# Utility maximization problem

For a less technical introduction, see Utility.

In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem.

## Basic setup

Suppose the consumer's consumption set, or the enumeration of all possible consumption bundles that could be selected if there were no budget constraint, has L commodities and is limited to positive amounts of consumption of each commodity. Let x be the vector x={xi;i=1,...L} containing the amounts of each commodity; then

$x \in \mathbb{R}^L_+ \ .$

Suppose also that the price vector (p) of the L commodities is positive,

$p \in \mathbb{R}^L_+ \ ,$

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

$B(p, w) = \{x \in \mathbb{R}^L_+ : \langle p , x \rangle \leq w\} \ ,$

where $\langle p , x \rangle$ is the dot product of p and x, or the total cost of consuming x of the products at price level p:

$\langle p , x \rangle=\sum_{i=1}^L p_i x_i .$

The consumer would like to buy the best affordable package of commodities.

It is assumed that the consumer has a cardinal utility function, called u. It is a real valued function with domain being the set of all commodity bundles, or

$u : \mathbb{R}^L_+ \rightarrow \mathbb{R}_+ \ .$

Then the consumer's optimal choice x(p, w) is the utility maximizing bundle of all bundles in the budget set, or

$x(p, w) = \operatorname{argmax}_{x^* \in B(p, w)} u(x^*)$.

Finding x(p, w) is the utility maximization problem. If u is continuous and no commodities are free of charge, then x(p, w) exists.[citation needed] If there is a unique maximizer for all values of the price and wealth parameters, 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.

### Nonunique solution

The solution x(p, w) need not be unique.[citation needed] If a consumer always picks an optimal package as defined above, then x(p, w) is called the Marshallian demand correspondence.

## Reaction to changes in prices

For a given level of real wealth, only relative prices matter to consumers, not absolute prices. If consumers reacted to changes in nominal prices and nominal wealth even if relative prices and real wealth remained unchanged, this would be an effect called money illusion. The mathematical first order conditions for a maximum of the consumer problem guarantee that the demand for each good is homogeneous of degree zero jointly in nominal prices and nominal wealth, so there is no money illusion.