economics 102 lecture 8 ways to measure utility rev
TRANSCRIPT
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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
($) Res.
Values
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
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Reservation Price Curve
Quantity
($) Res.
Values
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
Reservation Price Curve
Quantity
($) Res.
Values
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
p
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Reservation Price Curve
Quantity
($) 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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Reservation Price Curve
Half units
($) Res.
Values
1 2 3 4 5 6
r1
r3
r5
r7
r9
r11
7 8 9 10 11
Reservation Price Curve
Half units
($) 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
Reservation Price Curve
Half units
($) Res.
Values
1 2 3 4 5 6
r1
r3
r5
r7
r9
r11
7 8 9 10 11
pG
Consumers surplus
And if good 1 is available in one-quarter
units ...
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Reservation Price Curve
0
2
4
6
8
10
One quarter units
($) Res.
Values
1
Reservation Price Curve
0
2
4
6
8
10
One quarter units
($) Res.
Values
1
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Reservation Price Curve for Gasoline
0
2
4
6
8
10
($) 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
Reservation Price Curve
Good 1
(P) Res.
Prices
p
Reservation Price Curve
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1
Good 1
(P) Res.
Prices
p
Reservation Price Curve
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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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.
Q: What is the least extra income that, at the
new prices, just restores the consumers
original utility level?
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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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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x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1p1=p1p2 is fixed.
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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p1 rises.
Q: What is the least extra income that, at the
original prices, just restores the consumers
original utility level?
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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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" " "
m p x p x2 1 1 2 2' '" '"
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x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1p1=p1p2 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' '" '"
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
2/12/1
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.
2/12/1
2/12/1
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
An excise tax imposed at a rate of Pt
per traded unit destroys these gains.
pf
CS
Deadweight
Loss
So does a floor
price set at pf
PS
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QD, QS
Price
q0q1
An excise tax imposed at a rate of Pt
per traded unit destroys these gains.
pc
Deadweight
Loss
So does a floor
price set at pf,
a ceiling price set
at pc
PS
CS
QD, QS
Price
q0q1
An excise tax imposed at a rate of Pt
per traded unit destroys these gains.
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.