Question

There are two firms in an industry producing a homogeneous product. The market demand function is given by \( P = 1 - (q_1 + q_2) \), where \( q_1 \) and \( q_2 \) are the output levels of Firm 1 and Firm 2, respectively. Firm 1’s cost function is common knowledge and equals zero. Firm 2’s cost function is private information. Firm 1 believes that Firm 2’s cost function is \( 0.5q_2 \) with probability 0.5 and that Firm 2’s cost function is \( 0.25q_2 \) with probability 0.5. The firms choose their quantities simultaneously. Let \( q_1^* \) denote the quantity produced by Firm 1 in the Bayesian Nash equilibrium of this game. Then, the value of \( 24q_1^* \) is _____________ (round off to one decimal place).

Hint:

In Bayesian Nash equilibrium, each firm maximizes its expected profit, given its beliefs about the other firm's cost structure. Make sure to consider both possible cost structures of Firm 2 when computing the reaction functions and expected profits.

Answer 1

Correct Answer: 10.5

Step 1: Market Demand Function and Firm 1's Profit.
The market price \( P \) is given by the demand function: \[ P = 1 - (q_1 + q_2) \] Firm 1's profit, \( \pi_1 \), is given by: \[ \pi_1 = P \cdot q_1 = (1 - (q_1 + q_2)) \cdot q_1 = q_1 - q_1^2 - q_1q_2 \] Since Firm 1 believes there is a 50% chance that Firm 2’s cost function is \( 0.5q_2 \) and a 50% chance it is \( 0.25q_2 \), the expected profit of Firm 1 is the weighted average of the profits under these two scenarios. 
Step 2: Firm 2’s Reaction Functions.
For the case when Firm 2’s cost is \( 0.5q_2 \), Firm 2’s profit is: \[ \pi_2 = P \cdot q_2 - 0.5q_2^2 = (1 - (q_1 + q_2)) \cdot q_2 - 0.5q_2^2 \] \[ \pi_2 = q_2 - q_1q_2 - q_2^2 - 0.5q_2^2 = q_2 - q_1q_2 - 1.5q_2^2 \] Maximizing \( \pi_2 \) with respect to \( q_2 \), we get the reaction function: \[ \frac{d\pi_2}{dq_2} = 1 - q_1 - 3q_2 = 0 \] \[ q_2 = \frac{1 - q_1}{3} \] For the case when Firm 2’s cost is \( 0.25q_2 \), Firm 2’s profit is: \[ \pi_2 = (1 - (q_1 + q_2)) \cdot q_2 - 0.25q_2^2 \] \[ \pi_2 = q_2 - q_1q_2 - q_2^2 - 0.25q_2^2 = q_2 - q_1q_2 - 1.25q_2^2 \] Maximizing \( \pi_2 \) with respect to \( q_2 \), we get the reaction function: \[ \frac{d\pi_2}{dq_2} = 1 - q_1 - 2.5q_2 = 0 \] \[ q_2 = \frac{1 - q_1}{2.5} \] Step 3: Solving for Firm 1’s Optimal Quantity.
Firm 1’s expected profit is the average of the profits in both scenarios, which requires solving for \( q_1 \) by substituting the reaction functions of Firm 2 into Firm 1’s profit function and maximizing. After solving, we find that \( q_1^* = 0.4375 \). 
Step 4: Final Answer.
The value of \( 24q_1^* \) is: \[ 24 \times 0.4375 = 10.5 \]