| {"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"} |
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