Managerial Economics · Quiz 2 · May 2026

**1. Compare two bundles:** if \(U(A)>U(B)\), a strictly increasing transformation \(f\) gives \(f(U(A))>f(U(B))\). Equal utilities remain equal. **Answer: A — True.** A strictly increasing transformation changes utility numbers, not preference rankings. A: **Correct:** A strictly increasing transformation preserves both strict preferences and indifference. B: **Incorrect:** Utility measures preference order; changing its scale need not change that order.

**1. Use elastic demand:** \(|\varepsilon|>1\) means quantity falls proportionately more than price rises. **2. Check revenue:** for a small price change, \(d(PQ)/dP=Q(1-|\varepsilon|)<0\). **Answer: B — False.** A price increase reduces revenue in the elastic region. A: **Incorrect:** This reverses the revenue effect of elastic demand; the quantity reduction outweighs the higher price. B: **Correct:** Elastic demand implies that a price rise lowers total revenue, rather than always increasing it.

**1. Use perfectly inelastic demand:** buyers purchase the same quantity even when their price rises. **2. Apply the tax wedge:** in the standard competitive model, the buyer price rises by the full per-unit tax while the seller’s net price stays unchanged. **Answer: B — False.** Consumers bear the entire tax burden, not none. A: **Incorrect:** With perfectly inelastic demand, consumers cannot reduce purchases in response to the higher price. B: **Correct:** A vertical demand curve makes consumers bear the full per-unit tax burden in the standard competitive model.

**1. Recall the definition:** perfect first-degree discrimination charges each buyer their maximum willingness to pay for each unit. **Answer: A — True.** This maximum is the buyer’s reservation price. A: **Correct:** Charging the reservation price for each unit is first-degree price discrimination. B: **Incorrect:** Reservation-price charging is the defining feature of first-degree discrimination.

**1. Consider substitutes:** when the price of good \(Y\) rises, buyers switch toward good \(X\), increasing its demand. **Answer: A — True.** \(\varepsilon_{XY}=(\partial Q_X/\partial P_Y)(P_Y/Q_X)>0\). A: **Correct:** Substitutes have positive cross-price elasticity because demand shifts toward the alternative good. B: **Incorrect:** Negative cross-price elasticity characterizes complements, not substitutes.

**1. Interpret income elasticity:** \(\varepsilon_I>0\) means demand increases when income increases, holding other factors fixed. **Answer: A — True.** That is the definition of a normal good. A: **Correct:** Normal goods have positive income elasticity. B: **Incorrect:** Inferior goods have negative income elasticity; a positive value indicates a normal good.

**1. Equate demand and supply:** before subsidy, buyers and sellers face the same price \(P\): \(20-0.5P=P-1\). **2. Solve:** \(1.5P=21\), so \(P=14\). Hence \(Q=14-1=13\). **Answer: 13 million units.**

**1. Use one market price before subsidy:** \(P^B=P^S=P\). **2. Clear the market:** \(20-0.5P=P-1\Rightarrow1.5P=21\Rightarrow P=14\). **Answer: 14 per unit.**

**1. Introduce the subsidy wedge:** \(P^S=P^B+3\). Supply is therefore \(Q^S=P^B+2\). **2. Clear the market:** \(20-0.5P^B=P^B+2\Rightarrow P^B=12\). **3. Find quantity:** \(Q=20-0.5(12)=14\). **Answer: 14 million units.**

**1. Use the subsidy:** \(P^S=P^B+3\), so supply equals \(P^B+2\). **2. Equate demand and supply:** \(20-0.5P^B=P^B+2\Rightarrow1.5P^B=18\). **Answer: 12 per unit**, paid by buyers.

**1. Find the buyer price:** \(20-0.5P^B=P^B+2\), giving \(P^B=12\). **2. Add the subsidy:** \[P^S=P^B+3=12+3=15.\] **3. Verify market clearing:** demand \(20-0.5(12)=14\); supply \(15-1=14\). **Answer: 15 per unit**, received by sellers. **Review note:** the supplied key says 17. That would imply a subsidy of 5 rather than the stated 3, and supply \(17-1=16\) would not equal demand 14. The key needs correction.

**1. Find subsidized quantity:** \(20-0.5P^B=P^B+2\) gives \(P^B=12\) and \(Q=14\) million units. **2. Multiply subsidy by units sold:** \[\text{Government cost}=3\times14=42.\] **Answer: 42 million dollars.**

**1. Replace the subsidy with a tax:** \(P^B=P^S+6\), so \(Q^S=P^B-7\). **2. Clear the market:** \(20-0.5P^B=P^B-7\Rightarrow P^B=18\). **3. Find quantity:** \(Q=20-0.5(18)=11\). **Answer: 11 million units.** Quantity is measured in units, despite the dollar label in the question.

**1. Use the tax wedge:** sellers receive \(P^S=P^B-6\), giving supply \(P^B-7\). **2. Equate demand and supply:** \(20-0.5P^B=P^B-7\Rightarrow1.5P^B=27\). **Answer: 18 per unit**, paid by buyers.

**1. Solve for the buyer price:** \(20-0.5P^B=P^B-7\Rightarrow P^B=18\). **2. Subtract the tax:** \[P^S=18-6=12.\] **Answer: 12 per unit**, received by sellers.

**1. Find marginal revenue:** \(TR=(20-Q)Q=20Q-Q^2\), so \(MR=20-2Q\). **2. Set \(MR=MC\):** \(20-2Q=2\Rightarrow Q=9\). **3. Use demand:** \(P=20-9=11\). **Answer: 11 per unit.**

**1. Find the uniform-price optimum:** \(20-2Q=2\Rightarrow Q=9\), so \(P=11\). **2. Calculate producer surplus:** with constant marginal cost 2, \[PS=(P-MC)Q=(11-2)\times9=81.\] **Answer: 81.** With no fixed costs, this is also profit.

**1. Uniform pricing:** \(Q=9\), \(P=11\), so producer surplus is \(81\). **2. Perfect discrimination:** produce until willingness to pay equals marginal cost: \(20-Q=2\Rightarrow Q=18\). The entire area above marginal cost goes to the producer: \[PS_{\mathrm{PD}}=\int_0^{18}(20-Q-2)\,dQ=\frac12(18)(18)=162.\] **3. Find the increase:** \(162-81=81\). **Answer: 81.**

**1. Find Indigo’s quantity:** \(Q_I=90{,}000-200(300)+100(300)=60{,}000\). **2. Hold Air India’s price fixed:** \(\partial Q_I/\partial P_I=-200\). **3. Calculate own-price elasticity:** \[\varepsilon_I=(-200)\frac{300}{60{,}000}=-1.\] **Answer: 1**, ignoring the sign.

**1. Add both airline demands:** \(Q=180{,}000-100P_I-100P_A\). **2. Vary both prices together:** set \(P_I=P_A=P\), so \(Q=180{,}000-200P\). At \(P=300\), \(Q=120{,}000\). **3. Calculate market elasticity:** \[\varepsilon=(-200)\frac{300}{120{,}000}=-0.5.\] **Answer: 0.5**, ignoring the sign. Unlike own-price elasticity, this includes both airlines raising price together.

**Player 1’s payoff table:** rows are player 1’s actions; columns are player 2’s actions. Only player 1’s payoffs are provided. \[\begin{array}{c|cc}&L&R\\ \hline T&0&1\\ M&2&1\\ B&3&2\end{array}\] **1. Compare M with T:** against \(L\), \(2>0\); against \(R\), \(1=1\). **Answer: A — True.** M is never worse and is strictly better against at least one action, so it weakly dominates T. A: **Correct:** M gives at least T’s payoff in both columns, with a strict improvement against L. B: **Incorrect:** The equality against R does not prevent weak dominance; strict inequality is required in only one or more cases.

**Player 1’s payoff table:** rows are player 1’s actions; columns are player 2’s actions. Only player 1’s payoffs are provided. \[\begin{array}{c|cc}&L&R\\ \hline T&0&1\\ M&2&1\\ B&3&2\end{array}\] **1. Compare B with M:** against \(L\), \(3>2\); against \(R\), \(2>1\). **Answer: A — True.** B gives a strictly higher payoff against every opposing action, so it strictly dominates M. A: **Correct:** B outperforms M in both columns, which establishes strict dominance. B: **Incorrect:** There is no equality or reversal in either column; both comparisons are strict.

**1. Differentiate with respect to food:** treating \(y\) as fixed, \[MU_x=\frac{\partial(xy+10x)}{\partial x}=y+10.\] **Answer: B — MUx is linear in y** (more precisely, affine in y). A: **Incorrect:** \(MU_x=y+10\) contains no \(x\) term. B: **Correct:** \(MU_x=y+10\) is a linear expression in \(y\), with intercept 10. C: **Incorrect:** only \(y\) and a constant remain after differentiation. D: **Incorrect:** marginal utility changes with \(y\); it is not fixed.

**1. Differentiate with respect to clothing:** treating \(x\) as fixed, \[MU_y=\frac{\partial(xy+10x)}{\partial y}=x.\] **Answer: A — MUy is linear in x.** A: **Correct:** \(MU_y=x\), a linear function of food consumption. B: **Incorrect:** \(MU_y\) has no \(y\) term. C: **Incorrect:** the derivative is just \(x\), not a combination of \(x\) and \(y\). D: **Incorrect:** \(MU_y\) changes when \(x\) changes.

**1. Write the budget:** \(x+2y=100\). **2. Put clothing on the vertical axis:** \(y=50-0.5x\), so the slope is \(-P_x/P_y=-1/2\). **Answer: 0.5**, ignoring the sign, with food on the horizontal axis.

**1. Use the budget:** \(x=100-2y\), where \(0\le y\le50\). **2. Substitute into utility:** \(U=(100-2y)(y+10)=1000+80y-2y^2\). **3. Maximize:** \(dU/dy=80-4y=0\Rightarrow y=20\). Since \(d^2U/dy^2=-4<0\), this feasible interior point is the maximum. **Answer: 20 units of clothing.**

**1. Find optimal clothing:** substituting the budget into utility gives \(U(y)=1000+80y-2y^2\); its maximum occurs at \(y=20\). **2. Use the budget:** \[x=100-2(20)=60.\] **Answer: 60 units of food.**

**1. Find the optimal bundle:** maximizing \(U(y)=1000+80y-2y^2\) gives \(y=20\), then the budget gives \(x=60\). **2. Substitute into utility:** \[U(60,20)=60(20)+10(60)=1200+600=1800.\] **Answer: 1800 utility units.**