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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