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In microeconomic analysis, the concept of the indirect utility function plays a critical role in understanding consumer behavior under constraints. A consumer’s indirect utility function expresses the maximum level of utility that the consumer can achieve given their income and prevailing market prices. Unlike the ordinary utility function, which is a direct function of the quantities of goods consumed, the indirect utility function is expressed in terms of prices and income, providing a different but equally important perspective on consumption decisions. Consider a consumer with a utility function $U(x_1, x_2, \dots, x_n)$, where $x_i$ denotes the quantity of the $i$-th good consumed. The consumer faces prices $p_1, p_2, \dots, p_n$ and has a fixed income $M$. The consumer’s objective is to maximize utility subject to the budget constraint: $$ \sum_{i=1}^n p_i x_i \le M. $$ This is the classical utility maximization problem. Solving this problem involves determining the optimal consumption bundle $(x_1^*, x_2^*, \dots, x_n^*)$ that maximizes utility given the budget constraint. These optimal quantities are functions of prices and income: $x_i^* = x_i(p_1, p_2, \dots, p_n, M)$. Substituting these optimal quantities back into the ____ ________ ___ ______ _______ ___ ________ __________ _________ ____ ________.
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