In microeconomic theory, the utility maximization problem formalizes how a consumer allocates limited resources across different goods and services. The consumer is assumed to have well-defined preferences over all feasible bundles of goods and to be able to rank these bundles according to the level of utility they provide. Given a budget constraint determined by income and prices, the consumer chooses the most preferred bundle that is affordable. The utility maximization problem yields a systematic analysis of consumer demand and how it changes in response to changes in income or prices.

In microeconomics, a consumer is defined as an individual or a household consisting of one or more individuals. The consumer is the basic decision-making unit that determines which goods and services are purchased and in what quantities. Each day, millions of such choices are made, shaping the allocation of the trillions of dollars worth of goods and services produced annually in the world economy.

The utility maximization problem was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. It is formulated as follows: find the consumption bundle that maximizes the consumer's utility subject to his budget constraint.

A consumption bundle is an element ${\displaystyle x}$ in ${\displaystyle X}$ ${\displaystyle (x\in X)}$ where ${\displaystyle x\in R_{+}^{k}}$. That is, every element ${\displaystyle x}$ in ${\displaystyle X}$ is a nonnegative orthant in ${\displaystyle R^{k}}$. A consumption bundle takes the following form: ${\displaystyle x=(x_{1},x_{2},...,x_{k})}$ where ${\displaystyle x_{i}\geq 0}$ ${\displaystyle \forall i=1,..,k}$. In simple words, the consumer cannot consume a negative amount of good.

The consumer maximizes his utility subject to his budget constraint. The budget constraint is the most simple and intuitive constraint faced by a consumer. The consumer may face a time constraint (the act of consuming takes time), a constraint of both time and money, an intertemporal budget constraint and many more. The economic problem originates from scarcity, therefore, when formulating and economic problem we will usually see some formulation of a constraint.

Assume their is a price vector ${\displaystyle p}$ where ${\displaystyle p=(p_{1},...,p_{k})}$ and ${\displaystyle p_{i}>0\forall i=1,..,k}$. That is a price of a good is a positive number.

Furthermore, assume that the consumer's income is ${\displaystyle I}$. The budget set, or the set of all possible consumption bundles is:

${\displaystyle B(p,I)=\{x\in \mathbb {R} _{+}^{k}|\mathbb {\Sigma } _{i=1}^{k}p_{i}x_{i}\leq I\}\ }$.