4 Utility Maximization

The budget and the consumer’s choice

A utility-maximization problem asks a consumer to choose the most-preferred bundle they can afford. For two goods with quantities $x$ and $y$, prices $p_x$ and $p_y$, and income $m$, the consumer solves

The budget set consists of all bundles satisfying the constraint. Its boundary, the budget line, is given by $p_xx+p_yy=m$. The intercepts are $m/p_x$ and $m/p_y$, and its slope is $-p_x/p_y$. If preferences are locally nonsatiated, meaning more of a desirable good is preferred, an optimum uses the full budget. The consumer then chooses the highest attainable indifference curve.

Smooth interior optima

An interior solution has positive quantities of both goods, so $x^*>0$ and $y^*>0$. If utility is differentiable and the optimum is smooth, the indifference curve is tangent to the budget line. The condition is that the marginal rate of substitution equals the price ratio:

Here, $MU_x=\partial U/\partial x$ and $MU_y=\partial U/\partial y$. The marginal rate of substitution is the amount of $y$ the consumer is willing to give up for one more unit of $x$, holding utility constant. At the optimum, this willingness to trade matches the market trade-off represented by the price ratio.

The same condition says that marginal utility per dollar is equal across goods:

If one good provided more marginal utility per dollar, the consumer could shift spending toward it and increase utility. Tangency identifies a candidate interior optimum, but does not by itself guarantee a maximum; preferences and feasibility must also be checked.

For example, let $U(x,y)=\sqrt{xy}$, with income $m=12$, prices $p_x=2$, and $p_y=1$. Since $MU_x/MU_y=y/x$, tangency requires $y/x=2$, or $y=2x$. Substituting this into the budget equation $2x+y=12$ gives

The bundle spends $2(3)+1(6)=12$ and satisfies both the tangency condition and the budget constraint.

Corners and kinks

A corner solution occurs when the best affordable bundle contains none of at least one good, such as $x^*>0$ and $y^*=0$. It can arise when one good is a better use of the budget at the margin, or when the indifference curves do not permit an interior tangency. Solving only the tangency equation may therefore miss the optimum; boundary choices must also be compared with any interior candidate.

For example, let $U(x,y)=3x+y$, with income $m=12$, $p_x=2$, and $p_y=1$. One dollar spent on $x$ buys $1/2$ unit of $x$, worth $3/2$ utility, while one dollar spent on $y$ buys one unit, worth $1$ utility. Because $x$ provides more utility per dollar, the consumer spends all income on $x$, choosing $(x^*,y^*)=(6,0)$. This is a corner solution, not a tangency. If the goods provided equal utility per dollar, every affordable bundle along the budget line could be optimal.

Not every optimum is a smooth tangency. With perfect complements, preferences may be represented by $U(x,y)=\min\{x,y\}$. The optimum is typically at an indifference-curve kink, where a single smooth slope—and therefore the usual tangency condition—is unavailable.

A practical solution method

Use this procedure to identify the highest-utility affordable bundle:

1. Write the utility function, the nonnegativity restrictions, and the budget constraint.

2. Find any feasible interior candidate using the tangency condition and the budget line.

3. Check boundary candidates, including $x=0$ and $y=0$, and check kinks when preferences are nonsmooth.

4. Compare utility across the feasible candidates and choose the bundle with the highest utility.

The tangency condition is appropriate for a smooth interior optimum. At a corner or kink, also consider the relevant boundary or nonsmooth candidates.

• perfect substitutes