When Walras meets Vickrey
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Delacrétaz, David; Loertscher, Simon; Mezzetti, Claudio Article When Walras meets Vickrey Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Delacrétaz, David; Loertscher, Simon; Mezzetti, Claudio (2022) : When Walras meets Vickrey, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 17, Iss. 4, pp. 1803-1845, https://doi.org/10.3982/TE4296 This Version is available at: https://hdl.handle.net/10419/296400 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/
Theoretical Economics 17 (2022), 1803–1845 1555-7561/20221803 When Walras meets Vickrey David Delacrétaz Department of Economics, University of Manchester Simon Loertscher Department of Economics, University of Melbourne Claudio Mezzetti School of Economics, University of Queensland We consider general asset market environments in which agents with quasilinear payoffs are endowed with objects and have demands for other agents’ objects. We show that if all agents have a maximum demand of one object and are endowed with at most one object, the VCG transfer of each agent is equal to the largest net Walrasian price of this agent. Consequently, the VCG deficit is equal to the sum of the largest net Walrasian prices over all agents. Generally, whenever Walrasian prices exist, the sum of the largest net Walrasian prices is a nonnegative lower bound for the deficit, implying that no dominant-strategy mechanism runs a budget surplus while respecting agents’ ex post individual rationality constraints. Keywords. Asset markets, efficient trade, VCG deficit, largest net Walrasian prices. JEL classification. C72, D44, D47, D61. 1. Introduction The prices set by a Walrasian auctioneer, who by assumption knows the demand and supply functions, are the same for the buyer and the seller of any given object traded. They balance supply and demand by ensuring that the agents’ optimal trades lead to an efficient allocation. In other words, Walrasian prices satisfy complete-information incentive compatibility and individual rationality constraints for all agents while always balancing both supply and demand, as well as the budget. However, as they rest on the David Delacrétaz: [email protected] Simon Loertscher: [email protected] Claudio Mezzetti: [email protected] The paper has benefited from comments and suggestions by three reviewers. We are also grateful for comments and feedback from Leslie Marx, Ilya Segal and audiences at the 2019 Transatlantic Theory Workshop, the 2019 Lisbon Meetings, Stanford University, the University of Lausanne, and the University of Melbourne. Financial support through a visiting research scholar grant from the Faculty of Business and Economics at the University of Melbourne, the Australian Research Council Grant DP200103574, and the Samuel and June Hordern Endowment is gratefully acknowledged. Anand Kannan Bharadwaj and Bing Liu provided excellent research assistance. Mezzetti’s work was funded in part by Australian Research Council grant DP120102697. ©2022 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4296
1804 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) assumption that the market maker knows the agents’ supply and demand functions, a long standing criticism has been that they fail to provide the agents with the incentives to reveal the information about values that is required to set market clearing prices in the first place.1 The Vickrey–Clarke–Groves (VCG) mechanism achieves this feat by endowing all agents with dominant strategies to report their valuations truthfully. In general, VCG transfers are nonuniform and do not balance across agents that trade objects with each other. Moreover, for a large domain of problems, the VCG mechanism, while inducing an efficient allocation, also generates a deficit for the market maker. These fundamental differences between Walrasian and VCG prices are not surprising, given that they solve fundamentally different problems—market clearing under complete information about values and truthful revelation of values under private information, respectively. In this paper, we show that there is a deep and tight connection between Walrasian prices and VCG transfers. We study general trading environments in which agents with quasilinear payoffs may be endowed with objects that they value and have demands for other agents’ objects; hence, each agent may sell some objects and buy other ones. Define the net price of an agent in any Walrasian price vector as the sum of prices of the objects he sells minus the sum of prices of the objects he buys. If all agents are singleobject traders, that is, have a maximum demand of one object and are endowed with at most one object, our first main result—Theorem 1—states that the largest net price that an agent receives in any Walrasian price vector is equal to the VCG transfer he receives. As a consequence, the sum of the largest net Walrasian prices over all agents equals the VCG deficit. Intuitively, for each agent, the largest net Walrasian price corresponds to the best terms of trade offered by any Walrasian price vector. With single-object traders, each agent’s largest net Walrasian price is equal to his externality on the other agents, which by definition is his VCG transfer. Unless all agents have additive payoffs, Theorem 1 does not extend to environments with multiobject traders because Walrasian prices are individual to each object and, unlike VCG transfers, do not necessarily represent the social value of a bundle of objects. However, our second main result—Theorem 2—states that, as long as a Walrasian equilibrium exists, the relationship remains as a lower bound: each agent’s VCG transfer is weakly greater than his largest net Walrasian price; hence, the sum of largest net Walrasian prices over all agents constitutes a nonnegative lower bound for the VCG deficit. These general results have several insightful corollaries in more specialized settings. Consider first what, following Shapley and Shubik (1972), may be called two-sided allocation problems. These are problems in which every agent’s trading position is independent of types and determined a priori: agents without endowments either buy or do not trade and agents with endowments either sell or do not trade. Two-sided allocation problems include the problems that motivated the papers by Vickrey (1961)andMyerson and Satterthwaite (1983).2The bilateral trade problem of Myerson and Satterthwaite is the simplest possible setting in this domain. Assuming the buyer’s value and 1See, for example, Arrow (1959). 2Shapley and Shubik (1972)calltheseproblemstwo-sided market games, but as the term “two-sided market” now has a very specific and different meaning in the Industrial Organization literature, our terminology seems preferable.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1805 the seller’s value are elements of the same compact interval, the deficit under the VCG mechanism is equal to the difference between the buyer’s and the seller’s value whenever trade is ex post efficient.3Any price between the seller’s and the buyer’s value is a Walrasian price; hence, the deficit is equal to the difference between the largest and the smallest Walrasian prices. With a homogeneous good market (in which every agent sees all objects as identical) and multiple single-object buyers and sellers, this result generalizes; the deficit under the VCG mechanism is equal to the Walrasian price gap times the quantity traded.4 An implication of Theorem 1is that these insights generalize beyond the narrow confines of homogeneous good markets. Specifically, for two-sided allocation problems with single-object traders, the result implies that the deficit under the VCG mechanism is equal to the sum of the Walrasian gaps over the objects that are traded under efficiency. The reason is that in two-sided allocation problems the largest net Walrasian price of every buyer (seller) is equal to the lowest (highest) Walrasian price for the object he trades. Put differently, for these two-sided environments with single-object traders, the—extremal—Walrasian prices provide the traders with precisely the right incentives to reveal their valuations. The subtle but important twist is that incentive compatible information revelation requires the use of two different Walrasian prices for every object that is traded, one on each side of the market, thereby generating a deficit on every object that is traded. If we still assume two-sided allocation problems but allow for buyers to have demand for multiple objects and for sellers to be endowed with more than one object, Theorem 2implies that the sum of the Walrasian price gaps over the objects traded under efficiency is a lower bound for the deficit under VCG. The remainder of this paper is organized as follows. Section 2provides an illustrative example. Section 3presents the general setup and basic concepts such as asset markets and the deficit under the VCG mechanism. Section 4introduces the concept of largest net Walrasian prices. Sections 5and 6contain the main results for singleobject and multi-object traders, respectively. Section 7analyzes in detail two important special cases, namely two-sided allocation problems and homogeneous good markets. Section 8provides a comprehensive discussion of the related literature. Section 9con- cludes the paper. Proofs are in Appendix Aand additional background material is in Appendix B. 2. An illustrative example An example is useful to illustrate how largest net Walrasian prices are calculated and how they relate to VCG transfers. Suppose there are two agents, Leon and William. Leon owns a rare book and William is endowed with a collection of stamps. Leon’s value for the book is 5 and his value for the stamp collection is 7 while William’s value for the book 3See, for example, Krishna (2002) for a proof along these lines. Myerson and Satterthwaite (1983)implicitly noted an implication of this result when they observed that, with identical supports, the subsidy that would be required for efficiency is equal to the ex ante expected welfare under efficiency. 4Our results on homogeneous good markets in Section 7generalize those of Tatur (2005) and Loertscher and Mezzetti (2019).
1806 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) is 3 and his value for the stamp collection is 2. Neither of them gets additional value from a second object. Welfare is therefore maximized when the book is allocated to William and the stamp collection to Leon, which generates a welfare of 10. The situation is summarized in the following matrix. The endowment is shown in bold face and the efficient allocation is shown with square boxes. The VCG transfer made to Leon is the difference between the welfare and his value for the good he obtains under the efficient allocation, which is 10 minus 7, and the maximum welfare without him and his endowment, which is 2. Thus, the VCG transfer Leon obtains is 1. Applying the same logic, William receives a VCG transfer of 2. Hence, the resulting deficit is 3. Consider now the set of Walrasian prices. It is not hard to see that it takes the form depicted in Figure 1,wherep1is the price of the book and p2is the price of the stamp collection.5Leon’s largest net Walrasian price is the largest difference, among all the Walrasian price vectors, between the price of the book he sells (p1) and the price of the stamp collection he acquires (p2). In Figure 1, it is equal to the vertical (or equivalently the horizontal) distance between the lowest line of slope 1 that touches the set of Walrasian prices (displayed in red) and the 45-degree line, which is equal to 1. Likewise, William’s largest net Walrasian price is the largest difference between the price for the stamp collection William sells and the book he acquires. In Figure 1, Figure 1. Panel (a): The set of Walrasian price vectors (shaded). Panel (b): The largest net Walrasian prices (indicated by arrows). 5We generalize this in Section 5(Example 1) and provide full details in Appendix B.1. As we show there, p=(p1,p2)is a Walrasian price if and only if 0 ≤p1≤3 and max{0, p1−1}≤p2≤p1+2.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1807 William’s largest net Walrasian price is equal to the horizontal or vertical distance between the highest line of slope 1 that touches the set of Walrasian prices, which is displayed in blue, and the 45-degree line. This difference is 2. It follows that each agent’s largest net Walrasian price is equal to his VCG transfer, and consequently, the sum of the largest net Walrasian prices is equal to the deficit under VCG. 3. Preliminaries We consider an asset market with a finite set of agents Awith typical element aand a finite set of objects Owith typical element o.Anallocation X=(Xa)a∈Aassigns to each agent a∈Aabundle Xa⊆O. An allocation Xis feasible if a∈AXa⊆Oand Xa∩Xa=∅for any a,a∈Awith a= a.WedenotebyXthe set of all feasible allocations. Each object, or asset, is indivisible and is initially owned by an agent. Formally, an endowment E=(Ea)a∈Ais a feasible allocation such that, for every a∈A,Eais the bundle endowed to agent awith Esatisfying a∈AEa=O.Thatis,underEevery object is allocated to exactly one agent. Being endowed with Eameans that ahas complete property rights over the objects in Ea,sothatacan exclude all other agents from consuming these objects. For every agent a∈A,leta, with typical element θa,beagenta’s type space. Denote the type space by =× a∈Aa, with typical element θ.Thevaluation (or willingness to pay) of agent awith type θafor any bundle of objects Y⊆Ois denoted by va(Y,θa). We normalize the value of the empty bundle to zero, that is, va(∅,θa)=0 for every a∈A and every θa∈a, and assume that for each type θavaluations are monotone;thatis, for every a∈A,anyY,Z⊆Owith Y⊆Z,andanyθa∈a, va(Y,θa)≤va(Z,θa). This assumption is often referred to as “free-disposal,” as it captures the idea that agents can freely dispose of any unwanted objects. Because valuations are monotone, without loss of generality we can restrict, as do Gul and Stacchetti (1999), the set of feasible allocations Xto allocations Xin which each object is assigned to exactly one agent, that is, a∈AXa=O.6As each agent ahas complete property rights over the objects in Ea,it follows that va(Ea,θa)is the value of a’s outside option when a’s type is θa. We also assume that each agent a∈Ahas a sufficiently large amount of money, say more than maxθa∈ava(O,θa), and that payoffs are quasilinear in money: if ais allocated abundleY⊆Oand receives an additional money transfer t∈R,thenhispayoff is7 va(Y,θa)+t. 6All of our results would go through if we assumed instead, like Bikhchandani and Mamer (1997), that some objects may not be allocated. In a nutshell, the reason is that any object that is not allocated in a Walrasian equilibrium must have a zero Walrasian price, and by the monotonicity of valuations, must also have a zero Walrasian price if it has to be allocated to some agent. 7The assumption that each agent has a sufficiently large money endowment is standard; see, for example, Gul and Stacchetti (1999) and Bikhchandani and Mamer (1997). The latter observed that it guarantees that the initial endowment of objects to the agents is “irrelevant for the existence of market clearing prices.”
1808 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) Fixing a type vector θ∈,thewelfare created by the allocation X∈Xis W(X,θ)= a∈A va(Xa,θa). We denote by X∗(θ)=argmax X∈X W(X,θ) the set of efficient allocations.As Xis finite, the existence of an efficient allocation is guaranteed; however, it may not be unique. We denote a typical efficient allocation by X∗(θ)∈X∗(θ).If X∗(θ)contains multiple elements, then X∗(θ)may be chosen arbitrarily among them. We denote by W∗(θ)=WX∗(θ),θ the efficient level of welfare. When there is no risk of confusion, we drop the dependency on types and write va(Y)for the value that agent aassigns to bundle Y,X∗∈X∗ for a typical efficient allocation, and W∗for the efficient level of welfare. For any I⊆Aand any K⊆O,letW∗ −I,−Kdenote the level of welfare achieved among the agents in A\Iwhen the objects in O\Kare efficiently allocated to these agents. Then W∗−W∗ −I,−K represents the joint marginal contribution of the agents in Iand the objects in K. Amechanism is a pair (χ,t),whereχ:→Xis the allocation rule and t:→R|A| is the payment rule.Thus,givenreportsθ,χ(θ)is the allocation and each agent a∈A receives ta(θ), which may be positive or negative. The social planner incurs a deficit from mechanism (χ,t)equal to the sum of the transfers that the social planner makes to the agents, that is, the deficit is 8 D(χ,t)(θ)= a∈A ta(θ). A mechanism (χ,t)is efficient if it always selects an efficient allocation, that is, if χ(θ)is efficient for every θ∈,andex post individually rational (EIR) if every agent has an incentive to participate, that is, if for all θ∈and all a∈A, vaχa(θ),θa+ta(θ)≥va(Ea,θa) holds. A mechanism (χ,t)is dominant strategy incentive compatible (DIC) if every agent has a dominant strategy to report his true type; that is, for every agent a∈Awith true type θa∈a,everyreportˆ θa∈a, and every vector of reports θ−a∈−afrom other agents, vaχa(θa,θ−a),θa+ta(θa,θ−a)≥vaχa(ˆ θa,θ−a),θa+ta(ˆ θa,θ−a). 8The revenue to the social planner from the mechanism is then −D(χ,t)(θ). As the paper focuses on settings in which the deficit is positive (hence, the revenue is negative), we refer throughout to the deficit (rather than the revenue) for simplicity.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1809 Fixing a type vector θ,theVCG mechanism (χVCG,tVCG )selects an efficient allocation χVCG ∈X∗and makes a transfer to each agent equal to his externality on other agents, that is, for all a∈A, tVCG aχVCG a=W∗ −a,−χVCG a−W∗ −a,−Ea. When ais present, he is efficiently assigned the bundle of objects χVCG aand the remaining objects in O\χVCG aare efficiently allocated among the remaining agents in A\{a}. Therefore, the first term W∗ −a,−χVCG a represents the level of welfare that agents other than aachieve when ais present. When agent ais absent, so are the objects in his endowment, and the remaining objects in O\Eaare efficiently allocated among the remaining agents in A\{a}. Therefore, the second term W∗ −a,−Earepresents the level of welfare that agents other than aachieve when aand his endowment are absent. The difference between the two is a’s externality on other agents. In the VCG mechanism, the payoff of agent ais equal to his marginal contribution: va(χVCG a)+tVCG a(χVCG a)=W∗−W∗ −a,−Ea. Note that in the VCG mechanism, if adoes not trade, then χVCG a=Easo W∗ −a,−χVCG a = W∗ −a,−Eaand areceives a transfer of 0. If aonly sells, then χVCG a⊂Easo W∗ −a,−χVCG a ≥ W∗ −a,−Eaand areceives a weakly positive transfer. If aonly buys, then Ea⊂χVCG aso W∗ −a,−χVCG a ≤W∗ −a,−Eaand areceives a weakly negative transfer. Otherwise, the sign of the VCG transfer that areceives depends on whether the bundle that asells or the bundle that abuys has the larger value to other agents. It follows that the deficit under the VCG mechanism is a∈A tVCG aχVCG a= a∈AW∗ −a,−χVCG a−W∗ −a,−Ea. We make two assumptions, which guarantee that the type space is “sufficiently rich.” First, for each agent a,allθa,θ a∈a,andallλ∈[0, 1],thereexistsθ asuch that va(Y,θ a)=λva(Y,θa)+(1−λ)va(Y,θ a)for all Y⊆O. This implies that the set of valuations Va={va(·,θa)|θa∈a}, and hence, V=× a∈AVa, is convex so that we can apply Theorem 2 in Holmström (1979), which states that a mechanism is efficient and DIC if and only if it belongs to the class of Groves mechanisms (which includes the VCG mechanism). The second “richness” assumption we impose on the type space implies that the VCG mechanism is not only efficient and DIC, but also EIR and has the lowest lumpsum transfer to each agent compatible with efficiency, DIC and EIR: For every aand every θ−a∈−a,thereexistsatypeθa(θ−a)∈asuch that X∗ a(θa(θ−a),θ−a)=Ea. The vector of VCG transfers depends on which efficient allocation the mechanism picks because the transfer that areceives depends on the bundle he is allocated, opening the possibility that the VCG deficit depends on the efficient allocation chosen. However, our next result shows that this is not the case, and hence, we may denote the deficit by DVCG without reference to the efficient allocation selected.
1810 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) Claim 1. For any two efficient allocations X∗,X∈X∗, a∈AW∗ −a,−X∗ a−W∗ −a,−Ea= a∈AW∗ −a,−X a −W∗ −a,−Ea=DVCG. Claim 1is a simple consequence of the fact that the deficit can be written as the sum of the marginal values, which are independent of the allocation. 4. Largest net Walrasian prices In this section, we introduce the concept of the largest net Walrasian price for an agent a, which will play a fundamental role in the main results of this paper. Aprice vector p=(po)o∈Ois a |O|-dimensional vector that assigns a price to each object. The price vector p=(po)o∈Ois a Walrasian price vector if there is an allocation X=(Xa)a∈Asuch that, for all a∈Aand for all Y⊆O: va(Xa)− o∈Xa po≥va(Y)− o∈Y po. If this condition is satisfied, then Xis a Walrasian allocation, supported by the Walrasian price vector p=(po)o∈O. In other words, a Walrasian price vector is such that every agent finds it optimal to purchase the bundle that the agent is assigned under the Walrasian allocation. As shown in Proposition 1 of Bikhchandani and Mamer (1997), if Walrasian prices exist, then the Walrasian allocation is efficient. Thus, the Walrasian price vector p= (po)o∈Osupports an efficient allocation X∗∈X∗. We verify next that, should there be multiple efficient allocations, Walrasian prices do not depend on which one is chosen. Claim 2. If a price vector psupports an efficient allocation, then psupports all efficient allocations. To the best of our knowledge, Claim 2was first derived by Bikhchandani and Mamer (1997) as Corollary 1 of their main result. For the purpose of keeping the paper selfcontained, we provide a direct proof in Appendix A.Claim2implies that a Walrasian price vector can be equivalently defined to be a price vector that supports all efficient allocations. Given a type vector θ∈,wedenotebyPW(θ)the set of Walrasian price vectors. Note also that the initial ownership of the objects plays no role in determining the set of Walrasian price vectors, nor does it affect the efficient allocation(s) and welfare. However, the initial ownership will matter in the VCG mechanism because tVCG a(χVCG a)= W∗ −a,−χVCG a −W∗ −a,−Eadepends on Ea. Given a price vector p=(po)o∈O,thenet price received by agent a∈Ais o∈Ea\X∗ a po− o∈X∗ a\Ea po. That is, agent ais paid for the objects he sells and pays for the objects he buys; the net price he receives is the difference between the two (which may be positive or negative).
Theoretical Economics 17 (2022) When Walras meets Vickrey 1817 Summing up over all agents a= a, the left-hand side becomes the VCG transfer W∗ −a,−X∗ a−W∗ −a,−Ea, while the right-hand side becomes agent a’s net price at the Walrasian price vector p. Because the inequality holds for all Walrasian price vectors, it holds for the one that maximizes the right-hand side. Thus, we have tVCG a≥qa. Theorem 2provides a lower bound for each agent’s VCG transfer (hence, on the deficit) based on Walrasian prices, which holds as long as Walrasian prices exist, and thus applies to a wide range of settings. Gul and Stacchetti (1999) showed that the gross substitutes condition (a formal definition of which is provided in Appendix B.2)implies that the set of Walrasian price vectors is nonempty (in fact, it forms a nonempty complete lattice). In more general settings, whether the set of Walrasian prices is nonempty (hence, whether Theorem 2applies) depends on the realization of types; that is, the gross substitutes condition is sufficient but not necessary for the existence of a Walrasian price vector. In Example 3, the valuation of a2does not satisfy the gross substitutes condition;10 yet, the set of Walrasian prices is nonempty.11 7. Two-sided allocations and homogeneous good markets In this section, we consider two popular special cases of an asset market: two-sided allocations and homogeneous good markets. We will define these formally after defining ex post buyers and sellers and showing how the largest net Walrasian price simplifies for them. Given an efficient allocation X∗∈X∗,anobjectistraded if it is efficiently assigned to an agent different from the one who is endowed with it, that is, object o∈Ois traded if o∈Ea∩X∗ afor some a,a∈Awith a= a.Wedenoteby TX∗=o∈O:o∈Ea∩X∗ afor some a,a∈Awith a= a the set of objects that are traded under the efficient allocation X∗. For any traded object o∈Ea∩X∗ a(a,a∈A,a= a), we say that asells oand abuys o.Foranyagenta∈A,we say that atrades if he sells or buys at least one object, that is, if Ea= X∗ a. Consider an object o∈T(X∗)that is sold by a∈Aand bought by a∈A,thatis, o∈Ea∩X∗ a. We say that object o∈Ois traded vacuously if o’s marginal value to ais zero, that is, if va(X∗ a)=va(X∗ a\{o}), in which case we also say that asells ovacuously and abuys ovacuously. The term captures the idea that trading odoes not contribute to welfare. We denote the set of objects that are traded nonvacuously under the efficient allocation X∗by TX∗=o∈O:o∈Ea∩X∗ afor some a,a∈Awith a= aand vaX∗ a>v aX∗ a\{o}. For every agent a∈A, we say that atrades nonvacuously if he either buys or sells at least one object nonvacuously; formally, the set of agents who trade nonvacuously is AX∗=a∈A:Ea∪X∗ a∩ TX∗= ∅. 10As va2({o1,o2})>v a2({o1})+va2({o2}),a2’s valuation violates the submodularity condition, which is satisfied by all gross substitutes valuations (Gul and Stacchetti,1999, Lemma 5). 11See Baldwin and Klemperer’s (2019) unimodularity theorem for a necessary and sufficient condition for the existence of an equilibrium in a discrete economy.
1818 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) We say that ais an ex post buyer if he buys at least one object nonvacuously and either does not sell, or only sells objects vacuously. Analogously, we say that ais an ex post seller if he sells at least one object nonvacuously and either does not buy, or only buys objects vacuously. Formally, the sets of ex post buyers and ex post sellers are, respectively, BX∗=a∈A:Ea∩ TX∗=∅,X∗ a∩ TX∗= ∅and SX∗=a∈A:Ea∩ TX∗= ∅,X∗ a∩ TX∗=∅ . Given a type vector θ∈, for every object o∈O,denoteby po(θ)=min (pˆ o)ˆ o∈O∈PW(θ) poand po(θ)=max (pˆ o)ˆ o∈O∈PW(θ) po the smallest and largest prices of object oin any Walrasian price vector. We call the difference po(θ)−po(θ)the Walrasian price gap of object o.Thepricevectorsp(θ)= (po(θ))o∈Oand p(θ)=(po(θ))o∈Oconstitute a lower and an upper bound for the set of Walrasian price vectors in the sense that, for any Walrasian price vector p∈PW(θ), p(θ)≤p≤p(θ).Ifp(θ)is a Walrasian price vector (i.e., p(θ)∈PW(θ)), we call p(θ) the smallest Walrasian price vector. Similarly, we call p(θ)the largest Walrasian price vector if p(θ)∈PW(θ). A sufficient condition for p(θ)and p(θ)to be Walrasian price vectors is that all valuations satisfy the gross substitutes condition.12 We again drop the dependencies on types whenever there is no risk of confusion. We now present two results that focus on the largest net Walrasian prices of ex post buyers and sellers. Claim 4. If p∈PWthen, for every efficient allocation X∗∈X∗and every ex post buyer b∈ B,qb(X∗)=−o∈X∗ b\Ebpo. Claim 5. If p∈PWthen, for every efficient allocation X∗∈X∗and every ex post seller s∈ S,qs(X∗)=o∈Es\X∗ spo. An ex post seller only buys objects vacuously (if he buys at all). As the price of a vacuously traded object is zero in all Walrasian price vectors (see Lemma A.2 in Appendix A for a formal statement), an ex post seller’s net price is the sum of the prices of the objects he sells. If a largest Walrasian price vector exists, that sum is maximized by individually maximizing the price of each object. An analogous reasoning holds for buyers; however, the sum is negative and is maximized by individually minimizing the price of each object. 12See Appendix B.2 for a formal definition. As Gul and Stacchetti (1999, Corollary 1) show, if all valuations satisfy the gross substitutes condition, then the set of Walrasian price vectors forms a nonempty complete lattice, which implies that it contains extremal elements.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1819 Two-sided allocations We say that an efficient allocation X∗∈X∗is a two-sided efficient allocation if, under X∗, every agent who trades nonvacuously is either an ex post buyer or an ex post seller; formally, the set of two-sided efficient allocations is X∗=X∗∈X∗: BX∗∪ SX∗= AX∗. In general, whether or not an efficient allocation is two-sided depends on the realization of types. In fact, it may also depend on which efficient allocation is picked as some may be two-sided while others are not. Define a two-sided allocation problem as an asset market in which every agent is exogenously either a buyer, as he has an empty endowment, or a seller, as he derives zero value from any object that is not in his endowment. Clearly, in a two-sided allocation problem every efficient allocation X∗∈X∗is a two-sided efficient allocation and all results in this subsection apply. The next proposition follows from Claims 4and 5and Theorem 2. Proposition 2. Suppose that p,p∈PW. Then, for every two-sided efficient allocation X∗∈ X∗, DVCG ≥Q= o∈T(X∗) (po−po). Since a sufficient condition for the existence of a smallest and largest Walrasian price vector (i.e., for p,p∈PW) is that the valuation of every agent satisfies the gross substitutes condition, Proposition 2applies to all gross substitutes environments.13 Singleobject traders satisfy the gross substitutes condition.14 The following proposition shows that if all traders are single-object traders and the efficient allocation is two-sided, then the social planner can charge the buyer of any nonvacuously traded object his smallest Walrasian price, but has to pay the seller of that object his largest Walrasian price. Thus, on each traded object the social planner makes a deficit equal to that object’s Walrasian price gap. Proposition 3. Suppose that all agents are single-object traders. Then, for every twosided efficient allocation X∗∈ X∗and every object o∈ T(X∗)that is nonvacuously sold by an agent s∈Aand nonvacuously bought by an agent b∈A, tVCG sX∗=po,tVCG bX∗=−poand DVCG =Q= o∈T(X∗) (po−po). 13As the sum of the largest net Walrasian prices Qis the same under every efficient allocation (by Claim 3), Proposition 2implies that the sum of the Walrasian gaps over all objects traded is the same for every two-sided efficient allocation. 14The valuation of a single-object trader satisfies the unit demand condition. As noted by Gul and Stacchetti (1999), the unit demand condition is a special case of the strong no complementarities condition, which implies the gross substitutes condition.
1820 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) In the example in Section 2, the sum of the Walrasian gaps over all traded objects is (po1−po1)+(po2−po2)=(3−0)+(5−0)=8 and exceeds the VCG deficit, which is 3. The reason for the discrepancy is that the efficient allocation is not two-sided: each agent sells an object and buys the other. In Example 2, the efficient allocation is twosided: a1is a buyer and a2is a seller. Furthermore, agents’ valuations satisfy the gross substitutes condition and so a smallest and a largest Walrasian price vector exist: p= (3, 2)and p=(5, 7). Therefore, in line with Proposition 2, DVCG =8≥7=qa1+qa2=(po1−po1)+(po2−po2). Homogeneous good markets We now specialize the model to one with a homogeneous good. Although in principle agents can simultaneously buy and sell, with a homogeneous good there is always an efficient allocation in which each agent either only buys, only sells, or does not trade; that is, at least one two-sided efficient allocation exists. An asset market is a homogeneous good market if, for every agent a∈A,everytypeθa∈a, and any two bundles Y,Z⊆O with |Y|=|Z|,va(Y,θa)=va(Z,θa). In other words, in a homogeneous good market, agents care about the number of objects they are allocated but not about the identity of those objects. Given an efficient allocation X∗∈X∗, we say that agent a∈Ais a net buyer if |X∗ a|> |Ea|and a net seller if |X∗ a|<|Ea|.WedenotebyBN(X∗)⊆Athe set of net buyers and by SN(X∗)⊆Athe set of net sellers. For every net buyer b∈BN(X∗), we say that bbuys |X∗ b|−|Eb|units. Similarly, for every net seller s∈SN(X∗), we say that ssells |Es|−|X∗ s| units. We call every agent a∈A\(BN(X∗)∪SN(X∗)) aneutral agent; by definition, ais a neutral agent if |X∗ a|=|Ea|. As the number of objects allocated is the same under both X∗and E, the number of units bought by net buyers equals the number of units sold by net sellers. We denote that number by #(X∗): #X∗= b∈BN(X∗)X∗ b−| Eb|= s∈SN(X∗)|Es|−X∗ s. In a homogeneous good market, because agents do not care about the identity of the objects they are allocated, the smallest and largest price that an object can have in any Walrasian price vector must be the same across all objects. Therefore, the price vectors pand pare uniform in that each assigns the same price pand pto every object, that is, po=pand po=pfor all o∈O. A direct consequence of this uniformity is that, whenever Walrasian prices exist, each agent’s largest net Walrasian price can be expressed in terms of the net number of units of the homogeneous good that he buys or sells. Proposition 4. Consider a homogeneous good market in which p,p∈PWand any efficient allocation X∗∈X∗. Then, for every net buyer b∈BN(X∗),qb(X∗)=− (|X∗ b|−|Eb|)p; for every net seller s∈SN(X∗),qs(X∗)=(|Es|−|X∗ s|)p; and for every neutral agent a∈A\(BN(X∗)∪SN(X∗)),qa(X∗)=0. The sum of the largest net Walrasian prices is Q=#X∗(p−p).
Theoretical Economics 17 (2022) When Walras meets Vickrey 1821 There is a clear intuition behind Proposition 4: the net price of a net buyer is maximized by setting the price as low as possible and the net price of a net seller is maximized by setting the price as high as possible.15 To appreciate how generally Proposition 4applies, it is useful to consider conditions under which a smallest and largest Walrasian price vector exist. Recall that the existence of a smallest and largest Walrasian price vector is guaranteed as long as every agent’s valuation satisfies the gross substitutes condition. We show in Appendix B.2 that in a homogeneous good market an equivalent condition is that all agents have decreasing marginal values, that is, the marginal value of their nth unit is no smaller than the marginal value of their n+1-st unit.16 Therefore, Proposition 4applies to every homogeneous good market with decreasing marginal values. Beyond decreasing marginal values, the set of Walrasian prices may be empty. However, provided it is nonempty, Proposition 4ap- plies unless all objects are allocated to the same agent. Suppose that, under at least one efficient allocation, no agent is allocated all objects (i.e., there exists X∗∈X∗such that X∗ a= Ofor all a∈A). Any Walrasian price vector psupports X∗(by Claim 2); hence, as we formally show in Appendix A(Lemma A.4), pis uniform for otherwise an agent has an incentive to swap one of his objects for a cheaper one. Consequently, the order p≤ˆ p is complete, that is, for any p,ˆ p∈PW, either p≥ˆ por p≤ˆ pholds. Thus, as long as the set of Walrasian price vectors is nonempty, there exists a smallest Walrasian price vector pand a largest Walrasian price vector p, and both of them are uniform. Proposition 5. Consider a homogeneous good market and suppose that either (i) all agents have decreasing marginal values or (ii) PW= ∅ and there exists X∗∈X∗such that X∗ a= Ofor all a∈A. Then p,p∈PW. Proposition 5means that Proposition 4applies to “almost all” homogeneous good markets in which the set of Walrasian prices is nonempty. The only exception occurs when some marginal values are increasing and, under every efficient allocation, all objects are allocated to the same agent.17 The following example illustrates how Proposition 4may fail in this specific case. There are two agents, each of whom is endowed with one object and has the following valuations: 15As Qdoes not depend on which efficient allocation is chosen (by Claim 3), Proposition 4implies that, as long as there exist multiple Walrasian prices, #(X∗)=#(X)for any X∗,X∈X∗. This need not be the case in the presence of a unique Walrasian price vector. For example, suppose there are two agents and one object for which each agent has a value of 1. The unique Walrasian price is 1. One efficient allocation leaves the object with the agent to whom it is endowed (hence, no units are traded in this allocation) while the other efficient allocation gives the object to the other agent (hence, one unit is traded). 16Formally (see Definition B.2 in Appendix B.2), for every agent a∈Aand any bundles Y1,Y2,Y3⊆O with |Y1|+2=|Y2|+1=|Y3|,wehavethatva(Y2)−va(Y1)≥va(Y3)−va(Y2). 17We thank an anonymous referee for pointing out this special case to us.
1822 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) The unique efficient allocation has both objects allocated to a2and the set of Walrasian price vectors contains all price vectors whose sum lies between 1 and 2; hence, there are no smallest and largest Walrasian price vectors. The net price of a1is po1,whichis maximized by the vector (2, 0)so qa1=2. Similarly, the net price of a2is −po1,whichis maximized by the vector (0, 2)so qa2=0. As single-object traders have decreasing marginal values, Proposition 4applies to this setting and can be combined with Theorem 1to obtain the following corollary. Corollary 2. Consider a homogeneous good market and suppose that all agents are single-object traders. Then, for every efficient allocation X∗∈X∗, the VCG deficit on each unit traded is p−p, and hence, the VCG deficit is DVCG =#(X∗)(p−p). Corollary 2is a known result for two-sided allocation problems; see, for example, Tatur (2005). When all agents are single-object traders, each net buyer pays a transfer equal to the smallest Walrasian price for the unit he buys and every net seller receives a transfer equal to the largest Walrasian price for the unit he sells. Therefore, the social planner incurs a deficit on each unit traded equal to the Walrasian price gap. Combining Proposition 4with Theorem 2, we obtain the following corollary. Corollary 3. Consider a homogeneous good market in which p,p∈PW.Then,forevery efficient allocation X∗∈X∗,theVCGdeficitisDVCG ≥#(X∗)(p−p). Corollary 3generalizes Theorem 1 in Loertscher and Mezzetti (2019) in two ways. First, in our environment whether an agent is a net buyer or a net seller depends on the types whereas in Loertscher and Mezzetti (2019) agents’ trading positions are exogenously given. Second, Loertscher and Mezzetti assume that agents have decreasing marginal values, while Corollary 3applies beyond decreasing marginal values, as long as the set of Walrasian prices is nonempty and there exists an efficient allocation under which no agent is allocated all objects. A further implication of Theorem 2and Corollary 3is that if the deficit under VCG is zero in a homogeneous good market in which extremal Walrasian price vectors exist, then the Walrasian price gap has to be zero as well, that is, p=phas to hold. Note that the condition p=p, which is nongeneric in two-sided allocation problems with finitely many agents and, say, continuously distributed types, can naturally be satisfied in asset markets because the Walrasian price may need to make a single agent indifferent between buying and selling. This occurs, for example, if all agents have constant marginal values up to some maximum demands and if, under efficiency, one agent with a positive endowment less than his maximum demand consumes exactly the amount he is endowed.18 As noted by Loertscher and Marx (2020), in this case the VCG mechanism has a deficit of zero. However, the question under what more general conditions p=p implies a VCG deficit of zero remains open and is best left for future research. Related, one may wonder whether the VCG mechanism runs a budget surplus when Walrasian 18Perhaps the simplest environment has an odd number of agents, each agent with an endowment of one and a maximum demand of two. Then the Walrasian price is equal to the value of the median agent.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1823 prices fail to exist. While a comprehensive answer is beyond the scope of this paper, the following example shows that at least in some cases the answer is affirmative.19 Consider a homogeneous good market with three agents. Each agent ai,i=1, 2, 3, has an endowment of one, a value of zero for a single unit, and a value of vai>0fortwoor three units (i.e., the marginal value of a second unit is vaiand the marginal value of a third unit is 0). Assuming va1>v a2>v a3, efficiency requires that agent a1be allocated two units and the last unit be allocated to any of the three agents. The VCG mechanism runs a budget surplus as the transfer of agent a1is −va2and the transfer of the other two agents is 0. As Theorem 2implies, if the VCG mechanism runs a budget surplus, the set of Walrasian prices has to be empty. To see that this is indeed the case, note that if all three units are allocated to agent a1, their marginal value to him is zero; therefore, their price must also be zero. If one object is allocated to one of the other agents, by analogous reasoning the price of that unit must be zero. Then the price of the other two units must also be zero as otherwise agent a1would want to swap one of his units for the cheaper one. It follows that the only candidate for a Walrasian price vector is (0, 0, 0); however, this price vector creates excess demand as all agents want to keep their endowment and purchase a second unit. 8. Related literature This paper brings together different strands of the literature. The first strand uncovers a connection between Walrasian prices and the equilibrium prices in the Vickrey auction. Demange (1982)andLeonard (1983) study a one-sided assignment problem in which each agent must be assigned to a single object, or position. Positions can be viewed as “dummy agents” who do not need to be provided incentives for value revelation, and hence, play no role in the deficit calculation. By postulating that each dummy agent is endowed with an object, that actual agents are not endowed with any objects and adding the assumption that each dummy agent dhas no value for any good (i.e., vd(Y,θd)=0 for all Y⊆O,allθd,andalld), the assignment problem can be viewed as a special case of an asset market with single-object traders. Demange (1982)andLeonard (1983)show that in their setting the smallest Walrasian price vector coincides with the prices in the Vickrey auction and, as a consequence, the aggregate payment of buyers in a Walrasian equilibrium coincides with the revenue in a VCG auction. Their results can be viewed as an extension of the observation that, with a single seller and a single object, the price in a second-price auction coincides with the lowest Walrasian price (any price between the second highest and highest bidder’s value is a Walrasian price). Gul and Stacchetti (1999) study a more general setting in which buyers demand (i.e., have value for) multiple objects. They focus on the structural properties of the set of Walrasian equilibria when buyers’ preferences satisfy the gross substitutes condition. A by-product of their analysis (their Theorem 8) shows that the aggregate payment of buyers at the smallest Walrasian prices is an upper bound for the total revenue raised by the VCG mechanism; with multiunit demand, equality need not hold. In contrast to 19We are thankful to an anonymous referee for having proposed this example.
1824 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) the present paper, Demange (1982), Leonard (1983), and Gul and Stacchetti (1999)do not consider the issue of incentive compatibility for sellers, and thus provide no direct connection between Walrasian prices and the VCG deficit. The second strand of the literature focuses on a game-theoretic, mechanism-design conceptualization, and characterization of perfect competition. Makowski and Ostroy (1987) define an exchange economy with quasilinear preferences as perfectly competitive if no agent has price impact: with or without him, the Walrasian prices are the same. More precisely, an exchange economy is perfectly competitive if for any possible valuation of agents, there exists a Walrasian price vector that remains a Walrasian price vector if the valuation of a single agent changes.20 Under standard technical conditions, they show that an exchange economy is perfectly competitive if and only if the total money transfer each agent receives in a Walrasian equilibrium (i.e., using a Walrasian price vector) coincides with his transfer in the VCG mechanism.21 Thus, in a perfectly competitive economy the VCG mechanism is budget balanced. Section 2 of Gretsky, Ostroy, and Zame (1999) studies a generalization of the finite assignment model analyzed by Demange (1982)andLeonard (1983); besides buyers who value only one object and have no endowment, there are sellers. Each seller is endowed with an object and only has a positive value for the object he owns. Gretsky, Ostroy, and Zame (1999)providenecessary and sufficient conditions for the assignment economy to be perfectly competitive in the sense of Makowski and Ostroy (1987), and argue that while “most finite economies are imperfectly competitive, ...most continuum economies are perfectly competitive” (p. 60). In contrast, our paper focuses on imperfectly competitive economies and provides a connection between Walrasian prices in such economies and the VCG deficit. Furthermore, even with single-object traders our model is more general than the assignment model. In the assignment model, each buyer is matched with a seller and the largest net Walrasian price is a single price (for a buyer it is the lowest price of the object he buys and for a seller it is the largest Walrasian price of the object he sells). In our model, even with single-object traders, there could be trading chains of arbitrary length and the largest net Walrasian price can be the difference between two Walrasian prices. The payoff of each agent ain a VCG mechanism is equal to his social marginal product, defined as W∗−W∗ −a,−Ea.Makowski and Ostroy (1995) define the private marginal benefit of an agent in a Walrasian equilibrium as his equilibrium payoff, which is equal to the allocation valuation plus the net trade revenue (or, equivalently and as they write it, minus the net trade expenditure); in our notation and indivisible objects setting: va(X∗ a,θa)−(o∈X∗ apo−o∈Eapo).Makowski and Ostroy (1995) are interested in deriving and understanding the conceptual significance of the first welfare theorem for their notion of a perfectly competitive economy. Without being particularly interested in the VCG mechanism per se, or in finding a bound in the deficit it generates, their Theorem 1 is closely connected to our Theorem 2. It shows that for all Walrasian price 20They do not assume indivisible objects; an exchange economy with indivisible objects is what we call an asset market. 21Makowski and Ostroy (1987) call it the “full appropriation mechanism” to distinguish it from a VCG mechanism with added lump-sum transfers.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1825 vectors the social marginal product of an individual is at least as large as his private marginal product. Given that our setting is substantively different from theirs (e.g., we have indivisible goods and we do not have agents choosing occupations before trading), we provide a simple independent proof in Appendix A(see the paragraph immediately after Theorem 2for a sketch). That said, Theorem 2couldalsobeprovenintheset- ting of Makowski and Ostroy (1995) along the following lines: (i) one could choose for each agent athe Walrasian price vector that generates the largest net trade revenue; (ii) this net revenue would correspond to our largest net Walrasian price for agent a; (iii) by rearranging terms, the inequality in their Theorem 1, that the social marginal product exceeds the private marginal product, could then be stated as saying that the largest net Walrasian price for each agent is less than or equal to his VCG transfer, which is our Theorem 2. It is also worth mentioning that Theorem 2couldalsobeprovenbyadaptingthear- gument in the proof of Theorem 8 in Gul and Stacchetti (1999). To that end, consider an efficient allocation X∗, and a Walrasian price vector psupporting it. Pick any agent aand change his valuation to va(Y,θa)=o∈Ypofor all Y⊆O.Asais indifferent among all packages, the allocation X∗is still supported by pand so remains efficient with an associated welfare of o∈X∗ apo+W∗ −a,−X∗ a. An alternative is to allocate ahis endowment and efficiently allocate the remaining objects to the other agents, with an associated welfare of o∈Eapo+W∗ −a,−Ea.AsX∗is efficient, we have that o∈X∗ apo+W∗ −a,−X∗ a≥o∈Eapo+ W∗ −a,−Ea, which can be rearranged as W∗ −a,−X∗ a−W∗ −a,−Ea≥o∈Eapo−o∈X∗ apo.Then, as pis an arbitrary Walrasian price vector, tVCG a(X∗)≥qa(X∗).22 Third, dating back to the seminal contributions of Vickrey (1961)andMyerson and Satterthwaite (1983), there is a large literature on the (im)possibility of efficient, incentive compatible, and individually rational trade. General results that do not necessarily relate to markets (i.e., private goods) are in Makowski and Mezzetti (1993,1994), Williams (1999), and Segal and Whinston (2016). For a recent contribution in market settings and additional references see, for example, Delacrétaz, Loertscher, Marx, and Wilkening (2019). With the exceptions of Tatur (2005)andLoertscher and Mezzetti (2019), which study homogeneous good settings, this literature makes no explicit connection between Walrasian prices and the deficit under the VCG mechanism. Our paper’s contribution to this literature is an impossibility result for general private goods providing a link between the VCG deficit and Walrasian prices. 22Yet another way to derive our Theorem 2wouldbealongthelinesofSegal and Whinston (2016)by recognizing that Walrasian equilibria are in the core. Hence, given an agent a, an efficient allocation X∗, and a Walrasian price vector p,itmustbethat W∗−vaX∗ a+ o∈Ea\X∗ a po− o∈X∗ a\Ea po≥W∗ −a,−Ea. Rearranging and recalling that an agent’s payoff under the VCG mechanism is his marginal contribution, we obtain that vaX∗ a+tVCG aX∗=W∗−W∗ −a,−Ea≥vaX∗ a+ o∈Ea\X∗ a po− o∈X∗ a\Ea po. As pis an arbitrary Walrasian price vector, it follows that tVCG a(X∗)≥qa(X∗).
1826 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) Fourth and last, there is a small but growing literature in which agents’ trading positions in a homogeneous good market are endogenously determined as a function of their own values and the values of all other traders. Extending the setup of Cramton, Gibbons, and Klemperer (1987) to account for limited capacities (or demands) by the agents, Lu and Robert (2001) derive the profit-maximizing market mechanism, while Loertscher and Marx (2020) provide a trade sacrifice mechanism that either allocates efficiently or close to efficiently and never runs a deficit.23 In Bayesian settings with a homogeneous good such as those of Lu and Robert (2001)andCramton, Gibbons, and Klemperer (1987), the allocation problem is always ex post two-sided because every trading agent either only sells or only buys. In the general asset markets that we study in this paper, this is not the case as an agent may simultaneously buy some objects while selling others. 9. Conclusions For an asset market with quasilinear utilities, we show there is a tight connection between Walrasian prices and VCG transfers. We define an agent’s largest net Walrasian price to be the largest difference between the sum of the prices of the objects he sells and the sum of the prices of the objects he buys in any Walrasian price vector. When every agent is a single-object trader—that is, every agent has a maximum demand of one object and is endowed with at most one object—we show that each agent’s largest net Walrasian price is equal to his VCG transfer; hence, the deficit of the VCG mechanism is equal to the sum of the largest net Walrasian prices of all agents. Beyond single-object traders, we show that, whenever the set of Walrasian prices is nonempty, each agent’s largest net Walrasian price constitutes a lower bound for his VCG transfer; therefore, the sum of the largest net Walrasian prices constitutes a (nonnegative) lower bound for the deficit of the VCG mechanism (and any efficient, ex post individually rational, and dominant strategy incentive compatible mechanism). Because these results only require the existence of Walrasian prices, they are as general within this domain as possible. An interesting avenue for future research is to explore whether these results can be generalized to environments in which the set of Walrasian prices is empty. One could consider a divisible version of the market in which agents may be assigned fractions of bundles. Market clearing prices in this divisible market always exist and are sometimes called pseudo-equilibrium prices.24 To the best of our knowledge, it is an open question whether the VCG transfers are bounded below or connected in some way with some elements of this set of pseudo-equilibrium prices. 23See also Chen and Li (2018) for an analysis of dominant strategy foundations in the settings of Cramton, Gibbons, and Klemperer (1987) and Lu and Robert (2001). 24Bikhchandani and Mamer (1997) proved that the set of such market clearing prices is nonempty and coincides with the set of Walrasian prices if the latter set is also nonempty. The properties of these pseudoequilibrium prices were further investigated by Milgrom and Strulovici (2009).
Theoretical Economics 17 (2022) When Walras meets Vickrey 1833 We next add an agent ˜ asuch that, for every bundle Y⊆O, v˜ a(Y)=W∗(+˜ a,+o)−W∗(+˜ a,+o) ·,−oif o∈Y, 0ifo/∈Y. Agent ˜ ahas unit demand and only values o. Starting with an efficient allocation in the market in which ˜ aand ohave been added, we can obtain an efficient allocation in the market where ˜ ahas also been added by allocating the same bundle to every a∈Aand allocating the empty bundle to ˜ a. Therefore, an efficient allocation in this market is X such that X ˜ a=∅,X ˜ a={o},andX a=X∗ afor all a∈A. Let (pˆ o)ˆ o∈Obe a Walrasian price vector in the market in which ˜ a,˜ a,andthecopyof ohave been added. (The set of Walrasian price vectors in this market is nonempty since all agents are single-object traders. Moreover, as oand its copy are identical, their price in any Walrasian price vector is the same;27 therefore, we can define poto be the price of both oand its copy.) By Claim 2,(pˆ o)ˆ o∈Osupports X. Moreover, by construction, (pˆ o)ˆ o∈Osupports X∗in the original market, meaning that (pˆ o)ˆ o∈Ois a Walrasian price vector in the original market. Therefore, it remains to show that po−po≥W∗−W∗(·,+o) ·,−o. As (pˆ o)ˆ o∈Osupports X, when facing those prices it is optimal for ˜ anot to acquire any object—hence, po≥W∗(+˜ a,+o)−W∗(+˜ a,+o) ·,−o—and for ˜ ato acquire o—hence, po≤ W∗(·,+o)−W∗. Recalling that W∗(+˜ a,+o)=W∗(·,+o), we conclude that po−po≥W∗−W∗(+˜ a,+o) ·,−o. By Theorem 2 in Shapley (1962), an agent and an object are complements to each other: W∗(+˜ a,+o) ·,−o−W∗(·,+o) ·,−o+W∗(·,+o)−W∗(·,+o) ·,−o≤W∗(+˜ a,+o)−W∗(·,+o) ·,−o. Therefore, we have W∗(+˜ a,+o) ·,−o−W∗(·,+o) ·,−o≤W∗(+˜ a,+o)−W∗(·,+o)=0. It follows that W∗(+˜ a,+o) ·,−o≤W∗(·,+o) ·,−o, and hence, as required: po−po≥W∗−W∗(·,+o) ·,−o. Proof of Lemma A.3. By Lemma A.1,W∗=W∗ −a,−o+va({o})so we need to show that W∗(·,+o) ·,−o=W∗ −a,−o+va({o}). Let ˆ X∗be an efficient allocation in the original problem such that (i) every agent is allocated at most one object and (ii) ais allocated o. Such an allocation necessarily exists since all agents are single-object traders and abuys ononvacuously.28 For every a∈A,let ˆ o∗ a∈O∪{∅}be the object (if any) that ais allocated under ˆ X∗.ThenW∗= a∈Ava({o∗ a}). 27If the prices are different, both ˜ aand the agent who is allocated ounder Xonly demand whichever one of oor its copy is cheaper; hence, such a price vector does not support X. 28 ˆ X∗can be constructed by starting from X∗and, for each agent who is allocated multiple objects, reallocating all but one of them to agents who are allocated the empty bundle.
1834 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) Consider now the market in which a copy of o—which we denote by ˜ o—is added and ois removed. Toward a contradiction, suppose that there is no efficient allocation in this market under which ais allocated o=ˆ o∗ a. In this market, consider the efficient allocations ˆ Xsuch that, again, each agent is allocated at most one object. For every a∈A,wedenoteby ˆ o a∈O∪{∅}the object (if any) that ais allocated under ˆ X.Then W∗(·,+o) ·,−o=a∈Ava({ˆ o a}). As every agent is allocated one object, ˆ Xis defined by: (i)a chain of reallocations o0→a1→o1→a2→o2→···→an→on such that o0=˜ o,on=o,oi=ˆ o∗ ai,andoi−1=ˆ o aifor all i=1, ,n,and(ii)the property that all agents not in the chain are allocated the same object as in the efficient allocation ˆ X∗of the original problem: ˆ o∗ a=ˆ o afor all a∈A\{a1,,an}. By assumption, ˆ o a= ˆ o∗ a=o; therefore, there exists m=1, ,n−1suchthatam=a and om=o. Consider now the alternative allocation in which aiis allocated oifor all i=1, ,m,am+1is allocated o0=˜ o, and every remaining agent a∈A\{a1,,am+1} is allocated ˆ o a. That allocation is not efficient by assumption since it allocates oto a; therefore, the aggregate value it creates is strictly less than that created by ˆ X,which implies that m+1 i=1 vai{oi−1}>v am+1{o0}+ m i=1 vai{oi}. As om=oand o0=˜ o,wehavethatvam+1({om})=vam+1({o0})and va1({o0})=va1({om}). It follows that va1{om}+ m i=2 vai{oi−1}> m i=1 vai{oi} ⇔va1{om}+ m i=2 vai{oi−1}+ a∈A\{a1,,am} vaˆ o∗ a> a∈A vaˆ o∗ a, which contradicts the assumption that ˆ X∗is an efficient allocation in the original market. We conclude that, in the market in which a copy of ohas been added and ohas been removed, there exists an efficient allocation under which ais allocated o.Then, by Lemma A.1,W∗(·,+o) ·,−o=W∗ −a,−o+va({o}),asrequired. Proof of Proposition 1. For every object o∈Oand every k=1, ,|A|,letak o∈A be the agent with the kth highest valuation for o;thatis,va1 o({o})≥va2 o({o})≥ ··· ≥ va|A| o({o}). Construct an efficient allocation X∗by assigning each object to the agent who values it the most. Since valuations are additively separable, the welfare created by X∗is W∗=W(X∗)=o∈Ova1 o({o}). Consider now the allocation problem where some agent a∈Aand his endowment Eahave been removed. By an analogous reasoning, welfare is maximized by allocating
Theoretical Economics 17 (2022) When Walras meets Vickrey 1835 each object to the agent who values it the most. Therefore, each object o∈O\(X∗ a∪Ea)is allocated to a1 oand each object o∈X∗ a\Eais assigned to a2 o(since a1 o=ais unavailable). We conclude that W∗ −a,−Ea= o∈O\(X∗ a∪Ea) va1 o{o}+ o∈X∗ a\Ea va2 o{o}.(6) By Lemma A.1, the efficient level of welfare when aand his allocation X∗ aare removed is W∗ −a,−X∗ a= o∈O\X∗ a va1 o{o}= o∈O\(X∗ a∪Ea) va1 o{o}+ o∈Ea\X∗ a va1 o{o}.(7) Using (6)and(7), we find that the VCG transfer of any agent a∈Ais tVCG aX∗=W∗ −a,−X∗ a−W∗ −a,−Ea= o∈Ea\X∗ a va1 o{o}− o∈X∗ a\Ea va2 o{o}.(8) We next show that the set of Walrasian price vectors is PW=(po)o∈O∈R|O|:po∈va2 o{o},va1 o{o}for all o∈O.(9) Consider a price vector (po)o∈O. Suppose first that, for some ˆ o∈O,pˆ o<v a2 ˆ o.Then it is optimal for a2 ˆ oto pick ˆ owhen he faces (po)o∈O; therefore, (po)o∈Odoes not support X∗and is not a Walrasian price vector. Suppose next that, for some ˆ o∈O,pˆ o>v a1 ˆ o({ˆ o}). Then it not optimal for a1 ˆ oto pick ˆ owhen he faces (po)o∈O; again, (po)o∈Odoes not support X∗and is not a Walrasian price vector. Finally, suppose that, for all ˆ o∈O,pˆ o∈ [va2 ˆ o({ˆ o}),va1 ˆ o({ˆ o})]. Then, for all ˆ o∈O,whenagentsface(po)o∈O,itisoptimalfora1 ˆ oto pick ˆ oand optimal for all other agents not to pick ˆ o. We have therefore established (9). By definition, the largest net Walrasian price of agent a∈Ais qa=max (po)o∈O∈PW o∈Ea\X∗ a po− o∈X∗ a\Ea po, which combined with (9)yields qa= o∈Ea\X∗ a va1 o{o}− o∈X∗ a\Ea va2 o{o}. By (8), we obtain that tVCG a(X∗)=qa(X∗). As this holds for all a∈A,weconcludethat tVCG a(X∗)=qa(X∗). Then, by definition, we have that DVCG = a∈A tVCG aX∗= a∈A qaX∗=Q. Finally, Theorem 2yields Q≥0. Proof of Proposition 2.Theorem2yields DVCG ≥Q; hence, we need to show that Q=o∈T(X∗)(po−po)
1836 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) As X∗is a two-sided efficient allocation (i.e., X∗∈ X∗), B(X∗)∪ S(X∗)= A(X∗),so Q= a∈A\ A(X∗) qaX∗+ b∈ B(X∗) qbX∗+ s∈ S(X∗) qsX∗. By Lemma A.2, the price of a vacuously traded object is zero in every Walrasian price vector; therefore, qa=0 for all a∈A\ A(X∗)and we have that Q= b∈ B(X∗) qbX∗+ s∈ S(X∗) qsX∗. Using Claims 4and 5and rearranging, we obtain that Q= b∈ B(X∗)− o∈X∗ b\Eb po+ s∈ S(X∗) o∈Es\X∗ s po = o∈s∈ S(X∗)(Es\X∗ s) po− o∈b∈ B(X∗)(X∗ b\Eb) po. By Lemma A.2, for any object o∈T(X∗)\ T(X∗),po=po=0. It follows that Q= o∈ T(X∗)∩(s∈ S(X∗)(Es\X∗ s)) po− o∈ T(X∗)∩(b∈ B(X∗)(X∗ b\Eb)) po. (10) The set T(X∗)∩(s∈ S(X∗)(Es\X∗ s)) contains all the objects that are nonvacuously sold by an ex post seller and the set T(X∗)∩(b∈ B(X∗)(X∗ b\Eb)) contains all the objects that are nonvacuously bought by an ex post buyer. By construction, every object that is nonvacuously traded is sold by exactly one seller and bought by exactly one buyer; hence, we have that TX∗∩ s∈ S(X∗)Es\X∗ s= TX∗∩ b∈ B(X∗)X∗ b\Eb= TX∗. (11) Combining (10)and(11) and rearranging yields Q= o∈ T(X∗) po− o∈ T(X∗) po= o∈ T(X∗) (po−po). (12) Invoking Lemma A.2 again, we have po=po=0 for every vacuously-traded object o∈ T(X∗)\ T(X∗).By(12), we conclude that Q=o∈T(X∗)(po−po),asrequired. Proof of Proposition 3. We show that qs(X∗)=poand qb=−po, which implies the desired result by Theorem 1and Proposition 2. The largest net Walrasian price of agent sis qs=max (pˆ o)ˆ o∈O∈PW ˆ o∈Es\X∗ s pˆ o− ˆ o∈X∗ s\Es pˆ o.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1837 As sis a single-object trader, scannot sell any object other than oso Es\X∗ s={o}.As X∗is two-sided, sis an ex post seller so any object that he buys is traded vacuously and, by Lemma A.2, has a price of zero in any Walrasian price vector. It follows that qs=max(pˆ o)ˆ o∈O∈PWpo. As all agents are single-object traders, the set of Walrasian prices contains a largest element; therefore, qs=po. The largest net Walrasian price of agent bis qb=max (pˆ o)ˆ o∈O∈PW ˆ o∈Eb\X∗ b pˆ o− ˆ o∈X∗ b\Eb pˆ o. As bis a single-object trader, bbuys at most one object, object o, nonvacuously. As X∗is two-sided, bis an ex post buyer and any object he sells is traded vacuously. It follows that ois the only object that btrades nonvacuously. By Lemma A.2,qb=max(pˆ o)ˆ o∈O∈PW−po. As all agents are single-object traders, the set of Walrasian prices contains a smallest element; therefore, qb=−po. We next introduce a result that is useful to prove Propositions 4and 5. Lemma A.4. Consider a homogeneous good market and suppose that there exists an efficient allocation X∗∈X∗such that X∗ a= O, for all a∈A. Then every Walrasian price vector is uniform. Proof. Toward a contradiction, suppose there exists a Walrasian price vector p= (po)o∈Othat is not uniform. As not all objects are allocated to the same agent under X∗, there exist two agents ˆ aand aand two objects ˆ oand osuch that ˆ o∈X∗ ˆ a,o∈X∗ a, and pˆ o<p o. Then, as agents do not care about the identity of the objects they are assigned, we have that vaX∗ a\o∪{ˆ o}− o∈(X∗ a\{o})∪{ˆ o} po>v aX∗ a− o∈X∗ a po so pdoes not support the efficient allocation X∗,whichbyClaim2contradicts the assumption that pis a Walrasian price vector. Proof of Proposition 4. By assumption, p,p∈PWso PWis nonempty. Then the largest net Walrasian price of each agent a∈Ais well-defined and equal to qaX∗=max (po)o∈O∈PW o∈Ea\X∗ a po− o∈X∗ a\Ea po. Suppose first that there exists an efficient allocation under which not all objects are allocated to the same agent; that is, there exists X∗∈X∗such that X∗ a= O,foralla∈A. By Lemma A.4, all Walrasian price vectors are uniform so the largest net Walrasian price of each agent asimplifies to qaX∗=max p∈[p,p]Ea\X∗ ap−X∗ a\Eap,
1838 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) which is equivalent to qaX∗=max p∈[p,p]|Ea|−X∗ ap. (13) If ais a net buyer, |Ea|−|X∗ a|<0, then the maximization problem in (13)issolvedby setting pas low as possible, that is, p=p.Thenqa(X∗)=(|Ea|−|X∗ a|)p=− (|X∗ a|− |Ea|)p.Ifais a net seller, |Ea|−|X∗ a|>0, then the maximization problem in (13)issolved by setting p=pand qa(X∗)=(|Ea|−|X∗ a|)p.Ifais a neutral agent, |Ea|−|X∗ a|=0and the maximization problem in (13)issolvedbyanyp∈[p,p]and yields qa(X∗)=0. Suppose now that, under every efficient allocation, all objects are allocated to the same agent. Let X∗∈X∗be any efficient allocation, then there exists an agent bsuch that X∗ b=Oand X∗ s=∅for every agent s= b.AsX∗ b=O,agentbdoes not sell any object so his largest net Walrasian price is qbX∗=max (po)o∈O∈PW− o∈X∗ b\Eb po. By assumption, pis the smallest Walrasian price vector, and as we argued in the main text, it is uniform.29 Therefore, the largest net Walrasian price of agent bis qb(X∗)= −(|X∗ b|−|Eb|)p. For every agent s= b,X∗ s=∅so sdoes not buy any object and his largest net Walrasian price is qsX∗=max (po)o∈O∈PW o∈Es\X∗ s po. By assumption, pis the largest Walrasian price vector and, as we argued in the main text, it is uniform. Hence, the largest net Walrasian price of agent sis qs(X∗)=(|Es|− |X∗ s|)p. Proof of Proposition 5. Suppose first that all agents have decreasing marginal values. By Proposition B.1 in Appendix B.2, the valuation of every agent satisfies the gross substitutes condition; hence, by Corollary 1 of Gul and Stacchetti (1999), PWis a nonempty complete lattice, which implies that p,p∈PW. Suppose now that PW= ∅ and there exists X∗∈X∗such that X∗ a= Ofor all a∈A.By Lemma A.4, every Walrasian price vector is uniform, which implies that p,p∈PW. Appendix B: Background material B.1 Details of examples In this Appendix, we detail the computations of the largest net Walrasian prices and VCG transfers in our examples. 29Formally, if there exist two objects oand osuch that po<p o, then the vector (ˆ pˆ o)ˆ o∈Osuch that ˆ po=po,ˆ po=po, and ˆ pˆ o=pˆ ofor all ˆ o∈O\{o,o}is a Walrasian price vector, which contradicts the assumption that pis the smallest Walrasian price vector.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1839 Example 1A price vector (po1,po2)is a Walrasian price vector if it supports the efficient allocation (i.e., it is optimal for a1to choose o2and for a2to choose o1), which requires satisfying the following six conditions: va1{o2}−po2≥0a1weakly prefers {o2}to ∅ va1{o2}−po2≥va1{o1}−po1a1weakly prefers {o2}to {o1} va1{o2}−po2≥va1{o1,o2}−po1−po2a1weakly prefers {o2}to {o1,o2} va2{o1}−po1≥0a2weakly prefers {o1}to ∅ va2{o1}−po1≥va2{o2}−po2a2weakly prefers {o1}to {o2} va2{o1}−po1≥va2{o1,o2}−po1−po2a2weakly prefers {o1}to {o1,o2}. As agents are single-objects traders, va1({o1,o2})=max{va1({o1}),va1({o2})}so the third condition is equivalent to po1≥max{0, va1({o1})−va1({o2})}; hence, the third and fourth conditions are jointly equivalent to po1∈[max{0, va1({o1})−va1({o2})},va2({o1})]. Analogously, the first and last conditions are jointly equivalent to po2∈[max{0, va2({o2})−va2({o1})},va1({o2})]. Finally, it is easy to see that the second and fifth conditions are jointly equivalent to po1−po2∈[va1({o1})−va1({o2}),va2({o1})−va2({o2})]. Therefore, a price vector (po1,po2)is a Walrasian price vector if it satisfies the following three conditions: po1∈max0, va1{o1}−va1{o2},va2{o1} po2∈max0, va2{o2}−va2{o1},va1{o2} po1−po2∈va1{o1}−va1{o2},va2{o1}−va2{o2}. The price vector (po1,po2)=(max{0, va2({o1})−va2({o2})},max{0, va2({o2})−va2({o1})}) satisfies the first condition since va1({o1})−va1({o2})≤va2({o1})−va2({o2})≤va2({o1}) (as the efficient allocation assigns o1to a2and o2to a1,andva2({o2})≥0), the second condition since po2is equal to its lower bound, and the third condition since the difference po1−po2is equal to its upper bound va2({o1})−va2({o2}). Therefore, (po1,po2)is a Walrasian price vector, which means that there exists a Walrasian price vector for which the difference between the prices of o1and o2is va2({o1})−va2({o2}). As any price vector with a larger difference violates the third condition, we conclude that the largest net Walrasian price of a1is30 qa1=max (po1,po2)∈PW[po1−po2]=va2{o1}−va2{o2}. Analogous reasoning establishes that the largest net Walrasian price of a2is qa2=max (po1,po2)∈PW[po2−po1]=va1{o2}−va1{o1}. 30We omit the dependency of largest net Walrasian prices and VCG transfers on an allocation since there is a unique efficient allocation.
1840 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) The VCG transfer of a1is his externality on a2.Whena1is present, a2is allocated o1 while when a1is removed (with his endowment), a2is allocated o2; therefore, a1’s VCG transfer is tVCG a1=W∗ −a1,−o2−W∗ −a1,−o1=va2{o1}−va2{o2}=qa1. Analogously, the VCG transfer of a2is his externality on a1,whichis tVCG a2=W∗ −a2,−o1−W∗ −a2,−o2=va1{o2}−va1{o1}=qa2. Our illustrative example from Section 2is the special case of Example 1in which va1({o1})=5, va1({o2})=7, va2({o1})=3, and va2({o2})=2. The largest net Walrasian price and VCG transfer of a1(Leon) are 3 −2=1 while the largest net Walrasian price and VCG transfer of a2(William) are 7 −5=2. Example 2A price vector (po1,po2)is a Walrasian price vector if it supports the efficient allocation X∗, which requires satisfying the following six conditions: 12 −po1−po2≥0a1weakly prefers {o1,o2}to ∅ 12 −po1−po2≥5−po1a1weakly prefers {o1,o2}to {o1} 12 −po1−po2≥7−po2a1weakly prefers {o1,o2}to {o2} 0≥3−po1a2weakly prefers ∅to {o1} 0≥2−po2a2weakly prefers ∅to {o2} 0≥4−po1−po2a2weakly prefers ∅to {o1,o2}. The third and fourth conditions imply that po1∈[3, 5]. The second and fifth conditions imply that po2∈[2, 7]. The first and last conditions imply that po1+po2∈[4, 12]; however, the lower bounds po1≥3andpo2≥2 imply that po1+po2≥4andtheup- per bounds po1≤5andpo2≤7 imply that po1+po2≤12. Therefore, a price vector (po1,po2)is a Walrasian price vector if po1∈[3, 5]and po2∈[2, 7].Asa1buys both objects, his largest net Walrasian price is qa1=max (po1,po2)∈PW[−po1−po2]=−3−2=−5. As a2sells both objects, his largest net Walrasian price is qa2=max (po1,po2)∈PW[po1+po2]=5+7=12. The VCG transfer of a1is his externality on a2.Whena1is present, a2is not allocated any object while when a1is removed, a2is allocated both objects; therefore, a1’s VCG transfer is tVCG a1=W∗ −a1,−{o1,o2}−W∗ −a1,·=0−4=−4>−5=qa1.
Theoretical Economics 17 (2022) When Walras meets Vickrey 1841 Analogously, a1is allocated both objects when a2is present and none when a2is absent; hence, a2’s VCG transfer is tVCG a2=W∗ −a2,·−W∗ −a2,−{o1,o2}=12 −0=12 =qa2. Example 3A price vector (po1,po2)is a Walrasian price vector if it satisfies the following six conditions: 9−po2≥0a1weakly prefers {o2}to ∅ 9−po2≥3−po1a1weakly prefers {o2}to {o1} 9−po2≥12 −po1−po2a1weakly prefers {o2}to {o1,o2} 4−po1≥0a2weakly prefers {o1}to ∅ 4−po1≥4−po2a2weakly prefers {o1}to {o2} 4−po1≥9−po1−po2a2weakly prefers {o1}to {o1,o2}. The third and fourth conditions imply that po1∈[3, 4]. The first and last conditions imply that po2∈[5, 9]. The second and fifth conditions imply that po2−po1∈[0, 6], which is always satisfied when po1∈[3, 4]and po2∈[5, 9]. Therefore, a price vector (po1,po2)is a Walrasian price vector if po1∈[3, 4]and po2∈ [5, 9].Asa1sells o1and buys o2, his largest net Walrasian price is qa1=max (po1,po2)∈PW[po1−po2]=4−5=−1. As a2sells o2and buys o1, his largest net Walrasian price is qa2=max (po1,po2)∈PW[po2−po1]=9−3=6. The VCG transfer of a1is his externality on a2.Whena1is present, a2is allocated o2 while when a1is removed, a2is allocated o1; therefore, a1’s VCG transfer is tVCG a1=W∗ −a1,−o2−W∗ −a1,−o1=4−4=0>−1=qa1. Analogously, a1is allocated o2when a2is present and o1when a2is absent; hence, a2’s VCG transfer is tVCG a2=W∗ −a2,−o1−W∗ −a2,−o2=9−3=6=qa2. Example 4A price vector (po1,po2)is a Walrasian price vector if it satisfies the following six conditions:31 3−po2≥0a1weakly prefers {o2}to ∅ 31We consider here the efficient allocation in which a1is allocated o2and a2is allocated o1.Thecalculations are analogous for the other efficient allocation in which a1is allocated o1and a2is allocated o2, and as predicted by Claim 2, yield the same set of Walrasian price vectors.
1842 Delacrétaz, Loertscher, and Mezzetti Theoretical Economics 17 (2022) 3−po2≥3−po1a1weakly prefers {o2}to {o1} 3−po2≥4−po1−po2a1weakly prefers {o2}to {o1,o2} 4−po1≥0a2weakly prefers {o1}to ∅ 4−po1≥4−po2a2weakly prefers {o1}to {o2} 4−po1≥6+ε−po1−po2a2weakly prefers {o1}to {o1,o2}. The second and fifth conditions imply that po1=po2, the third and fourth conditions imply that po1∈[1, 4], and the first and last conditions imply that po2∈[2+ε,3 ].Therefore, the set of Walrasian price vectors contains all price vectors such that po1=po2∈ [2+ε,3 ].Asa2buys an object from a1, the largest net Walrasian prices are qa1=max (po1,po2)∈PWpo1=3andqa2=max (po1,po2)∈PW−po1=−2−ε. The VCG transfer of a1is his externality on a2.Whena1is present, a2is allocated one object while when a1is removed, a2is not allocated anything; therefore, a1’s VCG transfer is tVCG a1=W∗ −a1,−o2−W∗ −a1,−{o1,o2}=4−0=4>3=qa1. When a2is present, a1is allocated one object while when a2is removed, a1is allocated both objects; therefore, a2’s VCG transfer is tVCG a2=W∗ −a2,−o1−W∗ −a2,·=3−4=−1>−2−ε=qa2. The largest net Walrasian prices of both agents are strictly smaller than their VCG transfers, and as a result, the sum of the largest net Walrasian prices (3 −2−ε=1−ε) is strictly smaller than the VCG deficit (4 −1=3). B.2 Gross substitutes valuations For any agent aand any price vector p=(po)o∈O,let Da(p)=Y⊆O:va(Y)− o∈Y po≥va(Z)− o∈Z pofor all Z⊆O be the set of bundles that are optimal for ato pick when he faces the price vector p. Definition B.1 (Kelso and Crawford,1982). The valuation vaof agent a∈Asatisfies the gross substitutes condition if for any two price vectors p=(po)o∈Oand p=(p o)o∈O with p≥p,andanybundleY∈Da(p),thereexistsabundleZ∈Da(p)such that {o∈ Y:po=p o}⊆Z. Definition B.2. In a homogeneous good market, agent a∈Ahas decreasing marginal values if, for any bundles Y1,Y2,Y3⊆Owith |Y1|+2=|Y2|+1=|Y3|,wehavethat va(Y2)−va(Y1)≥va(Y3)−va(Y2).