Because relative prices determine the slope of the budget constraint, changes in relative prices will change its slope. Figure 4.15a demonstrates what happens to our example budget constraint if the price of lattes doubles to $10. The budget constraint rotates clockwise around the vertical axis. Because P_X doubles, P_X/P_Y doubles, and the budget constraint becomes twice as steep. If Sarah spends all her money on lattes, then the doubling of the price of lattes means she can buy only half as many lattes with the same income (the 5 lattes shown at A′, rather than 10 lattes as before). If, on the other hand, she spends all her money on burritos (point B), then the change in the price of a latte doesn’t affect the bundle she can consume. That’s because the price of burritos is still the same ($10). Notice that after the price increase, the set of feasible consumption bundles is smaller: There are now fewer combinations of goods that Sarah can afford with her income.

If, instead, the price of a burritos doubles to $20, but lattes remain at their original $5 price (as in Figure 4.15b), the budget constraint’s movement is reversed: It rotates counterclockwise around the horizontal axis, becoming half as steep. If Sarah wanted to buy only lattes, this wouldn’t affect the number of lattes she could buy. If she wants only burritos, now she can obtain only half as many (if you could buy a half a burrito; we’ll assume you can for now), at bundle B′. Notice that this price increase also shrinks the feasible set of bundles just as the lattes’ price increase did. Always remember that when the price rises, the budget constraint rotates toward the origin. When the price falls, it rotates away from the origin.

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Now suppose prices are stable but Sarah’s income falls by half (to $25). With only half the income, Sarah can buy only half as many lattes and burritos as she could before. If she spends everything on lattes, she can now buy only 5. If she buys only burritos, she can afford 2.5. But because relative prices haven’t changed, the tradeoffs between the goods haven’t changed. To buy 1 more burrito, Sarah still has to give up 2 lattes. Thus, the slope of the budget constraint remains the same. This new budget constraint is shown in Figure 4.15c.

Note that had both prices doubled while income stayed the same, the budget constraint would be identical to the new one shown in Figure 4.15c. We can see this more clearly if we plug in 2P_X and 2P_Y for the prices in the slope-intercept format of the budget constraint:

In both the figure and the equation, this type of change in prices decreases the purchasing power of the consumer’s income, shifting the budget constraint inward. The same set of consumption bundles is feasible in either case. If Sarah’s income had increased rather than decreased as in our example (or the prices of both goods had fallen in the same proportion), the budget constraint would have shifted out rather than in. Its slope would remain the same, though, because the relative prices of burritos and lattes would not have changed.

We’ve now considered what happens to the budget constraint in two situations: when income changes while prices stay constant and when prices change, holding income constant. If prices and income both go up proportionally (e.g., all prices double and income doubles), then the budget constraint doesn’t change at all. You have double the money but everything costs twice as much, so you can only afford the same bundles you could before. You can see this mathematically in the equation for the budget constraint above: If you multiply all prices and income by any positive number (call it k), all the ks will cancel out, leaving you with the original equation.

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