Question

Using a linear demand function and constant marginal cost function as illustration, explain the Cournot model with product differentiation. Derive the sufficient condition that all firms have no incentive to cheat.

Answer 1

Cournot model is a model where the firms decides the quantity simultaneously given other firms quantity decision.

Suppose there are 2 firms which are producing differentiated product. The demand curve for the commodity is given below:

P=a-bQ1-bQ2

The marginal cost for both firms are 0.

The cournot model equilibrium and sufficient condition is calculated below:

Profit1= (a-bQ1-bQ2)*Q1-0*Q1

Profit2= (a-bQ1-bQ2)*Q2-0*Q2

dProfit1/dQ1=(a-2bQ1-bQ2)

(a-2bQ1-bQ2)=0

Q1=(a-bQ2)/2b

dProfit2/dQ2=(a-2bQ2-bQ1)

(a-2bQ2-bQ1)=0

Q2=(a-bQ1)/2b

Solving the 2 equations, the value of quantites are :

Q1=(a)/3b

Q2=(a)/3b

P=a-bQ1-bQ2

P=a/3

Profit1=a2/9b

Profit2=a2/9b

The sufficient conditions for firms not deviating fron the equilibrium are :

d2Profit1/dQ12=-2b

d2Profit2/dQ22=-2b

Given b is positive, the second order is negative which implies the profit is maximised at the equilibrium quantity.

If any firm increases its quantity, the price will fall and the firm profit will fall. Therefore, no firm has incentive to deviate from the equilibrium