| {"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": "3.10/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": "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"} |
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