## Rules for Maximizing Utility

### 5. 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.

$\dfrac{MU_1}{P_1}=\dfrac{MU_2}{P_2}$

​​ 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

$\dfrac{22}{14}=\dfrac{11}{7}$

$\displaystyle 1.6=1.6$

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

#### ANOTHER RULE FOR MAXIMIZING UTILITY

The rule, $\dfrac{MU_1}{P_1}=\dfrac{MU_2}{P_2}$, 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 2:0 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 O 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:

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

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).