Utility maximisation must be seen as an optimisation problem regarding the *utility function* and the *budget constraint*. These two sides of the problem, define *Marshallian demand curves*.

An individual is therefore faced with the following problem: faced with a set of choices, or baskets of *goods*, and a fixed budget, how to choose the basket which maximises their utility?

If we know an individual’s utility function, and we know their budget, we have the two restrictions necessary to maximise their utility. This can be done graphically, with the point where budget and utility function meet defining an optimum, as shown in the adjacent figure.

It can be also done mathematically, through a *Lagrangian*, where the first derivatives determine a system of equations that can be resolved by submitting our utility function to the restriction presented by the budget:

**Video – Utility maximisation:**