Envy-free and efficient allocations in large public good economies

Economics Letters 36 (1991) 227-232 North-Holland

227

Envy-free and efficient allocations in large public good economies Dimitrios Diamantaras * Temple

Unioersity,

Philadelphia,

PA 19122,

USA

Received 31 January 1991 Accepted 29 April 1991

In an economy with private and public goods and a continuum of agents, we show that every envy-free and efficient allocation is generated by a public competitive equilibrium from equal endowments (not a Lindahl equilibrium from equal endowments).

1. Introduction

No-envy [Foley (1967)J is a prominent equity criterion. In this paper we focus on the properties of envy-free allocations in large economies with public goods. For other equity criteria and a general survey see Thomson (1989). The results we obtain are not mathematically involved, but they shed light on some rather overlooked issues in the theory of public good economies. We study a property first shown by Varian (1976) in the context of an exchange economy. Under appropriate conditions, any envy-free and efficient allocation in an exchange economy with a continuum of agents is a Walrasian equilibrium from equal endowments. This result, which relies on stronger assumptions than the core equivalence results, provides a characterization of Walrasian equilibrium from equal endowments when coupled with the observation that any Walrasian equilibrium from equal endowments is envy-free in an exchange economy [Foley (1967)J One might expect that the extension of these results to public good economies would be a straightforward, if perhaps tedious, exercise, with Lindahl equilibrium playing the role of Walrasian equilibrium, as in many studies. It comes then as a surprise to find out that this cannot be the case. As demonstrated by example in section 3, one can find very simple public good economies whose envy-free and efficient allocations are not Lindahl equilibria from equal incomes, although these economies satisfy conditions which are sufficient for the equivalence result in the exchange economy case. This example complements the result of Thomson (1987) that a Lindahl equilibrium from equal endowments may violate no-envy. * I am grateful to William Thomson, the members of the Game Theory reading group at the University of Rochester in 1987-1988, Alejandro Hemlndez, and Larry Kranich for their comments and discussion. Part of the research for this paper was supported by a Raymond N. Ball Dissertation Year Fellowship awarded by the College of Arts and Science of the University of Rochester. which is gratefully acknowledged. AI1 shortcomings of this paper are solely of my own creation. 0165-1765/91/$03.50

0 1991 - Elsevier Science Publishers B.V. (North-Holland)

228

D. Diamantoras

/ Envy-free and efficient allocations

Beyond the negative result, we show that there is an equilibrium concept which enjoys equivalence with the envy-free and efficient correspondence in large public good economies. This equilibrium concept is the public competitive equilibrium with proportional taxation, as defined by Greenberg (1977) [who modifies the original definition of a public competitive equilibrium, due to Foley (1967)]. The property of no-envy for public competitive equilibria with proportional taxation and from equal endowments in any (large or small) public good economy was first shown by Foley (1967). Our result allows us to compare the Lindahl equilibrium to the public competitive equilibrium with proportional taxation from a novel viewpoint. A more general way to view this result efficiency

imply

competitive efficient property

the

equality

equilibrium

of the

with proportional

allocations with could exist.

equal

is that in economies

expenditures taxation

expenditures

and section

goods

by

from equal endowments

on private

As a corollary to the main result, if the economy envy-free efficient allocations and public competitive significantly more relaxed The paper is organized

with ‘many’

on private goods,

but

agents,

the

The

is a mechanism

other

mechanisms

and

public

to obtain with

this

has only one private good, the equivalence of equilibria from equal endowments holds under

assumptions, and even if the economy has two agents. as follows. Section 2 sets up the notation, section 3 presents

the results,

4 concludes.

2. Definitions,

notation

Let (T, Y, p) be the measure space of agents, where T is an open, connected Euclidean space R”, Y is the Bore1 u-field of T, and p is a probability measure number

no-envy

agents.

of private

goods

is 12 1 and

the number

of (pure)

public

goods

subset of some on (T, F). The

is q 2 1. R/+4 is the

(I + q)-dimensional Euclidean space, R ‘+g + the non-negative orthant of this space. Let >> , > , 2 be the vector inequalities in this space and let ‘.’ denote the inner product. For any subset X of tR’+q, X, denotes the projection of X on R’, and X, denotes the projection of X on R4. Let X = Ryq denote the consumption space of each agent. [Since the origin has not been excluded, homothetic preference relations do not qualify as smooth [Mas-Cole11 (1985, p. 69)]. We can remedy this by removing the origin from the consumption space, which has no impact on the results.] The relation 1, c X X X represents the preferences of agent t E T, and is complete,

reflexive,

and

transitive. For any y E X,, for any 1 E T, let k,(y) be the restriction of 1, to X, given y. The space of preferences is T’, = (1, C XX X 1 for all y E R4+, LI (y) is continuous, monotone, C2, and differentiably strictly convex}. For The endowment of the economy goods for simplicity. Because p is aggregate quantities to be defined,

further details on this set, see Mas-Cole11 (1985). is w E R$+. We do not admit positive endowments of the public a probability measure, we interpret total endowments (and other such as profit) as averages. Thus, equal division of endowments

means that each t gets w. Each agent receives a share of the profit n equal to 8(t)m, where e(f) 2 0 for almost every t E T and j@(t) = 1. We set e(t) = 1 for p-almost every t E T, and so we omit further mention of 8. Summarizing, the space of admissible characteristics is &= qYr’,, with typical element a = (k,). We give J&’ the structure of a complete metric space as in Mas-Cole11 [1985]. The production set is Z c Iwltq . We assume that Z is closed, convex, contains the origin, and is such that Z n [ - Z] = (0). Z will be held fixed in the rest of the discussion. An economy is a pair (8, Z) where E: T +.B?. We assume that d is Bore1 measurable and that there is a compact

set KC..&

with

6(t)

E x

for all t E T. An

allocation is a pair (x,

y) where

229

D. Diamantaras / Envy-free and efficient allocations

x : T + X, and y E X,. Under an allocation (x, y), an agent t E T consumes (x(t), y) E X. ’ An allocation is attainable if (/,x(t) - w,, y) E Z. Let 3 denote the set of attainable allocations. Given an allocation (x, y), Bx,Y C R’+4 is the smallest closed set such that {t E T 1(x(t), y) E B,.,} is null. An allocation (x, y) is envy-free if for p-almost every t E T, (x(t), y) is k,-maximal on B, y. The price space is P = Ry? A price system is a pair (pt, p,), where p, E P, and p4 : T + P, is integrable. The notational shortcut used before allows the convenience of writing p E P. Given a price system p E P and an allocation (x, y), the profit rr(x, y ; p) is defined by x(x, y ; p) = (P,, pa). (Lx(f) - a, YL where PQ = /r~Jt). A Lindahl equilibrium (from equal endowments) is an attainable allocation (x, y) E 3 and a price system p E P, such that (L.1) For almost every t E T, (x(t), y) maximizes IP,. 0 + a(x, y ; PI}. (L.2) The vector (lrx(t) - w, y) maximizes (p,,

k1

ouerthebudgetset

{(x’,

y’)Ip,.x’+p,(t).y’

pQ). (z, u) subject to (z, u) E Z.

Let (x, y) E _Y be an attainable allocation and p E P a price system. A feasible tax system for (x, y) under p is a function a : T -+ R such that for almost every t E T, a(t) I p, . w + T(X, y ; p). A tax scheme for (8, Z) is a continuous function (Y: .%X P X T -+ Iw such that for all (x, y) E 22” and all p E P, a( x, y ; p ; . ) : T ---*Iw is a feasible tax system for (x, y ) under p. We will be particularly interested in a special tax scheme, proportional taxation, which is defined byaP’(x, Y;P;~)~(P,.~/~~PI.W)PQ.Y=PQ’Y. Let (Y be a tax scheme for the economy (8, Z). A public competitive equilibrium (from equal endowments) for (8, Z) relative to (Y is a pair of an attainable allocation (x, y) E 3 and a price system p E P such that (P.1) For almost every t E T, (x(t), +a(& y;p)-4x3 y;p;t)). (P.2) The vector ( /rx(t) (P.3)

y) maximizes

- w, y) maximizes

(p,,

k,(y)

pQ) .

over the budget set {(x’,

(z,

0)

y) I p/.x’


w

subject to (z, CJ)E Z.

There does not exist a feasible tax system c?# (Y and a feasible allocation (2, j) with J # y such that for almost every t E T(Y(t), 7) > t(x(t), y) and p,.n(t) + &(z, J; p; t) - sr(x’, y’; p) I p/e x(t)

+ 4x2

y ; p ; t) - m(x, y ; P>.

(p.4) /T(y(x, y; pi t)=pQ-Y. An attainable allocation (x, y) E .Y is Pareto efficient if there does not exist another attainable allocation (x’, y’) E? such that for p-almost every t E T, (x’(t), y’)~,(x(t), y), and (x’(t), y’) > ,(x( t), y) for t E T, where T c T is non-null.

3. Results We will show that in a continuum economy with public goods, satisfying assumptions similar to those in Kleinberg (1980) Champsaur and Laroque (1981) or Mas-Cole11 (1987), any envy-free and efficient allocation is a public competitive equilibrium from equal endowments. The following ’ This involves a slight abuse of notation, which we also use later. Strictly speaking, x(t) are projections.

+ y E X is true, because X, and X,

230

D. Diamantnras / Envy-free and efficient allocations

example shows that we cannot have a similar theorem for Lindahl equilibrium. (It is even easier to construct an example with only one private good. We present this example to dispel any possible suspicion that the condition I= 1 might be driving the result.) Let

Example.

parameterized

I = 2,

q = 1,

as follows:

T = (0, 1).

Let

I_L be

u : T X lR ‘, + + R defined

the

Lebesgue

measure.

Utility

functions

with a(f) = [(l + t)/3(10 - t)], P(t) = l/3, y(t) = l/3, for all t E T. Production takes place according to a linear production function with unitary coefficients, aggregate endowments of goods x,, Consider the allocation

defined

are

by

and the

x2 are w,, w2 such that w, > 0, w2 > 0, and w, + w2 = 41/2.

by Z,(t)=l+t;

It is envy-free

?-,(t)=lO-t;

and efficient

(the

equilibrium with equal incomes. To show this, let us denote ((_C,(t));;‘;7-,

easy proof the

income

tET.

is left to the reader). of

agent

tE T

by

However, w(t).

Then,

it is not a Lindahl at

the

allocation

we have

j),

w(t)

forall

j=19/2,

= 12,(t)

=

11

+

+ lZ-,( t) +

Y(f)

4(t)

qu(t; 4(t), -%(t>, 9) _ qp(t;

A

--my=ll+

2,(t),

i,(t),

Jqy

g+(1+t)=21-t.

P

Then w(t) # w(s) whenever

t # s, and the claim is proved.

We

result.

turn

Mas-Cole11

now to our main (1985),

The

following

two results,

taken

with slight

changes

from

will be used in the proof.

Let x be an envy-free and efficient allocation for an exchange economy with set of agents monotone, C2, and T = (0, 1) and preferences on the consumption space lR: that are continuous, differentiably strictly convex. If for almost all t E T and all x E W!+ the set { y 1xz,y } is compact, then there exists k > 0 such that for almost all t E TX(t) I ke, where e is the vector in W\ with 1 in

Fact I.

each component. Fact 2. efficient

(Proposition

7.5.2,

Let an exchange economy allocation for that economy

p. 300). be defined as in Fact 1. Then every bounded envy-free and is Walrasian from equal incomes. (Proposition 7.5.4, p. 302)

Theorem. Assume that suppgc s),‘,, and that for p-almost evev t E T and for every (X, j) E 3, the lower contour set of ,,( j) is compact. Then every envy-free and efficient allocation (x, y) for & can

D. Diamantaras

arise as a public f?(t) = 1.

competitive

equilibrium

231

/ Enuy-free and efficient allocations

with proportional

taxation

from

equal endowments

and

Proof _ Suppose that (x, y) is an envy-free and efficient allocation in (8, 2). We define a private good exchange economy b’ by ({?,(Y)}~~~, { Jr.~},ET), where each t receives the average of x as endowment and preferences are restricted to the private good subspace. Fact 1 applies to E’, because of our assumption on the lower contour sets of z,(y). This implies that x is bounded. Fact 2 can now be applied, because of the boundedness and because x is an envy-free and efficient allocation in b’, as is easily seen. This implies that, if p, E R’+ supports x in b’, then for almost every t E T, x(t)

maximizes 2,(y)

subject to

xIp,.xlp,.

(1)

and

(2)

p,.x(t)-p,.j;x=o. But we can rewrite (2) as follows (where p E P supports 2 at ( jTx(t) before): p,.X(t)-p,.W+pQ.y-pQ.y-p,.

p,.x(t)-p,*w,+C(x, (1) and (3) imply that (x(t),

/

y;p;t)-r(x,

- w, y) with the same p, as

Tx+P,-w=o-

y;p)=O.

(3)

y) satisfies (P.l). (P.2) IS automatically satisfied by the definition of •I of (x, y); finally, P.4) is satisfied by construction.

( p,, pa). (P.3) follows from the efficiency

The condition on the lower contour sets of the restricted preferences for the private goods may merit some discussion. It can be roughly restated as requiring that each indifference curve intersect all axes. It is a form of a desirability condition: under it, no good is indispensable; not consuming any quantity of some good can always be made up by a sufficiently high, but finite, consumption of the other goods. At the same time, all goods are desired enough that high enough consumption of any good can make up for zero consumption of all the others. Cobb-Douglas preferences do not satisfy this condition, but they do if translated down by any positive amount. Also, all CES preferences with elasticity of substitution greater than unity satisfy the condition, among other examples. If we have only one private good, then we have a similar theorem under significantly weaker assumptions: it is no longer important how large T is (it can even be finite of any size) and we only need continuity and monotonicity assumptions on the preferences.

4. Concluding

remarks

The approach taken here is a little limiting. Since we restrict to a single endowment vector, we cannot appeal to results which impose a richness condition on the economy, as in Mas-Cole11 (1987).

232

D. Diamantarm / Envy-free and efficient allocations

A topic for further research is to obtain a direct result for public good economies with an arbitrary finite number of goods. It may be possible to rely on weaker assumptions, and it may be possible to extend our theorem to more general spaces of agents. Our example reinforces this suspicion, because, although it uses preferences which violate the assumptions of the theorem, the envy-free and efficient allocation exhibited is a public competitive equilibrium with proportional taxation from equal endowments. (One can modify the example to satisfy the assumptions of the theorem by translating the preference maps.)

References Champsaur, P. and Laroque, G., 1981, Fair allocations in large economies, Journal of Economic Theory 25, 269-282. Foley, D., 1967, Resource allocation and the public sector, Yale Economic Essays 7, 45-98. Foley, D., 1970, Lindahl’s solution and the core of an economy with public goods, Econometrica 38, 66-72. Greenberg, J., 1977, Existence of an equilibrium with arbitrary tax schemes for financing local public goods, Journal of Economic Theory 16, 137-150. Kleinberg, N., 1980, Fair allocations and equal incomes, Journal of Economic Theory 23, 189-200. Mas-Colell, A., 1985, The theory of general economic equilibrium, a differentiable approach, Econometric Society monograph no. 9 (Cambridge University Press, Cambridge). Mas-Colell, A., 1987, On the second welfare theorem for anonymous net trades in exchange economies with many agents, in: T. Groves, R. Radner, S. Reiter, eds., Information, incentives, and economic mechanisms, Essays in honor of Leonid Hurwicz (University of Minnesota Press, Minneapolis, MN). Working paper no. 76 (Rochester Center for Economic Research, Thomson, W., 1987, Notions of equal opportunities, Rochester, New York). Thomson, W., 1989, Equity concepts in economics, Mimeo. (University of Rochester, Rochester). Varian, H., 1976, Two problems in the theory of fairness, Journal of Public Economics 5 (nos. 3, 4) 249-260.