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

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**Welfare measurement: individualCS, CV, EV and PS**(Cost Benefit Analysis DEC 51304) Z&D 5 R. Jongeneel**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)**Welfare function of individual**• Understanding ‘economic man’ • Utility: • Change:**Welfare function of individual**• Conversion of utils in money • Marginal utility of income:**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**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?**€ 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.**€ 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.**€ 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.**€ 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.**€ 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.**€ Equivalent Utility Gains**r1 r2 r3 r4 r5 r6 1 2 3 4 5 6**€ 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.**€ Equivalent Utility Gains (surplus)**€ value of net utility gains-to-trade r1 r2 r3 r4 pG r5 r6 1 2 3 4 5 6**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**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.**Compensating Variation**p1=p1’ p2 is fixed. x2 u1 x1**Compensating Variation**p1=p1’p1=p1” p2 is fixed. x2 u1 u2 x1**Compensating Variation**p1=p1’p1=p1” p2 is fixed. x2 u1 u2 x1**Compensating Variation**p1=p1’p1=p1” p2 is fixed. x2 u1 CV = m2 - m1. u2 x1**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.**Equivalent Variation**p1=p1’ p2 is fixed. x2 u1 x1**Equivalent Variation**p1=p1’p1=p1” p2 is fixed. x2 u1 u2 x1**Equivalent Variation**p1=p1’p1=p1” p2 is fixed. x2 u1 u2 x1**Equivalent Variation**p1=p1’p1=p1” p2 is fixed. x2 u1 EV = m1 - m2. u2 x1**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.**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.**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.**Consumer’s Surplus**Reservation price curve for gasoline (€) Ordinary demand curve for gasoline $ value of net utility gains-to-trade pG Gasoline**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**Consumer’s Surplus**The consumer’s utility function isquasilinear in x2. Take p2 = 1. Then the consumer’schoice problem is to maximize subject to**Consumer’s Surplus**That is, choose x1 to maximize The first-order condition is That is, This is the equation of the consumer’sordinary demand for commodity 1.**Consumer’s Surplus**Ordinary demand curve, p1 is exactly the consumer’s utility gain from consuming x1’ units of commodity 1. CS**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.**Consumer’s Surplus and P-change**p1 p1(x1) CS before**Consumer’s Surplus and P change**p1 p1(x1) CS after**Consumer’s Surplus and P change**p1 p1(x1), inverse ordinary demand curve for commodity 1. Lost CS**Consumer’s Surplus, Compensating Variation and Equivalent**Variation • Relationship 1: When the consumer’s preferences are quasilinear, all three measures are the same.**Consumer’s Surplus, Compensating Variation and Equivalent**Variation So when the consumer has quasilinearutility, CV = EV = DCS. But, otherwise, we have:Relationship 2: In size, EV < DCS < CV.**CV, EV and CS and price decline**p1 H(p,U1) H(p,U11) CS = A + B CV = A EV = A + B + C 1 C A B 2 M(p,m)**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**Producer’s Surplus**Output price (p) Marginal Cost y (output units)**Producer’s Surplus**Output price (p) Marginal Cost Revenue= y (output units)**Producer’s Surplus**Output price (p) Marginal Cost Variable Cost of producingy’ units is the sum of themarginal costs y (output units)**Producer’s Surplus**Revenue less VCis the Producer’sSurplus. Output price (p) Marginal Cost Variable Cost of producingy’ units is the sum of themarginal costs y (output units)**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**CBA in Individual consumer economy & Small project**a P1.0 d e P2.0 b P1.1 c q1.0 q1.1 q2.0 q2.1**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