Informed intermediaries

Onuchic, Paula Article Informed intermediaries Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Onuchic, Paula (2022) : Informed intermediaries, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 17, Iss. 1, pp. 57-87, https://doi.org/10.3982/TE4072 This Version is available at: https://hdl.handle.net/10419/253516 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 17 (2022), 57–87 1555-7561/20220057 Informed intermediaries Paula Onuchic Department of Economics, New York University I develop a theory of intermediation in a market where agents meet bilaterally to trade and buyers cannot commit to payments. Some agents observe the past trading history of traders in the market. These informed agents can secure trades by punishing traders who previously defaulted. The punishing strategy affects equilibrium prices and determines which trades are hindered by the risk of default. Intermediation is a robust equilibrium feature, generated by asymmetric punishing strategies that yield informed agents either more effective opportunities to trade or the ability to extract more surplus in trades. Keywords. Intermediation, limited commitment, bilateral trade, trade network. JEL classification. D83, D85. 1. Introduction In most decentralized markets, a buyer and seller meet and negotiate a price for a good, expecting that the seller will deliver the good and the buyer will pay the agreed-upon price. If either party fails to honor the terms, they face consequences from a legal system. In some markets, however, agents cannot rely on an exogenous authority to guarantee that contracts are honored. For obvious reasons, in markets for stolen goods or corruption markets, agents who fail to honor their debts cannot be prosecuted through formal means. Even in markets that are not illegal, legal fees can be prohibitively high or writing certain contracts may be infeasible. In extreme cases, an effective state or justice system may not exist. In a variety of empirical contexts, such markets are documented to have a hierarchical trading structure, where some central agents often trade goods not for their own use, but rather to profit from intermediating trades between other market participants. For example, Schneider (2005) interviews 50 “prolific burglars” and finds their most common method for disposal of burgled or shoplifted goods is selling them to fences, who then resell the goods to final consumers for a higher price. Della Porta and Vannucci (2016) also document the presence of brokers in corruption networks in Italy, and make a case for the importance of mafias as enforcers in the Italian market for corrupt Paula Onuchic: [email protected] I am grateful for the advice and guidance I received from Ricardo Lagos and Debraj Ray, and am indebted to Joshua Weiss and Samuel Kapon for long discussions and helpful comments. I also thank Paulo Arvate, three anonymous referees, the participants of the Search Theory Workshop at NYU, the 2018 Summer Workshop on Money, Banking, Payments and Finance, and the 2019 Summer School of the Econometric Society. This research was supported by NSF Grant SES-1629370 to Debraj Ray. ©2022 The Author. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4072 58 Paula Onuchic Theoretical Economics 17 (2022) exchange. Additionally, many papers document the hierarchical network structure of trade in different financial markets.1 In this paper, I propose a model where agents trade bilaterally and payments are not enforced by an outside authority. Rather, agents honor terms of trade because they wish to maintain a reputation of being trustworthy. In their 1990 paper, Milgrom, North, and Weingast write, “A good reputation can be an effective bond for honest behavior in a community of traders if members of the community know how others have behaved in the past—even if any particular pair of traders meets only infrequently.” Much in the spirit of the quote, a share of traders in the model are informed and observe others’ past history of trade. Informed agents emerge as tacit “police” who secure transactions. Consistent with empirical observations, I show in the model that a robust feature of equilibria is a hierarchical trade network in which informed agents are central and often intermediate trades between other market participants. The model is a variation of the over-the-counter market in Duffie, Garleanu, and Pedersen (DGP, (2005)). A mass of agents meet bilaterally and continuously trade due to differences in their valuation for an asset. Unlike in DGP, buyers have no exogenous ability to commit to payments: Sellers first transfer the asset; only then do buyers decide whether to make the agreed-upon payment. If a buyer does not pay, the seller has no other recourse. Some traders are informed and observe a record of all past meetings. Informed agents can secure trades by refusing to trade with past defaulters. When a potential seller meets a potential buyer, she makes a take-it-or-leave-it price offer to the buyer. Despite this protocol, the seller does not necessarily extract all of the buyer’s trade surplus, because she must choose a price that induces the buyer not to default. Aware of the punishment a defaulting buyer is subject to, the seller proposes the highest price that induces no default. For example, if a buyer is punished with a long period of exclusion from trade, he would not default even if the proposed price is high, and the seller indeed offers a high price. If, otherwise, a buyer is only lightly punished for defaulting, he can only be trusted to pay a low price. Through this channel, the strategy that informed agents use to punish defaulters shapes the terms of trade in equilibrium. In Proposition 1, I propose a first class of equilibria that can be sustained if agents are sufficiently patient and enough informed agents exist: all-trade equilibria, where the limited commitment friction is completely overcome and no trades are hindered by the risk of default. In all-trade equilibria, informed agents who default are punished lightly, but defaulting uninformed agents are punished harshly. As such, informed agents are able to buy assets at a cheaper price and sell them at a higher price than uninformed agents do. All-trade equilibria are efficient, because all trades in which the buyer values the asset more highly than the seller take place, despite the limited commitment friction. Moreover, intermediation trades also happen: informed agents trade with uninformed agents not for the consumption value of the asset, but rather for its future trade value. Because informed agents buy assets at a cheaper price and sell them at a higher price than uninformed agents, they profit from trading and retrading the asset. 1See, for example, Ashcraft and Duffie (2007), Bech and Atalay (2010), and Afonso and Lagos (2015)for the market for federal funds. Theoretical Economics 17 (2022) Informed intermediaries 59 Proposition 2proves the existence of a second equilibrium class: core–periphery equilibria. These equilibria are supported by a punishing strategy whereby uninformed buyers who default against uninformed sellers face no consequence. As such, uninformed traders are peripheral agents who do not trade with each other, but rather trade only with informed agents who form the core of the network. Because uninformed agents are constrained by the limited commitment friction, core–periphery equilibria are not efficient. In core–periphery equilibria, informed agents also act as intermediaries, precisely because they form the core of the trade network and have more trade opportunities than uninformed agents. As such, uninformed agents sell assets to informed agents at a discount because finding a final buyer on their own would take longer, and they buy assets for a higher price for a similar motive. This price spread makes intermediation profitable to core traders. Given the equilibrium multiplicity, Section 5proposes a refinement: I look for equilibria that maximize informed agents’ values. Because all equilibria rely on informed agents coordinating on punishing strategies to ensure trade, it is reasonable to expect informed agents’ preferred equilibria. The main results show that using punishing strategies that yield them better terms of trade (Proposition 5) and create a core–periphery trade network by preventing trade between uninformed agents (Proposition 4)canbenefit informed agents. Both of these channels that benefit informed agents also generate motives for equilibrium intermediation, and we can conclude that intermediation is connected to rewarding informed agents who “monitor” trades in the market. Proposition 5shows informed agents can receive a higher value without sacrificing efficiency, through more favorable terms of trade. However, a consequence of Proposition 4is that sacrificing efficiency by preventing trades between uninformed agents is even better from the point of view of informed agents. 1.1 Related literature This paper is related to recent literature on intermediation in over-the-counter markets. Initial models feature exogenously given middlemen who facilitate trade (e.g., Duffie et al. (2005), Lagos and Rocheteau (2009)). Many papers since tackle the question of the endogenous emergence of some agents as intermediaries. The papers that are closest to mine are Farboodi, Jarosch, and Shimer (2020) and Farboodi, Jarosch, Menzio, and Wiriadinata (2019).2The former finds that agents with faster meeting rates, and, hence, more opportunities to trade, become intermediaries, whereas the latter shows that intermediation arises purely due to differences in bargaining power across agents. I propose an alternative theory, which relies on agents’ limited commitment to future payments. This new theory can be seen as a microfoundation for the two previous explanations, because the punishing strategies that informed agents in my model use lead to some agents endogenously having better trade opportunities or surplus extraction. 2See also Afonso and Lagos (2015), Chang and Zhang (2019), and Bethune, Sultanum, and Trachter (2020). 60 Paula Onuchic Theoretical Economics 17 (2022) Babus and Hu (2017) study a repeated game played in a network and also show a link between intermediation and trade when agents cannot commit to payments.3In their model, without intermediation, trade fails as the number of agents becomes large, and with intermediation, trade can be sustained regardless of the size of the market. The first main difference between their model and mine is that I study the limit case with a continuum of agents. Second, intermediation emerges dynamically in my model, when agents trade for future resale purposes. In their framework, dynamic intermediation is not possible, because assets are not carried between periods. Finally, my paper highlights the effect of punishing strategies on equilibrium features. My paper also relates to the sizable literature on limited commitment in searchtheoretic models of liquidity. Generally, the main difference between those models and mine is that the assets agents trade in my environment are long-lived and might be traded for speculative motives, rather than a commodity that is consumed before the end of each period. This distinction is key for linking the limited commitment friction to intermediation.4 Finally, I contribute to the literature linking intermediation and trade efficiency. Previous work found that intermediation facilitates trade by minimizing transaction costs (Townsend (1978)), minimizing search frictions (Rubinstein and Wolinsky (1987), Duffie et al. (2005)), or reducing monitoring costs (Diamond (1984)). In terms of the monitoring motivation for intermediation, my paper is closely related to Diamond (1984) and the subsequent literature. In Diamond’s model, intermediation increases efficiency by lowering the total cost of monitoring. In my model, intermediation can be welfareimproving by providing incentives for informed agents to join the market and then monitor trades. 2. Model I study an economy where time is continuous and the horizon is infinite, and future utility flows are discounted by all agents at rate r>0. A unit measure of agents bilaterally meet to trade an indivisible asset with supply fixed at 1/2. In these meetings, a numeraire good is used as the medium of exchange. Preferences. At any time, each agent in the market either holds an asset or does not. Agents cannot accumulate assets or sell assets they do not yet have. An agent with idiosyncratic valuation v∈{L,H}receives flow value δvwhen holding an asset, where δH>δ L≥0. Agents’ valuations and asset holdings are observable to other market participants. Asset holdings are subject to shocks: At a Poisson rate η>0, an agent who is holding an asset loses it and, at that same rate, an agent who is not holding an asset receives one. 3Fainmasser (2019) is another paper that considers the link between intermediation and cooperation in repeated games in networks. 4My paper is close to Cavalcanti and Wallace (1999), where agents are either monitored, having all their previous trades recorded, or not monitored. Monitored agents can commit to future payments since they can be punished for defaults. In my paper, agents are heterogeneous in their access to a public record rather than in their being recorded or not. Other related papers are Carapella and Williamson (2015) and Bethume, Hu, and Rocheteau (2018). Theoretical Economics 17 (2022) Informed intermediaries 61 The difference in flow payoffs between high- and low-valuation agents, as well as the asset-holding shocks, implies agents wish to continuously trade and retrade the assets.5 Bilateral Trade and Limited Commitment. At rate λ>0, an agent meets another randomly selected agent. If one of the agents in the meeting has an asset and the other does not, they have an opportunity to trade. The agent who holds an asset (the potential seller) makes a take-it-or-leave-it price offer, in terms of the numeraire good, to the agent who does not have an asset (the potential buyer). If the offer is accepted, the asset is transferred from seller to buyer. After this transfer takes place, the buyer chooses to pay the agreed-upon price or to default.6 Despite the seller making a take-it-or-leave-it price offer, she does not necessarily extract all of the buyer’s surplus, because she must also make sure the buyer chooses not to default at the proposed price. The seller-optimal price offer is such that the buyer is made exactly indifferent between defaulting and not defaulting. Information and Trigger Punishing Strategies. A measure φof agents is informed (I) and upon meeting a potential trading partner, observe their past trading history, i.e., the identity of past trading partners and whether they defaulted. The other 1 −φagents are uninformed (U) and do not observe said history. Agents’ information types are independent of their valuation and are observable to other market participants. Informed agents use their knowledge of past trading history to punish defaulting agents. To that end, they use trigger punishing strategies: when a buyer defaults, a punishment regime is triggered with some probability. In that case, the buyer is forever excluded from trade with informed agents.7All informed agents use the same anonymous8trigger strategy. Formally, the set of trading histories plus the realization of a public randomization device is partitioned into elements that induce informed agents to trade (unflagged histories) and elements that induce informed agents not to trade (flagged histories). To fully describe the punishing strategy, I also need to assign punishments to informed agents who fail to punish when called upon to do so, that is, informed agents who trade with flagged agents. Imposing that informed agents who trade with flagged agents become flagged is sufficient. If a flagged agent wants to buy from an informed agent, she has no incentive to repay and, hence, the informed agent will not engage in this sale. If a flagged agent wants to sell to an informed agent, the informed agent already gets punished from engaging in this trade and, hence, has no incentive to pay. Anticipating that possibility of default, the flagged agent does not sell. 5In most of the literature stemming from DGP, this trading motive is achieved through shocks in agents’ valuations for the asset, rather than to the asset holdings as in my model. In the absence of limited commitment, these modeling choices are equivalent. 6The buyer has limited commitment and can choose to default, but the model could seamlessly be flipped to the case where the buyer first pays and then the seller chooses whether to transfer the good. 7Other non-trigger strategies also support the equilibria I find, and for some parameter values, other strategies might support equilibria that are not supported by the trigger class. However, the harshest trigger strategy, whereby a buyer is punished with probability 1 after default, is also the harshest punishment informed agents can inflict on defaulters across all the possible, even non-trigger, strategies. 8An informed agent who meets two agents with the same trading history chooses the same action in both meetings. 62 Paula Onuchic Theoretical Economics 17 (2022) 3. Equilibrium definition Let Vi va be the value to an unflagged agent of information type i∈{I,U},valuationv∈ {H,L}, and asset holding a∈{0, 1}. Accordingly, let ˜ Vi va be the value of that same agent when he is in the flagged regime. Sixteen potential trades occur between unflagged agents, and (is,vs,ib,vb)denotes a trade between a seller of type (is,vs)∈{I,U}×{H,L} andabuyeroftype(ib,vb)∈{I,U}×{H,L}. The surplus in this meeting is given by Vis vs0−Vis vs1+Vib vb1−Vib vb0, the value to the buyer of acquiring an asset minus the value to the seller of losing an asset. Punishing Strategy and Price Determination. Define the trigger punishing strategy τ:({I,U}×{H,L})2→[0, 1],whereτ(is,vs,ib,vb)is the probability that the buyer becomes flagged after defaulting on trade (is,vs,ib,vb). In meeting (is,vs,ib,vb),thebuyer chooses to pay the seller if the seller’s price offer psatisfies Vib vb1−p≥1−τ(is,vs,ib,vb)Vib vb1+τ(is,vs,ib,vb)˜ Vib vb1.(1) The left-hand side of condition (1) shows the value to the buyer of paying the price p requested by the seller: the buyer leaves the meeting with the asset and without triggering any punishment, and pays price p. The right-hand side show the value of defaulting: with probability (1−τ(is,vs,ib,vb)), the buyer leaves the meeting with the asset and does not trigger punishment, and with probability τ(is,vs,ib,vb), the buyer leaves the meeting with the asset but triggers the punishment regime. When set to equality, condition (1) determines the highest price the seller can charge while guaranteeing the buyer does not default. Thus, the seller’s optimal take-it-or- leave-it price offer in meeting (is,vs,ib,vb)is p(is,vs,ib,vb)=τ(is,vs,ib,vb)Vib vb1−˜ Vib vb1.(2) If participating in the market is valuable, Vib vb1is larger than ˜ Vib vb1and the seller can charge a positive price and guarantee the buyer does not default. Equation (2)shows that a harsher punishment for defaulting (larger τ) raises the maximum price the seller can charge, because it decreases the buyer’s deviation value. In the other limit, in the absence of punishment for defaulting (τ=0), the price is zero. It is convenient to define the buyer and seller surplus shares in each meeting. Let β:({I,U}×{H,L})2→R,whereβ(is,vs,ib,vb)is the proportion of the surplus that remains with the seller in meeting (is,vs,ib,vb): β(is,vs,ib,vb)=Vis vs0−Vis vs1+p(is,vs,ib,vb) Vis vs0−Vis vs1+Vib vb1−Vib vb0 .(3) Equilibrium Trades. Let I:({I,U}×{H,L})2→{0, 1}be such that I(is,vs,ib,vb)=1 indicates trade (is,vs,ib,vb)takes place in equilibrium, and let I(is,vs,ib,vb)=0 indicate it does not. A trade is mutually beneficial if both the seller and buyer retain a positive surplus. In equilibrium, only mutually beneficial trades take place. Additionally, all Theoretical Economics 17 (2022) Informed intermediaries 63 strictly beneficial trades must take place: I(is,vs,ib,vb)=1 ⇒β(is,vs,ib,vb)∈[0, 1]and Vis vs0−Vis vs1+Vib vb1−Vib vb0≥0, (4) β(is,vs,ib,vb)∈(0, 1)and Vis vs0−Vis vs1+Vib vb1−Vib vb0>0 ⇒I(is,vs,ib,vb)=1. (5) Value Functions. Given the surplus sharing rules defined above, we can write the unflagged and flagged agents’ value functions. To that end, also let {μi va}denote the stationary distribution of unflagged agents across valuations and asset holdings. I focus on stationary equilibria with no default on path, and so I refrain from adding notation for the stationary measure of flagged agents, which must be zero. Value functions consist of the flow value received if holding an asset, the value due to asset-holding shocks, and flows from trade. For unflagged and flagged agents, respectively, they are rV i v0=ηVi v1−Vi v0 +λ vs∈{L,H} μI vs1I(I,vs,i,v)1−β(I,vs,i,v)Vi v1−Vi v0+VI vs0−VI vs1 +λ vs∈{L,H} μU vs1I(U,vs,i,v)1−β(U,vs,i,v)Vi v1−Vi v0+VU vs0−VU vs1,(6) rV i v1=δv+ηVi v0−Vi v1 +λ vb∈{L,H} μI vb0I(i,v,I,vb)β(i,v,I,vb)VI vb1−VI vb0+Vi v0−Vi v1 +λ vb∈{L,H} μU vb0I(i,v,U,vb)β(i,v,U,vb)VU vb1−VU vb0+Vi v0−Vi v1,(7) r˜ Vi v0=η˜ Vi v1−˜ Vi v0+λ vs∈{L,H} μU vs1I(U,vs,i,v)˜ Vi v1−˜ Vi v0,(8) r˜ Vi v1=δv+η˜ Vi v0−˜ Vi v1 +λ vb∈{L,H} μU vb0I(i,v,U,vb)β(i,v,U,vb)VU vb1−VU vb0+˜ Vi v0−˜ Vi v1.(9) Agents in flagged agents do not trade with informed agents; thus (8)and(9)donotaccount for value of meeting with informed agents. Flagged agents can still trade with uninformed agents, who are not able to observe their past defaults. Moreover, because the flagged agent is already in the punishment regime and cannot be punished further, flagged buyers always default, which is also accounted for in (8). Because I work with equilibria with no default on path, these trades never take place. However, they still affect the value of the deviation. 64 Paula Onuchic Theoretical Economics 17 (2022) Stationary Distribution. The distribution of agents across types must satisfy the adding-up constraints  v∈{H,L} a∈{0,1} μI va =φ, (10)  v∈{H,L} a∈{0,1} μU va =1−φ, (11)  i∈{I,U} v∈{H,L} μi v1=1 2. (12) Finally, in any stationary equilibrium, the stationary distribution is such that the inflow to each state is equal to the outflow. For any v∈H,Land i∈{I,U}, μi v1η+ vb∈{H,L} ib∈{I,U} μib vb0I(i,v,ib,vb) =μi v0η+ vs∈{H,L} is∈{I,U} μis vs1I(is,vs,i,v)(13) must hold. Symmetry. I focus on symmetric equilibria, where an agent’s equilibrium trading behavior depends only on her information type and on whether their asset holdings are well aligned with their valuation.9Agents’ portfolios are misaligned if they have high valuation but no asset or if they hold an asset but have low valuation; their portfolios are well aligned otherwise. Symmetry requires a misaligned seller to have a trading pattern that mirrors that of a misaligned buyer of the same information type. Formally, I(is,vs,ib,vb)=I(ib,∼vb,is,∼vs), (14) β(is,vs,ib,vb)=1−β(ib,∼vb,is,∼vs). (15) Equilibrium Definition. A stationary equilibrium with no default is a set of unflagged value functions {Vi va}, flagged value functions {˜ Vi va}, trade indicator I, seller surplus shares β, punishment strategy τ, and stationary distribution {μi va}such that (3)–(15)are satisfied. 4. All-trade and core–periphery equilibria I propose two classes of equilibria. In the first class, all-trade equilibria, the limited commitment friction is completely overcome and no positive-surplus trades are hindered by the threat of default. In the second class, core–periphery equilibria, the threat of default prevents uninformed agents from directly trading with each other. In these equilibria, uninformed agents are peripheral traders, only trading with the informed core traders. Core traders trade among themselves and also with peripheral traders. 9Farboodi et al. (2020) similarly restrict attention to the set of symmetric equilibria in their model. Theoretical Economics 17 (2022) Informed intermediaries 71 Figure 3. Trade pattern supported in core–periphery equilibria. The upper left, lower right, and vertical arrows indicate portfolio-balancing trades. The upper right and lower left arrows indicate intermediation trades. The dashed arrow indicates a trade that does not occur in this pattern, due to limited commitment. where ˆαI=1 2r+2η+λ2ˆμUβI+ˆμI (23) and ˆαU=⎡ ⎢ ⎢ ⎣ r+2η+λβI2ˆμU+ˆμI+λ1−βIφ 2−ˆμI r+2η+λφ 21−βI ⎤ ⎥ ⎥ ⎦ ˆαI. (24) Because βI≥0.5, αU>α Iand agents’ values are ordered SU H>S I H>S I L>S U L. With these values, and the conjecture that positive-surplus trades take place if and only if they involve at least one informed trader, we can verify the trading pattern in core–periphery equilibrium, which is depicted in Figure 3. As in all-trade equilibria, informed agents engage in portfolio-balancing trades with both informed and uninformed trading partners. However, portfolio-balancing trades between two uninformed agents do not happen. Intermediation trades also occur in equilibrium, where informed sellers with high valuation sell to uninformed buyers with high valuation, and informed buyers with low valuation buy assets from uninformed buyers with low valuation. This equilibrium thus embeds a core–periphery trade network, whereby informed agents are core traders, who trade among themselves, as well as with the peripheral uninformed agents. Uninformed agents form the network periphery and do not trade with each other directly, but rather have their trades endogenously intermediated by informed agents in the network’s core. In all-trade equilibria, intermediation stems from the surplus-sharing rule that benefits informed agents. Although this channel is still present in core–periphery equilibria, an additional motive for intermediation exists: informed agents at the network’s core effectively have more trade opportunities and trade at a faster rate than uninformed agents in the periphery. 72 Paula Onuchic Theoretical Economics 17 (2022) The difference in opportunities to trade is not inherent to the agents, but rather a feature of the equilibrium conjecture that uninformed agents do not trade with each other even if the trade has positive surplus. This conjecture is sustained in equilibrium by a punishing strategy that does not punish uninformed buyers who default on uninformed sellers. As such, the threat of default prevents trades between peripheral uninformed agents. Core–Periphery Equilibrium with βI=0.5.When βI=0.5, surplus is shared evenly between the buyer and seller in all meetings that result in trade, even between informed and uninformed agents. In that case, we still have SU H>S I H>S I L>S U L.Toverifythis ordering, substitute βI=0.5 into (23)and(24) to find ˆαI=1 2r+2η+λˆμU+ˆμI and ˆαU=⎡ ⎢ ⎣ r+2η+λˆμU+λφ 4 r+2η+λφ 4 ⎤ ⎥ ⎦ˆαI, so that ˆαU>ˆαUeven if βI=0.5. This observation illustrates that in core–periphery equilibria, intermediation does not stem solely from the asymmetry in surplus sharing between informed and uninformed agents. Rather, the core–periphery structure of the trading network itself yields a strictly positive value to intermediation trades. Verifying Equilibrium Existence. To verify that a τthat supports the core–periphery equilibrium conjectures exists, I follow the same steps as with all-trade equilibria in Section 4.1, with one caveat, which I now explain. To ensure that no trade takes place between uninformed agents, even when trades have positive surplus, informed agents must set no punishment for default in trades between two uninformed agents. Formally, set τ(is,vs,ib,vb)=0whenever(is,ib)= (U,U). With this (non-) punishment in place, an uninformed seller foresees that an uninformed buyer will default on any price offer and, thus, refuses to sell. For all the trades (is,vs,ib,vb)that are conjectured to take place, I show that if ris sufficiently small and φis sufficiently large, a probability of punishment τ(is,vs,ib,vb) exists that exactly implements the conjectured surplus-sharing rule. Efficiency. Constrained efficiency is not achieved in core–periphery equilibria. Because meetings between uninformed agents do not result in trade, some opportunities to transfer assets from low- to high-valuation agents are missed. Consequently, the proportion of misaligned agents is higher than in all-trade equilibria,11 and the total flow payoff to the economy is not as large as it could be. 4.3 Other equilibria The complete equilibrium set is not exhausted by all-trade and core–periphery equilibria. Reverse-Intermediation Equilibria. In the equilibria proposed in Sections 4.1 and 4.2, informed agents are assigned higher shares of the surplus in meetings as well as 11This fact is verified in the Appendix. Theoretical Economics 17 (2022) Informed intermediaries 73 more opportunities to securely trade. As discussed, both channels lead to intermediation trades whereby informed agents act as the intermediaries. For each of the equilibria proposed, a reverse equilibrium exists where uninformed agents are the intermediaries. For example, “reverse” all-trade equilibria exist where βI<0.5, in which case uninformed agents keep a higher share of trade surplus than informed agents. In that case, reverse intermediation takes place. Equally, “reverse” core–periphery equilibria exist where βI<0.5 and informed agents are peripheral and do not trade with each other due to the risk of default. Again, reverse intermediation takes place. Such equilibria are not intuitive, as they rely on informed agents coordinating on uninformed agents’ larger shares of the trade surplus than that of informed agents. Indeed, in Section 5, I show that reverse-intermediation equilibria are not selected if informed agents can coordinate to maximize their own value. Other. Equilibrium trading is determined by the ordering of agents’ values (SI H,SU H, SI L,SU L)as well as whether, when agents meet, the trades are secured by the punishing strategy. Because agents with high valuation have a higher flow value from holding the asset, in any equilibrium Si H>S j Lfor i,j∈{I,U}. Moreover, the symmetry requirements in (14)and(15)imply(SI H−SU H)=− (SI L−SU L). Thus, in any equilibrium, either SU H≥SI H>S I L≥SU Lor SI H>S U H>S U L>S I L. Although the whole equilibrium set is not exhausted by the equilibria described so far, they do illustrate these two possibilities. They also illustrate that intermediation trades happen whenever SI H= SU Hand SI L= SU L, and that these differences can be generated either due to the equilibrium surplus sharing or due to the set of secured trade opportunities in an equilibrium. 5. Maximizing value to informed agents The equilibria shown in the last section demonstrate the channels that generate intermediation, determined by the punishing strategy. I now refine the set of equilibria by characterizing equilibria that informed agents prefer. This requirement is natural because informed agents are the ones who coordinate on punishing strategies that sustain trade in equilibrium. Define the objects VIand VU, the value of being informed and uninformed, respectively: VI≡VI H0+VI L0 2, (25) VU≡VU H0+VU L0 2. (26) The weighting in VIand VUreflects that half the agents have high valuation and half have low valuation, regardless of their information type. The definition of these values also considers agents who enter the market without holding an asset. It already accounts for the value of holding the asset, because these agents over time are hit by asset-holding shocks and also trade assets. The results in this section equally hold if we 74 Paula Onuchic Theoretical Economics 17 (2022) look at the opposite case, where the value accounts for an agent who enters the market holding an asset. Proposition 3, shows that although equilibria where uninformed agents act as intermediaries exist, they are not the equilibria that informed agents prefer. This result allows us to rule out reverse-intermediation equilibria, as in Section 4.3, and asserts that intermediation is typically performed by informed agents (as in all-trade and core–periphery equilibria). Proposition 3. If there exists an equilibrium featuring intermediation by uninformed agents, then another equilibrium exists, yielding a higher VI, in which uninformed agents are not intermediaries. The next result states that, to the informed agent, core–periphery equilibria are preferable to all-trade equilibria: preventing uninformed agents from trading with each other is beneficial to informed agents. One implication of Proposition 4is, thus, that informed agents select inefficient equilibria to maximize their own value. Proposition 4. For any βI∈[0.5, 1), the core–periphery equilibrium with surplus share βIyields a higher VIthan the all-trade equilibrium with that same surplus share βI. All-trade and core–periphery equilibria differ in that, in the latter, uninformed agents do not trade with each other. When trade between uninformed agents is shut down, they have fewer trade opportunities and are more eager to trade in meetings with informed agents. Therefore, the surplus in such meetings increases and informed agents can extract higher value from them. Finally, Proposition 5states that the value to informed agents is higher when informed agents keep a larger share of surplus in trades with uninformed agents. Proposition 5. The value to informed agents (VI) is increasing in βIin both all-trade and core–periphery equilibria. The proofs of Propositions 3,4,and5are provided in the Appendix. 5.1 Numerical exercise Propositions 4and 5allow us to compare informed agents’ values across different equilibria. However, they do not determine which equilibrium maximizes informed agents’ values among the equilibria that can be supported for a given set of parameters. In this numerical exercise, I compare all-trade and core–periphery equilibria that can be supported across different parameterizations. Figure 4displays three main results from numerical simulations.12 12The qualitative features of Figure 4described in Numerical Results 1,2, and 3are robust to other parameter specifications. Theoretical Economics 17 (2022) Informed intermediaries 75 Figure 4. In the top two panels, I plot the highest βIachievable in all-trade (lower lines) and core-periphery (higher lines) equilibria as a function of the share of informed agents φ(left panel, keeping r=0.005) and discount rate r(right panel, keeping φ=0.7). In the bottom two panels, I plot the highest VIachievable in all-trade (again, lower lines) and core–periphery (again, higher lines) equilibria as a function of the share of informed agents φand discount rate r. Other parameters are λ=2, η=0.2, δH=1, and δL=0. Numerical Result 1. If an all-trade equilibrium with surplus share βIexists, then a core–periphery equilibrium with surplus share of at least βIalso exists. This result, along with Proposition 4, implies that when an all-trade equilibrium exists, it is dominated—in terms of value to informed agents—by a core–periphery equilibrium that also exists. In Figure 4, I confirm that the highest value to informed agents achieved by an all-trade equilibrium is smaller than the highest value achieved by a core–periphery equilibrium. Numerical Result 2. Informed agents’ value is higher under the best available core– periphery equilibrium than under the best available all-trade equilibrium. Finally, I state a couple of comparative static results. First, the upper bound on βI mentioned in Propositions 1and 2is higher when agents are more patient and more informed agents exist. Second, informed agents’ value is highest at some interior value of φ. 76 Paula Onuchic Theoretical Economics 17 (2022) Numerical Result 3. In both all-trade and core–periphery equilibria (i) the highest supported βIis increasing in φand decreasing in r (ii) informed agents’ value is maximized at an interior φ∈(0, 1). One way to interpret the measure of informed agents φis as the underlying technology, where φ=1 is the frictionless benchmark where all agents access the recordkeeping technology. In that case, Numerical Result 3shows that informed agents attain the highest value in the presence of nonzero friction. In a first region, where φvalues are very low, no equilibrium exists. In a second region, where equilibria can be sustained, increasing φleads to two effects: first, the highest βIthat can be sustained is higher, which increases VI; second, the share of uninformed agents decreases, so the measure of agents from whom informed agents can extract high surplus shares or intermediate trades is lower, which decreases VI. 6. Conclusion I developed a dynamic model where agents meet bilaterally to trade and buyers cannot commit to payments. This limited commitment friction is present in many applications, such as markets for stolen goods or markets for corrupt exchange. A robust feature of equilibria in the model, also empirically observed in these markets, is the presence of intermediation, where some central agents trade goods not for their own use, but rather to profit from future trade value. In the model, equilibria with trade are supported by informed agents who punish traders who do not honor payments. In illegal markets, the presence of well connected (informed) groups is indeed important to ensure the “good” behavior of market participants. For example, Della Porta and Vanucci (2016) argue that the mafia, whose business leans on detailed knowledge of the behavior of individuals in a community, are important enforcers in the Italian market for corrupt exchange. In Section 3, I proposed two main classes of equilibria, which are supported by informed agents’ threats to punish defaulting agents. All-trade equilibria are efficient and all agents are able to trade despite the limited commitment friction, and core– periphery equilibria are inefficient because peripheral uninformed agents are unable to trade among themselves. One important result, stated in Section 5, is that these latter inefficient equilibria are robust, because they yield higher value to informed agents than the former, efficient, equilibria. Another feature of core–periphery equilibria is that informed agents do not use information about trades between uninformed agents. Consequently, they are also robust to variations in the information technology: if informed agents have access only to information about trades involving at least one informed agent, these equilibria are still supported. On the contrary, equilibria in which uninformed agents trade with each other require informed agents to have access to all the information available. Appendix A.1 Proof of Proposition 1(all-trade equilibria) I build the all-trade equilibria through a big guess and verify. Theoretical Economics 17 (2022) Informed intermediaries 77 Guesses. Surplus sharing: β(I,vs,U,vb)=βI≥1 2,β(U,vs,I,vb)=1−βI≤1 2, β(I,vs,I,vb)=1 2,β(U,vs,U,vb)=1 2. Trading pattern: I(is,vs,ib,vb)=1⇔vs=Land vb=H. Stationary Distribution. Given the guesses for I, the inflow equal to outflow equations for the stationary distribution become, for i∈{I,U}, μU L1η+λμI H0+μU H0+μI L0=ημU L0, (27) μU H0η+λμI L1+μU L1+μI H1=ημU H1, (28) μI L1η+λμU H0+μI H0=η+λμU L1μI L0, (29) μI H0η+λμU L1+μI L1=μI H1η+λμU H0. (30) In (27), substitute μU L0=1−φ 2−μU L1(which is true because half of uninformed agents have low valuation). Similarly, in (28), substitute μU H1=1−φ 2−μU H0. Then combine the two equations to get μU L12η+λμU H0+μI H0+λμI L0μU L1=μU H02η+λμU L1+μI L1+λμI H1μU H0 =η(1−φ) 2. (31) In (29), substitute μI L0=φ 2−μI L1(as before, this is true because half of informed agents have low valuation). Similarly, in (30), substitute μI H1=φ 2−μI H0. Then combine the two equations to get μI L12η+λμU H0+μI H0−λμI L0μU L1=μI H02η+λμU L1+μI L1−λμI H1μU H0 =ηφ 2. (32) Adding up (31)and(32)yields μI L1+μU L12η+λμU H0+μI H0=μI H0+μU H02η+λμU L1+μI L1=η 2(33) ⇒μI L1+μU L1=μI H0+μU H0. (34) Now substitute (33) back into (34)toget λμI L1+μU L12+2ημI L1+μU L1−η 2=0 ⇒μI L1+μU L1=μI H0+μU H0=−η λ+η λ2 +1 2 η λ. (35) 78 Paula Onuchic Theoretical Economics 17 (2022) Plug (35) back into (31)and(32)togetμU H0=μU L1=:μUand μI H0=μI L1=:μI,and,finally, μU=− η λ+φ 4+η λ2 +1 2 η λ+φ 42 , μI=−η λ−μU+η λ2 +1 2 η λ. Unflagged Values. Taking into account the guesses for Iand β, unflagged values are given by the system rV I H0=ηVI H1−VI H0+λμI L1 VI H1−VI H0+VI L0−VI L1 2+λμU L1βIVI H1−VI H0+VU L0−VU L1, rV I H1=δH+ηVI H0−VI H1+λμU H0βIVI H0−VI H1+VU H1−VU H0, rV I L1=δL+ηVI L0−VI L1+λμI H0 VI L0−VI L1+VI H1−VI H0 2 +λμU H0βIVI L0−VI L1+VU H1−VU H0, rV I L0=ηVI L1−VI L0+λμU L1βIVI L1−VI 00 +VU L0−VU L1, rV U H0=ηVU H1−VU H0+λμI L11−βIVU H1−VU H0+VI L0−VI L1 +λμI H11−βIVU H1−VU H0+VI H0−VI H1+λμU L1 VU H1−VU H0+VU L0−VU L1 2, rV U H1=δH+ηVU H0−VU H1, rV U L1=δL+ηVU L0−VU L1+λμI H01−βIVU L0−VU L1+VI H1−VI H0 +λμI L01−βIVU L0−VU L1+VI L1−VI L0+λμU H0 VU H1−VU H0+VU L0−VU L1 2, rV U L0=ηVU L1−VU L0. Using the result from the stationary distribution and writing the system in terms of the values of holding an asset, we get rSI H=δH−2ηSI H+λμUβISU H−SI H+λμISI L−SI H 2+λμUβISU L−SI H, rSI L=δL−2ηSI L+λμUβISU L−SI L+λμISI H−SI L 2+λμUβISU H−SI L, rSU H=δH−2ηSU H+λμI1−βISI L−SU H +λφ−2μI 21−βISI H−SU H+λμUSU L−SU H 2, rSU L=δL−2ηSU L+λμI1−βISI H−SU L Theoretical Economics 17 (2022) Informed intermediaries 79 +λφ−2μI 21−βISI L−SU L+λμUSU L−SU H 2. Add up the first two and the last two to get (r+2η)SI H+SI L=δH+δL+2λμUβISU H+SU L−2λμUβISI H+SI L, (r+2η)SU H+SU L=δH+δL+λφ 21−βISI H+SI L−λφ 21−βISU H+SU L. These imply (SI H+SI L)=(SU H+SU L)=δH+δL r+2η. Now, from the original system, subtract the second equation from the first and the fourth from the third to find (r+2η)SI H−SI L=δH−δL−λ2μUβI+μISI H−SI L, (r+2η)SU H−SU L=δH−δL−λφ 21−βISU H−SU L+λφ 2−μI1−βISI H−SI L −λμUSU H−SU L. Rearrange these to get the expressions SI H−SI L=αI(δH−δL), SI H−SI L=αU(δH−δL), (36) where αI=1 2r+2η+λ2μUβI+μI, αU=r+2η+λβI2μU+μI+λ1−βIφ/2−μI r+2η+λ(μU+φ/21−βIαI, which finally implies Si H=1 2(r+2η)(δH+δL)+αi 2(δH−δL), Si L=1 2(r+2η)(δH+δL)−αi 2(δH−δL). Punishing Strategy. To conclude that an all-trade equilibrium exists, all that is left to show is that there exists a punishing strategy τunder which the conjectured βsatisfies (3). For the trades for which I(is,vs,ib,vb)=0, which have nonpositive surplus, as found above, we can set any punishment level. For instance, let τ(is,vs,ib,vb)=0if I(is,vs,ib,vb)=0. Now define Di va =Vi va −˜ Vi va. The other conditions that need to be satisfied by τso as to support the conjectured βare τ(I,L,I,H)DI H1=SI H+SI L 2τ(U,L,I,L)DI L1=βISU L+1−βISI L, 80 Paula Onuchic Theoretical Economics 17 (2022) τ(U,L,I,H)DI H1=βISU L+1−βISI Hτ(I,H,U,H)DU H1 =βISU H+1−βISI H, (37) τ(I,L,U,H)DU H1=βISU H+1−βISI Lτ(U,L,U,H)DU H1=SU H+SU L 2. First, I solve for the values DI H1,DI L1,andDU H1. The system that defines {DI va}is rDI H1=η+λμUDI H0−DI H1, rDI L1=η+λμUDI L0−DI L1+λμISI H−SI L 2, rDI L0=η+λμUDI L1−DI L0−λμUβISU L+1−βISI L, rDI H0=η+λμUDI H1−DI H0+λμISI H−SI L 2−λμUβISI H+1−βISU L. Combining the first line with the fourth and the second line with the third yields rDI H1−DI H0=−2η+λμUDI H1−DI H0−λμISI H−SI L 2+λμUβISI H+1−βISU L, rDI L1−DI L0=−2η+λμUDI L1−DI L0+λμISI H−SI L 2+λμUβISU L+1−βISI L. Solve to find DI H1−DI H0=− λμI r+2η+2λμU SI H−SI L 2+λμU r+2η+2λμUβISI H+1−βISU L, DI L1−DI L0=λμI r+2η+2λμU SI H−SI L 2+λμU r+2η+2λμUβISU L+1−βISI L. Plug this back into the original system to get DI H1=η+λμU rr+2η+2λμUλμISI H−SI L 2−λμUβISI H+1−βISU L, (38) DI L1=r+η+λμU rr+2η+2λμUλμISI H−SI L 2 −λμUη+λμU rr+2η+2λμUβISU L+1−βISI L. (39) The system that defines {DU va}is rDU H1=ηDU H0−DU H1, rDU H0=η+λμUDU H1−DU H0 Theoretical Economics 17 (2022) Informed intermediaries 87 Fainmesser, Itay (2019), “Exclusive intermediation in unobservable networks.” Games and Economic Behavior, 113, 533–548. [60] Farboodi, Maryam, Gregor Jarosch, Guido Menzio, and Ursula Wiriadinata (2019), “Intermediation as rent extraction.” Working paper. [59] Farboodi, Maryam, Gregor Jarosch, and Robert Shimer (2020), “The emergence of market structure.” Working paper. 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