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**Welfare measurement: individual CS, CV, EV and PS**

(Cost Benefit Analysis DEC 51304) Z&D 5 R. Jongeneel

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**Lecture Plan Welfare function of individual**

Willingness to pay & willingness to accept Compensated & Equivalent Variation Consumer & Producer Surplus measures Approximation and accuracy BC-A in Single consumer economy Interdependent utility (social prefs)

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**Welfare function of individual**

Understanding ‘economic man’ Utility: Change:

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**Welfare function of individual**

Conversion of utils in money Marginal utility of income:

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**Willingness to Pay & Accept**

WTP: 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

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**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?

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**€ Equivalent Utility Gains (benefit)**

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.

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**€ 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.

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**€ Equivalent Utility Gains (benefit)**

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.

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**€ Equivalent Utility Gains (benefit)**

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.

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**€ Equivalent Utility Gains**

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.

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**€ Equivalent Utility Gains**

r1 r2 r3 r4 r5 r6 1 2 3 4 5 6

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**€ Equivalent Utility Gains (surplus)**

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.

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**€ Equivalent Utility Gains (surplus)**

€ value of net utility gains-to-trade r1 r2 r3 r4 pG r5 r6 1 2 3 4 5 6

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**Compensating Variation and Equivalent Variation**

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

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**Compensating 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.

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**Compensating Variation**

p1=p1’ p2 is fixed. x2 u1 x1

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**Compensating Variation**

p1=p1’ p1=p1” p2 is fixed. x2 u1 u2 x1

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**Compensating Variation**

p1=p1’ p1=p1” p2 is fixed. x2 u1 u2 x1

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**Compensating Variation**

p1=p1’ p1=p1” p2 is fixed. x2 u1 CV = m2 - m1. u2 x1

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**Equivalent Variation 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.

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Equivalent Variation p1=p1’ p2 is fixed. x2 u1 x1

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Equivalent Variation p1=p1’ p1=p1” p2 is fixed. x2 u1 u2 x1

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Equivalent Variation p1=p1’ p1=p1” p2 is fixed. x2 u1 u2 x1

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**Equivalent Variation EV = m1 - m2. p1=p1’ p1=p1” p2 is fixed. x2 u1 u2**

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**Relationship CV, EV and B, C**

Comp. 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

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Consumer’s Surplus 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.

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Footnote on WTP curve 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.

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**Footnote on WTP & Demand**

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.

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**Consumer’s Surplus (€) Reservation price curve for gasoline**

Ordinary demand curve for gasoline $ value of net utility gains-to-trade pG Gasoline

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**Consumer’s Surplus (€) Reservation price curve for gasoline**

Ordinary demand curve for gasoline $ value of net utility gains-to-trade Consumer’s Surplus pG Gasoline

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Consumer’s Surplus The consumer’s utility function is quasilinear in x2. Take p2 = 1. Then the consumer’s choice problem is to maximize subject to

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**Consumer’s Surplus That is, choose x1 to maximize**

The first-order condition is That is, This is the equation of the consumer’s ordinary demand for commodity 1.

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**Consumer’s Surplus Ordinary demand curve, p1 CS**

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

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Consumer’s Surplus 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.

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**Consumer’s Surplus and P-change**

p1(x1) CS before

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**Consumer’s Surplus and P change**

p1(x1) CS after

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**Consumer’s Surplus and P change**

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

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**Consumer’s Surplus, Compensating Variation and Equivalent Variation**

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

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**Consumer’s Surplus, Compensating Variation and Equivalent Variation**

So when the consumer has quasilinear utility, CV = EV = DCS. But, otherwise, we have: Relationship 2: In size, EV < DCS < CV.

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**CV, EV and CS and price decline**

H(p,U1) H(p,U11) CS = A + B CV = A EV = A + B + C 1 C A B 2 M(p,m)

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Producer’s Surplus 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

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Producer’s Surplus Output price (p) Marginal Cost y (output units)

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**Producer’s Surplus y Output price (p) Marginal Cost Revenue =**

(output units)

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**Producer’s Surplus y Output price (p) Marginal Cost**

Variable Cost of producing y’ units is the sum of the marginal costs y (output units)

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**Producer’s Surplus y Revenue less VC is the Producer’s Surplus.**

Output price (p) Marginal Cost Variable Cost of producing y’ units is the sum of the marginal costs y (output units)

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**CBA in Individual consumer economy & Small project**

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**CBA in Individual consumer economy & Small project**

Single individual Single price change / income kept constant No 2nd market price effects First-best economy (no other distortions) Marshallian CS good approximation PS good approximation

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**CBA in Individual consumer economy & Small project**

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

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**CBA in Individual consumer economy & Small project**

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

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**CBA in Individual consumer economy & Small project**

+ -- q1.0 q1.1 q2.1 q2.0

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**CBA in Individual consumer economy & Small project**

Change in welfare: dW continued . Small project: price changes small Total benefits-method => income effect- method

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**Interdependent Utilities & Welfare Measurement**

Social preferences: Altruism Envy Formally:

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**Interdependence & Welfare**

Household h’s utility change:

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**Double Interdependence & Welfare**

h cares about k and k cares about h Int.dep. multiplier:

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