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.

If a consumer always picks an optimal package as defined above, then x(p, w) is called the Marshallian demand correspondence. If there is always a unique maximizer, then it is called the Marshallian demand function. The relationship between utility and Marshallian demand in the Utility Maximization Problem mirrors the relationship between indirect utility and Hicksian demand in the Expenditure Minimization Problem.

References

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