1 Budget Constraints

Affordable bundles and the budget set

A budget constraint describes which combinations of goods a consumer can afford given income and market prices. It identifies feasible choices, but it does not indicate which of those choices the consumer prefers.

For two goods, let income be $I$, quantities be $x$ and $y$, and prices be $p_x$ and $p_y$. The budget set includes every nonnegative bundle whose total cost is no greater than income:

Bundles on or below the budget line are affordable; bundles beyond it cost more than the consumer can spend. If the consumer can freely save unspent income, bundles strictly inside the budget set are also possible.

Reading the budget line

The budget line marks bundles that spend all available income. For two goods, it is described by:

Solving for $y$ gives the line in intercept-slope form:

The vertical intercept, $I/p_y$, is the most of good $y$ the consumer could buy if they bought no $x$. The horizontal intercept, $I/p_x$, is the most of good $x$ they could buy if they bought no $y$. The slope, $-p_x/p_y$, represents the trade-off between the goods: how many units of $y$ must be given up to afford one more unit of $x$. This is the opportunity cost of $x$, measured in units of $y$.

For example, suppose a consumer has $\$20$, coffee costs $\$4$ per cup, and sandwiches cost $\$2$ each. Let $x$ be cups of coffee and $y$ be sandwiches. The budget set is described by:

The budget line is $y = 10 - 2x$. The consumer can afford at most $5$ coffees or $10$ sandwiches. Its slope is $-2$: one additional coffee requires giving up two sandwiches to remain on the budget line.

For $n$ goods, the same budget constraint is written as:

where each quantity $q_i$ is nonnegative and each price is $p_i$.

How income and prices change choices

Changes in income and prices alter the feasible set in different ways.

• Income changes while prices stay the same: Higher income shifts the budget line outward in parallel, and both intercepts rise proportionally. Lower income shifts it inward. The slope, which represents the relative-price trade-off, stays the same.

• The price of one good changes while income and other prices stay the same: The budget line pivots because the intercept for that good changes while the other intercept remains fixed. A higher price reduces how much of that good the consumer can afford and raises its opportunity cost in terms of the other good. A lower price has the reverse effect.

• All prices and income change by the same proportion: The budget set remains unchanged because each bundle has the same affordability as before.

In the coffee-and-sandwich example, if the coffee price rises from $\$4$ to $\$5$ while income and the sandwich price stay fixed, the maximum number of coffees falls from $5$ to $4$, while the sandwich intercept remains $10$. The line pivots inward around the sandwich intercept and becomes steeper. These changes describe the feasible set; they do not, on their own, determine the consumer’s eventual purchase.

Connecting feasibility and choice

The budget set contains every affordable bundle, while the budget line marks bundles that spend all available income. Income determines the overall purchasing range, and prices determine the intercepts and the trade-offs represented by the slope. Income changes shift the line, while a change in one good’s price pivots it. Preferences determine which feasible bundle the consumer chooses.