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Essays on sorting with financial securities Lam, Wing Tung 2017

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Essays on Sorting with FinancialSecuritiesbyWing Tung LamA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2017c© Wing Tung Lam 2017AbstractThis dissertation studies one-to-one matching between workers and as-sets in a market where financial securities are offered. The quality of anasset is publicly known, but a worker’s productivity is private information.The asset side first posts contracts, under which the payment is contingenton the realized output. Then the workers direct their search based on theoffers. Production exhibits complementarity so that the efficient allocationfeatures positive assortative matching (PAM).I consider a frictionless setting in the first chapter. First, I characterizethe sufficient and necessary conditions for decentralizing PAM. For any dis-tribution of types, these conditions ensure that the set of posted contractsnot only induces the workers to sort assortatively but also precludes the as-set owners from poaching. In comparison with the case of full information,the asset side’s share of the matching surplus is always greater and increaseswith the asset quality at a faster rate in equilibrium. Second, I show that allasset owners will always be better off if the feasible contracts are replacedwith steeper ones, which cost better workers more than weaker workers.The second chapter focuses on the class of output sharing contracts. Istudy how it affects the matching efficiency and sorting pattern in the pres-ence of search friction. The unique equilibrium features inefficient PAM.The matched pairs fully separate into a continuum of markets, where theiiAbstractqueue length in each market still maximizes the expected surplus given theworker’s equilibrium payoff. However, regardless of the distribution of types,all but the best workers pair up with better assets compared to the SecondBest allocation. There is either an excessive entry of workers or an insuffi-cient entry of assets. Sorting is inefficient because a reduction in the outputshare costs less to weaker workers than better workers. This handicaps theircompetition for better assets, driving up the output share of the best assets.These asset owners then induce an inefficiently long queue of workers toincrease their matching probability.iiiLay SummaryThis dissertation studies matching between parties on the two sides ofthe markets where financial securities are commonly offered. An exampleis that firms hire CEOs from outside and offer equity shares in the remu-neration packages. The agents can be ranked by their types, say firm sizeand candidate’s ability. Since both sides compete for better partners fromthe given pools, the equilibrium matching pattern and the divisions of thematching surpluses vary with the distribution of types. My contributions areto provide qualitative results which hold for all distributions of types. Thefirst chapter studies when the efficient allocation can always be decentralizedin a frictionless environment, and how the forms of financial securities avail-able affect the distribution of the surplus. The second chapter analyzes howthe offering of output shares affect the matching pattern and entry decisionsin the presence of search friction.ivPrefaceThis dissertation is my original, unpublished and solo work. All errorsare mine.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Decentralizing Assortative Matching With Financial Secu-rities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . 112.3 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . 142.4 Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.1 Two-sided matching . . . . . . . . . . . . . . . . . . . 192.4.2 Contingent contract and division of surplus . . . . . . 21viTable of Contents2.4.3 Market structure . . . . . . . . . . . . . . . . . . . . . 242.4.4 Equilibrium definition . . . . . . . . . . . . . . . . . . 262.4.5 Menu of contracts . . . . . . . . . . . . . . . . . . . . 322.5 First Best Allocation . . . . . . . . . . . . . . . . . . . . . . 352.5.1 First Best decentralization under full information . . 372.5.2 First Best decentralization in price competition . . . 392.6 Decentralization Of Positive Assortative Matching . . . . . . 412.6.1 Sorting of workers . . . . . . . . . . . . . . . . . . . . 422.6.2 Screening by asset owners . . . . . . . . . . . . . . . . 482.6.3 Conditions for Positive Assortative Matching . . . . . 572.6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 622.7 Conditions On Contracts And Production Complementarity 692.7.1 Applications to standard securities . . . . . . . . . . 722.7.2 Conditions on production complementarity . . . . . . 752.8 Feasible Contracts And Comparative Statics . . . . . . . . . 802.8.1 Steepness and division of surplus . . . . . . . . . . . . 822.8.2 Steepness and contract offering . . . . . . . . . . . . . 842.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 953 Inefficient Sorting Under Output Sharing . . . . . . . . . . 973.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . 1023.3 Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.3.1 Production . . . . . . . . . . . . . . . . . . . . . . . . 1043.3.2 Matching . . . . . . . . . . . . . . . . . . . . . . . . . 1053.3.3 Equilibrium definition . . . . . . . . . . . . . . . . . . 1073.3.4 Assortative matching . . . . . . . . . . . . . . . . . . 111viiTable of Contents3.4 Second Best Allocation . . . . . . . . . . . . . . . . . . . . . 1123.4.1 Price competition . . . . . . . . . . . . . . . . . . . . 1163.5 Equilibrium Characterization . . . . . . . . . . . . . . . . . . 1183.6 Matching Efficiency . . . . . . . . . . . . . . . . . . . . . . . 1243.6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 1293.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 1304 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134AppendicesA Appendix for chapter 2 . . . . . . . . . . . . . . . . . . . . . . 141A.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . 141A.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . 144A.3 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . 149A.4 Proof of Lemma 2-4 and Proposition 3 . . . . . . . . . . . . 151A.4.1 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . 153A.4.2 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . 154A.4.3 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . 156A.4.4 Proof for necessity of conditions . . . . . . . . . . . . 156A.5 Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . 161A.6 Proof of Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . 163A.7 Proof of Lemma 6 . . . . . . . . . . . . . . . . . . . . . . . . 163A.8 Proof of Remark 6 . . . . . . . . . . . . . . . . . . . . . . . . 165A.9 Proof of Lemma 7 . . . . . . . . . . . . . . . . . . . . . . . . 166A.10 Proof of Lemma 8 . . . . . . . . . . . . . . . . . . . . . . . . 166viiiTable of ContentsA.11 Proof of Remark 7 . . . . . . . . . . . . . . . . . . . . . . . . 167A.12 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . 168A.13 Proof of Proposition 5 . . . . . . . . . . . . . . . . . . . . . . 171B Appendix for chapter 3 . . . . . . . . . . . . . . . . . . . . . . 173B.1 Proof of Remark 8 . . . . . . . . . . . . . . . . . . . . . . . . 173B.2 Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . 175B.3 Proof of Proposition 8 . . . . . . . . . . . . . . . . . . . . . . 185ixList of Figures2.1 An example of PAM . . . . . . . . . . . . . . . . . . . . . . . 152.2 Spill-over effect of increased competition across assets . . . . 833.1 Properties of workers’ preferences . . . . . . . . . . . . . . . . 1183.2 Characterization of active markets . . . . . . . . . . . . . . . 1193.3 Deviations by assets below threshold type . . . . . . . . . . . 1223.4 Law of motion in a thought experiment . . . . . . . . . . . . 126xList of Symbolsp Worker’s productivityq Asset qualityy Output levelΩy Support of output distributionF (y|p, q) C.D.F. of conditional output distributionf(y|p, q) P.D.F. of conditional output distributionV Workers’ outside optionU Assets’ outside option{pl}Ll=1 Support of worker’s type in finite types setting{qk}Kk=1 Support of Asset quality in finite types settingP (pl) Measure of workers with productivity plin finite types settingQ(qk) Measure of assets with quality qkin finite types settingr Public belief about a worker’s type∆(X) Set of probability distributions defined over Xt(y) ContractΩt Set of feasible contractsT Sigma algebra for Ωtv(p, q, t) Worker’s expected payoffxiList of Symbolsu(q, p, t) Asset owner’s expected payoffSt Ordered set of securitiess Contract term indexing Stµ Tightness ratioη(µ) Matching Probability for workersv(p) Worker’s equilibrium payoffu(q) Asset owner’s equilibrium payoffψ(q) Set of contracts posted by owners of asset qΨ Set of active marketsW Measure of participating workersWpq(pl, qk) Measure of workers assigned to the match (pl, qk)Cpq(pl, qk) Measure of assets assigned to the match (pl, qk)T Direct revelation mechanismsΩDRMT (q) Feasible set of T for owners of asset ql Threshold type of workersk Threshold type of assetsrFBq (q) Distribution of workers who match with assetof quality q under PAM{(t(.; s˜k), qk)}k≥k Set of active markets supporting PAMF (p) Measure of workers of productivity below p incontinuous types settingf Derivative of FG(q) Measure of assets with qualities below q incontinuous types settingg Derivative of Gλ Queue lengthδ(λ) Matching probability for asset sidexiiList of SymbolsK(q, s) Measure of asset owners in markets (q′, s′) ≤ (q, s)L(p, q, s) Measure of workers with types p′ ≤ p in markets(q′, s′) ≤ (q, s)Λ(q, s;K,L) Belief of queue length in market (q, s)R(q, s;K,L) Belief of workers’ types in market (q, s)xSB Object of interest x in price competitionx˜ Object of interest x in equilibriumxiiiAcknowledgementsI am personally indebted to Wei Li and Hao Li for their continuous andunconditional support. I am grateful to Vitor Farinha Luz and Michael Pe-ters for their valuable guidance and suggestions. I thank university examin-ers and the external examiner for their comments and inputs. Additionally,I thank Nancy Gallini, Alvaro Parra, Anne-katrin Roesler, Sergei Severi-nov, Wing Suen, Yaniv Yedid-Levi as well as the participants of the 2017Asian Meeting of the Econometric Society, and the Macro and TEB lunchworkshops at the Vancouver School of Economics.xivChapter 1IntroductionThis dissertation studies two-sided one-to-one matching in markets wherethe offering of financial securities or, more generally, contingent contractsis common. In many circumstances, the two sides can be ranked by somecharacteristics, or simply their types. Since both sides are competing forbetter partners from the given pools, the set of equilibria varies with thedistribution of types. As the distribution of types is constantly changingand may not be observable to outsiders, qualitative results applicable to alldistributions of types are of particular interest.The seminal work of Becker (1973) shows that if the matching surplus ex-hibits supermodularity in types, positive assortative matching (PAM) max-imizes the total surplus, and therefore prevails in a frictionless competitivemarket.1 However, the agents on one side are often better informed abouttheir own types, or the surplus from a match.2 The forms of contingentpayments offered then determine how much an informed party gains from abetter partner, and more importantly, how such gain depends on his private1Suppose a better agent generates a larger increase in the matching surplus when pair-ing up with a better partner, then the matching surplus is said to exhibit supermodularity.An allocation features PAM if the matches consist of the highest types on both sides, thesecond highest types, and so on.2I use feminine pronouns for the uninformed side and masculine one for the informedside.1Chapter 1. Introductiontype. I study how the offering of contingent contracts affects matching andthe divisions of surpluses in equilibrium.I consider the following framework. There are continuums of asset ownersand workers. A worker’s productivity is privately known, whereas the qualityof an asset is publicly observable. Each worker may operate an asset. Thetypes on both sides determine the output distribution.3 The asset sidefirst posts contracts tied to the future outputs. The owners who have thesame asset quality and post the same contracts will gather and form a (sub-)market. Observing the contracts posted, each worker may visit at most onemarket. The participants on both sides of a (sub-)market pair up randomly.Those who end up unmatched will stay idle. A worker forms his belief abouthis peers’ choices, and hence his matching probability in each market. Whenan asset owner is deciding her contract offer, the contracts offered by herpeers restrict the distribution of workers it may attract. Specifically, shebelieves that a deviating offer may only attract the workers who accept thelowest matching probability, or equivalently, the greatest percentage gainrelative to their equilibrium payoff.In the literature on assortative matching, the environment described isclosest to the papers where the informed parties make up-front paymentsand select their partners based on the offered prices. As the informed sideassumes the residual claim, the incentives for both sides is the same as inthe full information case. Hence, price competition still decentralizes effi-cient allocations in a competitive market. The price competition has served3Note that the distribution of the output does not depend on the contract chosen.Here the sole purpose of a contract is to determine the split of the matching surplus.This simplification allows me to concentrate on the potential distortions in the matchingpattern.2Chapter 1. Introductionas the benchmark case in the existing literature. However, the feasibilityof the buyout arrangement may be undermined by the presence of wealthconstraints and incentive provisions for other stakeholders. They motivatethe use of contingent contracts.The difference between this dissertation and competitive screening lit-erature is also noteworthy. In the latter, only the principals compete forinformed agents but not the other way around. Specifically, the equilibriumanalysis can be conducted in a sequential game in which multiple principalssimultaneously post their offers, then one single agent decides among them.The lack of competition among agents implies that the set of the separatingequilibria and the associated distortion depend on neither the number ofagents nor the distribution of their types.4 This implication is the majorconvenience, but also the restriction, of the competitive screening models.Here both sides are competing for better partners from the exogenouslygiven pools of assets and workers respectively. Finding “distribution-free”results is a non-trivial task because of the competition on both sides andtheir interaction.Chapter 2 considers a benchmark environment, in which the participantsin a market pair up frictionlessly.5 It focuses on two questions: When PAMcan always be decentralized in an equilibrium, and in this case, how thedivisions of matching surpluses depend on the form of contingent paymentsavailable. Apart from theoretical interests, the conditions for decentralizingPAM provide guidance on how to restrict the set of contracts available to en-4This is because the support of the type distribution alone pins down the set of incentivecompatibility conditions.5That is, the short side, which has fewer participants, will get matched for sure. Onlythe long side, with more participants, will be rationed.3Chapter 1. Introductionsure matching efficiency and redistribute welfare. For empirical researchers,The comparative statics on the equilibrium payoffs produce testable im-plications on the presence of private information in these markets againstcompeting theories.Chapter 3 introduces search friction into the matching process and fo-cuses on the class of output sharing contracts such as equity shares. I iden-tify a novel source of inefficiency in such markets and analyze the form ofdistortion on the matching pattern and entry decisions. In particular, theequilibrium is unique and still features PAM.6 I compare the equilibrium al-location with the Second Best one and obtain qualitative conclusions whichare universal to all distributions of types.Chapter 4 concludes.6With search friction, PAM only requires that a better worker searches for weaklybetter assets. This is satisfied by infinitely many allocations, but only a subset of them isSecond Best.4Chapter 2Decentralizing AssortativeMatching With FinancialSecurities2.1 IntroductionIn many matching markets, financial securities is the prevailing formof contracts between partners. It specifies how the payment between thepartners depends on certain outcomes, such as the realized output, andimposes little restrictions on the actions taken by the partners. The followingapplications are some examples:Market for top management: Firms are hiring top executives, whosecontribution to the firm’s profit depends on his ability and the firm size.However, the candidates know their own abilities better than the hiringfirms. Nowadays, stock and stock warrant are major components of theremuneration packages. Frydman and Jenter (2010) look at the compositionof CEO pay in S&P 500 firms: During the period 2000 to 2008, base salarymakes up less than 20% of the remuneration, and over half of it are optiongrants and restricted stock grants.Market for venture capitals: Venture capitalists can be ranked by their52.1. Introductionreputation and the prospect of the entrepreneurs’ projects vary. Entrepreneursinitially know certain aspects of their projects better than the outsiders. Ka-plan and Stro¨mberg (2003) analyze 213 rounds of investments: In over 90%of financing rounds, venture capitalists obtain convertible preferred stock,sometimes along with other financial securities, from the entrepreneurs inreturn for their assistance and financing.The defining feature is the contingent nature of the payment, whichneeds not involve any direct financial claims or explicit contracts betweenthe partners. When forming business partnerships, contingent payments areoften implemented via the capital structure of a joint venture. Even whena seller of an asset demands a fixed price, any postponed installment is stillcontingent on the outcome. This is because the buyer is protected by limitedliability and may default if the business turns sour or the news reveals dimprospects. Clearly, a variety of forces can be at play in the above examples.The focus here is how the form of contingent payments available affects two-sided matching in the presence of information asymmetry. I will adopt theterm contingent contract to underline this focus. 7Under a contingent contract, the expected payment made by a workerdepends on his private type. More importantly, the asset owners becomeconcerned about the types of their partners and take screening into accountwhen deciding contract offer. They may attempt to poach better candidates.PAM, though efficient, needs not occur in decentralized markets using con-tingent contracts.To have a direct comparison with price competition, the analysis centerson a class of contracts, which are ranked by the division of surplus. Specif-7In particular, I sidestep the issue on control rights specified in the financial securities.62.1. Introductionically, these contracts can be indexed by a single contract term, for whichworkers of all types prefer a more generous term given the asset qualitywhereas the asset owners prefer a less generous term for a given worker.8Examples are equity contracts indexed by the percentage share and default-able debts indexed by the principal amount.I start by constructing a candidate equilibrium decentralizing PAM. Byvirtue of the order structure in the setup, the conditions for voluntary par-ticipation by both sides and the incentive compatibility for the workers pindown a unique set of contracts for generic distributions of types. This can-didate set of contracts requires the workers to accept a less generous termfor a better asset. More importantly, it defines indirect mappings from thedistribution of types to the equilibrium payoffs, and the deviating payoffsfor both sides. To obtain “distribution-free” results, the analysis then cen-ters on the relation of these mappings with the output distribution and thefeasible set of contracts.I first study whether a single worker or asset owner may profit froma deviation under the candidate set of contracts. The analysis culminateswith the necessary and sufficient condition for the decentralization of PAM.It can be decomposed into three conditions, which separately address theincentives of workers, the participating asset owners and the asset ownerswho shall take their outside option. These conditions are stated in terms of8Namely, the set of feasible contracts is an ordered set of securities introduced inDeMarzo, Kremer and Skrzypacz (2005).In the other extreme, Riordan and Sappington (1988) characterize the condition thatperfect screening can be achieved costlessly using contingent payments. In this scenario,the equilibrium allocation and payoffs are the same as in the full information case. Thediscussed order structure for the contract space precludes this possibility.72.1. Introductionthe worker’s induced preference over the contract term, his partner’s type,and his matching probability.9 They apply to arbitrary distributions oftypes. The conditions are the same when the uninformed side may postsingle contracts only or menus of contracts. For each of the conditionsviolated, I provide a procedure constructing distributions of types for whichthe corresponding group must profit from a deviation. This illustrates theforces against PAM.I also provide a unifying sufficient condition on the worker’s expectedpayoff: When switching to a better partner offering a less generous term,a better worker always sees a larger increase (or smaller reduction) in hisexpected payoff, measured by either amount or percentage. This increasingdifference condition, termed as Global ID, is sufficient for decentralizingPAM by serving two purposes. The first one is to induce workers to sortassortatively. The second purpose is to preclude poaching offers from assetowners. I first show that under the set of candidate contracts inducingPAM, an asset owner never profits from poaching weaker workers. She maystill attempt to poach better workers. To compete with better assets, theasset owner must offer a more favorable term to maintain its appeal. Thisgenerous offer will also interest workers of lower types. Since at most oneworker will be hired, the workers must face rationing when seeking for thisdeviating contract. They have to trade off a better asset and a highermatching probability against a more favorable contract term. Global IDensures that in the candidate equilibrium, the better workers always prefer9The output distribution and the forms of feasible contracts jointly induce preferencesfor both workers and assets workers. The conditions on the worker’s preference involvethe asset owner’s preference and the outside options for both sides indirectly.82.1. Introductionthe former while the weaker workers prefer the generous term.10 As a result,the poaching offer will fail to attract better workers.In some sense, Global ID is satisfied if the variation in the contingentpayments aligns with the form of production complementarity. Based on thisobservation, I propose two notions of production complementarity, whichmanifest as a shift in the output distribution toward higher levels. For eachof these conditions on the output distribution, I provide the correspondingsufficient condition for the set of feasible contracts. One condition appliesto mixtures of cash and securities. The other applies to securities such asdebt contracts, or stock options if the asset side makes the payment.The second part is to study the effect of changes in the set of feasiblecontracts. A contract is steeper than another if it costs more to the betterworkers but less to the lower types.11 The other contract is said to be flatter.Suppose that the entire set of feasible contracts is replaced by a steeper setand the equilibrium matching remains PAM. The use of steeper contractshandicaps the competition among workers. In particular, a low type workeris willing to accept a less favorable term in exchange for a better asset. Theintensified competition for better assets will drive up the asset side’s share ofthe surplus, despite the same allocation. While the first result is based on thecompetition among workers, the second result stems from the competitionamong asset owners. The asset side always prefers the flattest contracts10Such monotonicity in the preference is stronger than necessary. The poaching offerscan be deterred if they attract only the lowest type among all participating workers. Thenecessary and sufficient conditions for the asset side relaxes the above increasing differencecondition by exploiting this observation.11For example, DeMarzo, Kremer and Skrzypacz (2005) show that under the assumptionof MLRP, the upfront payment, defaultable debt, equity share, call option are in ascendingorder by their steepness.92.1. Introductionavailable, which are prone to attract better workers. Putting together, weobtain comparative statics on how the introduction (or exclusion) of steeper(or flatter) contracts affects the divisions of the matching surpluses underassortative matching.Since the inclusion of steeper contracts in the feasible set has no impactson the sets of equilibrium allocations and payoffs, all results can be extendedto larger sets of contracts, which are not fully ranked. In particular, Iconsider examples that the workers have wealth constraints and that theasset owners may misappropriate the outputs.This also leads to the main testable implication of the model here. Sup-pose a new regulation restricts the feasible set of contracts so that steepercontracts are offered, my result on comparative statics predicts that all as-set owners are better off under PAM. The opposite shall occur following aslash of regulation. Such test, albeit demanding, is powerful. In pure moralhazard or risk sharing models, a restriction in the feasible set would onlyresult in the offering of a suboptimal contract.I then provide examples demonstrating that when offered flatter con-tracts, better workers may benefit less from a match with a better assetor an increase in matching probability. As a result, inefficiency may ariseafter the introduction of the flatter contracts. In this sense, these exam-ples illustrate how restricting the feasible set of contracts may improve totalsurplus.This chapter is organized as follows: Section 2.2 discusses the relatedliterature and the contribution of the present work. I illustrate the mainelements of the analysis casually using an example in Section 2.3.Section 2.4 details the model setting and the equilibrium definition. Sec-tion 2.5 formally defines PAM, the First Best allocation in the Utilitarian102.2. Related Literatureframework. It always occurs under symmetric information or in price com-petition. Section 2.6 characterizes the conditions on the workers’ preferencefor PAM decentralization. Joint sufficient conditions on the feasible con-tracts and the distribution of output are provided in Section 2.7. Section2.8 discusses the effects of changes in the set of feasible contracts. Section2.9 concludes. All proofs are relegated to the Appendix A.2.2 Related LiteratureAssortative matching in directed searchThough PAM in various environments, including non-transferable utilityand random search, have been studied in the literature, the previous workwith similar environments has exclusively considered the case that the in-formed side makes up-front payments. Mailath, Postlewaite and Samuelson(2016) (and the references therein) study the agents’ decisions of privatelyobserved pre-investment when they pair up with uninformed partners in aprice competition afterward. Damiano and Li (2007) consider the rent ex-traction problem of a matchmaker who decides a menu of meeting places andadmission fees for agents with private types. My setting is closely related tothat in Eeckhout and Kircher (2010). The authors study price competitionusing a competitive search framework. They show that a stronger form ofproduction complementarity is required to support (imperfect) PAM in thepresence of search friction.The use of contingent contracts not only changes the sorting incentivesfor the informed but also gives rise to the screening problems for the un-informed side. The latter has never been studied in this literature. For anasset owner, the pool of workers attracted by a deviating offer varies with the112.2. Related Literaturedistribution of types indirectly through the set of contracts posted. Char-acterizing the conditions for deterring poaching offers is the main challengein my analysis.Competitive screening with bilateral matchingMy setting is related to the competitive screening models with bilateralmatching including Gale (1996) and Guerrieri, Shimer and Wright (2010).My equilibrium definition closely follows that the latter. They consider acompetitive search setting with free entry of principals, who have both con-tract and matching probability as screening instruments. Facing the offersposted, the matching probabilities for the agents are jointly determined withtheir choices of contracts.12 The authors characterize the equilibrium andstudy the form of distortion in various applications.My point of departure is to consider two-sided one-to-one matching inwhich both sides compete for partners from given pools.13 As a result, thedistribution of types determines the set of feasible allocations, and hence thesets of efficient and equilibrium allocations. In general, the distortion in two-sided matching varies with the distribution of types. Instead of targeting ata specific form of distortion, the policy recommendation here is restrictingthe feasible set of contracts to ensure efficient matching for any distributionof types. To this end, I characterize the conditions for decentralizing PAM.Security-bid auctionMy formulation of the contract space is closely related to DeMarzo, Kre-12This also implies that the introduction of bilateral matching changes the principals’deviating payoff in comparison with the textbook competitive screening models, ensuringthe existence of an equilibrium.13I also focus on a set of fully ordered contracts, so that any separation among work-ers’ types must be induced by some variation in the asset quality and their matchingprobability. This allows me to concentrate on the distortions in the matching pattern.122.2. Related Literaturemer and Skrzypacz (2005). They consider an auction of an asset, in whichthe buyers bid in the form of securities. The authors introduce the conceptof ordered sets of securities, for which the expected payment can be rankedunambiguously for all types of buyers. They compare different ordered setsof securities in terms of their steepness. The auctioneer can improve herrevenue by requiring the buyers to bid from a steeper ordered set of secu-rities, provided that the equilibrium allocation remains efficient.14 This isbecause steeper securities strengthen the linkage between the winner’s typeand the payment he makes, handicapping the competition among buyers.I adopt their definitions of an ordered set of securities and security steep-ness. The comparative statics on the divisions of matching surpluses can beattributed to the insight in DeMarzo, Kremer and Skrzypacz (2005). Mycontribution is to establish the connection between the security-bid auctionand assortative matching in this aspect. I further show that the intensifiedcompetition among workers for the same type of assets spills over to thecompetition for the better assets under PAM. Besides, the change in thefeasible contracts will affect how the partners divide the gain from produc-tion complementarity, potentially changing the equilibrium allocation. Thispaper provides conditions ensuring that PAM always occurs in equilibrium.Moral hazard and assortative matchingThis paper is related to the literature on how incentive provision affectsthe matching pattern. This strand of literature also considers the use ofcontracts in two-sided matching markets. Serfes (2005) studies the equi-librium matching pattern in a principal-agent setup. Legros and Newman14DeMarzo, Kremer and Skrzypacz (2005) assume that the payoff for the winner isstrictly log-supermodular in his type and the ranking of his security-bid. This ensuresthat the buyer of the highest type always outbids the others.132.3. An Illustrative Example(2007) discuss a related example. Kaya and Vereshchagina (2014) study howownership structure and production technology shape the cost of incentiveprovision in team production. In these models, the parties’ types are pub-licly observable. One side or both will take a private action after pairing up.Despite production complementarity, assortative matching needs not arisein equilibrium because of two channels. First, utility is not perfectly trans-ferable. When adjusting the term of a contract to transfer utility betweentwo sides, the conversion rate is not constant and dependent on types. Sec-ond, types affect productivity as well as the cost of incentive provision, sothe matching surplus does not inherit supermodularity from the productiontechnology.I consider information asymmetry when forming matches. To focus onthe potential distortion of the matching pattern, the choice of contract in mysetting affects only the division, but not the size, of the matching surplus.This ensures PAM if the types are publicly observable. Here inefficiencycan only arise because of both the use of contingent contracts and privateinformation. Furthermore, it allows me to obtain general results on howthe form of the contingent payments affects the divisions of surpluses inequilibrium.2.3 An Illustrative ExampleThis section illustrates the main elements of the analysis. I also casuallysketch out the equilibrium definition along the way. The formal setup willbe laid out in the next section.Consider an economy with two types of workers p ∈ {1, 2}. Each type hasa unit measure. There are three types of assets q ∈ {1, 2, 3} with 12q measure142.3. An Illustrative Examplerespectively. Both sides have the same outside option, say 1. E(Y |p, q)denotes the expected output from a match (p, q). The matching surplus issupermodular (SPM) in types, so the First Best allocation is PAM. Thatis, better workers pair up with better assets. In this example, all high typeworkers are allocated the best assets. Half of the low type workers matchwith the best assets, and the other half match with median quality assets.The owners for the remaining half of the median quality assets and thelowest quality assets take their outside option.Figure 2.1: An example of PAMFor simplicity, I only consider two classes of contracts. The workerseither pay a fixed price upfront or promise the asset owner a share of thefuture output.It is well known that PAM prevails in a competitive market under fullinformation. In such equilibrium, the owners of the median quality assetsmust be indifferent about their outside option, and the low type workers areindifferent about matches with the two types of assets. These indifference152.3. An Illustrative Exampleconditions uniquely pin down the division of matching surplus.1 = UFB(1) = UFB(2),E(Y |1, 2) = V FB(1) + UFB(2),E(Y |1, 3) = V FB(1) + UFB(3),E(Y |2, 3) = V FB(2) + UFB(3),where V FB(p) and UFB(q) denote the equilibrium payoffs for the workersand asset owners respectively. Observe thatV FB(p) + UFB(q) ≥ E(Y |p, q),no pairs will profit from switching their partners. Furthermore, the divisionof matching surplus is the same under the two classes of contracts. If upfrontpayment is feasible, the participating asset owners simply post the priceUFB(q). Under full information, an equity contract may function as a postedprice by making its term contingent on worker’s type to implement theintended transfer UFB(q).Now suppose workers have private types. If the asset side may postprices, the equilibrium allocation and payoffs remain the same. This isbecause the payment made by the worker does not depend on his privatetype, so the deviating payoff for both sides is always the same as in the fullinformation case. The class of fixed prices, or cash payment, is in fact theonly contracts for which the division of matching surplus is unaffected byinformation asymmetry.Now we turn to the case that only equity contracts are available. Lets˜q denote the equity shares for the assets of quality q. To induce PAM, theindifference conditions mentioned must continue to hold. They uniquely162.3. An Illustrative Exampledetermine the contracts offered, and hence the equilibrium payoffsU(1) = U(2) = s˜2E(Y |1, 2) = 1,V (1) = (1− s˜2)E(Y |1, 2) = (1− s˜3)E(Y |1, 3),V (2) = (1− s˜3)E(Y |2, 3),U(3) = (1− s˜3)[13E(Y |1, 3) + 23E(Y |2, 3)].(2.1)In this example, all but the highest types have the same equilibrium payoffas in the the full information case. Yet the competition for the high qualityassets has intensified because the equity shares cost the high type workersmore than the low type. As a result, the high type workers end up payingmore for the best assets,s˜3E(Y |2, 3) > UFB(3) = s˜3E(Y |1, 3).This in turn leads toV (2) + U(q) < E(Y |2, q), q = 1, 2.So far we have pin down the set of posted contracts and the equilibriumpayoffs. The next step is to investigate the conditions ensuring that noagents may profit from a deviation. The market structure and the beliefrestriction provide the foundation for the deviating payoffs.Facing the posted contracts s˜2 and s˜3, a high type worker may onlyswitch to the median quality assets, earning (1− s˜2)E(Y |2, 2). Substitutings˜2 and s˜3, he will not profit from such deviation ifE(Y |2, 3)E(Y |2, 2) >E(Y |1, 3)E(Y |1, 2) . (2.2)(2.2) is stronger than SPM of the matching surplus. SPM merely ensuresthat better workers benefit more from an improvement in the asset quality172.3. An Illustrative Exampleunder the same contract. Yet, they also suffer more from the reduction intheir share. So PAM requires high type workers to have a greater percentagegain from a better asset.Unlike workers, the deviating payoff for the asset side depends on thepool of workers attracted by a deviating offer. I adopt the belief restrictionthat such offer will only attract the workers, if any, who accept the lowestmatching probability, or equivalently, see the greatest percentage gain rel-ative to their equilibrium payoff. This restriction ensures that the pool ofthe workers attracted depends on the strength of complementarity in types.Take the lowest quality asset as an example. Suppose an owner posts a shares′, a worker will be interested in this offer only if (1− s′)E(Y |p, 1) > V (p).In this case, his percentage gain is simply (1−s′)E(Y |p,1)V (p) . Combining with theexpression V (p) = (1 − s˜3)E(Y |1, 3), any deviating offer will at most drawonly the low-type workers ifE(Y |2, 3)E(Y |1, 3) >E(Y |2, 1)E(Y |1, 1) . (2.3)However, the asset owners never profit from a partner weaker than the oneunder PAM in this framework.V (1) + U(q) ≥ E(Y |1, q), q = 1, 2, 3.One can check that the owners of the median and highest quality assets haveno profitable deviations as well. Hence, PAM can be supported.15Nevertheless, the conditions (2.2) and (2.3) only apply to the given distri-bution of types. The counterpart of the conditions (2.1) for general contractsdefines indirect mappings from a generic distribution of types to the set of15Furthermore, I will show that the asset owners can do no better by offering menus ofcontracts.182.4. Model Settingcontracts posted, hence the equilibrium payoffs and the deviating payoffs.These mappings and their relation to the model primitives will be the centralobject of the analysis.2.4 Model Setting2.4.1 Two-sided matchingProduction is carried out by a single worker using an asset. There arecontinuums of workers and asset owners. Each asset owner owns an asset.Assets can be ranked according to their publicly known qualities q ∈ [q, q].All workers are ex-ante homogeneous but differ in their actual productivityp ∈ [p, p], Every worker privately knows his productivity. All parties are riskneutral and have a quasi-linear preference.Production takes place after a worker pairs up with an asset. The out-put Y is stochastic and contractible. Given the pair of types (p, q), theconditional distribution of output Y |(p, q) has C.D.F. F (y|p, q) with com-mon support Ωy ⊆ R+. The outside options for the workers and assetsare given by V > 0 and U > 0 respectively, so the matching surplus isE(Y |p, q)− V − U.16Assumption (P). Y |(p, q) has the following properties:1. For any y ∈ Ωy, F (y|p, q) is continuous and strictly decreasing in pand q.2. E(Y |p, q) is strictly supermodular(SPM) in p and q.3. E(Y |p, q) = U + V .16Measurability and integrability are tacitly assumed whenever they are required.192.4. Model SettingAssumption (P) states that a higher worker’s productivity or asset qual-ity always improves the output distribution in a strict F.O.S.D. sense. Thematching surplus is strictly increasing and supermodular in types p and q.Furthermore, the matching surplus is positive for any pair. Therefore, thetotal surplus is maximized under PAM.For tractability, the type distribution for workers and assets assume tohave finite supports {pl}Ll=1 ⊆ (p, p] and {qk}Kk=1 ⊆ (q, q] respectively, whereL ≥ 2 and K ≥ 1.17 Higher types of worker and asset refer to greater l andk respectively, i.e. pl > pl−1 and qk > qk−1.The measure of workers with productivity pl is denoted by P (pl). r ∈∆({pl}Ll=1) denotes the public belief about a worker’s type p, where ∆({pl}Ll=1)is the set of probability distributions defined over {pl}Ll=1. Likewise, the mea-sure of assets with quality qk is Q(qk). Taking outside options by the assetowner and worker are referred as p0 and q0 respectively. Q(q0) and P (p0)are defined as +∞.In the analysis, types with subscripts, pl and qk, are used when consider-ing a particular distribution of types. Otherwise, the discussion is referringto general types in the type space.17It is more convenient to work with a finite distribution of types when discussing generalcontracts or menus of contracts.Notice that there are no workers of type p and assets of type q. Together withE(Y |p, q) = U + V , the matching surplus is always positive but can be arbitrarily smallfor some pairs. Section 2.6 characterizes the conditions for First Best decentralization.This property is exploited in establishing necessity of theses conditions. The conditionsremain sufficient if E(Y |p, q)) > U + V .202.4. Model Setting2.4.2 Contingent contract and division of surplusSuppose a worker pairs up with an asset. The two parties may enteran agreement on the contingent payments, denoted by t : Ωy → R. Whenthe production concludes, the worker receives the realized output y andmakes payment t(y) to the asset owner accordingly. The contingent paymentscheme t(y) will be referred as a contract. It inherently satisfies ex-postbudget balance. The argument y will be omitted from t(y) if no confusionwill arise. Only bilateral contracts are considered.All types of asset owners have access to the same set of feasible contracts,which is denoted by Ωt. T denotes a sigma algebra for Ωt. The set of feasiblecontracts captures various restrictions such as contract incompleteness andlimited liability.Assumption (C). For all t ∈ Ωt, t(y) and y−t(y) are not constant functionsover Ωy and weakly increasing in y.Assumption (C) rules out contracts specifying only a fixed payment toeither side.18 When the output level increases, one side must receive ahigher payment under ex-post budget balance. Assumption (C) furtherrequires that the asset owner’s and worker’s payoff are always increasingin the production outcome under any contract. This can be motivated bya threat of sabotage on both sides. It also implies that the payoffs forboth sides are continuous in the output level. Such double monotonicity iscommon in financial agreements.18This can easily be motivated by incentive provision for both sides. For example, theproduction may require both partners to make an arbitrarily small investment or effort,which are privately observed.212.4. Model SettingGive the pair of types, the contract determines the surplus division be-tween the two parties. The worker’s share of surplus is denoted byv(p, q, t) := E(Y − t(Y )|p, q),and that for the asset owner is denoted byu(q, p, t) := E(t(Y )|p, q).v(p, q, t) and u(q, p, t) will be referred as expected payoffs. Since a highertype always improves the output distribution in a strict F.O.S.D. sense andthe contingent payment is monotonic in output, an agent, who has a highertype himself or has a better partner, always enjoy a higher expected pay-ment.Remark 1. For any t ∈ Ωt, u(q, p, t) and v(p, q, t) are continuous andstrictly increasing in p and q.Assumption (P) and (C) jointly introduce a uniform and monotonic pref-erence for the partner’s type. This leads to competition for a better partneron both sides. More importantly, an asset owner concerns about the typeof workers attracted by her offer.When entering the same contract, a better worker not only pays outmore to his partner but also keeps a higher amount of residual payment.Consequentially, only the workers of the lowest type may take their outsideoptions. However, the same conclusion does not automatically hold for theasset side as an owner of a better asset may end up with a weaker worker.To facilitate the comparison with price competition and security-bid auc-tion, I adopt the convention that the informed party is entitled to the fulloutput and makes payment. This may differ from the default output di-vision in the application considered. One must make adjustments to the222.4. Model Settingforms of contracts accordingly, so that the payoffs at each output level arethe same as in the setup here.Ordered set of securities The analysis focuses on a special class ofcontracts, which has a complete order. The formulation here is built onDeMarzo, Kremer and Skrzypacz (2005).Definition. St is called an ordered set of securities if there exists a mappingt(.; .) : Ωy × [0, 1]→ R such that1. St = {t(.; s) : s ∈ [0, 1]} ⊆ Ωt, and2. t(.; s) is continuous in s with respect to supremum norm, and3. v(p, q, t(; s)) is strictly decreasing in s, whereas u(q, p, t(.; s)) is strictlyincreasing in s for any (p, q) ∈ [p, p]× [q, q], and4. For any (p, q) ∈ [p, p]×[q, q], v(p, q, t(.; 1)) ≤ V and u(q, p, t(.; 0)) ≤ U.An ordered set of securities is a subset of feasible contracts indexed by s,which will be referred as the contract term. v(p, q, t(.; s)) and u(q, p, t(.; s))are continuous in all arguments under the second condition. The thirdcondition states that for any pair of types, a higher value of s represents agreater share of surplus for the asset owner. For a given partner, workersof all types unanimously prefer a lower term s, whereas the asset ownersalways prefer the opposite. Many standard securities in practice can beranked in this manner. Examples of contract term s include the amountof cash payment, the equity share, the principal amount of debt and thestrike price for options. These examples satisfy the third condition becauset(.; sH) ≥ t(.; sL) whenever sH > sL.232.4. Model SettingNow consider a matched pair (p, q) who may only enter a contract inSt. Under the second and third condition, [v(p, q, t(.; 1)), v(p, q, t(.; 0))] isthe range of the feasible payoff for the worker and that for the asset owneris [u(q, p, t(.; 0), u(q, p, t(.; 1)]. The fourth condition then implies that anyvalue v′ between V and E(Y |p, q)− U can be achieved by a contract in St.v′ represents a split of matching surplus as the payoffs for both parties areabove their outside options. An example when the fourth condition holds isthat t(y; 0) = 0 and t(y; 1) ≥ y.In summary, an ordered set of securities shares two similarities withprices: a monotonic preference for all types and perfect transferability ofthe matching surplus under full information. On the other hand, it rendersthe expected payment from the worker dependent on his private type.Throughout the analysis, an ordered set of securities is always feasible,St ⊆ Ωt. As we shall see, this ensures that the equilibrium allocation underfull information is always the First Best.2.4.3 Market structureFor a given distribution of types, there are continuums of (sub-)marketsindexed by (t, q). An owner of asset quality qk may decide between her out-side option and one of the markets (t, qk) while a worker may take his outsideoption or participate in any one of the markets. The market structure isinterpreted as follows: Asset owners decide what contract they post.19 Own-ers of the same asset quality qk posting the same contract t gather into onemeeting place, which forms the (sub-)market (t, qk). When a worker decidesto accept the offer t posted by the owners of asset quality qk, he participates19Section 2.4.5 will address the possibility of a menu of contracts.242.4. Model Settingin the market (t, qk).The participants on the two sides of a market will pair up randomly. De-fine the tightness ratio µ ∈ [0,∞] as the ratio between the measure of assetsand that of participating workers in the market. A worker gets matchedwith probability η(µ) while the matching probability for an asset owner isη(µ)µ . The matching is frictionless, so η(µ) = min{µ, 1}. The payoffs for thoseleft unmatched are normalized to zero. Hence, the values of outside optionare the cost of participation for the two sides.Consider a market (t, q) with tightness ratio µ and the distribution ofparticipating workers is given by r ∈ ∆({pl}Ll=1). By a slight abuse of nota-tion, I denote the expected surplus for a matched asset owner byu(q, r, t) :=∑Ll=1u(q, pl, t)r(pl).When participating in this market, the expected payoff for an asset owneris given by η(µ)µ u(q, r, t) and that for a worker of type p is η(µ)v(p, q, t). Aworker is said to prefer the contract (t, q) to (t′, q′) if v(p, q, t) ≥ v(p, q′, t′).He prefers the market (t, q) to (t′, q′) if η(µ(t, q))v(p, q, t) ≥ η(µ(t′, q′))v(p, q′, t′).The timing of the events is as follows: In the contract posting stage,the asset owners make their participation decisions simultaneously. At thebeginning of acceptance stage, the workers observe the measure of asset own-ers across markets. They simultaneously make their participation decisions.Matches are then formed.A market (t, q) is active if if it is chosen by some asset owners in equi-librium. Otherwise, it is inactive. An active market clears if it has a unitytightness ratio. The workers are said to be rationed in the market (t, q) ifits tightness ratio is below unity. In the opposite case, the asset owners aresaid to be rationed.252.4. Model Setting2.4.4 Equilibrium definitionIn this subsection, I first propose a formal definition of an equilibrium.I then explain the terminologies and motivate the belief restriction.In the equilibrium definition, each market (t, q) is associated with a tight-ness ratio µ(t, q) and a distribution of participating workers r(t, q). r(pl|t, q)is the proportion of workers of type pl and the support is denoted as Ωp(t, q).Everyone takes µ and r as given.20 For the active markets, µ and r cap-ture the participation decision of the workers and asset owners. For aninactive market (t˜, q˜), µ(t˜, q˜) and r(t˜, q˜) are interpreted as the public beliefregarding the tightness ratio and the composition of participating workersin that market after an owner of asset quality q˜ deviates to it. This notationeliminates the need to distinguish between deviations to active markets orinactive markets by an asset owner. The equilibrium payoffs for both sidesare included in the definition to capture the optimality of the participationdecisions.Definition. A competitive matching equilibrium consists of the asset own-ers’ equilibrium payoff U : {qk}Kk=1 → R+, workers’ equilibrium payoffV : {pl}Ll=1 → R+, asset owners’ contract posting set ψ : {qk}Kk=1 →Ωt ∪ {p0}, the set of active markets Ψ ⊆ Ωt × {qk}Kk=1, the measure of par-ticipating workers W : T × P({qk}Kk=1)→ [0, 1], the distribution of workersr : Ωt×{qk}Kk=1 → ∆({pl}Ll=1) and market tightness µ : Ωt×{qk}Kk=1 → [0,∞]such that1. Asset Owners’ Optimal Contract Posting:20The tightness ratio is often interpreted as the market clearing price of the concernedmarket. All parties take µ as given, therefore the environment here is “competitive”.262.4. Model SettingFor all (t, q) ∈ Ωt×{qk}Kk=1, U(q) ≥ η(µ(t,q))µ(t,q) u(q, r(t, q), t) with equalityif t ∈ ψ(q).2. Workers’ Optimal Acceptance:i)For all (t, q) ∈ Ωt × {qk}Kk=1,V (p) ≥ η(µ(t, q))v(p, q, t) (2.4)with equality if p ∈ Ωp(t, q) and µ(t, q) <∞.ii)µ(t, q) =∞ if V (p) > v(p, q, t) for all p ∈ {pl}Ll=13. Active Markets:Ψ := {(t, q) ∈ Ωt × {qk}Kk=1 : t ∈ ψ(q)} is the support of W.4. Optimal Participation:i)U(q) ≥ U and V (p) ≥ V .ii)∫Ωt×{qk}Kk=1 r(pl|t, q)dW ≤ P (pl) with equality if V (pl) > V .iii)∫Ωtµ(t, qk)dW ≤ Q(qk) with equality if U(qk) > U.An allocation is denoted by (Wpq, Cpq) ∈ RK×L+ ×RK×L+ , whereWpq(pl, qk)and Cpq(pl, qk) denote the measure of the workers with type pl and that forthe assets of quality qk assigned to the match (pl, qk) respectively. For agiven equilibrium, the allocation is given byWpq(pl, qk) =∫Ωt×{qk}r(pl|t, qk)dW , andCpq(pl, qk) =∫Ωt×{qk}µ(t, qk)dW.272.4. Model SettingActive Markets ψ(qk) denotes the subset of contracts or outside optionchosen by the owners of asset quality qk. t′ ∈ ψ(qk) if and only if a positivemeasure of asset owners participate in the market (t′, qk). In equilibrium, theasset owners participate in a market only when they anticipate a positivemeasure of workers on the opposite side. Therefore, the set of active markets,denoted by Ψ, is the union of ψ(qk)× qk across all asset qualities, excludingthe outside option. For any subset of markets A, W (A) denotes the totalmeasure of participating workers on the equilibrium path. The support ofW must be Ψ by the same reasoning.For the active markets, r and µ have to be consistent with the partici-pation decisions for both sides. The feasibility constraints require that forany pl and qk,Q(qk) ≥∫Ωtµ(t, qk)dW, andP (pl) ≥∫Ωt×{qk}Kk=1r(pl|t, q)dW.The equality must hold for type qk if these asset owners strictly prefer theactive markets to their outside option in equilibrium, and likewise for theworkers’ side. These lead to the optimal participation condition. Withthis condition in place, one can recover the participation decisions for bothsides from the market tightness µ|Ψ, the workers’ composition r|Ψ, and themeasure of participating workers W , where g|Ψ denotes the restriction of gto the set Ψ.Since workers and asset owners may ensure themselves a payoff of Vand U respectively, so V (p) ≥ V and U(q) ≥ U are called participationconstraints for workers and asset owners.A worker will never get matched if he unilaterally switches to an inactivemarket. So workers only consider deviations to active markets or outside282.4. Model Settingoption. This is captured by the inequality (2.4) for the active markets andcondition ii) in Optimal Participation. Abusing the terminology, I will alsocall this set of conditions as incentive compatibility (IC) condition.In a worker’s perspective, the set of active markets Λ represents thecompetition between the asset owners. The competition from other workersis summarized by the matching probabilities η(µ(t, q)) in these markets.Belief restriction Since there are continuums of workers and assets, switch-ing between active markets or taking outside option by a single party hasnegligible impacts. The same is true when a worker unilaterally switches toan inactive market. The focus here is the deviation to some inactive marketby an asset owner, and her belief about the pool of workers who will, inresponse, participate in that market.The workers’ optimal acceptance condition impose restrictions on thisoff-equilibrium-path belief. In particular, the inequality (2.4) is required tohold for the inactive markets as well. The conditions implicitly require allparties to share the same belief off the equilibrium path.Suppose an owner of asset quality q˜ deviates to post a contract t˜. Thebelief restriction states that if V (pl) > v(pl, q˜, t˜) for all types, then no workerswill be attracted, and so µ(t˜, q˜) = ∞. No restrictions are imposed on ther(t˜, q˜), which has no bearing in such case. Now suppose that V (p) < v(p, q˜, t˜)for a subset of types. Then µ(t˜, q˜) is given by the greatest value of tightnessratio for which the inequality (2.4) holds for all types of workers. Put itdifferently, µ(t˜, q˜) is the lowest matching probability some types of workersare willing to endure. Furthermore, the support of r(t˜, q˜) contains only thosetypes. The remaining possibility is that V (p) ≥ v(p, q˜, t˜) holds for all typesof workers and with equality for some type. The inequality (2.4) requires292.4. Model Settingthat µ(t˜, q˜) ≥ 1. The restriction on r(t˜, q˜) in the preceding discussion stillapplies if µ(t˜, q˜) is finite. All results in this study are robust to additionalrestrictions on µ(t˜, q˜) in this case.The belief restriction is interpreted as follows: Suppose an owner ofasset quality q˜ is pondering a deviating offer (t˜, q˜). Workers of type pl isinterested in it if V (pl) < v(pl, q˜, t˜). If multiple workers are interested in thedeviating offer, the competition between them manifests as a reduction intheir matching probability, dissipating any gain from the offer. The worker,who ends up matching with her, must be among those who are willing toendure the lowest matching probability.In the view of a single asset owner, {V (pl)}Ll=1 reflects the competitionfrom other asset owners of various qualities. This in turns affects the compe-tition among the workers for the deviating offer, and hence the distributionof workers it attracts. These are captured by µ(t˜, q˜) and r(t˜, q˜) respectively.The belief restriction here is often motivated by the “subgame perfec-tion” in the competitive search literature. Suppose only -measure of theowners of asset quality q˜ deviate to some inactive market (t˜, q˜). Observingthe measure of the asset owners in every market, a worker has to anticipatehis matching probability in each of the markets accordingly. When → 0+,no types of workers can strictly gain from participating in the market (t˜, q˜)in the equilibrium of this “subgame”. Otherwise, workers of all such typeswill turn up in this market but only -measure of them will get matched,resulting in an expected payoff below their outside option. It follows thatthe participating workers in the market (t˜, q˜), if any, are willing to acceptthe lowest matching probability. By continuity, the workers’ payoff in theequilibrium of this “subgame” must converge to V (p). This justifies the302.4. Model Settingbelief restriction discussed.21Discussion Gale (1996) considers a continuum of markets indexed by allpossible contracts and defines a notion of competitive equilibrium in thepresence of adverse selection. In his definition of a “refined equilibrium”, therestriction on the type of informed parties attracted by an off-equilibrium-path contract is the same as here.22 The author suggests that this belief re-striction is analogous to the “Universal Divinity” in Banks and Sobel (1988).Eeckhout and Kircher (2010) define an equilibrium as a pair of mea-sures of workers and assets across the markets. The set of active marketsand the equilibrium payoffs are then derived from the pair of equilibriummeasures. Eeckhout and Kircher (2010) adopt the same restriction on themarket tightness for inactive markets. Since uninformed parties post pricesin their setting, they leave out the off-equilibrium-path belief on the worker’stype.This study focuses on an equilibrium supporting positive assortativematching. As I will show, the corresponding incentive compatibility con-dition for workers and optimal participation conditions pin down a uniquepair of measures of workers and assets across the markets for generic distri-butions of types. The remaining analysis is to verify that both sides have noprofitable deviations and study the comparative statics of the equilibriumpayoffs. To simplify the notation, I define an equilibrium using a set ofequilibrium conditions directly involving the equilibrium payoffs and other21The case that V (p) ≥ v(p, q˜, t˜) for all types of workers and equality holds for sometype, say p′, is intricate. In such case, the limit µ(t˜, q˜) may depend on the equilibriumstrategies of both the workers and the asset owners. Nevertheless, the inequality (2.4)remains valid and equality holds for type p′.22To be precise, Gale (1996) imposes the restriction only on r but not µ.312.4. Model Settingequilibrium objects of interest. This equilibrium definition closely followsthat of Guerrieri, Shimer and Wright (2010), which share the same beliefrestriction.I assume a finite distribution of types as a formal analysis for menusof general contracts with continuums of types invites substantial technicalcomplications. Therefore, I define an equilibrium with respect to the sup-port of a given finite distribution of types. The underlying arguments do nothinge on the assumption of a finite distribution. One may approximate anygiven distribution of types with a finite distribution close by. The resultshere apply all such finite distributions. On the other hand, the type spacetakes the form of an interval. This is because I have to vary the distribu-tion of types in a continuous manner when establishing the necessity of theconditions for decentralizing PAM.2.4.5 Menu of contractsThe baseline setting assumes that an asset owner may post only a singlecontract. Now consider a more general setting, in which an asset owner ofqk may post a menu of contracts specifying the asset quality q ≤ qk, theseparation probability pi ≤ 1 and the associated payment scheme t. Afterthe matching stage, the worker selects (pi′, q′, t′) from the menu. A lottery ofthe stated probability pi′ will be conducted publicly. If the lottery outcomeis separation, then two parties will get their unmatched payoff. If the lotteryoutcome is continuation, the asset owner will impair the asset quality to q′.The pair will then perform production and split the output according to thecontract t′. The key departure from the baseline setting is that the termof the contract, the asset quality and the matching probability for workers,322.4. Model Settingafter adjusted for the separation probability, can be made contingent on theworker’s type, subject to additional incentive compatibility conditions.It is without loss to focus on direct revelation mechanisms (DRM). Thesuperscript and subscript denote the reported type and true type respec-tively. T = {ql, pil, tl}Ll=1 denotes a DRM, and satisfies the incentive com-patibility conditions,(1− pil)v(pl, ql, tl) ≥ (1− pil′)v(pl, ql′ , tl′)for all l and l′. The set of DRM differs across asset qualities, and ΩDRMT (q)denotes that for owners of asset quality q.The continuum of markets is now indexed by (T, q). An owner of assetquality qk may take her outside option or participate in one of the marketsT ∈ ΩDRMT (qk). A worker may take his outside option or participate inany market. It is noteworthy that even for the same menu of contracts T,(T, q) and (T, q′) are two distinct markets in this formulation. The workers’participation decisions in these two markets are allowed to differ. It isstraightforward to modify the definition of equilibrium accordingly. Theformal definition is relegated to the appendix.Lemma 1 states that when the asset owners are allowed to post a menuof contracts, the set of equilibrium payoffs and allocations weakly expands.Lemma 1. For every competitive matching equilibrium, there exists an equi-librium using direct revelation mechanisms which supports the same alloca-tion and equilibrium payoffs, and the asset owners post only degenerate directrevelation mechanisms.This result stems from two features of the current model. First, thepair of types determines the output distribution once a match is formed.332.4. Model SettingAny post-matching or post-contracting messaging is a zero-sum game. Anasset owner may elicit a truthful report by posting a menu of contracts, butthis confers the worker information rent without improving the matchingsurplus. Therefore, a menu of contracts is useful only if it affects the workers’participation decisions before the matching stage.Second, an asset owner may pair up with at most one worker. Supposean asset owner deviates to post a menu of contracts. As she cannot con-tract with a continuum of workers, the competition between workers drivesdown their matching probability to the level that no workers will gain fromthe deviating offer. The workers, who are willing to remain despite beingrationed, are indifferent between the deviating offer and some other activemarkets. They may be of different types and select a different contract fromthe menu. When constructing an equilibrium using DRM, the asset owneris assumed to believe that only the lowest type among these workers will beattracted by her deviating offer.23 If she profits from contracting with suchtype of workers, she must also profit from posting only the contract chosenby that type. In the presence of capacity constraint, this single contract willresult in the same market tightness as the menu of contracts and attracta pool of workers of potentially higher types. Or put it differently, if anasset owner cannot profit from posting any contracts, deviation to a menucomprising these contracts neither improves her matching probability northe worst admissible belief about her partner’s type. Therefore, the set of23First, this off-equilibrium-path belief is adopted in constructing the correspondingequilibrium using DRM but not required for the equilibrium using single contracts.Second, the described belief is not the most pessimistic one allowed in the equilibriumdefinition. A contract chosen by workers of a higher type may provide the asset owner alower payment.342.5. First Best Allocationequilibria in the baseline model is robust to the modification that the assetowners can post a menu of contracts.24In the baseline setting, owners of the same asset quality may post severaldifferent contracts only if they are indifferent about these contracts in equi-librium. This is no longer true when the asset owners may post a menu ofcontracts. Even though an asset owner prefers some contracts in her postedmenu over the rest, she refrains from posting only her favoured contractsif doing so will lead to a deterioration of her partner’s type.25 As a re-sult, allowing menus of contracts potentially expands the set of equilibriumallocations and payoffs.For notational simplicity, we restriction our attention to the baselinesetting, where the asset owners post only a single contract, in the subsequentsections. All the results are robust to the introduction of the menus ofcontracts, including the necessary conditions for decentralizing PAM. Wewill revisit this issue at the end of Section 2.6.2.5 First Best AllocationI now formally define the First Best program. Since participation iscostly, it is without loss to assume that the Utilitarian planner will poolall workers and assets assigned to the same match (p′, q′) into one meeting24In fact, the argument underlying Lemma 1 applies to more general forms of menusbeyond the scope of the subsequent analysis.25This can be prevented by imposing some local sorting condition or stronger beliefrestriction. Though the choice of contract affects the matching surplus in Guerrieri, Shimerand Wright (2010), introducing menus of contracts has no effects in their setting becauseof a local sorting condition.352.5. First Best Allocationplace. The total surplus for an allocation is then given byTS(Wpq, Cpq) :=∑Kk=1∑Ll=1E(Y |pl, qk)Wpq(pl, qk)η( Cpq(pl, qk)Wpq(pl, qk))−∑Kk=1∑Ll=1[VWpq(pl, qk) + UCpq(pl, qk)] .Definition. Given the distribution of types (P,Q), a First Best allocation(WFBpq , CFBpq ) maximizes the total surplus TS(Wpq, Cpq) subject to the re-sources constraints:∑Kk=1Wpq(pl, qk) ≤ P (pl),∀l ∈ [1, L]∑Ll=1Cpq(pl, qk) ≤ Q(qk),∀k ∈ [1,K]The set of First Best allocations is well-defined and always exists. WFBpq =CFBpq because participation is costly. Hence, every active market clears inan equilibrium supporting a First Best allocation.The First Best allocation is indeed unique under Assumption (P). Sincethe matching surplus is always positive, min{∑Kk=1 P (pl),∑Ll=1Q(qk)}mea-sure of agents on both sides participate in matching. The rest take their out-side options. Only the highest types participate and pair up assortativelybecause the matching surplus is strictly increasing and SPM in types.Definition. Positive Assortative matching (PAM) is an allocation (Wpq, Cpq)satisfying Wpq = Cpq and for any l′ ≥ 1 and k′ ≥ 1,∑Ll≥l′∑Kk≥k′WFBpq (pl, qk) = min{∑Ll=l′P (pl),∑Kk=k′Q(qk)}. (2.5)Remark 2. Positive Assortative matching is the unique First Best alloca-tion.The lowest participating type on each side is referred as the threshold362.5. First Best Allocationtype. The threshold types are given byk = maxk′≥1{∑Kk=k′Q(qk) ≥ min{∑Ll=1P (pl),∑Kk=1Q(qk)}},l = maxl′≥1{∑Ll=l′P (pl) ≥ min{∑Ll=1P (pl),∑Kk=1Q(qk)}}.Note that k = 1 in the case∑Kk=1Q(qk) ≤∑Ll=1 P (pl) and l = 1 if itis the other way around. For any k ≥ k, define rFBq (qk) = {rFBq (pl|qk)}Ll=1whererFBq (pl|qk) =WFBpq (pl, qk)∑Ll′=1WFBpq (pl′ , qk). (2.6)rFBq (qk) is the distribution of workers that the asset owners of quality qkmatch with under the First Best allocation.2.5.1 First Best decentralization under full informationWhen the workers’ types are public, the contract posted by an assetowner not only specifies the contingent payment scheme, but also the typeof worker she commits to pair with. Owners of assets with the same qualitymay match with different type of workers, provided that they are indifferentbetween these contract offers. In this formulation, (t, p, q) indexes the con-tinuums of markets. The market (t, p′, q′) only opens to the workers of typep′ and the owners of asset quality q′. The equilibrium definition in Section2.4.4 is adapted to this market structure to define an equilibrium under fullinformation. The formal definition is provided in Appendix A.2.Proposition 1. Under full information, an allocation is supported by anequilibrium if and only if it is First Best.Proposition 1 does not require Assumption (P). It only relies on theavailability of an ordered set of securities, St ⊆ Ωt, so that any fraction ofmatching surplus can be transferred between partners.372.5. First Best AllocationLet V FB(pl) = V + ∆VFB(pl) and UFB(qk) = U + ∆UFB(qk), where∆V FB(pl) ≥ 0 and ∆UFB(qk) ≥ 0 are the shadow prices of the respec-tive resource constraint in the First Best program. Under Assumption (P),V FB(pl) and UFB(qk) are monotonic in types, and strictly increasing forthose above threshold types pl and qk. The first order conditions for theFirst Best program are given byV FB(pl) + UFB(qk) ≥ E(Y |pl, qk) (2.7)with equality if WFBpq (pl, qk) > 0, and that the corresponding resource con-straint must bind if V FB(pl) > V or UFB(qk) > U.A First Best allocation can be decentralized in the following manner.Asset owners of qk, with a measure of WFBpq (pl, qk) > 0 will post the con-tract t(.; slk) for the workers of type pl, where t(.; slk) provides the payoffu(qk, pl, t(.; slk)) = UFB(qk). If an asset owner posts a contract to attractworkers of type pl′ , she has to offer v(pl′ , qk, t′) ≥ V FB(pl′). This leaves herat most u(qk, pl′ , t′) ≤ UFB(qk) under the first order condition in (2.7). Inthis equilibrium, a worker is indifferent between any contracts available tohim. Therefore, no one can gain from deviation.Now consider an equilibrium under full information. Suppose E(Y |pl, qk) >V (pl) + U(qk), then an asset owner of qk will profit from offering the work-ers of pl a contract with payoff slightly above V (pl). Her contract will beaccepted and leaves her a payoff above U(qk). Since participation is costly,rationing necessarily results in E(Y |pl, qk) > V (pl) + U(qk) for any pair ofparticipants in that market. This immediately implies market clearing inevery active market. If the resource constraint is not binding for some typeof agents, say pl, the decision that some of these workers are taking theiroutside option indicates that V (pl) = V . Hence, the equilibrium payoffs and382.5. First Best Allocationallocation for every equilibrium conform with the first order conditions ofthe First Best program. It follows that the equilibrium allocation is a FirstBest.Corollary 1. Under full information, the First Best allocation can be de-centralized using any set of contracts with transfers UFB(qk) for qk abovethe threshold type. The equilibrium divisions of the surpluses are invariantto the set of feasible contracts.Suppose that the asset owners may post a menu of contracts, which thecontract term is contingent on the type of the worker. The markets areindexed by the pair of types and the menu of contracts. The meeting isbilateral. Proposition 1 can be extended to this setting using essentially thesame argument.2.5.2 First Best decentralization in price competitionI now return to the case of one-sided private information but omit As-sumption (C) in this subsection. I will consider the class of cash payment,or fixed prices, which is represented bytc(y; s) = sE(Y |p, q).Price competition refers to a setting where the class of cash payment isfeasible.Proposition 2 states that in price competition, the equilibrium allocationand payoffs are irrespective of whether the workers’ types are private orpublic. Therefore, price competition always leads to the First Best outcome.This result is not surprising at all. With cash payment, the workers receivethe residual claims, while the asset owners find the belief about the worker’s392.5. First Best Allocationtype irrelevant. As a result, the incentives for both sides are the same as inthe full information case.Proposition 2. When cash payment is feasible, an allocation is supportedby an equilibrium if and only if it is First Best. Furthermore, the equilibriumpayoffs in this equilibrium are the same as under full information.Like Proposition 1, Proposition 2 does not hinge on Assumption (P). AFirst Best allocation is decentralized in an equilibrium, in which owners ofquality qk post the cash payment UFB(qk). In this equilibrium, a workerof type pl receives E(Y |pl, qk) − UFB(qk) from a match with an asset ofquality qk. From the first order condition in (2.7), he can earn no morethan his equilibrium payoff V FB(pl) by deviating to other active markets.As in the full information, this first order condition also rules out prof-itable deviations on the asset side. The converse is also true. Specifically, ifE(Y |pl, qk) > V (pl) +U(qk), then an asset owner of qk will post a cash pay-ment slightly above V (pl), earning a payoff above u(qk). We then concludethat in every equilibrium, the allocation is a First Best and in particular,all active markets clear.Consider a generic distribution of types which satisfies∑Kk=k′ Q(qk) 6=∑Ll=l′ P (pl) for all k′ ≥ 1 and l′ ≥ 1. Under Assumption (P), the first orderconditions of the First Best program uniquely determine the equilibriumpayoffs, and hence the set of prices posted in equilibrium. The asset ownersof the threshold type, if participating, will post the cash paymentUFB(qk) = E(Y |pl, q1)− V ,U, if∑Kk=1Q(qk) <∑Ll=1 P (pl); andif∑Kk=1Q(qk) >∑Ll=1 P (pl).Let l(k) denote max{l ≥ 1 : WFBpq (pl, qk) > 0, qk ≥ qk}, the highesttype of workers pairing up with qk ≥ qk in the First Best allocation. l(k)402.6. Decentralization Of Positive Assortative Matchingis increasing in l under PAM. Generically, the First Best allocation alsoinvolves matches between workers of type pl(k) and the assets of qk+1. Fromthe first order condition in (2.7),V FB(pl(k)) = E(Y |pl(k), qk)− UFB(qk) = E(Y |pl(k), qk+1)− UFB(qk+1),which allows us to construct the set of posted prices recursively,UFB(qk+1)− UFB(qk) = E(Y |pl(k), qk+1)− E(Y |pl(k), qk). (2.8)2.6 Decentralization Of Positive AssortativeMatchingThis section studies whether the First Best allocation, defined in theequality (2.5), can be decentralized in an equilibrium. For exposition, I firstanalyze the case that the whole set of feasible contracts is fully ordered.26Assumption (S). The feasible set of contracts is an ordered set of securities,Ωt = St.I will construct a candidate equilibrium in the process. In general, theexpected payment between partners is not separable in the contract termand the expected output. So I characterize the existence conditions in termsof the worker’s expected payoff, representing his trade-off between the assetquality q, the contract term s and his matching probability η(µ). Theseconditions apply to any distribution of types. They are Condition Sorting-pwhich renders deviations to other active markets unprofitable for the work-ers, Condition Screening-q and Condition Entry-q which deter deviations by26Section 2.8 will relax this assumption and discuss what other contracts can be madefeasible without affecting the results.412.6. Decentralization Of Positive Assortative Matchingthe asset owners above and below the threshold quality respectively. In thesubsequent sections, I will explain how they address the potential deviationsby workers and asset owners in the candidate equilibrium. The argumentfor necessity is relegated to the appendix. This section culminates with aunifying sufficient condition, Global ID, for the decentralization of the FirstBest allocation. I will evaluate the robustness of the result at the end of thesection.2.6.1 Sorting of workersIt is instructive to start with the problem of implementing the First Bestallocation, in which the Utilitarian planner decides the contracts for eachtype of assets subject to their voluntary participation.27 The implementa-tion problem highlights the limitations due to the incentive compatibilityconstraints for the workers. Its solution then serves as the set of activemarkets in the candidate equilibrium supporting the First Best allocation.All workers, regardless of their types, share the same preference over anordered set of securities. The First Best allocation requires market clearingin every active market, and therefore owners of the same asset quality, ifparticipating, must post the same contract. Otherwise, either the assetowners posting the contract with the highest term s will be left unmatched orthe workers will be rationed in the market with the lowest term s. Hence, thesolution to the implementation problem can be written as {(t(.; sk), qk)}k≥k,where t(.; sk) denotes the contract for the owners of asset quality qk ≥ qk.27In the market structure here, it is straightforward to implement such policy by re-stricting the set of markets available to the asset owners.422.6. Decentralization Of Positive Assortative MatchingA worker has to decide among the active markets and his outside option,V (pl) = max{V , {v(pl, qk, t(.; sk))}k≥k}.The implementation of the First Best allocation entails two sets of in-centive compatibility (IC) conditions on the workers’ side. The first set ofconditions is given by V (pl) = V if∑Ll=1WFBpq (pl, qk) < P (pl), so that theseworkers are indifferent about taking their outside options. The second setconcerns the choice of contracts, requiring V (pl) = v(pl, qk, t(.; sk)) if (pl, qk)is in the support of WFBpq .The distributions of workers in the active markets are consistent with theFirst Best allocation, which are given by rFBq defined in equality (2.6). Vol-untary participation of the asset owners requires that u(qk, rFBq (qk), t(.; s˜k)) ≥U for qk ≥ qk. Furthermore u(qk, rFBq (qk), t(.; sk)) = U if∑Ll=1WFBpq (pl, qk) <Q(qk), so that the corresponding active market clears.Now consider a generic distribution of types with∑Kk=k′ Q(qk) 6=∑Ll=l′ P (pl)for all k′ ≥ 1 and l′ ≥ 1. In this case, the outlined conditions uniquely pindown the set of contracts offered . Let {(t(.; s˜k), qk)}k≥k denote this set ofcontracts.s˜k is chosen to ensure efficient participation of the threshold type on thelong side, which requires the following indifference condition.28 v(pl, qk, t(.; s˜k)) = V ,u(qk, rFBq (qk), t(.; s˜k)) = U,if∑Kk=1Q(qk) <∑Ll=1 P (pl); andif∑Kk=1Q(qk) ≥∑Ll=1 P (pl).(2.9)This avoids excessive participation of the threshold type on the long side byleaving them indifferent between participation and their outside options.28In equation (2.9), I assume that the workers receive the entire matching surplus whenboth sides are of equal measure. This is innocuous and not required for any of my results.432.6. Decentralization Of Positive Assortative MatchingRecall that pl(k) denotes the highest type of workers pairing up withthe assets of qk ≥ qk in the First Best allocation. For generic distributionsof types, the First Best allocation also involves matches between workersof type pl(k) and the assets of qk+1. As a result, these workers must beindifferent between the market (t(.; sk), qk) and (t(.; sk+1), qk+1), so that thelocal upward IC condition must hold with equality.v(pl(k), qk, t(.; s˜k)) = v(pl(k), qk+1, t(.; s˜k+1)). (2.10)For k > k, s˜k is defined recursively by the indifference condition in (2.10).29Remark 3. Give any generic distribution of types, the set of active marketsfor any equilibrium supporting the First Best allocation, if exists, is uniqueand given by {(t(.; s˜k), qk)}k≥k, where {s˜k}k≥k is determined by equations(2.9) and (2.10) recursively.{s˜k}k≥k is a strictly increasing sequence, representing the trade-off be-tween the asset quality and the contract term facing the workers. Moreimportantly, {(t(.; s˜k), qk)}k≥k define indirect mappings from the distribu-tion of types to the equilibrium payoffs, as well as the deviating payoffs forboth sides. Decentralization of the First Best allocation requires the for-mer to be always above the latter. The subsequent analysis studies howthese mappings depend on the properties of the worker’s and asset owner’sexpected payoff. For non-generic distributions of types, the First Best al-location can also be supported by a continuum of equilibria, including the29Equation (2.10) should not be interpreted as binding local upward IC conditions. Asubset of IC conditions is said to be binding only with respect to certain optimizationproblems, such as profit maximization or information rent minimization. The equality in(2.10) holds merely because the First Best allocation involves matches between workersof type pl(k) and the asset of qk+1 as well as those of qk.442.6. Decentralization Of Positive Assortative Matchingcandidate equilibrium with the active markets {(t(.; s˜k), qk)}k≥k. Hence, allresults apply to any distribution of types.Condition (Sorting-p). v(p, q, t(.; s)) satisfies Condition Sorting-p if forany pH > pL, qH > qL and sH > sL satisfying v(pL, qL, t(.; sL)) ≥ V andu(qL, pH , t(.; sL)) ≥ U , the following holds:v(pH , qH , t(.; sH)) ≥ (>)v(pH , qL, t(.; sL))if v(pL, qH , t(.; sH)) ≥ (>)v(pL, qL, t(.; sL)).(2.11)And v(p, q, t(.; s)) satisfies Condition strict Sorting-p if (2.11) is replaced byv(pH , qH , t(.; sH)) > v(pH , qL, t(.; sL)) if v(pL, qH , t(.; sH)) ≥ v(pL, qL, t(.; sL)).Condition Sorting-p is a single crossing property on the worker’s prefer-ence over asset quality and contract term, provided that he will get matched.Consider two contracts (t(.; sH), qH) and (t(.; sL), qL), the line (2.11) statesthat a high type worker must strictly prefer the contract (t(.; sH), qH) to(t(.; sL), qL) if a worker of lower type does so. Its contrapositive requiresthat a low type worker must strictly prefer the contract (t(.; sL), qL) to(t(.; sH), qH) if a worker of higher type does so. Nevertheless, only the con-tracts that may be posted in the candidate equilibrium have to satisfy thisproperty. This is achieved by the restrictions v(pL, qL, t(.; sL)) ≥ V andu(qL, pH , t(.; sL)) ≥ U, which ensure that the outside options for the work-ers and the asset owners are no more attractive than the contracts.30 Hence,Condition Sorting-p is weaker than the standard single crossing property,which applies to the entire domain. Condition strict Sorting-p is a stronger30For the worker’s type in Ωp(t(.; s˜k), qk), pl(k) plays the role of pH andu(qk, pl(k), s˜k) ≥ U by construction. Therefore, Condition Sorting-p imposes the restric-tion u(qL, pH , t(.; sL)) ≥ U.452.6. Decentralization Of Positive Assortative Matchingversion of Sorting-p. It further requires that a high type worker strictlyprefer the contract (t(.; sH), qH) to (t(.; sL), qL) when a worker of lower typeis indifferent between the two.It is the well-known that the single crossing property in Condition Sorting-p ensures the set of adjacent upward IC conditions in (2.10) are sufficientfor the workers’ incentive compatibility.The construction of {(t(.; s˜k), qk)}k≥k implies that the equilibrium pay-offs are monotonic in types, and hence the participation constraints for theboth sides are satisfied. Therefore, Condition Sorting-p allows the imple-mentation of the First Best allocation.31 As we will see, it is also necessaryfor the implementation for arbitrary distributions of types.32Lemma 2. Suppose Condition Sorting-p holds. Then the participation con-straints for both sides and the workers’ incentive compatibility condition arealways satisfied given the set of active markets {(t(.; s˜k), qk)}k≥k.Recall that in the candidate equilibrium, the distributions of workers inthe active markets are given byr(t(.; s˜k), qk) = rFBq (qk), qk ≥ qk.Hence, the equilibrium payoff for the asset owners is given byU(qk) = u(qk, rFBq (qk), t(.; s˜k)),U, if qk ≥ qk; andif qk < qk.On the other side, a participating worker pays the asset owner u(qk, pl, t(.; s˜k))in expectation, leaving himV (pl) = E(Y |pl, qk)− u(qk, pl, t(.; s˜k)).31The argument for Lemma 2 does not hinge on the equality in (2.10) and applies tonon-generic distribution of types.32The necessity will be established in the proof of Proposition 3.462.6. Decentralization Of Positive Assortative MatchingIn price competition, he pays UFB(qk) in equilibrium. Since u(qk, pl, t(.; s))strictly increases with the worker’s type, V (pl) must increase at a slowerrate than in price competition, and the opposite is true for the asset side.In both cases, the equilibrium payoff for the threshold type on the longside is the same as his outside option, henceu(qk, pl(k), t(.; s˜k)) ≥ UFB(qk).The adjacent upward IC conditions (2.8) and (2.10) in these two cases canbe rewritten asu(qk+1, pl(k), t(.; s˜k+1))− UFB(qk+1) = u(qk, pl(k), t(.; s˜k))− UFB(qk).As the payment made by the worker increases with his type, the workersall pay more in the candidate equilibrium when they are on the long side.When the workers are on the short side, those of types above pl(k) must paymore while their peers of the threshold type are better off.Remark 4. Comparing with the equilibria under full information or pricecompetition, in the candidate equilibrium,1. the equilibrium payoff for the asset owners is higher, and increaseswith their types at a faster rate, and2. the equilibrium payoff for the workers increases with their types at aslower rate, and is lower for those matching with assets of qualitystrictly above the threshold type.The shift in the equilibrium divisions of the matching surpluses is relatedto the linkage principle in DeMarzo, Kremer and Skrzypacz (2005). To seethe connection, consider a setting with a continuum of worker’s types. The472.6. Decentralization Of Positive Assortative Matchingcontract term s˜k+1, and hence the payment received by an asset owner ofqk+1, is determined by the local competition between the workers of pl(k)and those of slightly higher types for the same assets qk+1. This resemblesan auction of an asset of qk+1. Since the linkage between the worker’s typeand the expected payment he made is greater under a contingent contractthan a posted price, DeMarzo, Kremer and Skrzypacz (2005) states that thecompetition between workers will intensify, bidding up the payment to theasset owner.What is novel in assortative matching is the additional spillover effectto the competition for better assets. As the workers of pl(k+1) find thecontract term for assets qk+1 less favorable, they are willing to pay morefor an asset of qk+2, further intensifying the competition for such assets.This spillover effect keeps growing when moving up to better assets. As aresult, all asset owners are better off in comparison with price competition.33Section 2.8.1 will generalize this comparative statics for two different sets offeasible contracts.2.6.2 Screening by asset ownersIn this subsection, we assume that the IC conditions on the workers’ sideare all met and turn to the incentives for the asset side.34 An asset ownerchooses between her outside option and posting a contract. The First Bestallocation can be decentralized if no asset owners may profit from posting a33Note that the comparison with the full information case is due to the competitionamong the asset owners. The auctioneer will extract all the surplus if she knows thebidder’s types.34Even if Condition Sorting-p is not met, there are distributions of types for which theworkers’ IC conditions are satisfied in the candidate equilibrium. Condition Sorting-pplays no role in the analysis in this subsection.482.6. Decentralization Of Positive Assortative Matchingdeviating offer,U(qk) = max{U, {u(qk, r(t(.; s), qk), t(.; s), µ(t(.; s), qk))}s∈[0,1]}.Remark 5. An owner of asset qk ≥ qk will never profit from a match witha worker of type no higher than the lowest type in the support of rFBq (qk).Let pL denote the lowest type of workers whom the asset owner may pairwith in the First Best allocation. She does not gain from matching with suchworkers under full information because the decline in the expected outputoutweighs the savings in the payment for the worker,UFB(qk) ≥ E(Y |pL, qk)− V FB(pL) > E(Y |pl, qk)− V FB(pl).Remark 4 establishes that the equilibrium payoff for workers increases at aslower rate in the candidate equilibrium. Hence, the amount of savings iseven lower now,U(qk) ≥ E(Y |pL, qk)− V (pL) > E(Y |pl, qk)− V (pl).Yet the asset owner can receive at most E(Y |pl, qk) − V (pl) when pairingup with a worker of pl. Otherwise, the worker will not accept the offeredcontract. Therefore, an asset owner will never attempt to poach weakerworkers! On another hand, poaching a better worker is potentially profitablebecause they are willing to accept a lower payoff than in the full informationcase.The worker’s gain from a match v(p, qk, t(.; s)) depends on the worker’stype. As the quality of her asset is given, an asset owner may screen out bet-ter workers by varying both the contract term and the matching probability492.6. Decentralization Of Positive Assortative Matchingfor the workers. To induce rationing, the asset owner must accept a lowerterm s compensating the workers for their risk of leaving unmatched.35The distribution of workers attracted by a deviating offer depends onother options available to them, which are the active markets {(t(.; s˜k), qk)}k≥kfrom the preceding section. I now construct the off-equilibrium-path belief inthe candidate equilibrium. For any inactive market (t(.; s′), qk), no workerswill participate if the contract (t(.; s′), qk) provides them no more than theirequilibrium payoffs.36 In this case, r(t(.; s′), qk) can be set arbitrarily, saythe prior distribution for the worker’s side. Suppose certain types of work-ers strictly gain from the contract (t(.; s′), qk), competition among workerspushes down the tightness ratio until no one gains from participating in thatmarket,µ(t(.; s′), qk) = max{µ′ ≤ 1 : V (p′) ≥ µ′v(p′, qk, t(.; s′)), p′ ∈ {pl}Ll=1}.Our equilibrium definition restricts the support of r(t(.; s′), qk) to the work-ers who are willing to endure the lowest matching probability. In the candi-date equilibrium, r(t(.; s′), qk) is taken to be degenerate at the lowest typeamong these workers,min{p′ ∈ {pl}Ll=1 : V (p′) = µ(t(.; s′), qk)v(p′, qk, t(.; s′))}This is the most pessimistic belief allowed.I will address the screening incentive for an owner of asset quality qk ≥ qkand qk < qk in sequence. An owner of asset quality qk ≥ qk will facecompetition from other owners of the same or lower asset quality while35Section 2.6.4 provides further discussion on the screening instrument available to theasset owner.36In the candidate equilibrium, I take µ(t(.; s′), q′) = ∞ when V (pl) = v(pl, q′, t(.; s′))for some types. All results remain valid if µ(t(.; s′), q′) is finite in such case.502.6. Decentralization Of Positive Assortative Matchingan owner of asset quality qk < qk will compete only with those of higherasset quality, in particular the threshold type qk. As we are looking fornecessary and sufficient conditions for arbitrary distributions of types, themost substantial difference between the two cases lies in the restrictionsimposed on the set of active markets in the candidate equilibrium. Note thatfor any asset quality q ∈ (q, q), it is below the threshold type in the FirstBest allocation for some distributions of types, while above the thresholdfor other distributions of types.Let us consider the screening problem for the owner of asset qualityqk ≥ qk. Suppose an asset owner posts a contract with s > s˜k, this contractis dominated by the contract t(.; s˜k) posted by other owners of asset qualityqk. As the active market clears in the candidate equilibrium, the deviatingcontract will attract no workers. Now consider a contract with lower s < s˜k,competition among workers drives down their matching probability in themarket (t(.; s), qk), raising the possibility of screening.Condition (Strong Screening-q). v(p, q, t(.; s)) satisfies Strong Screening-qif for any q ∈ (q, q], pH > pL and sH > sL satisfying u(q, pH , t(.; sL)) >max{U, u(q, pL, t(.; sH))} and v(pL, q, t(.; sH)) ≥ V , thenv(pH , q, t(.; sH))v(pH , q, t(.; sL))>v(pL, q, t(.; sH))v(pL, q, t(.; sL)). (2.12)Condition Strong Screening-q concerns the worker’s preference over thecontract term and his matching probability for the same type of assets.The inequality (2.12) is an increasing difference (ID) condition (in ratio)on v(p, q, t(.; s)), under which an increase in the term s reduces the valueof a match for a low type worker proportionally more than a high typeworker. This in turn implies that when facing a trade-off between matchingprobability and contract term, a high type worker prefers a higher matching512.6. Decentralization Of Positive Assortative Matchingprobability whereas a low type worker prefers a more generous term. As aresult, rationing can never improve the distribution of workers.Fix a distribution of types, the contract (t(.; s˜k), qk) corresponds to(t(.; sH), q) in Condition Strong Screening-q. pL is the lowest type of work-ers whom the asset owner may match with in the First Best allocation.An owner of asset quality qk gains from posting a contract with sL < s˜konly if the deviating offer will be accepted by some workers of pH , satis-fying u(qk, pH , t(.; sL)) > U(qk). Condition Strong Screening-q ensures theworkers of pL are willing to endure a lower matching probability in the mar-ket (t(.; sL), qk), crowding out workers of pH . To see this, the equilibriumtightness ratio in the market (t(.; sL), qk) must satisfyV (pL) = v(pL, qk, t(.; s˜k)) ≥ µ(t(.; sL), qk)v(pL, qk, t(.; sL).The IC condition for workers of pH and the inequality (2.12) together yieldV (pH) ≥ v(pH , qk, t(.; s˜k)) > µ(t(.; sL), qk)v(pH , qk, t(.; sL),so no workers of pH will accept the deviating offer.Condition Strong Screening-q, though intuitive, is stronger than neces-sary. The reason is that it fails to capitalize on the presence of workersmatching with assets of other qualities. Condition Screening-q is weakerthan Condition Strong Screening-q in this regard. Condition Screening-qfactors in the possibility that there can be some workers of type pl ≤ pL, whoneeds not be participating in the market (t(.; s˜k), qk), are willing to endurea lower matching probability than those of pH in the market (t(.; sL), qk).Given the equilibrium payoffs, this happens if and only ifV (pH)V (pl)≥ v(pH , qk, t(.; sL))v(pl, qk, t(.; sL)).522.6. Decentralization Of Positive Assortative MatchingSince the equilibrium payoffs are endogenous, the challenge is to char-acterize when this will be the case for arbitrary distributions of types andworker’s expected payoff. PAM and the equalities conditions (2.9) and (2.10)restrict the type distributions for which (t(.; s˜k), qk) is an active market, andhence the active markets for assets of lower qualities. Condition Screening-qexploits this restriction. It is necessary and sufficient for preventing ownersof asset quality qk ≥ qk from deviating in the candidate equilibrium.Condition (Screening-q). v(p, q, t(.; s)) is said to satisfy Screening-q if forany q ∈ Ωq, pH > pL and sH > sL satisfying v(pL, q, t(.; sH)) ≥ V andu(q, pH , t(.; sL)) > max{U, u(q, pL, t(.; sH))}, then eitherv(pH , q, t(.; sH))v(pH , q, t(.; sL))≥ v(pL, q, t(.; sH))v(pL, q, t(.; sL)), (2.13)Or the followings hold:1. v(pL, q, t(.; sH)) ∈ (V ,E(Y |pL, q)− U ], and2. Let q′ ≤ q and s′ ≤ sH satisfy v(pL, q′, t(.; s′)) = v(pL, q, t(.; sH)). Ifv(p′, q′, t(.; s′)) = V or u(q′, p′, t(.; s′)) ≤ U for some p′ < pL, thenv(pH , q, t(.; sH))v(pH , q, t(.; sL))≥ v(p′, q′, t(.; s′))v(p′, q, t(.; sL)). (2.14)The inequality (2.13) merely replaces the strict inequality (2.12) in Con-dition Strong Screening-q with a weak one. In comparison with Condi-tion Strong Screening-q, Condition Screening-q allows the situation thatthe workers of pH are willing to endure a lower matching probability thanworkers of pL for the contract term sL in the candidate equilibrium. Suchexception is permitted only under two additional conditions, which jointlyguarantee that there are other workers of type below pL prepared to accepta lower matching probability than those of type pH .532.6. Decentralization Of Positive Assortative MatchingThe first condition ensures that some workers of types below pL areparticipating in other active markets. Recall that the contract (t(.; s˜k), qk)plays the role of (t(.; sH), q) and pL is the lowest type of workers participatingin this market. For any asset quality q′ below qk, define the contract t(.; ŝ(q′))such that a worker of type pL is indifferent between the contracts (t(.; s˜k), qk)and (t(.; ŝ(q′)), q′). Condition Screening-q requires v(pL, qk, t(.; s˜k) > V andu(q′, pL, t(.; ŝ(q′))) > U for any q′ < qk. By construction of s˜k, the very factthat some workers of pL are participating in the active market (t(.; s˜k), qk)in the candidate equilibrium implies that the First Best allocation involvesmatches between workers of types below pL and assets of quality below qk.The second condition ensures that workers of pH will not participatein the market (t(.; sL), qk). pl is the lowest type among the participatingworkers. If workers are on the short side, there always exists q′ < qk suchthat V (pl) ≤ v(pl, q′, t(.; ŝ(q′))) and u(q′, pl, t(.; ŝ(q′))) ≤ U . Conversely,there exists q′ < qk satisfying V (pl) = v(pl, q′, t(.; ŝ(q′))) = V if workers areon the long side. Given the continuity of the expected payoff, the existenceof q′ is an implication of the workers’ incentive compatibility conditions andthe construction of s˜k. In general, q′ is not in the support of the distributionof types and q′ 6= qk in particular.37Under Condition Screening-q, the inequality (2.14) implies that in thecandidate equilibrium,V (pl) ≤ v(pl, q′, t(.; ŝ(q′))) < v(pH , qk, t(.; sL)),andV (pH)V (pl)≥ v(pH , qk, t(.; sH))v(pl, q′, t(.; ŝ(q′)))≥ v(pH , qk, t(.; sL))v(pl, qk, t(.; sL)).37The support of type distribution Q is {qk}Kk=1, a finite subset of the types space (q, q],and q′ ∈ (q, q).542.6. Decentralization Of Positive Assortative MatchingIt follows that workers of pl accept a lower matching probability in the mar-ket (t(.; sL), qk) than their peers of pH , and hence the asset owner believesthat her deviating offer will attract no workers of pH . Her deviating offermay be accepted other workers of types above pL but she will not profitfrom such match.Lemma 3. Suppose participation constraints for both sides and the workers’incentive compatibility condition are met. Under Condition Screening-q, anowner of asset quality qk ≥ qk cannot profit from posting a contract t(.; s),where s 6= s˜k, in the candidate equilibrium.We now proceed to the participation decision of the asset owners ofqk < qk in the candidate equilibrium. For them, only the deviating offerswith sL < s˜k are relevant. Otherwise, the offer will be dominated by themarket (t(.; s˜k), qk). A direct consequence of Condition Sorting-p is that adeviating offer from an owner of asset quality qk < qk, if attracts any workersat all, will also interest the workers of the lowest type p1. The constructionof s˜k implies that workers of type p1 pay the asset owners of qk no morethan the latter’s outside option, so they will pay the deviating asset ownereven less under the contract t(.; sL). Consequently, the asset owner cannotprofit from such offer if it will attract only the workers of p1.Condition Entry-q builds on this observation. It is an increasing dif-ference condition concerning the worker’s preference over a more generouscontract term and improvements in both asset quality and his matchingprobability. Condition Entry-q is necessary and sufficient to render anydeviation by the owners of asset quality qk < qk unprofitable in the candi-date equilibrium. It shall be stressed that the sufficiency is irrespective ofwhether Condition Sorting-p holds.552.6. Decentralization Of Positive Assortative MatchingCondition (Entry-q). v(p, q, t(.; s)) is said to satisfy Entry-q if for anypH > pL, qH > qL and sH > sL satisfyingv(pH , qL, t(.; sL)) > v(pH , qH , t(.; sH)), andv(pL, qH , t(.; sH)) ≥ V , andu(qL, pH , t(.; sL)) > U ≥ u(qH , pL, t(.; sH)),(2.15)thenv(pH , qH , t(.; sH))v(pL, qH , t(.; sH))≥ v(pH , qL, t(.; sL))v(pL, qL, t(.; sL)). (2.16)Suppose that workers of some type pH prefer the contract (t(.; sL), qL) to(t(.; sH), qH). Under these two contracts, a worker of type pH pays more thanthe asset owner’s outside option, whereas a worker of some type pL does not.Then the inequality (2.16) in Condition Entry-q has two implications. First,workers of type pL also prefer the contract (t(.; sL), qL) to (t(.; sH), qH).38Second, for any pair of matching probabilities in the market (t(.; sH), qH)and (t(.; sL), qL), workers of type pL always prefer the latter market to theformer if their peers of type pH do so.Lemma 4. Suppose participation constraints for both sides and the work-ers’ incentive compatibility condition are met. Under Condition Entry-q, anowner of asset quality qk < qk cannot profit from posting any contract in thecandidate equilibrium.In the candidate equilibrium, pL and (t(.; sH), qH) correspond to p1 and(t(.; s˜k), qk) respectively. The contract (t(.; sL), qL) is a deviating offer byan owner of asset quality qL < qk. The construction of s˜k ensures that the38Though the inequality (2.16) implies the statement (2.11), the former.is only requiredto hold for a smaller set of types and contracts. Therefore, Condition Entry-q is consistentwith, but no stronger than, Condition Sorting-p.562.6. Decentralization Of Positive Assortative Matchingpreconditions in (2.15) are always met when the asset owners are on the longside. Under Condition Entry-q, the workers of p1 are willing to endure alower matching probability than any workers whose participation profits thedeviating asset owner. Hence, the asset owners of qL will take their outsideoptions instead of posting the contract t(.; sL).It is noteworthy that in both Condition Screening-q and Entry-q, workersof the threshold type pl, whose exact type depends on the distribution oftypes, play a key role in deterring deviations. Given the construction ofthe active markets {(t(.; s˜k), qk)}k≥k, a match with them always rendersthe concerned deviating offers unprofitable. This explains the sufficiency.The less obvious part is why the necessary and sufficient conditions pivoton them, but not their peers above the threshold type. This is becausethese workers choose the same contract in the candidate equilibrium fora larger subset of distributions of types. To see this, consider perturbingthe distribution of types above the threshold types on both sides, the pairsof threshold types (pl, qk) and the contract term s˜k they choose remainunchanged while the other active markets {(t(.; s˜k), qk)}k>k and the typesof their participants are potentially being affected. This feature stems fromthe fact that the set of active markets is determined recursively from bottomto top.2.6.3 Conditions for Positive Assortative MatchingProposition 3. The First Best allocation can be supported by an equilibriumfor any distribution of types if and only if Condition Sorting-p, Screening-qand Entry-q all hold.The preceding discussion covers the sufficiency of the conditions and572.6. Decentralization Of Positive Assortative Matchingthe construction of the candidate equilibrium.39 The proof for necessityis also constructive. For each of the conditions, I provide a procedure toconstruct a generic distribution of types, for which no equilibria can supportthe First Best allocation, if the condition is not met. The procedure identifiesthe corresponding profitable deviation in the process. This illustrates theincentives against assortative matching for an individual anticipating thatthe actions by the rest are consistent with the First Best allocation.Given the order structure in the First Best allocation and the prefer-ence of the two sides, it is well known that some kinds of single crossingor increasing difference conditions are sufficient for supporting assortativematching. The novelty here is to characterize the exact conditions for alldistributions of types, which involves two new complications.The first complication stems from the fact that the expected payoff forworkers is generally non-separable in the contract term and the matchingsurplus. As a result, the contract terms cannot be canceled out and willremain in the necessary and sufficient conditions. A prerequisite for suchconditions is then characterizing the set of active markets and the types ofthe participants in the candidate equilibrium for all distributions of typesbecause any restrictions inconsistent with this characterization are point-39 Condition Sorting-p, Screening-q and Entry-q should be viewed as three intersectingsubsets in the space of the expected payoff functions v(p, q, t(.; s)). None of the subsetscontains another. Each of them represents the exact subset of functions v for which thecorresponding type of deviations is not profitable in the candidate equilibrium. The threeconditions are separate in the sense that the arguments for sufficiency and necessity foreach of them do not rely on the other two conditions. In this perspective, one may inter-pret that the intersection of the subsets corresponds to a grand necessary and sufficientcondition, which Condition Sorting-p, Strong Screening-q and Entry-q are its decomposi-tion.582.6. Decentralization Of Positive Assortative Matchingless. This characterization must apply to a large class of expected payofffunctions, covering those indeed satisfying the necessary and sufficient con-ditions.In the setting here, the expected payoffs v(p, q, t(.; s)) and u(q, p, t(.; s))are monotonic and continuous in all arguments and there exists pairs yieldingarbitrarily small matching surplus, formally, E(Y |p, q) = U +V in Assump-tion (P).40 These two properties allow a closed-form characterization of theset of active markets and the types of associated participants, which is thenincorporated as preconditions. The inclusion of preconditions allows me todecompose the grand necessary and sufficient condition into three separateconditions, underscoring the difference in the incentives in supporting PAMfor various groups.The second complication lies in the analysis of the screening problem forthe uninformed side. The pool of workers attracted by a contract offer inthe candidate equilibrium is determined by the workers’ equilibrium payoffand their expected payoff function. This seems to suggest that finding outthe exact condition involves characterizing the set of active markets and theworkers’ equilibrium payoff, which are dependent on the expected payoff40The assumptions provide the following properties:1. Given any (p, qH , sH) and qL < qH , one can find sL such that v(p, qH , t(.; sH)) =v(p, qL, t(.; sL)).2. Given any term ŝ, one finds a term s′ arbitrarily close to ŝ such that there existseither a pair of types (p′, q′) satisfying v(p′, q′, t(.; s′)) = V and u(q′, p′, t(.; s′)) ≥ Uor a pair of types (p′′, q′′) satisfying v(p′′, q′′, t(.; s′)) ≥ V and u(q′′, p′′, t(.; s′)) < U.For each pair of types, I then find out the closure of the set of contract terms theymay choose in the candidate equilibrium for some distribution of types using these twoproperties.592.6. Decentralization Of Positive Assortative Matchingitself, for any distribution of types. I circumvent this task by introducing ahierarchy of preconditions. Given the distribution of types, the set of activemarkets must satisfy the equalities (2.9) and (2.10). Exploiting this systemof equalities and PAM, the preconditions are constructed to categorize thecandidate equilibrium into various scenarios. I then characterize the requiredproperty for the expected payoff for each type of scenario.I now provide a unifying sufficient condition for Condition Sorting-p,Screening-q and Entry-q. It is an increasing difference condition statingthat when switching to a better asset with a higher contract term s, abetter worker will benefit more or suffer less, in term of either amount orpercentage. Simplicity is its main advantage and allows me to provide suffi-cient conditions on the outcome distribution and contract space in Section2.7.Condition (Global ID). v(p, q, t(.; s)) satisfies Global ID if for any pH > pL,qH ≥ qL and sH > sL, at least one of the following conditions hold:v(pH , qH , t(.; sH))−v(pH , qL, t(.; sL)) ≥ v(pL, qH , t(.; sH))−v(pL, qL, t(.; sL)),Or there exist cH ≥ cL ≥ 0 such thatv(pH , qH , t(.; sH)) + cLv(pH , qL, t(.; sL)) + cL≥ v(pL, qH , t(.; sH)) + cHv(pL, qL, t(.; sL)) + cH.And v(p, q, t(.; s)) is said to satisfy strict Global ID if the weak inequali-ties in the two conditions are replaced with strict inequalities.Condition Global ID consists of two inequalities, one in level and theother in ratio. It does not require the same inequality to hold for every pairof {pH , pL} and {(qH , sH), (qL, sL)}. Instead, one of the inequalities musthold for any given pair of {pH , pL} and {(qH , sH), (qL, sL)}. In this sense,602.6. Decentralization Of Positive Assortative MatchingCondition Global ID are not composed of two separate increasing differenceconditions. Furthermore, the values of cH and cL may vary with the pair of{pH , pL} and {(qH , sH), (qL, sL)}. The constants cH and cL can be thoughtto present in the level inequality but cancel out each other.In the baseline setting, the workers and the asset owners receive nothingif they participate in matching but end up unmatched. This assumptionis innocuous as it merely normalizes the parties’ payoff with respect to theevent they are left unmatched. Other reference points, the absolute out-put level and the worker’s initial wealth for examples, are sometimes moreconvenient for the purpose of studying primitive conditions on the contractspace and the output distribution. Taking the quantities cH and cL as thenew workers’ payoff when they left unmatched, Condition Global ID appliesto the re-normalized worker’s expected payoff. This property will be helpfulin Section 2.7. The restriction cH ≥ cL reflects the requirement that theworker’s unmatched payoff is weakly decreasing in his type. The restric-tion cL ≥ 0 ensures that the worker’s expected payoff is always positive,otherwise the worker will simply take his outside option.Proposition 4. Condition Global ID implies all Condition Sorting-p, Screening-q and Entry-q. Condition strict Global ID implies all Condition StrictSorting-p, Strong Screening-q and Entry-q.It is trivial that the inequalities in Condition Global ID imply Condi-tion Sorting-p. Neither one of the inequalities is weaker than the other insupporting Condition Sorting-p. When a worker, be it high type or lowtype, finds the contract posted by the owner of higher quality asset moreattractive, then the level inequality is the weaker of the two. The oppositeis true if a worker prefers the contract posted by the owner of lower asset612.6. Decentralization Of Positive Assortative Matchingquality.Recall from the previous discussion, Condition Strong Screening-q andEntry-q concern the circumstance that some workers of pL are participatingin the active market (t(.; sH), qH) in the candidate equilibrium and bothworkers of types pH and pL strictly prefer the deviating offer (t(.; sL), qL)to (t(.; sH), qH).41 In this case, the level inequality implies the strict ratioinequality with cH = cL = 0, and hence the two mentioned conditions. Ifthe ratio inequality holds for some pairs of re-normalized expected payoffv(pH , q, t) + cL and v(pL, q, t) + cH , the worker’s marginal value of matchingprobabilities in the two markets also satisfies ratio inequality with cH =cL = 0. This is because a high type worker receives a lower payoff if he endsup unmatched.2.6.4 DiscussionMenu of contracts revisited Proposition 3 applies to the setting inwhich the asset owners may post a menu of contracts. The sufficiency di-rectly follows from Lemma 1 in section 2.4.5. The necessity merits somediscussion. For each of the conditions violated, the proof of Proposition 3details the construction of a generic distribution of types such that the cor-responding First Best allocation cannot be supported by any equilibrium inwhich the asset owners may post only a single contract. Fix this distributionof types and let us conjecture an equilibrium which supports the First Bestallocation using menus of contracts. In such equilibrium, the menu postedby an owner of asset quality qk ≥ qk must include the contract t(.; s˜k) andpotentially some other contracts which are never chosen by workers. The41Condition Strong Screening-q concerns the case that qH = qL.622.6. Decentralization Of Positive Assortative Matchingreason is that all workers will pick the same contract from a menu becausetheir preferences over the contracts are the same.42 As the distribution oftypes is generic, the adjacent IC conditions imply that {(t(.; s˜k), qk)}k≥k isthe set of contracts chosen from the menus. This in turn implies that whenan asset owner deviates to post a single contract, the most pessimistic beliefabout the pool of workers it attracts is the same as in an equilibrium which{(t(.; s˜k), qk)}k≥k is the set of active markets. Therefore, the correspondingdeviation identified in the proof of Proposition 3 remains profitable.Destruction never improves screening This section shows that anasset owner never benefits from a lottery for separation, or a commitmentto impair her asset under Condition Screening-q and Entry-q.43Fix an equilibrium and consider two inactive markets (1, q′, t′) and (pi′, q′, t′)with pi′ < 1. Suppose the market (pi′, q′, t′) is believed to attract someworkers. As these workers are willing to endure the matching probability(1− pi′)η(µ(pi′, q′, t′)), they must also accept the same matching probabilityin the market (1, q′, t′), so(1− pi′)η(µ(pi′, q′, t′)) = µ(1, q′, t′).It follows that any worker will anticipate the same payoff from these twoinactive markets, so the sets of admissible beliefs regarding the distribution42Asset owners of the same quality may still post different menus. Since the equilibriumsupports the First Best allocation, workers face no variation in their matching probabilitiesin the active markets. They must pick the same contract in every menu for a given assetquality.43According to Lemma 1, it is without loss to focus on the deviations to post a singlecontract. Therefore, the candidate equilibrium supporting the First Best allocation inProposition 3 is robust to the introduction of menus of contracts specifying the separationprobability pi ≤ 1, the asset quality q ≤ qk, and the contingent payment t.632.6. Decentralization Of Positive Assortative Matchingof worker’s type are the same for the two markets. An exogenous reductionin the matching probability for workers will not improve screening. Never-theless, the separation lottery exposes the asset owner to a higher risk ofending up unmatched. We can exclude such lottery from our consideration,irrespective of Condition Screening-q and Entry-q.We now turn to the option of asset impairment. Under Condition Screening-q and Entry-q, an asset owner cannot benefit from impairing her asset evenwhen it is costless. The key reason is that the equilibrium payoff for an assetowner is increasing in her asset quality in the candidate equilibrium. Sup-pose, to the contrary, that for some distribution of types, an asset owner of qk̂gains from posting a contract (t(.; s′), q′) in the candidate equilibrium, whereq′ ∈ [qk, qk̂). {qk}Kk≥k denotes the set of asset qualities. Let q′ ∈ [qk′ , qk′+1),then we modify the distribution of types by adding the same measure ofassets q′ and workers of type pl(k′). This construction leads to the followingproperties for the resulting candidate equilibrium. First, the equilibriumpayoffs for workers stay unchanged, so does the off-equilibrium-path belieffor the inactive market (t(.; s′), q′) in the new candidate equilibrium. Second,the owners of the asset qualities {qk}Kk≥k see no change in their equilibriumpayoff. Third, the new candidate equilibrium retains the same set of ac-tive markets {(t(.; s˜k), qk)}k≥k and includes a new active market (t(.; s˜′), q′),where s˜′ satisfies v(pl(k′), q′, t(.; s˜′)) = V (pl(k′)). Note that the owners of as-set quality q′ earn less than their peers of asset quality qk̂. Hence, an ownerof asset quality q′ must profit from posting t(.; s′) in the new candidate equi-librium. This is impossible under Condition Screening-q. Condition Entry-qrules out the case q′ < qk with a similar argument.44 Therefore, an asset44This covers the case that the owner may reduce the quality of her asset to a levelbelow q if Condition Entry-q is extended to hold for qL ≤ q.642.6. Decentralization Of Positive Assortative Matchingowner will never profit from a lower asset quality.It should be highlighted that Proposition 3 is robust only to the com-mitments for deterministic impairment of the asset quality. The result doesnot apply to the setting, where asset owners may post joint lotteries overthe asset quality and the contract term. For example, an asset owner maypost a lottery over (qH , sL) and (qL, sH), where qH > qL and sL ≤ sH . Inthe context of the above argument, this joint lottery corresponds to sidepayments between owners of assets qH and qL. Condition Screening-q andEntry-q are silent on this example as they apply only to cases that a higherasset quality is accompanied with a higher term s. The underlying reason isthat the analysis focuses on the workers’ preference over the asset quality,the contract term and his matching probability, but not the joint lotteriesover them.Properties of other equilibriaEquilibrium rationing Suppose that in some active market (t(.; s), qk),workers are being rationed. A contract of a slightly higher term sH will inter-est these workers. Among those are interested, Condition Strong Screening-q implies that the types of workers attracted to the contract t(.; sH) areno lower than those of the participants in the market (t(.; s), qk). An assetowner will profit from the contract t(.; sH) as it provides a greater divisionof surplus and attracts potentially better workers.Lemma 5. Under Condition Strong Screening-q, workers are never rationedin any equilibrium.In contrast, asset owners may be rationed in equilibrium. Define Ip(t, q) :={p ∈ {pl}Ll=1 : V (p) = η(µ(t, q))v(p, q, t)}. Ip(t, q) denotes the types of work-652.6. Decentralization Of Positive Assortative Matchingers, who obtain their equilibrium payoff if they participate in the market(t, q). It is non-empty for any active market. Ip(t, q) and Ip(t, q) denote thelowest and highest type in Ip(t, q).Suppose that there is an active market (t(.; s′), qk) with µ(t(.; s′), qk) > 1.Consider a contract t(.; sL), where sL < s′. The incentive compatibility con-ditions and Condition Strong Screening-q jointly imply that workers of typesgreater than Ip(t(.; s′), qk) will never accept such contract. Therefore, theasset owners have to trade off between a jump in matching probability anda less favorable distribution of partner’s type. When sL < s′ is sufficientlyclose to s′, only workers of type Ip(t(.; s′), qk) will be attracted. Such localdeviation is the most profitable one among all contracts in St. It imposes anupper bound on the tightness ratio through the following condition,U(qk) =η(µ(t(.; s′), qk))µ(t(.; s′), qk)u(qk, r(t(.; s′), qk), t(.; s′)) ≥ u(qk, Ip(t(.; s′), qk), t(.; s′)).(2.17)Note that the market (t(.; s′), qk) must clear if Ip(t(.; s′), qk) contains a singletype only.Assortative matching Lemma 5 has strong implications for the struc-ture of an equilibrium. Since the workers are never rationed in equilibrium,the properties of v(p, q, t(.; s)) determine their choices of active markets. Thefirst implication is that in any equilibrium, owners of the same asset quality,if participating, must post the same contract and the choice of contract terms increases with their asset quality. Second, workers’ participation and equi-librium payoff are monotonic in their types. Now consider an equilibriumwith active markets (t(.; sH), qH) and (t(.; sL), qL). Condition Sorting-p al-lows the possibility that a high type worker participates in (t(.; sL), qL) whilea low type worker participates in (t(.; sH), qH). This occurs when these two662.6. Decentralization Of Positive Assortative Matchingtypes of workers are indifferent between the two active markets. ConditionStrict Sorting-p rules out this case entirely. More importantly, ConditionStrict Sorting-p implies that the asset owners’ participation decision andthe equilibrium payoff are monotonic as well. To see this, suppose that(t(.; sL), qL) is an active market and pH is the highest type of participat-ing workers. Under Condition Strict Sorting-p, an owner of asset qualityqH > qL can find a contract t(.; sH), which gives workers of pH a payoff justabove what they may receive from the contract (t(.; sL), qL) and attracts noworkers of lower types. By posting this contract, her payoff will be strictlygreater than that of the owners of qL.Lemma 6. Under Condition Strict Sorting-p and Strong Screening-q, everyequilibrium has the following properties:1. Workers are not rationed.2. Owners of the same asset quality, if participating, post the same con-tract.3. Participation on both sides is monotonic in type.4. The equilibrium payoffs are monotonic in types for both sides.5. The types of workers matching with better assets must be no lower thanthose matching with assets of lower qualities.The characterization in Lemma 6 states that for any equilibrium, theset of active markets takes the form {(t(.; sk), qk)}k≥k̂ , where qk̂ is thelowest asset quality among the participating asset owners. The types ofworkers participating in an active market Ωp(t(.; sk), qk) are increasing in672.6. Decentralization Of Positive Assortative Matchingthe asset quality. Furthermore, only the highest type of workers in themarket (t(.; sk), qk) may also participate in the market (t(.; sk+1), qk+1).Therefore, Lemma 6 implies that under Condition Strict Sorting-p andStrong Screening-q, rationing of the asset owners is the only form of in-efficiency in the set of equilibrium allocations. Together with ConditionEntry-q, a continuum of such equilibria always exists for generic finite dis-tributions of types.45 They can be constructed by perturbing the candidateequilibrium.Under Condition Strict Sorting-p and Strong Screening-q, the equilib-rium characterization in Lemma 6 allows us to construct a upper boundoveru(qk,Ip(t(.;s′),qk),t(.;s′))u(qk,Ip(t(.;s′),qk),t(.;s′))for every active market. The inequality (2.17) thenyields the following bound on the tightness ratio of an active market,µ(t(.; s′), qk) ≤ max{u(qk, plH , t(.; s′))u(qk, plL , t(.; s′)):lH∑l′=lLP (pl′) ≤ Q(qk), L ≥ lH ≥ lL ≥ 1}.Note that when the distribution of types converges to a continuous one,plH − plL → 0 and hence µ(t(.; s′), qk)→ 1.Belief restriction In the equilibrium definition, a deviating offer will onlyattract the types of workers who see the greatest proportional increase intheir payoffs if they get matched. The belief restriction here is the same asin Guerrieri, Shimer and Wright (2010), who consider homogeneous princi-pals.46 With heterogeneity on both sides, this belief restriction is crucial forthe decentralization of assortative matching. It establishes a linkage between45Lemma 6 is silent on the existence of equilibria. The inclusion of Condition Entry-qensures the existence of the candidate equilibrium.46DeMarzo, Kremer and Skrzypacz (2005) also adopts a symmetric belief refinementwhen the informed party chooses the contract.682.7. Conditions On Contracts And Production Complementaritythe complementarity in types and the pool of workers attracted by a deviat-ing offer. Suppose that in the candidate equilibrium, an asset owner intendsto poach workers from her peers of higher asset qualities. The workers ac-cepting the deviating offer will suffer a reduction in the asset quality. Thebelief restriction implies that no workers of higher types will be attracted ifthey derive a sufficiently large gain from the complementarity in types. Theexact requirement is captured by Condition Screening-q and Entry-q.472.7 Conditions On Contracts And ProductionComplementarityThis section discusses primitive conditions on the ordered set of securi-ties and the conditional distribution of output, which give rise to ConditionGlobal ID. The classes of output distributions we consider conform with As-sumption (P). The type space and the values of outside options are assumedto satisfy E(Y |p, q) > V + U ≥ E(Y |p, q).48For notational simplicity, assume that the output is continuously dis-tributed on an interval with a lower bound y and an upper bound y ∈ (y,∞].Abusing the notation, Ωy = [y,∞) if y =∞. F (y|p, q) and f(y|p, q), respec-tively, denote the conditional distribution function and density function for47Consider an alternative belief restriction. Suppose the asset owner believes that anyworkers who benefit from the deviating offer will accept it, and the distribution of herpartner’s type is proportional to the prior distribution. In this case, the significance ofcomplementarity in types diminishes in the screening problem for the asset side. It isstraightforward to construct a distribution of types for which assortative matching cannotbe decentralized.48If V + U > E(Y |p, q), the type space [p, p]× [q, q] is truncated to [p˜, p]× [q˜, q], whereE(Y |p˜, q˜) = V + U. The choice of the pair (p˜, q˜) is not unique.692.7. Conditions On Contracts And Production ComplementarityY |(p, q). The results in this section can be extended to settings with discreteoutput distributions readily.The joint conditions on the feasible contracts and the distribution ofoutput are necessarily intertwined. I begin with conditions allowing decen-tralization of the First Best allocation if the assets are all homogeneous. Foreach of these conditions on the contingent contracts, I then proceed to thesufficient condition on the output distribution leading to Condition GlobalID.49Assumption (MLRP). Y |(p, q) satisfies strict conditional Monotone Like-lihood Ratio Property(MLRP). Given any asset quality q ∈ [q, q], for allyH > yL and pH > pL,f(yH |pH , q)f(yL|pH , q) >f(yH |pL, q)f(yL|pL, q) . (2.18)As is well known, a higher output level is a favorable signal for theworker’s type under Assumption (MLRP), regardless of the asset quality.50Remark 6. When the assets are homogeneous, there exists an equilibriumsupporting the First Best allocation if either of the following conditions hold:1. y − t(y; s) is SPM, or2. y − t(y; s) + c is non-negative and log-SPM for some constant c ≥ 0.In the case of homogeneous assets, Remark 1 already ensures monotonicparticipation on the workers’ side in any equilibrium. The First best allo-cation, which is still defined by the equality (2.5), can be decentralized if49This approach rests on the assumption that uninformed parties of all types have accessto the same set of feasible contracts.50The strictness is not needed in Section 2.7. However, it will be required for Lemma 9in Section 2.8.702.7. Conditions On Contracts And Production Complementaritya contract of a higher term s requires not only a greater expected transferfrom the worker but also his payoff to be increasing with the output at afaster rate, either in level or percentage. Under Assumption (MLRP), areduction of the term always benefits a low type more than a high type,preventing the asset owners from cream skimming. In the view of the un-informed side, screening and increasing transfer complement each other forsuch set of contracts. Formally, either of the conditions in Remark 6 ensuresthat Condition Global ID holds for qH = qL and cH = cL = c.Consider the condition y − t(y; s) is SPM. When the contract term sincreases, the worker has to pay the asset owner more on average but thecontingent payment increases with the output at a slower rate. A conse-quence of these two requirements is that the worker must make a higherpayment at the lowest output level under a contract of a higher term s.Presence of wealth constraint or limited liability may prevent the workerfrom making such a payment, and hence the SPM of his payoff as well.Log-SPM of y−t(y; s)+c is a promising alternative in this regard. It canbe met even if all contracts specify the same payment at the lowest outputlevel, and thus circumvents the wealth constraint. Log-SPM of y− t(y; s)+cis a weaker condition than SPM of y − t(y; s) if the contingent paymentuniformly increases with the contract term s at all output levels. In general,neither one of them implies the other because how t(y; s) changes with s isindefinite. As we shall see, a stronger notion of production complementarityis, nonetheless, needed for Condition Global ID under log-SPM of y−t(y; s)+c.712.7. Conditions On Contracts And Production Complementarity2.7.1 Applications to standard securitiesThe classes of standard securities introduced here satisfy Assumption(C). They also satisfy the definition of an ordered set of securities whenindexed by appropriate contract terms. Note that the two conditions inRemark 6 concern the worker’s payoff, which depends on both the initialdivision of output and the contract between the two parties. As in thebaseline setting, we first consider the convention that the worker receives theentire output. The output level is normalized to be non-negative throughoutthis section, y ≥ 0.I first discuss equity, debt and call option. These securities have theadditional properties that i) t(.; sH) ≥ t(.; sL) if sH > sL, and ii) y ≥t(y; s) ≥ 0. Thus, log-SPM of y − t(y; s) + c is a weaker requirement thanSPM.Example (Equity). An equity contract is represented by t(y) = αy. It canbe indexed by the output share α = s so thaty − tE(y; s) = (1− s)y.When the output level increases, the worker’s payoff increases by the samepercentage across all equity contracts. However, the increase in level issmaller if the asset owner is paid a greater share of output. Therefore,y − tE(y; s) is non-negative and log-SPM but not SPM. In fact, Global IDholds for equity contract whenever the expected output exhibits log-SPM.Example (Debt). A debt or bond is represented by t(y) = min{y, d}. It isindexed by the principal amount d = sŷ such thaty − tD(y; s) = max{0, y − sŷ}.722.7. Conditions On Contracts And Production Complementarityŷ can be simply set as y if y is finite. Otherwise, ŷ has to be large enoughso that E(min{Y, ŷ}|p, q) ≥ E(Y |p, q)−V . This ensures v(p, q, tD(y; 1)) ≤ V .Under the debt contract, the worker keeps the residual output after pay-ing out the principal amount in full. For any output level above the principalamount, the worker’s payoff increases with the output by the same amount.Nevertheless, the percentage increase will be greater if he pays a larger prin-cipal amount. Formally, y − tD(y; s) is non-negative and log-SPM but notSPM.Example (Call option). A call option is represented by t(y) = max{y−c, 0}.It is indexed using the strike price c = (1− s)ŷ, so thaty − tCO(y; s) = min{(1− s)ŷ, y}.Again, ŷ is chosen so that E(min{Y, ŷ}|p, q) ≥ E(Y |p, q) − U , ensuringU(q, p, tCO(y; 0)) ≤ U.When granting the asset owner a call option, the worker receives theoutput only up to the strike price of the option. His residual claim ceasesto increase with the output when the latter exceeds the strike price. When sincreases, the worker’s residual claim starts flattening out at a lower outputthreshold, and so increases at a slower rate with the output level. Therefore,y − tCO(y; s) is neither log-SPM nor SPM. So the results in the subsequentsection do not apply to the case that the uninformed parties are compensatedusing contracts within the class of call options.In the baseline setting, the informed party receives the output and makespayment to the uninformed party. In a number of applications such asexecutive compensation, the flow of payment goes the other way around andthe contingent payment is implemented using a different class of securities.732.7. Conditions On Contracts And Production ComplementarityAs an example, consider in our setting that the worker takes a debt ofprincipal d from the asset owner in exchange for the use of the asset. Theex-post surplus division in this case coincides with the arrangement that theasset owner is entitled to the output and compensates the worker with a calloption of strike price d. The previous discussion applies after re-indexingthe class of call option as follows,tCO(y; 1− s) = max{0, y − sŷ} = y − tD(y; s).Now suppose that the asset owner is entitled to the output and enjoyslimited liability. If the asset owners compete for workers by offering fixedwage contract w, this compensation scheme is equivalent to the arrangementthat the worker buys out the asset by granting its owner a call option withstrike price w. From the preceding analysis, Condition Global ID is not metin general. This problem is more acute for small business and startups, whoare likely to liquidate in case of a low revenue.Example (Mixture of cash and standard securities). Another popular ar-rangement in practice is a mixture of cash and standard securities such asequity or bond. In this case, the contract can be indexed by the portfolioweight of cash payment. Fix a particular securities t̂(y) satisfying Assump-tion (C), we can define an ordered set of securities as followstMIX(y; s|t̂) = s[E(Y |p, q)− V ] + (1− s)[t̂(y)− E(t̂(Y )|p, q) + U ].s[E(Y |p, q)− V ]− (1− s)[E(t̂(Y )|p, q)−U ] is interpreted as the amountof up-front cash payment and the rest is the security portion of the offer. Itis straightforward to verity that tMIX(y; s|t̂) satisfies Assumption (C) andthe conditions for an ordered set of securities. Since the security portion742.7. Conditions On Contracts And Production Complementaritydecreases with the term s, a contract of high term s renders the worker’spayoff more sensitive to the output level. Hence, y − tMIX(y; s|t̂) is SPM.So far, we have restricted attention to the circumstances, that the entireoutput produced by the match is contractible. The result can also be appliedto another extreme, which the outcome is non-contractible and the partiesexchange cash payment. Let ĥ(y) and y− ĥ(y) denote the non-contractible,possibly non-monetary, payoff for the asset owner and worker in the absenceof transfers respectively. Mailath, Postlewaite and Samuelson (2013) termthe pair E(ĥ(Y )|p, q)−U and E(Y −ĥ(Y )|p, q)−V as premuneration values.As the asset owners post prices, we define an ordered set of securities astPRE(y; s|ĥ) = ĥ(y)− E(ĥ(Y )|p, q) + U + s[E(Y |p, q)− V − U ].If ĥ satisfies Assumption (C), then tPRE(y; s|ĥ) satisfies Assumption (C)and the conditions for an ordered set of securities. Lemma 7 applies becausey − tPRE(y; s|ĥ) is SPM.2.7.2 Conditions on production complementarityCondition (Survival-SPM). For any y ∈ (y, y), F (y|p, q) is strictly de-creasing and weakly pairwise submodular in p and q, and F (y|p, q) is strictlypairwise submodular in p and q for some subinterval of (y, y).Condition Survival-SPM is stronger than supermodularity of expectedoutput. The interpretation of Condition Survival-SPM is that when thereare two types of agents on both sides, with types {pH , pL} and {qH , qL}, thedistribution of total output under positive assortative matching F.O.S.D.that under negative assortative matching.752.7. Conditions On Contracts And Production ComplementarityLemma 7. Condition Global ID holds if y − t(y; s) is SPM and ConditionSurvival-SPM holds.The SPM of y−t(y; s) implies that v(p, q, t(.; s)) is pairwise SPM in (p, s)while Condition Survival-SPM establishes that v(p, q, t(.; s)) is pairwise SPMin (p, q). Hence, the level inequality in Global ID always holds.51To save on space, ∨ and ∧ are used to denote maximum and minimumoperator respectively.Condition (Survival-logSPM). Given any pH ≥ pL and qH ≥ qL, for anyy and y′ in Ωy,[1− F (y ∨ y′|pH , qH)][1− F (y ∧ y′|pL, qL)] (2.19)≥ [1− F (y|pL, qH)][1− F (y′|pH , qL)],with strict inequality if y 6= y′ and (pH , qH) 6= (pL, qL).Compared to strict log-SPM of [1−F (y|p, q)], Condition Survival-logSPMis slightly weaker as it does not require strict pairwise log-SPM in (p, q).On the other hand, Condition Survival-logSPM is stronger than ConditionSurvival-SPM.52Under Condition Survival-logSPM, conditioning on the event that theoutput is above some level y > y, the public belief regarding the type of theworkers will be higher in F.O.S.D. sense if he is operating with an asset ofhigher quality.53 In this sense, pairing up with an asset of higher quality51Lemma 7 does not require Assumption (MLRP).52Note that by taking y = y and y′ = y, (2.19) implies that F (y|p, q) is strictly decreasingin p and q. Take y′ = y, [1− F (y|p, q)] is strictly increasing and log-SPM in (p, q). Thus,it must be strictly pairwise SPM in (p, q).53Abusing the notation, let P denote the random variable for the worker’s type. Forany y ∈ (y, y) and qH > qL, the ratio Pr(Y≥y,P≤p′|qH )Pr(Y≥y,P≤p′|qL) is increasing in p′. Thus Pr(P ≤p′|Y ≥ y, qH) ≤ Pr(P ≤ p′|Y ≥ y, qL).762.7. Conditions On Contracts And Production Complementaritymakes a high output level an even more favorable signal for the worker’stype.Notice that the expected output can be written asE(Y |p, q) = y +∫ yy1− F (y|p, q)dy.Since log-SPM for a non-negative function is preserved under integration,the expected output must be log-SPM under Condition Survival-logSPM.Lemma 8. Condition Global ID holds if Condition Survival-logSPM holdsand y − t(y; s) + c is non-negative and log-SPM for some constant c ≥ 0.Condition Survival-logSPM, together with Assumption (C), ensure thatv(p, q, t(.; s)) + c is pairwise log-SPM in (p, q). Assumption (MLRP) impliesthat v(p, q, t(.; s)) + c is pairwise log-SPM in (p, s). These two propertiesyield the desired condition.54Under Condition Survival-SPM and Survival-logSPM, production com-plementarity manifests as a shift in the entire output distribution toward theright. The worker’s payoff is always increasing with the output level becauseof Assumption (C). So a worker benefits more from an improvement in theasset quality than his peers of lower types under the same contract. Theconditions in Remark 6 further imply that the contracts posted in the can-didate equilibrium amplifies such difference between workers’ types. This isbecause the owners of higher asset quality demand a higher contract term,rendering the worker’s payoff increasing with the output at a higher rate.In general, the form of the production complementarity must align withthe worker’s compensation, improving the odds of the states in which he54For the function v(p, q, t(.; s)) + c, pairwise log-SPM in (p, s) is weaker than SPM in(p, s), whereas pairwise log-SPM in (p, q) is stronger than SPM in (p, q). This explainswhy Lemma 8 requires a stronger condition on the output distribuition.772.7. Conditions On Contracts And Production Complementarityis generously rewarded. As a result, a worker enjoys a greater raise inexpected payoff for a better partner than his peers of lower types for thesame contract. The catch is that such differential among workers must bepreserved after accounting for the difference in the contracts offered. Thisrequires further alignment between the forms of feasible contracts and theform of production complementarity.The well-known result that multiplication and integration preserve log-SPM for non-negative functions may be directly applied to establish a suf-ficient condition for log-SPM of v(p, q, t(.; s)), and hence Condition GlobalID. It is interesting to compare Lemma 8 with such condition. For [y −t(y; s) + c]f(y|p, q) to be log-SPM in all arguments, the sufficient conditionon the conditional distribution is given byf(y|pL, qH)f(y′|pH , qL) ≤ f(y ∨ y′|pH , qH)f(y ∧ y′|pL, qL). (2.20)When integrating the conditional density functions over their common sup-port, both sides of the inequality will be unity. This turns out to imposestrong restrictions on the conditional density functions satisfying the in-equality (2.20).Remark 7. Suppose a conditional density function f(y|p, q) satisfies As-sumption (MLRP) and the inequality (2.20), then f(y|p, q) is pairwise log-modular in (y, q) and (p, q).The set of conditions in Remark 7 is far more demanding than ConditionSurvival-logSPM. The monotonicity of the worker’s payoff is instrumentalfor the weaker condition required in Lemma 8.Examples of parametric distribution782.7. Conditions On Contracts And Production ComplementarityExample (Bernoulli distribution). Suppose Y |(p, q) is a Bernoulli distri-bution with support {y, y}. Assumption (P) and (MLRP) hold if and onlyif Pr(Y = y|p, q) is strictly increasing and strictly SPM. This already im-plies Condition Survival-SPM. Condition Survival-logSPM holds if and onlyif strict SPM is strengthened to log-SPM.Example (Exponential distribution). Suppose Y |(p, q) follows exponentialdistribution with mean µ(p, q), then Assumption (P) and (MLRP) hold ifand only if µ(p, q) is strictly increasing and strict SPM. Both ConditionSurvival-SPM and Survival-logSPM are equivalent to the requirement thatµ−1(p, q) is submodular.Assumption (MLRP) and Condition Survival-SPM and Survival-logSPMare preserved under monotonic transformation of random variable Y . Thisobservation leads to the following example.Example (Transformed geometric distribution). {∆n}n∈N is a non-negativedeterministic sequence. {Xn}n∈N is a sequence of non-degenerate Bernoullirandom variables with support {0, 1} and {Xn}n∈N|(p, q) are independent.Define Y = y +∑Mn=1 ∆n where M = min{n ≥ 1 : Xn = 0}. SupposePr(Xn = 1|p, q) = G(p, q;n) is strictly increasing in (p, q), then1. Assumption (MLRP) is met if G(pH ,q;n)1−G(pH ,q;n)1−G(pL,q;n)G(pL,q;n)is weakly decreas-ing in n for any q and pH > pL.2. Condition Survival-SPM is satisfied if G(p, q;n) is strictly SPM in(p, q) for all n.3. Condition Survival-logSPM is satisfied if G(p, q;n) is log-SPM in (p, q)for all n.792.8. Feasible Contracts And Comparative StaticsThe condition for Assumption (MLRP) is always met if G(p, q;n) canbe written as Gpq(p, q)Gn(n) where Gn(n) is weakly decreasing.The output distribution can be interpreted as follows: Once a match isformed, a base output y is produced immediately. A sequence of productionstages has to take place in succession. ∆n and G(p, q;n) are the outputand probability of success for the n-th stage. A failure is irrevocable andterminates the production. There may be a cap on the number of possibleproduction stages, say N. This is accommodated by letting ∆n = 0 andG(., .;n) = G(., .;N) for n > N. A continuous distribution counterpart forthe example of Transformed geometric distribution can be obtained readily.2.8 Feasible Contracts And Comparative StaticsThis section studies how a change in the feasible set of contracts willaffect the equilibrium allocation, and the divisions of the matching surplus.I will compare the contracts in term of their steepness, a partial order basedon DeMarzo, Kremer and Skrzypacz (2005).Definition. Given Y |(p, q), a contract ts is steeper than another contracttf if E(ts(Y )|p′, q′) = E(tf (Y )|p′, q′) for some (p′, q′) ∈ [p, p] × [q, q], thenfor all pH > p′ > pL, E(ts(Y )|pH , q′) > E(tf (Y )|pH , q′)E(ts(Y )|pL, q′) < E(tf (Y )|pL, q′) .Furthermore, tf is said to be flatter than ts.In essence, a contract is steeper if it costs more to workers of highertypes but less to those of lower types, regardless of the asset quality. Wesay a single contract t is steeper(flatter) than a set of contracts Φt if t is802.8. Feasible Contracts And Comparative Staticssteeper(flatter) than every contract from Φt. Likewise, a set of contracts issteeper(flatter) than another set of contracts if every member of the formeris steeper(flatter) than every member of the latter.Lemma 9 (DeMarzo, Kremer and Skrzypacz, 2005). A contract ts is steeperthan another contract tf if there exists some y∗ ∈ (y, y) such that ts(y) ≥tf (y) if y > y∗ and ts(y) ≤ tf (y) if y < y∗, with strict inequality for someinterval in (y, y).Under Assumption (MLRP), a higher output level is a favorable signalfor the worker’s type. Lemma 9 states that a contract ts is steeper than acontract tf if the former cuts the latter from below.55 For examples, Lemma9 can be applied to rank the classes of standard securities. Call option is thesteepest, followed by equity. Equity is steeper than debt. Cash is the flattestunder Assumption (C).The analysis of the comparative statics consists of two parts. The firstpart considers the case that only an ordered set of securities is feasible andit is replaced by another ordered set. I study the changes in the equilibriumdivisions of the surplus, provided that the equilibrium allocation remainsFirst Best. This comparative statics is driven by the sorting of the work-ers. It applies whenever the conditions in Section 2.7 are met. The secondpart relaxes Assumption (S) and considers the introduction of new contractsinto the feasible set. I study when the equilibrium allocation and payoffswill remain unchanged. Such invariance stems from the screening consider-ations by the asset side. This result also applies when contracts are madeunavailable.55An implication of Assumption (C) is that unless one contract always specifies a higherpayment than the other, any two contracts must intersect at some output level.812.8. Feasible Contracts And Comparative StaticsThe subsequent discussion assumes the baseline convention that the pay-ment flows from the worker to the asset owner. If it is the asset owner isentitled to the output and the flow of payment goes the other way around,then a flatter contract costs more to high type workers, and all results willbe flipped.2.8.1 Steepness and division of surplusProposition 5. With a steeper ordered set of securities, the equilibrium pay-off for the asset owners will be higher in the candidate equilibrium, whereasthe equilibrium payoff will be lower for the workers matching with assets ofquality strictly above the threshold type.Fix a distribution of types where the workers are on the short side,and every type of workers match with two types of assets in the First Bestallocation. When switching to a steeper ordered set of securities, considerthe following the thought experiment: First starts with the contract terms{sk}k≥1 keeping the same equilibrium payoff for the asset side. As the newcontracts are steeper, a worker of type p1 will pay less if he deviates to matchwith the asset of quality q2. To satisfy the IC condition for the workers oftype p1, the contract term s2 must increase, driving up the equilibriumpayoff for the owners of asset quality q2. This in turns makes the deviationto the market with q3 even more profitable for workers of type p2, resultingin a greater increase in s3.822.8. Feasible Contracts And Comparative StaticsFigure 2.2: Spill-over effect of increased competition across assetsInductively, all owners of asset quality above q2 must post higher contractterms.For any distribution of types, the above line of reasoning applies to theassets of quality above the threshold type and the workers they match with.For the workers matching with assets of the threshold type, the impact ontheir equilibrium payoff depends on the distribution of types. In particular,the workers of the threshold type are better off under a steeper ordered setof securities when the assets are on the long side. On another hand, allworkers will be weakly worse off if they are on the long side.Since the cash payment is the flattest ordered set of securities, Remark4 in Section 2.6.1 is a special case of Proposition 5. In fact, Proposition 5can be viewed as an extension of the Linkage principle in DeMarzo, Kremerand Skrzypacz (2005). The authors show that in the security-bid auction,a steeper ordered set of securities allows the auctioneer to extract moreinformation rent from the bidders, provided that the equilibrium allocationis efficient and remains unchanged. Under assortative matching, the spillover832.8. Feasible Contracts And Comparative Staticseffect across different types of assets further shifts the equilibrium divisionof matching surplus in the asset side’s favor. Hence, I establish an analogousresult in the context of assortative matching.2.8.2 Steepness and contract offeringThis subsection relaxes Assumption (S), which requires the feasible con-tracts to be fully ordered by a contract term, and analyzes the asset owners’choice of contract in a larger feasible set. The main conclusion is that all theresults remain valid if the asset owners may post steeper contracts. Sincea flatter contract costs the workers of higher types less than a steeper con-tract, the asset owners always prefer posting the former as it is less proneto attract the low type workers. Yet posting a flatter contract will has noeffects on the pool of workers in certain situations, so that the asset ownersare indifferent between the two contracts. Proposition 6 formalizes this ob-servation. It states that when steeper contracts are made available, the setof equilibria weakly expands. Nevertheless, the set of equilibrium allocationsand payoffs remain the same.56Proposition 6. Suppose St ⊆ Φt, and ts /∈ Φt is a contract steeper than St.1. For every equilibrium under the contract space Φt, there is a corre-sponding equilibrium under the contract space Φt ∪ {ts} with the sameequilibrium payoffs {U, V }, the same active markets Ψ ⊆ Φt and thesame distribution of participants in every active market.2. For every equilibrium under the contract space Φt ∪ {ts}, there mustbe an equilibrium under the same contract space, which supports the56Note that the cash payment satisfies Condition Global ID. Nevertheless, Proposition2 is not a corollary to Proposition 3 and Lemma 6 as it holds without Assumption (P).842.8. Feasible Contracts And Comparative Staticssame equilibrium payoffs and allocation, and the asset owners only postcontracts in Φt.Note that Proposition 6 allows Φt to contain other contracts flatter thanSt. With an ordered set of securities available, the introduction of the newcontract does not improve the transferability of surplus within the pair.From the preceding discussion, posting a steeper contract never benefits anasset owner. Therefore, any equilibrium is robust to the introduction of asteeper contract. It is obvious that the converse of the first statement is alsotrue.We proceed to the case that the ordered set of securities and a steepercontract ts are both available. Notice that for any match (p, q), there isa contract t(.; s(p, q)) in St providing both parties the same payoff as thesteeper contract ts. As the former contract is flatter, workers of lower typesall strictly prefer the contract ts to t(.; s(p, q)). With these flatter contractsavailable, the asset owners are willing to post the steeper contract in anequilibrium only when the corresponding market (ts, q) clears and attractsexactly a single type of workers.5757To see this, suppose multiple types of workers participate in this active market andpH is the highest type. an asset owner must profit from posting a contract t(.; s′) with aterm s′ slightly below s(pH , q). This is because the flatter contract will attract no workersof type below pH and weakly improve her matching probability. Hence, pH must be theonly type of workers attracted to the market (ts, q). This argument further implies thatworkers of higher types will be strictly worse off if they deviate to the market (ts, q), andworkers are not rationed in this market. Suppose the asset side is being rationed in themarket (ts, q), then an asset owner will deviate to a contract t(.; s) where s is slightlyabove s(pH , q). Posting such contract will lead to a jump in the matching probability andattracts only workers of pH .This argument actually does not hinge on the assumption of finite distribution of types.852.8. Feasible Contracts And Comparative StaticsNow suppose that all the asset owners and workers in the market (ts, q)switch to the market (t(.; s(p′, q)), q). As the contract (t(.; s(p′, q)), q) is flat-ter, it will not attract workers of lower types. It will not attract workers ofhigher types either, otherwise the asset owners would have deviated to postit after the first place. So all equilibrium conditions will still be satisfied.The equilibrium payoffs and the result allocation remain unchanged in thenew equilibrium. This argument holds irrespective of the presence of othercontracts in Φt.An immediate corollary is that it is without loss to focus on the orderedset of securities if all other feasible contracts are steeper.Corollary 2. Suppose that At is steeper than St. an allocation and a pairof equilibrium payoffs can be supported by an equilibrium under the contractspace St if and only if they can be supported by an equilibrium under thecontract space At ∪ St.DeMarzo, Kremer and Skrzypacz (2005) also consider informal auctions,in which the buyers may submit their bids from a larger set of securities andthe seller selects the winning bid based on her belief. A worker signals histype by bidding with a flatter security, which is costlier to the low types.In equilibrium, all buyers bid with the flattest securities available. Herethe competition among uninformed parties drives them to post the flattestsecurities available because of the screening incentive.Corollary 2 allows us to generalize the results to larger sets of contracts.Example (Linear compensation contracts). Consider Ωt = {t(y;α,w) =αy − w : α ∈ [0, 1], w ≥ 0}, which satisfies Assumption (C) and limitedliability of the worker. The worker’s payoff is given by y − t(y;α,w) =(1 − α)y + w, so Ωt represents a class of linear compensation contracts. If862.8. Feasible Contracts And Comparative Staticsa contract with w = 0 intersects with another one with w > 0, the formermust cut the latter from below. The subclass of contracts with w = 0, whichare effectively equity contracts, is flatter than any contracts with w > 0.Therefore, it is without loss to focus on the class of equity contracts tE(y; s).Example (Wealth constraint). Fix some pi ≥ 0, let Ωt be the set of contractssatisfying Assumption (C) and t(y) ≤ y + pi. pi is interpreted as the initialwealth level for the worker, so that the payment he made to the asset ownercannot exceed y + pi for any output level y. Take y as finite for simplicity.Under Assumption (C), any t ∈ Ωt is absolutely continuous. Hence, t(y)can be expressed as t(y) +∫ yy t′(z)dz, where t′(z) ∈ [0, 1]. t(y) ≤ y + pi isnecessary and sufficient for t(y) ≤ y + pi for any output level y ∈ [y, y].Define the following ordered set of securitiestD+pi(y; s) = min{y + pi, s(y + pi)}.For s ≤ piy+pi , the worker pays out cash up to his wealth level pi. For s > piy+pi ,the worker tops up the cash payment pi with a debt of principal amountsy+(1−s)pi. This ordered set of securities is flatter than any other contractsin Ωt.58 Therefore, it is without loss to focus on the class tD+pi(.; s). Sincey− tD+pi(y; s) + pi = max{0, y+ pi− s(y+ pi)} is non-negative and log-SPM,58Fix some s′ ∈ [0, 1], let t̂ be a contract in Ωt such that t̂ and tD+pi(.; s′) intersectsat least once somewhere in (y, y). Note that t̂(y) ≤ tD+pi(y; s′) = y + pi. For s′ ≤ piy+pi ,tD+pi(y; s′) = s′(y + pi). Since t̂(y) is increasing in y, t̂(y) must cut tD+pi(y; s′) frombelow at some y∗ ∈ (y, y). For s′ > piy+pi, t̂(y) increases no faster than tD+pi(y; s′) fory < s′(y + pi). Furthermore, tD+pi(y; s′) is constant in the region (s′′(y + pi), y]. Hence,t̂(y) must cut tD+pi(y; s′) from below at some y∗ ∈ (s′(y + pi), y).The above construction can be generalized to the constraint that t′(y) ∈ [α(y), α(y)] ⊆[0, 1] in the following manner: t(y; s) = y+pi, t′(y; s) = α(y) if y < s(y+pi) and t′(y; s) =α(y) if y > s(y + pi). An example for such case is the threat of diversion.872.8. Feasible Contracts And Comparative StaticsCondition Global ID holds under Condition Survival-logSPM.Example (A threat of misappropriation). Let Ωt be the set of contractssatisfying Assumption (C) and t′(y) ≥ κ. This can be motivated as follows:The asset owner receives the output and pays the worker compensation. Shemay underreport the output level and misappropriate the unreported portionat a unit cost 1 − κ. Incentive provision for the asset owner leads to theconstraint t′(y) ≥ κ. Take y as finite for simplicity.Define the following ordered set of securitiestκ(y; s) = κy + sy − (1− s)κE(Y |p, q).tκ(y; s) is essentially a cash payment topped with equity share κ. Any contractin Ωt, if intersecting at all, must cut tκ(y; s) from below. Therefore, itsuffices to consider only the class tκ(.; s). As y− tκ(y; s) is SPM, ConditionGlobal ID holds under Condition Survival-SPM.Testable Implications Proposition 5 and 6 together yield a testable im-plication regarding private types on one side.Corollary 3. Suppose an ordered set of securities is always available, thenunder assortative matching, exclusion of the flattest contracts increases theequilibrium payoff for the asset side.The preceding comparative statics predict an increase in the asset side’sequilibrium payoff when the prevailing form of contracts offered, presumablythe flattest available, switches to a steeper one, due to exogenous reasonssuch as financial regulation. It is hard to justify this prediction withoutinformation asymmetry when forming matches.882.8. Feasible Contracts And Comparative StaticsUnder full information, a contingent payment is merely an instrumentfor transferring the matching surplus. Proposition 1 states that the equilib-rium payoffs are invariant to changes in the feasible set of contracts. Nowsuppose at least one side is risk-averse, so the partners share the risk usingthe contingent contract. When contracts are excluded from the feasible set,the partners may either stay with the same contract or move to a subopti-mal one. Some asset owners above threshold type must not gain from theexclusion of the contracts. The same argument applies in the case of puremoral hazard. Restriction on the incentive contracts will not benefit all assetowners.Introduction Of Flatter Contracts Proposition 6 states that the in-troduction of steeper contracts have no effects on the set of equilibriumallocations and payoffs. When flatter contracts are introduced, the samecannot be said. If only particular contracts, say tf , are introduced, the as-set owners will gain from posting lotteries over tf and St, which effectivelyform a flatter ordered set of securities. Therefore, I only consider the intro-duction of a flatter ordered set of securities. From the previous section, weknow that assortative matching is still decentralized if the flatter ordered setof securities and the distribution of outputs satisfy the joint conditions inLemma 7 and Lemma 8. The following examples illustrate that inefficiencymay occur if the conditions are not all met. In other words, restricting thefeasible set of contracts can improve welfare in these examples.Introduction of a flatter ordered set of securities Consider the fol-lowing example: The type space is given by [p, p] = [q, q] = [0, 1]. Produc-tion may result in three outcomes, Ωy = {0, 12 , 1}. The output distribution892.8. Feasible Contracts And Comparative Staticsis given byf(y|p, q) = 18p(1 + 2q)12(1− )pq + 14q + 18, if y = 1; and, if y = 12 ,where  ∈ (0, 1).The output distribution has the following properties:1. F(y|p, q) is continuous and strictly decreasing in p and q, and2. f(y|p, q) satisfies Assumption(MLRP), and3. F(y|p, q) satisfies Condition Survival-SPM, and4. The expected output E(Y |p, q) = 14(p+ 12)(q + 12) is log-modular, andhence SPM.It is noteworthy that the survival function 1− F(y|p, q) is pairwise log-SPM in (p, q) and (p, y) but not (q, y), so Condition Survival-logSPM is notmet. The values of outside options can be chosen to satisfy V + U = 116 ,and hence Assumption (P).Suppose that only the class of equity contracts is feasible. For a given ∈ (0, 1), the worker’s expected payoff is denoted byv(p, q, tE(.; s)) =14(p+12)(q +12)(1− s).It immediately follows that v(p, q, tE(.; s)) satisfies Global ID, and hencethe First Best allocation can always be decentralized.Now suppose that the class of debt contracts is also made available. Sincethe class of debt contracts is flatter than that of equity, Corollary 2 states902.8. Feasible Contracts And Comparative Staticsthat it is without loss to assume the asset owners post only debt contracts.v(p, q, tD(.; s))= 12(p+ 12)[12(q + 12)− s(q + 14)]− 14(pq + 14)s18(1− s)p(1 + 2q), if s < 12 ; and, if s ≥ 12 .The key property is that workers of all types share the same preference over(q, s, η) in the region s ≥ 12 . From now on, we focus on the limiting case of → 0+ and v0+ denotes the worker’s expected payoff at the limit. All theresults hold when  is sufficiently small.For the workers side, v0+(p, q, tD(.; s)) satisfies Condition Sorting-p. Forthe asset side, it is easy to verify that for any  ∈ (0, 1) and sH ≥ 12 ,v(pH , qH , tD(.; sH))v(pL, qL, tD(.; sL))≥v(pL, qH , tD(.; sH))v(pH , qL, tD(.; sL)).(2.21)For the case that 12 ≥ sH > sL, the above inequality (2.21) holds at thelimit → 0+ if and only ifsH [12(qL +12)− sL(qL + 14)] ≥ sL[12(qH +12)− sH(qH + 14)].This is true whenever qH = qL, so Condition Screening-q is met.However, the inequality (2.21) is violated for a large range of (p, q, s)where qH > qL. Condition Entry-q is not satisfied for certain values ofoutside options. Hence, there are distributions of types for which owners ofasset quality below the threshold type profit from deviations in the candidateequilibrium. It is noteworthy that in this example, the First Best allocationcan be decentralized if the asset side is homogeneous.The concept of “steepness” does not concern about how the gain fromproduction complementarity is shared within pair. In the current example,912.8. Feasible Contracts And Comparative Staticsthe worker’s reward becomes more concentrated in the highest output levelunder the flatter class of contracts. While the highest output level is the mostinformative about the worker’s productivity, the probability of its occurrenceexhibits very weak complementarity between types. As a result, the workersbenefit less from production complementarity to the extent that ConditionEntry-q is no longer met, while Condition Sorting-p remains valid.Introduction of a slightly flatter ordered set of securities Considerthe following example: The type space is given by [p, p] = [q, q] = [0, 1].Y |(p, q) is a Bernoulli distribution with support {y, y}, where 1 > y > y ≥ 0.Pr(Y = y|p, q) = pq is the probability of the good state, so ConditionSurvival-logSPM is met. The values of outside options satisfy V + U = yand y2 ≥ U , and so Assumption (P).For the classes of equity contracts and debt contracts, the worker’s ex-pected payoffs are given byv(p, q, tE(.; s)) = (1− s)[y + (y − y)pq]v(p, q, tD(.; s)) = max(y − s, 0)pq[y + (y − y)pq]− s , if s ∈ [y, 1]; and, if s ∈ [0, y).For a given  ∈ (0, 1), define a mixture of debt and equity contracts byt(.; s) = (1− )sy + min(y, s),so thatv(p, q, t(.; s)) = (1− )v(p, q, tE(.; s)) + v(p, q, tD(.; s)). is the weight on the debt contract while 1−  is the weight on the equity922.8. Feasible Contracts And Comparative Staticscontract.59 Though [y − tE(y; s)] and [y − tD(y; s)] are log-SPM in (y, s),y − t(y; s) is neither log-SPM nor SPM for any  ∈ (0, 1).If only equity contracts are feasible, v(p, q, tE(.; s) satisfies Global ID, andhence the First Best allocation can always be decentralized. Now supposemixtures of debt and equity with a fixed  are introduced, it is withoutloss to consider only such mixtures because they are flatter than the equitycontracts.First, v(p, q, t(.; s)) satisfies Condition Sorting-p for any  ∈ (0, 1).The reason is that v(p, q, tE(.; s)) and v(p, q, tD(.; s)) both satisfy Condi-tion Sorting-p and are linear in p for any given pair of (q, s), so do anylinear combination of the two.For a given asset quality, all workers share the same preference over(η, s) in the region s ∈ [y, 1]. When  is sufficiently small, a high typeworker is willing to endure a lower matching probability for an incrementalreduction in contract term than a low type worker if s ∈ (y, y). The oppositehappens if s ∈ [0, y). This holds for all asset qualities. It can be shown thatv(p, q, t(.; s)) does not satisfy Condition Screening-q for sufficiently small using a more involved argument.6059When only equity contracts or debt contracts are available, the analysis is invariantto any order-preserving transformation of s in the definitions of tE(.; s) and tD(.; s). Thisis no longer true for a mixture of debt and equity contracts because the contract term sdetermines the pair of debt and equity contracts forming the mixture. The definition oft(.; s) here implies that the principal amount is capped at y for debt contracts for s > y.60Suppose, to the contrary, that Condition Screening-q is met. Consider y > sH >sL > y. q, pH and pL are chosen so that v(pL, q, t(.; sH)) ≥ V and u(q, pH , t(.; sL)) > U .u(q, pH , t(.; sL)) > u(q, pL, t(.; sH))} if sH and sL are close enough. As discussed, theinequality (2.13) does not hold. It follows that the inequality (2.14) must hold for someq′ ≤ q and s′ ≤ sH satisfying v(pL, q′, t(.; s′)) = v(pL, q, t(.; sH)). The inequality (2.14)implies that v(p′, q, t(.; sL)) > v(p′, q′, t(.; s′)), while Condition Sorting-p requires that932.8. Feasible Contracts And Comparative StaticsFor small values of , v(p, q, t(.; s)) indeed satisfies Condition Entry-q.Condition Entry-q has the pre-condition U ≥ u(qH , pL, t(.; sH)). At  = 0,y2 ≥ U ≥ sH [y + (y − y)pLqH ], and hence y > sH . For any pair of sH andsL in this range, v(p, q, tE(.; s)) is strictly pairwise log-SPM in (p, s). Sincev(p, q, tE(.; s)) is also strictly pairwise log-SPM in (p, q), we must havev(pH , qH , t(.; sH))v(pL, qH , t(.; sH))>v(pH , qL, t(.; sL))v(pL, qL, t(.; sL)),when → 0+.In summary, when a small component of the debt contract is introduced,there exist distributions of types for which owners of asset quality above thethreshold type have profitable deviations in the candidate equilibrium. Thisis in stark contrast to the case with only equity contracts. Under a debtcontract with term s ∈ (y, y), a worker will repay the principal in full only inthe good state. A tiny reduction in the contract term reduces the expectedpayment from a worker more if his productivity is higher. Yet, the workersof lower types still see a greater decline in percentage. However, if thecontract includes an overwhelming component of equity, it turns out thatthe reduction in contract term will result in a greater percentage gain forthe workers of high types. So the asset owners will succeed in poaching thebetter workers.v(p′, q′, t(.; s′)) ≥ v(p′, q, t(.; sH)). Since this holds for sL arbitrarily close to sH , it followsthat v(p′, q′, t(.; s′)) = v(p′, q, t(.; sH)). Suppose that the workers of pL are indifferentbetween the contract (t(.; sH), q) and (t(.; s′′), q′′), where q′′ ∈ (q, q′). Condition Sorting-p then requires that all workers of types between p′ and pL are also indifferent betweenthese two contracts. This is impossible because v(p, q, tE(.; s)) satisfies Condition strictSorting-p and v(p, q, t(.; s)) is a linear combination of v(p, q, tE(.; s)) and v(p, q, tD(.; s)).942.9. Concluding Remarks2.9 Concluding RemarksIn this chapter, I study how the use of the contingent payment affectsthe matching efficiency and the divisions of surpluses in a market where thetypes on one side are privately known. I propose a stylized framework toaddress these questions. To uncouple the potential sources of inefficiencies,I focus on an equilibrium decentralizing PAM. I analyze the sorting decisionfor the informed side, and the uninformed side’s choices of contracts insuch equilibrium. I characterize the conditions under which PAM can bedecentralized for any distribution of types. If these conditions are not allmet, I detail how to construct some distributions of types and identify theprofitable deviations by the corresponding group, illustrating the incentivesagainst assortative matching. The convenience, which is also its limitation,of this approach is that it leaves out the interaction among these incentives.Studying their interactions and the resulting allocation is left for futureresearch.I then provide a unifying sufficient condition, Global ID, which is intu-itive and easy to interpret. Its simplicity allows me to provide joint sufficientconditions on the contingent contracts and the form of production comple-mentarity. When these primitive conditions are not all met for the applica-tion at hand, one shall directly check whether and which of the necessaryand sufficient conditions on the expected payoff is violated. This points tothe groups who potentially have incentives against assortative matching. Iprovide examples illustrating how restricting the feasible set of contractscan align the incentives for such group, and hence improve the matchingefficiency in the decentralized market. In this light, this paper is a first steptoward how a benevolent planner mitigates the potential inefficiency caused952.9. Concluding Remarksby screening in the matching markets. This is a promising direction forfurther studies.Though the use of contingent contracts may leave the equilibrium al-location unchanged, it does affect the divisions of the matching surplusesbetween the two sides. Comparing with the full information case, the assetowners will enjoy higher payoffs at the expense of the workers above thethreshold types. Furthermore, the equilibrium payoffs for the workers in-crease with their types at a slower rate, and the opposite holds for the assetowners. Recall that the equilibrium payoff for the workers and firm ownersin the full information case represents their shadow value in a social plan-ner’s problem maximizing the total surplus. In this light, my result indicatesthere is a wedge between the social and private benefit of getting matched,and how the size of this wedge changes with the feasible set of contracts.This opens avenues for future research on how this wedge will interact withother channels such as pre-investment and search friction in richer models.96Chapter 3Inefficient Sorting UnderOutput Sharing3.1 IntroductionThis paper studies sorting in a frictional market where the two sidescan be ranked by some characteristics, or simply their types. One sidecompetes for partners by offering financial securities or contracts specifyinghow the payment between the partners is contingent on certain outcomes,say the realized output. An example which has received much attention isthe market for top executives. Firms are ordered by their size, while thecandidates are ranked by their productivity or talents. Gabaix and Landier(2008); Tervio¨ (2008) apply a frictionless assignment model to study howassortative matching accounts for the empirical distribution of the amountof CEOs pay among the largest publicly traded companies in the UnitedStates. Frydman and Jenter (2010) look at the composition of CEO payin S&P 500 firms and document that base salary makes up less than 20%of the remuneration, and over half of it are option grants and restrictedstock grants during the period 2000 to 2008. Other applications include thesorting between the entrepreneurs and venture capitals (Sørensen (2007)),or between the acquiring firms and target firms in M&A (Rhodes-Kropf and973.1. IntroductionRobinson (2008)).In many circumstances, the parties on one side, say candidates for theCEO positions, are better informed about their own types. Little is knownabout how such information asymmetry may interact with search frictionand contingent payment, and the overall effect on sorting.I address this question for the class of output sharing contracts in a com-petitive search framework. There are double continuums of types of assetsand workers. A worker’s productivity is privately known, whereas the qual-ity of an asset is publicly observable. Each worker may operate an asset.The types on both sides determine the output.61 The asset owners first postsharing contracts specifying the payment contingent on the future outputs.Then the workers decide which type of asset and contract they search for.The meeting is bilateral and subject to search friction. Production exhibitscomplementarity, so the Second Best allocation always features positive as-sortative matching (PAM) despite search friction.62 I identify a novel sourceof inefficiency in this environment and analyze the resulting distortion.To better understand the source of inefficiency, let us first consider thebenchmark result in Eeckhout and Kircher (2010). They study price compe-tition in the described environment, in which the asset side post fixed prices.Hence the informed workers, once matched, will pay the asset owners up-front and assume the residual claim. The authors show that the equilibriasupports Second Best allocations. Furthermore everyone receives her “social61Here the choice of contract determines the split of the output, while leaving its sizeunaffected. This simplification allows me to concentrate on the potential distortions inthe matching pattern.62In the presence of search friction, PAM occurs if better workers always search forbetter assets.983.1. Introductionvalue”, the shadow price in the Utilitarian planner’s problem, in equilibrium.However, wealth constraint of the workers and incentive provision for the as-set owners may undermine the feasibility of the buyout arrangement, callingfor the use of sharing contracts.The Second Best allocations can no longer be decentralized using thesharing contracts. A low-type worker pays less than a high-type workerwhen conceding a larger output share to the asset owner. So the offeringof sharing contracts handicaps the competition among workers for the sametype of assets, increasing the expected payoff for the asset owners. Thisshift in the divisions of matching surpluses can be attributed to the link-age principle in auction theory (DeMarzo, Kremer, and Skrzypacz, 2005)because the allocation is held unchanged. Assortative matching gives rise toan additional spillover effect. Since the workers pay more for their partnersin the Second Best allocation, they will find better assets more attractive,further intensifying the competition among workers for better assets. As aresult, the set of incentive compatible contracts supporting a Second Bestallocation must provide the asset side a larger slice of the matching surplusthan in price competition.Consequently, the private benefit for an asset to get matched is above thesocial benefit. The wedge is the largest at the top. Facing search friction,the owners of the best assets increase their matching probability by inducingan inefficiently long queue of workers. This leads to the unravelling of theSecond Best allocation. Inefficiency here is caused by the interplay betweenthree elements: sharing contracts, private types and search friction.Two questions naturally arise from the preceding discussion. Unlikefixed prices, the asset owners are now concerned about the types of theirpartners, which affect their expected payment under sharing contracts. An993.1. Introductionasset owner will take screening into account when deciding contract offer,and may attempt to poach better workers.63 The first question is whetherPAM can still be supported by an equilibrium. In such equilibria, the poolof workers left to the lower quality assets must deteriorate amid an increasein the queue length for the best assets. This presents a countervailing forceas an asset owner gains less from a match with a weaker worker, and mayinduce a shorter queue of workers instead. Hence, the distortions in thequeue lengths and the sorting pattern are intertwined. The second questionis what form of distortion arises in equilibrium.To distinguish the channel of inefficiency here from those in the searchand matching literature literature, I consider the setting that all workershave the same preference over the contract term and the matching probabil-ity the two given the asset quality. This property ensures that asset ownersnever use queue length as an instrument to screen out better workers. Thestylized setting yields a unique equilibrium, which still features PAM. Inthis equilibrium, the matched pairs of types fully separate into a continuumof (sub-)markets. The term of the contract offered in every market is givenby the Hosios (1990) condition, under which the equilibrium payoff of anagent is the reduction in the aggregate surplus if she is removed from thepopulation.64 The standard interpretation is that both sides fully internal-63An asset owner believes that an off-equilibrium-path contract will only attract thetypes of workers accepting the lowest matching probability, given all other contract offers.Guerrieri, Shimer, and Wright (2010) motivates this belief restriction using “subgameperfection” with bilateral matching.64The equilibrium allocation here is inefficient. Unlike the standard setting where bothsides are homogeneous, the equilibrium payoff for an agent is no longer the same as her“social value”, the maximum increase in the aggregate surplus a Utilitarian planner mayachieve from assigning a new agent of the same type. I will provide a precise interpretation1003.1. Introductionize their search externality on other participants in the same market. Sothe asset owners in every market induce a queue length maximizing the ex-pected surplus for the pair of types, subject to free-entry of the workers attheir equilibrium payoff. Nevertheless, the matching pattern and the work-ers’ equilibrium payoff are endogenously determined. The asset owners donot account for the effects of their contract offers on the information rentfor other types of workers and on the remaining pool of workers left to othertypes of assets. Therefore, the sorting inefficiency only arises in two-sidedmatching.The equilibrium and the Second Best allocation vary with the entiredistribution of types. The key contribution of this paper is the qualitativefeatures of the distortion which are universal for all distributions. Such“distribution-free” result is of theoretical interest as the argument illumi-nates general economic forces which are always at play.As one may have expected, the queue length for the best assets is alwaysinefficiently high. The surprising and novel result is that all but the bestassets will always pair up with weaker workers. Depending on the distribu-tion of types, there is either an excessive entry of workers or an insufficiententry of assets. As a result, the best workers will suffer while the weakestworkers gain from the offering of sharing contracts. The opposite is true forthe asset side.It is noted that here the comparative statics on the equilibrium payoffsdiffer from that in Proposition 5 in chapter 2. This is due to the adjustmentsin the equilibrium allocation following the change in the form of contingentpayments.of Hosios condition in the current setting with double continuums of types.1013.2. Related LiteratureSection 3.2 discusses the related literature. Section 3.3 details the modelsetting and the equilibrium definition. Section 3.4 covers the benchmarkresults from Eeckhout and Kircher (2010). That is, the Second Best allo-cations feature PAM and can be decentralized when asset side post prices.Section 3.5 characterizes the equilibria when asset side offers output shares.Section 3.6 studies the forms of distortion in equilibrium. Section 3.7 con-cludes. All proofs are relegated to the Appendix B.3.2 Related LiteratureThis paper is part of the literature on assortative matching. My settingis closely related to Eeckhout and Kircher (2010). The authors show thatin price competition, the n-root-supermodularity condition is necessary andsufficient for any equilibrium to feature PAM, regardless of the distributionof types. In addition, the Second Best allocations are supported by the equi-libria. The production and matching technology in my setting satisfy thiscondition. The offering of sharing contracts not only changes the sortingincentives for the workers but also renders poaching potentially profitablefor the asset side. I address how such arrangement distorts sorting in equi-librium.This paper also contributes to the literature on efficiency in search andmatching models. Hosios (1990) considers a market where both sides arehomogeneous. He provides the condition on the division of the matchingsurplus, under which the equilibrium queue length is constrained efficient.Albrecht, Navarro, and Vroman (2010) and Julien and Mangin (2016) showthat the Hosios condition no longer ensures constrained efficiency when mul-tiple types are pooled into a single market. This is because the participation1023.2. Related Literatureof an agent also affects the distribution of the partners for the other side.In my setting, every active market features one pair of types and meets theHosios condition. This distinguishes the channel of sorting inefficiency fromthe search externalities in the literature.Guerrieri (2008) studies dynamic efficiency in a directed search modelwhere a worker privately observes his match-specific productivity upon meet-ing. The worker then weighs the current match against his continuationvalue in the unemployed pool. As a result, the wage offers in the future pe-riods determine the probability of workers’ acceptance and their informationrent in the present period. Under free entry, firms do not account for theeffects of their offers on the markets in previous periods. The author showsthat the convergence to the steady state is inefficiently slow in equilibrium.The source of sorting inefficiency here shares the similarity that the assetowners do not internalize the effects of their offers on the markets for otherassets.Guerrieri, Shimer, and Wright (2010) study competitive screening ina competitive search framework. They assume free entry of homogenousprincipals. These principals have both contract and matching probabilityas screening instruments, and the latter is endogenously determined. Theauthors characterize the equilibrium and study the form of distortion in var-ious applications. I consider two-sided matching where both sides competefor partners from given pools.The key difference is that the distortion nowdepends on the distribution of types. I obtain “distribution-free” featuresof the form of distortion.1033.3. Model Setting3.3 Model Setting3.3.1 ProductionThere are continuums of workers and asset owners. Each asset ownerowns a unit of asset. Assets can be ranked according to their publicly knownqualities q ∈ [0, 1]. All workers are ex-ante homogeneous but differ in theiractual productivity p ∈ [0, 1]. Every worker privately knows his productivity.The values of outside options for workers and asset owners are given by Vand U respectively. ∅ denotes the choice of outside option. All parties arerisk neutral and have a quasi-linear preference. I shall use feminine pronounsfor asset owners and masculine one for the workers.Production takes place after a worker pairs up with an asset. The match-ing surplus for the pair of types (p, q) is the output they produce, denotedby y(p, q). y : [0, 1]2 → R++ is positive, strictly increasing and twice contin-uously differentiable (C2) in (p, q).Assumption (Y). The output y(p, q) is strictly log-supermodular (log-SPM)in p and q.Assumption (Y) has two important implications in a frictionless setting.First, it represents a stronger form of production complementarity thanstrict supermodularity (SPM). Without search friction, the total surplus ismaximized under perfect positive assortative matching. Second, log-SPM ofthe output y(p, q) is also necessary and sufficient for decentralizing PAM ina frictionless world, when the asset side may only post output shares.Example (O-ring production). Assumption (Y) is satisfied if the condi-tional distribution of output Y |(p, q) is a Bernoulli distribution with support{y, y}, where y > y > 0 and Pr(Y = y|p, q) = pq : Production is composed1043.3. Model Settingof two tasks. The probability of success for the first task is p, and that ofthe second task is q. The production yields high output y if both tasks aresuccessful. Otherwise, only base output y is produced.The types on both sides are continuously distributed with support [0, 1]2.F (p) denotes the measure of workers of productivity below p and G(q) isthe measure of assets with qualities below q. F and G are C2 and theirderivatives are denoted by f and g respectively. f and g are positive andbounded over [0, 1].Suppose a worker pairs up with an asset. The two parties may enter intoa sharing contract s ∈ [0, 1] where s and 1 − s are the shares of output forthe asset owner and worker respectively.3.3.2 MatchingThere are continuums of (sub-)markets indexed by (q, s) ∈ [0, 1]2. Anowner of asset quality q may participate in one of the markets (q, s) while aworker may participate in any one of the markets.The timing of the events is as follows: In the contract posting stage, theasset owners make their participation decisions simultaneously. Observingthe measure of asset owners in every market, the workers simultaneouslymake their participation decisions. Matches are then formed.The participants on the two sides of a market will pair up randomly.Define the queue length λ ∈ [0,∞] as the ratio of the workers to the assetowners in the market. A worker gets matched with probability η(λ) whilethe matching probability for a asset owner is δ(λ). Meeting is bilateral,so δ(λ) ≤ min{λ, 1} and λη(λ) = δ(λ). The payoffs for those who leftunmatched are normalized to zero.1053.3. Model Settingη is a strictly decreasing function. δ : [0,∞] → [0, 1] is C2, strictlyincreasing and strictly concave. These properties jointly imply the follow-ing: For any positive λ ∈ (0,∞), d ln δd lnλ ∈ (0, 1).65 and 1 > η(λ) > δ′(λ).limλ→∞ δ′(λ) = 0 because limλ→0+ δ(λ) = 0 and strictly concavity implythat δ(λ) > λδ′(λ) for all λ > 0.Assumption (M). d ln δd lnλ , the elasticity for δ(λ), is decreasing.Following Eeckhout and Kircher (2010), I assume a decreasing elasticityfor δ(λ).66 Since 1 = d ln δd lnλ− d ln ηd lnλ , the elasticity for η(λ) must be increasing.67The presence of search friction gives rise to an insurance motive against therisk of being unmatched. Assumption (M) states that an asset owner’smarginal gain in her matching probability from an increase in the queuelength is diminishing. Symmetrically, the workers see a diminishing marginalgain from a decrease in the queue length.Example (Random matching). Assumption (M) is satisfied for δ(λ) =λλ+1 : All participants on both sides are pooled together to form pairs ran-domly. The pair may carry out production only when it consists of a workerand an asset owner.Example (Urn-ball matching). Assumption (M) is satisfied for Urn-ballmatching function, δ(λ) = 1− exp(−λ) : Every worker approaches one asset65In particular, d ln δd lnλ= 1 + d ln ηd lnλ< 1 for λ ∈ (0,∞).66Note that Assumption (M) is equivalent to a unit upper bound on the elasticity ofsubstitution of the aggregate matching function. Suppose M(L,K) is the number ofmatches in a market with L workers and K assets. Then ML(λ,1)MK(λ,1)MLK(λ,1)M(λ,1)≤ 1 for anyλ ≥ 0.67Eeckhout and Kircher (2010) assume a strictly decreasing elasticity for δ(λ). Theirresults remain valid here because I strengthen the assumption on output y(p, q) to strictlog-SPM.1063.3. Model Settingowner without coordination. An asset is utilized if its owner is approachedby at least one worker.Participation in matching is costly because the agent has to forgo heroutside option. We assume that the total output from the matched pairsat the top can always cover the total opportunity costs of participation.Formally,maxλ≥0[δ(λ)y(1, 1)− λV − U ] > 0. (3.1)It ensures that for any distribution of types, it is always efficient to have thebest agents on both sides searching for partners.Example (Random matching). Given δ(λ) = λλ+1 , condition (3.1) is satis-fied if and only if y(1, 1) > (√U +√V )2.Example (Urn-ball matching). Given δ(λ) = 1− exp(−λ), condition (3.1)is satisfied if (1−√e)y(1, 1) > 12U + V .3.3.3 Equilibrium definitionK(q, s) is the measure of asset owners participating in the markets(q′, s′) ≤ (q, s). L(p, q, s) is the measure of workers with types p′ ≤ p par-ticipating in the markets (q′, s′) ≤ (q, s). The marginal distributions aredenoted with the corresponding variables as subscripts. (K,L) is feasible ifKq ≤ G and Lp ≤ F. G(q) − Kq(q) and F (p) − Lp(p) are respectively themeasures of assets of quality below q and workers with productivity belowp assigned to the outside option. The support of K is denoted by Ψ. Amarket is active if it is in Ψ. Otherwise it is inactive. Since participation iscostly, it is never optimal for workers to visit a market with no asset owners.Therefore, Lqs is required to be absolutely continuous w.r.t. K.6868This requirement ensures the Radon-Nikodym derivativedLqsdKis well-defined.1073.3. Model SettingThe equilibrium concept here follows the literature on large games (e.g.Mas-Colell, 1984). The payoff for every single agent depends on her owndecision, and the decisions of all others only through K and L. K and L inturn are consistent with the optimal decisions of all individual agents.Each market (q, s) is associated with a queue length Λ(q, s;K,L) anda distribution of participating workers R(q, s;K,L), where R(.|q, s;K,L)is the C.D.F. for worker’s type. The environment is competitive in thesense that everyone takes Λ and R as given. For the active markets, Λ isthe Radon-Nikodym derivative,dLqsdK and R is derived using Bayes’ law.69By participating in an active market (q, s), a worker of type p receives anexpected payoffη(Λ(q, s;K,L))(1− s)y(p, q), (3.2)while an asset owner receives an expected payoffδ(Λ(q, s;K,L))s∫y(p, q)dR(p|q, s;K,L). (3.3)I now extend the payoff functions to the inactive markets. I will elabo-rate on the belief restriction underlying the payoffs functions afterwards. Aworker will never get matched if visiting an inactive market. A worker oftype p can at most generateV (p;K,L) = sup{η(Λ(q, s;K,L))(1− s)y(p, q), (q, s) ∈ Ψ} ∪ {V }.V (.;K,L) then determines the deviating payoff for the asset owners. Forany inactive market,Λ(q, s;K,L) = inf{λ ∈ [0,∞] : V (p) ≥ η(λ)(1−s)y(p, q), ∀p ∈ [0, 1]}, (3.4)69Formally,∫ψΛ(q, s)dK =∫ψdLqs, and∫ψR(p′|q, s)dLqs =∫{p≤p′}×ψ dL for any mea-surable subset ψ ⊆ Ψ and any p′ ∈ [0, 1].1083.3. Model Settingand R(q, s;K,L) is degenerate atinf{p ∈ [0, 1] : V (p;K,L) ≤ η(Λ(q, s;K,L))(1− s)y(p, q)}. (3.5)The definition (3.4) and (3.5) represent two conditions. First, if V (p;K,L) >η(0)(1− s)y(p, q) for all p ∈ [0, 1], then Λ(q, s;K,L) = 0 and R(q, s;K,L) isdegenerate at p = 0 for such inactive market. Second, for any market, be itactive or inactive,V (p;K,L) ≥ η(Λ(q, s;K,L))(1− s)y(p, q)for all types of workers, and equality holds if p is in the support ofR(q, s;K,L)and Λ(q, s;K,L) > 0.Facing Λ(q, s;K,L) and R(q, s;K,L), an owner of asset quality q canreceiveU(q;K,L)= sup{δ(Λ(q, s;K,L))s∫y(p, q)dR(p|q, s;K,L), (q, s) ∈ [0, 1]2} ∪ {U}.Definition. An equilibrium is a pair of distributions (K,L) satisfying:• Asset owners’ optimal contract posting: (q, s) ∈ Ψ only if s maximizesthe asset owner’s expected payoff (3.3). K ′q(q) ≤ g(q) with equality ifU(q;K,L) > U.• Workers’ optimal acceptance: (p, q, s) is in the support of L only if(q, s) ∈ Ψ and maximizes the worker’s expected payoff (3.2). L′p(p) ≤f(p) with equality if V (p;K,L) > V .Fix an equilibrium (K,L), V (p;K,L) and U(q;K,L) are the equilibriumpayoff for workers and asset owners respectively. The argument K and Lwill be omitted from the equilibrium objects if no confusion arises.1093.3. Model SettingDefinition. A pair of distributions (K,L) is incentive compatible if it satis-fies workers’ optimal acceptance condition in the definition of an equilibrium.Definition. A pair of distributions (K,L) induces voluntary participationof the asset side if for any (q, s) ∈ Ψ, the asset owners’ expected payoff in(3.3) is no less than U .The notions of incentive compatibility and voluntary participation willbe useful in the discussion of the Utilitarian planner’s problem.Belief restriction Since there are continuums of workers and assets, switch-ing between active markets or taking outside option by a single party hasnegligible impacts. The same is true when a worker unilaterally switches toan inactive market. The focus here is the deviation to some inactive marketby an asset owner. For an inactive market (q, s), Λ(q, s) and R(q, s) areinterpreted as the public belief regarding the queue length and the composi-tion of the workers attracted to that market after an owner of asset qualityq deviates to it. An advantage of this notation is to eliminate the distinctionbetween deviations to active markets or inactive markets by an asset owner.Suppose an owner of asset quality q deviates to post a contract s. IfV (p) ≥ η(0)(1− s)y(p, q) for all types, then no workers will ever profit fromaccepting the deviating offer. The asset owner believes such offer will attractno workers and R(q, s), which has no bearing in such case, is degenerate atp = 0. Now consider the case that V (p) < η(0)(1− s)y(p, q) for some types.Then Λ(s, q) is uniquely determined by the lowest matching probability someworkers are willing to endure. The asset owner believes that only the lowesttype among these workers will be attracted.The restriction on the “off-equilibrium-path” belief here is often moti-1103.3. Model Settingvated by the “subgame perfection” on the workers’ side in the competitivesearch literature. Suppose only -measure of the owners of asset qualityq deviate to some inactive market (s, q). Observing the measure of assetowners in every market, a worker has to anticipate his matching probabil-ity in each of the markets and adjust his participation decision accordingly.When  → 0+, no types of workers can strictly gain from participating inthe market (s, q) in the equilibrium of this “subgame”. Otherwise, workersof all such types will turn up in this market but only -measure of them willget matched, resulting in an expected payoff below their outside option. Itfollows that any workers attracted to the market (s, q), if any, are those will-ing to endure the lowest matching probability. By continuity, the workers’payoff in the equilibrium of this “subgame” must converge to V (p). Thisjustifies the belief restriction discussed.In particular, the belief restriction here closely follows Guerrieri, Shimer,and Wright (2010). Eeckhout and Kircher (2010) adopt the same restrictionon the queue length. Since asset owners post prices in their setting, theyleave out the off-equilibrium-path belief on the worker’s type.3.3.4 Assortative matchingDefinition. A pair of distributions features positive assortative matching(PAM) if there exists a pair of threshold types (p, q) < (1, 1) and an in-creasing function κ : [p, 1] → [q, 1] such that κ(p) = q and Lpq(p, κ(p)) =F (p)− F (p).κ(p) denotes the quality of the asset assigned to a worker of type p.The above definition of PAM not only requires the participants to matchassortatively, but also every worker above the threshold type to participate1113.4. Second Best Allocationin matching. This is not restrictive because a better worker always gainsmore when entering the same market, so only the lowest types may taketheir outside options in equilibrium and in any efficient allocations.This conclusion does not automatically extend to the asset side. Evenwhen posting the same contract, an owner of a better asset may end upattracting weaker workers, gaining less from participation. In the subsequentsections, I will show that monotonic participation for both sides indeedoccurs in any efficient allocations and equilibria. In this case, κ is bijectiveand strictly increasing. The inverse of κ is well-defined and denoted byr : [q, 1] → [p, 1]. r(q) is the type of worker assigned to the asset of qualityq.3.4 Second Best AllocationSuppose that a Utilitarian planner, whose goal is to maximize the to-tal output, have complete information and may dictate the participationdecision for each type. Nevertheless, search friction remains present in thematching process. Her problem is given bymaxK,L∫η(Λ(q, s))y(p, q)dL+ [F (1)− Lp(1)]V + [G(1)−Kq(1)]Usubject toKq ≤ G,Lp ≤ F and Λ(q, s) = dLqsdK.The search friction introduces an insurance motive, which is conduciveto negative assortative matching. This is because the most efficient wayto increase the matching probability for high types is assigning them to amarket flooded with low types from the other side. The more responsivethe matching probabilities to a change in tightness ratio, the greater the1123.4. Second Best Allocationinsurance motive. The Utilitarian planner’s solution always features PAMonly if the production complementarity outweighs the insurance motive forall distributions of types.Theorem (Eeckhout and Kircher, 2010). Under Assumption (Y) and (M),Second Best allocations always feature PAM.The efficient allocation requires monotonic participation because thematching surplus is strictly increasing in types. The strict concavity ofδ(λ) implies that it is efficient to pool the same pairs of types into one mar-ket. The contract term s can be omitted as it does not affect the size ofthe matching surplus. As a result, the Utilitarian planner’s problem can besimplified asmaxp,q,r,λ∫ 1qδ(λ(q))y(r(q), q)dG(q) + F (p)V +G(q)Usubject tor(q) = p, r(1) = 1, (3.6)and for q ≥ q, ∫ 1qλ(q′)dG(q′) = F (1)− F (r(q)).Abusing the terminology, a solution, denoted by (rSB, λSB, pSB, qSB),is called a Second Best (SB) allocation.70 The Utilitarian planner’s prob-lem can be reformulated as an optimal control problem with r as the statevariable and λ as the control variable. The law of motion is given byr′(q) =g(q)f(r(q))λ(q). (3.7)70There is a continuum of (K,L) with the same matching pattern (r, λ, p, q) but differentdivisions of matching surpluses.1133.4. Second Best AllocationIntroducing the co-state variable τ , the Hamiltonian is given byH(q, r, λ, τ) = g(q)[δ(λ)y(r, q)− τ(q) λf(r)].Eeckhout and Kircher (2010) show that the first order conditions can bewritten asvSB(rSB(q)) = δ′(λSB(q))y(rSB(q), q), (3.8)∂vSB(p)∂p∣∣∣∣p=rSB(q)= η(λSB(q))∂y(p, q)∂p∣∣∣∣(p,q)=(rSB(q),q), (3.9)where τ(q) = f(rSB(q))vSB(rSB(q)). In particular, rSB and λSB are contin-uously differentiable, C1.vSB(p) is the shadow value for a worker of type p ≥ pSB. When com-paring vSB(p) and v′SB(p), it is noteworthy that δ′(λ) < η(λ) < 1 for λ > 0.The gap between δ′(λ) and η(λ) reflects the benefit of a better asset. Onecan derive the shadow value for an asset of quality q ≥ qSBby a symmetricapproach,uSB(q) = [δ(λSB(q))− λSB(q)δ′(λSB(q))]y(rSB(q), q), 71 (3.10)and uSB(q) is strictly increasing. The boundary conditions at the bottomare given byqSB[uSB(qSB)− U ] = pSB[vSB(pSB)− V ] = 0. (3.11)The above set of conditions, (3.6)-(3.11), defines a boundary value problemfor (pSB, qSB, rSB, λSB, vSB, uSB), which admits a unique solution underAssumption (Y) and (M). The assumption in (3.1) ensures participationat the top, pSB< 1 and qSB< 1. The shadow value of an agent below71Note that δ − λδ′ = dη(λ)dλ−1 .1143.4. Second Best Allocationthe threshold type is simply the value of her outside option. I extend thedefinition of vSB and uSB to their entire type space, with vSB(p) = V forp < pSBand uSB(q) = U for q < qSB.72Remark 8. The Second Best allocation is unique. For any p, q and λ,uSB(q) + λvSB(p) ≥ δ(λ)y(p, q) (3.12)with equality if and only if λ = λSB(q) and p = rSB(q).The inequality in (3.12) is the counterpart of the well-known conditionfor stable matching. Without search friction, δ(λ) = min{λ, 1}, the set ofinequalities (3.12) collapses to uSB(q) + λvSB(p) ≥ y(p, q).When defining the Second Best allocation, it is assumed that the Util-itarian planner knows the workers’ types. One may question if this is anappropriate benchmark when the workers’ types are privately known. Sup-pose the Utilitarian planner observes only the types of the assets and mayrestrict the set of markets available. In essence, she may dictate the sharingcontracts s(q) for each type of assets, subject to their voluntary participa-tion. The planner can induce the Second Best allocation by excluding allassets below qSBfrom participation, and imposing ŝ(q) for q ≥ qSB, whereŝ(q) = 1− δ′(λSB(qSB))η(λSB(q))exp(−∫ qqSB∂ ln y(p, q′)∂q∣∣∣∣p=rSB(q′)dq′).At q = qSB, ŝ(qSB) = 1− d ln δd lnλ∣∣λ=λSB(qSB), so that the expected payoffsfor the pair of threshold types are the same as their shadow values in (3.8)and (3.10). For q > qSB, ŝ(q) satisfiesddqln(1− ŝ(q))η(λSB(q))y(p, q)∣∣∣∣p=rSB(q)= 0.72By an abuse of notation, vSB and uSB denote their respective restrictions over [pSB, 1]and [qSB, 1] when referring to the solution of the boundary value problem.1153.4. Second Best AllocationUnder Assumption (Y), the workers of type rSB(q) strictly prefer the market(ŝ(q), q) to all other markets, receiving an expected payoff of V̂ (rSB(q)) =η(λSB(q))(1−ŝ(q))y(rSB(q), q). By construction, vSB(p) ≥ V̂ (p) ≥ vSB(pSB)for p ≥ pSBand equalities hold only at p = pSB.73 This ensures incentivecompatibility on the workers’ side and voluntary participation for asset own-ers of q ≥ qSB. Therefore, the Second Best allocation can be supported byŝ(q).743.4.1 Price competitionPrice competition refers to the benchmark setting that the asset ownersmay post prices, and the workers buy out the asset up front. When par-ticipating in a market with posted price w and queue length λ, a workerreceives an expected payoffη(λ)[y(p, q)− w],while an asset owner receives an expected payoff of δ(λ)w. An equilibriumin price competition can be defined analogously. It is essentially the equi-librium definition in Eeckhout and Kircher (2010).Theorem (Eeckhout and Kircher, 2010). Under Assumption (Y) and (M),the Second Best allocation can be decentralized in price competition.For any (pSB, qSB, rSB, λSB, vSB, uSB) satisfying (3.6)-(3.11), the au-thors construct an equilibrium which supports the Second Best allocation73This is because ∂∂pln vSB(p) >∂∂pln V̂ (p) > 0 for p ≥ pSB. It also implies that theschedule ŝ(q) ∈ [sSB(q), 1) is well-defined.74One can recover the corresponding pair of distributions (K,L) from the Second Bestallocation and the set of active markets{(q, ŝ(q)) : q ∈ [qSB, 1]}and check the conditionsformally. The construction of (K,L) mirrors that in the proof of Proposition 7 in theAppendix.1163.4. Second Best Allocation(rSB, λSB, pSB, qSB) and the equilibrium payoffs for the two sides are givenby vSB and uSB. Let wSB(q) denote the price posted by the owners of assetquality q ≥ qSBin equilibrium. wSB(q) is determined by the FOC (3.8),vSB(rSB(q)) = δ′(λSB(q))y(rSB(q), q) = η(λSB(q))[y(rSB(q), q)− wSB(q)],so thatwSB(q) =(1− d ln δd lnλ∣∣∣∣λ=λSB(q))y(rSB(q), q). (3.13)This is known as the Hosios condition, for which a worker’s share of thematching surplus is given by the elasticity of δ(λ) at the equilibrium queuelength he is facing. Furthermore, Eeckhout and Kircher (2010) show thatwSB(q) is increasing in q. The corresponding pair of distributions (K,L)can be again recovered from the Second Best allocation and wSB(q) readily.The FOC (3.9) ensures thatvSB(rSB(q)) = maxq′∈[qSB,1]{η(λSB(q′))[y(rSB(q), q′)− wSB(q′)]}.Together with the boundary condition(3.11), the incentive compatibility forworkers is met.Given the type of her potential partner rSB(q), an owner of asset qualityq cannot profit from adjusting the price if and only if the Hosios conditionholds. This is the standard result in settings with homogeneous workers.With heterogeneous workers,Λ(q, w) = infλ∈[0,∞]{vSB(p) ≥ η(λ)[y(p, q)− w], p ∈ [0, 1]},so a deviating offer may attract a longer queue of workers of other types.Yet Assumption (Y) and (M) ensure such offer will not be profitablefor the asset owners. The reason is that the price paid by the worker is1173.5. Equilibrium Characterizationindependent of his type. Rearranging the inequality (3.12),uSB(q) ≥ maxp,λ[δ(λ)y(p, q)− λvSB(p)] = maxwδ(Λ(q, w))w,and equality holds for q ≥ qSB.3.5 Equilibrium CharacterizationWe now turn to the set of equilibria when only output sharing contractsare feasible. The workers’ expected payoff can be separated into y(p, q)and η(λ)(1 − s), and only the former depends on the private type. Thishas two important implications. First, the workers’ preferences over q andη(1 − s) satisfy the strict single crossing property (SCP) as the matchingsurplus exhibits strict log-SPM. This reduces the multi-dimensional sortinginto a familiar single dimensional one. Second, workers of all types sharethe same preference over their matching probability and the contract termfor any given asset quality. The corresponding sets of indifference curves forworkers are illustrated in the Figure 3.1Figure 3.1: Properties of workers’ preferencesThe first property implies that if a worker prefers a market with betterassets to another market with lower quality assets, then all better workers1183.5. Equilibrium Characterizationstrictly prefer the one for better assets, vice versa. This holds regardless ofthe queue lengths and the contract terms in these markets. Therefore, theparticipants must match assortatively in any equilibrium.This property also ensures monotonic participation on the asset side inany equilibrium. Suppose some type of workers participate in an activemarket (qL, sL), an owner of a better asset qH can find a less generous con-tract sH leaving these workers indifferent about accepting the two contracts.Hence, the queue length in the market (qH , sH) will be no lower than that inthe active market (qL, sL). The strict SCP then implies that all weaker work-ers strictly prefer the latter market to the former. So posting the contractsH provides an asset owner of qH an expected payoff above the equilibriumpayoff for her peers of qL.Let (p˜, q˜, κ˜, r˜) denote PAM in the equilibrium under consideration.Figure 3.2: Characterization of active marketsThe indifference curves over q and η(1−s) of the participating workers, whichyield their equilibrium payoff, are plotted in the left panel of Figure 3.2. The1193.5. Equilibrium Characterizationlower envelope of all these indifference curves must be η(Λ(q, s))(1−s) for theset of active markets. Furthermore, the workers will have strict preferenceover the resulting set of active markets. That is, a worker of any type otherthan r˜(q′) will be strictly worse off if he deviates to the active market (q′, s′).Put it differently, the workers of r˜(q′) accept a lower matching probabilityin the market (q′, s′) than anybody else.The second property then implies that only the workers of r˜(q′) willaccept the lowest matching probability for any contract (q′, s). This is be-cause for a given asset, the share and the matching probability are perfectsubstitutes to the workers. The competition among workers of r˜(q′) alonewill result in an adjustment in their matching probability fully offsettingthe variation in the posted share. A deviating asset owner of q′ > q˜, ifgets matched, always pair up with the same type of workers in equilibrium.She only trades off between her matching probability and her output share.This is illustrated in the right panel of Figure 3.2. Suppose we fix the pairof types (r˜(q′), q′) and plot the indifference curves over η(λ) and (1 − s).Taking the workers’ equilibrium payoff as given, the tangent point of theindifferent curves of both sides is the optimal contract, and the associatedqueue length for the asset owner. This is exactly the Hosios condition. Inequilibrium, owners of the same asset quality q, if participating, will postthe same shares = 1− d ln δd lnλ∣∣∣∣λ=λ˜(q),where λ˜(q) is the resulting queue length. Hence, r˜ and λ˜ must satisfy thelaw of motion in (3.7).The Hosios condition can be rearranged asδ′(λ˜(q)) = η(λ˜(q))(1− s′),1203.5. Equilibrium Characterizationand thus,V (r˜(q)) = δ′(λ˜(q))y(r˜(q), q).This is the same condition in (3.8), which I will refer to as the Hosios con-dition as well. The violation of the Hosios condition is the reason why ŝ(q)in Section 3.4 cannot be supported by an equilibrium.The incentive compatibility (IC) condition for the workers above thethreshold type can also be rewritten asV (r˜(q)) = maxq′∈[q˜,1]δ′(λ˜(q′))y(r˜(q), q′). (3.14)After accounting for the contract posting decisions, sorting of workers isinduced by the variation in the queue length. The better the asset, thegreater the queue length in the active market. Under Assumption (M), theoutput shares posted by the asset owners increase with their asset quality.Apply the envelope theorem to (3.14), we obtain∂V (r˜(q))∂p= δ′(λ˜(q))∂y(r˜(q), q)∂p. (3.15)Under the strict SCP over q and η(1 − s), the conditions (3.14) and (3.15)are in fact equivalent. Abusing the terminology, I will call the latter as theworkers’ IC condition.The strict SCP also implies that the active market for asset quality q˜is the most profitable deviation for the worker of type below p˜. Therefore,the workers of the threshold type p˜ must be indifferent about participation.This yields the boundary condition for the workers side, p˜(V (p˜)− V ) = 0.The boundary condition for the asset side is more complicated as wehave to find out the deviating payoff for the owners of asset quality belowq˜. Suppose an owner of asset quality q′ < q˜ post a deviating offer s′. Weagain look at the lower envelope of the workers’ indifference curves over q1213.5. Equilibrium Characterizationand η(1−s) which yield their equilibrium payoff, including those below p˜, inFigure 3.3. The indifference curve for the workers attracted must be tangentto the lower envelope, which pins down η(Λ(q′, s′))(1− s′).Figure 3.3: Deviations by assets below threshold typeUnder the strict SCP, the workers of the threshold type p˜ is the highesttype a deviating offer may attract. So the asset owner can never make morethan her peers of the threshold type q˜. On the other hand, posting the sameoutput share provides an asset owner slightly below q˜ a deviating payoffclose to the equilibrium payoff of the threshold type q˜. The latter is givenby U(q˜) = [δ(λ˜(q˜))− δ′(λ˜(q˜)λ˜(q˜)]y(p˜, q˜) under the Hosios condition. So theboundary condition for the asset side, q˜(U(q˜)−U) = 0, mirrors that for theworkers. Notice that the boundary conditions at the bottom are the sameas those for the Second Best allocation in (3.11).The set of equilibrium conditions (3.6)-(3.8),(3.10), (3.11), and(3.15) de-fines a boundary value problem, for which the set of equilibria can be re-covered from the solutions. In the appendix, I will analyze this boundaryvalue problem. I establish the existence and the uniqueness of its solution,1223.5. Equilibrium Characterizationand hence the equilibrium. So Proposition 7 fully characterize the set ofequilibria.Proposition 7. There exists a unique equilibrium. This equilibrium sup-ports PAM and has the following properties:1. Asset owners (workers) participate if and only if q ≥ q˜ (p ≥ p˜), and2. Workers of p ≥ p˜ have equilibrium payoffs v˜(p), and3. The set of active markets Ψ is given by{(q, s) : q ∈ [q˜, 1], s = 1− d ln δd lnλ∣∣λ=λ˜(q)},with Λ(q, s) = λ˜(q) for (q, s) ∈ Ψ, and4. For any s ∈ (0, 1), R(q, s) is degenerate at r˜(q) if q ≥ q˜ and Λ(q, s) > 0.r˜ : [q˜, 1] → [p˜, 1], λ˜ : [q˜, 1] → R++, and v˜ : [p˜, 1] → R++ are all con-tinuously differentiable and strictly increasing. Together with the pair ofthreshold types (p˜, q˜), they satisfy the conditionsr˜(q˜) = p˜, r˜(1) = 1,0 = q˜[(δ(λ˜(q˜))− δ′(λ˜(q˜))λ˜(q˜))y(p˜, q˜)− U ],0 = p˜(v˜(p˜)− V ),r˜′(q) =g(q)f(r(q))λ˜(q),v˜(r˜(q)) = δ′(λ˜(q))y(r˜(q), q),∂v˜(p)∂p∣∣∣∣p=r˜(q)= δ′(λ˜(q))∂y(p, q)∂p∣∣∣∣(p,q)=(r˜(q),q).(3.16)1233.6. Matching Efficiency3.6 Matching EfficiencyAs explained in the introduction, the offering of output sharing contractsinevitably leads to an inefficient allocation. By comparing the set of condi-tions (3.16) in Proposition 7 with the set of conditions (3.6)-(3.11) for theSecond Best allocation, I establish the form of distortion in equilibrium forany distribution of types.Proposition 8. In comparison with the Second Best allocation, the equilib-rium allocation has the following features:1. The queue length for the best assets is greater, λSB(1) < λ˜(1); and2. All participating asset owners pair up with worse partners, rSB(q) >r˜(q) for q ∈ (q˜, 1); and3. Higher participation on the workers’ side, pSB≥ p˜, but lower partici-pation on the asset side, q˜ ≥ qSB; and4. The threshold type on one side remains unchanged only if that sidefeatures full participation, i.e., If pSB= p˜(q˜ = qSB), then pSB= p˜ =0(q˜ = qSB= 0).Corollary 4. In comparison with price competition,1. the best workers are strictly worse off, whereas the lowest types of theparticipating workers of the threshold type, p ∈ (p˜, pSB] strictly benefit;and2. the owners of the highest quality assets are strictly better off, whereasthose of the threshold asset quality q ∈ (qSB, q˜] must be strictly worseoff.1243.6. Matching EfficiencyTo better understand the form of distortion, it is instructive to start withan equilibrium in price competition. Suppose we replace the posted priceswith output shares sSB(q) = 1 − d ln δd lnλ∣∣λ=λSB(q), keeping the same divisionof the matching surplus for every matched pair. This set of contracts isnot incentive compatible for the workers. In comparison with a fixed price,a fixed share of output costs more to the better workers but less to thelow types. The workers will have a higher deviating payoff from the activemarkets for better assets. In particular, the workers above the thresholdtype pSBcan always profit from searching for slightly better assets.This must result in a longer queue of workers for the best assets.75 Inresponse, their owners will post a greater share to partially offset the increasein the queue length. These asset owners decide to retain a longer queuethan in the Second Best allocation because their private value of matchingprobability increases with their share of the surplus.76 On the other side,the best workers will suffer from the reductions in their share of surplus aswell as their matching probability.Under assortative matching, the pool of workers available to the lowerquality assets must deteriorate. The asset owners in the intermediate rangeface two counteracting forces. First, a sharing contract costs less to weakerworkers, intensifying the local competition among workers. Given the sametype of workers, a less generous term is required to maintain sorting. Witha greater share of the surplus, the asset owners gain by improving their75wSB(q) is monotonic in q but sSB(q) needs not be. So we can only deduce a higherqueue length for the best assets.76In a setting with a large but finite number of agents, the average type of workers willdecrease with the queue length. The owners of the best assets would still decide to retaina longer queue in equilibrium. Otherwise, a single asset owner will deviate to post a moregenerous term, drawing workers from other owners of the best assets.1253.6. Matching Efficiencymatching probability. On the other hand, the asset owners are left withweaker workers amid the increase in the queue length for the better assets.As they see a lower gain from a match, they have incentive to induce ashorter queue of workers instead.The relative strengths of the two forces depend on the distribution oftypes. Surprisingly, it turns out that all but the best assets must settlewith weaker partners, regardless of the distribution of types. By continuity,this must happen to the assets of second highest quality. Suppose, to thecontrary, that we move down from the top and find the owners of assetq̂ > q˜ pairing with the same type p̂ = r˜(q̂) = rSB(q̂) as in the SecondBest allocation. Since the workers slightly better than p̂ now match withbetter assets, the queue length for the assets q̂ must be below than in theSecond Best allocation, λSB(q̂) ≥ λ˜(q̂), under assortative matching. Figure3.4 depicts the situation.Figure 3.4: Law of motion in a thought experimentNow consider the thought experiment of removing all workers above p̂and assets above q̂. The Utilitarian planner still finds the original SecondBest allocation optimal for this truncated distribution of types. Otherwise,she would have improved upon it. The same set of equilibrium conditionsin Proposition 7 still applies to the truncated distribution of types. The1263.6. Matching Efficiencyunderlying reason is that the set of contracts posted is to deter the agentsbelow the threshold types from participating and the participating workersfrom deviating to match with slightly better assets. As p̂ and q̂ are now thehighest types, we have argued that the owners of asset quality q̂ will inducean inefficiently long queue of workers, λSB(q̂) < λ˜(q̂). This contradicts ourprevious claim!To understand the distortion in the threshold types, let us return to ourpreceding discussion on the hypothetical set of contracts sSB(q). SupposepSB> 0, the workers of the threshold type pSBare indifferent about theiroutside option and entering the market (sSB(qSB), qSB). Those with typebelow pSBnow pay less under the sharing contract sSB(qSB). Some of themwill be induced to participate. So the participation on the workers side mustincrease in equilibrium if pSB> 0.The result on the matching pattern only states that the workers with typepSBpair up with better assets in equilibrium, κ˜(pSB) ≥ κSB(pSB) = qSB.This raises the question on whether the workers with type p˜ may turn outmatching with assets below qSB. The answer is negative because for thesecond best allocation, the Utilitarian planner would keep assigning agentsinto participation until the expected surplus for the last pair of types declinesto zero. Suppose, to the contrary, that pSB> p˜ and qSB> q˜,0 = (uSB(qSB)− U) + λSB(qSB)(vSB(pSB)− V )= maxλ≥0[δ(λ)y(pSB, qSB)− λV − U ] > maxλ≥0[δ(λ)y(p˜, q˜)− λV − U ]!(3.17)This is impossible as one side of the threshold types (p˜, q˜) will be better offtaking outside option. Therefore, we conclude that the participation on theasset side can only be inefficiently low.7777Since κ˜(p) > κSB(p) for p ∈ (pSB, 1), the case pSB= p˜ and qSB> q˜ is impossible.1273.6. Matching EfficiencyIt remains to argue that some asset owner must be discouraged fromparticipating if qSB> 0. Suppose, to the contrary, that q˜ = qSB> 0, andhence u˜(q˜) = uSB(q˜) = U. Again, the Utilitarian planner would assign someassets to their outside options only if the pair of threshold types yields zeroexpected surplus or all the workers are exhausted, pSB= 0. Recall thatpSB> p˜ if pSB> 0. So the former case is impossible as the inequality(3.17) applies again. For the remaining possibility pSB= p˜ = 0, the queuelength for the pair of threshold types must also stay the same under theHosios condition. Since the workers near the lowest type pay less under thesharing contract, the asset owners slightly above the threshold quality facea longer queue than in price competition. This is exactly opposite to thecase in Figure 3.4. These asset owners must pair up with better workersthan in the Second Best allocation, contracting our previous claim! Withthe distortion of the threshold types, v˜(pSB) > vSB(pSB) and u˜(q˜) < uSB(q˜)follow from the boundary conditions and the Hosios condition.Notice that the above arguments only hinge on the property that thebetter workers always pay more under the sharing contracts.78 Nonetheless,we are still able to draw conclusions on how the matching pattern and theparticipation margin are distorted.Again, two counteracting forces affect the distortion of the queue lengthat the bottom. On one hand, the owners of assets slightly above q˜ nowbenefit less from a higher matching probability as they face weaker partners.On the other hand, their cost of increasing their matching probability mayalso fall. This happens when v˜(pSB) > vSB(pSB) > v˜(p˜). Therefore, thequeue length at the lower end can be distorted in either direction. This is78Formally, the expected payment received by the asset owner strictly increases withthe worker’s type for any contracts and asset quality.1283.6. Matching Efficiencyillustrated in the following example.A symmetric example Suppose types on the two sides are both uni-formly distributed over [0, 1]. The outside options for the two sides yield thesame payoff V = U. Consider a market with O-ring production y(p, q) =(y − y)pq + y and random matching technology δ(λ) = λλ+1 . The SecondBest allocation inherits the symmetry between both sides in the setup. It isgiven by rSB(q) = q, λSB(q) = 1, vSB(p) =14y(p, p) and uSB(q) =14y(q, q).pSB= qSBsatisfy 14 [(y−y)q2SB+y] = U if 14y ≤ U. Otherwise, pSB = qSB =0.We first consider the case 14y ≤ U. The boundary conditions at thebottom immediately imply that p˜q˜ > 0 and(1λ˜(q˜)+1)2[(y − y)p˜q˜ + y] = V = U =(λ˜(q˜)λ˜(q˜)+1)2[(y − y)p˜q˜ + y].It follows that λ˜(q˜) = 1 and p˜q˜ = pSBqSB. Applying the characterization inProposition 7 and 8, we conclude that p˜ < pSB= qSB< q˜ and the queuelength in almost every active market is inefficiently high, λ˜(q) > 1.For the case 14y > U, Proposition 8 states that 0 = p˜ = pSB = qSB < q˜.The boundary condition for q˜ > 0 implies that λ˜(q˜) < 1. Hence, there is athreshold asset quality q̂ such that all active markets for q > q̂ feature aninefficiently high queue length and the opposite occurs to the active marketsfor q < q̂.3.6.1 DiscussionThe recipe for inefficiency here contains three ingredients: output sharingcontracts, private types and search friction.1293.7. Concluding RemarksEeckhout and Kircher (2010) show that Second Best allocations canalways be decentralized in price competition. If the workers’ types are con-tractible, a menu of output shares may function as a posted price as its termcan be made contingent on types to implement the required transfer. So theSecond Best allocation will be decentralized.79Now consider an environment where the parties face no search friction.In the First Best allocation, the matching is perfectly assortative with aunit queue length for every matched pair. The First Best allocation alwaysprevails if the asset side may post prices. When we replace the posted priceswith output shares keeping the same division of the matching surplus, theworkers will again deviate to better assets, resulting in a longer queue forthe best assets. Without search friction, the owners of the best assets willincrease their posted share until the queue length restores to unity. Theasset owners still have a greater share of surplus upon matching, and hencea higher marginal value of matching probability. However, they cannotimprove their matching probability by distorting the queue length. Theirdecisions in turn leave the same pool of workers to the asset owners of thesecond highest quality. Inductively, the equilibrium allocation remains FirstBest amid higher equilibrium payoff for all participating asset owners.3.7 Concluding RemarksThis chapter studies how the output sharing arrangement affects sortingefficiency in a directed search framework where one side has private types. I79Under condition 3.12 in Remark 8, an asset owner cannot gain from posting a menuof output shares specifying different expected payment for different types. This is becausethe meeting is bilateral in the setting here.1303.7. Concluding Remarksconsider a stylized setting to disentangle the source of inefficiency from thewell-known channels. I characterize the unique equilibrium which featuresinefficient PAM. I provide qualitative features of the distortion in equilib-rium. These features applies to any distribution of types as the underlyingforces are always present. In particular, the unique equilibrium features fullseparation of types and the Hosios condition is met in every active market.I then provide qualitative features of the distortion in the matching patternand the participation thresholds.For other forms of securities or contingent contract, the preference overthe matching probability and the term of the contract differ across workers.In equilibrium, the matched pairs will not fully separate into a continuumof markets where the Hosios condition is satisfied. This is because in suchcase, the asset owners will distort the queue length to screen out betterworkers. 80 As a result, the channel of sorting inefficiency here will confoundwith the distortions associated with the screening by the asset owners aswell as the search externalities such as thick market effect, congestion effectand compositional effect. An avenue for future research is to study suchinteractions and the resulting form of distortion. The results in this chapterwill then serve as a useful benchmark.80Under Hosios condition, an incremental distortion of the queue length leads to asecond-order loss while an improvement in the worker’s type yields a first-order gain.131Chapter 4ConclusionIn two-sided one-to-one matching, the equilibrium matching pattern andthe divisions of surpluses vary with the distribution of types. A central ques-tion is to identify “distribution-free” qualitative features and their relationto the model primitives such as the production technology and the marketstructure. These results yield testable implications and policy recommenda-tion, which are robust to misspecification of the distribution of types. Thearguments underlying the results illuminate general economic forces.A recurrent focus in the literature has been the conditions for assortativematching, or lack thereof. This dissertation introduces private informationand contingent payment, which are present in a number of circumstances.The contribution is twofold.First, I provide economically meaningful conditions ensuring decentral-ization of efficient matching in a frictionless environment. The analysisextends our understanding of the potential forces against PAM from fixedprices to more general forms of contingent contracts. The policy implicationhere is to restrict the flattest contracts available so that Condition GlobalID is met. Roughly speaking, assortative matching occurs whenever thevariation in the contingent contracts aligns with the form of productioncomplementarity. Such consideration has been absent from the literature ofcontract theory and in particular, security design.132Chapter 4. ConclusionSecond, I provide two novel comparative statics. I show how the formof financial securities available affects the divisions of matching surplusesin a frictionless environment. 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A competitive matching equilibrium using direct revelation mech-anisms consists of the asset owners’ equilibrium payoff U : {qk}Kk=1 → R+,workers’ equilibrium payoff V : {pl}Ll=1 → R+, asset owners’ contract post-ing set ψT : {qk}Kk=1 → ∪Kk=1ΩDRMT (qk) ∪ {p0}, the set of active mar-kets ΨT ⊆ ∪Kk=1ΩDRMT (qk) × {qk}Kk=1, the measure of participating workersW T :∏Ll=1{B([q, q] × [0, 1]) × T } × P({qk}Kk=1) → [0, 1], the distribution ofworkers rT : ∪Kk=1ΩDRMT (qk) × {qk}Kk=1 → ∆({pl}Ll=1) and market tightnessµT : ∪Kk=1ΩDRMT (qk)× {qk}Kk=1 → [0,∞] such that1. Asset Owners’ Optimal Contract Posting:i)For all q ∈ {qk}Kk=1 and T ∈ ΩDRMT (q),U(q) ≥ η(µT (T, q))µT (T, q)L∑l=1rT (pl|T, q)(1− pil)u(ql, pl, tl)with equality if T ∈ ψT (q).ii)µT (T, q) = 0 and T /∈ ψT (q) if T /∈ ΩDRMT (q).141A.1. Proof of Lemma 12. Workers’ Optimal Acceptance:i)For all (T, q) ∈ ∪Kk=1ΩDRMT (qk)× {qk}Kk=1,V (pl) ≥ η(µT (T, q))(1− pil)v(pl, ql, tl) (A.1)with equality if rT (pl|T, q) > 0 and µT (T, q) <∞.ii)µT (T, q) =∞ if V (pl) > (1− pil)v(pl, ql, tl) for all p ∈ {pl}Ll=1.3. Active Markets:ΨT := {(T, q) ∈ ∪Kk=1ΩDRMT (qk)× {qk}Kk=1 : T ∈ ψT (q)} is the supportof W T .4. Optimal Participation:i)U(q) ≥ U and V (pl) ≥ V .ii)∫ΩDRMT ×{qk}Kk=1 rT (pl|T, q)dW T ≤ P (pl) with equality if V (pl) > V .iii)∫ΩDRMTµT (T, qk)dWT ≤ Q(qk) with equality if U(qk) > U.Notice that the inequality (A.1) in the workers’ optimal acceptance de-cision holds for both active and inactive markets. Therefore, the worker’soptimal acceptance condition captures the belief restriction in section 2.4.4.Suppose that {U, V, ψ,Ψ,W, r, µ} is a competitive matching equilibrium.Then there exists an equilibrium using direct revelation mechanisms{U, V, ψT ,ΨT ,W T , rT , µT } supporting the same equilibrium payoffs {U, V }and same allocation,∫ΩDRMT (qk)rT (pl|T, qk)dW T =∫Ωtr(pl|t, qk)dW∫ΩDRMT (qk)rT (pl|T, qk)µT (t, qk)dW T =∫Ωtr(pl|t, qk)µ(t, qk)dWfor all pl and qk. Furthermore, all active markets ΨT involve only degeneratedirect revelation mechanisms.142A.1. Proof of Lemma 1Proof. Fix an equilibrium {U, V, ψ,Ψ,W, r, µ}, I now construct a correspond-ing equilibrium {U, V, ψT ,ΨT ,W T , rT , µT } involving only degenerate DRM.The participation decisions are as follows: ψT (qk) = {T = {q′, pi′, t′}Ll=1 :(q′, pi′, t′) ∈ ψ(qk)} if {p0} /∈ ψ(qk). Otherwise, ψT (qk) = {T = {q′, pi′, t′}Ll=1 :(q′, pi′, t′) ∈ ψ(qk)} ∪ {p0}. As required in the equilibrium definition, ΨT ={(T, q) ∈ ΩDRMT ×{qk}Kk=1 : T ∈ ψT (q)}.W T is defined as follows. For everyset A,W T (A) = W ({(q′, pi′, t′) : {q′, pi′, t′}Ll=1 ∈ A ∩ΨT }).Consider T ′ ∈ ψT (qk). By construction, T ′ takes the form of {q′, pi′, t′}Ll=1,where (q′, pi′, t′) ∈ ψ(qk).Define rT (T ′, qk) = r((q′, pi′, t′), qk) and µT (T ′, qk) =µ((q′, pi′, t′), qk). {U, V, ψ,Ψ,W, r, µ} and {U, V, ψT ,ΨT ,W T , rT , µT } supportthe same allocation by construction.For infeasible menus T ′ /∈ ΩDRMT (qk), then µT (T ′, q) = 0 and rT (p1|T ′, qk) =1. Now consider an inactive market T ∈ ΩDRMT (qk)\ψT (qk). Define the mar-ket tightness byµT (T, qk) = sup{µ′ ∈ [0,∞] : V (pl) ≥ η(µ′)(1− pil)v(pl, ql, tl), L ≥ l ≥ 1}.Notice that µT (T, qk) = ∞ if V (pl) ≥ (1 − pil)v(pl, ql, tl) for all type pl.Otherwise, µT (T, qk) < 1 and rT (T, qk) is defined to be degenerate atmin{p ∈ {pl}Ll=1 : V (pl) = µT (T, qk)(1− pil)v(pl, ql, tl)}.The construction of ΨT and the equilibrium conditions for {U, V, ψ,Ψ,W, r, µ}immediately imply that deviations to active markets are never profitable.It is sufficient to check if an asset owner of qk cannot profit from postingT˜ ∈ ΩDRMT (qk)\ψT (qk). Consider T˜ = {q˜l, ˜l, t˜l}Ll=1 with µT (T˜ , qk) <∞. Byconstruction, µT (T˜ , qk) < 1 and rT (T˜ , qk) must be degenerate at some pl̂.143A.2. Proof of Proposition 1For any type pl,V (pl) ≥ µT (T˜ , qk)(1− ˜l)v(pl, q˜l, t˜l)≥ µT (T˜ , qk)(1− ˜l̂)v(pl, q˜ l̂, t˜l̂).The first inequality is merely the definition of µT (T˜ , qk) and the secondinequality is the IC condition for DRM. As they hold with equalities forpl = pl̂, it follows that µ((˜l̂, q˜ l̂, t˜l̂), qk) = µT (T˜ , qk) < 1. Any pl′ in thesupport of r((˜l̂, q˜ l̂, t˜l̂), qk) must satisfyV (pl′) = µT (T˜ , qk)(1− ˜l̂)v(pl′ , q˜ l̂, t˜l̂)The previous inequality immediately implies that V (pl′) = µT (T˜ , qk)(1 −˜l′)v(pl′ , q˜l′ , t˜l′) and hence, pl̂ ≤ pl′ by construction. From the optimalcontract posting condition for {U, V, ψ,Ψ,W, r, µ},U(qk) ≥ (1− ˜l̂)u(q˜ l̂, r((˜l̂, q˜ l̂, t˜l̂), qk), t˜l̂)≥ (1− ˜l̂)u(q˜ l̂, pl̂, t˜l̂).The last inequality is due to Remark (1). Hence, the asset owner has noprofitable deviations.A.2 Proof of Proposition 1I first define an equilibrium in the full information case. Each market isassociated with a tightness ratio µFI(t, p, q). The distribution of workers istrivial, and therefore omitted.Definition. A full information competitive matching equilibrium consistsof the asset owners’ equilibrium payoff UFI : {qk}Kk=1 → R+, workers’144A.2. Proof of Proposition 1equilibrium payoff V FI : {pl}Ll=1 → R+, asset owners’ contract posting setψFI : {qk}Kk=1 → Ωt × {pl}Ll=1 ∪ {p0}, the set of active markets ΨFI ⊆Ωt × {pl}Ll=1 × {qk}Kk=1, the measure of participating workers WFI : T ×P({pl}Ll=1 × {qk}Kk=1) → [0, 1], and market tightness µFI : Ωt × {pl}Ll=1 ×{qk}Kk=1 → [0,∞] such that1. Asset Owners’ Optimal Contract Posting:For all (t, p, q) ∈ Ωt×{pl}Ll=1×{qk}Kk=1, UFI(q) ≥ η(µFI(t,p,q))µFI(t,p,q)u(q, p, t)with equality if t ∈ ψFI(q).2. Workers’ Optimal Acceptance:For all (t, p, q) ∈ Ωt × {pl}Ll=1 × {qk}Kk=1,V FI(p) ≥ η(µFI(t, p, q))v(p, q, t) (A.2)with equality if V FI(p) ≤ v(p, q, t). Otherwise, µFI(t, p, q) =∞.3. Active Markets:ΨFI := {(t, p, q) ∈ Ωt × {pl}Ll=1 × {qk}Kk=1 : (t, p, q) ∈ ψFI(q)} is thesupport of WFI .4. Optimal Participation:i)UFI(q) ≥ U and V FI(p) ≥ V .ii)WFI(Ωt × {pl} × {qk}Kk=1) ≤ P (pl) with equality if V FI(pl) > V .iii)∫Ωt×{pl}Ll=1 µFI(t, p, qk)dWFI ≤ Q(qk) with equality if UFI(qk) > U.I am going to show that the set of First Best allocations coincides withthe set of equilibrium allocations under full information. The proof heredoes not require Assumption (P).145A.2. Proof of Proposition 1Every First Best allocation can be supported by an equilibrium under fullinformation.Lemma 10. Suppose (WFBpq , CFBpq ) is a First Best allocation and (∆VFB,∆UFB)is a set of associated Lagrange multipliers for the First Best program, where∆V FB(p) and ∆UFB(q) denote the shadow price of the corresponding re-source constraint respectively. Define V FB(p) := V+∆V FB(p) and UFB(q) :=U + ∆UFB(q). Then WFBpq = CFBpq and1. V FB(p) + UFB(q) ≥ E(Y |p, q). Equality holds if WFBpq (p, q) > 0.2. V FB(p) ≥ V . ∑Kk=1WFBpq (p, qk) = P (p) if V FB(p) > V .3. UFB(q) ≥ U . ∑Ll=1CFBpq (pl, q) = Q(q) if UFB(q) > U.Proof. For any (W ′pq, C ′pq), define min{W ′pq, C ′pq} = W ′′ where W ′′(p, q) :=min{W ′pq(p, q), C ′pq(p, q)} for all (p, q). If W ′pq 6= C ′pq, then TS(W ′pq, C ′pq) <TS(min{W ′pq, C ′pq},min{W ′pq, C ′pq}). ThereforeWFBpq = CFBpq . Substitute thisinto the First Best program, the Kuhn-Tucker conditions for the recastedprogram give rise to the remaining conditions.Fix a First Best allocation and a set of shadow prices. For all pairs (pl, qk)with WFBpq (pl, qk) > 0, Lemma 10 states that VFB(pl) ∈ [V ,E(Y |pl, qk) −U ]. Hence, there must exist some contract t(.; s′) ∈ St s.t. UFB(qk) =u(qk, pl, t(.; s′)) and V FB(pl) = v(pl, qk, t(.; s′)). Denote s′ by sFB(pl, qk).I now construct a candidate equilibrium with V FI = V FB and UFI =UFB. The participation decisions are defined as follows: If∑Ll=1WFBpq (pl, qk′) =Q(qk′),ψFI(qk′) = {(t, pl) : WFBpq (pl, qk′) > 0, t = t(.; sFB(pl, qk′))}.146A.2. Proof of Proposition 1If∑Ll=1WFBpq (pl, qk′) ∈ (0, Q(qk′)),ψFI(qk′) = {(t, pl) : WFBpq (pl, qk′) > 0, t = t(.; sFB(pl, qk′))} ∪ {p0}.ψFI(qk′) = {p0} if∑Ll=1WFBpq (pl, qk′) = 0.ΨFI = {(t, pl, qk) : WFBpq (pl, qk) > 0, t = t(.; sFB(pl, qk))}. WFI is degen-erate with ΨFI as support and WFI(t(.; sFB(pl, qk)), pl, qk) = WFBpq (pl, qk).µFI(t, pl, qk) = 1 for all (t, pl, qk) ∈ ΨFI . Workers’ optimal acceptancecondition pins down µFI for inactive markets. For any (t, pl, qk) /∈ ΨFI ,µFI(t, pl, qk)µFI(t, pl, qk) = sup{µ′ ∈ [0,∞] : V FI(pl) ≥ η(µ′)v(pl, qk, t)}.It remains to verify that ψFI is the set of optimal contracts for the assetowners. All other equilibrium conditions are met by construction. Sup-pose that for some (t̂, pl̂, qk̂) /∈ ΨFI , η(µFI(t̂,pl̂,qk̂))µFI(t̂,pl̂,qk̂)u(qk̂, pl̂, t̂) > UFI(qk̂). Itfollows that u(qk̂, pl̂, t̂) > UFI(qk̂) and µFI(t̂, pl̂, qk̂) > 0 so that v(pl̂, qk̂, t̂) ≥V FI(pl̂). These two inequalities jointly implies V FB(pl̂)+UFB(qk̂) < E(Y |pl̂, qk̂),contradicting the efficiency of the First Best allocation!Every equilibrium allocation is a First Best.Lemma 11. An allocation (Ŵpq, Ĉpq) satisfying resources constraints is aFirst Best if Ŵpq = Ĉpq and there exists a pair of functions Û : {qk}Kk=1 →R+ and V̂ : {pl}Kk=1 → R+ such that1. V̂ (p)+Û(q) ≥ E(Y |p, q) for all (p, q) ∈ {pl}Ll=1×{qk}Kk=1, and equalityholds if Ŵpq(p, q) > 0.2. V̂ (p) ≥ V . ∑Kk=1 Ŵpq(p′, qk) = P (p′) if V̂ (p′) > V .3. Û(q) ≥ U. ∑Ll=1 Ĉpq(pl, q′) = Q(q′) if Û(q′) > U .147A.2. Proof of Proposition 1Proof. For any (W ′pq, C ′pq), define min{W ′pq, C ′pq} = W ′′ where W ′′(p, q) :=min{W ′pq(p, q), C ′pq(p, q)} for all (p, q). Consider any allocation (Wpq, Cpq)satisfying the resource constraints,TS(Wpq, Cpq) ≤ TS(min{Wpq, Cpq},min{Wpq, Cpq})=∑Kk=1∑Ll=1[E(Y |pl, qk)− U − V ] min{Wpq(pl, qk), Cpq(pl, qk)}+V∑Ll=1P (pl) + U∑Kk=1Q(qk)≤∑Ll=1{[V̂ (pl)− V]∑Kk=1min{Wpq(pl, qk), Cpq(pl, qk)}+ P (pl)V}+∑Kk=1{[Û(qk)− U]∑Ll=1min{Wpq(pl, qk), Cpq(pl, qk)}+Q(qk)U}≤∑Ll=1P (pl)V̂ (p) +∑Kk=1Q(qk)Û(q) = TS(Ŵpq, Ĉpq).The first inequality stems from the specification of matching function. Thesecond inequality is obtained by substituting V̂ (p)+Û(q) ≥ E(Y |p, q). Since(Wpq, Cpq) satisfies the resource constraints, the second and third conditionfor V̂ and Û in Lemma 11 leads to the third inequality. The last equality isdue to the equality conditions for the first condition in Lemma 11. Hence,(Ŵpq, Ĉpq) is a First Best allocation.Now consider an equilibrium under full information. First, every ac-tive market clears. Suppose not, µFI(t′, pl′ , qk′) < 1 for some (t′, pl′ , qk′) ∈ΨFI . Then v(pl′ , qk′ , t′) > V FI(pl′) ≥ V and by ex-post budget balance,E(Y |pl′ , qk′))− v(pl′ , qk′ , t′) = UFI(qk′) ≥ U. There exist contracts t(.; s′′) ∈St yielding v(pl′ , qk′ , t(.; s′′)) arbitrarily close to but below v(pl′ , qk′ , t′), sothat µFI(t(.; s′′), pl′ , qk′) < 1 and u(qk′ , pl′ , t(.; s′′)) > UFI(qk′). an assetowner will gain from offering such contract. A symmetric argument rulesout µFI(t′, pl′ , qk′) > 1. This argument remains valid when V FI(pl′) = Vor UFI(qk′) = U. Second, VFI(pl) + UFI(qk) ≥ E(Y |pl, qk) for all pairs of148A.3. Proof of Proposition 2(pl, qk), and equality holds if (t′, pl, qk) ∈ ΨFI for some t′ ∈ Ωt. The equalityis a direct consequence of ex-post budget balance and µFI(t, pl, qk) = 1 forall (t, pl, qk) ∈ ΨFI . Now suppose V FI(pl′) + UFI(qk′) < E(Y |pl′ , qk′), thenan asset owner with quality qk′ will deviate to a market (t(.; s′), pl′ , qk′) ∈ St,in which v(pl′ , qk′ , t(.; s′)) is slightly above V FI(pl′)!Since µFI(t, pl, qk) = 1 for all (t, pl, qk) ∈ ΨFI , the equilibrium allocationis given by CFIpq (pl, qk) = WFIpq (pl, qk = WFI(Ωt × (pl, qk)). Optimal partici-pation conditions imply that CFIpq and WFIpq satisfy the resources constraints,with equality in the case UFI(qk) > U and VFI(pl) > V respectively.The equilibrium conditions also imply UFI(qk) ≥ U , V FI(pl) ≥ V andV FI(pl) + UFI(qk) = E(Y |pl, qk) whenever WFIpq (pl, qk) > 0. In summary,(WFIpq , CFIpq ) satisfies the conditions in Lemma 11 with (UFI , V FI) = (Û , V̂ ).A.3 Proof of Proposition 2I am going to prove that if Ct ⊆ Ωt, the set of First Best allocationscoincides with the set of equilibrium allocations in price competition. Theproof again does not require Assumption (P). The structure of this proofclosely follows that for Proposition 1.Every First Best allocation can be supported by an equilibrium under fullinformation.Proof. I now construct a candidate equilibrium with V = V FB and U =UFB, where V FB and UFB are defined in the proof of Proposition 1. Letψ(qk) = {p0} if∑Ll=1WFBpq (pl, qk) = 0. For any qk̂ satisfying∑Ll=1WFBpq (pl, qk̂) >0, Lemma 10 implies that UFB(qk̂) ∈ [U,maxk{E(Y |pl, qk̂) − V }]. Hence,these exists sk̂∈ [0, 1] s.t. tc(.; sk̂) = UFB(qk̂). Let ψ(qk̂) = {tc(.; sk̂)} if149A.3. Proof of Proposition 2∑Ll=1WFBpq (pl, qk̂) = Q(qk̂) and ψ(qk̂) = {tc(.; sk̂)}∪{p0} if∑Ll=1WFBpq (pl, qk̂) <Q(qk̂). Hence, Ψ = {(tc(.; sk), qk) :∑Ll=1WFBpq (pl, qk) > 0}. W is degeneratewith Ψ as support and W (tc(.; sk), qk) =∑Ll=1WFBpq (pl, qk). For an activemarket (tc(.; sk), qk) ∈ Ψ, µ(tc(.; sk), qk) = 1 and r(tc(.; sk), qk) = rFBq (qk).Workers’ optimal acceptance condition pins down µ and r for the inactivemarkets. For any (t′, q′) /∈ Ψ,µ(t′, q′) = sup{µ′ ∈ [0,∞] : V FB(p) ≥ η(µ′)v(p, q′, t′)}.If µ(t′, q′) <∞, then r(t′, q′) is degenerate atmin{p ∈ {pl}Ll=1 : V FB(p) = µ(t′, q′)v(p, q′, t′)}.If µ(t′, q′) = ∞, r(t′, q′) has no bearings and is assumed to be the same asprior.I first show that ψ(qk) is optimal for the owners of asset quality qk.Suppose that for some (t˜, q˜) /∈ Ψ, η(µ(t˜,q˜))µ(t˜,q˜)u(q˜, r(t˜, q˜), t˜) > UFB(q˜). By con-struction, µ(t˜, q˜) < ∞ and r(t˜, q˜) is degenerate at some type p˜. It followsthat u(q˜, p˜, t˜) > UFB(q˜) and v(p˜, q˜, t˜) ≥ V FB(p˜). These two inequalitiesjointly implies V FB(p˜) + UFB(q˜) < E(Y |p˜, q˜), contradicting Lemma 10!Second, the workers cannot gain by deviating to other active markets. Forany p ∈ {pl}Ll=1 and (tc(.; sk), qk) ∈ Ψ, V (p) ≥ E(Y |p, qk) − UFB(qk) =v(p, qk, tc(.; sk)}). The inequality is due to Lemma 10 and the last equalitymakes use of the property of cash payment. It suffices to check for these twotypes of deviations. All other equilibrium conditions are met by constructionand Lemma 10.Every equilibrium allocation is a First Best.Proof. Fix an equilibrium. First, every active market clears. Suppose not,µ(t′, q′) > 1 for some (t′, q′) ∈ Ψ and let r(p′|t′, q′) > 0. Then u(q′, r(t′, q′), t′) >150A.4. Proof of Lemma 2-4 and Proposition 3U(q′) ≥ U. Then there exists contract tc(.; s′′) yielding v(p′, q′, tc(.; s′′))arbitrarily close to but above v(p′, q′, t′), so that µ(tc(.; s′′), q′) < 1, andu(q′, r(tc(.; s′′), q′), tc(.; s′′)) > U(q′) for arbitrary r(tc(.; s′′), q′). An assetowner will gain from offering such contract. A symmetric argument rulesout µ(t′, q′) < 1. Second, V (p) + U(q) ≥ E(Y |p, q) for all pairs of (p, q),and equality holds if (t′, q) ∈ Ψ for some t′ ∈ Ωt. The equality is a di-rect consequence of µ(t, q) = 1 for all (t, q) ∈ Ψ. Now suppose V (p′) +U(q′) < E(Y |p′, q′). There is some contract tc(.; s′) in which v(p′, q′, tc(.; s′))is slightly above V (p′), so that µ(tc(.; s′), q′) < 1, and u(q′, p′′, tc(.; s′)) =tc(.; s′) > U(q′) for any p′′ ∈ {pl}Ll=1. An asset owner with quality q′ willgain from offering such contract.The allocation is given byW ′pq(pl, qk) = C ′pq(pl, qk) =∫Ωtr(pl|t, qk)dW (t, pl, qk).Since µ(t, q) = 1 in the support of W , optimal participation conditions implythat C′pq and W′pq satisfy the resources constraints, with equality in the caseU(q) > U and V (p) > V respectively. In addition, V (p) +U(q) = E(Y |p, q)whenever W ′pq(p, q) > 0. Applying Lemma 11 with (U, V ) = (Û , V̂ ), theequilibrium allocation (W ′pq, C ′pq) is a First Best.A.4 Proof of Lemma 2-4 and Proposition 3The sufficiency part in Proposition 3 directly follows from Lemma 2-4,which I will establish in sequence.I first reiterate the candidate equilibrium for reference. The set of ac-tive markets is given by Ψ = {(t(.; s˜k), qk)}k≥k where {s˜k}k≥k satisfies the151A.4. Proof of Lemma 2-4 and Proposition 3equalities (2.9) and (2.10).W ({t, q}) =Q(qk)∑Ll=1WFBpq (pl, qk)0if {t, q} = {(t(.; s˜k), qk)} and qk > qk;if {t, q} = {(t(.; s˜k), qk)};Otherwise.ψ(qk) ={(t(.; s˜k), qk)}{(t(.; s˜k), qk)} ∪ {p0}{p0}if qk > qk;if qk = qk and∑Ll=1WFBpq (pl, qk) < Q(qk);if qk < qk.The equilibrium payoffs for workers and asset owners are given byV (pl) = max{v(pl, qk, t(.; s˜k))}k≥k,V , if pl ≥ pl; andif pl < pl.U(qk) = u(qk, rFBq (qk), t(.; s˜k)),U, if qk ≥ qk; andif qk < qk.For the active markets, r(t(.; s˜k), qk) = rFBq (qk) and µ(t(.; s˜k), qk) = 1.For an inactive market (t(.; s′), qk′) /∈ Ψ,µ(t(.; s′), qk′) = sup{µ′ ∈ [0,∞] : V (p′) ≥ η(µ′)v(p′, qk′ , t(.; s′)), p′ ∈ {pl}Ll=1}.If µ(t(.; s′), qk′) <∞, r(t(.; s′), qk′) is degenerate atmin{p′ ∈ {pl}Ll=1 : V (p′) = µ(t(.; s′), qk′)v(p′, qk′ , t(.; s′)))}.This is the most pessimistic belief allowed in our equilibrium definition. Ifµ(t(.; s′), qk′) = ∞, r(t(.; s′), qk′) has no bearings and is assumed to be thesame as prior.152A.4. Proof of Lemma 2-4 and Proposition 3A.4.1 Proof of Lemma 2Lemma 12. Suppose Condition Sorting-p holds. For any k′ ≥ k, V ≥v(pl, qk′ , t(.; s˜k′)) if pl ≤ pl and v(pl, qk, t(.; s˜k)) ≥ v(pl, qk′ , t(.; s˜k′)) if (pl, qk)is in the support of WFBpq .Proof. To save on space, I will denote v(pl, qk, t(.; s˜k)) by g(l, k) for k ≥ k.l(k) denotes min{l ≥ 1 : WFBpq (pl, qk) > 0, qk ≥ qk}, the lowest type pairingup with qk ≥ qk. For generic distributions of types, l(k+1) = l(k). Obviously,v(pl, qk, t(.; s˜k)) ≥ V for pl ≥ pl and u(qk, pl, t(.; s˜k)) ≥ U for pl ≥ pl(k).For any n ≥ 1, g(l(k+n−1), k+n−1) ≥ g(l(k+n−1), k+n) by (2.10).Condition Sorting-p implies that g(l(k), k + n − 1) ≥ g(l(k), k + n). Thisargument holds for any n ≥ 1. Inductively, g(l(k), k) ≥ g(l(k), k + n) for alln ≥ 1. For any n ∈ [1, k− k + 1], g(l(k − n), k− n) ≥ g(l(k − n), k− n− 1).By a symmetric argument, g(l(k), k) ≥ g(l(k), k − n) for n ∈ [1, k − k].Now consider pl in the support of rFBq (qk) and kH > k ≥ k. FromCondition Sorting-p, g(l, k) ≥ g(l, kH) because g(l(k), k) ≥ g(l(k), kH). Forthe case k > kL ≥ k, g(l(k), k) ≥ g(l(k), kL), and so g(l, k) ≥ g(l, kL).Putting together, we show that if WFBpq (pl, qk) > 0, then g(l, k) ≥ g(l, l′) forany k′ ≥ k.For pl ≤ pl, v(pl, qk, t(.; s˜k)) ≤ v(pl, qk, t(.; s˜k)) ≤ v(pl, qk, t(.; s˜k) = V .The first inequality is due to Remark 1 and the preceding argument estab-lishes the second inequality. The last equality is from the construction ofs˜k.For workers of type pl ≥ pl, V (pl) = v(pl, qk, t(.; s˜k)) for (pl, qk) in thesupport of WFBpq . For workers of type pl < pl cannot gain from participat-ing in any active markets. For any pl ≥ pl, V (pl) ≥ V (pl) ≥ V . Hence,153A.4. Proof of Lemma 2-4 and Proposition 3the participation constraints for workers hold. Therefore, workers have noprofitable deviations in the candidate equilibrium.For qk ≥ qk,U(qk+1)− U(qk) ≥ u(qk+1, pl(k), t(.; s˜k+1))− u(qk, pl(k), t(.; s˜k))= E(Y |pl(k), qk+1)− E(Y |pl(k), qk) > 0The first inequality is due to Remark (1). The equality (2.10) gives rise tothe equality on last line. The last inequality is due to Assumption (P). Ourconstruction of s˜k ensures that U(qk) ≥ U. The participation constraints forthe asset owners are met.A.4.2 Proof of Lemma 3Proof. For any qk ≥ qk, V (pl) ≥ v(pl, qk, t(.; s˜k)). These asset owners willnot get matched if posting a contract with s > s˜k. I now show that own-ers of asset quality qk ≥ qk have no incentive in posting a contract withs < s˜k. Suppose not, an asset owner of qk̂ ≥ qk profits from postingsL̂ < s˜k̂. By construction, r(t(.; sL̂), qk̂) is degenerate at some worker’stype, say pH . pL denotes the lowest type in Ωp(t(.; s˜k), qk). It is trivial thatu(qk̂, pH , t(.; sL̂)) > U(qk̂) ≥ max{U, u(qk̂, pL, t(.; s˜k̂))} and pH > pL. Fur-thermore,v(pH , qk̂, t(.; s˜k̂)) ≤ V (pH) = µ(t(.; sL̂), qk̂)v(pH , qk̂, t(.; sL̂)),v(pL, qk̂, t(.; s˜k̂)) = V (pL) > µ(t(.; sL̂), qk̂)v(pL, qk̂, t(.; sL̂)),which impliesv(pH , qk̂, t(.; s˜k̂))v(pH , qk̂, t(.; sL̂))<v(pL, qk̂, t(.; s˜k̂))v(pL, qk̂, t(.; sL̂)).For all q ≤ qk̂, define ŝ(q) by v(pL, q, t(.; ŝ(q))) = v(pL, qk̂, t(.; s˜k̂)).Condition Screening-q implies that V (pL) = v(pL, qk̂, t(.; s˜k̂)) > V and154A.4. Proof of Lemma 2-4 and Proposition 3u(q, pL, ŝ(q)) > U for any q ≤ qk̂. Our construction of {s˜k}k≥k implies thatk̂ > k ≥ 1 and pL > pl. Note thatU(qk̂) > u(qk̂, pL, t(.; sL̂) > u(qk̂, pl, t(.; sL̂).From the workers’ IC condition, s˜k ≥ ŝ(qk) for k ≤ k ≤ k̂ − 1 becausev(pL, qk, t(.; ŝ(qk))) = v(pL, qk̂, t(.; s˜k̂)) ≥ v(pL, qk, t(.; s˜k).Consider the case that∑Kk=1Q(qk) <∑Ll=1 P (pl), then v(pl, q1, t(.; ŝ(q1)) ≥v(pl, q1, t(.; s˜1)) = V (pl) = V ≥ v(pl, qk̂, t(.; ŝ(qk̂))). The last inequality isdue to Lemma 12. By continuity of v(p, q, t(.; s)) in q and s, there mustexist q′ ∈ [q1, qk̂] such that v(pl, q1, t(.; s˜1)) = v(pl, q′, t(.; ŝ(q′)) = V . Recallthat V (pH) ≥ v(pH , qk̂, t(.; s˜k̂)). Putting together, Condition Screening-qimplies thatV (pH)V (pl)≥ v(pH , qk̂, t(.; s˜k̂))v(pl, q′, t(.; ŝ(q′))≥ v(pH , qk̂, t(.; sL̂))v(pl, qk̂, t(.; sL̂)).Therefore, V (pl) ≤ µ(t(.; sL̂), qk̂)v(pl, qk̂, t(.; sL̂)). Our construction of off-equilibrium-path belief implies r(t(.; sL̂), qk̂) is degenerate at pl < pH !Consider the case that∑Kk=1Q(qk) ≥∑Ll=1 P (pl). Recall that s˜k ≥ŝ(qk), so U ≥ u(qk, p1, t(.; s˜k)) ≥ u(qk, p1, t(.; ŝ(qk))) and v(p1, qk, t(.; ŝ(qk)) ≥v(p1, qk, t(.; s˜k)) = V (p1). Combining with V (pH) ≥ v(pH , qk̂, t(.; s˜k̂)), Con-dition Screening-q impliesV (pH)V (p1)≥ v(pH , qk̂, t(.; s˜k̂))v(p1, qk, t(.; ŝ(qk))≥ v(pH , qk̂, t(.; sL̂))v(p1, qk̂, t(.; sL̂)).Therefore, V (p1) ≤ µ(t(.; sL̂), qk̂)v(p1, qk̂, t(.; sL̂)), and hence r(pl|t(.; sL̂), qk̂) =1!155A.4. Proof of Lemma 2-4 and Proposition 3A.4.3 Proof of Lemma 4Proof. Suppose to the contrary that an owner of asset quality q̂ < qk maygain from posting some contract ŝ. It follows that ŝ < s˜k. Otherwise,no workers will accept such contract because V (pl) ≥ v(pl, qk, t(.; s˜k)) >v(pl, q̂, t(.; ŝ)). By construction, r(t(.; ŝ), q̂) is degenerate at some worker’stype, say pH . The deviation is profitable only if u(q̂, pH , t(.; ŝ)) > U. The con-struction of s˜k ensures that U ≥ u(qk, p1, t(.; s˜k)), and hence pH > p1. Fromthe worker’s IC condition, v(pH , q̂, t(.; ŝ)) > V (pH) ≥ v(pH , qk, t(.; s˜k)). Theinequality (2.16) in Condition Entry-q implies thatV (p1) = v(p1, qk, t(.; s˜k)) < v(p1, q̂, t(.; ŝ)),andV (pH)V (p1)≥ v(pH , qk, t(.; s˜k))v(p1, qk, t(.; s˜k))≥ v(pH , q̂, t(.; ŝ))v(p1, q̂, t(.; ŝ)).The equilibrium condition requires V (p1) ≥ µ(t(.; ŝ), q̂)v(p1, q̂, t(.; ŝ)). Itfollows that V (pH) ≥ µ(t(.; ŝ), q̂)v(pH , q̂, t(.; ŝ)). Our construction of off-equilibrium-path belief implies r(pH |t(.; ŝ), q̂) = 0!A.4.4 Proof for necessity of conditionsIf Condition Sorting-p fails, then there exists some distribution of types(P ,Q) such that the First Best allocation cannot be supported by an equilib-rium.Proof. Suppose Condition Sorting-p fails, there exist p̂H > p̂L, q̂H > q̂L andŝH > ŝL s.t. v(p̂L, q̂L, t(.; ŝL)) ≥ V , u(q̂L, p̂H , t(.; ŝL)) ≥ U and v(p̂H , q̂H , t(.; ŝH)) ≤ v(p̂H , q̂L, t(.; ŝL)),v(p̂L, q̂H , t(.; ŝH)) ≥ v(p̂L, q̂L, t(.; ŝL)), (A.3)156A.4. Proof of Lemma 2-4 and Proposition 3where at least one of the inequalities in (A.3) is strict. Notice that equal-ity cannot hold for both v(p̂L, q̂L, t(.; ŝL)) ≥ V and u(q̂L, p̂H , t(.; ŝL)) ≥U . By continuity, ŝL can be chosen such that v(p̂L, q̂L, t(.; ŝL)) > V ,u(q̂L, p̂L, t(.; ŝL)) 6= U , and both inequalities in (A.3) are strict. Note thatu(q̂H , p̂H , t(.; ŝH)) > U. Define ŝ(q) by v(p̂L, q̂L, t(.; ŝL)) = v(p̂L, q, t(.; ŝ(q)))for all q ≤ q̂L. ŝ(.) is continuous and increases with ŝL. Furthermore,ŝL can be chosen so that v(p, q, t(.; ŝ(q))) 6= V almost everywhere when(p, q)→ (p, q).The first case is that u(q̂L, p̂L, t(.; ŝL)) < U. In this case, ŝL can be chosensuch that u(q̂L, p̂H , t(.; ŝL)) > U . Consider the distribution of types withsupport {p1, p2} = {p̂L, p̂H} and {q1, q2} = {q̂L, q̂H}. Moreover, P (p̂H) +P (p̂L) < Q(q̂H) + Q(q̂L), P (p̂H) > Q(q̂H) and u(q̂L, rFBq (q̂L), t(.; ŝL)) = U.In the First Best allocation, workers of type p̂H will match with assets ofquality q̂H and q̂L. In any equilibrium supporting the First Best, there mustbe two active markets (t(.; ŝL), q̂L) and (t(.; s˜H), q̂H) s.t. v(p̂H , q̂H , t(.; s˜H)) = v(p̂H , q̂L, t(.; ŝL))v(p̂L, q̂L, t(.; ŝL)) ≥ v(p̂L, q̂H , t(.; s˜H)) (A.4)Comparing with (A.3), the first equality requires s˜H < ŝH while the secondequality requires s˜H > ŝH !!!Consider the case v(p̂L, q̂L, t(.; ŝL)) ∈ [E(Y |p̂L, q)−U,E(Y |p̂L, q̂L)−U).It follows that v(p̂L, q̂L, t(.; ŝL)) = E(Y |p̂L, q′)− U for some q′ ∈ (q, q̂L). Inthis case, consider the distribution of types with support {p1, p2} = {p̂L, p̂H}and {q1, q2, q3} = {q′, q̂L, q̂H}. Moreover,Q(q̂H) +Q(q̂L) +Q(q′) > P (p̂H) + P (p̂L) > Q(q̂H) +Q(q̂L)and P (p̂H) > Q(q̂H). In any equilibrium supporting the First Best, theremust be three active markets (t(.; ŝ(q′)), q′), (t(.; ŝL), q̂L) and (t(.; s˜H), q̂H),157A.4. Proof of Lemma 2-4 and Proposition 3where u(q′, p̂L, t(.; ŝ(q′))) = U and v(p̂L, q̂L, t(.; ŝL)) = v(p̂L, q′, t(.; ŝ(q′))).(A.4) must be satisfied and the previous argument applies.The last case is v(p̂L, q̂L, t(.; ŝL)) ∈ (V ,E(Y |p̂L, q) − U ]. Recall thatE(Y |p, q)) = V +U, our choice of ŝL ensures that when (p, q) is sufficientlyclose to (p, q), v(p, q, t(.; ŝ(q))) /∈ [V ,E(Y |p, q) − U ]. Either there exists(p′, q′) < (p̂L, q̂L) satisfying v(p′, q′, t(.; ŝ(q′)) = V and u(q′, p′, t(.; ŝ(q′)) >U or there exists (p′′, q′′) < (p̂L, q̂L) such that v(p′, q′, t(.; ŝ(q′)) > V andu(q′, p′, t(.; ŝ(q′)) < U.For the first possibility, consider the distribution of types with support{p1, p2, p3} = {p′, p̂L, p̂H} and {q1, q2, q3} = {q′, q̂L, q̂H}.P (p̂H) + P (p̂L) + P (p′) > Q(q̂H) +Q(q̂L) +Q(q′)> P (p̂H) + P (p̂L) > Q(q̂H) +Q(q̂L),and P (p̂H) > Q(q̂H). In any equilibrium supporting the First Best, theremust be three active markets (t(.; ŝ(q′)), q′), (t(.; ŝL), q̂L) and (t(.; s˜H), q̂H).(A.4) must be satisfied and the previous argument applies again.For the remaining possibility, consider the distribution of types withsupport {p1, p2, p3} = {p′′, p̂L, p̂H} and {q1, q2, q3} = {q′′, q̂L, q̂H} satisfy-ing Q(q̂H) + Q(q̂L) + Q(q′′) > P (p̂H) + P (p̂L) + P (p′′), P (p̂H) + P (p̂L) >Q(q̂H) + Q(q̂L), P (p̂H) > Q(q̂H) and u(q′′, rFBq (q′′), t(.; ŝ(q′′)) = U. In anyequilibrium supporting the First Best, there must be three active markets(t(.; ŝ(q′′)), q′′), (t(.; ŝL), q̂L) and (t(.; s˜H), q̂H). The previous argument ap-plies again.Suppose Entry-q fails, then there exists some distribution of types (P ,Q)such that the First Best allocation cannot be supported by an equilibrium.Proof. Suppose that Condition Entry-q fails for some q̂H > q̂L, p̂H > p̂L158A.4. Proof of Lemma 2-4 and Proposition 3and ŝH > ŝL satisfyingv(p̂H , q̂L, t(.; ŝL)) > v(p̂H , q̂H , t(.; ŝH)) > v(p̂L, q̂H , t(.; ŝH)) ≥ V ,u(q̂L, p̂H , t(.; ŝL)) > U ≥ u(q̂H , p̂L, t(.; ŝH)), andv(p̂H , q̂H , t(.; ŝH))v(p̂L, q̂H , t(.; ŝH))<v(p̂H , q̂L, t(.; ŝL))v(p̂L, q̂L, t(.; ŝL)).Since u(q̂H , p̂L, t(.; ŝH)+v(p̂L, q̂H , t(.; ŝH)) > U+V , ŝH can be chosen sothat u(q̂H , p̂L, t(.; ŝH)) < U. Consider the distribution of types with support{p1, p2} = {p̂L, p̂H} and {q1, q2} = {q̂L, q̂H}. Moreover, P (p̂H) + P (p̂L) <Q(q̂H) and u(q̂L, rFBq (q̂H), t(.; ŝH)) = U. In the First Best allocation, allworkers will match with assets of quality q̂H and the asset owners of q̂L willtake their outside options. In any equilibrium supporting the First Best,there is a single market (t(.; ŝH), q̂H). Consider the deviation that an assetowner of q̂L posts the contract t(.; ŝL). SinceV (p̂H)V (p̂L)=v(p̂H , q̂H , t(.; ŝH))v(p̂L, q̂H , t(.; ŝH))<v(p̂H , q̂L, t(.; ŝL))v(p̂L, q̂L, t(.; ŝL)),It follows that V (p̂L) > µ(t(.; ŝL), q̂L)v(p̂L, q̂L, t(.; ŝL)) and r(p̂H |t(.; ŝL), q̂L) =1. Such contracts yield the asset owner u(q̂L, p̂H , t(.; ŝL)) > U = U(q̂L)!Suppose Screening-q fails. There exists some distribution of types (P ,Q)such that the First Best allocation cannot be supported by an equilibrium.Proof. Suppose that Condition Screening-q fails for some q̂, p̂H > p̂L andŝH > ŝL s.t. v(p̂L, q̂, t(.; ŝH)) ≥ V , u(q̂, p̂H , t(.; ŝL)) > max{U, u(q̂, p̂L, t(.; ŝH))}andv(p̂H , q̂, t(.; ŝH))v(p̂L, q̂, t(.; ŝH))<v(p̂H , q̂, t(.; ŝL))v(p̂L, q̂, t(.; ŝL)).Define ŝ(q) by v(p̂L, q̂, t(.; ŝH)) = v(p̂L, q, t(.; ŝ(q))) for q ≤ q̂. ŝH can be cho-sen such that v(p̂L, q̂, t(.; ŝH)) > V , u(q̂, p̂L, t(.; ŝH)) 6= U , and v(p, q, ŝ(q)) 6=V almost everywhere when (p, q)→ (p, q).159A.4. Proof of Lemma 2-4 and Proposition 3The first case is u(q̂, p̂L, t(.; ŝH)) < U. Consider the distribution of typeswith support {p1, p2} = {p̂L, p̂H} and {q1} = {q̂}. P (p̂H)+P (p̂L) < Q(q̂) andu(q̂, rFBq (q̂), t(.; ŝH)) = U. In the First Best allocation, workers of both typesmatch with assets of quality q̂. In any equilibrium supporting the First Best,there is only one active market (t(.; ŝH), q̂) with µ(t(.; ŝH), q̂) = 1. Considerthe deviation that an owner of asset quality q̂ posts the contract t(.; ŝL).SinceV (p̂H)V (p̂L)=v(p̂H , q̂, t(.; ŝH))v(p̂L, q̂, t(.; ŝH))<v(p̂H , q̂, t(.; ŝL))v(p̂L, q̂, t(.; ŝL)), (A.5)It follows that V (p̂L) > µ(t(.; ŝL), q̂)v(p̂L, q̂, t(.; ŝL)) and r(p̂H |t(.; ŝL), q̂) =1. By posting the contract t(.; ŝL), the asset owner can earn a payoffu(q̂, p̂H , ŝL) > U = U(q̂).For the case v(p̂L, q̂, t(.; ŝH)) ∈ [E(Y |p̂L, q)−U,E(Y |p̂L, q̂)−U), there ex-ists some q′ ∈ (q, q̂L) such that u(q′, p̂L, t(.; ŝ(q′))) = U and v(p̂L, q̂, t(.; ŝH)) =v(p̂L, q′, t(.; ŝ(q′))). Consider the distribution of types with support {p1, p2} ={p̂L, p̂H} and {q1, q2} = {q′, q̂}. Moreover,Q(q̂) +Q(q′) > P (p̂H) + P (p̂L) > Q(q̂) > P (p̂H).In the First Best allocation, workers of both types match with assets ofquality q̂ and only low type workers match with assets of quality q′. Inany equilibrium supporting the First Best, there must be two active mar-kets (t(.; ŝ(q′)), q′) and (t(.; ŝH), q̂). (A.5) holds in equilibrium, so thatr(p̂H |t(.; ŝL), q̂) = 1. Posting t(.; ŝL) is a profitable deviation for the ownersof asset quality q̂.The remaining case is v(p̂L, q̂, t(.; ŝH)) ∈ (V ,E(Y |p̂L, q) − U ]. Supposethat for some p̂1 < p̂L and q′ ≤ q̂, v(p̂1, q′, t(.; ŝ(q′))) = V butv(p̂H , q̂, t(.; ŝH))v(p̂H , q̂, t(.; ŝL))<v(p̂1, q′, t(.; ŝ(q′)))v(p̂1, q̂, t(.; ŝL)). (A.6)160A.5. Proof of Proposition 4I will restrict the attention to the case q′ < q̂. This also covers the caseq′ = q̂ because of continuity. Consider the distribution of types with support{p1, p2, p3} = {p̂1, p̂L, p̂H} and {q1, q2} = {q′, q̂}. Moreover,P (p̂H) + P (p̂L) + P (p̂1) > Q(q̂) +Q(q′) > P (p̂H) + P (p̂L) > Q(q̂) > P (p̂H).In any equilibrium supporting the First Best, there must be two activemarkets (t(.; ŝ(q′)), q′) and (t(.; ŝH), q̂). By construction, v(p̂L, q̂, t(.; ŝH)) =v(p̂L, q′, t(.; ŝ(q′))). By construction,V (p̂H)V (p̂1)=v(p̂H , q̂, t(.; ŝH))v(p̂1, q′, t(.; ŝ(q′)))<v(p̂H , q̂, t(.; ŝL))v(p̂1, q̂, t(.; ŝL)).Together with (A.5), we obtain r(p̂H |t(.; ŝL), q̂) = 1. Posting t(.; ŝL) is aprofitable deviation for the owners of asset quality q̂.A similar construction applies to the case u(q′, p̂1, t(.; ŝ(q′))) ≤ U and(A.6) holds. The only change is that Q(q̂)+Q(q′) > P (p̂H)+P (p̂L)+P (p̂1),P (p̂H) + P (p̂L) > Q(q̂) > P (p̂H) and u(q′, rFBq (q′), t(.; ŝ(q′))) = U.A.5 Proof of Proposition 4It is trivial that Condition Global ID gives rise to Condition Sorting-p.The proof here is to establish the followingCondition (ID-q). For any pH > pL, qH ≥ qL and sH > sL satisfy-ing v(pL, qH , t(.; sH)) ≥ V , u(qL, pH , t(.; sL)) > U and v(pH , qL, t(.; sL)) >v(pH , qH , t(.; sH)). Thenv(pH , qH , t(.; sH))v(pL, qH , t(.; sH))≥ v(pH , qL, t(.; sL))v(pL, qL, t(.; sL)). (A.7)Note that when qH = qL = q̂, v(pH , q̂, t(.; sL)) > v(pH , q̂, t(.; sH)) isalways true. Condition ID-q implies both Condition Screening-q and Entry-161A.5. Proof of Proposition 4q. Condition strong Screening-q holds if (A.7) always holds with strictinequality.Proof. Fix the pairs {pH , pL} and {(qH , sH), (qL, sL)}, for any cH ≥ cL ≥ 0,denote∆product(v; cH , cL) = [v(pH , qH , t(.; sH)) + cL][v(pL, qL, t(.; sL)) + cH ]−[v(pH , qL, t(.; sL)) + cL][v(pL, qH , t(.; sH) + cH ]∆sum(v) = v(pL, qL, t(.; sL)) + v(pH , qH , t(.; sH))−v(pH , qL, t(.; sL))− v(pL, qH , t(.; sH).Notice that∆product(v; cH , cL)−∆product(v; 0, 0)=cH∆sum(v)− (cH − cL)[v(pL, qL, t(.; sL))− v(pL, qH , t(.; sH)].Recall the pre-condition v(pL, qH , t(.; sH)) ≥ V > 0 and v(pH , qL, t(.; sL)) >v(pH , qH , t(.; sH)) in Condition ID-q. Global ID immediately implies thatv(pL, qL, t(.; sL)) > v(pL, qH , t(.; sH)). The first case is that ∆sum(v) ≥ 0,then∆product(v; 0, 0)≥∆product(v; cH , cL) + (cH − cL)[v(pL, qL, t(.; sL))− v(pL, qH , t(.; sH)]−[v(pL, qH , t(.; sH)) + cH ]∆sum(v)=∆product(v;−v(pL, qH , t(.; sH)),−v(pL, qH , t(.; sH)))=[v(pH , qH , t(.; sH))− v(pL, qH , t(.; sH))][v(pL, qL, t(.; sL))− v(pL, qH , t(.; sH))]>0For the case that ∆sum(v) < 0, ∆product(v; 0, 0) ≥ 0 if ∆product(v; cH , cL) ≥ 0and ∆product(v; 0, 0) = 0 if and only if cH = cL = 0.162A.6. Proof of Lemma 5Putting together, if ∆product(v; cH , cL) ≥ 0, then ∆product(v; 0, 0) ≥ 0regardless of the sign for ∆sum(v). Hence, Condition Global ID implies that∆product(v; 0, 0) ≥ 0. Furthermore, ∆product(v; 0, 0) > 0 if ∆sum(v) ≥ 0 or∆product(v; cH , cL) ≥ 0 for some cH > 0.A.6 Proof of Lemma 5Proof. Suppose not, (t(.; s′), q′) is an active market with µ(t(.; s′), q′) < 1.Let pH be the highest type in Ip(t(.; s′), q′). Pick a sufficiently small  > 0such that v(pH , q′, t(.; s′+) > µ(t(.; s′), q′)v(pH , q′, t(.; s′)). µ(t(.; s′+), q′) <1 and satisfiesV (pH) = µ(t(.; s′), q′)v(pH , q′, t(.; s′)) ≥ µ(t(.; s′ + ), q′)v(pH , q′, t(.; s′ + )For all p′ < pH , Condition Strong Screening-q and the worker’s IC condition,V (p′) ≥ µ(t(.; s′), q′)v(p′, q′, t(.; s′)) > µ(t(.; s′ + ), q′)v(p′, q′, t(.; s′ + .)Hence, r(p′|t(.; s′+ ), q′) = 0 only if p′ < pH . Posting this contract gives theasset owner a payoff u(q′, r(t(.; s′+), q′), t(.; s′+)) ≥ u(q′, pH , t(.; s′+)) >u(q′, pH , t(.; s′)) ≥ U(q˜)!A.7 Proof of Lemma 6Proof. Fix an equilibrium {U, V, ψ,Ψ,W, r, µ}. From Lemma 5, µ(t(.; s′), qk) ≥1 if (t(.; s′), qk) ∈ Ψ. A worker of pl obtains v(pl, qk, t(.; s′)) from partic-ipating in this active market. It follows that for any s′′ > s′, V (pl) ≥v(pl, qk, t(.; s′)) > v(pl, qk, t(.; s′′)) for all pl, so that µ(t(.; s′′), qk) = 0.(t(.; s′), qk) must be only active market for asset quality qk. This showsthat owners of the same quality , if participating, post the same contract.163A.7. Proof of Lemma 6For any equilibrium, the set of active markets can be written as Ψ ={(t(.; sk), qk)}k≥k̂. The equilibrium payoff is V (pl) = max{V , v(pl, qk, t(.; s)) :(t(.; s), qk) ∈ Ψ}, which is increasing in p. It also follows that worker’s par-ticipation is monotonic.Now consider an active market (t(.; s′), qk) and pl is the highest type inΩp((t(.; s′), qk). Then V (pl) = v(pl, qk, t(.; s′)) and V (pL) ≥ v(pL, qk, t(.; s′))for any pL < pk. For any qH > qk, there exists s′′ > s′ satisfying v(pl, qH , t(.; s′′))= v(pl, qk, t(.; s′)). Condition Strict Sorting-p implies that v(pL, qH , t(.; s′′)) <v(pL, qk, t(.; s′)) ≤ V (pL) for all pL < pk. For sufficiently small  > 0,V (pl) < v(pl, qH , t(.; s′′ − ) and V (pL) > v(pL, qH , t(.; s′′ − )). Hence,µ(t(.; s′′ − ), qH) < 1 and pL /∈ Ωp((t(.; s′′ − ), qH) for any pL < pl. Notethatu(qH , pk, t(.; s′′)) = [E(Y |pl, qH)− E(Y |pl, qk)] + u(qk, pl, t(.; s′))> u(qk, pl, t(.; s′)) ≥ U(qk) ≥ UFor sufficiently small  > 0, posting t(.; s′′ − ) provides the asset owner ofqH a payoff u(qH , pk, t(.; s′′ − )) > U(qk). This establishes that U(qH) >U(qk) ≥ U, so the asset owner’s participation is monotonic and their equi-librium payoff is increasing in q.Consider two active markets (t(.; sH), qH) and (t(.; sL), qL) where qH >qL. µ(t(.; sH), qH) ≥ 1 and µ(t(.; sL), qL) ≥ 1 requires sH > sL. Now sup-pose that there exists pH > pL such that pL ∈ Ωp(t(.; sH), qH) and pH ∈Ωp(t(.; sL), qL). IC condition for workers requires both v(pH , qL, t(.; sL)) ≥v(pH , qH , t(.; sH)) and v(pL, qL, t(.; sL)) ≤ v(pL, qH , t(.; sH)). This contra-dicts Condition Strict Sorting-p. Therefore, if p ∈ Ωp(t(.; sH), qH) andp′ ∈ Ωp(t(.; sL), qL), then p ≥ p′.164A.8. Proof of Remark 6A.8 Proof of Remark 6First, consider the case that [y − t(y; s)] is SPM, which is the sameas [t(y; sL) − t(y; sH)] is weakly increasing in y. Since Y |(pH , q) F.O.S.D.Y |(pL, q),v(pH , q, t(.; sH)) + v(pL, q, t(.; sL))− v(pH , q, t(.; sL))− v(pL, q, t(.; sH))=∫ yy[t(y; sL)− t(y; sH)]d[F (y|pH , q)− F (y|pL, q)] ≥ 0.Second, suppose that [y − t(y; s) + c] is non-negative and log-SPM. To-gether with Assumption (MLRP), [y−t(y; s)+c]f(y|p, q) is also non-negativeand log-SPM in (y, s, p). It is well known that these two properties are jointlypreserved under integration w.r.t. y, so that[v(pH , q, t(.; sH)) + c][v(pL, q, t(.; sL)) + c]≥[v(pH , q, t(.; sL)) + c][v(pL, q, t(.; sH)) + c].When either of these conditions hold, the inequalities in Condition GlobalID hold for qH = qL. This is a special case of the proof for Lemma 4 andhence,v(pH , q, t(.; sH))v(pL, q, t(.; sH))≥ v(pH , q, t(.; sL))v(pL, q, t(.; sL))> 1.This ensures the First Best allocation is supported by the candidate equi-librium with a single active market (t(.; s˜1), q1) in the case of homogeneousassets.165A.9. Proof of Lemma 7A.9 Proof of Lemma 7Proof. Under Assumption (C) and Condition Survival-SPM,v(pH , qH , t(.; s)) + v(pL, qL, t(.; s))− v(pH , qL, t(.; s))− v(pL, qH , t(.; s))= −∫ yy[F (y|pH , qH) + F (ypL, qL)− F (y|pH , qL)− F (y|pL, qH)]d[y − t(y; s)]≥ 0.Recall from the proof of Remark 6, SPM of [y − t(y; s)] impliesv(pH , q, t(.; sH)) + v(pL, q, t(.; sL)) ≥ v(pH , q, t(.; sL)) + v(pL, q, t(.; sH)).Combining the two inequalities together yields Global ID.A.10 Proof of Lemma 8Proof. Under Assumption (C), [y − t(y; s) + c] is absolutely continuous,weakly increasing and strictly increasing for some interval of [q, q].81 Byintegration by parts,v(p, q, t(.; s)) + c = [y − t(y; s) + c] +∫ yy[1− F (y|p, q)]d[y − t(y; s) + c].The log-SPM of the survival function in Condition Survival-SPM is pre-served under integration. v(p, q, t(.; s)) + c then inherits log-SPM from∫ yy [1− F (y|p, q)]d[y − t(y; s) + c], so that[v(pH , qH , t(.; s)) + c][v(pL, qL, t(.; s)) + c]≥[v(pH , qL, t(.; s)) + c][v(pL, qH , t(.; s)) + c].81For any interval (yL, yH), yH − yL ≥ [yH − t(yH ; s)] − [yL − t(yL; s)] ≥ 0. Hence,y − t(y; s) + c is absolutely continuous in y.166A.11. Proof of Remark 7Recall from the proof of Remark 6, log-SPM of [y− t(y; s) + c] and Assump-tion (MLRP) jointly imply[v(pH , q, t(.; sH)) + c][v(pL, q, t(.; sL)) + c]≥[v(pH , q, t(.; sL)) + c][v(pL, q, t(.; sH)) + c].The two inequalities together yield Global ID.A.11 Proof of Remark 7Proof. To prove this remark, I will make use of a result based on Theorem2.1 in Karlin and Rinott (1980). Fix the pairs of {pH , pL} and {qH , qL}. Tosaves on space, denote f(y|pH , qH), f(y|pL, qL), f(y|pL, qH) and f(y|pH , qL)by fHH(y), fLL(y), fLH(y) and fHL(y) respectively. For any y, y′ ∈ Ωy, wehavefHH(y ∨ y′)fLL(y ∧ y′) ≥ fLH(y)fHL(y′). (A.8)Since∫f(y|p, q)dy = 1, we have the identity0 =∫fHH(y)dy∫fLL(z)dz −∫fLH(y)dy∫fHL(z)dz=∫{yH>yL}{fHH(yH)fLL(yL) + fHH(yL)fLL(yH)− fLH(yH)fHL(yL)− fLH(yL)fHL(yH)}d(yH , yL).Fix a pair of (yH , yL). Denote [fHH(yH)fLL(yL)− fLH(yH)fHL(yL)] by A,[fHH(yH)fLL(yL)− fLH(yL)fHL(yH)] by B andC = fHH(yH)fLL(yL)fHH(yL)fLL(yH)−fLH(yH)fHL(yL)fLH(yL)fHL(yH).Applying the inequality (A.8), we obtain that A, B and C are all non-negative. Since fHH(yH)fLL(yL) is positive throughout the support, the167A.12. Proof of Proposition 6integrand can be expressed as1fHH(yH)fLL(yL)[AB + C)] ≥ 0.It follows that AB = C = 0 for any pair of (yH , yL).82 The equality C = 0and the inequality (A.8) imply that for y ∈ {yH , yL},fHH(y)fLL(y) = fLH(y)fHL(y).Hence, f(y|p, q) is pairwise log-modular in (p, q). Assumption (MLRP),together with the above equality imply that A > 0, and hence B = 0. Theequality B = 0 again impliesfHH(yH)fHL(yL) = fHH(yL)fHL(yH).Hence, f(y|p, q) is pairwise log-modular in (y, q).A.12 Proof of Proposition 6{U, V, ψ,Ψ,W, r, µ} is an equilibrium under the contract space Φt if andonly if there exists a corresponding equilibrium {U, V, ψ,Ψ,W ′, r′, µ′} underthe contract space Φt ∪ {ts} with the same equilibrium payoff {U, V }, thesame active markets Ψ ⊆ Φt and the same distribution of participants inevery active market, {µ|Ψ, r|Ψ,W |Ψ} = {µ′|Ψ, r′|Ψ,W ′|Ψ}.82Alternatively, Condition Global ID can be obtained by replacing the inequality (A.8)with[y ∨ y′ − t(y ∨ y′; sH) + c][y ∧ y′ − t(y ∧ y′; sL) + c]f(y ∨ y′|pH , qH)f(y ∧ y′|pL, qL)≥ [y − t(y; sH) + c][y′ − t(y′; sL) + c]f(y|pL, qH)f(y′|pH , qL).The same equality condition still applies when sH → sL.168A.12. Proof of Proposition 6Proof. The if part is trivial because Ψ ⊆ Φt. For only if part, I now constructa corresponding equilibrium.For the markets (ts, q) ∈ {ts} × {qk}Kk=1,µ(ts, q) = sup{µ′ ∈ [0,∞] : V (p) ≥ η(µ′)v(p, q, ts)}.If µ(ts, q) <∞, then r(ts, q) is degenerate at the typemin{p ∈ {pl}Ll=1 : V (p) = µ(ts, q)v(p, q, ts)}.It is sufficient to show that an asset owner cannot gain from a posting thecontract ts. Suppose not, an owner of asset quality qk′ profits from postingts. By construction, r(ts, q) is degenerate at some pl′ , where v(pl′ , qk′ , ts) ≥V (pl′) and u(qk′ , pl′ , ts) > U(qk′). s′ can be chosen s.t. v(pl′ , qk′ , t(.; s′))is slightly above v(pl′ , qk′ , ts) and hence µ(t(.; s′), qk′) < min{µ(ts, qk′), 1}.Since ts is steeper than t(.; s′), r(pl|t(.; s′), qk′) = 0 if pl < pl′ .83 Thiscontradicts with the equilibrium condition that the asset owner cannot gainfrom posting t(.; s′) asU(qk′) <η(µ(ts, qk′))µ(ts, qk′)u(qk′ , pl′ , ts) < u(qk′ , r(t(.; s′), qk′), t(.; s′)).For every equilibrium {U, V, ψ,Ψ,W, r, µ} under the contract space Φt ∪{ts}, there also exists an equilibrium {U, V, ψ′,Ψ′,W ′, r′, µ′} under the samecontract space, for which the asset owners only contracts in Φt and support-ing the same allocation,∫Ωt×{qk} r(pl|t, qk)dW =∫Ωt×{qk} r′(pl|t, qk)dW ′ and∫Ωt×{qk} µ(t, qk)dW =∫Ωt×{qk} µ′(t, qk)dW ′ for all pl and qk.83This result does not depend on the assumption of finite distribution of types. s′ canbe chosen such that r(t(.; s′), q′) contains types in some local neighborhood of p′, whichsatisfies u(q′, p, t(.; s′)) > U(q′)169A.12. Proof of Proposition 6Proof. I will first establish some properties of an equilibrium {U, V, ψ,Ψ,W, r, µ}with ts ∈ ψ(qk) for some qk.The first property is that if ts ∈ ψ(qk′), then Ωp(ts, qk′) is a singleton.Suppose to the contrary that Ωp(ts, qk′) contains more than one type. Let pl′be the highest type in Ωp(ts, qk′) and so u(qk′ , pl′ , ts) > u(qk′ , r(ts, qk′), ts).There exists s˜ such that E(t(Y ; s˜)|pl′ , qk′) = E(ts(Y )|pl′ , qk′). It follows thatfor all p < pl′ , v(p, qk′ , ts) > v(p, qk′ , t(.; s˜)) because ts is steeper than St.For sufficiently small  > 0,V (pl′) ≤ η(µ(ts, qk′))v(pl′ , qk′ , ts) < η(µ(ts, qk′))v(pl′ , qk′ , t(.; s˜− )),but V (pl) > η(µ(ts, qk′))v(pl, qk′ , t(.; s˜− )) for all pl < pl′ in the support ofP .84 The last inequality makes use of the incentive compatibility conditionfor workers with pl < pl′ . Hence, µ(t(.; s˜ − ), qk′) < min{µ(ts, qk′), 1} andr(pl|t(.; s˜− ), qk′) = 0 if pl < pl′ . Hence,U(qk′) <η(µ(ts, qk′))µ(ts, qk′)u(qk′ , r(ts, qk′), ts) < u(qk′ , r(t(.; s˜− ), qk′), t(.; s˜− )).and an asset owner will gain from posting a contract t(.; s˜− ).The second property is that µ(ts, qk′) = 1. Now let Ωp(ts, qk′) = {pl′}.Thecase for µ(ts, qk′) > 1 follows a similar argument. For sufficiently small > 0, V (pl′) < v(pl′ , qk′ , t(.; s˜ − )) but V (pl) > v(pl, qk′ , t(.; s˜ − )) forall pl < pl′ . Hence, µ(t(.; s˜ − ), qk′) < 1 and r(p|t(.; s˜ − ), qk′) > 0 onlyif p ≥ pl′ . an asset owner will gain from posting t(.; s˜ − ) for sufficientlysmall . Now consider µ(ts, qk′) < 1. If a worker with type p is indifferentbetween participating in (ts, qk′) and a market (t′, qk′) with some tightnessratio µ′, then define µ(p|t′, qk′) = µ′. Otherwise, µ(p|t′, qk′) =∞. Note thatµ(pl′ |t(.; s˜), qk′) = µ(ts, qk′). For sufficiently small  > 0, µ(pl|t(.; s˜), qk′) >84This result does not depend on the assumption of finite distribution of types.170A.13. Proof of Proposition 5µ(pl′ |t(.; s˜+), qk′) > µ(ts, qk′) whenever pl < pl′ . Hence, µ(t(.; s˜+), qk′) < 1and r(pl|t(.; s˜ + ), qk′) > 0 only if pl ≥ pl′ . an asset owner will gain fromposting t(.; s˜+ ) for sufficiently small .It follows that U(qk′) = u(qk′ , pl′ , t(.; s˜)) and V (pl′) = v(pl′ , qk′ , t(.; s˜)).V (pl) > v(pl, qk′ , t(.; s˜)) for all pl 6= pl′ . This holds for pl < pl′ becauseV (pl) ≥ v(pl, qk′ , ts) and ts is steeper than t(.; s˜). The part for pl > pl′ comesfrom the fact that an owner of asset quality qk′ cannot profit from the de-viation of posting t(.; s˜). Note that if ψ(qk′) also contains t(.; s˜), it must bethat Ωp(t(.; s˜), qk′) = {pl′} and µ(t(.; s˜), qk′) = 1. Otherwise, the asset own-ers will not be indifferent between the two markets. Putting all together, allequilibrium conditions will still be met if those asset owners of qk′ posting tsand the workers participating in (ts, qk′) all switch to the market (t(.; s˜), qk′).In the new equilibrium, Ωp(t(.; s˜), qk′) = {pl′} and µ(t(.; s˜), qk′) = 1. It istrivial that the resulting allocation is unchanged.A.13 Proof of Proposition 5Consider two ordered set of securities Sst and Sft , where Sst is steeperthan Sft . Fix a distribution of types, the contracts ts(.; s) and tf (.; s) areindexed in a manner that same set of contract terms {s˜k}k≥k are posted inthe two candidate equilibriums under the contract space Sst and Sft .Proof. First, notice that v(pl(k), qk, ts(.; s˜k)) ≤ v(pl(k), qk, tf (.; s˜k)). If work-ers are on the long side, then v(pl, q1, ts(.; s˜1)) = v(pl, q1, tf (.; s˜1)) = V .The above inequality holds because tf (.; s˜1) is flatter. If workers are onthe short side, then u(qk, rFBq (qk), ts(.; s˜k)) = u(qk, rFBq (qk), tf (.; s˜k)) = U.Suppose, to the contrary, that v(pl(k), qk, ts(.; s˜k)) > v(pl(k), qk, tf (.; s˜k)) =171A.13. Proof of Proposition 5v(pl(k), qk, ts(.; s′)). Then v(pl, qk, tf (.; s˜k)) ≥ v(pl, qk, ts(.; s′)) whenever l(k) ≥l ≥ 1. By ex-post budget balance,u(qk, rFBq (qk), ts(.; s˜k)) > u(qk, rFBq (qk), ts(.; s′)) ≥ u(qk, rFBq (qk), tf (.; s˜k))!!!The indifference condition in (2.10) implies thatv(pl(k), qk+1, ts(.; s˜k+1)) = v(pl(k), qk, ts(.; s˜k))≤ v(pl(k), qk, tf (.; s˜k)) = v(pl(k), qk+1, tf (.; s˜k+1)).Hence, there exists v(pl(k), qk+1, ts(.; s˜k+1)) = v(pl(k), qk+1, tf (.; s′′)) for somes′′ ≥ s˜k+1. Now consider any higher type l ≥ l(k), in particular l(k + 1),v(pl, qk+1, ts(.; s˜k+1)) ≤ v(pl, qk+1, tf (.; s′′)) ≤ v(pl, qk+1, tf (.; s˜k+1)).Hence, u(qk+1, rFBq (qk+1), ts(.; s˜k+1)) ≥ u(qk+1, rFBq (qk+1), tf (.; s˜k+1)).By induction, it follows that under the contract space Sst , U(qk) is higherwhereas V (pl) is lower for pl ≥ pl(k).172Appendix BAppendix for chapter 3B.1 Proof of Remark 8Eeckhout and Kircher (2010) show that the boundary value problem forthe Second Best allocation admits a solution.85 Fix (pSB, qSB, rSB, λSB, vSB, uSB),I will first show that it satisfies the inequality (3.12), and use the inequalityto establish uniqueness of the Second Best allocation. DefineÛ(p, q) = maxλ≥0{δ(λ)y(p, q)− λvSB(p)}.Since δ(λ) is strictly concave, the unique maximizer, denoted by λ̂(p, q), isdetermined by the FOC,δ′(λ̂(p, q))y(p, q) = vSB(p).We first consider p ≥ pSB. From the condition (3.8),vSB(p) = δ′(λSB(κSB(p)))y(p, κSB(p)).Therefore, λ̂(p, q) > (<)λSB(κSB(p)) if q > (<)κSB(p). By envelope theo-85Though Eeckhout and Kircher (2010) assume the values of outside options to be zero,their proof is readily extended to cover the case with positive outside options.173B.1. Proof of Remark 8rem,∂∂pÛ(p, q)= δ(λ̂(p, q))∂∂py(p, q)− λ̂(p, q) ∂∂pvSB(p)= δ(λ̂(p, q))∂∂py(p, q)− λ̂(p, q)η(λSB(κSB(p))) ∂∂py(p, κSB(p))= λ̂(p, q)vSB(p)[η(λ̂(p, q))δ′(λ̂(p, q))∂ ln y(p, q)∂p− η(λSB(κSB(p)))δ′(λSB(κSB(p)))∂ ln y(p, κSB(p))∂p]The second inequality obtained by substituting the condition (3.9). Un-der Assumption (Y) and (M), ∂∂p Û(p, q) > (<)0 if q > (<)κSB(p). Forp < pSB, vSB(p) = V = vSB(pSB), so Û(p, q) < maxλ≥0{δ(λ)y(pSB, q) −λvSB(pSB)} = Û(pSB, q).Putting together, for q ≥ qSBand any p ∈ [0, 1]uSB(q) = δ(λSB(q))y(rSB(q), q)− λSB(q)vSB(rSB(q))= Û(rSB(q), q) = maxλ≥0{δ(λ)y(p, q)− λvSB(p)}For q < qSB, the boundary condition requires thatuSB(q) = U = maxλ≥0{δ(λ)y(p, qSB)−λvSB(p)} > maxλ≥0{δ(λ)y(p, q)−λvSB(p)}.This establishes the inequality (3.12).Let supp(L) denote the support of measure L. The total surplus for(K,L) is given by∫supp(L)η(dLqsdK)y(p, q)dL+ [F (1)− Lp(1)]V + [G(1)−Kq(1)]U≤∫supp(L)dKdLqsuSB(q) + vSB(p)dL+ [F (1)− Lp(1)]V + [G(1)−Kq(1)]U≤∫uSB(q)dG(q) +∫vSB(p)dF (p)=∫ 1qδ(λSB(q))y(rSB(q), q)dG(q) + F (p)V +G(q)U174B.2. Proof of Proposition 7The first inequality is due to the inequality (3.12) and the second inequalitystems from the boundary conditions for the Second Best allocation. Theabove inequality holds with equality if and only if (K,L) features PAMwith (pSB, qSB, κSB) anddLqsdK = λSB(q) almost everywhere in the supportof L.B.2 Proof of Proposition 7In this proof, I first show the properties listed in Proposition 7. I thenproceed to show that the boundary value problem in system (3.16) admitsa unique solution. Note that (r˜, λ˜, v˜) in any solution must be continu-ously differentiable and strictly increasing. In particular, ∂ ln v˜(p)∂p∣∣∣p=r˜(q)=∂ ln y(p,q)∂p∣∣∣(p,q=(r˜(q),q), so the gain from matching with a better asset must beoffset by a reduction in δ′(λ˜(q)). Hence, λ˜ is strictly increasing.Characterization of the equilibriaAny equilibrium satisfies the properties in Proposition 7.“Only if”First fix an equilibrium (K,L). The assumption in (3.1) ensures thatthe set of active markets is non-empty. Suppose not, consider the inactivemarket (1, s′) where s′ and λ′ satisfy V = δ′(λ′)y(1, 1) = η(λ′)(1−s′)y(1, 1).Since η(λ′)(1−s′)y(p, 1) < V if p < 1, only the best workers will be attractedto this inactive market, and the resulting queue length is λ′. The deviatingpayoff for an owner of asset quality q is [δ(λ′) − δ′(λ′)λ′]y(1, 1) > U ! Sinceparticipation is costly, every active market must have a positive finite queuelength, and s ∈ (0, 1). Furthermore, the participation on the workers sidemust be monotonic.Step 1: All equilibria feature PAM175B.2. Proof of Proposition 7Suppose not, then there must exist {(qH , s1), (qL, s0)} ∈ Ψ, where qH >qL, and pH > pL where pL and pH are in the support of R(qH , s1) andR(qL, s0) respectively. The workers’ acceptance decisions are optimal onlyifη(Λ(qL, s0))(1− s0)y(pH , qL) ≥ η(Λ(qH , s1))(1− s1)y(pH , qH), andη(Λ(qH , s1))(1− s1)y(pL, qH) ≥ η(Λ(qL, s0))(1− s0)y(pL, qL),This implies y(pH , qL)y(pL, qH) ≥ y(pH , qH)y(pL, qL), which contradictsstrict log-SPM of y(p, q)!Since the equilibrium allocation features PAM, the threshold types (p, q)and κ(p) are well-defined. κ is strictly increasing because the distribution oftypes is atomless and every active market must have a positive finite queuelength.Step 2: U(q) is strictly increasing for q ≥ qSuppose (qL, s0) ∈ Ψ and qL = κ(pL). Fix q̂ > qL. Consider the inactivemarket (q̂, ŝ), where ŝ is given by(1− ŝ)y(pL, q̂) = (1− s0)y(pL, qL).Λ(q̂, ŝ) ≥ Λ(qL, s0) because V (pL) ≥ η(Λ(q̂, ŝ))(1 − ŝ)y(pL, q̂). For allp < pL,V (p) ≥ η(Λ(qL, s0))(1− s0)y(p, qL) > η(Λ(q̂, ŝ))(1− ŝ)y(p, q̂).The strict inequality follows from log-SPM of y(p, q). It follows that thesupport of R(q̂, ŝ) contains no type below pL. An owner of asset q̂ can ensureherself a payoff ofδ(Λ(q̂, ŝ))ŝ∫y(p, q̂)dR(q̂, ŝ) > U(qL).176B.2. Proof of Proposition 7Therefore, the participation of the asset side is monotonic, and the func-tion r(q) is well-defined. By definition,V (p) = η(Λ(q, s))(1− s)y(p, q) if q = κ(p) and (q, s) ∈ Ψ.Step 3: For any p ∈ [0, 1] and (q, s) ∈ Ψ,V (p) > η(Λ(q, s))(1− s)y(p, q) if q 6= κ(p).Consider two active markets (qH , s1) and (qL, s0), where qH > qL. Sup-pose a worker of type r(qH) > 0 is indifferent between these two markets.His acceptance decision is optimal only ifη(Λ(qL, s0))(1− s0)y(r(qH), qL) = η(Λ(qH , s1))(1− s1)y(r(qH), qH)≥ η(Λ(q, s′))(1− s′)y(r(qH), q)for all (q, s′) ∈ Ψ where q ∈ (qL, qH). Strict log-SPM of y(p, q) implies thatfor any p < r(qH),η(Λ(qL, s0))(1− s0)η(Λ(qH , s1))(1− s1) =y(r(qH), qH)y(r(qH), qL)>y(p, qH)y(p, qL)andη(Λ(qL, s0))(1− s0)η(Λ(q′, s′))(1− s′) ≥y(r(qH), q′)y(r(qH), qL)>y(p, q′)y(p, qL),for all (q′, s′) ∈ Ψ where q′ ∈ (qL, qH). PAM then implies that Λ(q′, s′) = 0if (q′, s′) ∈ Ψ and q′ ∈ (qL, qH). Hence, U(q′) = U ≤ U(qL), contradictingour previous claim!Suppose 1 > qL = k(pL) and (qL, s0) ∈ Ψ. A symmetric argument rulesout the case that a worker of type pL is indifferent between (qL, s0) andanother active market (qH , s1) where qH > qL.Step 4:Characterize active marketsΨ.177B.2. Proof of Proposition 7Lemma 13. Suppose (q, s′) ∈ Ψ and for any p ∈ [0, 1],V (p) ≥ η(Λ(q, s′))(1− s′)y(p, q),with equality if and only if q = κ(p). Then for any s ∈ [0, 1), R(q, s) isdegenerate at r(q) if Λ(q, s) > 0. Furthermore, an owner of asset quality qhas no profitable deviations if and only if Λ(q, s′) satisfiesδ′(Λ(q, s′)) = η(Λ(q, s′))(1− s′). (B.1)Proof. For s ∈ [0, 1) and p 6= r(q),V (p)V (r(q))>y(p, q)y(r(q), q)=η(Λ(q, s))(1− s)y(p, q)η(Λ(q, s))(1− s)y(r(q), q) .Suppose Λ(q, s) > 0, then V (p) = η(Λ(q, s))(1 − s)y(p, q) if and only ifp = r(q), and hence R(q, s) is degenerate at r(q). In this case, Λ(q, s) isdetermined byV (r(q)) = η(Λ(q, s′))(1− s′)y(r(q), q) = η(Λ(q, s))(1− s)y(r(q), q).An asset owner has no profitable deviations if and only ifU(q) ≥ δ(Λ(q, s))sy(r(q), q),with equality at s = s′. This can further simplified asΛ(q, s′) ∈ arg maxλ∈[0,∞]δ(λ)− λη(Λ(q, s′))(1− s′).Since δ(λ) is strictly concave and Λ(q, s′) ∈ (0,∞), the above holds if andonly if the equality (B.1) holds.Step 5: Establish the boundary value problem (3.16).There exists a function λ˜ : [0, 1]→ (0,∞) such thatΨ = {(q, s) : q ∈ [q, 1], s = 1− d ln δd lnλ∣∣∣∣λ=λ˜(q)},178B.2. Proof of Proposition 7and Λ(q, s) = λ˜(q) for (q, s) ∈ Ψ. (λ˜, V, r) is continuously differentiable andsatisfies the differential equation system in (3.16) along with q.Recall that the participation is monotonic on both sides, so there is atleast one active market (q′, s′) for any q′ ∈ [q, 1]. Consider the workers ofp ≥ p. Substitute the equality (B.1), the expression of their equilibriumpayoff can be expressed asV (r(q′)) = δ′(Λ(q′, s′))y(r(q′), q′),and incentive compatibility requiresV (r(q′)) = max(q,s)∈Ψ{δ′(Λ(q, s))y(r(q′), q)}.The envelope theorem implies that V is continuously differentiable (C1).Since δ′(λ) and y(p, q) are C1, there must exist a C1 function λ : [0, 1] →(0,∞) such that∂V (p)∂p∣∣∣∣p=r(q)= δ′(λ(q))∂y(p, q)∂p∣∣∣∣(p,q)=(r(q),q).This also establishes that for each asset quality q ≥ q, there is exactly oneactive market (q′, s′) with s′ = 1− d ln δd lnλ∣∣λ=λFurthermore, Λ(q′, s′) = λ(q′).Given full participation for p ∈ [p, 1] and q ∈ [q, 1], λ and r must satisfythe law of motion (3.7).It remains to show the boundary conditions for the threshold types.Suppose p > 0. Since y(p, q) is strictly increasing in p, the workers p < p alltake their outside option only if V (p) = V . Now suppose q > 0, (q, s) ∈ Ψand U(q) > U. If p = 0, an owner of q′ slightly below q can secure a payoffδ(λ(q))s′y(0, q′) > U by posting a share s′ satisfying η(λ(q))(1−s′)y(0, q′) =V (0). We turn to the case p > 0 so that V (p) = η(λ(q))(1− s)y(p, q) = V .179B.2. Proof of Proposition 7By continuity, for q′ slightly below q, there must exist s′ < s and p′ < psatisfying bothδ(λ(q))s′y(p′, q′) > Uη(λ(q))(1− s′)y(p′, q′) = VIt follows that Λ(q′, s′) ≥ λ(q), and η(Λ(q′, s′))(1 − s′)y(p, q′) < V for anyp < p′. Hence, the expected payoff for an asset owner to participate in (q′, s′)must be above U , rendering the deviation profitable. Therefore, U(q) = Uif q > 0.The preceding analysis verifies all properties in Proposition 7.“If”Fix a solution (p˜, q˜, r˜, λ˜, v˜) to the boundary value problem in system(3.16). One can recover a unique candidate equilibrium (K˜, L˜) satisfyingthe properties in listed Proposition 7. Let κ˜ denote the inverse of r˜. Defines˜(q) = 1− d ln δd lnλ∣∣λ=λ˜(q). Since λ˜(q) is continuous and strictly increasing, s˜(q)is also continuous and increasing in q under Assumption (M). K˜(q′, s′) = 0 ifq′ ≤ q˜ or s′ ≤ s˜(q˜). Otherwise, K˜(q′, s′) = G(sup{q ≤ q′ : s˜(q) ≤ s′})−G(q˜).L˜(p′, q′, s′) = F (sup{p ≤ p′ : κ˜(p) ≤ q′, s˜(κ˜(p)) ≤ s′})− F (p˜) if p > p˜, q′ > q˜and s′ > s˜(q˜). Otherwise, L˜(p′, q′, s′) = 0.I first verify that the workers’ acceptance decision is optimal. V (p) =v˜(p) > V for p > p˜. Combining the conditions (3.8) and (3.14),∂ ln v˜(p)∂p=∂ ln y(p, q)∂p∣∣∣∣q=κ˜(p), p ≥ p˜.180B.2. Proof of Proposition 7Consider any p0 ≥ p˜ and p1 6= p0,lnV (p1)− ln δ′(κ˜(p0))y(p1, κ˜(p0))= [lnV (p1)− ln v˜(p0)]− [ln y(p1, κ˜(p0))− ln y(p0, κ˜(p0)]=∫ p1p0∂ lnV (p)∂p− ∂ ln y(p, q)∂p∣∣∣∣q=κ˜(p0)dp≥∫ max{p1,p˜}p0∂ ln y(p, q)∂p∣∣∣∣q=κ˜(p)− ∂ ln y(p, q)∂p∣∣∣∣q=κ˜(p0)dp > 0.The last strict inequality is due to the strict log-SPM of y(p, q) and κ˜ isstrictly increasing. Recall that η(Λ(q, s))(1 − s) = δ′(λ˜(q)) holds for anyactive market (q, s). Therefore, a worker of p = r˜(q) receives his highestpayoff only at (q, s) ∈ Ψ, and the outside option is optimal for the workersof p < p˜.We now turn to the asset side. For q ≥ q˜,U(q) = [δ(λ˜(q))− λ˜(q)δ′(λ˜(q))]y(r˜(q), q) ≥ U.Together with Lemma 13, the contract posting decision is optimal for ownersof q ≥ q˜.Now suppose q˜ > 0. Consider an inactive market (qL, s′) where qL < q˜.For any pH > p˜,R(pH |qL, s′) = 0 becauseV (pH)V (p˜)>δ′(λ˜(q˜))y(pH , q˜)δ′(λ˜(q˜))y(p˜, q˜)>(1− s′)y(pH , qL)(1− s′)y(p˜, qL) .Hence, R(qL, s′) is degenerate at some pL ≤ p˜. The case Λ(qL, s′) = 0is trivial. For the case Λ(qL, s′) > 0, Λ(qL, s′) satisfies η(Λ(qL, s′))(1 −s′)y(pL, qL) = V (pL). If deviating to the market (qL, s′), an asset owner will181B.2. Proof of Proposition 7receiveδ(Λ(qL, s′))s′y(pL, qL) = δ(Λ(qL, s′))y(pL, qL)− Λ(qL, s′)V (pL)≤ maxλ[δ(λ)y(pL, qL)− λV (pL)]= maxλ[δ(λ)y(pL, qL)− λV (p˜)] < U(q˜) = U.The second equality holds because of the boundary condition p˜(V (p˜)−V ) =0. So it is never optimal for an owner of q < q˜ to participate.Analysis of the boundary value problemBy differentiating the Hosios condition w.r.t. q and subtracting it withthe expression ∂ ln v˜(p)∂p∣∣∣p=r˜(q), we obtain∂ ln δ′(λ˜(q))∂q= −∂ ln y(r˜(q), q)∂qThere exists a unique pair of λ and λ satisfying [δ(λ)−δ′(λ)λ]y(1, 1) = Uand δ′(λ)y(1, 1) = V respectively. Note that λ > λ. Consider the followinginitial value problem (IPV-λ(1)):r′(q) =g(q)f(r(q))λ(q),∂ ln δ′(λ(q))∂q= −∂ ln y(r(q), q)∂q,where the initial values are given by r(1) = 1 and λ(1) = λ1 ∈ [λ, λ].Since the differential equation system is locally Lipschitz, Picard’s existencetheorem ensures that IPV-λ(1) (in the downward direction) admits a uniquesolution {r(q;λ1), λ(q;λ1)} over the interval [q(λ1), 1], where q(λ1) is the firstlevel of q where either of the following cases occurs:0 = r(q;λ1)[δ′(λ(q;λ1))y(r(q;λ1), q)− V ] = 0, or (B.2)0 = q[(δ(λ(q;λ1))− δ′(λ(q;λ1))λ(q;λ1))y(r(q;λ1), q)− U ]. (B.3)182B.2. Proof of Proposition 7Furthermore, q(λ1) and p(λ1) := r(q(λ1);λ1) are continuous in λ1. Forq > q(λ1), λ(q;λ1) and r(q;λ1) are strictly increasing.For p ∈ [p(λ1), 1], κ(p;λ1) denote the inverse of r(q;λ1). Also definev(p;λ1) = δ′(λ(κ(p;λ1);λ1))y(p, κ(p;λ1)), p ∈ [p(λ1), 1], andu(q;λ1) = [δ(λ(q;λ1))− δ′(λ(q;λ1))λ(q;λ1)]y(r(q;λ1), q ∈ [q(λ1), 1].Notice that for q > q(λ1), r(q;λ1)[v(r(q;λ1);λ1)−V ] and q[u(q;λ1)−U ] arepositive and strictly increasing in q.86Existence of a solutionThe boundary value problem has a solution if there exists some λ1 suchthat the solution to the IPV-λ(1) with λ(1) = λ1 satisfies both condition(B.2) and (B.3) at q = q(λ1). By construction, condition (B.2) holds atq = q(λ) and condition (B.3) holds at q = q(λ). Considerλ̂ = inf{λ′ ≥ λ : [v(p(λ1);λ1)− V ]p(λ1) = 0,∀λ1 ≥ λ′}.By continuity, [v(p(λ̂); λ̂)− V ]p(λ̂) = 0. If λ̂ = λ, then we have argued thatcondition (B.3) also holds at q = q(λ̂). Suppose λ̂ > λ, the constructionof λ̂ ensures that there is a convergent sequence {λ1n} with limit λ̂ suchthat λ1n < λ̂ and only condition (B.3) holds at q = q(λ1n) for λ1 = λ1n.By continuity, q(λ̂)[u(q(λ̂); λ̂) − U ] = 0 must hold as well. Therefore thesolution to the IPV-λ(1) with λ(1) = λ̂ solves the boundary value problem.Uniqueness of the solutionSuppose λH > λL. For p ∈ [max{p(λH), p(λL)}, 1), κ(p;λH) > κ(p;λL),v(p;λH) < v(p;λL) and λ(κ(p;λH);λH) > λ(κ(p;λL);λL).Proof. Since r′(1;λH) > r′(1;λL), κ(p;λH) > κ(p;λL), λ(κ(p;λH);λH) >86v(r(q;λ1);λ1) is stricly increasing in q because ∂ ln δ′(λ(q;λ1))∂q+ ∂ ln y(r(q;λ1),q)∂q= 0.183B.2. Proof of Proposition 7λ(κ(p;λL);λL) and v(p;λH) < v(p;λL) must hold in some neighborhood ofp = 1.Consider the case that κ(.;λH) and κ(.;λL) intersects somewhere in[max{p(λH), p(λL)}, 1). pκ = max{p < 1 : κ(p;λH) = κ(p;λL)} is thenwell-defined and by construction, κ(p;λH) > κ(p;λL) for all p ∈ (pκ, 1). Itfollows that v(.;λH) and v(.;λL) must intersect somewhere in [pκ, 1). Oth-erwise, for all p ∈ (pκ, 1],δ′(λ(κ(p;λH);λH))y(p, κ(p;λH)) = v(p;λH)< v(p;λL) = δ′(λ(κ(p;λL);λL))y(p, κ(p;λL)),and hence λ(κ(p;λH);λH) > λ(κ(p;λL);λL). This contradicts the law ofmotion,0 >∫ 1pκ1λ(κ(p;λH);λH)− 1λ(κ(p;λL);λL)dF= [G(1)−G(κ(pκ;λH))]− [G(1)−G(κ(pκ;λL))] = 0!Consider the case that v(.;λH) and v(.;λL) intersects somewhere in[max{p(λH), p(λL)}, 1). Define pv = max{p < 1 : v(p;λH) = v(p;λL)}.Since v(p;λH) < v(p;λL) for p > pv,∂ ln y(p, q)∂p∣∣∣∣(p,q)=(pv ,κ(pv ;λH))=∂ ln v(pv;λH)∂p≤ ∂ ln v(pv;λL)∂p=∂ ln y(p, q)∂p∣∣∣∣(p,q)=(pv ,κ(pv ;λL)).Assumption (Y) implies that κ(pv;λH) ≤ κ(pv;λL). So κ(.;λH) and κ(.;λL)intersects somewhere in [pv, 1].Putting together, it must be that κ(p;λH) > κ(p;λL) and v(p;λH) <v(p;λL) throughout [max{p(λH), p(λL)}, 1). Otherwise, pv and pκ are well-defined satisfying pv > pκ and pκ ≥ pv. λ(κ(p;λH);λH) > λ(κ(p;λL);λL)then follows from the definition of v(p;λH).184B.3. Proof of Proposition 8There is a unique initial value of λ(1) for which the solution to the IPV-λ(1)satisfying both condition (B.2) and (B.3) at q = q(λ1).Suppose not, the solutions to the IPV-λ(1) with λ(1) = λH and λ(1) =λL satisfy both condition (B.2) and (B.3) at q = q(λ1), where λH > λL.Consider the case p(λL) > p(λH). Since p(λL) > 0, V = v(p(λL);λL) >v(p(λL);λH). Condition (B.2) cannot be met at p(λH)!Now consider the case p(λL) ≤ p(λH), then q(λH) = κ(p(λH);λH) >κ(p(λH);λL) ≥ q(λL) and λ(q(λH);λH) > λ(κ(p(λH);λL);λL). This isimpossible because q(λH) > 0 impliesU = u(q(λH);λH) > u(κ(p(λH);λL);λL)Condition (B.3) cannot be met at q(λL)!B.3 Proof of Proposition 8Denote the equilibrium payoff for the asset owners byu˜(q) = (δ(λ˜(q))− δ′(λ˜(q))λ˜(q))y(r˜(q), q), q ≥ q˜.The equilibrium allocation and the Second Best allocation (or the equilib-rium allocation in price competition) can be respectively recovered from(p˜, q˜, r˜, λ˜, v˜, u˜) and (rSB, λSB, pSB, qSB, vSB, uSB), which both satisfy thefollowing conditionsBoundary conditions: r(q) = p, r(1) = 1, q[u(q)− U ] = p(v(p)− V ) = 0,Law of motion: r′(q) = g(q)f(r(q))λ(q),Hosios condition: v(r(q)) = δ′(λ(q))y(r(q), q).185B.3. Proof of Proposition 8The only difference is in the worker’s IC conditions, which are given by∂ ln v˜(p)∂p∣∣∣∣p=r˜(q)=∂ ln y(p, q)∂p∣∣∣∣(p,q)=(r˜(q),q),∂ ln vSB(p)∂p∣∣∣∣p=rSB(q)=η(λSB(q))δ′(λSB(q))∂ ln y(p, q)∂p∣∣∣∣(p,q)=(rSB(q),q).The listed set of conditions defines two boundary value problems, and weare going to compare their solutions.Step 1: For any p̂ > max{pSB, p˜} and q̂ > 0 satisfying q̂ = κ˜(p̂) = κSB(p̂),then λSB(q̂) < λ˜(q̂).Suppose, to the contrary that, λSB(q̂) ≥ λ˜(q̂). Then vSB(p̂) ≤ v˜(p̂)and ∂ ln v˜(p̂)∂p <∂ ln vSB(p̂)∂p . There must exist some  > 0 such that for allp ∈ (p̂− , p̂), vSB(p) < v˜(p) and κSB(p) > κ˜(p).87Consider the case that κ˜ and κSB intersect in [max{pSB, p˜}, p̂). pκ de-notes the first intersection point of κ˜ and κSB in [max{pSB, p˜}, p̂), so thatκSB(p) > κ˜(p) for all p ∈ (pκ, p̂). Then vSB and v˜ must intersect some-where in between pκ and p̂. Otherwise, for all p ∈ (pκ, p̂), the Hosioscondition implies δ′(λ˜(κ˜(p)))y(p, κ˜(p)) > δ′(λSB(κSB(p)))y(pv, κSB(p)), andhence λ˜(κ˜(p)) < λSB(κSB(p)). This contradicts the law of motion,0 <∫ p̂pκ1λ˜(κ˜(p))− 1λSB(κSB(p))dF= [G(q̂)−G(κ˜(pκ))]− [G(q̂)−G(κSB(pκ))] = 0!Consider the case that vSB and v˜ intersect in [max{pSB, p˜}, p̂). Let pvbe the first intersection point of vSB and v˜ in [max{pSB, p˜}, p̂), so thatvSB(p) < v˜(p) for all p ∈ (pv, p̂). Then κ˜ and κSB must intersect at somepoint between pv and p̂. Suppose not, the continuity of κ˜ and κSB imply87For the case λSB(q̂) = λ˜(q̂), one can show∂ ln δ′(λSB(q̂))∂q> ∂ ln δ′(λ˜(q̂))∂qby differentiatingHosios condition v(r(q)) = δ′(λ(q))y(r(q), q) w.r.t. q and combining it with ∂ ln v(p)∂p∣∣∣p=r(q).186B.3. Proof of Proposition 8that κSB(p) > κ˜(p) for p ∈ (pv, p̂). The Hosios condition again requiresλ˜(κ˜(p)) < λSB(κSB(p)) for all p ∈ (pκ, p̂). Under Assumption (Y) and (M),for all p ∈ (pκ, p̂),∂ ln vSB(p)∂p=η(λSB(κSB(p)))δ′(λSB(κSB(p)))∂ ln y(p, κSB(p))∂p>η(λ˜(κ˜(p)))δ′(λ˜(κ˜(p)))∂ ln y(p, κ˜(p))∂p>∂ ln v˜(p)∂p.Hence, vSB and v˜ cannot intersect at pv.It follows that vSB(p) < v˜(p) and κSB(p) > κ˜(p) for p ∈ [max{pSB, p˜}, p̂).Otherwise, pv and pκ will co-exist, satisfying p̂ > pκ > pv and p̂ > pv > pκ!Again λ˜(κ˜(p)) < λSB(κSB(p)) throughout [max{pSB, p˜}, p̂) because of theHosios condition.These conclusions cannot be consistent with the boundary conditions.Suppose p˜ > pSB, then the boundary condition for p˜ > 0 requires V =v˜(p˜) > vSB(p˜)! Suppose p˜ ≤ pSB, then qSB = κSB(pSB) > κ˜(pSB) ≥ q˜ andλSB(qSB) > λ˜(κ˜(pSB)). The boundary condition for qSB > 0 then requiresU = uSB(qSB) > u˜(κ˜(pSB)) ≥ u˜(q˜)!Corollary: λSB(1) < λ˜(1), vSB(1) > v˜(1) and uSB(1) < u˜(1).Step 2: κ˜(p) > κSB(p) for any p ∈ (max{pSB, p˜}, 1).Since λSB(1) < λ˜(1), the law of motion implies r′SB(1) < r˜′(1), and henceκ˜(p) > κSB(p) for sufficient large p. Suppose κ˜(.) and κSB(.) intersectssomewhere in (max{pSB, p˜}, 1). Consider the first intersection point p̂ =max{p ∈ (0, 1) : κ˜(p) = κSB(p)}. Let q̂ = κ˜(p̂) = κSB(p̂). The previousclaim states that λSB(q̂) < λ˜(q̂). However, rSB(q) > r˜(q) for q > q̂ byconstruction. From the law of r′SB(q̂) ≥ r˜′(q̂) only if λSB(q̂) ≥ λ˜(q̂)!Step 3: pSB≥ p˜ and q˜ ≥ qSB, and one of the inequalities must be strict.First, suppose, to the contrary that, p˜ > pSB≥ 0. Then q˜ = κ˜(p˜) ≥187B.3. Proof of Proposition 8κSB(p˜) > κSB(pSB) = qSB. The boundary conditions must be violated ifp˜ > pSBand q˜ > qSB.U = u˜(q˜) + λ˜(q˜)[v˜(p˜)− V ]= maxλ≥0[δ(λ)y(p˜, q˜)− λV ] > maxλ≥0[δ(λ)y(pSB, qSB)− λvSB(pSB)] (B.4)= uSB(qSB)!The first equality is due to the boundary conditions for q˜ > 0 and p˜ > 0while the second equality and the last equality come from their FOCs andthe Hosios condition. Interchanging the role of (pSB, qSB) and (p˜, q˜) inthe inequality (B.4), the case pSB> p˜ and qSB> q˜ is also ruled out. Acontinuity argument rules out the case pSB= p˜ and qSB> q˜. For any p′slightly above p˜, κ˜(p′) > κSB(p′) > qSB and κ˜(p˜) = q˜ < qSB. κ˜ must bediscontinuous at p˜! Therefore, we establish that pSB≥ p˜ and q˜ ≥ qSB.Step 4: pSB= p˜ only if pSB= p˜ = 0 and v˜(0) > vSB(0). q˜ = qSB only ifq˜ = qSB= 0 and uSB(0) > u˜(0).First, consider the case pSB= p˜ and q˜ = qSB. Differentiating the Hosioscondition ln v(r(q)) = ln δ′(λ(q)) + ln y(r(q), q) and subtracting it with therespective expressions of ∂ ln vSB(p)∂p∣∣∣p=rSB(q)and ∂ ln v˜(p)∂p∣∣∣p=r˜(q), we obtain∂ ln δ′(λSB(q˜))∂q >∂ ln δ′(λ˜(q˜))∂q . It follows that λ˜(q˜) < λSB(q˜). Otherwise, λ˜(q′) >λSB(q′) for q′ slightly above q˜. The law of motion in turn implies that r˜(q′) >rSB(q′), contradicting the previous claim in Step 2! We can immediately ruleout the cases with pSB= p˜ > 0 or q˜ = qSB> 0. This is because the boundaryconditions and the Hosios condition in such case require λ˜(q˜) = λSB(q˜)! Theremaining possibility is that pSB= p˜ = q˜ = qSB= 0. v˜(0) > vSB(0) anduSB(0) > u˜(0) because λ˜(0) < λSB(0).Consider the case pSB= p˜ and q˜ > qSB. From the boundary conditions188B.3. Proof of Proposition 8and Hosios condition,maxλ≥0[δ(λ)y(pSB, qSB)− λvSB(pSB)] = uSB(qSB)≥ U = u˜(q˜) = maxλ≥0[δ(λ)y(pSB, q˜)− λv˜(pSB)].This immediately implies that v˜(pSB) > vSB(pSB) ≥ V , and hence pSB =p˜ = 0. The case pSB> p˜ and q˜ = qSBfollows from a symmetric argument.Corollary: v˜(pSB) > vSB(pSB) and uSB(q˜) > u˜(q˜)The previous claim establishes the cases of pSB= p˜ or q˜ = qSB. SupposepSB> p˜, the boundary condition immediately implies v˜(pSB) > vSB(pSB) =V . Similarly, uSB(q˜) > u˜(q˜) = U if q˜ > qSB.Notice that the above arguments only require the workers’ IC conditionto satisfy ∂v˜(p)∂p∣∣∣p=r˜(q)< η(λ˜(q)) ∂y(p,q)∂p∣∣∣(p,q)=(r˜(q),q)for v˜(p) ≥ V .189

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