## A Special Case: Corner Solutions

Up to this point, we have analyzed situations in which the consumer optimally consumes some of both goods. This assumption usually makes sense if we think that utility functions have the property that the more you have of a good, the less you are willing to give up of something else to get more. Because the first little bit of a good provides the most marginal utility in this situation, a consumer will typically want at least some amount of a good.

corner solution

A utility-maximizing bundle located at the “corner” of the budget constraint where the consumer purchases only one of two goods.

interior solution

A utility-maximizing bundle that contains positive quantities of both goods.

Depending on the consumer’s preferences and relative prices, however, in some cases a consumer will not want to spend any of her money on a good. When consuming all of one good and none of the other maximizes a consumer’s utility given her budget constraint, the solution to the optimization problem is called a corner solution. (Its name comes from the fact that the optimal consumption bundle is at the “corner” of the budget line, where it meets the axis.) If the utility-maximizing bundle has positive quantities of both goods (like the cases we’ve looked at so far in this chapter), it is called an interior solution.

Figure 4.20 depicts a corner solution. Greg, our consumer, has an income of \$240 and is choosing his consumption levels of mystery novels and Bluetooth speakers. Let’s say a hardcover mystery novel costs \$20, and a Bluetooth speaker costs \$120. Because speakers are more expensive than mystery novels, Greg can afford up to 12 mysteries but only 2 speakers. Nonetheless, the highest utility that Greg can obtain given his income is bundle A, where he consumes all speakers and no mystery novels.

Figure 4.20: Figure 4.20 A Corner Solution
Figure 4.20: A corner solution occurs when the consumer spends all his money on one good. Given Greg’s income and the relative prices of mystery novels and Bluetooth speakers, Greg is going to consume two speakers and zero mystery novels at his optimal consumption bundle (A). All other feasible consumption bundles, such as point B, correspond to indifference curves with lower utility levels than the indifference curve U2 at point A. Greg cannot afford consumption bundles at a higher utility level, such as U3, with his current income.

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How do we know A is the optimal bundle? Consider another feasible bundle, such as B. Greg can afford it, but it is on an indifference curve U1 that corresponds to a lower utility level than U2. The same logic would apply to bundles on any indifference curve between U1 and U2. Furthermore, any bundle that offers higher utility than U2 (i.e., above and to the right of U2) isn’t feasible given Greg’s income. So U2 must be the highest utility level Greg can achieve, and he can do so only by consuming bundle A, because that’s the only bundle he can afford on that indifference curve.

∂ The online appendix further explains the mathematics of corner solutions. (http://glsmicro.com/appendices)

In a corner solution, then, the highest indifference curve touches the budget constraint exactly once, just as with the interior solutions we discussed earlier. The only difference with a corner solution is that bundle A is not a point of tangency. The indifference curve is flatter than the budget constraint at that point (and everywhere else). That means Greg’s MRS—the ratio of his marginal utility from mystery novels relative to his marginal utility from Bluetooth speakers—is less than the price ratio of the two goods rather than equal to it. In other words, even when he’s consuming no mysteries, his marginal utility from them is so low, it’s not worth paying the price of a book to be able to consume one. The marginal utility he’d have to give up with fewer speakers would not be made up for by the fact that he could spend some of his money on novels.

### figure it out 4.5

A pizza chain recently offered the following special promotion: “Buy one pizza at full price and get your next three pizzas for just \$5 each!” Assume that the full price of a pizza is \$10, your daily income \$40, and the price of all other goods \$1 per unit.

1. Draw budget constraints for pizza and all other goods that reflect your situations both before and during the special promotion. (Put the quantity of pizzas on the horizontal axis.) Indicate the horizontal and vertical intercepts and the slope of the budget constraint.

2. How is this special offer likely to alter your buying behavior?

Solution:

1. To draw your budget constraint, you need to find the combinations of pizza and all other goods that are available to you before and during the promotion. The starting place for drawing your budget constraint is to find its x- and y-intercepts.

Before the promotion, you could afford 4 pizzas a day (\$40/\$10) if you spent all of your income on pizza. This is the x-intercept (Figure A). Likewise, you could afford 40 units of all other goods per day (\$40/\$1) if you purchased no pizza. This is the y-intercept. The budget constraint, shown in Figure A, connects these two points and has a slope of –40/4 = –10. This slope measures the amount of other goods you must give up to have an additional pizza. Note that this is also equal to –Px/Py = –\$10/\$1 = –10.

Once the promotion begins, you can still afford 40 units of all other goods if you buy no pizza. The promotion has an effect only if you buy some pizza. This means the y-intercept of the budget constraint is unchanged by the promotion. Now suppose you buy 1 pizza. In that case, you must pay \$10 for the pizza, leaving you \$30 for purchasing all other goods. This bundle is point A on the diagram. If you were to buy a second pizza, its price would be only \$5. Spending \$15 on 2 pizzas would allow you to purchase \$25 (\$40 – \$15) worth of other goods. This corresponds to bundle B. The third and fourth pizzas also cost \$5 each. After 3 pizzas, you have \$20 left to spend on other goods, and after 4 pizzas, you are left with \$15 for other goods. These are points C and D on the diagram.

A fifth pizza will cost you \$10 (the full price) because the promotion limits the \$5 price to the next 3 pizzas you buy. That means if you choose to buy 5 pizzas, you will spend \$35 on pizza and only \$5 on other goods, as at bundle E. Now that you have again purchased a pizza at full price, you are eligible to receive the next 3 at the reduced price of \$5. Unfortunately, you only have enough income for one more \$5 pizza. Therefore, if you would like to spend all of your income on pizza, you can buy 6 pizzas instead of just 4.

As a result of the promotion, then, your x-intercept has moved out to 6, and your budget line has pivoted out (in a somewhat irregular way because of all the relative price changes corresponding to purchasing different numbers of pizzas) to reflect the increase in your purchasing power due to the promotion.

2. It is likely that the promotion will increase how much pizza you consume. Most of the new budget constraint lies to the right of the initial budget constraint, increasing the number of feasible bundles available to you. Because more is preferred to less, it is likely that your optimal consumption bundle will include more pizza than before.

3. If your indifference curves are very flat, you have a strong preference for other goods relative to pizza. For example, look at UA (Figure B). The slope of this indifference curve is relatively small (in absolute value). This means that the marginal rate of substitution of pizza for other goods is small. If your indifference curves look like this, you are not very willing to trade other goods for more pizza, and your optimal consumption bundle will likely lie on the section of the new budget constraint that coincides with the initial budget constraint. The promotion would cause no change in your consumption behavior; pizza is not a high priority for you, as indicated by your flat indifference curve.

On the other hand, if your indifference curves are steeper, like UB, your marginal rate of substitution is relatively large, indicating that you are willing to forgo a large amount of other goods to consume an additional pizza. This promotion will then, more than likely, cause you to purchase additional pizzas.

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