Question

Suppose that the market demand curve is given by q=10-p and that production costs are zero for each of four oligopolists. (a) Determine the level of output for each of the four oligopolists according to the Cournot model. (b) What general rule can you deduce from your answer to part (a)?

Answer 1

A).

Consider the given problem here the market demand curve is given by, “P=a-Q”, where “a=10” and “Q=q1+q2+q3+q4”. So, the profit function of “firm1” is given by.

=> A1 = P*q1 – C1= (a-q1-q2-q3-q4)*q1 = a*q1 - q1^2 - q2*q1 - q3*q1 - q4*q1.

=> A1= a*q1 - q1^2 - q2*q1 - q3*q1 - q4*q1. Similarly the profit functions of other firms are given by.

=> A2 = a*q2 - q1*q2 - q2^2 - q3*q2 - q4*q2, for “firm2”.

=> A3 = a*q3 - q1*q3 - q2*q3 - q3^2 - q4*q3, for “firm3”.

=> A4 = a*q4 - q1*q4 - q2*q4 - q3*q4 - q4^2, for “firm4”.

So, the FOC require “dA1/dq1= dA2/dq2 = dA3/dq3 = dA4/dq4 = 0”.

=> dA1/dq1= 0, => a - 2*q1 - q2 - q3 - q4 = 0, => q1 = a/2 - q2/2 - q3/2 - q4/2, be the reaction function of “firm1”. Similarly, by “dA2/dq2 = 0”, we have “q2 = a/2 - q1/2 - q3/2 - q4/2”, be the reaction function of “firm2”. Now, by “dA3/dq3 = 0”, we have “q3 = a/2 - q1/2 - q2/2 - q4/2”, be the reaction function of “firm3” and by “dA4/dq4 = 0”, we have “q4 = a/2 - q1/2 - q2/2 - q3/2”, be the reaction function of “firm4”. So, here we can see that all the reaction functions are identical, => if we solve all these simultaneously, => then at the equilibrium “q1=q2=q3=q4”.

So, the output produce by individual firms are given by, “q1=q2=q3=q4=a/5=10/5=2” at the equilibrium.

B).

So, here there are 4 identical firms with zero cost case are competing in cournot duopoly, => at the equilibrium each will produce “a/5”. So, here we can generalize the above result for “N firms”, => if there “N” firms everything remains same, => at the equilibrium each will produce “a/n+1”.