3 Budget-Constrained Consumer Choice

The budget constraint and affordable choices

Suppose a consumer chooses quantities $x$ and $y$ of two goods, with prices $p_x$ and $p_y$ and spending income $m$. The budget constraint is

The budget set contains every affordable bundle. The budget line contains bundles that use all available income:

Solving the budget-line equation for $y$ gives

The vertical intercept, $\frac{m}{p_y}$, is the maximum amount of $y$ the consumer can buy, and the horizontal intercept is $\frac{m}{p_x}$. The slope, $-\frac{p_x}{p_y}$, is the opportunity cost of $x$ in units of $y$: gaining one more unit of $x$ requires giving up $\frac{p_x}{p_y}$ units of $y$ while keeping spending unchanged.

If prices do not change, a rise in income shifts the budget line outward in parallel. A change in one price rotates the line around the intercept for the other good. Either change affects the consumer’s affordable choices.

Preferences and utility

Preferences rank alternative bundles. A utility function, written $U(x,y)$, represents this ranking: a bundle with a higher utility value is preferred to one with a lower value. Utility is generally ordinal, meaning it represents an ordering of preferences rather than objectively measurable units of satisfaction.

An indifference curve joins bundles that provide the same utility. With the usual assumption that both goods are desirable, curves farther from the origin represent preferred bundles.

The marginal utility of a good is the change in utility from a small increase in its consumption while holding the other good fixed. The marginal rate of substitution (MRS) describes how much $y$ the consumer is willing to give up for a small increase in $x$ while remaining equally satisfied:

Choosing the best affordable bundle

The consumer chooses the bundle that maximizes utility among affordable bundles:

If more of either good is always preferred, the consumer spends all available income, so the budget constraint binds. At an interior optimum, where positive amounts of both goods are purchased, the highest attainable indifference curve is tangent to the budget line. The condition is

or, equivalently,

The second expression says that the last dollar spent on each good yields the same marginal utility. If one good instead yielded more utility per dollar, shifting spending toward it could raise utility. This tangency condition applies to an interior solution, not necessarily to a corner solution, where the consumer buys none of one good. At a corner, compare utility across feasible boundary choices to identify the optimum.

Solving a Cobb–Douglas choice

Consider Cobb–Douglas preferences represented by

For an interior optimum, marginal utilities are

Therefore,

Set this equal to the price ratio and combine it with the budget equation:

Solving gives the optimal quantities:

For example, if $\alpha=\frac{1}{2}$, income is $m=\$120$, and prices are $p_x=\$4$ and $p_y=\$2$, then $x^*=15$ and $y^*=30$. The bundle costs $\$4(15)+\$2(30)=\$120$, so it uses the full budget.

From consumer choice to individual demand

An individual demand function describes the quantity of a good a consumer chooses at different prices while holding income, other prices, and preferences fixed. It is derived by solving the consumer’s optimization problem for each relevant set of prices and income.

For the Cobb–Douglas example, demand for $x$ is

With $\alpha$, $m$, and $p_y$ fixed, a higher $p_x$ reduces the chosen quantity of $x$ in this example. Repeating the optimization for different values of $p_x$ gives the consumer’s demand curve for $x$. More generally, a price change alters relative prices and purchasing power, so the consumer re-optimizes; the resulting price–quantity choices trace out individual demand.