Managerial Economics · Quiz 2 · Jan 2025

**1. Recall the definition of quasi-linear preferences:** A utility function representing **quasi-linear preferences** over two goods $(x, y)$ has the form: $$U(x, y) = v(x) + by$$ where $b > 0$ is a constant and $v(x)$ is a non-linear function (such as $\ln x$ or $\sqrt{x}$). Thus, quasi-linear preferences are **linear in one good** (the numeraire good $y$) and **non-linear in the other good** ($x$), NOT non-linear in both goods. **Answer: Option B** A: **Incorrect:** Quasi-linear utility is linear in one of the goods and non-linear in the other. B: **Correct:** Because quasi-linear utility is linear in one good and non-linear in the other, the statement is FALSE.

**1. Check the effect of a monotonic transformation on marginal utility:** Suppose $V(x, y) = f(U(x, y))$ is a strictly increasing transformation ($f'(U) > 0$) of the utility function $U(x, y)$. By the chain rule, the new marginal utility of good $x$ is: $$MU_x^V = \frac{\partial V}{\partial x} = f'(U(x, y)) \cdot \frac{\partial U}{\partial x} = f'(U(x, y)) \cdot MU_x^U$$ Unless $f'(U) = 1$, the marginal utility value **changes** under a monotonic transformation. Thus, the statement is **TRUE**. **Answer: Option A** A: **Correct:** Multiplying or applying a nonlinear monotonic transformation scales the partial derivatives by $f'(U)$, changing the marginal utilities. B: **Incorrect:** While the ratio of marginal utilities (MRS) is invariant, individual marginal utilities change under monotonic transformations.

**1. Check the effect of a strictly increasing transformation on the Marginal Rate of Substitution (MRS):** Let $V(x, y) = f(U(x, y))$ with $f'(U) > 0$. Then: $$MRS_{xy}^V = \frac{\partial V / \partial x}{\partial V / \partial y} = \frac{f'(U)\, (\partial U / \partial x)}{f'(U)\, (\partial U / \partial y)} = \frac{\partial U / \partial x}{\partial U / \partial y} = MRS_{xy}^U$$ Because the factor $f'(U)$ cancels out in the ratio, the **MRS does not change** under a positive monotonic transformation of the utility function. Thus, the statement is **FALSE**. **Answer: Option B** A: **Incorrect:** The chain-rule factor $f'(U)$ cancels out when dividing $MU_x$ by $MU_y$, leaving MRS unchanged. B: **Correct:** MRS represents the slope of the indifference curve, which is invariant under any positive monotonic transformation.

**1. Recall the tax incidence (pass-through) formula:** The fraction of a per-unit quantity tax borne by buyers is given by: $$\text{Buyers' share} = \frac{E_s}{E_s - E_d} = \frac{E_s}{E_s + |E_d|} = \frac{1}{1 + |E_d| / E_s}$$ where $E_s > 0$ is the price elasticity of supply and $|E_d| > 0$ is the magnitude of the price elasticity of demand. **2. Analyze what happens as $E_s$ increases:** As supply becomes more own-price elastic ($E_s$ increases), the ratio $|E_d| / E_s$ decreases, so the fraction $\frac{1}{1 + |E_d| / E_s}$ **rises**. Therefore, the statement is **TRUE**. **Answer: Option A** A: **Correct:** When supply is more elastic relative to demand, sellers can more easily adjust quantity, shifting a larger fraction of the tax burden onto buyers. B: **Incorrect:** The pass-through fraction $\frac{E_s}{E_s + |E_d|}$ strictly increases with $E_s$.

**1. Use the tax incidence formula for buyers:** $$\text{Buyers' share} = \frac{E_s}{E_s + |E_d|}$$ As demand becomes more own-price elastic ($|E_d|$ increases), the denominator $E_s + |E_d|$ increases, so the fraction of the tax paid by buyers **falls** (buyers are more price-sensitive and escape the tax burden, shifting more of it onto sellers). Thus, the statement is **FALSE**. **Answer: Option B** A: **Incorrect:** More elastic demand reduces, rather than raises, the fraction of the tax paid by buyers. B: **Correct:** When demand is more elastic, buyers reduce quantity demanded sharply in response to price increases, so their share of the tax falls.

**1. Derive the monopolist's first-order condition for profit maximization:** A monopoly's profit is $\pi(Q) = TR(Q) - TC(Q)$. Differentiating with respect to output $Q$ gives the necessary condition for an interior maximum: $$\frac{d\pi}{dQ} = \frac{d(TR)}{dQ} - \frac{d(TC)}{dQ} = MR(Q) - MC(Q) = 0 \implies MR(Q) = MC(Q)$$ Thus, the statement is **TRUE**. **Answer: Option A** A: **Correct:** Profit maximization requires producing the output level where marginal revenue equals marginal cost ($MR = MC$). B: **Incorrect:** If $MR \ne MC$, the monopolist can increase profit by adjusting output.

**1. Define a natural monopoly:** A **natural monopoly** arises when the cost structure of an industry exhibits strong economies of scale (subadditivity of costs) over the entire relevant range of market demand, so that **a single firm can supply the entire market output at a lower total cost than two or more competing firms**. Thus, the statement is **TRUE**. **Answer: Option A** A: **Correct:** Subadditivity of the cost function (where one firm can produce total market output more cheaply than multiple firms) is the exact definition of a natural monopoly. B: **Incorrect:** This is the standard economic definition of a natural monopoly.

**1. Use the relationship between Marginal Revenue ($MR$) and price elasticity of demand ($|E_d|$):** $$MR = P\left(1 - \frac{1}{|E_d|}\right)$$ Since a profit-maximizing monopolist equates $MR = MC$ and marginal cost is non-negative ($MC \ge 0$): $$P\left(1 - \frac{1}{|E_d|}\right) \ge 0 \implies 1 - \frac{1}{|E_d|} \ge 0 \implies |E_d| \ge 1$$ Thus, a monopolist **always produces in the elastic (or unit-elastic) region** of the market demand curve, never in the inelastic region ($|E_d| < 1$, where $MR < 0$). The statement is **FALSE**. **Answer: Option B** A: **Incorrect:** In the inelastic region ($|E_d| < 1$), marginal revenue is negative, so reducing output both raises revenue and lowers cost. B: **Correct:** A monopolist always operates where $|E_d| \ge 1$ (the elastic region), making the statement FALSE.

**1. Compare Player 1's payoffs from strategy $M$ vs. strategy $T$:** From the payoff matrix for Player 1: - If Player 2 plays $L$: $u_1(M, L) = 2 > u_1(T, L) = 1$. - If Player 2 plays $R$: $u_1(M, R) = 0 \ge u_1(T, R) = 0$. Since $M$ gives a **strictly higher** payoff than $T$ when Player 2 plays $L$ ($2 > 1$) and **at least as high** a payoff when Player 2 plays $R$ ($0 = 0$), strategy $M$ **weakly dominates** strategy $T$. Thus, the statement is **TRUE**. **Answer: Option A** A: **Correct:** Payoffs $(2, 0)$ from $M$ are component-wise $\ge (1, 0)$ from $T$ with strict inequality in the first column. B: **Incorrect:** $M$ satisfies the definition of weak dominance over $T$.

**1. Compare Player 1's payoffs from strategy $B$ vs. strategy $M$:** From the payoff matrix for Player 1: - If Player 2 plays $L$: $u_1(B, L) = 2 \ge u_1(M, L) = 2$. - If Player 2 plays $R$: $u_1(B, R) = 1 > u_1(M, R) = 0$. Since $B$ gives at least as high a payoff as $M$ against $L$ ($2 = 2$) and a strictly higher payoff against $R$ ($1 > 0$), strategy $B$ **weakly dominates** strategy $M$. Thus, the statement is **TRUE**. **Answer: Option A** A: **Correct:** Payoffs $(2, 1)$ from $B$ are component-wise $\ge (2, 0)$ from $M$ with strict inequality in the second column. B: **Incorrect:** $B$ satisfies the definition of weak dominance over $M$.

**1. Apply the monopolist's inverse-elasticity markup formula:** At the profit-maximizing price $P$ where $MR = MC$: $$MR = P\left(1 - \frac{1}{|E_d|}\right) = MC \implies P = \frac{MC}{1 - \frac{1}{|E_d|}}$$ **2. Substitute $MC = \$1$ and $|E_d| = 1.25$:** $$P = \frac{1}{1 - \frac{1}{1.25}} = \frac{1}{1 - 0.8} = \frac{1}{0.2} = \$5$$ **Answer: 5**

**1. Evaluate both economic statements:** - **Statement 1:** The $\$1.5\text{ billion}$ and $10\text{ years}$ spent on R&D have already been incurred and cannot be recovered, so they are **sunk costs** and do not affect marginal cost or the profit-maximizing price. (**True**) - **Statement 2:** Once the patent on Ageless expires, generic competitors can enter and offer close substitutes, which makes the demand faced by your brand **more price-elastic** (the magnitude of elasticity rises). (**True**) Since both statements are correct, **Option C** is the right choice. **Answer: Option C** A: **Incorrect:** While true, Statement B is also true, making Option C the complete answer. B: **Incorrect:** While true, Statement A is also true, making Option C the complete answer. C: **Correct:** R&D expenditures are sunk costs that do not enter $MR = MC$, and patent expiration increases the availability of close substitutes, raising demand elasticity. D: **Incorrect:** Both statements are true.

**1. Set up the monopsonist's Marginal Value ($MV$) and Marginal Expenditure ($ME$):** - **Marginal Value (Demand for TAs):** $$MV(n) = 30,000 - 125n$$ - **Average Expenditure (Supply of TAs):** $$W(n) = 1,000 + 75n$$ - **Total Expenditure ($TE$)** on $n$ TAs: $$TE(n) = W(n) \cdot n = 1,000n + 75n^2$$ - **Marginal Expenditure ($ME$):** $$ME(n) = \frac{d(TE)}{dn} = 1,000 + 150n$$ **2. Equate $ME(n) = MV(n)$ to find the optimal number of TAs ($n^*$):** $$1,000 + 150n = 30,000 - 125n \implies 275n = 29,000 \implies n = \frac{29,000}{275} \approx 105.45 \approx 105\text{ TAs}$$ **Answer: Option A** A: **Correct:** Setting $ME = 1000 + 150n$ equal to $MV = 30000 - 125n$ gives $n = 29000/275 \approx 105$ TAs. B: **Incorrect:**At $n = 120$, $ME = 19,000 > MV = 15,000$. C: **Incorrect:** At $n = 125$, $ME = 19,750 > MV = 14,375$. D: **Incorrect:** At $n = 115$, $ME = 18,250 > MV = 15,625$.

**1. Substitute the monopsony employment level $n^* = 105$ into the TA supply curve:** A monopsonist pays the wage on the labor supply curve corresponding to the number of TAs hired ($n^* = 105$): $$W = 1,000 + 75n = 1,000 + 75(105) = 1,000 + 7,875 = \$8,875$$ **Answer: Option B** A: **Incorrect:** Does not lie on the supply curve $W = 1000 + 75(105)$. B: **Correct:** Substituting $n = 105$ into the supply curve $W = 1000 + 75n$ gives $W = 8875$. C: **Incorrect:** Arithmetic error. D: **Incorrect:** Arithmetic error.

**1. Apply the competitive firm's profit-maximizing rule $P = MC(q)$:** Given marginal cost $MC(q) = 3 + 2q$ and market price $P = 9$: $$3 + 2q = 9 \implies 2q = 6 \implies q^* = 3$$ **Answer: Option B** A: **Incorrect:** At $q = 2$, $MC(2) = 7 < P = 9$, so producing more increases profit. B: **Correct:** Setting $MC(q) = 3 + 2q = 9$ gives $q = 3$. C: **Incorrect:** At $q = 4$, $MC(4) = 11 > P = 9$. D: **Incorrect:** At $q = 5$, $MC(5) = 13 > P = 9$.

**1. Compute Producer Surplus ($PS = TR - VC$) at $q^* = 3$ and $P = 9$:** - Total Revenue: $TR = P \times q^* = 9 \times 3 = 27$. - Variable Cost: $VC(3) = \int_0^3 MC(q)\,dq = \int_0^3 (3 + 2q)\,dq = \left[3q + q^2\right]_0^3 = 9 + 9 = 18$. - Producer Surplus: $$PS = TR - VC = 27 - 18 = 9$$ (Equivalently, the area of the triangle between $P = 9$ and $MC(q) = 3 + 2q$ is $\frac{1}{2}(9 - 3)(3) = 9$.) **Answer: Option C** A: **Incorrect:** $27 - 18 = 9$, not $7$. B: **Incorrect:** $27 - 18 = 9$, not $8$. C: **Correct:** $PS = TR - VC = 9(3) - (3(3) + 3^2) = 27 - 18 = 9$. D: **Incorrect:** $6$ is the economic profit after subtracting fixed cost $FC = 3$, not the producer surplus.

**1. Calculate the firm's economic profit $\pi$:** - From Question 238, at $P = 9$, the firm produces $q^* = 3$, earning Total Revenue $TR = 9 \times 3 = 27$. - With $AVC(q) = 3 + q$, Variable Cost at $q = 3$ is $VC = q \cdot AVC(3) = 3(3 + 3) = 18$. - Given Fixed Cost $FC = 3$, Total Cost is $TC = VC + FC = 18 + 3 = 21$. - Economic Profit: $$\pi = TR - TC = 27 - 21 = 6 > 0$$ Since $\pi = 6 > 0$, the firm earns a **positive economic profit**. **Answer: Option A** A: **Correct:** Since $TR = 27$ exceeds $TC = 18 + 3 = 21$, the firm earns a positive economic profit of ₹6. B: **Incorrect:** Profit is $+6$, which is positive, not negative. C: **Incorrect:** Profit is $+6 \ne 0$. D: **Incorrect:** All necessary revenue and cost information is provided.

**1. Compute the marginal utilities of $x$ and $y$ from $U(x, y) = xy + 10x$:** - Marginal utility of $x$: $$MU_x = \frac{\partial U}{\partial x} = y + 10$$ Notice that $MU_x = y + 10$ does not contain $x$, so **marginal utility of $x$ is independent of $x$**. - Marginal utility of $y$: $$MU_y = \frac{\partial U}{\partial y} = x$$ Notice that $MU_y = x$ does not contain $y$, so **marginal utility of $y$ is independent of $y$**. Therefore, **both** statements are correct (**Option C**). **Answer: Option C** A: **Incorrect:** While true, Statement B is also true, making Option C the complete choice. B: **Incorrect:** While true, Statement A is also true, making Option C the complete choice. C: **Correct:** $MU_x = y + 10$ is independent of $x$, and $MU_y = x$ is independent of $y$. D: **Incorrect:** Both statements A and B are true.

**1. Compare the marginal utility per dollar spent on food ($x$) and clothing ($y$):** - Budget constraint: $P_x x + P_y y = x + 2y = 10$ (with $x \ge 0, y \ge 0$, so $0 \le x \le 10$ and $0 \le y \le 5$). - Marginal utility per dollar of food ($x$): $$\frac{MU_x}{P_x} = \frac{y + 10}{1} = y + 10 \ge 10$$ - Marginal utility per dollar of clothing ($y$): $$\frac{MU_y}{P_y} = \frac{x}{2} \le \frac{10}{2} = 5$$ **2. Identify the corner solution:** For every feasible bundle on the budget line, $\frac{MU_x}{P_x} \ge 10 > 5 \ge \frac{MU_y}{P_y}$ (or equivalently, $MRS = \frac{y + 10}{x} \ge 1 > \frac{1}{2} = \frac{P_x}{P_y}$). Thus, Narendra spends all of his $\$10$ income on food ($x$) and buys zero clothing ($y$), giving the optimal bundle $(x^*, y^*) = (10, 0)$: - It lies on the **X-axis**, not in the interior (Option A is false) and not on the Y-axis (Option B is false). - At $(10, 0)$, the indifference curve slope is $-1$ while the budget line slope is $-0.5$, so the budget line is **not tangent** to the indifference curve (Option C is false). Hence, **None** of the statements A, B, C is true (**Option D**). **Answer: Option D** A: **Incorrect:** Because $MU_x/P_x > MU_y/P_y$ everywhere on the budget line, the optimum is a corner solution $(10, 0)$, not an interior point. B: **Incorrect:** The optimal bundle $(10, 0)$ lies on the X-axis, not the Y-axis. C: **Incorrect:** At the corner solution $(10, 0)$, $MRS = 10/10 = 1 \ne P_x/P_y = 1/2$, so the budget line is not tangent to the indifference curve. D: **Correct:** The optimal bundle is the corner solution $(10, 0)$ on the X-axis where $MRS > P_x/P_y$, so none of A, B, or C is true.

**1. Find the optimal quantity of food ($x^*$):** From Question 242, since $\frac{MU_x}{P_x} = y + 10 \ge 10 > 5 \ge \frac{x}{2} = \frac{MU_y}{P_y}$ across the entire budget line $x + 2y = 10$, Narendra spends his entire income $I = \$10$ on food ($P_x = \$1$): $$x^* = \frac{I}{P_x} = \frac{10}{1} = 10$$ (Check: setting $y = \frac{10 - x}{2}$ in $U(x, y) = x(y + 10) = x\left(15 - \frac{x}{2}\right)$ gives $\frac{dU}{dx} = 15 - x > 0$ for all $x \in [0, 10]$, so $U$ is maximized at $x^* = 10$.) **Answer: 10**

**1. Find the optimal quantity of clothing ($y^*$):** Since Narendra spends all $\$10$ of his income on $x^* = 10$ units of food at $P_x = \$1$, the remaining income for clothing is $\$0$: $$y^* = 0$$ **Answer: 0**

**1. Equate pre-tax demand and supply:** - Demand: $P = 10 - Q$ - Supply: $P = Q - 4$ Set $10 - Q = Q - 4$: $$14 = 2Q \implies Q^* = 7\text{ (thousand units)}$$ **Answer: 7**

**1. Substitute $Q^* = 7$ into the demand (or supply) equation:** $$P^* = 10 - Q^* = 10 - 7 = 3\text{ INR}$$ (Check: $P^* = 7 - 4 = 3\text{ INR}$.) **Answer: 3**

**1. Impose the per-unit tax of $t = 1\text{ INR}$:** With a tax of $t = 1$ per unit, the price paid by buyers $P^B = 10 - Q$ and the price received by sellers $P^s = Q - 4$ satisfy: $$P^B - P^s = 1$$ Substitute the demand and supply equations: $$(10 - Q) - (Q - 4) = 1 \implies 14 - 2Q = 1 \implies 2Q = 13 \implies Q^t = 6.5\text{ (thousand units)}$$ **Answer: 6.5**

**1. Substitute the post-tax quantity $Q^t = 6.5$ into the demand curve:** $$P^B = 10 - Q^t = 10 - 6.5 = 3.5\text{ INR}$$ **Answer: 3.5**

**1. Calculate the net price received by sellers ($P^s$):** Subtract the tax $t = 1\text{ INR}$ from the buyer's price $P^B = 3.5\text{ INR}$ (or plug $Q^t = 6.5$ into the supply curve $P = Q - 4$): $$P^s = P^B - 1 = 3.5 - 1 = 2.5\text{ INR}$$ **Answer: 2.5**

**1. Impose the per-unit subsidy of $s = 1\text{ INR}$:** With a subsidy of $1\text{ INR}$ per unit, the price received by sellers $P^s = Q - 4$ exceeds the price paid by buyers $P^B = 10 - Q$ by $1$: $$P^s - P^B = 1$$ Substitute the supply and demand equations: $$(Q - 4) - (10 - Q) = 1 \implies 2Q - 14 = 1 \implies 2Q = 15 \implies Q^s = 7.5\text{ (thousand units)}$$ **Answer: 7.5**

**1. Substitute the post-subsidy quantity $Q^s = 7.5$ into the demand curve:** $$P^B = 10 - Q^s = 10 - 7.5 = 2.5\text{ INR}$$ **Answer: 2.5**

**1. Calculate the price received by sellers including the subsidy ($P^s$):** Add the subsidy of $1\text{ INR}$ to the buyer's price $P^B = 2.5\text{ INR}$ (or substitute $Q^s = 7.5$ into the supply curve $P = Q - 4$): $$P^s = P^B + 1 = 2.5 + 1 = 3.5\text{ INR}$$ **Answer: 3.5**