Question

There are two auto producers in Glassboro, F1 and F2. The cars they produce are essentially identical. The market inverse demand curve is given by p=a-bQ, where p is price (in thousands of dollars); Q market output (in thousands of units); and a and b are parameters. It is estimated that a=25, b=0.1. Both F1 and F2 have a marginal cost of 10 thousand dollars per car. Competition in the Glassboro auto market works as follows. At the beginning of each year, both firms simultaneously and independently decide how many cars to produce. Then the market price adjusts so that supply equals demand.

(a) Determine F1’s best response mapping.

Answer 1

The best response mapping for F1 is determined by maximizing its profit based on the market price and quantity of cars produced. F1's profit function is calculated by subtracting its cost from its revenue. By taking the derivative of the profit function with respect to its quantity decision and solving for the quantity, F1's best response to F2's quantity decision can be determined.

Step-by-step explanation:

The best response mapping for F1 in this scenario can be determined by considering its profit-maximizing strategy. F1's profit is equal to its revenue minus its cost. Since the cars produced by F1 and F2 are essentially identical, the market price will be the same for both firms. F1's revenue can be calculated by multiplying the market price by the quantity of cars it produces, which is Q1. Its cost is equal to the marginal cost per car multiplied by the quantity of cars it produces.

Therefore, F1's profit function can be expressed as: Profit1 = (p - 10) * Q1

To maximize its profit, F1 will choose the quantity of cars that gives it the highest profit. This can be determined by taking the derivative of the profit function with respect to Q1, setting it equal to zero, and solving for Q1. The resulting quantity is F1's best response to F2's quantity decision.

By doing the math, F1's best response mapping can be expressed as:

• If Q2 < 140, F1's best response is Q1 = 0
• If Q2 = 140, F1's best response is any quantity greater than or equal to 0
• If Q2 > 140, F1's best response is Q1 = Q2 - 140