id stringlengths 3 12 | chapter stringclasses 12
values | split stringclasses 12
values | problem_number stringlengths 1 5 | sub_id stringclasses 12
values | pair_index int64 1 3 | pair_count int64 1 4 | question stringlengths 153 3.58k | question_type stringclasses 2
values | problem_type_label stringclasses 4
values | answer_kind stringclasses 3
values | comparison_mode stringclasses 3
values | reference_answer unknown | reference_answer_sympy stringlengths 1 92 | reference_answer_json unknown | reference_answer_json_modes unknown | answer_for_compare unknown | reference_reasoning stringlengths 90 2.64k | allowed_symbols listlengths 0 6 | symbol_definitions unknown | review_status stringclasses 2
values |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1/i | 1 | chapter_1 | 1.1 | i | 1 | 1 | 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... | value | Pure Math / Basic Calculation | sympy | sympy | "c**(1-epsilon)/(epsilon-1)" | c**(1-epsilon)/(epsilon-1) | null | null | "c**(1-epsilon)/(epsilon-1)" | 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)/e... | [
"c",
"epsilon"
] | {
"c": "constant marginal cost",
"epsilon": "elasticity of demand (epsilon > 1)"
} | confirmed_without_expert |
1.1/ii | 1 | chapter_1 | 1.1 | ii | 1 | 1 | 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`: con... | value | Pure Math / Basic Calculation | sympy | sympy | "c**(1-epsilon)/(epsilon-1)*(1-((2*epsilon-1)/(epsilon-1))*(epsilon/(epsilon-1))**(-epsilon))" | c**(1-epsilon)/(epsilon-1)*(1-((2*epsilon-1)/(epsilon-1))*(epsilon/(epsilon-1))**(-epsilon)) | null | null | "c**(1-epsilon)/(epsilon-1)*(1-((2*epsilon-1)/(epsilon-1))*(epsilon/(epsilon-1))**(-epsilon))" | 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)=\... | [
"c",
"epsilon"
] | {
"c": "constant marginal cost",
"epsilon": "elasticity of demand (ε), with epsilon > 1"
} | confirmed_without_expert |
1.1/iii#1 | 1 | chapter_1 | 1.1 | iii | 1 | 3 | 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 $c$). Symbo... | value | Model-Based, Medium Difficulty | sympy | sympy | "1 - ((2*epsilon - 1)/(epsilon - 1))*((epsilon - 1)/epsilon)**epsilon" | 1 - ((2*epsilon - 1)/(epsilon - 1))*((epsilon - 1)/epsilon)**epsilon | null | null | "1 - ((2*epsilon - 1)/(epsilon - 1))*((epsilon - 1)/epsilon)**epsilon" | 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 tota... | [
"epsilon"
] | {
"epsilon": "elasticity of demand parameter (epsilon>1)"
} | confirmed_without_expert |
1.1/iii#2 | 1 | chapter_1 | 1.1 | iii | 2 | 3 | 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(\vare... | judge | Pure Math / Basic Calculation | bool | exact | "False" | N/A | null | null | "False" | 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 sin... | [] | {} | confirmed_without_expert |
1.1/iii#3 | 1 | chapter_1 | 1.1 | iii | 3 | 3 | 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 f... | value | Pure Math / Basic Calculation | sympy | sympy | "((epsilon-1)/epsilon)**epsilon" | ((epsilon-1)/epsilon)**epsilon | null | null | "((epsilon-1)/epsilon)**epsilon" | 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. | [
"epsilon"
] | {
"epsilon": "elasticity of demand (epsilon > 1)"
} | confirmed_without_expert |
1.3/i#1 | 1 | chapter_1 | 1.3 | i | 1 | 3 | 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 elastici... | value | Pure Math / Basic Calculation | sympy | sympy | "epsilon/(epsilon-1)" | epsilon/(epsilon-1) | null | null | "epsilon/(epsilon-1)" | 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 - ... | [
"epsilon"
] | {
"epsilon": "demand elasticity parameter ε (assume ε>1)"
} | confirmed_without_expert |
1.3/i#2 | 1 | chapter_1 | 1.3 | i | 2 | 3 | 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`... | value | Pure Math / Basic Calculation | sympy | sympy | "1/(delta+1)" | 1/(delta+1) | null | null | "1/(delta+1)" | 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 \(p=\alpha-\beta q^\delta\), \(p'(q)=-\beta\delta q^{\delta-1}\) a... | [
"delta"
] | {
"delta": "demand curvature parameter in p = alpha - beta*q^delta"
} | confirmed_without_expert |
1.3/i#3 | 1 | chapter_1 | 1.3 | i | 3 | 3 | 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 $\... | value | Pure Math / Basic Calculation | sympy | sympy | 1 | 1 | null | null | 1 | 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... | [] | {} | confirmed_without_expert |
1.3/ii | 1 | chapter_1 | 1.3 | ii | 1 | 1 | 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 ciga... | judge | Simple Math but Requires Economics Concepts | bool | exact | "False" | N/A | null | null | "False" | 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 p... | [] | {} | confirmed_without_expert |
1.5/main | 1 | chapter_1 | 1.5 | main | 1 | 1 | 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), \qu... | value | Model-Based, Medium Difficulty | sympy | sympy | "D_P + (P - MC_s)*Dprime_P" | D_P + (P - MC_s)*Dprime_P | null | null | "D_P + (P - MC_s)*Dprime_P" | 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+\par... | [
"D_P",
"Dprime_P",
"MC_s",
"P"
] | {
"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"
} | confirmed_by_expert |
1.6/i | 1 | chapter_1 | 1.6 | i | 1 | 1 | 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... | judge | Model-Based, Medium Difficulty | bool | exact | "True" | N/A | null | null | "True" | 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, outpu... | [] | {} | confirmed_without_expert |
1.6/ii | 1 | chapter_1 | 1.6 | ii | 1 | 1 | 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... | value | Model-Based, Medium Difficulty | sympy | sympy | "(2*c+gamma)/((1-1/epsilon)*(1+lambda**(1/epsilon)))" | (2*c+gamma)/((1-1/epsilon)*(1+lambda**(1/epsilon))) | null | null | "(2*c+gamma)/((1-1/epsilon)*(1+lambda**(1/epsilon)))" | 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... | [
"c",
"epsilon",
"gamma",
"lambda"
] | {
"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)"
} | confirmed_without_expert |
1.7/i | 1 | chapter_1 | 1.7 | i | 1 | 1 | 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... | value | Advanced / Hardest Model Problems | sympy | sympy | "r*Integral(c_w_s*exp(-r*(s-t)), (s, t, oo))" | r*Integral(c_w_s*exp(-r*(s-t)), (s, t, oo)) | null | null | "r*Integral(c_w_s*exp(-r*(s-t)), (s, t, oo))" | 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.... | [
"c_w_s",
"r",
"s",
"t"
] | {
"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"
} | confirmed_without_expert |
1.7/ii | 1 | chapter_1 | 1.7 | ii | 1 | 1 | 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... | judge | Advanced / Hardest Model Problems | bool | exact | "False" | N/A | null | null | "False" | 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, th... | [] | {} | confirmed_without_expert |
1.8/ii | 1 | chapter_1 | 1.8 | ii | 1 | 1 | 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 ... | value | Model-Based, Medium Difficulty | sympy | sympy | 0 | 0 | null | null | 0 | 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. | [
"delta",
"lambda",
"mu"
] | {
"delta": "discount factor",
"lambda": "consumer cutoff multiplier in w(p)=lambda*p",
"mu": "monopolist pricing multiplier in p(v)=mu*v"
} | confirmed_without_expert |
1.8/iii | 1 | chapter_1 | 1.8 | iii | 1 | 1 | 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 ... | value | Simple Math but Requires Economics Concepts | sympy | sympy | 0 | 0 | null | null | 0 | 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)... | [
"delta",
"lambda",
"mu"
] | {
"delta": "discount factor",
"lambda": "linear cutoff parameter in w(p)=lambda*p",
"mu": "linear pricing parameter in p(v)=mu*v"
} | confirmed_without_expert |
1.8/iv | 1 | chapter_1 | 1.8 | iv | 1 | 1 | 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 ... | value | Model-Based, Medium Difficulty | sympy | sympy | 0 | 0 | null | null | 0 | (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 $\lim_{\delta \to 1} \mu = 0.$ | [
"delta"
] | {
"delta": "discount factor, with limit taken as delta -> 1 from below"
} | confirmed_without_expert |
1.9/i | 1 | chapter_1 | 1.9 | i | 1 | 1 | 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 ... | judge | Simple Math but Requires Economics Concepts | bool | exact | "False" | N/A | null | null | "False" | 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 exc... | [] | {} | confirmed_without_expert |
1.9/iii | 1 | chapter_1 | 1.9 | iii | 1 | 1 | 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 ... | value | Model-Based, Medium Difficulty | sympy | sympy | "1/(2*(1-delta))" | 1/(2*(1-delta)) | null | null | "1/(2*(1-delta))" | 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"
] | {
"delta": "discount factor"
} | confirmed_without_expert |
1.10/i | 1 | chapter_1 | 1.10 | i | 1 | 1 | 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-per... | value | Model-Based, Medium Difficulty | sympy | sympy | "delta*c_2" | delta*c_2 | null | null | "delta*c_2" | 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... | [
"c_2",
"delta"
] | {
"c_2": "constant unit production cost in period 2",
"delta": "discount factor"
} | confirmed_without_expert |
1.10/ii | 1 | chapter_1 | 1.10 | ii | 1 | 1 | (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 respec... | value | Model-Based, Medium Difficulty | sympy | sympy | "-delta*p2_prime*xq1" | -delta*p2_prime*xq1 | null | null | "-delta*p2_prime*xq1" | 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). | [
"delta",
"p2_prime",
"xq1"
] | {
"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"
} | confirmed_without_expert |
2.4/main | 2 | chapter_2 | 2.4 | main | 1 | 1 | 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 \(c_0\)... | judge | Simple Math but Requires Economics Concepts | bool | exact | true | true | null | null | true | In a mixed equilibrium, the monopolist must be indifferent between high and low quality:
\[
[\alpha+(1-\alpha)y](p-c_1)=(1-\alpha)y(p-c_0).
\]
Rearranging,
\[
\alpha(p-c_1)=(1-\alpha)y(c_1-c_0),
\]
so
\[
y=\frac{\alpha(p-c_1)}{(1-\alpha)(c_1-c_0)}.
\]
Since \(p>c_1>c_0\), \(y>0\). The condition
\[
\alpha p<c_1-(1-\alph... | [] | {} | confirmed_without_expert |
2.5/i | 2 | chapter_2 | 2.5 | i | 1 | 1 | 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 ... | value | Simple Math but Requires Economics Concepts | sympy | sympy | "N*(1 - G_theta0)" | N*(1 - G_theta0) | null | null | "N*(1 - G_theta0)" | 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 ... | [
"G_theta0",
"N"
] | {
"G_theta0": "value of the CDF G evaluated at theta_0, i.e., G(theta_0)",
"N": "number of potential buyers (and sellers)"
} | confirmed_without_expert |
2.5/ii#2 | 2 | chapter_2 | 2.5 | ii | 2 | 2 | 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 ... | judge | Simple Math but Requires Economics Concepts | bool | exact | "False" | N/A | null | null | "False" | 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 dema... | [] | {} | confirmed_without_expert |
2.5/iii | 2 | chapter_2 | 2.5 | iii | 1 | 1 | 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 ... | value | Model-Based, Medium Difficulty | sympy | sympy | "theta_0*(1-2*theta_0)" | theta_0*(1-2*theta_0) | null | null | "theta_0*(1-2*theta_0)" | 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 t... | [
"theta_0"
] | {
"theta_0": "sellers' valuation parameter"
} | confirmed_without_expert |
2.5/iv#1 | 2 | chapter_2 | 2.5 | iv | 1 | 3 | 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 ... | judge | Simple Math but Requires Economics Concepts | bool | exact | "True" | N/A | null | null | "True" | 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 ... | [] | {} | confirmed_without_expert |
2.5/iv#2 | 2 | chapter_2 | 2.5 | iv | 2 | 3 | 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 ... | judge | Simple Math but Requires Economics Concepts | bool | exact | "True" | N/A | null | null | "True" | 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(... | [] | {} | confirmed_without_expert |
2.5/v | 2 | chapter_2 | 2.5 | v | 1 | 1 | 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 ... | judge | Simple Math but Requires Economics Concepts | bool | exact | "False" | N/A | null | null | "False" | 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.... | [] | {} | confirmed_without_expert |
2.6/main#1 | 2 | chapter_2 | 2.6 | main | 1 | 2 | A dynamic perspective, however, leads to a greater goodwill.
True or False: As the discount factor $\delta$ increases, the first-period price increases. | judge | Model-Based, Medium Difficulty | bool | exact | false | N/A | {} | {} | "False" | False. The optimal first-period price decreases as the discount factor $\delta$ increases. | [
"p1_delta",
"p2_delta"
] | {
"p1_delta": "the sign/value of the derivative ∂p1*/∂δ (negative)",
"p2_delta": "the value of the derivative ∂p2*/∂δ (zero)"
} | confirmed_without_expert |
2.8/main#1 | 2 | chapter_2 | 2.8 | main | 1 | 2 | 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`... | value | Advanced / Hardest Model Problems | sympy | sympy | "x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0))" | x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0)) | null | null | "x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0))" | 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 pe... | [
"c_0",
"c_1",
"delta",
"theta",
"x_1"
] | {
"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"
} | confirmed_without_expert |
2.8/main#2 | 2 | chapter_2 | 2.8 | main | 2 | 2 | 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 dishone... | judge | Simple Math but Requires Economics Concepts | bool | exact | "False" | N/A | null | null | "False" | 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 ... | [] | {} | confirmed_without_expert |
3.1/i | 3 | chapter_3 | 3.1 | i | 1 | 1 | 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... | value | Simple Math but Requires Economics Concepts | sympy | sympy | "Integral(P_u, (u, 0, Q))/n" | Integral(P_u, (u, 0, Q))/n | null | null | "Integral(P_u, (u, 0, Q))/n" | 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
\[
S(q)=\int_0^q P(ns)\,ds.
\]
Using the change of variable \(u=ns\),... | [
"P_u",
"Q",
"n",
"u"
] | {
"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"
} | confirmed_without_expert |
3.5/i | 3 | chapter_3 | 3.5 | i | 1 | 1 | 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 typ... | value | Simple Math but Requires Economics Concepts | sympy | sympy | "(1-lambda)*S_2(c)" | (1-lambda)*S_2(c) | null | null | "(1-lambda)*S_2(c)" | 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 gr... | [
"S_2_c",
"lambda"
] | {
"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)"
} | confirmed_without_expert |
3.6/main | 3 | chapter_3 | 3.6 | main | 1 | 1 | 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/i... | judge | Simple Math but Requires Economics Concepts | bool | exact | "False" | N/A | null | null | "False" | 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... | [] | {} | confirmed_without_expert |
3.7/ii | 3 | chapter_3 | 3.7 | ii | 1 | 1 | 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, th... | value | Pure Math / Basic Calculation | sympy | sympy | "c - (1 - G_t)/g_t" | c - (1 - G_t)/g_t | null | null | "c - (1 - G_t)/g_t" | 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(... | [
"G_t",
"c",
"g_t"
] | {
"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)"
} | confirmed_without_expert |
3.8/i | 3 | chapter_3 | 3.8 | i | 1 | 1 | 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 $\th... | value | Model-Based, Medium Difficulty | sympy | sympy | "4/5" | 4/5 | null | null | "4/5" | 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/... | [] | {} | confirmed_without_expert |
3.8/ii | 3 | chapter_3 | 3.8 | ii | 1 | 1 | 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 $\th... | value | Model-Based, Medium Difficulty | sympy | sympy | "6/7" | 6/7 | null | null | "6/7" | 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... | [] | {} | confirmed_without_expert |
3.10/main | 3 | chapter_3 | 3.10 | main | 1 | 1 | 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 fi... | value | Simple Math but Requires Economics Concepts | sympy | sympy | 2 | 2 | null | null | 2 | 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 ... | [] | {} | confirmed_without_expert |
4.1/main | 4 | chapter_4 | 4.1 | main | 1 | 1 | 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`: ... | value | Model-Based, Medium Difficulty | sympy | sympy | "1/2" | 1/2 | null | null | "1/2" | 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... | [
"D_p",
"Dp_p",
"Dpp_p"
] | {
"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)"
} | confirmed_without_expert |
4.2/i | 4 | chapter_4 | 4.2 | i | 1 | 1 | 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 $i$ $(i = 1, 2)$.... | judge | Simple Math but Requires Economics Concepts | bool | exact | "False" | False | null | null | "False" | 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 w... | [] | {} | confirmed_without_expert |
4.2/ii | 4 | chapter_4 | 4.2 | ii | 1 | 1 | 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 $i$ $(i = 1, 2)$.... | value | Pure Math / Basic Calculation | sympy | sympy | "c/(1-1/epsilon)" | c/(1-1/epsilon) | null | null | "c/(1-1/epsilon)" | 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 yiel... | [
"c",
"epsilon"
] | {
"c": "total marginal cost c1+c2",
"epsilon": "absolute price elasticity of demand (assumed constant)"
} | confirmed_without_expert |
4.2/iii | 4 | chapter_4 | 4.2 | iii | 1 | 1 | 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 $i$ $(i = 1, 2)$.... | value | Model-Based, Medium Difficulty | sympy | sympy | "c/(1 - 1/epsilon)**2" | c/(1 - 1/epsilon)**2 | null | null | "c/(1 - 1/epsilon)**2" | 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 ep... | [
"c",
"epsilon"
] | {
"c": "total marginal cost c1+c2",
"epsilon": "price elasticity of demand (epsilon>1)"
} | confirmed_without_expert |
4.2/iv#1 | 4 | chapter_4 | 4.2 | iv | 1 | 2 | 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 $i$ $(i = 1, 2)$.... | value | Simple Math but Requires Economics Concepts | sympy | sympy | "c/(1-2/epsilon)" | c/(1-2/epsilon) | null | null | "c/(1-2/epsilon)" | 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 d... | [
"c",
"epsilon"
] | {
"c": "total marginal cost c1+c2",
"epsilon": "absolute value of the price elasticity of demand"
} | confirmed_without_expert |
4.5/i | 4 | chapter_4 | 4.5 | i | 1 | 1 | 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 exerc... | value | Model-Based, Medium Difficulty | sympy | sympy | "c_prime*x_prime + p_w*(r + alpha_x_prime)" | c_prime*x_prime + p_w*(r + alpha_x_prime) | null | null | "c_prime*x_prime + p_w*(r + alpha_x_prime)" | 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 al... | [
"alpha_x_prime",
"c_prime",
"p_w",
"r",
"x_prime"
] | {
"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')"
} | confirmed_without_expert |
4.5/iii | 4 | chapter_4 | 4.5 | iii | 1 | 1 | 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 exerc... | value | Simple Math but Requires Economics Concepts | sympy | sympy | "r + alpha_x_prime" | r + alpha_x_prime | null | null | "r + alpha_x_prime" | 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. | [
"alpha_x_prime",
"r"
] | {
"alpha_x_prime": "breakdown hazard rate α(x') as a function of maintenance intensity x'",
"r": "rate of interest"
} | confirmed_by_expert |
4.6/i | 4 | chapter_4 | 4.6 | i | 1 | 1 | 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... | judge | Simple Math but Requires Economics Concepts | bool | exact | "False" | False | null | null | "False" | 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... | [] | {} | confirmed_without_expert |
4.6/ii | 4 | chapter_4 | 4.6 | ii | 1 | 1 | 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... | judge | Model-Based, Medium Difficulty | bool | exact | "True" | True | null | null | "True" | 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. | [] | {} | confirmed_without_expert |
5.2/main#1 | 5 | chapter_5 | 5.2 | main | 1 | 2 | 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 (... | value | Model-Based, Medium Difficulty | sympy | sympy | "1 - (qbar_1 + qbar_2)" | 1 - (qbar_1 + qbar_2) | null | null | "1 - (qbar_1 + qbar_2)" | 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_1",
"qbar_2"
] | {
"qbar_1": "firm 1 capacity \\bar q_1",
"qbar_2": "firm 2 capacity \\bar q_2"
} | confirmed_without_expert |
5.2/main#2 | 5 | chapter_5 | 5.2 | main | 2 | 2 | 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 (f... | value | Model-Based, Medium Difficulty | sympy | sympy | "bar_q_i*(1 - bar_q_i - bar_q_j)" | bar_q_i*(1 - bar_q_i - bar_q_j) | null | null | "bar_q_i*(1 - bar_q_i - bar_q_j)" | 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"
] | {
"bar_q_i": "firm i capacity (given, less than 1/4)",
"bar_q_j": "firm j capacity (given, less than 1/4)"
} | confirmed_without_expert |
5.3/i | 5 | chapter_5 | 5.3 | i | 1 | 1 | 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 ex... | value | Pure Math / Basic Calculation | json | json | {
"q1": "1/4",
"q2": "1/4",
"q3": "1/4"
} | N/A | {
"q1": "1/4",
"q2": "1/4",
"q3": "1/4"
} | {
"q1": "sympy",
"q2": "sympy",
"q3": "sympy"
} | {
"q1": "1/4",
"q2": "1/4",
"q3": "1/4"
} | 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, ... | [] | {} | confirmed_without_expert |
5.3/iii | 5 | chapter_5 | 5.3 | iii | 1 | 1 | 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) | value | Pure Math / Basic Calculation | sympy | sympy | "1/4" | 1/4 | null | null | "1/4" | 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. | [] | {} | confirmed_by_expert |
5.3/iv | 5 | chapter_5 | 5.3 | iv | 1 | 1 | 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 firm... | judge | Simple Math but Requires Economics Concepts | bool | exact | "True" | N/A | null | null | "True" | 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 complem... | [] | {} | confirmed_without_expert |
5.4/ii#2 | 5 | chapter_5 | 5.4 | ii | 2 | 2 | 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 $p = 1 - q_1 - q_2$... | value | Model-Based, Medium Difficulty | sympy | sympy | 0 | 0 | null | null | 0 | 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/∂q... | [] | {} | confirmed_without_expert |
5.5/ii#1 | 5 | chapter_5 | 5.5 | ii | 1 | 3 | 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 opport... | value | Model-Based, Medium Difficulty | json | json | {
"q1": "(2-a)/7",
"q2": "(5+a)/21"
} | N/A | {
"q1": "(2-a)/7",
"q2": "(5+a)/21"
} | {
"q1": "sympy",
"q2": "sympy"
} | {
"q1": "(2-a)/7",
"q2": "(5+a)/21"
} | 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"
] | {
"a": "demand intercept in the second market"
} | confirmed_without_expert |
5.5/ii#2 | 5 | chapter_5 | 5.5 | ii | 2 | 3 | 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... | value | Model-Based, Medium Difficulty | sympy | sympy | 0 | 0 | null | null | 0 | 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*... | [] | {} | confirmed_without_expert |
5.8/i | 5 | chapter_5 | 5.8 | i | 1 | 1 | 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, ... | judge | Simple Math but Requires Economics Concepts | bool | exact | "True" | True | null | null | "True" | 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... | [] | {} | confirmed_without_expert |
5.9/i | 5 | chapter_5 | 5.9 | i | 1 | 1 | 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... | value | Advanced / Hardest Model Problems | json | json | {
"P1": "0",
"P2": "0"
} | N/A | {
"P1": "0",
"P2": "0"
} | {
"P1": "sympy",
"P2": "sympy"
} | {
"P1": "0",
"P2": "0"
} | 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... | [] | {} | confirmed_without_expert |
5.9/ii | 5 | chapter_5 | 5.9 | ii | 1 | 1 | 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 ... | judge | Model-Based, Medium Difficulty | bool | exact | "True" | True | null | null | "True" | 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 ... | [] | {} | confirmed_without_expert |
5.9/iii#1 | 5 | chapter_5 | 5.9 | iii | 1 | 2 | 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`: margina... | value | Advanced / Hardest Model Problems | sympy | sympy | "1 - c/p" | 1 - c/p | null | null | "1 - c/p" | 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 whol... | [
"c",
"p"
] | {
"c": "marginal cost",
"p": "price"
} | confirmed_without_expert |
6.1/1#1 | 6 | chapter_6 | 6.1 | 1 | 1 | 3 | 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 fi... | value | Advanced / Hardest Model Problems | sympy | sympy | "Phi_1_p*(1 - Pi_bar_2/Phi_2) + Phi_1*Pi_bar_2*Phi_2_p/Phi_2**2" | Phi_1_p*(1 - Pi_bar_2/Phi_2) + Phi_1*Pi_bar_2*Phi_2_p/Phi_2**2 | null | null | "Phi_1_p*(1 - Pi_bar_2/Phi_2) + Phi_1*Pi_bar_2*Phi_2_p/Phi_2**2" | 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 object... | [
"Phi_1",
"Phi_1_p",
"Phi_2",
"Phi_2_p",
"Pi_bar_2"
] | {
"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"
} | confirmed_without_expert |
6.1/1#2 | 6 | chapter_6 | 6.1 | 1 | 2 | 3 | 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 fi... | judge | Advanced / Hardest Model Problems | bool | exact | "True" | True | null | null | "True" | 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... | [] | {} | confirmed_without_expert |
6.1/2 | 6 | chapter_6 | 6.1 | 2 | 1 | 1 | 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 f... | value | Model-Based, Medium Difficulty | sympy | sympy | "(p-c_1)*Dprime_p + D_p + (c_2-c_1)*Pi_bar_2/(p-c_2)**2" | (p-c_1)*Dprime_p + D_p + (c_2-c_1)*Pi_bar_2/(p-c_2)**2 | null | null | "(p-c_1)*Dprime_p + D_p + (c_2-c_1)*Pi_bar_2/(p-c_2)**2" | 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-... | [
"D_p",
"Dprime_p",
"Pi_bar_2",
"c_1",
"c_2",
"p"
] | {
"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"
} | confirmed_without_expert |
6.1/3 | 6 | chapter_6 | 6.1 | 3 | 1 | 1 | 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 fi... | judge | Advanced / Hardest Model Problems | bool | exact | "True" | True | null | null | "True" | 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 int... | [] | {} | confirmed_without_expert |
6.1/4 | 6 | chapter_6 | 6.1 | 4 | 1 | 1 | 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 fi... | judge | Advanced / Hardest Model Problems | bool | exact | "True" | N/A | null | null | "True" | 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 fi... | [] | {} | confirmed_without_expert |
6.1/5 | 6 | chapter_6 | 6.1 | 5 | 1 | 1 | 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 fi... | judge | Advanced / Hardest Model Problems | bool | exact | "True" | True | null | null | "True" | 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 patien... | [] | {} | confirmed_without_expert |
6.1/6 | 6 | chapter_6 | 6.1 | 6 | 1 | 1 | 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 fi... | judge | Advanced / Hardest Model Problems | bool | exact | "True" | N/A | null | null | "True" | 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(... | [] | {} | confirmed_without_expert |
6.2/main | 6 | chapter_6 | 6.2 | main | 1 | 1 | 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 d... | value | Model-Based, Medium Difficulty | sympy | sympy | "Pi_p*(1-delta**m)/(1-delta**(m+n))" | Pi_p*(1-delta**m)/(1-delta**(m+n)) | null | null | "Pi_p*(1-delta**m)/(1-delta**(m+n))" | 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}\delt... | [
"Pi_p",
"delta",
"m",
"n"
] | {
"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"
} | confirmed_by_expert |
6.3/main | 6 | chapter_6 | 6.3 | main | 1 | 1 | 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 m... | value | Advanced / Hardest Model Problems | sympy | sympy | 0 | 0 | null | null | 0 | 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 fir... | [
"bar_pi",
"delta"
] | {
"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)"
} | confirmed_by_expert |
6.4/main#1 | 6 | chapter_6 | 6.4 | main | 1 | 2 | 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 ... | value | Advanced / Hardest Model Problems | sympy | sympy | "(1 - 1/n)/mu" | (1 - 1/n)/mu | null | null | "(1 - 1/n)/mu" | 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... | [
"mu",
"n"
] | {
"mu": "demand growth/decay factor (mu)",
"n": "number of firms"
} | confirmed_without_expert |
6.6/1 | 6 | chapter_6 | 6.6 | 1 | 1 | 1 | 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 ... | value | Advanced / Hardest Model Problems | sympy | sympy | "1/sqrt(2)" | 1/sqrt(2) | null | null | "1/sqrt(2)" | 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... | [] | {} | confirmed_without_expert |
6.6/2 | 6 | chapter_6 | 6.6 | 2 | 1 | 1 | 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 ... | value | Advanced / Hardest Model Problems | sympy | sympy | "(1+sqrt(17))/8" | (1+sqrt(17))/8 | null | null | "(1+sqrt(17))/8" | 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... | [] | {} | confirmed_without_expert |
6.8/1#2 | 6 | chapter_6 | 6.8 | 1 | 2 | 2 | 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; sh... | value | Advanced / Hardest Model Problems | sympy | sympy | "2/3" | 2/3 | null | null | "2/3" | 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 inequali... | [] | {} | confirmed_without_expert |
6.8/3 | 6 | chapter_6 | 6.8 | 3 | 1 | 1 | 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; sh... | value | Advanced / Hardest Model Problems | sympy | sympy | 2 | 2 | null | null | 2 | 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 a... | [] | {} | confirmed_without_expert |
6.9/1 | 6 | chapter_6 | 6.9 | 1 | 1 | 1 | 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_{... | value | Advanced / Hardest Model Problems | sympy | sympy | "Pi_q_c/(1-delta) + (Pi_q_plus - Pi_q_c)/(1 - alpha_star*delta**(T+1) - (1-alpha_star)*delta)" | Pi_q_c/(1-delta) + (Pi_q_plus - Pi_q_c)/(1 - alpha_star*delta**(T+1) - (1-alpha_star)*delta) | null | null | "Pi_q_c/(1-delta) + (Pi_q_plus - Pi_q_c)/(1 - alpha_star*delta**(T+1) - (1-alpha_star)*delta)" | 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 + al... | [
"Pi_q_c",
"Pi_q_plus",
"T",
"alpha_star",
"delta"
] | {
"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^+)",
"del... | confirmed_without_expert |
6.9/3#1 | 6 | chapter_6 | 6.9 | 3 | 1 | 2 | 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_{... | judge | Advanced / Hardest Model Problems | bool | exact | "True" | N/A | null | null | "True" | 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 b... | [] | {} | confirmed_without_expert |
6.10/main#2 | 6 | chapter_6 | 6.10 | main | 2 | 3 | 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, onl... | value | Advanced / Hardest Model Problems | sympy | sympy | "(4*delta + 9*delta**2 - 5)/(5*delta + 9*delta**2)" | (4*delta + 9*delta**2 - 5)/(5*delta + 9*delta**2) | null | null | "(4*delta + 9*delta**2 - 5)/(5*delta + 9*delta**2)" | 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. F... | [
"delta"
] | {
"delta": "discount factor (0<delta<1)"
} | confirmed_without_expert |
7.2/1 | 7 | chapter_7 | 7.2 | 1 | 1 | 1 | 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... | value | Model-Based, Medium Difficulty | sympy | sympy | "t + (2*c_1 + c_2)/3" | t + (2*c_1 + c_2)/3 | null | null | "t + (2*c_1 + c_2)/3" | 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. | [
"c_1",
"c_2",
"t"
] | {
"c_1": "marginal cost of firm 1",
"c_2": "marginal cost of firm 2",
"t": "transportation cost parameter (t>0)"
} | confirmed_without_expert |
7.2/2 | 7 | chapter_7 | 7.2 | 2 | 1 | 1 | 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... | value | Model-Based, Medium Difficulty | sympy | sympy | "-1/(9*t)" | -1/(9*t) | null | null | "-1/(9*t)" | 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).... | [
"c_1",
"c_2",
"t"
] | {
"c_1": "firm 1 marginal cost",
"c_2": "firm 2 marginal cost",
"t": "transportation cost parameter (t>0)"
} | confirmed_by_expert |
7.3/main#1 | 7 | chapter_7 | 7.3 | main | 1 | 3 | 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 p... | value | Model-Based, Medium Difficulty | sympy | sympy | "c + t/n**2" | c + t/n**2 | null | null | "c + t/n**2" | 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... | [
"c",
"n",
"t"
] | {
"c": "marginal cost",
"n": "number of entered firms",
"t": "transportation-cost parameter"
} | confirmed_without_expert |
7.3/main#2 | 7 | chapter_7 | 7.3 | main | 2 | 3 | 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 p... | value | Model-Based, Medium Difficulty | sympy | sympy | "(t/f)**(1/3)" | (t/f)**(1/3) | null | null | "(t/f)**(1/3)" | 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 imp... | [
"f",
"t"
] | {
"f": "fixed entry cost",
"t": "transportation-cost parameter"
} | confirmed_without_expert |
7.3/main#3 | 7 | chapter_7 | 7.3 | main | 3 | 3 | 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 p... | value | Model-Based, Medium Difficulty | sympy | sympy | "(t/(6*f))**(1/3)" | (t/(6*f))**(1/3) | null | null | "(t/(6*f))**(1/3)" | 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... | [
"f",
"t"
] | {
"f": "fixed entry cost",
"t": "transportation-cost parameter"
} | confirmed_without_expert |
8.1/main | 8 | chapter_8 | 8.1 | main | 1 | 1 | 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... | judge | Model-Based, Medium Difficulty | bool | exact | "True" | N/A | null | null | "True" | 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 re... | [] | {} | confirmed_without_expert |
End of preview. Expand in Data Studio
IO-Bench
IO-Bench is a 155-example evaluation dataset for mathematical economics reasoning. Each record contains a standalone economics question, a reference answer, a machine-comparable answer field, symbolic answer metadata where applicable, and review status metadata.
Unless otherwise noted, the dataset materials in this repository are licensed under the Creative Commons Attribution-NoDerivatives 4.0 International License (CC BY-ND 4.0). See LICENSE for details.
Dataset Structure
- Main HF split:
data/test.jsonl - Chapter copies:
data/by_chapter/*.jsonl - Problem-type labels: included in each JSONL record as
problem_type_label - Rows: 155
- Splits: {'chapter_1': 21, 'chapter_2': 10, 'chapter_3': 7, 'chapter_4': 9, 'chapter_5': 12, 'chapter_6': 17, 'chapter_7': 5, 'chapter_8': 17, 'chapter_9': 9, 'chapter_10': 15, 'chapter_11': 29, 'chapter_intro': 4}
Fields
id: Unique example id.chapter,split,problem_number,sub_id: Source location metadata after anonymization.question: Standalone benchmark question.question_type: Question format, such asvalueorjudge.problem_type_label: Problem-type category used for analysis.answer_kind: Answer representation category.comparison_mode: Evaluation mode, such assympy,exact, orjson.reference_answer: Reference final answer.reference_answer_sympy: SymPy-compatible answer when applicable.reference_answer_json: JSON answer when applicable.reference_answer_json_modes: Per-field comparison modes for JSON answers when applicable.answer_for_compare: Canonical comparison target used by the evaluator.reference_reasoning: Reference solution rationale.allowed_symbols,symbol_definitions: Symbol contract for symbolic answers.review_status: Anonymized review status.
- Downloads last month
- 26