The maximum utility level a consumer can achieve expressed as a function of prices and income. Consider a consumer choosing the quantities x1 and x2 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
x1 = d1(p1, p2, M)
x2 = d2(p1, p2, M).
Substituting the optimal choices back into the utility function gives the maximized level of utility as
V(p1, p2, M)≡ U(d1(p1,p2, M),d2(p1,p2, M).
The function V(p1, p2, M) is the indirect utility function. Denote the marginal utility of income by α. Roy's identity states that ∂V/∂pi = −αxi, a result that is useful for calculating the welfare consequences of a price change. See also expenditure function.