## Quick Reference

A theorem determining the effect of a differential change in a parameter on the outcome of a maximization problem. Formally, consider the problem of choosing *x* to maximize a function *f*(*x*; *a*), where *a* is a parameter. Denote the solution to this maximization by *x*(*a*) and define the maximum value function *V*(*a*) by*V*(*a*)=*max*_{{x}}*f*(*x*;*a*)=*f*(*x*(*a*);*a*).The envelope theorem states thatso that the derivative of the value function is the partial derivative of the objective evaluated at the solution to the optimization.

*V*(*a*)=*max*_{{x}}*f*(*x*;*a*)=*f*(*x*(*a*);*a*).

*Subjects:*
Economics.