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
Famous quotes containing the words calculating the, calculating and/or equilibrium:
“[The] elderly and timid single gentleman in Paris ... never drove down the Champs Elysees without expecting an accident, and commonly witnessing one; or found himself in the neighborhood of an official without calculating the chances of a bomb. So long as the rates of progress held good, these bombs would double in force and number every ten years.”
—Henry Brooks Adams (1838–1918)
“Sin in this country has been always said to be rather calculating than impulsive.”
—Frank Moore Colby (1865–1925)
“There is a relation between the hours of our life and the centuries of time. As the air I breathe is drawn from the great repositories of nature, as the light on my book is yielded by a star a hundred millions of miles distant, as the poise of my body depends on the equilibrium of centrifugal and centripetal forces, so the hours should be instructed by the ages and the ages explained by the hours.”
—Ralph Waldo Emerson (1803–1882)