Fair allocations and equal incomes

JOURNAL

OF ECONOMIC

THEORY

23, 189-200

Fair Allocations NORMAN

(1980)

and Equal Incomes* L. KLEINBERG

Couranf Institute of Mathematical New York University, New York, Received

June 5. 1979: revised

Sciences, N. Y. 10012

October

15, 1979

1. INTRODUCTION In the context of a pure exchange economy with fixed resources, a fair allocation of goods may loosely be defined as a feasible allocation for which no agent prefers what any other agent receives (equity), and for which there exists no alternative, “better” feasible allocation. “Better” here means that every individual is at least as satisfied with the alternative as with the original distribution, and at least one strictly prefers it. An equal income allocation is simply one that gives every individual a combination of goods which, when evaluated at “market prices,” yields the same value.

Fair Allocations Since equal distribution of income is often used as a yardstick against which to measure the degree of economic equity, it would be useful to see when the two concepts mentioned above coincide. That is, if we make correspond fair allocation with equitable allocation, we are led to ask when a fair allocation must be an equal income allocation (the converse is trivial, in most cases). Now it turns out that, at least in the relatively simple economic model considered here, a quite satisfactory answer can be given, and in fact this entire question is intimately tied to the purely mathematical notions of Lipschitz condition and rectifiable set. More precisely, we show explicitly below that: (1) if the distribution of tastes, as embodied in the function which assigns to each agent a utility function, satisfies a pointwise Lipschitz condition at a point, or agent, z, then any fair allocation must satisfy a pointwise Lipschitz condition at z as well; * This paper is a condensed version of Chapter I of the author’s Massachusetts Institute of Technology, January, 1978.

Ph. D. thesis

done

at

189 0022.0531/80/‘050189-12SO2.00/0 Copyright C 1980 by Academic Press, Inc. All rights of reproduction m any form reserved.

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NORMAN L.KLElNBERG

(2) if a fair allocation satisfies a pointwise Lipschitz condition at all agents in the economy, then it must be an equal income allocation. We note in passing that the considerations above are not empty, since it has been shown that fair allocations do exist in the types of economies considered here (see Varian [3, Theorem 2.3 I). In Section 2 we detail the notations used and present the structure of our economic model. Section 3 contains the proof of (2), Section 4 contains the proof of (1). and Section 5 presentssome known extensions to these results.

2. THE ECONOMIC

MODEL

In this paper, the set of agents will be represented by A, an open-convex subset of R”, M > 1. We do not require that A be bounded. Fixed, henceforth, will be the number of goods under consideration at, say, N> 2. The commodity space may then be identified with P, the strictly positive orthant of N-dimensional Euclidean space, and a point of P will then correspond to a specific commodity bundle. The notation 1.x will denote the appropirate Euclidean norm for x, depending on whether x is a scalar, a vector, or a matrix. In particular: forxER,Ixl=x,ifx>O,

=-x,ifx
for X E Rk, with X = (X1 ,..., xk), /XI =

2 Xf

( i=l

i

for x E Rk2, with x = [Xij], 1x1 = ( i$, Xi)

I”.

l/Z ;

For aER” and E>O, U(U,E)-{x:1x--al<&}, B(a,&)-{x:1x---al<&}. S = P n {x: Ix]= 1}. Pk denotes the outer Lebesguemeasure on Rk, and for T c Rk, Cl(T) denotes the closure of T in R k. We shall assumein all that follows that there exists an open set W c P with Cl(w) c P, such that each agent a E A has, in W, preferences representable by a suitably smooth utility function. More specifically, let C’,(w) denote the linear space of all bounded, real-valued functions on W with bounded partial derivatives through the second order. If we let a = (a, ,..., a,,,) be an ordered N-tuple of non-negative integers ai, we call a a multi-index and associate with each a the differential operator D” = (a /3~,)~l (a /a~,)~~~ whose order is Ial =a1 + .,. + aN. If 1aI= 0, Oaf =J: Then, if fE C&(w), there exists a K < co such that supXGw1D”f(x)l < K for all a with 1011 < 2. Denote now, for each a, /ID”fll=

FAIR ALLOCATIONS

ANDEQUALINCOMES

191

sup,,a, 1D”f(x)l, and then define a norm on Ci#+‘J by llfll= * I’* cc ,a,c*II~afll ) . Now let 3( I+‘) denote the subset of CL(w) which consists of those functions which are non-negative, strictly concave, and whose first-order partial derivatives are strictly positive: then give Y(W) the relative topology of C:,(W). Our assumptions on the preferences of the agents can then be summarized by the statement: each agent a E A is assumedto have preferencesin W representable by a 17,E Y( I%‘). We now define an economy 8(u) as a continuous mapping U: A -+ .U( I+‘); an allocation for 8(u) is then just a map w: A + W. DEFINITION

1. An allocation w is fair for an economy Z(u) if and only

if: (a)

u,(w(a)) > U,(w(a’)) for all a’, a E A (equity);

(b)

Du,(w(a)) = cap, some c, > 0, some fixed p E S,

for all a E A (efficiency). Part (b) can be shown to be equivalent to the requirement that w be a strongly efficient allocation. Strongly efficient here meansthat there exists no alternative, “better” feasible allocation, “better” being defined in the second sentenceof Section 1. In this paper, the efficiency price associatedwith any fair allocation w will always be denoted by p.

3. POINTWISE LIPSCHITZ

FAIR ALLOCATIONS

Our purpose in the next two sections is to provide a reasonable set of restrictions on an economy 8’(u) so that any fair allocation for a(U) must be an equal income allocation. If these restrictions are suitably general (within the context of the particular types of preferences which we have hypothesized), we may then say that, in a “large” number of economies, if incomes are not equal, then the allocation of goods is not a desirable one; in particular, it is not fair. In this section we will demonstrate that, if a fair allocation w for an economy Z’(u) possessescertain strong continuity properties, then w must give the same income to every agent. The key result we will use (Lemma 2) is due to H. Varian and shows that, if the fair allocation is differentiable at a point a of the closed unit interval, then income is differentiable at a and its derivative is zero there. We may then combine this result with the implications of strong continuity to infer that the .Y’ measureof the image of any line segmentin A, under p . w, is zero. Under our assumption that A is convex, this of course will imply that p w is constant on A. We now define precisely two notions:

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NORMAN L.KLEINBERG

DEFINITION

2.

Let f: B + Y, B c R”, Y a n.1.s.with norm ]] &.. Then:

IS said to satisfy a Lipschitz condition (LC) on B if there is an s > O’Zchfthat

IIf

-f(Y)llY < s Ix -45

for all

x,y E B:

the number s is then called a Lipschitz constant off. Every such mapf then has a least Lipschitz constant, denoted Lip(f). (b) For B an open set, f is said to be pointwise Lipschitz (PL) at a point b E B if there is a a(f, b) < 03 such that

lim sup IV(x) -f(Y>II* < o(f b) x-b

Iff

Ix-b1

’

’

is (PL) at every point of B, we will say that f is (PL) on B.

We note that, even if B is convex, a function which is (PL) on B need not satisfy an (LC), even locally (however, see Lemma 1 below). In this paper, we will employ Definition 2 in one of two cases: for B = A, Y = RN, f = w, and for B=A, Y=Ck(W),f= U. Now let w be a fair allocation for some economy a(U), and supposethat w is (PL) on A. We explicitly do not require that o(w, a) be continuous, bounded, or even locally bounded on A; just finite. In fact, however, the property of being (PL) on an open set is sufficient already to imply a certain degree of regularity: LEMMA 1. Iff: B + RN, ifB is an open subset of R”, and iff is (PL) on B, then f is dtflerentiable a.e. in B. Further, B is the union of a countable family of IPM measurable subsets such that the restriction off to each member of the family satisfies an (LC).

Proof.

See Federer ] 1, 3.1.8, 3.1.91.

For the moment, consider the case M = 1. We then have available LEMMA 2. Suppose that w is a fair allocation for an economy Z’(U); if w is differentiable at a point a E A, then D(p w)(a) = 0.

Proof: See Varian [4, Theorem], which uses the closed unit interval as the set of agents. Now, if w is (PL) on A, the same is true of p w; in light of Lemmas 1 and 2, we may infer the existence of a set T c A, with 9’(A - T) = 0, such that D(p w)(a) = 0 for all a E T. The following lemma insures both that the range of incomes over T, and the range of incomes over A - T, have measurezero.

FAIR ALLOCATIONS LEMMA

193

ANDEQUALINCOMES

3. Let f map B c RM into R”. Then:

(a> p”(fP))

< (Lip(f))“’ y”(B);

(b)

if B is an open set, if C is a set of critical points off, PM(C) < 00, then Ip”‘(f(C)) = 0.

and if

Proof: For (a), see Federer [ 1, 2.10.111. For (b), see Sard [2, Theorem 11. Now suppose that the measure of p . w(T) > 0. Then, by defining Z(n) 5 Tn (x: -n < x < n} and considering Y’( p w(Z(r))) for finite r, we infer the existence of an s < co such that Y’( p . w(Z(s))) > 0 (note: we do not require that the sets {p w(Z(r))} be measurable, since !Y’ is a regular measure). This contradicts Lemma 3(b), and thus we conclude that Y’(p w(T)) = 0. As for the set A - T, we note that by Lemma 1 there is a sequence ( V, } of ip’ measurable subsets of A such that U V, = A and p w restricted to {A - T) n V,, satisfies an (LC). By Lemma 3(a), w(A - 7’)) = 0. Thus ip’(p w((A N T] n V,)) = 0, for all n, so that Y’(p Y’(p w(A)) < 9’(p w(A - 7’)) + P(p . w(T)) = 0. Noting that p . w(A) is connected as the continuous image of the connected set A, we infer finally that p w is constant on A. For A an open-convex subset of R”, M > 1, we let a, a’ be two points of A, and consider any open straight line segment containing them and contained completely in A. There then exists a 4: (0, 1) -+OntoL satisfying an (LC). Then w 0 d is (PL) on (0, 1); since w o 4 is obviously a fair allocation for 8(U 0 #), by what have just shown 4p’(p . (w 0 #){(O, l)}) = 0; i.e., income is constant on (0, 1). Thus income is constant on L: the convexity of A then immediately yields that income is constant on A. We have thus demonstrated: THEOREM 1. Let A be an open-convex subset of R”, and let w: A --) RN be a fair allocation for an economy 8(U). Zf W is (PL) on A, then w is an equal income allocation.

4. FAIR ALLOCATIONS

AND POINTWISE

LIPSCHITZ

PREFERENCES

It thus remains to derive conditions on the map U which will imply that any fair allocation for 8’(U) must be (PL) on A. Now brief reflection should indicate the direction in which we are heading. For if an allocation fails to be (PL) on A it must, in general, treat agents which are close in the metric of A relatively differently. If that allocation is moreover fair, then it follows that those agents which are treated relatively differently must have preferences which are relatively different. It is of course understood that our use of the 642/23/2-5

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NORMAN

L. KLEINBERG

term “relative” corresponds to comparison with metric distance, and thus conforms to the notion of pointwise Lipschitz condition. Thus it will turn out that the formally simple requirement that the map U be (FL) on A will suffice to insure that any allocation for Z’(U) is, as well. We first introduce the notion of the Signum function: sgn: RN - {O} --+RN := sgn(x) =x/lx\. This function is of use precisely becauseof our definition of efficiency; i.e., Definition l(b) just says that, at any point w(a) lying in the image of a fair allocation w, sgn o DU,(w(a)) =p. Thus the properties of the mappings sgn o DU, and D(sgn o DU,) will play an important role. First, it is easy to show that, on every subset B c RN such that inf,,, 1x1~ m(B) > 0, sgn satisfies an (LC); in fact Lip(sgnl,) ,< 2(m(B))-‘. Following immediately from this is: LEMMA 4. Zf U: A -+ Y(W) is (PL) at a E A, a + sgn 0 DUO(x) is (PL) at a, for every Jixed x E W.

then

the

map

ProojI Fix x E W and pick E > 0 such that B(a, E) c A. Then, since #:y- DU,,(x) is a continuous map of A into RN, #(B(a, E)) is a compact subset of RN. Since iJ, E Y(w>, for ail y E B(a, E), #(B(a, E)) does not contain the origin. Thus: a(w

0 DU. (x), a) < 2(m(#(B(a, &))I)- ’ o(DU. (xl, a) < 2(m(W(a,

E))))- ’ o(U. (xl, a) < 03;

where the last two inequalities follow by consideration of the norm on cDvv. I There is one other result concerning the Signum function we will need. The reader can easily convince himself that, if a point w(y) in the image of some fair allocation w lies on an indifference surface of agent y which is flat, there is little reason to believe that that allocation will be (PL) at y. Now of course we have ruled out such surfaces by our assumption that each U, be strictly concave. To make the concept of (PL) applicable, however, we will have to derive a quantifiable lower bound to the degree of flatness along certain hyperplanes in RN. In particular let, for q E S and a E A, r(q, a) = (sgn o DU,)-‘(q); i.e., just the locus of points in RN at which the differential of U, points in the direction q. Note that, by efficiency, if w is a fair allocation for a(v), then w(a) E T(p, a), for all a E A. By our assumptions on .Y( IV), r(q, a) is a C’ curve in RN and thus possessesa tangent vector at w(a), call it u(q, a). We shall be concerned with the local flatness of U,, at w(a), along the hyperplane orthogonal to u(q, a). Let then g(v): RN -+ R N-’ denote the orthogonal projection whose kernel is

FAIR

ALLOCATIONS

AND

EQUAL

the vector U, and thus whose image (Img(u)), to U. We then have: LEMMA

5.

195

INCOMES

is the hyperplane orthogonal

Let t E r(q, a): then there exist positive

scalars

A, y, such

that minimum lP(w IX/= lJelmn(o~q,a))

0 DU,W’)l

xl > Y,

for all

t’&B(t, A).

Proof. Let v(q, a) = U. We note first that in fact sgn is continuously differentiable (at least) on its domain, so that D(sgn 0 DU,)(t’) exists as a linear operator on RN, for every t’ E W. Now D”U,(t) is non-singular, since U, is strictly concave: also [D”U,(t)] u E L(q), the one-dimensional subspace spanned by the vector q, because the image of r(q, a) under DU, is clearly contained in L(q), which is its own tangent cone. Thus only vectors in L(u) are mapped into L(q) by D’U,(t); in particular, D’U,(t) restricted to Img(u) does not intersect L(q). But the kernel of the linear map D(sgn 0 DUO)(t) is precisely L(q). Thus D(sgn o DUO)(t) restricted to Im g(u) is non-singular as an operator on RN- ‘. Hence there is a y’ > 0 such that min,x,=l.xElmg(v) I[D(w 0DU,N>l 4 > Y’. But D’U, is continuous at t, and Dsgn is continuous at DUO(t), so that there is a A > 0 such that B(t, A) c W and t’ E B(t, A) implies that D(sgn 0 DU,)(t’) restricted to Im g(v) is non-singular. Since B(t, A) is compact, and the map 4: B(t, A) -+ R := #(t’) = ,x,~l~~mK(c)

I[D(w

0 Du&t’)l

. XI

is continuous and never zero, there exists a y > 0 such that $(t’) > y, for all I’ E B(t, A), as claimed. 1 The result we are seeking now follows: THEOREM 2. Let A be an open-convex subset of R”, and suppose that U: A -+ 9(W) is (PL) on A. Then, if w is a fair allocation for a(U), w is (PL) on A. Hence, w is an equal income allocation.

Fix a E A, let 1 be such that B(a, A) c A, SUP,,~ 1D’Un,(x)J < A Proof. for all a’ E B(a, A), and let E be such that E < A and B(w(a), E) c W. Let t = w(a), and set a(U., a) = u. Now it can be shown that, when U is a continuous function, any allocation which is fair for Z(v) is a continuous function 14, Lemma]. In the sequel, we therefore take for granted the continuity of w. First we note that, say by Engel’s aggregation condition, v( p, a) is not orthogonal to p. Choose then u in the tangent cone of T(p, a) at r (i.e., just a

196

NORMAN

L. KLEINBERG

scalar multiple of v(p, a)) such that 1u 1= 1 and p ?J3 6 > 0, and let g(v) = g. Consider now the X-section X(t, r, ker g, s) = RNn

{x: s-’

Ig(x - t)l < Ix - tI < r},

r>O,O
1.

We will accomplish the proof in four steps. I. We show that there exist suitably small scalars 1’ 0, by the continuity of the inner product in RN, there is

an s’, 0 < s’ < 1, and a 6’ > 0, such that if Ix I = 1, if v x > 0, and if Ig(x)l < s’, then p x > 6’. Then, if y E X(t, r, ker g, s’) and if v (y - t) > 0, we have Ig(sgn(y-t))l=Iy-tl-‘Ig(y-t)l (ly-tl-‘s’ly-tj=s’, so that p sgn(y - t) > 6’ or p . (y - t) > 6’ I y - tl. Thus, for every r > 0, P (Y - t) > 6’ IY -

4,

for every y E X(t, r, ker g, s’) such that v (y-t)

> 0.

(4.1)

We now note that mino,,,,,,,,(min,,,,,,,,, lDU,,(t’)l) = m > 0, since closed balls are compact and convergence in the topology of 9(W) induces uniform convergence of the differentials. Pick E’ < E so small that As’ < 6’m

(4.2)

and then choose A’ < 1 so small that w(U(a, A’)) c U(t, E’). Given any a’ E B(a, A’) we assume that in fact t’ = w(a’) E X(t, E’, ker g, s’) and show that this leads to a contradiction. Suppose first that v (t’ - t) > 0; then U,(t’)

- u,(t) = DUa(t)

(t’ - t) + f(t’ - t)‘D’u,(t)(t’

- t),

where t is some convex combination of t and t’, hence r E B(t, E’). But = c,p, some c, > 0, by the efftciency of w, and c, > m, by our definition of m so that, in light of (4.1) we have DU,(t)

U,(t’)

- U,(t) > 6’m It’ - tI - It(t’ - t)‘D’U,(s)(t

- t)l.

Noting that It’ - tl < a’, (4.2) implies U,(t’)

- U,(t) > 6’m I t’ - t I - (m&/2)

Since this contradicts

) t’ - t I = (mS’/2) It’ - t 1 > 0.

the equity of the allocation

w, it is impossible

for

t’ E X(t, E’, ker g, s’), if v . (t’ - t) > 0. Now assume that v (t’ - t) < 0. Since the X-section is symmetric about implies that 2t - t’ E X(t, E’, ker g, s’). But t, t’ E X(t, E’, ker g, s’)

FAIR

ALLOCATIONS

u.((2t-t’)-t)=v.(t-t’)=-v.(t’-t)>O. Taylor expansion of U,, c,, > m, we arrive, exactly Since this again contradicts

AND

EQUAL

197

INCOMES

Then considering the about t’, and noting that DU,, = caCp, some as above, at the conlusion U,(t) - U,,(t’) > 0. the assumed equity of W, we have shown I.

II. We next establish the existence of scalars 6’ and s”, 0 < E” < E’, 0 < s” < s’, such that if x E T(p, a) - t, and / v (x - t)l < E”, then x E X(t, E’, ker g, s”). Pick any s”, 0 < s” < s’, and observe that, by virtue of the facts that v is an element of the tangent cone of T(p, a) at t, and T(p, a) is a C’ curve, there is an E” < E’( I - (.s”)~)“~ such that if x E T(p, a)- t and if 1u (x - t)l < E”, then I g(x - t)l < s”( 1 - (s”)~))“~ I u (x - t)l. Hence, breaking the vector x - t into its components along ZJand in Im g, we obtain the relation I g(x - t)l < s” Ix - tl and the restriction Ix - t ( ( E’, so that in fact x E X(t) E’, ker g, s”). This demonstrates II. III. Now I and II combine to yield the existence of a 1” 0 such that a” E B(a, A”) implies that 1v (w(a”) - w(a))1 < ICIa” - al. Choose A” < A’ such that w(U(a, A”)) c U(t, E”), and let a” E U(a, A”); we may assume without loss of generality that w(a”) E t” ft. Let O(P) E {t” + Im g} fl T(p, a); note e(t”) E X(t, E’, ker g, s”) U {t}, by the result II above. Since we have chosen E < A, by Lemma 5, 1sgn o DU,(t”)

- sgn 0 DU,(B(t”))l

= Isgn 0 DU,(s)(t”

- B(t”))l > y It” - 8(t”)I,

(4.3 1

for r some convex combination of t” and B(t”), hence 7 E B(t, E). Also, by Lemma 4, / sgn 0 DU,(t”) - sgn 0 DU,,,(t”)( < 2m- ‘a /a” - a /. Finally, we observe that, by the efficiency of w, and the definition of T(p, a), sgn o DU,,,(t”) = sgn 0 DU,(B(t”)) =p. Then, using (4.3), y 1t” - B(t”)I < /sgn 0 DU,(t”) = 1sgn o DU,(t”)

- sgn 0 DU,(tY(t”))i - sgn 0 DUGPS(t”)j < 2m-‘a

1a” - al,

so that )t” - O(t”)l < 2(my)- ‘a Ia” - a I. Now, becauset” - B(t”) E Im g, and g(x) = x for any x E Im g, we have

1t” - e(tq > 1g(t’f - ty - 1g(e(tq - t)l. But t” @X(t, E’, ker g, s’) U {t}, by I, thus

( g(t” - t)l > s’ It” - t I

(4.4 )

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NORMAN

L. KLEINBERG

which implies, again by using orthogonal

components,

( g(t” - t)l(1 - (s’)*)i’*(s’))’

that

> 1U (t” - t)l.

(4.5)

Now, as we have noted above, #(t”) E X(t, E’, ker g, s”): B(t”) # t. Then, using once more orthogonal components: 1v (c” - t)l* = Iv

(e(t”) - t)l’ > (1 - (s”)~)(s”)-*

Thus, in the case rY(t”) #

t, by

assume first that

1g(8(t”)

- t)i2. (4.6)

using (4.4), (4.5), (4.6) we obtain

] C” - e(r’)l > ((s’)( 1 - (s’)Z)-

iI2 - (S”)( 1 - (s”)2)

1’2) 1U (t” - r)l,

or rewriting Iv

(w(a”)

- w(u))l

-(S”)(l

< 2o(my)-‘((s’)(l

- (s’)*)-i’*

-(S’~)~)-~‘*)-~~u”--a~~K~a”--al.

If e(P’) = t, t” E t + Im g, so that 1~ (~(a”) - w(u))1 = 0, and the above relation is valid in this case also. This shows III. IV. We observe that, if a” E B(a, L”), then there is a K’ such that I g(ffr- t)l < Ic’ 1a” - a 1.

1g(t’f - t)l G 1g(t” - e(t’f))l + I g(e(ty - t)l. Now,

(4.7)

from III, I g(t” -

e(q)1 =

I t” -

e(tq

< 2(my)- IO 1a” - a /.

(4.8)

On the other hand, since e(P) E x(t, d, ker g, s”), Ig(B(P) - t)/ < s” )e(t”) - t 1, so that, using the orthogonal components of e(t”) - t along v and in Img, we obtain 1 g(e(ty

-

t)l

<

(dy(i

-

(df)*)-

1’2

1v

(f/f

-

t)l

< (S”K)( 1 - (s”)2) - I’* Ia” - a 1.

(4.9)

Hence; using (4.7) (4.8), (4.9), we have I g(t"

-

t)l < (2(r7&'0

+ (s"K)(~

-

(f)*)-I'*)

Ia"

-

Ul 3

K’

1Un

-

Uj.

This shows IV. From III and IV, and recalling that I = w(u), 1” = ~(a”), we conclude I w(d)

as desired. 1

-

w(u)1

<

(K*

-I- K’*)‘/*

1a”

-

a 1,for all a” E B(u, A”),

FAIR

ALLOCATIONS 5.

SOME

AND

EQUAL

199

INCOMES

EXTENSIONS

We should remark that several of the hypotheses made in connection with Theorems 1 and 2, are strictly speaking, overly strong. First, it is clear that we can ignore behavior on countable subsets of A. That is, if we let Tc A be any countable subset, the conclusion of Theorem 1 would follow if w were assumed (PL) only on A - T. Further, Theorem 2 would follow if U were assumed (PL) only on A - T. Also, we nowhere used, in our proof of Theorem 2, the fact that the maps Qa: A -+ supXp w ]D”f(x)], for 1a ] = 2, are (PL) on A. Hence the space of C* utility functions could have been viewed instead as a subset of a space such as C&(w), with the correspondingly weaker induced topology. However, we then would have required the additional hypothesis, in the statement of Theorem 2, that each #,, for ]a] = 2, be locally bounded on A. Finally, as the local nature of our arguments shows, we did not need the assumption that the utility functions have bounded partial derivatives over W. The assumed continuity of these derivatives implies already that they are bounded on compact subsets of W, and this is all we required for our proofs. The global boundedness merely made the metric on our function space simpler to obtain. I also believe that the requirement that each U, be strictly concave is unnecessary. If the restriction of the Hessian matrix of U,(t) to the hyperplane orthogonal to DU,(t) were negative definite, Theorem 2 should follow, although I have not proved it in this case. However, it can be shown, in the case A = [0, 11, that (1) If a sequence of finite agent economies “approaches” an economy with domain A, and if the associated sequence of taste distributions satisfies a uniform Lipschitz condition, then any corresponding sequence of fair allocations must converge to an equal income allocation. (2)

Any two-good

fair allocation is an equal income allocation.

(3) The image of any fair allocation in an economy of N goods is an (N - I)-rectifiable set. (4) If the function U: A * Y( IV) is an open mapping, then any fair allocation for 8’(U) is an equal-income allocation. In particular, if the economy is of N goods, the image of the fair allocation has an open projection into (N - l)-dimensional space. (5) There do exist non-equal income fair allocations four goods or more. Note that (3) is a partial converse to Theorem almost immediately from step I of the proof of that however, that true converses to Theorems 1 and 2 do may easily exist equal income fair allocations which

in economies of

2. In fact, (3) follows Theorem. We remark, not exist. That is, there are not (PL) on A, and

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NORMAN L.KLEINBERG

the fact that there exists a fair allocation for 8’(U) which is an equal income allocation does not insure that U is (PL) on A (cf. (4) above). In fact, this restriction on U is not guaranteed even if every fair allocation for C?(U)must be an equal income allocation (cf. (2) above). We note finally that (3) above can be extended to yield a generalization of Lemma 2. More specifically, the function mapping a bounded subsetof RN-’ onto the image of the fair allocation must be relatively differentiable at every point of its domain, and the relative differential of income is identically zero. The remark (4) above’ follows from this observation since in that case the domain of that function is an open set.

ACKNOWLEDGMENTS J wish to acknowledge

the helpful

suggestions

of an anonymous

reference.

REFERENCES Springer-Verlag, New York, 1969. 1. H. FEDERER, “Geometric Measure Theorey,” 2. A. SARD, Images of critical sets, Ann. of Math. 68 (1958), 247-259. 3. H. VARIAN, Equity, envy, and efficiency, J. Econ. Theory 9 (1974), 63-91. 4. H. VARIAN, Two problems in the theory of fairness, J. Public .&on. 5 (1976), 249-260.