In [[microeconomics]], the '''Utility Maximization Problem''' is the problem consumers face: how should they spend their money in order to maximize their [[utility]]?
Suppose theretheir [[consumption set]] <math>\textbf R^LL_+</math> has L commodities. If the prices of the L commodities are <math>p \in \textbf R^L_+</math> and the consumer's wealth is w, then the set of all affordable packages, the [[budget set]] is <math>B(p, w) = \{x \in \textbf R^L_+ : p \cdot x \leq w\}</math>. The consumer would like to buy the best package of commodities it can afford. If <math>u : \textbf R^L_+ \rightarrow R</math> is the consumer's utility function, then the consumers optimal choices x(p, w) are
