Let's tackle your question step by step, using the information provided and standard microeconomic tools. ## Given Data - **Two identical firms** (Firm 1 and Firm 2) - **Marginal cost (MC):** $800 per apartment - **Demand for Firm 1:** \( D_1(P_1, P_2) = 10,000 (1 - e^{-P_2}) \) - **Demand for Firm 2:** \( D_2(P_1, P_2) = 10,000 (1 - e^{-P_1}) \) - \( P_1, P_2 \) = Prices set by Firm 1 and Firm 2, respectively --- ## (a) **Best Response Function for Each Firm** Each firm chooses its price to maximize profit, given the other firm's price. ### **Profit for Firm 1:** \[ \pi_1 = (P_1 - 800) \cdot D_1(P_1, P_2) = (P_1 - 800) \cdot 10,000 (1 - e^{-P_2}) \] **Notice:** The demand for Firm 1 depends only on \( P_2 \), not \( P_1 \)! This is very unusual. Let's check: - \( D_1 \) does **not** depend on \( P_1 \), only on \( P_2 \). - Similarly, \( D_2 \) depends only on \( P_1 \), not \( P_2 \). This means: - Firm 1's sales are entirely determined by the price set by Firm 2. - This is not a standard Bertrand model, but let's proceed as per the question. ### **Maximize profit for Firm 1 wrt \( P_1 \):** \[ \frac{\partial \pi_1}{\partial P_1} = 10,000 (1 - e^{-P_2}) \] So unless \( 1 - e^{-P_2} = 0 \) (i.e., \( P_2 \to -\infty \)), Firm 1's profit **increases with \( P_1 \) without bound**. But this can't make sense in practice; at a very high price, Firm 1 would sell nothing. But, as per the demand function, **Firm 1's demand does not depend on its own price!** (Which is odd, but let's proceed.) The logic is: **Whatever price Firm 1 charges, it sells the same amount (given \( P_2 \)).** Thus, **Firm 1's best response is to set \( P_1 \to \infty \)** (or as high as possible). Symmetrically, **Firm 2's best response is to set \( P_2 \to \infty \)**. **Best response functions:** \[ \boxed{ \text{Firm 1: } P_1^* \to \infty \\ \text{Firm 2: } P_2^* \to \infty } \] But this is clearly not a plausible model for real-world pricing. Let's check if perhaps the demand functions are mixed up or if there's a typo. In a typical Bertrand duopoly, \( D_1 \) would depend on both \( P_1 \) and \( P_2 \), and so would \( D_2 \). --- ## (b) **Bertrand Equilibrium Prices and Quantities** If both firms set prices as high as possible, demand is: - For Firm 1: \( D_1(P_1, P_2) = 10,000(1 - e^{-P_2}) \) - As \( P_2 \to \infty \), \( e^{-P_2} \to 0 \), so \( D_1 \to 10,000 \). - For Firm 2: Symmetrically, as \( P_1 \to \infty \), \( D_2 \to 10,000 \). But **at these prices, no consumer would pay, so the model is inconsistent.** ### **Alternative approach:** Perhaps the intended demand functions are: \[ D_1(P_1, P_2) = 10,000 (1 - e^{-P_2}) \text{ if } P_1 < P_2, \text{ else } 0 \] But it's not specified. Alternatively, perhaps the demand functions are *mistyped*, and should be: \[ D_1(P_1, P_2) = 10,000 (1 - e^{-P_1}) \] i.e., each firm's demand depends on its own price. If so, then: \[ \pi_1 = (P_1 - 800) \cdot 10,000 (1 - e^{-P_1}) \] **Set first-order condition:** \[ \frac{d\pi_1}{dP_1} = 10,000 (1 - e^{-P_1}) + (P_1 - 800) \cdot 10,000 e^{-P_1} = 0 \] \[ 1 - e^{-P_1} + (P_1 - 800) e^{-P_1} = 0 \] \[ 1 + (P_1 - 800) e^{-P_1} = 0 \] \[ (P_1 - 800) e^{-P_1} = -1 \] This can be solved numerically. But with the demand as given, this is not possible. So, **based on the demand functions as written, the Bertrand equilibrium is for both firms to set infinite prices**. --- ## (c) **Rent Control at $1,000 per unit** If the city imposes a **maximum price of $1,000** (i.e., \( P \leq 1,000 \)), then both firms must set \( P_1 = P_2 = 1,000 \). **Demand for each firm:** \[ D_1(1000, 1000) = 10,000 (1 - e^{-1000}) \] Since \( e^{-1000} \approx 0 \), \( D_1 \approx 10,000 \). So, total apartments supplied: **20,000.** --- ## (d) **Optimal Rent Control to Maximize Market Surplus** **Market surplus** is maximized when price equals marginal cost (\( P^* = MC = 800 \)), which is the competitive equilibrium. - At \( P = 800 \): \[ D_1(800, 800) = 10,000 (1 - e^{-800}) \approx 10,000 \] Total supplied = 20,000 units. Therefore, **the optimal rent control level to maximize market surplus is $800 per unit**. --- ## **Summary Table** | Part | Answer | |------|--------| | (a) | **Best response for each firm:** Set price as high as possible (\( P \to \infty \)), given the demand functions as written. This is not realistic; likely a typo in the demand functions. | | (b) | **Bertrand equilibrium:** Both firms set price as high as possible, sell 10,000 units each if the other does so. This is not plausible; see above. | | (c) | **With rent control at $1,000/unit:** Each firm supplies 10,000 units, total 20,000 units. | | (d) | **Optimal rent control to maximize market surplus:** $800 per unit (i.e., P = MC). | --- If you **clarify the demand functions**, I can give you a more realistic and detailed solution using standard Bertrand competition. As written, the model is not internally consistent with standard economic theory.