## The Slope of the Budget Constraint

The relative prices of the two goods determine the slope of the budget constraint. Because the consumer spends all her money when she is on the budget constraint line, if she wants to buy more of one good and stay within her budget, she has to buy less of the other. Relative prices pin down the rate at which purchases of the two goods can be traded off. If she wants to buy 1 more burrito (a cost of $10), for example, she’ll have to buy 2 fewer lattes at $5.

We can see the equivalence between relative prices and the slope of the budget constraint by rearranging the budget constraint:

Income = *P*_{X}Q_{X} + *P*_{Y}Q_{Y}

*P*_{Y}Q_{Y} = Income – *P*_{X}Q_{X}

The equation shows if *Q*_{X}—the quantity purchased of good *X*—increases by 1 unit, the quantity of good *Y*, or *Q*_{Y}, that can be bought falls by *P*_{X}/*P*_{Y}. This ratio of the price of good *X* relative to the price of good *Y* is the negative of the slope of the budget constraint. It makes sense that this price ratio determines the slope of the constraint. If good *X* is expensive relative to good *Y* (i.e., *P*_{X}/*P*_{Y} is large), then buying additional units of *X* will mean you must give up a lot of *Y* and the budget constraint line will be steep. If good *X* is relatively inexpensive, you don’t have to give up a lot of *Y* to buy more *X*, and the constraint line will be flat.

We can use the equation for the budget constraint (Income = *P*_{X}Q_{X} + *P*_{Y}Q_{Y}) to find its slope and intercepts. Using the budget constraint (50 = 5*Q*_{lattes} + 10*Q*_{burritos}) shown in Figure 4.14, we get

Dividing each side by 10—the price of *Q*_{Y}—yields a slope of –1/2:

As we noted earlier, if Sarah spends all of her income on lattes, she will buy 10 lattes (the *x*-intercept), while she can purchase 5 burritos (the *y*-intercept) using all of her income. These relative prices and intercepts are shown in Figure 4.14.

As will become clear when we combine indifference curves and budget constraints in the next section, the slope of the budget constraint plays an incredibly important role in determining which consumption bundles maximize consumers’ utility levels.