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Unformatted text preview: CHAPTER 1 4 CONSUMER’S
SURPLUS In the preceding chapters we have seen how to derive a consumer’s demand
function from the underlying preferences or utility function. But in prac
tice we are usually concerned with the reverse problem~how to estimate
preferences or utility from observed demand behavior. We have already examined this problem in two other contexts. In Chap—
ter 5 we showed how one could estimate the parameters of a utility function
from observing demand behavior. In the Cobvaouglas example used in
that chapter, we were able to estimate a utility function that described
the observed choice behavior simply by calculating the average expendi
ture share of each good. The resulting utility function could then be used
to evaluate changes in consumption. In Chapter 7 we described how to use revealed preference analysis to
recover estimates of the underlying preferences that may have generated
some observed choices. These estimated indifference curves can also be
used to evaluate changes in consumption. In this chapter we will consider some more approaches to the problem
of estimating utility from observing demand behavior. Although some of
the methods we will examine are less general than the two methods we 248 CONSUMER’S SURPLUS (Ch. 14) examined previously, they will turn out to be useful in several applications
that we will discuss later in the book. We will start by reviewing a special case of demand behavior for which
it is very easy to recover an estimate of utility. Later we will consider more
general cases of preferences and demand behavior. 14.1 Demand for a Discrete Good Let us start by reviewing demand for a discrete good with quasilinear
utility, as described in Chapter 6. Suppose that the utility function takes
the form 12(30) + y and that the x—good is only available in integer amounts.
Let us think of the ygood as money to be spent on other goods and set its
price to 1. Let p be the price of the x—good. We saw in Chapter 6 that in this case consumer behavior can be described
in terms of the reservation prices, 11 2 11(1) — 11(0), r2 : 11(2) — "0(1), and
so on. The relationship between reservation prices and demand was very
simple: if 11 units of the discrete good are demanded, then 1“,, 2 p 2 Tn+1. To verify this, let’s look at an example. Suppose that the censumer
chooses to consume 6 units of the x—good when its price is p. Then the
utility of consuming (6, m — 6p) must be at least as large as the utility of
consuming any other bundle (:19, m — p110): 11(6)+m—6p21)(:v)+m—p$. (14.1)
In particular this inequality must hold for a: = 5, which gives us
11(6) +m—6p 2 12(5) +m—5p. Rearranging, we have v(6) — 11(5) = r6 2 p.
Equation (14.1) must also hold for a: = 7. This gives us 1;(6)+m—6p212(7)+m—7p, which can be rearranged to yield p > 11(7) — 12(6) = r7. This argument shows that if 6 units of the Xgood is demanded, then the
price of the xgood must lie between 116 and 1‘7. In general, if n units of
the x—good are demanded at price p, then Tn 2 p 2 Tn+1, as we wanted to
show. The list of reservation prices contains all the information necessary to
describe the demand behavior. The graph of the reservation prices forms a
“staircase” as shown in Figure 14.1. This staircase is precisely the demand
curve for the discrete good. CONSTRUCTING UTILlTY FROM DEMAND 249 14.2 Constructing Utility from Demand We have just seen how to construct the demand curve given the reservation
prices or the utility function. But we can also do the same Operation in
reverse. If we are given the demand curve, we can construct the utility
function—at least in the special case of quasilinear utility. At one level, this is just a trivial operation of arithmetic. The reservation
prices are deﬁned to be the difference in utility: r1 = v(1) ~— 21(0)
r2 : 11(2) — 11(1)
73 = 11(3) — 12(2) If we want to calculate 71(3), for example, we simply add up both sides of
this list of equations to ﬁnd m + m + T3 = 11(3) — 71(0). It is convenient to set the utility from consuming zero units of the good
equal to zero, so that v(0) : 0, and therefore v(n) is just the sum of the
ﬁrst n reservation prices. This construction has a nice geometrical interpretation that is illustrated
in Figure 14.1A. The utility from consuming 73 units of the discrete good is
just the area of the ﬁrst n bars which make up the demand function. This
is true because the height of each bar is the reservation price associated
with that level of demand and the width of each bar is 1. This area is
sometimes called the gross beneﬁt or the gross consumer’s surplus
associated with the consumption of the good. Note that this is only the utility associated with the consumption of
good 1. The ﬁnal utility of consumption depends on the how much the
consumer consumes of good 1 and good 2. If the consumer chooses 11. units
of the discrete good, then he will have m — pn dollars left over to purchase
other things. This leaves him with a total utility of v(n) + m — pm. This utility also has an interpretation as an area: we just take the area
depicted in Figure 14.1A, subtract off the expenditure on the discrete good,
and add m. The term v(n) ~ pn is called consumer’s surplus or the net con
sumer’s surplus. It measures the net beneﬁts from consuming 71. units of
the discrete good: the utility v(n) minus the reduction in the expenditure on consumption of the other good. The consumer’s surplus is depicted in
Figure 14.1B. 250 CONSUMER’S SURPLUS (Ch. 14) 12 3 4 5 6 QUANTiTY 12 3 4 5 6 QUANTITY
A Cross surplus 3 Net surplus Reservation prices and consumer’s surplus. The gross
beneﬁt in panel A is the area under the demand curve. This
measures the utility from consuming the x—good. The con
sumer’s surplus is depicted in panel B. It measures the utility
from consuming both goods when the ﬁrst good has to be pur
chased at a constant'price p. ' 14.3 Other Interpretations of Consumer’s Surplus There are some other ways to think about consumer’s surplus. Suppose
that the price of the discrete good is p. Then the value that the consumer
places on the ﬁrst unit of consumption of that good is m, but he only has
to pay p for it. This gives him a “surplus” of T1 ~ p on the ﬁrst unit of
consumption. He values the second unit of consumption at 7‘2, but again
he only has to pay 19 for it. This gives him a surplus of 7"; — p on that unit.
If we add this up over all 72 units the consumer chooses, we get his total
consumer’s surplus: CS=r1,—p+r2—p+~+rn—p=r1++’rn—np Since the sum of the reservation prices just gives us the utility of consump
tion of good 1, we can also write this as CS 2 v(n) — pm. We can interpret consumer’s surplus in yet another way. Suppose that a
consumer is consuming n units of the discrete good and paying pn dollars QUASlLlNEAR UTILITY 251 to do so. How much money would he need to induce him to give up his
entire consumption of this good? Let R be the required amount of money.
Then R must satisfy the equation v(0)+m+R=v(n)+m—pn.
Since 21(0) : 0 by deﬁnition, this equation reduces to
R = v(n) — pn, which is just consumer’s surplus. Hence the consumer’s surplus measures
how much a consumer would need to be paid to give up his entire con—
sumption of some good. 14.4 From Consumer’s Surplus to Consumers' Surplus Up until now we have been considering the case of a single consumer. If sev
eral consumers are involved we can add up each consumer’s surplus across
all the consumers to create an aggregate measure of the consumers’ sur
plus. Note carefully the distinction between the two concepts: consumer’s
surplus refers to the surplus of a single consumer; consumers’ surplus refers
to the sum of the surpluses across a number of consumers. Consumers’ surplus serves as a convenient measure of the aggregate gains
from trade, just as consumer’s surplus serves as a measure of the individual
gains from trade. 14.5 Approximating a Continuous Demand We have seen that the area underneath the demand curve for a discrete
good measures the utility of consumption of that good. We can extend this
to the case of a good available in continuous quantities by approximating
the continuous demand curve by a staircase demand curve. The area under
the continuous demand curve is then approximately equal to the area under
the staircase demand. See Figure 14.2 for an example. In the Appendix to this chapter we show
how to use calculus to calculate the exact area under a demand curve. 14.6 Quasilinear Utility It is worth thinking about the role that quasilinear utility plays in this
analysis. In general the price at which a consumer is willing to purchase 252 CONSUMER’S SURPLUS (Ch. 14) PRICE x QUANTITY X QUANTITY A Approximation to gross surplus B Approximation to net surplus Approximating a continuous demand. The consumer’s
surplus associated with a continuous demand curve can be ap— proximated by the consumer’s surplus associated with a discrete
approximation to it. some amount of good 1 will depend on how much money he has for con—
suming other goods. This means that in general the reservation prices for
good 1 will depend on how much good 2 is being consumed. But in the special case of quasilinear utility the reservation prices are
independent of the amount of money the consumer has to spend on other
goods. Economists say that with quasilinear utility there is “no income
effect” since changes in income don’t affect demand. This is what allows
us to calculate utility in such a simple way. Using the area under the
demand curve to measure utility will only be exactly correct when the
utility function is quasilinear. But it may often be a good approximation. If the demand for a. good
doesn’t change very much when income changes, then the income effects
won’t matter very much, and the change in consumer’s surplus will be a
reasonable approximation to the change in the consumer’s utility.1 14.7 Interpreting the Change in Consumer’s Surplus We are usually not terribly interested in the absolute level of consumer’s
surplus. We are generally more interested in the change in consumer’s 1 Of course, the change in consumer’s surplus is only one way to represent a change in
utility#the change in the square root of consumer’s surplus would be just as good.
But it is standard to use consumer’s surplus as a standard measure of utility. INTERPRETING THE CHANGE lN CONSUMER’S SURPLUS 253 surplus that results from some policy change. For example, suppose the
price of a good changes from p’ to p”. How does the consumer’s surplus
change? In Figure 14.3 we have illustrated the change in consumer’s surplus as—
sociated with a change in price. The change in consumer’s surplus is the
difference between two roughly triangular regions and will therefore have
a roughly trapezoidal shape. The trapezoid is further composed of two
subregions, the rectangle indicated by R and the roughly triangular region
indicated by T. Demand curve Change in
consumer's
surplus X" X!  X Change in consumer’s surplus. The change in consumer’s
surplus will be the difference between two roughly triangular
areas, and thus will have a roughly. trapeindal. shape. The rectangle measures the loss in surplus due to the fact that the con—
sumer is now paying more for all the units he continues to consume. After
the price increases the consumer continues to consume 93” units of the good,
and each unit of the good is now more expensive by p” — p’. This means he
has to spend (30” — p’):1:” more money than he did before just to consume
at” units of the good. But this is not the entire welfare loss. Due to the increase in the price
of the x—good, the consumer has decided to consume less of it than he was
before. The triangle T measures the value of the lost consumption of the
x—good. The total loss to the consumer is the sum of these two effects: R
measures the loss from having to pay more for the units he continues to
consume, and T measures the loss from the reduced consumption. 254 CONSUMER’S SURPLUS (Ch. 14) EXAMPLE: The Change in Consumer’s Surplus Question: Consider the linear demand curve 19(1)) 2 20 — 2p. When the
price changes from 2 to 3 what is the associated change in consumer’s
surplus? Answer: When p = 2, D(2) = 16, and when p = 3, D(3) = 14. Thus we
want to compute the area of a trapezoid with a height of 1 and bases of 14
and 16. This is equivalent to a rectangle with height 1 and base 14 (having an area of 14), plus a triangle of height 1 and base 2 (having an area of 1).
The total area will therefore be 15. 14.8 Compensating and Equivalent Variation The theory of consumer’s surplus is very tidy in the case of quasilinear
utility. Even if utility is not quasilinear, consumer’s surplus may still be
a reasonable measure of consumer’s welfare in many applications. Usually
the errors in measuring demand curves outweigh the approximation errors
from using consumer’s surplus. But it may be that for some applications an approximation may not
be good enough. In this section we’ll outline a way to measure “utility
changes” without using consumer’s surplus. There are really two separate
issues involved. The ﬁrst has to do with how to estimate utility when we
can observe a number of consumer choices. The second has to do with how
we can measure utility in monetary units. We’ve already investigated the estimation problem. We gave an example
of how to estimate a CobbDouglas utility function in Chapter 6. In that
example we noticed that expenditure shares were relatively constant and
that we could use the average expenditure share as estimates of the Cobb
Douglas parameters. If the demand behavior didn’t exhibit this particular
feature, we would have to choose a more complicated utility function, but
the principle would be just the same: if we have enough observations on
demand behavior and that behavior is consistent with maximizing some—
thing, then we will generally be able to estimate the function that is being
maximized. , Once we have an estimate of the utility function that describes some
observed choice behavior we can use this function to evaluate the impact
of proposed changes in prices and consumption levels. At the most funda—
mental level of analysis, this is the best we can hope for. All that matters
are the consumer’s preferences; any utility function that describes the con—
sumer’s preferences is as good as any other. However, in some applications it may be convenient to use certain mon
etary measures of utility. For example, we could ask how much money we COMPENSATING AND EQUIVALENT VARIATION 255 would have to give a consumer to compensate him for a change in his con
sumption patterns. A measure of this type essentially measures a change
in utility, but it measures it in monetary units. What are convenient ways
to do this? Suppose that we consider the situation depicted in Figure 14.4. Here
the consumer initially faces some prices (p’{, 1) and consumes some bundle
(331*, $3). The price of good 1 then increases from p’{ to 151, and the consumer
changes his consumption to (501, :32). How much does this price change hurt
the consumer? 'I Optimal
bundle at
' price pf x1 Slope=aﬁ1 I _
B The compensating and the equivalent variations. Panel
A shows the compensating variation (CV), and panel B shows
the equivalent variation (EV). One way to answer this question is to ask how much money we would
have to give the consumer after the price change to make him just as
well off as he was before the price change. In terms of the diagram, we
ask how far up we would have to shift the new budget line to make it tan
gent to the indifference curve that passes through the original consumption
point (ILJJE). The change in income necessary to restore the consumer to
his original indifference curve is called the compensating variation in
income, since it is the change in income that will just compensate the con—
sumer for the price change. The compensating variation measures how
much extra money the government would have to give the consumer if it
wanted to exactly compensate the consumer for the price change. Another way to measure the impact of a price change in monetary terms
is to ask how much money would have to be taken away from the consumer 256 CONSUMER’S SURPLUS (Ch. 14) before the price change to leave him as well off as he would be after the
price change. This is called the equivalent variation in income since it
is the income change that is equivalent to the price change in terms of
the change in utility. In Figure 14.4 we ask how far down we must shift
the original budget line to just touch the indifference curve that passes
through the new consumption bundle. The equivalent variation measures
the maximum amount of income that the consumer would be willing to pay
to avoid the price change. In general the amount of money that the consumer would be willing
to pay to avoid a price change would be different from the amount of
money that the consumer would have to be paid to compensate him for
a price change. After all, at different sets of prices a dollar is worth a
different amount to a consumer since it will purchase different amounts of
consumption. In geometric terms, the compensating and equivalent variations are just
two different ways to measure “how far apart” two indifference curves are.
In each case we are measuring the distance between two indifference curves
by seeing how far apart their tangent lines are. In general this measure
of distance will depend on the slope of the tangent linesithat is, on the
prices that we choose to determine the budget lines. However, the compensating and equivalent variation are the same in one
important caseithe case of quasilinear utility. In this case the indifference
curves are parallel, so the distance between any two indifference curves is
the same no matter where it is measured, as depicted in Figure 14.5. In
the case of quasilinear utility the compensating variation, the equivalent
variation, and the change in consumer’s surplus all give the same measure
of the monetary value of a price change. EXAMPLE: Compensating and Equivalent Variations r 1
Suppose that a consumer has a utility function u(:i:1,;c2) = $12 3322. He originally faces prices (1, 1) and has income 100. Then the price of good 1
increases to 2. What are the compensating and equivalent variations? We know that the demand functions for this Cobb—Douglas utility func
tion are given by m
(171 = —
2P1
m
352 =1 —.
2192 Using this formula, we see that the consumer’s demands change from
($1333) 2 (50,50) to (531,332) = (25, 50). To calculate the compensating variation we ask how much money would
be necessary at prices (2,1) to make the consumer as well off as he was
consuming the bundle (50,50)? If the prices were (2,1) and the consumer COMPENSATING AND EQUIVALENT VARIATION 257 Indifference indifference" curves Utility
differ
ence Quasilinear preferences. With quasilinear preferences, the
distance between two indifference curves is independent of the
slope of the budget lines. had income m, we can substitute into the demand functions to ﬁnd that
the consumer would optimally choose the bundle (m / 4, m / 2). Setting the
utility of this bundle equal to the utility of the bundle (50, 50) we have 1 1
m E m 5 l _1_
(r) (a) =502502< m = rum/i m 141. Solving for m gives us Hence the consumer would need about 141 ~ 100 = $41 of additional money
after the price change to make him as well off as he was before the price
change. In order to calculate the equivalent variation we ask how much money
would be necessary at the prices (1,1) to make the consumer as well off
as he would be consuming the bundle (25,50). Letting m stand for this
amount of money and following the same logic as before, m: Flux/ﬁe 70. Solving for m gives us Thus if the consumer had an income of $70 at the original prices, he would
be just as well off as he would be facing the new prices and having an income of $100. The equivalent variation in incom...
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 Fall '09
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