~__ JOURNALOF PUBLIC ECONOMICS

ELSE-¢IER Journal of Public Economics 61 (1996) 247-262

Suppes-Sen dominance, generalised Lorenz donfinance and the welfare economics of competitive

equilibrium: Some examples Paul Mac[de

School of Economic Studies, University of Manchester, Manchester M13 9PL. UK

Received October 1994; final version received October 1995

Abstract

This paper studies two quasi-orderings in the context of three simple equilibrium models; a general equilibrium model with identical consumers, a model of an artisan economy, and an education model. The first quasi-ordering is induced by Suppes- Sen dominance (or first-degree stochastic dominance) amongst utility vectors of individuals, and the r, ecood evolves from generalised Lorenz dominance (second- degree stochastic dominance) similarly applied. It is shown how Pareto-efficient competitive equilibria may or may not be Suppes-Sen optima/generalised Lorenz optima, and it is suggested that this information is valuable for welfare analysis.

Keywords: Dominance; Competitive equilibrium; Suppes-Sen; Generalised Lorenz

J E L classification: D63

1. Introduction

A Pareto-optimal (or efficient) allocation is a feasible allocation that cannot be strictly Pareto dominated (making someone better off and no one worse off) by any other feasible allocation. The familiar Pareto-dominance relation induces a 'quasi-ordering" on the set of feasible allocations, and, of course, analysis of Pareto optimality is the cornerstone of the welfare economics of general equilibrium models and competitive equilibrium, inter alia. The objective of this paper is to suggest that the analysis of two other quasi-orderings and their optima can lead to useful insights into welfare

0047-2727/96/$15.00 ~ 1996 Elsevier Science S.A. All rights reserved SSDI 0047-2727(95)01560-4

248 P ,Madden / Journal of Public Economics 61 (1996) 247-262

economics of competitive equilibrium, in particular. Unlike the Pareto quasi-ordering, our alternatives assume some degree of interpersonal comparability of utility and embody some, albeit small. "equity' content. We study a variety of simple models and investigate when their Pareto-optima and competitive equilibria are or are not optima under the alternative quasi-orderings.

The first quasi-ordering we study derives from Suppes" "grading principle' (Suppes, 1966; see also Sen, 1970, ch. 9), and is based on the "fundamental dominance' relation studied by Kolm (1971, 1989), referred to as 'domi- nance" by Blackorby and Donaldson (1977); we follow Saposnik (1983) and use the term Suppes-Sen dominance. A feasible allocation, A, say, Suppes- Sen dominates another feasible allocation B if the vector of utilities at A, or some permutation of this vector, Pareto dominates the vector of utilities at B; this is the same as first-degree stochastic dominance between the utility vectors. Our other quasi-ordering compares utility vectors by second-degree stochastic dominance, or the convex hull of dominance in Blackorby and Donaldson (1977); following Shorrocks (1983) we use the term generalised Lorenz dominance. A Suppes-Sen (resp., generalised Lorenzl optimum is then a feasible allocation that cannot be Suppes-Sen (resp., generalised Lorenz) dominated by another feasible allocation.

Now a generalised Lorenz optimum will be a Suppes-Sen optimum that will be a Pareto optimum, but not vice versa, in general. However, the additional demands of Suppes-Sen optimality compared with Pareto are small - the maximisation of any increasing welfare function that is symmet- ric produces Suppes-Sen optimality, for instance. Somewhat more is needed for generalised Lorenz opt imal i ty- maximisation of increasing, symmetric, quasi-concave welfare functions now suffices, for instance. The particular equilibrium theme we take is to show how Pareto-optimal competitive equilibria may fail to satisfy the "small" equity demands of Suppes-Sen or generalised Lorenz optimality, and thus in a sense are strongly inequitable. We study three examples, chosen to illustrate a range of results. First, a standard general equilibrium model with identical agents generates competi- tive equilibria that are Suppes-Sen optima (as well as Pareto optima) but typically generalised Lorenz dominated. Secondly, an artisan economy (similar to Mirrlees, 1974; Dasgupta and Hammond, 1980; Dasgupta, 1982) is shown to have competitive equilibria that are 5uppes-Sen dominated (but Pareto efficient). Thirdly, an education model (evolving from Arrow, 1971) is also shown to have competitive equilibria that are Pareto efficient but Suppes-Sen dominated in some cases. An incomplete information story suggests that the inequity of the education equilibrium is a deeper problem than in the artisan economy, since the equilibrium is there Suppes-Sen dominated by an implementable allocation, which is not true in the artisan model, at least under the natural incomplete information assumptions, following Dasgupta and Hammond (1980).

P. Madden / Journal of Public Economics 61 (1996) 247-262 249

Of course, the Lorenz and generalised Lorenz quasi-orderings have been applied extensively to the analysis of income distributions (Atkinson, 1970; Dasgupta et al., 1973; Kolm, 1968, 1977; Lambert, 1989; Rothschild and Stiglitz, 1973; Shorrocks, 1983); the empirical application of Suppes-Sen dominance to income distribution comparison has been taken up by Bishop et al. (1991), under the name, rank dominance. It should be noted that our application of the dominance criteria to utility vectors produces essentially a one-dimensional problem, different from Atkinson and Bourguignon (1980) who apply stochastic dominance to inequality in multi-dimensional dis- tributions. Foster et al. (1990) do compare equilibrium utility vectors, using Lorenz and generalised Lorenz quasi-orderings to investigate whether dominance amongst utility vectors is reflected in a corresponding dominance amongst equilibrium income distribu'.ions. With our focus on optima, perhaps the most direct predecessor of this paper is Kolm (1989), who analyses various assignment problems in the marriage and labour market contexts. In some cases, the competitive outcome is fundamentally domi- nant (or a Suppes-Sen optimum). Our objective is to add to Kolm's (1989) lead and suggest, by further examples, that systematic investigation of Suppes-Sen, and generalised Lorenz, optimality in the context of equilib- rium models is possible and fruitful.

The paper is organised a~ follows: Section 2 presents definitions and eharacterisations of Suppes-Sen and generalised Lorenz dominance. Section 3 looks at the identical agent model, Section 4 at the artisan economy, and Section 5 at the education model. Section 6 concludes.

2. Definitions and characterisations

The primitive concepts in this section are the utility levels u = (u~ . . . . . un) of n agents, and a set S of feasible utility vectors. This is for brevity of notation - it should be understood that utility will depend on the consumption of one or more goods in the applications. The dominance definitions that follow are those of strict dominance, as this is the most convenient form for later results; however, we drop the "strict" description, again for convenience.

Defini t ion 1. Suppes-Sen dominance. The vector li S u p p e s - S e n dominates ('SS dominates') the vector u iff fi ~> H u and t't ~ H u for some permutation matrix H.

There is a variety of equivalent definitions. Theorem 1 contains those of interest here and Theorem 2 will collect the corresponding results for generalised Lorenz. These results can be found in the literalure (Atkinson, 1970; Dasgupta et al., 1973; Koim, 1968, 1977; Lambert. 1989; Rothschild and Stiglitz. 1973: Saposnik, 1981. 1983; and Shorrocksl 19S3)" see Mosler

250 P Madden / Journal of Public Economics 61 (1996) 247-262

(1994) for a recent survey. A tailored proof of Theorems 1 and 2 is available on request from the author.

Theorem 1 (Suppes-Sen dominance characterisations). ~ SS dominates u i f f either,

(a) w( ~ ) >1 w(u) for all increasing w (i.e. w( dt ) >~ w(u) i f d~ >~ u), which are symmetric (i.e. w(llu) = w(u) for any u and any permutation matrix H), with strict inequality for some such w, or

(b) XT-I F(tii)~>ET-t F(ui), for all increasbzg functions F, with strict inequality for some such F, or

(c) h i ~> u,, i = l . . . . . n with strict inequality for some i when t'* and u are re-ordered in order o f increasing size.

This result characterises Suppes-Sen dominance in terms of social welfare functions (a) which are separable (b); (b) is also the first-degree stochastic dominance equivalence. The statement (e) provides a useful comparison with the generalised Lorenz curves, which come out of the corresponding statement for generalised Lorenz dominance below. A Suppes-Sen optimum ( 'SS optimum') is then a feasible utility vector u E S that is not SS dominated by any other feasible utility vector.

When applying the Suppes-Sen concept we shall assume that individual utilities are quasi-concave, increasing functions of vectors of goods. Clear|y, the Suppes-Sen comparison requires some interpersonal comps:ability of utilities. The quasi-ordering on the set of feasibJe allocations of goods induced by the Suppes-Sen relation is easily seen to be invariant to any common, increasing transformation of individual utilities. Thus, we are assuming ordinal interpersonal comparability in applying this concept. As such, the concept is different from, and apparently anrelated to, the concept of 'fairness" (see, for example, Baumol, 1986), which eschews such c,om- parisons.

Definition 2. Generalised Lorenz dominance. The vector t~ generalised Lorenz dominates ( 'GL dominates') the vector u iff t~ ~> Bu for some bistochastic matrix B with ~ ~ Bu if /~ is a permutation matrix.

Again, there are weU-known alternative statements.

Theorem 2. (Generalised Lorenz dominance characterisation), fi GL domi- nates u iff, either

(a) w(t~)>/w(u) for all increasing, S-concave w, with strict meqv.ality for s o m e s u c h W, Or

(b) E~_ t F(sJi)>~XT= t F(u,) for all increasing, concave F, with strict inequality for some such F, or

p Madden / Journal o f Public Economics 61 (1996) 247-262 25t

(C) k . k t wi th s tr ic t i nequa l i t y f o r s o m e k , w h e n t't ~'i~,l Ui ~ ~i_i If, k = 1 . . . . . n a n d u are r e - o r d e r e d in o r d e r o f i nc reas ing s i z e .

Again (a) and (b) characterise in terms of social welfare functions, and (b) is the second-degree stochastic dominance equivalence. If u, is the "income' of i then (c) says that the generalised Lorenz curve (Shorrocks, 1983) for ' income distribution' ~ lies nowhere below, and somewhere above, that for u; when t~ and u have the same mean the generalised Lorenz curve becomes the traditional Lorenz curve. And, of course, a g e n e r a l i s e d L o r e n z o p t i m u m ( 'GL opt imum') is a feasible utility vector u E S , which is not GL dominated by any other feasible utility vector.

In applying the generalised Lorenz concept, individual utilities will be assumed to be concave, increasing functions of goods allocations. The generalised Lorenz quasi-ordering on the feasible set of goods allocations is invariant to any common, increasing, and linear transformation of individual utilities, so that we will be assuming cardinal interpersonal comparability in applying this concept.

3. General equillbrium with identical consumers

In our first application we consider a special case of the standard general equilibrium model in which all consumers have identical convex consump- tion sets and identical, strictly increasing, strictly quasi-concave utility functions. There are n consumers, labelled i = 1 . . . . . n , m firms labeled j = 1 , . . . , m and ( commodities labelled h = 1 . . . . . ( with prices p = (p~ . . . . . p l ) E R ~. X C R ¢ + + denotes the consumption set of consumer i = 1 . . . . . n with typical element x i, and u ( x ' ) is i ' s utility function. For consumer i the initial endowment is w i ~ X , profit shares are 0i= (0,~ . . . . . 0,m) where E i0 , ,=1 for all j = l . . . . . m and net trades are denoted z ' = x ~ - w~; notice w' and 0, may vary across consumers. For firm j , Y~ C R ~ is the convex production set, yi C y i denotes a typical production plan, ap.d H j = E, PhY~, denotes profit:s.

A feasible allocation for the economy described is (x ~ . . . . . x , ~ ; y t . . . . . ym) = (x ,y) such that y ~ YJ for allj~ x ' E X for all i and E~ x~ <~ E i y~ + ~ w h for all h , leading to the set of feasible utility vectors S = {(u I . . . . . u , ) ]u, = u ( x ' ) , i = 1 . . . . . n and there exists a feasible allocation (x j . . . . . x"; y~ . . . . . y ' ) } . A competitive equilibrium is a feasible allocation (x , y ) and a set of prices p such that, given p . each y, is profit maximising given the technology, each x, is utility maximising given the consumption set and budget constraint, and a!! markets clear. Concepts of Pareto, SS and GL dominance and optimality can now be applied.

252 P. Madden / Journal of Public Economics" 61 (1096) 247-262

The assumption that utility functions (and consumption sets) are identical implies that the set S is symmetric (if u E S with feasible allocation (x, y), then ( l l x , y) is also feasible for any permutation matrix H and produces utility Hu). Hence, the set of SS optima coincides with the set of Pareto optima. And in this standard model a competitive equilibrium allocation is Pareto efficient, of course.

Proposi t ion I. For the general equil ibrium mode l with identical consumers an allocation is a S u p p e s - S e n op t imum i f and only i f it is a Pareto op t imum. In particular, a compedt ive equil ibrium allocation is an SS op t imum.

Assuming that utility is concave, straightforward and quite different results emerge for Lorenz. In fact, any feasible allocation with unequal utilities must be GL dominated - suppose (x, y) is such an allocation where u 1 = u(x t) ~ u: = u(x z) <~--. -< u, = u(x"). Convexity assumptions ensure that the aRocation where each consumer receives f i = u ( ( l / n ) r~,x ') is feasible. Concavity o~ u ensures that fi >/(1/n) ZT_~ u,. Given the ordering of u [ s , (t/n)E",=, u, >1 ( 1 / k ) E~= , u,, k = 1 . . . . . n with strict inequality for some k since u [ s differ. Thus, ktJ ~>E,*_~ u,, k = 1 . . . . . n with strict inequality for some k , and ,~ GL dominates u. Conversely, trivially, the equal utility allocation is a GL optimum. Hence,

Proposi t ion 2. For the general equil ibrium mode l with identical consumers and concave utilities, a Pareto-efficient allocation is a generalised Lorenz optim,~m i f and only i f # gives all cons,tracts the same utility. In particular, a compet i t ive equil ibrium allocation is a generalised Lorenz op t imum i f and on ly i f it gives all consumers equal utility; such equilibria occur i f endow- ments and shares are equally distributed.

The results can be applied to income distribution comparisons. Let Y = (Yt . . . . . Yn) and 37 = (y, . . . . . f ) be two (scalar) income distributions with possibly different means, and suppose the utility function for all i = 1 . . . . . n is u ( y i ) = y I. If y and 37 have the same mean, then the GL compai~son reduces to the standard Lorenz curve comparison; if only distributions with the same mean are feasible, then Proposition 2 indicates a unique GL opt imum that is complete equality, or the 45 ° standard Lorenz curve. However. the income distribution 37 SS dominates y if and only if 33,/>y,, i = 1 . . . . . n with strict inequality for some i, where )3 and y have been re-ordered in order of increasing size. Amongst distributions with the same mean, SS dominance has no f o r c e - f r o m Proposition 1 any dis- tribution is an SS optimum. However. this is not so if means differ. Indeed, Bishop et al. (1991) show that SS dominance (their rank dominance) is

I~ Madden ~' Journal of Public Economics 61 (1996) 247-2ti2 253

almost as successful in ranking international income distributions as (3L dominance.

The results of this section provide a benchmark for the next two sections where we look at some simple models witl~ non-identical consumers to investigate how the relation between their equilibria and SS and GL dominance changes.

4. An artisan economy

We now study a simplified version of a model of Mirrlees (1974), analysed further by Dasgupta and Hammond (1980) and Dasgupta (1982). where it is referred to as the "artisan economy' . Consumers are not identical n o w - their labour productivities differ. There are two goods, leisure and a

consumption good. and for simplicity we assume two consumers. The utility function of consumer i is u ( T - ~ , c , ) where (~ is labour supply, c, is the quantity of the consumption good and T is the endowment of time. u is C'-. strictly quasi-concave and increasing in both arguments: also. indifference curves are interior in that if T - ¢ I > 0 , c ,>O and u ( T - ( , , g , ) > u ( T - /',, c,), then T - i ) > 0 and ?, > 0. Production of the consumption good by artisan i is governed by the technology c<~.~,(, where ,~ > A 2 > 0 , so that artisan 1 is more productive. A feasible allocation is (¢, c) = (¢t, ~ . c~. c:) . where 0 ~< ¢, ~< T. c, ~> 0. i = i, 2 and c ~ + c 2 ~< ~ (~ + A 2 t" z, and is an interior allocation if T - (, > 0 and c, > 0. i = 1.2.

A first result is that any allocation where the more productive artisan supplies less labour ((~ < ¢,) is SS d o m i n a t e d - swapping allocations so that artisan 1 gets (C, c2) and artisan 2 gets (¢i. c~ ) is feasible and leads to extra production of the consumption good. allowing SS domination of the original allocation. Hence.

Proposition 3. In the artisan economy, a feasible allocation in which the more productive artisan works strictly less than the less productive artisan is SS dominated.

A stronger result emerges if leisure is a normal good (as in Mirrlees. 1974). it then follows that any allocation where the better-off individual (utility u i, say) works more is SS dominated - the normality ensures that the reduction in consumption good needed to take the better-off individual down to the other utility level (uj. say) exceeds the extra needed to increase j ' s utility to u,, with tc~, ¢2 remaining fixed. This reasoning, plus Proposition 3, allows us to show the following proposition (a proof of Proposition 4, and subsequent propositions are given in the appendix where necessary).

254 P. Madden / Journal of Public Economics 61 (1996) 247-262

Proposition 4. in the artisan economy, i f leisura is a normal good, a feasible, interior, Pareto-efficient allocation is an SS optimum if and only i f the less productive artisan is at least as well of]" as the more productive artisan.

That SS optima favour the less productive is consistent with the Mirrlees (1974) utilitarian conclusion. However, a competitive equilibrium will favour the more productive. If w is the wage per efficiency unit of labour, the budget constraint facing i in a competitive equilibrium is c, <~ wA,~, and the constant returns assumed ensure that w = 1 so that artisan 1 is strictly bet ter off than artisan 2. The interior indifference curve assumption implies that the equilibrium allocation is interior and, of course, it is Pareto efficient. Hence, from Proposition 4:

ProFosition 5. In the artisan ecGnomy, i f leisure is a normal good, the competitive equilibrium allocation is SS dominated.

Government action to rectify the inequity in Proposition 5 is feasible if the government can levy lump-sum taxes and is completely informed. However. if, following Dasgupta and Hammond (1980) and Dasgupta (1982), the government does not know which individual i~ the more productive, then the only implementahle allocations are those where the more productive artisan is at least as well off as the less productive, assuming that the more productive artisan can mimic the lower productivity, but not vice versa. Thus , the allocations that SS dominate the competitive equilibrium cannot be implemented, reducing the force of Proposition 5 and suggesting the definition: a feasible allocation is a constrained SS optimum if it cannot be SS dominated by an allocation that is implementable under the above incomplete information. We summarise in the proposition that follows.

Proposition 6. bl the artisan economy, i f leisure is a normal good, the competitive equilibrium allocation is a constrained SS optimum.

In the next section, an education model is studied where the relation between a competitive equilibrium and SS optimality is quite different.

5. An education model

Inspired by Arrow (1971) and the subsequent literature (Bruno, 1976; Hare, 1988, Hare and Ulph, 1979. 1980, 1981; Johnson, 1984; and Ulph, 1977), we study the welfare economics of an economy in which educational resources are distributed to individuals, providing them with human capital

P. Madden 1 Journal of Public Economics 61 (1996) 247-262 255

b. la Becker-Schulz (Becker. 1964; Schulz, 1961 ). Following Becker( 1964) - see also Arrow (1993) - we assume that as well as an investment motive tor human capital acquisition Cmoney income" in Becket, 1964) there is also a consumption motive ('psychic income') from education entering directly and positively into the utility function (see Lazear. 1977, for some empirical questioning of this assumption). As in earlier sections, we focus primarily on the market mechanism for the distribution of educational resources. This mechanism will produce Pareto optima in the model: the question is whether it produces SS/GL optima, constrained or otherwise.

The economy to be studied has two dates and, for brevity, two in- dividuals. A given amount of educational resource e > 0 is available at date 1 and individual i receives q; of this, where q~ + qz ~< e, As a result, i acquires human capital h; =f~(q;), where .1";, is i's human capital production function; other" possible inputs (e.g. time, effort . . . . ) are ignored, ¢, denotes the inverse o f ~ ; we assume for each i, ¢i: R , - - ~ R . is C 2, strictly convex and increasing with ~Pi(0)= 0 and ¢~(0)= 0. The function ~i embodies i 's 'ability" to benefit from education. Individual 1 is assumed to be the more 'able ' , in both total and marginal terms, so that, for all h > 0, ¢~(h)< ~_,(h) and g,'l(h) < ~ ' (h ) .

At date 2, individual i reaps the investment benefit from education, supplying h i units of efficiency labour that produces h i of a consumption good, of which i consumes ci, i = 1, 2; here, we are ignoring any disutility of labour supply. Both individuals have the same preferences, which can be represented by the utility function u(hi. ci). i = l , 2, which is C 2, strictly quasi-concave, strictly increasing in both arguments and exhibit interior indifference curves; if h i > 0 , c i > 0 and u(/~ i, ci) ~> u(hi, ci), then /~i > 0 , 6i > 0. The presence of hi as an argument reflects the consumption benefit of education to i, which thus s tems from the human capital (skills, educational at tainment . . . . ) achieved rather than, for example, the quantity of educa- tion consumed. A feasible allocation for the education economy is (h, c) = [(h t, cl) , (h 2, c2) ] E R4. such that ~l(h) + ¢~(h I ) <~ e and c I + c z ~< h~ + h 2. An allocation is interior if c i > 0 , h i > 0 , i = 1, 2. Interior, Pareto-efficient allocations are easily seen to be characterised by the following marginal conditions:

(a) ~,(h,) + @(hz)=e; (b) [1 + ~(h~, c~)]/~(h~) = [1 + tr(hz, c,.)l/~'(h,_), where tr(h, c) is the

marginal rate of substitution Ou/Oh/au/ac at (h, c). Analogous to Proposition 3 for the artisan economy, an allocation is now

SS dominated if it gives the more able individual so much less of the educational resource that he ends up with less human capital than his less able colleague.

Proposition 7. In the education economy, a feasible allocation in which the

256 P Madden / Journal of Public Economh's 61 (1996) 247-262

more able" individaal ends up with less human capital than the less able individual is SS dominated.

However, if the distribution of educational resources favours the more able so much that not only is h~ > h 2 but also q~'(h~) > ~0'(h_.) (so that the marginal educational resource would give more benefits to the less able), then the allocation is GL dominated.

Proposition 8. bz the education economy with concave utilities, a feasible allocation in. which q ' (h l ) > q ( h : ) (i.e. tile marginal edacation resource would give more benefit to the less able) is generalised Lorenz dominated.

Propositions 7 and 8 place bounds on the distribution of educational resources that are necessary for SS and GL optimality. For the subsequent study of the relation between market equilibria and SS optimality the following analogue of Proposition 4 is most useful. Suppose human capital is a normal good. Then, exactly as in the artisan economy, any allocation where the better-off individual receives less of the normal good (leisure before, now human capital) is SS dominated. However, whereas before the productivity considerations in Proposition 3 required that the more able (= more productive) receive (weakly) the lesser leisure allocation for SS optimality, and hence from normality (weakly) the lesser utility, the situation is now reserved. Productivity considerations now mean that the more able must end up with at least as much human capital as the less able for SS optimality (Preposition 7), and so from normality they must receive (weakly) the greater utility. Loosely, SS optimality requires that the more able actually produce more in either case; in the artisan economy this means that the more able work more, which makes them worse off, while in the education economy it means that the more able receive more education, which makes them better off.

Proposition 9. In the education economy, i f human capital is a normal good, a feasible, interior, Pareto-efficient allocation is an SS optimum if and only i f the more able individual is at least as well o f f as the less able individual.

Remark 1. In the terminology of UIph (1977), an allocation is welfare regressive (resp., progressive) if u ( h t , c l ) > ( r e s p . , <)u(h: , c2) and output regressive (resp., progressive) if h I >(resp. , <)h_,. In this terminology, welfare regressivity is necessary and sufficient for an efficient allocation to be SS optimal (Proposition 9), whilst output progressivity guarantees SS dominance (Proposition 7) and sufficient output regressivity produces GL dominance (Proposition 8).

P. Madden I Journal of Public Economics 61 (1996) 247-262 2 5 7

R e m a r k 2. In a somewhat different education model, Bruno (1976) reports in his Proposition 2 a result that is similar in style to Proposition 8 here.

We now study a market mechanism tor the distribution of educational resources. The resource e is now produced at date 1 from a single input (capital) under unit constant returns. Individual i is endowed with e i units of this physical capital and E e, = e. Let p denote the price of physical capital and, from unit constant returns, the price of a unit of educational resources: for a similar reason, let both efficiency labour and consumption good be numeraire. Assume that there is a complete set of perfect markets. The consumer problem faced by individual i at price p is then:

C P ( i , p ) max u ( h , c ) s.t. p ~ i ( h , ) + c, <~ pe , + h , . c , , h, ~ O . ( h r c l ) t r

A c o m p e t i t i v e e q u i l i b r i u m is a price p and an allocation (h , c) E R ~ such that (h i, c,) solves C P ( i , p), i = 1,2 and ~t(h~ ) + ,p_,(h:) = e.

Our assumptions ensure that competitive equilibrium allocations will be interior. It is easy to check that competitive equilibria will be Pareto efficient. Of course, this has been engineered by assumptions such as perfect capital markets and the absence of externalities. Now, if human capital is a normal good. Proposition 9 says that a competitive equilibrium will be SS dominated if individual 2 is better off in equilibrium than individual !. Sufficient conditions for this are easily derived, in a competitive equilib- r ium, l ' s human capital level (and that of 2) cannot exceed f~(e) . If, for all p > 0, 2"s budget constraint lies "above" that of I for any h ~<f~(e), 2 must be better off in equilibrium. Agent 2"s budget constraint lies above that of 1 in this manner if:

(1) p e z + h - p,pz (h ) > p e 1 + h - p ~ l ( h ) for all h <~ f l ( e ) and p > 0 , or (2) e,_ - e] > ,p2(h) - ~l(b) for all h <- f l (e ) .

Since ,p~(h) - ,~l(h) is increasing in h, (2) is true if and only if: (3) e 2 - e I > ~,_( f t (e)) - e. Condition (3) implies a "negative correlation' between endowments of

human and non-human wealth - the RHS of (3) is a measure of individual l ' s human wealth advantage over individual 2, while the LHS measures individual 2"s non-human wealth advantage over individual 1. To make the RHS of (3) 'small" requires 'not too much" difference in the abilities of I and 2. However, large values of e allow choices of e2 (near e) and e t (near zero), which make the LHS of (3) 'large'. A sufficient condition for the existence of initial endowment distributions that produce SS-dominated competitive equilibria is then:

(4) e > ~ z ( f I ( e ) ) - e. This has proved

258 P. Madden / Journal of Public Economics 61 (19961 247-262

Proposition 10. In the education economy where human capital is a normal good, an)' initial endowment distribution that satisfies (3) leads to SS- dominated competitive equilibria. Such initial endowment distributions exist i f (4) is satisfied.

For the intuitive reasons given earlier, the education model is quite different from the artisan economy in that SS-dominated equilibria now occur when the more able individual is worse off. If we impose the incompleteness of information that the government does not know which individual is more able and that the more able can mimic the less able bat not vice versa, then the efficient allocations that can be implemented are those where the more able consumer is at least as well off as the less able. Thus , allocations that SS-dominate equilibria are now implementable.

Proposition 11. In the education economy where human capital is a normal good, and where endowments satisfy (3), the competitive equilibrium alloca- tion is Suppes-Sen dominated by an implementable allocation.

Thus, the inequity of SS dominance generated by the market in education can now be feasibly correctcd by an incompletely informed government , and thus is accordingly stronger than its analogue in the artisan economy. The emerging lesson may be that use of the market to allocate goods that provide individuals with both positive consumption benefits and enhanced productivity (e.g. education) can generate inequity in a strong sense.

6. Conclusions

The paper has applied concepts of Suppes-Sen and generalised Lorenz dominance to some simple equilibrium models. The main claims are that equilibria that are Pareto optimal may or may not be Suppes-Sen or generalised Lorenz optimal, that definite results in answer to these questions can be had and that these answers are helpful. In particular, we have shown via examples how Pareto-optimal equilibria may fail, in various ways, to satisfy the weak equity demands of Suppes-Sen optimality or those (somewhat stronger) of generalised Lorenz optimality.

Acknowledgements

I am very grateful to the following who have advised/assis ted/com- mented/cor responded on previous versions: Alvin Birdi, Tapan Biswas, Peter Dolton, Marc Fleurbaey, Paul Hare, Peter Hammond , Serge-Christ-

P. Madden / Journal of Public Economics 61 (1996) 247-262 259

ophe Kolm, Peter Lambert, Anthony Ogus, John Roemer, Tony Shorrocks, David Ulph, seminar participants at Exeter University, Leicester University. LSE, Manchester University, QMW College, the 1992 Royal Economics Society Conference and the 1994 ESEM conference, i have not yet been able to incorporate all suggestions, and so the usual caveat applies more than usually.

Appendix A

Proof o f Proposition 4. "only if" We show that any feasible, interior allocation where u~ > u z

cannot be an SS optimum. Let us suppose that (~, c) is a feasible interior allocation where u , = u ( T - ( I , c l ) > u ( T - ~ , c z ) = u 2. If tel<(2, then Proposition 3 suffices; so we suppose ##, >~ ~ , in which case it must be that c~ > c z. If t e, = ~ , then swapping individual allocations so that 1 gets (te~_, cz) with utility uz, and 2 gets (te~,cl) with utility u, , is clearly feasible. Moreover, tel = te., > 0 since (C c) is Pareto efficient. The marginal con- ditions for the Pareto efficiency of an allocation where 0 < (, < T and c i > 0. i = 1, 2, require that l ' s (resp., 2's) marginal rate af substitution of consump- tion for leisure be A t (resp., Az), which is inconsistent with leisure as a normal good since A~ >A 2. Thus, the 'swapped' allocation is Pareto in- efficient, and thc original allocation is, therefore, SS dominated. Finally, if tel >tez (and c I >c2) define A by u ( T - ~ j , c , - A ) = u 2 and B by u ( T - tez, c2 + B) = u,. From normality of leisure, A > B, so the utility allocation (u 2, ut) can be Pareto dominated. Hence (u~, u_,) is SS dominated.

"if" Clearly, any Pareto-efficient allocation with equal utilities is an SS optimum. So we suppose (te, c) is an interior Pareto-efficient allocation, where up = u(T - tel, cl ) < u(T - te.,, cz) = u_,, and suppose this allocation is SS dominated. Then there exists a feasible allocation (~ ,? ) such that (a) fil = u ( T - [ , , c l) >~ u,. and (b) tJ z = u ( T - ~ , ~)>~ u, , with at least one of these inequalities being strict; from the interior indifference curve assump- tion it follows that (& ~) is also an interior allocation. In addition, fi~ > fiz, otherwise ~z~>fi~ >~uz>u t and the original allocation cannot be Pareto efficient. Hence, from the 'only if' proof (te, ~) is SS dominated and so there exists a feasible allocation (t 7, ?) such that (c) fit = u(T - ~7 t , cl) >~ ~i_, and (d) ~ z = u ( T - ~ , ~ ) > ~ with at least one of these two inequalities being strict. From (a)-(d) it follows that tT~ ~>u~ and 17_, ~>u: with at least one of these inequalities being strict, contradicting the Pareto efficiency of ( C c L which is, therefore, an SS optimum. []

Proof o f Proposition Z Let us suppose that (h, c) is feasible with h: > h~.

2@) F. Madden " Journal of Publw Economics 61 (1996) 247-262

Let q, = q~(hi), i = 1, 2 and qi = 'Pi(hj), i = I, j = 2 and i = 2, j = 1. F rom our assumpt ions , ¢'1 exceeds ~'2 on the interval ( t q , h : ) . Hence, w l ( h , ) - ~l (h I ) < ~.(h . ) - ~ ( h I )" New. q l + q-" = ~1 (h_,) + ~p_,(h I ) < ~Pt (h I ) + ~°z(h,_ ) = q~ + q, . This establishes that the allocation where 1 receives (h z, cz) and 2 gets (h t, c l ) is certainly feasible, leaving an excess of educational resources of q~ + q , - qt - t12. Distributing this excess to either individual shows that the original allocation (where 1 receives ( h i , q ) and 2 (hz,cz)) is SS dominated . []

Proo f o f Proposition 8. Let (h, c) be feasible with ~'l(hl)>~'2(h2) and so h~ > h z. Take a small amoun t (~) of educational resources from individual 1, reducing h, by ,Ah where O < A h < h t - h 2. Since q~i (h l )>q~(hz) , for • small enough there is 7 /< E such that an increase of "q in 2"s educational resources will increase h e by the same Ah. Thus, the allocation [ ( h ~ - Ah, c j ) , (h: + Ah, cz) ] is certainly feasible, leaving e - r t educational re- sources unused. Now we exchange the consumpt ion good so that 1 gets ( /~ , c t ) = (hi - Ah, (~) and 2 gets (/~2, 62) = (h2 + Ah, ( , ) , where (j + (2 = c t + c , and both individual allocations are convex combinat ions of (h~, q ) and (/~z, c2). It is easy to check that this is possible with:

and

and

(fit, ~,) = a (h , , q ) + (1 - a ) (h> c , ) .

A = (h I - h z - Ah) / (h j - h2) .

(he, e:) = (1 - a ) (h , , c , ) + a(h z- ca) .

Let F be any concave increasing function and write U for F(u) for short ; U will be concave, so:

U ( h , . ~ , ) + U(f~ 2,c2) = U [ A ( h , . c , ) + (1 - a ) ( h 2 ,Q) ]

+ U[(1 - A)(h> ct) + A(h z,c2) ]

>>- A U ( h t . c , ) + (1 - a ) U ( h > c 2) + (1 - A)U(hp c,) + aU(h 2,c z)

= U(hl . cl) + U(h> c_.).

Distributing the unused educational resources of e -r~ allows as to find a feasible allocation (/~, F) such that

U( fl,, ~, ) + U( fzz, ~,) > U(h, , C I ) -[- U(h 2, c, ) .

Since this is true for any concave, increasing F, it follows that (h, c) is GL dominated . []

P. Madden / Journal of Public Economics 61 (1996) _47-_6_ 26I

P r o o f o f Propos i t i on 9, "only if" We sh:)w that any feasible, interior allocation, where u z > u : ,

cannot be an SS opt imum. We suppose that (h ,c ) is feasible and interior with u I = u ( h ~ , c l ) < u ( h 2 , c : ) = u :. If h l < h 2, Proposition 7 proves ihe result. So. let us suppose h ~/> h 2: since u ~ < u , . it must be that c 2 > c ~. Then we suppose, first, that h~ > h , . We define A by u(h 2, c 2 - A) = u~ and B by u(h~ ,c~ + B) = u, . Since h is a normal good, it follows that A > B . So the increase in consumpt ion good needed by 1 to attain utility level u 2 is less than the reduction in consumpt ion good that would cause 2"s utility to fall to u~. Thus , transferring B of the consumption good from 2 to 1 is feasible and leads to SS dominat ion of the original utility allocation. Let us consider, finally, the case where hj = h 2 = h. Since c 2 > c j , the normality of h impiiei that o'(h 2, c z ) > tr(h~, c~). Switching c~ and c, around is feasible and gives individual 1 utility level u_, and individual 2 utility level u t . Since ~ : ( h ) < ~¢~(h), the "switched' allocation cannot satisfy the marginal conditions for Pareto efficiency, and so cannot be Pareto efficient, Thus, the original utility allocation (u, u2) can be SS dominated.

"if" Let us suppose that (h , c) is an interior Pareto-efficient allocation, where u~ = u(h I. c l ) >~ u (h z, c~)= u_,, and suppose that (h, c) is SS domi- nated. It must be that u~ > u 2. since a Pareto-efficient. equal utility allocation must be SS optimal. Also, there exists a feasible allocation (h. ? ) such fitat (a) i~ i = u ( h i , k l ) ->" ::_, and (b) t;,: = u(/~:~ ¢::) >~ u ~ wlrh at lea~t one of these inequalities being strict; it follows from the interior indifference curves that (/~.6) is an interior allocation. It must also be that ti: > ~ ; otherwise , ht~>/ta~>u t > u ~ and ( h . c ) cannot be Pareto efficient. Now, f rom the proof of 'only if', ( t i , ? ) is SS dominated. Hence. there exists a feasible allocation ( h , ? ) such that (c) tT~ = u( t '~ .?~ )>~t i , and (d) if_, = u ( / ~ , ~ ) > ~ h ~ , with at least one of these two inequalities being strict. It follows from (a ) - (d) that ti~ I> u~ and if2 ~> u2 with at least one of these two inequalities being strict, which contradicts the Pareto efficiency of (h .c ) . Thus , (h ,c ) is an SS opt imum. []

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