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?