welfare measurement: individual cs, cv, ev and ps (cost benefit analysis dec 51304) z&d 5 r....

Welfare measurement: individual

CS, CV, EV and PS

(Cost Benefit Analysis DEC 51304)

Z&D 5

R. Jongeneel

Lecture PlanWelfare function of individualWillingness to pay & willingness to

acceptCompensated & Equivalent VariationConsumer & Producer Surplus

measuresApproximation and accuracyBC-A in Single consumer economy Interdependent utility (social prefs)

Welfare function of individual

Understanding ‘economic man’

Utility:

Change:

n21hh ,......xx,xUU

1

nh

h ii i

UdU dx

x

Welfare function of individual

Conversion of utils in money

Marginal utility of income:

1 1

n nh h

h i i i ii ih h

U UdU p dx p dx

Y Y

ih

h

i

h pY

U

x

U

Willingness to Pay & AcceptWTP: the amount one is prepared to

pay in order to avoid a cost

WTA: the amount one would like to have to forego a good

Related to CV & EV measures of welfare

WTP and Gains-to-Trade

You can buy as much gasoline as you wish at €1 per gallon once you enter the gasoline market.

Q: What is the most you would pay to enter the market?

Suppose gasoline can be bought only in lumps of one gallon.

Use r1 to denote the most a single consumer would pay for a 1st gallon -- call this her reservation price for the 1st gallon.

r1 is the euro equivalent of the marginal utility of the 1st gallon.

€ Equivalent Utility Gains (benefit)

Now that she has one gallon, use r2 to denote the most she would pay for a 2nd gallon -- this is her reservation price for the 2nd gallon.

r2 is the euro equivalent of the marginal utility of the 2nd gallon.


Generally, if she already has n-1 gallons of gasoline then rn denotes the most she will pay for an nth gallon.

rn is the euro equivalent of the marginal utility of the nth gallon.


r1 + … + rn will therefore be the euro equivalent of the total change to utility from acquiring n gallons of gasoline at a price of €0.

So r1 + … + rn - pGn will be the euro equivalent of the total change to utility from acquiring n gallons of gasoline at a price of €pG each.


A plot of r1, r2, … , rn, … against n is a reservation-price curve. This is not quite the same as the consumer’s demand curve for gasoline.

€ Equivalent Utility Gains

€ Equivalent Utility GainsReservation Price Curve for Gasoline

0

2

4

6

8

10

Gasoline (gallons)

(€) Res.Values

1 2 3 4 5 6

r1

r2

r3

r4

r5

r6

The euro equivalent net utility gain for the 1st gallon is €(r1 - pG)

and is €(r2 - pG) for the 2nd gallon, and so on, so the euro value of the

gain-to-trade is€(r1 - pG) + €(r2 - pG) + …

for as long as rn - pG > 0.

€ Equivalent Utility Gains (surplus)

€ Equivalent Utility Gains (surplus)Reservation Price Curve for Gasoline

0

2

4

6

8

10

Gasoline (gallons)

€

1 2 3 4 5 6

r1

r2

r3

r4

r5

r6

pG

€ value of net utility gains-to-trade

Two ‘true’ monetary measures of the total utility change caused by a price change and/or income change are Compensating Variation and Equivalent Variation.

Principle: measure difference between utility indifference curves

Compensating Variation and Equivalent Variation

Assume in a market price p1 rises.Q: What is the least extra income

that, at the new prices, just restores the consumer’s original utility level?

A: The Compensating Variation.

Compensating Variation


x2

x1x1

'

u1

x2'

p1=p1’ p2 is fixed.

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


x2

x1x1

'

x2'

x1"

x2"

u1

u2

p1=p1’p1=p1”

p2 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


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

CV = m2 - m1.

Assume in a market price p1 rises.Q: What is the least extra income

that, at the original prices, just restores the consumer’s original utility level?

A: The Equivalent Variation.

Equivalent Variation


x2

x1x1

'

u1

x2'

p1=p1’ p2 is fixed.

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


x2

x1x1

'

x2'

x1"

x2"

u1

u2

p1=p1’p1=p1”

p2 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 ' '" '"


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 ' '" '"

EV = m1 - m2.

Relationship CV, EV and B, CComp. Var. Equiv. Var

Definition Leave one as well off as before

Leave one as well off as after

Welfare gain (benefit)

WTP for the change

WTA to forego the change

Welfare loss (cost)

WTA as compensation for change

WTP to avert the change

Approximating the net utility gain area under the reservation-price curve by the corresponding area under the ordinary demand curve gives the Consumer’s Surplus measure of net utility gain.

Consumer’s Surplus

A consumer’s reservation-price curve is not quite the same as her ordinary demand curve. Why not?

A reservation-price curve describes sequentially the values of successive single units of a commodity.

An ordinary demand curve describes the most that would be paid for q units of a commodity purchased simultaneously.

Footnote on WTP curve

The difference between the consumer’s reservation-price and ordinary demand curves is due to income effects.

But, if the consumer’s utility function is quasilinear in income then there are no income effects and Consumer’s Surplus is an exact € measure of gains-to-trade.

Footnote on WTP & Demand


Gasoline

Reservation price curve for gasolineOrdinary demand curve for gasoline

pG

$ value of net utility gains-to-trade

(€)


Gasoline

Reservation price curve for gasolineOrdinary demand curve for gasoline

pG

$ value of net utility gains-to-tradeConsumer’s Surplus

(€)


U x x v x x( , ) ( )1 2 1 2

The consumer’s utility function isquasilinear in x2.

Take p2 = 1. Then the consumer’schoice problem is to maximize

U x x v x x( , ) ( )1 2 1 2 subject to

p x x m1 1 2 .

Consumer’s SurplusThat is, choose x1 to maximize

v x m p x( ) .1 1 1 The first-order condition is

v x p'( )1 1 0

That is, p v x1 1 '( ).

This is the equation of the consumer’sordinary demand for commodity 1.


Ordinary demand curve,p1p v x1 1 '( )

x1*x1

'

p1'

CSis exactly the consumer’s utility gain from consuming x1’ units of commodity 1.

CS v x dx p xx '( ) ' ''

1 1 1 101

v x v p x( ) ( )' ' '1 1 10

Consumer’s Surplus is an exact euro measure of utility gained from consuming commodity 1 when the consumer’s utility function is quasilinear in commodity 2.

Otherwise Consumer’s Surplus is an approximation.


Consumer’s Surplus and P-change

p1

x1*x1

'

CS before

p1(x1)

p1'

Consumer’s Surplus and P change

p1

x1*x1

'

CS afterp1"

x1"

p1(x1)

p1'

Consumer’s Surplus and P change

p1

x1*x1

'x1"

Lost CS

p1(x1), inverse ordinary demand curve for commodity 1.

p1"

p1'

Relationship 1: When the consumer’s preferences are quasilinear, all three measures are the same.

Consumer’s Surplus, Compensating Variation and Equivalent Variation

Consumer’s Surplus, Compensating Variation and Equivalent Variation

So when the consumer has quasilinearutility,

CV = EV = CS.

But, otherwise, we have:

Relationship 2: In size, EV < CS < CV.

CV, EV and CS and price declinep1

x1*

x1' x1

"

M(p,m)p1"

p1'

A BC

H(p,U1)H(p,U11) CS = A + B

CV = A

EV = A + B + C1

2

Changes in a firm’s welfare can be measured in dollars much as for a consumer.

Producer surplus is quasi-rent and not ‘real surplus’.

Amount for remunerating quasi-fixed factors

Producer’s Surplus


y (output units)

Output price (p)

Marginal Cost

p'

y'


y (output units)

Output price (p)

Marginal Cost

p'

y'

Revenue= p y' '


y (output units)

Output price (p)

Marginal Cost

p'

y'

Variable Cost of producingy’ units is the sum of themarginal costs


y (output units)

Output price (p)

Marginal Cost

p'

y'

Variable Cost of producingy’ units is the sum of themarginal costs

Revenue less VCis the Producer’sSurplus.

CBA in Individual consumer economy & Small project


Single individualSingle price change / income kept

constantNo 2nd market price effectsFirst-best economy (no other

distortions)Marshallian CS good approximationPS good approximation


P1.0

P1.1

P2.0

q1.0 q1.1 q2.1 q2.0

d ea

bc


Change in CS : P10 a b P11

Income constant : change in spending is zero!

q10 c b q11 – P10 a c P11 – q21 d e q 20 = 0Change in welfare (dW)dW = P10 a b P11 = P10 a c P11 + abc

dW = q10 a b q11 + q21 d e q 20

W


P1.0

P1.1

P2.0

q1.0 q1.1 q2.1 q2.0

d ea

bc

+ --


Change in welfare: dW continued

.

Small project: price changes small

.

11

101 1 1

2

( )Nq

i iqi

dW P x dx P x

1

N

i ii

dW P x

Total benefits-method => income effect- method

Interdependent Utilities & Welfare Measurement

Social preferences:

Altruism Envy

Formally:

knhh UxxxUU ,,....., 21

0

k

h

U

U0

k

h

U

U

kn

kk

hn

hh xxUxxU ,....,,.... 11

Interdependence & Welfare

Household h’s utility change:

1 1

1 1

1

n nh kh h k

h i ih ki ii k i

n nh k

i i kh i ii i

nh k

i i kh ii

U U UdU dx dx

x U x

p dx W p dx

p dx W dx

Double Interdependence & Welfare

h cares about k and k cares about h

Int.dep. multiplier:

hkkhkh WW

M

1

1

kiikhkh

n

k

n

i h

hh dxpWM

Y

UdU

1 1

welfare measurement: individual cs, cv, ev and ps (cost benefit analysis dec 51304) z&d 5 r....

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