The fundamental theorems of welfare economics without proper preferences

Journal of Mathematical Economics 17 (1988) 41-54. North-Holland

THE FUNDAMENTAL THEOREMS OF WELFARE ECONOMICS WITHOUT PROPER PREFERENCES* C.D. ALIPRANTIS and 0. BURKINSHAW IUPVI,

Indianapolis, IN 66223, USA

Submitted October 1986, accepted October 1987 We present versions of the fundamental theorems of welfare economics in a very genera1 setting without assuming that the positive cone has a non-empty interior or that preferences are uniformly proper. We establish that an allocation in a production economy is weakly Pareto optimal if and only if it can be approximately price supported in the sense that consumers’ expenditures are approximately minimized and producers’ profits are approximately maximized.

1. Introduction Debreu (1954) was the first to study in a systematic manner the supportability by prices of Pareto optimal allocations. He established that if preferences were strictly convex, then every price supported allocation was a Pareto optimum (the first fundamental theorem of welfare economics). In the converse direction, he proved that if the positive cone of the commodity space has a non-empty interior, then every Pareto optimum can be supported by a price system (the second fundamental theorem of welfare economics). The reader can find detailed accounts of the fundamental theorems of welfare economics in Debreu (1959, ch. 6), Mas-Cole11 (1985, ch. 4) and Nikaido (1968, ch. 5). In economies with infinite dimensional commodity spaces, Mas-Cole11 (1986a, b) considered the problem of supporting weakly Pareto optimal allocations. In order to cope with commodity spaces whose positive cones had empty interiors, he introduced the notion of uniform properness for preferences and production sets. Under the assumption of uniform properness for preferences and production sets, Mas-Cole11 (1986a, b) was able to establish that indeed a weakly Pareto optimal allocation can be supported by a price system. Since non-emptiness of the interior of the positive cone implies uniform properness, Mas-Cole113 results can be considered as generalizations of the corresponding results of Debreu. In the absence of uniform properness, the best one can expect is that a *Research supported in part by NSF grant DMS 83-19594. 03044068/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)

42

CD. Aliprantis and 0. Burkinshaw, Welfare economics and proper preferences

weakly Pareto optimal allocation can be approximately supported by a price. This was first demonstrated in the work of Khan and Vohra (1985). The idea of supporting allocations approximately by prices made its debut in the paper by Aliprantis et al. (1987a). Recently, Khan and Vohra (1987) presented some interesting approximate versions of the second fundamental theorem of welfare economics in locally convex spaces. The purpose of this paper is to establish that in the setting of Riesz spaces the notion of ‘approximate supportability’ characterizes the weakly Pareto optimal allocations. 2. The economic model The basic concept associated with our economic model will be that of a Riesz dual system. Economic models based upon Riesz dual systems were introduced for the first time in Aliprantis and Brown (1983). Subsequently, they were used extensively in Aliprantis et al. (1987a, b). A Riesz dual system is a dual system (E, E’), where E is a Riesz space and E’ is an ideal of its order dual E” that separates the points of E. If the order intervals of E are all weakly compact, then the Riesz dual system (E,E’) is referred to as a symmetric Riesz dual system. Here are some examples of symmetric Riesz dual systems:

(an, 3”)

and

(ca(SL), Cal(Q)).

If the measure p is o-finite, then

are also symmetric Riesz dual systems. The reader will find the basic properties of Riesz dual systems in Aliprantis and Burkinshaw (1978, 1985). The characteristics of our economic model are described as follows. duality is given by (A) The commodity-price duality. The commodity-price a Riesz dual system (E,E’); E is the commodity space and E’ is the price space. If x E E and p E E’, then the evaluation (x, p) will be denoted by p. x, i.e., p.x=(x,p). (I?) Consumers.

There are m consumers indexed by i such that:

(1) Each consumer i has an initial endowment Oi>O and his consumption set Xi is a weakly closed convex subset of E+ with wi~Xi and OeXia

C.D. Aliprantis and 0. Burkinshaw,

Welfare economics

and proper preferences

43

(2) The aggregate endowment of the consumers will be denoted by w, i.e., O=Cy=r Oi* (3) Each consumer i has a convex preference ti on Xi. (4) There is a locally convex-solid topology z on E consistent with (E,E’) such that each preference ti is continuous (i.e., for each i and each x E Xi the sets { y E Xi: y ti x} and {z E Xi: x ti z} are both r-closed). (C) Producers. We assume that there are k production firms indexed by j. The production of each producer j is described by its production possibility set 5, the elements of which are called the production plans for the j producer. For a production plan y = y+ - y- E q the negative part y- of y is interpreted as the input and the positive part y+ as the output. The production sets are assumed to satisfy the following properties: (a) each Yj is a weakly closed convex subset of E containing zero; and (b) for each j we have 3 n E+ = (0). Our economy is now defined as follows. Definition 1.

An economy 8’ is a 3-tuple,

s=((E,E’),

{(Xi,Wi,ti):i=l,...,M},

where agents’ characteristics

{~:j=l,...,k}),

satisfy properties (A), (B) and (C) above.

3. Supporting allocations by prices An allocation is an (m+ k)-tuple (x,, . . . ,x,, yl,. . . , yk) such that XiE Xi for all i, yj~ 5 for all j and

Definition 2.

A non-zero price p E E’ supports an allocation (x1,. . . , x,, y,, . . . , yk)

whenever (a) x ti xi in Xi implies p. x 2 p. xi (cost minimization by consumers); and (b) p. yj 2 p. y holds for all y E 5 (profit maximization by producers). y,J be an allocation. We shall use the letter e to Let CQ,...,X,,Y~,..., designate the total commodity assigned by the allocation, i.e., el=i~lXi=

f

i=l

Oi+

i j=l

yj.

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C.D. Aliprantis and 0. Burkinshaw, Welfare economics and proper preferences

If the prevailing price vector is p, then p.e represents the total wealth of the consumers with respect to the allocation (xi,. . . , x,,y,, . . . , yk). The concept that an allocation is approximately price supported, in the sense that expenditures are approximately minimized and profits are approximately maximized, will now be made precise. As it will become clear, the total wealth should always be positive. This leads us to normalize prices with respect to the total wealth. Definition 3. An allocation (xi,. . . ,x,, y,,. . . , y,) is said to be approximately supported by prices whenever for each E>O and each a E E+ there exists a price p such that:

(1) pee= 1 (the total wealth of the consumers is positive); (2) x ~ixi in Xi implies P’x~~*x~-EE; and (3) p.yj~sup{p.y:yE~andy~a}-sforallj. It is natural to assume that at any given time period the actual production of the economy (due to technological limitations and other factors) is bounded, although the production sets may be unbounded. Therefore, the element a E E+ appearing in Definition 3 can be interpreted as expressing the limitations of the production lkrns at a given time period. It should be noted that if some Xi equals Ef and the corresponding preference ki is also monotone (i.e., x, YE Xi and x 2 y imply x >iy), then any price that satisfies property (2) above must be a positive price. If E is a Banach lattice, then the normalization in (1) above is not the same as the normalization over the norm. The next result clarifies the situation. Lemma 1. Assume that the commodity space E is a Banach lattice (and E’ its norm dual). If an allocation (x,,. . .,x,, y,,. . ., yk) is approximately supported by prices, then for each E > 0 and each a E Ef there exists a price p E E’ such that: (1) llpll= 1 and p*e>O; (2) x kixi in Xi implies p*xzp*xi-s; and (3) p.yj~sup(p.y:y~~ and ySa}-&for allj. Proof: Let (xi ,... ,x~, y,,.. ., y,J be an allocation approximately supported by prices. Fix E> 0 and a E E+. Put 6 = .s//el[, and then select a non-zero price qEE’ such that:

(a) q.e=l; (b) x&xi in Xi implies q.xBq.xi-66; (c) q.yj~SUp{q.y:yE q and ySa)-8

and for all j.

C.D. Alipmntis and 0. Burkinshaw, Welfare economics and proper preferences

Now consider the non-zero price p=q//q]],

45

and note that

Thus, using (b) above, we see that if x ~iXi holds in Xi, then

Similarly, if YE 5, then from (c) we have

and the proof is finished.

0

We are now in the position to show that for finite-dimensional commodity spaces the notions of price supportability and approximate price supportability coincide. Recall that an economy is said to be a free disposal economy whenever - E + E xi”= 1 E;. holds. Theorem

1.

If the commodity space of a free disposal economy is finite-

dimensional and w is strictly positive, then an allocation is price supported if and only

zfit

is approximately price supported.

Proof.

allocation. Assume that Let (x1 ,..., xm,yl ,..., yk) be an yk) is supported by a price q. By free disposal, it is easy to (x l,...,Xm,Ylr..., see that q>O, and hence q.w>O. On the other hand, we have q .yjZO for each j, and so q.e=q.o+

i

q.y,>O.

j=l

Now note that the price p = q/q. e satisfies p. e = 1 and supports the allocation. For the converse assume that (x,,.. .,x,,,,yl,. . .,y,) is approximately supported by prices. Fix a strictly positive vector DEE and let a, = nu. By Lemma 1, for each n there exists a price p. E E’ such that: (1) ((P.((= 1; (2) x ki xi in Xi implies p, . x 2 p.. xi - l/n; and (3) pn’yjzsup{pn.y:y~I;. and ySa,}-l/n for all j.

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C.D. Aliprantis and 0. Burkinshaw, Weljbre economics and proper preferences

Since the closed unit ball of E’ is norm compact, we can assume (by passing to a subsequence) that pn+p holds in E’. Clearly, /p(( = 1, and so p ~0. We claim that p supports (x1,. . . , x,,,, y,, . . . , yk). Indeed, note first that if x ti xi holds in Xi, then pn’ xzp; xi- l/n holds for all n, which implies that p’ x 2~. Xi. On the other hand, if y E q then ys a, holds eventually for all n, and so p,, . yj~p,. y- l/n also holds eventually for all n, from which it follows that p +yj 2 p. y. The proof of the theorem is now complete. 0 When the commodity space is infinite-dimensional, approximate supportability does not imply supportability. The next example [which is inspired by the example in Mas-Cole11 (1986a)] presents an allocation approximately supported by prices that fails to be supported by prices. Example

1. We consider a two-consumer pure exchange economy with respect to the symmetric Riesz dual system (/,,e,). The initial endowments of the consumers are given by

UI=%=

1 1 1 77’T5,...,22n+l’...

(

, >

and so o = w1 +o, is a strictly positive element of L’,. Both consumers will have the same utility function U that was constructed by Mas-Cole11 (1986a, Example 1, p. 1041) as follows. First, for each natural number n, let u,(t) = 29

=---1 2”

1 +t 22”

1

if

tl--22n’

if

r>&.

Now define Mas-Colell’s utility function U:L’: +%? by

W) =

“J%W>

x=(x,,x,,...)Ee:.

Clearly, the utility function U is concave, strictly monotone and norm continuous. The rest of the discussion is devoted to establishing that (w,,02) is a Pareto optimal allocation approximately supported by prices which cannot be supported by any non-zero price. (1) The allocation

(ol, w2) is Pareto

optimal.

Observe that if x=(x1,x2,.

. .)

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C.D. Aliprantis and 0. Burkinshaw, Welfare economics and proper preferences

and Y=(Y~,Y~,...) 05x,+yn52-*”

are two arbitrary for all n, then

positive

sequences

of 1, satisfying

U(n +y) = U(x) + U(y). Now assume that (x1,x2) is an allocation satisfying x,>ro, and n,~,w,. From x1 +x2 =wr +w, and the above observation, we see that U(x,) + U(x,) = U(x, +x,) = U(w, +w,) = U(q) + U(0,). From U(x,)z U(o,) and U(x,)z U(w,), we infer that U(x,)= U(o,) and U(x,) = U(w,). This implies that the allocation (ol, w2) is Pareto optimal. (2) The allocation (oI,02) cannot be supported by prices. Let pee, be a price that supports (01,w2). By the monotonicity of the utility function U, it follows that p 20. Denote by eL the sequence whose kth component is one and every other zero, i.e., ek =(O,. . . , 0, l,O, 0,. . .). Now a direct computation shows that (1 -2-k)o,

+2-2kek >1 ol,

and so by the supportability (1 -2-‘)p-w,

of p we get

+2-2kp’6?,&‘w,.

This implies

for all k. Therefore, p. co1 = 0, and since all components of w1 are positive, we see that p=O. Consequently, (wl,w2) cannot be supported by any non-zero price. (3) The allocation (co,, w2) is approximately supported by prices. Fix some k such that czSk 2-”
q=(2,22 )...) 2k,1,1,1 )... ). The price q satisfies the following properties. (a) For each i we have U(OJ 24. Oi-

Let E>O.

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C.D. Aliprantis and 0. Burkinshaw, Welfare economics and proper preferences

To see this, note that U(OJ=

F $iG

ll=l

(b) Ifx=(xl,xZ,...

n=l

1

&+n=$+l Zn+l=4’@i*

2

) E 8:) then q *x 2 U(x) - s/2 holds.

To see this, fix x=(x1, x2,. . .) EC!:. First, we claim that 2”x, 2 u,(x,)

for

each n.

Indeed, for x,5 l/2’” we have u.(xJ=2”x,. then note that

On the other hand, if x,> 1/22”,

2”x.=x.+(T-l)x.>x.+(2”-l)&=x.+&&=u.(x.). Moreover, an easy argument shows that 1 x, +2” 2 u,(x,)

for

all n.

Now note that the above inequalities yield q.x=

i 2”x,+ II=1

= i n=l

2

x,

n=k+l

2”x,+“=$+l (xn+&J-“~~+,~ n=k+l

= U(r) - ;,

(c) If x kiwi, then q’x~q.wi-E/2. Let x ti Oi, i.e., U(x) 2 U(Oi). Using (b) and then (a), we see that

C.D. Aliprantis and 0. Burkinshaw, Welfnre economics and proper prefmences

49

Finally, to see that (w1,02) is approximately supported by prices, let p = q/q *CD,where w = o1 + w2. Then p. w = 1 and, moreover, if x 2i wi, then by (c) and the fact that 2q. w > 1 we have

Therefore, (q, wJ can be approximately

supported by prices.

It should be noticed that the preceding example can be modified in an obvious manner to incorporate an arbitrary finite number of consumers. 4. The first and second welfare theorems We start allocations.

this section

with the definition

of weakly

Pareto

optimal

Definition 4.

An allocation (xi,. . .,xm,yl,. . .,yJ is said to be weakly Pareto optimal whenever there is no other allocation (fi, . . . , f,, g,, . . . , gk) satisfying L>ixi for all i.

As we have mentioned before, the main objective of this paper is to establish that the notion of approximate supportability characterizes the weakly Pareto optimal allocations. We start by establishing a version of the first fundamental theorem of welfare economics. Theorem 2. If an allocation is approximately supported by prices, then it is weakly Pareto optimal. Proof. Let (x1 ,..., xm,y, ,..., ,y,J be an allocation approximately supported by prices. Assume by way of contradiction that there exists another allocation (fi, . . . , f,,gl,. . . , g,) with fi>ixi for each i. Choose O< 6 < 1 such that 6fi>ixi holds for each i. (Since zero belongs to the convex set Xi for each i and preferences are continuous such a 6 always exists.) Now let E> 0. Put a =c& 1 lgjl and then pick a price p such that:

(a) p.e=l, where e=Cr=i xi; (b) x tixi implies p’x~p.xi-&E; and (c) p.yj~sup{p.y:y~~ and ysaa)-s

for all j.

From afi>ixi and (b), we infer that p - (SA) 2 p eXi-e,

and

SO

(1)

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C.D. Aliprantis and 0. Burkinshaw, W&are economics and proper preferences

Using (1) and taking into account (c), we infer that

+j=li =P’

(

izIf,

)

p,gj--ke-me

-(k+m)e.

Hence, (k+m)cZ(1-6)p*(~YZ1 jJ, and so from (l), it follows that (k+m)& 2 6)( 1 -me)/& i.e., 6(k + m)E2 (1 - 6)( 1 - ms). This implies

(1 -

Sz 1 -(m+6k)e.

Since E>O is arbitrary, the latter inequality shows that 6 2 1, which is impossible. This contradiction shows that (x1,.. .,x,, y,,.. .,yk) is weakly Pareto optimal. 0 To continue our discussion we need a simple result from the theory of Riesz spaces which appeared first as Lemma 4.5 in Aliprantis et al. (1987a). We include it here for completeness. Lemma 2. Assume that (E, E’) is a Riesz dual system, z is a consistent locally convex-solid topology on E, and {xa> and {y,} are two nets satisfying 05x, 5 y, +x for all a and some fixed XE E+. If y, A 0 and [0,x] is weakly compact, then the net {xa} has a weakly convergent subnet.

Proof: Since y, It 0 implies y,’ A 0, replacing {y,} by {y,‘}, we can assume that y, 2 0 holds for all a. By the Riesz Decomposition Property we can write x, = w, + u, with 05 w, g y, and 0 j v,sx for all a. Then w, A 0, and since [0,x] is weakly compact, we see that {Q} has a weakly convergent subnet. Therefore, {x~} has also a weakly convergent subnet. fJ In the sequel the following notation shall put Yy={yE

5: ysa}

(j=l,...,k)

will be employed. If aE E+, then we

C.D. Aliprantis and 0. Burkinshaw, Welfare economics and proper preferences

51

and r.=y”,+...+y;. To establish the converse of Theorem 2 we need a technical result. Lemma 3. Assume that the Riesz dual system for the economy is symmetric and let xicXi (i=l , . . . , m) be consumption bundles. If Fi = {x E Xi: x ti xi}, then for each aE E+ the convex set

F=F,+...+F,-Y" is weakly closed.

Proof: Fix aEE+, and let z be in the weak closure of F. Since F is convex, z also belongs to the r-closure of F. Thus, there exists a net {z,> of F with z, 1* z. Write za= t

xi-

i= 1

i j=

(2)

y’,, 1

where xi E Xi and y’, E 5 satisfy ~6 ki xi and y’, 5 a. Clearly, (y’,)’ s a and so, since [0, a] is weakly compact, every subnet of {(y’,)‘} has a weakly convergent subnet. Now from 05

i

XL+ f

i=l

(yh)-=z,+

j=t

(yJ6)+Sz,+ka,

f j=l

we see that OsxLSz,+ka

and Os(y’,)- sz,+ka hold for all i and j. Thus, by Lemma 2, we can assume (by passing to appropriate subnets) that all nets are weakly convergent. That is, we can assume that xi3vi and v’,r yj hold for all i and j. Since the Fi and Y; are weakly closed sets, we see that VIEFi (i= 1,. . . , m) and YjE Yy (j= 1,. . . , k). Now taking weak limits in (2), we see that $I

z=limz,=lim a

a

Xb-j$l

(

and the proof is finished.

pi

=izl >

0

Vi-

i j=l

yjEF,

52

Wel$are economics and proper preferences

C.D. Aliprantis and 0. Burkinshaw,

Recall that a bundle v E E’ is said to be strongly desirable by a consumer i whenever for all x E Xi and all tl > 0 we have (a) x + au EX~, and (b) x + au >i x. We are now in the position to present the main result of the paper; namely to present an approximation version of the second fundamental theorem of welfare economics. Theorem 3. Assume that the Riesz dual system for the economy is symmetric and that the aggregate endowment is strongly desirable by each consumer. Then every weakly Pareto optimal allocation is approximately supported by prices. Proof Let (x1,. . . ,x,, y,,. . . , yk) be a weakly Pareto optimal allocation and let aE E+ and E> 0 be fixed. We have to show that there exists some price p that satisfies the three properties of Definition 3. To this end, start by letting b =a +c;= 1 lyj( and Fi= {xEX~: x 2 xi} for all i. Next, choose some 0~ t < 1 such that O< 1 -t
is weakly closed. We claim that O$ F. To see this assume by way of contradiction OEF. Then there exist UiEFi (i=l,...,m) and hj~~ (j=l,...,k) ~~=I ui-c&r hi-to=O. This implies Vi+-

l-t m

o

= ~ i=l

Or+

~

that with

hj.

j=l

Since o is strongly desirable by each consumer, we see that ui +( 1 -t)/ for each i, which contradicts the weak Pareto optimality of (x 1,..., x,,y, ,..., y,).HenceO$F. Now by the classical separation theorem [see, for example, Aliprantis and Burkinshaw (1985, Theorem 9.12, p. 136)] there exists a price q E E’ satisfying q.f >Ofor each f EF. From (m)o hi vi &xi

(l-t)e=

f i=l

xi-

i

tyj-twEF,

j=l

we see that q. e>O. Put p=q/q. eE E’ and note that p. f >O holds for all The rest of the proof is devoted to proving that the price p approximately supports the allocation (x,, . . .,x,, y,, . . . , y,). f E F.

(I) p-e = 1. This is immediate from the definition of p.

C.D. Aliprantis and 0. Burkinshaw, Weljhre economics and proper preferences

53

(2) Zf x&q, then p *x 2 p. X,-E holds. To see this, let x E X, satisfy X&X,. Define Zi~Xi by Zi=x if i=r and Zi=xi if i#r. Then X-Xx,+(1-t)e=

and so p.[x-x,+(1-t)e]>O.

F zi-

i=l

f yj-w+(l-t)e j=l

Therefore, p*xBp+x,-E.

(3) Zf YEY~,

then p*yS2p*y-e holds. To see this, let YE Y, satisfy ysu. Define hjE Yj by hj=y if j=s and hj=yj if j#s. Then we have

and so p. [(l - t)e+ t(y,-y)] >O. This implies p.(y,-y) 2 -(l -Q/t > -E, and so p *yS2 p *y-e. The proof of the theorem is now complete. Cl Finally, invoking Theorems 2 and 3, the following combination fundamental theorems of welfare economics holds.

of the two

Theorem 4. Assume that the Riesz dual system for the economy is symmetric and that the aggregate endowment is strongly desirable by each consumer. Then an allocation is weakly Pareto optimal if and only if it is approximately supported by prices.

References Aliprantis, C.D. and D.J. Brown, 1983, Equilibria in markets with a Riesz space of commodities, Journal of Mathematical Economics 11, 189-207. Aliprantis, C.D. and 0. Burkinshaw, 1978, Locally solid Riesz spaces (Academic Press, New York-London). Aliprantis, C.D. and 0. Burkinshaw, 1985, Positive operators (Academic Press, New YorkLondon). Aliprantis, C.D., D.J. Brown and 0. Burkinshaw, 1987a, Edgeworth equilibria, Econometrica 55, 1109-1137. Aliprantis, C.D., D.J. Brown and 0. Burkinshaw, 1987b, Edgeworth equilibria in production economies, Journal of Economic Theory 43, 252-291.

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C.D. Aliprantis and 0. Burkinshaw, Welfare economics and proper preferences

Debreu, Cl., 1954, Valuation equilibrium and Pareto optimum, Proceedings of the National Academy of Sciences of the U.S.A. 40, 588-592. Debreu, G. 1959, Theory of value (Yale University Press, New Haven, CT). Khan, M.A. and R. Vohra, 1985, Approximate equilibrium theory in economies with infinitely many commodities, Working paper no. 85-29 (Brown University, Providence, RI). Khan, M.A. and R. Vohra, 1987, On approximate decentralization of Pareto optimal allocations in locally convex spaces, Journal of Approximation Theory, forthcoming. Mas-Colell, A., 1985, The theory of general economic equilibrium: A differentiable approach (Cambridge University Press, New York-London). Mas-Colell, A., 1986a, The price equilibrium problem in topological vector lattices, Econometrica 54, 1039-1053. Mas-Colell, A., 1986b, Valuation equilibrium and Pareto optimum revisited in: W. Hildenbrand and A. Mas-Colell, eds., Contributions to mathematical economics (North-Holland, New York), 317-331. Nikaido, H., 1968, Convex structures and economic theory (Academic Press, New YorkLondon).