Nonconvexity and Pareto optimality in large markets

We formulate a "limit economy" as a Non-standard Exchange Economy and prove some results on the pricing out of approximate optimal allocations of this economy.. We can then use these res[r]

Economics Department of the University of Pennsylvania Institute of Social and Economic Research Osaka University Nonconvexity and Pareto Optimality in Large Markets Author(s): M Ali Khan and S Rashid Source: International Economic Review, Vol 16, No (Feb., 1975), pp 222-245 Published by: Blackwell Publishing for the Economics Department of the University of Pennsylvania and Institute of Social and Economic Research Osaka University Stable URL: http://www.jstor.org/stable/2525895 Accessed: 28/03/2011 21:16 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive We use information technology and tools to increase productivity and facilitate new forms of scholarship For more information about JSTOR, please contact support@jstor.org Blackwell Publishing, Economics Department of the University of Pennsylvania, Institute of Social and Economic Research Osaka University are collaborating with JSTOR to digitize, preserve and extend access to International Economic Review http://www.jstor.org INTERNATIONAL ECONOMIC REVIEW Vol 16, No 1, February, 1975 NONCONVEXITY AND PARETO OPTIMALITY IN LARGE MARKETS* BY M ALI KHAN AND S RASHID1 INTRODUCTION IT IS WELL KNOWN THAT if tastes and technology are convex and, in Arrow's terms, markets are universal, then any Pareto-efficient allocation can be achieved as a comiipetitiveequilibrium by a suitable reallocation of initial resources, see ([1, 2, 8]) It is also known that this proposition remains approximately true if convexity is replaced by the "weakened convexity assumption that there are no indivisibilities large relative to the economy." Thus in [11], Farrell writes, "We have broadened the basis of the theory in that it is now valid if either the convexity assumption or the multiplicity assumption holds." Unfortunately, unlike the question of the existence of approximate equilibria in large, nonconvex markets, the proposition of approximate allocative efficiency has not been rigorously investigated The only exception seems to be the work of Hildenbrand [13], who answered the question in the context of economies with an atomless measure space of agents However, as he himself was to write later on, "atomless economies have not been accepted as a meaningful economic concept by some economists Our interest in these ideal economies is proportional to how much new information can be derived for large but finite economies." This is the task we set ourselves and our paper can be seen as a continuation of an investigation originally begun by Hildenbrand Alternatively, it can also be viewed as an asymptotic interpretation of his work In the next section we present the model and results for large exchange economies Section is devoted to the proofs Sections and show how the results can be extended to economies with production and indivisibilities LARGE BUT FINITE EXCHANGE ECONOMIES A finite exchange economy, Em, consists of a set of m traders, {Tm}, where m is a finite natural number, whose initial endowments and preferences are restricted to the commodity space QD,1 being a finite natural number of commodities We will denote the endowment of the t-th trader by I(t) and his preference Our objective is to formalize some approximate Pareto-Optimal relation by >t notions for E,s * Manuscript received November 29, 1973; revised February 15, 1974 A preliminary version of this paper was presented at the New York Meeting of the Econometric Society held in December, 1973 We would like to acknowledge the very helpful comments of Don Brown, Carl Christ and Cindy Lewis who was also the discussant of this paper in New York We would also like to thank an anonymous referee for his suggestions Our final thanks to Ilma Rosskopf for her painstaking typing Errors are solely ours 222 NONCONVEXITY AND PARETO OPTIMALITY 223 Under the classical definition, an allocation X is said to be Pareto-Optimal if there does not exist any other allocation Y which makes at least one trader better off without making any trader worse off Thus our search for approximate notions has to revolve around a formalization of one or all of the following factors: (a) an allocation (b) a trader being made better off (c) a measure of the set of traders, B, being made better off (d) a measure of the set of traders, C, being made worse off We can easily see that under the classical definition the above are made precise respectively as (a) Et CTmi X( t) - E t ETillI(t), (b) Y(t) >-t X(t), (c) IB 1, where IB I denotes the number of traders in B, (d) IC| = O Under Hildenbrand's formalization for economies with an atomless measure space of agents we have (a) jAX(t)dl t I(t)dt, (b) Y(t) >-t X(t), (c) p(B) > 0, (d) p(C) _ O where (A, p) is the atomless measure space, X and I are ,-integrable functions We now make precise the concepts we shall be using :2 D X is an s-allocation if it is a function from T,,, to Ql such that , t C Till X(t) < , t C Tm I(t) + ITn,lse D.2 An allocation X is an s-allocation with = D.3 An allocation X is said to be Pareto Optimal for En, if there does not exist any allocation Y such that for all t in Tm, Y(t) >-t X(t) with strict preference for at least one t D.4 An allocation X is said to be s-near optimal for E,n if there does not exist any allocation Y such that for all t in T, S(Y(t), s) >-t X(t) and X(t) > 8e if there does not exist any D.5 An allocation Xis said to be ss-optimal for Emn s-allocation Y such that for all t in T, Y(t) >-t X(t), Y(t) - X(t) > se D.6 If in D.5, Y is required to be an allocation, we shall call Xa s-optimal allocation for Em In what follows, for all x, y C Q1, x > y, x > y and x > y will have their usual meani-ing Except where explicitly stated, we will always take e to be an arbitrary small positive number e will denote a vector all of whose components are unity S(x, e) will denote an open ball with radius e and center x Finally, N and R will denote the set of positive integers and positive numbers respectively 224 M ALI KHAN AND S RASHID Note that in all of the above three formalizations of approximately optimal allocations we have modified factors (b) and (c) In D.5 we have also modified (a) It should also be noted that for any > 0, Pareto Optimal allocations are contained in s-near-optimal and s-optimal allocations Thus any results on the "pricing out" of such allocations also apply to Pareto Optimal allocations Another formalization of an approximately optimal allocation is given by D.7 An allocation Xis said to be sC-optimal for Em if there does not exist an allocation Y such that for any K c Tm, IKI < , Y(t) > X(t) ITMI t for all t in Tm - K with strict preference for at least one t in Tm - K In this definition we are modifying factor (d) This notion comes closest to that used by Hildenbrand [13] Whereas he neglected sets of measure zero, we are neglecting sets of traders of "size" less than or equal to S At first sight this definition is appealing because the set of 8C-optimal allocations is contained in the 8C-core of E,t, the latter being the set of allocations that cannot be blocked by allocations of "size" greater than or equal to (1 - s) We can thus apply the results of [21] on the "pricing out" of the 8C-core to the "pricing out" of the 8C-optimal allocations Unfortunately, unless we fix the set K, the SCoptimal allocation does not exist.3 However, by fixing K we just reduce our economy from Tmto Tm - K We now define some approximate price equilibria for E,n These are essentially variants of those used in Khan [17] We will denote an efficiency equilibrium by e equilibrium.4 D.8 (p, X) is an s-e equilibriumif p C RI, p > 0, such that IVP(X)I < ITmI where VP(X) = {t E TmI(gy E Q')(py < pX(t)-8) A (Y >- X(t))} Note that an efficiency equilibrium as used by Arrow [1], Debreu [8], and subsequently others is an s-e equilibrium with = and pO0 For the subsequent definitions, assume p E R' D.9 (p, X) is a weak s-e equilibriumif in D.8, p / and VP(X) is substituted for VP(X) where VP(X) {t E TmI(3yEQ') (py < pX(t) -) (y >- X(t)) A (Y ? X(t) ? } Note that under the above interpretations the work of the Continuum School can also be seen as dealing with approximate notions, where admittedly their approximations are much finer than ours Note that e here has nothing to with the vector e all of whose comoonents are unity NONCONVEXITY AND PARETO OPTIMALITY 225 and M a large positive number.' D.10 (p, X) is an s-ball e equilibriumif p > O such that IWWP(X)I<8 Tm where WP,(X)= {t C TmI(gy C Q') (py < pX(t)) A (Vz C S(y, e), z >- x(t)} D 11 (p, X) is a weak e-ball e equilibrium if in D.9, WVP(X)is substituted for WP,(X) where WP(X) {t C TmI(gYEQ') (py < pX(t)) A (VzsS(Y, 8) A (z > X(t)) A y < X(t) + M)} FIGURE and M a large positive number.' Let us now try and give an interpretation of these definitions A commodity bundle is efficient at a set of prices p if any other commodity bundle preferred to it costs more Thus in Figure 1, X(t) is efficient We will say that a commodity bundle is 6-efficientif all preferred commodity bundles cost more or not cost "very much" less than it Thus in Figure 1, Xe(t) is e-efficient If preferred commodity bundles are restricted to come from some cube and they not cost "very much" less than Xe(t), the latter is said to be weak e efficient In Figure 2, Xe(t) is weak 6-efficient and the cube is of M units Under D.8, X(t) is an efficient commodity bundle for all traders except for a deviant set, the size of which is smaller than D.9 uses weak 6-efficiency rather than s-efficiency D.10 and D.11 are analogous to D.8 and D.9 with the difference Strictly speaking these notions should be called weak sM - e equilibrium and weak s-ball M e equilibrium Fear of increasingirritationon the part of the readerpreventsus from doing so Of course, if X(t) is strictly positive nothing is gained by introducingM 226 M ALI KHAN AND S RASHID M M FIGURE being that we consider commodity bundles that are preferred to X(t) in a stronger sense as formalized by an s-ball Let W = {Em}m,N be an unbounded sequence of such exchange economies It is unbounded in the sense that for any natural number, however large, there exists an economy with the number of traders in it equal to that number This formalization of a "large" exchange economy dates back to Edgeworth who obtained such a sequence by replication His method was made precise by Debreu-Scarf [9], and has been used both by the Continuum School (see the work of Kannai [16], and Hildenbrand [14], among others), and those working with Nonstandard Limit Economies We will assume that Q always satisfies (A) where (A) (3rER)(VEm3E )[ I E I(t) > re] In addition we shall have to assume for our various results that W satisfies some of the following assumptions: ASSUMPTION 2.1 (Vt E Tm)(VEm E 0) [>t is reflexive] (VE,nE W2)(37z E Tm) [X(z-) is a non-satiation consumption for Z].6 X(z) is a non-satiation consumption for z if OyE Ql such that y >-, X(z) ASSUMPTION 2.2 (Vt E Tm)(VEmE W) [>t reflexiveand monotonic] monotonic if x > y implies x >-t y st is 2.3 The family of all traders' preference relations is equiconASSUMPTION tinuous on Q', i.e., (Vx > O)(Vy > 0)(36 > O)(VnE N)(Vt E Em) [z E S(x + y, 3) =# z >- x] An example of such a family is the set of preference relations defined by a finite family of continuous and strongly monotonic utility functions, ASSUMPTION 2.4 Let NONCONVEXITY AND PARETO OPTIMALITY 227 F(t) = {x E QlIx > X(t)} Then (3c E N)(Vt E Tm)(VEmE W) [R(F(t) < c].' R(S) measures the "degree of non-convexity" of a set S See Khan [18] We can now state our results THEOREM1 If ' satisfies Assumption 2.1, then for all s > there exists m E N such that correspondingto any Pareto Optimal allocation X of Ei (El E EEj > m) there exists p such that (p, X) is a weak s-e equilibrium COROLLARY1 If W satisfies Assumption 2.2 instead of 2.1, then p can be shown to be semi-positive THEOREM2 If 9? satisfies Assumptions 2.2 and 2.3, then for all e > there exists m E N that correspondingto any 1/m-near optimal allocation X of Ej(Ej E ?, IES!> m) there exists p such that (p, X) is a weak s-ball e equilibrium If X is chosen from a uniformly boundedsequence of 1/m-near COROLLARY optimal allocations, then there exists p such that (p, X) is an s-ball e equilibrium THEOREM3 If W satisfies Assumptions 2.2 and 2.4 then for all s > there exists m E N such that correspondingto any (1 /m) (1 /m)-optimal allocation X of Ej(Ej E , IEiI > m) there exists p such that (p, X) is a s-e equilibrium If W satisfies Assumption 2.1, then for all s > there exists COROLLARY m E N such that corresponding to any 1/m-optimal allocation of EI(El E EiI > m) there exists p such that (p, X) is a weak s-e equilibrium We now attempt to motivate these results We know that in general we cannot sustain a Pareto Optimal allocation of a finite non-convex economy as an efficiency equilibrium Theorem says that the Pareto Optimal allocations of such an economy can be sustained as an approximate efficiency equilibrium Further, the larger the economy, the finer the approximation Note that we only assume that the preferences are reflexive and that the Pareto Optimal allocation is not a bliss-point for at least one trader It is difficult to interpret Theorem in terms of price decentralization of a Pareto Optimal allocation The problem has to with the fact that at the prices p and a fictitious allowance to each trader of pX(t) units, each agent may buy a commodity bundle very different from X(t) Indeed, the fact that p can have zero coordinates makes matters worse However, this difficulty is also present for convex markets It is only when the preferences of each trader are strictly convex that the planner can expect each trader to pick the Pareto Optimal commodity bundle when he gives out the prices p Note that we are making the assumptionsof non-satiety and of bounded non-convexities dependent on X(t) In each of the theorems that we use these assumptions,we will take X(t) to be the correspondingoptimal allocation, 228 M ALI KHAN AND S RASHID An alternative interpretation can be given Consider a situation in which every trader holds a Pareto Optimal commodity bundle This outcome may have been achieved by some planning process which is not our concern The question we are interested in is whether there exist some set of prices under which this outcome will be "stable" in the sense that none of the traders will want to get rid of their Pareto Optimal bundles The answer of Theorem is yes, provided there is also a cost to trading This cost may be in the form of a lump-sum tax on trading, a transaction cost or whatever Under this cost no trader, except a small deviant set, will have an incentive to sell his Pareto Optimal commodity bundle X(t) and buy another There is, however, the proviso that the preferred commodity bundles a trader can buy are restricted to be not "unboundedly different" from X(t) In other words we are also giving out output quotas to each trader in addition to the shadow prices and the lump-sum tax However, the smaller the taxes and the larger the output quotas that we give out, the larger the economy should be for the stability property to hold Of course, if the world is one with transaction costs, the planners need only give out output quotas in addition to the shadow prices In summary note that s is parametrizing three factors: the size of the deviants, the output quotas and the lump-sum taxes or alternatively, transaction costs Thus the two principal weaknesses of Theorem are that p is just non-zero and that the set of preferred commodity bundles that is available to a trader is a bounded set By the assumption of monotonicity we can guarantee that p is non-negative However, it can still have some zero coordinates Theorem strengthens p to be strictly positive for a set of allocations larger than the set of Pareto Optimal allocations Further we have to assume that the preferences of the traders are not "too dissimilar." The particular formalization used is rather stringent as the only example of a family of preferences satisfying it is a finite one However, the approximate equilibria is different and does not require any transactions costs for a "stability" interpretation The second difficulty alluded to above persists however If the near optimal allocation is "equitable" in the sense that each trader gets a bounded bundle, then Corrollary shows that we not require the imposition of output quotas or alternatively that the set of preferred commodity bundles be bounded Theorem is another attempt to remedy this boundedness difficulty but without recourse to Assumption 2.3 Instead we assume that a measure of the "degree of non-convexity" of the preferences of each trader are bounded However, we now have to work with a notion of approximate optimality based on s-allocations, i.e., an allocation which sums to more than the total endowment Corollary tries to get rid of this feature and is a step backwards in that it tries to see how far we can get towards pricing-out this modified notion assuming only Assumption 2.1 It should be noted that Theorem is a straightforward deduction from Corollary Before getting on to the proofs it is well to consider whether these Theorems NONCONVEXITY AND PARETO OPTIMALITY ?29 could be directly derived from results on the "pricing-out" of approximate core allocations (See [17].) The argument is as follows.7 Suppose we have an allocation X which is Pareto Optimiialwith respect to an initial endowmnentL As in the definition of Pareto Optimnalityonly the aggregate endowment, ZtETm 1(t), is relevant, we can treat X as an initial endowment Then not only is X a Pareto Optimal allocation with respect to itself but it is also in the core with respect to itself Thus the results on the "pricing out" of core allocations apply imiimediately to Pareto Optimal allocations The only problem with this approach is that in this paper we make no assumptions on the boundedness of initial endowments Thus we not rule out "large" traders Indeed this is one of the principal characteristics of a Pareto Optimal allocation For the difficulties in the "pricing out" of core allocations in the presence of "large" traders (See Khan [19].) These remarks of course not apply to Corollary where we assume that X is chosen from a uniformly bounded sequence When we come to consider economies with production, this approach has added difficulties It is worth pointing out that in the context of economies with a continuum of agents a similar approach was suggested by Arrow to Hildenbrand (See Theorem in Hildenbrand [15].) In conclusion it is worth pointing out the nature of the boundedness difficulty to those of our readers who not want to go through the proofs The essential idea behind the proofs is to consider the set of commodity bundles preferred by each trader to his optimal allocation, optimal being interpreted in the various senses Each of these sets is not convex but their average suIm is convex provided (a) the sets are bounded, or (b) the sets have bounded non-convexities (a) gives us Theorem and (b) gives us Theoremn3 Thus technically both Theorems and are bounding the non-convexities However their economic interpretations are of course very different In Theorem this is done as a mnatter of policy and in Theorem it is part of the data of the problem PROOFS The methodology of this section is due to the seminal work of BrownRobinson [6] We formulate a "limit economy" as a Non-standard Exchange Economy and prove some results on the pricing out of approximate optimal allocations of this economy We can then use these results to prove the theorems of the previous section in a straightforward manner It is worth emphasizing that the nonstalndardeconomy plays the same role for us as the economy with an atomless measure space of agents plays in the work of Kannai [16] and Hildenbrand [14] We shall be working in *RI, the nonstandard extension of R', where denotes a standard number of commodities The non-negative orthant of *RI will be denoted by *Ql For x and y in *RI, x > y, x > y and x > y will have their usual meaning Further, we shall take x - y to mean x differs by an infinitesi7 We are indebtedto discussionswith Don Brown as regardsthis paragraph 230 M ALI KHAN AND S RASHID mal from y in all coordinates; x > y to mean x is non-infinitesimally greater than y in at least one coordinate and not less in any other; x t y to mean x is non-infinitesimally greater than y in all coordinates; x ? y to mean x - y or x >y Let *N be the non-standard extension of N, the set of positive integers, and T C *N be an internal set where ITI, the number of elements in T, is co, some infinite natural number That is, T = (1, 2, * * *, w) and ws*N - N T is to be subsequently interpreted as the set of traders in the economy We now formally define the limit economy A Non-standard Exchange Economy, F,, consists of a pair of functions I and P, satisfying A(i) to A(iii), where I:T -+*Ql and P: T _7(*Ql X *QI) and 7(l*QI) denotes the power set of *QI We shall denote the functions I and P respectively as (I(t))1t= and ( =)2=i where I(t) is to be interpreted as the initial endowment of the t-th trader and st as his preference relation over *QI A(i) to A(iii) are as follows: A( i ) the function indexing the initial endowments, I(t), is internal A(ii) (1/@) EteT I(t) - A(iii) The relation Q = {(t, >)It t T, >-E*QI t X *QI} is internal For the various results we shall have to assume that F,, satisfies some of the following assumptions: ASSUMPTION 3.1 >Zt is reflexivefor all t in T and there exists z in T such that X(r) is a non-satiation consumption for r ASSUMPTION 3.2 For all t in T, ?-t is reflexive and monotonic, i.e., x > y implies x >t Y ASSUMPTION 3.3 (t C T)(Vx, yC *QI; x, y finite) [x > O, co - x + yH ASSUMPTION 3.4 o t y] Let F(t) = {x C *Ql/x >- X(t)} t Then (gc C N)(Vt C T) [R(F(t)) < c] We will need to apply some of the concepts of Section to F<, These are essentially notions transferred from RI to *Rl and we will define them as before except for the understanding that they refer only to internal entities, e.g., an 231 NONCONVEXITY AND PARETO OPTIMALITY allocation Y is an internal function from T into *Ql such that E, tET Y(t) < E, I(t) tET However, we shall need the following additional concepts D A ,-allocation, Y, is an internal function from T into *Ql such that - I E Y(t) E (t) (D tET (O tET D.2 An allocation X is said to be ,-optimal if there does not exist any allocation Y such that for all t in T, ,(Y(t)) >-t X(t) Note that fJ(Y(t)) = {x(t) C *QlIx(t) Y(t)} D.3 An allocation X is said to be ppa-optimalif there does not exist any L-allocation Y such that for all t in T, Y(t) >-t X(t), Y(t) f) X(t) If in D.3 above, Y is required to be an allocation, we shall call X a weak ,e,allocation In what follows we shall speak of a property being true for almost all t in T By this we mean that the property is true for an internal set of traders K c T such that IKI/w - We can now state our results THEOREMla If F,,, satisfies Assumption 3.1, then correspondingto any Pareto optimal allocation X of F, there exists a standard vector p * such that for almost all t in T, px(t) ? pX(t) Vx(t) >- X(t), x(t) finite COROLLARY la If F,, satisfies Assumption 3.2 instead of 3.1, then p X THEOREM2a If F satisfies Assumption 3.3, then corresponding to any ,optimal allocation X of F, there exists p t 0, such that for almost all t in T, (a) px(t) X pX(t) for all finite x(t) such that ,(x(t) >-, X(t) 2a IJ COROLLARY (3M E N) (Vt C T) [X(t) < M] , then (a) can be strengthenedto px(t) >; pX(t) for all x(t) such that ,5(x(t) >?tX(t) 2a.2 If F satisfies Assumptions 3.2 and 3.4 instead of 3.3, then COROLLARY correspondingto any fe-optimal allocation X of F, there exists p > such that px(t) ? pX(t) Vfe(x(t)) THEOREM3a If >t X(t) , satisfies Assumptions 3.2 and 3.4, then correspondingto any fif-optimal allocation X of F, there exists p > such that for almost all t in T, 232 M ALI KHAN AND S RASHID px(t) X pX(t) for all x(t) >- X(t), x(t) e)- X(t) COROLLARY 3a.1 If ?,, satisfies Assumption 3.2, then corresponding to any weak ,Le4-optimalallocation X of F, there exists p > such that for almost all t in T, px(t) ? pX(t) for all finite x(t) > X(t), x(t) X(t) la Let X be a Pareto Optimal allocation Define the PROOF OF THEOREM following sets (Vt C T, t # T) {x(t) C *Ql/x(t) >? X(t)} -{X(t)} F(t) F(r) = {x(T) C *Ql/x(TQ)>- X()} -{X(T)} t B(t) = {x(t) C *QlIx(t) finite} Bn(t) = {x(t) C *QlIx(t) < X(t) + ne}, some n in N E (F(t) nB(t)) tET C _ n- (o E CtET (F(t) nfBn(t)) E tET G(t) nG(t) GCD tET Under Assumption 3.1, ?n is nonemnptyby Lemma of the Appendix Since f, C ? (Vn C N), C is nonempty By Lemma of the Appendix, is S-convex We can now show that M S-Int (-?) Suppose not, i.e., there exists h: T - *Ql and a > such that h(t) C F(t) n B(t) and - (V E tET h(t) = (-)e Let Y(t) == h(t) + X(t) Then Y(t) ? X(t) for all t in T with strict preference for r and E 0) tET Y(t)= 1) X(t) - ae <K- tET Thus X is not Pareto Optimal, a contradiction E (O tET I(t) AND PARETO OPTIMALITY NONCONVEXITY 233 Thus by the S-Separation Lemma, (see Appendix), there exists a standard Thus vector p + such that px > Vx E px > O (Vx C Y1) (Vn C N)C Then for any fixed n, application of Lemma of the Appendix gives Us Z InfpG,P(t) E > Inf px x C~ '1 (CO tECT Let {t C TIInfpG, (t) Sil < } for m C N Certainly Sm is an internal set We can also assert that (IS,nI/O) If not, then < E InfpQ(t) _ co tET ISmI + I E InfpGn(t) (w t ET-S co m Since for all t C T, InfpGn(t) < 0, we get O, E InfpGn(t) co tcT a contradiction Now consider the mapping f:N l(T) where f(m) stil This mapping need not be internal but since we are working in a compresuch that hensive enlargement, there exists an internal mapping g: *N f(T) f(m) = g(m) for all m in N Consider the internal set {m C*N + Ig(m) !,:g(m ) I Either it is empty or has a first element, say (v + 1) Since Sm cS,Qm+l for all N and thus g(m) g(m + 1) for all m < v (If the set is mC N, v C*N empty, the above assertion is trivially true.) Now consider the internal sequence It satisfies the hypothesis of Robinson's Theorem, (see Appen{Ig(m) IJ/)}enE*N dix), and thus there exists p C *N - N such that Ig(n)I/wo for all n < p Let r = Min (v, p) Then U g(m)Cg(O) USm mEN mnEN where g(() is internal and Ig(() |/lo Thus Inf pGn(t) for all t in T -Vn where Vn( = g(()), is an internal set such that (IVIfO) Now consider the mapping h:N j9(T) such that h(n) = T - Vn Note that for all n C N, T - Vn is internal, IT-VnI c) and 234 M ALI KHAN AND S RASHID) Even though the mapping h need not be internal, as before there exists an internal mapping k: *N > 9(T) such that k(n) = h(n) (Vn C N) Again as before we can show that there exists a E *N k(n) k(n - N such that 1) - for all n < a Now consider the internal sequence Ik(n)I - Ct) } nE*N As before we can assert that there exists i C *N N such that - Ik(n)II c) for all n < Let = Min (a, i) Then n nEN n T-Vvn= niEN k(n) D k(T,) Since T-Vn = {t C T1 InfpGn(t) }0 Thus 0O Vt C k(T,) InfpG(t) where k(;,) is an internal set such that Ik(r)1/1c- Thus for almost t in T, px(t) ;> pX(t), Vx(t) >- X(t), X(t) finite Q.E.D PROOF OF COROLLARY la The only place we make use of nonsatiety is to guarantee that F(r) / Under Assumption 3.2, this follows directly, and thus we can appeal to Theorem la We can now show that p > Suppose not, i.e., say Pi < Pick t C T such that px(t) ? pX(t) Vx(t) >- X(t) , x(t) finite t By monotonicity X(t) + (M, 3,, * d X(t) where MG N and a > Thus piM+ a > pi ?0 i=2 By choosing M large enough and a small enough, we can get a contradiction Q.E.D NONCONVEXITY AND PARETO OPTIMALITY 235 Remark Note that Assumption 3.4 does not allow us to strengthen the conclusion of Theorem la to px(t) ;> pX(t) vx(t) X(t), for almost all t in T The reason for this lies in the fact that we would have to separate (1/(w) Et>T I(t) from (1/() EtcT F(t) The latter however is not near-standard and thus the S-Separation Lemma cannot be applied To apply the Q-Separation Lemma, we have to show that ? w E I(t)6Q"con tET (I @ E F(t)) tCT leads to a contradiction We cannot show this since we are working with allocations rather than with jp-allocations as in the proof of Theorem 3a below Note that in the proof of Corollary 2a, we can convert the ,u-allocation h to an allocation This is again impossible here PROOFOFTI-IEOREM 2a Let X be a je-optimal allocation such that for all t in T, X(t) se Definite following sets G(t) = {x(t) C *Qlj,p(x(t)) >- X(t), x(t) finite} Gn(t) = {x(t) C *QlIS(x(t) {X(t)} (Vt C T) - ) > X(t), x(t) < X(t) + ne} - {X(t)} (VtC T) Let Y and ,2n be as in the proof of Theorem la Under Assumption 3.3, 2n is nonempty Thus, as before, ? is nonempty, S-convex and X S-Int (Y) Just by reworking the argument of the proof of Theorem la, we can show that there exists a standard vector p # such that, for almost all t in T, px(t) > 0, Vx(t) C G,(t), C *N - N Then, as in the proof of Corollary la, we can show that p > However, even this can be strengthened to p > Suppose not, i.e., for some coordinate, say the first, Pi = Since p > 0, there exists at least one coordinate, say the second, such that P2 > By Assumption 3.3, for all t in T, ,u(Xt) + 0* (1, O, * )) >- X(t) Thus, by Lemma of the Appendix, for all t in T, there exists c, such that O, ,u(X(t) + (1, -8 at 0* 0* O))>- X(t) Thus for some t, pX(t) + PI - 6tP2 ? pX(t), M ALI KHAN AND S RASHID 236 a contradiction Since G(t) c Gjt), we have so far shown that there exists p > such that; for almost all t in T, px(t) > pX(t) Vx(t) C (G(t) + X(t)) Suppose that for some y(t) E G(t) + X(t), py(t) pX(t) Usinig the argument above, we can show that there exists (y(t) - de) C G(t) d > such that + X(t) Thus py(t) - pYe pX(t) which implies - pde > 0, a contradiction Thus, for almost all t in T, px(t) ? pX(t) for all finite x(t) such that Q.E.D p(x(t) >- X(t)) PROOF OF COROLLARY 2a.1 If for all t in T, X(t) < M for some M C N, then the set {x(t) C *Ql Ipx(t) ? pX(t)} is also standardly bounded because of the strict positivity of p Thus, for almost all t in T, px(t) > pX(t) for all x(t) such that p(x(t) >-t X(t) Suppose not, i.e., py(t) < pX(t) for some y(t) such that pce(y(t))>-, X(t) Then y(t) is not finite because of Theorem 2a, a contradiction PROOFOF THEOREM3a Let X be a c1-optimal allocation Define for all n C N, (1/n) ?c, Fn(t) = x(t) C *QlIx(t) >- X(t), x(t) -X(t) Jn e (Vt C T) =-EFn(t) Ct)tCT Under Assumption 3.2, ,n is nonempty by Lemma of the Appendix Thus Q-Con (Y72) is an internal, nonempty, Q-convex set We can assert that E I(t) a Q-Con((2')X COtET Suppose not, i.e., there exists h: T *Ql such that for all t in T, h(t) CF,(t) and by Lemmas and of the Appendix, 1- C tET h(t) - E It) CO tET Thus X is not a ,-optimal allocation, a contradiction Thus by the Q-SeparationLemma, (see Appendix), there exists p,, C *RI, p,, # such that 237 AND PARETO OPTIMALITY NONCONVEXITY plx >? E Pnl(t) > Ct) tET I p,1X(t) (VX C E tl) - co tET Define =C RI,p : An 1O | pX> Co.tE T pX(t) (VxcG?n)} By defining Y2 for all n in *N, Y is a mapping from *N to the power set of *Rl Certainly Y,, is internal for all n in *N Consider the set #- = 01 {nC*NJin This is an internal set of integers If _rKlis nonempty, it has a first element, say v + Certainly J C *N - N If not we have a contradiction Thus #D# Note that if X_f' is empty, , * is trivially true Thus there exists p, C *RI, pD, / such that PDX > - E, Ct tET > ? (vx P(X(t)) D C Thus () E px(x(t) - tET X(t)) > (vx(t) C F,(t), Vt C T) But by Lemma of the Appendix 0O Inf [PDX Xe Y.)D E P X(t)] E- Infimum pj(x(t) - Infilurn-i px(x(t) - Ct) t CT x (t) CFD)(t) COt GT X(t)) But X(t)) < x(t) EFv(t) ) for all t in T We can now follow the argument of the proof of Theerem la to show that for almost all t in T pvx(t) ? pDX(t) vx(t) >- X(t), x(t) t X(t) 2J X Thus P"X(t) >P"X(t) vx(t) >- t X(t) x(t) X(t)- As in the proof of Corollary la, we can show that pD > Note that now p, Q.E.D is not a standard vector but we can choose ME *N, PROOF OF COROLLARY3a The argument is the same as in the proof of Theorem la, Just redefine the correspondence F as 238 M ALI KHAN AND S RASHID F(t) = {x(t) C *QlI x(t) >- X(t), x(t) -* X(t)} t for all t in T with the other sets unchanged 2a.2 Redefine the correspondence Fn as PROOFOF COROLLARY >- X(t)} (Vt C T) Fn(t) = {x(t) C *QI S(x(t), I) is internal, nonempty and Q-con- As in the proof of Theorem 3a, Q-con (n) vex We can also show that Z Co)teT I(t) a Q-con (5?n)a Only the following additional step is needed Let [ tET k < Eh(t)]- EZI(t) Ct)ET where k is some infinitesimal vector Define g(t) = h(t) - k(vt C T) Then (Vt C T)p(g(t)) >- X(t) and we obtain a contradiction Thus there exists pp > such that for almost all t in T, PVx(t) _> PDX(t) VP(X(t)) >- X(t) Q.E.D Note that we cannot strengthen p to be strictly positive We can now furnish the proofs of the Theorems of Section The methodology behind these proofs is due to Brown-Robinson [6, 7] Informally the idea is as follows: suppose a particular theorem is false implying its negation is true Then the non-standard extension of this negation is true which contradicts a corresponding theorem for the non-standard limit economy We illustrate the method by proving Theorem For a further illustration see [7] We shall need the following Lemma Lemma (Vm C *N - N) [1/m-near optimal ( t,,)c p-optimal ( f,)] where 1/m-near optimal (f,,t) denotes the set of 1/m-near optimal allocations of WX,,etc (a) Then there exists an allocation Y such X(t) Then certainly for all t in T, PROOF Suppose Xc p-optimal that for all t in T, p( Y(t)) >-t NONCONVEXITY AND PARETO OPTIMALITY 239 S( Y(t), 1/m) >- X(t) Thus XXj 1//m-optimal (a) Q.E.D PROOFOF THEOREM2 Suppose the theorem is false Then there exists a > such that 1Ei1> m)(3XE 1/m-near optimal (Ei)) (im E N)(3Ei E W, {VpE S21 p =,J0, I Wa(X)[> d Hence by transfer the following sentence is a true statement about * standard extension of ?: for some positive real number 6, (Ym E *N)(3Ej E * ', the non- lEi > m)(3XE 1//m-nearoptimal (Ei)) I I* Wa(X) IT3I > d E *S21p * o, {Vp {YPC*Ql,PVZO,~~~~ Pick m E *N - N in the above statement From the above lemma, 1/m-near optimal allocations of any non-standard exchange economy is also a p-optimal allocation of that economy Now if ? satisfies Assumption 2.3 of Section 2, all the non-standard economies in *0? satisfy Assumption 3.3 of this section, Q.E.D thereby giving us a contradiction to Theorem 2a The other Theorems of Section can be similarly proved ECONOMIESWITH MANY PRODUCERS In this section we extend the results of Section to economies with "many" consumers and "many" producers We conclude by an attempt at a justification of such an approach We now define a finite economy, Emn, as consisting of m consumers and n producers We shall denote the set of consumers by T, and the set of producers by Tf Each consumer has in addition to his endowment I(t) E RI, a consumption set 2(t) C R' on which is defined his preference relation >-t Each producer has a production set 3/(t) Our sign convention on production and consumption plans is the same as in Debreu [8] We need to redefine some of the concepts of Section D.1 (X, Y) is an allocation if X: T, -> R' and Y: Tf (a) (Vt E T) (X(t) E (t)) (b) (Vt E Tf) ( Y(t) E 3/ (t)) Y(t) + (c) [ Z X(t)]?< [ tETc teTr -> RI such that ZI(t)] teTc D.2 (X, Y) is an e-allocation if it is an allocation with (c) of D changed to 240 M ALI KHAN AND S RASHID | D.3 {Z Y(t)-l Z|TX(t) ITCItETc Z I(t)} + ce I~TcItETf tETc An allocation (X, Y) is said to be Pareto Optimal if there does not exist any other allocation (X', Y') such that for all t in Tc, X'(t) >-t X(t) with strict preference for at least one t The corresponding approximate be similarly can notions optimality defined for Esnns D.4 (p, X, Y) is an e-e equilibriumif p E RI, p I v'(X))I ITC I where IcVP(X) I is obtained fVP(Y) = We can similarly of producers Now <e such that IfV (Y)I and ITCI by substituting < for Ql in VP(X) 29(t) of D.8 and (t)) (py ? p Y(t) + e)} {t E Tf (3 y E modify let # the other price equilibria for the presence to allow = {EmnmE}fmEN nEN in be an unbounded sequence of such economies Note that it is unbounded the sense that for any pair of natural numbers (i, j) however large, there exists an economy with the number of consumers in it equal to i and the number of producers equal to j Note that we can also consider sequences X? in which Thus for exthe number of producers depends on the number of consumers ample ,' = {Em,61M}meN We can now modify analogues of Theorems Optimal allocations A and 2.1 to 2.4 of Section Assumptions to We will only sketch the argument to prove for Pareto THEOREM1' If a satisfies Assumption 2.1, then for all s > there exist (m, n) E N X N such that correspondingto any Pareto Optimalallocation (X, Y) of E Eij (EijM1C, ITcI > m, ITfI > n) there exists p such that (p, X, Y) is a weak e-e equilibrium limit economy a non-standard To prove this theorem we have to formulate in the same We can then use results for this limit economy with production We will not we used Theorem la to prove Theorem way as, for example, go into these steps and trust that they are obvious to the reader SKETCHOF PROOFOF THEOREMl'a of the proof of Theorem t= - la The argument is a routine modification Let (X, Y) be a Pareto Optimal Z (F(t) nB(t)) Co t ETc - - E G() t ETf allocation (F(t) nB(t)) Define NONCONVEXITY AND PARETO OPTIMALITY 241 where for all t E Tf, F(t) = The other sets are as defined be(t)-{Y(t)} fore We can again show that C is nonempty, S-convex and X S-Int (2) The other part of the argument of Theorem la is essentially unchanged by which we can show that for almost all t in (TCU Tf), InfpG(t) Thus for almost all t in Tc, px(t) > pX(t), vx(t) >-t X(t), x(t) finite, and for almost all t in Tf Q.E.D py(t) pY(t), Vy(t) E ?(t), y(t) finite Note that the heart of the argument is the fact that the average production set of a large number of productinn sets is convex provided (a) the individual production sets are bounded; (b) the individual production sets have bounded non-convexities Thus we are assuming either that the increasing returns are small in some sense or that all producers are instructed to produce within some upper limits.8 This is the precise interpretation we can give to Bator's remark [4], "To be sure nonconvexities which are insignificant are insignificant Unfortunately, in the real world non-con vexities appear to be anything but insignificant In truth, the central difficulty with markets in a stationary context, is that the world is full of substantial non-convexities which break down multiplicity." Thus, in a different context, Hildenbrand [13], "hesitates to assume that the restriction p/Ap is atomless (pure competition among producers) whichwouldimply by the Theorem of Liapunov that the total production set is convex." Hahn [12], writes " as far as households are concerned I have no great difficulties in accepting that a coin tinuum or a very large number of them is a satisfactory idealization But the same is not true of firms for there are logical difficulties in accounting for the existence of agents called firms unless we allow there to be increasing returns of some sort But when there are increasing returns it may be wrong to think of a very large number of firms." It is our contention that the above are telling arguments but for a descriptive exercise such as a proof of the existence of equilibrium with "umany"firms.9 In this paper we are interested in the sustaining of approximately optimal allocations by competitive behavior, i.e., our concerns are essentially prescriptive or planning oriented In this context it would be a policy objective to prevent the "breaking down of multiplicity." Put differently, given any tolerable degree of approximation, the results tell us the minimum number of producers needed Further, the interpretation of (a) above also makes sense in this context Thus a relaxation of (b) to allow us to consider technologies with a "larger" degree of increasing returns to scale would be worth having For preliminary work on this, see Khan [20] EXTENSION TO INDIVISIBILITIES Recently Emerson [10], has investigated the question of allocative efficiency for economies with indivisible commodities His concern is with finite exchange This is just a transfer to production sets of the remarks at the end of Section Even for this, however, there are precedents See [3 (Chapter 7)] 242 M ALI KHAN AND S RASHID economies and he assumes that the preferences are "discretely convex." We just point out here that the results of Sections and (and indeed their miodifications for economies with production) carry over to the case of indivisible comnmodities Note that the equilibrium notions we work with are rather crude and thus we need no assumptions on the continuity of preferences apart from Assumption 2.3 This however was used only to insure strict positivity of p and a consequent slight strengthening of the equilibrium notion As before the heart of the argument will be Lemmas and of the Appendix which carry over to sets which are totally disconnected The Johns Hopkins University, U.S.A and Yale University, U.S.A APPENDIX Consider a correspondence G: T -> ?A(*Ql) We shall denote its integral, C, by (1/o) EteT G(t) which is defined equal to {xC*Q lx= Eg(t) for allgin 5G} (O tET where _7G is the set of all internal selections from HteT G(t) G is said to be internal if the graph of G, {(t, x) ItE T, x E G(t)} is internal G is said to be internally bounded if there exists an internal function g such that for all t in T and for all x in G(t), lixjl < g(t) and g(t)/wo LEMMA If for all t in T, G(t) is internal and nonempty, a # PROOF.By transfer LEMMA2 If G is internally bounded, (1/o) EtcT G(t) is S-convex PROOF See Khan [18], which reproduces a proof essentially due to Brown Note that a set B in *RI is said to be S-convex if for all x, y E B and all - 2)y B is said to be Qconvex if - is replaced by = in the above definition The Q-convex hull of B, Q-con (B), is the set of all internal star-finite convex combinations of points in B We now consider correspondences which are not internally bounded but whose "degree of non-convexity" is bounded Let the radius of a set S, rad (S), be the radius of the smallest ball containing S, i.e., i E *(0, 1), there exists z E B such that zz x + (1 rad (S) = Inf r(x) xES where r(x) = {rIB(x, r) D S} Let x C Q-con (S) and 243 NONCONVEXITY AND PARETO OPTIMALITY {(x, S) = {A C S, xC Q-con (A)} Then, following Starr, the inner radius of a set S, R(S) is Supremum Inf rad (A) AE / x EQ-con (S) LEMMA3 If G is an internal correspondencesuch that (gc C N) (Vt E T) (R(G(t)) < c) , then (1/(O) EtET G(t) is S-convex PROOF See Khan [18] LEMMA4 If G is an internal correspondencesuch that there exists Mc *N and for all t in T, G(t) Me, G(t) # 0, then Infpy I - E InfpG(t) 0) tET YCt wherep E *RI and p # PROOF See Khan [17] LEMMA Let A be an internal S-convex set; then for any y C Q-con (A), there exists z E A such that z - y PROOF, See Khan [18] Let X(t) > ee for some e k and p(X(t)) with <, a < e such that p(X(t) - de) >-tY LEMMA a >-, y Then there exists PROOF Let Bn - {x *QIjx > y, (Vx E S(Z(t), 2) VZ(t) E S(X(t), I))} If Bn - for some n in N, the lemma is proved Thus suppose that for all n in N, Bn =A0 Bn D Bn+l and Bn is an internal set Since we are working in a comprehensive enlargement, there exists an internal infinite sequence {Xnn}E*N where for all n in N, xn C Bn and F-Lim Xn = X(t) Therefore by a Theorem due to Robinson, (see [22, (110)]), there exists v E *N - N such that (3x c- S(Z(t, ))s (Z(t E- SX(t),9 ))[x > y], a contradiction to the fact that p(x(t)) >-ty Q.E.D Q-SEPARATIONLEMMA Let A be an internal Q-convex set in *RI and y Then there exists p C *RI, p # such that px > py (Vx G A) PROOF By transfer of Theorem on page 384 in Arrow-Hahn [3] X A 244 M ALI KHAN AND S RASHID S-SEPARATION LEMMA If A is a nearstandard, S-convex set in *RI and SInt (A), then there exists a standard vector p # such that for all x C A, px > PROOF.See the modification of the original proof by Brown-Robinson [5], in Khan [17] THEOREM(ROBINSON) Let (Sn) be an internal sequencesuch that Sn is infinitesimal for all finite n Then there exists an infinite natural number v such that sn is infinitesimal for all n < v PROOF See [22, (65)] REFERENCES L1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] ARROW, K J., "An Extension of the Basic Theorems of Classical Welfare Economics, in Jerzy Neyman, ed., Proceedings of the Second Symposium on Mathematical Statistics and Probability (Berkeley: University of California Press, 1951) ,'Political and Economic Evaluation of Social Effects and Externalities," in Michael Intriligator, ed., Frontiers of Quantitative Economics (Amsterdam: North-Holland Publishing Company, 1971) AND F HAHN, General Competitive Analysis (San Francisco: Holden-Day, Inc., 1972) BATOR, F M., "On Convexity, Efficiency and Markets," Journal of Political Economy, LXIX (October, 1961), 430-483 BROWN, D J AND A ROBINSON, "Nonstandard Exchange Economies," to appear in Econometrica , 'A Limit Theorem on the Cores of Large Standard Exchange AND Economies," Proceedings of the National Academy of Science, U.S.A., LXIX, 1258-1260 A Correction in LXIX, 3068 , 'The Cores of Large Standard Exchange Economies," Cowles AND Foundation Discussion Paper No 326, to be revised DEBREU, GERARD, Theory of Value (New York: John Wiley and Sons, 1959) AND H E SCARF, "A Limit Theorem on the Core of an Economy," International Economic Review, IV (September, 1963), 236-246 EMERSON, R D., "Optima and Market Equilibria with Indivisible Commodities," Journal of Economic Theory, V (October, 1972), 177-188 FARRELL, M J., "The Convexity Assumption in the Theory of Competitive Markets," Journal of Political Economy, LXVII (August, 1959), 377-391 HAHN, FRANK H., On the Notion of Equilibrium in Economics, Inaugural Lecture (Cambridge: Cambridge University Press, 1973) HILDENBRAND, W., "Pareto Optimality for a Measure Space of Economic Agents," International Economic Review, X (October, 1969), 363-372 , "On Economies with Many Agents," Journal of Economic Theory, II (June, 1970) 161-188 , "The Core of an Economy with a Measure Space of Economic Agents," Review of Economic Studies, XXXV (October, 1968), 443-452 KANNAI, Y., "Continuity Properties of the Core of a Market," Econometrica, XXXVIII (November, 1970), 791-815 A correction in XL (September, 1972), 955-958 KHAN, M ALI, "Some Equivalence Theorems," to apper in The Review of Economic Studies , "Approximately Convex Average Sums of Unbounded Sets," to appear in Pro ceedings of the American Mathematical Society , "Exchange Economies with Large Traders," presented at the Econometric Society's Meetings at Toronto, December 1972, Also Johns Hopkins Working Paper No, 57 Baltimore, Maryland, NONCONVEXITY AND PARETO OPTIMALITY 245 , "On Sets with Unbounded Non-convexities," mimeographed [20] , "Some Remarks on the Core of a 'Large' Economy," to appear in Econometrica [211 [22] ROBINSON, ABRAHAM, Nonstandard Analysis (Amsterdam: North-Holland Publishing Company, 1965)