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In this chapter we will examine how the demand for a good changes as prices and income change.. 6.1 Normal and Inferior Goods We start by considering how a consumer’s demand for a good

The consumer’s demand functions give the optimal amounts of each

of the goods as a function of the prices and income faced by the consumer

We write the demand functions as

£1 = £1(P1,p2,M)

#a = #2(m, pa, m)

The left-hand side of each equation stands for the quantity demanded The right-hand side of each equation is the function that relates the prices and income to that quantity

In this chapter we will examine how the demand for a good changes as prices and income change Studying how a choice responds to changes in the economic environment is known as comparative statics, which we first described in Chapter 1 “Comparative” means that we want to compare

two situations: before and after the change in the economic environment

“Statics” means that we are not concerned with any adjustment process that may be involved in moving from one choice to another; rather we will only examine the equilibrium choice

In the case of the consumer, there are only two things in our model that affect the optimal choice: prices and income The comparative statics questions in consumer theory therefore involve investigating how demand changes when prices and income change

6.1 Normal and Inferior Goods

We start by considering how a consumer’s demand for a good changes

as his income changes We want to know how the optimal choice at one income compares to the optimal choice at another level of income During this exercise, we will hold the prices fixed and examine only the change in demand due to the income change

We know how an increase in money income affects the budget line when prices are fixed—it shifts it outward in a parallel fashion So how does this affect demand?

We would normally think that the demand for each good would increase when income increases, as shown in Figure 6.1 Economists, with a singular lack of imagination, call such goods normal goods If good I is a normal

good, then the demand for it increases when income increases, and de-

creases when income decreases For a normal good the quantity demanded always changes in the same way as income changes:

INCOME OFFER CURVES AND ENGEL CURVES 97

We have seen that an increase in income corresponds to shifting the budget line outward in a parallel manner We can connect together the demanded bundles that we get as we shift the budget line outward to construct the income offer curve This curve illustrates the bundles of goods that are demanded at the different levels of income, as depicted in Figure 6.3A The income offer curve is also known as the income expansion path If both goods are normal goods, then the income expansion path will have a positive slope, as depicted in Figure 6.3A

For each level of income, m, there will be some optimal choice for each

of the goods Let us focus on good 1 and consider the optimal choice at each set of prices and income, x; (p1,p2,™m) This is simply the demand function for good 1 If we hold the prices of goods 1 and 2 fixed and look

at how demand changes as we change income, we generate a curve known

as the Engel curve The Engel curve is a graph of the demand for one of the goods as a function of income, with all prices being held constant For

an example of an Engel curve, see Figure 6.3B

choices

x

An inferior good Good 1 is an inferior good, which means

that the demand for it decreases when income increases

How demand changes as income changes The income of-

fer curve (or income expansion path) shown in panel A depicts the optimal choice at different levels of income and constant

prices When we plot the optimal choice of good 1 against in-

come, m, we get the Engel curve, depicted in panel B

x2

` Indifference curves

Typical

budget

Perfect substitutes The income offer curve (A) and an Engel curve (B) in the case of perfect substitutes

Since the demand for good 1 is 21 = m/p, in this case, the Engel curve will be a straight line with a slope of p1, as depicted in Figure 6.4B (Since

mm is on the vertical axis, and x; on the horizontal axis, we can write

m = p,21, which makes it clear that the slope is p;.)

Perfect Complements

The demand behavior for perfect complements is shown in Figure 6.5 Since the consumer will always consume the same amount of each good, no matter

For the case of Cobb-Douglas preferences it is easier to: look ‘at the algebraic

form of the demand functions to see what the grapHš will:look like, If u(wy,£2) = 2%2}—*, the Cobb-Douglas demand for good: 1 has the form

#ị = am/p\ For a fixed value of p;, this is a linear function of m Thus doubling m will double demand, tripling m will triple demand, and.so on

In fact, multiplying m by any positive number ¢ will just multiply demand

by the same amount

The demand for good 2 is r2 = (1—a)m/pe, and this is also clearly linear

The fact that the demand functions for both goods are Imear functions

of income means that the income expansion paths will be straight lines through the origin, as depicted in Figure 6.6A The Engel curve for good 1 will be a straight line with a slope of p,/a, as depicted in ‘Figure 6.6B

SOME EXAMPLES 101

All of the income offér curves and Engel curves that we have seen up to now

have been straightforward—in fact they’ye been straight: lines! This has

happened because-eur examples have been so simple Real Engel curves do

not have-to be straight lines In general, when iricome goes up, the demand

for & good cd érease more or less rapidly than income increases If the

5 goes up by a greater:proportion.than income, we say

good, and: if it goes up by a lesser proportion than

Suppose that the-consumer’s preferences only depend on the ratio of

good 1 to good 2 This means that if the consumer prefers (21,22) to

(41,2), then she.automatically prefers (221, 2x2) to (20:, 202), (821,322)

to (391, 3y2), and so’on, since the ratio of good 1 to good 2 is the same for

all of these bundles In fact, the consumer prefers (tx, tra) to (ty1, ty2) for

any positive value of t Preferences that have this property are known as

homothetic preferences It is not hard to show that the three examples

of preferences given above—perfect substitutes, perfect complements, and

Cobb-Dougias—are all homothetic preferences

If the consumer has homothetic preferences, then the income offer curves are all straight lines through the origin, as shown in Figure 6.7 More specifically, if preferences are homothetic, it means that when income is scaled up or down by any amount ¢ > 0, the demanded bundle scales up

or down by the same amount This can be established rigorously, but it is fairly clear from looking at the picture If the indifference curve is tangent

to the budget line at (a7, 23), then the indifference curve through (¢z], tz3)

is tangent to the budget line that has t times as much income and the same prices This implies that the Engel curves are straight lines as well If you double income, you just double the demand for each good

A Income offer curve B Engel curve

Homothetic preferences An income offer curve (A) and an Engel curve (B) in the case of homothetic preferences,

Homothetic preferences are very convenient since the income effects are

so simple Unfortunately, homothetic preferences aren’t very realistic for

the same reason! But they will often be of use in our examples

Quasilinear Preferences

Another kind of preferences that generates a special form of income offer curves and Engel curves is the case of quasilinear preferences Recall the definition of quasilinear preferences given in Chapter 4 This is the case where all indifference curves are “shifted” versions of one indifference curve

SOME EXAMPLES 103

as in Figure 6.8 Equivalently, the utility function for these preferences

takes the form u(#,,22) = v(v,) +22 What happens if we shift the budget

line outward? In this case, if an indifference curve is tangent to the budget

line at a bundle (xj,23), then another indifference curve must also be

tangent at (xi, 25-+k) for any constant k Increasing income doesn’t change the demand for good 1 at all, and all the extra income goes entirely to the consumption of good 2 If preferences are quasilinear, we sometimes say that there is a “zero income effect” for good 1 Thus the Engel curve for good 1 is a vertical line—as you change income, the demand for good 1 remains constant

Quasilinear preferences An income offer curve (A) and an Engel curve (B) with quasilinear preferences

What would be a real-life situation where this kind of thing might occur? Suppose good 1 is pencils and good 2 is money to spend on other goods Initially I may spend my income only on pencils, but when my income gets large enough, I stop buying additional pencils—all of my extra income

is spent on other goods Other examples of this sort might be salt or toothpaste When we are examining a choice between all other goods and some single good that isn’t a very large part of the consumer’s budget, the quasilinear assumption may well be plausible, at least when the consumer’s income is sufficiently large

6.4 Ordinary Goods and Giffen Goods

Let us now consider price changes Suppose that we decrease the price of good 1 and hold the price of good 2 and money income fixed Then what can happen to the quantity demanded of good 1? Intuition tells us that the quantity demanded of good 1 should increase when its price decreases Indeed this is the ordinary case, as depicted in Figure 6.9

As it turns out, the answer is no It is logically possible to find well- behaved preferences for which a decrease in the price of good 1 leads to a reduction in the demand for good 1 Such a good is called a Giffen good,

ORDINARY GOODS AND GIFFEN GOODS — 105

\ | Indifference

\ \ curves

A Giffen good Good 1 is a Giffen good, since the demand

for it decreases when its price decreases

a week, you will have money left over with which you can purchase more milk In fact, with the extra money you have saved because of the lower price of gruel, you may decide to consume even more milk and reduce your consumption of gruel The reduction in the price of gruel has freed up some extra money to be spent on other things—but one thing you might want to

do with it is reduce your consumption of gruel! Thus the price change is to some extent like an income change Even though money income remains constant, a change in the price of a good will change purchasing power, and thereby change demand

So the Giffen good is not implausible purely on logical grounds, although Giffen goods are unlikely to be encountered in real-world behavior Most goods are ordinary goods—when their price increases, the demand for them declines We’ll see why this is the ordinary situation a little later

Incidentally, it is no accident that we used gruel as an example of both

an inferior good and a Giffen good It turns out that there is an intimate relationship between the two which we will explore in a later chapter But for now our exploration of consumer theory may leave you with the impression that nearly anything can happen: if income increases the demand for a good can go up or down, and if price increases the demand can

go up or down Is consumer theory compatible with any kind of behavior?

Or are there some kinds of behavior that the economic model of consumer behavior rules out? It turns out that there are restrictions on behavior imposed by the maximizing model But we’ll have to wait until the next chapter to see what they are

6.5 The Price Offer Curve and the Demand Curve

Suppose that we let the price of good 1 change while we hold pz and income fixed Geometrically this involves pivoting the budget line We can think of connecting together the optimal points to construct the price offer curve

as illustrated in Figure 6.11A This curve represents the bundles that would

be demanded at different prices for good 1

The price offer curve and demand curve _Panel A contains

a price offer curve, which-depicts the optimal choices as the price

of good 1 changes: Panel.B.-contains the associated demand curve, which depicts: a plot of the optimal choice of good 1.as a function of its price

a negative slope In terms of rates of change, we would normally have

Ax,

Ap,

which simply says that demand curves usually have a negative slope However, we have also seen that in the case of Giffen goods, the demand for a good may decrease when its price decreases Thus it is possible, but not likely, to have a demand curve with a positive slope

In order to find the demand curve, we fix the price of good 2 at some price pS and graph the demand for good 1 versus the price of good 1 to get the shape depicted in Figure 6.12B

Perfect Complements

The case of perfect complements—the right and left shoes example—is depicted in Figure 6.13 We know that whatever the prices are, a consumer will demand the same amount of goods 1 and 2 Thus his offer curve will

be a diagonal line as depicted in Figure 6.13A

We saw in Chapter 5 that the demand for good 1 is given by

m pi+ pe

If we fix m and po and plot the relationship between x, and pi, we get the curve depicted in Figure 6.13B

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