Question

A market has the demand Q=200-P in which Q=qi was the aggregate quantity. There were N greater than or equal to 1 firms in this market, and the marginal cost for each firm i was a constant ci. Suppose firms in this market compete choosing the level of output, and marginal cost is the same across firms. ci=30. What is each firm's profit in equilibrium?

Answer 1

Consider the given problem here the market demand curve is given by, “P = 200 – Q”, where “Q” be the sum of the individual outputs, => Q = ?qi. Now, let’s assume that there are 4 identical firm having same marginal cost function “ci = 30”. The profit function of the 1^st firm is given below.

=> ?1 = P*q1 – c1*q1 = (200 – q1 – q2 – q3 – q4)*q1 – 30*q1

=> ?1 = 200*q1 – q1^2 – q2*q1 – q3*q1 – q4*q1 – 30*q1.

=> ?1 = 170*q1 – q1^2 – q2*q1 – q3*q1 – q4*q1. Similarly the profit function of the ith firm is given below.

=> ?i = 170*qi – qi^2 – ?_jqjqi. So, here all the firms want to maximize their profit with respect to their quantity production. FOC require, “??i/?qi = 0”, known as reaction function.

=> ??i/?qi = 0, => 170 – 2*qi – ?qj = 0, => qi = 85 – (1/2)*?qj.

Now, we can see that here all the firm having identical cost structure and all of them are facing same market demand curve, => at the optimum “qi=qj”.

=> qi = 85 – (1/2)*?qj, => qi = 85 – (3/2)*qi, => 2.5*qi = 85, => qi = 85/2.5 = 34.

=> qi = 34, i = 1, 2, 3, 4. So, the total production is “Q = 4*qi = 4*34 = 136”. So, the corresponding price level is given by, “P = 200 – Q = 200 – 136 = 64.

So, the profit of the ith firm is given by, ?i = (P – 30)*qi = (64 – 30)*34 = 34*34 = 1156.

=> here all the firm are getting the same profit which is given by, “?i = 1156”.