{"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"}