In the preceding sections, we analyzed how a consumer’s preferences can be described by a utility function, why indifference curves are a convenient way to think about utility, and how the slope of the indifference curve—known as the marginal rate of substitution—captures the relative utility that a consumer derives from different goods at the margin. Our ultimate goal in this chapter is to understand how consumers maximize their utility by choosing a bundle of goods to consume. Because consumers have a limited amount of money and because goods are not free, consumers must make tradeoffs when deciding how much of each good to consume. The decision depends not only on the utility consumers get from each good, but also on how much money they have to spend and on the prices of the goods. We have to analyze the interaction among all of these factors.

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We start looking at the interactions of utility, income, and prices by making some assumptions. To keep things simple, we continue to focus on a model with only two goods.

To incorporate prices and the consumer’s income into our model of consumer behavior, we use a budget constraint. This constraint describes the entire set of consumption bundles that a consumer can purchase by spending all of her money. For instance, let’s go back to the example of Sarah and her burritos and lattes. Suppose Sarah has an income of $50 to spend on burritos (which cost $10 each) and lattes ($5 each). Figure 4.14 shows the budget constraint corresponding to this example. The number of lattes is on the horizontal axis; the number of burritos is on the vertical axis. If Sarah spends her whole income on lattes, then she can consume 10 lattes (10 lattes at $5 each is $50) and no burritos. This combination is point A in the figure. If instead Sarah spends all her money on burritos, she can buy 5 burritos and no lattes, a combination shown at point B. Sarah can purchase any combination of burritos and lattes that lies on the straight line connecting these two points. For example, she could buy 3 burritos and 4 lattes. This is point C.

The mathematical formula for a budget constraint is

Income = P_XQ_X + P_YQ_Y

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where P_X and P_Y are the prices for 1 unit of goods X and Y (lattes and burritos in our example) and Q_X and Q_Y are the quantities of the two goods. The equation simply says that the total expenditure on the two goods (the per-unit price of each good multiplied by the number of units purchased) equals the consumer’s total income.

Any combination of goods on or below the budget constraint (i.e., any point between the origin on the graph and the budget constraint, including those on the constraint itself) is feasible, meaning that the consumer can afford to buy the bundle with her income. Any points above or to the right of the budget line are infeasible. These bundles are beyond the consumer’s reach even if she spends her entire income. Figure 4.14 shows the feasible and infeasible bundles for the budget constraint 50 = 5Q_lattes + 10Q_burritos.

The budget constraint in Figure 4.14 is straight, not curved, because we assumed Sarah can buy as much as she wants of a good at a set price per unit. Whether buying the first latte or the tenth, we assume the price is the same. As we see later, if the goods’ prices change with the number of units purchased, the budget line will change shape depending on the amount of the goods purchased. That’s an unusual case, though.

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_XQ_X + P_YQ_Y

P_YQ_Y = Income – P_XQ_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_XQ_X + P_YQ_Y) to find its slope and intercepts. Using the budget constraint (50 = 5Q_lattes + 10Q_burritos) shown in Figure 4.14, we get

50 = 5Q_X + 10Q_Y

10Q_Y = 50 – 5Q_X

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.

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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.

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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In all the examples so far, the budget constraint has been a straight line. There are some cases in which the budget constraint would be kinked instead.

Quantity Discounts Suppose Alex spends his $100 income on pizzas and phone calls. A pizza costs $10; if he spends everything on pizzas, he can buy 10. If the price of phone minutes is constant at 10 cents per minute, he can buy as many as 1,000 minutes of phone time. Figure 4.16 portrays this example graphically. The budget constraint in the case where minutes are priced at a constant 10 cents is given by the solid section of the line running from zero minutes and 10 pizzas up to 1,000 minutes and zero pizzas.

Phone plans often offer quantity discounts on goods such as phone minutes. With a quantity discount, the price the consumer pays per unit of the good depends on the number of units purchased. If Alex’s calling plan charges 10 cents per minute for the first 600 minutes per month and 5 cents per minute after that, his budget constraint will have a kink. In particular, because phone minutes become cheaper above 600 minutes, the actual budget constraint has a kink at 600 minutes and 4 pizzas. Because the price of the good on the y-axis (phone minutes) becomes relatively cheaper, the constraint rotates clockwise at that quantity, becoming steeper. To find where the budget constraint intercepts the vertical axis, we have to figure out how many minutes Alex can buy if he purchases only cell phone time. This total is 1,400 minutes [(600 × $0.10) + (800 × $0.05) = $100]. In Figure 4.16, the resulting budget constraint runs from 10 pizzas and zero minutes to 4 pizzas and 600 minutes (part of the solid line) and then continues up to zero pizzas and 1,400 minutes (the dashed line). It’s clear from the figure that the lower price for phone time above the 600-minute threshold means that Alex can afford a set of phone minute and pizza combinations (the triangle above the initial budget constraint and below the dashed line) that he could not afford when phone minutes had a constant price of 10 cents.

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Quantity Limits Another way a budget constraint can be kinked is if there is a limit on how much of one good can be consumed. When the newest iPhone model comes out, Apple often sets a limit of one per customer. Similarly, during World War II in the United States, certain goods like sugar and butter were rationed. Each family could buy only a limited quantity. And during the oil price spikes of the 1970s, gas stations often limited the amount of gasoline people could buy. These limits have the effect of creating a kinked budget constraint.9

9 Note that limits on how much a consumer can purchase are a lot like the quotas we learned about in Chapter 3, except now they apply to a single consumer, rather than to the market as a whole.

Suppose that the government (or maybe his parents, if he lives at home) dictates that Alex can talk on the phone for no more than 600 minutes per month. In that case, the part of the budget constraint beyond 600 minutes becomes infeasible, and the constraint becomes horizontal at 600 minutes, as shown by the solid line in Figure 4.17. Note that neither Alex’s income nor any prices have changed in this example. He still has enough money to reach any part in the area below the dashed section of the budget constraint that is labeled infeasible. He just isn’t allowed to spend it. Consequently, for the flat part of the budget constraint, he will have unspent money left over. As we see in the next section, you will never actually want to consume a bundle on the flat part of the budget constraint. (You might want to see if you can figure out why yourself, before we tell you the answer.)

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