Question

Consider a two-stage context where the incumbent invests in capacity of K1 in stage one. For a demand function of Q = 50 – 0.5P, where Q is the market quantity, P is price, and there is one incumbent and one potential entrant in the market, show how to find the incumbent’s best response function in stage two. Per-unit labour and capital costs equal 20 and fixed costs equal 100 for both firms.

Answer 1

Answer: the incumbent's best response function in stage two is QI = 2P - 40.

Explanation: To find the incumbent's best response function in stage two, we need to consider the strategic interaction between the incumbent and the potential entrant in the market.

In stage two, the potential entrant observes the incumbent's capacity investment decision in stage one and decides whether to enter the market or not. The incumbent, anticipating the potential entrant's decision, then determines its optimal quantity to maximize profits.

Given the demand function Q = 50 - 0.5P and the cost structure, we can proceed as follows:

Calculate the potential entrant's profit function:

The potential entrant's profit function can be expressed as:

πE = (P - 20)Q - 100, where Q is the quantity produced.

Set up the incumbent's profit function:

The incumbent's profit function is given by:

πI = (P - 20)QI - 100, where QI is the incumbent's quantity.

Determine the best response:

To find the incumbent's best response function, we differentiate the profit function with

respect to QI and set the derivative equal to zero:

dπI / dQI = P - 20 - 0.5QI = 0

Solving for QI, we get:

QI = 2P - 40

Therefore, the incumbent's best response function in stage two is QI = 2P - 40. This represents the quantity that the incumbent will produce in response to a given price level, considering the potential entrant's decision and the cost structure.