Calculating The Equilibrium
In very general terms, let the price function for the (duopoly) industry be and firm i have the cost structure . To calculate the Nash equilibrium, the best response functions of the firms must first be calculated.
The profit of firm i is revenue minus cost. Revenue is the product of price and quantity and cost is given by the firm's cost function, so profit is (as described above): . The best response is to find the value of that maximises given, with, i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of with respect to is to be found. First take the derivative of with respect to :
Setting this to zero for maximization:
The values of that satisfy this equation are the best responses. The Nash equilibria are where both and are best responses given those values of and .
Read more about this topic: Cournot Competition
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