Mapping the Utility Function

In Chapter 10 we introduced the concept of a utility function, which determines a consumer’s total utility given his or her consumption bundle. In Figure 10-1 we saw how Cassie’s total utility changed as we changed the quantity of fried clams consumed, holding fixed the quantities of other items in her bundle. That is, in Figure 10-1 we showed how total utility changed as consumption of only one good changed. But we also learned in Chapter 10, from our example of Sammy, that finding the optimal consumption bundle involves the problem of how to allocate the last dollar spent between two goods, clams and potatoes. In this appendix we will extend the analysis by learning how to express total utility as a function of consumption of two goods. In this way we will deepen our understanding of the trade-off involved when choosing the optimal consumption bundle and of how the optimal consumption bundle itself changes in response to changes in the prices of goods. In order to do that, we now turn to 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 in the number of rooms, 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 10A-1 shows a three-dimensional “utility hill.” The distance along the horizontal axis measures the quantity of housing Ingrid consumes in terms of numbers 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, lies on the contour line labeled 450. So the total utility Ingrid receives from consuming 3 rooms and 30 restaurant meals is 450 utils.

Ingrid’s Utility Function The 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. Every point on a given contour line generates the same level of utility. So point B, corresponding to 6 rooms and 15 restaurant meals, generates the same level of utility as point A, 450 utils, since they lie on the same contour line.

A three-dimensional picture like Figure 10A-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 representing the utility function. In Figure 10A-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 10A-1, drawn at a total utility of 450 utils. This curve shows all the consumption bundles that yield a total utility of 450 utils. 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.

An Indifference Curve An 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.

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

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

A contour line that maps consumption bundles yielding 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 10A-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.

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 10A-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 yields. Because bundles A and B generate the same number of utils, 450, they lie on the same indifference curve, 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, a lower total utility than A or B. So Ingrid prefers consumption bundles A and B to bundle C. This is represented by the fact that C is on indifference curve I1, and I1 lies below I2. Bundle D, though, generates 519 utils, a higher total utility than A and B. It is on I3, an indifference curve that lies above I2. Clearly, Ingrid prefers D to either A or B. And, even more strongly, she prefers D to C.

An Indifference Curve Map The 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 10A-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.

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 10A-4:

Properties of Indifference Curves Panel (a) represents two general properties that all indifference curve maps 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.

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 and see the key role that diminishing marginal utility plays for them.