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The question is as follow:

Here is how we can think of N-firm Cournot competition. Assume all the firms have the same marginal cost C > 0. Firm 1 chooses Q1, Firm 2 chooses Q2, and so on. The market price P = A – (Q1 + Q2 + … + QN). Assume A > C.

*a) Solve for the Cournot (pure strategy) equilibrium. (Hint: the firms are all the same, so you should expect the equilibrium to be symmetric, that is, Q*1 = Q*2 = …= Q*N.)*

b) Based on your answer to a), show whether the equilibrium profit of a firm increases or decreases in the number of firms, N.

My answer to a is: Let Q be the market output, qi be output firm i,

P = A - Q for Q = ∑qi i=1

Marginal cost of firm i = C which C> 0 and A>C

πi = (A - Q - C)qi

take the derivative,

π'i = A - Q - qi - C since Q = qi + qj + ... + qn, Q' = 1

π'i = A - [(Nqi) + qi] - C since they are identical firms, therefore Q = Nqi

qi* = (A-C)/(N+1)

Since the equilibrium is symmetric, thus

P = A - Nqi*

P = A - N(A-C)/(N+1)

P = (A + NC)/(N+1)

For the profit of individual firm is

π = (A + NC)/(N+1) * (A-C)/(N+1) - C(A-C)/(N+1)

= (A-C)/(N+1) * [ (A + NC)/(N+1) - C ]

= (A-C)/(N+1) * (A-C)/(N+1)

= [(A-C)/(N+1)] ^ 2

For this reason, when N increase, the economic profit of a firm would decrease and vice versa.

Am i doing the right thing so far?

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