A simulation of the non-cooperative Nash equilibrium model

The present analysis is an application of the continuous time replicator dynamic to a market equilibrium model. Let us consider that there are two automobile corporations that make two types of automobiles, namely deluxe and ordinary cars. Each corporation has its own production limits for these two types of automobiles, and each produces two types of automobiles so as to maximize profits, calculated to include conjectural variations between the two firms. Let us define x(i)(P) as the quantity of automobiles for corporation p (p=1,2) and type i (i=1,2). The non-cooperative Nash equilibrium solution is obtained after assuming the profit maximization behavior for each corporation under the conditions of normalized constraints as : x(1)(1)+x(2)(1)=1 and x(1)(2) +x(2)(2)= 1 and non-negative constraints, x(i)(P) greater than or equal to 0 (p,i=1,2). To get the Nash equilibrium point, the profit function of corporation p is specified as : E-P(x(1),x(2))=Sigmaf(i)(p))-SigmaSigmatheta(ij)(p)x(j)(p)x(j)(q) (p=1,2,i,j=1,2(i#j,p#q)), where theta(ij)(p) is the conjectural variations. The replicator dynamic for corporation 1 is specified as : dx(1)(1)(t)/dt=x(1)(1)(t)x(2)(1)(t){df(1)(1)(x(1)(1)(t)/dx(1)(1)-df(2)(1)(x(2)(1)(t))/dx(2)(1)-Sigma(theta(1j)(1)-theta(2j)(1))x(j)(2)} and dx(2)(1)(t)/dt= x(1)(1)(t)x(2)(1)(t) {df(2)(1)(X-2(1)(t) /dX(2)(1)-df(1)(1)(x(1)(1)(t))/dx(1)(1)-Sigma(theta(2j)(1)-theta(1j)(1))x(j)(2)}. Changes in the values of theta(ij)(P) and the parameters included in the profit functions make possible many alternative production mixes between deluxe and ordinary cars for the two corporations.