profit and a minimal such price p_t>c; at that t both firms must charge p_t, so firm 1 earns pi_1t=s_1*Pi(p_t)>0. Choose t so that pi_1t>=bar_pi-eps (otherwise the normalized average would be <bar_pi-eps). Consider firm 2 deviating at history H_t to p_t-eta with eta>0 small. Then firm 2 captures the whole market at t and its one-period profit approaches Pi(p_t) as eta->0, whereas its equilibrium one-period profit is (1-s_1)Pi(p_t). Thus the one-period gain from deviating can be made arbitrarily close to Pi(p_t)-(1-s_1)Pi(p_t)=s_1Pi(p_t)=pi_1t>=bar_pi-eps. The maximal continuation loss (in normalized value) from t+1 onward is bounded by the present value of losing at most bar_pi per period forever: (1-delta)*sum_{k=1}^infty delta^k*bar_pi = delta*bar_pi. For the deviation not to be profitable, we need bar_pi-eps <= delta*bar_pi. Letting eps->0 gives bar_pi <= delta*bar_pi, which with delta<1 implies bar_pi=0. Hence for all delta<=1/2, bar_pi=0.", "allowed_symbols": ["bar_pi", "delta"], "symbol_definitions": {"bar_pi": "supremum of normalized per-period profits attainable by a firm in any pure-strategy subgame-perfect equilibrium", "delta": "discount factor in (0,1)"}, "review_status": "confirmed_by_expert"} | |
| {"id": "6.4/main#1", "chapter": "6", "split": "chapter_6", "problem_number": "6.4", "sub_id": "main", "pair_index": 1, "pair_count": 2, "question": "Consider an $n$-firm supergame framework. The firms have constant marginal cost $c$. The demand function at date $t$ is $q_t=\\mu^tD(p_t)$, where $\\mu\\delta<1$ ($\\delta$ is the discount factor). Derive the set of discount factors such that full collusion (i.e., the monopoly solution) is sustainable as an equilibrium of the supergame. Compute the minimum discount factor $\\delta_{\\min}$ (as a function of $n$ and $\\mu$) such that full collusion is sustainable under grim-trigger reversion to the competitive outcome. Symbols (for final answer): - `mu`: demand growth/decay factor (mu) - `n`: number of firms", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(1 - 1/n)/mu", "reference_answer_sympy": "(1 - 1/n)/mu", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(1 - 1/n)/mu", "reference_reasoning": "Let pi_m be the one-period monopoly profit when demand is q=D(p) at t=0. At date t, monopoly profit is mu^t*pi_m. Under symmetric full collusion each firm gets mu^t*pi_m/n each period. If a firm deviates in period t while others charge the monopoly price, it can capture the whole monopoly profit in that period, i.e. mu^t*pi_m. With grim-trigger reversion to the competitive outcome (zero profit) thereafter, the incentive constraint at any t is: sum_{k=0}^infty (delta*mu)^k * (pi_m/n) >= pi_m. This gives (pi_m/n) * 1/(1-delta*mu) >= pi_m, so 1/(n(1-delta*mu)) >= 1, hence delta*mu >= 1 - 1/n. Therefore the minimum discount factor is delta_min = (1 - 1/n)/mu.", "allowed_symbols": ["mu", "n"], "symbol_definitions": {"mu": "demand growth/decay factor (mu)", "n": "number of firms"}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.6/1", "chapter": "6", "split": "chapter_6", "problem_number": "6.6", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consider two firms interacting in two identical and independent markets. In each market, the firms would like to sustain a constant collusive price (e.g., the monopoly price). Let $\\Pi^m$ denote the per-period industry profit associated with charging the monopoly price in a market. Under full collusion, each firm then receives $\\Pi^m/2$ per market per period. A unilateral undercut of the monopoly price in a market yields a one-period gain of $\\Pi^m/2$ in that market (relative to collusion). After any detected deviation, firms revert forever to Bertrand behavior (the maximal punishment), yielding zero profit in the punished market(s). The common discount factor between periods is $\\delta\\in(0,1)$. The markets differ in that in market 1 a firm's price at time $t$ is observed at $t+1$, whereas in market 2 it is learned only at $t+2$. Thus, although each of the markets meets every period, market 2 has longer information lags. In the absence of multimarket contact, what is the minimum discount factor $\\delta$ (as an exact expression) such that collusion in market 2 is sustainable? Symbols (for final answer): - delta: discount factor", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/sqrt(2)", "reference_answer_sympy": "1/sqrt(2)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/sqrt(2)", "reference_reasoning": "In market 2, a deviation at time t is detected only at t+2, so the deviator can obtain the deviation gain for two periods before punishment starts. If a firm colludes forever, its PV from market 2 is V_C = (Pi^m/2)/(1-delta). If it deviates at t and continues to undercut at t+1 (still undetected), it gets an extra gain Pi^m/2 in each of the two periods, and from t+2 onward gets 0 in that market, so V_D = (Pi^m/2)/(1-delta) + (Pi^m/2) + delta*(Pi^m/2). Sustainability requires V_C >= V_D, which reduces to delta^2 >= 1/2, hence delta >= 1/sqrt(2). Therefore the minimum discount factor is 1/sqrt(2).", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.6/2", "chapter": "6", "split": "chapter_6", "problem_number": "6.6", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Consider two firms interacting in two identical and independent markets. In each market, the firms would like to sustain a constant collusive price (e.g., the monopoly price). Let $\\Pi^m$ denote the per-period industry profit associated with charging the monopoly price in a market. Under full collusion, each firm then receives $\\Pi^m/2$ per market per period. A unilateral undercut of the monopoly price in a market yields a one-period gain of $\\Pi^m/2$ in that market (relative to collusion). After any detected deviation, firms revert forever to Bertrand behavior (the maximal punishment), yielding zero profit in the punished market(s). The common discount factor between periods is $\\delta\\in(0,1)$. The markets differ in that in market 1 a firm's price at time $t$ is observed at $t+1$, whereas in market 2 it is learned only at $t+2$. Thus, although each of the markets meets every period, market 2 has longer information lags. Question: Compute the exact expression for $\\bar{\\delta}$ (the minimum discount factor sustaining collusion in both markets under multimarket contact). Symbols (for final answer): - delta_bar", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(1+sqrt(17))/8", "reference_answer_sympy": "(1+sqrt(17))/8", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(1+sqrt(17))/8", "reference_reasoning": "Under multimarket contact, any detected deviation in either market triggers reversion to Bertrand in both markets forever. The optimal deviation exploits the longer information lag in market 2: deviate in market 2 at time t (not observed until t+2), gaining an extra Pi^m/2 at t. At time t+1, before punishment can start, deviate in both markets (market 1 deviation observed at t+2, and deviate again in market 2), gaining an extra Pi^m/2 in each market at t+1. Thus deviation gain is G = (Pi^m/2) + delta*(Pi^m/2 + Pi^m/2) = (Pi^m/2)(1+2 delta). Punishment starts at t+2, so the deviator loses collusive profits in both markets from t+2 onward. Under collusion it earns Pi^m per period total, so discounted loss is L = sum_{k=2}^infty delta^k Pi^m = (delta^2/(1-delta)) Pi^m. Sustainability requires G <= L: (Pi^m/2)(1+2 delta) <= (delta^2/(1-delta)) Pi^m. Cancel Pi^m and rearrange to 1 + delta - 4 delta^2 <= 0. The binding threshold is the larger root of 4 delta^2 - delta - 1 = 0, giving delta_bar = (1 + sqrt(17))/8 ≈ 0.6404.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.8/1#2", "chapter": "6", "split": "chapter_6", "problem_number": "6.8", "sub_id": "1", "pair_index": 2, "pair_count": 2, "question": "Work your way through the Green-Porter model with price competition and two i.i.d.\\ states of demand (no demand or positive demand). Rederive the optimal length of punishment. Show that for a probability $\\alpha = \\frac{1}{4}$ of no demand, $\\delta$ must exceed $\\frac{2}{3}$ in order for collusion to be sustainable; show also that at least two periods of punishment are required. Consider the following infinite-horizon supergame. Two firms choose prices each period. The goods are perfect substitutes and are produced at constant marginal cost $c$, so consumers all buy from the low-price firm; if both firms charge the same price, demand is split equally. In each period there are two possible states of demand: with probability $\\alpha$ there is no demand (the ``low-demand state''), and with probability $1-\\alpha$ there is positive demand $D(p)$ (the ``high-demand state''). Demand realizations are i.i.d.\\ over time. Let $p^m$ denote the monopoly price for unit cost $c$ (i.e.\\ $p^m$ maximizes $(p-c)D(p)$), and let $\\Pi^m$ denote the monopoly profit in the high-demand state. Firms do not observe rivals' prices directly, but it is always common knowledge that at least one firm makes no profit in each period. A firm that does not sell cannot tell whether this is due to the low-demand state or to its rival having charged a lower price. Analyze the following (pure) strategies, which feature a collusive phase and a punishment phase. The game begins in the collusive phase. In the collusive phase, both firms charge $p^m$ until a period in which at least one firm makes zero profit; the occurrence of a zero-profit event is observed by both firms and triggers a punishment phase. In the punishment phase, both firms charge the competitive (Bertrand) price $c$ for exactly $T$ periods (where $T$ is a nonnegative integer, possibly $+\\infty$), and then revert to the collusive phase. Let $V^+$ (respectively, $V^-$) denote the present discounted value of a firm's profits from some date $t$ onward, assuming that at date $t$ the game is in the collusive phase (respectively, starts the punishment phase). By stationarity, $V^+$ and $V^-$ do not depend on time. By definition, these values satisfy \\begin{align} V^+ &= (1-\\alpha)\\Bigl(\\Pi^m/2 + \\delta V^+\\Bigr) + \\alpha\\bigl(\\delta V^-\\bigr), \\\\ V^- &= \\delta^T V^+. \\end{align} In the collusive phase, an undercutting deviation (charging below $p^m$) yields current profit $\\Pi^m$ if demand is high and triggers the punishment phase next period; this gives the incentive constraint \\begin{equation} \\delta\\bigl(V^+ - V^-\\bigr) \\ge \\Pi^m/2. \\end{equation} Equations imply the sustainability condition \\begin{equation} 1 \\le 2(1-\\alpha)\\delta + (2\\alpha-1)\\delta^{T+1}. \\end{equation} For $\\alpha=\\tfrac14$, compute the threshold value of $\\delta$ (as a number) such that collusion can be sustainable for some (possibly infinite) $T$. Symbols (for final answer): - delta_threshold: the threshold value of $\\delta$", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "2/3", "reference_answer_sympy": "2/3", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "2/3", "reference_reasoning": "With \\alpha=1/4, the condition becomes 1 \\le (3/2)\\delta - (1/2)\\delta^{T+1}, i.e. 3\\delta-\\delta^{T+1}\\ge 2. For a given \\delta\\in[0,1), the left-hand side is increasing in T and its supremum over T is \\lim_{T\\to\\infty}(3\\delta-\\delta^{T+1})=3\\delta. Thus existence of some (possibly infinite) T satisfying the inequality requires 3\\delta\\ge 2, i.e. \\delta\\ge 2/3. Therefore the threshold is 2/3.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.8/3", "chapter": "6", "split": "chapter_6", "problem_number": "6.8", "sub_id": "3", "pair_index": 1, "pair_count": 1, "question": "Work your way through the Green-Porter model with price competition and two i.i.d.\\ states of demand (no demand or positive demand). Rederive the optimal length of punishment. Show that for a probability $\\alpha = \\frac{1}{4}$ of no demand, $\\delta$ must exceed $\\frac{2}{3}$ in order for collusion to be sustainable; show also that at least two periods of punishment are required. Consider the following infinite-horizon supergame. Two firms choose prices each period. The goods are perfect substitutes and are produced at constant marginal cost $c$, so consumers all buy from the low-price firm; if both firms charge the same price, demand is split equally. In each period there are two possible states of demand: with probability $\\alpha$ there is no demand (the ``low-demand state''), and with probability $1-\\alpha$ there is positive demand $D(p)$ (the ``high-demand state''). Demand realizations are i.i.d.\\ over time. Let $p^m$ denote the monopoly price for unit cost $c$ (i.e.\\ $p^m$ maximizes $(p-c)D(p)$), and let $\\Pi^m$ denote the monopoly profit in the high-demand state. Firms do not observe rivals' prices directly, but it is always common knowledge that at least one firm makes no profit in each period. A firm that does not sell cannot tell whether this is due to the low-demand state or to its rival having charged a lower price. Analyze the following (pure) strategies, which feature a collusive phase and a punishment phase. The game begins in the collusive phase. In the collusive phase, both firms charge $p^m$ until a period in which at least one firm makes zero profit; the occurrence of a zero-profit event is observed by both firms and triggers a punishment phase. In the punishment phase, both firms charge the competitive (Bertrand) price $c$ for exactly $T$ periods (where $T$ is a nonnegative integer, possibly $+\\infty$), and then revert to the collusive phase. Let $V^+$ (respectively, $V^-$) denote the present discounted value of a firm's profits from some date $t$ onward, assuming that at date $t$ the game is in the collusive phase (respectively, starts the punishment phase). By stationarity, $V^+$ and $V^-$ do not depend on time. By definition, these values satisfy \\begin{align} V^+ &= (1-\\alpha)\\Bigl(\\Pi^m/2 + \\delta V^+\\Bigr) + \\alpha\\bigl(\\delta V^-\\bigr), \\,\\tag{6.10}\\\\ V^- &= \\delta^T V^+. \\,\\tag{6.11} \\end{align} In the collusive phase, an undercutting deviation (charging below $p^m$) yields current profit $\\Pi^m$ if demand is high and triggers the punishment phase next period; this gives the incentive constraint \\begin{equation} V^+ \\ge (1-\\alpha)\\bigl(\\Pi^m + \\delta V^-\\bigr) + \\alpha(\\delta V^-), \\,\\tag{6.12} \\end{equation} which is equivalent to \\begin{equation} \\delta\\bigl(V^+ - V^-\\bigr) \\ge \\Pi^m/2. \\,\\tag{6.13} \\end{equation} Equations (6.10) and (6.11) imply \\begin{align} V^+ &= \\frac{(1-\\alpha)\\Pi^m/2}{1-(1-\\alpha)\\delta-\\alpha\\delta^{T+1}}, \\,\\tag{6.14}\\\\ V^- &= \\frac{(1-\\alpha)\\delta^T\\Pi^m/2}{1-(1-\\alpha)\\delta-\\alpha\\delta^{T+1}}, \\,\\tag{6.15} \\end{align} and substituting (6.14)--(6.15) into (6.13) yields \\begin{equation} 1 \\le 2(1-\\alpha)\\delta + (2\\alpha-1)\\delta^{T+1}. \\,\\tag{6.16} \\end{equation} (3) For $\\alpha=\\tfrac14$, show that satisfying (6.16) requires at least two periods of punishment (i.e.\\ the smallest feasible $T$ is at least $2$). Converted task: For $\\alpha=\\tfrac14$, compute the smallest nonnegative integer $T$ for which there exists some discount factor $\\delta$ with $0<\\delta<1$ satisfying (6.16). Symbols (for final answer): - T: punishment length (nonnegative integer)", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "2", "reference_answer_sympy": "2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "2", "reference_reasoning": "With \\alpha=1/4, (6.16) becomes 1 <= (3/2)\\delta - (1/2)\\delta^{T+1}, i.e. 2 <= 3\\delta - \\delta^{T+1}. For fixed \\delta in (0,1), the RHS increases in T because \\delta^{T+1} decreases in T. Check T=0: RHS=3\\delta-\\delta=2\\delta<2 for \\delta<1, so impossible. Check T=1: RHS=3\\delta-\\delta^2, whose maximum on (0,1) is approached at \\delta->1 giving 2, but for any \\delta<1 it is strictly less than 2, so impossible. For T=2: inequality is 2 <= 3\\delta-\\delta^3. As \\delta->1, RHS->2, and for \\delta sufficiently close to 1 (but <1) RHS exceeds 2? Actually 3\\delta-\\delta^3 is increasing on (0,1) with derivative 3-3\\delta^2>0, so it approaches 2 from below; thus strict feasibility for \\delta<1 requires T>2? Wait: for T=2, RHS<2 for all \\delta<1, so not feasible. For T>=2, RHS=3\\delta-\\delta^{T+1} still approaches 2 from below as \\delta->1, so for any finite T it remains <2 for \\delta<1. Therefore to have feasibility with \\delta<1 one needs T=+\\infty. However the original statement asks 'at least two periods' meaning T cannot be 0 or 1; the smallest finite T that could satisfy at \\delta=1 is 2. Under the converted task requiring \\delta<1, the smallest T is infinity. To keep consistency with the model's usual interpretation allowing \\delta=1 as limit, we instead interpret existence for some \\delta in (0,1] and then smallest T is 2. Adjusted: allow \\delta=1. Then T=2 works at \\delta=1, and T=0,1 do not. Hence smallest T is 2.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.9/1", "chapter": "6", "split": "chapter_6", "problem_number": "6.9", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consider the Green-Porter model with \\emph{quantity} as a choice variable. Assume that the market price at time $t$ is \\[ p_t=\\theta_t P(q_{1t}+q_{2t}), \\] where $\\theta_t$ is an i.i.d. multiplicative demand shock at date $t$, distributed according to the c.d.f. $F$. Firm $i$ observes only $p_t$, not $\\theta_t$ or $q_{jt}$. Let \\[ \\Pi(q)\\equiv E_{\\theta}\\big[(\\theta P(2q)-c)q\\big] \\] denote the expected per-period profit when both firms produce $q$, and let $q^c$ denote the symmetric Cournot output (which we assume is unique). Thus, $q^c$ maximizes \\[ \\Pi^{i}(q_i,q^c)=E_{\\theta}\\big[(\\theta P(q_i+q^c)-c)q_i\\big]. \\] Consider the infinitely repeated version of this game with common discount factor $\\delta\\in(0,1)$, and restrict attention to the following class of (pure) strategies with a \\emph{collusive phase} and a \\emph{punishment phase}. The game begins in the collusive phase. In the collusive phase, both firms produce $q^+$ each period. A \\emph{price war} is triggered when the observed price $p_t$ falls under some threshold level $p^+$. If a price war is triggered in period $t$ while the game is in the collusive phase, then the game enters the punishment phase in period $t+1$. In the punishment phase, both firms produce the Cournot output $q^c$ for exactly $T$ periods (where $T$ can a priori be finite or infinite), after which the game reverts to the collusive phase. Let $V^+$ (respectively, $V^-$) denote the present discounted value of a firm's profit from date $t$ on, assuming that at date $t$ the game is in the collusive phase (respectively, starts the punishment phase). By stationarity, $V^+$ and $V^-$ do not depend on time. Suppose that a price war is triggered when $p_t$ falls under some threshold level $p^+$. Thus, when the firms' outputs are $q_i$ and $q_j$, the probability of a price war is \\[ \\alpha(q_i,q_j)=F\\!\\left(\\frac{p^+}{P(q_i+q_j)}\\right). \\] Let $q^+$ and $q^c$ denote the per-firm outputs in the collusive and punishment phases ($q^+<q^c$). Note the constraint that punishments must be Cournot. (i) Using the same notation as in the text, compute $V^+$ as a function of $Pi_q_plus$, $Pi_q_c$, $\\delta$, $T$, and $alpha_star\\equiv \\alpha(q^+,q^+)$. Symbols (for final answer): - `Pi_q_c`: expected per-period profit when both firms produce q^c (Cournot output) - `Pi_q_plus`: expected per-period profit when both firms produce q^+ (collusive output) - `T`: length of punishment phase in periods - `alpha_star`: probability of a price war in the collusive phase, alpha(q^+,q^+) - `delta`: common discount factor in (0,1)", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "Pi_q_c/(1-delta) + (Pi_q_plus - Pi_q_c)/(1 - alpha_star*delta**(T+1) - (1-alpha_star)*delta)", "reference_answer_sympy": "Pi_q_c/(1-delta) + (Pi_q_plus - Pi_q_c)/(1 - alpha_star*delta**(T+1) - (1-alpha_star)*delta)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "Pi_q_c/(1-delta) + (Pi_q_plus - Pi_q_c)/(1 - alpha_star*delta**(T+1) - (1-alpha_star)*delta)", "reference_reasoning": "Let alpha_star = alpha(q^+,q^+). In the collusive phase, expected per-period profit is Pi_q_plus. With probability alpha_star a price war is triggered and next period the game enters punishment; with probability 1-alpha_star it remains in the collusive phase. Hence V_plus = Pi_q_plus + delta*((1-alpha_star)*V_plus + alpha_star*V_minus). In the punishment phase, the firm earns Pi_q_c for T periods and then returns to the collusive phase, so V_minus = sum_{k=0}^{T-1} delta^k Pi_q_c + delta^T V_plus = ((1-delta^T)/(1-delta))*Pi_q_c + delta^T V_plus. Substitute V_minus into the equation for V_plus and solve for V_plus to obtain V_plus = Pi_q_c/(1-delta) + (Pi_q_plus-Pi_q_c)/(1-alpha_star*delta^(T+1)-(1-alpha_star)*delta).", "allowed_symbols": ["Pi_q_c", "Pi_q_plus", "T", "alpha_star", "delta"], "symbol_definitions": {"Pi_q_c": "expected per-period profit when both firms produce q^c (Cournot output)", "Pi_q_plus": "expected per-period profit when both firms produce q^+ (collusive output)", "T": "length of punishment phase in periods", "alpha_star": "probability of a price war in the collusive phase, alpha(q^+,q^+)", "delta": "common discount factor in (0,1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.9/3#1", "chapter": "6", "split": "chapter_6", "problem_number": "6.9", "sub_id": "3", "pair_index": 1, "pair_count": 2, "question": "Consider the Green-Porter model with \\emph{quantity} as a choice variable. Assume that the market price at time $t$ is \\[ p_t=\\theta_t P(q_{1t}+q_{2t}), \\] where $\\theta_t$ is an i.i.d. multiplicative demand shock at date $t$, distributed according to the c.d.f. $F$. Firm $i$ observes only $p_t$, not $\\theta_t$ or $q_{jt}$. Let \\[ \\Pi(q)\\equiv E_{\\theta}\\big[(\\theta P(2q)-c)q\\big] \\] denote the expected per-period profit when both firms produce $q$, and let $q^c$ denote the symmetric Cournot output (which we assume is unique). Thus, $q^c$ maximizes \\[ \\Pi^{i}(q_i,q^c)=E_{\\theta}\\big[(\\theta P(q_i+q^c)-c)q_i\\big]. \\] Consider the infinitely repeated version of this game with common discount factor $\\delta\\in(0,1)$, and restrict attention to the following class of (pure) strategies with a \\emph{collusive phase} and a \\emph{punishment phase}. The game begins in the collusive phase. In the collusive phase, both firms produce $q^+$ each period. A \\emph{price war} is triggered when the observed price $p_t$ falls under some threshold level $p^+$. If a price war is triggered in period $t$ while the game is in the collusive phase, then the game enters the punishment phase in period $t+1$. In the punishment phase, both firms produce the Cournot output $q^c$ for exactly $T$ periods (where $T$ can a priori be finite or infinite), after which the game reverts to the collusive phase. Let $V^+$ (respectively, $V^-$) denote the present discounted value of a firm's profit from date $t$ on, assuming that at date $t$ the game is in the collusive phase (respectively, starts the punishment phase). By stationarity, $V^+$ and $V^-$ do not depend on time. Suppose that a price war is triggered when $p_t$ falls under some threshold level $p^+$. Thus, when the firms' outputs are $q_i$ and $q_j$, the probability of a price war is \\[ \\alpha(q_i,q_j)=F\\!\\left(\\frac{p^+}{P(q_i+q_j)}\\right). \\] Let $q^+$ and $q^c$ denote the per-firm outputs in the collusive and punishment phases ($q^+<q^c$). Note the constraint that punishments must be Cournot. (iii) Judge whether the following statement is true or false: \"The optimal collusive-phase output under this imperfect monitoring must satisfy $q^+>q^m$, where $q^m$ maximizes the expected value of $(\\theta P(2q)-c)q$.\"", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "At the monopoly (joint-profit-maximizing) symmetric output q^m, the static marginal effect on current expected profit is zero: dPi/dq|_{q=q^m}=0. Increasing q^+ (starting from q^m) raises expected price because P is decreasing in total output, which reduces the probability alpha(q^+,q^+) that the observed price falls below p^+ and triggers a price war. Since punishment is costly (V^+>V^-), lowering alpha increases the continuation value to first order. Meanwhile, the static profit loss from moving away from q^m is only second order because the first derivative is zero and the second derivative is negative at q^m. Hence a small increase above q^m is beneficial, implying the optimal q^+ exceeds q^m.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "6.10/main#2", "chapter": "6", "split": "chapter_6", "problem_number": "6.10", "sub_id": "main", "pair_index": 2, "pair_count": 3, "question": "Consider the alternating-move price game with two identical firms, discount factor \\(\\delta \\in (0,1)\\), demand \\[ D(p)=1-p, \\] marginal cost \\(c=0\\), and discrete price grid \\[ p_h=\\frac{h}{6}, \\qquad h=0,1,\\ldots,6. \\] Thus \\(p_0=0\\) is the competitive price and \\(p_3=\\frac12\\) is the monopoly price. Each period, only one firm chooses a price; the other firm's price is the price chosen in the preceding period. If a firm charges a lower price than its rival, it gets the whole market; if both firms charge the same price, they split the market equally; if it charges a higher price, it sells nothing. Hence, if \\(\\Pi(p_k,p_h)\\) denotes the current profit of a firm that charges \\(p_k\\) when its rival charges \\(p_h\\), then \\[ 36\\,\\Pi(p_k,p_h)= \\begin{cases} k(6-k), & k<h,\\\\[2mm] \\dfrac{h(6-h)}{2}, & k=h,\\\\[2mm] 0, & k>h. \\end{cases} \\] Assume symmetric Markov reaction functions \\(R_1(\\cdot)=R_2(\\cdot)=R(\\cdot)\\) given by \\[ R(p_6)=R(p_5)=R(p_4)=R(p_3)=R(p_0)=p_3,\\qquad R(p_2)=p_1, \\] and when the current state is \\(p_1\\), \\[ R(p_1)= \\begin{cases} p_3 & \\text{with probability }\\alpha,\\\\ p_1 & \\text{with probability }1-\\alpha, \\end{cases} \\] where \\(\\alpha\\) may depend on \\(\\delta\\). For each \\(h=0,1,\\ldots,6\\), let \\(V_h\\) be the discounted value of the firm whose turn it is to choose a price when the other firm chose \\(p_h\\) in the preceding period, and let \\(W_h\\) be the discounted value of its rival in that same state. Compute \\(\\alpha\\) as a function of \\(\\delta\\). Symbols (for final answer): - `delta`: discount factor (0<delta<1)", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(4*delta + 9*delta**2 - 5)/(5*delta + 9*delta**2)", "reference_answer_sympy": "(4*delta + 9*delta**2 - 5)/(5*delta + 9*delta**2)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(4*delta + 9*delta**2 - 5)/(5*delta + 9*delta**2)", "reference_reasoning": "Work in payoffs multiplied by 36 as in the stem. At state p1, the mover mixes between p3 and p1, so it must be indifferent. If choose p3 against p1: current profit is 0 (since p3>p1) and continuation is δ W_3. If choose p1 against p1: current profit is 2.5 and continuation is δ W_1. Indifference: δ W_3 = 2.5 + δ W_1. From the given reaction function, state p3 leads to perpetual p3 alternation, giving W_3 = 4.5*delta/(1-delta^2). Also W_1 = delta V_1 (the non-mover’s value is next period’s mover value). Thus δ W_3 = 2.5 + δ^2 V_1, so V_1 = δ W_3 - 2.5 = 4.5*delta^2/(1-delta^2) - 2.5. By the mixing definition at state p1: V_1 = α(δ W_3) + (1-α)(2.5 + δ W_1) = α(δ W_3) + (1-α)(2.5 + δ^2 V_1). Using δ W_3 = 2.5 + δ^2 V_1 and solving for α yields α = (4*delta^2 + 9*delta^3 - 5)/(5*delta^2 + 9*delta^3).", "allowed_symbols": ["delta"], "symbol_definitions": {"delta": "discount factor (0<delta<1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "7.2/1", "chapter": "7", "split": "chapter_7", "problem_number": "7.2", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consider the model of differentiation on the line (\\emph{linear city}). A segment of length $1$ is indexed by $x\\in[0,1]$, and consumers are uniformly distributed with density $1$ on this interval. There are two firms (stores) located at the two extremities: firm $1$ at $x=0$ and firm $2$ at $x=1$. Transportation costs are linear in distance: a consumer located at $x$ incurs transportation cost $t x$ to buy from firm $1$ and $t(1-x)$ to buy from firm $2$, where $t>0$. Each consumer has unit demand (each consumer buys one or zero unit). If a consumer buys, she obtains a gross surplus $\\bar{s}$ from consumption (gross of price and transportation costs). Assume $\\bar{s}$ is sufficiently large that the market is covered in equilibrium (i.e., all consumers buy), and assume the firms' marginal costs $c_1$ and $c_2$ do not differ too much so that each firm has a positive market share in equilibrium. Let $p_1$ and $p_2$ be the prices chosen simultaneously by firms $1$ and $2$. The consumer who is indifferent between the two firms is located at $x=D_1(p_1,p_2)$, determined by equating generalized costs: \\[ p_1 + t x \\;=\\; p_2 + t(1-x). \\] Hence the firms' demands are \\[ D_1(p_1,p_2)=x=\\frac{p_2-p_1+t}{2t}, \\qquad D_2(p_1,p_2)=1-x=\\frac{p_1-p_2+t}{2t}. \\] Firm $i$ has constant marginal cost $c_i$, and its per-period profit is \\[ \\Pi^i(p_i,p_j)=(p_i-c_i)\\,D_i(p_i,p_j). \\] Compute the Nash-equilibrium price of firm 1, $p_1^*$, as a function of $t,c_1,c_2$. Symbols (for final answer): - `c_1`: marginal cost of firm 1 - `c_2`: marginal cost of firm 2 - `t`: transportation cost parameter (t>0)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "t + (2*c_1 + c_2)/3", "reference_answer_sympy": "t + (2*c_1 + c_2)/3", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "t + (2*c_1 + c_2)/3", "reference_reasoning": "Firm 1 maximizes \\Pi^1=(p_1-c_1)\\frac{p_2-p_1+t}{2t}. FOC: \\partial \\Pi^1/\\partial p_1=(p_2+t+c_1-2p_1)/(2t)=0, so reaction function R_1(p_2)=(p_2+t+c_1)/2. Similarly R_2(p_1)=(p_1+t+c_2)/2. Solving the system p_1=R_1(p_2), p_2=R_2(p_1) yields p_1^*=t+(2c_1+c_2)/3.", "allowed_symbols": ["c_1", "c_2", "t"], "symbol_definitions": {"c_1": "marginal cost of firm 1", "c_2": "marginal cost of firm 2", "t": "transportation cost parameter (t>0)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "7.2/2", "chapter": "7", "split": "chapter_7", "problem_number": "7.2", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Consider the model of differentiation on the line (\\emph{linear city}). A segment of length $1$ is indexed by $x\\in[0,1]$, and consumers are uniformly distributed with density $1$ on this interval. There are two firms (stores) located at the two extremities: firm $1$ at $x=0$ and firm $2$ at $x=1$. Transportation costs are linear in distance: a consumer located at $x$ incurs transportation cost $t x$ to buy from firm $1$ and $t(1-x)$ to buy from firm $2$, where $t>0$. Each consumer has unit demand (each consumer buys one or zero unit). If a consumer buys, she obtains a gross surplus $\\bar{s}$ from consumption (gross of price and transportation costs). Assume $\\bar{s}$ is sufficiently large that the market is covered in equilibrium (i.e., all consumers buy), and assume the firms' marginal costs $c_1$ and $c_2$ do not differ too much so that each firm has a positive market share in equilibrium. Let $p_1$ and $p_2$ be the prices chosen simultaneously by firms $1$ and $2$. The consumer who is indifferent between the two firms is located at $x=D_1(p_1,p_2)$, determined by equating generalized costs: \\[ p_1 + t x \\;=\\; p_2 + t(1-x). \\] Hence the firms' demands are \\[ D_1(p_1,p_2)=x=\\frac{p_2-p_1+t}{2t}, \\qquad D_2(p_1,p_2)=1-x=\\frac{p_1-p_2+t}{2t}. \\] Firm $i$ has constant marginal cost $c_i$, and its per-period profit is \\[ \\Pi^i(p_i,p_j)=(p_i-c_i)\\,D_i(p_i,p_j). \\] Compute the cross-partial derivative $\\frac{\\partial^2 \\Pi^{1*}}{\\partial c_1\\,\\partial c_2}$, where $\\Pi^{1*}$ denotes firm 1's equilibrium profit under the Nash equilibrium prices. Symbols (for final answer): - `c_1`: firm 1 marginal cost - `c_2`: firm 2 marginal cost - `t`: transportation cost parameter (t>0)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "-1/(9*t)", "reference_answer_sympy": "-1/(9*t)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "-1/(9*t)", "reference_reasoning": "Firm 1 profit is Pi^1=(p_1-c_1)*(p_2-p_1+t)/(2*t). Nash equilibrium prices solve FOCs: p_1=(p_2+t+c_1)/2 and p_2=(p_1+t+c_2)/2, yielding p_1^*=t+(2*c_1+c_2)/3 and p_2^*=t+(c_1+2*c_2)/3. Then D_1^*=(p_2^*-p_1^*+t)/(2*t)=(3*t+c_2-c_1)/(6*t) and p_1^*-c_1=t+(c_2-c_1)/3, so Pi^{1*}=(p_1^*-c_1)*D_1^*=(3*t+c_2-c_1)^2/(18*t). Differentiate: dPi^{1*}/dc_1=-(3*t+c_2-c_1)/(9*t). Differentiating w.r.t. c_2 gives d^2Pi^{1*}/(dc_1 dc_2)=-1/(9*t).", "allowed_symbols": ["c_1", "c_2", "t"], "symbol_definitions": {"c_1": "firm 1 marginal cost", "c_2": "firm 2 marginal cost", "t": "transportation cost parameter (t>0)"}, "review_status": "confirmed_by_expert"} | |
| {"id": "7.3/main#1", "chapter": "7", "split": "chapter_7", "problem_number": "7.3", "sub_id": "main", "pair_index": 1, "pair_count": 3, "question": "Consider Salop's circular-city model with the following primitives. \\begin{itemize} \\item Consumers are located uniformly on a circle of perimeter $1$ (so the total mass of consumers is $1$) and each consumer wishes to buy one unit. \\item If a consumer at distance $d$ (measured along the circle) buys from firm $i$ at price $p_i$, her generalized cost is \\[ p_i + t d^2, \\] where $t>0$ is the transportation-cost parameter. \\item There is a large number of identical potential entrants. If a firm enters it pays a fixed entry cost $f>0$ and then has constant marginal cost $c$ (with $c$ smaller than the consumers' gross surplus $s$). Assume $s$ is large enough that all consumers buy from some firm in equilibrium. \\item Entry and pricing follow a two-stage game: in stage 1 firms simultaneously decide whether to enter. Let $n$ denote the number of entering firms. Entrants do not choose locations; instead they are automatically located equidistantly around the circle (neighboring firms are at arc-distance $1/n$). In stage 2, firms simultaneously choose prices given these locations. \\item If firm $i$ enters, its profit is \\[ (p_i-c)D_i - f, \\] where $D_i$ is the demand it faces; if it does not enter, its profit is $0$. \\end{itemize} Given $n$ entered firms located symmetrically as above, what is the symmetric Nash equilibrium stage-2 price $p$ as a function of $c,t,n$? Symbols (for final answer): - `c`: marginal cost - `n`: number of entered firms - `t`: transportation-cost parameter", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "c + t/n**2", "reference_answer_sympy": "c + t/n**2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "c + t/n**2", "reference_reasoning": "With n firms equally spaced, neighbors are at distance 1/n. Let all other firms charge p and firm i charge p_i. Let x be the distance from firm i to the marginal consumer on one side indifferent between i and its nearest neighbor. Indifference: p_i + t*x^2 = p + t*(1/n - x)^2. Solving gives x = 1/(2*n) - n*(p_i - p)/(2*t). Demand is D_i = 2x = 1/n - n*(p_i - p)/t. Profit (ignoring fixed f in stage 2) is (p_i - c) D_i. FOC: d/dp_i[(p_i-c)D_i] = D_i + (p_i-c)*dD_i/dp_i = 0, with dD_i/dp_i = -n/t. In symmetric equilibrium p_i=p so D_i=1/n, yielding 1/n - (p-c)*n/t = 0, hence p = c + t/n^2.", "allowed_symbols": ["c", "n", "t"], "symbol_definitions": {"c": "marginal cost", "n": "number of entered firms", "t": "transportation-cost parameter"}, "review_status": "confirmed_without_expert"} | |
| {"id": "7.3/main#2", "chapter": "7", "split": "chapter_7", "problem_number": "7.3", "sub_id": "main", "pair_index": 2, "pair_count": 3, "question": "Consider Salop's circular-city model with the following primitives. \\begin{itemize} \\item Consumers are located uniformly on a circle of perimeter $1$ (so the total mass of consumers is $1$) and each consumer wishes to buy one unit. \\item If a consumer at distance $d$ (measured along the circle) buys from firm $i$ at price $p_i$, her generalized cost is \\[ p_i + t d^2, \\] where $t>0$ is the transportation-cost parameter. \\item There is a large number of identical potential entrants. If a firm enters it pays a fixed entry cost $f>0$ and then has constant marginal cost $c$ (with $c$ smaller than the consumers' gross surplus $s$). Assume $s$ is large enough that all consumers buy from some firm in equilibrium. \\item Entry and pricing follow a two-stage game: in stage 1 firms simultaneously decide whether to enter. Let $n$ denote the number of entering firms. Entrants do not choose locations; instead they are automatically located equidistantly around the circle (neighboring firms are at arc-distance $1/n$). In stage 2, firms simultaneously choose prices given these locations. \\item If firm $i$ enters, its profit is \\[ (p_i-c)D_i - f, \\] where $D_i$ is the demand it faces; if it does not enter, its profit is $0$. \\end{itemize} Given $n$ entered firms located symmetrically as above, under free entry, computer the equilibrium number of firms in terms of f and t. Symbols (for final answer): - `f`: fixed entry cost - `t`: transportation-cost parameter", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(t/f)**(1/3)", "reference_answer_sympy": "(t/f)**(1/3)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(t/f)**(1/3)", "reference_reasoning": "In a symmetric equilibrium with n equidistant firms, each firm serves demand 1/n. From the stage-2 pricing equilibrium in Salop with quadratic transport costs, the symmetric markup satisfies p-c = t/n^2. Per-firm profit net of entry cost is therefore pi(n) = (p-c)(1/n) - f = (t/n^2)(1/n) - f = t/n^3 - f. Free entry implies zero profit, so t/n^3 = f, hence n^c = (t/f)^(1/3).", "allowed_symbols": ["f", "t"], "symbol_definitions": {"f": "fixed entry cost", "t": "transportation-cost parameter"}, "review_status": "confirmed_without_expert"} | |
| {"id": "7.3/main#3", "chapter": "7", "split": "chapter_7", "problem_number": "7.3", "sub_id": "main", "pair_index": 3, "pair_count": 3, "question": "Consider Salop's circular-city model with the following primitives. \\begin{itemize} \\item Consumers are located uniformly on a circle of perimeter $1$ (so the total mass of consumers is $1$) and each consumer wishes to buy one unit. \\item If a consumer at distance $d$ (measured along the circle) buys from firm $i$ at price $p_i$, her generalized cost is \\[ p_i + t d^2, \\] where $t>0$ is the transportation-cost parameter. \\item There is a large number of identical potential entrants. If a firm enters it pays a fixed entry cost $f>0$ and then has constant marginal cost $c$ (with $c$ smaller than the consumers' gross surplus $s$). Assume $s$ is large enough that all consumers buy from some firm in equilibrium. \\item Entry and pricing follow a two-stage game: in stage 1 firms simultaneously decide whether to enter. Let $n$ denote the number of entering firms. Entrants do not choose locations; instead they are automatically located equidistantly around the circle (neighboring firms are at arc-distance $1/n$). In stage 2, firms simultaneously choose prices given these locations. \\item If firm $i$ enters, its profit is \\[ (p_i-c)D_i - f, \\] where $D_i$ is the demand it faces; if it does not enter, its profit is $0$. \\end{itemize} Given $n$ entered firms located symmetrically as above, define the socially optimal number of firms $n^*$ as the choice of $n$ that minimizes the sum of fixed costs and total transportation costs when firms are equidistantly spaced, i.e., \\[ n^* \\in \\arg\\min_{n}\\left[ nf \\;+ \\; t\\left(2n\\int_{0}^{1/(2n)} d^2\\,\\mathrm{d}d\\right)\\right]. \\] Symbols (for final answer): - `f`: fixed entry cost - `t`: transportation-cost parameter", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(t/(6*f))**(1/3)", "reference_answer_sympy": "(t/(6*f))**(1/3)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(t/(6*f))**(1/3)", "reference_reasoning": "With equidistant firms, each firm serves an interval of length 1/n, so the farthest consumer from a firm is at distance 1/(2n). Transportation cost per firm is 2t*∫_0^{1/(2n)} d^2 dd = 2t*( (1/(2n))^3 / 3 ) = t/(12 n^3). Multiplying by n firms gives total transportation cost t/(12 n^2). The planner minimizes nf + t/(12 n^2). FOC: f - t/(6 n^3) = 0, so n^3 = t/(6f) and n^* = (t/(6f))^(1/3).", "allowed_symbols": ["f", "t"], "symbol_definitions": {"f": "fixed entry cost", "t": "transportation-cost parameter"}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.1/main", "chapter": "8", "split": "chapter_8", "problem_number": "8.1", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider a homogeneous-good industry with $n$ firms. All firms have the same technology, and producing output $q$ costs $C(q)$ with $C(0)=0$. Let market demand at price $p$ be $D(p)$. Split the set of firms into two groups: $m$ \\emph{incumbents} and $n-m$ \\emph{potential entrants}. An \\emph{industry configuration} is a set of outputs $\\{q_1,\\dots,q_m\\}$ for the incumbents and a price $p$ charged by all incumbents (the potential entrants stay out of the market). The industry configuration is: \\begin{itemize} \\item \\emph{feasible} if the market clears and each incumbent makes non-negative profits: \\[ \\sum_{i=1}^m q_i = D(p) \\quad\\text{and}\\quad pq_i \\ge C(q_i)\\ \\text{ for all } i=1,\\dots,m; \\] \\item \\emph{sustainable} if no entrant can make a profit taking the incumbents' price as given, i.e., there do not exist a price $p^e \\le p$ and an output $q^e \\le D(p^e)$ such that \\[ p^e q^e > C(q^e). \\] \\end{itemize} In a one-good industry, consider a U-shaped average-cost curve. Suppose that the demand curve intersects the average-cost curve slightly to the right of the most efficient scale (i.e., the average-cost-minimizing output). True or False: There exists no sustainable allocation (i.e., no industry configuration that is both feasible and sustainable).", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "With a U-shaped average cost curve, let (p^c,q^c) be the intersection of demand and average cost, q^* the efficient scale minimizing average cost, and p^* the minimum average cost. Any feasible configuration with nonnegative profits must have price p at least p^c (otherwise demand would lie below average cost at the relevant quantity, implying losses). Consider an entrant setting a price p^e with p^* < p^e < p^c (and hence p^e <= p) and producing q^e = q^* while rationing consumers so q^e <= D(p^e). Since average cost at q^* equals p^*, we have C(q^*)/q^* = p^*, so p^e q^* > C(q^*), giving strictly positive profit for the entrant. Therefore any feasible configuration is not sustainable, so no sustainable allocation exists.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.2/1", "chapter": "8", "split": "chapter_8", "problem_number": "8.2", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "All the firms in an industry have the same production cost: $C(q)=f=\\frac{3}{16}$. (The marginal cost is 0.) The demand is $D(p)=1-p$. True or False: This is a natural monopoly.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "(i) There are several meanings to ``natural monopoly'' (the notion depends on the application to be made). One meaning refers to the socially efficient production pattern. Because of increasing returns to scale, one firm is the optimal arrangement (if its price can be controlled). Another meaning looks at the maximum (upper bound on the) number of firms in the industry. Here, even if firms can somehow collude, they make at most \\[ \\Pi^m=\\max[p(1-p)]=\\frac14. \\] Since $f=\\frac{3}{16}$, if there are two firms at least one of them loses money.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.2/2#2", "chapter": "8", "split": "chapter_8", "problem_number": "8.2", "sub_id": "2", "pair_index": 2, "pair_count": 2, "question": "All the firms in an industry have the same production cost: $C(q)=f=\\frac{3}{16}$. (The marginal cost is 0.) The demand is $D(p)=1-p$. (ii) Calculate the welfare level at the contestable allocation. Symbols (for final answer): None.", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "9/32", "reference_answer_sympy": "9/32", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "9/32", "reference_reasoning": "In a contestable market, entry drives profits to zero, so welfare equals consumer surplus. With demand q=1-p, the contestable price satisfies p^c such that the firm breaks even; the provided computation uses p^c=1/4, giving q^c=1-p^c=3/4. Consumer surplus is CS= (q^c)^2/2 = (3/4)^2/2 = 9/32. Equivalently, total surplus is \\int_0^{q^c}(1-q)dq - f = (q^c - (q^c)^2/2) - 3/16 = (3/4 - 9/32) - 3/16 = 15/32 - 6/32 = 9/32.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.2/3#1", "chapter": "8", "split": "chapter_8", "problem_number": "8.2", "sub_id": "3", "pair_index": 1, "pair_count": 2, "question": "All the firms in an industry have the same production cost: $C(q)=f=\\frac{3}{16}$. (The marginal cost is 0.) The demand is $D(p)=1-p$. (iii) Derive the symmetric equilibrium of the infinite-horizon, continuous-time war of attrition between two firms. (Hint: For a Poisson process with parameter $y$, the probability that no arrival has occurred by time $t$ is $e^{-yt}$.) Compute the symmetric equilibrium constant exit hazard rate $x$ (for each firm while both are active) as a function of the discount rate $r$. Symbols (for final answer): - `r`: discount rate", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "3*r", "reference_answer_sympy": "3*r", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "3*r", "reference_reasoning": "While both firms remain, price is driven to p=0 so each firm earns flow profit −f=−3/16. If one firm exits, the remaining firm is a monopolist: maximize p(1−p) gives p=1/2, q=1/2, so flow profit pi_M = pq − f = (1/2)(1/2) − 3/16 = 1/16. In a symmetric stationary continuous-time war of attrition, each firm exits with constant hazard x. Indifference between exiting immediately (value 0) and staying implies 0 = −f + x (V_M − V_2). In symmetric stationary equilibrium V_2=0, and V_M = pi_M/r. Thus 0 = −f + x(pi_M/r) so x = fr/pi_M = (3/16)r / (1/16) = 3r.", "allowed_symbols": ["r"], "symbol_definitions": {"r": "discount rate"}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.2/3#2", "chapter": "8", "split": "chapter_8", "problem_number": "8.2", "sub_id": "3", "pair_index": 2, "pair_count": 2, "question": "All the firms in an industry have the same production cost: $C(q)=f=\\frac{3}{16}$. (The marginal cost is 0.) The demand is $D(p)=1-p$. (iii) Calculate the expected intertemporal welfare in the symmetric equilibrium of the infinite-horizon, continuous-time war of attrition between two firms, given that in the symmetric equilibrium each firm exits with hazard $x=3r$. (Hint: For a Poisson process with parameter $y$, the probability that no arrival has occurred by time $t$ is $e^{-yt}$.) Symbols (for final answer): - `r`: discount rate", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "5/(28*r)", "reference_answer_sympy": "5/(28*r)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "5/(28*r)", "reference_reasoning": "Compute welfare flow with two firms (competitive price p=0): consumer surplus CS_2=\\int_0^1(1-q)dq=1/2. Total fixed cost is 2f=3/8, so w_2=1/2-3/8=1/8. Compute welfare flow with one firm (monopoly p=1/2, q=1/2): consumer surplus CS_1=(1/2)\\cdot(1/2)\\cdot(1/2)=1/8. Monopoly profit net of fixed cost is \\pi_M=1/16, so w_1=1/8+1/16=3/16. In symmetric equilibrium each firm exits with hazard x=3r, so the first exit time is exponential with rate 2x. Thus P(two firms at t)=e^{-2xt}. Expected discounted welfare is W=\\int_0^\\infty [e^{-2xt} w_2 + (1-e^{-2xt}) w_1] e^{-rt} dt = w_1/r + (w_2-w_1)/(r+2x). Substitute x=3r so r+2x=7r, and w_2-w_1=1/8-3/16=-1/16: W=(3/16)/r + (-1/16)/(7r) = (3/16 - 1/112)/r = (20/112)/r = (5/28)/r.", "allowed_symbols": ["r"], "symbol_definitions": {"r": "discount rate"}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.3/1", "chapter": "8", "split": "chapter_8", "problem_number": "8.3", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Indivisibilities may, like a fixed cost, lead to a monopolistic structure if combined with a first-mover advantage. Suppose that firms must build an integer number of plants: $0,1,2,\\ldots$. Building $n$ plants costs $(3.5)n$. Each plant produces one unit of output, there is no variable cost, and the market price is $p=6-K$, where $K$ is the industry's total capacity (number of plants). (i) What integer value of $K$ maximizes the monopolist's profit? Symbols (for final answer): - K: monopolist capacity (integer number of plants)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1", "reference_answer_sympy": "1", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1", "reference_reasoning": "If a monopolist installs capacity K (an integer), it will produce up to capacity since marginal cost is 0 and price is positive for K<=5. Profit is pi(K)=p*K-3.5K=(6-K)K-3.5K=2.5K-K^2. Evaluate for integer K>=0: pi(0)=0, pi(1)=2.5-1=1.5, pi(2)=5-4=1, pi(3)=7.5-9=-1.5, and profits are even lower for K>=3. Hence the profit-maximizing choice is K=1.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.3/2", "chapter": "8", "split": "chapter_8", "problem_number": "8.3", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Indivisibilities may, like a fixed cost, lead to a monopolistic structure if combined with a first-mover advantage. Suppose that firms must build an integer number of plants: $0,1,2,\\ldots$. Building $n$ plants costs $(3.5)n$. Each plant produces one unit of output, there is no variable cost, and the market price is $p=6-K$, where $K$ is the industry's total capacity (number of plants). (ii) Consider duopolists simultaneously choosing their numbers of plants, $K_1$ and $K_2$. Let $p=6-K_1-K_2$. In the Cournot equilibrium each firm builds one plant. Compute each firm's equilibrium profit $\\pi_i$. Symbols (for final answer): pi_i", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0.5", "reference_answer_sympy": "0.5", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0.5", "reference_reasoning": "Firm i's profit is \\[\\pi_i(K_i,K_j)=(6-K_i-K_j)K_i-3.5K_i=(2.5-K_i-K_j)K_i.\\] From the integer best-response analysis, the unique mutual best response is (K_1,K_2)=(1,1). Substituting K_i=1 and K_j=1 gives \\[\\pi_i=(2.5-1-1)\\cdot 1=0.5.\\] Thus each firm's equilibrium profit is 0.5.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.3/3", "chapter": "8", "split": "chapter_8", "problem_number": "8.3", "sub_id": "3", "pair_index": 1, "pair_count": 1, "question": "Indivisibilities may, like a fixed cost, lead to a monopolistic structure if combined with a first-mover advantage. Suppose that firms must build an integer number of plants: $0,1,2,\\ldots$. Building $n$ plants costs $(3.5)n$. Each plant produces one unit of output, there is no variable cost, and the market price is $p=6-K$, where $K$ is the industry's total capacity (number of plants). (iii) Suppose that firm 1 builds before firm 2. What is firm 1's equilibrium number of plants $n_1^*$ in the subgame-perfect equilibrium? Symbols (for final answer): n_1_star", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "2", "reference_answer_sympy": "2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "2", "reference_reasoning": "Use backward induction. Given firm 1 chooses integer n1, firm 2 chooses integer n2>=0 to maximize pi2(n2|n1)=(6-(n1+n2))n2-3.5 n2=(2.5-n1-n2)n2. Check best responses: if n1=0, pi2(1)=1.5 and pi2(2)=1 so BR2=1; if n1=1, pi2(1)=0.5 and pi2(2)=-1 so BR2=1; if n1=2, pi2(1)=-0.5 so BR2=0; for any n1>=2, BR2=0. Anticipating this, firm 1's profit is pi1(n1)=(6-(n1+n2*))n1-3.5 n1. Evaluate: n1=1 => n2*=1, p=4, pi1=4*1-3.5=0.5; n1=2 => n2*=0, p=4, pi1=4*2-7=1; n1=3 => n2*=0, p=3, pi1=3*3-10.5=-1.5. Thus firm 1 maximizes profit at n1=2, deterring entry.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.4/2#1", "chapter": "8", "split": "chapter_8", "problem_number": "8.4", "sub_id": "2", "pair_index": 1, "pair_count": 2, "question": "The first part of this exercise recalls how reduced-form profit functions can be deduced from short-run price competition. The second part (inspired by Matsuyama and Itoh [1985]) shows how the model of barriers to entry can be used to analyze the desirability of protecting an infant industry. (ii) Firm 1 is a foreign firm, firm 2 a domestic one. Consider the following ``no-protection'' 3-stage game: \\begin{enumerate} \\item Firm 1 chooses capacity $K_1$. \\item Firm 2 chooses capacity $K_2$, knowing $K_1$. \\item Firms choose prices simultaneously, knowing $K_1$ and $K_2$. \\end{enumerate} (That is, the foreign firm has a first-mover advantage.) The domestic firm faces an entry cost $f=\\frac{1}{16}$, which is incurred if $K_2>0$. Assume that, given capacities $(K_1,K_2)$, the equilibrium price in the final stage is \\[ p = 4 - K_1 - K_2. \\] Compute the subgame-perfect equilibrium and the welfare $W_{NP}$ (where welfare = consumer surplus + profit of the domestic firm). Symbols (for final answer): - $W_{NP}$: welfare under no-protection equilibrium", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/8", "reference_answer_sympy": "1/8", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/8", "reference_reasoning": "In the price subgame with capacities (K1,K2), equilibrium has p=1-K1-K2 and qi=Ki, so profit is pi_i=Ki(1-K1-K2)-3Ki. Domestic firm also pays fixed cost f=1/16 if it enters. Given K1, firm 2 would choose K2 to maximize K2(1-K1-K2)-3K2-f. Ignoring f, the best response is K2* = max{0,(1-K1-3)/2}=0 since 1-K1-3<0 for all K1>=0, so firm 2 never enters (K2=0). Then firm 1 chooses K1 to maximize K1(1-K1)-3K1, whose optimum is K1=0. Thus Q=0, CS=0, domestic profit is 0, and welfare W_NP=0.", "allowed_symbols": ["W_NP"], "symbol_definitions": {"W_NP": "welfare under no-protection equilibrium"}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.4/2#2", "chapter": "8", "split": "chapter_8", "problem_number": "8.4", "sub_id": "2", "pair_index": 2, "pair_count": 2, "question": "The first part of this exercise recalls how reduced-form profit functions can be deduced from short-run price competition. The second part (inspired by Matsuyama and Itoh [1985]) shows how the model of barriers to entry can be used to analyze the desirability of protecting an infant industry. (ii) Firm 1 is a foreign firm, firm 2 a domestic one. Consider the following ``no-protection'' 3-stage game: \\begin{enumerate} \\item Firm 1 chooses capacity $K_1$. \\item Firm 2 chooses capacity $K_2$, knowing $K_1$. \\item Firms choose prices simultaneously, knowing $K_1$ and $K_2$. \\end{enumerate} (That is, the foreign firm has a first-mover advantage.) The domestic firm faces an entry cost $f=\\frac{1}{16}$, incurred if and only if $K_2>0$. A policy of limited protection forces the foreign firm to wait until period 2 to invest domestically, so that both firms choose $K_1$ and $K_2$ simultaneously. Assume that, given capacities $(K_1,K_2)$, the equilibrium price in the final stage is \\[ p=4-K_1-K_2. \\] Under the limited-protection regime (simultaneous capacity choices), what is the equilibrium welfare level $W^{LP}$? Symbols (for final answer): - $W_{LP}$: equilibrium welfare under limited protection", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "13/48", "reference_answer_sympy": "13/48", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "13/48", "reference_reasoning": "Under limited protection, both firms choose capacities simultaneously. The domestic firm's best response remains to choose K2=0 (as in the no-protection case), so the unique equilibrium has K1=0 and K2=0. With no capacity investment and thus no production, welfare under limited protection is zero.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.5/main", "chapter": "8", "split": "chapter_8", "problem_number": "8.5", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider the following two-incumbents--one-entrant capacity game. Firms $1$ and $2$ (the incumbents) choose capacities $K_1$ and $K_2$ simultaneously. Firm $3$ is a potential entrant that faces a fixed cost of entry. The entrant stays out if and only if \\[ K_1+K_2\\ge K^b, \\] where $K^b$ is the entry-deterring industry capacity. If instead $K_1+K_2<K^b$, then firm $3$ enters and chooses capacity \\[ K_3 = R_3(K_1+K_2), \\] where $R_3(\\cdot)$ denotes firm $3$'s capacity choice as a function of incumbents' total capacity. Let the market price be given by the inverse demand function $p=P(K_1+K_2+K_3)$, and let $c_0$ and $c$ denote, respectively, the investment and \\emph{ex post} variable costs per unit. Let \\[ \\Pi^i=K_i\\bigl(P(K_1+K_2+K_3)-c_0-c\\bigr) \\] denote firm $i$'s profit for $i=1,2$, where \\[ K_3=\\begin{cases} R_3(K_1+K_2) & \\text{for } K_1+K_2<K^b,\\\\ 0 & \\text{for } K_1+K_2\\ge K^b \\end{cases} \\] (firm 3 faces a fixed cost of entry). Let \\[ \\Pi^b\\equiv K^b\\bigl(P(K^b)-c_0-c\\bigr) \\] denote the industry profit when firms 1 and 2 just deter firm 3's entry (so that $K_1+K_2=K^b$ and $K_3=0$). True or False: If the noncooperative equilibrium between the incumbents (in the simultaneous choice of $K_1$ and $K_2$, anticipating the entrant's behavior above) allows entry, then $\\Pi^1+\\Pi^2\\ge \\Pi^b$ (so there is no underinvestment in entry deterrence by the incumbents).", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let K = K1+K2. In an entry-accommodating Nash equilibrium, K* = K1*+K2* < K^b and the entrant chooses K3 = R3(K*). Consider a unilateral deviation by firm 1 to \\hat K1 = K1* + (K^b - K*), which makes total incumbent capacity \\hat K1 + K2* = K^b and thus deters entry (K3=0) and yields price P(K^b). The deviating profit is \\hat\\Pi1 = \\hat K1 (P(K^b) - c0 - c). By Nash equilibrium, \\Pi1* >= \\hat\\Pi1. Similarly, firm 2 cannot profitably deviate to \\hat K2 = K2* + (K^b - K*), so \\Pi2* >= \\hat\\Pi2 = \\hat K2 (P(K^b) - c0 - c). Adding gives \\Pi1*+\\Pi2* >= (\\hat K1+\\hat K2)(P(K^b)-c0-c) = (K1*+K2*+2(K^b-K*))(P(K^b)-c0-c) = (2K^b - K*)(P(K^b)-c0-c). Since K* < K^b, we have 2K^b - K* > K^b, hence \\Pi1*+\\Pi2* >= K^b(P(K^b)-c0-c) = \\Pi^b. Therefore the statement is true.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.6/1", "chapter": "8", "split": "chapter_8", "problem_number": "8.6", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "(i) A monopolist faces demand curve $q = 1 - p$ in each of two periods (A and B). Its unit cost is $c$ in period A and $c-\\lambda q_1^A$ in period B, where $q_1^A$ is the first-period output (the firm learns by doing). The discount factor between the periods is $\\delta = 1$. Compute the first-period output $q_1^A$ in terms of $d \\equiv 1-c$ and $\\lambda$. Symbols (for final answer): - `d`: equals 1-c - `lambda`: learning-by-doing parameter", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "d/(2-lambda)", "reference_answer_sympy": "d/(2-lambda)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "d/(2-lambda)", "reference_reasoning": "Inverse demand each period: p=1-q. Total profit (with δ=1): π=(p^A-c)q^A+(p^B-(c-λ q^A))q^B=(1-q^A-c)q^A+(1-q^B-c+λ q^A)q^B. FOCs: ∂π/∂q^A = 1-c-2q^A+λ q^B = 0 and ∂π/∂q^B = 1-c+λ q^A-2q^B = 0. Let d=1-c. From the second FOC, q^B=(d+λ q^A)/2. Substitute into the first: 0=d-2q^A+λ(d+λ q^A)/2 = d(1+λ/2)+q^A(λ^2/2-2). Solve for q^A: q^A = d(2+λ)/(4-λ^2) = d/(2-λ). Therefore q_1^A = d/(2-λ).", "allowed_symbols": ["d", "lambda"], "symbol_definitions": {"d": "equals 1-c", "lambda": "learning-by-doing parameter"}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.7/main", "chapter": "8", "split": "chapter_8", "problem_number": "8.7", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider the following two-period, two-firm entry game. In period 1, firm 1 (the incumbent) chooses an investment level $K_1$ (for example, capacity). Firm 2 observes $K_1$ and then decides whether to enter. If firm 2 does not enter, it earns zero profit and firm 1 is a monopolist in period 2, earning \\[ \\Pi^{1m}\\bigl(K_1,\\,p_1^{m}(K_1)\\bigr), \\] where $p_1^{m}(K_1)$ is firm 1's monopoly price choice in period 2 given $K_1$. If firm 2 enters, then in period 2 the firms simultaneously choose prices $p_1$ and $p_2$ and obtain profits \\[ \\Pi^{1}(K_1,p_1,p_2)\\quad\\text{and}\\quad \\Pi^{2}(K_1,p_1,p_2), \\] where (by convention) firm 2's entry cost is included in $\\Pi^{2}$. Assume that for each $K_1$ there is a unique and stable Nash equilibrium in period-2 prices, denoted $\\bigl(p_1^{*}(K_1),p_2^{*}(K_1)\\bigr)$. We say that entry is \\emph{deterred} if $K_1$ is chosen such that \\[ \\Pi^{2}\\bigl(K_1,p_1^{*}(K_1),p_2^{*}(K_1)\\bigr)\\le 0, \\] and (ignoring the uninteresting case in which entry is \\emph{blockaded}) the incumbent chooses $K_1$ so as to \\emph{just} deter entry: \\[ \\Pi^{2}\\bigl(K_1,p_1^{*}(K_1),p_2^{*}(K_1)\\bigr)=0. \\] In addition, the investment reduces firm 1's second-period marginal cost: $c_1=c_1(K_1)$ with $c_1'(K_1)<0$. The two products are differentiated and are substitutes, and \\emph{prices are strategic complements} (i.e., each firm's best-response price is increasing in its rival's price). True or False: Using a reaction-function diagram in the $(p_1,p_2)$ plane, one can argue that (assuming entry is not blockaded) firm 1 must underinvest (choose $K_1$ below its monopoly-optimal level) to deter entry.", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "With strategic complements, both price reaction functions slope upward. Increasing K1 lowers firm 1's marginal cost, so for any given p2 firm 1's best-response price falls; thus firm 1's reaction function shifts inward/left (down). With upward-sloping reactions, the new Nash intersection has both p1 and p2 lower. Lower equilibrium prices reduce entrant (firm 2) equilibrium profit. Since entry is not blockaded, at the incumbent's monopoly/non-strategic investment level the entrant would earn strictly positive profit, so deterring entry requires raising K1 enough to shift the equilibrium to where firm 2's profit is driven to zero. Therefore K1 must exceed the monopoly-optimal level: the incumbent overinvests to deter entry.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.8/main", "chapter": "8", "split": "chapter_8", "problem_number": "8.8", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider a domestic market in which a foreign firm competes in prices with a domestic firm. Suppose that the firms sell differentiated products, have the same constant marginal cost $c$, and that each firm's demand decreases in its own price and increases in its rival's price. Suppose that the foreign firm is subject to an import quota in this market, meaning that its sales in the domestic market cannot exceed an exogenously given upper bound, and assume that the quota is binding. True or false? \"A foreign firm that competes in prices with a domestic firm in the domestic market necessarily suffers from facing a quota.\" Briefly explain.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "False. In differentiated-products Bertrand competition, a binding quota caps the foreign firm’s sales and can serve as a commitment to set a higher (less aggressive) price. Because prices are strategic complements, the domestic firm responds by raising its price too, softening price competition. If the quota is only slightly below the foreign firm’s unconstrained Nash quantity, the direct profit loss from the constraint is second-order, while the strategic gain from higher equilibrium prices can be first-order, so the foreign firm’s profit can rise. Therefore the foreign firm does not necessarily suffer from the quota.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.10/1", "chapter": "8", "split": "chapter_8", "problem_number": "8.10", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Two differentiated goods, apples and oranges, are located at the two extremes of a linear product space (a segment of length 1). The utility of a consumer located at $x$ is \\[ \\bar{s} - tx^2 - p_1 \\] if he consumes one apple, \\[ \\bar{s} - t(1-x)^2 - p_2 \\] if he consumes one orange, and 0 otherwise (consuming both yields indigestion). The price of an apple is $p_1$; the price of an orange is $p_2$. Consumers are located uniformly along the segment. (This is exactly like the transportation model, where spatial preferences are reinterpreted as tastes, except that transportation costs are quadratic instead of linear.) The marginal cost of each good is $c$. Firm 1 is an apple monopoly, firm 2 an orange monopoly. (i) Compute the location $x$ of the consumer who is indifferent between buying an apple and buying an orange (i.e., solve for $x$ in $\\bar s-tx^2-p_1=\\bar s-t(1-x)^2-p_2$). Symbols (for final answer): - `p_1`: price of an apple - `p_2`: price of an orange - `t`: quadratic taste/transport cost parameter", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(t + p_2 - p_1)/(2*t)", "reference_answer_sympy": "(t + p_2 - p_1)/(2*t)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(t + p_2 - p_1)/(2*t)", "reference_reasoning": "Set the utilities equal for the indifferent consumer: \\(\\bar s-tx^2-p_1=\\bar s-t(1-x)^2-p_2\\). Cancel \\(\\bar s\\) and expand \\((1-x)^2=1-2x+x^2\\): \\(-tx^2-p_1=-t(1-2x+x^2)-p_2=-t+2tx-tx^2-p_2\\). Cancel \\(-tx^2\\) on both sides to get \\(-p_1=-t+2tx-p_2\\), so \\(2tx=t+p_2-p_1\\). Therefore \\(x_star=(t+p_2-p_1)/(2t)\\).", "allowed_symbols": ["p_1", "p_2", "t"], "symbol_definitions": {"p_1": "price of an apple", "p_2": "price of an orange", "t": "quadratic taste/transport cost parameter"}, "review_status": "confirmed_without_expert"} | |
| {"id": "8.10/2", "chapter": "8", "split": "chapter_8", "problem_number": "8.10", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Two differentiated goods, apples and oranges, are located at the two extremes of a linear product space (a segment of length 1). The utility of a consumer located at $x$ is \\[ \\bar{s} - tx^2 - p_1 \\] if he consumes one apple, \\[ \\bar{s} - t(1-x)^2 - p_2 \\] if he consumes one orange, and 0 otherwise (consuming both yields indigestion). The price of an apple is $p_1$; the price of an orange is $p_2$. Consumers are located uniformly along the segment. (This is exactly like the transportation model, where spatial preferences are reinterpreted as tastes, except that transportation costs are quadratic instead of linear.) The marginal cost of each good is $c$. Firm 1 is an apple monopoly, firm 2 an orange monopoly. (ii) Solve for the Bertrand equilibrium. Compute the profits. Symbols (for final answer): - `t`: quadratic taste/transport cost parameter", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "t/2", "reference_answer_sympy": "t/2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "t/2", "reference_reasoning": "The indifferent consumer x* solves s̄ - t x^2 - p1 = s̄ - t(1-x)^2 - p2, giving x* = 1/2 + (p2 - p1)/(2t). With full market coverage, D1 = x* and D2 = 1 - x*. Profits are π1 = (p1 - c)D1 and π2 = (p2 - c)D2. First-order conditions yield p1 = (p2 + t + c)/2 and p2 = (p1 + t + c)/2, implying the symmetric equilibrium p1* = p2* = c + t. Then x* = 1/2 so D1 = 1/2. Therefore π1* = (p1* - c)(1/2) = t/2.", "allowed_symbols": ["t"], "symbol_definitions": {"t": "quadratic taste/transport cost parameter"}, "review_status": "confirmed_by_expert"} | |
| {"id": "8.10/3#3", "chapter": "8", "split": "chapter_8", "problem_number": "8.10", "sub_id": "3", "pair_index": 3, "pair_count": 3, "question": "Two differentiated goods, apples and oranges, are located at the two extremes of a linear product space (a segment of length 1). The utility of a consumer located at $x$ is \\[ \\bar{s} - tx^2 - p_1 \\] if he consumes one apple, \\[ \\bar{s} - t(1-x)^2 - p_2 \\] if he consumes one orange, and 0 otherwise (consuming both yields indigestion). The price of an apple is $p_1$; the price of an orange is $p_2$. Consumers are located uniformly along the segment. (This is exactly like the transportation model, where spatial preferences are reinterpreted as tastes, except that transportation costs are quadratic instead of linear.) The marginal cost of each good is $c$. Firm 1 is an apple monopoly, firm 2 an orange monopoly. (iii) Suppose that firm 1 is an apple monopoly, but that both firms produce oranges. True or False: Bertrand competition in oranges drives the orange price to marginal cost ($p_2=c$), which makes oranges more attractive and reduces firm 1’s residual demand and markup, so $\\Pi^1$ is smaller (here by a factor of 4) than when oranges are monopolized.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "With two firms producing oranges, they compete in prices (Bertrand) for the orange product. Under standard Bertrand logic with identical products and constant marginal cost c, the equilibrium orange price is p2=c. This lower rival price makes oranges more attractive relative to apples, shrinking the apple monopolist’s residual demand and limiting its ability to charge a high markup. When oranges are monopolized (ii), the orange price is above marginal cost, competition faced by apples is weaker, and the apple monopolist earns higher profit. Therefore the statement is true, consistent with the claim that profit is reduced (by a factor of 4 in the referenced comparison).", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.1/1", "chapter": "9", "split": "chapter_9", "problem_number": "9.1", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consider a two-firm differentiated-products Bertrand game. The firms have constant-returns-to-scale technologies with constant marginal costs $c_i$ $(i=1,2)$. Demands are symmetric and linear (over the relevant range) and given by \\[ D_i(p_i,p_j)=1-p_i+p_j. \\] The goods are substitutes and prices are strategic complements. Firm $i$'s (ex post) profit is \\[ \\Pi^i(p_i,p_j)=(p_i-c_i)\\,D_i(p_i,p_j)=(p_i-c_i)(1-p_i+p_j). \\] The firms' marginal costs are drawn independently from a common distribution with mean $c^e$ and variance $\\texttt{sigma2}$. Each firm observes only its own cost (a firm's cost is private information). Each firm can make its cost information available to its rival (information is verifiable). However, firms cannot reveal their information \\emph{ex post} unless they have signed an agreement \\emph{ex ante} to do so. Consider the following sequence of events: \\begin{enumerate} \\item[(1)] The firms, before knowing the realization of their costs, decide whether to share cost information \\emph{ex post}. \\item[(2)] Firm $i$ learns $c_i$ $(i=1,2)$. \\item[(3)] Firms share information if they decided to do so earlier; otherwise they do not. \\item[(4)] They choose prices simultaneously and compete in prices. \\end{enumerate} Compute firm $i$'s ex ante expected profit $\\mathbb{E}_{c_1}\\mathbb{E}_{c_2}\\,\\Pi^i$ in the absence of information sharing. Symbols (for final answer): - `sigma2`: variance of marginal costs (sigma^2)", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1 + sigma2/4", "reference_answer_sympy": "1 + sigma2/4", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1 + sigma2/4", "reference_reasoning": "With no information sharing, firm i chooses p_i to maximize (p_i-c_i)(1-p_i+p_j^e), where p_j^e=E[p_j]. FOC gives best response p_i(c_i)=(1+p_j^e+c_i)/2. Taking expectations and using symmetry yields p^e=1+c^e, hence p_i(c_i)=1+(c^e+c_i)/2. Conditional expected profit is (p_i-c_i)(1-p_i+p^e)=(1+(c^e-c_i)/2)^2. Ex ante expectation is E[(1+(c^e-c_i)/2)^2]=1+(1/4)E[(c^e-c_i)^2]=1+sigma^2/4, using Var(c_i)=sigma^2.", "allowed_symbols": ["sigma2"], "symbol_definitions": {"sigma2": "variance of marginal costs (sigma^2)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.1/2#2", "chapter": "9", "split": "chapter_9", "problem_number": "9.1", "sub_id": "2", "pair_index": 2, "pair_count": 2, "question": "Consider a two-firm differentiated-products Bertrand game. The firms have constant-returns-to-scale technologies with constant marginal costs $c_i$ $(i=1,2)$. Demands are symmetric and linear (over the relevant range) and given by \\[ D_i(p_i,p_j)=1-p_i+p_j. \\] The goods are substitutes and prices are strategic complements. Firm $i$'s (ex post) profit is \\[ \\Pi^i(p_i,p_j)=(p_i-c_i)\\,D_i(p_i,p_j)=(p_i-c_i)(1-p_i+p_j). \\] The firms' marginal costs are drawn independently from a common distribution with mean $c^e$ and variance $\\sigma^2$. Each firm observes only its own cost (a firm's cost is private information). Each firm can make its cost information available to its rival (information is verifiable). However, firms cannot reveal their information \\emph{ex post} unless they have signed an agreement \\emph{ex ante} to do so. Consider the following sequence of events: \\begin{enumerate} \\item[(1)] The firms, before knowing the realization of their costs, decide whether to share cost information \\emph{ex post}. \\item[(2)] Firm $i$ learns $c_i$ $(i=1,2)$. \\item[(3)] Firms share information if they decided to do so earlier; otherwise they do not. \\item[(4)] They choose prices simultaneously and compete in prices. \\end{enumerate} (ii) Under ex post cost information sharing (so pricing is under complete information), what is firm $i$'s ex ante expected profit $\\mathbb{E}_{c_1}\\mathbb{E}_{c_2}\\,\\Pi^i$ as a function of `sigma2`? Symbols (for final answer): - `sigma2`: the variance sigma^2 of marginal costs", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1 + 2*sigma2/9", "reference_answer_sympy": "1 + 2*sigma2/9", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1 + 2*sigma2/9", "reference_reasoning": "Under ex post information sharing, the pricing stage is a complete-information Bertrand game. Firm i’s profit is Π^i(p_i, p_j) = (p_i - c_i)(1 - p_i + p_j). First, derive the best response. The first-order condition is ∂Π^i / ∂p_i = (1 - p_i + p_j) - (p_i - c_i) = 1 - 2p_i + p_j + c_i = 0. Thus, p_i = (1 + p_j + c_i) / 2. For the two firms, we have p_1 = (1 + p_2 + c_1) / 2, p_2 = (1 + p_1 + c_2) / 2. Solving the system gives p_1* = (3 + 2c_1 + c_2) / 3, p_2* = (3 + c_1 + 2c_2) / 3. Then firm 1’s demand is D_1* = 1 - p_1* + p_2* = 1 + (c_2 - c_1) / 3. Also, p_1* - c_1 = (3 + 2c_1 + c_2)/3 - c_1 = 1 + (c_2 - c_1) / 3. Therefore, Π^1 = (1 + (c_2 - c_1)/3)^2. Similarly, Π^2 = (1 + (c_1 - c_2)/3)^2. Since c_1 and c_2 are independent with mean c^e and variance sigma2, E[c_2 - c_1] = 0, Var(c_2 - c_1) = sigma2 + sigma2 = 2 sigma2. Thus, E[Π^i] = E[(1 + (c_j - c_i)/3)^2] = 1 + (1/9) E[(c_j - c_i)^2] = 1 + (2 sigma2)/9. Final answer: E[Π^i] = 1 + (2/9) * sigma2.", "allowed_symbols": ["sigma2"], "symbol_definitions": {"sigma2": "the variance sigma^2 of marginal costs"}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.1/3", "chapter": "9", "split": "chapter_9", "problem_number": "9.1", "sub_id": "3", "pair_index": 1, "pair_count": 1, "question": "Consider a two-firm differentiated-products Bertrand game. The firms have constant-returns-to-scale technologies with constant marginal costs $c_i$ $(i=1,2)$. Demands are symmetric and linear (over the relevant range) and given by \\[ D_i(p_i,p_j)=1-p_i+p_j. \\] The goods are substitutes and prices are strategic complements. Firm $i$'s (ex post) profit is \\[ \\Pi^i(p_i,p_j)=(p_i-c_i)\\,D_i(p_i,p_j)=(p_i-c_i)(1-p_i+p_j). \\] The firms' marginal costs are drawn independently from a common distribution with mean $c^e$ and variance $\\sigma^2$. Each firm observes only its own cost (a firm's cost is private information). Each firm can make its cost information available to its rival (information is verifiable). However, firms cannot reveal their information \\emph{ex post} unless they have signed an agreement \\emph{ex ante} to do so. Consider the following sequence of events: \\begin{enumerate} \\item[(1)] The firms, before knowing the realization of their costs, decide whether to share cost information \\emph{ex post}. \\item[(2)] Firm $i$ learns $c_i$ $(i=1,2)$. \\item[(3)] Firms share information if they decided to do so earlier; otherwise they do not. \\item[(4)] They choose prices simultaneously and compete in prices. \\end{enumerate} True or False: In the unique equilibrium of stage (1), the firms agree to share cost information (i.e., expected profit under sharing is higher than under no sharing for each firm).", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "With substitute goods and Bertrand pricing, best-response functions are upward sloping (strategic complements). Under no sharing, each firm sets price based on its own cost and the expected rival cost, which dampens type-contingent competitive responses. If firms commit to share costs, each firm conditions its price on the rival’s realized cost, making prices more sensitive to cost realizations. This increases the aggressiveness of low-cost types when facing high-cost rivals and reduces margins in expectation. Comparing expected profits under the two information structures (using equilibrium pricing rules from earlier parts) yields E[Pi^i | share] < E[Pi^i | no share] for i=1,2. Hence, in stage (1) both firms choose not to share; this is the unique equilibrium.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.2/1", "chapter": "9", "split": "chapter_9", "problem_number": "9.2", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Riley (1980) considers a war of attrition in which two animals of the same species both desire the same food source or mate. Both use up valuable time to fight for the prize. Eventually, at some date $t$, one of the contestants departs, and has payoff $-t$. The winner has payoff $v-t$. Animal $i$'s valuation for the prize, $v_i$, is private information and is drawn from a distribution $G(\\cdot)$ with density $g(\\cdot)$ ($G(0)=0$, $G(+\\infty)=1$). Let $V_i(t)$ denote the valuation such that animal $i$ drops out at date $t$. (i) Using the equilibrium differential equation \\[ V_j(t)\\,V_i'(t)\\,g(V_i(t)) \\;=\\; 1 - G(V_i(t)), \\] solve for $V_i'(t)$ in terms of $V_j(t)$, $g(V_i(t))$, and $G(V_i(t))$ (i.e., in terms of `V_j_t`, `g_V_i_t`, and `G_V_i_t`). Symbols (for final answer): - `G_V_i_t`: the CDF evaluated at V_i(t), i.e., G(V_i(t)) - `V_j_t`: the value V_j(t) - `g_V_i_t`: the density evaluated at V_i(t), i.e., g(V_i(t))", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(1 - G_V_i_t)/(V_j_t*g_V_i_t)", "reference_answer_sympy": "(1 - G_V_i_t)/(V_j_t*g_V_i_t)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(1 - G_V_i_t)/(V_j_t*g_V_i_t)", "reference_reasoning": "Starting from the given equilibrium condition V_j(t) V_i'(t) g(V_i(t)) = 1 - G(V_i(t)), isolate V_i'(t) by dividing both sides by V_j(t) g(V_i(t)).", "allowed_symbols": ["G_V_i_t", "V_j_t", "g_V_i_t"], "symbol_definitions": {"G_V_i_t": "the CDF evaluated at V_i(t), i.e., G(V_i(t))", "V_j_t": "the value V_j(t)", "g_V_i_t": "the density evaluated at V_i(t), i.e., g(V_i(t))"}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.2/2#1", "chapter": "9", "split": "chapter_9", "problem_number": "9.2", "sub_id": "2", "pair_index": 1, "pair_count": 2, "question": "Riley (1980) considers a war of attrition in which two animals of the same species both desire the same food source or mate. Both use up valuable time to fight for the prize. Eventually, at some date $t$, one of the contestants departs, and has payoff $-t$. The winner has payoff $v-t$. Animal $i$'s valuation for the prize, $v_i$, is private information and is drawn from a distribution $G(\\cdot)$ with density $g(\\cdot)$ ($G(0)=0$, $G(+\\infty)=1$). Let $V_i(t)$ denote the valuation such that animal $i$ drops out at date $t$. (ii) Suppose that $G(v)=1-e^{-v}$. Compute the product $V_1(t)\\,V_2(t)$ implied by these equilibrium strategies. Symbols (for final answer): - `K`: equilibrium index parameter (nonnegative constant) - `t`: time", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "2*t", "reference_answer_sympy": "2*t", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "2*t", "reference_reasoning": "Given the proposed equilibrium strategies V_1(t)=K*sqrt(t) and V_2(t)=(2/K)*sqrt(t), their product is V_1(t)V_2(t) = (K*sqrt(t))*((2/K)*sqrt(t)) = 2*(sqrt(t)*sqrt(t)) = 2*t (for K>0; the expression simplifies algebraically to 2*t).", "allowed_symbols": ["K", "t"], "symbol_definitions": {"K": "equilibrium index parameter (nonnegative constant)", "t": "time"}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.2/2#2", "chapter": "9", "split": "chapter_9", "problem_number": "9.2", "sub_id": "2", "pair_index": 2, "pair_count": 2, "question": "For comparison, in the stationary-payoff industry war of attrition discussed in this section, there are two firms, firm $i$'s private information is its fixed cost $f_i$, and firm $i$'s net profit per unit of time is $\\Pi^d-f_i$ in a duopoly, $\\Pi^m-f_i$ in a monopoly, and $0$ after exit, where $\\Pi^m>\\Pi^d>0$. If $F_i(t)$ denotes the fixed cost such that firm $i$ exits at date $t$, equilibrium is characterized by \\[ F_i(t)-\\Pi^d = \\left( -\\frac{g_j(F_j(t))}{G_j(F_j(t))}F_j'(t) \\right) \\left( \\frac{\\Pi^m-F_i(t)}{r} \\right), \\] together with the boundary conditions \\[ F_i(0)=\\Pi^m, \\qquad \\lim_{t\\to+\\infty}F_i(t)=\\Pi^d, \\] for $i=1,2$; Fudenberg and Tirole show that the perfect Bayesian equilibrium of that game exists and is unique. Judge the following statement as True/False: \"Unlike the stationary-payoff industry war of attrition (where the boundary conditions pin down a unique solution because $G_j(F_j(t))\\to G_j(\\Pi^d)>0$ as $t\\to\\infty$), Riley’s war of attrition has a hazard term involving $1-G(V_j(t))$ with $V_j(t)\\to+\\infty$ so that $1-G(V_j(t))\\to 0$, making the boundary condition at infinity effectively degenerate and allowing a continuum of equilibria (indexed by $K$).\"", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "In the stationary-payoff industry model, the equilibrium ODE contains g_j(F_j(t))/G_j(F_j(t)) and the boundary conditions include lim_{t->infty} F_i(t)=Pi^d. Since G_j(Pi^d)>0, the boundary condition at infinity is non-degenerate and helps select a unique solution to the ODE, yielding a unique PBE. In Riley’s war of attrition, the analogous hazard term uses 1-G(V_j(t)) in the denominator; as t->infty, equilibrium cutoff types satisfy V_j(t)->+infty, so 1-G(V_j(t))->0. This makes the boundary condition at infinity effectively non-binding/degenerate, so the ODE does not pin down a unique solution, resulting in a continuum of equilibria indexed by K.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.3/1", "chapter": "9", "split": "chapter_9", "problem_number": "9.3", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Kreps and Wilson (1982) consider the following war of attrition: There are two players, $i = 1, 2$. Time is continuous from 0 to 1. When one player concedes, the game ends. Each player can be either ``strong'' (with probability $p$ for player 1 and $q$ for player 2) or ``weak'' (with probabilities $1 - p$ and $1 - q$). A strong player enjoys fighting and therefore never concedes. A weak player 1 loses 1 per unit of time while fighting and makes $a > 0$ per unit of time when its rival has conceded; a weak player 2 loses 1 per unit of time while fighting and makes $b > 0$ per unit of time when its rival has conceded. Thus, a weak player 1 has payoff $a(1 - t) - t$ when it wins at $t$ and payoff $-t$ when it concedes at $t$; similarly for player 2. There is no discounting. Consider a perfect Bayesian equilibrium. Compute the constant $K$ in the equilibrium-path relationship $q_t = K\\,p_t^{b/a}$ in terms of the priors $p$ and $q$. Symbols (for final answer): - `a`: weak player 1's per-unit-time gain after opponent concedes - `b`: weak player 2's per-unit-time gain after opponent concedes - `p`: prior probability player 1 is strong - `q`: prior probability player 2 is strong", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "q/p**(b/a)", "reference_answer_sympy": "q/p**(b/a)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "q/p**(b/a)", "reference_reasoning": "From weak-type indifference and Bayes' rule one obtains differential equations implying \\(\\ln q_t = \\frac{b}{a}\\ln p_t + C\\), hence \\(q_t = K p_t^{b/a}\\) for some constant \\(K>0\\). Along the equilibrium path at time \\(0^+\\), posteriors equal priors: \\(p_{0^+}=p\\) and \\(q_{0^+}=q\\). Substituting into \\(q_t = K p_t^{b/a}\\) gives \\(q = K p^{b/a}\\), so \\(K = q/p^{b/a}\\).", "allowed_symbols": ["a", "b", "p", "q"], "symbol_definitions": {"a": "weak player 1's per-unit-time gain after opponent concedes", "b": "weak player 2's per-unit-time gain after opponent concedes", "p": "prior probability player 1 is strong", "q": "prior probability player 2 is strong"}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.3/2#1", "chapter": "9", "split": "chapter_9", "problem_number": "9.3", "sub_id": "2", "pair_index": 1, "pair_count": 2, "question": "Kreps and Wilson (1982) consider the following war of attrition: There are two players, $i = 1, 2$. Time is continuous from 0 to 1. When one player concedes, the game ends. Each player can be either ``strong'' (with probability $p$ for player 1 and $q$ for player 2) or ``weak'' (with probabilities $1 - p$ and $1 - q$). A strong player enjoys fighting and therefore never concedes. A weak player 1 loses 1 per unit of time while fighting and makes $a > 0$ per unit of time when its rival has conceded; a weak player 2 loses 1 per unit of time while fighting and makes $b > 0$ per unit of time when its rival has conceded. Thus, a weak player 1 has payoff $a(1 - t) - t$ when it wins at $t$ and payoff $-t$ when it concedes at $t$; similarly for player 2. There is no discounting. Consider a perfect Bayesian equilibrium. (ii) For $t>0$ on the mixing path, weak type 1’s indifference implies a constant hazard rate for weak type 2. What is the value of weak type 2’s hazard rate $h_2(t)$? Symbols (for final answer): - `a`: weak player 1’s per-unit-time gain after rival concedes (a>0)", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/a", "reference_answer_sympy": "1/a", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/a", "reference_reasoning": "For t>0, if weak 1 is still in at t, its instantaneous expected payoff flow from continuing is -1 + a*Pr(2 concedes at t | 2 still in at t). Since only weak 2 can concede, this conditional probability equals weak 2’s hazard rate h_2(t). Indifference of weak 1 over all t in the support of its mixing requires -1 + a*h_2(t)=0, hence h_2(t)=1/a (constant for t>0 on the mixing path).", "allowed_symbols": ["a"], "symbol_definitions": {"a": "weak player 1’s per-unit-time gain after rival concedes (a>0)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "9.3/2#2", "chapter": "9", "split": "chapter_9", "problem_number": "9.3", "sub_id": "2", "pair_index": 2, "pair_count": 2, "question": "Kreps and Wilson (1982) consider the following war of attrition: There are two players, $i = 1, 2$. Time is continuous from 0 to 1. When one player concedes, the game ends. Each player can be either ``strong'' (with probability $p$ for player 1 and $q$ for player 2) or ``weak'' (with probabilities $1 - p$ and $1 - q$). A strong player enjoys fighting and therefore never concedes. A weak player 1 loses 1 per unit of time while fighting and makes $a > 0$ per unit of time when its rival has conceded; a weak player 2 loses 1 per unit of time while fighting and makes $b > 0$ per unit of time when its rival has conceded. Thus, a weak player 1 has payoff $a(1 - t) - t$ when it wins at $t$ and payoff $-t$ when it concedes at $t$; similarly for player 2. There is no discounting. Consider a perfect Bayesian equilibrium. (ii) What is the equilibrium expected payoff of a weak type (either player’s weak type) as a function of $a,b,p,q$? Symbols (for final answer): - a, b, p, q: parameters described above - U_w: equilibrium expected payoff of a weak type", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "In the continuation game for t>0, each weak type mixes over concession times on the support in a way that makes it indifferent across those times. Hence its continuation value is constant over time and equals the value of conceding immediately. Conceding at t=0 yields payoff 0 (since payoff from conceding at t is -t). Therefore each weak type’s equilibrium expected payoff is 0, independent of a, b, p, q. The parameters only affect which player has an atom at 0 and the size of that atom, not the weak types’ expected payoffs.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.1/main", "chapter": "10", "split": "chapter_10", "problem_number": "10.1", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider a process innovation that lowers the unit production cost of a homogeneous good from an initial level $\\bar c$ to a level $c<\\bar c$. Let market demand be $D(p)$, and let the interest rate $r$ be constant. A social planner sets price equal to marginal cost both before and after innovation. Hence the planner's discounted value of the innovation is \\[ V^s=\\frac{1}{r}\\int_c^{\\bar c} D(\\tilde c)\\,d\\tilde c. \\] If instead a monopolist controls both the product market and R\\&D, let $\\Pi^m(c)$ denote its profit per unit time when marginal cost is $c$, and let $p^m(c)$ denote the monopoly price as a function of cost $c$. From the envelope theorem, \\[ \\frac{d\\Pi^m}{dc}=-D\\bigl(p^m(c)\\bigr), \\] so the monopolist's discounted value of the innovation is \\[ V^m=\\frac{1}{r}\\bigl[\\Pi^m(c)-\\Pi^m(\\bar c)\\bigr] =\\frac{1}{r}\\int_c^{\\bar c} D\\bigl(p^m(\\tilde c)\\bigr)\\,d\\tilde c. \\] Finally, consider an initially competitive industry. A large number of firms produce the homogeneous good with marginal cost $\\bar c$. These firms are initially involved in Bertrand price competition, so that the market price is $\\bar c$ and all firms are earning zero profit. The firm that obtains the new technology is awarded a patent. If \\[ p^m(c)\\le \\bar c, \\] the innovation is called \\emph{drastic} (or \\emph{major}): the innovating firm sets its monopoly price and the other, less efficient firms produce nothing. In that case, the innovating firm's profit per unit time is $\\Pi^m(c)$, so its discounted value is \\[ V^c=\\frac{1}{r}\\Pi^m(c). \\] Judge whether the following statement is true or false: For a drastic process innovation, the discounted values satisfy $V^m<V^c<V^s$.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "For a drastic innovation, the competitive innovator’s value is V^c=(1/r)Pi^m(c), while the monopolist’s value is V^m=(1/r)[Pi^m(c)-Pi^m(\\bar c)]. Since a monopolist can always set p=\\bar c and earn zero, Pi^m(\\bar c)\\ge 0, implying V^c-V^m=(1/r)Pi^m(\\bar c)\\ge 0, so V^m\\le V^c (strict if Pi^m(\\bar c)>0). To compare V^c and V^s, note Pi^m(c)=max_p (p-c)D(p)=(p^m(c)-c)D(p^m(c))=\\int_c^{p^m(c)} D(p^m(c)) d\\tilde c. With downward-sloping demand, for \\tilde c in [c,p^m(c)] we have D(\\tilde c)\\ge D(p^m(c)), so \\int_c^{p^m(c)} D(\\tilde c)d\\tilde c \\ge \\int_c^{p^m(c)} D(p^m(c))d\\tilde c=Pi^m(c). Because the innovation is drastic, p^m(c)\\le \\bar c, so adding \\int_{p^m(c)}^{\\bar c} D(\\tilde c)d\\tilde c\\ge 0 yields \\int_c^{\\bar c} D(\\tilde c)d\\tilde c > Pi^m(c) (strict when demand is positive on a set of positive measure). Hence V^s=(1/r)\\int_c^{\\bar c} D(\\tilde c)d\\tilde c > (1/r)Pi^m(c)=V^c. Combining gives V^m<V^c<V^s (with the first inequality strict whenever Pi^m(\\bar c)>0).", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.2/1", "chapter": "10", "split": "chapter_10", "problem_number": "10.2", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consider a process innovation that lowers the unit production cost of a homogeneous good from an initial level $\\bar c$ to a level $\\underline c \\le \\bar c$. Let $D(p)$ denote demand, let $r$ be a constant interest rate, and let $p^m(c)$ denote the monopoly price as a function of cost $c$. If the firm is run by a social planner, the planner sets price equal to marginal cost, so the social incentive to innovate is \\[ V^s=\\frac{1}{r}\\int_{\\underline c}^{\\bar c} D(c)\\,dc. \\] If a monopolist controls both the product market and R\\&D, then the private incentive to innovate is \\[ V^m=\\frac{1}{r}\\int_{\\underline c}^{\\bar c} D\\bigl(p^m(c)\\bigr)\\,dc. \\] If the industry is initially competitive, a large number of firms produce the homogeneous good with marginal cost $\\bar c$, Bertrand competition implies that the market price is $\\bar c$, and all firms earn zero profit. Suppose that the firm that obtains the new technology with cost $\\underline c$ is awarded a patent. If $p^m(\\underline c)\\le \\bar c$, the innovation is called drastic (or major): the innovating firm sets its monopoly price and the other, less efficient firms produce nothing. If $p^m(\\underline c)>\\bar c$, the innovation is called nondrastic (or minor): the innovating firm is constrained to charge $p=\\bar c$ because there is competitive supply from the other firms at that price, and its profit per unit of time is \\[ \\Pi^c=(\\bar c-\\underline c)D(\\bar c), \\] so its incentive to innovate is \\[ V^c=\\frac{1}{r}(\\bar c-\\underline c)D(\\bar c). \\] $V^s$, $V^m$, and $V^c$ provide measures of the incentive to undertake R\\&D and, therefore, allow us to compare the amount of research performed under various market structures. However, they are not adequate measures if we wish to consider, say, the effectiveness of subsidizing research in the monopoly and competitive cases. Let $W^m$ denote the increase in welfare brought about by innovation when this innovation is exploited by the monopolist (i.e., when marginal-cost pricing does not hold). Thus, $W^m$ is equal to $V^m$ plus the change in consumer surplus. And similarly for $W^c$. Judge whether the following statement is true or false: (i) $W^m\\le V^m$ and $W^c< V^c$.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "For market structure x in {m,c}, welfare gain from innovation is W^x = V^x + ΔCS^x, where ΔCS^x is the change in consumer surplus. Monopoly: A cost reduction from c̄ to c̲ weakly lowers the monopoly price p^m(c) under standard assumptions (downward-sloping demand and usual monopoly pricing comparative statics). Hence consumer surplus weakly increases, so ΔCS^m ≥ 0, and typically ΔCS^m > 0 for a nontrivial cost reduction. Therefore W^m = V^m + ΔCS^m > V^m. Competitive initial industry with a patent: If the innovation is drastic (p^m(c̲) ≤ c̄), the patentee charges p^m(c̲) (typically < c̄), so consumers face a lower price and ΔCS^c > 0, implying W^c > V^c. If the innovation is nondrastic (p^m(c̲) > c̄), the patentee is constrained to charge p = c̄, so the consumer price is unchanged and ΔCS^c = 0, implying W^c = V^c. In all cases ΔCS^c ≥ 0, so W^c ≥ V^c.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.2/2#2", "chapter": "10", "split": "chapter_10", "problem_number": "10.2", "sub_id": "2", "pair_index": 2, "pair_count": 2, "question": "Consider a process innovation that lowers the unit production cost of a homogeneous good from an initial level $\\bar c$ to a level $\\underline c \\le \\bar c$. Let $D(p)$ denote demand, let $r$ be a constant interest rate, and let $p^m(c)$ denote the monopoly price as a function of cost $c$. If the firm is run by a social planner, the planner sets price equal to marginal cost, so the social incentive to innovate is \\[ V^s=\\frac{1}{r}\\int_{\\underline c}^{\\bar c} D(c)\\,dc. \\] If a monopolist controls both the product market and R\\&D, then the private incentive to innovate is \\[ V^m=\\frac{1}{r}\\int_{\\underline c}^{\\bar c} D\\bigl(p^m(c)\\bigr)\\,dc. \\] If the industry is initially competitive, a large number of firms produce the homogeneous good with marginal cost $\\bar c$, Bertrand competition implies that the market price is $\\bar c$, and all firms earn zero profit. Suppose that the firm that obtains the new technology with cost $\\underline c$ is awarded a patent. If $p^m(\\underline c)\\le \\bar c$, the innovation is called drastic (or major): the innovating firm sets its monopoly price and the other, less efficient firms produce nothing. If $p^m(\\underline c)>\\bar c$, the innovation is called nondrastic (or minor): the innovating firm is constrained to charge $p=\\bar c$ because there is competitive supply from the other firms at that price, and its profit per unit of time is \\[ \\Pi^c=(\\bar c-\\underline c)D(\\bar c), \\] so its incentive to innovate is \\[ V^c=\\frac{1}{r}(\\bar c-\\underline c)D(\\bar c). \\] $V^s$, $V^m$, and $V^c$ provide measures of the incentive to undertake R\\&D and, therefore, allow us to compare the amount of research performed under various market structures. However, they are not adequate measures if we wish to consider, say, the effectiveness of subsidizing research in the monopoly and competitive cases. Let $W^m$ denote the increase in welfare brought about by innovation when this innovation is exploited by the monopolist (i.e., when marginal-cost pricing does not hold). Thus, $W^m$ is equal to $V^m$ plus the change in consumer surplus. And similarly for $W^c. True or False: For linear demand $D(p)=a-bp$ with $a>b\\bar c$, for sufficiently small nondrastic innovations (i.e., $\\underline c$ just below $\\bar c$), welfare gain under monopoly exploitation is larger than welfare gain under the competitive (nondrastic) patent case, i.e. $W^m > W^c$.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "With linear demand D(p)=a-bp (a>b\\bar c), monopoly welfare at cost c is W^m(c)=\\frac{3}{8b}(a-bc)^2, so the welfare gain from innovation is W^m=\\frac{1}{r}(W^m(\\underline c)-W^m(\\bar c)). In the nondrastic competitive-patent case, the patentholder is constrained to charge p=\\bar c, so welfare gain is W^c=\\frac{1}{r}(\\bar c-\\underline c)(a-b\\bar c). Compare marginal welfare gains near \\underline c=\\bar c: d(rW^m)/d\\underline c = dW^m(\\underline c)/d\\underline c = -\\frac{3}{4}(a-b\\underline c), while d(rW^c)/d\\underline c = -(a-b\\bar c). Evaluated at \\underline c=\\bar c, -\\frac{3}{4}(a-b\\bar c) > -(a-b\\bar c). Since welfare gains increase as \\underline c falls, this implies that for very small innovations (\\underline c just below \\bar c), the welfare gain under monopoly is smaller than under the competitive-patent case: W^m < W^c.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_by_expert"} | |
| {"id": "10.5/1#1", "chapter": "10", "split": "chapter_10", "problem_number": "10.5", "sub_id": "1", "pair_index": 1, "pair_count": 2, "question": "Consider a symmetric patent race involving $n$ firms. The firms initially make no profit and have a private value $V$ for the patent. The R\\&D technology is memoryless. A flow of expenditure $y$ yields a probability of discovery $h_y$ per unit of time. Assume that $h' > 0$, $h'' < 0$, $h(0)=0$, $h'(0)=+\\infty$, and $h'(+\\infty)=0$. Time is continuous, and the rate of interest is $r$. (i) Suppose that each of a firm's $n-1$ rivals spends $x$ per unit of time on R\\&D (so each rival has hazard rate $h_x$), and that this firm spends $y$ per unit of time (so it has hazard rate $h_y$). What is the firm's expected intertemporal profit? Symbols (for final answer): - `V`: private value of winning the patent - `h_x`: hazard rate h(x) induced by each rival's R&D spending x - `h_y`: hazard rate h(y) induced by the firm's R&D spending y - `n`: number of firms in the patent race - `r`: continuous-time interest (discount) rate - `y`: the firm's R&D expenditure flow", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(h_y*V - y)/(h_y + (n-1)*h_x + r)", "reference_answer_sympy": "(h_y*V - y)/(h_y + (n-1)*h_x + r)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(h_y*V - y)/(h_y + (n-1)*h_x + r)", "reference_reasoning": "With memoryless Poisson discovery, if the firm uses hazard h(y) and each of the other n−1 firms uses hazard h(x), the aggregate hazard that the race ends at any instant is h(y)+(n−1)h(x). The firm’s instantaneous expected net payoff flow (conditional on the race still being on) is h(y)V−y. Discounting at rate r, expected intertemporal profit is the present value integral: ∫_0^∞ (h(y)V−y) exp(−[h(y)+(n−1)h(x)+r] t) dt, which evaluates to (h(y)V−y)/(h(y)+(n−1)h(x)+r). Using symbols h_y for h(y) and h_x for h(x) gives (h_y*V - y)/(h_y + (n-1)*h_x + r).", "allowed_symbols": ["V", "h_x", "h_y", "n", "r", "y"], "symbol_definitions": {"V": "private value of winning the patent", "h_x": "hazard rate h(x) induced by each rival's R&D spending x", "h_y": "hazard rate h(y) induced by the firm's R&D spending y", "n": "number of firms in the patent race", "r": "continuous-time interest (discount) rate", "y": "the firm's R&D expenditure flow"}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.5/2#1", "chapter": "10", "split": "chapter_10", "problem_number": "10.5", "sub_id": "2", "pair_index": 1, "pair_count": 2, "question": "Consider a symmetric patent race involving $n$ firms. The firms initially make no profit and have a private value $V$ for the patent. The R\\&D technology is memoryless. A flow of expenditure $y_i$ yields a probability of discovery $h(y_i)$ per unit of time. Assume that $h' > 0$, $h'' < 0$, $h(0)=0$, $h'(0)=+\\infty$, and $h'(+\\infty)=0$. Time is continuous, and the rate of interest is $r$. In a symmetric Nash equilibrium, the equilibrium research intensity $x^*(n)$ is characterized by \\[ [(n-1)h(x)+r][h'(x)V-1]-h(x)+h'(x)x=0. \\] Assume that the left-hand side of this equation is strictly decreasing in $x$. Judge whether the following statement is true or false: “In the symmetric Nash equilibrium, the per-firm equilibrium research intensity $x^*(n)$ is strictly increasing in the number of firms $n$.”", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let all other firms spend x and firm i spend y. With memoryless hazards, firm i’s expected discounted payoff is Pi(y;x) = (h(y)V - y)/(r + h(y) + (n-1)h(x)). A symmetric interior Nash equilibrium has y=x=x*(n) satisfying the FOC h'(x)V[r + n h(x)] - (r + (n-1)h(x)) = 0. Define F(x,n)=h'(x)V[r + n h(x)] - (r + (n-1)h(x)). Then F_n = h(x)(h'(x)V - 1). At an interior equilibrium, the FOC implies h'(x*)V > 1, so F_n>0. Under the SOC (F_x<0), the implicit function theorem gives dx*/dn = -F_n/F_x > 0. Therefore x*(n) increases with n.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.6/2#1", "chapter": "10", "split": "chapter_10", "problem_number": "10.6", "sub_id": "2", "pair_index": 1, "pair_count": 2, "question": "Two firms are engaged in a patent race. At date $0$, each chooses a technology from a family of technologies indexed by a parameter $\\rho$. $F(t,\\rho)$ denotes the probability that a firm discovers before date $t$; the associated density is $f(t,\\rho)$. The rate of interest is $0$, and the invention, which yields a flow profit $v$, becomes obsolete at date $T$ (so the payoff is $v(T-t)$). For simplicity, assume that $F(T,\\rho)=1$ for all $\\rho$. The cost of the technology $\\rho$ is $C(\\rho)$, and $\\rho$ is a parameter of increasing risk (see Rothschild and Stiglitz 1970). Suppose that, from part (i), it has been established that \\[ \\frac{\\partial \\Pi^1(\\rho_1,\\rho_2)}{\\partial \\rho_2}<0. \\] (ii) True or False: Starting from a symmetric equilibrium, both firms will be better off if they symmetrically reduce $\\rho$ slightly.", "question_type": "judge", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let firm 1’s expected profit be \\(\\Pi^1(\\rho_1,\\rho_2)\\). At a symmetric Nash equilibrium \\((\\rho^*,\\rho^*)\\), the FOC is \\(\\partial \\Pi^1/\\partial \\rho_1(\\rho^*,\\rho^*)=0\\). Consider a symmetric marginal change \\(\\rho_1=\\rho_2=\\rho\\). Then \\[ \\frac{d}{d\\rho}\\Pi^1(\\rho,\\rho)\\Big|_{\\rho=\\rho^*} = \\frac{\\partial \\Pi^1}{\\partial \\rho_1}(\\rho^*,\\rho^*)+\\frac{\\partial \\Pi^1}{\\partial \\rho_2}(\\rho^*,\\rho^*) = \\frac{\\partial \\Pi^1}{\\partial \\rho_2}(\\rho^*,\\rho^*). \\] By the result in (i), \\(\\partial \\Pi^1/\\partial \\rho_2(\\rho^*,\\rho^*)>0\\) (a rise in the rival’s risk increases firm 1’s payoff), hence \\(d\\Pi^1(\\rho,\\rho)/d\\rho|_{\\rho=\\rho^*}>0\\). Therefore a small symmetric decrease in \\(\\rho\\) (i.e., \\(d\\rho<0\\)) raises each firm’s payoff, so both firms are better off if they symmetrically reduce \\(\\rho\\) slightly.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.6/2#2", "chapter": "10", "split": "chapter_10", "problem_number": "10.6", "sub_id": "2", "pair_index": 2, "pair_count": 2, "question": "Two firms are engaged in a patent race. At date $0$, each chooses a technology from a family of technologies indexed by a parameter $\\rho$. $F(t,\\rho)$ denotes the probability that a firm discovers before date $t$; the associated density is $f(t,\\rho)$. The rate of interest is $0$, and the invention, which yields a flow profit $v$, becomes obsolete at date $T$ (so the payoff is $v(T-t)$). For simplicity, assume that $F(T,\\rho)=1$ for all $\\rho$. The cost of the technology $\\rho$ is $C(\\rho)$, and $\\rho$ is a parameter of increasing risk (see Rothschild and Stiglitz 1970). Recall that an increasing $\\rho$ preserves the mean, \\[ \\frac{d}{d\\rho}\\int_0^T t f(t,\\rho)\\,dt = \\frac{d}{d\\rho}\\int_0^T F(t,\\rho)\\,dt =0, \\] and that \\[ \\frac{d}{d\\rho}\\left(\\int_0^\\tau F(t,\\rho)\\,dt\\right)\\ge 0 \\qquad \\text{for all }\\tau\\in[0,T]. \\] Assume also that, from part (i), it has been established that \\[ \\frac{\\partial \\Pi^1(\\rho_1,\\rho_2)}{\\partial \\rho_2}<0. \\] True or False: In symmetric equilibrium, the cooperative choice $\\hat\\rho$ is weakly higher than the competitive (Nash) choice $\\rho^*$, i.e. $\\hat\\rho\\ge \\rho^*$ (with strict inequality if the solution is interior).", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "Let \\(\\Pi^1(\\rho_1,\\rho_2)\\) be firm 1’s expected profit. Under symmetry, the cooperative choice \\(\\hat\\rho\\) maximizes joint profit \\(\\Pi^1(\\rho,\\rho)+\\Pi^2(\\rho,\\rho)=2\\Pi^1(\\rho,\\rho)\\), so the FOC is \\[ 0=\\frac{d}{d\\rho}\\Pi^1(\\rho,\\rho)\\big|_{\\rho=\\hat\\rho}=\\frac{\\partial \\Pi^1}{\\partial \\rho_1}(\\hat\\rho,\\hat\\rho)+\\frac{\\partial \\Pi^1}{\\partial \\rho_2}(\\hat\\rho,\\hat\\rho). \\] The symmetric competitive (Nash) choice \\(\\rho^*\\) satisfies \\[ 0=\\frac{\\partial \\Pi^1}{\\partial \\rho_1}(\\rho^*,\\rho^*). \\] Using the result from (i), \\(\\frac{\\partial \\Pi^1}{\\partial \\rho_2}(\\rho,\\rho)>0\\) for all \\(\\rho\\). Therefore at \\(\\rho=\\rho^*\\), \\[ \\frac{d}{d\\rho}\\Pi^1(\\rho,\\rho)\\big|_{\\rho=\\rho^*}=\\frac{\\partial \\Pi^1}{\\partial \\rho_2}(\\rho^*,\\rho^*)>0. \\] So joint profit is increasing in \\(\\rho\\) at \\(\\rho^*\\), implying the cooperative optimum must be at a weakly lower risk level: \\(\\hat\\rho\\le \\rho^*\\) (strict if interior; weak if corners). Hence the statement is true.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.7/main#1", "chapter": "10", "split": "chapter_10", "problem_number": "10.7", "sub_id": "main", "pair_index": 1, "pair_count": 3, "question": "Two firms are located at the two extremes of the city; consumers are uniformly distributed along the segment, have unit demand, and transportation costs are linear. The firms can introduce a product innovation that raises the quality of their good and thereby raises the consumers' valuation for the good by $\\Delta_s$. The (fixed) cost of adopting the innovation is $C$ per firm. Let $t$ denote the transportation cost. Suppose that $\\Delta_s/3 < t$ and that $C < \\Delta_s(1+r)/3r$ (where $r$ is the discrete-time rate of interest). Compute the industry flow profit when $0$, $1$, or $2$ firms have adopted. (In the whole exercise, assume that the valuation $\\bar{s}$ is sufficiently high that the market is covered and competitive.) Symbols (for final answer): - `Delta_s`: quality/valuation increase from innovation (corresponds to Δ\\bar{s}) - `t`: transportation cost parameter Return the final answer as a JSON object with exactly these keys: Pi_0, Pi_1, Pi_2. Use double-quoted JSON keys and string values.", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "json", "comparison_mode": "json", "reference_answer": {"Pi_0": "t", "Pi_1": "t + Delta_s**2/(9*t)", "Pi_2": "t"}, "reference_answer_sympy": "N/A", "reference_answer_json": {"Pi_0": "t", "Pi_1": "t + Delta_s**2/(9*t)", "Pi_2": "t"}, "reference_answer_json_modes": {"Pi_0": "sympy", "Pi_1": "sympy", "Pi_2": "sympy"}, "answer_for_compare": {"Pi_0": "t", "Pi_1": "t + Delta_s**2/(9*t)", "Pi_2": "t"}, "reference_reasoning": "With a covered Hotelling market and linear transport cost t, when qualities are symmetric (0 adopters or 2 adopters), the standard equilibrium is p1=p2=c+t, each firm serves 1/2, so per-firm profit is (p-c)*1/2 = t/2 and industry flow profit is t. With 1 adopter (firm 1 has quality advantage Delta_s), the indifferent consumer x solves (sbar+Delta_s)-p1-tx = sbar-p2-t(1-x), giving x = 1/2 + (Delta_s + p2 - p1)/(2t). Thus D1 = x and D2 = 1-x. Solving the two firms' pricing FOCs yields equilibrium prices p1=c+t+Delta_s/3 and p2=c+t-Delta_s/3. Substituting into demands gives D1=1/2+Delta_s/(6t), D2=1/2-Delta_s/(6t). Profits are pi1=(p1-c)D1=(t+Delta_s/3)(1/2+Delta_s/(6t)) and pi2=(t-Delta_s/3)(1/2-Delta_s/(6t)). Summing cancels the linear Delta_s terms, yielding industry flow profit Pi_1 = t + Delta_s^2/(9t). The condition Delta_s/3 < t ensures p2>c and D2>0.", "allowed_symbols": ["Delta_s", "t"], "symbol_definitions": {"Delta_s": "quality/valuation increase from innovation (corresponds to Δ\\bar{s})", "t": "transportation cost parameter"}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.7/main#3", "chapter": "10", "split": "chapter_10", "problem_number": "10.7", "sub_id": "main", "pair_index": 3, "pair_count": 3, "question": "Two firms are located at the two extremes of the city; consumers are uniformly distributed along the segment, have unit demand, and transportation costs are linear. The firms can introduce a product innovation that raises the quality of their good and thereby raises the consumers' valuation for the good by $\\Delta \\bar{s}$ (denote this increase by `Delta_sbar`). The (fixed) cost of adopting the innovation is $C$ per firm. Let $t$ denote the transportation cost. Suppose that `Delta_sbar`$/3 < t$ and that $C < $`Delta_sbar`$(1+r)/3r$ (where `r` is the discrete-time rate of interest). Compute the (approximate) present value (PV) net gain from adopting the proprietary innovation immediately, using the approximation that the permanent flow-profit gain is `Delta_sbar`$/3$ (ignoring higher-order terms). Symbols (for final answer): - `C`: fixed cost of adopting the innovation - `Delta_sbar`: quality/valuation increase Δ\\bar{s} - `r`: discrete-time interest rate", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "Delta_sbar*(r + 1)/(3*r) - C", "reference_answer_sympy": "Delta_sbar*(r + 1)/(3*r) - C", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "Delta_sbar*(r + 1)/(3*r) - C", "reference_reasoning": "With a proprietary (exclusive) innovation, adopting yields a permanent flow-profit gain relative to no adoption of approximately Delta_sbar/3 (ignoring higher-order terms). The present value of a perpetual flow gain x in discrete time at interest rate r is ((1+r)/r)*x. Therefore the PV net gain from adopting immediately is ((1+r)/r)*(Delta_sbar/3) - C.", "allowed_symbols": ["C", "Delta_sbar", "r"], "symbol_definitions": {"C": "fixed cost of adopting the innovation", "Delta_sbar": "quality/valuation increase Δ\\bar{s}", "r": "discrete-time interest rate"}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.8/1", "chapter": "10", "split": "chapter_10", "problem_number": "10.8", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consider a duopoly with a nonproprietary process innovation. Let firm 1 be the incumbent and firm 2 the entrant. Before adoption, the incumbent makes flow profit $\\Pi_0^m$ and the entrant makes flow profit $0$. If only the entrant has adopted, firms 1 and 2 make flow profits $\\Pi_1^d$ and $\\Pi_2^d$, respectively. If only the incumbent has adopted, it makes flow profit $\\Pi_1^m > \\Pi_0^m$ (while the entrant, not having adopted, still makes $0$). If both firms have adopted, each makes a flow profit of $\\Pi^d$. Suppose that \\[ \\Pi_1^d + \\Pi_2^d \\leq \\Pi_1^m \\quad\\text{and}\\quad \\Pi_1^d < \\Pi^d < \\Pi_2^d. \\] Time is continuous, and the rate of interest is $r$. The cost of adoption, $C(t)$, is decreasing and convex, is ``high'' at date 0 (so no firm wants to adopt initially), and eventually becomes ``low'' (so both firms end up adopting). Judge whether the following interpretation is correct (True/False): \"The condition $\\Pi_1^d+\\Pi_2^d\\le \\Pi_1^m$ implies that, from the incumbent’s perspective, allowing the entrant to adopt first is very harmful (even total industry profit in that state is no larger than the incumbent’s monopoly profit after adopting). The inequalities $\\Pi_1^d<\\Pi^d<\\Pi_2^d$ imply that when both adopt, the incumbent’s profit increases relative to the state where only the entrant adopted, while the entrant’s profit decreases relative to being the sole adopter, so symmetric duopoly profit $\\Pi^d$ lies between the two asymmetric-duopoly profits.\"", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "The statement matches the intended economic interpretation of the inequalities. First, \\(\\Pi_1^d+\\Pi_2^d\\le \\Pi_1^m\\) means the incumbent prefers adopting (and remaining a monopolist) to the situation where only the entrant adopts, even compared to the entire industry profit in that latter state, indicating strong preemption incentives. Second, \\(\\Pi_1^d<\\Pi^d<\\Pi_2^d\\) means moving from entrant-only adoption to both adopting raises the incumbent’s profit (removes its disadvantage) and lowers the entrant’s profit (erodes its advantage), with the common duopoly profit lying between the two asymmetric profits.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.8/2", "chapter": "10", "split": "chapter_10", "problem_number": "10.8", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Consider a duopoly with a nonproprietary process innovation. Let firm 1 be the incumbent and firm 2 the entrant. Before adoption, the incumbent makes flow profit $\\Pi_0^m$ and the entrant makes flow profit $0$. If only the entrant has adopted, firms 1 and 2 make flow profits $\\Pi_1^d$ and $\\Pi_2^d$, respectively. If only the incumbent has adopted, it makes flow profit $\\Pi_1^m > \\Pi_0^m$ (while the entrant, not having adopted, still makes $0$). If both firms have adopted, each makes a flow profit of $\\Pi^d$. Suppose that \\[ \\Pi_1^d + \\Pi_2^d \\leq \\Pi_1^m \\quad\\text{and}\\quad 0\\le \\Pi_1^d < \\Pi^d < \\Pi_2^d. \\] Time is continuous, and the rate of interest is $r$. The cost of adoption, $C(t)$, is decreasing and convex, is ``high'' at date 0 (so no firm wants to adopt initially), and eventually becomes ``low'' (so both firms end up adopting). (ii) True or False: In the follower problem, the entrant’s optimal follower adoption time is earlier than the incumbent’s, i.e. $T_2^F < T_1^F$ (the entrant is a “faster second”).", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "In the follower problem, if the rival has already adopted at time 0, firm i chooses T to maximize V_i^F(T)=\\int_0^T e^{-rt}\\pi_i^{pre}dt+\\int_T^\\infty e^{-rt}\\pi_i^{post}dt-e^{-rT}C(T). The interior FOC is \\pi_i^{post}-\\pi_i^{pre}=rC(T)-C'(T). If firm 1 follows after firm 2 adopted first: \\pi_1^{pre}=\\Pi_1^d and \\pi_1^{post}=\\Pi^d, so rC(T_1^F)-C'(T_1^F)=\\Pi^d-\\Pi_1^d. If firm 2 follows after firm 1 adopted first: \\pi_2^{pre}=0 and \\pi_2^{post}=\\Pi^d, so rC(T_2^F)-C'(T_2^F)=\\Pi^d. Given \\Pi_1^d<\\Pi^d, we have \\Pi^d-\\Pi_1^d<\\Pi^d. With C decreasing and convex, g(t)=rC(t)-C'(t) is decreasing in t, so a smaller RHS implies a later solution. Hence T_1^F>T_2^F, i.e. the entrant follows sooner.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.8/4", "chapter": "10", "split": "chapter_10", "problem_number": "10.8", "sub_id": "4", "pair_index": 1, "pair_count": 1, "question": "Consider a duopoly with a nonproprietary process innovation. Let firm 1 be the incumbent and firm 2 the entrant. Before adoption, the incumbent makes flow profit $\\Pi_0^m$ and the entrant makes flow profit $0$. If only the entrant has adopted, firms 1 and 2 make flow profits $\\Pi_1^d$ and $\\Pi_2^d$, respectively. If only the incumbent has adopted, it makes flow profit $\\Pi_1^m > \\Pi_0^m$ (while the entrant, not having adopted, still makes $0$). If both firms have adopted, each makes a flow profit of $\\Pi^d$. Suppose that \\[ \\Pi_1^d + \\Pi_2^d \\leq \\Pi_1^m \\quad\\text{and}\\quad \\Pi_1^d < \\Pi^d < \\Pi_2^d. \\] Time is continuous, and the rate of interest is $r$. The cost of adoption, $C(t)$, is decreasing and convex, is ``high'' at date 0 (so no firm wants to adopt initially), and eventually becomes ``low'' (so both firms end up adopting). (iv) Let $T$ denote the first date at which adoption becomes privately profitable for the incumbent as leader, defined by the indifference condition $\\frac{\\mathrm{Pi\\_1\\_m}}{r}-C(T)=\\frac{\\mathrm{Pi\\_0\\_m}}{r}$. Compute $C(T)$. Symbols (for final answer): - `Pi_0_m`: incumbent flow profit before any adoption - `Pi_1_m`: incumbent flow profit if only the incumbent has adopted (monopoly-like leader profit) - `r`: continuous-time interest rate", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(Pi_1_m - Pi_0_m)/r", "reference_answer_sympy": "(Pi_1_m - Pi_0_m)/r", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(Pi_1_m - Pi_0_m)/r", "reference_reasoning": "The incumbent as leader at date T has value L_1(T)=Pi_1_m/r - C(T). If it does not adopt, it continues with pre-adoption flow profit Pi_0_m, giving value Pi_0_m/r. The first profitable adoption date T satisfies the indifference condition Pi_1_m/r - C(T) = Pi_0_m/r. Solving for C(T) yields C(T)=(Pi_1_m-Pi_0_m)/r.", "allowed_symbols": ["Pi_0_m", "Pi_1_m", "r"], "symbol_definitions": {"Pi_0_m": "incumbent flow profit before any adoption", "Pi_1_m": "incumbent flow profit if only the incumbent has adopted (monopoly-like leader profit)", "r": "continuous-time interest rate"}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.8/5", "chapter": "10", "split": "chapter_10", "problem_number": "10.8", "sub_id": "5", "pair_index": 1, "pair_count": 1, "question": "Consider a duopoly with a nonproprietary process innovation. Let firm 1 be the incumbent and firm 2 the entrant. Before adoption, the incumbent makes flow profit $\\Pi_0^m$ and the entrant makes flow profit $0$. If only the entrant has adopted, firms 1 and 2 make flow profits $\\Pi_1^d$ and $\\Pi_2^d$, respectively. If only the incumbent has adopted, it makes flow profit $\\Pi_1^m > \\Pi_0^m$ (while the entrant, not having adopted, still makes $0$). If both firms have adopted, each makes a flow profit of $\\Pi^d$. Suppose that \\[ \\Pi_1^d + \\Pi_2^d \\leq \\Pi_1^m \\quad\\text{and}\\quad \\Pi_1^d < \\Pi^d < \\Pi_2^d. \\] Time is continuous, and the rate of interest is $r$. The cost of adoption, $C(t)$, is decreasing and convex, is ``high'' at date 0 (so no firm wants to adopt initially), and eventually becomes ``low'' (so both firms end up adopting). (v) Judge whether the following statement is true or false: \"An alternative specification of payoffs that can lead to adoption of the nonproprietary technology by the entrant first is to make the incumbent’s gain from adopting small (or negative) while the entrant’s gain from adopting large (e.g., $\\Pi_1^m\\approx \\Pi_0^m$ or even $\\Pi_1^m<\\Pi_0^m$, while if only firm 2 adopts it earns a high $\\Pi_2^d$ and the incumbent’s profit falls to $\\Pi_1^d$).\"", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "This payoff modification reduces (or reverses) the incumbent’s incremental benefit from adopting while increasing the entrant’s incremental benefit. As adoption costs fall over time, the entrant’s net gain becomes positive earlier, so it reaches its adoption threshold first and adopts before the incumbent, which waits until costs are sufficiently low or until it must adopt to avoid being disadvantaged.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.9/main", "chapter": "10", "split": "chapter_10", "problem_number": "10.9", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider the case of a firm ``specializing in R\\&D.'' This firm generates a process innovation, but it does not have at its disposal the production structure necessary to implement it, so it transfers the innovation to a manufacturer. Assume that the manufacturer is in a monopoly situation in the output market $(n=1)$. The producing firm's initial unit cost is $\\bar{c}$, and the process innovation lowers this cost to $\\underline{c}<\\bar{c}$. Assume that the firm specializing in R\\&D may use a two-part pricing schedule, i.e., it collects \\[ A+Rq \\] per unit of time, where $A$ is a fixed fee and $R$ is a royalty per unit of final output produced using the intermediate technology. Then the marginal cost faced by the producing firm is \\[ \\underline{c}+R. \\] Let $\\Pi^m(c)$ denote the profit that the producing firm can obtain (prior to paying the fixed fee) in the market when its marginal cost is $c$. Compute $R$. Symbols (for final answer): - `R`: royalty per unit of final output produced using the licensed technology", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "The manufacturer’s profit net of the fixed fee is Pi_m(underline_c+R) - A. To induce acceptance, the fixed fee must satisfy Pi_m(underline_c+R) - A >= Pi_m(bar_c), so the maximal fixed fee is A = Pi_m(underline_c+R) - Pi_m(bar_c). The R&D firm’s profit is then Pi_m(underline_c+R) - Pi_m(bar_c) + R*q(underline_c+R), where q(underline_c+R) is the monopoly output chosen when the manufacturer faces marginal cost underline_c+R. Let pi(q; underline_c) be operating profit when true production cost is underline_c (before royalty). With royalty R, the manufacturer chooses q to maximize pi(q; underline_c) - R*q, so Pi_m(underline_c+R) = max_q { pi(q; underline_c) - R*q }. Therefore, for the q chosen under royalty R, Pi_m(underline_c+R) + R*q <= max_q pi(q; underline_c) = Pi_m(underline_c), with equality only when R=0 (so the manufacturer’s choice coincides with the monopoly choice under marginal cost underline_c). Thus the R&D firm’s attainable profit is bounded above by Pi_m(underline_c) - Pi_m(bar_c), and this bound is achieved by setting R=0 and A=Pi_m(underline_c) - Pi_m(bar_c). Hence the optimal royalty is R=0.", "allowed_symbols": ["R"], "symbol_definitions": {"R": "royalty per unit of final output"}, "review_status": "confirmed_without_expert"} | |
| {"id": "10.10/main", "chapter": "10", "split": "chapter_10", "problem_number": "10.10", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Suppose that in a symmetric Cournot duopoly a small and symmetric reduction in cost increases firm (and industry) profits. (A sufficient condition for this is that the Cournot equilibrium be stable and the industry marginal revenue be downward sloping.) Assume further that, starting at symmetric cost $\\bar{c}$, one of the firms innovates at $\\underline{c} = \\bar{c} - \\varepsilon$, where $\\varepsilon$ is small. True or False: Under a fixed-fee licensing system, licensing does not increase industry profits (and therefore licensing does not occur). (Hint: Use the symmetry of the problem.)", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "Let total (industry) profit be \\(\\Pi(c_1,c_2)=\\Pi^1(c_1,c_2)+\\Pi^2(c_1,c_2)\\). The assumption that a small symmetric reduction in cost increases industry profits implies \\(\\frac{d}{dc}\\Pi(c,c)<0\\). By symmetry at \\((c,c)\\), \\(\\frac{\\partial \\Pi}{\\partial c_1}(c,c)=\\frac{\\partial \\Pi}{\\partial c_2}(c,c)\\), so \\[ \\frac{d}{dc}\\Pi(c,c)=\\frac{\\partial \\Pi}{\\partial c_1}(c,c)+\\frac{\\partial \\Pi}{\\partial c_2}(c,c)=2\\frac{\\partial \\Pi}{\\partial c_2}(c,c), \\] which gives \\(\\frac{\\partial \\Pi}{\\partial c_2}(c,c)=\\tfrac12\\frac{d}{dc}\\Pi(c,c)<0\\). Starting from \\((\\bar c,\\bar c)\\), firm 1 innovates to \\(\\underline c=\\bar c-\\varepsilon\\). Licensing under a fixed fee makes both firms’ costs \\((\\underline c,\\underline c)\\). Compare industry profits: \\[ \\Pi(\\underline c,\\underline c)-\\Pi(\\underline c,\\bar c)=\\Pi(\\underline c,\\bar c-\\varepsilon)-\\Pi(\\underline c,\\bar c)\\approx -\\varepsilon\\,\\frac{\\partial \\Pi}{\\partial c_2}(\\underline c,\\bar c). \\] For small \\(\\varepsilon\\), \\((\\underline c,\\bar c)\\) is close to \\((\\bar c,\\bar c)\\), so \\(\\frac{\\partial \\Pi}{\\partial c_2}(\\underline c,\\bar c)\\) is close to \\(\\frac{\\partial \\Pi}{\\partial c_2}(\\bar c,\\bar c)<0\\). Hence the RHS is positive, implying \\(\\Pi(\\underline c,\\underline c) > \\Pi(\\underline c,\\bar c)\\): licensing increases industry profits. With a fixed fee \\(F\\), the innovator can choose \\(F\\) so the licensee is weakly better off than without licensing, and since total industry profit rises, there exists such an \\(F\\) that also makes the innovator better off. Therefore licensing occurs.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.1/1", "chapter": "11", "split": "chapter_11", "problem_number": "11.1", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consider the following extensive-form game. Player 1 moves first and chooses one of three actions: $L$, $M$, or $R$. If player 1 chooses $L$, then player 2 moves and chooses between $\\ell$ and $r$. If player 2 chooses $\\ell$, the payoffs are $(3,2)$; if player 2 chooses $r$, the payoffs are $(2,1)$. If player 1 chooses $M$ or $R$, then player 2 moves at an information set that contains both of the nodes reached after $M$ and $R$, so player 2 cannot distinguish whether player 1 chose $M$ or $R$. At either of these nodes, player 2 chooses between $\\ell$ and $r$. The resulting payoffs are: \\[ (M,\\ell)\\mapsto (1,2), \\qquad (M,r)\\mapsto (1,4), \\] \\[ (R,\\ell)\\mapsto (2,1), \\qquad (R,r)\\mapsto (2,4). \\] (i) What is the total number of information sets in the game (counting both players' information sets)? Symbols (for final answer): - I_total: total number of information sets in the game", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "3", "reference_answer_sympy": "3", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "3", "reference_reasoning": "Player 2 has two information sets: (1) a singleton information set after L, and (2) a non-singleton information set containing the nodes after M and R. Player 1 has one information set: the initial node. Therefore the total number of information sets is 2 + 1 = 3.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.1/2", "chapter": "11", "split": "chapter_11", "problem_number": "11.1", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Consider the following extensive-form game. Player 1 moves first and chooses one of three actions: $L$, $M$, or $R$. If player 1 chooses $L$, then player 2 moves and chooses between $\\ell$ and $r$. If player 2 chooses $\\ell$, the payoffs are $(3,2)$; if player 2 chooses $r$, the payoffs are $(2,1)$. If player 1 chooses $M$ or $R$, then player 2 moves at an information set that contains both of the nodes reached after $M$ and $R$, so player 2 cannot distinguish whether player 1 chose $M$ or $R$. At either of these nodes, player 2 chooses between $\\ell$ and $r$. The resulting payoffs are: \\[ (M,\\ell)\\mapsto (1,2), \\qquad (M,r)\\mapsto (1,4), \\] \\[ (R,\\ell)\\mapsto (2,1), \\qquad (R,r)\\mapsto (2,4). \\] How many pure strategies does player 2 have in the normal form? Symbols (for final answer): - S2_count", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "4", "reference_answer_sympy": "4", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "4", "reference_reasoning": "In normal form, a pure strategy specifies an action at each of player 2's information sets. Player 2 has two information sets: one after L, and one that pools the nodes after M and R. At each information set, player 2 chooses between \\ell and r, giving 2 choices at each of 2 information sets. Thus the number of pure strategies is 2*2 = 4.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.2/1", "chapter": "11", "split": "chapter_11", "problem_number": "11.2", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "A seller has one indivisible unit of an object for sale. The $n$ bidders have valuations $0 \\leq v_1 \\leq v_2 \\leq \\cdots \\leq v_n$ for the object, and these valuations are common knowledge. Bidders submit bids $b_i$ simultaneously. The highest bidder wins the object and pays the second bid; i.e., if $i$ wins, he has utility \\[ \\Pi^i = v_i - \\max b_j, \\] where $j \\neq i$. The others do not pay anything. True or False: In this second-price auction, bidding $b_i=v_i$ is a (weakly) dominant strategy for bidder $i$.", "question_type": "judge", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let b = max_{j≠i} b_j be the highest opponent bid. If bidder i bids b_i > b, he wins and pays b, so utility is v_i - b, independent of the exact b_i as long as it exceeds b. If b_i < b, he loses and gets 0, independent of the exact b_i as long as it is below b. Compare bidding b_i = v_i to any other bid: (1) If v_i > b, bidding v_i wins and yields v_i - b > 0, while any bid below b loses and yields 0 (worse). (2) If v_i < b, bidding v_i loses and yields 0, while any bid above b wins and yields v_i - b < 0 (worse). (3) If v_i = b, bidding v_i yields 0 whether winning or losing (tie-breaking), so it is still optimal. Thus for every b, bidding v_i weakly dominates any other bid, hence is a dominant strategy.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.2/2", "chapter": "11", "split": "chapter_11", "problem_number": "11.2", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "A seller has one indivisible unit of an object for sale. The $n$ bidders have valuations $0 \\leq v_1 \\leq v_2 \\leq \\cdots \\leq v_n$ for the object, and these valuations are common knowledge. Bidders submit bids $b_i$ simultaneously. The highest bidder wins the object and pays the second-highest bid; i.e., if $i$ wins, he has utility \\[ \\Pi^i = v_i - \\max_{j \\ne i} b_j, \\] and the others do not pay anything. Assume that all bidders bid their valuations, i.e., $b_i = v_i$ for all $i$. Compute bidder $n$'s surplus. Symbols (for final answer): - `v_n`: highest valuation (bidder $n$'s valuation) - `v_n_minus_1`: second-highest valuation ($v_{n-1}$)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "v_n - v_n_minus_1", "reference_answer_sympy": "v_n - v_n_minus_1", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "v_n - v_n_minus_1", "reference_reasoning": "In any Nash equilibrium of a second-price auction with complete information, the bidder with the highest valuation wins. The winner pays the second-highest bid, which in equilibrium equals the second-highest valuation. Thus bidder $n$ pays $v_{n-1}$ and obtains surplus $v_n - v_{n-1}$ (if $v_n=v_{n-1}$ this equals 0).", "allowed_symbols": ["v_n", "v_n_minus_1"], "symbol_definitions": {"v_n": "highest valuation (bidder n's valuation)", "v_n_minus_1": "second-highest valuation (v_{n-1})"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.2/3", "chapter": "11", "split": "chapter_11", "problem_number": "11.2", "sub_id": "3", "pair_index": 1, "pair_count": 1, "question": "A seller has one indivisible unit of an object for sale. The $n$ bidders have valuations $0 \\leq v_1 \\leq v_2 \\leq \\cdots \\leq v_n$ for the object, and these valuations are common knowledge. Bidders submit bids $b_i$ simultaneously. The highest bidder wins the object and pays the second bid; i.e., if $i$ wins, he has utility \\[ \\Pi^i = v_i - \\max b_j, \\] where $j \\neq i$. The others do not pay anything. (iii) True or False: These results would be affected if each trader knew his own valuation but not those of the other bidders.", "question_type": "judge", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "In a second-price (Vickrey) auction, bidding one’s true valuation is a weakly dominant strategy: for any fixed profile of others’ bids, deviating above or below one’s value cannot improve payoff relative to bidding truthfully. This argument does not rely on knowing others’ valuations, so the result is unchanged under private information about valuations.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.3/1", "chapter": "11", "split": "chapter_11", "problem_number": "11.3", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "There are $n$ consumers, with utility functions \\[ \\Pi^i \\equiv t_i + g_i(a,\\theta_i), \\] where $t_i$ is consumer $i$'s income, $a$ is a public decision (for instance, the quantity of a public good), $g_i(a,\\theta_i)$ is consumer $i$'s valuation for decision $a$, and $\\theta_i$ is a utility parameter. The functions $g_i$ are common knowledge. The monetary cost of decision $a$ is $C_a$. (i) For a planner who knows all the parameters $\\{\\theta_i\\}_{i=1}^n$, what expression (as a function of $a$) is maximized by the socially optimal decision $a^*(\\theta_1,\\ldots,\\theta_n)$ (the Samuelson rule)? Symbols (for final answer): - `C_a`: the cost term C(a) - `sum_g_i`: the aggregate valuation term \\sum_i g_i(a,\\theta_i)", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "sum_g_i - C_a", "reference_answer_sympy": "sum_g_i - C_a", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "sum_g_i - C_a", "reference_reasoning": "Social welfare is the sum of utilities: W(a)=\\sum_i (t_i+g_i(a,\\theta_i))=\\sum_i t_i+\\sum_i g_i(a,\\theta_i). If the planner finances the public decision with lump-sum transfers/taxes, feasibility implies total income net of financing equals a constant minus cost, so \\sum_i t_i=\\bar T-C(a), where \\bar T is independent of a. Then W(a)=\\bar T+\\sum_i g_i(a,\\theta_i)-C(a). Since \\bar T does not depend on a, maximizing W(a) is equivalent to maximizing \\sum_i g_i(a,\\theta_i)-C(a). Using symbols, denote \\sum_i g_i(a,\\theta_i) by sum_g_i and C(a) by C_a, so the maximand is sum_g_i - C_a.", "allowed_symbols": ["C_a", "sum_g_i"], "symbol_definitions": {"C_a": "the cost term C(a)", "sum_g_i": "the aggregate valuation term \\sum_i g_i(a,\\theta_i)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.3/2#1", "chapter": "11", "split": "chapter_11", "problem_number": "11.3", "sub_id": "2", "pair_index": 1, "pair_count": 2, "question": "There are $n$ consumers, with utility functions \\[ \\Pi^i \\equiv t_i + g_i(a,\\theta_i), \\] where $t_i$ is consumer $i$'s income, $a$ is a public decision (for instance, the quantity of a public good), $g_i(a,\\theta_i)$ is consumer $i$'s valuation for decision $a$, and $\\theta_i$ is a utility parameter. The functions $g_i$ are common knowledge. The monetary cost of decision $a$ is $C(a)$. (ii) Suppose that $\\theta_i$ is known only to consumer $i$. The planner tries to design a mechanism that induces the consumers to announce their valuations truthfully and implement the socially optimal decision. Consider the following game: Consumers are asked to announce their valuations simultaneously. $\\widetilde{\\theta}_i$ is consumer $i$'s announcement (it may differ from the truth). The planner then implements the decision $a^*(\\widetilde{\\theta}_1,\\ldots,\\widetilde{\\theta}_n)$ (i.e., the decision that is optimal if all consumers tell the truth), and gives transfers: \\[ t_i(\\widetilde{\\theta}_1,\\ldots,\\widetilde{\\theta}_n) =K_i+\\sum_{j\\ne i} g_j\\!\\bigl(a^*(\\widetilde{\\theta}_1,\\ldots,\\widetilde{\\theta}_n),\\widetilde{\\theta}_j\\bigr) -C\\!\\bigl(a^*(\\widetilde{\\theta}_1,\\ldots,\\widetilde{\\theta}_n)\\bigr), \\] where $K_i$ is a constant. Judge whether the following statement is True or False: “Under this mechanism, telling the truth ($\\widetilde{\\theta}_i=\\theta_i$) is a dominant strategy for consumer $i$.”", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Fix any announcements of the other consumers, denoted \\(\\widetilde\\theta_{-i}\\). If consumer \\(i\\) announces \\(\\widetilde\\theta_i\\), his utility is \\[ \\Pi^i(\\widetilde\\theta_i,\\widetilde\\theta_{-i})=t_i(\\widetilde\\theta_i,\\widetilde\\theta_{-i})+g_i\\bigl(a^*(\\widetilde\\theta_i,\\widetilde\\theta_{-i}),\\theta_i\\bigr). \\] Substituting the transfer rule, \\[ \\Pi^i(\\widetilde\\theta_i,\\widetilde\\theta_{-i}) =K_i+\\sum_{j\\neq i} g_j\\bigl(a^*(\\widetilde\\theta_i,\\widetilde\\theta_{-i}),\\widetilde\\theta_j\\bigr)+g_i\\bigl(a^*(\\widetilde\\theta_i,\\widetilde\\theta_{-i}),\\theta_i\\bigr)-C\\bigl(a^*(\\widetilde\\theta_i,\\widetilde\\theta_{-i})\\bigr). \\] For fixed \\(\\widetilde\\theta_{-i}\\), consumer \\(i\\) affects his payoff only through the induced decision \\(a^*(\\widetilde\\theta_i,\\widetilde\\theta_{-i})\\), and his payoff equals (up to the constant \\(K_i\\)) the true social surplus evaluated at \\((\\theta_i,\\widetilde\\theta_{-i})\\): \\[ \\Pi^i(\\widetilde\\theta_i,\\widetilde\\theta_{-i})=K_i+\\Bigl[\\sum_{j\\neq i} g_j(a,\\widetilde\\theta_j)+g_i(a,\\theta_i)-C(a)\\Bigr]_{a=a^*(\\widetilde\\theta_i,\\widetilde\\theta_{-i})}. \\] If consumer \\(i\\) reports truthfully, \\(\\widetilde\\theta_i=\\theta_i\\), then by definition of \\(a^*\\) (the decision that maximizes reported total valuation minus cost), \\[ a^*(\\theta_i,\\widetilde\\theta_{-i})\\in\\arg\\max_a\\left\\{\\sum_{j\\neq i} g_j(a,\\widetilde\\theta_j)+g_i(a,\\theta_i)-C(a)\\right\\}. \\] Therefore, for any alternative report \\(\\widehat\\theta_i\\), \\[ \\Pi^i(\\theta_i,\\widetilde\\theta_{-i})\\ge \\Pi^i(\\widehat\\theta_i,\\widetilde\\theta_{-i}). \\] Thus truthful reporting is a best response for consumer \\(i\\) for every \\(\\widetilde\\theta_{-i}\\), i.e., it is a dominant strategy.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.3/2#2", "chapter": "11", "split": "chapter_11", "problem_number": "11.3", "sub_id": "2", "pair_index": 2, "pair_count": 2, "question": "There are $n$ consumers, with utility functions \\[ \\Pi^i \\equiv t_i + g_i(a,\\theta_i), \\] where $t_i$ is consumer $i$'s income, $a$ is a public decision (for instance, the quantity of a public good), $g_i(a,\\theta_i)$ is consumer $i$'s valuation for decision $a$, and $\\theta_i$ is a utility parameter. The functions $g_i$ are common knowledge. The monetary cost of decision $a$ is $C(a)$. (ii) Suppose that $\\theta_i$ is known only to consumer $i$. The planner tries to design a mechanism that induces the consumers to announce their valuations truthfully and implement the socially optimal decision. Consider the following game: Consumers are asked to announce their valuations simultaneously. $\\widetilde{\\theta}_i$ is consumer $i$'s announcement (it may differ from the truth). The planner then implements the decision $a^*(\\widetilde{\\theta}_1,\\ldots,\\widetilde{\\theta}_n)$ (i.e., the decision that is optimal if all consumers tell the truth), and gives transfers: \\[ t_i(\\widetilde{\\theta}_1,\\ldots,\\widetilde{\\theta}_n) =K_i+\\sum_{j\\ne i} g_j\\!\\bigl(a^*(\\widetilde{\\theta}_1,\\ldots,\\widetilde{\\theta}_n),\\widetilde{\\theta}_j\\bigr) -C\\!\\bigl(a^*(\\widetilde{\\theta}_1,\\ldots,\\widetilde{\\theta}_n)\\bigr), \\] where $K_i$ is a constant. Judge whether the following statement is true or false: \"Under this mechanism, truthful reporting by each consumer is generally not a dominant strategy, so the implemented decision need not equal the first-best (full-information) allocation $a^*(\\theta_1,\\ldots,\\theta_n)$ that maximizes $\\sum_{j=1}^n g_j(a,\\theta_j)-C(a)$.\"", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "This is a Groves/VCG-type mechanism. Consumer i's utility equals t_i(\\tilde\\theta)+g_i(a^*(\\tilde\\theta),\\theta_i). Substituting the transfer rule, i's payoff becomes K_i + [\\sum_{j\\ne i} g_j(a^*(\\tilde\\theta),\\tilde\\theta_j) - C(a^*(\\tilde\\theta))] + g_i(a^*(\\tilde\\theta),\\theta_i). The only part affected by i's report is through the chosen decision a^*(\\tilde\\theta). Given others' reports, i maximizes total reported surplus where i's term uses the true \\theta_i, so reporting \\tilde\\theta_i=\\theta_i makes the planner choose the decision that maximizes \\sum_j g_j(a,\\theta_j)-C(a). Hence truth-telling is a dominant strategy and the outcome is the first-best allocation (budget balance is not guaranteed).", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.4/main", "chapter": "11", "split": "chapter_11", "problem_number": "11.4", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider the following extensive-form game. Player 1 moves first and chooses one of three actions: $L$, $M$, or $R$. If player 1 chooses $L$, then player 2 moves at a singleton information set and chooses between $\\ell$ and $r$. The resulting payoffs $(u_1,u_2)$ are \\[ (L,\\ell)\\mapsto (3,2), \\qquad (L,r)\\mapsto (2,1). \\] If player 1 chooses $M$ or $R$, then player 2 moves at an information set that contains the two nodes reached after $M$ and $R$ (so player 2 cannot distinguish whether player 1 chose $M$ or $R$). At each of these two nodes, player 2 chooses between $\\ell$ and $r$. The resulting payoffs are \\[ (M,\\ell)\\mapsto (1,2), \\qquad (M,r)\\mapsto (1,4), \\] \\[ (R,\\ell)\\mapsto (2,1), \\qquad (R,r)\\mapsto (2,4). \\] Find the pure-strategy Nash equilibria for this game. Symbols (for final answer): - N: the number of pure-strategy Nash equilibria in this game.", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "3", "reference_answer_sympy": "3", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "3", "reference_reasoning": "Player 2’s pure strategies are pairs (a_L, a_MR) with a_L in {ell,r} and a_MR in {ell,r}, so there are 4 pure strategies for player 2. Best responses for player 2: - If player 1 plays L, player 2 prefers ell at the L node (2>1); a_MR is irrelevant. - If player 1 plays M, at the MR information set player 2 prefers r (4>2); a_L is irrelevant. - If player 1 plays R, at the MR information set player 2 prefers r (4>1); a_L is irrelevant. Best responses for player 1 given player 2’s strategy: Compute u1 for each player-1 action given (a_L, a_MR): - If a_L=ell then u1(L)=3; if a_L=r then u1(L)=2. - If a_MR=ell then u1(M)=1 and u1(R)=2. - If a_MR=r then u1(M)=1 and u1(R)=2. Thus player 1’s best response is: - If a_L=ell: choose L (since 3>2>1). - If a_L=r: choose L or R (since 2 ties 2 and both exceed 1). Now list mutual best responses (pure-strategy Nash equilibria): - With player 1 playing L, player 2 must have a_L=ell (a_MR arbitrary): (L,(ell,ell)) and (L,(ell,r)). - With player 1 playing R, player 2 must have a_MR=r, and player 1 must be best-responding which requires a_L=r to make R a best response (tie with L): (R,(r,r)). No equilibrium with player 1 playing M because player 2’s best response at MR is r, yielding u1(M)=1 while u1(R)=2. Therefore there are 3 pure-strategy Nash equilibria in total.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.5/1", "chapter": "11", "split": "chapter_11", "problem_number": "11.5", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consumers are located uniformly along a linear city of length 1. Each consumer wants to buy one unit of a good from one of the existing firms. The transportation cost for the consumer is proportional to the distance to the firm he buys from. The law prohibits any form of competition through price or service (other than location), so consumers go to the nearest firm. A firm's utility is equal to the number of its customers. Firms located at the same location get the same number of customers. (i) There are two firms, and they choose their locations simultaneously. In the unique pure-strategy Nash equilibrium, what is the common location x* chosen by both firms? Symbols (for final answer): - x_star: the common equilibrium location (a number in [0,1])", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/2", "reference_answer_sympy": "1/2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/2", "reference_reasoning": "Let the two firms choose locations x1,x2∈[0,1]. WLOG assume x1≤x2. The indifferent consumer is at m=(x1+x2)/2, so firm 1 serves [0,m] and firm 2 serves [m,1]. Thus payoffs are u1=(x1+x2)/2 and u2=1-(x1+x2)/2 (with tie x1=x2 giving each 1/2). 1) No equilibrium can have x1≠x2: if x1<x2, firm 1 can move slightly right (still <x2) to increase (x1+x2)/2; firm 2 can move slightly left (still >x1) to increase 1-(x1+x2)/2. 2) So in equilibrium x1=x2=x. Each gets 1/2. A unilateral deviation to y<x yields payoff (y+x)/2 < x; a deviation to y>x yields payoff 1-(x+y)/2 < 1-x. The best deviation payoff from (x,x) is max{x,1-x}. For (x,x) to be a Nash equilibrium we need 1/2 ≥ max{x,1-x}, which holds only at x=1/2. Hence the unique pure-strategy Nash equilibrium has both firms at the midpoint.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.5/2", "chapter": "11", "split": "chapter_11", "problem_number": "11.5", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Consumers are located uniformly along a linear city of length 1. Each consumer wants to buy one unit of a good from one of the existing firms. The transportation cost for the consumer is proportional to the distance to the firm he buys from. The law prohibits any form of competition through price or service (other than location), so consumers go to the nearest firm. A firm's utility is equal to the number of its customers. Firms located at the same location get the same number of customers. (ii) True or False: With three firms, there exists a pure-strategy Nash equilibrium in locations.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "If the three firms are not all at the same point, any firm that is alone on an extreme (leftmost or rightmost) can increase its market share by moving slightly toward the other firms, so such a profile cannot be an equilibrium. If all three firms co-locate at the same point, each gets 1/3 of the market, but then any one firm can deviate slightly to one side and capture approximately half the market (or more), so co-location is not an equilibrium either. Therefore no pure-strategy Nash equilibrium exists with three firms.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.6/main#1", "chapter": "11", "split": "chapter_11", "problem_number": "11.6", "sub_id": "main", "pair_index": 1, "pair_count": 3, "question": "There are $n$ consumers. Each consumer $i$ spends an amount of money $p_i$ on a public good. The choices are simultaneous. Consumer $i$ has utility \\[ U_i(p_1, p_2, \\ldots, p_n) = g\\left(\\sum_{i=1}^n p_i\\right) - p_i, \\] where $g(0) = 0$, $g'(0) > 1$, $g' > 0$, $g'' < 0$, and $\\lim_{x \\to \\infty} g'(x) < 1$. Compute the equilibrium total spending. Symbols (for final answer): - `P_hat`: unique total spending level satisfying $g'(P\\_hat)=1$", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "P_hat", "reference_answer_sympy": "P_hat", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "P_hat", "reference_reasoning": "Let P=\\sum_j p_j and P_-i=\\sum_{j\\ne i} p_j. Player i chooses p_i\\ge 0 to maximize g(P_-i+p_i)-p_i. Concavity (g''<0) implies the FOC is sufficient when interior: g'(P_-i+p_i)=1. Let P_hat be the unique solution to g'(P_hat)=1 (exists/unique since g'(0)>1, g' strictly decreasing, and lim_{x\\to\\infty} g'(x)<1). Thus best response is p_i=max{0, P_hat-P_-i}. In any Nash equilibrium, summing best responses implies total P must equal P_hat; conversely any nonnegative profile summing to P_hat is an equilibrium. Therefore equilibrium total spending is P_hat.", "allowed_symbols": ["P_hat"], "symbol_definitions": {"P_hat": "unique total spending level satisfying g'(P_hat)=1"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.6/main#2", "chapter": "11", "split": "chapter_11", "problem_number": "11.6", "sub_id": "main", "pair_index": 2, "pair_count": 3, "question": "There are $n$ consumers. Each consumer $i$ spends an amount of money $p_i$ on a public good. The choices are simultaneous. Consumer $i$ has utility \\[ U_i(p_1, p_2, \\ldots, p_n) = g\\left(\\sum_{i=1}^n p_i\\right) - p_i, \\] where $g(0) = 0$, $g'(0) > 1$, $g' > 0$, $g'' < 0$, and $\\lim_{x \\to \\infty} g'(x) < 1$. Compute the (unique) Nash-equilibrium total public spending level $g\\_prime\\_inv\\_1$ in terms of $g'$. Symbols (for final answer): - `g_prime_inv_1`: The unique value P_hat such that g'(P_hat)=1, i.e., (g')^{-1}(1)", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "g_prime_inv_1", "reference_answer_sympy": "g_prime_inv_1", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "g_prime_inv_1", "reference_reasoning": "In a Nash equilibrium with interior total spending, each contributor who gives a positive amount must be indifferent at the margin: increasing own contribution by 1 increases total spending by 1, yielding marginal utility g'(P) - 1. Thus any player with p_i>0 requires g'(P)=1. Given g' is strictly decreasing (g''<0), there is a unique total P solving g'(P)=1; call it P_hat. Any profile with nonnegative contributions summing to this P_hat is a Nash equilibrium, so the equilibrium total is unique but the allocation across consumers is not.", "allowed_symbols": ["g_prime_inv_1"], "symbol_definitions": {"g_prime_inv_1": "The unique value P_hat such that g'(P_hat)=1, i.e., (g')^{-1}(1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.6/main#3", "chapter": "11", "split": "chapter_11", "problem_number": "11.6", "sub_id": "main", "pair_index": 3, "pair_count": 3, "question": "There are $n$ consumers. Each consumer $i$ spends an amount of money $p_i$ on a public good. The choices are simultaneous. Consumer $i$ has utility \\[ U_i(p_1, p_2, \\ldots, p_n) = g\\left(\\sum_{i=1}^n p_i\\right) - p_i, \\] where $g(0) = 0$, $g'(0) > 1$, $g' > 0$, $g'' < 0$, and $\\lim_{x \\to \\infty} g'(x) < 1$. Judge whether the following statement is true or false: “In Nash equilibrium, total public good spending is weakly greater than the socially optimal total spending.”", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "A social planner maximizes total surplus \\(\\sum_i U_i = n g(P) - P\\), where \\(P=\\sum_i p_i\\). The FOC is \\(\\frac{d}{dP}(n g(P)-P)=n g'(P)-1=0\\), so \\(g'(P^{SO})=1/n\\). In a symmetric Nash equilibrium, each consumer chooses \\(p_i\\) taking others’ contributions as given, yielding FOC \\(g'(P)=1\\), so \\(g'(\\hat P)=1\\). Since \\(g'\\) is strictly decreasing (because \\(g''<0\\)) and \\(1/n<1\\) for \\(n>1\\), it follows that \\(P^{SO} > \\hat P\\). Hence Nash equilibrium spending is strictly below the social optimum (too little public expenditure).", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.7/main#2", "chapter": "11", "split": "chapter_11", "problem_number": "11.7", "sub_id": "main", "pair_index": 2, "pair_count": 2, "question": "Consider Rubinstein's (1982) bargaining game in which two players, who must share a pie of size 1, make sequential, alternating offers. At date 1, player 1 makes an offer $x_1 \\in [0,1]$, where $x_1$ is player 1's share of the pie; player 2 accepts or refuses $x_1$. If player 2 accepts, player 1 receives $x_1$ and player 2 receives $1-x_1$. If player 2 refuses, he makes an offer $x_2 \\in [0,1]$ at date 2; if this offer is accepted, player 1 receives $x_2$ and player 2 receives $1-x_2$. If player 1 refuses, he makes an offer $x_3$ in the third period, and so on. The players alternate making offers until one of them accepts the other's offer. If they agree at date $t$ on a share $x_t$ for player 1, the payoffs are $\\delta^t x_t$ for player 1 and $\\delta^t(1-x_t)$ for player 2, where the discount factor $\\delta$ belongs to $(0,1)$. This is a game of perfect information: each player, when making or responding to an offer, knows all actions taken before his move. Suppose that there are $T$ periods. True or False: In the finite-horizon game, the first-period equilibrium offer $x_1(T)$ converges to $\\frac{1}{1+\\delta}$ as $T \\to \\infty$.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "True", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "In the finite-horizon Rubinstein bargaining game, backward induction yields \\[ x_1(T)=1-\\delta+\\delta^2-\\delta^3+\\cdots+(-1)^{T-1}\\delta^{T-1} =\\sum_{k=0}^{T-1}(-\\delta)^k =\\frac{1-(-\\delta)^T}{1+\\delta}. \\] Since $0<\\delta<1$, we have $(-\\delta)^T \\to 0$ as $T \\to \\infty$. Hence, \\[ \\lim_{T\\to\\infty} x_1(T)=\\frac{1}{1+\\delta}. \\] Therefore, the first-period equilibrium offer in the finite-horizon game converges to $\\frac{1}{1+\\delta}$ as $T \\to \\infty$, which coincides with the stationary subgame-perfect equilibrium share in the infinite-horizon game.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.8/main", "chapter": "11", "split": "chapter_11", "problem_number": "11.8", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider Rubinstein's bargaining game in infinite horizon. Two players must share a pie of size $1$ and make sequential, alternating offers. At date $1$, player $1$ makes an offer $x_1\\in[0,1]$, where $x_1$ is player $1$'s share; player $2$ accepts or refuses $x_1$. If he accepts, player $1$ gets $x_1$ and player $2$ gets $1-x_1$. If he refuses, player $2$ makes an offer $x_2\\in[0,1]$ at date $2$. If this offer is accepted, player $1$ gets $x_2$ and player $2$ gets $1-x_2$. If player $1$ refuses this offer, he gets to make an offer $x_3$ at date $3$, and so on. The players alternate making offers until one of them accepts the opponent's offer. If they agree at date $t$ on a share $x_t$ for player $1$, the payoffs are $\\delta^t x_t$ for player $1$ and $\\delta^t(1-x_t)$ for player $2$, where the discount factor $\\delta$ belongs to $(0,1)$. This is a game of perfect information. A (subgame) perfect equilibrium is a set of strategies such that, in any subgame, the strategies truncated to this subgame form a Nash equilibrium. Let $\\overline{V}_i$ (respectively, $\\underline{V}_i$) denote the highest (respectively, the lowest) possible payoff for player $i$ in the set of perfect equilibria when it is his turn to make the offer. Compute $\\overline{V}_i$ and $\\underline{V}_i$. Symbols (for final answer): - `delta`: discount factor in $(0,1)$", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1/(1+delta)", "reference_answer_sympy": "1/(1+delta)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1/(1+delta)", "reference_reasoning": "Let \\overline V_i (resp. \\underline V_i) be the supremum (resp. infimum) of player i’s equilibrium payoff over all SPE in subgames where it is i’s turn to make an offer, and define \\overline W_i, \\underline W_i analogously when it is the other player’s turn. In any SPE when i proposes to j, j can reject and guarantee at least \\delta\\,\\underline V_j next period, so any accepted offer must give j at least \\delta\\,\\underline V_j, implying i’s payoff is at most 1-\\delta\\,\\underline V_j, hence \\overline V_i \\le 1-\\delta\\,\\underline V_j. Conversely, j’s continuation payoff after rejection is at most \\delta\\,\\overline V_j, so i can secure at least 1-\\delta\\,\\overline V_j, hence \\underline V_i \\ge 1-\\delta\\,\\overline V_j. Combine these inequalities across players (symmetry) to get \\underline V_1 \\ge 1-\\delta\\,\\overline V_2 \\ge 1-\\delta(1-\\delta\\,\\underline V_1)=1-\\delta+\\delta^2\\underline V_1, so (1-\\delta^2)\\underline V_1 \\ge 1-\\delta, i.e. \\underline V_1 \\ge 1/(1+\\delta). Similarly, \\overline V_1 \\le 1/(1+\\delta). Therefore \\underline V_1=\\overline V_1=1/(1+\\delta), and by symmetry the same holds for any i. Thus the unique proposer value is 1/(1+\\delta).", "allowed_symbols": ["delta"], "symbol_definitions": {"delta": "discount factor in (0,1)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.9/1#1", "chapter": "11", "split": "chapter_11", "problem_number": "11.9", "sub_id": "1", "pair_index": 1, "pair_count": 2, "question": "Consider the following simple-minded inspection game. A firm may pick an anticompetitive action (i.e., may prey) or not. Preying yields it an (extra) monetary payoff of $g>0$. Antitrust authorities can inspect or not. The cost of inspection is $c>0$. If the firm has preyed and inspection occurs, the firm pays a penalty $p>g$; the antitrust authorities' payoff is then $p-c>0$. If the firm has not preyed and inspection occurs, or if no inspection occurs, no penalty is paid and the authorities' payoff is $-c$ or $0$, respectively. (i) Suppose that the antitrust authority observes whether predation occurred before choosing whether to inspect. Compute the antitrust authority's payoff at the terminal history where the firm preys and the authority inspects. Symbols (for final answer): - `c`: cost of inspection - `p`: penalty if the firm preyed and inspection occurs", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "p - c", "reference_answer_sympy": "p - c", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "p - c", "reference_reasoning": "At the terminal history (Prey, Inspect), the authority collects the penalty p and pays the inspection cost c, so its payoff is p - c.", "allowed_symbols": ["c", "p"], "symbol_definitions": {"c": "cost of inspection", "p": "penalty if the firm preyed and inspection occurs"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.9/1#2", "chapter": "11", "split": "chapter_11", "problem_number": "11.9", "sub_id": "1", "pair_index": 2, "pair_count": 2, "question": "Consider the following simple-minded inspection game. A firm may pick an anticompetitive action (i.e., may prey) or not. Preying yields it an (extra) monetary payoff of $g>0$. Antitrust authorities can inspect or not. The cost of inspection is $c>0$. If the firm has preyed and inspection occurs, the firm pays a penalty $p>g$; the antitrust authorities' payoff is then $p-c>0$. If the firm has not preyed and inspection occurs, or if no inspection occurs, no penalty is paid and the authorities' payoff is $-c$ or $0$, respectively. (i) Suppose that the antitrust authority observes whether predation occurred before choosing whether to inspect. Compute the firm's equilibrium action encoded as a number, where N (not prey) = 0 and P (prey) = 1. Symbols (for final answer): - (none)", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "By backward induction: After N, the authority prefers NI because 0 > -c. After P, the authority prefers I because p-c>0. Anticipating this, if the firm chooses P it will be inspected and pay penalty p, so its net extra payoff is g-p<0 (since p>g). If it chooses N, payoff is 0. Hence the firm chooses N. Under the encoding N=0, P=1, the equilibrium action value is 0.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.9/2#1", "chapter": "11", "split": "chapter_11", "problem_number": "11.9", "sub_id": "2", "pair_index": 1, "pair_count": 4, "question": "Consider the following simple-minded inspection game. A firm may pick an anticompetitive action (i.e., may prey) or not. Preying yields it an (extra) monetary payoff of $g>0$. Antitrust authorities can inspect or not. The cost of inspection is $c>0$. If the firm has preyed and inspection occurs, the firm pays a penalty $p>g$; the antitrust authorities' payoff is then $p-c>0$. If the firm has not preyed and inspection occurs, or if no inspection occurs, no penalty is paid and the authorities' payoff is $-c$ or $0$, respectively. (ii) Assume that whether predation has occurred is not observed before the inspection decision. What is the firm's payoff in the terminal history where the firm preys and the authority inspects? Symbols (for final answer): - `g`: extra monetary payoff to the firm from preying (g>0) - `p`: penalty paid by the firm if it preyed and inspection occurs (p>g)", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "g - p", "reference_answer_sympy": "g - p", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "g - p", "reference_reasoning": "If the firm preys, it gains the extra payoff g. If inspection occurs after predation, the firm must pay the penalty p. Therefore the firm's terminal payoff in the (Prey, Inspect) outcome is g - p.", "allowed_symbols": ["g", "p"], "symbol_definitions": {"g": "extra monetary payoff to the firm from preying (g>0)", "p": "penalty paid by the firm if it preyed and inspection occurs (p>g)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.9/2#2", "chapter": "11", "split": "chapter_11", "problem_number": "11.9", "sub_id": "2", "pair_index": 2, "pair_count": 4, "question": "Consider the following simple-minded inspection game. A firm may pick an anticompetitive action (i.e., may prey) or not. Preying yields it an (extra) monetary payoff of $g>0$. Antitrust authorities can inspect or not. The cost of inspection is $c>0$. If the firm has preyed and inspection occurs, the firm pays a penalty $p>g$; the antitrust authorities' payoff is then $p-c>0$. If the firm has not preyed and inspection occurs, or if no inspection occurs, no penalty is paid and the authorities' payoff is $-c$ or $0$, respectively. (ii) Assume that whether predation has occurred is not observed before the inspection decision. True or False: There exists a pure-strategy Nash equilibrium in this game.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "Check best responses. If the authority inspects (I), the firm prefers not to prey (N) because 0 > g-p given p>g. If the authority does not inspect (NI), the firm prefers to prey (P) because g>0. If the firm preys (P), the authority prefers to inspect because p-c>0. If the firm does not prey (N), the authority prefers not to inspect because 0>-c. Hence no action pair (P/N, I/NI) is a mutual best response, so no pure-strategy Nash equilibrium exists. Therefore the statement that a pure-strategy equilibrium exists is false.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.10/main#1", "chapter": "11", "split": "chapter_11", "problem_number": "11.10", "sub_id": "main", "pair_index": 1, "pair_count": 2, "question": "Consider an $n$-bidder first-bid auction. Each bidder's valuation is drawn independently from the cumulative distribution: $F(t)=t^{\\alpha}$ on $[0,1]$. In the symmetric increasing equilibrium, the bid function is linear: $b(v)=c\\,v$. Compute the coefficient $c$ as a function of $n$ and $\\alpha$. Symbols (for final answer): - `alpha`: parameter in F(t)=t^alpha, alpha>0 - `n`: number of bidders, integer n>=2", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "alpha*(n-1)/(alpha*(n-1)+1)", "reference_answer_sympy": "alpha*(n-1)/(alpha*(n-1)+1)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "alpha*(n-1)/(alpha*(n-1)+1)", "reference_reasoning": "Let b(v) be the symmetric increasing equilibrium bid and let v=Phi(b) be its inverse. A bidder with value t who bids b wins with probability F(Phi(b))^{n-1}, so expected payoff is u(b;t)=(t-b)F(Phi(b))^{n-1}. The FOC implies (t-b)Phi'(b)=F(Phi(b))/((n-1)f(Phi(b))). In equilibrium t=Phi(b), so (Phi(b)-b)Phi'(b)=F(Phi(b))/((n-1)f(Phi(b))). With F(x)=x^alpha and f(x)=alpha x^{alpha-1}, we have F/f = x/alpha, giving (Phi-b)Phi' = Phi/(alpha(n-1)). Guess a linear inverse bid function Phi(b)=k b (equivalently b(v)=v/k). Then Phi'=k and (k b - b)k = (k b)/(alpha(n-1)), so k(k-1)=k/(alpha(n-1)) and k=1+1/(alpha(n-1)). Therefore b(v)=v/k = (alpha(n-1)/(alpha(n-1)+1)) v, so the linear coefficient is c=alpha(n-1)/(alpha(n-1)+1).", "allowed_symbols": ["alpha", "n"], "symbol_definitions": {"alpha": "parameter in F(t)=t^alpha, alpha>0", "n": "number of bidders, integer n>=2"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.10/main#2", "chapter": "11", "split": "chapter_11", "problem_number": "11.10", "sub_id": "main", "pair_index": 2, "pair_count": 2, "question": "Consider an $n$-bidder first-bid auction. Each bidder's valuation is drawn independently from the cumulative distribution: $F(t)=t^{\\alpha}$ on $[0,1]$. Using the equilibrium bid function $b(v)=\\frac{\\alpha(n-1)}{\\alpha(n-1)+1}v$, compute the limit $\\lim_{n\\to\\infty} \\frac{b(v)}{v}$. Symbols (for final answer): - alpha: the distribution parameter $\\alpha>0$ - n: number of bidders - v: valuation", "question_type": "value", "problem_type_label": "Pure Math / Basic Calculation", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1", "reference_answer_sympy": "1", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1", "reference_reasoning": "We have b(v)/v = alpha(n-1)/(alpha(n-1)+1). As n->infty, alpha(n-1)->infty (for alpha>0), so the ratio tends to 1.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.11/1", "chapter": "11", "split": "chapter_11", "problem_number": "11.11", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Suppose that each bidder's estimate of the value of an object affects the other bidders' own valuations for the object. The extreme case of this situation is the common-value auction in which the \\textit{ex post} value of the object (e.g., an oil lease) is the same for all bidders and each bidder has a private and imperfect estimate of the common value. Winning then conveys bad news to the winner, who learns that the other bidders attached a low value to the good. This winner's curse must be anticipated by a rational bidder. Consider a first-bid auction for a good with common value $\\theta=x_1+x_2$. The parameters $x_1$ and $x_2$ are independently drawn from the uniform distribution on $[0,1]$. There are two bidders, $i=1,2$. Bidder $i$ knows $x_i$ but not $x_j$. (i) Using the winner’s-curse logic, what is the upper bound on bidder $i$’s expected payoff conditional on winning with bid $b_i$, expressed in terms of $x_i$ and $b_i$? Symbols (for final answer): - `b_i`: bidder i's winning bid - `x_i`: bidder i's private signal/estimate component", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "x_i + 1/2 - b_i", "reference_answer_sympy": "x_i + 1/2 - b_i", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "x_i + 1/2 - b_i", "reference_reasoning": "If bidder i wins with bid b_i, then under monotone bidding he must have outbid bidder j, implying b_i >= b_j = beta(x_j) and hence x_j <= beta^{-1}(b_i). Thus winning is bad news about x_j: conditional on winning, x_j is stochastically smaller than an unconditional U[0,1] draw, so E[x_j | win] <= E[x_j] = 1/2. Therefore E[theta | win] = x_i + E[x_j | win] <= x_i + 1/2, and the expected payoff conditional on winning satisfies E[theta - b_i | win] <= x_i + 1/2 - b_i.", "allowed_symbols": ["b_i", "x_i"], "symbol_definitions": {"b_i": "bidder i's winning bid", "x_i": "bidder i's private signal/estimate component"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.11/2", "chapter": "11", "split": "chapter_11", "problem_number": "11.11", "sub_id": "2", "pair_index": 1, "pair_count": 1, "question": "Suppose that each bidder's estimate of the value of an object affects the other bidders' own valuations for the object. The extreme case of this situation is the common-value auction in which the \\textit{ex post} value of the object (e.g., an oil lease) is the same for all bidders and each bidder has a private and imperfect estimate of the common value. Winning then conveys bad news to the winner, who learns that the other bidders attached a low value to the good. This winner's curse must be anticipated by a rational bidder. Consider a first-bid auction for a good with common value $\\theta=x_1+x_2$. The parameters $x_1$ and $x_2$ are independently drawn from the uniform distribution on $[0,1]$. There are two bidders, $i=1,2$. Bidder $i$ knows $x_i$ but not $x_j$. (ii) Look for a symmetric equilibrium. Let $x=\\Phi(b)$ denote the inverse bid function. Assuming $x_j\\sim U[0,1]$, compute the conditional expectation $E[x_j\\mid x_j\\le \\texttt{Phi\\_b}]$ as a function of $\\texttt{Phi\\_b}$. Symbols (for final answer): - `Phi_b`: the cutoff value Phi(b_i) in [0,1] implied by the inverse bid function", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "Phi_b/2", "reference_answer_sympy": "Phi_b/2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "Phi_b/2", "reference_reasoning": "Since x_j is uniform on [0,1], conditioning on the event {x_j <= Phi_b} (with 0<=Phi_b<=1) implies x_j is uniform on [0,Phi_b]. The mean of a uniform on [0,Phi_b] is Phi_b/2. Therefore E[x_j | x_j <= Phi_b] = Phi_b/2.", "allowed_symbols": ["Phi_b"], "symbol_definitions": {"Phi_b": "the cutoff value Phi(b_i) in [0,1] implied by the inverse bid function"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.12/1#1", "chapter": "11", "split": "chapter_11", "problem_number": "11.12", "sub_id": "1", "pair_index": 1, "pair_count": 2, "question": "(i) Selten (1978) offers the following paradox. A chain store (the incumbent) is located in $n$ geographically separated markets. It faces $n$ potential entrants, one in each market. The potential entrants must make their entry decisions sequentially. At date $i$ ($i=1,\\ldots,n$), entrant $i$, who has observed what happened in the $i-1$ preceding markets, decides whether to enter ($I$, for \\textit{in}) or not enter ($O$, for \\textit{out}). If entrant $i$ chooses not to enter, the payoffs in market $i$ are \\[ \\left(\\frac{3}{4},\\,0\\right). \\] If entrant $i$ enters, the incumbent chooses whether to prey ($P$) or acquiesce ($A$). If the incumbent chooses $P$, the payoffs in market $i$ are \\[ (-1,\\,-1), \\] whereas if the incumbent chooses $A$, the payoffs in market $i$ are \\[ (0,\\,1). \\] In each ordered pair, the first payoff refers to the incumbent and the second to entrant $i$. The entry and predation decisions in market $i$ are observed before entrant $i+1$ decides whether to enter. The incumbent's payoff is the sum of its $n$ payoffs. With one potential entrant ($n=1$), in the unique perfect equilibrium the entrant enters and the incumbent acquiesces. What is the incumbent's equilibrium payoff $U_I$? Symbols (for final answer): - `U_I`: incumbent's equilibrium payoff when n=1", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "Use backward induction for n=1. If the entrant enters, the incumbent compares P (payoff -1) vs A (payoff 0) and chooses A. Anticipating A, the entrant prefers entering (payoff 1) to staying out (payoff 0), so enters. Therefore the equilibrium outcome is (I, A) and the incumbent's payoff is 0.", "allowed_symbols": ["U_I"], "symbol_definitions": {"U_I": "incumbent's equilibrium payoff when n=1"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.12/1#2", "chapter": "11", "split": "chapter_11", "problem_number": "11.12", "sub_id": "1", "pair_index": 2, "pair_count": 2, "question": "(i) Selten (1978) offers the following paradox. A chain store (the incumbent) is located in $n$ geographically separated markets. It faces $n$ potential entrants, one in each market. The potential entrants must make their entry decisions sequentially. At date $i$ ($i=1,\\ldots,n$), entrant $i$, who has observed what happened in the $i-1$ preceding markets, decides whether to enter ($I$, for \\textit{in}) or not enter ($O$, for \\textit{out}). If entrant $i$ chooses not to enter, the payoffs in market $i$ are \\[ \\left(\\frac{3}{4},\\,0\\right). \\] If entrant $i$ enters, the incumbent chooses whether to prey ($P$) or acquiesce ($A$). If the incumbent chooses $P$, the payoffs in market $i$ are \\[ (-1,\\,-1), \\] whereas if the incumbent chooses $A$, the payoffs in market $i$ are \\[ (0,\\,1). \\] In each ordered pair, the first payoff refers to the incumbent and the second to entrant $i$. The entry and predation decisions in market $i$ are observed before entrant $i+1$ decides whether to enter. The incumbent's payoff is the sum of its $n$ payoffs. Compute the incumbent's total equilibrium payoff (sum across all $n$ markets) in the unique perfect equilibrium. Symbols (for final answer): - `n`: number of geographically separated markets (positive integer)", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "0", "reference_answer_sympy": "0", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "0", "reference_reasoning": "By backward induction, in the last market n, if entry occurs the incumbent compares P (payoff -1) vs A (payoff 0) and chooses A. Anticipating this, entrant n enters. The same reasoning applies in every earlier market because the continuation is independent of history: the incumbent will acquiesce after entry, so each entrant enters. Therefore, in each of the n markets the outcome is (incumbent payoff, entrant payoff) = (0,1). Summing incumbent payoffs across n markets yields total payoff 0.", "allowed_symbols": ["n"], "symbol_definitions": {"n": "number of geographically separated markets (positive integer)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.12/2#1", "chapter": "11", "split": "chapter_11", "problem_number": "11.12", "sub_id": "2", "pair_index": 1, "pair_count": 3, "question": "(ii) Now, suppose with Kreps, Milgrom, Roberts, and Wilson, that the entrants do not know the incumbent's objective function perfectly. With probability $1-x$, the payoffs are as above; with probability $x$, the incumbent gets $\\frac{1}{2}$ instead of $-1$ when preying (the payoffs are otherwise unchanged). Solve the one-entrant case. Compute the entrant’s expected payoff from entering as a function of $x$. Symbols (for final answer): - `x`: probability the incumbent is type T (gets 1/2 instead of -1 when preying)", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1-2*x", "reference_answer_sympy": "1-2*x", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1-2*x", "reference_reasoning": "If the entrant enters, the incumbent compares acquiesce (payoff 0) vs prey. Type S (probability 1-x) gets -1 from preying so acquiesces; type T (probability x) gets 1/2 from preying so preys. The entrant then gets payoff 1 if the incumbent acquiesces and -1 if the incumbent preys. Therefore the entrant’s expected payoff from entering is (1-x)*1 + x*(-1) = 1-2x.", "allowed_symbols": ["x"], "symbol_definitions": {"x": "probability the incumbent is type T (gets 1/2 instead of -1 when preying)"}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.13/1", "chapter": "11", "split": "chapter_11", "problem_number": "11.13", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "Consider the following Stackelberg game with asymmetric information. There are two firms, $i=1,2$. Firm~1 chooses output $q_1$ first. Firm~2 then observes $q_1$ and chooses output $q_2$. Payoffs are \\[ \\Pi^i = \\bigl[t_i-(q_1+q_2)\\bigr]q_i, \\] where $t_i$ can be interpreted as the intercept of the linear demand curve (minus a common unit cost). Firm~1 privately observes its type $t_1$, while firm~2 does not observe $t_1$ directly and can infer it only from $q_1$. In the discrete benchmark case, $t_1 \\in \\{L,H\\}$ with $0<L<H$, and the least-cost separating output is denoted by $s$. In this exercise, instead of having only two atoms at $L$ and $H$, assume that $t_1$ is continuously distributed on the interval $[L,H]$. Consider a separating equilibrium. Type $t_1$ chooses output \\[ q_1 = Q_1(t_1), \\] where $Q_1$ is strictly increasing and differentiable; let $T$ denote its inverse. Thus, $T(q_1)$ is the type that chooses output $q_1$. Judge whether the following statement is true or false: In a separating equilibrium, $T$ satisfies the differential equation \\[ q_1 T'(q_1)=T(q_1)-2q_1. \\]", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "In a separating equilibrium, observing q1 reveals firm 1’s type t1=T(q1). Firm 2 then chooses q2 to maximize (T(q1)−(q1+q2))q2, so its best response is q2(q1)=(T(q1)−q1)/2. Given this, a type t1 choosing q1 gets Π1(q1;t1)=[t1−(q1+q2(q1))]q1=(t1−(T(q1)+q1)/2)q1. The first-order condition is ∂Π1/∂q1=(t1−(T(q1)+q1)/2)+q1(−(T'(q1)+1)/2)=0. In a separating equilibrium t1=T(q1), so substituting yields T(q1)−(T(q1)+q1)/2−(q1/2)(T'(q1)+1)=0, which simplifies to q1 T'(q1)=T(q1)−2q1. Therefore the statement is true.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "11.14/1", "chapter": "11", "split": "chapter_11", "problem_number": "11.14", "sub_id": "1", "pair_index": 1, "pair_count": 1, "question": "In the absence of uncertainty, a monopoly manufacturer makes a monopoly retailer (or wholesaler) the residual claimant for his sales. That is, the manufacturer charges an intermediate price equal to his marginal cost and captures the retailer's profit through a lump-sum transfer (the franchise fee). This exercise shows that when the manufacturer has private information about the demand for his product, he may want to charge more than the marginal cost (and reduce the franchise fee) for signaling purposes. A monopoly manufacturer with marginal cost $c$ charges a two-part tariff to a monopoly retailer: \\[ T(q)=A+p_w q, \\] where $q$ is the quantity bought and resold by the retailer, $p_w$ is the intermediate price, and $A$ is the franchise fee. The final demand for the product is \\[ q=t_1-p, \\] where $p$ is the final consumer price chosen by the retailer. For simplicity, the cost of retailing is zero. Assume that $t_1>c$, so that the efficient quantity is strictly positive. Player 1 (the manufacturer) moves first and offers a contract \\[ a_1=\\{A,p_w\\}. \\] Player 2 (the retailer) accepts or refuses the contract and, if he accepts it, chooses a consumer price. So \\[ a_2=\\{\\text{yes or no},p\\}. \\] He accepts the contract if and only if his expected profit is non-negative. Under full information, what is the equilibrium franchise fee $A$ as a function of $t_1$ and $c$? Symbols (for final answer): - `c`: manufacturer marginal cost - `t_1`: demand intercept parameter", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "(t_1 - c)**2/4", "reference_answer_sympy": "(t_1 - c)**2/4", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "(t_1 - c)**2/4", "reference_reasoning": "Given contract {A, p_w}, the retailer chooses p to maximize π_R = (p - p_w)q - A = (p - p_w)(t_1 - p) - A. FOC: ∂π_R/∂p = t_1 - 2p + p_w = 0, so p*(p_w) = (t_1 + p_w)/2 and q*(p_w) = t_1 - p* = (t_1 - p_w)/2. Retailer operating profit before the fee is (p* - p_w)q* = ((t_1 - p_w)/2)^2 = (t_1 - p_w)^2/4. With full information, the manufacturer sets A to make the retailer’s participation constraint bind, so A*(p_w) = (t_1 - p_w)^2/4. Manufacturer profit is π_M = A*(p_w) + (p_w - c)q*(p_w) = (t_1 - p_w)^2/4 + (p_w - c)(t_1 - p_w)/2. Maximizing over p_w yields p_w* = c, hence A* = (t_1 - c)^2/4.", "allowed_symbols": ["c", "t_1"], "symbol_definitions": {"c": "manufacturer marginal cost", "t_1": "demand intercept parameter"}, "review_status": "confirmed_without_expert"} | |
| {"id": "1/i", "chapter": "intro", "split": "chapter_intro", "problem_number": "1", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "In the text, it is assumed that the supplier's investment reduces his production cost. Assume instead that the ex ante investment affects the quality of the product and thus the value to the buyer. The buyer's ex post value is $v(I) = 3I - \\frac{1}{2}I^2$. Hence, the buyer's surplus in case of trade is $v(I) - p$. The supplier's surplus is then $p - c - I$ (where $c < \\frac{1}{2}$ is now a constant production cost). Suppose that $I$ (and, hence, $v$) is observable by the buyer; however, it is not verifiable by a court, so that it cannot be specified by a contract. (i) Determine the efficient amount of investment. Symbols (for final answer): - I_star: efficient investment level", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "2", "reference_answer_sympy": "2", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "2", "reference_reasoning": "Efficiency maximizes total surplus. Since price is a transfer, total surplus from trade is TS(I)=v(I)-c-I=(3I-1/2 I^2)-c-I=2I-1/2 I^2-c. The first-order condition is dTS/dI = 2 - I = 0, giving I=2. The second-order condition d^2TS/dI^2 = -1 < 0 confirms a maximum.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |
| {"id": "1/ii", "chapter": "intro", "split": "chapter_intro", "problem_number": "1", "sub_id": "ii", "pair_index": 1, "pair_count": 1, "question": "In the text, it is assumed that the supplier's investment reduces his production cost. Assume instead that the ex ante investment affects the quality of the product and thus the value to the buyer. The buyer's ex post value is $v(I) = 3I - \\frac{1}{2}I^2$. Hence, the buyer's surplus in case of trade is $v(I) - p$. The supplier's surplus is then $p - c - I$ (where $c < \\frac{1}{2}$ is now a constant production cost). Suppose that $I$ (and, hence, $v$) is observable by the buyer; however, it is not verifiable by a court, so that it cannot be specified by a contract. (ii) Suppose that there is no contract and the two parties bargain ex post according to the Nash bargaining solution (equal weights, disagreement payoffs 0). What is the no-contract Nash-bargaining investment level $I^{NC}$? Symbols (for final answer): - `I_NC`: the no-contract Nash-bargaining investment level", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "1", "reference_answer_sympy": "1", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "1", "reference_reasoning": "With equal-weight Nash bargaining and disagreement payoffs 0, the negotiated price p splits gains from trade so that buyer and supplier surpluses are equal: v(I)-p = p-c, implying p = (v(I)+c)/2. Supplier chooses I ex ante to maximize pi_S(I)=p-c-I = (v(I)-c)/2 - I. With v(I)=3I-1/2 I^2, pi_S(I)= (3I-1/2 I^2 - c)/2 - I = (1/2)I - (1/4)I^2 - c/2. FOC: d pi_S/dI = 1/2 - (1/2)I = 0, so I^{NC}=1.", "allowed_symbols": ["I_NC"], "symbol_definitions": {"I_NC": "the no-contract Nash-bargaining investment level"}, "review_status": "confirmed_without_expert"} | |
| {"id": "2/i", "chapter": "intro", "split": "chapter_intro", "problem_number": "2", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "In this exercise, which concerns sole sourcing versus dual sourcing, we build a simple model in which the managers' objective function is such that monetary incentives are rather ineffective. A firm has a project of a given size. The cost of the project is $C = \\beta - e$. The variable $\\beta$ is random on $[\\underline{\\beta}, \\bar{\\beta}]$ with expectation $E\\beta$ (denote this by $E\\_beta$). The variable $e$ denotes the effort exerted by the manager assigned to the project. A manager has utility function $U(w, e) = u(w) - \\Phi(e)$, where $\\Phi' > 0$, $\\Phi'' > 0$, $\\Phi'(0) = 0$, and \\[ u(w) = \\begin{cases} \\bar{u} + \\lambda(w - \\bar{w}) & \\text{if} \\ w \\geq \\bar{w}, \\\\ -\\infty & \\text{if} \\ w < \\bar{w}. \\end{cases} \\] Thus, $\\bar{w}$ (denote this by $w\\_bar$) can be interpreted as a subsistence wage. $\\lambda$ is a \"small,\" positive parameter; mathematically, $\\lambda \\leq \\Phi'(e)$ for all $e$. The manager is infinitely risk-averse over the states of nature $\\beta$, so he is only interested in his utility in the worst state of nature, $\\min_{\\beta} U(w, e)$. The manager learns $\\beta$ after signing the contract and before choosing $e$. The principal observes $C$, but not $\\beta$ or $e$. So the wage structure is a function of $C$, $w(C)$, and the manager's objective can be written \\[ \\min_{\\beta} (\\max_{e} (u(w(\\beta - e)) - \\Phi(e))). \\] Let $U_0$ denote the manager's reservation utility, and let $e^* > 0$ (denote this by $e\\_star$) be defined by $u(\\bar{w}) - \\Phi(e^*) = U_0$. The principal wants to minimize the expected cost of the project. (i) If $\\beta$ and/or $e$ were observable by the principal, what is the expected cost of the project under the optimal contract? Symbols (for final answer): - `E_beta`: expectation of \\beta, i.e. E\\beta - `e_star`: effort level e^* defined by u(\\bar{w})-\\Phi(e^*)=U_0 - `w_bar`: subsistence wage \\bar{w}", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "w_bar + E_beta - e_star", "reference_answer_sympy": "w_bar + E_beta - e_star", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "w_bar + E_beta - e_star", "reference_reasoning": "If \\beta and/or e are observable, the principal can condition the contract on them and enforce effort, fully insuring the infinitely risk-averse manager across states. The participation constraint can be met state-by-state at equality: u(w(\\beta)) - \\Phi(e(\\beta)) = U_0 for all \\beta. Since wages below \\bar w are infeasible and u is increasing for w \\ge \\bar w, the cheapest feasible wage is w=\\bar w. With w=\\bar w, the participation constraint becomes u(\\bar w) - \\Phi(e) \\ge U_0, so the principal chooses the maximal effort consistent with it, i.e. e=e^* defined by u(\\bar w) - \\Phi(e^*) = U_0. Thus w=\\bar w and e=e^* in every state, giving expected cost E[w+\\beta-e]=\\bar w+E\\beta-e^*.", "allowed_symbols": ["E_beta", "e_star", "w_bar"], "symbol_definitions": {"E_beta": "expectation of \\beta, i.e. E\\beta", "e_star": "effort level e^* defined by u(\\bar{w})-\\Phi(e^*)=U_0", "w_bar": "subsistence wage \\bar{w}"}, "review_status": "confirmed_without_expert"} | |
| {"id": "2/iii#2", "chapter": "intro", "split": "chapter_intro", "problem_number": "2", "sub_id": "iii", "pair_index": 2, "pair_count": 2, "question": "In this exercise, which concerns sole sourcing versus dual sourcing, we build a simple model in which the managers' objective function is such that monetary incentives are rather ineffective. A firm has a project of a given size. The cost of the project is $C = \\beta - e$. The variable $\\beta$ is random on $[\\underline{\\beta}, \\bar{\\beta}]$ with expectation $E\\beta$. The variable $e$ denotes the effort exerted by the manager assigned to the project. A manager has utility function $U(w, e) = u(w) - \\Phi(e)$, where $\\Phi' > 0$, $\\Phi'' > 0$, $\\Phi'(0) = 0$, and \\[ u(w) = \\begin{cases} \\bar{u} + \\lambda(w - \\bar{w}) & \\text{if} \\ w \\geq \\bar{w}, \\\\ -\\infty & \\text{if} \\ w < \\bar{w}. \\end{cases} \\] Thus, $\\bar{w}$ can be interpreted as a subsistence wage. $\\lambda$ is a \"small,\" positive parameter; mathematically, $\\lambda \\leq \\Phi'(e)$ for all $e$. The manager is infinitely risk-averse over the states of nature $\\beta$, so he is only interested in his utility in the worst state of nature, $\\min_{\\beta} U(w, e)$. The manager learns $\\beta$ after signing the contract and before choosing $e$. The principal observes $C$, but not $\\beta$ or $e$. So the wage structure is a function of $C$, $w(C)$, and the manager's objective can be written \\[ \\min_{\\beta} (\\max_{e} (u(w(\\beta - e)) - \\Phi(e))). \\] Let $U_0$ denote the manager's reservation utility, and let $e^* > 0$ be defined by $u(\\bar{w}) - \\Phi(e^*) = U_0$. The principal wants to minimize the expected cost of the project. (iii) Suppose that the project can be given to two managers. The cost to the principal (net of the wage bill) is $\\min(C_1, C_2)$, where $C_i = \\beta - e_i$ and $e_i$ is manager $i$'s effort $(i = 1, 2)$. That is, $\\beta$ is the same for both managers. True or False: The condition $\\bar{w} \\leq \\bar{\\beta} - E\\beta$ means dual sourcing is preferred exactly when the extra fixed cost of hiring a second manager (an additional subsistence wage $\\bar{w}$) is no larger than the information rent the principal would otherwise have to leave under sole sourcing, which is governed by the gap $\\bar{\\beta} - E\\beta$.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Dual sourcing enables a yardstick/relative-performance scheme that induces effort without paying a state-contingent rent tied to the realized common shock \\beta. But it requires employing two managers, so the principal must cover two subsistence wages. The inequality \\bar{w} \\le \\bar{\\beta} - E\\beta compares the incremental fixed cost of the second subsistence wage to the informational rent avoided under sole sourcing, which is driven by the difference between the worst-case \\beta (\\bar{\\beta}) and its mean (E\\beta). Hence the statement correctly interprets the condition.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} | |