Question

3. Consider the Cournot Duopoly model.

a. Set up the model in general form using qi , qj instead of q1 , q2

b. Solve for the equilibrium quantity that each firm would produce. (This means as a function of exogenous variables - so not just the reaction functions.)

c. Solve for the comparative static - how does each firm’s quantity change for a change in their own marginal cost and how does it change for a change in the other firm’s marginal cost?

Answer 1

A, B).

Consider the given problem here the market demand function is “P=a-Q”, where “Q=qi+qj”. So, the profit function of “firm i” is given below.

=> Ai = P*qi – Ci, => Ai = (a-qi-qj)*qi - ci*qi, => Ai = a*qi - qi^2 - qj*qi - ci*qi. Similarly, the profit function of “firm j” is given by.

=> Aj = a*qj - qj^2 - qj*qi - cj*qj. The FOC for profit maximization require “dAi/dqi = dAj/dqj = 0”.

=> dAi/dqi = 0, => a – 2*qi - qj – ci = 0, => qi = (a-ci)/2 - qj/2, be the reaction function of “firm i”.

=> dAj/dqj = 0, => a – 2*qj - qi – cj = 0, => qj = (a-cj)/2 - qi/2, be the reaction function of “firm j”.

Now, by solving these two reaction function we get the solution, => the solution of the problem is given below.

=> “qi = (a-2ci+cj)/3” and “qj = (a-2cj+ci)/3”.

C).

Now, let’s assume that “ci” increases, => the marginal cost of producing “qi” increases, => (a-2ci+cj) decreases, => numerator of “qi” decreases implied the optimum production of “qi” decreases. Now, let’s assume that “cj” increases, => the marginal cost of producing “qj” increases, => (a-2ci+cj) increases, => numerator of “qi” increases implied the optimum production of “qi” increases.

So, if the own marginal cost increases, => profit maximizing output decreases. if rival firms marginal cost increases implied the profit maximizing output increases.