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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