High School

Consider an industry with an incumbent (firm 1) and potential entrant (firm 2). Demand for homogenous good is given by p = 1 -Q. Potential entrant has marginal cost c-0.1, and a fixed sunk cost F. Incumbent marginal cost might be low (c= 0) or high (c = 0.1). Incumbent know the cost and entrant believes that it is low with a probability Pr(c = 0) = x. In the first period, firm 1 chooses its output. In the second period, firm 2 decides whether to enter or not. If firm 2 enters, it sinks its cost F and competes in quantity with the incumbent firm. Should Entrant know the incumbent's marginal cost, entrant will not enter if the incumbent firm is low cost and will enter if the incumbent is high cost. a) What is the Cournot equilibrium output when the incumbent firm is high cost? What is the Cournot equilibrium output when the incumbent firm is low cost? b) What is the maximum sunk cost for the entrant to enter if the incumbent firm is high cost? c) In a separating equilibrium, what is the equilibrium output for the incumbent firm in the first period? Does the incumbent firm successfully deter entry?

Answer :

The incumbent firm successfully deters entry because the potential entrant believes that the incumbent has a low cost and therefore does not enter the market.

a) Cournot equilibrium output when the incumbent firm is high cost:

In the Cournot model, firms choose their quantities simultaneously, taking into account their rivals' quantities. In this case, when the incumbent firm has a high cost (c = 0.1), the Cournot equilibrium output can be determined as follows:

Let Q1 represent the output of the incumbent firm and Q2 represent the output of the potential entrant. The total market quantity is Q = Q1 + Q2. To find the Cournot equilibrium output, we need to solve for the Nash equilibrium, where each firm maximizes its profits given the other firm's output.

The incumbent firm's profit function is:

π1 = (1 - Q)Q1 - cQ1

Taking the derivative with respect to Q1 and setting it equal to zero, we can find the first-order condition for firm 1's output:

∂π1/∂Q1 = 1 - 2Q - c = 0

Solving this equation, we find Q1 = (1 - c)/2. Substituting the value of c = 0.1, we get Q1 = 0.45.

The Cournot equilibrium output when the incumbent firm is high cost (c = 0.1) is Q1 = 0.45.

Cournot equilibrium output when the incumbent firm is low cost:

When the incumbent firm has a low cost (c = 0), the Cournot equilibrium output can be determined using the same steps as above. However, in this case, the incumbent firm's profit function will be different:

π1 = (1 - Q)Q1

Solving the first-order condition, we find Q1 = 0.5.

The Cournot equilibrium output when the incumbent firm is low cost (c = 0) is Q1 = 0.5.

b) Maximum sunk cost for the entrant to enter if the incumbent firm is high cost:

To determine the maximum sunk cost (F) for the entrant to enter the market when the incumbent firm is high cost (c = 0.1), we need to compare the profits of entering and not entering.

If the entrant enters, their profit will be:

π2 = (1 - Q)Q2 - (c - 0.1)Q2 - F

If the entrant does not enter, their profit will be zero.

To find the maximum sunk cost for entry, we need to set the entrant's profit equal to zero:

(1 - Q)Q2 - (c - 0.1)Q2 - F = 0

Simplifying the equation, we find:

Q2 = F / (1 - c + 0.1)

Substituting the values c = 0.1, we get:

Q2 = F / 0.9

Therefore, the maximum sunk cost (F) for the entrant to enter when the incumbent firm is high cost (c = 0.1) is F ≤ 0.9Q2.

c) In a separating equilibrium, the equilibrium output for the incumbent firm in the first period is:

In a separating equilibrium, the incumbent firm can signal its low cost (c = 0) to deter entry. The equilibrium output for the incumbent firm in the first period will be its Cournot equilibrium output when it has a low cost, which is Q1 = 0.5 (as determined in part a).

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