# Utility maximisation subject to a budget constraint

For a consumer with a limited income (and that is all of us once our work-leisure choices have been made), the best or efficient allocation of spending is the one which gives the greatest total utility. From all combinations of goods that are feasible to purchase given the budget constraint, the individual seeks that combination which will give the highest level of utility. We illustrate a solution to this problem in Figure 3.7. At the point , utility is maximised at the level U=. To confirm that this is true, consider any point other than which also satisfies the budget constraint. An indifference curve passing through such a point must lie below the highest attainable one, and so would yield lower utility. Note that the individual would prefer to be at a point on a higher indifference curve (such as U= ) but that is not possible as it would require that spending exceed available income.

One characteristic of this solution is of particular interest. Provided that indifference curves are smooth and continuous, and are bowed inwards in the way we have illustrated them (that is, they are convex from below), then the utility maximising combination of x and y purchased will lie at a point of tangency between the budget constraint and an indifference curve.

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