on the difference between the compensating and equivalent variations due to a change in an...

Journal of Public Economics 73 (1999) 135–145

On the difference between the compensating andequivalent variations due to a change in an exogenously

determined commodity

Roger LathamDepartment of Economics, York University, 4700 Keele Street, North York, Ontario M3J 1P3,

Canada

Received 31 March 1995; received in revised form 31 August 1996; accepted 31 August 1998

Abstract

This paper derives necessary and sufficient conditions for the difference between thecompensating and equivalent variations due to a change in an exogenously determinedcommodity to be positive, zero, or negative. These conditions are expressed both in terms ofthe forms of the conditional expenditure and indirect utility functions and in terms of theexogenously determined commodity’s notional aggregate elasticity of substitution for theother commodities, its notional income elasticity, and its notional expenditure share. This isaccomplished without resorting to a Taylor approximation and the consideration of bounds. 1999 Elsevier Science S.A. All rights reserved.

Keywords: Compensating and equivalent variations; Exogenously determined commodity;Conditional; Notional

JEL classification: D61; H41

1. Introduction

The purpose of this paper is to determine those factors which affect thedifference between the compensating and equivalent variations due to a change inan exogenously determined commodity. This difference has been the subject of aconsiderable amount of theoretical and empirical work by policy analysts wantingto measure the welfare effects of changes in the levels of public goods,

0047-2727/99/$ – see front matter 1999 Elsevier Science S.A. All rights reserved.PI I : S0047-2727( 98 )00105-4

136 R. Latham / Journal of Public Economics 73 (1999) 135 –145

1externalities, and amenities. Hopefully, the subsequent theoretical results will helpto explain the observed differences between such variations.

The utility maximization model employed here is a special case of the rationingmodel introduced by Tobin and Houthakker (1950), and analysed using dualitymethods by Latham (1980) and Neary and Roberts (1980), with the market priceof a rationed commodity set equal to zero. It has been used before by Randall andStoll (1980) and, more recently, by Hanemann (1991). Randall and Stoll (1980)derive bounds on the difference between the compensating and equivalent

`variations due to a change in an exogenously fixed commodity, a la Willig (1976).Hanemann (1991) provides a sufficient condition for these variations to be equaland then proceeds, following Randall and Stoll (1980), to examine bounds on thepossible difference between them.

By contrast, the present paper obtains an exact expression for the differencebetween the compensating and equivalent variations and then derives necessaryand sufficient conditions for this difference to be positive, zero, or negative. Theseconditions are expressed both in terms of the forms of the conditional expenditureand indirect utility functions and in terms of the exogenously determinedcommodity’s notional aggregate elasticity of substitution for the other com-modities, its notional income elasticity, and its notional expenditure share. This isaccomplished without resorting to a Taylor approximation and the consideration ofbounds. Inter alia, it is shown that the smaller is the exogenously fixedcommodity’s notional aggregate substitutability for the other commodities, thelarger the absolute value of its notional income elasticity, and the larger itsnotional expenditure share, ceteris paribus, the greater will be the differencebetween the compensating and equivalent variations, independently of its sign.

The rest of the paper is organized as follows. The basic framework isestablished in Section 2 and then, in Section 3, the necessary and sufficientconditions in terms of the forms of the conditional expenditure and indirect utilityfunctions are derived. The notional elasticities are introduced in Section 4. InSection 5, the necessary and sufficient conditions in terms of these notionalelasticities are obtained. Finally, in Section 6, there is a brief summary andconclusion.

2. Preliminaries

The individual’s ordinal utility function is u 5 u(x, q), where x is an n-dimensional vector of commodities and q is an additional commodity, the level ofwhich is exogenously determined. This utility function is assumed to be continu-

1See, for example, Cummings et al. (1986) and Peterson et al. (1988).

R. Latham / Journal of Public Economics 73 (1999) 135 –145 137

2ous, strongly monotonic, and quasi-concave (for the most part, strictly) in all ofits arguments. Solving

max u(x, q) subject to px # y (1)x

yields the conditional Marshallian demand functions x5h( p, q, y) and theconditional indirect utility function v( p, q, y);u(h( p, q, y), q). This indirect utilityfunction has the usual properties with respect to p and y and, by the envelopetheorem, is strictly increasing in q. Solving

min px subject to u(x, q) $ u (2)x

gives the conditional Hicksian demand functions x5g( p, q, u) and the conditionalexpenditure function m( p, q, u);pg( p, q, u). This expenditure function has theusual properties with respect to p and u and, by the envelope theorem, is strictlydecreasing in q. Given that they exist and that u5v and y5m, the solutions to theproblems (Eqs. (1) and (2)) are identical and, therefore, u;v( p, q, m ( p, q, u))and y;m( p, q, v( p, q, y)).

0 1 1 0Suppose that q changes from q to q , q .q , while p and y remain constant.Since v( p, q, y) is strictly increasing in q, it follows that the individual’s optimal

0 0 1 1utility level rises from u 5v( p, q , y) to u 5v( p, q , y). The compensating andequivalent variational measures of this change, denoted by C and E respectively,are defined implicitly by

1 0v( p, q , y 2 C) 5 v( p, q , y) (3)

and

1 0v( p, q , y) 5 v( p, q , y 1 E) (4)

and explicitly by

1q

0 0 1 0 0C 5 m( p, q , u ) 2 m( p, q , u ) 5 2E m ( p, s, u ) ds (5)q

0q

and

1q

0 1 1 1 1E 5 m( p, q , u ) 2 m( p, q , u ) 5 2E m ( p, s, u ) ds (6)q

0q

2This ensures local nonsatiation. Note that in the case of a ‘bad’ like pollution, the exogenouslydetermined commodity q is defined to be pollution reduction.


3where C[(0, y) and E[(0, `). Thus, in general, C may be greater than, equal to,or less than E.

3. The difference between C and E

Conditions, which are both necessary and sufficient for the difference betweenC and E to be positive, zero, or negative, are derived. Straightaway, therelationship between C2E and the dependency of m , the conditional shadowu

price of utility, on q is established by means of the following:

Lemma. C5E if and only if m is independent of q; and C.(,)E if and only ifu

m is strictly increasing (strictly decreasing) in q.u

0 1Proof. Given that m ( p, s, u) is differentiable with respect to u for all u[(u , u ),q4 0 1it follows from the mean-value theorem that there exists a u*[(u , u ) such that

1 0 1 0m ( p, s, u )5m ( p, s, u )1m ( p, s, u*)(u 2u ). Using this and the fact thatq q qu 1q0m ( p, s, u*)5m ( p, s, u*), it follows from Eq. (6) that E52e m ( p, s, u )qu uq q

1 0q q1 0ds2e m ( p, s, u*)(u 2u ) ds. From this and Eq. (5),uq

0q

1q

*C 2 E 5E m ( p, s, u )(u 2 u ) dsuq 1 0 (7)0q

* *5 m ( p, q , u ) 2 m ( p, q , u ) (u 2 u )f gu 1 u 0 1 0

1 0 1Therefore, since u .u , C2E ., 5, or ,0 according as m ( p, q , u*) ., 5, oru0 0 1 1 0 0

,m ( p, q , u*). Hence, C5E for all q and q , q .q , and resulting u*[(u ,u1 0 1 1 0u ), if and only if m is independent of q; and C.(,) E for all q and q , q .q ,u

0 1and resulting u*[(u , u ), if and only if m is strictly increasing (strictlyu

decreasing) in q.Q.E.D.

Remark. Note that Eq. (7) provides an exact expression for the differencebetween C and E.

First, it can be stated that

Proposition 1. C5E if and only if

3 ¨These definitions have their origin in Maler (1974). Following Hanemann (1991), C and E aredefined as the negative of the quantities appearing in Willig (1976) and Randall and Stoll (1980) so that

1 0sign C5sign E5sign(u 2u ).4See Apostol (1974), p. 110.


m( p, q, u) 5 a( p, u) 2 b( p, q) (8)

or, correspondingly,

v( p, q, y) 5 w( p, y 1 b( p, q)) (9)

Proof. Necessity: from the preceding lemma, C5E requires that m ( p, q, u) beu

independent of q. Therefore, letting m ( p, q, u)5a ( p, u) (.0), it follows thatu u

m( p, q, u)5a( p, u)2b( p, q), where [2b( p, q)] is an arbitrary function of thevariables kept constant during the integration. Thus, the conditional expenditurefunction must take the form Eq. (8). Since a ±0, taking a partial inverse of Eq.u

(8) with respect to u yields the corresponding conditional indirect utility function(Eq. (9)).

Sufficiency: given that the conditional expenditure function takes the form Eq. (8),1 0it follows from Eqs. (5) and (6) that C5b( p, q )2b( p, q )5E.Q.E.D.

Thus, C5E if and only if the conditional expenditure function is additivelyseparable in u and q.

Remark. Note that the conditional indirect utility function (Eq. (5)) in Hanemann(1991), which forms the basis for his Proposition 1 (a sufficiency statement), is aspecial case of Eq. (9).

Second, it can be stated that

Proposition 2. C.E if and only if

m( p, q, u) 5 c( p, q, u) 2 d( p, q), c strictly increasing [ q (10)u


v( p, q, y) 5 z( p, q, y 1 d( p, q)) (11)

Proof. Necessity: from the lemma, C.E requires that m ( p, q, u) be strictlyu

increasing in q. Therefore, letting m ( p, q, u)5c ( p, q, u) (.0), it follows thatu u

m( p, q, u)5c( p, q, u)2d( p, q), where c is strictly increasing in q and [2d( p, q)]u

is an arbitrary function of the variables kept constant during the integration. Thus,the conditional expenditure function must take the form (Eq. (10)). Since c ±0,u

taking a partial inverse of Eq. (10) with respect to u yields the correspondingconditional indirect utility function Eq. (11).

Sufficiency: given that the conditional expenditure function takes the form Eq.1 1 0(10), it follows from Eqs. (5) and (6) that C2E5[c( p, q , u )2c( p, q ,

0u1 1 0 0 0 1 0 1 0u )]2[c( p, q , u )2c( p, q , u )]5e [c ( p, q , r)2c ( p, q , r)] dr. Since q .qu u

0u


1 0 0and c is strictly increasing in q, c ( p, q , r).c ( p, q , r) (.0) at each r[[u ,u u u1u ] and, hence, C.E.Q.E.D.Third, it can also be stated that

Proposition 3. C,E if and only if

m( p, q, u) 5 e( p, q, u) 2 f( p, q), e strictly decreasing in q (12)u


v( p, q, y) 5 t( p, q, y 1 f( p, q)) (13)

Proof. This is similar, mutatis mutandis, to that of Proposition 2 and is, therefore,omitted.

4. Notional elasticities

Now consider the corresponding well-known hypothetical problem

maxu(x, q) subject to px 1 pq # y (14)x,q

in which p is the parametric price that would induce the individual to choose thegiven level of q. Solving Eq. (14) yields the notional Marshallian demand

qˆ ˆfunctions x 5 h( p, p, y), q 5 h ( p, p, y), and the notional indirect utility functionqˆ ˆv̂( p, p, y) ; u(h( p, p, y), h ( p, p, y), which has the usual properties with respect

to p, p, and y. Solving

min px 1 pq subject to u(x, q) $ u (15)x,qqˆ ˆgives the notional Hicksian demand functions x 5 g( p, p, u), q 5 g ( p, p, u), and

qˆ ˆ ˆthe notional expenditure function m( p, p, u);pg( p, p, u)1pg ( p, p, u), whichhas the usual properties with respect to p, p, and u. By the envelope theorem

ˆ(Shephard’s lemma), m hp( p, p, u)5q. This implicitly defines p, the virtual pricep

of q, as a function of p, q, and u, i.e.

ˆp 5 p( p, q, u) (16)

21 21 21ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆwhere p 5 2 m (m ) , i51,..., n, p 5 (m ) , and p 5 2 m (m ) .p p p pp q pp u p u ppi i

The conditional and notional expenditure functions are related via

ˆ ˆ ˆm( p, q, u) ; m( p, p( p, q, u), u) 2 p( p, q, u)q (17)

ˆ ˆTherefore, m ( p, q, u);2p( p, q, u) and, hence, m ( p, q, u);2p ( p, q, u).q qu u

Using Eq. (16) and the fact that m 5m , it follows from this that:qu uq

21ˆ ˆm ; m (m ) (18)uq p u pp


qˆThe commodity q is said to be notionally normal or inferior according as h . oryqˆ ˆ ˆ,0. Partial differentiation of h ( p, p, y) 5 m ( p, p, v( p, p, y)) with respect to ypq q21ˆ ˆˆ ˆ ˆ ˆ ˆ ˆyields h 5 m v . Since v 5 (m ) (.0), it follows that m 5 m h . There-y p u y y u p u u y

fore:

21 ˆˆ ˆ ˆm 5 m p u h (19)p u u q q

q21 21ˆ ˆˆ ˆ ˆwhere u 5 pq(m ) . 0 and h 5 mh q are the notional expenditure share andq q y

ˆnotional income elasticity of q, respectively. By definition, h .or,0 according asq

q is notionally normal or inferior.ˆ ˆ ˆFrom the concavity of m( p, p, u) in ( p, p), it follows that m #0. Since m( p,pp

p, u) is homogeneous of degree one in ( p, p), applying Euler’s theorem givesn

ˆ ˆ ô m p 1 m p 5 mp i pi ni51ˆ ˆPartial differentiation of this with respect to p gives o m p 1 m p 5 0sop p i ppn ii5121ˆ ˆ ˆthat m 5 2 p o m p # 0. This can be expressed as m 5 2pp p p i ppii51n

21 21ˆˆ ˆ ˆ ˆ ˆ ˆqp o s u # 0, where s 5 m m(m m ) is the notional partial elasticityiq i iq p p p pi ii51of substitution or complementarity between q and the ith other commodity (in the

21ˆ ˆHicks–Allen sense) and u 5 p x (m ) . 0 is the notional expenditure share of thei i inˆ îth commodity. Since o u 5 1 2 u ( . 0), this can be rewritten asi q

i51

21 ˆˆ ˆm 5 2 qp (1 2 u )s # 0 (20)pp q q

n n 21ˆ ˆˆwhere s 5 o s u o u is the notional aggregate elasticity of substitutionS Dq iq i ii51 i51 5 ˆbetween q and the other commodities. Note that s must be $0.q

ˆ ˆ ˆPartial differentiation of Eq. (17) with respect to u yields m ; (m 2 q)p 1u p u

ˆ ˆ ˆm . Since m 5 q, it follows that m ; m ( . 0). Given this, Eqs. (19) and (20)u p u u

can be substituted into Eq. (18) to yield:

2121 ˆ ˆˆ ˆm ; 2 m q u h 1 2 u s (21)fs d guq u q q q q

i.e. the elasticity of the conditional shadow price of utility with respect to a change2121 ˆ ˆˆ în q, m q(m ) , is identical to 2 u h 1 2 u s . Thus, the effect of ah fs d g juq u q q q q

change in q on the conditional shadow price of utility, m , can be expressed inuqˆ ˆ ˆterms of q’s notional expenditure share, u , q 9s notional income elasticity, h , andq q

the notional aggregate elasticity of substitution between q and the other com-0

ˆ ˆ ˆ ]modities, s . Note, for use below, that if s 5 0 and h is finite, then m 5`, ,q q q uq 0ˆ ˆ ôr 2` according as h ,, 5, or .0; if s [(0, `) and h is finite, then m .,q q q uq

ˆ ˆ ˆ5, or ,0 according as h ,, 5, or .0; and if s 5` and h is finite, thenq q q

m 50.uq

5See Diewert (1974), p. 16.


5. The difference between C and E in terms of notional elasticities

Corresponding to Proposition 1, it can be stated that

Proposition 4. C5E if and only if either (i) q is a notional aggregate substitutefor the other commodities and its notional income elasticity is zero, or (ii) q is anotional perfect substitute for at least one of the other commodities and itsnotional income elasticity is finite.

Proof. From the lemma in Section 3, C5E if and only if m is independent of q,u

ˆ î.e. m 50. From Eq. (21), m 50 if and only if either (i) s [(0, `) and h 50,uq uq q q

ˆ ˆ ôr (ii) s 5` and h is finite. Also, by definition, s 5` if and only if at least oneq q q

ôf the values of s 5`.Q.E.D.iq

Remark. This proposition provides a necessary and sufficient condition for C5Ein notional terms. By contrast, the Hanemann (1991) statement that ‘If at least oneprivate market good is a perfect substitute for q, then C5E’ (his Proposition 1) isin conditional terms, i.e. based upon his conditional indirect utility function (Eq.(5)).


Proposition 5. C.E if and only if q is a notional aggregate substitute for theother commodities and its notional income elasticity is negative (i.e. q isnotionally inferior).

Proof. From the lemma, C.E if and only if m is strictly increasing in q, i.e.u

ˆ ˆm .0. From Eq. (21), m .0 if and only if s [(0, `) and h ,0.Q.E.D.uq uq q q

This result is depicted in Fig. 1. Since the partial inverse of the notionalqˆHicksian demand function g ( p, p, u) with respect to p is identical to 2m ( p, q,q

0 1 0 1u), it follows from Eqs. (5) and (6) that C5area q bdq .E5area q agq . Notethat in this diagram, although q is notionally inferior, it is not notionally Giffen;

qôtherwise, the notional Marshallian demand function h ( p, p, y) would beupward-sloping.

Únder the regime of Proposition 5, it follows from Eqs. (21) and (7) that theˆˆ ˆsmaller is s , the more negative is h , and the larger is u , ceteris paribus, theq q q

greater will be m and, hence, C2E (.0). In other wordsuq

Corollary 1. Given Proposition 5, the smaller is q’s notional aggregate sub-stitutability for the other commodities, the more negative its notional incomeelasticity, and the larger its notional expenditure share, ceteris paribus, thegreater will be the positive difference between C and E.



Fig. 1.

Proposition 6. C,E if and only if q is a notional aggregate substitute for theother commodities and its notional income elasticity is positive (i.e. q is notionallynormal).

Proof. From the lemma, C,E if and only if m is strictly decreasing in q, i.e.u

ˆ ˆm ,0. From Eq. (21), m ,0 if and only if s [(0, `) and h .0.Q.E.D.uq uq q q

This result is illustrated in Fig. 2. In this case, it follows from Eqs. (5) and (6)0 1 0 1that C5area q bdq ,E5area q agq .´Under the regime of Proposition 6, it follows from Eqs. (21) and (7) that the

ˆˆ ˆsmaller is s , the more positive is h , and the larger u , ceteris paribus, the greaterq q q

will be (2m ) and, hence, E2C (.0). In other words,uq

Corollary 2. Given Proposition 6, the smaller is q’s notional aggregate sub-stitutability for the other commodities, the more positive its notional incomeelasticity, and the larger its notional expenditure share, ceteris paribus, thegreater will be the negative difference between C and E.

Combining Corollaries 1 and 2, it can be concluded that

Corollary 3. Given Propositions 5 and 6, the smaller is q’s notional aggregatesubstitutability for the other commodities, the larger the absolute value of its


Fig. 2.

notional income elasticity, and the larger its notional expenditure share, ceterisparibus, the greater will be the absolute value of the difference between C and E.

Remark. By contrast, the results in Hanemann (1991) are approximate andconcerned with the determination of bounds on the possible difference between Cand E (see, in particular, his Proposition 3 and expressions Eqs. (15) and (17)).

6. Conclusion

In this paper, necessary and sufficient conditions for the difference between thecompensating and equivalent variations due to a change in an exogenouslydetermined commodity to be greater than, equal to, or less than zero have beenderived. The difference is positive, zero, or negative if and only if the conditionalexpenditure and indirect utility functions take the forms Eqs. (10) and (11), Eqs.(8) and (9), or Eqs. (12) and (13), respectively.

Concomitantly, it has been shown that the crucial determinants of this differenceare the exogenously fixed commodity’s notional aggregate elasticity of substitutionfor the other commodities, its notional income elasticity, and its notionalexpenditure share. The difference is zero if and only if either the exogenouslyfixed commodity is a notional aggregate substitute for the other commodities and


notionally income inelastic or the exogenously fixed commodity is a notionalperfect substitute for at least one of the other commodities and its notional incomeelasticity is finite. The difference is positive (negative) if and only if theexogenously fixed commodity is a notional aggregate substitute for the othercommodities and notionally inferior (normal). Finally, the smaller is the exogen-ously fixed commodity’s notional aggregate substitutability for the other com-modities, the larger the absolute value of its notional income elasticity, and thelarger its notional expenditure share, ceteris paribus, the greater will be thedifference between the compensating and equivalent variations, whatever its sign.

Thus, the difference between the compensating and equivalent variations arisingfrom a change in an exogenously determined commodity has been completelycharacterized in both conditional and notional terms. These results are intended tohelp clarify those studies which use these variations in order to measure thewelfare effects of policy changes in certain areas of public and environmentaleconomics.

Acknowledgements

The author is grateful to two anonymous referees and Tony Atkinson for helpfulcomments but the usual caveat applies.

References

Apostol, T.M., 1974. Mathematical Analysis. Addison-Wesley, London.Cummings, R.G., Brookshire, D.S., Schulze, W.D., 1986. Valuing Public Goods: An Assessment of the

Contingent Valuation Method. Rowman and Allanheld, Totowa, NJ.Diewert, W.E., 1974. A note on aggregation and elasticities of substitution. Canadian Journal of

Economics 7, 12–20.Hanemann, W.M., 1991. Willingness to pay and willingness to accept: how much can they differ?.

American Economic Review 81, 635–647.Latham, R., 1980. Quantity constrained demand functions. Econometrica 48, 307–313.

¨Maler, K.-G., 1974. Environmental Economics: A Theoretical Inquiry. Johns Hopkins University Press,Baltimore.

Neary, J.P., Roberts, K.W.S., 1980. The theory of household behaviour under rationing. EuropeanEconomic Review 13, 25–42.

Peterson, G.L., Driver, B.L., Gregory, R. (Eds.), 1988. Amenity Resource Valuation: IntegratingEconomics with Other Disciplines. Venture, State College, PA.

Randall, A., Stoll, J.R., 1980. Consumer’s surplus in commodity space. American Economic Review71, 449–457.

Tobin, J., Houthakker, H.S., 1950. The effects of rationing on demand elasticities. Review of EconomicStudies 18, 140–153.

Willig, R., 1976. Consumer’s surplus without apology. American Economic Review 66, 589–597.

on the difference between the compensating and equivalent variations due to a change in an...

Documents