Benefit-Cost Analysis

Transcription

1 Chapter 12 Beneft-Cost Analyss Utlty Possbltes and Potental Pareto Improvement Wthout explct nstructons about how to compare one person s benefts wth the losses of another, we can not expect beneft-cost analyss to tell us whether a publc project should or should not be adopted. The best we can hope for from beneft-cost analyss s to learn whether a project s potentally Pareto mprovng. Consder a communty wth two selfsh people, one prvate good and m publc goods. An allocaton s determned by the quanttes of prvate good for persons 1 and 2 and the vector y of publc goods that s avalable. The two people are endowed wth a total of W unts of prvate good, whch s to be allocated between consumpton for person 1, consumpton for person 2, and nputs for the producton of publc good. Suppose that the amount of prvate goods needed to produce the vector y of publc goods s C(y). Then f the supply of publc goods s y, the total amount of prvate goods to be dvded between 1 and 2 s W C(y). Gven the vector y of publc goods, each feasble dstrbuton of prvate goods determnes a dstrbuton of utltes between persons 1 and 2. We defne the graph of such utlty dstrbutons to be the y-contngent utlty possblty fronter. One such curve s shown as UP (y) n Fgure Suppose that a publc project wll ncrease the amounts of publc goods from y to y at a cost of some reducton n total prvate consumpton. Wth the project n place, there s a new utlty possblty fronter UP (y ). In Fgure 12.1, nether of the two curves, UP (y) and UP (y ) les entrely beneath the other. Thus there s no unambguous way to determne whch s the better outcome. Some utlty dstrbutons are attanable only f the project s mplemented and others are attanable only f t s not.

2 2 Chapter 12. Beneft-Cost Analyss Fgure 12.1: Utlty Possbltes and beneft-cost U 2 UP (y) UP (y ) A U 1 Let us suppose that ntally the amount of publc goods s y and that the dstrbuton of prvate goods corresponds to the utlty allocaton marked A n the fgure. We see that the curve UP (y ) ncludes ponts that are above and to the rght of A. Ths mples that t s possble to change the supply of publc goods from y to y and stll have enough prvate goods left over so that both ndvduals can be made better off than they were at A. When ths s the case, we say that the project s potentally Pareto mprovng. These deas extend naturally to the case of more than two consumers. Where x s prvate consumpton of consumer, the set of all feasble allocatons s {(x 1,..., x n, y) x + C(y) W }, where C(y) s the cost n terms of prvate goods of producng the vector y of publc goods. The y-contngent utlty possblty fronter n n dmensons s then defned n the obvous way. Defnton 1 Suppose that the ntal allocaton of prvate and publc goods s A = (x 1,..., x n, y). A change n the amount of publc goods from y to y s potentally Pareto mprovng f there exsts a feasble allocaton A = (x 1,..., x n, y ) such that A s Pareto superor to A. A Beneft-cost Test Where the ntal allocaton s A = (x 1,..., x n, y), we defne an ndvdual s wllngness-to-pay for changng the vector of publc goods from y to y as the quantty of prvate goods that she would be wllng to sacrfce n return for ths change. Ths number could be ether postve or negatve, dependng on whether prefers y to y or vce versa.

3 3 Defnton 2 If the ntal allocaton s (x, y) = (x 1,..., x n, y), then ndvdual s wllngness to pay for changng the amount of publc goods to y s w where w solves the equaton U (x w, y ) = U (x, y). For an economy wth selfsh ndvduals, a smple beneft-cost test determnes whether a change s potentally Pareto mprovng. Theorem 1 If ndvduals are selfsh and the ntal allocaton s (x, y) where x + C(y) = W, then a change n the amount of publc goods from y to y s potentally Pareto mprovng f and only f the sum of ndvdual wllngnesses-to-pay for the change exceeds the dfference n cost C(y ) C(y). A detaled proof of Theorem 1 s found n the Appendx. But the dea behnd the proof s qute smple. If the sum of wllngnesses-to-pay exceeds total cost, then t s possble to change the amount of publc goods from y to y and pay for ths change by collectng an amount from each ndvdual that s smaller than her wllngness to pay. Dong so consttutes a Pareto mprovement. Conversely, f the change s potentally Pareto mprovng, there must be a way to dstrbute the costs of the change so that nobody s worse off after the project s mplemented and cost shares are assgned. By defnton, ths mples that each ndvdual s share of the cost s smaller than hs wllngness-to-pay for the project. Snce the cost shares add to the total cost of the project t follows that the project would pass the beneft-cost test. A Calculus-Based Necessary Condton Theorem 1 s very general n the sense that t does not depend on preferences beng convex or contnuous and t apples whether the change beng consdered s large or nfntesmal. But the weakness of ths theorem s that t only gves us a way to evaluate projects one at at tme. Applyng the beneft-cost crteron n ths theorem would requres a separate survey of consumers to determne the merts of every possble publc project. If we are wllng to assume that preferences are smooth and convex, then calculus-based methods allow us to make more sweepng judgments about the drecton of potental Pareto mprovements, whch depend smply on a comparson of margnal costs and margnal wllngnesses to pay. The frst order necessary condton for effcent provson of publc goods was elucdated by Samuelson [7] and s known as the Samuelson condton. Ths condton requres that at an nteror Pareto optmum, for each publc good,

4 4 Chapter 12. Beneft-Cost Analyss the sum of all consumers margnal rates of substtuton between that publc good and the prvate goods s equal to the margnal cost of that publc good. Whle the Samuelson condton can be used to determne whether an exstng allocaton s Pareto optmal, ths s not exactly the result that s needed for beneft-cost analyss. The task of beneft-cost studes s to determne whether specfc changes n the amount of publc goods could be fnanced n a way that the outcome s Pareto mprovng. In a convex envronment, t turns out that a smple extenson of Samuelson s result allows one to determne whether an ncrease or decrease n the amount of publc goods s potentally Pareto mprovng. Defnton 3 At a feasble allocaton (x, y), we say that an ncrease n the amount of publc good j passes (fals) the Samuelson test f at the allocaton (x, y) the sum of margnal rates of substtuton between publc good j and the prvate good s greater than (less than) the margnal cost C j (y) of publc good j. Under approprate convexty assumptons, the Samuelson test gves us smple necessary condtons for an ncrease or for a decrease n the amount of a publc good to be potentally Pareto mprovng. A proof of the followng result s found n the Appendx. Theorem 2 If preferences of each ndvdual are selfsh and convex and f the cost functon C(y) s convex n y, then a necessary condton for an ncrease n the amount of publc goods to be potentally Pareto mprovng s that an ncrease passes the Samuelson test. A necessary condton for a decrease n the amount of publc goods to be potentally Pareto mprovng s that an ncrease fals the Samuelson test. Beneft-Cost as a Frst Step An attractve feature of beneft-cost analyss s that t seems to allow analysts to make polcy recommendatons about specfc publc expendtures wthout takng a stand on questons of ncome dstrbuton. Ths ndependence of dstrbutonal consderatons s acheved by focussng on potental rather than actual Pareto mprovements. Efforts to construct a useful beneft cost analyss n the absence of dstrbutonal judgments has a long hstory n economcs. In 1939, Kaldor [5] and Hcks [4] artculated the vew that has come to be known as the New Welfare Economcs, and whch centers on the compensaton prncple. The compensaton prncple states that an nsttutonal change consttutes an mprovement f t s possble for the

5 5 ganers to compensate the losers for ther losses and stll be better off after the change, whether or not the redstrbuton actually takes place. 1 The modern consensus s that the case for dstrbuton-ndependent project evaluaton s much weaker than was orgnally hoped. Samuelson [6] demonstrated that f the utlty possblty sets correspondng to alternatve polces are not nested, then the socal orderngs mpled by these crtera are hghly unsatsfactory. Chpman and Moore pont out that n economes wth several prvate goods and no publc goods, nestng of the utlty possblty sets requres essentally that preferences be dentcal and homothetc, or wth some further qualfcatons, of the Gorman polar form. Bergstrom and Cornes [1] show that for an economy wth publc goods, utlty possblty sets correspondng to amounts of publc goods wll be nested only under specal crcumstances that are formally dual to the Gorman polar form. It s noteworthy that Hcks foundng manfesto of the New Welfare Economcs [4], does not appear to advocate acceptance of reforms that pass the compensaton test f compensaton s not actually pad. Hcks states that The man practcal advantage of our lne of approach s that t fxes attenton on compensaton. Every smple economc reform nflcts a loss on some people... Yet when such reforms have been carred through n hstorcal fact, the advance has usually been made amd the clash of opposng nterests, so that compensaton has not been gven, and economc progress has accumulated a roll of vctms, suffcent to gve all sound polcy a bad name. [4], p. 711 Much of the controversy surroundng the use of the crteron of potental Pareto mprovement seems to be avodable f we thnk of beneft-cost analyss as only a frst step n a project evaluaton. For example, f an ncrease n the amount of a publc good passes the margnal Samuelson test, then we know that some ncrease n the amount of the publc good s potentally Pareto mprovng. Of course ths does not mean that every possble way of mplementng the project s Pareto mprovng. All we know s that there would be some way of ncreasng the amount of ths publc good and dvdng costs so that everyone benefts. The next step n evaluaton of the project s to consder alternatve ways of fnancng ths project and estmatng who wll then be the wnners and the losers. Incentve problems wll 1 An elegant and enlghtenng ntellectual hstory of the compensaton prncple and the New Welfare Economcs s found n Chpman [3]. Chpman and Moore [2] present a rgorous treatment of these ssues, usng modern technques.

6 6 Chapter 12. Beneft-Cost Analyss normally prevent polcy makers from knowng exactly how much each ndvdual values the project and so estmates of the dstrbuton of wnners and losers wll be statstcal and not exact. It s usually not reasonable to expect that lterally everyone wll be better off after the change, but may be possble to use avalable nformaton to ensure that a very large fracton of the populaton benefts from the polcy and that very few ndvduals are sgnfcantly harmed. There s an nterestng asymmetry n the results of a margnal beneftcost tests. If an ncrease n the amount of a publc good fals the test, then t must be that there s no way to dvde the costs of the project n such a way to acheve a Pareto mprovement. The project can reasonably be descrbed as specal nterest legslaton. To make a case n favor of a project that fals the test, one would need to argue that mplementng ths project and payng for t wth a specfed tax scheme s lkely to acheve redstrbutve goals that for some reason could not be more effcently acheved through redstrbuton of prvate goods. Appendx Proof of Theorem 1 Let (x, y) be the ntal allocaton and let w be s wllngness to pay for the change from y to y. Suppose that the sum of wllngnesses to pay exceeds C(y ) C(y). Then there exsts ɛ > 0 such that (w ɛ) > C(y ) C(y). For each, let x = x (w ɛ). From the defnton of w, t follows that for each, U (x, y ) > U (x, y). Now x = x (w ɛ) < x ( C(y ) C(y) ) (12.1) Rearrangng terms n 12.1, we have x + C(y ) < x + C(y) W (12.2) Expresson 12.2 mples that that the allocaton (x, y ) s feasble. Therefore f the sum of wllngnesses to pay for the movement from y to y exceeds C(y ) C(y), then the change from y to y s potentally Pareto mprovng. Conversely, suppose that the change from y to y s potentally Pareto mprovng. Then there exsts a feasble allocaton (x, y ) such that U (x, y ) U (x, y) for all wth strct nequalty for some. From the defnton of

7 APPENDIX 7 s wllngness to pay, w, t follows that x x w for all wth strct nequalty for some Therefore w > (x x ) (12.3) Snce x + C(y) = W and snce feasblty of (x, y ) mples that x + C(y ) W, t must be that (x x ) C(y ) C(y) (12.4) Then from Expressons 12.3 and 12.4 t follows that w > C(y ) C(y).

8 8 Chapter 12. Beneft-Cost Analyss

9 Bblography [1] Theodore Bergstrom and Rchard Cornes. Independence of allocatve effcency from dstrbuton n the theory of publc goods. Econometrca, 51(6): , November [2] John Chpman and James Moore. The new welfare economcs Internatonal Economc Revew, 19(3): , October [3] John Chpman. The New Palgrave A Dctonary of Economcs, chapter Compensaton Prncple, pages Macmllan Press, London, [4] John R. Hcks. The foundatons of welfare economcs. Economc Journal, 49(196): , December [5] Ncholas Kaldor. Welfare propostons of economcs and nterpersonal comparsons of utlty. Economc Journal, 49(195): , September [6] Paul Samuelson. Evaluaton of real natonal ncome. Oxford Economc Papers, 2(1):1 29, January [7] Paul Samuelson. The pure theory of publc expendtures. Revew of Economcs and Statstcs, 36(4): , 1954.