| {"id": "2.4/main", "chapter": "2", "split": "chapter_2", "problem_number": "2.4", "sub_id": "main", "pair_index": 1, "pair_count": 1, "question": "Consider a monopolist selling an experience good at a fixed price \\(p\\) to a unit mass of consumers. The monopolist chooses the quality of the good after consumers make their purchase decisions. There are two possible qualities: high quality, produced at unit cost \\(c_1\\), and low quality, produced at unit cost \\(c_0\\), where \\(p>c_1>c_0\\).\n\nA fraction \\(\\alpha\\in(0,1)\\) of consumers are informed, meaning that they observe the quality before purchasing. The remaining fraction \\(1-\\alpha\\) are uninformed and do not observe quality before purchasing.\n\nConsumers have unit demand. If a consumer buys the good, the payoff is \\(v-p\\) if quality is high and \\(-p\\) if quality is low, where \\(v>p\\). Thus, informed consumers buy if and only if quality is high.\n\nUninformed consumers hold a belief \\(\\mu\\in[0,1]\\) that the good is of high quality. Their expected payoff from purchasing is\n\\[\n\\mu(v-p)+(1-\\mu)(-p)=\\mu v-p,\n\\]\nand they purchase if and only if \\(\\mu v\\ge p\\). Let \\(y\\in[0,1]\\) denote the fraction of uninformed consumers who purchase.\n\nIf the monopolist chooses high quality, its profit is\n\\[\n\\pi_H=(p-c_1)[\\alpha+(1-\\alpha)y],\n\\]\nwhile if it chooses low quality, its profit is\n\\[\n\\pi_L=(p-c_0)(1-\\alpha)y.\n\\]\n\nAn equilibrium is defined as a Perfect Bayesian equilibrium. Judge whether the following statement is true or false:\n\nIf\n\\[\n\\alpha p<c_1-(1-\\alpha)c_0,\n\\]\nthen in equilibrium the monopolist randomizes between high and low quality, and only a fraction \\(y\\in(0,1)\\) of uninformed consumers purchase.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "true", "reference_answer_sympy": "true", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "true", "reference_reasoning": "In a mixed equilibrium, the monopolist must be indifferent between high and low quality:\n\\[\n[\\alpha+(1-\\alpha)y](p-c_1)=(1-\\alpha)y(p-c_0).\n\\]\nRearranging,\n\\[\n\\alpha(p-c_1)=(1-\\alpha)y(c_1-c_0),\n\\]\nso\n\\[\ny=\\frac{\\alpha(p-c_1)}{(1-\\alpha)(c_1-c_0)}.\n\\]\nSince \\(p>c_1>c_0\\), \\(y>0\\). The condition\n\\[\n\\alpha p<c_1-(1-\\alpha)c_0\n\\]\nis equivalent to\n\\[\n\\alpha(p-c_1)<(1-\\alpha)(c_1-c_0),\n\\]\nwhich implies \\(y<1\\). Hence \\(y\\in(0,1)\\), so only a fraction of uninformed consumers purchase. Uninformed consumers mix only if they are indifferent between buying and not buying. Since their expected payoff from buying is\n\\[\n\\mu v-p,\n\\]\nindifference requires\n\\[\n\\mu v-p=0,\n\\]\nso\n\\[\n\\mu=\\frac{p}{v}.\n\\]\nBecause \\(v>p\\), this belief is feasible. Therefore, the statement is true.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} |
| {"id": "2.5/i", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "i", "pair_index": 1, "pair_count": 1, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (i) Determine the efficient volume of trade (i.e., the volume of trade under symmetric information). Express your answer using $G_{\\theta_0} \\equiv G(\\theta_0)$ and $N$. Symbols (for final answer): - `G_theta0`: value of the CDF G evaluated at theta_0, i.e., G(theta_0) - `N`: number of potential buyers (and sellers)", "question_type": "value", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "N*(1 - G_theta0)", "reference_answer_sympy": "N*(1 - G_theta0)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "N*(1 - G_theta0)", "reference_reasoning": "Under symmetric information, trade of a car with quality s to a buyer of type theta is efficient iff the buyer's value exceeds the seller's value: theta*s - theta0*s >= 0. Since s>0 on [s_min,s_max], this is equivalent to theta >= theta0. Therefore, the efficient number of trades equals the number of buyers with theta >= theta0, which is N times the upper-tail probability 1-G(theta0). Denote G(theta0) by G_theta0.", "allowed_symbols": ["G_theta0", "N"], "symbol_definitions": {"G_theta0": "value of the CDF G evaluated at theta_0, i.e., G(theta_0)", "N": "number of potential buyers (and sellers)"}, "review_status": "confirmed_without_expert"} |
| {"id": "2.5/ii#2", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "ii", "pair_index": 2, "pair_count": 2, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (ii) True or False: The demand curve is necessarily downward sloping in price $p$ (given no income effects).", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "False. With asymmetric information, a higher price induces more (and higher-quality) sellers to enter, raising the expected quality of cars offered/traded \\bar s(p)=E[s|s\\le p/\\theta_0]. Then the buyer cutoff type is \\theta=p/\\bar s(p), which need not rise with p; it can fall if \\bar s(p) rises sufficiently. Hence demand D(p)=N[1-G(p/\\bar s(p))] can increase with p over some range, so demand is not necessarily downward sloping even without income effects.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} |
| {"id": "2.5/iii", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "iii", "pair_index": 1, "pair_count": 1, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (iii) For the case $\\theta_0<1/2$, what is the competitive equilibrium price $p^*$ as a function of $\\theta_0$? Symbols (for final answer): - `theta_0`: sellers' valuation parameter", "question_type": "value", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "theta_0*(1-2*theta_0)", "reference_answer_sympy": "theta_0*(1-2*theta_0)", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "theta_0*(1-2*theta_0)", "reference_reasoning": "With s,theta ~ U[0,1]. At price p, a seller sells iff p >= theta_0*s, i.e. s <= p/theta_0, so offered quality support is [0, \\bar s] with \\bar s = min{p/theta_0,1}. If p<theta_0 then \\bar s=p/theta_0 and E[s|trade]=\\bar s/2 = p/(2 theta_0). A buyer buys iff theta*E[s|trade] >= p, i.e. theta >= p / (p/(2 theta_0)) = 2 theta_0, so demand is N(1-2 theta_0) when theta_0<1/2. Supply is N p/theta_0. Market clearing N p/theta_0 = N(1-2 theta_0) gives p^* = theta_0(1-2 theta_0). This p^* satisfies p^*<theta_0 when theta_0<1/2, so the case is consistent.", "allowed_symbols": ["theta_0"], "symbol_definitions": {"theta_0": "sellers' valuation parameter"}, "review_status": "confirmed_without_expert"} |
| {"id": "2.5/iv#1", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "iv", "pair_index": 1, "pair_count": 3, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (iv) True or False: In general, there can exist multiple equilibria because the buyers’ willingness to pay depends on the expected quality of cars offered for sale, and that expected quality depends on the price, so the market-clearing condition can have more than one solution.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "With asymmetric information, at price p sellers with quality s sell iff p >= theta_0 s, so the set of cars offered is s <= p/theta_0 (capped at s_max). The average quality of cars offered, \\bar s(p)=E[s | s <= p/theta_0], increases with p. Buyers then buy iff theta \\bar s(p) >= p, so demand depends on p through p/\\bar s(p). Since \\bar s(p) changes with p, the mapping p -> p/\\bar s(p) need not be monotone, making the (inverse) demand potentially non-monotone. Therefore the market-clearing equation S(p)=D(p) can have multiple solutions, implying multiple equilibria in general.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} |
| {"id": "2.5/iv#2", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "iv", "pair_index": 2, "pair_count": 3, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (iv) Then assume that this is the case. Judge whether the following statement is true or false: “If there are two equilibria with prices $p_L<p_H$, then the allocation associated with the higher-price equilibrium $p_H$ Pareto dominates the allocation associated with the lower-price equilibrium $p_L$.”", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "True", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "True", "reference_reasoning": "Let p_L<p_H be two equilibria. A higher price weakly increases the set of sellers willing to sell, so S(p_H)≥S(p_L), implying weakly higher average traded quality \\bar s(p_H)≥\\bar s(p_L). Market clearing gives D(p)=S(p), so D(p_H)≥D(p_L). With cutoff buyer type \\theta^*(p)=p/\\bar s(p), we have \\theta^*(p_H)=p_H/\\bar s(p_H) ≤ p_L/\\bar s(p_L)=\\theta^*(p_L). Hence at p_H at least as many trades occur and expected traded quality is weakly higher. Sellers are weakly better off because any seller trading at p_L can also trade at p_H and receives a higher price; non-trading sellers are unchanged. Buyers who traded at p_L remain willing to trade at p_H since \\theta^*(p_H)≤\\theta^*(p_L), and higher expected quality with market clearing implies their expected surplus \\theta\\bar s(p)-p is weakly higher. Therefore the p_H allocation Pareto dominates the p_L allocation.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} |
| {"id": "2.5/v", "chapter": "2", "split": "chapter_2", "problem_number": "2.5", "sub_id": "v", "pair_index": 1, "pair_count": 1, "question": "Potential sellers and buyers of used cars are in equal number, $N$ (where $N$ is \"large\"). The distribution of qualities in the population of potential sellers is given by the c.d.f. $F(s)$ with density $f(s)$ on $[s_{\\min}, s_{\\max}]$. All sellers have surplus $\\{\\theta_0 s\\}$ when keeping a car with quality $s$, and $p$ when selling it (so all sellers are identical but for the quality of their car). A buyer has surplus $\\{\\theta s - p\\}$ when buying a car of quality $s$ and $0$ when not buying. Buyers are heterogeneous: $\\theta$ is distributed on $[\\theta_{\\min}, \\theta_{\\max}]$ according to the c.d.f. $G(\\theta)$ and the density $g(\\theta)$. Assume that $\\theta_{\\min} < \\theta_0 < \\theta_{\\max}$. Sellers know the quality of their cars, but buyers do not (asymmetric information). (v) True or False: An enforceable minimum quality standard $s_0>0$ can never improve welfare in this market; restricting trade to $s\\ge s_0$ cannot raise buyers’ rational-expectations expected quality of traded cars and therefore cannot restore positive trade relative to a no-trade adverse-selection equilibrium.", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "With asymmetric information, adverse selection can lead to unraveling and a no-trade rational-expectations equilibrium (e.g., price collapses so no seller participates), yielding zero market surplus. An enforceable minimum quality standard (MQS) s0 restricts the set of qualities that can be offered for sale to s >= s0. This truncation increases the expected quality of cars conditional on trade, which increases buyers’ willingness to pay. If the restricted market admits an equilibrium with positive trade (some sellers with s >= s0 sell and some buyers with sufficiently high theta buy), then total surplus from those trades is strictly positive because trades occur only when theta > theta0, generating gains (theta - theta0)s > 0. Therefore, relative to the no-trade outcome, an enforceable MQS can strictly increase welfare.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} |
| {"id": "2.6/main#1", "chapter": "2", "split": "chapter_2", "problem_number": "2.6", "sub_id": "main", "pair_index": 1, "pair_count": 2, "question": "A dynamic perspective, however, leads to a greater goodwill.\n\nTrue or False: As the discount factor $\\delta$ increases, the first-period price increases.", "question_type": "judge", "problem_type_label": "Model-Based, Medium Difficulty", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": false, "reference_answer_sympy": "N/A", "reference_answer_json": {}, "reference_answer_json_modes": {}, "answer_for_compare": "False", "reference_reasoning": "False. The optimal first-period price decreases as the discount factor $\\delta$ increases.", "allowed_symbols": ["p1_delta", "p2_delta"], "symbol_definitions": {"p1_delta": "the sign/value of the derivative ∂p1*/∂δ (negative)", "p2_delta": "the value of the derivative ∂p2*/∂δ (zero)"}, "review_status": "confirmed_without_expert"} |
| {"id": "2.8/main#1", "chapter": "2", "split": "chapter_2", "problem_number": "2.8", "sub_id": "main", "pair_index": 1, "pair_count": 2, "question": "Compute if $\\delta(\\theta - c_0) > c_1 - c_0 > \\delta(\\theta x_1 - c_0)$, the dishonest monopolist establishes a reputation for the second period with probability \\alpha. Compute $\\alpha$ as a symbolic expression. Symbols (for final answer): - `c_0`: unit cost of low quality - `c_1`: unit cost of high quality - `delta`: discount factor - `theta`: consumer valuation parameter - `x_1`: prior probability the monopolist is honest", "question_type": "value", "problem_type_label": "Advanced / Hardest Model Problems", "answer_kind": "sympy", "comparison_mode": "sympy", "reference_answer": "x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0))", "reference_answer_sympy": "x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0))", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "x_1*(delta*(theta - c_0) - (c_1 - c_0))/((1 - x_1)*(c_1 - c_0 + delta*c_0))", "reference_reasoning": "Let the prior probability the monopolist is honest be x_1. The honest type always supplies high quality. Let alpha be the probability the dishonest type supplies high quality in period 1. If high quality is observed in period 1, Bayes’ rule gives the posterior probability of honesty x_2 = x_1/(x_1+(1-x_1)*alpha). In period 2, given belief x_2, consumers’ willingness to pay is theta*x_2. Under the stated parameter restriction, the dishonest type mixes, so he is indifferent between choosing high and low quality in period 1, implying c_1-c_0 = delta*(theta*x_2 - c_0). Solving gives x_2 = (c_1-c_0+delta*c_0)/(delta*theta). Equating this to the Bayes expression and solving for alpha yields alpha = x_1*(delta*(theta-c_0)-(c_1-c_0))/((1-x_1)*(c_1-c_0+delta*c_0)).", "allowed_symbols": ["c_0", "c_1", "delta", "theta", "x_1"], "symbol_definitions": {"c_0": "unit cost of low quality", "c_1": "unit cost of high quality", "delta": "discount factor", "theta": "consumer valuation parameter", "x_1": "prior probability the monopolist is honest"}, "review_status": "confirmed_without_expert"} |
| {"id": "2.8/main#2", "chapter": "2", "split": "chapter_2", "problem_number": "2.8", "sub_id": "main", "pair_index": 2, "pair_count": 2, "question": "If $\\delta(\\theta - c_0) > c_1 - c_0 > \\delta(\\theta x_1 - c_0)$, the dishonest monopolist establishes a reputation for the second period with probability \\[ \\alpha = \\frac{x_1[\\delta(\\theta - c_0) - (c_1 - c_0)]}{(1 - x_1)(c_1 - c_0 + \\delta c_0)}. \\] True/False: As $x_1 \\to 0$, the reputational benefit to the dishonest monopolist from choosing high quality in period 1 remains substantial, so he still often finds it worthwhile to incur the extra cost to build a reputation (i.e., $\\alpha$ does not approach 0).", "question_type": "judge", "problem_type_label": "Simple Math but Requires Economics Concepts", "answer_kind": "bool", "comparison_mode": "exact", "reference_answer": "False", "reference_answer_sympy": "N/A", "reference_answer_json": null, "reference_answer_json_modes": null, "answer_for_compare": "False", "reference_reasoning": "When the prior probability of an honest monopolist x_1 is very small, observing high quality in period 1 barely increases consumers’ posterior belief that the monopolist is honest. Hence the expected future (period-2) gain from mimicking honesty is negligible for the dishonest type, so he rarely pays the extra cost of high quality to build a reputation, implying alpha tends to 0 as x_1 tends to 0.", "allowed_symbols": [], "symbol_definitions": {}, "review_status": "confirmed_without_expert"} |
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