Lindahl Equilibrium as a Collective Choice Rule
aa r X i v : . [ ec on . T H ] A ug Lindahl Equilibrium as a Collective Choice Rule † Faruk Gul and Wolfgang PesendorferPrinceton UniversityAugust 2020
Abstract
A collective choice problem is a finite set of social alternatives and a finite set ofeconomic agents with vNM utility functions. We associate a public goods economy witheach collective choice problem and establish the existence and efficiency of (equal income)Lindahl equilibrium allocations. We interpret collective choice problems as cooperativebargaining problems and define a set-valued solution concept, the equitable solution (ES).We provide axioms that characterize ES and show that ES contains the Nash bargainingsolution. Our main result shows that the set of ES payoffs is the same a the set of Lindahlequilibrium payoffs. We consider two applications: in the first, we show that in a large classof matching problems without transfers the set of Lindahl equilibrium payoffs is the same asthe set of (equal income) Walrasian equilibrium payoffs. In our second application, we showthat in any discrete exchange economy without transfers every Walrasian equilibrium payoffis a Lindahl equilibrium payoff of the corresponding collective choice market. Moreover,for any cooperative bargaining problem, it is possible to define a set of commodities so thatthe resulting economy’s utility possibility set is that bargaining problem and the resultingeconomy’s set of Walrasian equilibrium payoffs is the same as the set of Lindahl equilibriumpayoffs of the corresponding collective choice market. † This research was supported by grants from the National Science Foundation. . Introduction
Consider an organization that must decide among several alternatives that affect thewelfare of its members. The goal is to find an equitable and efficient solution to the problemwhen no transfers among members are possible. Examples include a community’s decisionon how to allocate infrastructure investments among neighborhoods, the allocation of officespace among groups within an organization, or the assignment of college roommates.In this paper, we analyze the following mechanism. Each group member is givenan equal budget of fiat money and confronts a price for each of the relevant alternativesunder consideration. As in standard consumer theory, members choose an alternative thatmaximizes their utility subject to the budget constraint. The organization acts as anauctioneer and implements an alternative that maximizes revenue. We allow choices to bestochastic; that is, agents choose lotteries over social outcomes. We also allow the prices forthe various social outcomes to be individual-specific. Thus, our mechanism is analogousto a Lindahl mechanism (Lindahl (1919)) in a public goods setting. It differs from thestandard Lindahl equilibrium in that the objects of choice are social outcomes (allocations)rather than individual consumptions. We refer to this mechanism as a collective choicemarket and to its equilibria as Lindahl equilibria.As an illustration, suppose a town with n inhabitants considers implementing projectA or project B. The status quo yields zero utility for all inhabitants who are divided equallybetween A and B-supporters. Each inhabitant receives utility 1 if her favored alternativeis implemented and zero otherwise. In a collective choice market, every agent is given oneunit of fiat money and prices for the two alternatives; she must choose an optimal lotteryover the two alternatives subject to her budget constraint. In the Lindahl equilibrium,B-supporters must pay 2 for alternative B and zero for A; A-supporters must pay 2 for Aand zero for B. At these prices, it is optimal for all agents to choose the lottery that yields A and B with equal probabilities. This lottery also maximizes auctioneer revenue and isthus a Lindahl equilibrium.The town’s decision problem can also be described as an n-person bargaining problemin which the attainable utility profiles correspond to the inhabitants’ utilities for lotteries1ver A and B . More generally, every collective choice market can be mapped to an n -person bargaining problem. We introduce a new solution concept for n -person bargainingproblems - the equitable solution . Unlike standard solution concepts such as the Nash bar-gaining solution or the Kalai-Smorodinsky solution, it is set valued. That is, it associateswith every bargaining problem a set of equitable solutions. Our main result (Theorem 2)relates Lindahl equilibrium outcomes and equitable outcomes of the associated bargaininggame.As the following example illustrates, it is often the case that several solutions to abargaining game may qualify as equitable. Suppose Ann and Bob must divide two cakes, apeanut butter cake and a chocolate cake. Utilities are linear but Bob is allergic to peanutswhile Ann likes the peanut butter cake just as much as the chocolate cake. One equitabledivision might be to give Ann the peanut butter cake and divide the chocolate cake equallybetween Ann and Bob. To justify this outcome, Ann could argue as follows: Since Bob hasno use for the peanut butter cake, from his perspective the situation is as if we only hadthe chocolate cake and in that situation it’s obviously equitable to divide the chocolate cakeequally.
Perles and Maschler (1981) provide an axiomatic foundation for Ann’s argument.The Perles-Maschler solution to the bargaining game would give Ann the peanut buttercake and divide the chocolate cake equally between the two players. On the other hand,Bob could make the following argument:
If I were not allergic to peanuts we would each getone cake. Since Ann is indifferent between the chocolate cake and the peanut butter cake itmakes sense that she gets the peanut butter cake and I get the chocolate cake.
Nash (1951)provides an axiomatic foundation for Bob’s argument. In the Nash bargaining solution ofthis game, Ann gets the peanut butter cake and Bob gets the chocolate cake. Of course,in-between solutions can also be justified. For example, the Kalai-Smorodinsky solutionsuggests giving 2/3 of the chocolate cake to Bob and 1/3 to Ann while Ann gets all of thepeanut butter cake.To define the equitable solution, we first identify a subclass of bargaining problemsthat have a self-evidently fair outcome. In those bargaining problems agents have linearutility over a fixed surplus to be divided among them and all standard bargaining solutions2gree that equal division is the only plausible equitable outcome. For a general bargainingproblem; that is, when a fair solution is not self-evident, the equitable solution picksall those outcomes which are the obvious fair outcome of some dominating bargainingproblem. In Theorem 1 we provide an axiomatic foundation for the equitable solution.Our main result, Theorem 2, shows that the set of Lindahl equilibrium payoffs coin-cides with the equitable outcomes of the corresponding bargaining game. One direction,every Lindahl equilibrium is equitable, can be interpreted as an equity-analogue of the firsttheorem of welfare economics. It shows that our axioms capture the sense in which equi-librium outcomes of collective choice markets are equitable. The second direction, everyequitable outcome can be achieved as a Lindahl equilibrium, is analogous to the secondtheorem of welfare economics. It shows that collective choice markets are flexible enoughto implement any outcome that is equitable.We first apply our results to matching. A group of individuals must decide whomatches with whom as, for example, in a roommate assignment problem or in the clas-sic two-sided matching problem (Gale and Shapley (1962)). We show that in matchingproblems without transfers the Lindahl equilibrium outcomes of a collective choice marketcoincide with the outcomes of Walrasian equilibria in a market for partners. In a collectivechoice market, consumers express their demands for social outcomes (that is, allocations)while in a Walrasian equilibrium, consumers express their demand for potential partners.In both cases, utility is non-transferable and consumers have the same budget of fiat money.Our result implies that the Walrasian equilibria of matching markets with non-transferableutility and equal budgets are equitable.In our second application, we consider the allocation of non-divisible goods when con-sumers have identical budgets of fiat money. Every Walrasian equilibrium of this exchangeeconomy is a Lindahl equilibrium of the corresponding collective choice market. However,the set of Walrasian equilibria is sensitive to the exact specification of the traded goods(that is, how property rights are defined) while Lindahl equilibria are not. More precisely, See Thomson (1994) for a survey of bargaining solutions. For our setting, the relevant bargainingsolutions are ordinal, that is, solutions that are invariant to affine transformation of utilities. The Nashbargaining solution (Nash (1950)), the Kalai-Smorodinsky solution (Kalai and Smorodinsky (1975)) andthe Perles-Maschler solution (Perles and Maschler (1981)) are examples of ordinal solutions. Bargaining problem A dominates problem B if for every utility profile in B there is a better utilityprofile in A and for every utility profile in A there is a worse utility profile in B . Our paper is related to the extensive literature on axiomatic bargaining theory (seeThomson 1994) for a survey). Our axiomatic treatment is related to Nash (1950) andwe discuss this relationship in detail after the statement of Theorem 1 below. For 2-person bargaining, our solution concept includes the Kalai-Smorodinsky solution (Kalaiand Smorodinsky (1975)) and the Perles-Maschler solution (Perles and Maschler (1981))in the set of equitable outcomes.Hylland and Zeckhauser (1979) were the first to propose Walrasian equilibria as solu-tions to stochastic allocation problems. Gul, Pesendorfer and Zhang (2020) extend Hyllandand Zeckhauser from unit demand preferences to general gross-substitutes preferences. Col-lective choice markets allow for arbitrary preferences, public goods and externalities andhence provide a further generalization of the environment considered in these papers.Foley (1967), Schmeidler and Vind (1972) and Varian (1974) associate equity withenvy-freeness. Walrasian equilibria with equal budgets are envy free and, thus, theseauthors establish a connection between competitive outcomes and equity. In a publicgoods setting, two agents may contribute different amounts to the same public good andhence it is not straightforward to adapt the notion of envy freeness to Lindahl equilibria.Moreover, as we show in section 5.2, there are multiple ways to commodify each collective4hoice market. The same utility profile may be envy free for one specification but fail envy-freeness for another. Thus, collective choice markets do not lend themselves to a coherentdefinition of envy-freeness. Instead, we provide a definition of equitable outcomes basedon the associated cooperative bargaining problem and show its equivalence to Lindahlequilibria.
2. Collective Choice Markets and Lindahl Equilibrium
Consider the following collective choice problem: n agents, i ∈ { , . . . , n } , must decideon one of k social outcomes, j ∈ K = { , . . . , k } . A random outcome, q ∈ Q := { ˆ q ∈ IR k + | P j ∈ K ˆ q j = 1 } , is a probability distribution over the social outcomes. Agents areexpected utility maximizers; i ’s utility if outcome j ∈ K occurs is u ji ≥ u i =( u i , . . . , u ki ) denotes i ’s utility index. In addition to the k outcomes described above, thereis a disagreement outcome that yields zero utility to every agent. We dismiss all agentswho have no stake in the collective decision; that is, we assume that for every utility u i there is some j such that u ji >
0. The vector u = ( u , . . . , u n ) denotes a profile of utilities.We will define a social choice rule by identifying the Lindahl equilibria of a corre-sponding market economy, the collective choice market . This market has n + 1 agents,the n described above, now called consumers, and one firm. Consumer i has one unitof fiat money and can purchase quantity q j ≥ j ∈ { , . . . , k } atprice p ji ≥
0. Outcome 0 is identified with not purchasing any good (and therefore hasprice 0). Let e = (1 , . . . ,
1) be the k − dimensional unit vector. Consumer i faces prices p i = ( p i , . . . , p ki ) and solves the following maximization problem: U i ( p ) = max q u i · q subject to p i · q ≤ , e · q ≤ q is an minimal-cost solution to the maximization problem above if q solvesthat problem and p · ˆ q ≤ p · q for every other solution ˆ q . The price paid for outcome j depends on the identity of the consumer. We let p i = ( p i , . . . , p ki ) ∈ IR kn + be consumer i ’s Sato (1987) adapts the notion of envy-freeness to Lindahl equilibria by assuming that agent i converts j ’s actual consumption of the public good into a virtual quantity based on j ’s utility of that good. In effect,Sato identifies a commodity space and associated utility functions for which envy freeness of Lindahlequilibria with equal budgets is satisfied. p = ( p , . . . , p n ) be a price profile. The firm chooses q to maximize profit, thatis, to solve R ( p ) = max q n X i =1 p i · q subject to e · q = 1 (2) Definition:
The pair ( p, q ) is a Lindahl equilibrium (LE) for the collective choice marketif q is a minimal-cost solution to every consumer’s maximization problem at prices p i andsolves the firm’s maximization problem at prices p . As is well known, a Lindahl equilibrium can be re-interpreted as a Walrasian equilib-rium of a suitably modified alternative economy. To do so, we identify i ’s consumption ofpublic good j with a distinct private good ( i, j ) and assume the firm can meet demand( q j . . . . , q nj ) with supply q if max ≤ i ≤ n q ij ≤ q j for all j . This connection to Walrasianequilibrium means that existence of a Lindahl equilibrium can be shown by adapting stan-dard results. The proof of Lemma 1 and subsequent results are in the appendix. Lemma 1:
Every collective choice market has a Lindahl equilibrium; all Lindahl equi-libria are Pareto efficient.
A broad range of applications including all discrete allocation and matching problemscan be modeled as collective choice markets. For example, if the aggregate endowmentconsists of a collection of indivisible goods, then the set of outcomes is simply the set ofall allocations.Similarly, to map a two-sided matching problem into a collective choice market, wepartition “consumers” into two groups, N , N and let the outcomes be the set of allmatchings; that is, one-to-one functions j : N ∪ N → N ∪ N such that j ( j ( i )) = i and[ i ∈ N l implies j ( i ) = i or j ( i ) / ∈ N l ]. Hence, a member of group l is either unmatched( j ( i ) = i ) or is matched with someone from the other group ( j ( i ) / ∈ N l ). In section 4, weanalyze these applications and relate Lindahl equilibrium outcomes to standard Walrasianoutcomes of these economies.Let L ( u ) ⊂ IR n + be the set of utility profiles that can be supported as a LE of the corre-sponding collective choice market u ; that is, L ( u ) = { ( u · q, . . . , u n · q ) | ( p, q ) is a LE of u } .6 . The Bargaining Problem In this section, we define and characterize the equitable solution, a cooperative solutionconcept. For any x, y ∈ IR n , we write x ≤ y to mean x i ≤ y i all i . For any bounded set X , we let d ( X ) (the disagreement point of X ) denote the infimum of X in IR n ; that is, (i) d ( X ) ≤ x for all x ∈ X and (ii) y ≤ x for all x ∈ B implies y ≤ d ( X ). Similarly, we let b ( X ) (the bliss point X ) denote the supremum of any bounded X ; that is, b ( B ) = − d ( − B ).For any finite, bounded set X ⊂ IR n , let conv X denote its convex hull and comp X denote its comprehensive hull of X ; that is,comp X := { x ∈ IR n | d ( X ) ≤ x ≤ y for some y ∈ X } Then, the set coco X := comp conv X is it convex comprehensive hull; that is, the smallest(in terms of set inclusion) convex and comprehensive set that contains X .For any a ∈ IR n and x , let a ⊗ x = ( a · z , . . . , a n · z n ), a ⊗ B = { a ⊗ x | x ∈ B } and B + z = { x + z | x ∈ B } . Let e i denote unit vector with zeros in every coordinate except i , o := (0 , . . . , ∈ IR n , e := (1 , , . . . , ∈ IR n and ∆ = conv { o, e , . . . , e n } .We consider bargaining problems in which the set of attainable utility profiles is afull dimensional, comprehensive polytope. Let B denote the set of all such polytopes and B o = { B ∈ B | o = d ( B ) } denote the set of all polytopes with disagreement point o ;that is, the set of normalized polytopes. A (set-valued) bargaining solution is a mapping S : B → IR n \∅ such that S ( B ) ⊂ B . Hence, a bargaining solution chooses a nonempty setof alternatives from every bargaining problem. Below, we define a set-valued bargainingsolution, which we call the equitable solution (ES) and provide axioms that characterize it.Equal division is the natural fair outcome if the bargaining set is the unit simplex.Since von Neumann-Morgenstern utilities are unique only up to positive affine transforma-tions, any affine transformation of the unit simplex should yield the corresponding affinetransformation of the fair outcome. Let D be the bargaining problems that are positiveaffine transformations of the unit simplex, that is, B ∈ D ⊂ B if there exists a, z ∈ IR n such that a i > i and B = a ⊗ ∆ + z . For B = a ⊗ ∆ + z ∈ D the fair outcomeis x = n a ⊗ e + z . Hence, only bargaining problems that are equivalent, up to a positiveaffine transformation of utilities, to ∆ have fair outcomes, and the fair outcome is the7orresponding affine transformation of to the unique symmetric and efficient outcome of∆. Let F ( B ) be the set of fair outcomes of B ∈ B with the convention that F ( B ) = ∅ if B
6∈ D . Definition:
Let
A, B ∈ B . Then, A ≥ B if for every x ∈ A, y ∈ B , there exist x ′ ∈ A , y ′ ∈ B such that x ′ ≥ y and x ≥ y ′ . Thus, A ≥ B if for every utility vector is in B there is a corresponding utility vectorin A that dominates it and, conversely, for every utility vector in A there is a correspond-ing utility vector in B that is dominated by it. The equitable solution consists of thoseoutcomes of B that coincide with the fair outcome of a simplex A ≥ B . Definition:
The equitable solution is E ( B ) := { x ∈ B ∩ F ( A ) | A ≥ B } . With some abuse of terminology, we also refer to the mapping from bargaining prob-lems to their equitable sets as ES. Below, we provide axioms on the bargaining solution S that characterize the equitable solution. The first axiom, scale-invariance, is familiarfrom other bargaining solutions including the Nash bargaining solution and the Kalai-Smorodinsky solution. It asserts that positive affine transformations of utilities do notchange the set of chosen (physical) outcomes. Scale Invariance: S ( a ⊗ B + z ) = a ⊗ S ( B ) + z whenever a i > for all i . Our symmetry axiom applies only to the bargaining problem ∆. It ensures that theunique symmetric and efficient outcome of ∆ is the only one chosen.
Symmetry: S (∆) = { n · e } . The following axiom is similar to independence of irrelevant alternatives (IIA) of theNash bargaining solution.
Consistency: B ≤ A implies S ( A ) ∩ B ⊂ S ( B ) . One important difference between consistency and IIA is that the former is applicableeven if d ( A ) = d ( B ). The latter replaces ≤ with ⊂ and requires d ( A ) = d ( B ). Thenext axiom, justifiability, is our main assumption. We discuss its interpretation followingTheorem 1 below. Justifiability: x ∈ S ( B ) implies B ≤ A and { x } = S ( A ) for some A . Theorem 1:
The only bargaining solution that satisfies the four axioms above is ES.
It is easy to verify that E satisfies the axioms. For the converse, let S be a bargainingsolution that satisfies the axioms and note that scale-invariance and symmetry imply that S ( a ⊗ ∆ + z ) = n { a ⊗ e + z } = F ( a ⊗ ∆ + z ) for all a ∈ IR n ++ and z . Then, consistency,ensures E ( B ) ⊂ S ( B ) while justifiability implies x / ∈ S ( B ) whenever x / ∈ S ( B ) for all B .To see how our axioms relate to the axioms for the Nash bargaining solution (Nash1950), consider a single-valued solution. In that case, adding d ( A ) = d ( B ) to consistencyand justifiability implies that the Nash bargaining solution is characterized by either ofthese two axioms together with scale invariance and symmetry. We can get this resulteven if we replace ≤ with ⊂ while maintaining d ( A ) = d ( B ) in both consistency andjustifiability.The equitable solution is a permissive solution concept that, in the case of two agents,includes all the standard scale invariant bargaining solutions. In particular, the Nash bar-gaining solution (Nash (1950)), the Kalai-Smorodinsky solution (Kalai and Smorodinsky(1975)) and the Perles-Maschler solution (Perles and Maschler (1981)) are contained in theequitable solution when n = 2.To see the motivation for justifiability consider again the example in the introduction:Ann and Bob must divide two cakes, a peanut butter cake and a chocolate cake. Utilitiesare linear but Bob is allergic to peanuts while Ann likes the peanut butter cake just asmuch as the chocolate cake. The corresponding bargaining set B is the convex hull of thepoints { (0 , , (1 , , (1 / , , (0 , } . Ann could argue as follows: I should get the peanutbutter cake since you don’t have any use for it. We both like the chocolate cake and soit makes sense to divide it equally. So, we should choose x = (3 / , / as our solution. Ann’s argument leads to the Perles-Maschler solution of this bargaining game. On theother hand, Bob could make the following argument:
If I were not allergic to peanuts wewould each get one cake. Since you are indifferent between the chocolate and the peanutbutter cakes it makes sense that you get the peanut butter cake and I get the chocolatecake. So we should choose y = (1 / , as our solution. Bob’s argument leads to theNash-bargaining solution of this bargaining game.9ustifiability says that both Ann and Bob have valid arguments and, thus, permitsboth solutions as possible outcomes. More precisely, Ann can justify her favored outcomeby noting that the bargaining game A = conv { (1 / , , (1 , , (1 / , } is a translationof a simplex and its fair outcome is x . Since A ≥ B , her proposed alternative x isjustified. Bob can justify his favored outcome by pointing to the bargaining game A ′ =conv { (0 , , (1 , , (0 , } ≥ B which has y as its fair outcome.The following lemma characterizes the equitable solution for the 2-player case andyields as a straightforward corollary the fact that the three solutions above are containedin the equitable solution. Lemma 2: If n = 2 , then E ( B ) consists of all Pareto efficient utility profiles x ∈ B suchthat x ≥ d ( B ) + ( b ( B ) − d ( B )) / . Lemma 2 shows that in the case of two agents the equitable solution consists of allPareto efficient outcomes that satisfy midpoint domination . The midpoint corresponds tothe outcome that would be achieved if a coin is flipped to decide which agent is made adictator. In his survey of bargaining solutions, Thomson (1994, p. 1254) points out thatmost bargaining solutions satisfy midpoint domination. In particular, the three solutionsmentioned above satisfy it.As Lemma 2 shows, the two agent case allows a simple characterization of the set ofequitable outcomes. However, when there are more than 2 agents this is no longer thecase. In particular, while the equitable solution continues to satisfy midpoint domination,the converse is no longer true. Not every Pareto efficient outcome that satisfies midpointdomination can be equitable. For example consider the 3-person bargaining problem B =coco { (0 , , , (1 , , / , (1 / , / , } . Then, x = (1 / , / ,
1) is a Pareto efficient outcomethat satisfies midpoint domination (since each agent receives at least 1 /
3) but it is not anequitable solution since there is no A ∈ D , A ≥ B that has x as its fair outcome. Our mainresult, Theorem 2 below, characterizes the set of equitable outcomes as the set of Lindahlequilibrium outcomes of the corresponding collective choice market.Before doing so, we relate the equitable solution to the Nash bargaining solution. Forany B ∈ B , let f B ( x ) := X i log( x i − d i ( B ))10he Nash bargaining solution of B , η ( B ), is the unique outcome that maximizes f B ( · ) in B . Definition:
The outcome x ∈ B is Nash-sustainable if there exists A ∈ B , A ≥ B suchthat x = η ( A ) . Note that the fair outcome coincides with the Nash bargaining solution if the bar-gaining set is a positive affine transformation of a simplex. Therefore, any outcome in theequitable solution must be Nash sustainable. Lemma 3, below, shows that the converse istrue as well. We let N ( B ) denote the set of Nash-sustainable outcomes of the bargainingproblem B . Lemma 3:
For all B ∈ B , E ( B ) = N ( B ) . Lemma 3 shows that Nash sustainability is an alternative characterization of theoutcome function that satisfies our axioms.
4. The Equitable Solution and Lindahl Equilibria
Any collective choice problem u , corresponds to a unique bargaining problem B u := coco ( { ( u j , . . . , u jn ) | j ∈ K } ∪ { o } )Hence, B u is the set of feasible utilities in the collective choice problem. Theorem 2 below isour main result. It shows that the equitable solution coincides with the Lindahl equilibriumpayoffs.One direction, every Lindahl equilibrium is equitable, shows that the equitable so-lution captures the notion of fairness embodied in Lindahl equilibria of collective choicemarkets. Just as the first welfare theorem clarifies the notion of efficiency embodied incompetitive models, Theorem 2 shows that Lindahl equilibria with equal budgets representoutcomes that are justifiable in the sense of our final axiom above. The other direction,every equitable solution is a Lindahl equilibrium, shows that all equitable solutions can beimplemented via a collective choice market. Thus, this direction of Theorem 2 is analogousto the second welfare theorem albeit without the need to vary consumer’s wealth.11 heorem 2: The set of Lindahl equilibrium payoffs is the same as the equitable solutionof the corresponding bargaining problem: L ( u ) = E ( B u ) for all u . To see how every equitable outcome can be made into a Lindahl equilibrium payoff,take any equitable a ∈ B u . By definition, there exists some bargaining set A that (i)is a positive affine transformation of a simplex; (ii) has a as its fair outcome; and (iii)satisfies A ≥ B u . Clearly, a must be on the Pareto-frontier of B u . In general, d ( A ) neednot be the origin but in the special case where it is we have B u ⊂ A and A = na ⊗ ∆.Consider that special case. Since a is Pareto efficient, there is a distribution q over Paretoefficient outcomes that delivers the utility vector a . The vector q is the Lindahl equilibriumallocation. To find the Lindahl equilibrium prices, let y j = ( u j , . . . , u jn ) be the utility vectorcorresponding to outcome j . Since B u ⊂ A the vector y must be a convex combinationof the extreme points of A = na ⊗ ∆. Hence, there are z ji ≥ P i z ji = 1 and P i na i z ji · e i = y j for all j . Then, set p ji = nz ji to be the price agent i pays for outcome j . It is then not difficult to verify that ( p, q ) is a Lindahl equilibrium. Our proof that E ( B u ) ⊂ L ( u ) for all u generalizes this argument to cover the case in which d ( A ) = o when we have B u ≤ A but not B u ⊂ A .For the converse; that is, to see that every Lindahl equilibrium payoff is equitable, let( p, q ) be a Lindahl equilibrium and recall that q solves the linear program below for everyconsumer i : max q ′ u i · q ′ subject to p i · q ′ ≤ , e · q ′ ≤ α i be the shadow price of the constraint p i · q ′ ≤ c i be the shadow price of theconstraint e · q ′ ≤
1. Then, the constraint of the dual of the above linear program requiresthat for all i, j , α i p ji ≥ u ji − c i (3)Moreover, optimality of q implies that inequality (3) holds with equality if q j >
0. Suppose α i , the shadow price of the budget constraint, is strictly positive for every consumer. Then, p i · q = 1 and, since α i p ji = u ji − c i if q j >
0, we have α i + c i = u i · q for all i ; that is, α + c , where α = ( α , . . . , α n ) and c = ( c , . . . , c n ), is the payoff vector associated with theequilibrium ( p, q ). Note that α + c is the fair outcome of the bargaining game nα ⊗ ∆ + c .12he key step of the proof establishes that B u ≤ nα ⊗ ∆ + c which then implies that α + c is an equitable outcome of B u .Theorem 2 shows that Lindahl equilibria are equitable, which, by Lemma 3, impliesthat every Lindahl equilibrium is Nash sustainable. Thus, if B u is the bargaining gamecorresponding to the collective choice problem then each Lindahl equilibrium payoff isthe payoff in the Nash bargaining solution of a bargaining game A such that A ≥ B u .In Corollary 1, below, we rely on this connection to characterize all Lindahl equilibriumallocations in terms of the Nash bargaining solution.For any collective choice market u , the allocation q that yields the Nash bargainingoutcome is the unique solution to the maximization problem: N ∗ = max q ∈ Q X i log( u i · q ) ( N )We refer to the solution of (N) as the Nash allocation for u . Note that if q is the Nashallocation of u , then N ( B u ) = u · q = ( u · q, . . . , u n · q ).To relate Lindahl equilibria to Nash allocations, we define a family of utilities derivedfrom the original utility u of the collective choice problem. Let C i ( u i ) = [0 , max j u ji ) and C ( u ) = Q ni =1 C i ( u i ). For c i ∈ C i ( u i ), define¯ u ji ( c i ) := max { u ji − c i , } Since c ∈ C ( u ), ¯ u ji ( c i ) > j and hence ¯ u i ( c i ) = ( u i , . . . , u ki ) is a utility. Let¯ u ( c ) = (¯ u ( c ) , . . . , ¯ u n ( c n )) be the corresponding profile of utilities. Definition:
Let q be the Nash allocation for ¯ u ( c ) for some c ∈ C ( u ) . Then, c is admis-sible if u ji ≥ c i for all j such that q j > . Thus, a vector c is admissible if the corresponding Nash allocation only chooses alter-natives that give each consumer a utility no less than c i (according to the original utility u i ). Corollary 1: If q is the Nash allocation for ¯ u ( c ) and c is admissible, then ( p, q ) suchthat p i = ¯ u i ( c i ) / (¯ u i ( c i ) · q ) is a Lindahl equilibrium for u . Conversely, if q is a Lindahlallocation for u , then there is an admissible c such that q is a Nash allocation for ¯ u ( c ) . u ( c ) for some admissible c . To understand the need for admissibility,let q be a solution to (N) for an arbitrary c ∈ C ( u ). For q to be a Lindahl equilibrium, thepayoff of agent i must be ¯ u i ( c ) · q + c i ; note that ¯ u i ( c ) · q + c i ≥ u i · q but feasibility requiresthat this inequality holds with equality. Equality holds if and only if c is admissible.Above we have interpreted c i as a parameter that modifies consumer i ’s utility. Alter-natively, we can interpret the c i ’s as modifying the disagreement points in the bargaininggame B u . The original disagreement point of B u is the origin since we assume all con-sumers get utility zero if no agreement is reached. For c ≥
0, let B u ( c ) := { u ∈ B u | u i ≥ c i } be the bargaining game B u modified so that the disagreement point is c instead. We canre-state the definition of admissibility, above, to say that c is admissible if the Nash bar-gaining solution of B u ( c ) can be implemented with an allocation that only puts weight onoutcomes that give utility at least c i to consumer i . Corollary 1 shows that every Lindahlequilibrium corresponds to a Nash bargaining solution with an admissible disagreementpoint c and, conversely, every Nash bargaining solution with an admissible disagreementpoint c can be implemented as a Lindahl equilibrium of the collective choice market.
5. Applications
In this section, we analyze two applications of the Lindahl collective choice rule. Inthe first, we consider one-to-one matching problems without transfers; in the second, weconsider general allocation problems with indivisible goods and no transfers.
A group of agents must decide who matches with whom as, for example, in the problemof choosing roommates. A matching, is a bijection j from the set of all agents to itself suchthat j ( j ( i )) = i for all i . If j ( i ) = i , then i is said to be unmatched. Some matchings maybe infeasible. For example, agents could be workers and firms and firms matching withanother firms may be disallowed. The set K = { , . . . , k } represents the feasible matchingsand let N i = { m | j ( i ) = m, for some j ∈ K } be the feasible matches of agent i and assumethat N i = ∅ for all i . 14et w mi be the utility of agent i when she matches with agent m . We normalizeagents’ utilities so that being unmatched yields 0 utility for every agent and assume that w mi > m ∈ N i . We eliminate all matchings that are not individually rationaland therefore, we assume w j ( i ) i ≥ i and j . For notational convenience, we set w mi = 0 if m N i .We identify the following competitive market with this matching problem: each agenthas one unit of fiat money and agent i must pay price π mi for matching with agent m .The price of remaining unmatched π ii is zero for all agents. For notational convenience, wealso set π mi = 0 for m N i . Let π i = ( π i , . . . , π ni ) be the corresponding price vector andlet π = ( π , . . . , π n ). A commodity, in this setting, is an ordered pair ( i, m ) denoting i ’smatch with m . Since we allow randomization, i ’s consumption of ( i, m ) can vary between0 and 1. Thus, i ’s consumption set is: Z i = n ξ i ∈ IR n + (cid:12)(cid:12)(cid:12) ξ i · e ≤ ξ mi > m ∈ N i o Then, given prices π , each agents solves the following maximization problem: W i ( p ) = max ξ i ∈ Z i w i · ξ i subject to π i · ξ i ≤ ξ i is a minimum cost solution to the consumer’s problem if it solves theabove maximization problem and no other solution costs less than ξ i at prices π i .From the firm’s perspective, the revenue of matching j is the sum of the prices ofthe individual matches; that is, P ni =1 π j ( i ) i . Therefore, the firm solves the maximizationproblem: R ( π ) = max q ≥ X j ∈ K n X i =1 π j ( i ) i · q j subject to q · e ≤ ξ = ( ξ , . . . , ξ n ) be a vector of consumer choices. The triple ( π, ξ, q ) is a (strong) Wal-rasian equilibrium of the matching market v if ξ i is a minimum cost solution to consumer i ’s problem (M1), q solves the firm’s maximization problem (M2) and for all i, mξ mi = X { j | j ( i )= m } q j (M3)15quation (M3) is the market clearing condition; that is, the requirement that consumer i ’s demand for matches with m are met by the firm’s supply of such matches.The competitive economy differs from the collective choice market in the way con-sumers express their demands. In the competitive economy, consumers specify demandsfor private goods (partners m ∈ N i ) while in a collective choice market, consumers specifytheir desired collective goods (matches j ∈ K ). To map the matching market into a col-lective choice market, define u i such that u ji = w j ( i ) i . A Lindahl equilibrium ( p, q ) specifiesa distribution over matches q and a price p ji that consumer i must pay for matching j .Below, we relate Walrasian equilibria of the competitive economy to the Lindahl equilibriaof the collective choice market.For any price p , let π mi ( p ) = (cid:26) min { j : j ( i )= m } p ji if m ∈ N i π i ( p ) be the corresponding price vector and note that it assigns each partner m ∈ N i the price of the least cost match j that yields this partner. For any allocation q , define ξ i ( q ) such that ξ mi ( q ) = P { j : j ( i )= m } q j . Thus, ξ i ( q ) is the partner assignment implied bythe allocation q . Theorem 3: If ( p, q ) is a Lindahl equilibrium collective choice market, then ( π ( p ) , ξ ( q ) , q ) is a Walrasian equilibrium of the competitive economy. If ( π, ξ, q ) is a Walrasian equilib-rium of the competitive economy, then ( p, q ) such that p ji = π j ( i ) i is a Lindahl equilibriumof the collective choice market. To see why the second part of Theorem 3 is true, note that every agent’s optimizationproblem in the collective choice market is equivalent to their corresponding optimizationproblem in the Walrasian setting. Condition (M3) implies that q achieves the desireddistribution of partners. Moreover, since ξ i is a least cost solution for every consumer inthe competitive setting, so is q in the collective choice setting.To see why the first part of Theorem 3 is true, note that in a Lindahl equilibriumconsumers choose minimum cost solutions to their optimization problems. Therefore, j ( i ) = j ′ ( i ) and p ji > p j ′ i implies that q j = 0. It follows that ξ mi ( q ) must be a least cost16olution to the i ’s optimization problem at prices π i ( p ). For the same reason, the firmrevenue of any matching j such that q j > j such that q j = 0, the firm revenue is no greater than in the collectivechoice market and, therefore, q must be optimal for the firm.Since we have established the existence of a Lindahl equilibrium, Theorem 3 impliesthat Walrasian equilibria of the matching market exist. Corollary 1, above, shows that theNash bargaining solution of the collective choice market can be implemented as a Lindahlequilibirum and, therefore, Theorem 3 implies that the Nash bargaining solution can beimplemented as a Walrasian equilibrium of the competitive economy. In the discrete good allocation problem, there is a finite set of goods H = { , . . . , r } to be distributed to the n agents. There is no divisible good. Agent’s i utility for bundle M is v i ( M ); we assume that v i ( ∅ ) = 0 and v i ( L ) ≤ v i ( M ) whenever L ⊂ M . We write v i ( h ) instead of v i ( { h } ). Let G ( H ) denote this general set of all such utility functions. Autility function, v i , is additive if v i ( L ∪ M ) = v i ( L ) + v i ( M )whenever L ∩ M = ∅ . The discrete allocation problem can be transformed into an exchangeeconomy by endowing each agent with one unit of fiat money and permitting randomconsumption. Let Θ i be the set of all possible random consumptions; that is, probabilitydistributions over 2 H and let P = { p : 2 H → IR + | p ( ∅ ) = 0 } be the set of all prices. Hence, we allow nonadditive (or combinatorial or package) prices.The price p is additive if p ( L ∪ M ) = p ( L ) + p ( M )whenever L ∩ M = ∅ . 17hen, given any price p , a consumer’s budget is B ( p ) := n θ i ∈ Θ i (cid:12)(cid:12)(cid:12) X M p ( M ) θ i ( M ) ≤ o and her utility maximization problem is: V i = max θ i ∈ B ( p ) X M v i ( M ) θ i ( M )A random consumption θ i is a minimal-cost solution to the utility maximization problemif it is a solution to the maximization problem above and no other solution costs less.Let ˚ A ∗ be the set of all partitions of H . The firms maximization problem is: R ( p ) = max a ∗ ∈ A ∗ X M ∈ a ∗ p ( M )A (feasible) allocation is a = ( M , . . . , M n ) ⊂ H r such that M i ∩ M l = ∅ whenever i = l ;a random allocation is a probability distribution over allocations. For any allocation, a = ( M , . . . , M n ), a i denotes the i ’th entry of a ; that is, M i . Let K be the set of allallocations and Θ be the set of all random allocations. For any random allocation θ ∈ Θ,let θ i denote the i ’th marginal of θ ; that is, θ i ( M ) = X a : a i = M θ ( M )The pair ( p , θ ) is a (strong) Walrasian equilibrium if for all i , θ i is a minimal-costsolution to consumer i ’s utility maximization problem at prices p and P i ( p ( a i )) = R ( p )for all a such that θ ( a ) >
0. It is easy to show that an arbitrary exchange economy, v ∈ G n ( H ) may have no Walrasian equilibrium in additive prices. Gul, Pesendorfer andZhang (2020) show that every gross substitutes economy v = ( v , . . . , v n ) ∈ GS n ( H ) has aWalrasian equilibrium with additive prices. Let W( v ) be the set of Walrasian equilibriumutility vectors for v .The bargaining problem associated with the discrete allocation economy v is B v :=coco ( { ( v ( a ) , . . . , v n ( a )) | a ∈ K } ∪ { o } ). Clearly, each allocation problem yields a unique Gross substitutes utilities are a class that includes additive utilities. n agents are all endowed with nothing other than a unit of fiat money. Nevertheless, theset of Walrasian equilibrium payoffs depends on how the goods are defined. The examplebelow illustrates this point. Office allocation example:
Three agents must decide on an office allocation. One officewill be equipped with a premium desk, one with a good desk, and one with a standard desk.We will offer two descriptions for this allocation problems that yield the same bargainingproblem but different Walrasian outcomes. The second description will yield all Lindahlequilibria as a Walrasian equilibrium outcome.In the first version, the corresponding allocation problem has three goods and allagents have unit-demands; that is, for M = ∅ , we have v i ( M ) = max h ∈ M v i ( h ) where v (1) = 10 , v (2) = 4 , v (3) = 2 v (1) = 10 , v (2) = 7 , v (3) = 3 v (1) = 10 , v (2) = 5 , v (3) = 1Thus, good 1 is the office with the premium desk, good 2 is the office with the good deskand good 3 is the office with the standard desk. This economy has a unique Walrasianequilibrium in which good 2 is allocated to agent 2 and agents 1 and 3 each have an equalchance at getting good 1 or 3. Hence, the equilibrium utilities are v = 6 , v = 7 , v = 5 . v ( { , } ) = 10 , ˆ v ( { , } ) = 4 , ˆ v (1) = 2ˆ v ( { , } ) = 10 , ˆ v ( { , } ) = 7 , ˆ v (2) = 3ˆ v ( { , } ) = 10 , ˆ v ( { , } ) = 5 , ˆ v (3) = 1This economy has many Walrasian equilibria and it can be shown, by appealing to Corollary1, that the set of Walrasian equilibrium payoffs of this economy is the same its set of Lindahlequilibrium payoffs. Notice that B ˆ v = B v and, thus, the two descriptions above representthe same economy in terms of achievable utility profiles. This example show that it ispossible to commodify a bargaining problem B in two different ways ( B v = B ˆ v = B )leading to two distinct sets of Walrasian equilibria ( W ( v ) = W (ˆ v )).Theorem 4, below, generalizes the example. It shows that the set of Walrasian equi-librium payoffs of any exchange economy are contained in the set of Lindahl equilibriumpayoffs of the corresponding collective choice market. Moreover, every bargaining prob-lem can be commodified (with additive utilities if n = 2) so that the set of Walrasianequilibrium payoffs with additive prices and the set of Lindahl equilibrium payoffs are thesame. Theorem 4: (i) W ( v ) ⊂ L ( v ) for all v ∈ G n ( H ) . (ii) For all B ∈ B o , there is v ∈ G n ( H ) such that B v = B and W ( v ) = L ( v ) ; for n = 2 , the preceding statement holds with additiveutilities and additive prices. Theorem 4 and the example preceding it, reveal an advantage of Lindahl equilibriaover Walrasian equilibria. Lindahl equilibria depend only on the bargaining game while twodifferent exchange economies that yield the same bargaining game may have two differentsets of Walrasian equilibria. At the same time, Walrasian equilibria are simpler thanLindahl equilibria because the former involve many fewer prices. This is so because thenumber of allocations typically exceeds the number of goods and because Lindahl prices arepersonal while Walrasian prices are not. However, if the commodity space is rich enough,as is the commodity space we construct in the proof of Theorem 3, the distinction betweenLindahl equilibrium and Walrasian equilibrium disappears.20ote that commodification does not modify the firm’s technology. Though we al-low non-linear pricing, we retain the assumption of an exchange economy. By contrast,the standard mapping used in the proof of Lemma 1 introduces a new technology thatconstrains the firm so that only feasible allocations can be “produced.”
6. Conclusion
To achieve equitable outcomes it is often necessary to randomize. In practice, thisrandomization often occurs ex ante to give priority to agents while the mechanism remainsdeterministic. For example, when allocating offices, an organization may first randomlydetermine a priority order and then ask members to sequentially choose their preferredoffice. As Hylland and Zeckhauser (1979) point out, such mechanisms lead to ex anteinefficiency. As an alternative, they propose a market mechanism in which agents aregiven a budget of fiat money and choose lotteries over the available offices. The Walrasianmechanism proposed by Hylland and Zeckhouser is efficient and, if agents have identicalbudget, equitable but limited in its applicability. Gul, Pesendorfer and Zhang (2019)extend Hylland and Zeckhauser’s approach from unit demand to multi unit demand withgross substitutes utilities. As we show in that paper, demand complementarities createexistence problems for the standard market mechanism. Moreover, externalities and publicgoods render it inefficient. By contrast, the collective choice markets proposed in thecurrent paper are broadly applicable to all discrete allocation problems and always efficient.In a collective choice market, each agent expresses her demand for social alternativesrather than private outcomes. On the one hand, this allows us to deal with a muchbroader range of applications but, on the other hand, the number of social alternative canbe large and, therefore, the collective choice market may be too unwieldy to implement inpractice. For the case of matching markets, we show that Lindahl equilibria coincide withstandard competitive equilibria and, therefore, each agent need only consider the set ofpossible partners and not the set of possible matchings (allocations) when formulating herdemand. An important direction for future research is to examine other circumstances inwhich a smaller set of markets (and prices) suffices to implement Lindahl equilibria.21 . Appendix
Consider the following modified economy with private goods: let X = { ( x, β ) = ( x , . . . , x k , . . . , x n , . . . , x kn , β ) ∈ IR kn +1 } be the commodity space. There are n + 1 agents and every agent has endowment zero.Agent i ’s, for i = 1 , . . . , n , consumption set is D i = { ( x, β ) ∈ X | x ≥ , x jl = 0 , l = i, β ≥ − } and her utility function is U i ( x, β ) = P kj =1 u ji x ji . Agent n + 1 has the consumption set D n +1 = n ( x, β ) ∈ X (cid:12)(cid:12)(cid:12) n X j =1 min i x ji ≥ − , β ≥ o and her utility function is U n +1 ( x, β ) = β .An allocation ( x , β , . . . , x n +1 , β n +1 ) is feasible if ( x i , β i ) ∈ D i for all i , P n +1 i =1 x i ≤ P n +1 i =1 β i ≤
0. It is easy to verify that the set of feasible allocations is compact,the consumption sets are convex and bounded below, and the utility functions are quasi-concave, continuous and locally non-satiated. Let ˜ p be any price such that the utility ofevery agent is bounded. Then, ˜ p β > p ji > u ji >
0. It is straightforward toverify that at any ˜ p that satisfies these conditions there exists, for every i = 1 , . . . , n + 1,( x i , β i ) , ( x ′ i , β ′ i ) ∈ D i such that ˜ p · ( x i , β i ) < ˜ p · ( x ′ i , β ′ i ) ≤
0. Thus, the conditions of theequilibrium theorem (pg 12) in Gale and Mas-Colell (1975) are satisfied and the economyhas a Walrasian equilibrium. In that Walrasian equilibrium ˜ p β > p β = 1. Local non-satiation of utilities implies that theequilibrium allocation is Pareto efficient and that utility maximizing consumptions areleast-cost. Pareto efficiency implies that P kj =1 min i x jin +1 = −
1. If x jmm + min i x jin +1 < m ∈ { , . . . , n } , then utility maximization of agent n + 1 implies ˜ p jm = 0 and,therefore, we can modify the allocation so that x jmm = x jii for all j, m ∈ { , . . . , n } and22aintain all equilibrium conditions. Setting q = ( x , . . . , x k ) , p i = ( ˜ p i , . . . , ˜ p ki ) thenyields the desired Lindahl equilibrium. Proof of Lemma 2:
Let x ∈ E ( B ). It follows that there is a simplex A ≥ B suchthat the fair solution of A is x . Since A ≥ B it follows that x is Pareto efficient and that d ( A ) ≥ d ( B ) and b ( A ) ≥ b ( B ). Thus, x = d ( A )+( b ( A ) − d ( A )) / ≥ d ( B )+( b ( B ) − d ( B )) / x ∈ B be Pareto efficient and satisfy x ≥ d ( B ) + ( d ( B ) − b ( B )) / E satisfies scale invariance, we may assume d ( B ) = o and b ( B ) = (1 , x is Pareto efficient and B ∈ B , there is a = ( a , a ) > o such that a · x ≥ a · y forall y ∈ B . Since x > o , we have ax > a x , ax > a x . Moreover, we must have(2 a x ≥ ax ≥ a x ) or (2 a x ≥ ax ≥ a x ). Without loss of generality, assume thelatter holds. Let d = ax/a − x and note that d ∈ [0 , x ). Then, let d = ( d , b = ( a · ( x − d ) /a , a · ( x − d ) /a ) + d and A = conv { d, ( b , , ( d , b ) } . It is straightforwardto verify that A ≥ B and F ( A ) = x and, therefore, x ∈ E ( B ), as desired.Call A ∈ B a simplex if A = a × ∆ + b for some a such that a i > i and b suchthat b i ≥ i . We say that the simplex A supports B at x with (exterior normal) w if x ∈ B ⊂ A , d ( B ) = d ( A ) and wy ≤ wx for all y ∈ A . Lemma A1: (i) For all B there is a simplex A that supports B at η ( B ) with ∇ f B ( x ) .(ii) For any simplex A , η ( A ) = n b ( A ) + n − n d ( A ) . Proof:
The proof of (ii) is straightforward and omitted. Assume d ( B ) = o , let x = η ( B )and let w = ∇ f B ( η ( B )). Since B is convex it follows that wy ≤ wx for all y ∈ B . Since x i > i it follows that w i = Q k = i x k > i . Let wx/w i = nx i . Then, thesimplex A = conv { o, nx e , . . . , nx n e n } supports B at η ( B ) with ∇ f B ( η ( B )).For arbitrary B , let C = B − d ( B ). By the preceding argument, there is a simplex A that supports C at η ( C ) with w = ∇ f C ( x ). Then, note that η ( B ) = η ( C ) + d ( B ), ∇ f B ( η ( B )) = ∇ f C ( η ( C )) and therefore, A + d ( B ) supports B at η ( B ) with ∇ f B ( η ( B )). Proof of Lemma 3:
First, we show that N ( B ) ⊂ E ( B ): take x ∈ N ( B ). Hence, x ∈ B and x = η ( A ) for some A such that B ≤ A . By Lemma A1, there is a simplex C that23upports A at x = η ( A ) with ∇ A f ( η ( A )). Since C is a simplex, { x } = { η ( A ) } = F ( C ).Hence, B ≤ A ≤ C and x ∈ F ( C ), therefore, x ∈ E ( B ).Next, we show that E ( B ) ⊂ N ( B ). If x ∈ E ( B ), then there is A such that { x } = F ( A ).Hence, x = η ( A ) and B ≤ A , therefore x ∈ N ( B ). Proof of Theorem 1:
That E satisfies scale invariance is obvious. If A and B aresimplices such that B ≤ A and F ( A ) ⊂ B , then A = B . Therefore E ( A ) = F ( A ) for everysimplex A . Since F (∆) = { n e } , it follows that E (∆) = { n e } , as desired.To prove consistency, let B ≤ A and x ∈ E ( A ) ∩ B . Then, there exists C suchthat A ≤ C and F ( C ) = { x } . Hence, B ≤ C and therefore x ∈ E ( B ), as desired. Toprove justifiability, assume x ∈ E ( B ). Then, there is A such that B ≤ A and x ∈ F ( A ).Hence, A is a simplex and therefore E ( A ) contains a single element x . That is, E satisfiesjustifiability.Next, assume that S satisfies the axioms. We will show that S = E . First, note thatscale invariance and symmetry imply that S ( A ) = F ( A ) for every simplex A . It followsthat any x ∈ E ( B ) is justifiable, that is, for any x ∈ E ( B ) there is B ≤ A such that { x } = S ( A ). Since E satisfies scale invariance, symmetry and consistency, we concludethat E ( B ) ⊂ S ( B ) for all B ∈ B . Suppose x ∈ S ( B ). Then, { x } = S ( A ) for some A such that B ≤ A . Since E ( A ) ⊂ S ( A ), we must have E ( A ) = { x } and since E satisfiesconsistency, x ∈ E ( B ) as desired. Let p be a price. In a collective choice market the consumer solves U i ( p ) = max q u i · q subject to p i · q ≤ , e · q ≤ µ ,µ ≥ µ + µ subject to µ e + µ p i ≥ u i (D)24he vector ( q, µ , µ ) is feasible if q satisfies the constraints of (P) and ( µ , µ ) satisfies theconstraints of (D). A feasible vector ( q, µ , µ ) is optimal (that is, q solves (P) and µ , µ solves (D)) if and only if µ ( q · e −
1) = 0 µ ( q · p i −
1) = 0for all j, q j ( µ + µ p ji − u ji ) = 0 (CS)Let J ( q ) = { j | q j > } . For any utility u i and c i < max j u ji , let ¯ u ji ( c i ) = max { , u ji − c i } .Note that ¯ u i ( c i ) is a utility. Lemma A2:
Let q ∈ Q be a minimal cost solution to consumer i ’s maximization problemfor utility u i and prices p such that q · p i = 1 . There are c ≥ and α > such that αp ji ≥ u ji − c for all j and αp ji = u ji − c for j ∈ J ( q ) . Moreover, q is a minimal cost solutionto consumer i ’s maximization problem for utility ¯ u i ( c ) and prices p . Proof:
Let µ , µ be the associated solution of the dual (D). First, consider the case inwhich u ji = u mi for some j, m ∈ J ( q ). Then, (CS) implies µ >
0. Set c = µ , α = µ .Feasibility and (CS) imply that αp ji ≥ u ji − c with equality if j ∈ J ( q ), as desired.It remains to show that q is a minimal cost solution to the consumer’s maximizationproblem given utility ¯ u i ( c ). Since ( q, µ , µ ) is feasible (for utility u i ), we have αp ji ≥ u ji − c for all j . We also have αp ji ≥ ¯ u i ( c ) for all j , since the left-hand side is non-negative.The first part of the Lemma implies u ji − c = ¯ u ji ( c ) for all j ∈ J ( q ) and, therefore, q j ( αp ji − ¯ u i ( c )) = 0 for all j . Hence, ( q, µ , µ ) such that µ = 0 , µ = α is a feasiblesolution for (P) and (D) that satisfies (CS) for utility ¯ u ( c ); that is, q solves consumer i ’smaximization problem given utility ¯ u ji ( c ). Since u ji = u mi for some j, m ∈ J ( q ), we have¯ u ji ( c ) = u ji − c = u mi − c = ¯ u mi ( c ) and therefore, this solution must be minimal cost.Second, consider the case where β = u ji = u mi for all j, m ∈ J ( q ). Since q is aminimal cost solution and q · p i = 1, we must have p ji = p mi = 1 for all j, m ∈ J ( q ). Let µ , µ be the associated solution of the dual (D). If µ >
0, set α = µ and repeat theargument above to conclude that q is a solution maximization problem given utility ¯ u i ( c )and ¯ u ji ( c ) = u ji − c = αp ji = α > j ∈ J ( q ). Then, since q is a minimal cost solutionfor u i implies q is a minimal cost solution for ¯ u i ( c ).25f µ = 0, set c = max { j | p j < } u ji if { j | p j < } 6 = ∅ and c = 0 otherwise and set α = β − c . Note that β − c > q is a minimal cost solution. Then, c + αp ji ≥ c ≥ u ji if p ji < c + αp ji ≥ µ ≥ u ji if p ji ≥
1. Therefore, αp ji ≥ u ji − c for all j and withequality if j ∈ J ( q ). Then, note that ( q, c, α ) is a feasible vector that satisfies (CS) andhence q is a solution to the consumer’s optimization problem. Since β − c > q is aminimal cost solution to the optimization problem for u i , it must also be a minimal costsolution for ¯ u i ( c ). Lemma A3:
If there is λ = ( λ , . . . , λ n ) ≥ o such that v i = u i + ( λ i , . . . , λ i ) for all i ,then L ( u ) + λ ⊂ L ( v ) . Proof:
Suppose such a λ exists and choose x ∈ L ( u ) and let ( p, q ) be a Lindahl equilibriumof u that yields the payoff vector x . Then, for all i and ˆ q such that e · ˆ q ≤ u i · ˆ q ≤ u i · q = v i · q − λ i and v i · ˆ q − λ i ≤ u i · ˆ q . It follows that q is a minimal cost solution to consumer i ’smaximization problem in v . Clearly, q is a solution to the firm’s maximization problem inthe collective choice market v and hence ( p, q ) is a Lindahl equilibrium of v and therefore, x + λ ∈ L ( v ). Proof of Theorem 2:
First, we show that x ∈ E ( B u ) implies x ∈ L ( u ). Let x ∈ E ( B u ).Then, there is a simplex A = na ⊗ ∆ + d such that B u ≤ A and x = a + d ∈ B u . Assumefor now, that d = o so that d ( A ) = d ( B u ) = o . Then, B u ⊂ A . For any j ∈ K , let y j = ( u j , . . . , u jn ). Since y j ∈ B u ⊂ A , it is a unique convex combination of the extremepoints of A . Let z ji be the weight of na i e i in that convex combination and set p ji = nz ji .Note that z ji > p ji > j ∈ K such that p ji >
0, we have u ji /p ji = a i . Since x ∈ B u , it is a convex combination of the extreme points of B u . Let q j be the weight of y j in that convex combination. By construction, u i · q = a i . Note thatfor any ˆ q such that ˆ q · p i ≤ u i · ˆ q = K X j =1 a i p ji · ˆ q j ≤ a i with equality only if ˆ q · p i = 1. Therefore, q is a minimal cost solution to the consumer’sproblem and ( p, q ) is a Lindahl equilibrium.26f o = d ( B u ) = d ; that is, d ( B u ) ≤ d ( A ), then repeat the above construction for A ′ = A − d to show that x − d is a Lindahl equilibrium payoff vector of the economy v such that v ji = u ji − d i . Then, Lemma A3 ensures that x is a Lindahl equilibrium payoffvector for the economy u .Next, we show that x ∈ L ( u ) implies x ∈ E ( B u ). Let x ∈ L ( u ) and let ( p, q ) be thecorresponding Lindahl equilibrium outcome. First, consider the case in which p i · q = 1 forall i . By Lemma A2, there is, for each i , some c i ≥ α i > u ji − c i ≥ α i p ji for all j with equality for j ∈ J ( q ). Furthermore, Lemma A2 implies that ( p, q ) is also aLindahl equilibrium for the collective choice market (¯ u i ( c i )) ni =1 . Since ¯ u ji ( c i ) = α i p ji for all j ∈ J ( q ) and p i · q = 1, we have ¯ u i ( c i ) · q = α i p i · q = α i and x i = c i + α i . Let z ∈ B u andlet ˆ q be such that z i = u i · ˆ q .Since α i p ji ≥ u ji − c i for all i, j , we have α i p i · ˆ q ≥ z i − c i for all i or p i · ˆ q ≥ ( z i − c ) /α i .Note that the firm’s profit cannot be greater than n , the aggregate endowment of money.Therefore, firm optimality implies n ≥ P ni =1 p i · q ≥ P ni =1 p i · ˆ q . Thus, we obtain n ≥ n X i =1 z i − cα i (1)for all z i ∈ B u . Let d = ( c , . . . , c n ), a = ( α , . . . , α n ) and A = na ⊗ ∆ + d and note that(1) implies A ≥ B u . Clearly, F ( A ) = a + d = x and, therefore, x ∈ E ( B u ).Finally, consider the case in which p i · q < i . Let I be the set of all suchagents. Let J ∗ i = { j | u ji ≥ u mi ∀ m } be the bliss outcomes for i . If p i · q < i ∈ I , then J ( q ) ⊂ J ∗ i since otherwise i is not choosing a utility maximizing plan. Furthermore, since q is a minimal cost solution for consumer i ∈ I , p ji = p mi for all i ∈ I and j, m ∈ J i ( q ).Define ¯ p = ( ¯ p , . . . , ¯ p n ) as follows: ¯ p ji = 1 if i ∈ I and j ∈ J i and ¯ p ji = p ji otherwise.It is easy to see that ( ¯ p, q ) is a Lindahl equilibrium. Moreover, every consumer satisfies¯ p i · q = 1 for all i . Thus, we can apply the argument above to show that x ∈ E ( B u ). Proof of Corollary 1:
For c ∈ C ( u ), define H i ( c i ) = conv { e j | u ji ≥ c i , j ∈ K } andnote that ¯ u ji ( c ) · q + c = u ji · q if and only if q ∈ H i ( c i ). Let H ( c ) = T ni =1 H i ( c i ). Let q bethe Nash allocation for ¯ u ( c ). Note that c is admissible if and only if q ∈ H ( c ).To prove the first statement of the corollary, assume c is admissible and q is a Nashallocation for ¯ u ( c ). Hence, q ∈ H ( c ) and for all i , ¯ u ji ( c i ) > j and, therefore, the27onstraint q · e ≤ λ > n X i =1 ¯ u ji ( c i )¯ u i ( c i ) · q ≤ λ ( F )and n X i =1 ¯ u ji ( c i ) q j ¯ u i ( c i ) · q = λq j Then, q · e = 1 implies λ = n . Set µ i = c i , µ i = ¯ u i ( c i ) · q and p i = ¯ u i ( c i ) / (¯ u i ( c i ) · q ).Hence, p i · q = 1 and q ∈ H i ( c i ) and therefore, µ i + µ i p ji ≥ u ji for all j . Moreover, equalityholds for all j such that q j >
0. That is, ( µ i , µ i , q ) is a feasible solution that satisfies(CS) and, therefore, is optimal. It remains to show that q solves the firm’s problem atprices ( p , . . . , p n ). Since λ = n , inequality (F) implies that P ni =1 p ji e ji ≤ n for all j and,therefore, firm profit cannot exceed n . Since q yields profit n , it maximizes profit.To prove the second statement of the corollary, assume that ( p, q ) is a Lindahl equi-librium. Then, consumer optimality implies that there are µ i , µ i such that µ i + µ i p ji ≥ u ji for all j where equality holds for all j such that q j >
0. Therefore, µ i + µ i = u i · q and µ i ≥ ¯ u ji ( µ i ) for all j with equality if q j >
0. Summing over all i we obtain n X i =1 ¯ u ji ( c i ) µ i = n X i =1 ¯ u ji ( c i )¯ u i ( µ i ) · q ≤ n ( F )with equality if q j >
0. Thus, q is the Nash allocation for ¯ u ( c ). Moreover, if e j H i ( µ i )then µ i + µ i p ji ≥ µ i > u ji and, therefore, q j = 0. Hence q ∈ H i ( µ i ) = H i ( c i ) for all i . Lemma A1(i) yields a simplex A = conv { z e , . . . , z n e n } ( z i > i ) that supports B w at η ( B w ) with ∇ f B w ( η ( B w )). Hence, η ( A ) = η ( B w ) and d ( B ) = d ( A ) = o and therefore, by Lemma A1(ii), η ( B w ) = n e ⊗ z . Let q be anyprobability distribution over K such that P j q j ( w j , . . . , w jn ) = η ( B w ).Each x ∈ B w and is a unique convex combination of the extreme point of A . Thatis, x i = λ i · z i for λ , . . . , λ n ≥ P i λ i ≤
1. Let λ i ( x ) be the weight of z i in that convex combination. For x = ( w j , . . . , w jn ), let p ji = λ i ( w ji ) and let π mi = p ji for28ome j such that j ( i ) = m . Clearly, λ ( w ji ) = λ ( w j ′ i ) whenever j ( i ) = j ′ ( i ). Hence, π mi iswell-defined. Verifying that ( π, q ) is a WE and ( p, q ) is a LE is straightforward.To conclude the proof, we will show that any WE is an LE. Let ( π, q ) be an WE andlet p ji = π j ( m ) i . Then, clearly, ( p, q ) is a WE. Proof of Theorem 4:
Let ( p , θ ) be a WE and let K = { a (1) , . . . , a ( k ) } be the set of allfeasible allocations. Then, for every feasible allocation a ( j ), let p ji = X h ∈ a i p h Then, it is easy to verify that ( p, θ ) is an LE.To prove the second assertion, let n = 2 and y = ( y , y ) , . . . , y k = ( y k , y k ) be all ofthe vertices on the Pareto frontier of B ∈ B o ordered so that y < y < . . . < y k . Firstconsider the case in which y = 0 and y k = 0. Then, let H = { , . . . , k − } , v ( h ) = y h +11 − y h and v ( h ) = y h − y h +12 for h ∈ H . For ∅ 6 = M ⊂ H , let v i ( M ) = P h ∈ M u i ( h )for i = 1 ,
2. Hence, v i is additive.Since f ( h ) = y h − y h +12 y h +11 − y h is an increasing function, in any efficient allocation of theeconomy v , there is l such that agent 2 consumes M ( l ) = { , . . . , l } and agent 1 consumes M ( l ) = { l + 1 , . . . k − } . Hence, B v = B . A random allocation is Pareto efficient ifand only if it has at most two elements in its support, and hence is of the form a ( l ) =( a ( l ) , a ( l )) = ( M ( l ) , M ( l )) with probability γ ∈ (0 ,
1] and a ( l + 1) = ( a ( l + 1) , a ( l +1)) = ( M ( l + 1) , M ( l + 1)) with probability 1 − γ .Let x ∈ B v be a Lindahl equilibrium payoff vector. Hence, q l the probability ofthe allocation a ( l ), is γ and q l +1 = 1 − γ for a ( l ) and a ( l + 1) as above. By Lemma 2, x ≥ b ( B v ) /
2; in particular, x ≥ y k /
2. Since x is Pareto efficient, there is a ∈ IR suchthat ax ≥ az for all z ∈ B v . We must have a x ≥ a x or a x ≥ a x . Without loss ofgenerality, assume the latter holds. Then, let p h = v ( h ) /x for all h ≤ l if γ = 1 and forall h ≤ l + 1 (if γ < a ( l ) with probability γ and a ( l + 1) with probability 1 − γ is just affordable for agent 1. Moreover, agent 1’sutil per price of the goods she is to consume is constant at 1 /x .29ote that since x ≥ y k /
2, if we were to set the price, p h , of each of the remaininggoods to v ( h ) /x , agent 2 could purchase the random consumption that yields a ( l ) withprobability γ and a ( l + 1) for less than 1, her endowment of fiat money.Conversely, if we were to set p h for each of the remaining goods to a v ( h ) /a x , agent2 would not be able to afford the random consumption that yields a ( l ) with probability γ and a ( l + 1) if a x > a x . Let I h = h v ( h ) x , a v ( h ) a x i . Since a /a is the slope of thetangent line through the point x , I h = ∅ for all h > l + 1 if γ < h > l if γ = 1.Note also that if γ <
1, the interval I l +1 is a single point. Hence, we can choose p h ∈ I h for all such h ≥ l of γ < h ≥ l + 1 if γ = 1 so that agent 2 can just afford therandom consumption ( γ, a ( l ); 1 − γ, a ( l + 1)). Moreover, agent 2’s util per price of thegoods he consumes in no greater than a /a x .Since the slope of f is increasing, agent 1’s util per price for goods h ≥ l + 1 is greaterthan 1 /x while agent 2’s util per price of goods h ≤ l is greater than a v /a x . It followsthat ( γ, a i ( l ); 1 − γ, a i ( l + 1)) is a minimal cost solution to the maximization problem ofagent i . Hence, x ∈ W ( v ).If y k − >
0, then let H = { , . . . , k } and define the utility of the new good k as follows: v ( k ) = 0 and v ( k ) = y k − . If y >
0, then add a new good 0, (or another new good 0,if k has already been added) to H such that v (0) = y and v (0) = 0. Then, repeat theconstruction above, after extending each v i to all subsets of H additively to derive, onceagain, a WE that yields x as its payoffs.For the general case; that is, for n ≥ B ∈ B o , we define thecommodity space as follows: let X be the set of Pareto efficient extreme points of B .Then, let X i be the projection of X to the i ’the coordinate; that is, X i = { x i | x ∈ X }\{ } and define the set of possible partial payoff vectors as X = ( X ∪ {− } ) × ( X ∪ {− } ) × · · · × ( X n ∪ {− } )For y ∈ X , x i ∈ X i let ( y − i , x i ) ∈ X be the vector in which the i − coordinate of y isreplaced by x i . Define the set of consistent partial payoff vectors and minimally inconsistentpartial payoff (MIPP) vectors as follows: X + = { x ∈ X | there is y ∈ X such that x i ∈ {− , y i } for all i } X − = { x ∈ X \ X + | there is i such that ( x − i , − ∈ X + } x ∈ X − , let δ ( x ) be the cardinality of the set { i | x i = − } minus 1 and let Y ( x )be a set that contains exactly δ ( x ) copies of x . Define the set of goods as follows: H = [ i X i ∪ [ x ∈ X − Y ( x )Hence, H contains one copy, y i ∈ X i , of every possible payoff for each agent and δ ( x )copies of every MIPP x ∈ X − ; that is, one fewer copy than the number of possible payoffsin the MIPP vector x (i.e., entries that are not − M ⊂ H , define theutility of agent i for the bundle M as follows: for all y i ∈ X i , g i ( M, y i ) = 0 if there isˆ y i ≤ y i such that ˆ y i ∈ X i \ M or there is x ∈ X − such that x i = y i and Y ( x ) ∩ M = ∅ ;otherwise g i ( M, y i ) = 1. Then v i ( M ) = max { y i | g i ( M, y i ) = 1 } Next, we will show that B v = B . Take y ∈ X and for every i and x ∈ X − suchthat x i = y i , choose a distinct x l i ∈ Y ( x ). Then, let M i be the set consisting of eachsuch x l i i and all ˆ y i ≤ y i in X i . Clearly, v ( M i ) = y i . To see that this collection of M i ’s, for i = 1 , . . . , n , is feasible, note that if such a collection were not feasible, there would be some x ∈ X − such that the cardinality of the set { i | x i = y i } is greater than the cardinality ofthe set Y ( x ). But this contradicts the definition of δ ( x ).Thus we have shown X ⊂ B v and therefore coco ( X ∪ { o } ) = B ⊂ B v . To see that B v ⊂ B , take any collection of disjoint subsets M , . . . M n , ⊂ H . Let y i = v i ( M i ) for all i and assume that y = ( y , . . . , y n ) / ∈ B . Hence, there is x ∈ X − such that x i = y i for all i such that x i = −
1. Then, each M i must contain an element of Y ( x ), contradicting thedefinition of δ ( x ).Finally, we will show that L ( v ) ⊂ W ( v ). Let K be the set of all possible allocationsof goods for the competitive economy v . For any allocation a ∈ K , let u a i = v i ( a i ). Takeany LE random outcome q . By Corollary 1, there is an LE price vector p for q such that p ji = p li whenever consumer i receives the same bundle in allocation j and l . Hence, forany M ⊂ H , let p ( M ) = p a i if M is a minimal subset of a i such that v i ( M ) = v i ( a i ). Forall remaining M ⊂ H , let p ( M ) be the maximum of p ( L ) among all L ⊂ M for which p ( L ) has been defined above. 31hen, let θ i ( q ) be the random consumption of player i implied by the random outcome(i.e., random allocation) q and let θ ( q ) = ( θ ( q ) , · · · θ n ( q )). Note that consumer i gets thesame utility with any ˆ q in the collective choice problem as she gets with consumption θ i and the two cost the same. Moreover, for every random consumption ˆ θ i in the competitiveeconomy, there is a random outcome ˆ q that yields the same utility and costs the same.Then, the minimal-cost optimality of θ i follows from the minimal cost optimality of q .Suppose there is some collection { M , . . . , M l } ∈ A ∗ such that P kk =1 p ( M l ) > R ( p ).First, we note that from the definition of v i it follows that k = m implies v i ( M k ) · v i ( M m ) =0 for all i . Hence, there is a collection { M , . . . M n } ⊂ { M , . . . , M l } ∪ {∅} such that forany M k not in this collection, v i ( M k ) = 0 for all i . Hence, p ( M k ) = 0 for all such M k , a := ( M , . . . , M l ) ∈ A and P i p ( M i ) > R ( p ). But, p ( M i ) = p a i for all i , contradictingthe firm-optimality of ( p, q ) in the Lindahl equilibrium.32 eferences Fujishige, S. and Z. Yang (2003) “A Note on Kelso and Crawford’s Gross SubstitutesCondition,”
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