## 7.2 Utility Maximization and Demand

### Learning Objectives

1. Derive an individual demand curve from utility-maximizing adjustments to changes in price.
2. Derive the market demand curve from the demand curves of individuals.
3. Explain the substitution and income effects of a price change.
4. Explain the concepts of normal and inferior goods in terms of the income effect.

Choices that maximize utility—that is, choices that follow the marginal decision rule—generally produce downward-sloping demand curves. This section shows how an individual’s utility-maximizing choices can lead to a demand curve.

## Deriving an Individual’s Demand Curve

Suppose, for simplicity, that Mary Andrews consumes only apples, denoted by the letter A, and oranges, denoted by the letter O. Apples cost $2 per pound and oranges cost$1 per pound, and her budget allows her to spend $20 per month on the two goods. We assume that Ms. Andrews will adjust her consumption so that the utility-maximizing condition holds for the two goods: The ratio of marginal utility to price is the same for apples and oranges. That is, Equation 7.4 $MUA 2 = MUO 1$ Here MUA and MUO are the marginal utilities of apples and oranges, respectively. Her spending equals her budget of$20 per month; suppose she buys 5 pounds of apples and 10 of oranges.

Now suppose that an unusually large harvest of apples lowers their price to $1 per pound. The lower price of apples increases the marginal utility of each$1 Ms. Andrews spends on apples, so that at her current level of consumption of apples and oranges

Equation 7.5

$MUA 1 > MUO 1$

Ms. Andrews will respond by purchasing more apples. As she does so, the marginal utility she receives from apples will decline. If she regards apples and oranges as substitutes, she will also buy fewer oranges. That will cause the marginal utility of oranges to rise. She will continue to adjust her spending until the marginal utility per $1 spent is equal for both goods: Equation 7.6 $MUA 1 = MUO 1$ Suppose that at this new solution, she purchases 12 pounds of apples and 8 pounds of oranges. She is still spending all of her budget of$20 on the two goods [(12 x $1)+(8 x$1)=$20]. Figure 7.2 Utility Maximization and an Individual’s Demand Curve Mary Andrews’s demand curve for apples, d, can be derived by determining the quantities of apples she will buy at each price. Those quantities are determined by the application of the marginal decision rule to utility maximization. At a price of$2 per pound, Ms. Andrews maximizes utility by purchasing 5 pounds of apples per month. When the price of apples falls to $1 per pound, the quantity of apples at which she maximizes utility increases to 12 pounds per month. It is through a consumer’s reaction to different prices that we trace the consumer’s demand curve for a good. When the price of apples was$2 per pound, Ms. Andrews maximized her utility by purchasing 5 pounds of apples, as illustrated in Figure 7.2 "Utility Maximization and an Individual’s Demand Curve". When the price of apples fell, she increased the quantity of apples she purchased to 12 pounds.

Notice that, in this example, Ms. Andrews maximizes utility where not only the ratios of marginal utilities to price are equal, but also the marginal utilities of both goods are equal. But, the equal-marginal-utility outcome is only true here because the prices of the two goods are the same: each good is priced at $1 in this case. If the prices of apples and oranges were different, the marginal utilities at the utility maximizing solution would have been different. The condition for maximizing utility—consume where the ratios of marginal utility to price are equal—holds regardless. The utility-maximizing condition is not that consumers maximize utility by equating marginal utilities. ## From Individual to Market Demand The market demand curves we studied in previous chapters are derived from individual demand curves such as the one depicted in Figure 7.2 "Utility Maximization and an Individual’s Demand Curve". Suppose that in addition to Ms. Andrews, there are two other consumers in the market for apples—Ellen Smith and Koy Keino. The quantities each consumes at various prices are given in Figure 7.3 "Deriving a Market Demand Curve", along with the quantities that Ms. Andrews consumes at each price. The demand curves for each are shown in Panel (a). The market demand curve for all three consumers, shown in Panel (b), is then found by adding the quantities demanded at each price for all three consumers. At a price of$2 per pound, for example, Ms. Andrews demands 5 pounds of apples per month, Ms. Smith demands 3 pounds, and Mr. Keino demands 8 pounds. A total of 16 pounds of apples are demanded per month at this price. Adding the individual quantities demanded at $1 per pound yields market demand of 40 pounds per month. This method of adding amounts along the horizontal axis of a graph is referred to as summing horizontally. The market demand curve is thus the horizontal summation of all the individual demand curves. Figure 7.3 Deriving a Market Demand Curve The demand schedules for Mary Andrews, Ellen Smith, and Koy Keino are given in the table. Their individual demand curves are plotted in Panel (a). The market demand curve for all three is shown in Panel (b). Individual demand curves, then, reflect utility-maximizing adjustment by consumers to various market prices. Once again, we see that as the price falls, consumers tend to buy more of a good. Demand curves are downward-sloping as the law of demand asserts. ## Substitution and Income Effects We saw that when the price of apples fell from$2 to $1 per pound, Mary Andrews increased the quantity of apples she demanded. Behind that adjustment, however, lie two distinct effects: the substitution effect and the income effect. It is important to distinguish these effects, because they can have quite different implications for the elasticity of the demand curve. First, the reduction in the price of apples made them cheaper relative to oranges. Before the price change, it cost the same amount to buy 2 pounds of oranges or 1 pound of apples. After the price change, it cost the same amount to buy 1 pound of either oranges or apples. In effect, 2 pounds of oranges would exchange for 1 pound of apples before the price change, and 1 pound of oranges would exchange for 1 pound of apples after the price change. Second, the price reduction essentially made consumers of apples richer. Before the price change, Ms. Andrews was purchasing 5 pounds of apples and 10 pounds of oranges at a total cost to her of$20. At the new lower price of apples, she could purchase this same combination for $15. In effect, the price reduction for apples was equivalent to handing her a$5 bill, thereby increasing her purchasing power. Purchasing power refers to the quantity of goods and services that can be purchased with a given budget.

To distinguish between the substitution and income effects, economists consider first the impact of a price change with no change in the consumer’s ability to purchase goods and services. An income-compensated price changeAn imaginary exercise in which we assume that when the price of a good or service changes, the consumer’s income is adjusted so that he or she has just enough to purchase the original combination of goods and services at the new set of prices. is an imaginary exercise in which we assume that when the price of a good or service changes, the consumer’s income is adjusted so that he or she has just enough to purchase the original combination of goods and services at the new set of prices. Ms. Andrews was purchasing 5 pounds of apples and 10 pounds of oranges before the price change. Buying that same combination after the price change would cost $15. The income-compensated price change thus requires us to take$5 from Ms. Andrews when the price of apples falls to $1 per pound. She can still buy 5 pounds of apples and 10 pounds of oranges. If, instead, the price of apples increased, we would give Ms. Andrews more money (i.e., we would “compensate” her) so that she could purchase the same combination of goods. With$15 and cheaper apples, Ms. Andrews could buy 5 pounds of apples and 10 pounds of oranges. But would she? The answer lies in comparing the marginal benefit of spending another $1 on apples to the marginal benefit of spending another$1 on oranges, as expressed in Equation 7.5. It shows that the extra utility per $1 she could obtain from apples now exceeds the extra utility per$1 from oranges. She will thus increase her consumption of apples. If she had only $15, any increase in her consumption of apples would require a reduction in her consumption of oranges. In effect, she responds to the income-compensated price change for apples by substituting apples for oranges. The change in a consumer’s consumption of a good in response to an income-compensated price change is called the substitution effectThe change in a consumer’s consumption of a good in response to an income-compensated price change.. Suppose that with an income-compensated reduction in the price of apples to$1 per pound, Ms. Andrews would increase her consumption of apples to 9 pounds per month and reduce her consumption of oranges to 6 pounds per month. The substitution effect of the price reduction is an increase in apple consumption of 4 pounds per month.

The substitution effect always involves a change in consumption in a direction opposite that of the price change. When a consumer is maximizing utility, the ratio of marginal utility to price is the same for all goods. An income-compensated price reduction increases the extra utility per dollar available from the good whose price has fallen; a consumer will thus purchase more of it. An income-compensated price increase reduces the extra utility per dollar from the good; the consumer will purchase less of it.

In other words, when the price of a good falls, people react to the lower price by substituting or switching toward that good, buying more of it and less of other goods, if we artificially hold the consumer’s ability to buy goods constant. When the price of a good goes up, people react to the higher price by substituting or switching away from that good, buying less of it and instead buying more of other goods. By examining the impact of consumer purchases of an income-compensated price change, we are looking at just the change in relative prices of goods and eliminating any impact on consumer buying that comes from the effective change in the consumer’s ability to purchase goods and services (that is, we hold the consumer’s purchasing power constant).

### Case in Point: Found! An Upward-Sloping Demand Curve

The fact that income and substitution effects move in opposite directions in the case of inferior goods raises a tantalizing possibility: What if the income effect were the stronger of the two? Could demand curves be upward sloping?

The answer, from a theoretical point of view, is yes. If the income effect in Figure 7.5 "Substitution and Income Effects for Inferior Goods" were larger than the substitution effect, the decrease in price would reduce the quantity demanded below q1. The result would be a reduction in quantity demanded in response to a reduction in price. The demand curve would be upward sloping!

The suggestion that a good could have an upward-sloping demand curve is generally attributed to Robert Giffen, a British journalist who wrote widely on economic matters late in the nineteenth century. Such goods are thus called Giffen goods. To qualify as a Giffen good, a good must be inferior and must have an income effect strong enough to overcome the substitution effect. The example often cited of a possible Giffen good is the potato during the Irish famine of 1845–1849. Empirical analysis by economists using available data, however, has refuted the notion of the upward-sloping demand curve for potatoes at that time. The most convincing parts of the refutation were to point out that (a) given the famine, there were not more potatoes available for purchase then and (b) the price of potatoes may not have even increased during the period!

A recent study by Robert Jensen and Nolan Miller, though, suggests the possible discovery of at least one Giffen good. They began their search by thinking about the type of good that would be likely to exhibit Giffen behavior and argued that, like potatoes for the poor Irish, it would be a main dietary staple of a poor population. In such a situation, purchases of the item are such a large percentage of the diet of the poor that when the item’s price rises, the implicit income of the poor falls drastically. In order to subsist, the poor reduce consumption of other goods so they can buy more of the staple. In so doing, they are able to reach a caloric intake that is higher than what can be achieved by buying more of other preferred foods that unfortunately supply fewer calories.

Their empirical work shows that in Hunan province in southern China rice is a Giffen good for poor consumers. Rice provides calories at a relatively low cost and dominates the diet, while meat is considered the tastier but higher cost-per-calorie food. In order to look at individual household decision making, they conducted a field experiment in which randomly selected poor households were given vouchers, redeemable with local merchants, for price reductions of varying sizes on the staple good. Households and merchants were given explicit instructions that selling the vouchers for cash or reselling the staple good would result in dismissal from the program and audits of the program seemed to confirm that participants were conforming to the ground rules. Overall about 1,300 households participated. Households also completed a detailed questionnaire reporting what they ate and drank, as well as other characteristics of the family on income, employment, other expenditures, and the like. They then divided the households into two categories: 1) those who were so poor that, prior to the experiment, almost all of their calories were from the staple good (Households in this category would not be expected to show Giffen behavior because their extreme poverty gives them no choice but to consume less of the staple when its price rises.) and 2) those who were somewhat less poor in the sense that, prior to the experiment, they got at least 20% of their calories from sources other than the staple good. Households in this “poor-but-not-too-poor” group exhibited Giffen behavior. In particular, they estimated that a 1% increase in the price of rice leads to a 0.45% increase in rice consumption.

A similar experiment by the authors on wheat consumption in Gansu province in northern China showed less evidence of its being a Giffen good, probably because there are more substitutes available for the specific form of wheat—wheat flour used to make wheat-based foods in the home—that was the subject of the experiment. In Gansu, people also consume wheat noodles at restaurants or road-side stands or buy wheat-based products from stores in prepared forms. A study by David McKenzie tested whether tortillas were a Giffen good for poor Mexicans. He found that they were an inferior good but not a Giffen good and similarly speculated that the availability of substitutes was the likely reason.

Jensen and Miller argue that despite the fact that their research is the first to uncover a real example of a Giffen good, other examples are likely waiting to be discovered in areas of the world where the population is poor but not-too-poor and where there are few substitutes for the staple good.

### Answer to Try It! Problem

One hundred fifty dollars is the income that allows Ms. Drakulic to purchase the same items as before, and thus can be used to measure the substitution effect. Looking only at the income-compensated price change (that is, holding her to the same purchasing power as in the original relative price situation), we find that the substitution effect is 3 more CDs (from 5 to 8). The CDs that she buys beyond 8 constitute her income effect; it is 2 CDs. Because the income effect reinforces the substitution effect, CDs are a normal good for her and her demand curve is similar to that shown in Figure 7.4 "The Substitution and Income Effects of a Price Change".