# Utility Maximization Problem

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In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?"

Suppose their consumption set

$\textbf R^L_+$

has L commodities. If the prices of the L commodities are

$p \in \textbf R^L_+$

and the consumer's wealth is w, then the set of all affordable packages, the budget set, is

$B(p, w) = \{x \in \textbf R^L_+ : p \cdot x \leq w\}$.

The consumer would like to buy the best package of commodities it can afford. If

$u : \textbf R^L_+ \rightarrow \textbf R$

is the consumer's utility function, then the consumer's optimal choices x(p, w) are

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

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 $\textbf R$. By the Heine-Borel theorem, every compact set contains its maximum, so we can conclude that u(B(p, w)) has a maximum and hence there must be a package in B(p, w) that maps to this maximum.

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 the utility function and Marshallian demand in the Utility Maximization Problem mirrors the relationship between the expenditure function and Hicksian demand in the Expenditure Minimization Problem.

In practice, a consumer may not always pick an optimal package. For example, it may require too much thought. Bounded rationality is a theory that explains this behaviour with satisficing - picking packages that are suboptimal but good enough.