We now have in place all the pieces necessary to figure out what a consumer will buy. We know the consumer’s preferences for bundles of goods from the utility function and its associated indifference curves. We know the consumer’s budget constraint and which bundles are feasible given her income and the goods’ prices.

As we mentioned in the introduction to this chapter, the choice of how much to consume (like so many economic decisions) is a constrained optimization problem. There is something you want to maximize (utility, in this case), and there is something that limits how much of the good thing you can get (the budget constraint, in this case). And as we see in the next chapter, the constrained optimization problem forms the basis of the demand curve.

Before we try to solve this constrained optimization problem, think for a minute about what makes it tricky: We must compare things (e.g., income and prices) measured in dollars and things (e.g., consumer utility) measured in imaginary units like utils that don’t directly translate into dollars. How can you know whether you’re willing to pay $3 to get an extra unit of utility? Well, you can’t, really. What you can figure out, however, is whether spending an extra dollar on, say, golf balls gives you more or less utility than spending an extra dollar on something else, like AAA batteries. It turns out that figuring out this choice using indifference curves and budget constraints makes solving the consumer’s optimization problem straightforward.

Look at the axes we use to show indifference curves and budget constraints. They are the same: The quantity of some good is on the vertical axis, and the quantity of some other good is on the horizontal axis. This arrangement is extremely important, because it means we can display indifference curves and the budget constraint for two goods in the same graph, making the consumer’s problem easier to solve.

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Figure 4.18 presents an example that shows a combination of indifference curves and a budget constraint. Remember, the consumer wants to get as much utility as possible from consuming the two goods, subject to the limits imposed by her budget constraint. What bundle will she choose? The bundle at point A. That’s the highest indifference curve she can reach given her budget line.

Why is A the utility-maximizing consumption bundle? Compare point A to another feasible bundle, such as B. Point B is on the budget constraint, so the consumer could afford it. However, because B is on a lower indifference curve (U₁) than A is (U₂), bundle B provides less utility than A. Bundles C and D, too, are feasible but worse than A in terms of utility provided because they are on the same indifference curve as B. The consumer would love to consume bundle E because it’s on an indifference curve (U₃) that corresponds to a higher utility level than U₂. Unfortunately, she can’t afford E. It’s outside her budget constraint.

A look at the consumer’s optimal consumption bundle A in Figure 4.18 shows that it has a special feature: The indifference curve, U₂, touches the budget constraint once and only once, exactly at A. Mathematically speaking, U₂ and the budget constraint are tangent at point A. As long as the original assumptions we made about utility hold, no other indifference curve we can draw will have this feature. Any other indifference curve will not be tangent, and therefore will cross the budget constraint twice or not at all. If you can draw another indifference curve that is tangent, it will cross an indifference curve shown in Figure 4.18, violating the transitivity assumption. (Give it a try—it is a useful exercise.)

This single tangency is not a coincidence. It is, in fact, a requirement of utility maximization. To see why, suppose an indifference curve and the budget constraint never touched. Then no point on the indifference curve is feasible, and by definition, no bundle on that indifference curve can be the way for a consumer to maximize her utility given her income.

Now suppose the indifference curve crosses the budget constraint twice. This implies there must be a bundle that offers the consumer higher utility than any on this indifference curve and that the consumer can afford. For example, the shaded region between indifference curve U₁ and the budget constraint in Figure 4.18 reflects all of the bundles that are feasible and provide higher utility than bundles B, C, D, or any other point on U₁. That means no bundle on U₁ maximizes utility; there are other bundles that are both affordable and offer higher utility. This means that only at a point of tangency are there no other bundles that are both (1) feasible and (2) offer a higher utility level. This tangency is the utility-maximizing bundle for the consumer.

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Mathematically, the tangency of the indifference curve and budget constraint means that they have the same slope at the optimal consumption bundle. This has a very important economic interpretation that is key to understanding why the optimal bundle is where it is. In Section 4.2, we defined the negative of the slope of the indifference curve as the marginal rate of substitution, and we discussed how the MRS_XY reflects the ratio of the marginal utilities of the two goods. In Section 4.3, we saw that the slope of the budget constraint equals the negative of the ratio of the prices of the two goods. Therefore, the fact that the consumer’s utility-maximizing bundle is at a tangency between an indifference curve and the budget constraint gives us this key insight: When the consumer spends all her income and maximizes her utility, her optimal consumption bundle is the one at which the ratio of the goods’ marginal utilities exactly equals the ratio of their prices.

This economic idea behind utility maximization can be expressed mathematically. At the point of tangency,

Slope of indifference curve = Slope of budget constraint

–MRS_XY = –MU_X/MU_Y = –P_X/P_Y

MU_X/MU_Y = P_X/P_Y

Why are the marginal utility and price ratios equal when the consumer maximizes her utility level? If they were not equal, she could do better by shifting consumption from one good to the other. To see why, let’s say Meredith is maximizing her utility over bottles of Gatorade and protein bars. Suppose bottles of Gatorade are twice as expensive as protein bars, but she is considering a bundle in which her marginal utilities from the two goods are not 2 to 1 as the price ratio is. Say she gets the same amount of utility at the margin from another bottle of Gatorade as from another protein bar, so that the ratio of the goods’ marginal utilities is 1. Given the relative prices, she could give up 1 bottle of Gatorade and buy 2 more protein bars and doing so would let her reach a higher utility level. Why? Because those 2 extra protein bars are worth twice as much in utility terms as the lost bottle of Gatorade.

Now suppose that a bottle of Gatorade offers Meredith 4 times the utility at the margin as a protein bar. In this case, the ratio of Meredith’s marginal utilities for Gatorade and protein bars (4 to 1) is higher than the price ratio (2 to 1), so she could buy 2 fewer protein bars in exchange for 1 more bottle of Gatorade. Because the Gatorade delivers twice the utility lost from the 2 protein bars, Meredith will be better off buying fewer protein bars and more Gatorade.

It is often helpful to rewrite this optimization condition in terms of the consumer’s marginal utility per dollar spent:

Here, the utility-maximization problem can be restated as finding the consumption bundle that gives the consumer the most bang for her buck. This occurs when the marginal utility per dollar spent (MU/P) is equal across all goods. If this is not the case, the consumer can improve her utility by adjusting her consumption of good X and good Y.10

10 Interestingly, the same consumer utility-maximization problem can be solved as a cost-minimization problem instead and it will give exactly the same answer. In this case, a consumer tries to minimize the cost of reaching a target level of utility. Economists call this the “dual” to the utility-maximization problem. The appendix to this chapter gives the mathematics behind both approaches and shows why they give the same answer.

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The marginal-utility-ratio-equals-price-ratio result has another implication for the economy as a whole that can initially be surprising. Even if two consumers have very different preferences between two goods, they will have the same ratio of marginal utilities for the two goods, because utility maximization implies that MRS equals the ratio of the prices.11

11 Technically, this is true only of consumers who are consuming a positive amount of both goods, or who are at “interior” solutions in economists’ lingo. We’ll discuss this issue in the next section.

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This might seem odd. First of all, we have said that you can’t compare utility levels across people. Second, even if you could—if, say, Jack consumes 9 packs of gum and 1 iTunes download, while Meg consumes 9 downloads and only 1 pack of gum—it would seem that Jack likes gum a lot and would therefore be willing to pay more for another pack of gum (and a lot less for iTunes) than Meg.

This assertion would be true if both Jack and Meg had to consume the same bundle, but they don’t have to. They can choose how much of each good they want to consume. Because Jack likes gum a lot, he will consume so much of it that he drives down his marginal utility until, by the time he and Meg both reach their utility-maximizing consumption bundles, they will both end up placing the same relative marginal utilities on the two goods. This outcome occurs because the relative value they place on any two goods (on the margin) comes from the relative prices. Because Meg and Jack face the same prices, the goods they consume will end up with the same relative marginal utilities.

Figure 4.19 shows this outcome. To keep the picture simple, we assume Jack and Meg have the same incomes. Because they also face the same relative prices, their budget constraints are the same. Jack really likes gum relative to iTunes downloads, so his indifference curves tend to be flat: He has to be given a lot of iTunes to make up for any loss of gum. Meg has the opposite tastes. She has to be given a lot of gum to make up for giving up an iTunes download, so her indifference curves are steep. Nevertheless, both Jack and Meg’s utility-maximizing bundles are on the same budget line, and they will pick consumption points where their marginal rates of substitution at those bundles are the same (and match the relative prices). We’ve drawn the indifference curves for Jack (U_J) and Meg (U_M) so that they are tangent to the budget line and therefore contain the utility-maximizing bundle.

Even though Jack and Meg have the same MRS_XY, the amounts of each good they consume are different. Jack, the gum lover, maximizes his utility by choosing a bundle (J) with a lot of gum and not many iTunes downloads. Meg’s optimal consumption bundle (M), on the other hand, has a lot of iTunes downloads and little gum. Again, both consumers end up with the same MRS in their utility-maximizing bundles because each consumes a large amount of the good for which he or she has the stronger preference. By behaving in this way, Jack and Meg drive down the relative marginal utility of the good they prefer until their marginal utility ratio equals the price ratio.

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Note that these two indifference curves cross. Although earlier we learned that indifference curves can never cross, the “no crossing” rule applies only to the indifference curves of one individual. Figure 4.19 shows indifference curves for two different people with different preferences, so the transitivity rule we talked about before doesn’t hold. If you like gum more than iTunes, and your friend likes iTunes more than, say, coffee, that doesn’t imply you have to like gum more than coffee. So the same consumption bundle (say, 3 packs of gum and 5 iTunes downloads) can offer different utility levels to different people.

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.

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.

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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.

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.

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