lecture 11 consumer surplus

8/2/2019 Lecture 11 Consumer Surplus

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Lecture 11: Consumer Surplus c 2009 Jerey A. Miron

Outline

1. Introduction

2. Consumer Welfare from a Discrete Good under Quasilinear Preferences

3. Consumer Surplus

4. Compensating and Equivalent Variation

5. Cost-Benet Analysis

1 Introduction

Our analysis in the rst eight lectures aimed at understanding what choices a con-sumer would make when faced with a given set of prices and income. That is onequestion of interest in many contexts. In particular, it provides a positive analysis,meaning one that describes what actions the consumer will take. This is exactlywhat individuals, or rms, or policy makers might want to know in some circum-

stances.

An additional question, however, concerns to what degree dierent prices or poli-cies make a consumer better or worse o. We have already examined whetherchanges in the economic environment raise or lower consumer welfare whether theconsumer ends up on a higher or lower indierence curve but we have not dis-cussed any way to quantify these changes. In many circumstances, it is importantto provide a magnitude, not just a direction, for the welfare implications.

This lecture discusses several approaches to quantifying consumer welfare. Eachis related to the others, and in practice they do not dier radically. They do

dier conceptually, however, and it is useful to understand each approach and itslimitations.

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2 Consumer Welfare from a Discrete Good with

Quasilinear Preferences

Consider rst the case where one good is discrete, meaning it can only be purchasedin lumpy increments. For example, one can buy one car, or two cars, and so on,but not 30% of a car.

Assume also that preferences are quasilinear. In particular, they are linear insome composite good that we can think of as money spent on all other goods. Thus,the utility function is

u(x; y) = v(x) + y

where x is the number of units of the discrete good and y is spending on all othergoods. The price of the discrete good is p and the price of the composite good is 1:

It is useful to describe the consumers demand for the discrete good in termsof reservation prices. A reservation price is dened as the price at which theconsumer is just indierent between consuming or not consuming the good. Thus,the reservation price this consumer would assign to consuming one unit of the discretegood is

r1 = v(1) v(0);

the reservation price this consumer would assign to consuming a second unit of thegood is

r2 = v(2) v(1);

and so on.

The relation between these reservation prices and the quantity of the discretegood demanded is as follows. Ifn units are demanded, then it must be that

rn p rn+1

That is, the actual price of the good must be less than or equal to the reservation pricefor purchasing an nth unit of the good and greater than or equal to the reservationprice for purchasing an (n + 1)st unit of the good.

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To see this more clearly, consider an example. Suppose the consumer chooses toconsume 6 units of the discrete good when its price is p. Then it must be that

u(6; m 6p) u(x; mpx).

for all x. In words, this says that the utility from consuming 6 units of the good andspending the rest of the consumers income on other goods must be greater than theutility from choosing any other number of units and spending the rest of income onother goods. This means, given the quasilinear utility function assumed here, that

v(6) + m 6p v(x) + mpx

for any x. For example, this must hold for x = 5:

v(6) + m 6p v(5) + m 5p

which implies

v(6) v (5) = r6 p.

Since the general equation must also hold with x = 7, we get

v(6) + m 6p v(7) + m 7p

which can be rearranged to give

p v(7) v(6) = r7

This shows that if 6 units are demanded, then the price of the discrete good mustbe between r6 and r7.

Thus, the list of reservation prices contains all the information necessary to de-scribe the demand behavior.

We can see that the graph of the reservation prices forms a staircase; this is thedemand curve for a discrete good.

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Graph: Reservation Prices and Consumer Surplus

QUANTITY

PRICE

r1

r2

r3

r4

r5

r6

1 2 3 4 5 6

3 Constructing Utility from Demand

We have seen how to construct the demand curve given reservation prices or theutility function.

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 this special case.

At one level, this is just a trivial operation of arithmetic. Since we know that

r1 = v(1) v(0)

r2 = v(2) v(1)

r3 = v(3) v(2)

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and so on, we could calculate, say, v(3); by adding up both sides to get

v(3) v(0) = r1 + r2 + r3

It is then convenient to set the utility from zero units to zero, so

v(3) = r1 + r2 + r3.

More generally, v(n) is just the sum of the rst n reservation prices. We can illustratethis graphically:

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Graph: Reservation Prices and Gross Consumer Surplus

QUANTITY

PRICE

r1

r2

r3

r4

r5

r6

1 2 3 4 5 6

The utility from n units is the area of the rst n bars.

This is the gross benet or gross consumer surplus from consuming x, the dis-crete good. In most applications, we want to know something related but dierent.The expression

v(n) np

is known as net consumer surplus, or, more commonly, just consumer surplus(CS). It measures the net benets from consuming n units of the discrete good. Inwords, CS is the utility from consuming the discrete good, v(n), minus the reductionin expenditure on consumption of other goods, pn. We can illustrate this as below:

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Graph: Reservation Prices and Net Consumer Surplus

QUANTITY

PRICE

r1

r2

r3

r4r5r6

1 2 3 4 5 6

p -----

Thus, CS is the area under the demand curve but above price.

Two other interpretations ofCS are useful. Suppose that the price of the discretegood is p. Then the value that the consumer places on the rst unit of consumptionof that good is r1, but he only has to pay p for it. This gives him a surplus of

r1 p

from consumption of the rst unit.

The consumer values the second unit of the discrete good at r2, but again onlyhas to pay p for it. This second unit therefore gives him a surplus of

r2 p

and so on through all the units of the discrete good purchased. Adding this up over

the n units purchased implies that

CS = r1 + r2 + : : : + rn np

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Since the sum of the rst n reservation prices gives us the utility of consuming nunits, we can also write this as

CS = v(n) np,

as before. Thus, we can think of CS as the sum of the reservation prices, minusexpenditure on the good.

We can interpret consumer surplus in still another way. Suppose the consumeris consuming n units of the discrete good and paying np dollars to do so. How muchmoney 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

v(0) + m + R = v(n) + m np

Since v(0) = 0 by assumption, this reduces to

R = v(n) np

which is the same expression for consumer surplus that we derived above. Thus,CS can be thought of as the amount of money necessary to get the consumer to giveup his entire consumption of the good.

Three further thoughts about CS:

1. In many cases we care not just about the CS of a given consumer but about

the CS of a group of consumers. In that case we could add up the CS of eachconsumer to get a measure of consumers surplus, rather than a consumers surplus.

2. We have shown that the area underneath the demand curve for a discrete goodmeasures the utility from the consumption of that good. One possible limitation ofthis approach is that many goods are not discrete. For example, one can buy anyamount of gasoline, not just a tank full or none at all.

In many applications, however, the discrete approach might be a good approx-imation to a continuous demand curve. It is an empirical question as to whetherthis approximation is good enough, and for what purposes.

3. A dierent limitation of the CS approach is that it assumes quasilinear utility.This functional form means that the price at which a consumer is willing to buy goodx does not depend on the amount of money he has left over to spend on good y.

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This does not hold for general utility functions. Instead, the reservation price forgood 1 will depend on how much of good 2 is being consumed. Quasilinear utilityeliminates any income eect, which simplies the analysis. It also means, however,

that if utility is not quasilinear, then the formulas above are only an approximation.

OPTIONAL MATERIAL: The assumption of a discrete good is not necessaryif we use calculus to characterize CS. Assume the consumer solves

max v(x) + y

subject to px + y = m

Substituting out for y using the constraint and then taking the derivative with respectto x gives us the FOC

v0(x) = p,

and this implies an inverse demand function

p(x) = v0(x).

This just says that the price the consumer is willing to pay for x units of the goodis determined by the marginal utility function for the good.

Now consider the following. Given that v(0) = 0 by assumption, we can write

v(x) = v(x) v(0) = Rx0 v0(t)dt = Rx0 p(t)dtIn words, this says that the utility from consuming x units of the good, given quasi-linear utility, is the area under the demand curve. So, the usual denition ofCS isexactly right under these assumptions. END OPTIONAL MATERIAL

4 Compensating and Equivalent Variation

The CS approach to quantifying the eect of price changes on consumer well-beingrelies on various assumptions or approximations. For these reasons, it is useful toexamine other approaches.

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Assume that the consumer initially faces the prices (p1; p

2) and consumes thebundle (x1; x

2). The price of good 1 then increases from p

1 to bp1, and the consumerchanges his consumption to (bx1;bx2).

How much does this price increase hurt the consumer? It turns out we cananswer this in two ways.

The rst asks how much money we would have to give the consumer after theprice change to make him just as well o as he was before the price change.

This is illustrated in the following gure:

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GRAPH: Compensating Variation

x1

x2

CV[m*

C Optimal bundleat price p1^

Slope = -p1

Slope = -p1^

(x1*, x2*)

(x1^, x2^)

A

The idea is that we take the new slope of the budget line as given and ask howfar up we would have to shift this line to make it tangent to the indierence curvethat passes through the original consumption point (x1; x

2).

This change in income is called the compensating variation (CV); it is the

amount of money it would take to compensate the consumer for the price change.

The second way to answer the question (of how much the price increase harmsthe consumer) is to ask how much money would have to be taken away from theconsumer before the price change to leave him as well o as he ends up beingafter the price change.

This is illustrated in the following gure:

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GRAPH: Equivalent Variation

x1

x2

EV[m*

E

Optimal bundleat price p1*

Slope = -p1*

Slope = -p1^

(x1*, x2*)

(x1^, x2^)

B

The idea is that we nd the shift in the original budget line that makes it tangentto the indierence curve that the consumer ends up on as the result of the pricechange.

This change in income is called the equivalent variation (EV); it is the amount

of money that is equivalent to the loss in income the consumer suered as the resultof the price increase.

We can also think of the EV as the maximum amount of income the consumerwould pay to avoid the price change.

Note that in general the CV and the EV do not have to be the same. In words,the amount of money the consumer would be willing to pay to avoid a price changecould be dierent from the amount of money the consumer would have to be paid tocompensate him for that price change.

The reason is that a dollar is worth dierent amounts to the consumer at dierentsets of prices, since the consumer will purchase dierent amounts of consumption atdierent prices.

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In geometric terms, the CV and EV are dierent ways to measure the distancebetween two indierence curves.

In each case we are measuring the distance by seeing how far apart their tangentlines are. In general this depends on the slope; that is, on the prices that we chooseto determine the budget lines.

However, the CV and EV are the same in one case: quasilinear utility. In thiscase the indierence curves are parallel, which means distance is the same no matterwhere measured.

In fact, in the quasilinear case,

CV = EV = CS

We can see this explicitly as follows. Suppose the consumer has the utility function

v(x1) + x2.

In this case we know from previous material that the demand for good 1 can bewritten as

x1(p1).

That is, demand for good 1 depends only on the price of good 1, not on the price ofgood 2 or income.

Now let the price increase from p1 to bp1. We are going to compute the CV andEV.

We know that at p1 the consumer chooses

x1(p

1)

and gets utility

v (x1) + mp

1x

1

And, at bp1 the consumer choosesbx1 = x1(bp1)

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and gets utility

v (bx1) + mbp1bx1Then the CV, which equals the amount of extra money the consumer would needafter the price change to make him as well of as he was before the price change, mustsatisfy

v (bx1) + m + CV bp1bx1 = v (x1) + mp1x1,so

CV = v (x1) v (bx1) + bp1bxp1x11.Similar reasoning shows that the equivalent variation, which is the amount of money

we would have to take away from the consumer before the price change to leave himwith the same utility that he would have after the price change, must satisfy

v (x1) + mEV p

1x

1 = v (bx1) + mbp1bx1which implies

EV = v (x1) v (bx1) + bp1bx1 p1x1.This is the same expression as that for the CV. One useful exercise is to try thiswith a utility function that is not quasilinear and show that in general the CV and

EV are not the same.

The CS from the price change is just the utility of the consumers choice underthe initial price minus the utility of the consumers choice under the new price, whichis again the same expression.

5 Cost-Benet Analysis

One key application of the material we have just examined is analyzing the welfareimplications of various policies. We will consider many examples later in the course,but it is useful to consider a simple case here. We will use CS rather than CV or

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EV, since CS is approximately right under reasonable assumptions and by far themost commonly used approach in practice.

We will also use the concept of producer surplus, even though we will not deneit carefully until after developing the theory of the rm. For now, it is exactly whatyou learned in EC 10: the area above the supply curve but below price.

The example we want to consider is rent control. Assume we have standarddemand and supply curves for apartments. In the absence of rent control theequilibrium of this market would look like the following graph:

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GRAPH: The Supply and Demand for Apartments

x

p

p*

x*

Consumer's surplus

Producer's surplus

S

D

The equilibrium quantity is q0 and the equilibrium price is p0.

Now assume that a government body imposes a maximum rent that can becharged, pc, with

pc < p0.

What happens to the welfare of consumers and producers in this market?

One possible answer is based on the following diagram:

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GRAPH: Supply and Demand with Rent Control

x

p

p0

q0

CS

PS

S

D

pe

pc

qc = qeO

A

B

C

E

This graph shows that at the controlled price, the supply of apartments is qc, whichis less than q0:

Consumer surplus declines from the triangle AEp0 to the trapezoid ABCpc.

Producer surplus declines from the triangle OEp0 to the triangle OCpc.

Under one key assumption, the overall eect on welfare is therefore a loss equalto the triangle BE C. This assumption is that the people who get the apartmentsat the controlled price are the ones who value them the most. In practice, this isunlikely, so the loss in welfare is probably greater than the diagram would suggest.Some people who end up with rent-controlled apartments do not value them thathighly, while some who value them a lot do not get them.

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lecture 11 consumer surplus

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