On the interiors of production sets in infinite dimensional spaces

Journal

of Mathematical

Economics

18 (1989) 29-39.

North-Holland

ON THE INTERIORS OF PRODUCTION SETS IN INFINITE DIMENSIONAL SPACES* M. Ali KHAN Uniuersity of Illinois, Champaign, IL 61820, USA Johns Hopkins University, Baltimore, MD 21218, USA

N.T. PECK University of Illinois, Urbana, IL 61801, USA Submitted

July 1987, accepted

May 1988

We show that ‘bounded marginal rates of substitution’ as formalized a closed, convex production set with ‘free disposal’ has a non-empty for Banach spaces but false for more general locally convex spaces.

by Khan-Vohra imply that interior. This result is true

1. Introduction The importance of the Hahn-Banach Theorem for the second fundamental theorem of welfare economics on the decentralization of Pareto optimal allocations as price equilibria is, by now, well understood. In the setting of economies with a finite dimensional commodity space, the essential hypothesis on preferences and production sets is that of convexity, as in Arrow (1951) and Debreu (1951). However, in the infinite dimensional case, this needs to be supplemented by an interiority condition on the set to be supported. If the commodity space is equipped with an order structure and is one whose positive cone has a non-empty interior, the interiority condition follows from economically innocuous assumptions such as ‘free disposal’ or ‘desirability’; see Debreu (1954) and Bewley (1972). If the positive cone does not have a non-empty interior, as in LQ), co >pz 1, or in the space of regular measures on a compact Hausdorff space, additional assumptions have to be made on preferences and production sets if one is to avoid assuming the interiority condition at the outset. For economies with production, as is our concern here, a variety of such assumptions have been made on production sets; see the work of Aliprantis, Brown and Burkinshaw (1987), Khan and Vohra (1988), Mas-Cole11 (1986b), Richard (1988) and Zame (1987). These assumptions have been seen as formalizations of *This research was supported by an N.S.F. Grant to the University of Illinois. Khan would like to acknowledge helpful conversations with Rajiv Vohra. Errors are solely the authors’. 0304-4068/89/%3.50

J.Math-

B

0

1989, Elsevier

Science

Publishers

B.V. (North-Holland)

30

M. Ali Khan and N.T Peck, Interiors of production sets

‘bounded marginal rates of substitution’ in production and in the case of Aliprantis et al. (1987), Richard (1988), Mas-Cole11 (1986b) and Zame (1987) have been directly inspired by Mas-Colell’s (1986a) concept of ‘properness’. In this note, we focus on the condition proposed in Khan and Vohra (1988). This is an obvious formalization of bounded marginal rates of substitution and simply requires that the set of supporting functionals to a production set be bounded below, in terms of the induced order, by a nonzero positive functional. Note that the condition does not require the production set to have any support points and it is vacuously fulfilled for sets without any such points. Under this condition on a single production set, Khan and Vohra (1988) present an infinite dimensional version of the Arrow-Debreu second fundamental theorem. However, Khan and Vohra do not present any example of a production set with ‘free disposal’, which satisfies their condition but does not possess a non-empty interior. Our first result is that in an ordered Banach space such sets do not exist. Our result is a consequence of the BishopPhelps Theorem and can be restated to say that in an ordered Banach space, order boundedness from below of supporting functionals forces closed, convex sets containing the negative orthant to have a non-empty interior. We also present an extension of our result to weak * closed convex sets in a dual Banach space. This extension prompted us to ask whether our result itself generalizes to locally convex spaces. We present three examples in spaces of interest in mathematical economics for which this conjecture is false. These three counterexamples are the second contribution of this note. It is worth stating that, in terms of the economics, our result can be more constructively viewed as providing a sufficient condition for the validity of Debreu’s (1954) theorem in ordered Banach spaces whose positive orthant has an empty interior. Furthermore, our examples bring out that the second welfare theorem presented in Khan and Vohra (1988) has to be evaluated in the context of ordered locally convex spaces which are not Banach spaces. Finally, it is of some interest that our result allows us to make the observation that the technology in Zame’s (1987) fourth example has a nonempty interior. The result is presented in the next section and the examples in section 3. Section 4 concludes the paper. 2. The result Let X be a Banach space over the reals and with X* its dual. The norm in X and in X* will be denoted by I[*[/. W e sh a 11assume that X is ordered by 2 and denote the positive cone by X,. X$ will denote the positive cone of X* with the order on X* induced by 2. For any x EX and any f’ E X*, we shall denote the value by f(x).

M.

Ali Khan and N.T Peck, Interiors of production sets

31

We shall denote the weak topology on X by cr(X,X*) and the weak * topology on X* by a(X*,X). For any non-empty subset C of X (or of X*) and for any f EX* (or of X), let

4f C)= SUPf (4. 9

xec

For any Cc X, Co will denote the norm-interior We can now present

of C.

Theorem 1. Let X be an ordered Banach space with X, C be a closed convex subset of X such that (i) -X+cC, (ii) 3grzX:, g#O implies f 28.

such that f EX*,

its positive cone. Let

f supports C, llfll=l

and a(f,C)
Then Co # 121.

Proof

By the Bishop-Phelps

Theorem [Bishop and Phelps (1963, Corollary

211 C= n H,, JES S=(f

where

EX*: a(f,C)
Hf={x~X:

)Ifl)=l

and f supports C],

f(x)srx(f,C)}.

Since g#O, there exists x0 E X + such that g(xo) >O. Let E=g(x,)/2. We shall show that {YEX: II-x0 - yll =E. By linearity f(y) 5 -E. Since OE C, a(f, C) 2.0. Hence f(y) < tL(f, C) or y E H,. We are done. 0 Remark 1.

Note that f ES * f 20. (Suppose there exists x0 EX, such that f (x0) ~0. Then f ( - x0) > 0. Since - X + c C, we can contradict a( f, C) < co.) However, we do not use this fact in the proof. Theorem 1 admits of the following extension.

Theorem 2. Theorem 1 is true with X and X, and weak * closed substituted for closed.

interchanged with X* and X*,

M. Ali Khan and N.II: Peck, Interiors

32

Proof:

Simply use Phelps’ Theorem the BishopPhelps Theorem. 0

of production

sets

[Phelps (1964, Corollary 2)] instead of

Theorem 2 leads us to the following conjecture. Conjecture. Let X* be a dual Banach space with X its predual and with X*, and X, their respective positive cones. Let C be a weak * closed, convex set in X* such that (i) -X*,cC, (ii) 3gEX+, g#O such that JEX, weak * interior C# @.

and cl(f,C)
llfll=l

Then

In the next section we shall exhibit three examples for which this is false. Note that unlike the theorem, the conjecture does not even require that f support C. 3. Three examples

We now present examples of three production sets for each of which the conjecture is false. Furthermore, the sets also satisfy the economic assumption of ‘irreversibility’, i.e., Y n { - Y> = (0). 3.1. Counterexample

I

For our first counterexample

~={yEl’:

yl+yi~O},

Y,={yd:

y&O),

we work in the dual pair [l’,cJ.

Let

i=2,3,...,

(For any YEI’, yi denotes the ith coordinate.) Claim 1.

Y has an empty a(l’,c,)-interior.

Proof. Suppose to the contrary that e is such an interior point. Then there exists a weak * open set containing e and contained in Y Hence there exists a positive integer k, positive numbers E, and f” E cO (a = 1,. . . , k) such that {yell:

(f”(y-e)lSE,,

c1= l,..., k}c

E:

M. Ali Khan

and N.7: Peck, Interiors

of production

33

sets

Certainly, e, ~0. Indeed, if e, =0 pick JE I’ such that

yi=f?i

(i# 11,

where M = max llf”/.

E= min E,, a

a:

Then ))j-e/1=~/2M. Hence for any CI,If”(y-e)l s(lf”ll~/2Msc/2. But j$ Y,, and hence j$ Y Now pick AEN such that for any u, lf;l< (- s/4e,). Since ff converge to zero and since there are only a finite number of f”, such an h can be found. Since e E I,, 1 Ie iI < cc. Hence we can find rie N such that leAIc( -2e,). Let n = max (A,fi) and

y,*= -2e,.

Since e E I’, y* E I’. Also y* q!Y, and therefore y* $ Y since y: + y.*= -e, > 0. For any CI, I~“(Y* -4l=

IfiX-2el -41

5 lfX-2el)l+

IfXeJl

5 E/2 + &f2 5 E,. We have a contradiction. Next we have Claim 2.

Y is a a(l’, c,)-closed,

convex

set such that

( - 1:) c Y and

Yn(-Y)={O}.

Proof: Since the intersection of convex sets is convex, convexity is obvious. Each x(i # 1) is a(/‘, c,)-closed since it is a closed half-space defined by an element of q,. Again, as intersection of weak * closed sets, Y is weak * closed. The fact that (- 1:) c Y is obvious. Finally, pick ye Y Then ye x for all i. Hence yi +y, SO. If y E( - Y), then -(yi + yi) SO. Hence yi 5 -y, and

34

M. Ali Khan and N.T Peck, Interiors of production sets

yiz -y, and therefore yi=yl Hence y = 0. 0 Claim 3.

{EC,, llfll=l,

for all i. But y E Y and y~( - Y) implies y, =O.

fx(f,Y)
implies fz(l,O,O

,..., 0).

Proof Suppose fr = 0. Then there exists j E N such that fi> 0 where N denotes the set of positive integers. Let j;=o

G#.i, l),

yy=-n

and

j$=n

for

nEN.

Certainly 9” E Y for all n E N. f(y) =nfi. By choosing n large enough we can show j-(9”) > a(f, Y). Now suppose fr < 1. Pick E>O such that fi < 1 --E. Then there exists jE N such that fj> 1 --E. Now let

yl=-fj fl

3

y._l-c J

7~

(i#1,j).

Yi=O

Since y,+yj=(l/f,)(l-s-&)
Since YEY,, and also y~x(i#l,j), (1--E--fr)>O. Since Y is a cone,

3.2. Counterexample 2 We shall work in the dual pair [lm, I’]. Bewley (1972) was the first to work with the pair [L”(p),L’(p)]. Let Y be as defined in Counterexample 1 but regarded as a subset of 1”. Claim 1.

Y has an empty (~(l~,I’)-interior.

Proof. Suppose to the contrary that e is an interior point of Y Then there exist a positive integer k, positive numbers E, and f” E I’(cr= 1,. . . , k) such that {~EZ~: If”(y-e)l
cr=l,...,

k)cY

Certainly e, SO. If e, =O, pick JE I” such that j1=&/2M and yi=ei(i#l) where s=minas, and M=max,Ilf”/. Th en for any a, If”@--e)l=(~/2M)(f”,I SE/~<&,. But jl>O and hence 34 Y,. Since f” E I’, there exists n E N such that 1.” If91 < s/3/ellm for all a. Hence, for all a, certainly Ifi1 <.5/3llell~.

M. Ali Khan and N.T. Peck, Interiors

Now let yi=ei(i#n) and y,=-2e,. and hence y $ I: For any 01,

of production

sets

Since yI+yn=el-2e,=-e,>O,

35

y$x

(f”(y--)J=Jf~(-2e,-e,)) 5 Ifj( -2eJl+

Ifj (4

We are done. Y is weak * closed, convex and contains ( -I:)

Claim 2.

Proof:

and Y n ( - Y) = 0.

As easy as the proof of Claim 2 of Counterexample

Claim 3.

J-El,, JJfJJ=l, supxoYf(x)=af
1.

implies fh(),O,...,O).

Proof. Pick an f as in Claim 3. Suppose fr =O. Follow the argument as in Counterexample 1. Suppose Oo. 1

2

1

Since YE Y + ;ly E Y for all ,I 20, f(nj) = n{(j) and hence cr(f, Y) < cc can be contradicted. 0 3.3. Counterexample

3

Let JZ denote the space of signed regular Bore1 measures on [0, l] and %? the space of (bounded) continuous real-valued functions on [0,11. We shall work in the dual pair [.&,$?I. For any f E%? and t E [O, 11, we shall denote the value of f at t by f(t) and use the notation f[p], ,uE.~’ for the canonical pairing. 6,,, will denote the Dirac measure at 1x1. It is worth pointing out that Mas-Cole11 (1975) was the first to work with a commodity space modelled on the space of signed regular Bore1 measures on a compact Hausdorff space and endowed with the weak * topology. We consider the following set: Y={~EA:

f[cl]50

whenever Ilfll=l,

j-20, f(l)=l).

36

M. Ali Khan and N.T Peck, Interiors of production sets

Claim 1.

Y has an empty a(JY,%)-interior.

Proof: Suppose to the contrary that e is a weak * interior point of Y; Then there exist a positive integer k, positive numbers E, and f” E V (a:= 1,. . . , k) such that

~EW={~LEA: j-Q]<&,,

o!=l,...,

k}

implies (p + e) E Y As above, we shall denote min, E, by E and M = max, llf”ll. Certainly M > 0. If M = 0, then f”(t) = 0 for all t E [0, l] and for all a. But in this case (&(,,)E W for all positive integers n. Since (&(,,+e)E I: we obtain e( [0,11) + n 5 0 for all n which is an absurdity. Observe that e([O, 11) < 0. Since e E I: [0,1lde SO. Hence e([O, 11) $0. Suppose e([O, 11) = 0. Let (T= (E/M)~{~). Then i f”[o] ) = I(e/M)f=( 1)l 0, which is a contradiction to the fact that (e+cr)E Y Now pick any t E [0,11, t # 0,l and let j?=l+e-(t)+e-(1), where e+ and e- are respectively the positive and negative parts of e, i.e., e=e+ -e-. Since f” is a continuous function, we can find an open set V” in [0, l] which contains t and is such that f”( V”)cf”(t) +B,,,(O). (B,(O) is the open c-ball around 0.) Let I/= A V”. Certainly I/ is open and contains t. Pick s E K s # t and let p = fi(6,,, - a,,,). Certainly p E J%. Furthermore, for any a,

which shows that p E lY Let T. = B,,,(t) u (I&,,( 1) n [0,11). Certainly there exists a positive integer E such that s# T, for all ngii. Henceforth, n will refer to integers greater than fi. For any A c [0, 11, let A” denote the complement of A in [0, 11. Then T;=(B,,,(t))‘n [0, 1 -(l/n)]. Certainly T; is a closed set disjoint from the closed set {t} u { 11. We can construct the following ‘broken-line’ continuous non-negative function f. such that llf.ll= 1 and f,(l) = 1.

=nx+(l

-n)

(1-(l/n))sxsl,

=nx+(l-nnt)

(t--(l/n))5xs4

= -nx+(l+nt)

tsxs(t+(l/n)).

M. Ali Khan and N.7: Peck, Interiors

of productionsets

31

We can now show that for large enough n the set Y does not contain (~+e). Towards this end, note to begin with, T,=~T,,+,~~~~({t}u{l}) and hence e-(T,)-re-({t} u {l})=e-(t)+e-(1). Now,

=8+ J .L(x)de =l+e-(t)+e-(l)+

zl+

s f,(x)de’r”

j f,(x)der,

j f,(x)de++e-(t)+e-(1)-e-(7”). T”

For large enough n, we obtain f,[p + e] > 0 and hence a contradiction fact that (P+e)E Y 0

to the

Next we have Claim

2.

Y is a a(&,%)-closed,

convex

set such that (-A+)

c Y and

Yn(-Y)=(O). Proof. We only prove the third assertion. Suppose p and ( -_cL) are in I: Then for all f such that Ilfll=l, J-20, f(l)=l, f[p]sO and f[--~150 which implies f[p] = 0. We first claim that ~((1)) =O. Indeed, let fne %?,02 f, g 1, f( 1) = 1, and f, converges pointwise to the characteristic function of {l}. Then for each n, f&] = 0; on passage to the limit, p{ l} = 0. Now let A be any closed subset of [0, l] not containing 1. Using Urysohn’s Lemma [Dugundji (1966, p. 146)], construct a sequence (f,) in %?, Osf. 5 1, f, E 1 on A u {l}, and f, converges pointwise to the characteristic function of A u { 1). Then for each n, f,[p] =O; on passage to the limit, 0 p(A) = 0 (using that p{ l} = 0). This proves that p E 0. Finally we have Claim 3. Proof:

f(l)
fog,

llfll= 1, cr(f, Y) < cc implies f( 1) = 1.

Suppose there exists f~%? such that llfll= 1, a(f, Y)< co and As in Remark 1 in section 2, certainly ~“20. Since llfll= 1, there

M. Ali Khan and N.T Peck, Interiors of production sets

38

exists t in [0, l] such that f(t) = 1. Let p = -&, +&. Certainly ,uE I: But f[p] =(f(t) - f( 1)) E p > 0. Since p E Y implies k,u E Y for all k > 0, f[kp] = k/?. This contradicts the fact that a(f, Y) < co and completes the proof of the claim. Cl 4. Concluding remarks Zame (1987) has presented a theorem on the existence of competitive equilibrium under a condition on each of the production sets that can be seen as an alternative formalization of ‘bounded marginal rates of substitution’. Zame also presents four examples, the first three of which do not satisfy his condition and in which there does not exist any equilibrium. It is of some interest that the production sets in these examples also do not satisfy the condition studied in this paper. We leave it to the interested reader to establish this for himself and limit ourselves to the fourth example of Zame which satisfies his condition and in which an equilibrium does exist. It is of interest that the production set in this example also satisfies our condition and has a non-empty interior by virtue of our Theorem 1. To see this, recall that this example is set in [I’, P] and that T

Y=

y~l’: yT+‘s i

Claim.

1 y(t)yT-’ for all 7; O
I

Let f E I”, f 20, llfll= 1 and ~l(f, Y) < co. Then fi = 1.

Proof: Suppose 0 5 fi < 1. Then there exists i # 1 such that fi >O. Let YE I’, j, = 1 (t s i), yi=O (j > i). Certainly YE Y and f(J) > 0. Since Y is a cone, a(f, Y) < co is contradicted. 0 Remark

2.

By Theorem 1, Y has a non-empty norm interior.

References Aliprantis, C.D., D.G. Brown and 0. Burkinshaw, 1987, Edgeworth equilibria in production economies, Journal of Economic Theory 43, 252-291. Arrow, K.J., 1951, An extension of the basic theorems of classical welfare economics, in: J. Neyman, ed., Proceedings of the second Berkeley symposium on mathematical statistics (University of California Press, Berkeley, CA) 507-532. Bewley, T.F., 1972, Existence of equilibria in economies with infinitely many commodities, Journal of Economic Theory 4, 514-540. Bishop, E. and R.R. Phelps, 1963, The support functionals of a convex set, in: American Mathematical Society, Proceedings of a symposium on pure mathematics, Vol. 7 (American Mathematical Society, Providence, RI) 27-35. Debreu, G., 1951, The coefficient of resource utilization, Econometrica 19, 273-292. Debreu, G., Valuation equilibrium and Pareto optimum, 1954, Proceedings of the National Academy of Sciences of the U.S.A. 40, 588-592.

M. Ali Khan and N.7: Peck, Interiors

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Debreu, G., 1959, Theory of value (Wiley, New York). Dugundji, J., 1966, Topology (Allyn and Bacon Inc., Boston, MA). Khan, M. Ali and R. Vohra, 1988, On approximate decentralization of Pareto optimal allocations in locally convex spaces, Journal of Approximation Theory 52, 149-161. Mas-Colell, A., 1975, A model of equilibrium with differentiated commodities, Journal of Mathematical Economics 2, 263-296. Mas-Colell, A., 1986a, The price equilibrium existence problem in topological vector lattices, Econometrica 54, 1039-1053. Mas-Colell, A., 1986b, Valuation equilibrium and Pareto optimum revisited, in: A. Mas-Cole11 and W. Hildenbrand, eds., Contributions to mathematical economics (North-Holland, New York, NY) 317-331. Phelps, R.R., 1964, Weak * support points of convex sets in E *, Israel Journal of Mathematics 2, 177-182. Richard, S.F., 1988, A new approach to production equilibria in vector lattices, Journal of Mathematical Economics, this issue. Zame, W., 1987, Equilibria in production economies with an infinite dimensional commodity space, Econometrica 55, 1075-l 108.