A Graphical Depiction of Hicksian Partial-Equilibrium Welfare Analysis

Transcription

1 A rahical eiction of Hicksian Partial-quilibrium Welfare Analysis Keir. Armstrong eartment of conomics Carleton University Ottawa, ON KS 5B6 May 22, 23

2 Abstract An inescaable conclusion to be drawn from the literature on the measurement of welfare is that the use of consumer s surlus is a bad idea. This is esecially true in the light of the fact that modern comuting ower facilitates the straightforward calculation of equivalent variation, the oerational welfare indicator that is most strongly justified by economic theory. Consequently, a correct welfare analysis of a rice-income change of the sort discussed in virtually every cost-benefit tet should be rincially in terms of Hicksian demand curves, not ordinary (Marshallian) ones. The resent aer elains how to construct a grahical deiction of such an analysis, in artial equilibrium, which may be adoted in teaching the rinciles of cost-benefit analysis to graduate and advanced undergraduate students. This is done by way of several detailed eamles covering the scoe of alicability of the technique.

3 . Introduction More than two decades have now assed since the lack of validity of consumer s surlus as a welfare measure was demonstrated rigorously and reasonable alternatives were roosed. Chiman and Moore (976, 98) showed that restrictive and emirically untenable assumtions on references are necessary in order to make consumer s surlus a valid measure. McKenzie and Pearce (976) and Vartia (983) rovided rocedures that facilitate the calculation of a generalized version of Hick s (942) equivalent variation an oerational welfare indicator that resolves the concetual roblem of welfare measurement (Slesnick, 998,. 22) without the imosition of restrictions on the form of the ordinary demand function other than the standard integrability conditions. esite these advances, and desite the fact that modern comutational ower is more than u to the task of handling their inherent comleity, consumer s surlus continues to be the overwhelming choice as a welfare indicator [in] emirical cost-benefit analyses (Slesnick, 998,. 2). Part of the blame for this state of affairs must be assigned to the tets used to teach cost-benefit analysis (CBA). While most of these tets acknowledge the roblems with using consumer s surlus as a welfare measure, they go ahead and do so anyway after either minimizing the attendant roblems or claiming that aroriate adjustments can be made when necessary. For eamle, Boardman et al. (2,. 64) states that the biased estimate of [equivalent variation] that results from using Marshallian rather than Hicksian demand schedules to measure [welfare change] deends on the size of the income effect associated with a rice change. Usually this income effect and, hence, the bias are small and can be safely ignored in CBA. [mhasis in the original.] ramlich (99,. 57), on the other hand, asserts that when measuring the utility gain to consumers from [a] rice reduction the income effect must be ecluded. To measure the utility gain eactly, an eerimenter might ask the consumer [how much comensation he would be willing to accet in lieu of] the rice change. While most tets discuss the concet of equivalent variation, none to my knowledge actually use it as the basis for welfare analysis. A useful first ste towards rectifying this situation might be the facility to construct a grahical deiction of a 2

4 welfare analysis of a rice-income change that is rincially in terms of Hicksian demand curves as oosed to ordinary (Marshallian) ones. The resent aer rovides such a facility in the contet of a two-good artial-equilibrium framework. It does so by way of several detailed eamles, beginning with a base case eamle in Section 2, and then roceeding to modifications of that eamle along articular qualitative dimensions in Section 3. The resulting cases cover the scoe of alicability of the technique. 2. An amle Following Sugden and Williams (978, ), consider a governmentoerated rail service between a city and one of its suburbs together with a cometitive market for suburban rental housing. For the sake of simlicity, assume that the total demands for these goods are generated by a reresentative (utility-maimizing) consumer. A roject is roosed that would reduce the rice of rail tris by some amount. Assuming that rail tris and housing are normal goods and comlements in consumtion (and therefore, by the Slutsky equation, gross comlements as well), such a rice reduction would shift the ordinary demand curve for housing to the right. This would induce an increase in the rice of housing that would, in turn, cause the ordinary demand curve for rail tris to shift to the left. Sugden and Williams (978) argue that the consumer s surlus in the rail market should be calculated as the area between the initial and final rail rices and to the left of the observed demand curve obtained by considering a rice ath that kees the housing market in equilibrium as the rice of rail tris is reduced in a monotonic sequence of very small stes. Since the consumer s and roducers surluses in the housing market offset one another eactly along this rice ath, the overall (Marshallian) welfare measure is simly the consumer s surlus in the rail market. A correct welfare analysis of the rail-rice-reduction roject would be in terms of equivalent variation areas to the left of Hicksian (or comensated) demand curves for rail tris and housing at the after-roject utility level. Since comensated rice effects are symmetric, the line integral that defines such a Hicksian surlus measure is ath indeendent. This means that the analyst is free to choose any rice ath connecting the 3

5 initial and final rices of the two goods as the basis for determining the overall welfare effect. The simlest choices involve changing one rice comletely and then the other. In order to deict an equivalent variation analysis in grahical terms, it is necessary to be able to show the ositions of articular Hicksian demand curves in ricequantity sace. Since such curves are not directly observable, this requirement boils down to showing how the Hicksian demand curves are related to their ordinary counterarts, which are observable in rincile. Critical means to this end are rovided by the Slutsky equation. Mas-Colell et al. (995,. 7) states the Slutsky equation in the contet of a continuous utility function reresenting a locally nonsatiated and strictly conve reference relation defined on the [m-dimensional uclidean sace] as, for all commodity-rice vectors and total eenditures w, h (, u) (, w) = + k k (, w) w k (, w), k {,, m}, where u = v(, w), v( ) denotes the associated indirect utility function, and h l ( ) and l ( ) denote the associated Hicksian and ordinary demand functions for commodity l. Since both housing and rail tris are normal goods, this equation (with m = 2) tells us that, at the oint of intersection between the ordinary and Hicksian demand curves for one of the goods, holding the rice of the other good fied, the latter curve is steeer than the former. Since utility is a decreasing function of rice along any ordinary demand curve, other things being equal, the corresonding family of Hicksian demand curves for a normal good is such that utility decreases from right to left in the relevant grah. If the chosen rice ath moves the rice of housing from its initial value to its final value after the rice of rail tris has been so moved, then the correct welfare analysis of the Sugden and Williams (978, ) eamle is illustrated by Figure. Resectively, subscrits and 2 denote rail tris and housing, and suerscrits and denote before- and after-roject values. On the rice ath that moves to (< ) and then 2 to 2 (> 2 ), the aggregate (consumer s lus roducers ) equivalent variation with resect to the housing market is the negative of the triangular area BCH bounded by the suly curve y 2 ( ), the Hicksian demand curve h 2 (,, u ) 4

6 that asses through the after-roject equilibrium oint (given by the intersection of the suly curve and the shifted ordinary demand curve (, ; w)), and a horizontal line drawn at the level of the initial housing rice 2. The relevant Hicksian demand curve in the market for rail tris is the one given by the initial rice of housing and the after-roject utility level. This curve must lie to right of the one given by the final rice of housing and the after-roject utility level since housing and rail tris are comlements, and to the left of the one given by the initial rice of housing and the (indirect) utility level at that rice and the final rice of rail tris since this utility level must be higher than the after-roject utility level (on account of the fact that indirect utility functions are non-increasing in each commodity rice). In algebraic terms, h (, 2, u ) must lie to the right of h (, 2, u ) and to the left of h (, 2, u*), where u* := v(, 2 ) > v(, 2 ) =: u. Both of these reference Hicksian demand curves are easy to locate in the grah of the market for rail tris: the former asses through the after-roject equilibrium oint (); the latter intersects the initial demand curve at the final rice of rail tris (oint ). The equivalent variation with resect to the market for rail tris is the area F to the left of the relevant Hicksian demand curve between the initial and final rices of rail tris. Note that because this curve lies to the right of Sugden and Williams (978) observed demand curve (through oints A and ), it is unclear from a qualitative grahical illustration such as Figure whether the overall aggregate equivalent variation F BCH is larger or smaller than its Marshallian counterart ( A ). If, instead of being cometitive, the suly of housing were controlled by a single roducer with a constant marginal cost of roduction c 2, the equivalent variation with resect to the market for rail tris would be (qualitatively) the same, but the aggregate equivalent variation with resect to the housing market would be the rectangular area BIJK bounded by the marginal cost curve, a horizontal line drawn at the level of the initial housing rice, and vertical lines drawn at the levels of the before- and after-roject quantities of housing echanged ( 2 and ), less the triangular area CHI bounded by the Hicksian demand curve h 2 (,, u ) that asses through the after-roject equilibrium oint (given by the intersection of the monooly eansion ath y 2 *( ) and the shifted ordinary demand curve (, ; w)), the horizontal line drawn at the level of the initial 5

7 housing rice, and the vertical line drawn at the level of the after-roject quantity of housing echanged. This variant of the Sugden and Williams (978, ) eamle is illustrated by Figure Modifications The two-good artial-equilibrium model analyzed above can be modified to some etent with resect to the qualitative characteristics of the goods without changing its essential structure. The Slutsky equation rovides four rules that govern the relationshis among these characteristics: a good that is both a comlement and normal must also be a gross comlement (C N C); a good that is both a gross comlement and inferior must also be a comlement (C I C); a good that is both a substitute and inferior must also be a gross substitute (S I S); and a good that is both a gross substitute and normal must also be a substitute (S N S). The symmetry roerty of comensated rice effects rovides a fifth rule: the two goods are either both substitutes or both comlements. These five rules imly eighteen distinct cases: (i) (ii), (iii) (iv), (v) (vi) (i) () (i), (ii) (iii), (iv) (v) (viii) C (N, N) (C, C) C (N, I) (C, C) (C, S) C (I, N) (C, C) (S, C) C (I, I) (C, C) (C, S) (S, C) (S, S) S (I, I) (S, S) S (I, N) (S, S) (S, C) S (N, I) (S, S) (C, S) S (N, N) (S, S) (S, C) (C, S) (C, C) ach of these cases can be characterized by a combination of one of eight ossible grahs for the first good ( rail tris ) together with one of four ossible grahs for the second good ( housing ). The eight ossible grahs for the first good differ with resect to the direction of the shift of the initial ordinary demand curve (L or R), whether the ordinary demand curves are flatter than their Hicksian counterarts (F) or vice versa (V), 6

8 and the osition of the relevant Hicksian demand curve (either to the left or to the right of one of the two reference Hicksian demand curves henceforth called and after the names of the critical oints they ass through). The four ossible grahs for the second good differ with resect to the direction of the shift of the initial ordinary demand curve and whether or not the ordinary demand curves are flatter than their Hicksian counterarts. More secifically, the twelve grahs and the cases they hel to characterize are as follows: L-F-L (i), (ii) L-F-R (iii), (v), (viii) L-V-L (), (i) L-V-R (iv), (vi), (i) R-F-L (iii) R-F-R (iv), (vi), (vii) R-V-L (ii) R-V-R (v), (vii), (viii) R-F (i), (iv), (v), (ii), (vi), (viii) R-V (ii), (vi), (viii) L-F (i), (v), (vii) L-V (iii), (vii), (i), (), (iii), (iv) In the receding list, the notation that the relevant Hicksian demand curve lies to the left of one of the reference curves (L or L) means that it also lies to the right of the other (R or R), and the notation that it lies to the right of or means that it is situated even further to the right of or. The twelve grahs are shown in Figures and 3 through and are ordered by the cases they characterize: Figure shows the two case-(i) grahs (discussed in the receding section); Figure 3 shows the grah of good 2 required to characterize case (ii) (along with the grah of the good in Figure ); Figure 4 shows the two case-(iii) grahs; 7

9 Figure 5 shows the grah of good required to characterize case (iv) (along with the grah of good 2 in Figure ); etc. 8

10 h (,, u ) 2 A F h (,, u ) 2 h (, 2, u*) h 2 (,, u ) B C H y ( ) 2 2 (, ) (, ) Figure. Case (i) C (Ν, Ν) (C, C) [rahs L-F-L and R-F] y 2 *( ) y ( ) c 2 B K C I J H MC (, ) (, ) 2 2 B C H Figure 2. Case (i*) Figure 3. rah R-V 9

11 y ( ) 2 2 F A H C B (, ) (, ) Figure 4. Case (iii) C (Ν, Ι) (C, S) [rahs R-F-L and L-V] F A F A Figure 5. rah L-V-R Figure 6. rah R-V-R

12 y ( ) 2 2 F A H C B (, ) (, ) 2 2 Figure 7. rah L-V-L Figure 8. rah L-F F A (, 2 ) (, 2 ) A F Figure 9. rah R-V-L Figure. rah L-F-R

13 A F Figure. rah R-F-R References Boardman, Anthony., avid H. reenberg, Aidan R. Vining, and avid L. Weimer. Cost-Benefit Analysis: Concets and Practice. 2nd ed. Uer Saddle River, N.J.: Prentice Hall, 2. Chiman, John S., and James Moore. The Scoe of Consumer s Surlus Arguments. In volution, Welfare and Time in conomics: ssays in Honour of Nicholas eorgescu- Roegen. d. A.M. Tang et al. Leington: Heath-Leington Books, 976, Chiman, John S., and James Moore. Comensating Variation, Consumer s Surlus, and Welfare. American conomic Review, 7, No. 5 (98), ramlich, dward M. A uide to Benefit-Cost Analysis. 2nd ed. Prosect Heights, Ill.: Waveland Press, 99. Hicks, John R. Consumers Surlus and Inde-Numbers. Review of conomic Studies, 9, No. 2 (942), McKenzie, eorge W., and Ivor Pearce. act Measures of Welfare and the Cost of Living. Review of conomic Studies, 43, No. 3 (976), Mas-Colell, Andreu, Michael. Whinston, and Jerry R. reen. Microeconomic Theory. New York: Oford University Press, 995. Slesnick, aniel T. mirical Aroaches to the Measurement of Welfare. Journal of conomic Literature, 36, No. 4 (998), Sugden, Robert, and Alan Williams. The Princiles of Practical Cost-Benefit Analysis. Oford: Oford University Press,

14 Vartia, Yrjö O. fficient Methods of Measuring Welfare Change and Comensated Income in Terms of Ordinary emand Functions. conometrica, 5, No. (983),