## Rules for Maximizing Utility

### Learning Objectives

• Explain why maximizing utility requires that the last unit of each item purchased must have the same marginal utility per dollar
• Calculate the utility-maximizing choice

The problem of finding consumer equilibrium, that is, the combination of goods and services that will maximize an individual’s total utility, comes down to comparing the trade-offs between one affordable combination (shown by a point on the budget line in Figure 1, below) with all the other affordable combinations.

Most people approach their utility-maximizing combination of choices in a step-by-step way. This step-by-step approach is based on looking at the tradeoffs, measured in terms of marginal utility, of consuming less of one good and more of another. You can think of this step-by-step approach as the “biggest bang for the buck” principle. For example, say that José starts off thinking about spending all his money on T-shirts and choosing point P, which corresponds to four T-shirts and no movies, as illustrated in Figure 1. Figure 1. A Choice between Consumption Goods. José has income of $56. Movies cost$7 and T-shirts cost $14. The points on the budget constraint line show the combinations of movies and T-shirts that are affordable. José chooses this starting point randomly; he has to start somewhere. Then he considers giving up the last T-shirt, the one that provides him the least marginal utility, and using the money he saves to buy two movies instead. Table 1 tracks the step-by-step series of decisions José needs to make (Key: T-shirts cost$14, movies cost $7, and Jose’s income is$56).

Table 1. A Step-by-Step Approach to Maximizing Utility
Try Which Has Total Utility Marginal Gain and Loss of Utility, Compared with Previous Choice Conclusion
Choice 1: P 4 T-shirts and 0 movies 81 from 4 T-shirts + 0 from 0 movies = 81      –      –
Choice 2: Q 3 T-shirts and 2 movies 63 from 3 T-shirts + 31 from 0 movies = 94 Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13 Q is preferred over P
Choice 3: R 2 T-shirts and 4 movies 43 from 2 T-shirts + 58 from 4 movies = 101 Loss of 20 from 1 less T-shirt, but gain of 27 from two more movies for a net utility gain of 7 R is preferred over Q
Choice 4: S 1 T-shirt and 6 movies 22 from 1 T-shirt + 81 from 6 movies = 103 Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a net utility gain of 2 S is preferred over R
Choice 5: T 0 T-shirts and 8 movies 0 from 0 T-shirts + 100 from 8 movies = 100 Loss of 22 from 1 less T-shirt, but gain of 19 from two more movies, for a net utility loss of 3 S is preferred over T

### DECISION MAKING BY COMPARING MARGINAL UTILITY

José could use the following thought process (if he thought in utils) to make his decision regarding how many T-shirts and movies to purchase:

Step 1. From Table 1, José can see that the marginal utility of the fourth T-shirt is 18. If José gives up the fourth T-shirt, then he loses 18 utils.

Step 2. Giving up the fourth T-shirt, however, frees up $14 (the price of a T-shirt), allowing José to buy the first two movies (at$7 each).

Step 3. José knows that the marginal utility of the first movie is 16 and the marginal utility of the second movie is 15. Thus, if José moves from point P to point Q, he gives up 18 utils (from the T-shirt), but gains 31 utils (from the movies).

Step 4. Gaining 31 utils and losing 18 utils is a net gain of 13. This is just another way of saying that the total utility at Q (94 according to the last column in Table 1) is 13 more than the total utility at P (81).

Step 5. So, for José, it makes sense to give up the fourth T-shirt in order to buy two movies.

José clearly prefers point Q to point P. Now repeat this step-by-step process of decision making with marginal utilities. José thinks about giving up the third T-shirt and surrendering a marginal utility of 20, in exchange for purchasing two more movies that promise a combined marginal utility of 27. José prefers point R to point Q. What if José thinks about going beyond R to point S? Giving up the second T-shirt means a marginal utility loss of 21, and the marginal utility gain from the fifth and sixth movies would combine to make a marginal utility gain of 23, so José prefers point S to R.

However, if José seeks to go beyond point S to point T, he finds that the loss of marginal utility from giving up the first T-shirt is 22, while the marginal utility gain from the last two movies is only a total of 19. If José were to choose point T, his utility would fall to 100. Through these stages of thinking about marginal tradeoffs, José again concludes that S, with one T-shirt and six movies, is the choice that will provide him with the highest level of total utility. This step-by-step approach will reach the same conclusion regardless of José’s starting point.

This approach to finding consumer equilibrium is somewhat tedious. Next, we’ll turn to a quicker and more intuitive approach.

## A Rule for Maximizing Utility

This process of decision making described previously suggests a rule to follow when maximizing utility. Since the price of T-shirts is not the same as the price of movies, it’s not enough to just compare the marginal utility of T-shirts with the marginal utility of movies. Instead, we need to control for the prices of each product. We can do this by computing and comparing marginal utility per dollar of expenditure for each product. Marginal utility per dollar is the amount of additional utility José receives given the price of the product.

$\begin{array}{rcl}\text{marginal utility per dollar}& =& \frac{\text{marginal utility}}{\text{price}}\end{array}$

For José’s T-shirts and movies, the marginal utility per dollar is shown in Table 2.

Table 2. Marginal Utility per Dollar
Quantity of T-Shirts Total Utility Marginal Utility Marginal Utility per Dollar Quantity of Movies Total Utility Marginal Utility Marginal Utility per Dollar
1 22 22 22/$14=1.6 1 16 16 16/$7=2.3
2 43 21 21/$14=1.5 2 31 15 15/$7=2.14
3 63 20 20/$14=1.4 3 45 14 14/$7=2
4 81 18 18/$14=1.3 4 58 13 13/$7=1.9
5 97 16 16/$14=1.1 5 70 12 12/$7=1.7
6 111 14 14/$14=1 6 81 11 11/$7=1.6
7 123 12 12/$14=1.2 7 91 10 10/$7=1.4

### A Rule for maximizing Utility

If a consumer wants to maximize total utility, for every dollar that they spend, they should spend it on the item which yields the greatest marginal utility per dollar of expenditure.

## Applying the Rule

José’s first purchase will be a movie. Why? José’s choices are to purchase either a T-shirt or a movie. Table 1 shows that the marginal utility per dollar spent on the first T-shirt is 1.6 compared with 2.5 for the first movie. Because the first movie gives José more marginal utility per dollar than the first T-shirt, and because the movie is within his budget, he will purchase a movie first.

José will continue to purchase the good which gives him the highest marginal utility per dollar until he exhausts the budget. José will keep purchasing movies because they give him a greater “bang for the buck” until the sixth movie is equivalent to a T-shirt purchase. José can afford to purchase that T-shirt. So José will choose to purchase six movies and one T-shirt. That combination, six movies and one T-shirt, is his consumer equilibrium

## Another Rule for Maximizing Utility

Since the price of T-shirts is twice as high as the price of movies, to maximize utility the last T-shirt chosen needs to provide exactly twice the marginal utility (MU) of the last movie. If the last T-shirt provides less than twice the marginal utility of the last movie, then the T-shirt is providing less “bang for the buck” (i.e., marginal utility per dollar spent) than if the same money were spent on movies. If this is so, José should trade the T-shirt for more movies to increase his total utility. Marginal utility per dollar measures the additional utility that José will enjoy given what he has to pay for the good.

If the last T-shirt provides more than twice the marginal utility of the last movie, then the T-shirt is providing more “bang for the buck” or marginal utility per dollar, than if the money were spent on movies. As a result, José should buy more T-shirts. Notice that at José’s optimal choice the marginal utility from the first T-shirt, of 22 is exactly twice the marginal utility of the sixth movie, which is 11. At this choice, the marginal utility per dollar is the same for both goods. This is a tell-tale signal that José has found the point with highest total utility.

This argument can be written as another rule: the utility-maximizing choice between consumption goods occurs where the marginal utility per dollar is the same for both goods, and the consumer has exhausted his or her budget.

$\displaystyle\frac{MU_1}{P_1}=\frac{MU_2}{P_2}$

A sensible economizer will pay twice as much for something only if, in the marginal comparison, the item confers twice as much utility. Notice that the formula for the table above is

$\displaystyle\frac{22}{14}=\frac{11}{7}$

$1.6=1.6$

The following feature provides step-by-step guidance for this concept of utility-maximizing choices.

### Another Rule for Maximizing Utility

The rule, $\displaystyle\frac{MU_1}{P_1}=\frac{MU_2}{P_2}$, means that the last dollar spent on each good provides exactly the same marginal utility. So:

Step 1. If we traded a dollar more of movies for a dollar more of T-shirts, the marginal utility gained from T-shirts would exactly offset the marginal utility lost from fewer movies. In other words, the net gain would be zero.

Step 2. Products, however, usually cost more than a dollar, so we cannot trade a dollar’s worth of movies. The best we can do is trade two movies for another T-shirt, since in this example T-shirts cost twice what a movie does.

Step 3. If we trade two movies for one T-shirt, we would end up at point R (two T-shirts and four movies).

Step 4. Choice 4 in Table 3 shows that if we move to point S, we would lose 21 utils from one less T-shirt, but gain 23 utils from two more movies, so we would end up with more total utility at point S.

Table 3. A Step-by-Step Approach to Maximizing Utility
Try Which Has Total Utility Marginal Gain and Loss of Utility, Compared with Previous Choice Conclusion
Choice 1: P 4 T-shirts and 0 movies 81 from 4 T-shirts + 0 from 0 movies = 81      –      –
Choice 2: Q 3 T-shirts and 2 movies 63 from 3 T-shirts + 31 from 0 movies = 94 Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13 Q is preferred over P
Choice 3: R 2 T-shirts and 4 movies 43 from 2 T-shirts + 58 from 4 movies = 101 Loss of 20 from 1 less T-shirt, but gain of 27 from two more movies for a net utility gain of 7 R is preferred over Q
Choice 4: S 1 T-shirt and 6 movies 22 from 1 T-shirt + 81 from 6 movies = 103 Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a net utility gain of 2 S is preferred over R
Choice 5: T 0 T-shirts and 8 movies 0 from 0 T-shirts + 100 from 8 movies = 100 Loss of 22 from 1 less T-shirt, but gain of 19 from two more movies, for a net utility loss of 3 S is preferred over T

In short, the rule shows us the utility-maximizing choice.

There is another, equivalent way to think about this. The rule can also be expressed as the ratio of the prices of the two goods should be equal to the ratio of the marginal utilities. When the price of good 1 is divided by the price of good 2, at the utility-maximizing point this will equal the marginal utility of good 1 divided by the marginal utility of good 2. This rule can be written in algebraic form:

$\displaystyle\frac{P_1}{P_2}=\frac{MU_1}{MU_2}$

Along the budget constraint, the total price of the two goods remains the same, so the ratio of the prices does not change. However, the marginal utility of the two goods changes with the quantities consumed. At the optimal choice of one T-shirt and six movies, point S, the ratio of marginal utility to price for T-shirts (22:14) matches the ratio of marginal utility to price for movies (of 11:7).

### Watch It

This video applies these same concepts to a graph. It demonstrates how there will come a point when the marginal utility per dollar of one good goes down enough so that it makes more sense to buy another good.

## Measuring Utility with Numbers

This discussion of utility started off with an assumption that it is possible to place numerical values on utility, an assumption that may seem questionable. You can buy a thermometer for measuring temperature at the hardware store, but what store sells an “utilimometer” for measuring utility? However, while measuring utility with numbers is a convenient assumption to clarify the explanation, the key assumption is not that utility can be measured by an outside party, but only that individuals can decide which of two alternatives they prefer.

To understand this point, think back to the step-by-step process of finding the choice with highest total utility by comparing the marginal utility that is gained and lost from different choices along the budget constraint. As José compares each choice along his budget constraint to the previous choice, what matters is not the specific numbers that he places on his utility—or whether he uses any numbers at all—but only that he personally can identify which choices he prefers.

In this way, the step-by-step process of choosing the highest level of utility resembles rather closely how many people make consumption decisions. We think about what will make us the happiest; we think about what things cost; we think about buying a little more of one item and giving up a little of something else; we choose what provides us with the greatest level of satisfaction. The vocabulary of comparing the points along a budget constraint and total and marginal utility is just a set of tools for discussing this everyday process in a clear and specific manner. It is welcome news that specific utility numbers are not central to the argument, since a good utilimometer is hard to find. Do not worry—while we cannot measure utils, by the end of the next module, we will have transformed our analysis into something we can measure—demand.

### Try It

These questions allow you to get as much practice as you need, as you can click the link at the top of the first question (“Try another version of these questions”) to get a new set of questions. Practice until you feel comfortable doing the questions.

### Try It

These questions allow you to get as much practice as you need, as you can click the link at the top of the first question (“Try another version of these questions”) to get a new set of questions. Practice until you feel comfortable doing the questions.

### Try It

Click through the buttons in this simulation to think about applications of utility.

First, you’ll imagine that you are desperately searching for food and water on a hot day. Given the heat and your time constraints, you’ll need to decide what to obtain first based on its utility. Then, you can see these combinations on a graph and consider what the utility of each choice would be.

Be sure to choose the option to “learn more about marginal utility per dollar spent” so that you can see how marginal-decision making comes into play when choosing to forage for either apples or berries.

### Glossary

consumer equilibrium:
the combination of goods and services that will maximize an individual’s total utility
marginal utility per dollar:
the additional satisfaction gained from purchasing a good given the price of the product; MU/Price