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 is

The solution is described by the two Marshallian demand functions x₁ = d₁(p₁, p₂, M) and x₂ = d₂(p₁, p₂, M). Substituting the optimal choices back into the utility function gives the maximized level of utility as

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/∂p_i = −αx‎_i, a result that is useful for calculating the welfare consequences of a price change. See also expenditure function.