Characterizing Pareto Optima: Sequential Utilitarian Welfare Maximization
Yeon-Koo Che, Jinwoo Kim, Fuhito Kojima, Christopher Thomas Ryan
CCharacterizing Pareto Optima: SequentialUtilitarian Welfare Maximization Jinwoo Kim Fuhito Kojima Christopher Thomas Ryan August 26, 2020
Abstract
We characterize Pareto optimality via sequential utilitarian welfare maximization:a utility vector u is Pareto optimal if and only if there exists a finite sequence of non-negative (and eventually positive) welfare weights such that u maximizes utilitarianwelfare with each successive welfare weights among the previous set of maximizers.The characterization can be further related to maximization of a piecewise-linear con-cave social welfare function and sequential bargaining among agents `a la generalizedNash bargaining. We provide conditions enabling simpler utilitarian characterizationsand a version of the second welfare theorem. Keywords:
Pareto optima, weighted utilitarian welfare maximization, sequential util-itarian welfare maximization, (eventual) exposure of extreme faces, the second-welfaretheorem.
JEL Numbers:
C60, D60, D50. We are grateful to Timothy Y Chan, Atsushi Kajii, Yuichiro Kamada, and Michihiro Kandori for theirhelpful comments and conversations, as well as to seminar audiences at Carlo Alberto, Carlos III, KAIST,Stanford, and the International Conference on Game Theory at Stony Brook. We are especially gratefulto Ludvig Sinander and Gregorio Curello for a question that led to this project. We acknowledge researchassistance from Xuandong Chen, Jiangze Han, Kevin Li, and Yutong Zhang. Yeon-Koo Che is supported byNational Science Foundation Grant SES-1851821. Christopher Thomas Ryan is supported by the NaturalSciences and Engineering Research Council of Canada Discovery Grant RGPIN-2020-06488 and the UBCSauder Exploratory Grants Program. This work was supported by the Ministry of Education of the Republicof Korea and the National Research Foundation of Korea (NRF-2020S1A5A2A03043516) Department of Economics, Columbia University Department of Economics, Seoul National University Department of Economics, Stanford University Operations and Logistics Division, UBC Sauder School of Business, University of British Columbia a r X i v : . [ ec on . T H ] A ug Introduction
Pareto optimality is a central concept in economics. When the utility possibility set isclosed and convex, it is natural to associate each Pareto optimum with utilitarian welfaremaximization under a suitably chosen welfare weights of agents. Yet, such a characterizationhas been so far elusive.It is well-known that, given a closed and convex utility possibility set, which we assumethroughout, every Pareto optimal utility vector maximizes some nonnegatively weighted sumof utilities of agents (see Mas-Colell, Whinston, and Green (1995) Proposition 16.E.2). Butthe converse is false: not every such maximizer is Pareto optimal. To see this, suppose asociety consists of two agents, 1 and 2, and the utility possibility set is given by U in Figure 1.All points on the frontier including the vertical segment maximize suitably weighted sums U uu (cid:48) u (cid:48)(cid:48) Figure 1: Utilitarian welfare maximization need not yield a Pareto optimum.of agents’ utilities within U , but not all of them are Pareto optimal. In particular, thepoints on the vertical segment strictly below u all maximize the utility sum with weights φ = (1 , u (cid:48) is Pareto optimal and obtained by utilitarian welfaremaximization with strictly positive weights, but u and u (cid:48)(cid:48) , which are Pareto optimal, cannot.While positive welfare weights do not rationalize points like u in Figure 1, one mayconjecture that they may in the limit; for instance, u is a limit of welfare-maximizing utilityvectors with positive weights (1 , /n ), as n → ∞ . Indeed, Arrow, Barankin, and Blackwell(1953) show that every Pareto optimal vector is a limit of a sequence of utility vectors thatmaximize some positively-weighted sum of utilities—a result known as the ABB theorem. Propositions 16.E.2 in Mas-Colell, Whinston, and Green (1995) is stated in the context of an exchangeeconomy. However, it is straightforward to see that the relationship holds generally as long as the utilitypossibility set is convex. This theorem has spawned a series of extensions and generalizations to spaces more general than Eu-clidean space. See Daniilidis (2000) for a survey of ABB theorems. = (0 , , S = ( α, β, u u u V = (0 , , Figure 2: The “tilted cone” adapted from Arrow, Barankin, and Blackwell (1953) and Bitranand Magnanti (1979). The set is the convex hull of the portion of the unit disk centeredat the origin in the u - u plane from the point K to the point S (where α + β = 1 with α ∈ (0 , V = (0 , , again its converse is false—namely, a limit point of such a sequence may not bePareto optimal. To see this, suppose there are three agents, 1, 2, and 3, with possible utilityprofiles depicted in Figure 2. The point K is a limit of the sequence of points maximizinga positively-weighted sum of utilities (see the arrow) but is Pareto dominated, say by thepoint V . Figure 3 depicts Pareto optima in relationship with alternative notions of utilitarianwelfare maximization.This paper provides an exact characterization of Pareto optima in terms of utilitarianwelfare maximization. In particular, our characterization views each Pareto optimal vector u as a result of multiple rounds of utilitarian welfare maximization. The characterizationis easy to explain with the example in Figure 1, reproduced in Figure 4(a). In the firstround, utilities are maximized within U with weights φ , which is maximized by the thickvertical segment containing u as explained before. One can interpret this as the socialplanner first maximizing the utility of agent 1 while disregarding the welfare of the otherindividual completely. Since agent 1 is indifferent among all points as long as he receives the When there are two agents, the limit u ∈ U of any sequence { u k } of utilities u k ∈ U maximizing apositively-weighted sum of utilities is Pareto optimal, where U is the utility possibility set, assumed to beclosed and convex. Let { φ k } be the sequence of positive weights, normalized to be in the simplex, suchthat u k maximizes (cid:80) i =1 φ ki u ki , and let φ denote its limit (say of a convergent subsequence). Clearly, u mustmaximize (cid:80) i =1 φ i u i . If φ and φ are both positive, then u is Pareto optimal, so assume without loss φ = 1and φ = 0. Suppose for contradiction u is not Pareto optimal. Then, there must exist v such that v = u and v > u . Since u k ’s are all Pareto optimal, this means that u k ≤ v = u and u k ≥ v > u for all k , so u k never converges to u , a contradiction. ++ U P U + cl( U ++ ) Figure 3: Alternative notions of utilitarian welfare maximization in relationship with Paretooptimality. The set U P consists of Pareto optimal utility vectors. The set U + consists ofutility vectors that maximizing a nonnegatively-weighted sum of utilities. The set U ++ con-sists of utility vectors that maximize a positively-weighted sum of utilities. The containment U ++ ⊂ U P ⊂ U + is Proposition 16.E.2 in Mas-Colell, Whinston, and Green (1995). The con-tainment U P ⊂ cl( U ++ ) is from Arrow, Barankin, and Blackwell (1953). The containmentcl( U ++ ) ⊂ U + is straightforward. U u φ = (1 , (a) First round u φ = (1 , (b) Second round Figure 4: Rationalizing a Pareto optimal point in two rounds of sequential utilitarian welfaremaximization. 4aximum utility, the social planner seeks to engage in further optimization. In the secondand last round, utilities are again maximized but only within the vertical segment, now with(arbitrary) positive weights φ . The weights φ “rationalize” u as the unique maximizer, asillustrated in Figure 4(b). More generally, our Theorem 1 asserts that a utility vector u is Pareto optimal if and onlyif there exists a finite sequence of nonnegative welfare weights, with the terminal weightsbeing strictly positive for all agents, such that in each round t , u maximizes the round- t weighted sum of utilities out of those surviving from round t − Indeed, Che, Kim, and Kojima (2019) utilizethe exact characterization of the current paper to provide conditions for Pareto optima tovary monotonically with agents’ preferences. Second, Pareto optimality is sometimes invokedto rationalize utilitarianism, which may otherwise be problematic due to the ordinal natureof utilities and the difficulty with interpersonal comparisons of utility (see Yaari (1981)).For this purpose, an exact characterization of Pareto optima is more appropriate than anapproximate one. Our characterization builds on notions of convex geometry. Of particular interest is aspecial class of subsets of closed convex sets called (extreme) faces and the property of It is worth emphasizing that, in general, the terminal step need not identify a unique element of U butrather a set. Also, in this example, many normals work for the second-step maximization: in particular, φ = (0 ,
1) also works. This latter choice makes this sequential maximization procedure feel reminiscentof serial dictatorship under strict preferences (Svensson, 1999), which implements every Pareto optimalallocation of agents to indivisible objects sequentially (in a one-to-one manner) by maximizing agents’ welfareone at a time according to a suitably chosen serial order. Although the idea is similar in spirit, there are acouple of differences. First, a collection of individuals’ joint (weighted) welfare is maximized in each roundhere instead of a single agent’s welfare. More importantly, the sequential procedure, while practically useful,is unnecessary to find an Pareto optimal allocation in the setting of matching agents to indivisible objects.A one-round maximization of a weighted sum of utilities finds every Pareto optimum if the weights are setsufficiently differently across agents to “reflect” their serial orders. To illustrate the issue, consider two utility possibility sets U and V , corresponding to two parameters—or “before” and “after” a change of environment—and let U P and V P denote the sets of Pareto optima.Typically, comparing U P and V P directly is difficult—hence, the need for “optimization-based” character-ization. Suppose for instance that V + dominates U + according to a relevant set order. This still does notimply that V P dominates U P according to the same set order. The same problem arises when using U ++ or cl( U ++ ) for the comparison. In fact, Yaari (1981) weakens Pareto optimality to weak
Pareto optimality—i.e., no agent should bemade strictly better off from reallocation—to achieve an exact characterization. This weaker notion ofPareto optimality, what he calls
Pareto Principle , is “exactly” characterized by all utility vectors maximizing nonnegatively -weighted sums of utilities. Weak Pareto optimality, it should be noted, is not as satisfactoryas Pareto optimality from a normative viewpoint. eventual) exposure . Importantly, it is known that any face is “eventually exposed,” thatis, the face coincides with the set of points that sequentially maximize possibly negatively- weighted sum of utilities. While serving as a crucial step toward the proof, this result isnot sufficient for our characterization because we require the weights to be nonnegative andeventually positive. We prove that nonnegative and eventually positive weights can be foundif and only if the face consists of Pareto optimal points. The proof is nontrivial.Our characterization has several economic interpretations. First, we provide a sense inwhich our characterization “reveals” preferences of the social planner. Formally, for eachPareto optimal vector u , it maximizes a piecewise-linear concave social welfare functionwhose linear pieces are specified by the sequence of welfare weights that rationalize u , aresult reminiscent of a characterization of individual choices by Afriat (1967). Second, weestablish that the sequence of welfare weights identified by our characterization result can beinterpreted as the relative bargaining power of agents in a multi-round variant of generalizedNash bargaining.With the main characterization at hand, we next ask when our characterization reducesto a simple (that is, one-round) utilitarian welfare maximization. First, if individuals’ util-ities are concave and monotonic in an exchange economy setting, then Pareto optimality ischaracterized by utilitarian welfare maximization with nonnegative weights. In this case,the two sets U P and U + in Figure 3 coincide. Second, we identify a convex geometric condi-tion under which Pareto optimality is characterized by utilitarian welfare maximization withstrictly positive weights and find that the condition is met when individuals have piecewise-linear concave utility functions over a choice set that forms a convex polyhedron. In thiscase, the two inner sets in Figure 3 coincide. Last, we employ our methodology to generalizethe second welfare theorem, showing that it holds for all Pareto optimal allocations includingthose in which not all types of goods are consumed by all individuals (which are excluded inthe existing theorems) in exchange economies when individuals have piecewise-linear concaveutility functions that satisfy a mild monotonicity property.The remainder of the paper is organized as follows. Section 2 states the problem formallyand presents our characterization and its economic interpretations. Section 3 proves thecharacterization result. Section 4 explores conditions that enable simple characterizationsvia one-round utilitarian welfare maximization. Section 5 presents a version of the second-welfare theorem. Section 6 concludes. The proofs that are not provided in the main textcan be found in the appendix. Let I = { , , . . . , n } denote a finite set of agents and U ⊂ R n the set of possible utilityprofiles they may attain, or utility possibility set . We assume that U is closed and convex.If U stems from an underlying choice space X via utility functions ( u i ) i ∈ I : X → R n , then See Theorem 12.7 in Soltan (2015), reproduced as Lemma 3 below. Throughout, we use “ ⊂ ” to mean weak inclusion, or ⊆ , and likewise ⊃ means ⊇ . Strict inclusion willbe indicated by (cid:40) and (cid:41) .
6e let U = { ( u i ) i ∈ I ∈ R n | ∃ x ∈ X, ∀ i ∈ I, u i ∈ [ u i , u i ( x )] } (1)where u i = inf x ∈ X u i ( x ). That U is closed and convex is arguably a mild assumptionthat is satisfied if, for instance, U is induced by utility functions ( u i ) i ∈ I that are uppersemicontinuous and concave on a choice set X that is compact and convex. For any u, v ∈ U , we write v ≥ u if v i ≥ u i , ∀ i ∈ I , v > u if v ≥ u and v (cid:54) = u , and v (cid:29) u if v i > u i , ∀ i ∈ I . We say a point u in U is Pareto optimal , or maximal if there does notexist a v ∈ U with v > u . Let U P ⊂ U denote the set of all Pareto optimal points.For any φ ∈ R n , consider the optimization problem: β φ := max u ∈ U (cid:104) φ, u (cid:105) , (2)where (cid:104) φ, u (cid:105) := (cid:80) ni =1 φ i u i . We call φ a normal vector since it is the normal vector ofthe hyperplane { u ∈ R n | (cid:104) φ, u (cid:105) = β φ } . Throughout the paper, we only consider nonzeronormal vectors (i.e., φ (cid:54) = 0). We say a point u ∈ U maximizes the normal vector φ over U (or simply maximizes φ ) if u is an optimal solution to (2). We call a normal vector φ nonnegative if φ > positive if φ (cid:29)
0. For any vector v ∈ R n , the support of v is the setof indices where v is nonzero; i.e., supp v := { i ∈ I | v i (cid:54) = 0 } . A positive φ has full support;i.e., supp φ = I .Of particular interest are the points maximizing utilitarian welfare with nonnegativeweights: U + := { u ∈ U | ∃ φ > u ∈ arg max u (cid:48) ∈ U (cid:104) φ, u (cid:48) (cid:105)} and those maximizing utilitarian welfare with positive weights: U ++ := { u ∈ U | ∃ φ (cid:29) u ∈ arg max u (cid:48) ∈ U (cid:104) φ, u (cid:48) (cid:105)} . As noted in Figure 3, we have U ++ ⊂ U P ⊂ cl( U ++ ) ⊂ U + . Definition 1 (Sequential utilitarian welfare maximization) . We say u ∈ U sequentiallymaximizes a tuple Φ = ( φ , φ , . . . , φ T ) of normals over U if u ∈ U t := arg max u (cid:48) ∈ U t − (cid:104) φ t , u (cid:48) (cid:105) , for each t = 1 , . . . , T, (3)where U = U . We say u ∈ U sequentially maximizes utilitarian welfare over U ifthere is a tuple Φ = ( φ , φ , . . . , φ T ) of T ≤ n nonnegative normals (i.e., φ t >
0) with φ T (cid:29) u sequentially maximizes Φ. To be precise, the utility possibility set is often defined as { u ∈ R n | u = ( u i ( x )) i ∈ I for some x ∈ X } ,which differs from (1). However, the two sets share the same set of maximal points since those points areon the frontier. Thus, formulating the set U either way makes no difference for our results while the currentformation facilitates our analysis. Note that compactness and convexity of the choice set X are satisfied if, for instance, all lotteries ofsocial outcomes are feasible. heorem 1. Let U be a closed convex set. Then u is Pareto optimal if and only if u sequentially maximizes utilitarian welfare over U .This theorem states that even though simple utilitarian welfare maximization may notcharacterize Pareto optima, sequential utilitarian welfare maximization does. As discussedin the introduction, this characterization is useful in its own right—as exemplified by Che,Kim, and Kojima (2019)—but it also allows for interesting interpretations of Pareto optimaas follows.First, even though the maximization of a utilitarian welfare function does not charac-terize Pareto optimal points, our sequential characterization identifies a nonlinear welfarefunction that a Pareto optimal point maximizes. In fact, the tuple of normals Φ identifiedby Theorem 1 parameterizes a piecewise-linear concave (PLC) social welfare function thePareto optimal point maximizes. Corollary 1.
Let u ∈ U be a Pareto optimal point. Then u ∈ arg max u (cid:48) ∈ U W ( u (cid:48) ) , where W ( u (cid:48) ) := min t ∈{ ,...,T } (cid:104) φ t , ( u (cid:48) − u ) (cid:105) with φ t > , ∀ t , and φ T (cid:29)
0, where Φ is a tupleof normals identified in Theorem 1. Moreover, any point in arg max u (cid:48) ∈ U W ( u (cid:48) ) is Paretooptimal. Proof.
By Theorem 1, there is a tuple ( φ , . . . , φ T ) with which u sequentially maximizesutilitarian welfare over U . To show the first part, let u (cid:48) ∈ U and U , . . . , U T be the setsdefined in Definition 1. If u (cid:48) ∈ U T , then (cid:104) φ t , u (cid:48) (cid:105) = (cid:104) φ t , u (cid:105) for every t , so W ( u (cid:48) ) = 0 = W ( u ). If u (cid:48) (cid:54)∈ U T , then there exists t ∈ { , . . . , T } such that (cid:104) φ t , u (cid:48) (cid:105) < (cid:104) φ t , u (cid:105) , so W ( u (cid:48) ) =min t ∈{ ,...,T } (cid:104) φ t , ( u (cid:48) − u ) (cid:105) < W ( u ) . Therefore u ∈ arg max u (cid:48) ∈ U W ( u (cid:48) ).To show the second part, let v ∈ arg max u (cid:48) ∈ U W ( u (cid:48) ). From the proof of the first part, v ∈ U T . Therefore, by Theorem 1, v is Pareto optimal.Second, one can think of each Pareto optimum as emerging from a sequence of negoti-ations among individuals, where φ t shapes the agents’ relative bargaining power in round t negotiation. This idea can be fleshed out in the following way. First, we assume that eachagent has a disagreement utility, normalized as zero, that is less than any Pareto optimalutility: u ∈ U P means u (cid:29)
0. For a partition I = { I , . . . , I T } of I , imagine that the agentsengage in a sequence of bargaining: in round 1, agents in I bargain from U to a set V ⊂ U ,and in round t = 2 , . . . , T , agents in set I t bargain from V t − to a set V t . The bargainingprotocol in each round t is a generalized Nash bargaining game (Kalai, 1977) in which eachagent i ∈ I t has a bargaining power ψ i > (cid:80) i ∈ I t ψ i = 1 and a disagreement payoff This interpretation is reminiscent of Afriat’s theorem that also constructs a PLC function to rationalizeobserved consumer choices. In Afriat’s theorem, however, every normal vector used in the construction isstrictly positive since it coincides with the price vector associated with each choice. Also, in Afriat’s theorem,each choice associated with each normal vector is rationalized by the PLC utility function whereas, in ourresult, it is only the choices in the last step (i.e., U T ) that are rationalized by the constructed PLC function.
8. More specifically, for each t ≥ V t := arg max u ∈ V t − (cid:81) i ∈ I t u ψ i i and let the solution v of the bargaining be defined by v i = v ti for each i ∈ I t where v t is an arbitrary elementof V t (it turns out that v ti = w ti for every i ∈ I t if v t , w t ∈ V t ). We call such a bargainingprotocol a sequential generalized Nash bargaining game . Corollary 2.
Let u ∈ U be a Pareto optimal point and Φ = ( φ , . . . , φ T ) be the normalswith which u sequentially maximizes utilitarian welfare over U . Then, u is a solution to asequential generalized Nash bargaining game, where the round- t bargainers are I t = supp φ t \ ( (cid:83) t (cid:48) ≤ t − I t (cid:48) ) (with I = ∅ ) and their bargaining powers are ψ i = φ ti u i (cid:80) j ∈ It φ tj u j for each i ∈ I t . Proof.
Since the sets of bargainers are disjoint across rounds, the game is separated into acollection of generalized Nash bargaining games. We argue inductively that V t = { u (cid:48) ∈ U | u (cid:48) i = u i , ∀ i ∈ ∪ t − (cid:96) =0 I (cid:96) } ⊂ U t − , where U t − is defined in (3).To prove the claim, consider round t = 1. We wish to prove that V = { u (cid:48) ∈ U | u (cid:48) i = u i , ∀ i ∈ I } forms a solution to the round-1 generalized Nash bargaining game, or morespecifically, it solves:[ N B ] max u (cid:48) ∈ U W ( { u (cid:48) i } i ∈ I ) := (cid:89) i ∈ I ( u (cid:48) i ) ψ i . If | I | = 1, then u i , { i } = I , must maximize u (cid:48) i within U , and V is clearly the set ofoptimal solutions for [ N B ]. We thus assume | I | >
1. To solve [
N B ] for this case, considera relaxed program:[ N B (cid:48) ] max u (cid:48) ∈ R n W ( { u (cid:48) i } i ∈ I ) s.t. (cid:104) φ , u (cid:48) (cid:105) ≤ (cid:104) φ , u (cid:105) . If V is the set of optimal solutions for [ N B (cid:48) ], then V must also be the set of optimalsolutions for [ N B ] since V ⊂ U ⊂ { u (cid:48) ∈ R n | (cid:104) φ , u (cid:48) (cid:105) ≤ (cid:104) φ , u (cid:105)} . Since [ N B (cid:48) ] has astrictly concave objective function and a linear constraint with respect to ( { u (cid:48) i } i ∈ I ), first-order conditions completely characterize its optimal solution. They yield, for any i, j ∈ I , φ i φ j = ψ i u (cid:48) j ψ j u (cid:48) i ⇒ u (cid:48) i u (cid:48) j = u i u j . Suppose u (cid:48) j (cid:54) = u j for some j ∈ I . Then, either u (cid:48) i > u i for all i ∈ I or u (cid:48) i < u i forall i ∈ I . In the former case, the solution violates (cid:104) φ , u (cid:48) (cid:105) ≤ (cid:104) φ , u (cid:105) . In the latter case, W ( { u (cid:48) i } i ∈ I ) < W ( { u i } i ∈ I ) since W is strictly increasing. It follows that the optimalsolutions have u (cid:48) i = u i for all i ∈ I —i.e., V forms the optimal solutions for [ N B ].Since the first round bargaining pins down the utilities for all i ∈ I , the second roundbargaining by I occurs over a feasible set V = { u (cid:48) ∈ U | u (cid:48) i = u i , ∀ i ∈ I } . The argumentfrom this point onwards is analogous, noting that V t ⊂ V t − ⊂ U t − . Remark 1.
As mentioned earlier (in Footnote 4), there can be different sets of normals thatcharacterize the same maximal point: in Figure 4 for instance, φ = (1 ,
0) and φ = (0 ,
1) also9 u u (1 , ,
0) (0 , , , , φ = (1 , , , , − U U u = (1 / , / , − / (a) First round u u u U (0 , , φ = (0 , , φ = (2 , , u = (1 / , / , − / U (1 , , − (b) Second round Figure 5: Theorem 1 cannot be modified so that the supports of the normals in Φ partition I .work, selecting the vertical edge containing u from U and then u from this edge, respectively.This characterization has a special appeal since it lends a serial-dictatorship interpretationto Pareto optimal choice: in the example of Figure 4, agents 1 and 2 play as a dictator inthe first and second step, respectively. One way to generalize such a characterization beyondthe two-agent case would be to construct a finite set of normals Φ = ( φ , . . . , φ T ) wherethe supports of φ t for t = 1 , . . . , T partition I , so agents in supp φ t play jointly as step- t dictators.However, such a partition characterization of Pareto optimality does not hold when thereare more than two agents. Consider the maximal point u = (1 / , / , − /
2) of the closedconvex set U in Figure 5. In Step 1, the point u must maximize nonnegative normals over U of the form φ = ( α, α,
0) where α > α = 1), which entails U (shaded face weakly below the thick line segment) as the set of maximizers. For the normalsto partition I , the Step 2 normal should be of the form (0 , , β ) for some β > β = 1 in the normal ˜ φ in the figure). However, u does not maximize such a normal outof U in Step 2. Any normal maximized by u in Step 2 must assign positive weights to atleast one of the first two components, violating the partition structure. However, a strictlypositive normal φ = (2 , ,
1) is maximized by u among the points in U (indeed, every pointin the thick line segment U maximizes φ ). 10 Proof of Theorem 1
We now prove Theorem 1. The “if” direction is rather straightforward:
Proof of the “if ” part of Theorem 1.
Let u sequentially maximize utilitarian welfare over U .Then, there is a tuple ( φ t ) Tt =1 that is sequentially maximized by u and satisfies φ t > , ∀ t and φ T (cid:29)
0. Suppose to the contrary that u is not maximal so there is a point v ∈ U such that v > u . Observe then that (cid:104) φ t , v (cid:105) ≥ (cid:104) φ t , u (cid:105) , ∀ t , which implies v ∈ U t , ∀ t , since u ∈ U t , ∀ t .In particular, u, v ∈ U T − . However, φ T (cid:29) v > u imply (cid:104) φ T , v (cid:105) > (cid:104) φ T , u (cid:105) , whichcontradicts that u ∈ U T .The “only if” direction of the proof of Theorem 1 is nontrivial. We begin with somepreliminaries necessary for the proof. Only statements of results in this section are given.Proofs are either found in standard references (such as Soltan (2015)) or placed in theappendix when not shown elsewhere. Let us first introduce a few concepts that are crucial for our analysis. A face of U is anonempty convex subset F of U with the property that if u ∈ F and u = αv + (1 − α ) w for some 0 < α < v, w ∈ U then it must be that v, w ∈ F . That is, F is a face of aconvex set if none of its elements are convex combinations of elements that lie outside of F .A proper face of U is a face of U that is a proper subset of U . A face F is an exposedface of U if there is a normal φ ∈ R n such that F = arg max u ∈ U (cid:104) φ, u (cid:105) . In this case, we saythat φ exposes F out of U . A face need not be exposed, as can be seen in Figure 1, where u forms a singleton face (and is thus also an extreme point) that is not exposed. The face U in Figure 5 is an example of a higher-dimensional non-exposed face.For any convex subset G of U , its relative interior ri( G ) is the set of all u ∈ G such thatfor every u (cid:48) ∈ G there exists λ > u + λ ( u − u (cid:48) ) ∈ G .The following lemma shows a face structure of a convex set that is interesting in itselfand useful for our analysis. Lemma 1 (Corollary 11.11(a) in Soltan (2015)) . For a convex set U ⊆ R n , the collection ofrelative interiors of faces—that is, { ri( F ) : F is a face of U } —forms a partition of U .The next lemma shows that maximal points “come in faces.” Lemma 2.
Let u be a maximal point of a closed convex set U in the relative interior of aface F of U . Then, every point in F is maximal.Accordingly, we say a face is maximal if all of its elements are maximal. Importantlyfor our purpose, Lemma 1 and Lemma 2 imply that every maximal point of U belongs to arelative interior of a unique maximal face of U (possibly U itself).11he next result provides a key step of our argument: every face, possibly non-exposed,is eventually exposed. Lemma 3 (Theorem 12.7 in Soltan (2015)) . Let U ⊂ R n be a convex set and F be anonempty proper face of U . There is a sequence of convex sets ( G t ) Tt =0 such that F = G T ⊂ G T − ⊂ · · · ⊂ G ⊂ G = U, (4)where G t is a nonempty proper exposed face of G t − for each t = 1 , . . . , T .This lemma, which will be a crucial element of our proof, is already illustrated in theIntroduction. In Figure 4, the singleton face u is exposed in two steps: the vertical segmentis exposed first by a normal (1 , u is exposed by normal (1 ,
1) (among manyothers) out of that vertical segment. This lemma is not enough for our result, however,as it is silent about any additional properties on the normals that expose the sequence offaces. Crucially, our characterization requires the normals to be nonnegative and eventuallypositive.For these additional features, we need to introduce a set of analytical tools. Let J beany subset of the index set I and let χ J denote the vector whose i -th coordinate is equal to1 for every i ∈ J and equal to 0 for every i / ∈ J . When J is the singleton { i } we simplify χ { i } to χ i . A convex set U is downward closed in coordinates J ⊂ I if, for all u ∈ U and all τ ≥ u − τ χ K ∈ U for any subset K of J . A convex set that is downward closedin coordinates I is simply called downward closed . The downward closure of a closedconvex set U is the downward closed set dc( U ) := (cid:83) u ∈ U ( u − R n + ). It is straightforward tosee that dc( U ) is closed and convex if U is closed and convex.One useful feature of downward closure is that it preserves maximal elements and thusmaximal faces. Lemma 4.
The set of maximal elements of a closed convex set coincides with that of itsdownward closure. If F is a maximal face of U then F is a maximal face of dc( U ).Crucially for our arguments, supporting hyperplanes of downward-closed sets must havenonnegative normals. Lemma 5.
For any closed convex set U that is downward closed in coordinates J ⊂ I , anysupporting hyperplane of U has a normal φ with φ j ≥ , ∀ j ∈ J .The next lemma shows that downward closedness is preserved under maximization forthe coordinates to which the normal assigns zero weights. Lemma 6.
Let F be a face of a closed convex set U that is downward closed in coordinates J ⊂ I . If φ exposes F out of U , then F is downward closed in coordinates J \ supp φ .Armed with these preliminary observations, we are now ready to prove the “only if”direction of Theorem 1. Theorem 5 of Lopomo, Rigotti, and Shannon (2019) proves the same result for singleton faces F ; i.e.,extreme points. .2 Proof of the “only if ” direction Fix any maximal point u of U . We wish to show that u sequentially maximizes utilitarianwelfare over U . The proof consists of several steps. Step 1.
There exists a unique face F of dc( U ) such that u ∈ ri( F ). All points of F aremaximal in dc( U ). Proof.
By Lemma 4, u is a maximal point of dc( U ). By Lemma 1 there is a unique face F of dc( U ) which contains u in ri( F ). By Lemma 2, every point of F is maximal in dc ( U ), asdesired. Step 2.
The face F (containing u ) is a proper face of dc( U ). Proof.
If not, we must have F = dc( U ). Pick any u (cid:48) ∈ dc( U ). Then u (cid:48)(cid:48) = u (cid:48) − (cid:15) (1 , , . . . , U ) by the downward closure property. Clearly, u (cid:48)(cid:48) is not a maximal point ofdc( U ) and cannot belong to F by Step 1, a contradiction. Step 3.
There exists a sequence of convex sets ( G t ) Tt =0 of dc( U ) such that G t is a properexposed face of G t − for t = 1 , . . . , T , where G = dc( U ), G T = F , and T ≤ n . Proof.
Since F is a proper face of dc( U ) by Step 2, the result follows from Lemma 3. Forany set V , let dim( V ) denote its dimension. If V (cid:48) is a proper face of convex set V ,then dim( V (cid:48) ) < dim( V ) by Theorem 11.4 in Soltan (2015). Thus, we have T ≤ n sincedim( G t − ) < dim( G t ) and since dim( G ) = dim(dc( U )) = n . Step 4.
There exists a tuple Φ = ( φ , . . . , φ T ) such that for each t = 1 , . . . , T , G t = arg max x ∈ G t − (cid:104) φ t , x (cid:105) , where φ t > φ T (cid:29)
0, and supp φ t ⊃ supp φ t − . Proof.
By Step 3, there exists a sequence of normals Ψ = ( ψ , . . . , ψ T ) such that, for each t = 1 , . . . , T , ψ t exposes G t out of G t − . We construct Φ = ( φ , . . . , φ T ) with the statedproperties.The construction is recursive. First, since G = dc( U ), by Lemma 5, φ := ψ is non-negative. For an inductive hypothesis, suppose that there are φ k , k = 1 , .., t −
1, with thestated properties and that for each k = 1 , . . . , t − G k is downward-closed in coordinates J k := { i ∈ I | φ ki = 0 } = I \ supp φ k . Note that J t − ⊂ J t − ⊂ · · · ⊂ J := I . From now, weconstruct φ t and show that G t is downward-closed in coordinates J t = { i ∈ I | φ ti = 0 } .First, since ψ t is a normal for the supporting hyperplane of G t − and G t − is downward-closed in coordinates J t − , Lemma 5 implies that ψ tj ≥ j ∈ J t − . Consider The dimension dim( V ) of a convex subset V of U , including one of U ’s faces, is defined by the dimensionof its affine hull: aff( V ) := { (cid:80) kj =1 α j v j | k ∈ N , v j ∈ V, α j ∈ R , (cid:80) kj =1 α j = 1 } . i ∈ supp φ t − = I \ J t − . For such i , it is indeed possible for ψ ti to be negative. Butnoting φ t − i > i , we define φ t = λ t φ t − + ψ t , where λ t > max i ∈ supp φ t − | ψ ti | /φ t − i is a (sufficiently large) positive scalar. Given this con-struction, φ ti ≥ i ∈ I and φ ti > i ∈ supp φ t − , that is, supp φ t ⊃ supp φ t − .Let us show that φ t exposes G t out of G t − . To this end, let M t := max x ∈ G t − (cid:104) φ t − , x (cid:105) .For all x ∈ G t − , we have (cid:104) φ t , x (cid:105) = λ t (cid:104) φ t − , x (cid:105) + (cid:104) ψ t , x (cid:105) = λM t + (cid:104) ψ t , x (cid:105) , since (cid:104) φ t − , x (cid:105) = M t for all x ∈ G t − . Henceforth,arg max x ∈ G t − (cid:104) φ t , x (cid:105) = arg max x ∈ G t − (cid:104) ψ t , x (cid:105) = G t . Since G t − is downward-closed in coordinates J t − and φ t exposes G t out of G t − , Lemma 6implies that G t is downward-closed in coordinates J t − \ supp φ t = ( I \ supp φ t − ) \ supp φ t = I \ supp φ t = J t , where the penultimate equality holds since supp φ t − ⊂ supp φ t .It remains to show that φ T is positive. Supposing not, there must be some i ∈ I suchthat φ ti = 0 for all t = 1 , . . . , T , so i ∈ J t for all t = 1 , . . . , T . Then, Lemma 6 implies thatfor all t = 1 , . . . , T , G t is downward-closed in coordinate i , which contradicts the fact that G T = F is maximal.We have so far shown that u sequentially maximizes welfare over dc( U ). We now provethe main result: u sequentially maximizes welfare over U . To this end, the following laststep suffices. Step 5. u sequentially maximizes utilitarian welfare over U . Proof.
Recall a sequence of normals Φ from Step 4. Let U , U , . . . , U T be convex subsets of U such that, for each t = 1 , . . . , T , U t is the face of U t − exposed by normal φ t , i.e., U t = arg max x ∈ U t − (cid:104) φ t , x (cid:105) , where U := U . It suffices to prove that U T = F , as this will prove that u sequentiallymaximizes utilitarian welfare over U .To this end, it suffices to prove F ⊂ U t ⊂ G t for each t = 0 , , , . . . , T . We proceedinductively for the proof. First, note that the claim is trivially true for t = 0 because U := U ⊂ dc ( U ) := G and F ⊂ U = U by definition. Now, suppose that the claim holdsfor t . We show (i) F ⊂ U t +1 and (ii) U t +1 ⊂ G t +1 as follows.For (i), fix any point v in F . Then, since F ⊂ G t +1 and φ t +1 exposes G t +1 out of G t , wehave (cid:104) φ t +1 , v (cid:105) ≥ (cid:104) φ t +1 , w (cid:105) for every w ∈ G t . Because U t ⊂ G t by the inductive assumption, (cid:104) φ t +1 , v (cid:105) ≥ (cid:104) φ t +1 , w (cid:105) (5)14or every w ∈ U t . Moreover, v ∈ U t by the assumption that F ⊂ U t . This fact, combinedwith (5), implies that φ t +1 is maximized by v over U t and so v ∈ U t +1 , since φ t +1 exposes U t +1 out of U t . This holds for every v ∈ F and so F ⊂ U t +1 , implying (i) holds for t + 1.As for (ii), fix any point v in U t +1 . By (i), we know that (cid:104) φ t +1 , v (cid:105) = (cid:104) φ t +1 , w (cid:105) (6)for any w ∈ F , since U t +1 is exposed by φ t +1 and F is a subset of U t +1 . Also, by definitionof G t +1 and the fact F ⊂ G t +1 by construction, we know that (cid:104) φ t +1 , w (cid:105) ≥ (cid:104) φ t +1 , z (cid:105) (7)for any w ∈ F and z ∈ G t . Combining (6) and (7) implies that (cid:104) φ t +1 , v (cid:105) ≥ (cid:104) φ t +1 , z (cid:105) for any z ∈ G t . This, and the fact that v ∈ G t (which immediately follows from v ∈ U t +1 ⊂ U t ⊂ G t ),means that v ∈ G t +1 . Since this holds for any v ∈ U t +1 , we can conclude that U t +1 ⊂ G t +1 ,so (ii) holds for t + 1.This completes the induction and establishes the result. The previous section provided a precise rationalization of Pareto optimality in terms ofsequential utilitarian welfare maximization. The question remains, however, as to whenPareto optimality coincides with the simpler notions of utilitarianism: nonnegative andpositive. In particular, this section explores when U P coincides with either U + or U ++ .These conditions follow naturally from our characterization in Theorem 1. U P ) and Nonnegative Utilitarianism ( U + ) One case where U P = U + is rather well-understood in the literature; the case with strict convexity. Formally, we say that a set U ⊂ R n is strictly convex if u, v ∈ U and λ ∈ (0 , λu + (1 − λ ) v ∈ int( U ), where int denotes the interior of a set. It is well-known thatif U is closed and strictly convex, then U P = U + . Another setting of interest is the following exchange economy environment. Let there be m types of goods with some integer m >
0. For each k ∈ { , . . . , m } , let ¯ e k > k goods in the environment. Let ¯ e denote the vector (¯ e k ) mk =1 . Each alternative x = ( x i ) i ∈ I , x i = ( x ki ) mk =1 ∈ R m + , specifies consumption bundle x i for each i ∈ I . A profileof consumption bundles x is said to be feasible if and only if (cid:80) i ∈ I x i ≤ ¯ e . In this context, This result appears to be a folk result, and we do not know who first made this observation. Forcompleteness, we provide a proof sketch here. First, the fact U P ⊂ U + follows from Theorem 1. To provethe set inclusion relationship in the opposite direction, suppose for contradiction that u ∈ U maximizesa nonnegative normal φ but there exists v ∈ U such that v > u . Then, for any λ ∈ (0 , w := λu + (1 − λ ) v satisfies (cid:104) φ, w (cid:105) = λ (cid:104) φ, u (cid:105) + (1 − λ ) (cid:104) φ, v (cid:105) ≥ (cid:104) φ, u (cid:105) . Because U is strictly convex, w ∈ int ( U )and hence there exists x ∈ U such that x i > w i for every i ∈ I . Therefore we obtain (cid:104) φ, x (cid:105) > (cid:104) φ, w (cid:105) ≥ (cid:104) φ, u (cid:105) ,a contradiction. X is defined as the set of all feasible profiles of consumption bundles. Eachindividual i ∈ I is endowed with a utility function u i : R m + → R .Suppose that, for each i ∈ I , the utility function u i : R m + → R is concave. We say u i ( · )is strictly monotonic if u i ( x i ) > u i ( y i ) for every x i , y i ∈ R m + with x i > y i . Without loss ofgenerality, we normalize u i (0) = 0 for all i ∈ I : note that u i (0) = min x i ∈ R m + u i ( x i ) if u i ( · ) isstrictly monotonic because 0 ∈ R m + is the smallest element of R m + . The utility possibility set U is then defined as in (1) with u i = 0 , ∀ i ∈ I . Let us refer to a tuple E := ( I, ¯ e, ( u i ( · )) i ∈ I )as an economy. Theorem 2.
Let E be an economy where u i ( · ) is strictly monotonic for each i ∈ I . Theset of Pareto optimal points coincides with those that maximize nonnegative normals, i.e., U P = U + .We note a subtle but crucial difference between this result and existing results in generalequilibrium theory. In the latter, it is customary to restrict attention to a subset of Paretooptimal points that are supported by an alternative with the additional restriction thatevery individual receives a strictly positive amount of every type of good, i.e., x ki > i ∈ I and k ∈ { , . . . , m } (see Argenziano and Gilboa (2015) for instance). Theorem 2,by contrast, does not make any such restriction and characterizes the entire set of Paretooptimal points.It is worth noting that strict monotonicity of utility function differs from local nonsatia-tion, a commonly assumed condition in general equilibrium theory (see, for instance, Section16.C in Mas-Colell, Whinston, and Green (1995)). We say that a utility function u i : R m + → R is locally nonsatiated if, for any x i ∈ R m + and (cid:15) >
0, there exists y i ∈ { y i ∈ R m + | | y i − x i | < (cid:15) } with u i ( y i ) > u i ( x i ). The following example shows that the characterization in Theorem 2does not hold if we weaken the strict monotonicity to local nonsatiation. Example 1.
Suppose that there are two individuals, 1 and 2, as well as two types of divisiblegoods 1 and 2 with unit supply each, i.e., ¯ e = (1 , u ( x , x ) = x ,u ( x , x ) = √ x + x . Note that these utility functions satisfy local nonsatiation, but u ( · ) fails strict monotonicityas it is constant in x . The utility possibility set coincides with U in Figure 1, so U P doesnot coincide with U + . U P ) and Positive Utilitarianism ( U ++ ) The goal of this subsection is to discover natural conditions for U P to coincide with U ++ .The following corollary, which follows easily from the proof of Theorem 1, is the key to ourinvestigation. 16 orollary 3. If u is a maximal element of U that lies in the relative interior of an exposedface of dc( U ) then u maximizes a positive normal over U . Proof.
In the proof of the “only if” part of Theorem 1 in Section 3.2, if u is a maximalelement of U that lies in the relative interior of an exposed face of dc( U ), then T = 1 inStep 3 and by Step 4 we know φ is positive. Hence, Φ = ( φ ) and so by Step 5, we concludethat u maximizes the positive normal φ over U .This corollary allows us to prove the following characterization of when U P = U ++ . Theproof uses the concept of a normal cone and some of its properties, details of which are foundin Appendix C. Theorem 3.
Let U be a closed convex set. Then U P = U ++ if and only if every maximalelement of U belongs to some exposed maximal face of dc( U ).We now discuss a few of the nuances in the statement of Theorem 3. First, the conditioncannot be weakened so that every maximal element of U simply lies in a (potentially non-maximal) exposed face of dc( U ). Consider our canonical example in Figure 1. The point u lies on an exposed face of dc( U ) but this face is not a maximal face of dc( U ).Figure 1 also demonstrates that it is not sufficient for a point to lie on a maximal exposedface of U (as opposed to dc( U )) to guarantee it maximizes a positive normal. Consider thepoint u (cid:48)(cid:48) , which is a maximal exposed extreme point of U , but clearly does not maximizeany positive normal over U . However, u (cid:48)(cid:48) does not lie on a maximal exposed face of dc( U )and so does not contradict the theorem.Given the above nuance, a simpler sufficient condition may be useful. Consider the settingwhere all maximal faces of dc( U ) are exposed. Corollary 4. If U is a closed convex set such that all maximal faces of dc( U ) are exposed,then U P = U ++ . Proof.
Note that every maximal element of U lies in a maximal face of dc( U ) by Lemma 4.This and the hypothesis imply that every maximal element of U belongs to some exposedmaximal face of dc( U ). Applying Theorem 3, we obtain the desired conclusion.However, the converse of Corollary 4 is false, as illustrated by the example in Figure 6.One sufficient condition for the hypothesis of Corollary 4 to hold is that U is a polyhedron.In that case, all faces of U are all exposed (Theorem 13.21 of Soltan (2015)); moreover, itsdownward closure of a polyhedron is also a polyhedron (Theorem 13.20 of Soltan (2015)),so all of its faces are exposed. Utility possibility sets that arise as polyhedra is not an This cannot be derived easily from Arrow, Barankin, and Blackwell (1953). To see this, recall thatthey establish U ++ ⊂ U P ⊂ cl( U ++ ). This implies that if U ++ is closed then U P = U ++ . However, in the“tilted cone” in Figure 2, U ++ is not closed since the point K does not lie in U ++ but is the limit pointof elements in U ++ (indicated by the line in the figure). However, it is straightforward to check that U P and U ++ coincide. One can also check that all maximal faces of dc( U ) for U in Figure 2 are exposed, thecondition of Corollary 4. u φu (cid:48)(cid:48) φ (cid:48)(cid:48) φ Figure 6: The maximal extreme point u is not exposed while U P = U ++ .uncommon phenomenon. For example, the following special class of utility functions givesrise to such a case.Let X be a polyhedral subset of R m + (possibly R m + itself). The utility function u i : X → R is piecewise-linear concave (PLC) if there exist finite index set K i and affine functions u i,k : R m + → R for each k ∈ K i such that u i ( x ) = min k ∈ K i u i,k ( x ) for all x ∈ X .PLC utility functions have appeared elsewhere in the literature. For instance, Afriat’stheorem (Afriat (1967)) shows that they arise naturally in the context of revealed preferences.Also, the PLC case features prominently in the computer science literature on questions ofhardness in computing equilibria (see Chen, Dai, Du, and Teng (2009) and Garg, Mehta,Vazirani, and Yazdanbod (2017) for instance). Moreover, it is well-known that concavefunctions can be approximated arbitrarily well by PLC functions with sufficiently manypieces (see, for instance, Bronshteyn and Ivanov (1975); Ghosh, Pananjady, Guntuboyina,and Ramchandran (2019)). Proposition 1.
If each agent has a PLC utility function defined on a polyhedron X and U is defined according to (1), then dc( U ) is a polyhedron.The following is obtained immediately from Corollary 4 and Proposition 1, and the factthat all faces of polyhedra are exposed. It is a clean economic setting where U P and U ++ coincide. Theorem 4.
If each agent has a PLC utility function defined on a polyhedron X and U isdefined according to (1), then U P = U ++ . It is worth noting that the ABB theorem provides an alternative proof of this result. Recall that itsuffices to argue U ++ is closed in order to conclude U P = U ++ . Clearly, the elements of U ++ comes infaces, and a polyhedron has finitely-many faces. Since faces of a polyhedron are closed, and a finite union ofclosed sets is closed, this implies that U ++ is closed. Second Welfare Theorem with Piecewise Linear Con-cave Utility Functions
The notions of exposed face and normal vector play crucial roles for our characterization ofa Pareto optimal utility profile as a welfare-maximizing point. Recall that the normal vectoralso plays an important role in the second theorem of welfare economics in identifying aprice vector that supports a Pareto optimal allocation as a competitive equilibrium outcome.Unlike in our characterization, the idea of a normal vector in the second welfare theoremapplies to the space of goods, not the space of utility profiles. However, the fact that thetwo spaces are closely connected hints at the possibility of establishing the second welfaretheorem using the machinery we have developed so far. We do so in the current sectionunder a set of assumptions on the agent preferences and endowments that generalize theexisting welfare theorem in a certain direction.To begin, consider an exchange economy with n agents (index by i ) and m goods (indexedby k ) introduced in Section 4.1. Suppose that each agent i is endowed with a vector of goods e i ∈ R m + \{ } and let ¯ e = (cid:80) i ∈ I e i . A vector p ∈ R m is referred to as a price profile. A pair( p, x ) of a price profile p and a profile x = ( x i ) i ∈ I of consumption bundles is a Walrasianequilibrium if1. (cid:80) i ∈ I x i = ¯ e, and2. x i ∈ arg max y i ∈ B i ( p ) u i ( y i ) for each i ∈ I , where B i ( p ) := { y i ∈ R m + | (cid:104) p, y i (cid:105) ≤(cid:104) p, e i (cid:105)} is the budget set of i .We consider a case where utility functions of all players are piecewise-linear concave(PLC), as defined in Section 4.2. PLC utility functions may appear somewhat restrictive,but as noted earlier any concave function can be approximated arbitrarily closely by a PLCutility function. Meanwhile, we make a weaker assumption in another dimension—preferencemonotonicity. The existing second welfare theorem assumes agents’ utility functions to bestrictly monotonic. We invoke a weaker form of monotonicity. Say that an allocation ( x i ) i ∈ I is strictly feasible for good k if it is feasible and satisfies (cid:80) i ∈ I x ki < ¯ e k . We assume that theagent preferences are monotonic under limited resources in the following sense: for anyallocation ( x i ) i ∈ I that is strictly feasible for good k , there exist an agent j and ˜ x j ∈ R m + suchthat u j (˜ x j ) > u j ( x j ) while ˜ x k (cid:48) j = x k (cid:48) j , ∀ k (cid:48) (cid:54) = k , ˜ x kj > x kj , and ˜ x kj + (cid:80) i (cid:54) = j x kj ≤ ¯ e k . That is,given any allocation that does not exhaust the endowment of good k , there exists an agentwho gets better off by consuming more of that good within its endowment. This condition isfairly weak. For instance, it allows for agents to consider certain good indifferently or evenas bads (rather than goods), as long as there is at least one agent who likes to consume thatgood. We are now ready to prove the second welfare theorem under the above assumptions. Theorem 5.
Consider the exchange economy described above. If ( u i ( e i )) i ∈ I is Pareto op-timal, then there exists a positive price vector p (cid:29) p, ( e i ) i ∈ I ) is a Walrasianequilibrium.In addition to the weakening of preference monotonicity, we also dispense with the typicalassumption required by the existing second-welfare theorem that every consumer have a19ositive endowment for every type of good (i.e., e i (cid:29) , ∀ i ∈ I ). The positive endowmentassumption can be quite restrictive, excluding many realistic situations. In fact, relaxingthe same assumption was an important motivation behind Arrow’s generalization of the firstwelfare theorem. At the same time, the theorem assumes PLC utility functions. This assumption guar-antees that the “upper contour set” of the target allocation—or the set of goods weaklypreferred to ( e i ) i ∈ I —is a polyhedron. Meanwhile, preference monotonicity and Pareto-optimality of ( u i ( e i )) i ∈ I ensure that the vector ¯ e is a (minimal) face of this set. InvokingTheorem 4, ¯ e is then exposed by a positive normal (or price vector) that supports ( e i ) i ∈ I asa competitive equilibrium allocation. In this paper, we characterized Pareto optimality via a new notion of sequential utilitarianwelfare maximization. We established a strong connection between Pareto optimality and thegeometric concept of exposed faces, where sequentiality was tied to the notion of “eventual”exposure. We used these insights to obtain conditions for characterizations by simpler,nonsequential utilitarian welfare maximizations and highlighted implications for polyhedralsets of utility vectors (and their associated PLC utility functions) whose faces are all exposed.This connection allowed us to establish a second welfare theorem in economies with PLCutilities under weaker regularity conditions than those in the existing literature.The application of our methodology to the second welfare theorem suggests two relatedareas of exploration for future work. The first relates to further exploration of how our mainresults drive implications for problems stated in the choice space X , as opposed to the utilitypossibility space U . Indeed, examining the structure of what points in the choice set give riseto Pareto optima has been a major focus in the multi-objective optimization literature. Anearly contribution in that literature is Charnes and Cooper (1967), who showed an equiva-lence between the problem of finding Pareto-optimal solutions (in the choice set X ) and thatof solving a constrained nonlinear programming problem. Following their contribution, tech-niques in nonlinear programming were utilized to characterize Pareto optima under variousconditions (Ehrgott, 2005; Ben-Israel, Ben-Tal, and Charnes, 1977; Van Rooyen, Zhou, andZlobec, 1994; Glover, Jeyakumar, and Rubinov, 1999; Ben-Tal, 1980) all of which requiresome form of differentiability of the utility functions. We believe further investigation intoour approach may have the potential to add to this literature in at least two aspects. First,our characterization does not assume any form of differentiability. Indeed, the subtlety in-volving non-exposure of maximal faces often arises when utility functions are not smooth(e.g., Example 1). Our methods may suggest ways to handle Pareto optimality when dif-ferentiability fails. Second, our methods may suggest a bridge between existing results inthe domain space and results in the utility possibility space, where notions of (sequential) “While listening to a talk about housing by Franko Modigliani, Arrow realized that most people consumenothing of most goods (for example living in just one particular kind of house), and thus that the prevailingefficiency proofs assumed away all the realistic cases,” according to Geanakoplos (2019). A Proofs of preliminary results for Theorem 1
A.1 Proof of Lemma 2
The stated result is immediate in the case F is a singleton, so we may assume that F isnot a singleton. Suppose for contradiction that F contains a nonmaximal element u (cid:48) . Thus,there exists a v ∈ U such that v > u (cid:48) . Since u ∈ ri( F ), there exists λ > w (cid:48) = u + λ ( u − u (cid:48) ) ∈ F . Now let z = αw (cid:48) + (1 − α ) v , where α = λ or α (1 + λ ) = 1. Notethat z ∈ U since U is convex. Also, z = α ( u + λ ( u − u (cid:48) )) + (1 − α ) v = u − αλu (cid:48) + (1 − α ) v = u + (1 − α )( v − u (cid:48) ) > u, contradicting the maximality of u . A.2 Proof of Lemma 4
Let U be a closed convex set and dc( U ) its downward closure. Let u be a maximal elementof dc( U ); that is, ( u + R n + ) ∩ dc( U ) = { u } . If u ∈ U then this implies ( u + R n + ) ∩ U = { u } since U ⊂ dc( U ) and so u is a maximal element of U . Note that if u ∈ dc( U ) \ U then itcannot be maximal. Indeed, this implies that u = v − w for some v ∈ U and nonzero w ∈ R n + and so v > u and so u is not maximal.Conversely, we prove the contrapositive. Suppose u ∈ dc( U ) is not a maximal element.This implies that there exists a w (cid:54) = u with w ∈ dc( U ) and w ≥ u . But then we can find a v ≥ w ≥ u and v (cid:54) = u and v ∈ U . This implies that u is not a maximal element of U .We next prove the second statement. To see that F is a face of dc( U ), consider any x, y ∈ dc( U ) and λ ∈ (0 ,
1) such that z = λx + (1 − λ ) y ∈ F . We need to show that21oth x and y belong to F . We first show that x and y are both maximal. Suppose forcontradiction that x is not maximal. Then, we must have some x (cid:48) ∈ dc( U ) such that x (cid:48) > x .Let z (cid:48) = λx (cid:48) + (1 − λ ) y and observe that z (cid:48) ∈ dc( U ), z (cid:48) ≥ z , and z (cid:48) (cid:54) = z , which contradicts themaximality of z . Given that x and y are both maximal, we must have x, y ∈ U since thereis no maximal point in dc( U ) \ U . That F is a face of U then implies x, y ∈ F as desired. A.3 Proof of Lemma 5
Suppose there exists a supporting normal φ (i.e., there exists a β φ such that (cid:104) φ, u (cid:105) ≤ β φ for all u in U ) with a negative component φ j for some j ∈ J . Let v be an arbitraryelement of U . Since U is downward closed in coordinates J , we also have v − λχ j ∈ U for any λ ≥
0, where χ j is the unit vector with 1 in component j . However, observe that (cid:104) φ, v − λχ j (cid:105) = (cid:104) φ, v (cid:105) − λ (cid:104) φ, χ j (cid:105) = (cid:104) φ, v (cid:105) − λφ j . But (cid:104) φ, v (cid:105) − λφ j → ∞ as λ → ∞ since φ j < φ is a supporting normal. A.4 Proof of Lemma 6
Take any j ∈ K := J \ supp φ and set u (cid:48) = u − (cid:15)χ j for some u ∈ F and (cid:15) >
0. Since U is downward closed in coordinates J and j ∈ J , we have u (cid:48) ∈ U . Moreover, (cid:104) φ, u (cid:48) (cid:105) = (cid:104) φ, u − (cid:15)χ j (cid:105) = (cid:104) φ, u (cid:105) − (cid:15) (cid:104) φ, χ j (cid:105) = (cid:104) φ, u (cid:105) − (cid:15)φ j = (cid:104) φ, u (cid:105) since φ j = 0 when j ∈ K since noelement of K lies in supp φ . But then u (cid:48) ∈ F since (cid:104) φ, u (cid:48) (cid:105) = (cid:104) φ, u (cid:105) = max v ∈ U (cid:104) φ, v (cid:105) and F = max v ∈ U (cid:104) φ, v (cid:105) since F is exposed by φ . B Proof of Theorem 2
Proof.
The relationship U P ⊂ U + follows from Theorem 1 because if u is Pareto optimal,then it maximizes a sequential set of normals Φ = ( φ , . . . , φ T ), and hence u maximizes anonnegative normal φ . In the remainder of this proof, we will show the relation U + ⊂ U P .Suppose for contradiction that the desired conclusion does not hold. Then there exists u ∈ U + \ U P . More specifically, there exists some u (cid:48) ∈ U such that u (cid:48) > u while u ∈ arg max v ∈ U (cid:104) φ, v (cid:105) for some nonnegative normal φ . We first note that φ j = 0 for every j ∈ I such that u (cid:48) j > u j . This is because otherwise φ j > u (cid:48) j > u j , but this and the factthat u (cid:48) ≥ u imply (cid:104) φ, u (cid:48) (cid:105) > (cid:104) φ, u (cid:105) , contradicting the assumption that u ∈ arg max v ∈ U (cid:104) φ, v (cid:105) .Now, fix j ∈ I with φ j = 0 and u (cid:48) j > u j : Note that there exists such j ∈ I because u (cid:48) > u .Also, fix j (cid:48) ∈ I such that φ j (cid:48) >
0; note that such j (cid:48) exists because φ is a nonnegative normaland that j (cid:48) (cid:54) = j because φ j = 0. Then, because u j ≥ u (cid:48) j > u j ≥
0. Let x = ( x i ) i ∈ I ∈ X be such that u (cid:48) ≤ ( u i ( x i )) i ∈ I : Such x exists by the definition of U and the assumption that u (cid:48) ∈ U . Thenit follows that u j ( x j ) ≥ u (cid:48) j >
0, so x j ≥ x j (cid:54) = 0 because x j ∈ R m + and u j (0) = 0. Thenconsider an alternative consumption profile y ∈ R m + defined as y j = 0 , y j (cid:48) = x j (cid:48) + x j , and x i = y i for all i (cid:54) = j, j (cid:48) ; Note that y ∈ X (this is because (cid:80) i ∈ I y i = (cid:80) i ∈ I x i by definition of y and (cid:80) i ∈ I x i ≤ ¯ e by the assumption that x ∈ X ) and hence u ( y ) ∈ U . Then, because the22tility function u j (cid:48) ( · ) is strictly monotonic by assumption while x j ≥ x j (cid:54) = 0, we have u j (cid:48) ( y j (cid:48) ) > u j (cid:48) ( x j (cid:48) ) ≥ u (cid:48) j (cid:48) ≥ u j (cid:48) . Moreover, by construction we have u i ( y i ) = u i ( x i ) ≥ u (cid:48) i ≥ u i for every i (cid:54) = j, j (cid:48) . Therefore, because φ j = 0 and φ j (cid:48) >
0, we have (cid:104) φ, u ( y ) (cid:105) > (cid:104) φ, u (cid:105) , whichis a contradiction to the assumption that u ∈ arg max v ∈ U (cid:104) φ, v (cid:105) . C Proof of Theorem 3
The proof uses results from the following three lemmas:
Lemma C.1 (Line Segment Principle, see Proposition 1.3.1 in Bertsekas (2009)) . Let U bea closed convex set. If u ∈ ri( U ) and v ∈ U , then [ u, v ) ∈ ri( U ), where [ u, v ) := { u (cid:48) ∈ U | u (cid:48) = λu + (1 − λ ) v, ∃ λ ∈ (0 , } .The normal cone of U at a point u ∈ U is the set N U ( u ) = { φ ∈ R n | (cid:104) φ, u (cid:105) ≥ (cid:104) φ, v (cid:105) for all v ∈ U } . If φ ∈ N U ( u ) then u is a maximizer of the linear function (cid:104) φ, u (cid:105) over the set U . Lemma C.2.
Let F be a face of a convex set U . Then every point in the relative interiorof F has the same normal cone. Proof.
Let u, u (cid:48) be distinct in the relative interior of F and suppose N U ( u ) contains anelement φ not in N U ( u (cid:48) ). This implies (cid:104) φ, u (cid:105) > (cid:104) φ, u (cid:48) (cid:105) . Since u is the relative interior,the point v = u + λ ( u − u (cid:48) ) lies in F for a sufficiently small positive λ . But, (cid:104) φ, v (cid:105) = (cid:104) φ, u (cid:105) + λ (cid:104) φ, u − u (cid:48) (cid:105) > (cid:104) φ, u (cid:105) , violating the assumption that φ is in N U ( u ).The above result lets us define the normal cone of a face F of U , denoted N U ( F ), as thenormal cone of each of its relative interior points.Next, let us consider the relative boundaryof F , defined as F \ ri( F ). As the next result shows, the relative boundary points of a face F must contain N U ( F ) and additional normal vectors. Lemma C.3.
Let F be a face of a convex set U . Then every relative boundary point u of F has N U ( u ) ⊃ N U ( F ). Proof.
Let u be in the relative boundary of F . Suppose there is a normal φ in N U ( v ) (where v is any relative interior element of F ) that is not in N U ( u ). That is, (cid:104) φ, u (cid:105) (cid:54) = (cid:104) φ, v (cid:105) . (8)By the Line Segment Principle, we can get an element of relative interior of F arbitrarilyclose to u , which yield a contradiction of the continuity of (cid:104) φ, ·(cid:105) because of (8). Proof of Theorem 3. ( ⇐ ) Observe that U ++ ⊂ U P is immediate from Proposition 16.E.2 inMas-Colell, Whinston, and Green (1995). It remains to show U P ⊂ U ++ . Let u ∈ U P . If u lies in the relative interior of an exposed face of dc( U ), then u ∈ U ++ from Corollary 3.23he remaining case is where u lies on the relative boundary of a maximal exposed face F of dc( U ). Since F is a maximal exposed face, then an element v in its relative interiormaximizes a positive normal φ , again by Corollary 3. By Lemma C.2, this implies that thenormal cone N U ( F ) of face F contains φ and so, by Lemma C.3, the normal cone N U ( u ) ofthe point u contains φ . In other words, u maximizes the positive normal φ . This completesthe proof.( ⇒ ) Let u be a maximal element of U . By the equivalence of U P and U ++ , u maximizesa positive normal φ . Let F = arg max v ∈ U (cid:104) φ, v (cid:105) . We claim that F is a maximal exposedface of dc( U ), which clearly contains u . The fact that F is maximal in dc( U ) follows sinceProposition 16.E.2 in Mas-Colell, Whinston, and Green (1995) (along with Lemma 2) implies F is maximal in U and thus maximal in dc( U ) by Lemma 4. Suppose to the contrary that F is not exposed in dc( U ). Then, there must exist an element u (cid:48) ∈ dc( U ) \ U that maximizes φ but is not in F . However, since u (cid:48) in dc( U ) \ U there must exist a u (cid:48)(cid:48) ∈ U such that u (cid:48) ≤ u (cid:48)(cid:48) and u (cid:48) i < u (cid:48)(cid:48) i for some index i . But, this implies that (cid:104) φ, u (cid:105) ≥ (cid:104) φ, u (cid:48)(cid:48) (cid:105) > (cid:104) φ, u (cid:48) (cid:105) , where theweak inequality holds by the definition of F and the strict inequality holds since φ is positive.This yields a contradiction and so we conclude that F is an exposed face of dc( U ). D Proof of Proposition 1
For each k ∈ K i , let X i,k = { x ∈ X | u i,k ( x ) ≤ u i,k (cid:48) ( x ) , ∀ k (cid:48) ∈ K i } . Since X is a polyhedronand all functions ( u i,k ) k ∈ K i are affine, X i,k is an intersection of finitely many polyhedra andthus a polyhedron.Now let K = { k = ( k i ) i ∈ I | k i ∈ K i for all i } . For each k ∈ K , let X k = ∩ i ∈ I X i,k i andobserve that X k is a polyhedron. Also, all functions u ( · ) , . . . , u I ( · ) are affine on X k since foreach i ∈ I , u i ( x ) = u i,k i ( x ) , ∀ x ∈ X k . Then, by Theorem 13.21 of Soltan (2015), the set U k = { ( u i ( x )) i ∈ I | x ∈ X k } is a polyhedron. Observe that U = { ( u i ( x )) i ∈ I | x ∈ X } = ∪ k ∈K U k .While we do not know whether the set U , which is a union of polyhedra, is a polyhedron,Theorem 13.19 of Soltan (2015) shows that U := cl(conv ∪ k ∈K U k ) is a polyhedron, where cland conv denote the closure and convex hull, respectively.Next, we show that dc( U ) = dc( U ). By definition of U , dc( U ) ⊂ dc( U ) is clear. Toshow dc( U ) ⊂ dc( U ), consider any ˜ u ∈ conv ∪ k ∈K U k so that ˜ u = (cid:80) k ∈K λ k ˜ u k for someweight ( λ k ) k ∈K and ˜ u k ∈ ∪ k (cid:48) ∈K U k (cid:48) . Also, for each ˜ u k , we can find ˜ x k ∈ X k such that( u i (˜ x k )) i ∈ I = ˜ u k . Letting x = (cid:80) k ∈K λ k ˜ x k , observe that x ∈ X by the convexity of X andthat for all i ∈ I , u i ( x ) ≥ (cid:80) k ∈K λ k u i (˜ x k ) = ˜ u i by the concavity of u i ( · ), which means that˜ u ∈ dc( U ). Thus, conv ∪ k ∈K U k ⊂ dc( U ), implying that cl(conv ∪ k ∈K U k ) ⊂ dc( U ) sincedc( U ) is closed, from which dc( U ) ⊂ dc( U ) follows, as desired.Lastly, observe that dc( U ) = U + R n − and that both U and R n − are polyhedra, whichimplies (by Theorem 13.20 of Soltan (2015)) that dc( U ) = dc( U ) is a polyhedron.24 Proof of Theorem 5
The proof of Theorem 5 uses the following preliminary results.
Lemma E.1.
The following properties on polyhedra hold:(i) Let P , P , . . . , P n be a finite collection of polyhedra in R m . The Cartesian product P × P × · · · × P n is a polyhedron in R mn .(ii) Let π : R d → R m be a affine map and let P be a polyhedron in R d . Then π ( P )is a polyhedron.(iii) All faces of a polyhedron are exposed.(iv) The downward closure of a polyhedron is also a polyhedron. Proof. (i) Consider two polyhedra in R m , P and P . Letting Q := P × R m and Q := R m × P , each Q k is a polyhedron in R m , so P × P = ∩ k =1 , Q k is a polyhedron in R m . Theresult follows from applying this argument repeatedly. (ii) This is Theorem 13.21 in Soltan(2015). (iii) This is Corollary 13.12 in Soltan (2015). (iv) This follows since dc( P ) = P + R n − where R n − is the nonpositive orthant of R n and applying Theorem 13.20 of Soltan (2015).Let A i := { x ∈ R m + | u i ( x ) ≥ u i ( e i ) } for each agent i . Observe that each A i is apolyhedron since it is an intersection of two polyhedra, { x ∈ R m | x ≥ } and { x ∈ R m | u i ( x ) ≥ u i ( e i ) } = ∩ k ∈ K i { x ∈ R m | u i,k ( x ) ≥ u i ( e i ) } Consider the set A = (cid:8) x ∈ R m + | ∃ x ∈ A , x ∈ A , . . . , x n ∈ A n s.t. x = (cid:80) i ∈ I x i (cid:9) . Ob-serve that A is the image of the set A × A × · · · × A n under the affine mapping π thatmaps ( x i ) i ∈ I to (cid:80) i ∈ I x i . By Lemma E.1(i) and (ii), A itself is a polyhedron.Next, we argue that ¯ e is a minimal element of the set A . Suppose for contradiction thatthere exists an element x ∈ A where x < ¯ e where x k < ¯ e k for some good k . Since x ∈ A ,there exists an allocation ( y i ) i ∈ I where y i ∈ A i such that x = (cid:80) i ∈ I y i . Since this allocationis strictly feasible for the good k , the monotone preference under limited resources impliesthat there are some agent j and ˜ y j ∈ R m + such that u j ( y j ) < u j (˜ y j ) while ˜ y k (cid:48) j = y k (cid:48) j , ∀ k (cid:48) (cid:54) = k ,˜ y kj > y kj , and ˜ y kj + (cid:80) i (cid:54) = j y ki ≤ ¯ e k . Now consider an alternative allocation ( z i ) i ∈ I , whichis identical to ( y i ) i ∈ I except that z j = ˜ y j . Note that this allocation is feasible under theendowment ¯ e and that u j ( z j ) > u j ( y j ) ≥ u j ( e j ) while u i ( z i ) = u i ( y i ) ≥ u i ( e i ) , ∀ i (cid:54) = j , whichcontradicts the Pareto optimality of ( e i ) i ∈ I .That ¯ e is a minimal element of A implies that − ¯ e is a maximal element of − A . ByLemma 4, this implies that − ¯ e is a maximal element of dc( − A ). Moreover, by Lemma E.1(iv)dc( − A ) is a polyhedron and so by Lemma E.1(iii) all of its faces are exposed. Thus, byCorollary 3, there exists a supporting hyperplane of − A through the point − ¯ e with a positivenormal φ . The same normal p := φ can define a supporting hyperplane to A through thepoint ¯ e ; that is, (cid:104) p, y (cid:105) ≥ (cid:104) p, ¯ e (cid:105) , ∀ y ∈ A, where p is a strictly positive vector of prices.It remains to show that the positive price vector p just constructed supports the allocation( e i ) i ∈ I as a Walrasian equilibrium. For this, it suffices to show that each e i maximizes u i ( · )25nder the prices p and the budget (cid:104) p, e i (cid:105) . To do so, we take any x i with u i ( x i ) > u i ( e i ) andwill show that agent i cannot afford x i .By continuity of u i , the inequality u i ( x i ) > u i ( e i ) implies that for some λ < u i ( λx i ) > u i ( e i ), so by definition we have λx i ∈ A i . Thisimplies that λx i + (cid:80) j (cid:54) = i e j ∈ A . Since (cid:104) p, λx i + (cid:80) j (cid:54) = i e j (cid:105) ≥ (cid:104) p, (cid:80) i ∈ I e i (cid:105) , we must also have (cid:104) p, λx i (cid:105) ≥ (cid:104) p, e i (cid:105) . Dividing through by λ gives (cid:104) p, x i (cid:105) ≥ (cid:104) λ p, e i (cid:105) > (cid:104) p, e i (cid:105) where the strictinequality holds since e i is nonnegative and nonzero while p is strictly positive. References
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