Optimal Consumption

Let’s consider Figure 10-4. As in Figure 10-3, we can measure both the quantity of clams and the quantity of potatoes on the horizontal axis due to the budget constraint. Along the horizontal axis of Figure 10-4—also as in Figure 10-3—the quantity of clams increases as you move from left to right, and the quantity of potatoes increases as you move from right to left. The curve labeled MUC/PC in Figure 10-4 shows Sammy’s marginal utility per dollar spent on clams as derived in Table 10-3. Likewise, the curve labeled MUP/PP shows his marginal utility per dollar spent on potatoes. Notice that the two curves, MUC/PC and MUP/PP, cross at the optimal consumption bundle, point C, consisting of 2 pounds of clams and 6 pounds of potatoes.

Marginal Utility per Dollar Sammy’s optimal consumption bundle is at point C, where his marginal utility per dollar spent on clams, MUC/PC, is equal to his marginal utility per dollar spent on potatoes, MUP/PP. This illustrates the utility-maximizing principle of marginal analysis: at the optimal consumption bundle, the marginal utility per dollar spent on each good and service is the same. At any other consumption bundle on Sammy’s budget line, such as bundle B in Figure 10-3, represented here by points BC and BP, consumption is not optimal: Sammy can increase his utility at no additional cost by reallocating his spending.

Moreover, Figure 10-4 illustrates an important feature of Sammy’s optimal consumption bundle: when Sammy consumes 2 pounds of clams and 6 pounds of potatoes, his marginal utility per dollar spent is the same, 2, for both goods. That is, at the optimal consumption bundle MUC/PC = MUP/PP = 2.

PITFALLS: THE RIGHT MARGINAL COMPARISON

PITFALLS

THE RIGHT MARGINAL COMPARISON
Marginal analysis solves “how much” decisions by weighing costs and benefits at the margin: the benefit of doing a little bit more versus the cost of doing a little bit more. However, as we noted in Chapter 9, the form of the marginal analysis can differ, depending upon whether you are making a production decision that maximizes profits or a consumption decision that maximizes utility. Let’s review that difference again to make sure that it’s clearly understood.
In Chapter 9, Alex’s decision was a production decision because the problem he faced was maximizing the profit from years of schooling. The optimal quantity of years that maximized his profit was found using marginal analysis: at the optimal quantity, the marginal benefit of another year of schooling was equal to its marginal cost. Alex did not face a budget constraint because he could always borrow to finance another year of school.
But if you were to extend the way we solved Alex’s production problem to Sammy’s consumption problem without any change in form, you might be tempted to say that Sammy’s optimal consumption bundle is the one at which the marginal utility of clams is equal to the marginal utility of potatoes, or that the marginal utility of clams was equal to the price of clams. But both those statements would be wrong because they don’t properly account for the budget constraint and the fact that consuming more of one good requires consuming less of another.
In a consumption decision, your objective is to maximize the utility that your limited budget can deliver. And the right way to find the optimal consumption bundle is to set the marginal utility per dollar equal for each good in the consumption bundle. When this condition is satisfied, the “bang per buck” is the same across all the goods and services you consume. Only then is there no way to rearrange your consumption and get more utility from your budget.

This isn’t an accident. Consider another one of Sammy’s possible consumption bundles—say, B in Figure 10-3, at which he consumes 1 pound of clams and 8 pounds of potatoes. The marginal utility per dollar spent on each good is shown by points BC and BP in Figure 10-4. At that consumption bundle, Sammy’s marginal utility per dollar spent on clams would be approximately 3, but his marginal utility per dollar spent on potatoes would be only approximately 1. This shows that he has made a mistake: he is consuming too many potatoes and not enough clams.

How do we know this? If Sammy’s marginal utility per dollar spent on clams is higher than his marginal utility per dollar spent on potatoes, he has a simple way to make himself better off while staying within his budget: spend $1 less on potatoes and $1 more on clams. We can illustrate this with points BC and BP in Figure 10-4. By spending an additional dollar on clams, he gains the amount of utility given by BC, about 3 utils. By spending $1 less on potatoes, he loses the amount of utility given by BP, only about 1 util.

Because his marginal utility per dollar spent is higher for clams than for potatoes, reallocating his spending toward clams and away from potatoes would increase his total utility. But if his marginal utility per dollar spent on potatoes is higher, he can increase his utility by spending less on clams and more on potatoes. So if Sammy has in fact chosen his optimal consumption bundle, his marginal utility per dollar spent on clams and potatoes must be equal.

According to the utility-maximizing principle of marginal analysis, the marginal utility per dollar spent must be the same for all goods and services in the optimal consumption bundle.

This is a general principle, which we call the utility-maximizing principle of marginal analysis: when a consumer maximizes utility in the face of a budget constraint, the marginal utility per dollar spent on each good or service in the consumption bundle is the same. That is, for any two goods C and P the optimal consumption rule says that at the optimal consumption bundle:

It’s easiest to understand this rule using examples in which the consumption bundle contains only two goods, but it applies no matter how many goods or services a consumer buys: in the optimal consumption bundle, the marginal utilities per dollar spent for each and every good or service in that bundle are equal.

ECONOMICS in Action: Buying Your Way Out of Temptation

Buying Your Way Out of Temptation

For many consumers, paying extra for portion control is worth it.
Jeff Greeenberg/Alamy

It might seem odd to pay more to get less. But snack food I companies have discovered that consumers are indeed willing to pay more for smaller portions, and exploiting this trend is a recipe for success. A company executive explained why small packages are popular—they help consumers eat less without having to count calories themselves. “The irony,” said David Adelman, a food industry analyst, “is if you take Wheat Thins or Goldfish, buy a large-size box, count out the items and put them in a Ziploc bag, you’d have essentially the same product.” He estimates that snack packs are about 20% more profitable for snack makers than larger packages.

It’s clear that in this case consumers are making a calculation: the extra utility gained from not having to worry about whether they’ve eaten too much is worth the extra cost. As one shopper said, “They’re pretty expensive, but they’re worth it. It’s individually packaged for the amount I need, so I don’t go overboard.” So it’s clear that consumers aren’t being irrational here. Rather, they’re being entirely rational: in addition to their snack, they’re buying a little hand-to-mouth restraint.

Quick Review

  • According to the utility-maximizing principle of marginal analysis, the marginal utility per dollar—the marginal utility of a good divided by its price—is the same for all goods in the optimal consumption bundle.

  • Whenever marginal utility per dollar is higher for one good than for another good, the consumer should spend $1 more on the good with the higher marginal utility per dollar and $1 less on the other. By doing this the consumer will move closer to his or her optimal consumption bundle.

10-3

  1. Question 10.5

    In Table 10-3 you can see that marginal utility per dollar spent on clams and marginal utility per dollar spent on potatoes are equal when Sammy increases his consumption of clams from 3 pounds to 4 pounds and his consumption of potatoes from 9 pounds to 10 pounds. Explain why this is not Sammy’s optimal consumption bundle. Illustrate your answer using the budget line in Figure 10-3.

  2. Question 10.6

    Explain what is faulty about the following statement, using data from Table 10-3: “In order to maximize utility, Sammy should consume the bundle that gives him the maximum marginal utility per dollar for each good.

Solutions appear at back of book.