The Representative Consumer’s Preferences

To begin, we consider the behavior of a single representative consumer, who acts as a stand-in for all of the consumers in the economy. We show how to represent a con- sumer’s preferences over the available goods in the economy and how to represent the consumer’s budget constraint, which tells us what goods are feasible for the consumer to purchase given market prices. We then put preferences together with the budget constraint to determine how the consumer behaves given market prices, and how he or she responds to a change in nonwage income and to a change in the market wage rate.

The Representative Consumer’s Preferences

It proves simplest to analyze consumer choice and is just right for the issues we want to address in this chapter and the next, to suppose that there are two goods that con- sumers desire. The first is a physical good, which we can think of as an aggregation of all consumer goods in the economy, or measured aggregate consumption. We call this the consumption good. The second good is leisure, which is any time spent not working in the market. In terms of our definition, therefore, leisure could include recreational activities, sleep, and work at home (cooking, yardwork, housecleaning).

For macroeconomic purposes, it proves convenient to suppose that all consumers in the economy are identical. In reality, of course, consumers are not identical, but for many macroeconomic issues diversity among consumers is not essential to addressing the economics of the problem at hand, and considering it only clouds our thinking. Identical consumers, in general, behave in identical ways, and so we need only analyze the behavior of one of these consumers. Further, if all consumers are identical, the economy behaves as if there were only one consumer, and it is, therefore, convenient to write down the model as having only a single representative consumer. We must recognize, however, that the representative consumer in our macroeconomic model plays the role of a stand-in for all consumers in the economy.

A key step in determining how the representative consumer makes choices is to show how we can capture the preferences of the representative consumer over leisure and consumption goods by a utility function, written as

U(C, l),

where U is the utility function, C is the quantity of consumption, and l is the quantity of leisure. We refer to a particular combination of consumption and leisure—for example,

(C1, l1), where C1is a particular consumption quantity and l1is a particular quantity of

ranks different consumption bundles. That is, suppose that there are two different consumption bundles, representing different quantities of consumption and leisure,

denoted (C1, l1) and (C2, l2). We say that (C1, l1) is strictly preferred by the consumer

to (C2, l2) if

U(C1, l1)7 U(C2, l2);

(C2, l2) is strictly preferred to (C1, l1) if

U(C1, l1)6 U(C2, l2);

and the consumer is indifferent between the two consumption bundles if

U(C1, l1)= U(C2, l2).

It is useful to think of U(C, l) as giving the level of happiness, or utility, that the con- sumer receives from consuming the bundle (C, l). The actual level of utility, however, is irrelevant; all that matters for the consumer is what the level of utility is from a given consumption bundle relative to another one.

To use our representation of the consumer’s preferences for analyzing macroeco- nomic issues, we must make some assumptions concerning the form that preferences take. These assumptions are useful for making the analysis work, and they are also consistent with how consumers actually behave. We assume that the representative consumer’s preferences have three properties: more is preferred to less; the consumer likes diversity in his or her consumption bundle; and consumption and leisure are normal goods. We discuss each of these in turn.

1. More is always preferred to less. A consumer always prefers a consumption bundle that contains more consumption, more leisure, or both. This may appear unnat- ural, because it seems that we can get too much of a good thing. For example, consuming too much of one good may sometimes make one worse off, as when we overeat. In terms of general consumption goods, however, the average con- sumer in the United States today consumes far more than the average consumer 200 years ago would have dreamed possible, and it certainly seems that the aver- age consumer today in the United States would like to consume more if it were feasible. Indeed, even the extremely wealthy appear to desire more than they have.

2. The consumer likes diversity in his or her consumption bundle. To see that this is a natural property of consumer preferences, consider a consumer who, instead of consuming consumption goods and leisure, is making a decision about where to eat lunch during the week. Lynn can go to one of two restaurants to eat lunch, one of which serves only hamburgers, while the other serves only tuna sandwiches. One choice open to Lynn is to eat a hamburger for lunch on each day of the week, and another choice is to eat tuna sandwiches all week. Suppose that Lynn is indifferent between these two choices. If she has a preference for diversity, Lynn would prefer to alternate between restaurants during the week rather than eat at one place every day. In the case of our representative consumer, who is choosing among consumption bundles with different combinations of consumption goods and leisure, a preference for diversity means that, if the consumer is indifferent between two consumption bundles, then some mixture of the two consumption

bundles is preferable to either one. At the extreme, suppose that the consumer is indifferent between a consumption bundle that has six units of consumption and no leisure and another bundle that has no consumption goods and eight units of leisure. Then, a preference for diversity implies that the consumer would prefer a third consumption bundle, consisting of half of each of the other bun- dles, to having either of the other consumption bundles. This preferable third consumption bundle would have 3 units of consumption goods and 4 units of leisure.

3. Consumption and leisure are normal goods. A good is normal for a consumer if the quantity of the good that he or she purchases increases when income increases. For example, meals at high-quality restaurants are a normal good for most peo- ple; if our income increases, we tend to eat out more in good places. In contrast, a good is inferior for a consumer if he or she purchases less of that good when income increases. An example of an inferior good is food from Bob Evans; most people would tend to eat less at Bob Evans as their income increases. In our model, then, given that consumption and leisure are normal goods, the rep- resentative consumer purchases more consumption goods and increases his or her leisure time when income increases. This seems intuitively appealing; if, for example, you received a windfall increase in your income, perhaps through an inheritance, you would probably want to consume more goods as well as taking more vacation time (leisure). In practice, the behavior of consumers is consistent with consumption and leisure being normal goods.

While we postpone discussion of property (3) of the representative consumer’s preferences until we have more machinery to analyze how the consumer behaves, our next step is to show how we represent properties (1) and (2) graphically. It is helpful to consider the representative consumer’s preferences using a graphical representation of the utility function, called the indifference map. The indifference map is a family of indifference curves.

DEFINITION 1

Figure 4.1 shows two indifference curves. In the figure, I1is an indifference curve,

and two points on the indifference curve are (C1, l1) (point B) and (C2, l2) (point D).

Because these two consumption bundles lie on the same indifference curve, we must

have U(C1, l1)= U(C2, l2). That is, being indifferent implies that the consumer receives

the same level of happiness from each consumption bundle. Another indifference curve

is I2. Because indifference curve I2lies above indifference curve I1, and we know more

is preferred to less, consumption bundles on I2 are strictly preferred to consump-

tion bundles on I1. For example, consider point A, which represents a consumption

bundle with the same quantity of leisure as at point B, but with a higher quantity of the consumption good. Because more is preferred to less, A is strictly preferred to B.

Figure 4.1 Indifference Curves

The figure shows two indifference curves for the consumer. Each indifference curve represents a set of consumption bundles among which the consumer is indifferent. Higher indifference curves represent higher welfare for the consumer. I2 I1 l1 l2 C1 C2 A B D Leisure, l Consumption, C

An indifference curve has two key properties:

1. An indifference curve slopes downward.

2. An indifference curve is convex, that is bowed-in toward the origin.

Because the indifference map is just the graphical representation of preferences, it should not be surprising that the properties of the indifference curve are related to the properties of preferences, (1) and (2), described above. In fact, property (1) of an indifference curve follows from property (1) of preferences (more is always preferred to less), and property (2) of an indifference curve follows from property (2) of preferences (the consumer likes diversity in his or her consumption bundle).

To see why the fact that indifference curves slope downward follows from the assumption that more is preferred to less, consider Figure 4.2. At point A, consumption

is C1 and leisure is l1. Suppose that we now consider holding the quantity of leisure

constant for the consumer at l1 and reduce the consumer’s quantity of consumption

to C2, so that the consumer now has the consumption bundle represented by point

D. Because more is preferred to less, point D must be on a lower indifference curve

(indifference curve I2) than is point A (on indifference curve I1). Now we can ask

how much leisure we would have to add to l1, holding consumption constant at C2,

Figure 4.2 Properties of Indifference Curves

Indifference curves are downward-sloping because more is preferred to less. A preference for diversity implies that indifference curves are convex (bowed-in toward the origin). The slope of an indifference curve is the negative of the marginal rate of substitution.

Leisure, l Consumption, C I1 l1 C1 C2 A B D Slope = –MRSl,C l2 I2

and B. Point B must lie below and to the right of point A because, if we are taking consumption goods away from the consumer, we need to give him or her more leisure.

Thus, the indifference curve I1is downward-sloping because more is preferred to less.

To understand why the convexity of the indifference curve follows from the preference of the representative consumer for diversity, we introduce the following concept.

DEFINITION 2

We have

MRSl,C = -[the slope of the indifference curve passing through (C, l)].

To see why the marginal rate of substitution is minus the slope of the indifference curve, consider consumption bundles A and B in Figure 4.2. There, the rate at which

the consumer is willing to substitute leisure for consumption in moving from A to B

is the ratio C1-C2

l2-l1 , or minus the slope of the line segment AB. Minus the slope of AB

tells us how much consumption we need to take away for each unit of leisure added as we move from A to B, with the consumer being just indifferent between A and B.

If we imagine choosing a point like point B on the indifference curve I1 below point

A but closer and closer to A, then as the distance between that point and A becomes small, the rate at which the consumer is willing to substitute leisure for consumption between A and the chosen point is the marginal rate of substitution, which is minus the slope of the indifference curve at point A (or minus the slope of a tangent to the indifference curve at A).

Suppose, for example, that Krystyna can choose how many weeks of vacation to take each year, and that she currently works 50 weeks in a year and takes 2 weeks of vacation, so that her leisure time is 2 weeks. To keep things simple, sup- pose Krystyna consumes only coconuts, so that we can measure her consumption in coconuts. Currently, she eats 500 coconuts per year. If Krystyna were to take one more week of vacation per year, she would be just as happy as she is now if she were to give up 50 coconuts per year. This implies that Krystyna’s marginal rate of substitution of leisure for consumption, given her current consumption bundle of 500 coconuts of consumption and 2 weeks of leisure, is 50 coconuts per week.

Stating that an indifference curve is convex [property (2) of the indifference curve] is identical to stating that the marginal rate of substitution is diminishing. That is, note that the indifference curve in Figure 4.2 becomes flatter as we move down the indifference curve from left to right, that is, as the consumer receives more leisure and less of the consumption good. Thus, minus the slope of the indifference curve becomes smaller as leisure increases and consumption decreases. In other words, the marginal rate of substitution is diminishing. This is because, as we increase the quantity of leisure and reduce the quantity of consumption, the consumer needs to be compen- sated more and more in terms of leisure time to give up another unit of consumption. The consumer requires this extra compensation because of a preference for diversity.

To give a concrete example of a preference for diversity in terms of a consumption– leisure choice, suppose that Allen sleeps 8 hours in every 24-hour period. He therefore has 112 hours per week to split between work and leisure. Consider two situations. In the first, Allen takes 10 hours of leisure per week and works 102 hours, and in the second he takes 102 hours of leisure per week and works 10 hours. In the first circumstance, Allen is willing to give up much more consumption expenditure in exchange for one extra hour of leisure than in the second case.