Volume 29, Issue 3. On the definition of nonessentiality. Udo Ebert University of Oldenburg

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1 Volume 9, Issue 3 On the definition of nonessentiality Udo Ebert University of Oldenburg Abstrat Nonessentiality of a good is often used in welfare eonomis, ost-benefit analysis and applied work. Various definitions of this property are presented in the literature on publi and environmental eonomis. This note larifies their relationship. I thank an Editor of the Bulletin for helpful omments. Citation: Udo Ebert, (009) ''On the definition of nonessentiality'', Eonomis Bulletin, Vol. 9 no.3 pp Submitted: Jul Published: August 0, 009.

2 Introdution A good is nonessential (for a onsumer) if there are situations under whih the onsumer is willing to do without it. E.g., luxury goods and goods whih are not vitally neessary an be nonessential. In the literature three different definitions of this onept have been presented and are used independently. This note investigates the onnetions between them. Two definitions are based on demand funtions. In this ase a good is alled nonessential if the demand for it an be zero. The demand funtion an be either unompensated (Marshallian) or ompensated (Hiksian). A third definition proposed by Willig requires that the onsumer an attain any (feasible) utility level without onsuming the nonessential good. Nonessentiality of a good is often introdued and disussed in theoretial and applied welfare eonomis. If demand an be zero, the demand urve intersets the prie axis and the orresponding onsumer surplus (CS) measure is always well defined. The property is therefore important whenever welfare measures are to be alulated sine it guarantees 1 the finiteness of the Marshallian CS and, respetively, the Hiksian measures. Moreover, the property of nonessentiality is employed in publi and environmental eonomis when the onept of weak omplementarity is used. A private good and a nonmarket good are weakly omplementary if the marginal willingness to pay for the nonmarket good is zero when the private good is not onsumed (f. e.g. Freeman 003). In this ase the private good must be nonessential. The onept allows to evaluate hanges in nonmarket goods (like environmental and publi goods) by observations on the related private good. This note reonsiders the alternative definitions of nonessentiality and ompletely larifies their relationship. It turns out that the definitions are not equivalent. The strongest one is based on the ompensated demand funtion. It implies the other ones whih are in turn independent from one another.. Alternative definitions of nonessentiality The framework and the notation are introdued first. We assume that there are only two (private) goods, and. Here we denote by the good whih is nonessential and by a Hiksian omposite whih ould be replaed by m ommodities without diffiulties in the analysis below: A ommodity bundle is denoted by (, ). We onsider one onsumer possessing a preferene ordering over bundles (, ). Its domain D is a subset of, i.e. the ordering is also defined for bundles ( ) ontains and 0, where. It is assumed that ( 0,0) D. The ordering is supposed to be ontinuous and onvex on D and inreasing in both ommodities on. Furthermore it is assumed that the demand orrespondene is single-valued and the ordering is represented by a (diret) utility funtion U, whih is ontinuous, inreasing and onave on D. Pries are given by ( ) (, ) and the range of U by ( ): (, ) (, ) { } p p R U = U D. Let M be the onsumer s (exogenous) inome. The ordinary (Marshallian) and, respetively, ompensated,, p, p, M and (Hiksian) demand funtions are denoted by ( p p M ), ( ) ( p, p, u ), ( p, p, u ) where u R ( U) is a utility level. 1 If the demand urve does not interset the prie axis CS an be finite or infinite depending on the underlying preferene ordering.

3 - - Given this framework we are able to introdue the onept of nonessentiality preisely. We present three different definitions: Definition a) Good is Marshall-nonessential (M-nonessential) : For all ( p, M ) there is a minimal prie p ( p, M) suh that ( ( ) ) p p, M, p, M = 0. (M) b) Good is Hiks-nonessential (H-nonessential) p, u U : For all ( ) R ( ) there is a minimal prie p ( p, u) p ( p, u), p, u = 0. ( ) ) Good is Willig-nonessential (W-nonessential), D, suh that ( ) : For all ( ) there is a quantity ( ) suh that U (, ) U 0, (, ) (H) =. (W) The minimal prie p and, respetively, p for whih onsumption of is zero is often alled hoke prie. The alternative definitions are suggested in the literature: A good is M-nonessential if one an find a hoke prie under whih its onsumption is driven to zero (Just, Hueth, and Shmitz 004, p. 1). The hoke prie p orresponds to the prie where the (Marshallian) demand urve intersets the prie axis. A good is alled H-nonessential if there is a (minimum) prie for it suh that the Hiksian demand equals zero (Bokstael and Kling 1988, p. 656 and Freeman 003, p. 11). Then the ompensated demand urve for the nonessential good intersets the prie axis at p. H-nonessentiality of a good is equivalent to a geometrial property of the preferene ordering: all indifferene urves interset the = 0 -axis. W-nonessentiality of a good requires that it an be omitted from the onsumption bundle without ompletely deimating the onsumer (Willig 1978, p. 30), i.e. that any bundle inluding the nonessential good an be mathed by a bundle exluding it (Johansson (1987), p. 46). In this ase there is a quantity of the other good(s) whih will ompensate the onsumer for the absene of (or loss of aess to) the nonessential good (Bokstael, MConnell, and Strand 1991, p. 39, and Bokstael and MConnell 1993, p. 148). The referenes mentioned demonstrate that nonessentiality is a relevant property in welfare eonomis and environmental eonomis. 3. Disussion and onlusion Now we want to investigate the relationship between the alternative definitions. At first we show that these properties an be satisfied simultaneously: (1) Existene: There exist preferene orderings satisfying (M), (H), and (W). Proof: Consider Example 1: The preferene ordering is represented by ( ) ( ) 1 1 U, = 1 and implies the demand funtions ( p, p, M) = ( M p 1) and (,, ) ( ) 1 p p u = p p u 1. It is obvious that p ( p, M) = M, p ( p, u) = pu and (, ) = ( 1). But the definitions presented are not equivalent as the following analysis demonstrates. In order to distinguish between the alternative onepts we extend nonessential by an appropriate term.

4 - 3 - () Impliations of (M): (M) does not imply (H) or (W). Proof: We introdue Example : In this ase the preferene ordering is represented by U(, ) = ln( 1) 1. It leads to the demand funtion (,, ) M p ( ( M p ) p ) p and the hoke prie p ( p, M) = M p (M) is satisfied. On the other hand U( 0, ) 1 obvious that (W) is violated sine there is no (, ) whenever U (, ) 0 p p M =. Thus = is always stritly negative. Therefore it is. For the same reason, for u 0 the (ompensated) demand for annot beome zero. Thus (H) is also not satisfied. In Example some indifferene urves ut the = 0 -axis (those orresponding to a stritly negative utility level), others do not (the remaining ones). Therefore (H) and (W) are not fulfilled. (3) Impliations of (W): (W) does not imply (M) or (H). Proof: We present Example 3: Consider U(, ) =. We get (, ) = ( ), i.e. (W) is satisfied. The orresponding demand funtions are given by ( p, p, M) = ( p p ) M ( p p) and ( p,, ) ( 1 ) p u u p p and R ( ) = =. Demand for good is always stritly positive sine p, M U. Therefore the properties (M) and (H) are violated. In Example 3 every indifferene urve uts the = 0 -axis, but the marginal willingness to pay for the first unit of is infinite if is equal to zero (the marginal rate of substitution MRS, = ). Thus there is no finite prie system between and is equal to ( ) (, ) p p for whih the onsumer hooses = 0. Finally we examine the impliations of (H): (4a) Impliation of (H): (H) implies (M). Obviously (H) and (M) are not independent. One would expet this result for the ase in whih is a normal good. Sine then the Hiksian demand urve intersets the Marshallian demand urve from above. But normality is not used in the proof: The result is also true if is (loally) inferior. Proof: Choose any U satisfying (H). Then define ˆ : = 0, ˆ : M p ˆ ˆ ˆ u: = U, for given ( p, M ). (H) implies that there is p (, ˆ p u) suh that p p, uˆ, p, uˆ = 0= p p, uˆ, p, uˆ = = M p. Then ( ( ) ) ˆ and ( ) (, ) = (, ˆ) ( ) ˆ =, and ( ) p p M p p u. (4b) Impliation of (H): (H) implies (W). The properties (H) and (W) are also related. Proof: In order to prove this laim we hoose any U satisfying (H). Then define uˆ : U,, D. Setting ˆ : 1 pˆ : = MRS, we obtain = ( ) for any ( ) (,, ), (,, ) ( ) U pˆ pˆ uˆ pˆ pˆ uˆ uˆ p = and ( ) =. Beause of (H) there exists p ( pˆ, uˆ) leading to

5 - 4 - ( ( ) ) and ( ) = = =. Thus (, ) = ˆ ˆ : = p p ˆ, u ˆ, p ˆ, u ˆ = 0 ( ˆ, ˆ) ( 0, ˆ) ˆ (, ) U U u U ( ) ˆ : = p pˆ, uˆ, pˆ, uˆ. Then. These investigations larify the relationship between (M), (H), and (W). Colleting the results we obtain Proposition (H) implies (M) and (W). Neither (M) nor (W) implies (H). (M) and (W) are independent. It turns out that Hiks-nonessentiality is the strongest property. It has to be emphasized that the proposition holds for all preferene orderings satisfying the basi assumptions. No further properties, like e.g. normality or inferiority of good, are required. Given the main result Hiks-nonessentiality seems to be preferable to Marshall- and Willignonessentiality sine it implies (M) and (W). But more important is the fat that given (H) any kind of welfare measurement with respet to hanges of the nonessential good an be performed: The areas behind the Marshallian and Hiksian demand funtions whih measure the ompensation required by the onsumer for an elimination of this good are always finite. Thus onsumer surplus and the Hiksian welfare measures are also well-defined and finite for an arbitrary hange in (the prie of) the nonessential good, i.e. they an be employed without any diffiulties in applied work. Referenes Bokstael, N.E. and C.L. Kling (1988) Valuing environmental quality: weak omplementarity with sets of goods Amerian Journal of Agriultural Eonomis 70, Bokstael, N.E. and K.E. MConnell (1993) Publi goods as harateristis of non-market ommodities Eonomi Journal 103, Bokstael, N.E., K.E. MConnell, and I.E. Strand (1991) Rereation in Measuring the demand for environmental quality by J.B. Braden and C.D. Kolstad, Eds., Elsevier Siene Publishers, Amsterdam, Chapter 3, Freeman, A.M. III (003) The measurement of Environmental and Resoure Values, nd ed., Resoures for the Future: Washington D.C. Johansson, Per-Olov (1987) The eonomi theory and measurement of environmental benefits, Cambridge University Press: Cambridge. Just, R.E., D.L. Hueth, and A. Shmitz (004) The welfare eonomis of publi poliy, Edward Elgar: Cheltenham. Willig, R.D. (1978) Inremental onsumer's surplus and hedoni prie adjustment Journal of Eonomi Theory 17, 7-53.