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14 © 2010 W. W. Norton & Company, Inc. Consumer’s Surplus

Monetary Measures of Gains-to. Trade u You can buy as much gasoline as you wish at \$1 per gallon once you enter the gasoline market. u Q: What is the most you would pay to enter the market? © 2010 W. W. Norton & Company, Inc. 2

Monetary Measures of Gains-to. Trade u A: You would pay up to the dollar value of the gains-to-trade you would enjoy once in the market. u How can such gains-to-trade be measured? © 2010 W. W. Norton & Company, Inc. 3

Monetary Measures of Gains-to. Trade u Three such measures are: – Consumer’s Surplus – Equivalent Variation, and – Compensating Variation. u Only in one special circumstance do these three measures coincide. © 2010 W. W. Norton & Company, Inc. 4

\$ Equivalent Utility Gains u Suppose gasoline can be bought only in lumps of one gallon. u Use r 1 to denote the most a single consumer would pay for a 1 st gallon -- call this her reservation price for the 1 st gallon. u r 1 is the dollar equivalent of the marginal utility of the 1 st gallon. © 2010 W. W. Norton & Company, Inc. 5

\$ Equivalent Utility Gains u Now that she has one gallon, use r 2 to denote the most she would pay for a 2 nd gallon -- this is her reservation price for the 2 nd gallon. u r 2 is the dollar equivalent of the marginal utility of the 2 nd gallon. © 2010 W. W. Norton & Company, Inc. 6

\$ Equivalent Utility Gains u Generally, if she already has n-1 gallons of gasoline then rn denotes the most she will pay for an nth gallon. u rn is the dollar equivalent of the marginal utility of the nth gallon. © 2010 W. W. Norton & Company, Inc. 7

\$ Equivalent Utility Gains u r 1 + … + rn will therefore be the dollar equivalent of the total change to utility from acquiring n gallons of gasoline at a price of \$0. u So r 1 + … + rn - p. Gn will be the dollar equivalent of the total change to utility from acquiring n gallons of gasoline at a price of \$p. G each. © 2010 W. W. Norton & Company, Inc. 8

\$ Equivalent Utility Gains u. A plot of r 1, r 2, … , rn, … against n is a reservation-price curve. This is not quite the same as the consumer’s demand curve for gasoline. © 2010 W. W. Norton & Company, Inc. 9

\$ Equivalent Utility Gains r 1 r 2 r 3 r 4 r 5 r 6 1 2 © 2010 W. W. Norton & Company, Inc. 3 4 5 6 10

\$ Equivalent Utility Gains u What is the monetary value of our consumer’s gain-to-trading in the gasoline market at a price of \$p. G? © 2010 W. W. Norton & Company, Inc. 11

\$ Equivalent Utility Gains u The dollar equivalent net utility gain for the 1 st gallon is \$(r 1 - p. G) u and is \$(r 2 - p. G) for the 2 nd gallon, u and so on, so the dollar value of the gain-to-trade is \$(r 1 - p. G) + \$(r 2 - p. G) + … for as long as rn - p. G > 0. © 2010 W. W. Norton & Company, Inc. 12

\$ Equivalent Utility Gains r 1 r 2 r 3 r 4 r 5 r 6 p. G 1 2 © 2010 W. W. Norton & Company, Inc. 3 4 5 6 13

\$ Equivalent Utility Gains r 1 r 2 r 3 r 4 r 5 r 6 p. G 1 2 © 2010 W. W. Norton & Company, Inc. 3 4 5 6 14

\$ Equivalent Utility Gains r 1 r 2 r 3 r 4 r 5 r 6 \$ value of net utility gains-to-trade p. G 1 2 © 2010 W. W. Norton & Company, Inc. 3 4 5 6 15

\$ Equivalent Utility Gains u Now suppose that gasoline is sold in half-gallon units. u r 1, r 2, … , rn, … denote the consumer’s reservation prices for successive half-gallons of gasoline. u Our consumer’s new reservation price curve is © 2010 W. W. Norton & Company, Inc. 16

\$ Equivalent Utility Gains r 1 r 3 r 5 r 7 r 9 r 11 1 2 3 4 5 6 7 8 9 10 11 © 2010 W. W. Norton & Company, Inc. 17

\$ Equivalent Utility Gains r 1 r 3 r 5 r 7 r 9 r 11 p. G 1 2 3 4 5 6 7 8 9 10 11 © 2010 W. W. Norton & Company, Inc. 18

\$ Equivalent Utility Gains r 1 r 3 r 5 r 7 r 9 r 11 \$ value of net utility gains-to-trade p. G 1 2 3 4 5 6 7 8 9 10 11 © 2010 W. W. Norton & Company, Inc. 19

\$ Equivalent Utility Gains u And if gasoline is available in onequarter gallon units. . . © 2010 W. W. Norton & Company, Inc. 20

\$ Equivalent Utility Gains 1 2 3 4 5 6 7 8 9 10 11 © 2010 W. W. Norton & Company, Inc. 21

\$ Equivalent Utility Gains p. G 1 2 3 4 5 6 7 8 9 10 11 © 2010 W. W. Norton & Company, Inc. 22

\$ Equivalent Utility Gains \$ value of net utility gains-to-trade p. G © 2010 W. W. Norton & Company, Inc. 23

\$ Equivalent Utility Gains u Finally, if gasoline can be purchased in any quantity then. . . © 2010 W. W. Norton & Company, Inc. 24

\$ Equivalent Utility Gains (\$) Res. Prices Reservation Price Curve for Gasoline © 2010 W. W. Norton & Company, Inc. 25

\$ Equivalent Utility Gains (\$) Res. Prices Reservation Price Curve for Gasoline p. G Gasoline © 2010 W. W. Norton & Company, Inc. 26

\$ Equivalent Utility Gains (\$) Res. Prices Reservation Price Curve for Gasoline \$ value of net utility gains-to-trade p. G Gasoline © 2010 W. W. Norton & Company, Inc. 27

\$ Equivalent Utility Gains u Unfortunately, estimating a consumer’s reservation-price curve is difficult, u so, as an approximation, the reservation-price curve is replaced with the consumer’s ordinary demand curve. © 2010 W. W. Norton & Company, Inc. 28

Consumer’s Surplus u. A consumer’s reservation-price curve is not quite the same as her ordinary demand curve. Why not? u A reservation-price curve describes sequentially the values of successive single units of a commodity. u An ordinary demand curve describes the most that would be paid for q units of a commodity purchased simultaneously. © 2010 W. W. Norton & Company, Inc. 29

Consumer’s Surplus u 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. © 2010 W. W. Norton & Company, Inc. 30

Consumer’s Surplus (\$) Reservation price curve for gasoline Ordinary demand curve for gasoline Gasoline © 2010 W. W. Norton & Company, Inc. 31

Consumer’s Surplus (\$) Reservation price curve for gasoline Ordinary demand curve for gasoline p. G Gasoline © 2010 W. W. Norton & Company, Inc. 32

Consumer’s Surplus (\$) Reservation price curve for gasoline Ordinary demand curve for gasoline \$ value of net utility gains-to-trade p. G Gasoline © 2010 W. W. Norton & Company, Inc. 33

Consumer’s Surplus (\$) Reservation price curve for gasoline Ordinary demand curve for gasoline \$ value of net utility gains-to-trade Consumer’s Surplus p. G Gasoline © 2010 W. W. Norton & Company, Inc. 34

Consumer’s Surplus (\$) Reservation price curve for gasoline Ordinary demand curve for gasoline \$ value of net utility gains-to-trade Consumer’s Surplus p. G Gasoline © 2010 W. W. Norton & Company, Inc. 35

Consumer’s Surplus u The difference between the consumer’s reservation-price and ordinary demand curves is due to income effects. u 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. © 2010 W. W. Norton & Company, Inc. 36

Consumer’s Surplus The consumer’s utility function is quasilinear in x 2. Take p 2 = 1. Then the consumer’s choice problem is to maximize subject to © 2010 W. W. Norton & Company, Inc. 37

Consumer’s Surplus The consumer’s utility function is quasilinear in x 2. Take p 2 = 1. Then the consumer’s choice problem is to maximize subject to © 2010 W. W. Norton & Company, Inc. 38

Consumer’s Surplus That is, choose x 1 to maximize The first-order condition is That is, This is the equation of the consumer’s ordinary demand for commodity 1. © 2010 W. W. Norton & Company, Inc. 39

Consumer’s Surplus p 1 Ordinary demand curve, CS © 2010 W. W. Norton & Company, Inc. 40

Consumer’s Surplus p 1 Ordinary demand curve, CS © 2010 W. W. Norton & Company, Inc. 41

Consumer’s Surplus p 1 Ordinary demand curve, CS © 2010 W. W. Norton & Company, Inc. 42

Consumer’s Surplus p 1 Ordinary demand curve, CS © 2010 W. W. Norton & Company, Inc. is exactly the consumer’s utility gain from consuming x 1’ units of commodity 1. 43

Consumer’s Surplus u Consumer’s Surplus is an exact dollar measure of utility gained from consuming commodity 1 when the consumer’s utility function is quasilinear in commodity 2. u Otherwise Consumer’s Surplus is an approximation. © 2010 W. W. Norton & Company, Inc. 44

Consumer’s Surplus u The change to a consumer’s total utility due to a change to p 1 is approximately the change in her Consumer’s Surplus. © 2010 W. W. Norton & Company, Inc. 45

Consumer’s Surplus p 1(x 1), the inverse ordinary demand curve for commodity 1 © 2010 W. W. Norton & Company, Inc. 46

Consumer’s Surplus p 1(x 1) CS before © 2010 W. W. Norton & Company, Inc. 47

Consumer’s Surplus p 1(x 1) CS after © 2010 W. W. Norton & Company, Inc. 48

Consumer’s Surplus p 1(x 1), inverse ordinary demand curve for commodity 1. Lost CS © 2010 W. W. Norton & Company, Inc. 49

Consumer’s Surplus x 1*(p 1), the consumer’s ordinary demand curve for commodity 1. Lost CS measures the loss in Consumer’s Surplus. p 1 © 2010 W. W. Norton & Company, Inc. 50

Compensating Variation and Equivalent Variation u Two additional dollar measures of the total utility change caused by a price change are Compensating Variation and Equivalent Variation. © 2010 W. W. Norton & Company, Inc. 51

Compensating Variation u p 1 rises. u Q: What is the least extra income that, at the new prices, just restores the consumer’s original utility level? © 2010 W. W. Norton & Company, Inc. 52

Compensating Variation u p 1 rises. u Q: What is the least extra income that, at the new prices, just restores the consumer’s original utility level? u A: The Compensating Variation. © 2010 W. W. Norton & Company, Inc. 53

Compensating Variation x 2 p 1=p 1’ p 2 is fixed. u 1 x 1 © 2010 W. W. Norton & Company, Inc. 54

Compensating Variation x 2 p 1=p 1’ p 1=p 1” p 2 is fixed. u 1 u 2 x 1 © 2010 W. W. Norton & Company, Inc. 55

Compensating Variation x 2 p 1=p 1’ p 1=p 1” p 2 is fixed. u 1 u 2 x 1 © 2010 W. W. Norton & Company, Inc. 56

Compensating Variation x 2 p 1=p 1’ p 1=p 1” p 2 is fixed. u 1 u 2 CV = m 2 - m 1. x 1 © 2010 W. W. Norton & Company, Inc. 57

Equivalent Variation u p 1 rises. u Q: What is the least extra income that, at the original prices, just restores the consumer’s original utility level? u A: The Equivalent Variation. © 2010 W. W. Norton & Company, Inc. 58

Equivalent Variation x 2 p 1=p 1’ p 2 is fixed. u 1 x 1 © 2010 W. W. Norton & Company, Inc. 59

Equivalent Variation x 2 p 1=p 1’ p 1=p 1” p 2 is fixed. u 1 u 2 x 1 © 2010 W. W. Norton & Company, Inc. 60

Equivalent Variation x 2 p 1=p 1’ p 1=p 1” p 2 is fixed. u 1 u 2 x 1 © 2010 W. W. Norton & Company, Inc. 61

Equivalent Variation x 2 p 1=p 1’ p 1=p 1” p 2 is fixed. u 1 u 2 EV = m 1 - m 2. x 1 © 2010 W. W. Norton & Company, Inc. 62

Consumer’s Surplus, Compensating Variation and Equivalent Variation u Relationship 1: When the consumer’s preferences are quasilinear, all three measures are the same. © 2010 W. W. Norton & Company, Inc. 63

Consumer’s Surplus, Compensating Variation and Equivalent Variation u Consider first the change in Consumer’s Surplus when p 1 rises from p 1’ to p 1”. © 2010 W. W. Norton & Company, Inc. 64

Consumer’s Surplus, Compensating Variation and Equivalent Variation If © 2010 W. W. Norton & Company, Inc. then 65

Consumer’s Surplus, Compensating Variation and Equivalent Variation If then and so the change in CS when p 1 rises from p 1’ to p 1” is © 2010 W. W. Norton & Company, Inc. 66

Consumer’s Surplus, Compensating Variation and Equivalent Variation If then and so the change in CS when p 1 rises from p 1’ to p 1” is © 2010 W. W. Norton & Company, Inc. 67

Consumer’s Surplus, Compensating Variation and Equivalent Variation If then and so the change in CS when p 1 rises from p 1’ to p 1” is © 2010 W. W. Norton & Company, Inc. 68

Consumer’s Surplus, Compensating Variation and Equivalent Variation u Now consider the change in CV when p 1 rises from p 1’ to p 1”. u The consumer’s utility for given p 1 is and CV is the extra income which, at the new prices, makes the consumer’s utility the same as at the old prices. That is, . . . © 2010 W. W. Norton & Company, Inc. 69

Consumer’s Surplus, Compensating Variation and Equivalent Variation © 2010 W. W. Norton & Company, Inc. 70

Consumer’s Surplus, Compensating Variation and Equivalent Variation So © 2010 W. W. Norton & Company, Inc. 71

Consumer’s Surplus, Compensating Variation and Equivalent Variation u Now consider the change in EV when p 1 rises from p 1’ to p 1”. u The consumer’s utility for given p 1 is and EV is the extra income which, at the old prices, makes the consumer’s utility the same as at the new prices. That is, . . . © 2010 W. W. Norton & Company, Inc. 72

Consumer’s Surplus, Compensating Variation and Equivalent Variation © 2010 W. W. Norton & Company, Inc. 73

Consumer’s Surplus, Compensating Variation and Equivalent Variation That is, © 2010 W. W. Norton & Company, Inc. 74

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. © 2010 W. W. Norton & Company, Inc. 75

Producer’s Surplus u Changes in a firm’s welfare can be measured in dollars much as for a consumer. © 2010 W. W. Norton & Company, Inc. 76

Producer’s Surplus Output price (p) Marginal Cost y (output units) © 2010 W. W. Norton & Company, Inc. 77

Producer’s Surplus Output price (p) Marginal Cost y (output units) © 2010 W. W. Norton & Company, Inc. 78

Producer’s Surplus Output price (p) Marginal Cost Revenue = y (output units) © 2010 W. W. Norton & Company, Inc. 79

Producer’s Surplus Output price (p) Marginal Cost Variable Cost of producing y’ units is the sum of the marginal costs y (output units) © 2010 W. W. Norton & Company, Inc. 80

Producer’s Surplus Output price (p) Revenue less VC is the Producer’s Surplus. Marginal Cost Variable Cost of producing y’ units is the sum of the marginal costs y (output units) © 2010 W. W. Norton & Company, Inc. 81

Benefit-Cost Analysis u 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? u Yes, by using measures such as the Consumer’s Surplus and the Producer’s Surplus. © 2010 W. W. Norton & Company, Inc. 82

Benefit-Cost Analysis Price The free-market equilibrium Supply p 0 Demand q 0 © 2010 W. W. Norton & Company, Inc. Q D, Q S 83

Benefit-Cost Analysis Price The free-market equilibrium and the gains from trade generated by it. Supply CS p 0 PS Demand q 0 © 2010 W. W. Norton & Company, Inc. Q D, Q S 84

Benefit-Cost Analysis Price The gain from freely trading the q 1 th unit. Supply Consumer’s gain CS p 0 PS Producer’s gain q 1 © 2010 W. W. Norton & Company, Inc. q 0 Demand Q D, Q S 85

Benefit-Cost Analysis Price p 0 The gains from freely trading the units from q 1 to q 0. Consumer’s gains CS PS Producer’s gains q 1 © 2010 W. W. Norton & Company, Inc. q 0 Supply Demand Q D, Q S 86

Benefit-Cost Analysis Price p 0 The gains from freely trading the units from q 1 to q 0. Consumer’s gains CS PS Producer’s gains q 1 © 2010 W. W. Norton & Company, Inc. q 0 Supply Demand Q D, Q S 87

Benefit-Cost Analysis Price Consumer’s gains CS p 0 PS Producer’s gains q 1 © 2010 W. W. Norton & Company, Inc. q 0 Any regulation that causes the units from q 1 to q 0 to be not traded destroys these gains. This loss is the net cost of the regulation. Q D, Q S 88

Benefit-Cost Analysis Price pb An excise tax imposed at a rate of \$t per traded unit destroys these gains. Deadweight Loss CS Tax Revenue t ps PS q 1 © 2010 W. W. Norton & Company, Inc. q 0 Q D, Q S 89

Benefit-Cost Analysis Price pf An excise tax imposed at a rate of \$t per traded unit destroys these gains. Deadweight Loss CS So does a floor price set at pf PS q 1 © 2010 W. W. Norton & Company, Inc. q 0 Q D, Q S 90

Benefit-Cost Analysis Price An excise tax imposed at a rate of \$t per traded unit destroys these gains. Deadweight Loss CS pc So does a floor price set at pf, a ceiling price set at pc PS q 1 © 2010 W. W. Norton & Company, Inc. q 0 Q D, Q S 91

Benefit-Cost Analysis Price pe pc CS PS An excise tax imposed at a rate of \$t per traded unit destroys these gains. Deadweight Loss So does a floor price set at pf, a ceiling price set at pc, and a ration scheme that allows only q 1 units to be traded. q 1 q 0 Q D, Q S Revenue & Company, Inc. by holders of ration coupons. © 2010 W. W. Norton received 92