## Nonstandard Budget Constraints

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.

Figure 4.16: Figure 4.16 Quantity Discounts and the Budget Constraint
Figure 4.16: When the price of phone minutes is constant at 10 cents per minute, Alex’s budget constraint for phone minutes and pizza has a constant slope, as represented by the solid line. If the phone company offers a quantity discount on phone minutes, however, Alex’s budget constraint will be kinked. Here, Alex’s calling plan charges 10 cents per minute for the first 600 minutes per month and 5 cents per minute after that, resulting in the kink at 600 phone minutes shown by the dashed line. The triangle above the initial budget constraint and below the dashed line represents the set of phone minute and pizza combinations Alex can afford under the new pricing scheme that he could not have purchased at a constant price of 10 cents per minute.

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.)

Figure 4.17: Figure 4.17 Quantity Limits and the Budget Constraint
Figure 4.17: When there is a limit on how much of a good a person can consume, a budget constraint will be kinked. When Alex is limited to 600 minutes on the phone per month, his budget constraint is horizontal at that quantity. The triangle above the horizontal section of the budget constraint and below the dashed line represents the set of phone minutes and pizzas that are now infeasible for Alex to buy. Note that Alex can still afford these sets since his income and the prices have not changed, but the restrictions on how much he can purchase dictate that he cannot buy them.

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