上财中级微观经济学课件Ch14Consumersurplus.ppt
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Chapter FourteenConsumer’s Surplus2024/8/161Intermediate Micro: Consumer Surplus Monetary Measures of Gains-to-Traden nYou can buy as much gasoline as you wish at $1 per gallon once you enter the gasoline market. n nQ: What is the most you would pay to enter the market?2024/8/162Intermediate Micro: Consumer Surplus n nA: You would pay up to the dollar value of the gains-to-trade you would enjoy once in the market.n nHow can such gains-to-trade be measured?Monetary Measures of Gains-to-Trade2024/8/163Intermediate Micro: Consumer Surplus n nThree such measures are:n nConsumer’s SurplusConsumer’s Surplusn nEquivalent Variation, andEquivalent Variation, andn nCompensating Variation.Compensating Variation.n nOnly in one special circumstance do these three measures coincide.Monetary Measures of Gains-to-Trade2024/8/164Intermediate Micro: Consumer Surplus n nSuppose gasoline can be bought only in lumps of one gallon.n nUse r1 to denote the most a single consumer would pay for a 1st gallon -- call this her reservation price for the 1st gallon.n nr1 is the dollar equivalent of the marginal utility of the 1st gallon.$ Equivalent Utility Gains2024/8/165Intermediate Micro: Consumer Surplus n nNow 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.n nr2 is the dollar equivalent of the marginal utility of the 2nd gallon.$ Equivalent Utility Gains2024/8/166Intermediate Micro: Consumer Surplus n nGenerally, if she already has n-1 gallons of gasoline then rn denotes the most she will pay for an nth gallon.n nrn is the dollar equivalent of the marginal utility of the nth gallon.$ Equivalent Utility Gains2024/8/167Intermediate Micro: Consumer Surplus n nr1 + … + rn will therefore be the dollar equivalent of the total change to utility from acquiring n gallons of gasoline at a price of $0.n nSo r1 + … + rn - pGn will be the dollar equivalent of the total change to utility from acquiring n gallons of gasoline at a price of $pG each.$ Equivalent Utility Gains2024/8/168Intermediate Micro: Consumer Surplus n nA 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 Gains2024/8/169Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains123456r1r2r3r4r5r62024/8/1610Intermediate Micro: Consumer Surplus n nWhat is the monetary value of our consumer’s gain-to-trading in the gasoline market at a price of $pG?$ Equivalent Utility Gains2024/8/1611Intermediate Micro: Consumer Surplus n nThe dollar equivalent net utility gain for the 1st gallon is $(r1 - pG)n nand is $(r2 - pG) for the 2nd gallon, n nand so on, so the dollar value of the gain-to-trade is$(r1 - pG) + $(r2 - pG) + …for as long as rn - pG > 0.$ Equivalent Utility Gains2024/8/1612Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains123456r1r2r3r4r5r6pG2024/8/1613Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains123456r1r2r3r4r5r6pG2024/8/1614Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains123456r1r2r3r4r5r6pG$ value of net utility gains-to-trade2024/8/1615Intermediate Micro: Consumer Surplus n nNow suppose that gasoline is sold in half-gallon units.n nr1, r2, … , rn, … denote the consumer’s reservation prices for successive half-gallons of gasoline.n nOur consumer’s new reservation price curve is$ Equivalent Utility Gains2024/8/1616Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains1 2 3 4 5 6r1r3r5r7r9r117 8 9 10 112024/8/1617Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains1 2 3 4 5 6r1r3r5r7r9r117 8 9 10 11pG2024/8/1618Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains1 2 3 4 5 6r1r3r5r7r9r117 8 9 10 11pG$ value of net utility gains-to-trade2024/8/1619Intermediate Micro: Consumer Surplus n nAnd if gasoline is available in one-quarter gallon units ...$ Equivalent Utility Gains2024/8/1620Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains1 2 3 4 5 6 7 8 9 10 112024/8/1621Intermediate Micro: Consumer Surplus $ Equivalent Utility Gains1 2 3 4 5 6 7 8 9 10 11pG2024/8/1622Intermediate Micro: Consumer Surplus $ Equivalent Utility GainspG$ value of net utility gains-to-trade2024/8/1623Intermediate Micro: Consumer Surplus n nFinally, if gasoline can be purchased in any quantity then ...$ Equivalent Utility Gains2024/8/1624Intermediate Micro: Consumer Surplus $ Equivalent Utility GainsGasoline($) Res.PricesReservation Price Curve for Gasoline2024/8/1625Intermediate Micro: Consumer Surplus $ Equivalent Utility GainsGasoline($) Res.PricespGReservation Price Curve for Gasoline2024/8/1626Intermediate Micro: Consumer Surplus $ Equivalent Utility GainsGasoline($) Res.PricespGReservation Price Curve for Gasoline$ value of net utility gains-to-trade2024/8/1627Intermediate Micro: Consumer Surplus n nUnfortunately, estimating a consumer’s reservation-price curve is difficult,n nso, as an approximation, the reservation-price curve is replaced with the consumer’s ordinary demand curve.$ Equivalent Utility Gains2024/8/1628Intermediate Micro: Consumer Surplus n nA consumer’s reservation-price curve is not quite the same as her ordinary demand curve. Why not?n nA reservation-price curve describes sequentially the values of successive single units of a commodity.n nAn ordinary demand curve describes the most that would be paid for q units of a commodity purchased simultaneously.Consumer’s Surplus2024/8/1629Intermediate Micro: Consumer Surplus n nApproximating 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 Surplus2024/8/1630Intermediate Micro: Consumer Surplus Consumer’s SurplusGasoline($)Reservation price curve for gasolineOrdinary demand curve for gasoline2024/8/1631Intermediate Micro: Consumer Surplus Consumer’s SurplusGasolineReservation price curve for gasolineOrdinary demand curve for gasolinepG($)2024/8/1632Intermediate Micro: Consumer Surplus Consumer’s SurplusGasolineReservation price curve for gasolineOrdinary demand curve for gasolinepG$ value of net utility gains-to-trade($)2024/8/1633Intermediate Micro: Consumer Surplus Consumer’s SurplusGasolineReservation price curve for gasolineOrdinary demand curve for gasolinepG$ value of net utility gains-to-tradeConsumer’s Surplus($)2024/8/1634Intermediate Micro: Consumer Surplus Consumer’s SurplusGasolineReservation price curve for gasolineOrdinary demand curve for gasolinepG$ value of net utility gains-to-tradeConsumer’s Surplus($)2024/8/1635Intermediate Micro: Consumer Surplus n nThe difference between the consumer’s reservation-price and ordinary demand curves is due to income effects.n nBut, 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 Surplus2024/8/1636Intermediate Micro: Consumer Surplus Consumer’s SurplusThe consumer’s utility function isquasilinear in x2.Take p2 = 1. Then the consumer’schoice problem is to maximizesubject to2024/8/1637Intermediate Micro: Consumer Surplus Consumer’s SurplusThe consumer’s utility function isquasilinear in x2.Take p2 = 1. Then the consumer’schoice problem is to maximizesubject to2024/8/1638Intermediate Micro: Consumer Surplus Consumer’s SurplusThat is, choose x1 to maximizeThe first-order condition is That is,This is the equation of the consumer’sordinary demand for commodity 1.2024/8/1639Intermediate Micro: Consumer Surplus Consumer’s SurplusOrdinary demand curve,p1CS2024/8/1640Intermediate Micro: Consumer Surplus Consumer’s SurplusOrdinary demand curve,p1CS2024/8/1641Intermediate Micro: Consumer Surplus Consumer’s SurplusOrdinary demand curve,p1CS2024/8/1642Intermediate Micro: Consumer Surplus Consumer’s SurplusOrdinary demand curve,p1CSis exactly the consumer’s utility gain from consuming x1 units of commodity 1.2024/8/1643Intermediate Micro: Consumer Surplus n nConsumer’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. n nOtherwise Consumer’s Surplus is an approximation.Consumer’s Surplus2024/8/1644Intermediate Micro: Consumer Surplus n nThe change to a consumer’s total utility due to a change to p1 is approximately the change in her Consumer’s Surplus. Consumer’s Surplus2024/8/1645Intermediate Micro: Consumer Surplus Consumer’s Surplusp1p1(x1), the inverse ordinary demand curve for commodity 12024/8/1646Intermediate Micro: Consumer Surplus Consumer’s Surplusp1CS beforep1(x1)2024/8/1647Intermediate Micro: Consumer Surplus Consumer’s Surplusp1CS afterp1(x1)2024/8/1648Intermediate Micro: Consumer Surplus Consumer’s Surplusp1Lost CSp1(x1), inverse ordinary demand curve for commodity 1.2024/8/1649Intermediate Micro: Consumer Surplus Consumer’s Surplusp1LostCSx1*(p1), the consumer’s ordinary demand curve for commodity 1.measures the loss in Consumer’s Surplus.2024/8/1650Intermediate Micro: Consumer Surplus n nTwo additional dollar measures of the total utility change caused by a price change are Compensating Variation and Equivalent Variation.Compensating Variation and Equivalent Variation2024/8/1651Intermediate Micro: Consumer Surplus n np1 rises.n nQ: What is the least extra income that, at the new prices, just restores the consumer’s original utility level?Compensating Variation2024/8/1652Intermediate Micro: Consumer Surplus n np1 rises.n nQ: What is the least extra income that, at the new prices, just restores the consumer’s original utility level?n nA: The Compensating Variation.Compensating Variation2024/8/1653Intermediate Micro: Consumer Surplus Compensating Variationx2x1u1p1=p1 p2 is fixed.2024/8/1654Intermediate Micro: Consumer Surplus Compensating Variationx2x1u1u2p1=p1 p1=p1 p2 is fixed.2024/8/1655Intermediate Micro: Consumer Surplus Compensating Variationx2x1u1u2p1=p1 p1=p1 p2 is fixed.2024/8/1656Intermediate Micro: Consumer Surplus Compensating Variationx2x1u1u2p1=p1 p1=p1 p2 is fixed.CV = m2 - m1.2024/8/1657Intermediate Micro: Consumer Surplus n np1 rises.n nQ: What is the least extra income that, at the original prices, just restores the consumer’s original utility level?n nA: The Equivalent Variation.Equivalent Variation2024/8/1658Intermediate Micro: Consumer Surplus Equivalent Variationx2x1u1p1=p1 p2 is fixed.2024/8/1659Intermediate Micro: Consumer Surplus Equivalent Variationx2x1u1u2p1=p1 p1=p1 p2 is fixed.2024/8/1660Intermediate Micro: Consumer Surplus Equivalent Variationx2x1u1u2p1=p1 p1=p1 p2 is fixed.2024/8/1661Intermediate Micro: Consumer Surplus Equivalent Variationx2x1u1u2p1=p1 p1=p1 p2 is fixed.EV = m1 - m2.2024/8/1662Intermediate Micro: Consumer Surplus n nRelationship 1: When the consumer’s preferences are quasilinear, all three measures are the same.Consumer’s Surplus, Compensating Variation and Equivalent Variation2024/8/1663Intermediate Micro: Consumer Surplus n nConsider first the change in Consumer’s Surplus when p1 rises from p1 to p1.Consumer’s Surplus, Compensating Variation and Equivalent Variation2024/8/1664Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent VariationIfthen2024/8/1665Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent VariationIfthenand so the change in CS when p1 risesfrom p1’ to p1” is2024/8/1666Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent VariationIfthenand so the change in CS when p1 risesfrom p1’ to p1” is2024/8/1667Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent VariationIfthenand so the change in CS when p1 risesfrom p1’ to p1” is2024/8/1668Intermediate Micro: Consumer Surplus n nNow consider the change in CV when p1 rises from p1’ to p1”.n nThe consumer’s utility for given p1 isand CV is the extra income which, at the new prices, makes the consumer’s utility the same as at the old prices. That is, ...Consumer’s Surplus, Compensating Variation and Equivalent Variation2024/8/1669Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent Variation2024/8/1670Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent VariationSo2024/8/1671Intermediate Micro: Consumer Surplus n nNow consider the change in EV when p1 rises from p1’ to p1”.n nThe consumer’s utility for given p1 isand EV is the extra income which, at the old prices, makes the consumer’s utility the same as at the new prices. That is, ...Consumer’s Surplus, Compensating Variation and Equivalent Variation2024/8/1672Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent Variation2024/8/1673Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent VariationThat is,2024/8/1674Intermediate Micro: Consumer Surplus Consumer’s Surplus, Compensating Variation and Equivalent VariationSo when the consumer has quasilinearutility,CV = EV = D DCS.But, otherwise, we have:Relationship 2: In size, EV < D DCS < CV.2024/8/1675Intermediate Micro: Consumer Surplus Compensating and Equivalent Variationn nthis presentation takes you through the calculation of compensating and equivalent variation using the Cobb-Douglas utility function as an example.n nFirst we calculate the demands for x and y before and after the price change and then the two welfare measures2024/8/1676Intermediate Micro: Consumer Surplus The demand functions when utility is Cobb-DouglasWe know that for Cobb-Douglas Utility functions with U = xayb that the demand function are: 2024/8/1677Intermediate Micro: Consumer Surplus Suppose we assume that M = 100, p1=1 & p2=1then with U(x,y) = x1/2 y1/2, that is a=b=1/22024/8/1678Intermediate Micro: Consumer Surplus So the initial demands for x and y with the price of x =1 ,price of y=1 and M= 100 are: xd = 50 and yD = 502024/8/1679Intermediate Micro: Consumer Surplus 5050xyThe Cobb-Douglas Utility Function with M=100 and px=py=12024/8/1680Intermediate Micro: Consumer Surplus n nSuppose we increase the price of x, what will the demand for x and y be now?2024/8/1681Intermediate Micro: Consumer Surplus Using our Cobb-Douglas demand functions we can calculate the new demand for x (and y) If px rises to 2then x = 25and y = 502024/8/1682Intermediate Micro: Consumer Surplus Successive clicks of the mouse reveals the effect of increasing the price of x. x5050y252024/8/1683Intermediate Micro: Consumer Surplus Compensating Variationn nTo calculate this we need to calculate the income required to ensure that we can reach the original utility at the new set of prices.n nThe original utility was U(50,50)=501/2 501/2n nU(50,50)=(50.50)1/2 (50 2 )1/2 = 502024/8/1684Intermediate Micro: Consumer Surplus 5050y25CV {Next we need to find the income that must be given back to a consumer to compensate him or her for the rise in prices. This is the compensating variation, and it can be seen by clicking on the mouse.2024/8/1685Intermediate Micro: Consumer Surplus What level of Income gives Utility =50 at the new prices?n nTo find this out we can use the demand functions we have derived for x and y.n nNext we substitute these demands (using the new prices (2,1) directly into the utility function and set it equal to U02024/8/1686Intermediate Micro: Consumer Surplus Applying the specific Cobb-Douglas utility function we get:2024/8/1687Intermediate Micro: Consumer Surplus Compensating Variation2024/8/1688Intermediate Micro: Consumer Surplus Equivalent VariationWE find this in a similar manner to compensating WE find this in a similar manner to compensating variation except that now we want to find the variation except that now we want to find the income required to ensure that we are at the new income required to ensure that we are at the new level of utility but with the old set of prices.level of utility but with the old set of prices.The new level of utility is: U(25,50) = 25The new level of utility is: U(25,50) = 251/21/250501/21/22024/8/1689Intermediate Micro: Consumer Surplus 5050y25EV{Clicking the mouse shows the amount of income we need to take away from the consumer at the old set of prices to bring them to the new utility level.Uo=35.5U1=502024/8/1690Intermediate Micro: Consumer Surplus Equivalent VariationOnce again we find the Income which will give us this Utility at the old prices.2024/8/1691Intermediate Micro: Consumer Surplus ResultnCompensating Variation = 141nEquivalent Variation =70.7nand since price has risen CV>EV as we saw in lectures2024/8/1692Intermediate Micro: Consumer Surplus n nChanges in a firm’s welfare can be measured in dollars much as for a consumer.Producer’s Surplus2024/8/1693Intermediate Micro: Consumer Surplus Producer’s Surplusy(output units)Output price (p)Marginal Cost2024/8/1694Intermediate Micro: Consumer Surplus Producer’s Surplusy(output units)Output price (p)Marginal Cost2024/8/1695Intermediate Micro: Consumer Surplus Producer’s Surplusy(output units)Output price (p)Marginal CostRevenue= 2024/8/1696Intermediate Micro: Consumer Surplus Producer’s Surplusy(output units)Output price (p)Marginal CostVariable Cost of producingy units is the sum of themarginal costs2024/8/1697Intermediate Micro: Consumer Surplus Producer’s Surplusy(output units)Output price (p)Marginal CostVariable Cost of producingy units is the sum of themarginal costsRevenue less VCis the Producer’sSurplus.2024/8/1698Intermediate Micro: Consumer Surplus 。
