Economics 102 Lecture 8 Ways to Measure Utility Rev

7/30/2019 Economics 102 Lecture 8 Ways to Measure Utility Rev

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Demand for a Discrete Good demand for a discrete good with a quasilinear utility.

utility function takes the form:

the x good is the discrete good and the y good is the

money to be spent on other goods (price is 1).

Consumer behavior was described in terms ofreservation prices:

yxvu )(

)....1()2(

)0()1(

2

1

vvr

vvr

Relationship between reservation prices and demand: if n

units was demanded, then:

Example: If 6 units are consumed at price p, utility of

consuming (6,m-6p) must be at least as large as consuming

any other bundle (x, m-px):

1 nn rpr

7

6

)6()7(

grearrangin

7)7(6)6(

:also

)5()6(

grearrangin

5)5(6)6(

rvvp

pmvpmv

prvv

pmvpmv


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Constructing Utility from Demand Since the reservation prices are just the differences in

utility, add up the reservation prices to come with totalutility associated with consumption of x units.

If we set v(0) is equal to zero, the utility associated withconsuming n units = sum of the first n reservationprices.

321

3

2

1

)0()3(

)2()3(

)1()2(

)0()1(

rrrvv

vvr

vvr

vvr

A plot of r1, r2, , rn, against n is a

reservation-price curve.


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Gross benefit or gross consumers surplus associatedwith the first n consumed of good 1 is therefore the areaof the first n bars which make up the demand function.

Consumers surplus or net consumers surplusmeasures the net benefits from consuming the n unitsof the discrete good. final utility of consumption depends on both good 1 and good 2

Therefore, in the discrete good example, the consumption ofgood 2 is m-pn. The total utility is therefore v(n)-m-pn.

is called the consumers surplus or net consumerssurplus.

It measures the utility v(n) minus the reduction in expenditure onthe consumption of the other good.

pnnv

)(

Reservation Price Curve

Quantity

($) Res.

Values

1 2 3 4 5 6

r1

r2

r3

r4

r5

r6


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Quantity

($) Res.

Values

1 2 3 4 5 6

r1

r2

r3

r4

r5

r6

pG


Quantity

($) Res.

Values

1 2 3 4 5 6

r1

r2

r3

r4

r5

r6

p


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Quantity

($) Res.

Values

1

2 3

4

5

6

r1

r2

r3

r4

r5

r6

p

Consumers surplus

Consumer surplus can be interpreted as theexcess of value placed on a unit ofconsumption over the price that he has to payfor it:

Adding up over all n units that the consumer

chooses, the total consumer surplus is:

Since the sum of the reservation prices just givesthe utility, then this can be rewritten as:

pr

nprrrCS n ...21

pnnvCS )(


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Consumers surplus can also be interpreted asthe amount of money that a consumer has tobe paid in order for him to give up his entireconsumption of that good Let R be that amount of money, R must satisfy:

Since v(0)=0, then by definition, the equationreduces to:

pnmnvRmv )()0(

pnnvR )(

Consumers surplus can be generated fromconsumers surplus. The former refers tothe sum of surpluses across a number ofconsumers. Thus if we have measures ofsingle consumer surpluses, then we canadd these all up to come up with anaggregate measure.


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Extend the case of a discrete good to thecase of continuous quantities byapproximating the continuous demandcurve by a staircase demand curve.

The area under the continuous demand curveis therefore approximately equal to the areaunder the staircase demand.

Suppose that good 1 is sold in half-units.

r1, r2, , rn, denote the consumers

reservation prices for successive unit of x1.

Our consumers new reservation price curve

is


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Half units

($) Res.

Values

1 2 3 4 5 6

r1

r3

r5

r7

r9

r11

7 8 9 10 11


Half units

($) Res.

Values

1 2 3 4 5 6

r1

r3

r5

r7

r9

r11

7 8 9 10 11

p


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1


Half units

($) Res.

Values

1 2 3 4 5 6

r1

r3

r5

r7

r9

r11

7 8 9 10 11

pG

Consumers surplus

And if good 1 is available in one-quarter

units ...


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0

2

4

6

8

10

One quarter units

($) Res.

Values

1


0

2

4

6

8

10

One quarter units

($) Res.

Values

1


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1

Reservation Price Curve for Gasoline

0

2

4

6

8

10

($) Res.

Values

p

P value of net utility gains-to-trade

Finally, if good 1 can be purchased in any

quantity then ...


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1

Good 1

(P) Res.

Prices


Good 1

(P) Res.

Prices

p



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1

Good 1

(P) Res.

Prices

p


Consumers surplus

However, using the area under the demand curve as

a measure of utility is only exactly correct when the

utility function is quasilinear.

reservation prices are independent of the amount of

money the consumer has to spend on other goods -

there is no income effect it is a good approximation if demand for a good doesnt

change very much when income changes.

In general though, reservation prices for good 1 will

depend on how much good 2 is being consumed.


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Changes in Consumers Surplus

Changes in consumers surplus are helpful in

analyzing policy changes.

For instance, if we want to analyze a price

change owing to some government project.

How will the consumers surplus change?

The change to a consumers total utility due

to a change in p1 is approximately the

change in her Consumers Surplus.


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Suppose, prices increase from p to p. Difference is roughly the difference between two

triangular areas. The difference is roughlytrapezoidal in shape with two regions R and T.

R represents the loss in surplus because theconsumer is paying more for all the units that heconsumes. This is the area (p-p)x.

T represents the value of lost consumptionbecause the consumer has decided to consumeless because of the price increase.

The total loss to the consumer is the sum ofthese two effects: the loss from having to paymore for the units he consumes and the lossfrom reduced consumption

p1

x1*x1

'

p1'

p1(x1), the inverse ordinary demand

curve for commodity 1


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1

p1

x1*x1

'

CS before

p1(x1)

p1'

p1

x1*x1

'

CS afterp1"

x1"

p1(x1)

p1'


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1

R

p1

x1*x1

'x1"

Change in CS= R + T

p1(x1), inverse ordinary demandcurve for commodity 1.

p1"

p1' T

Example: A linear demand curve:

When the price changes from 2 to 3, what is the

change in consumers surplus?

When p=2, D=16 and when p=3, D=14.

Compute an area of a trapezoid with a height of 1

and base of 14. This is the sum of a rectangle with

a height of base 1 and base 14 and of a rectangle

with a height of 1 and base of 2.

The area is therefore 15.

ppD 220)(


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Alternative measure of utility changes without using

consumers surplus.

Two issues involved:

How to estimate utility when a number of consumer choices

are observed.

How to measure utility in money units

First issue:

With enough observations on demand behavior and that

behavior is consistent with maximizing something, then

we will generally be able to estimate the function that isbeing maximized, e.g. Cobb-Douglas

Use the function to evaluate the impact of proposed

changes in prices and consumption levels.

Use monetary measures of utility convenient in

some applications.

Expressed as how much money would have to be given

to a consumer in order to compensate him for a change

in his consumption patterns.

Two ways are usually utilized: compensating variation

and equivalent variation


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Defined as the change in income necessary

to restore the consumer to his original

indifference curve.

How much money would have to be given to

the consumer after the price change to

make him just as well off as he was before

the price change

p1 rises.

Q: What is the least extra income that, at the

new prices, just restores the consumers

original utility level?


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p1 rises.


new prices, just restores the consumers


A: The Compensating Variation.

x2

x1x1'

u1

x2'

p1=p1 p2 is fixed.

m p x p x1 1 1 2 2' ' '


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2

x2

x1x1'

x2'

x1"

x2"

u1

u2

p1=p1p1=p1p2 is fixed.

m p x p x1 1 1 2 2' ' '

p x p x1 1 2 2" " "

x2

x1x1'

u1

u2

x1"

x2"

x2'

x2'"

x1'"

p1=p1

p1=p1

p2 is fixed.

m p x p x1 1 1 2 2' ' '

p x p x1 1 2 2" " "

'"22

'"1

"12 xpxpm


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2

x2

x1x1'

u1

u2

x1"

x2"

x2'

x2'"

x1'"


m p x p x1 1 1 2 2' ' '

p x p x1 1 2 2" " "

'"22

'"1

"12 xpxpm

CV = m2 - m1.

Measures the maximum amount of income that theconsumer would be willing to pay to avoid the pricechange.

How much money would have to be taken away from theconsumer before the price change to leave him as well

off as he would be after the price change.

Depicted as how far down we must shift the originalbudget line to just touch the indifference curve thatpasses through the new consumption bundle.Fig.4


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2

p1 rises.


original prices, just restores the consumers


A: The Equivalent Variation.

x2

x1x1'

u1

x2'

p1=p1 p2 is fixed.

m p x p x1 1 1 2 2' ' '


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2

x2

x1x1'

x2'

x1"

x2"

u1

u2


m p x p x1 1 1 2 2' ' '

p x p x1 1 2 2" " "

x2

x1x1'

u1

u2

x1"

x2"

x2'

x2'"

x1'"

p1=p1

p1=p1

p2 is fixed.

m p x p x1 1 1 2 2' ' '

p x p x1 1 2 2" " "

m p x p x2 1 1 2 2' '" '"


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2

x2

x1x1'

u1

u2

x1"

x2"

x2'

x2'"

x1'"


m p x p x1 1 1 2 2' ' '

p x p x1 1 2 2" " "

m p x p x2 1 1 2 2' '" '"

EV = m1 - m2.

CV vs EV

Amount of money that consumer would be willing to pay toavoid a price change amount of money that would have tobe paid to a consumer to compensate him for a price change

Reason: a peso is worth differently to a consumer at differentsets of prices because it will purchase different amounts ofconsumption.

CV and EV are just two ways of measuring the distancebetween indifference curves by seeing how far apart thetangent lines are. Since the distance depend on the tangentlines, it matters from what price level you are proceeding.

CV, EV and consumers surplus are equal in the case ofquasilinear utility.


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Example:

Suppose:

prices are (1,1) and income is 100. Let the price of

good 1 increase to 2.

Demand for these Cobb-Douglas u functions are:

the original demanded bundle is (50,50) and the

new bundle is (25,50)

2/1

2

2/1

121 ),( xxxxu

2211 2/and2/ pmxpmx

Compensating variation: How much money would be

necessary at prices (2,1) to make him as well off as he

was consuming bundle (50,50)?

At the new prices and at the level of income with the

compensating variation, m, the consumer would be

consuming (m/4,m/2).

Setting the utility of this bundle with the original bundle, we

can solve for m.

Solving for m, m is approximately 141. Therefore, the

consumer would need about 41 pesos to make him as well

off as before.

2/12/1

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505024

mm


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Equivalent variation: How much moneywould be necessary at prices (1,1) to makethe consumer as well off as he would beconsuming bundle (25,50).

Letting m be the amount of money, andfollowing the same logic:

Solving for m gives m approximately equal to

70. The equivalent variation is therefore 30.

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502522

mm

Example: CV and EV using quasilinear

preferences

Suppose a quasilinear function:

Demand for good 1 depends only on the prices of

good 1:

Suppose prices change from .

Demands and utilities are:

21)( xxv

)( 111 pxx

1

*

1 to

pp

11111

*

1

*

1

*

1

*

1

*

1

)(),(

)(),(

xpmxvpx

xpmxvpx


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Compensating variation, C, amount of

money that consumer would need after

the price change to make him as well off:

Solving for C:

*

1

*

1

*

1111)()( xpmxvxpCmxv

*

1

*

1111*

1 )()( xpxpxvxvC

Equivalent variation, E, satisfies the

equation:

Solving for E:

*

1

*

1

*

1111 )()( xpEmxvxpmxv

*

1

*

1111*

1 )()( xpxpxvxvE


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Thus C=E.

They are also equal to the change in net

consumers surplus:

Recall:

])([])([ 111*

1

*

1

*

1

xpxvxpxvCS

pnnvCS )(

Supply curve measures the amount that will besupplied at each price. Area above the supply curveis the producers surplus. It measures the surplusenjoyed by the suppliers of the good.

Conduct the analysis in terms of the producers

inverse supply curve ps(x). This function measureswhat the price would have to be to get the producerto supply x units of the good.

Analogue would be the analysis for a discrete good. Whilethe producer is willing to supply the first n units at a supplyreservation price, what he actually gets is higher. Theexcess is the producers surplus.


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Changes in a firms welfare can be measured

in pesos much as for a consumer.

y (output units)

Output price (p)

Marginal Cost


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y (output units)

Output price (p)

Marginal Cost

p'

y'

y (output units)

Output price (p)

Marginal Cost

p'

y'

Revenue=

p y' '


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y (output units)

Output price (p)

Marginal Cost

p'

y'

Variable Cost of producing

y units is the sum of the

marginal costs

y (output units)

Output price (p)

Marginal Cost

p'

y'

Variable Cost of producingy units is the sum of the

marginal costs

Revenue less VC

is the Producers

Surplus.


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Net producers surplus=difference between theminimum amount that she would be willing tosell the x* units for and the amount she actuallysells the unit for.

What is the change in the surplus when there isa price increase from p to p. R measures the

gain from selling the units previously sold at ahigher price. T measures the gain from selling

the extra units at a higher price.

Can we measure in money units the net

gain, or loss, caused by a market

intervention; e.g., the imposition or the

removal of a market regulation?

Yes, by using measures such as theConsumers Surplus and the Producers

Surplus.


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QD, QS

Price

Supply

Demand

p0

q0

The free-market equilibrium

CS

QD, QS

Price

Supply

Demand

p0

q0

The free-market equilibrium

and the gains from trade

generated by it.

PS


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CS

QD, QS

Price

Supply

Demand

p0

q0

PS

q1

Consumers

gain

Producers

gain

The gain from freely

trading the q1th unit.

CS

QD, QS

Price

Supply

Demand

p0

q0

The gains from freely

trading the units from

q1 to q0.

PS

q1

Consumers

gains

Producers

gains


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CS

QD, QS

Price

Supply

Demand

p0

q0

The gains from freely

trading the units from

q1 to q0.

PS

q1

Consumers

gains

Producers

gains

CS

QD, QS

Price

p0

q0

PS

q1

Consumers

gains

Producers

gains

Any regulation that

causes the units

from q1 to q0 to be

not traded destroys

these gains. This

loss is the net cost

of the regulation.


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Tax

Revenue

QD, QS

Price

q0

PS

q1

An excise tax imposed at a rate of Pt

per traded unit destroys these gains.

ps

pb

t

CS

Deadweight

Loss

QD, QS

Price

q0q1



pf

CS

Deadweight

Loss

So does a floor

price set at pf

PS


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QD, QS

Price

q0q1



pc

Deadweight

Loss

So does a floor

price set at pf,

a ceiling price set

at pc

PS

CS

QD, QS

Price

q0q1



pc

Deadweight

Loss

So does a floor

price set at pf,

a ceiling price set

at pc, and a ration

scheme that

allows only q1

units to be traded.

PS

pe

CS

Revenue received by holders of ration coupons.


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Calculating Gains and Losses

Estimates of the demand functions for households orrepresentative households would enable us tocalculate the impact of policy changes on eachhousehold in terms of equivalent or compensatingvariation.

This sort of analysis would therefore enable us toaddress distributional issues, ie., the question of whogains and who losses from the proposal.

Documents

Economics 102 Lecture 8 Ways to Measure Utility Rev