ENRICHMENT MODULE D: Indifference Curves and Consumer Choice

Mapping the Utility Function

Earlier we introduced the concept of a utility function, which determines a consumer’s total utility, given his or her consumption bundle. Here we will extend the analysis by learning how to express total utility as a function of the consumption of two goods. This will deepen our understanding of the trade-offs involved when choosing the optimal consumption bundle. We will also see how the optimal consumption bundle itself changes in response to changes in the prices of goods. We begin by examining a different way of representing a consumer’s utility function, based on the concept of indifference curves.

Indifference Curves

Ingrid is a consumer who buys only two goods: housing, measured by the number of rooms in her house or apartment, and restaurant meals. How can we represent her utility function in a way that takes account of her consumption of both goods?

One way is to draw a three-dimensional picture. Figure D.1 shows a three-dimensional “utility hill.” The distance along the horizontal axis measures the quantity of housing Ingrid consumes in terms of the number of rooms; the distance along the vertical axis measures the number of restaurant meals she consumes. The altitude or height of the hill at each point is indicated by a contour line, along which the height of the hill is constant. For example, point A, which corresponds to a consumption bundle of 3 rooms and 30 restaurant meals, and point B, which corresponds to a bundle of 6 rooms and 15 restaurant meals, lie on the contour line labeled 450. So the total utility Ingrid receives from consuming either of these bundles is 450 utils.

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Figure 1.1: FIGURE D.1 Ingrid’s Utility FunctionThe three-dimensional hill shows how Ingrid’s total utility depends on her consumption of housing and restaurant meals. Point A corresponds to consumption of 3 rooms and 30 restaurant meals. That consumption bundle yields Ingrid 450 utils, corresponding to the height of the hill at point A. The lines running around the hill are contour lines, along which the height is constant. So point B and every other point along the same contour line as point A generates the same level of utility.

A three-dimensional picture like Figure D.1 helps us think about the relationship between consumption bundles and total utility. But anyone who has ever used a topographical map to plan a hiking trip knows that it is possible to represent a three-dimensional surface in only two dimensions. A topographical map doesn’t offer a three-dimensional view of the terrain; instead, it conveys information about altitude solely through the use of contour lines.

The same principle can be applied to the representation of a utility function. In Figure D.2, Ingrid’s consumption of rooms is measured on the horizontal axis and her consumption of restaurant meals on the vertical axis. The curve here corresponds to the contour line in Figure D.1, drawn at a total utility of 450 utils. This curve shows all the consumption bundles that yield a total utility of 450 utils As we’ve seen, one point on that contour line is A, a consumption bundle consisting of 3 rooms and 30 restaurant meals. Another point on that contour line is B, a consumption bundle consisting of 6 rooms but only 15 restaurant meals. Because B lies on the same contour line, it yields Ingrid the same total utility—450 utils—as A. We say that Ingrid is indifferent between A and B: because bundles A and B yield the same total utility level, Ingrid is equally well off with either bundle.

Figure 1.2: FIGURE D.2 An Indifference CurveAn indifference curve is a contour line along which total utility is constant. In this case, we show all the consumption bundles that yield Ingrid 450 utils. Consumption bundle A, consisting of 3 rooms and 30 restaurant meals, yields the same total utility as bundle B, consisting of 6 rooms and 15 restaurant meals. That is, Ingrid is indifferent between bundle A and bundle B.

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An indifference curve shows all the consumption bundles that yield the same amount of total utility for an individual.

A contour line that represents consumption bundles that give a particular individual the same amount of total utility is known as an indifference curve. An individual is always indifferent between any two bundles that lie on the same indifference curve. For a given consumer, there is an indifference curve corresponding to each possible level of total utility. For example, the indifference curve in Figure D.2 shows consumption bundles that yield Ingrid 450 utils; different indifference curves would show consumption bundles that yield Ingrid 400 utils, 500 utils, and so on.

The entire utility function of an individual can be represented by an indifference curve map—a collection of indifference curves, each of which corresponds to a different total utility level.

A collection of indifference curves that represents a given consumer’s entire utility function, with each indifference curve corresponding to a different level of total utility, is known as an indifference curve map. Figure D.3 shows three indifference curves—I₁, I₂, and I₃—from Ingrid’s indifference curve map, as well as several consumption bundles, A, B, C, and D. The accompanying table lists each bundle, its composition of rooms and restaurant meals, and the total utility it provides. Because bundles A and B generate the same number of utils, 450, they lie on the same indifference curve, I₂.

Figure 1.3: FIGURE D.3 An Indifference Curve MAPThe utility function can be represented in greater detail by increasing the number of indifference curves drawn, each corresponding to a different level of total utility. In this FIGURE, bundle C lies on an indifference curve corresponding to a total utility of 391 utils. As in Figure D.2, bundles A and B lie on an indifference curve corresponding to a total utility of 450 utils. Bundle D lies on an indifference curve corresponding to a total utility of 519 utils. Ingrid prefers any bundle on I₂ to any bundle on I₁, and she prefers any bundle on I₃ to any bundle on I₂.

Although Ingrid is indifferent between A and B, she is certainly not indifferent between A and C: as you can see from the table, C generates only 391 utils, fewer than A or B. So Ingrid prefers consumption bundles A and B to bundle C. This is evident from the graph because C is on indifference curve I₁, and I₁ lies below I₂. Bundle D, though, generates 519 utils, more than A and B. So bundle D is on indifference curve I₃, which lies above I₂. Clearly, Ingrid prefers D to either A or B. And, even more strongly, she prefers D to C.

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Are Utils Useful?

In the table that accompanies Figure D.3, we give the number of utils achieved on each of the indifference curves shown in the figure. But is this information actually needed?

The answer is no. As you will see shortly, the indifference curve map tells us all we need to know in order to find a consumer’s optimal consumption bundle. That is, it’s important that Ingrid has higher total utility along indifference curve I₂ than she does along I₁, but it doesn’t matter how much higher her total utility is. In other words, we don’t have to measure utils in order to understand how consumers make choices.

Economists say that consumer theory requires an ordinal measure of utility—one that ranks consumption bundles in terms of desirability—so that we can say that bundle X is better than bundle Y. The theory does not, however, require cardinal utility, which actually assigns a specific number to the total utility yielded by each bundle.

So why introduce the concept of utils at all? The answer is that it is much easier to understand the basis of rational choice by using the concept of measurable utility.

Properties of Indifference Curves

No two individuals have the same indifference curve map because no two individuals have the same preferences. But economists believe that, regardless of the person, every indifference curve map has two general properties. These are illustrated in panel (a) of Figure D.4 on the next page.

Figure 1.4: FIGURE D.4 Properties of Indifference CurvesPanel (a) represents two general properties that all indifference curves share. The left diagram shows why indifference curves cannot cross: if they did, a consumption bundle such as A would yield both 100 and 200 utils, a contradiction. The right diagram of panel (a) shows that indifference curves that are farther out yield higher total utility: bundle B, which contains more of both goods than bundle A, yields higher total utility. Panel (b) depicts two additional properties of indifference curves for ordinary goods. The left diagram of panel (b) shows that indifference curves slope downward: as you move down the curve from bundle W to bundle Z, consumption of rooms increases. To keep total utility constant, this must be offset by a reduction in quantity of restaurant meals. The right diagram of panel (b) shows a convex-shaped indifference curve. The slope of the indifference curve gets flatter as you move down the curve to the right, a feature arising from diminishing marginal utility.

Indifference curves never cross. Suppose that we tried to draw an indifference curve map like the one depicted in the left diagram in panel (a), in which two indifference curves cross at A. What is the total utility at A? Is it 100 utils or 200 utils? Indifference curves cannot cross because each consumption bundle must correspond to one unique total utility level—not, as shown at A, two different total utility levels.
The farther out an indifference curve lies—the farther it is from the origin—the higher the level of total utility it indicates. The reason, illustrated in the right diagram in panel (a), is that we assume that more is better—we consider only the consumption bundles for which the consumer is not satiated. Bundle B, on the outer indifference curve, contains more of both goods than bundle A on the inner indifference curve. So B, because it generates a higher total utility level (200 utils), lies on a higher indifference curve than A.

Furthermore, economists believe that, for most goods, consumers’ indifference curve maps also have two additional properties. They are illustrated in panel (b) of Figure D.4:
Indifference curves slope downward. Here, too, the reason is that more is better. The left diagram in panel (b) shows four consumption bundles on the same indifference curve: W, X, Y, and Z. By definition, these consumption bundles yield the same level of total utility. But as you move along the curve to the right, from W to Z, the quantity of rooms consumed increases. The only way a person can consume more rooms without gaining utility is by giving up some restaurant meals. So the indifference curve must slope downward.
Indifference curves have a convex shape. The right diagram in panel (b) shows that the slope of each indifference curve changes as you move down the curve to the right: the curve gets flatter. If you move up an indifference curve to the left, the curve gets steeper. So the indifference curve is steeper at A than it is at B. When this occurs, we say that an indifference curve has a convex shape—it is bowed-in toward the origin. This feature arises from diminishing marginal utility, a principle we discussed in Module 51. Recall that when a consumer has diminishing marginal utility, consumption of another unit of a good generates a smaller increase in total utility than the previous unit consumed. Next we will examine in detail how diminishing marginal utility gives rise to convex-shaped indifference curves.

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Goods that satisfy all four properties of indifference curve maps are called ordinary goods. The vast majority of goods in any consumer’s utility function fall into this category. In the next section we will define ordinary goods more precisely and see the key role that diminishing marginal utility plays for them.