Consumer Surplus Revisited

Transcription

1 Consume Suplus Revsted Danel Schaffa Unvesty of Rchmond School of Law Novembe 3, 2018 Economsts have long studed how changes n pces affect consume wellbeng. Despte the consdeable pogess made towads esolvng ths fundamental economc nquy, cental questons eman. Ths pape evsts a topc that has not eceved much attenton n ecent yeas: consume suplus. Duput 1844 s cedted wth fst notng that demand functons could be used nfe the changes n consume welfae due to pce changes. 1 It dd not take long fo ctcs to suface. Bodas 1847 noted that consume suplus as Duput descbed t could not account fo changes n ethe ncome o othe pces. Thus the queston of how to thnk about consume suplus n a mult-good settng was ased fo the fst tme. Hotellng 1938 was the fst to suggest that the natual genealzaton [to n commodtes of the ntegal epesentng total beneft, of whch consumes suplus s a pat, s [a lne ntegal. Hotellng also noted that the lne ntegal depended on the path of ntegaton. Auebach 1985 povdes the most famla fomulaton of the agument that the change n consume suplus fom one pce vecto to anothe wll depend on the path though pce space that ths change takes. Ths, howeve, s nconsstent wth the defnton of consume suplus as the aea unde the demand cuve fom the cuent pce to an nfntely hgh pce. The aea unde a cuve s a mathematcal obect that depends only on the the cuve and the bounds of ntegaton. The change n consume suplus s then the dffeence between two aeas. Because both aeas ae functons of a pce vecto and not a pce path, the dffeence between these aeas must also be ndependent of path. dschaffa@gmal.com. I thank Jm Hnes, Dan Jaqua, and attendees of the Unvesty of Mchgan Publc Fnance Semna fo helpful comments. I gatefully acknowledge suppot fom the NIA tanng gant to the Populaton Studes Cente at the Unvesty of Mchgan T32 AG Any eos ae my own. 1 See Ekelund and Hebet 1985; Hnes 1999 fo futhe hstocal detal on the development of consume suplus.

2 Whle the aea unde the demand cuve does not suffe fom path-dependence, t s nonetheless mpefect. Fo some utlty functons, ncludng CES wth 0, the change n consume suplus s an adequate stand-n fo change n utlty, but thee ae also cases n whch change n consume suplus and change n utlty have dffeent sgns. Consume suplus as the aea unde a cuve also ases conceptual ssues. In many cases consume suplus s nfnte. In some cases the change n consume suplus s nfnte. Fo seveal utlty functons, nceasng all pces and ncome by the same facto can change consume suplus. Most fundamentally, consume suplus does not lend tself to an obvous economc ntepetaton. Whle consume suplus may have lttle use as an analytcal tool, t stll beas coectng the msconcepton that the aea unde the cuve can be path dependent. 1 Consume suplus as the aea unde a cuve Consume suplus s defned as the aea unde the demand cuve fom a gven pce to a pce of nfnty. In one dmenson: CSp = Whch can be gaphcally epesented. p xtdt 1 p xp apples If the pce of the good changes, then the consume suplus changes. If, say, the pce of apples nceases, the new consume suplus s the yellow lghte gey aea and the change n consume suplus s the ed dake gay aea. 2 2 I use subscpts to ndex goods and supescpts to ndex pce vectos. 2

3 p 1 p 0 xp apples In the sngle good case, the change n consume suplus can be ewtten as a sngle ntegal, but ths does not genealze to 2 o moe dmensons. CSp 1 CSp 0 = p 0 p 1 xtdt xtdt = xtdt 2 p 0 p 1 A lne ntegal n 2 o moe dmensons s not a change. To undestand how consume suplus may be computed wth abtay goods, consde fst the two-good case. Just as wth the sngle-good case, consume suplus changes wth own pce changes, but consume suplus also changes wth coss pce changes. As coss-pce changes shft the demand cuve, consume suplus changes. x 1 p 1, p 1 2 x 1 p 1, p 0 2 apples The blue dak gey aea s the ncease n consume suplus. Of couse t s possble fo both own- and coss-pce changes to take place smultaneously. 3

4 p 1 1 x 1 p 1, p 1 2 x 1 p 1, p 0 2 apples Note that the ode of the pce change does not affect ethe the ntal o fnal consume suplus. p 1 1 x 1 p 1, p 1 2 x 1 p 1, p 0 2 apples CS 0 1 yellow and CS 1 1 blue stpped ae the same egadless of whch pce changes fst; CS s same egadless of path. If p 1 s changed fst, then the paallelogam fomed between the two demand cuves s not ncluded n the CS fo ethe good. If p 2 s changed fst, then the paallelogam fomed between the two demand cuves s ncluded n the CS fo both goods and s theeby cancelled out. The ntal consume suplus fom x 1 s CS 0 1 = x 1 t 1, p 0 2dt 1 Afte the pce change, the consume suplus fom x 1 s CS 1 1 = The change n consume suplus fom x 1 s p 1 1 x 1 t 1, p 1 2dt 1 4

5 CS 1 = CS 1 1 CS 0 1 = x 1 t 1, p 1 2dt 1 p 1 1 x 1 t 1, p 0 2dt 1 whch cannot be epesented as a sngle ntegal bounded by and p 1 1 because p 2 has also changed. Pevous wok, summazed n Auebach 1985, fals to account fo the coss-pce effect on the change n consume suplus. It does show that 1 x 1 p 1, p 0 2dp p 0 2 x 2 p 1 1, p 2 dp 2 2 p 0 2 x 2, p 2 dp x 1 p 1, p 1 2dp 1 Howeve, these sets of ntegals fal to account fo changes n consume suplus due to cosspce changes outsde the ange of own-pce changes. Shown gaphcally p 1 1 p 1 2 x 1 p 1, p 1 2 x 1 p 1, p 0 2 p 0 2 x 2 p 1 1, p 2 x 2, p 2 apples oanges geen: 1 x 1 p 1, p 0 2dp p 0 2 x 2 p 1 1, p 2 dp 2 stpped ed: 2 p 0 2 x 2, p 2 dp x 1 p 1, p 1 2dp 1 When the coss pce effect s ncluded, the ode of the pce change does not matte, as seen below. 5

6 p 1 1 p 1 2 x 1 p 1, p 1 2 x 1 p 1, p 0 2 p 0 2 x 2 p 1 1, p 2 x 2, p 2 apples oanges The sold aeas ae nceases, and the stped aeas ae deceases. The geen aeas ae changes nduced by the the pce of apples, and the blue aeas ae changes nduced by the pce of oanges. These aeas wll be the same egadless of the ode, o path, of pce changes. CS = x p 1 1,..., p,..., p 1 ndp p 1 p 0 p 1 x p 1 1,..., p,..., p 1 ndp p 1 x,..., p,..., p 0 ndp = x,..., p,..., p 0 ndp x,..., p,..., p 0 ndp = p 0 [ x p 1 1,..., p,..., p 1 n x,..., p,..., p 0 n dp p 1 } {{ } suplus change fom coss-pce change Summng ove all pce changes p 0 x,..., p,..., p 0 ndp }{{} suplus change fom own-pce change CS = [ x p 1 1,..., p,..., p 1 ndp p 1 p 0 x,..., p,..., p 0 ndp It s woth notng that the decomposton of the change n suplus s path dependent even though the total s not. 6

7 CS = [ x p 1 1,..., p,..., p 1 n x,..., p,..., p 0 n dp p 1 } {{ } suplus change fom coss-pce change, own-pce fst x,..., p,..., p 0 ndp p 0 }{{} suplus change fom own-pce change, own-pce fst = [ x p 1 1,..., p,..., p 1 n x,..., p,..., p 0 n dp p 0 } {{ } suplus change fom coss-pce change, coss-pce fst Because x p 1 1,..., p,..., p 1 ndp p 0 }{{} suplus change fom own-pce change, coss-pce fst 1 x 1 p 1, p 0 2dp p 0 2 x 2 p 1 1, p 2 dp 2 2 p 0 2 x 2, p 2 dp x 1 p 1, p 1 2dp 1 2 The welfae popetes of consume suplus That the change n consume suplus s not path dependent makes t a lkele canddate fo use n welfae analyss. Ideally economsts would obseve ndect utlty functons, n whch case they would be able to know the pecse welfae mplcatons of any change n pce. Howeve, utlty functons and not obsevable, so economsts make assumptons about the undelyng utlty functons to daw welfae conclusons fom quantty and pce data. If changes n consume suplus ae addtve 3 and always have the same sgn as changes n ndect utlty, then the change n consume wll elay all the elevant nfomaton about welfae. Consume suplus has the advantage of equng only an estmate of the demand system to be computable, and the change n consume suplus s always addtve. Howeve, fo many utlty 3 Meanng that the change n consume suplus fom pce vecto p to pce vecto q s equal to the change n consume suplus fom p to added to the change n consume suplus fom to q,.e. CSp, q = CSp, + CS, q 7

8 functons, the change n consume suplus does not have a suffcently clea elatonshp to the change n ndect utlty to be an adequate substtute. An excepton s CES utlty wth 0. ux = 1 ln α x Fo that utlty functon, the change n ndect utlty caused by movng fom pce p to p s 4 v = 1 ln 1 p 1 1 p 1 The change n consume suplus caused by movng fom pce p to p s lm z y 1 ln z 1 α p p z 1 1 α 1 p p 1 1 Fo all CES utlty functons wth weakly complementay goods, 0, takng the lmt yelds ny1 ln p 1 p Thus ny v, meanng that computng the change n consume suplus s suffcent to make welfae clams. Howeve, when goods ae substtutable, the change n consume suplus may have the opposte sgn as the change n ndect utlty. Consde two pefect substtutes wth pces such that the consume s ndffeent between between them. The aea unde the both demand cuves s 0 because any ncease n pce leads to 0 quantty demanded. Thus consume suplus s 0 fo both goods. If ethe pce nceases, then thee wll be postve consume suplus fo the coss-good. Howeve, ndect utlty wll decease because the consume can only affod a less pefeed bundle at the hghe pce. Othe poblems ase fo seveal utlty functons. Fo Stone-Geay utlty, ux = β ln x γ, consume suplus s CSp = lm z 4 All devatons n appendx. [1 β γ z + p x 1 β p γ ln z 1 β γ p p x 1 β p γ ln p 8

9 whch s unbounded. Even moe poblematc, the change n consume suplus wll be unbounded fo pce changes such that 1 β p γ p γ 0. lm z 1 β p γ p γ ln z 1 β γ p γ p p x 1 β p γ ln p +p x 1 β p γ ln p 3 Concluson Because consume suplus s only useful as a means to measue the change n welfae fo some utlty functons, t has lttle use as an analytcal tool Indeed, makng the assumpton that consume suplus wll be an adequate suogate fo ndect utlty functon essentally assumes the utlty functon. Dong so makes dect computaton of the change n utlty pefeed to computng consume suplus. Nonetheless, f we agee as a dscplne that consume suplus s the aea unde a cuve, then the change n consume suplus wll not be path dependent. To the extent that ou atcles, books, and lesson plans state othewse, we should coect them. 9

10 Refeences Auebach, A. J., The theoy of excess buden and optmal taxaton. Handbook of Publc Economcs 1, Bodas, L., De la mesue de l utlté des tavaux publcs. In: Annales des Ponts et Chaussées. Vol. 13. pp Duput, J., On the measuement of the utlty of publc woks. Intenatonal Economc Papes , Ekelund, R., Hebet, R., Consume suplus: The fst hunded yeas. Hstoy of Poltcal Economy 17 3, Hnes, J. R., Thee sdes of habege tangles. Jounal of Economc Pespectves 13 2, Hotellng, H., The geneal welfae n elaton to poblems of taxaton and of alway and utlty ates. Econometca: Jounal of the Econometc Socety, A n good Constant elastcty ux = α x 1 ux = 1 ln α x such that L = 1 ln p x = y α x + λ y p x FOC 10

11 α x 1 1 α x 1 k α = λp kx k α x 1 λp α k x k k α x 1 = λp α k x k k = α x 1 p = α x 1 p λp α k x k k α 1 1 x p 1 1 p = x p α 1 1 x p 1 α 1 1 x p 1 1 x = vp = 1 ln vp = 1 ln vp = 1 ln vp = ln y ln 1 p 1 1 y p y p p α 1 1 p 1 1 p α p 1 1 vp = ln y = 1 y 1 1 p 1 1 p 1 = y y 1 α p x [ + 1 ln [ ln ln [ 1 1 p 1 1 α p 1 α 1 1 p 1 p 1 11

12 [ v = 1 ln CS = CS = y 1 v = 1 y z p z p p 1 1 ln 1 ln 1 p 1 1 p 1 1 t 1 1 t 1 1 t p α [ 1 1 t 1 1 p 1 1 p 1 dt 1 dt x b x b+1 + y = 1 b + 1 x b x b+1 /b y/b + 1 = 1 b + 1 lnxb+1 /b y/b + 1 [ 1 ln z 1 CS = 1 = 1 z p z p ln CS = y 1 1 t 1 1 t t 1 t 1 p 1 ln y 1 + p 1 1 t y 1 CS = 1 ln 1 1 z 1 p 1 ln z 1 1 p 1 dt p p 1 1 p dt z p p + p p p

13 y 1 ln z p p ln z p p y 1 ln z p p z 1 1 p p 1 1 Note that s pefect complements; 0 s Cobb-Douglas, and 1 s pefect substtutes. > 1 o < 0 mples that whch wll have the same sgn as 0, 1 = 1 < 0, so 1 y 1 > 0, so ln ny 1 ny1 v = 1 ln ln ln p 1 p p 1 1 p 1 p 1 p 1 1 p 1 1 p y 1 ln z y p p ln 1 1 p p z 1 1 p p 1 p p

14 y1 ny1 ln ln y 1 y 1 p 1 p 1 ln ln p 1 p p p 1 1 p p ln y p 1 p ln p p 1 p 1 p p 1 1 p y 1 ln p y p 1 + p 1 ln 1 p 1 1 p ln p ln p p 1 p 1 + p B n good Cobb-Douglas ux = α ln x such that L = p x = y α ln x + λy p x FOC 14

15 Thus α x = λp λp α x = λp α x p x α = p x α p x = α y check x = α y p vp = α y p = p α y α ln = p α y = y α ln α + ln y ln p v = α ln α + ln y ln p α ln α + ln y ln p v = α ln p CS = z p α y dt t CS = y α ln z ln p y α ln z ln p y α ln z ln p y α ln p 15

16 C n good Stone-Geay ux = x γ β OR ux = β ln x γ such that p x = y Note: ths assumes that all goods ae postvely consumed. L = β ln x γ + λ y p x FOC β x γ = λp β β λp = λp x γ x γ p β x γ = p β x γ p β x p β γ = p β x p β γ β p x β p γ = p x β p γ assumng β = 1 β y β p γ = p x p γ β 16

17 x = γ + β y β p γ p p x = γ + β y p β p γ β p p p γ p x = p γ + y p γ p γ β β vp = vp = p x p γ + p γ = y p γ β β β ln β [ln γ + β y p y β p γ γ p p γ + ln β ln p v = β [ln y p γ + ln β ln p β [ln y p γ + ln β ln p v = β [ln y p γ + ln β ln p ln y p γ ln β + ln p let c = y p γ v = β [ ln c ln p v = ln c β β ln p v = ln c β ln p CSp = z p [ γ + β y t β t γ β t t p γ dt 17

18 CSp = z p [ 1 β γ + β t y p γ dt CSp = [1 β γ z + β y p γ ln z 1 β γ p β y p γ ln p CSp = [ p 1 β γ z + β x p γ + p γ ln z 1 β γ p β y p γ ln p β β CSp = [1 β γ z + p x 1 β p γ ln z 1 β γ p p x 1 β p γ ln p [1 β γ z + p x 1 β p γ ln z 1 β γ p p x 1 β p γ ln p [1 β γ z + p x 1 β p γ ln z 1 β γ p p x 1 β p γ ln p 1 β γ z 1 β γ z + p x 1 β p γ ln z p x 1 β p γ ln z 1 β γ p 1 β γ p p x 1 β p γ ln p p x 1 β p γ ln p p x p x 1 β p γ p γ ln z 18

19 1 β γ p γ p p x 1 β p γ ln p p x 1 β p γ ln p p + p x + x p x 1 β γ p ln z 1 β γ p p + p x + x 1 β p + p γ ln p 1 + p x 1 β p γ ln p 19