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Axiomatic foundations of Hicksian measures of welfare change
Journal
of Public
Economics
AXIOMATIC
33 (1987) 115-124.
North-Holland
FOUNDATIONS OF HICKSIAN WELFARE CHANGE
MEASURES
OF
Udo EBERT* University Received October
of Bonn, 5300 Bonn I, FRG
1984, revised version
received
The aggregate Hicksian measures ~&fZ~ and ~I,C~ are properties. One property concerns the ranking of two states. second property expresses the postulate that welfare changes are third one deals with redistributions of incomes. Distributional account by employing variable distributional weights li.
February
1987
characterized by three simple The measures are Paretian. A to be evaluated in money. The considerations are taken into
1. Introduction’ This paper explains why the Hicksian measures of change in welfare are so popular in public economics. The weighted sum of the equivalent variations x&Eq and the sum of the compensating variations x+Zr/; play an important role in cost-benefit analysis. Mishan (1976) discusses their properties. King (1983) applies these measures to cross section data of individual households in order to evaluate tax reforms.2 Just, Hueth and Schmitz (1982) recommend the use of c Eq and 1 Cl$ Ng accepts them as indicators of the magnitude of welfare change. This list of citations is by far not exhaustive. Thus we may ask why these measures are so attractive. In section 3 it is shown that x&Ev can be characterized by three simple properties3 which are rather weak. One property concerns the ranking of two states. The measure is Paretian. A second property expresses the postulate that welfare changes are to be evaluated in money. The third one deals with redistributions of incomes. Moreover the properties take into account distributional considerations by means of (variable) distributional weights ;li. The axiomatization of a weighted sum of equivalent variations in general holds only locally, i.e. for relatively small changes of prices and incomes. This result is ‘This paper is an extended and completely revised version of ‘Why to choose the Hicksian measures of change in welfare’, Discussion Paper 157, SFB 21, University of Bonn, 1984. I thank Hans-Dieter Stolper, two editors of this Journal, and three anonymous referees for most helpful comments and suggestions. ‘King’s definition of equivalent and compensating gain is identical to the concept of EV and CK respectively, used below. sAlready Mohring (1971) has specified a list of desiderata for a procedure to measure welfare change quantitatively and discussed common welfare measurement techniques.
0047-2727/87/$3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland).
116
U. Ebert, Hicksian measures of weljare change
reasonable since the distributional weights may depend on the old income or welfare distribution. Everyone who requires that a welfare measure possess these respective three properties has no choice: he must take C&E& Since these properties are desiderata in cost-benefit analysis this result explains the popularity of this measure. Section 4 discusses the case of c /1,Cv. Ev and Cv are only different with regard to the state which is stressed - the old one or the new one. Thus a similar characterization of c I.&v can easily be derived. But here the welfare weights have to depend on the new situation. Therefore it seems that x&E& is more suited to the measurement of welfare. As a byproduct we get an axiomatization of the unweighted sums c E v and 1 Cq which play a role in connection with compensation tests. This paper does not deal with the derivation of Ev or Cy from empirical data. For this problem the reader is referred to the results of Hause (1975) Willig (1976) Hausman (198 l), and Vartia (1983). Furthermore no characterizations of welfare measures of one individual are developed. These can be found in Ebert (1984).
2. Notation We consider n individuals and k commodities. Each individual i possesses an income mi>=O (i=l,..., n) and faces the price system p E IQ’!++. His or her preference ordering is represented by the ordinal indirect utility function v(p, mi) and the expenditure function Ei(ui, p), respectively, satisfying the usual properties [cf. Blackorby et al. (1978)]. A situation SE rWv: is described by a price vector p and all incomes m, (i= 1,. . . ,n). Y denotes the set of all possible situations. Let I/(S) be the . . . ,m,) we can convector (V,(p, m,), . . . , V,(p, m,)). Given a situation S=(p,m,, sider individual i’s ‘personal’ situation Si =(p, mi) E R”,+,‘. With the help of this notation we define individual i’s equivalent variation E<(S,+S,) and compensating variation CK (Si-+si) for the change from SET to SEY by
respectively. The ordering 5 in R” is defined by: x5 y (Vx=(x,, y,) E Ii?) if and only if xi 5 yi for all i, 15 i 5 n. y=(y,,..., 3. Characterization
. . . , x,),
of 1 liEVi
The aim of this section
is to characterize
a well-known
welfare
measure,
U. Ebert, Hicksian measures of welfare change
117
namely the weighted sum of the individual equivalent variations. At tirst we have to define this notion. Definition
A measure of change in welfare ([or shortly] welfare measure) I is defined by a function I:Y x Y+Iw’.
The meaning of a welfare measure I is the following. For any situations SEY and SEY, Z(S,3) evaluates the change in welfare for society if the individuals are in situation S before the change and in situation 3 after the change. In this regard the first argument of Z denotes the reference situation and the second one the new situation. If Z(S,$ is positive, welfare increases from S to 5. One important aspect of this definition is that welfare measures are applied to situations, i.e., the price systems and the individuals’ incomes are taken into account. Thus the severe restrictions implied by priceindependent welfare measurement are avoided a priori [cf. Roberts (1980)]. The magnitude of a welfare measure should reflect the changes in the individuals’ utilities. This requirement is expressed for a very special case by Property P (Pareto)
(P) When all individuals unanimously (weakly) prefer situation 3 to S the change in welfare from S to L!?must not be smaller than the change from S to S, i.e., if two situations S and g are both compared to the same status quo S and if s’ is preferred to S by all individuals (s is a Pareto improvement) the welfare measure should indicate this fact. This postulate is rather weak. Even if Z is constant, property P is satisfied. But there is an important implication of (P). I(&$) does not depend on the specific new situation s, but only on the individuals’ utilities in g. We have Proposition Z Zf the welfare measure Z satisfies Property P there exists a function J:Y x R(V)-+R’ such that Z(S,$)=J(S, V(g) for all S,SEY’, where R(V) is the range of V(S). Proof
Take any 3, ,!?EY
s.t.
V($) = I’(S (5. By (P) we have
Z(S,3) = Z(S,$. From this, Proposition
1 follows immediately.
0
This result may be compared with what is called welfarism in social choice theory [cf. Sen (1977)]. But there are two differences. In this context we
U. Ehert, Hicksian measures of welfare change
118
explore orderings which are not necessarily transitive. Furthermore we cannot derive an analogous result for the lirst argument of I from Property P. For most practical purposes one is interested in a monetary measure of welfare changes. In the framework given here we only have information about the price systems and incomes in different situations. This suggests using these data to normalize a measure I. Again we will consider a very specific case. We compare two situations S and 3 which differ only with respect to individual i,‘s income. The prices and the incomes of the other individuals do not change. The change in welfare can be measured by the difference in individual i,‘s income. In this special case Z(S,!?) coincides with i,‘s equivalent and compensating variation of income. This conforms with Harberger’s view of cost-benefit analysis [Harberger (1971,1978)], namely costs of benefits are evaluated ‘without regard to the individual(s) to whom they accrue’ (1971, p. 785). But in the following distributional considerations will be taken into account, as is postulated by most economists [e.g. Mishan (1976), Sen (1979), Hammond (1983,1984)]. Therefore we introduce i(S), a vector of positive distributional weights (i,(S), . . .,3.,(S)), where E.,(S) specifies the weight for i’s welfare gain or loss. 2 may depend on S which will be the status quo if we aim at the weighted sum of the equivalent variations. It may be desirable to eliminate or to diminish differences in income - in addition to a (general) welfare improvement. In this case it is reasonable to give more weight to the gains or losses of a lower income individual (e.g. ii(S) = 2) than to those of a higher income individual (e.g. L,(S) = 1); i.e. li generally varies in inverse proportion to the magnitude of individual i’s income. In the literature [cf. e.g. Atkinson (1970), Feldstein (1972), Pearce and Nash (1981), Brent (1984)] one specific type of weights is often proposed4:
I:= i m, is the average income and y 20 is a single constant where fi=n-’ elasticity. But of course many different types of weights can be imagined. For example ii(S) may be a function of the rank order of individual i’s income and/or utility (if the possibility of comparing the utility levels of all individuals is supposed [cf. Sen (1976), Hammond (1978), Ebert (1987)]). Moreover one could derive the weights &(S) from a social welfare function. However in this case it must be asked whether one is interested in developing an aggregate welfare measure when one has a social welfare function at hand. We consider this vector A(S) as given below. Employing distributional weights we postulate 4For an axiomatization
of these weights
see Ebert (1986).
119
U. Ebert, Hicksian measures OJwelfare change
WN (Normalization
using weights) (WN)
The welfare gain or loss of individual i, is multiplied by the corresponding weight liO, which may depend on the status quo S. This is a natural requirement since one needs a distribution of incomes or welfare levels as a basis of distributional weights. On the other hand weights which depend on the initial distribution are only valid or reasonable for relatively small changes of income or welfare - a fact which will be taken into account below. Property WN is compatible with the measurement of welfare changes for one individual [cf. Ebert (198431. We note _= Proposition 2 Assume I satisfies (WN): For all S, SEY s.t. p=p (if i,)I coincides with the weighted equivalent and compensating individual i,
and tii=rEi variation of
Z(S,S)=&&S)Eq)(Si,+Si,)=~i,(s) cigs,,4,,). The properties P and WN do not imply anything in case we have conflicting evaluations for different individuals, e.g. if individual i gains and j loses. Of course, one has to compare such situations by means of a welfare measure, too. The following property deals with a simple type of redistribution. Property
WR (Redistribution _= S,SEY::=Jk_=
Z(S,S)=Z for all i, j=l,...,
((
according
s, p’,A,
)...)
to weights)
ri&+L ~i(s)....'~j-~,...'~"
a >>
n, i# j, and O~a~~,(S)Gj
The main assumption of WR is again the condition that prices do not change. Given this WR allows for a redistribution of income in the new situation. Of course one has to take account of the distributional weights. The welfare change when going from s to 3 remains constant if in the new situation individual j’s income is decreased by the amount cc/A,(S) and individual i’s income is increased by a/&(S). It is easy to see that in this case the weights must be put into the denominator. If individual i gets an additional amount a/n,(S) welfare is increased by n,(S) *a/L,(S) =a. At the same time welfare is decreased by L,(S). a/L,(S). Therefore both changes compensate.
V. Ehert, Hicksian measures of welfare change
120
A further look at property WR demonstrates why it is reasonable to allow the distributional weights to depend only on the initial situation. Otherwise (if it were based on the new situation as well) a property like WR could not be formulated since different vectors of weights had to be used.5 Moreover the amount c( (which is to be redistributed if 2XS) = 1) has to be positive and less than 2,(S)5ij. This restriction prevents us from giving the following theorem in a global form. But we can prove a ‘local’ result. Theorem 3 and only if
A welfare measure I satisfies the properties P, WN, and WR if
Z(S,S)= ~ ~i(S)Er/;(Si-*Si) i=l
in a neighbourhood
of S.
Proof. ‘=s’ Choose any S and SEY’. At first we define the new situation - l%(&& r,. . . ,rTiJ by fi: =p and 6ii: = Ei( q(p, r&),p). We have V(f) = V(R) and _= conclude I(S,S)=I(S,g) by proposition 1. By repeated application of WR we get Z(S, S) = I($ (p, 6i ~,...,cl,+m,-%“))
=I
((s,
These redistributions
‘&s-l,+
i:2?@(6i-mi),ti2 )...) m, c 1
are allowed _
c
h+ i=~+,~(fii-6ii)
)I .
if for k=l,...,n-1
is nonnegative. Possibly one can secure this condition if one chooses a different ordering for the redistributions. Now we get by property WN and definition of EV
‘Hammond situation.
(1983,
1984) employs
welfare
weights
which
also
use information
of the new
U. Ehert, Hicksian measures of welfare change
‘e’ The converse is obvious.
121
0
At first sight, it looks as if the additional qualification ‘in a neighbourhood of s’ is a severe restriction in Theorem 3. But it must be stressed that it is a reasonable condition. If we admit all possible new situations one must not use the distributional weights n(S) depending on S. These weights can only be sensible in the proximity of the old situation S. 4. Extensions Of course one can characterize the x&CI$criterion similarly. Equivalent and compensating variations are almost symmetrical concepts: E~(Si~Si) = -Cx($+$). The equivalent variation stresses the old situation, the compensating variation the new one. Therefore we have to consider properties which are counterparts to P, WN, and WR, respectively. We define for example Property P*
t/s, s,SEY: V(s) 2 V(S)=z-Z(S, S) 5 Z(S, S).
(p*)
_= This property can be interpreted in the same manner as (P). Z(S, S) measures the change in welfare as before (situation S is the reference situation). But if we compare the magnitude of Z(S,S) and Z(f,S) for two situations S and S satisfying F’(S)2 I’($) then the welfare change is measured in relation to the new situation S. Let us suppose for instance V(S)? V(S)2 V(s) then it is clear that Z(S,S) should be less than I($, S). Analogously we formulate WN* and WR*. In both properties the distributional weights have to depend on the new situation f. Moreover WR* deals with the redistribution of incomes in the old situation. We get Theorem 4 A wetfare measure Z satisfies the properties P*, WN*, and WR* if and only if
Z(S,S)= i
;li(~)C~(Si-*Si)
i=l
in a neighhourhood off.
Theorem 3 and Theorem 4 demonstrate the differences between x&Ev and IniCK. In most cases the properties P, WN, WR will be preferred to P*,
U. Ehert, Hicksian measures 01 welfare change
122
WN*, WR*. P requires a ranking of situations usually needed if one wants to compare different new situations to a common (old) status quo. Moreover the dependence of the welfare weights in WN* and WR* on the new situation is strange, but can not be avoided in this framework.6 Finally we consider the special case where all distributional weights are chosen to be constant and equal one. Then 1 ,$Ev SC Ey and C~~iCr/;.-C Cq d e g enerate to an ‘unweighted’ measure of welfare.’ The distribution of incomes or utilities is not explicitly taken into account. These measures were connected with compensation tests. But now it is well known that there is no unique relationship between the sign of CEq or CC< and potential Pareto improvements or compensation tests [Boadway (1974) Ebert (1985)]. Let us denote by WN,, WR,, and WR,* the respective versions of the properties WN, WR, and WR* where all distributional weights equal one. The following result is proved by the method used in the proof of Theorem 3. Corollary 5. A welfare measure and WR,* if and only if
I satisfies
the properties
P[P*],
WN,,
WR,,
Z(S,S)= k EtqSi-Ji) i=l
Z(S,f)= t cqsi+si) [
i=l
1 .
Moreover this axiomatization even holds globally. (The weights are constant!). The difference between CEV, and CCq the kind of ranking property (P or P*).
distributional is reduced to
One further comment should be added. The law of one price [Slivinski (1983)] can be weakened. It is not difficult to adapt a slightly more general concept of state SE Y which allows personalized prices and to carry out the same analysis.
5. Conclusion In this paper the aggregate Hicksian measures of change in welfare are derived by means of an axiomatic approach. Usually they are defined by formulae or by verbal definitions. Characterizations by axioms facilitate choosing between different measures because one knows a respective set of %f. the discussion about WR. Furthermore remember that situation is used in the definition of the compensating variation. ‘Of course even specifying Ai G 1 is based on value judgements.
the price
system
of the
new
U. Ebert, Hicksian measures of welfare change
123
essential properties. For example, we can conclude that in general x&EI/;. is more suited for the measurement of welfare changes than ~L,C~. Furthermore the analysis reveals the implications and the power of assumptions like the Pareto property (P), and the properties dealing with normalization (WN, WN*) and redistribution (WR, WR*).
References Atkinson, A.B., 1970, On the measurement of inequality, Journal of Economic Theory 2, 244263. Blackorby, D., D. Primont and R.R. Russell, 1978, Duality, separability, and functional structure (North-Holland, Amsterdam). Boadway, R.W., 1974, The welfare foundations of cost-benefit analysis, Economic Journal 84, 926939. Brent, R.J., 1984, Use of distributional weights in cost-benefit analysis: A survey of schools, Public Finance Quarterly 12, 213-230. Ebert, U., 1984, Exact welfare measures and economic index numbers, Zeitschrift fur NationalGkonomie 44,27-38. Ebert, U., 1985, On the relationship between the Hicksian measures of change in welfare and the Pareto Principle, Social Choice and Welfare 1, 263-272. Ebert, U., 1986, Equity and distribution in cost-benefit analysis, Zeitschrift fur Nationalokonomie, Supplementurn 5 on ‘Second Best Welfare Economics’, 67-78. Ebert, U., 1987, Measurement of inequality: An attempt at unification and generalization, Social Choice and Welfare, forthcoming. Feldstein, M., 1972, Distributional equity and the optimal structure of public prices, American Economic Review 62, 32-36. Hammond, P.J., 1978, Economic welfare with rank order price weighting, Review of Economic Studies 45, 381-384. Hammond, P.J., 1983, Approximate measures of the social welfare benetits of large projects, in: S.B. Dahiya, ed., Theoretical Foundations of Development Planning. Hammond, P.J., 1984, Approximate measures of social welfare and the size of tax reform, in: D. Biis, M. Rose and Ch. Seidl, eds., Beitrage zur neueren Steuertheorie (Springer-Verlag, Berlin). Harberger, A.C., 1971, Three basic postulates for applied welfare economics: An interpretive essay, Journal of Economic Literature 9, 7855797. Harberger, A.C., 1978, On the use of distributional weights in social cost-benefit analysis, Journal of Political Economy 86, S87-S120. Hause, J.C., 1975, The theory of welfare cost measurement, Journal of Political Economy 83, 114551182. Hausman, J.A., 1981, Exact consumer’s surplus and dead-weight loss, American Economic Review 71, 662-676. Just, R.E., D.L. Hueth and A. Schmitz, 1982, Applied welfare economics and public policy (Prentice-Hall, Englewood Cliffs). King, M.A., 1983, Welfare analysis of tax reforms using household data, Journal of Public Economics 21, 183-214. Mishan, E.J., 1976, Cost-benefit analysis (Praeger, New York). Mohring, H., 1971, Alternative welfare gain and loss measures, Western Economic Journal 9, 349-368. Ng, Y.K., 1979, Welfare economics (Macmillan, London). Pearce, D.W. and C. Nash. 1981. The social auuraisal of nroiects. A text in cost-benefit analysis __ _ _ (Macmillan, London). Roberts, K. 1980, Price-independent welfare prescriptions, Journal of Public Economics 13, 277-297.
124
U. Ehert,
Hicksian measures of weljare change
Sen, A.K., 1976, Real national income, Review of Economic Studies 43, 19-39. Sen, A.K., 1977, On weights and measures: Informational constraints in social welfare analysis, Econometrica 45, 1539-1572. Sen, A.K., 1979, The welfare basis of real income comparisons: A survey, Journal of Economic Literature 17, l-45. Slivinski, A.D., 1983, Income distribution evaluation and the law of one price, Journal of Public Economics 20, 103-l 12. Vartia, Y.O., 1983, Eflicient methods of measuring welfare change and compensating income in terms of ordinary demand functions, Econometrica 51, 79-98. Willig, R.D., 1976, Consumer’s surplus without apology, American Economic Review 66, 5899597.
of Public
Economics
AXIOMATIC
33 (1987) 115-124.
North-Holland
FOUNDATIONS OF HICKSIAN WELFARE CHANGE
MEASURES
OF
Udo EBERT* University Received October
of Bonn, 5300 Bonn I, FRG
1984, revised version
received
The aggregate Hicksian measures ~&fZ~ and ~I,C~ are properties. One property concerns the ranking of two states. second property expresses the postulate that welfare changes are third one deals with redistributions of incomes. Distributional account by employing variable distributional weights li.
February
1987
characterized by three simple The measures are Paretian. A to be evaluated in money. The considerations are taken into
1. Introduction’ This paper explains why the Hicksian measures of change in welfare are so popular in public economics. The weighted sum of the equivalent variations x&Eq and the sum of the compensating variations x+Zr/; play an important role in cost-benefit analysis. Mishan (1976) discusses their properties. King (1983) applies these measures to cross section data of individual households in order to evaluate tax reforms.2 Just, Hueth and Schmitz (1982) recommend the use of c Eq and 1 Cl$ Ng accepts them as indicators of the magnitude of welfare change. This list of citations is by far not exhaustive. Thus we may ask why these measures are so attractive. In section 3 it is shown that x&Ev can be characterized by three simple properties3 which are rather weak. One property concerns the ranking of two states. The measure is Paretian. A second property expresses the postulate that welfare changes are to be evaluated in money. The third one deals with redistributions of incomes. Moreover the properties take into account distributional considerations by means of (variable) distributional weights ;li. The axiomatization of a weighted sum of equivalent variations in general holds only locally, i.e. for relatively small changes of prices and incomes. This result is ‘This paper is an extended and completely revised version of ‘Why to choose the Hicksian measures of change in welfare’, Discussion Paper 157, SFB 21, University of Bonn, 1984. I thank Hans-Dieter Stolper, two editors of this Journal, and three anonymous referees for most helpful comments and suggestions. ‘King’s definition of equivalent and compensating gain is identical to the concept of EV and CK respectively, used below. sAlready Mohring (1971) has specified a list of desiderata for a procedure to measure welfare change quantitatively and discussed common welfare measurement techniques.
0047-2727/87/$3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland).
116
U. Ebert, Hicksian measures of weljare change
reasonable since the distributional weights may depend on the old income or welfare distribution. Everyone who requires that a welfare measure possess these respective three properties has no choice: he must take C&E& Since these properties are desiderata in cost-benefit analysis this result explains the popularity of this measure. Section 4 discusses the case of c /1,Cv. Ev and Cv are only different with regard to the state which is stressed - the old one or the new one. Thus a similar characterization of c I.&v can easily be derived. But here the welfare weights have to depend on the new situation. Therefore it seems that x&E& is more suited to the measurement of welfare. As a byproduct we get an axiomatization of the unweighted sums c E v and 1 Cq which play a role in connection with compensation tests. This paper does not deal with the derivation of Ev or Cy from empirical data. For this problem the reader is referred to the results of Hause (1975) Willig (1976) Hausman (198 l), and Vartia (1983). Furthermore no characterizations of welfare measures of one individual are developed. These can be found in Ebert (1984).
2. Notation We consider n individuals and k commodities. Each individual i possesses an income mi>=O (i=l,..., n) and faces the price system p E IQ’!++. His or her preference ordering is represented by the ordinal indirect utility function v(p, mi) and the expenditure function Ei(ui, p), respectively, satisfying the usual properties [cf. Blackorby et al. (1978)]. A situation SE rWv: is described by a price vector p and all incomes m, (i= 1,. . . ,n). Y denotes the set of all possible situations. Let I/(S) be the . . . ,m,) we can convector (V,(p, m,), . . . , V,(p, m,)). Given a situation S=(p,m,, sider individual i’s ‘personal’ situation Si =(p, mi) E R”,+,‘. With the help of this notation we define individual i’s equivalent variation E<(S,+S,) and compensating variation CK (Si-+si) for the change from SET to SEY by
respectively. The ordering 5 in R” is defined by: x5 y (Vx=(x,, y,) E Ii?) if and only if xi 5 yi for all i, 15 i 5 n. y=(y,,..., 3. Characterization
. . . , x,),
of 1 liEVi
The aim of this section
is to characterize
a well-known
welfare
measure,
U. Ebert, Hicksian measures of welfare change
117
namely the weighted sum of the individual equivalent variations. At tirst we have to define this notion. Definition
A measure of change in welfare ([or shortly] welfare measure) I is defined by a function I:Y x Y+Iw’.
The meaning of a welfare measure I is the following. For any situations SEY and SEY, Z(S,3) evaluates the change in welfare for society if the individuals are in situation S before the change and in situation 3 after the change. In this regard the first argument of Z denotes the reference situation and the second one the new situation. If Z(S,$ is positive, welfare increases from S to 5. One important aspect of this definition is that welfare measures are applied to situations, i.e., the price systems and the individuals’ incomes are taken into account. Thus the severe restrictions implied by priceindependent welfare measurement are avoided a priori [cf. Roberts (1980)]. The magnitude of a welfare measure should reflect the changes in the individuals’ utilities. This requirement is expressed for a very special case by Property P (Pareto)
(P) When all individuals unanimously (weakly) prefer situation 3 to S the change in welfare from S to L!?must not be smaller than the change from S to S, i.e., if two situations S and g are both compared to the same status quo S and if s’ is preferred to S by all individuals (s is a Pareto improvement) the welfare measure should indicate this fact. This postulate is rather weak. Even if Z is constant, property P is satisfied. But there is an important implication of (P). I(&$) does not depend on the specific new situation s, but only on the individuals’ utilities in g. We have Proposition Z Zf the welfare measure Z satisfies Property P there exists a function J:Y x R(V)-+R’ such that Z(S,$)=J(S, V(g) for all S,SEY’, where R(V) is the range of V(S). Proof
Take any 3, ,!?EY
s.t.
V($) = I’(S (5. By (P) we have
Z(S,3) = Z(S,$. From this, Proposition
1 follows immediately.
0
This result may be compared with what is called welfarism in social choice theory [cf. Sen (1977)]. But there are two differences. In this context we
U. Ehert, Hicksian measures of welfare change
118
explore orderings which are not necessarily transitive. Furthermore we cannot derive an analogous result for the lirst argument of I from Property P. For most practical purposes one is interested in a monetary measure of welfare changes. In the framework given here we only have information about the price systems and incomes in different situations. This suggests using these data to normalize a measure I. Again we will consider a very specific case. We compare two situations S and 3 which differ only with respect to individual i,‘s income. The prices and the incomes of the other individuals do not change. The change in welfare can be measured by the difference in individual i,‘s income. In this special case Z(S,!?) coincides with i,‘s equivalent and compensating variation of income. This conforms with Harberger’s view of cost-benefit analysis [Harberger (1971,1978)], namely costs of benefits are evaluated ‘without regard to the individual(s) to whom they accrue’ (1971, p. 785). But in the following distributional considerations will be taken into account, as is postulated by most economists [e.g. Mishan (1976), Sen (1979), Hammond (1983,1984)]. Therefore we introduce i(S), a vector of positive distributional weights (i,(S), . . .,3.,(S)), where E.,(S) specifies the weight for i’s welfare gain or loss. 2 may depend on S which will be the status quo if we aim at the weighted sum of the equivalent variations. It may be desirable to eliminate or to diminish differences in income - in addition to a (general) welfare improvement. In this case it is reasonable to give more weight to the gains or losses of a lower income individual (e.g. ii(S) = 2) than to those of a higher income individual (e.g. L,(S) = 1); i.e. li generally varies in inverse proportion to the magnitude of individual i’s income. In the literature [cf. e.g. Atkinson (1970), Feldstein (1972), Pearce and Nash (1981), Brent (1984)] one specific type of weights is often proposed4:
I:= i m, is the average income and y 20 is a single constant where fi=n-’ elasticity. But of course many different types of weights can be imagined. For example ii(S) may be a function of the rank order of individual i’s income and/or utility (if the possibility of comparing the utility levels of all individuals is supposed [cf. Sen (1976), Hammond (1978), Ebert (1987)]). Moreover one could derive the weights &(S) from a social welfare function. However in this case it must be asked whether one is interested in developing an aggregate welfare measure when one has a social welfare function at hand. We consider this vector A(S) as given below. Employing distributional weights we postulate 4For an axiomatization
of these weights
see Ebert (1986).
119
U. Ebert, Hicksian measures OJwelfare change
WN (Normalization
using weights) (WN)
The welfare gain or loss of individual i, is multiplied by the corresponding weight liO, which may depend on the status quo S. This is a natural requirement since one needs a distribution of incomes or welfare levels as a basis of distributional weights. On the other hand weights which depend on the initial distribution are only valid or reasonable for relatively small changes of income or welfare - a fact which will be taken into account below. Property WN is compatible with the measurement of welfare changes for one individual [cf. Ebert (198431. We note _= Proposition 2 Assume I satisfies (WN): For all S, SEY s.t. p=p (if i,)I coincides with the weighted equivalent and compensating individual i,
and tii=rEi variation of
Z(S,S)=&&S)Eq)(Si,+Si,)=~i,(s) cigs,,4,,). The properties P and WN do not imply anything in case we have conflicting evaluations for different individuals, e.g. if individual i gains and j loses. Of course, one has to compare such situations by means of a welfare measure, too. The following property deals with a simple type of redistribution. Property
WR (Redistribution _= S,SEY::=Jk_=
Z(S,S)=Z for all i, j=l,...,
((
according
s, p’,A,
)...)
to weights)
ri&+L ~i(s)....'~j-~,...'~"
a >>
n, i# j, and O~a~~,(S)Gj
The main assumption of WR is again the condition that prices do not change. Given this WR allows for a redistribution of income in the new situation. Of course one has to take account of the distributional weights. The welfare change when going from s to 3 remains constant if in the new situation individual j’s income is decreased by the amount cc/A,(S) and individual i’s income is increased by a/&(S). It is easy to see that in this case the weights must be put into the denominator. If individual i gets an additional amount a/n,(S) welfare is increased by n,(S) *a/L,(S) =a. At the same time welfare is decreased by L,(S). a/L,(S). Therefore both changes compensate.
V. Ehert, Hicksian measures of welfare change
120
A further look at property WR demonstrates why it is reasonable to allow the distributional weights to depend only on the initial situation. Otherwise (if it were based on the new situation as well) a property like WR could not be formulated since different vectors of weights had to be used.5 Moreover the amount c( (which is to be redistributed if 2XS) = 1) has to be positive and less than 2,(S)5ij. This restriction prevents us from giving the following theorem in a global form. But we can prove a ‘local’ result. Theorem 3 and only if
A welfare measure I satisfies the properties P, WN, and WR if
Z(S,S)= ~ ~i(S)Er/;(Si-*Si) i=l
in a neighbourhood
of S.
Proof. ‘=s’ Choose any S and SEY’. At first we define the new situation - l%(&& r,. . . ,rTiJ by fi: =p and 6ii: = Ei( q(p, r&),p). We have V(f) = V(R) and _= conclude I(S,S)=I(S,g) by proposition 1. By repeated application of WR we get Z(S, S) = I($ (p, 6i ~,...,cl,+m,-%“))
=I
((s,
These redistributions
‘&s-l,+
i:2?@(6i-mi),ti2 )...) m, c 1
are allowed _
c
h+ i=~+,~(fii-6ii)
)I .
if for k=l,...,n-1
is nonnegative. Possibly one can secure this condition if one chooses a different ordering for the redistributions. Now we get by property WN and definition of EV
‘Hammond situation.
(1983,
1984) employs
welfare
weights
which
also
use information
of the new
U. Ehert, Hicksian measures of welfare change
‘e’ The converse is obvious.
121
0
At first sight, it looks as if the additional qualification ‘in a neighbourhood of s’ is a severe restriction in Theorem 3. But it must be stressed that it is a reasonable condition. If we admit all possible new situations one must not use the distributional weights n(S) depending on S. These weights can only be sensible in the proximity of the old situation S. 4. Extensions Of course one can characterize the x&CI$criterion similarly. Equivalent and compensating variations are almost symmetrical concepts: E~(Si~Si) = -Cx($+$). The equivalent variation stresses the old situation, the compensating variation the new one. Therefore we have to consider properties which are counterparts to P, WN, and WR, respectively. We define for example Property P*
t/s, s,SEY: V(s) 2 V(S)=z-Z(S, S) 5 Z(S, S).
(p*)
_= This property can be interpreted in the same manner as (P). Z(S, S) measures the change in welfare as before (situation S is the reference situation). But if we compare the magnitude of Z(S,S) and Z(f,S) for two situations S and S satisfying F’(S)2 I’($) then the welfare change is measured in relation to the new situation S. Let us suppose for instance V(S)? V(S)2 V(s) then it is clear that Z(S,S) should be less than I($, S). Analogously we formulate WN* and WR*. In both properties the distributional weights have to depend on the new situation f. Moreover WR* deals with the redistribution of incomes in the old situation. We get Theorem 4 A wetfare measure Z satisfies the properties P*, WN*, and WR* if and only if
Z(S,S)= i
;li(~)C~(Si-*Si)
i=l
in a neighhourhood off.
Theorem 3 and Theorem 4 demonstrate the differences between x&Ev and IniCK. In most cases the properties P, WN, WR will be preferred to P*,
U. Ehert, Hicksian measures 01 welfare change
122
WN*, WR*. P requires a ranking of situations usually needed if one wants to compare different new situations to a common (old) status quo. Moreover the dependence of the welfare weights in WN* and WR* on the new situation is strange, but can not be avoided in this framework.6 Finally we consider the special case where all distributional weights are chosen to be constant and equal one. Then 1 ,$Ev SC Ey and C~~iCr/;.-C Cq d e g enerate to an ‘unweighted’ measure of welfare.’ The distribution of incomes or utilities is not explicitly taken into account. These measures were connected with compensation tests. But now it is well known that there is no unique relationship between the sign of CEq or CC< and potential Pareto improvements or compensation tests [Boadway (1974) Ebert (1985)]. Let us denote by WN,, WR,, and WR,* the respective versions of the properties WN, WR, and WR* where all distributional weights equal one. The following result is proved by the method used in the proof of Theorem 3. Corollary 5. A welfare measure and WR,* if and only if
I satisfies
the properties
P[P*],
WN,,
WR,,
Z(S,S)= k EtqSi-Ji) i=l
Z(S,f)= t cqsi+si) [
i=l
1 .
Moreover this axiomatization even holds globally. (The weights are constant!). The difference between CEV, and CCq the kind of ranking property (P or P*).
distributional is reduced to
One further comment should be added. The law of one price [Slivinski (1983)] can be weakened. It is not difficult to adapt a slightly more general concept of state SE Y which allows personalized prices and to carry out the same analysis.
5. Conclusion In this paper the aggregate Hicksian measures of change in welfare are derived by means of an axiomatic approach. Usually they are defined by formulae or by verbal definitions. Characterizations by axioms facilitate choosing between different measures because one knows a respective set of %f. the discussion about WR. Furthermore remember that situation is used in the definition of the compensating variation. ‘Of course even specifying Ai G 1 is based on value judgements.
the price
system
of the
new
U. Ebert, Hicksian measures of welfare change
123
essential properties. For example, we can conclude that in general x&EI/;. is more suited for the measurement of welfare changes than ~L,C~. Furthermore the analysis reveals the implications and the power of assumptions like the Pareto property (P), and the properties dealing with normalization (WN, WN*) and redistribution (WR, WR*).
References Atkinson, A.B., 1970, On the measurement of inequality, Journal of Economic Theory 2, 244263. Blackorby, D., D. Primont and R.R. Russell, 1978, Duality, separability, and functional structure (North-Holland, Amsterdam). Boadway, R.W., 1974, The welfare foundations of cost-benefit analysis, Economic Journal 84, 926939. Brent, R.J., 1984, Use of distributional weights in cost-benefit analysis: A survey of schools, Public Finance Quarterly 12, 213-230. Ebert, U., 1984, Exact welfare measures and economic index numbers, Zeitschrift fur NationalGkonomie 44,27-38. Ebert, U., 1985, On the relationship between the Hicksian measures of change in welfare and the Pareto Principle, Social Choice and Welfare 1, 263-272. Ebert, U., 1986, Equity and distribution in cost-benefit analysis, Zeitschrift fur Nationalokonomie, Supplementurn 5 on ‘Second Best Welfare Economics’, 67-78. Ebert, U., 1987, Measurement of inequality: An attempt at unification and generalization, Social Choice and Welfare, forthcoming. Feldstein, M., 1972, Distributional equity and the optimal structure of public prices, American Economic Review 62, 32-36. Hammond, P.J., 1978, Economic welfare with rank order price weighting, Review of Economic Studies 45, 381-384. Hammond, P.J., 1983, Approximate measures of the social welfare benetits of large projects, in: S.B. Dahiya, ed., Theoretical Foundations of Development Planning. Hammond, P.J., 1984, Approximate measures of social welfare and the size of tax reform, in: D. Biis, M. Rose and Ch. Seidl, eds., Beitrage zur neueren Steuertheorie (Springer-Verlag, Berlin). Harberger, A.C., 1971, Three basic postulates for applied welfare economics: An interpretive essay, Journal of Economic Literature 9, 7855797. Harberger, A.C., 1978, On the use of distributional weights in social cost-benefit analysis, Journal of Political Economy 86, S87-S120. Hause, J.C., 1975, The theory of welfare cost measurement, Journal of Political Economy 83, 114551182. Hausman, J.A., 1981, Exact consumer’s surplus and dead-weight loss, American Economic Review 71, 662-676. Just, R.E., D.L. Hueth and A. Schmitz, 1982, Applied welfare economics and public policy (Prentice-Hall, Englewood Cliffs). King, M.A., 1983, Welfare analysis of tax reforms using household data, Journal of Public Economics 21, 183-214. Mishan, E.J., 1976, Cost-benefit analysis (Praeger, New York). Mohring, H., 1971, Alternative welfare gain and loss measures, Western Economic Journal 9, 349-368. Ng, Y.K., 1979, Welfare economics (Macmillan, London). Pearce, D.W. and C. Nash. 1981. The social auuraisal of nroiects. A text in cost-benefit analysis __ _ _ (Macmillan, London). Roberts, K. 1980, Price-independent welfare prescriptions, Journal of Public Economics 13, 277-297.
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Hicksian measures of weljare change
Sen, A.K., 1976, Real national income, Review of Economic Studies 43, 19-39. Sen, A.K., 1977, On weights and measures: Informational constraints in social welfare analysis, Econometrica 45, 1539-1572. Sen, A.K., 1979, The welfare basis of real income comparisons: A survey, Journal of Economic Literature 17, l-45. Slivinski, A.D., 1983, Income distribution evaluation and the law of one price, Journal of Public Economics 20, 103-l 12. Vartia, Y.O., 1983, Eflicient methods of measuring welfare change and compensating income in terms of ordinary demand functions, Econometrica 51, 79-98. Willig, R.D., 1976, Consumer’s surplus without apology, American Economic Review 66, 5899597.