Managerial Economics · Quiz 2 · Jan 2026

**1. Recall the form:** Quasilinear utility can be written \(U(x,y)=v(x)+ay\), linear in one good and generally nonlinear in the other. **Answer: B — False.** A: **Incorrect:** Linear in both goods would describe linear perfect-substitute utility, not the general quasilinear form. B: **Correct:** Quasilinear preferences are linear in only one argument in their usual form.

**1. Differentiate the transformed utility:** If \(V=f(U)\) with \(f^{\prime}>0\), then \(V_x=f^{\prime}(U)U_x\). **2. Compare:** The derivative is multiplied by \(f^{\prime}(U)\), so marginal utility need not remain unchanged even though preferences do. **Answer: B — False.** A: **Incorrect:** Monotone transformations preserve rankings rather than marginal-utility levels. B: **Correct:** The chain rule changes each marginal utility by the same pointwise factor.

**1. Apply the chain rule:** A differentiable increasing transformation gives \(V_x=f^{\prime}(U)U_x\) and \(V_y=f^{\prime}(U)U_y\). **2. Take their ratio:** \[\frac{V_x}{V_y}=\frac{f^{\prime}(U)U_x}{f^{\prime}(U)U_y}=\frac{U_x}{U_y}.\] Thus the MRS is unchanged wherever the derivatives and ratio are defined. **Answer: A — True.** **Convention note:** This statement refers to differentiable strictly increasing transformations at points where the MRS is defined. A: **Correct:** The common positive transformation factor cancels in the MRS. B: **Incorrect:** Marginal utilities can change, but their ratio does not.

**1. Write the buyer’s tax share:** It is \(E_S/(E_S+|E_D|)\). **2. Raise supply elasticity:** With demand elasticity fixed, a larger \(E_S\) raises the share borne by buyers. **Answer: A — True.** A: **Correct:** More elastic supply places a larger part of the tax burden on buyers. B: **Incorrect:** Increasing supply elasticity raises, rather than lowers, the buyer share.

**1. Use the incidence formula:** The buyer’s share is \(E_S/(E_S+|E_D|)\). **2. Make demand more elastic:** Increasing \(|E_D|\) enlarges the denominator and lowers the buyer’s share. **Answer: B — False.** A: **Incorrect:** More elastic demand lets buyers avoid more of the tax burden. B: **Correct:** The buyer share falls as demand becomes more elastic.

**1. Check marginal revenue:** On a strictly inelastic demand segment, \(MR=P(1+1/\epsilon)<0\). **2. Compare output choices:** Reducing output raises revenue and does not raise production cost when marginal cost is nonnegative. Such a point cannot maximize monopoly profit. **Answer: B — False.** **Convention note:** The standard result assumes nonnegative marginal cost; a zero-cost optimum may be at unit elasticity. A: **Incorrect:** A standard profit-maximizing monopoly does not choose the strictly inelastic segment. B: **Correct:** The claim reverses the usual monopoly condition.

Player 1 owns rows; player 2 owns columns. Only player 1’s payoffs are supplied. \[\begin{array}{c|cc} & L & R \\ T & 1 & 0 \\ M & 2 & 0 \\ B & 2 & 1\end{array}\] **1. Compare M with T:** Against L, M pays 2 instead of 1; against R, both pay 0. **2. Apply weak dominance:** M never pays less and pays strictly more against L, so M weakly dominates T. **Answer: A — True.** A: **Correct:** M is at least as good in both columns and better in one. B: **Incorrect:** Equality in one column is compatible with weak dominance.

Player 1 owns rows; player 2 owns columns. Only player 1’s payoffs are supplied. \[\begin{array}{c|cc} & L & R \\ T & 1 & 0 \\ M & 2 & 0 \\ B & 2 & 1\end{array}\] **1. Compare B with M:** Against L both pay 2; against R, B pays 1 and M pays 0. **2. Distinguish dominance:** B weakly dominates M, but does not strictly dominate it because the L payoff is equal. **Answer: B — False.** A: **Incorrect:** Strict dominance requires a strict inequality in every column. B: **Correct:** The tie against L prevents strict dominance.

**1. Differentiate Anita’s utility:** For \(U(F,C)=FC+F\), \(MU_F=C+1\) and \(MU_C=F\). **2. Inspect dependence:** Food’s marginal utility depends on clothing, and clothing’s marginal utility depends on food. Neither independence claim holds. **Answer: D — None.** A: **Incorrect:** Food’s marginal utility is C+1 and therefore depends on C. B: **Incorrect:** Clothing’s marginal utility equals F. C: **Incorrect:** Both independence claims are false. D: **Correct:** Neither listed independence claim holds.

**1. Set up the budget and tangency:** \(F+2C=22\), and \((C+1)/F=P_F/P_C=1/2\). Hence \(F=2C+2\). **2. Solve the two conditions:** \(2C+2+2C=22\), so \(C=5\) and \(F=12\). **3. Verify optimality:** Substituting \(C=(22-F)/2\) gives \(U=12F-F^2/2\), a concave quadratic maximized at F=12. Utility is \(12(5)+12=72\), and MRS is \((5+1)/12=0.5\). **Answer: C — Budget line is a tangent to Anita’s indifference curve at optimal bundle.** A: **Incorrect:** The optimum has positive amounts of both goods. B: **Incorrect:** The optimum has positive amounts of both goods. C: **Correct:** The optimum is interior and satisfies MRS=price ratio. D: **Incorrect:** The tangency claim is correct.

**1. Set up the budget and tangency:** \(F+2C=22\), and \((C+1)/F=P_F/P_C=1/2\). Hence \(F=2C+2\). **2. Solve the two conditions:** \(2C+2+2C=22\), so \(C=5\) and \(F=12\). **3. Verify optimality:** Substituting \(C=(22-F)/2\) gives \(U=12F-F^2/2\), a concave quadratic maximized at F=12. Utility is \(12(5)+12=72\), and MRS is \((5+1)/12=0.5\). **Answer: 12.**

**1. Set up the budget and tangency:** \(F+2C=22\), and \((C+1)/F=P_F/P_C=1/2\). Hence \(F=2C+2\). **2. Solve the two conditions:** \(2C+2+2C=22\), so \(C=5\) and \(F=12\). **3. Verify optimality:** Substituting \(C=(22-F)/2\) gives \(U=12F-F^2/2\), a concave quadratic maximized at F=12. Utility is \(12(5)+12=72\), and MRS is \((5+1)/12=0.5\). **Answer: 5.**

**1. Set up the budget and tangency:** \(F+2C=22\), and \((C+1)/F=P_F/P_C=1/2\). Hence \(F=2C+2\). **2. Solve the two conditions:** \(2C+2+2C=22\), so \(C=5\) and \(F=12\). **3. Verify optimality:** Substituting \(C=(22-F)/2\) gives \(U=12F-F^2/2\), a concave quadratic maximized at F=12. Utility is \(12(5)+12=72\), and MRS is \((5+1)/12=0.5\). **Answer: 72.**

**1. Set up the budget and tangency:** \(F+2C=22\), and \((C+1)/F=P_F/P_C=1/2\). Hence \(F=2C+2\). **2. Solve the two conditions:** \(2C+2+2C=22\), so \(C=5\) and \(F=12\). **3. Verify optimality:** Substituting \(C=(22-F)/2\) gives \(U=12F-F^2/2\), a concave quadratic maximized at F=12. Utility is \(12(5)+12=72\), and MRS is \((5+1)/12=0.5\). **Answer: 0.5.**

**1. Equate demand and supply:** Without a policy wedge, \(P_d=P_s=P\), so \(10-0.5P=-2+P\). **2. Solve:** \(1.5P=12\), giving \(P=8\) and \(Q=6\) million units. **Answer: 6.**

**1. Equate demand and supply:** Without a policy wedge, \(P_d=P_s=P\), so \(10-0.5P=-2+P\). **2. Solve:** \(1.5P=12\), giving \(P=8\) and \(Q=6\) million units. **Answer: 8.**

**1. Introduce the subsidy wedge:** Producers receive \(P_s=P_d+3\). Thus \(10-0.5P_d=-2+(P_d+3)\). **2. Solve:** \(1.5P_d=9\), so \(P_d=6\), \(P_s=9\) and \(Q=7\) million. **3. Compute government spending:** \(3\times7=21\) million dollars. **Answer: 7.**

**1. Introduce the subsidy wedge:** Producers receive \(P_s=P_d+3\). Thus \(10-0.5P_d=-2+(P_d+3)\). **2. Solve:** \(1.5P_d=9\), so \(P_d=6\), \(P_s=9\) and \(Q=7\) million. **3. Compute government spending:** \(3\times7=21\) million dollars. **Answer: 6.**

**1. Introduce the subsidy wedge:** Producers receive \(P_s=P_d+3\). Thus \(10-0.5P_d=-2+(P_d+3)\). **2. Solve:** \(1.5P_d=9\), so \(P_d=6\), \(P_s=9\) and \(Q=7\) million. **3. Compute government spending:** \(3\times7=21\) million dollars. **Answer: 9.**

**1. Introduce the subsidy wedge:** Producers receive \(P_s=P_d+3\). Thus \(10-0.5P_d=-2+(P_d+3)\). **2. Solve:** \(1.5P_d=9\), so \(P_d=6\), \(P_s=9\) and \(Q=7\) million. **3. Compute government spending:** \(3\times7=21\) million dollars. **Answer: 21.**

**1. Replace the subsidy with a tax:** Buyers pay \(P_d=P_s+6\). There is no remaining subsidy. Equilibrium requires \(10-0.5(P_s+6)=-2+P_s\). **2. Solve:** \(7-0.5P_s=-2+P_s\) gives \(P_s=6\), \(P_d=12\), and \(Q=4\) million. **Answer: 4.**

**1. Replace the subsidy with a tax:** Buyers pay \(P_d=P_s+6\). There is no remaining subsidy. Equilibrium requires \(10-0.5(P_s+6)=-2+P_s\). **2. Solve:** \(7-0.5P_s=-2+P_s\) gives \(P_s=6\), \(P_d=12\), and \(Q=4\) million. **Answer: 12.**

**1. Replace the subsidy with a tax:** Buyers pay \(P_d=P_s+6\). There is no remaining subsidy. Equilibrium requires \(10-0.5(P_s+6)=-2+P_s\). **2. Solve:** \(7-0.5P_s=-2+P_s\) gives \(P_s=6\), \(P_d=12\), and \(Q=4\) million. **Answer: 6.**

**1. Use the price-taking condition:** \(P=MC\) gives \(9=3+2q\), so \(q=3\). **2. Check the direction of adjustment:** MC rises with output; producing less than 3 leaves profitable additional units, while producing more has MC above price. **Answer: B — 3 units.** **Convention note:** The usual competitive-firm solution assumes no shutdown constraint binds; the implied AVC at q=3 is 6, below price 9. A: **Incorrect:** At q=2, MC=7 is below price, so another unit raises profit. B: **Correct:** At q=3, MC equals the market price 9. C: **Incorrect:** At q=4, MC=11 exceeds price. D: **Incorrect:** At q=5, MC=13 exceeds price.

**1. Find output:** \(9=3+2q\) gives \(q=3\). **2. Calculate variable cost:** Normalizing variable cost to zero at zero output, \(VC=\int_0^3(3+2q)\,dq=3(3)+3^2=18\). **3. Subtract from revenue:** \(PS=TR-VC=9(3)-18=9\). Fixed cost is not subtracted from producer surplus. **Answer: C — 9.** A: **Incorrect:** The area above MC and below price is 9, not 7. B: **Incorrect:** The area is 9, not 8. C: **Correct:** Revenue minus variable cost is 27−18=9. D: **Incorrect:** This is AVC at the chosen output, not total producer surplus.

**1. Find marginal expenditure:** Labour supply is \(w=2+2L\). Total labour expenditure is \(wL=2L+2L^2\), giving \(ME=2+4L\). **2. Equate with marginal revenue product:** Since \(Q=5L\) and output price is 10, \(MRP_L=10(5)=50\). Thus \(2+4L=50\), giving \(L=12\) thousand hours per week. **3. Read the wage from supply:** \(w=2+2(12)=26\) dollars per hour. **Answer: 12000 hours per week.**

**1. Find marginal expenditure:** Labour supply is \(w=2+2L\). Total labour expenditure is \(wL=2L+2L^2\), giving \(ME=2+4L\). **2. Equate with marginal revenue product:** Since \(Q=5L\) and output price is 10, \(MRP_L=10(5)=50\). Thus \(2+4L=50\), giving \(L=12\) thousand hours per week. **3. Read the wage from supply:** \(w=2+2(12)=26\) dollars per hour. **Answer: $26 per hour.**