Center for Economic Research. No CONSTRAINED SUBOPTIMALITY WHEN PRICES ARE NON-COMPETITIVE

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1 Center for Economic Research No CONSTRAINED SUBOPTIMALITY WHEN PRICES ARE NON-COMPETITIVE By P. Jean-Jacques Herings and Aexander Konovaov November 2 ISSN

2 Constrained Suboptimaity when Prices Are Non-competitive P. Jean-Jacques Herings 1 Aexander Konovaov 2 November 13, 2 1 Department of Economics, Universiteit Maastricht, P.O. Box 616, 62 MD Maastricht, The Netherands. E-mai: P.Herings@agec.unimaas.n 2 Department of Econometrics, Tiburg University, P.O. Box 9153, 5 LE Tiburg, The Netherands. E-mai: aexk@kub.n

3 Abstract The paper addresses the foowing question: how ecient is the market system in aocating resources if trade takes pace at prices that are not necessariy competitive? Even though there are many partia answers to this question, an answer that stands comparison to the rigor by which the rst and second wefare theorems are derived is acking. We rst prove a"fok Theorem" on the generic suboptimaity of equiibria at non-competitive prices. The more interesting probem is whether equiibria are constrained optima, i.e. ecient reative to a aocations that are consistent with prices at which trade takes pace. We give a necessary condition, caed the separating property, for constrained optimaity: each constrained househod shoud be constrained on each constrained market. If the number of commodities is ess than or equa to two, then this necessary condition is aso sucient. In that case equiibria are constrained optima. In a other cases, this necessary condition is typicay not sucient and equiibria are genericay constrained suboptima. Key words: Non-competitive prices, wefare, Pareto improvement. JEL cassication numbers: D45, D51, D61.

4 1 Introduction More than two centuries ago, Adam Smith described how the pursuit of sef-interest can promote the interest of society. Since then, economists have devoted much of their time in providing rigorous foundations to this caim, which nay resuted in the rst and second fundamenta wefare theorems. These theorems are vaid ony in ideaized circumstances, among which the requirement that a trade takes pace at competitive prices, incuding trade in contracts contingent on a imaginabe future events. The case where the assumption of compete nancia markets is reaxed has received much attention in the recent iterature. When markets are incompete, then a competitive equiibrium is typicay suboptima. The appropriate question to ask, however, is not whether competitive equiibria are optima, but whether competitive aocations are optima reative to the restrictions imposed by market incompeteness. When a fuy informed centra panner, who takes into account the impications of market incompeteness, is abe to improve upon a competitive aocation, then competitive equiibria are said to be constrained suboptima. Geanakopos and Poemarchakis (1986) show that competitive equiibria are typicay constrained suboptima, by showing that Pareto improvements can be obtained by making the appropriate redistributions in househods' initia asset portfoios and next restricting a trade in asset markets. More recenty, simiar resuts have been obtained that show the possibiity of generating Pareto improvements by introducing new nancia assets, see Cass and Citanna (1998), or Citanna, Kajii and Vianacci (1998) for a more genera perspective, and the possibiity of generating Pareto improvements by price reguation, see Dreze and Goier (1993) and Herings and Poemarchakis (1999). The assumption that a trade takes pace at competitive prices has aso been reaxed. During the ast quarter of the 2th century, traditiona Warasian theory has accommodated in a genera equiibrium setting the possibiity of suggish price adjustment, short-run price rigidities, and, as a consequence, non-cearing markets. For semina contributions, see Benassy (1975), Dreze (1975) and Younes (1975). Attention has been focused on issues of equiibrium existence, and on expanations why prices and wages may not adjust freey to equate suppy and demand. Instances of the atter are cases with information imperfectness, menu costs, renegotiation costs and so on. Many empirica studies show that quantity constraints, ike invountary unempoyment in the abor market, and infrequent price adjustments, ike nomina wage rigidities, are common in the rea word, see Romer (1996). Other exampes where the anaysis of non-cearing markets is reevant are situations invoving market power, panned economies and markets for agricutura products, see Benassy (1993). More generay, appication of standard toos from pubic choice theory shows that governments have incentives to intervene in the price formation process to gain votes, see Herings (1997) and Tuinstra (1998). 1

5 Most economists share the strong conviction that imperfections in the price formation process, and in particuar trade at non-competitive prices, has strongy negative wefare consequences. Given the strength of this conviction, it is surprising that most of its foundations come from partia settings. No rigorous genera resuts that stand comparison to the rst and second wefare theorems, or the constrained suboptimaity resuts in the case of market incompeteness, are avaiabe. It is therefore that we abe the caim on the detrimenta eects of trade at non-competitive prices as a Fok Theorem. The paper addresses the foowing question: how ecient isthemarket system in aocating resources when prices are not competitive. To get an answer, we anayse the equiibria of the ceanest xed price mode avaiabe, the one of Dreze (1975). In his semina paper, Dreze introduced the concept of quantity rationing in a genera equiibrium mode with price rigidities. In this approach a househod chooses that commodity bunde which is most preferred by it, subject to both the budget constraint and the quantity constraints on net trades. The quantity rationing may aect either suppy or demand of a commodity,but it never aects both simutaneousy to reect the transparency of markets. The rst main resut we show is the Fok Theorem on the genericay suboptima aocation of resources when prices are non-competitive. Inspired by the incompete markets iterature, we continue our investigation by anayzing a concept of constrained optimaity, that takes into account the restrictions imposed by trading at fase prices. Suppose that trade takes pace at prices p and that an aocation is ecient reative to the set of physicay feasibe aocations for which the net trades of a househods have vaue zero at the price vector p: Such an aocation is said to be p-optima. Bohm and Muer (1977) give an exampe of an economy, whose equiibria are not p- optima. Maskin and Tiroe (1984) observe that if a traders have stricty positive weights in a wefare program, then non-competitive p-optima aocations invoving trade in a markets are never vountary, that is impy forced trade, and, therefore, are not equiibria. However, satiation or non-constrained maximization is typica for a xed price mode, see Aumann and Dreze (1986), which impies that p-optima aocations need not be soutions to wefare programs with stricty positive weights. The question rises whether trade at non-competitive prices eads typicay to constrained suboptima aocations. A househod is said to be constrained if it is subject to quantity rationing in at east one market. A market is said to be constrained if at east one househod faces constraints in that market. We give an easy to verify necessary condition for equiibria to be p-optima: each constrained househod shoud be constrained on each constrained market. If the number of commodities is ess than or equa to two, then this necessary condition is aso sucient. In that case equiibria are constrained optima. This case is not entirey without interest as it is the genera equiibrium equivaent of the partia equiibrium textbook picture that 2

6 anayzes the eects of a minimum or a maximum price on a singe good. In cases with more than two goods, this necessary condition is not sucient and generic constrained suboptimaity of equiibria is obtained. The paper has been organized as foows. Section 2 exposes a mode of an exchange economy where trade takes pace at non-competitive prices. Section 3 shows that in such an economy, equiibria are typicay suboptima. Section 4 shows that in the two commodity case constrained optimaity hods. Section 5 derives the necessary condition that a constrained househods be constrained on each constrained market for constrained optimaity to hod. It is shown that this criterion is typicay not met when the number of commodities is greater than or equa to three. Finay, Section 6 concudes. 2 The mode We consider an exchange economy denoted by E =< N L fx i u i w i g i2n >:Here N = f1 ::: Ng is the set of househods, indexed by i, andl = f 1 ::: Lg is the set of commodities, indexed by : Each househod is characterized by a consumption set X i a subset of IR L+1 a utiity function u i dened on X i and a vector of initia endowments w i in X i : Price systems beong to the set P = fp 2 IR L+1 j p =1 pg: Commodity serves as a numeraire commodity. To evauate the wefare consequence of trade at non-competitive prices, we x a price system p in P at which trade is supposed to take pace. In genera, since prices p might be not compatibe with a competitive equiibrium, traders wi face quantity constraints on suppy and demand. The description of the market mechanism is now extended in the sense that the information transmitted by it is no onger ony the price system, but aso the maxima amount a househod is abe to suppy of every commodity, caed the rationing scheme on suppy, and the maxima amount a househod is abe to demand of every commodity, caed the rationing scheme on demand. In this we foow the approach and formuation of Dreze (1975). A exchange takes pace against the numeraire commodity, which is not rationed. A househod faces rationing z 2 IR ; and z 2 IR + in the market for each commodity 6= which represents the minima and the maxima amount ofgood househod i is abe to trade. For the sake of simpicity we ony consider uniform rationing, which means that z and z are the same for every househod i: Given a price system p and a rationing scheme (z z) 2 IR L ;IR L + a househod maximizes its utiity function u i over its constrained budget set dened by B i (z z p) =fx i 2 X i j px i pw i and z x i ; w i z =1 ::: Lg: Throughout the paper we wi use the foowing assumptions with respect to the economy E : 3

7 A1 For every househod i 2N X i =IR L+1 ++ : A2 For every househod i 2N the utiity function u i is C 2 on X i u i is dierentiaby stricty increasing, i.e., Du i (x i ) for a x i 2 X i and u i is dierentiaby stricty quasiconcave, i.e., the Gaussian curvature of I x i = fy i 2 X i j u i (y i )=u i (x i )g is dierent from zero for any x i 2 X i : Moreover, u i satises the boundary condition, that is, for every x i 2 X i the set fy i 2 X i j u i (y i ) u i (x i )g is cosed reative to IR L+1 : A3 For every househod i 2N w i 2 IR L+1 ++ : The demand function of individua i is dened by d i (z z p) = argmax x i 2B i (z z p) ui (x i ): Assumptions A1{A3 guarantee that d i is a function indeed. Given (z z p) 2 IR L ; IR L + P househod i is said to be constrained on its suppy in market if for any ~z 2 IR L ; such that ~z k = z k for k 6= and ~z = z ; " for some positive " u i (d i (~z z p)) >u i (d i (z z p)): If househod i is constrained on its suppy in the market for commodity then it foows from the strict quasiconcavity of the utiity function that d i (z z p) ; w i = z : A househod that is constrained on its suppy in market can improve its utiity if the rationing on suppy in market is reaxed. A simiar denition is made with respect to rationing on demand. The market for commodity is said to be constrained if there is at east one househod constrained on it, either on suppy or on demand. p 2 P: Foowing Dreze (1975), we introduce a Dreze equiibrium of the economy E at prices Denition 2.1 A Dreze equiibrium at prices p 2 P of an economy E is an aocation (x 1 ::: x N ) 2 Q i2n X i such that there exists (z z) 2 IR L ; IR L + satisfying the foowing conditions: (i) for a i 2N x i maximizes u i on B i (z z p) (ii) P i2n x i = P i2n w i (iii) for every 2Lnfg x i ; w i = z for some i 2N impies x i ; w i >z for a i 2N x i ; w i = z for some i 2N impies x i ; w i < z for a i 2N: The rst two conditions of the denition are standard, they state that every househod behaves optimay given the price system and the rationing scheme, and that a markets 4

8 cear. Condition (iii) guarantees that markets are transparent. Constraints are on one side of the market at most. The requirement that(z z) beongtoir L ; IR L + impies that there is no forced trading. Nothing precudes that the prices p are competitive. A competitive equiibrium is indeed a specia case of a Dreze equiibrium,itisadreze equiibrium without binding rationing. Notice that the case with two commodities, L =1 is the exact genera equiibrium anaogue of the standard textbook anaysis of partia equiibrium, where, for instance, a minimum price is imposed in the market for commodity 1 which isexchanged against commodity : If at the minimum price suppy exceeds demand, which is the case with the standard upward soping suppy curves and downward soping demand curves, then the quantity actuay traded is determined by the short side of the market, the tota demand for commodity 1: Some of the suppiers wi be constrained. They are ony abe to suppy part of their preferred suppy. Now consider the genera equiibrium set-up with L = 1 and suppose that at prices p tota net suppy exceeds tota net demand. A Dreze equiibrium wi necessariy invove ony rationing on the suppy side, so z 1 equas a number sucienty arge not to aect the househods' decision probems. Tota net demand is not aected by rationing on the suppy side, so when z 1 is such that constrained net suppy equas tota net demand for commodity 1, the unique Dreze equiibrium is obtained. 3 Suboptimaity of equiibrium Our rst aim is to provethefok Theorem that, given a tupe of utiity functions u 1 ::: u N and prices p 2 P for amost a initia endowments (w 1 ::: w N ) 2 IR N (L+1) ++ every Dreze equiibrium is suboptima. An aocation (x 1 ::: x N ) is said to be feasibe if x i 2 X i i 2N and P i2n x i = P i2n w i : Denition 3.1 A feasibe aocation (x 1 ::: x N )isoptima if there is no feasibe aocation y such that u i (y i ) u i (x i ) for a i 2N with at east one inequaity strict. As a consequence of the boundary condition on the utiity function, an aocation (x 1 ::: x N ) is optima if and ony x iu i (x i x i u i (x i ) u i (x ) i x i u (x ) i i for every i i 2N: 5

9 Lemma 3.2 Suppose that (x 1 ::: x N ) is a Dreze equiibrium at prices p 2 P of an economy E for a rationing scheme (z z) 2 IR L ; IRL + : Then there exist i 2 IR ++ i i 2 IR + i2n 2Lnfg such x iu i (x i x i u i (x i ) = p ; i ; i i 2N 2 Lnfg i i x i u i (x i ) i > impies xi ; wi = z, and i > impies xi ; wi = z : Proof. The concusion of the emma foows from the rst-order conditions for the maximization probem of househod i: max x i 2X i ui (x i ) s.t. px i ; pw i z x i ; wi z 2 Lnfg: The Lagrangian of the probem is L i (x i i i i )=u i (x i ) ; i (px i ; pw i ) ; LX =1 i (z ; x i + w i )+ i (;z + x i ; w i ) : The derivatives of L i with respect to x i are equa to zero at x i x i L i x i u i (x i ) ; i x i L i x iu i (x i ) ; i p + i ; i = 6= : Notice that i is never equa to since the numeraire commodityisaways desirabe. The rst part of Lemma 3.2 is now straightforward. Moreover, the Kuhn-Tucker conditions impy that i (zi ; x i + w i ) = i (z i ; x i + w i ) = which gives the second part of the emma. 2 The interpretation of the emma is very natura. The margina rate of substitution between good and the numeraire equas the price of good if the househod is unconstrained in market : It is ess than p if the househod is constrained on its suppy in market and it is greater than p if the househod is constrained on its demand in market : 6

10 Dreze equiibria may be optima, for instance, if the initia aocation of resources is optima. This is necessariy the case if there is just one commodity or just one househod. But when the number of commodities and househods is greater than one, we show this situation to be rather exceptiona. When L 2 and N 2 we can construct an exampe of an economy with an optima Dreze equiibrium at non-competitive prices p but an inecient initia distribution of resources. Suppose that there are two househods and three commodities, and competitive equiibrium prices p are such that the excess demand for commodity 2 is zero for both househods, so ony two goods are traded in non-zero amounts. Consider prices p such that p 2 <p 2 and p = p, = 1: It is possibe to choose utiity functions such that the demand for good 2 becomes positive for both househods under the price system p: In fact, this is the natura case. Choose a rationing scheme (z z) with z 2 = and other components of (z z) non-binding. Strict quasi-concavity of preferences impies that z 2 = is binding in an optima soution to the househod's decision probem. But then the competitive aocation is a Dreze equiibrium of the economy E at prices p which meansthat the Dreze equiibrium is optima. For L = 1 such an exampe cannot be constructed. If L = 1 and the initia distribution of resources is inecient, then a Dreze equiibrium at non-competitive prices p is necessariy inecient. Suppose L = 1 and et x be an optima Dreze equiibrium at non-competitive prices p of an economy E whereas the initia resources of E are distributed inecienty. There is at east one househod i who is rationed, and at east one househod i, who is not. The former hods because p is non-competitive. The atter because trade is needed to go from an inecient initia distribution of resources to an optima aocation. Condition (iii) of Denition 2.1 impies that one side of the market of commodity 1 is unconstrained. By Lemma 3.2 it hods x iu i (x i x i u i (x i ) u i (x ) i x i u (x ) i i which contradicts the optimaity of x: The foowing proposition gives a usefu characterization of optima Dreze equiibria. It caims that each optima Dreze equiibrium coincides with a competitive equiibrium aocation. Proposition 3.3 ADreze equiibrium x at prices p of an economy E for a rationing scheme (z z) is optima if and ony if x corresponds to a competitive equiibrium aocation. 7

11 Proof. Let x be an optima Dreze equiibrium at prices p and a rationing scheme (z z): Optimaity together with Assumption A1 impies x iu i (x i x i u i (x i ) i u x i (x ) x i u (x ) for any i i 2N: i i Then it foows from Lemma 3.2 that ( i ; i )= i does not depend on i: Together with Condition (iii) in the denition of a Dreze equiibrium, it impies that for each market 2Lnfg either every i is positive, or every i is positive, or both these mutipiers are equa to zero. The rst two cases mean that everyone in market is constrained, so from market cearing it foows that x i = wi for a i 2N: The ast possibiity isequivaent to the situation of a free market without rationing. Consider a vector of prices p 2 P such that p = p =1 p = p ; i ; i i 6= : Since prices p are dierent fromp ony for markets without trade, x satises the budget condition under the price system p i. e. p x i = p w i i2n: It foows immediatey thatx is a competitive equiibrium aocation at competitive prices p which proves the \ony if" part of the proposition. It is immediate that a Dreze equiibrium x which corresponds to a competitive equiibrium aocation, is optima. 2 The next step is to show the generic suboptimaity of Dreze equiibria. Theorem 3.4 Fix any price system p 2 P and utiity functions u 1 ::: u N satisfying Assumption A2. Then there isanopen set of fu Lebesgue measure of initia endowments in IR N (L+1) ++ such that every Dreze equiibrium at prices p of the economy E is suboptima. Proof. By Proposition 3.3, an optima Dreze equiibrium corresponds to a competitive equiibrium aocation. It foows from the resuts in Laroque (1978) that for an open set of fu Lebesgue measure of initia endowments, for every competitive equiibrium aocation x x i ; w i 6= for every househod i and every commodity : Therefore, using Lemma 3.2, genericay in initia endowments, cases where a i > and a i > are excuded. Genericay in 8

12 endowments, an optima Dreze equiibrium consists of competitive equiibrium prices and a competitive equiibrium aocation. To compete the proof we need to show that for generic w the price system p is not competitive. Let z(p w) denote aggregate excess demand at prices p and endowments w =(w 1 ::: w N ): Let F (p w) be equa to z (p w) for =1 ::: L and dene F L+1 (p w) =p L ;p L : If (p w)issuchthatz(p w) = then the rank of the w z(p w) is L see Mas-Coe (1985), p Therefore, the rank of the w pl F (p w z(p pl z(p w) 1 is equa to L +1 if F (p w) =: By the Transversaity ~p F w (p) has fu rank for amost a w 2 IR N (L+1) ++ if F w (p) = where F w (p) =F (p w) ~p =(p 1 ::: p L ): Since the rank ~p F w (p) can at most be L genericay in initia endowments, z(p w) = and p L = p L has no soution. Consequenty, genericay in initia endowments, the price system p is not competitive. We concude that for a set of endowments of fu Lebesgue measure, a Dreze equiibria are suboptima. We now show thatwecanchoose the set of initia endowments of fu Lebesgue measure for which a Dreze equiibria are suboptima to be open. Notice that the set of (p w) 2 P IR N (L+1) ++ such that F (p w) = is cosed due to the continuity off: It foows from Baasko (1988), page 89, that the natura projection function, which maps(p w) into w is proper. This impies that the set of initia endowments, for which some competitiveequiibrium price has its ast component equa to p L is cosed, since the image of a cosed set under a proper function is cosed. The compement of this set is open and of fu Lebesgue measure. The intersection of this set with the open set of fu Lebesgue measure for which there is trade for every househod for every commodity at a competitive equiibrium, is open and of fu measure and contains ony endowments with suboptima Dreze equiibria. 2 The theorem gives a rigorous statement of the Fok Theorem on the generic suboptimaity of equiibria at non-competitive prices. The next step is whether we can even strengthen this concusion to generic constrained suboptimaity. 4 Constrained optimaity when L =1 It is apparent that as ong as prices are not competitive, fu optimaity istoomuch to be expected. The appropriate criterion in this case is the one of constrained optimaityor 9

13 p-optimaity, that is optimaity reative to a aocations for which trades of a househods have zero vaue at an admissibe price system p. The notion of p-optimaity was introduced for the rst time in Younes (1975). Denition 4.1 Fix a price system p 2 P: A feasibe aocation (x 1 ::: x N ) 2 Q i2n X i is p-optima if there is no aocation (y 1 ::: y N ) 2 Q i2n X i such that (i) 8i 2N py i = pw i (ii) P y i = P w i i2n i2n (iii) 8i 2N u i (y i ) u i (x i ) with strict inequaity for at east one i 2N: We start by anayzing the case with two commodities, so L = 1: Strict quasi-concavity of utiity functions impies that the preferences of househods over the set of a attainabe amounts of good 1 given xed prices and the budget constraint are singe-peaked. In this case it is possibe to show uniqueness and constrained optimaity of a Dreze equiibrium. Proposition 4.2 If L =1 then a Dreze equiibrium (x 1 ::: x N ) at prices p 2 P of an economy E is unique and p-optima. Proof. If L =1 then quantity constraints are present ony on the market of commodity 1: As far as an anaysis of equiibrium is concerned, there is no oss of generaityby indexing a reevant rationing schemes as a function of q q 2 [ 1] : z 1 (q) =;q( X i2n w i 1 + ") z 1 (q) =q( X i2n w i 1 + ") where " is an arbitrariy chosen positive rea number. The aggregate excess demand for commodity 1 at prices p and rationing parameters q q is given by z 1 (q q) = X i2n d i 1(z 1 (q) z 1 (q) p) ; X i2n w i 1 : Since househods face constraints either on demand, or on suppy, but not on both of them, it is possibe to represent a reevant rationing schemes by a singe parameter q 2 [ 1] as foows: ~z 1 (q) =z 1 (minf2q 1g minf2 ; 2q 1g): 1

14 Here q = corresponds to the fu rationing on suppy, q = 1 corresponds to the fu rationing on demand, and when q =1=2 there is no rationing at a. It is immediate that ~z 1 () and ~z 1 (1) : The function ~z 1 is continuous and is easiy shown to be weaky decreasing in q: Proposition 4.2 is ceary true if p is competitive. If p is non-competitive, assume that ~z 1 (1=2) < : Let q s be the minima vaue of q such that ~z 1 (q s )=ez 1 (1=2): Since the rationing scheme z 1 (2q) is binding for at east one househod for a q 2 [ q s ) it is easy to see that the function ~z 1 is stricty decreasing on the interva [ q s ]: Moreover, ~z 1 () and ~z 1 (q s )=~z 1 (1=2) < : This impies that a Dreze equiibrium exists and is unique. Househods on the short side of the market, the demand side in this case, are not rationed and get the most preferred consumption bunde they can reach under the given xed prices, househods on the ong side cannot improve without making some other househod worse o. Therefore, a Dreze equiibrium is p-optima. A simiar argument appies when ez 1 (1=2) > : 2 Without any doubt the case L = 1 is specia. We think it has some importance, as it is the case that is typicay anayzed in textbooks. In Section 3 we have argued that optimaity of equiibrium typicay fais when L = 1: More precisey,we have argued above Proposition 3.3 that an equiibrium at non-competitive prices p is ecient for L = 1 if and ony if the initia distribution of resources is ecient. Proposition 4.2 makes cear that a weaker notion of optimaity, p-optimaity hods for a equiibria. 5 Constrained suboptimaity when L 2 If L is greater than or equa to 2 the situation becomes dierent. Then a Dreze equiibrium is not necessariy p-optima. Two counter-exampes can be found in Bohm and Muer (1977). Using a modied Edgeworth box diagram, they showed that equiibria and constrained optima constitute two disjoint sets. At the same time, robust exampes of constrained optima Dreze equiibria can be easiy found as we. Figure 1 shows such an exampe. There are three goods in an economy and two househods endowed with the same amount of initia resources. The big triange CDE corresponds to the set fx i 2 X i j px i = pw i g where the price system p is xed. The second househod consumes its most preferred consumption bunde on CDE: The triange x 1 AB corresponds to the constrained budget set of the rst househod. This househod faces ower bounds on the net trade in the market for both commodities 1 and 2. An indierence curve through x 1 is depicted. It is easiy veried that (x 1 x 2 )isap-optima Dreze equiibrium. Our next aim 11

15 is to provide mid conditions that rue out exampes ike this one. D A AA A A AA A AA A AA ; AA x 2 ; AA ; w 1 = w A ; AA B x A AA A A E ; x 1 ; 6 x 2 Figure 1: Exampe of a p-optima equiibrium. Let p 2 P be a xed price system. To study the matter of constrained optimaity, consider a \transformed" economy ~E with the same set of traders N as in the origina economy E and the set of goods ~L = Lnfg: The economy ~E is derived from E by using the budget equaity to eiminate commodity : Initia endowments, consumption sets and utiity functions of househod i 2N are specied as foows: ~w i = w i ; ~X i = f(~x i 1 ::: ~xi L) 2 IR L ++ j ~p(~x i ; ~w i ) w i g ~u i (~x i ) = u i (w i ; ~p(~x i ; ~w i ) ~x i 1 ::: ~xi L) where ~p =(p 1 ::: p L ): The rst order conditions for an optimum of the economy ~E give the foowing characterization of a p-optima aocation for E: It hods that (x 1 ::: x N ) 2 Q i2n X i is a p-optimum if and ony if there exist some q 2 IR L nfg and 2 IR N + such that (@ x iu i (x i ) ; x i u i (x i )) =1 ::: L = i q: 12

16 As we have seen before, at any Dreze equiibrium aocation x iu i (x i ) ; x i u i (x i )=; i + i for some non-negative rea i i such that i i = : Let N C N be the set of a constrained househods, given a Dreze equiibrium (x 1 ::: x N ): Then i = impies that househod i consumes its most preferred eement of the set fx i 2 X i j px i = pw i g: For each i 2N C, i 6=: Therefore, if a Dreze equiibrium is p-optima, then q = ) i = i = 8i 2N C q > ) i > 8i 2NC q < ) i > 8i 2NC : The above aows us to formuate a necessary condition for a Dreze equiibrium to be p-optima. We ca this condition the separating property. Proposition 5.1 (Separating property) If a Dreze equiibrium is p-optima, then every constrained househod is constrained in every constrained market. The separating property is quite powerfu since it is stated in observabe data ony. Whenever there are two househods that face constraints, but in dierent markets, constrained suboptimaity is the case. The separating property is a very strong requirement, so very stringent conditions are needed to achieve constrained optimaity. Notice that in the exampe in Figure 1, the separating property is satised. The separating property is triviay satised if L =1 or if there is ony one constrained househod. The rst case has been anayzed in Section 4, where it has been concuded that constrained optimaity resuts in the case with two commodities. The vector q is aso caed a vector of coupons prices in the iterature, see Dreze and Muer (198). Note the one to one correspondence between the side of rationing and the sign of a component of a coupons price vector for p-optima Dreze equiibria. We aready argued that the separating condition is strong, and if satised, ony a necessary condition. The next resut gives conditions for the separating condition to be sucient forp-optimaity. Theorem 5.2 Any Dreze equiibrium at prices p of an economy E with the number of constrained househods or the number of constrained markets ess than or equa to one, is p-optima. Proof. In the case where the number of constrained househods or the number of constrained markets equas zero, the Dreze equiibrium corresponds to a competitive equiibrium aocation, so optimaity and therefore p-optimaity foows. 13

17 Suppose the number of constrained househods equas one. Then dene i =1and q = ; i + i =1 ::: L with i i the Lagrange mutipiers corresponding to the rationing constraints of the constrained househod i: It foows that the condition for p- optimaity is satised. Suppose the number of constrained markets equas one, say market is constrained. Then dene i = ; i + i i2n and dene q to be the -th unit vector in IRL : Again, it foows that the condition for p-optimaity is satised. 2 Our na caim is that under weak conditions, the separating property is typicay not a sucient condition for constrained optimaity. More precisey, Dreze equiibria for which the set of constrained househods N C and the set of constrained markets L C consist of more than one eement each, are genericay not p-optima. Theorem 5.3 Fix any price system p 2 P and utiity functions u 1 ::: u N satisfying Assumption A2. There isanopen set of fu Lebesgue measure of initia endowments in IR N (L+1) ++ such that every Dreze equiibrium at prices p of the economy E with the number of constrained househods and the number of constrained markets greater than or equa to two, is not p-optima. Proof. It is hepfu to introduce a vector r that describes the state of the markets. The vector r is an eement of R = fr 2 IR L jr = ;1 or 1g where r = ;1 if there is suppy rationing in market r = if there is no rationing in market and r = 1 if there is demand rationing in market : We asointroduce a vector s that describes whether a househod is rationed or not. The vector s is an eement of S = fs 2 IR N js i = or 1g where s i = 1 if and ony if i beongs to the set of constrained househods N C. Denote by c i (x i ) the derivative with respect to x i of the \transformed" utiity function ~u i we used before: c i (x i )=@ x iu i (x i ) ; x i u i (x i ): We know from Lemma 3.2 that for any Dreze equiibrium (x 1 ::: x N ) 2 Q i2n X i c i (xi )=; i + i 14

18 for some non-negative i i the Lagrange mutipiers corresponding to the rationing constraints. If a Dreze equiibrium x is p-optima, then for any 2L i i 2N c i (xi ) c i (xi ) c i (xi ) c i (xi ) = ; i + i ; i + i ; i + i + i ;i For given r 2 R and s 2 S consider the sets = i q i q i q i q =: M rs = f( i ) i2n C 2 IR jn C jjl Cj j i < ifr = ;1 i > ifr =1g Z r = fz 2 IR jlc j j z <"if r = ;1 z > ;" if r =1g where " is some given positive number. By the Kuhn-Tucker theorem, if (x 1 ::: x N )isadreze equiibrium that satises the separating property, then there exists r 2 R, s 2 S, 2 IR N ++, 2 M rs, and z 2 Z r such that 1 px i ; pw i = i 2N x i u i (x i ) ; i = i 2N x iu i (x i ) ; i p ; i = i 2N 2Lnfg (3) X (x i ; w) i = 2 Lnfg (4) i2n x i ; w i ; z = i 2N C 2L C : (5) The number of unknowns in this system equas N(L +2)+jN C jjl C j + jl C j which is ess than the number of equations N(L +2)+jN C jjl C j + L or is equa to it if jl C j = L: Since there is a nite number of constrained markets and househods, there is a nite number of such systems. Since a nite intersection of open sets of fu Lebesgue measure is open and of fu Lebesgue measure, it is enough to restrict attention to an arbitrary xed r and s. Suppose that (x 1 ::: x N )isap-optima Dreze equiibrium. Without oss of generaity, f1 2g N C and f1 2g L C : By the separating property and the equation in determinants derived before, ; =: (6) Thus, p-optima Dreze equiibria satisfy a system of n equations, where n =(N +1)(L +2)+jN C jjl C j;1 1 Notice that a Dreze equiibrium aways eads to a soution to the system of equations. The other way around is not necessariy true, since non-binding inequaity constraints have been omitted, and the denition of Z r impies that a imited amount (") of forced trading is not excuded in a soution to the system of equations. 15

19 which is at east one more than there are unknowns. Let (w x z ) be the function dened as the eft-hand side of the equations (1) { (6). It is dened on Q i2n X i Q i2n X i Z r IR N ++ M rs and takes its vaues in IR (N +1)(L+2)+jN C jjl C j;1 : The key eement of the proof is the fact that is transversa to the origin, i.e. whenever (w x z ) = its Jacobian is of fu rank. Suppose that there exists y 2 IR n such that y (:)=: We write y =(y 1 ::: y 6 ) where y 1 2 IR N y 2 2 IR N y 3 2 IR NL y 4 2 IR L y 5 2 IR jn C jjl C j and y 6 2 IR: Then, in particuar, y w i (:) =;p y 1 i = so y 1 =: If 2 LnL C then taking into account the previous expression one gets y w 1 (:) =;y 4 =: For 2L C wehave and y w i y w i y z (:) =; X (:) = ;y 4 ; y 5 i = for i 2N C (:) = ;y 4 = for i 2NnN C i2n C y 5 i =: Thus, P i2n C y 5 i = jn C jy 4 so y 4 = 2L C : Therefore, y 5 i = i2n C 2N C : Moreover, y 1 1 (:) =y = so y 6 =: To compete the proof of reguarity itissucient to show that the matrix (@ 2 u i (x i ) ;p > ) has a fu row rank for every i 2N: This foows from the dierentiabe strict quasiconcavity of the utiity function, Proposition of Mas-Coe (1985), and the possibiity to cover a consumption set by a countabe number of compacts. By the Transversaity w (x z ) (x z ) has fu rank for amost a w 2 IR N (L+1) ++ if w (x z ) = where w (x z ) = (w x z ): Therefore, genericay in w the inverse image of fg has the same co-dimension as zero, which impies that w (x z ) = has no soution. We concude that for a set of endowments of fu Lebesgue measure, any Dreze equiibrium is constrained suboptima. 16

20 Denote by S the set Q i2n X i Z r IR N ++ M rs : To showthatwe can choose the set of initia endowments of fu Lebesgue measure for which Dreze equiibria are constrained suboptima to be open, consider the set of a (w x z ) 2 IR N (L+1) ++ S such that (w x z )=: This set is cosed by continuity of : Moreover, it is not dicut to see that the set f(w x z ) 2 j w 2 Kg is compact for any compact subset K of IR N (L+1) ++ : The atter means that the natura projection function :! IR N (L+1) ++ that maps (w x z )tow is proper. Therefore, the set of a w for which the concusion of the theorem does not hod, is cosed as the image of a cosed set by a proper function. Its compement is open and, as has been shown before, contains a set of fu Lebesgue measure. 2 The condition that the number of constrained markets is greater than or equa to two may be omitted from the statement of Theorem 5.3, since it is a generic property when L>1: The proof of this fact goes aong the same ines as the proof of the theorem above. Thus, we have the foowing coroary. Coroary 5.4 Fix any price system p 2 P and utiity functions u 1 ::: u N satisfying Assumption A2. There isanopen set of fu Lebesgue measure of initia endowments in IR N (L+1) ++ such that every Dreze equiibrium at prices p of the economy E with the number of constrained househods greater than or equa to two, is not p-optima. It is not possibe to caim that the number of constrained househods is genericay greater than one. For any tupe of utiity functions u 1 ::: u N satisfying Assumption A2, there is an open set of initia endowments IR N (L+1) ++ such thatforevery w 2 there isadreze equiibrium with ony one constrained househod. The exampe is constructed in such away that the constrained househod is constrained on its suppy in a markets, whereas a other househods have sma net demands for a non-numeraire commodities. Consider any tupe of utiity functions u 1 ::: u N satisfying Assumption A2 and x an arbitrary price system p 2 P: Let x i be the unconstrained demand of househod i at prices p when it has initia endowments e where e is the vector of a ones in IR L+1 : Pick initia endowments for househod 1 and a rationing scheme (z z) such that househod 1, at prices p and rationing scheme (z z) is constrained on its suppy in each market for non-numeraire commodities and ;z is smaer than x i for i =2 ::: N: To achieve this, one may take w 1 2fw2 X 1 jpw = peg such that w 1 ; x 1 ; so the unconstrained demand of househod 1 at prices p equas x 1 and househod 1 prefers to suppy a non-numeraire commodities. Take z 1 1 =x 1 1 ; w " 1 with " 1 a sma positive number. By continuity, the demand ~x 1 of househod 1 when taking z 1 into account is cose to x 1 in particuar a non-numeraire commodities are sti suppied by househod 1, and rationing in the market 17

21 for commodity 1 is binding. Take z 2 =~x 1 2 ; w " 2 with " 2 a positive number that is sma enough for rationing in the market for commodity 1 to remain binding. This construction is repeated unti the rationing scheme z is obtained that induces rationing on the suppy of househod 1 in the markets for a non-numeraire commodities. By taking w 1 sucienty cose to x 1 the requirement that;z be smaer than x i i=2 ::: N can be fued as we. The rationing scheme z is chosen as never to aect the choice of any househod. For i =2 ::: N initia endowments are taken such that househod i is on the short side of each market, w i =x i ; LP =1 p z =(N ; 1) ;z 1 =(N ; 1). ;z L =(N ; 1) 1 C A : Since a househods, but househod 1, get their most preferred commodity bunde at prices p it foows that (d 1 (z z p) x 2 ::: x N )isap-optima Dreze equiibrium. If we sighty perturb the initia endowments, tota net demand of househods excuding househod 1 changes sighty,andadreze equiibrium is obtained by rationing the suppy of househod 1 by this amount. A other househods remain unconstrained. The property of p-optimaity is obviousy kept. It foows that there is an open set of initia endowments with p-optima Dreze equiibria. 6 Concusions Notwithstanding the strong conviction of most economists that trade at non-competitive prices has detrimenta wefare consequences, it is not based on foundations derived with equa rigor as the rst and second wefare theorems. This paper provides these foundations. We show that the Fok Theorem hods that equiibria are typicay suboptima when trade at non-competitive prices occurs. The more appropriate question to answer is whether equiibria are perhaps not even constrained optima. In this paper we formaized the notion of constrained optimaity as optimaity among aocations where budget constraints of a househods at the noncompetitive prices are met. A necessary condition for constrained optimaity to prevai is the separabiity condition that a constrained househods be constrained on a constrained markets. This condition is of interest in itsef as it is formuated in terms that are observabe. If the number of commodities is ess than or equa to two, then this necessary condition is aways satised. We show that in this case it is aso a sucient condition, so constrained optimaity hods in the two commodity case. When there are three or more 18

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