Consumer s Surplus Molly W. Dahl Georgetown University Econ 101 Spring 2009 1
Inverse Demand Functions Taking quantity demanded as given and then asking what the price must be describes the inverse demand function of a commodity. Usually we ask Given p 1 what is the quantity demanded of? But we could also ask the inverse question Given that the quantity demanded is, what must p 1 be? 2
Inverse Demand Functions p 1 Given p 1, what quantity is demanded of commodity 1? Answer: units. p 1 * 3
Inverse Demand Functions p 1 p 1 Given p 1, what quantity is demanded of commodity 1? Answer: units. The inverse question is: Given units are demanded, what is the price of x * commodity 1? 1 Answer: p 1 4
Consumer s Surplus p 1 p 1 CS Consumer s surplus is the consumer s utility gain from consuming units of commodity 1. * 5
Change in Consumer s Surplus The change to a consumer s total utility due to a change to p 1 is approximately the change in her Consumer s Surplus. 6
Change in Consumer s Surplus p 1 p 1 ( ), the inverse ordinary demand curve for commodity 1 p 1 * 7
Change in Consumer s Surplus p 1 p 1 ( ) p 1 CS before * 8
Change in Consumer s Surplus p 1 p 1 ( ) p 1 " CS after p 1 " * 9
Change in Consumer s Surplus p 1 p 1 ( ) p 1 " p 1 Lost CS " * 10
In Class: Calculating Consumer Surplus 11
Producer s Surplus Changes in a firm s welfare can be measured in dollars much as for a consumer. 12
Producer s Surplus Output price (p) S = Marginal Cost y (output units) 13
Producer s Surplus Output price (p) S = Marginal Cost p y y (output units) 14
Producer s Surplus Output price (p) S = Marginal Cost p Revenue = py y y (output units) 15
Producer s Surplus Output price (p) p y S = Marginal Cost Variable Cost of producing y units is the sum of the marginal costs y (output units) 16
Producer s Surplus Output price (p) Revenue less VC is the Producer s Surplus. S = Marginal Cost p Variable Cost of producing y units is the sum of the marginal costs y y (output units) 17
Cost-Benefit Analysis 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 the Consumer s Surplus and the Producer s Surplus. 18
Cost-Benefit Analysis Price The free-market equilibrium Supply p 0 Demand q 0 Q D, Q S 19
Cost-Benefit Analysis Price The free-market equilibrium and the gains from trade generated by it. Supply CS p 0 PS Demand q 0 Q D, Q S 20
Cost-Benefit Analysis Price CS The gain from freely trading the q 1 th unit. Consumer s gain Supply p 0 PS Producer s gain Demand q 1 q 0 Q D, Q S 21
Cost-Benefit Analysis Price The gains from freely trading the units from q 1 to q 0. Consumer s gains CS Supply p 0 PS Producer s gains Demand q 1 q 0 Q D, Q S 22
Cost-Benefit Analysis Price The gains from freely trading the units from q 1 to q 0. Consumer s gains CS Supply p 0 PS Producer s gains Demand q 1 q 0 Q D, Q S 23
Cost-Benefit Analysis Price p 0 CS PS Consumer s gains Producer s gains 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 1 q 0 Q D, Q S 24
Cost-Benefit Analysis t Price p b CS Tax Revenue An excise tax imposed at a rate of $t per traded unit destroys these gains. Deadweight Loss p s PS q 1 q 0 Q D, Q S 25
Cost-Benefit Analysis Price p f CS An excise tax imposed at a rate of $t per traded unit destroys these gains. Deadweight Loss So does a floor price set at p f PS q 1 q 0 Q D, Q S 26
Cost-Benefit Analysis Price CS An excise tax imposed at a rate of $t per traded unit destroys these gains. Deadweight Loss So does a floor price set at p f, a ceiling price set at p c p c PS q 1 q 0 Q D, Q S 27
Cost-Benefit Analysis Price p e p c 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 p f, a ceiling price set at p c, and a ration scheme that allows only q 1 units to be traded. q 1 q 0 Q D, Q S Revenue received by holders of ration coupons. 28
Compensating Variation and Equivalent Variation Two additional dollar measures of the total utility change caused by a price change are Compensating Variation and Equivalent Variation. 29
Compensating Variation p 1 rises. Q: What is the extra income that, at the new prices, just restores the consumer s original utility level? Or, after the policy has been implemented, how much must you be compensated to reach the same utility as before the policy? A: The Compensating Variation. 30
Compensating Variation p 1 =p 1 p 2 is fixed. 1 1 1 2 2 m = p x + p x u 1 31
Compensating Variation " p 1 =p 1 p 1 =p 1 u 1 p 2 is fixed. 1 1 1 2 2 = px " 1 " 1 + p " 2 m = p x + p x " u 2 32
Compensating Variation " " p 1 =p 1 p 1 =p 1 p 2 is fixed. 1 1 1 2 2 = px " 1 " 1 + p " 2 m = p x + p x " " 2 = p1x1 p2x u 1 m + " 2 u 2 " " 33
Compensating Variation " " p 1 =p 1 p 1 =p 1 p 2 is fixed. 1 1 1 2 2 = px " 1 " 1 + p " 2 m = p x + p x " " 2 = p1x1 p2x u 1 m + " 2 u 2 CV = m 2 -m 1. " " 34
Equivalent Variation p 1 rises. Q: What is the extra income that, at the original prices, just restores the consumer s original utility level? Or, how much would you pay to avoid moving to the new policy? A: The Equivalent Variation. 35
Equivalent Variation p 1 =p 1 p 2 is fixed. 1 1 1 2 2 m = p x + p x u 1 36
Equivalent Variation " p 1 =p 1 p 1 =p 1 u 1 p 2 is fixed. 1 1 1 2 2 = px " 1 " 1 + p " 2 m = p x + p x " u 2 37
Equivalent Variation " " p 1 =p 1 p 1 =p 1 u 1 p 2 is fixed. 1 1 1 2 2 = px " 1 " 1 + p " 2 " " 2 = 1 1 + 2 2 m = p x + p x m p x p x u 2 " " 38
Equivalent Variation " p 1 =p 1 p 1 =p 1 p 2 is fixed. 1 1 1 2 2 = px " 1 " 1 + p " 2 " " 2 = 1 1 + 2 2 m = p x + p x m p x p x " u 2 u 1 EV = m 1 -m 2. " " 39
Consumer s Surplus, Compensating Variation and Equivalent Variation When the consumer has quasilinear utility, CV = EV = CS. Why? There are no income effects with quasilinear utility. Otherwise, EV < CS < CV. 40