Question

Three firms produce ethanol and compete in a Cournot oligopoly. Each firm has the same cost function, MC(Q) = 1/2Q.Market demand is given by Q=600-P.The quantities produced by the three firms are respectively Q1,Q2,Q3

(a) what is frim 1's optimal quantity as a function of Q2 and Q3

(b) what is the total quantity produced by all three firms?

Answer 1

Solution:

The cost function for each of the firm is: MC(q) = (1/2)*q or 0.5*q

Market demand function: Q = 600 - P ; where Q = q1 + q2 + q3 (sum of production by each firm)

a) For any firm i, profit will be maximized where the marginal revenue will equal the marginal cost.

Total revenue = P*qi = (600 - (q1 + q2 + q3))*qi = 600qi - (q1 + q2 + q3)*qi

Then, marginal revenue becomes = dTR/dqi = 600 - d*((q1 + q2 + q3)*qi)/dqi

So, for firm 1, marginal revenue = 600 - 2*q1 - q2 - q3

Then, for optimal point of production, MR of firm 1 = MC(q1)

600 - 2*q1 - q2 - q3 = 0.5*q1

So, 2.5*q1 = 600 - q2 - q3

q1 = (600 - q2 - q3)/2.5 = 240 - 0.4*q2 - 0.4*q3

So, firm 1's optimal quantity as a function of q2 and q3 is: q1*(q2, q3) = 240 - 0.4*q2 - 0.4*q3

b) Doing it the easier, shorter way: Due to similar cost function, and same weightage for demand function, there are two things to be noted:

1. The optimal quantity production of a firm with respect to other firms' production would be similar as well:

So, q2*(q1, q3) = 240 - 0.4*q1 - 0.4*q3

And q3*(q1, q2) = 240 - 0.4*q1 - 0.4*q2

2. Now, we have three equations in three variables so, can easily solve for q1, q2, q3, and so Q. But it can further be reduced to lower working: due to reason mentioned in explanation (similar cost function and such demand function), q1 = q2 = q3 (= q, let's say) in optimal

Then, using any one of the above optimal equations then: q = 240 - 0.4*q - 0.4*q

q + 0.4q + 0.4q = 240

1.8q = 240

So, q = 240/1.8

Then, total quantity, Q = q1 + q2 + q3

Q = q + q + q = 3*q

Q = 3*(240/1.8) = 400 units

(You could solve the long way, as in solving using first order conditions to reach the optimal qantity equations and verify with the mentioned ones. Then, you may solve the three linear equations in three variables, and verify the result).