Utility maximization problem

In microeconomics, the Utility Maximization Problem is the problem consumers face: how should they spend their money in order to maximize their utility?

Suppose their consumption set ${\displaystyle {\textbf {R}}_{+}^{L}}$ has L commodities. If the prices of the L commodities are ${\displaystyle p\in {\textbf {R}}_{+}^{L}}$ and the consumer's wealth is w, then the set of all affordable packages, the budget set is ${\displaystyle B(p,w)=\{x\in {\textbf {R}}_{+}^{L}:p\cdot x\leq w\}}$. The consumer would like to buy the best package of commodities it can afford. If ${\displaystyle u:{\textbf {R}}_{+}^{L}\rightarrow R}$ is the consumer's utility function, then the consumers optimal choices x(p, w) are

${\displaystyle x(p,w)=argmax_{x^{*}\in B(p,w)}u(x^{*})}$.

Finding x(p, w) is the Utility Maximization Problem. The solution x(p, w) need not be unique. If u is continuous and no commodities are free of charge, then x(p, w) is nonempty. Proof: B(p, w) is a compact space. So if u is continuous, then the Weierstrass theorem implies that u(B(p, w)) is a compact subset of ${\displaystyle {\textbf {R}}}$, and hence contains an upper bound.

References

Mas-Colell, Andreu; Whinston, Michael; & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 0195073401