## Quick Reference

The maximum utility level a consumer can achieve expressed as a function of prices and income. Consider a consumer choosing the quantities *x*_{1} and *x*_{2} of two goods to maximize utility subject to a budget constraint. The utility maximization problem isThe solution is described by the two Marshallian demand functions

*x*_{1} = *d*_{1}(*p*_{1}, *p*_{2}, *M*)

and

*x*_{2} = *d*_{2}(*p*_{1}, *p*_{2}, *M*).

Substituting the optimal choices back into the utility function gives the maximized level of utility as

*V*(*p*_{1}, *p*_{2}, *M*)≡ *U*(*d*_{1}(*p*_{1},*p*_{2}, *M*),*d*_{2}(*p*_{1},*p*_{2}, *M*).

The function *V*(*p*_{1}, *p*_{2}, *M*) is the indirect utility function. Denote the marginal utility of income by *α*. Roy's identity states that ∂*V*/∂*p*_{i} = −*αx*_{i}, a result that is useful for calculating the welfare consequences of a price change. See also expenditure function.

*Subjects:*
Economics.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.