Equal Marginal Utility per Dollar

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Maximizing Utility

In some instances, relative satisfactions from goods can be approximated by looking at utility in terms of money. We cannot scientifically ascertain whether one person gains more than someone else from an extra dollar of income or extra syrup on a waffle. We may be fairly sure, however, that the personal marginal utilities of each person from particular goods are roughly proportional to the goods’ prices, because people adjust their spending patterns whenever relative prices and marginal utilities are not in balance.

Suppose that you allocate $12 from your weekly budget for ice cream, your favorite snack. Table 1 lists your total and marginal utilities from macadamia nut crunch and chocolate ice cream cones. How many of each will you buy at $1 apiece to maximize your satisfaction? Eating five chocolate and seven crunch cones maximizes your total satisfaction at 46 utils per week. No other $12 combination yields more utility.

Let’s see how this purchasing pattern develops. Table 1 shows that your first macadamia nut cone generates 8 utils while your first chocolate cone yields only 5 utils. Thus, your first purchase will be a macadamia nut cone. What flavor would you buy second? Each cone is $1, and a second macadamia nut yields extra satisfaction of 7 utils. Your third choice will also be macadamia nut, which increases your satisfaction by 6 utils. Now, your fourth macadamia nut cone only yields 4 utils compared to 5 utils for your first chocolate cone. Thus, your fourth cone will be chocolate. You will ultimately spend the $12 on seven macadamia nut and five chocolate cones per week. Notice that the decision for each purchase hinges on the flavor with the greatest marginal utility per dollar spent.

Total (TU) and Marginal (MU) Utilities for Ice Cream Cones When Price = $1 per Cone
Macadamia Nut Crunch			Chocolate
Quantity	TU	MU	Quantity	TU	mu
1	8	8	1	5	5
2	15	7	2	9	4
3	21	6	3	12	3
4	25	4	4	14	2
5	28	3	5	15	1
6	30	2	6	15	0
7	31	1
8	31	0

Balancing Marginal Utilities

In the example shown, your purchasing pattern suggested that you would look at the next cone, whether chocolate or macadamia nut, and determine which flavor provided you with the largest increase in utility.

The principle of equal marginal utilities per dollar: A consumer maximizes utility when the last dollar spent on any good generates the same satisfaction as the last dollar spent on every other good.

Your purchasing pattern becomes stable only when the last dollar spent on ice cream yields the same satisfaction as the last dollar spent on lemonade, clothes, books, or housing.

Satisfaction from the last spending on a good is calculated by dividing its marginal utility by its price. For example, if the last chocolate cone was $1 and its marginal utility was 1, then MUch/Pch = 1/1 = 1, and the marginal utility per dollar of the last cone was 1. Individuals are in equilibrium when

where a, b, ..., z are the various goods purchased. In our ice cream cone example, the marginal utilities per dollar equaled one for both macadamia nut crunch and chocolate cones. By equating marginal utilities per dollar, utility is maximized. No other allocation of resources will result in higher satisfaction. A little introspection should confirm that your own spending pattern conforms to this principle of equal marginal utilities per dollar.

A corollary of the principle of equal marginal utilities per dollar is that consumer equilibrium requires marginal benefits from every good to be proportional to their relative market prices. Market prices and the subjective demand prices discussed in Chapter 1 are different ways of viewing opportunity costs. Demand prices can be thought of as the ratios of the marginal utilities of various goods. Only if market prices do not exceed demand prices will you buy. In equilibrium, these subjective price ratios must equal the relative market prices for all the goods you choose to purchase. That is, MUa/MUb = Pa/Pb for any two goods we choose to label a and b, respectively.1 Focus 1 addresses the issue of whether economic assumptions about how people process information are reasonable.

Price Adjustments and Marginal Utility

Let’s use this format to describe how quantities demanded adjust as relative prices change. First, consider what would happen to your equilibrium purchases of ice cream cones (from the first table) if a worldwide macadamia nut crop failure boosted the price of macadamia nut crunch to $2 per cone. Your utility schedules along with the new prices are shown in Table 2. You now buy only four macadamia nut crunch cones, and your consumption of chocolate cones drops to four per week. Marginal utilities per dollar now equal 2. Adjusting your spending pattern to various possible prices for macadamia nut ice cream traces out the demand curve shown in Figure 2.

To summarize, the higher-priced goods you buy uniformly generate more marginal utility than your lower-priced purchases: the more you pay for a good, the more it is worth to you at the margin. You are in equilibrium when the marginal utilities per dollar are equal for all goods. Ultimately, the last dollar you spend on any good yields the same satisfaction as the last dollar spent on any other good. Finally, if you choose not to spend all of your money, the satisfaction you gain from your saving or holding each dollar must equal the satisfaction you would gain from spending it on some other good.

Table 2 Total (TU) and Marginal (MU) Utilities for Ice Cream Cones When the Price of Macadamia Nut Crunch Rises to $2 per Cone
Macadamia Nut Crunch				Chocolate
Quantity	TU	MU	MU/P	Quantity	TU	mu	MU/P
1	8	8	4.0	1	5	5	5.0
2	15	7	3.5	2	9	4	4.0
3	21	6	3.0	3	12	3	3.0
4	25	4	2.0	4	14	2	2.0
5	28	3	1.5	5	15	1	1.0
6	30	2	1.0	6	15	0	0
7	31	1	0.5
8	31	0	0

The Law of Diminishing Marginal Utility

The declining marginal utility from lemonade as you drink more and more during a given time interval occurs with virtually all goods and all people. Observing that similar reactions were common, classical economists generalized this behavior into

The law of diminishing marginal utility: The marginal utility from consuming equal units of a good eventually declines as the amount consumed increases.

The word “eventually” is important. Horror movie fanciers might enjoy their first movie this week immensely and their second film even more. But they almost certainly will not enjoy their sixteenth movie of the week as much as their third. Benjamin Franklin’s observation in Poor Richard’s Almanac, “Fish and visitors stink in three days,” says far more about the diminishing marginal utility of visitors than about the deterioration of fish.

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Author: Ralph Byrns

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