Consider a Cournot model with two firms and fixed costs. Firms simultaneously ch

Question

Consider a Cournot model with two firms and fixed costs. Firms simultaneously choose their capacity of production q_1 and q_2. The cost of producing zero is zero. The cost of producing q_i, > 0 is F + q_i where F > 0 is the cost of building a factory. When the total capacity of production is Q = q_1 + q_2 the market clearing price is P(Q) = max{0.2 - Q}. Firms only care about their own expected profits. (1)Formally describe a normal-form game that captures the situation above. (2)Find all the Nash equilibria for each value of F greaterthanorequalto 0.

Explanation / Answer

a)

Given is a Cournot model with two firms and fixed costs. Each firm has a cost function F + qiwhere F>0. This implies that the marginal cost is qi for ith firm

The market clearing price is P(Q) = max(0, 2— Q) which can be defined as:

P(Q) = max[0, 2 – Q] = 2 Q if 0<Q 2

= 0 if Q > 2

For firm 1, it has a marginal cost of q1. For a total of q1 units, the total cost is (q1)2. The price it charges can be traced from the demand function:

Payoff to firm 1 is 1(q1, q2)

= q1(P(q1 + q2) c)

= q1(2 q1 q2 q1)

=2q1 (q1)2 q1q2                     if if q1 < 2 – q2

Or     

= (q1)2                            if q1 > 2 - q2

Similarly for firm 2,

Payoff to firm 2 is 2(q1, q2)

= q2(P(q1 + q2) c)

= q2(2 q2 q2 q1)

=2q2 (q2)2 q1q2                     if if q2 < 2 – q1

Or      = (q2)2                            if q2 > 2 – q1

b)

Game has unique Nash equilibrium.

For this firm, c1 = q1, c2 = q2, then

Output level of firm 1 = q1* = (2-q2)/3

Output level of firm 2 = q1* = (2-q1)/3

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