## The Analysis of Consumer Choice

### 2. The Concept of Utility

#### 2.3. Maximizing Utility

Economists assume that consumers behave in a manner consistent with the maximization of utility. To see how consumers do that, we will put the marginal decision rule to work. First, however, we must reckon with the fact that the ability of consumers to purchase goods and services is limited by their budgets.

##### The Budget Constraint

The total utility curve in Figure 7.1 "Total Utility and Marginal Utility Curves" shows that Mr. Higgins achieves the maximum total utility possible from movies when he sees six of them each month. It is likely that his total utility curves for other goods and services will have much the same shape, reaching a maximum at some level of consumption. We assume that the goal of each consumer is to maximize total utility. Does that mean a person will consume each good at a level that yields the maximum utility possible?

The answer, in general, is no. Our consumption choices are constrained by the income available to us and by the prices we must pay. Suppose, for example, that Mr. Higgins can spend just $25 per month for entertainment and that the price of going to see a movie is $5. To achieve the maximum total utility from movies, Mr. Higgins would have to exceed his entertainment budget. Since we assume that he cannot do that, Mr. Higgins must arrange his consumption so that his total expenditures do not exceed his budget constraint: a restriction that total spending cannot exceed the budget available.

Suppose that in addition to movies, Mr. Higgins enjoys concerts, and the average price of a concert ticket is $10. He must select the number of movies he sees and concerts he attends so that his monthly spending on the two goods does not exceed his budget.

Individuals may, of course, choose to save or to borrow. When we allow this possibility, we consider the budget constraint not just for a single period of time but for several periods. For example, economists often examine budget constraints over a consumer's lifetime. A consumer may in some years save for future consumption and in other years borrow on future income for present consumption. Whatever the time period, a consumer's spending will be constrained by his or her budget.

To simplify our analysis, we shall assume that a consumer's spending in any one period is based on the budget available in that period. In this analysis consumers neither save nor borrow. We could extend the analysis to cover several periods and generate the same basic results that we shall establish using a single period. We will also carry out our analysis by looking at the consumer's choices about buying only two goods. Again, the analysis could be extended to cover more goods and the basic results would still hold.

##### Applying the Marginal Decision Rule

Because consumers can be expected to spend the budget they have, utility maximization is a matter of arranging that spending to achieve the highest total utility possible. If a consumer decides to spend more on one good, he or she must spend less on another in order to satisfy the budget constraint.

The marginal decision rule states that an activity should be expanded if its marginal benefit exceeds its marginal cost. The marginal benefit of this activity is the utility gained by spending an additional $1 on the good. The marginal cost is the utility lost by spending $1 less on another good.

How much utility is gained by spending another $1 on a good? It is the marginal utility of the good divided by its price. The utility gained by spending an additional dollar on good X, for example, is

This additional utility is the marginal benefit of spending another $1 on the good.

Suppose that the marginal utility of good X is 4 and that its price is $2. Then an extra $1 spent on X buys 2 additional units of utility ( MU_{X} / P_{X} =4/2=2 ). If the marginal utility of good X is 1 and its price is $2, then an extra $1 spent on X buys 0.5 additional units of utility ( MU_{X} / P_{X} =1/2=0.5 ).

The loss in utility from spending $1 less on another good or service is calculated the same way: as the marginal utility divided by the price. The marginal cost to the consumer of spending $1 less on a good is the loss of the additional utility that could have been gained from spending that $1 on the good.

Suppose a consumer derives more utility by spending an additional $1 on good X rather than on good Y:

Equation 7.1

The marginal benefit of shifting $1 from good Y to the consumption of good X exceeds the marginal cost. In terms of utility, the gain from spending an additional $1 on good X exceeds the loss in utility from spending $1 less on good Y. The consumer can increase utility by shifting spending from Y to X.

As the consumer buys more of good X and less of good Y, however, the marginal utilities of the two goods will change. The law of diminishing marginal utility tells us that the marginal utility of good X will fall as the consumer consumes more of it; the marginal utility of good Y will rise as the consumer consumes less of it. The result is that the value of the left-hand side of Equation 7.1 will fall and the value of the right-hand side will rise as the consumer shifts spending from Y to X. When the two sides are equal, total utility will be maximized. In terms of the marginal decision rule, the consumer will have achieved a solution at which the marginal benefit of the activity (spending more on good X) is equal to the marginal cost:

Equation 7.2

We can extend this result to all goods and services a consumer uses. Utility maximization requires that the ratio of marginal utility to price be equal for all of them, as suggested in Equation 7.3:

Equation 7.3

Equation 7.3 states the utility-maximizing condition: Utility is maximized when total outlays equal the budget available and when the ratios of marginal utilities to prices are equal for all goods and services.

Consider, for example, the shopper introduced in the opening of this chapter. In shifting from cookies to ice cream, the shopper must have felt that the marginal utility of spending an additional dollar on ice cream exceeded the marginal utility of spending an additional dollar on cookies. In terms of Equation 7.1, if good X is ice cream and good Y is cookies, the shopper will have lowered the value of the left-hand side of the equation and moved toward the utility-maximizing condition, as expressed by Equation 7.1.

##### The Problem of Divisibility

If we are to apply the marginal decision rule to utility maximization, goods must be divisible; that is, it must be possible to buy them in any amount. Otherwise we cannot meaningfully speak of spending $1 more or $1 less on them. Strictly speaking, however, few goods are completely divisible.

Even a small purchase, such as an ice cream bar, fails the strict test of being divisible; grocers generally frown on requests to purchase one-half of a $2 ice cream bar if the consumer wants to spend an additional dollar on ice cream. Can a consumer buy a little more movie admission, to say nothing of a little more car?

In the case of a car, we can think of the quantity as depending on characteristics of the car itself. A car with a compact disc player could be regarded as containing "more car" than one that has only a cassette player. Stretching the concept of quantity in this manner does not entirely solve the problem. It is still difficult to imagine that one could purchase "more car" by spending $1 more.

Remember, though, that we are dealing with a model. In the real world, consumers may not be able to satisfy Equation 7.3 precisely. The model predicts, however, that they will come as close to doing so as possible.