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
MUXPX
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 (MUX/PX=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 (MUX/PX=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:
MUXPX>MUYPY |
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 Table 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:
MUXPX=MUYPY |
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 Table 7.3:
MUAPA=MUBPB=MUCPC=...=MUnPn |
Table 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 Table 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 Table 7.1.
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