Intermediation in markets for goods and markets for assets

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ScienceDirect Journal of Economic Theory 183 (2019) 876–906 www.elsevier.com/locate/jet

Intermediation in markets for goods and markets for assets ✩ Ed Nosal a , Yuet-Yee Wong b , Randall Wright c,d,∗ a FRB Atlanta, United States of America b Binghamton University, United States of America c Zhejiang University, China d University of Wisconsin-Madison, United States of America

Received 15 August 2017; final version received 28 February 2019; accepted 1 July 2019 Available online 11 July 2019

Abstract We analyze agents’ decisions to act as producers or intermediaries using equilibrium search theory. Extending previous analyses in various ways, we ask when intermediation emerges and study its efficiency. In one version of the framework, meant to resemble retail, middlemen hold goods, which entails (storage) costs; that model always displays uniqueness and simple transition dynamics. In another version, middlemen hold assets, which entails negative costs, i.e., positive returns; that model can have multiple equilibria and complicated belief-based dynamics. These results are consistent with the venerable view that intermediation in financial markets is more prone to instability than in goods markets. © 2019 Elsevier Inc. All rights reserved. JEL classification: G24; D83 Keywords: Middlemen; Intermediation; Search; Bargaining; Multiplicity

✩ An earlier version of this paper circulated as “Who Wants To Be A Middleman?” For input we thank Karl Shell, Alberto Trejos, Steve Williamson, Gregor Jarosch, Giorgia Piacentino and seminar participants at the Chicago Fed, Minneapolis Fed, Philadelphia Fed, Cornell, Simon Fraser, UNC, Wisconsin, the Canadian Macro Study Group, and the St. Louis Fed/Tsinghua Monetary Policy Conference in Beijing. We especially thank Yu Zhu, who helped with the numerical work. Wright acknowledges support from the Ray Zemon Chair in Liquid Assets at the Wisconsin School of Business. The usual disclaimers apply. * Corresponding author at: University of Wisconsin-Madison, 975 University Ave., Madison WI 53706, United States of America. E-mail address: [email protected] (R. Wright).

https://doi.org/10.1016/j.jet.2019.07.001 0022-0531/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction This paper studies intermediation in markets for goods and markets for assets, building on the search-and-bargaining framework of Rubinstein and Wolinsky (1987), hereafter RW. The analysis extends existing versions of RW on several dimensions, and, in particular, endogenizes market composition by letting agents choose to act as either middlemen or producers. A version of the model designed to resemble goods markets, in a simple but reasonable way, has a unique equilibrium under standard assumptions; a version designed to resemble asset markets, under similar assumptions, can have multiple equilibria that are ranked in terms of efficiency. It is shown that for this result we must have the sets of producers and middlemen endogenous – if they are exogenous uniqueness obtains in markets for assets as well as goods. As background, first note that the original RW environment has no cost of production or search, equal numbers of producers and consumers, symmetric bargaining, and a fixed number of middlemen. In equilibrium middlemen participate in the market iff they have a better search technology than producers, as is efficient. In Nosal et al. (2015) we extended this to allow more general bargaining and costs, but that is relatively straightforward because, following RW, we maintained linear utility, only considered steady states, restricted inventories to {0, 1}, and kept the sets of producers and middlemen fixed. This paper relaxes all of those restrictions, with the key generalization being that agents decide to act as producers or middlemen, and that requires a rather different approach.1 We then distinguish between markets for goods and markets for assets as follows: as in retail markets, holding inventories of goods entails a storage cost; holding inventories of assets instead entails a positive return, or a negative cost, as is potentially the case with houses, art, productive capital, etc. Now there can also be costs to safely storing assets, but it is worth considering the case where the net return is positive, because it turns out that is necessary (not sufficient) for generating multiple steady states and endogenous dynamics. Of course, it is well known that one can generate multiplicity and dynamics in search theory with a variety of other devices, e.g., increasing returns in matching or production technologies (Diamond, 1982; Mortensen, 1999). We eschew those devices to concentrate on something new, complementarities in decisions by middlemen to adopt either buy-and-hold or buy-and-sell strategies. Our producers generate goods, or assets like capital, and trade them to end users. They may also trade them to middlemen, who may or may not trade them to end users. With goods as defined here (storage has a cost) the trading decision of a middleman meeting an end user is trivial, because buy-and-hold strategies cannot be optimal with negative returns. With assets as defined here (storage has a positive return) the decision is not trivial. To motivate why this is interesting, our asset market intermediaries can be interpreted in a stylized way as financial institutions, acquiring capital from originators and choosing when to pass it on. Our results thus provide support for the notion that financial institutions are less stable than other intermediaries, 1 For experts in search theory we can explain why it requires a different approach, although this can be skipped without loss of continuity. First, previous RW models take the arrival rates α = {αij } as exogenous, where αij is the rate at which type i meets type j . Under certain conditions that is legitimate because there exists a distribution of types, say n = {ni }, consistent with α, uniform random matching, and the identities implied by bilateral meetings, ni αij = nj αj i . While it is convenient to take α as fixed, we cannot do so when n is endogenous. Hence, we use uniform random matching, αij = αi nj , where αi is a baseline arrival rate for type i. But now the relevant identities imply αi = α ∀i, and in particular αpc = αmc , so we must abandon RW’s idea that middlemen are useful when αmc > αpc . Fortunately, other factors here take over for arrival rates, including bargaining powers and storage costs or returns.

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since we get multiplicity and belief-based volatility in intermediated markets for assets but not goods.2 For the intuition, suppose first that middlemen pass capital inventories on to end users. Then lots of middlemen will be searching for new capital, leading to high producer profit and hence many producers. With more producers in the market it is easier for middlemen to get capital, thus rationalizing their decision to trade it away and making active intermediation an equilibrium for some parameters. Now suppose middlemen keep capital for themselves. Then they are more likely to already have capital, leading to lower profits and fewer producers. This makes it harder for middlemen to get capital, thus rationalizing their decision to not trade it away in another equilibrium for the same parameters. Moreover, the equilibria can be welfare ranked – having middlemen trade with end users is better. Again, this can only arise when the return on inventories is positive and when the composition of the market is endogenous. The rest of the paper is organized as follows. Section 2 describes a general benchmark environment. Then we analyze markets for goods and markets for assets separately, the former in Sections 3-4, and the latter in Sections 5-7. Section 8 discusses robustness, including, in particular, an extension that allows general inventories, while the baseline model restricts inventories to {0, 1}.3 Section 9 contains concluding remarks. Extensions and proofs are in the Appendices.4 2. Environment There is a continuum of infinitely-lived agents of different types. Measure nc of them are end users, or consumers, labeled C. The rest choose to be producers, middlemen or nonparticipants, labeled P , M or N , with measure np , nm or nn . As discussed below, the market is active if type P agents produce, in which case they trade with C, and might trade with M, when they meet. They meet bilaterally in continuous time, with the Poisson rate at which any type i meets type j given by αij = αnj /h nh . Note that this displays constant returns since, e.g., doubling 2 The notion that financial intermediation engenders instability/fragility is commonly associated with names like Minsky, Kindleberger and Keynes. As regards banking, in particular, Rolnick and Weber (1986) say “Historically, even some of the staunchest proponents of laissez-faire have viewed banking as inherently unstable and so requiring government intervention.” Now this paper is not about unstable banking or credit arrangements (see Vives, 2016 and Gu et al., 2013 on those issues); it is about instability in intermediated asset markets, but that seems at least as relevant. Also, in terms of theory, the relevance of having returns positive or negative has been recently stressed by Han et al. (2019). 3 Having inventories in {0, 1} is special, although as in search theory going back to Diamond (1982) it allows one to make some salient points succinctly. In addition to studies of middlemen like RW, examples include models of monetary exchange (e.g., Kiyotaki and Wright, 1993), banking (e.g., Cavalcanti and Wallace, 1999), OTC financial markets (e.g., Duffie et al., 2005), unemployment (e.g., Pissarides, 2000) and partnership formation (e.g., Burdett and Coles, 1997). Still, we go beyond {0, 1} to see if our main results hinge on it; they do not. 4 Here we say more about the literature, although it can be skipped if one prefers to see the theory first. Related papers include Biglaiser (1993), Wright (1995), Li (1998), Camera (2001), Johri and Leach (2002), Shevchenko (2004), Smith (2004), Masters (2007, 2008), Tse (2009) and Watanabe (2010); see Wright and Wong (2014) for more on this work. In subsequent work, Farboodi et al. (2017, 2018) have technically similar models, but the applications are quite different. See Hugonnier et al. (2019) and references therein for more on OTC market intermediation. As a referee suggested, we mention that in those models one’s buyer/seller status changes over time – one can buy a house, car, bond... and sell it later. We instead have permanent buyer/seller types, as in RW, but changing that may be interesting in future work. It was also suggested that we put up front a comparison to the monetary theory surveyed by Lagos et al. (2017) or Rocheteau and Nosal (2017). Those models have multiplicities ostensibly related to ours, but the mechanism is actually different. There, a seller’s decision to accept an asset in exchange depends on what others accept; here the crucial element is strategic interaction in middlemen’s decision to adopt buy-and-hold or buy-and-trade strategies. Also, our effect requires endogenous market composition, which is not the case in monetary theory.

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nj ∀j doubles the number of meetings and leaves αij the same. To ease notation, we normalize   nc + np + nm + nn = 1 and α = 1, with no loss of generality, and write n = nc , np , nm , nn . There are two tradeable objects: x, which is indivisible and storable; and y, which is divisible but not storable. Type P produces x at cost 0, which for our applications is without loss of generality. Type M does not produce but can acquire x from P . Type C get payoff u from acquiring x, which can be interpreted as them consuming it in markets for goods or holding/investing it in markets for assets. Under either interpretation, P and M can hold x, but only 0 or 1 unit at a time (this is relaxed below). The costs of holding/storing x for P and M are γp and γm , which are positive in markets for goods and negative in markets for assets. In the latter case we use ρm = −γm > 0 to denote the flow return on asset holdings. The other tradeable object y is used as a payment instrument when acquiring x. It can be produced by anyone at unit cost and consumed by anyone for utility U (y). As a benchmark we set U (y) = y, which means transferable utility; the general case is studied in the Appendix. Type P agents always have 1 unit of x, while M can have 0 or 1 in inventory. Let μ be the fraction of M holding x. This increases at rate np nm (1 − μ) (the measure of P meeting M without x) and decreases at rate nc nm μτ , where τ is the probability M trades x to C (the measure of C meeting M with x and trading). Hence μ˙ = np nm (1 − μ) − nc nm μτ , and in steady state5 μ=

np . n p + nc τ

(1)

For now we focus on steady states; dynamics are studied in Section 7. Bargaining determines the terms of trade: agents i and j split the total surplus with θij denoting the share (bargaining power) of i, and θj i = 1 − θij , as follows from various common solution concepts (e.g., Nash or Kalai bargaining). Hence when they trade C pays ycp or ycm to P or M, while M pays ymp to P . The surplus when C trades with P , e.g., is u − ycp = θcp u, since ycp = θpc u, given that for both the continuation values and outside options cancel.6 To  keep track of payments, let y = ycp , ymp , ycm . Next let Vp be P ’s value function, let V0 or V1 be M’s value function  when he has 0 or 1 unit of x, let Vc and Vn = 0 be C’s and N ’s value functions, and let V = Vp , V0 , V1 , Vc , Vn . Eliminating the y’s from the V ’s using the bargaining solution, we get standard dynamic programming equations rVp = nc θpc u + nm (1 − μ)θpm (V1 − V0 ) − γp + V˙p rV0 = np θmp (V1 − V0 ) + V˙0

(2)

rV1 = nc τ θmc (u + V0 − V1 ) − γm + V˙1

(4)

rVc = np τ θcp u + nm μθcm (u + V0 − V1 ) + V˙c ,

(5)

(3)

where in steady state V˙j = 0. In words, (2) says: the flow value rVp is the rate at which P meets C times his share of the surplus; plus the rate at which he meets M without x times his share of 5 In deriving (1), we proceed as if P and M trade whenever they meet, which is not valid if γ > 0 is very big; but in m that case nm = 0, so the conditions are still correct. The situation is simpler when P and C meet, since C always wants x and P can always produce another unit, so they trade. The situation is more complicated when M and C meet, which is why we need τ . 6 This is because our all agents stay in the market forever, different from the original RW specification, where M stays but C and P exit after one trade; Nosal et al. (2015) argue that having them all stay reduces the algebra without affecting the results too much.

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that surplus; minus storage cost; plus (out of steady state) a capital gain V˙p . Similar stories apply to (3)-(5). Next, for the decision of a nonconsumer to act as type P or M, we assume that if he chooses M he starts without x for payoff V0 ; one interpretation is that if he produced x as a type P , he must sacrifice it to acquire the middleman technology. In any case, this implies   np > 0 ⇒ Vp ≥ max {V0 , 0} and nm > 0 ⇒ V0 ≥ max Vp , 0 (6) and, in particular, Vp = V0 if np , nm > 0. A steady state equilibrium is defined as a nonnegative list μ, V, n such that: μ satisfies (1); V satisfies (2)-(5); and n satisfies (6). Given μ, V, n we can compute payments y, the spread s = ycm − ymp , the stock of middleman inventories nm μ, etc. 3. Goods market equilibrium When γj > 0 it is immediate that τ = 1 (M with x always trades with C). Hence there are three possible outcomes. A class 0 equilibrium is one where np = nm = 0 and nn = 1 − nc , so the market shuts down. A class 1 equilibrium is one where np = 1 − nc and nm = nn = 0, with production but no intermediation. A class 2 equilibrium is one where np > 0, nm > 0 and nn = 0, with production and intermediation. In principle we can have nn > 0, np > 0 and nm > 0, too, but it only occurs in a measure 0 set of parameters, so it is ignored. We consider the other possibilities in turn, with detailed proofs in the Appendix. Consider first class 0 equilibrium, with np = nm = 0. This requires Vp ≤ 0 and V0 ≤ 0. When nm = 0, Vp ≤ 0 iff γp ≥ γ¯p ≡ nc θpc u, and V0 = 0 for all parameters. So class 0 equilibrium exists iff γp ≥ γ¯p , and obviously there are not multiple class 0 equilibria. However, unless parameters satisfy the condition in Lemma 1, a class 0 equilibrium violates subgame perfection and is ignored: Lemma 1. A (subgame perfect) class 0 equilibrium exists iff γp ≥ γ¯p and γm ≥ g(γp ), where g is defined in (9) below. When it exists it is unique. Consider next candidate class 1 equilibrium, with np = 1 − nc and nm = 0. This requires Vp ≥ 0 and Vp ≥ V0 , so that type P agents do not want to deviate and become type N or M. It is easy to check Vp ≥ 0 iff γp ≤ γ¯p , and Vp ≥ V0 iff γm ≥ f (γp ) ≡ γ¯m −

r + nc θmc + (1 − nc ) θmp (γ¯p − γp ), (1 − nc ) θmp

(7)

where γ¯m ≡ nc θmc u. Since (2)-(5) are linear, there cannot be multiple class 1 equilibria. Lemma 2. A class 1 equilibrium exists iff γp ≤ γ¯p and γm ≥ f (γp ), where f is defined in (7). When it exists it is unique. Consider finally class 2, with np , nm > 0. Here it is convenient to proceed using μ and later recover n. We need μ ∈ (0, μ), ¯ where μ¯ = 1 − nc . Now routine algebra reduces Vp = V0 to Q(μ) = 0, where Q(μ) = κ1 μ2 + κ2 μ + κ3 is obtained by replacing np and nm with their values in terms of μ, and the coefficients are

(8)

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Fig. 1. Equilibrium outcomes in (γm , γp ) space.

κ1 = θpm (γ¯m − γm ) κ2 = −[2(1 − nc )θpm + nc ](γ¯m − γm ) − (r + nc θmc − nc θmp )(γ¯p − γp ) κ3 = (1 − nc )θpm (γ¯m − γm ) + (r + nc θmc )(γ¯p − γp ). We seek μ ∈ (0, μ) ¯ such that Q(μ) = 0 and V0 ≥ 0. Now V0 ≥ 0 iff γm ≤ γ¯m , which implies κ1 > 0, and hence Q (μ) is convex. As shown by the curves Qa , Qb and Qc in the right panel of Fig. 1, there are three ways Q(μ) can have a solution in (0, μ): ¯ (a) one root with Q(0) < 0 < Q(μ); ¯ (b) one root with Q(0) > 0 > Q(μ); ¯ or (c) two roots. The Appendix rules out cases (a) and (c): Lemma 3. A class 2 equilibrium exists iff Q(0) > 0 > Q(μ). ¯ To see when the conditions in Lemma 3 hold, note that Q(μ) ¯ < 0 iff γm < f (γp ) where f is defined above, while Q(0) > 0 iff γm < g(γp ) where g(γp ) ≡ γ¯m +

r + nc θmc (γ¯p − γp ). (1 − nc )θpm

(9)

Also, since there is exactly one μ ∈ (0, μ) ¯ with Q (μ) = 0, and again (2)-(5) are linear, there cannot be multiple class 2 equilibria. Lemma 4. A class 2 equilibrium exists iff γm < f (γp ) and γm < g(γp ). When it exists it is unique. The results are shown in the left panel of Fig. 1, drawn with f (0) < 0, although f (0) > 0 is also possible. They accord well with intuition – e.g., intermediation requires γm not too high, naturally, but also requires γp neither too high nor too low, since P does not produce when γp is very high and produces but does not need M when γp is very low. The equilibrating force is this: When np increases, P is less likely to meet M, and when he does it is more likely M already has x. Hence raising np lowers Vp , which is why we get uniqueness. Intermediation can be essential in the sense used by monetary theorists: an institution like money is said to be essential if the set of outcomes that can be supported as equilibria expands when money is introduced. For money or intermediation the concept is nontrivial, since both are inessential in standard general equilibrium theory. Here, in the region where class 2 equilibrium

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exists with γp > γ¯p , production depends on middlemen: if we were to eliminate M, say by taxation, the market would shut down. We summarize these results as follows: Proposition 1. With γj > 0 equilibrium exists and is generically unique, as shown in Fig. 1. For some parameters intermediation is essential. Additional insights come from changing parameters in class 2 equilibrium. To that end the following is useful: Lemma 5. An increase in γp shifts Q (μ) down; an increase in γm shifts Q (μ) down if γp < γ¯p and up if γp > γ¯p . Based on this it is immediate that ∂np ∂μ ∂nm < 0, < 0 and > 0. ∂γp ∂γp ∂γp This accords well with intuition: if γp is higher, we get fewer producers, and so middlemen hold x with lower probability. Less intuitive is this: ∂np ∂μ > 0, > 0 and ∂γm ∂γm ∂np ∂μ γp > γ¯p ⇒ < 0, < 0 and ∂γm ∂γm γp < γ¯p ⇒

∂nm <0 ∂γm ∂nm > 0. ∂γm

The case γp > γ¯p is surprising: how can we get more middlemen when γm is higher? This is answered in Section 4. 4. Goods market efficiency Consider the planner problem with objective function rW = max np (nc u − γp ) + μnm (nc u − γm ) st np + nm = 1 − nc , μ,np ,nm

(10)

where the first (second) term is the gain from direct (indirect) trade.7 The Appendix shows: Proposition 2. The efficient outcome exists and is generically unique. It has nop = nom = 0 iff γp ≥ nc u and γm ≥ g o (γp ); nop > 0 = nom iff γp ≤ nc u and γm ≥ f o (γp ); and nop , nom > 0 iff γm < g o (γp ), f o (γp ); where g o (γp ) ≡

nc (u − γp ) −n2c u + γp and f o (γp ) ≡ . 1 − nc 1 − nc

(11)

Fig. 2 shows f o and g o , as well as the analogs from the equilibrium analysis f and g, for two examples. Notice the region where nom > 0 lies strictly below the 45o line, so for intermediation to be optimal we need γm < γp . In contrast, γm < γp may or may not entail nm > 0 in equilibrium, 7 Note that (10) can be obtained by summing payoffs over agents in steady state, but maximizing it is equivalent to solving the dynamic planner problem and then letting r → 0 (see Nosal et al., 2015 for a discussion in a related model).

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Fig. 2. Comparing equilibrium and efficient outcomes.

since the decision to become a middlemen depends on the ability to extract rents, captured by bargaining power. Perhaps more importantly, from these examples we know that equilibrium can have too many or too few middleman: equilibrium can have nm > 0 when the planner wants nm = 0 or vice versa; and even when the equilibrium and optimum both have middlemen, we do not get nm = nom in general. o = 1 and θ o = 1 to avoid Unsurprisingly, efficiency depends on the θ ’s. First we need θpc mc holdup problems associated with the costs γp and γm , which are sunk when P and M meet C. o there is also a holdup problem when P meets M, but now other forces arise. One is this: For θpm when agents act as type P they neglect making it harder for other P ’s to meet M’s and easier for M’s to meet P ’s, as usual in search theory. Another, more novel, force is this: higher np implies higher μ, making it harder for P to trade when they meet M. Balancing these forces leads to a version of efficiency results going back to Mortensen (1982) and Hosios (1990) applicable to our three-sided market. Proposition 3. Let S0o , S1o and S2o be the sets of γ ’s where the efficient outcome is class 0, class o = θ o = 1 and: (i) (γ , γ ) ∈ S o ⇒ θ o = 1; 1 and class 2, resp. Equilibrium is efficient iff θpc p m mc pm 0 o o (ii) (γp , γm ) ∈ S1 ⇒ θpm = 0; and (iii) (γp , γm ) ∈ S2o ⇒ o θpm =

(1 − μo )(1 − nc − μo ) ∈ (0, 1). (1 − μo )(1 − nc − μo ) + μo nc [1 − (nc u − γp )/(nc u − γm )]

Next consider how the optimum varies with γj . Similar to equilibrium, we have the natural results ∂μo /∂γp < 0, ∂nop /∂γp < 0 and ∂nom /∂γp > 0, plus the surprising results ∂nop ∂μo > 0, > 0 and ∂γm ∂γm ∂nop ∂μo < 0, < 0 and γ p > nc u ⇒ ∂γm ∂γm γp < n c u ⇒

∂nom <0 ∂γm ∂nom > 0. ∂γm

How can higher γm lead to more middlemen? To answer that we use this: Lemma 6. For all parameters, ∂(nom μo )/∂γm < 0.

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This says that an increase in γm always reduces the stock of inventories held by middlemen, nom μo , but there are two ways to make that happen. One is to reduce nom , which in steady state means higher μo ; the other is to reduce μo , which means higher nom . When γp < nc u it is optimal to use the extensive margin and reduce nom ; when γp > nc u it is optimal to use the intensive margin and reduce μo , which means higher nom . This explains the planner’s choices; the idea is similar for equilibrium. 5. Asset market equilibrium Suppose now that storing x is profitable. As a colorful example, suppose P is a painter, C is a collector and M is an art dealer, and that ρj = −γj > 0 because paintings generate payoffs simply from gazing at them, or from charging admission to M’s gallery. Will M trade x to C? Of course the answer depends on fundamentals – i.e., who values it more – but also on the ease or difficulty with which inventory can be replaced and that depends on equilibrium considerations. Indeed, one might say that liquidity is a key factor: the market can be said to be more liquid when it is easier to get x, which is the case when np is bigger, which is the case when τ is bigger, because that makes Vp higher. This is relevant for any asset with positive net returns, not only art. As a financial application, suppose P produces or otherwise obtains capital suitable for investment by either C or M. Then a type M agent with capital may not want to pass it on to C, again depending on fundamentals like ρm , but also on the strategies of others. This interpretation allows us to investigate the view (recall fn. 2) that financial institutions may be more unstable/fragile that other intermediaries, such as those in retail goods markets. As another application, x can be a interpreted as housing that provides utility as shelter, in which case M must decide whether to keep it as a residence or flip it to C. With this interpretation we can investigate the idea that real estate markets are susceptible to “hot and cold spells” depending on the speculations of flippers. Of course, the model is quite abstract, but we think it still provides insights into these kinds of applications. Moreover, our position is not that the only relevant distinction between goods or assets is γm > 0 or γm < 0; our position is that it is a distinction that is interesting because of the way it affects results.8 While τ = 1 is immediate for γm > 0, it is not for γm < 0, and so there are more candidate equilibria. We call nm = 0 and τ = 1 a class 1T equilibrium, with T indicating M trades x to C (in fact there are no type M agents on the equilibrium path, but off this path type M with x would trade it to C). Similarly, nm = 0 and τ = 0 is a class 1K equilibrium, with K indicating M keeps x. Also, nm = 0 and τ ∈ (0, 1) is a class 1R equilibrium, with R indicating M randomizes, but we ignore it because it can only be an equilibrium for nongeneric parameters. We also ignore nm = 1 − nc , which is uninteresting, because there is no production, and unnecessary for our purposes. Thus, in addition to 1K and 1T , the other relevant candidates all have nm ∈ (0, 1 − nc ) 8 It is clear that what matters is ρ > 0, not ρ > 0, but for symmetry here we assume both. Also, an editor suggested m p we mention that in reality there are primary and secondary asset markets: the former have buyers trading with originators, captured by P in the model, while the latter have them trading with others who previously bought the asset, captured by M. While some assets trade exclusively or mainly in secondary markets, we let C potentially get x from either P or M, the way one can get, e.g., T-Bills directly from the government or indirectly through a security dealer. Gong (2019) analyzes in greater depth a related model that determines endogenously whether C can buy from only P , from only M, or from both.

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Table 1 Candidate equilibria with ρj = −γj > 0. nm \τ

0

[0, 1]

1

0 (0, 1 − nc ) 1 − nc

1K 2K ×

× 2R ×

1T 2T ×

Fig. 3. Outcomes in (γp , γm ) and in (ρp , ρm ) space.

and either: τ = 1, which is a class 2T equilibrium; τ = 0, which is a class 2K equilibrium; or τ ∈ (0, 1), which is a class 2R equilibrium. See Table 1. Lemma 7 states some results formally, but they are perhaps better understood from Fig. 3, which in one panel has the negative quadrant of (γp , γm ), space and in the other the positive quadrant of (ρp , ρm ) space, which we find easier to interpret. While there are various issues that can be discussed using the results – e.g., as in the model with γj > 0, one can ask about the parameters that are more likely to make intermediation an equilibrium outcome – but we highlight the following: Suppose ρp is not too high, which is necessary for nm > 0. Then for ρm very high we get τ = 0 and for ρm very low we get τ = 1, naturally, but for ρm neither too high nor too low there coexist a 2K equilibrium with τ = 0, a 2T equilibrium with τ = 1, and a 2R equilibrium with τ ∈ (0, 1). Lemma 7. Define γm ≡ −u[r + (1 − nc )θmp ]  r + (1 − nc )θmp f(γp ) ≡ −(γ p − γp ) (1 − nc )θmp k(γp ) ≡ −ru − γ p + γp .

(12) (13) (14)

Also define  k(γp ) to be the lower root of the quadratic given in the proof in the Appendix. Then class 1T equilibrium exists iff γm ≥ max{ γm , f (γp )}; class 1K exists iff f(γp ) ≤ γm ≤  γm ; class

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2K exists iff γm ≤ min{k(γp ), f(γp )}; class 2R exists iff  k(γp ) < γm < k(γp ); and class 2T exists iff  k(γp ) ≤ γm ≤ f (γp ). We conclude that γj < 0 implies the liquidity of the market, and in particular whether x is passed on to C or “hoarded” by M, can be a self-fulfilling prophecy, i.e., not pinned down by fundamentals. This constitutes part (i) of Proposition 4. Part (ii) says that if n is fixed we get back uniqueness even with γj < 0, confirming that multiplicity requires both γj < 0 and endogenous np . Proposition 4. (i) With γj < 0 equilibrium exists. As shown in Fig. 3, ∀γp < γ˜p where γ˜p < 0, and γm neither too high nor too low class 2K , 2T and 2R equilibria coexist. (ii)If nm and np are fixed equilibrium is unique: ρm = −γm > r + np θmp u ⇒ τ = 0 and ρm < r + np θmp u ⇒ τ = 1. 6. Asset market efficiency One can emulate the method in Section 4 for the planner, but now we have to choose τ as well as nm , so the details are relegated to a Supplemental Appendix (see Appendix C for details), while here we report a few findings.9 First, depending on parameters nm can be too big or too small, as in the model with γj > 0. However, different from that model, with γj < 0 it is not possible to find θ ’s such that equilibrium is efficient even if we select the best equilibrium when there is multiplicity. To see this, we can show (in the Supplemental Appendix) that the borders between regions are the same for the equilibrium and planner iff θpc = θmc = θmp = 1, but with those θ ’s the values of nm in the 2T region are different. While in equilibrium nm can be too big or too small, there is no analogous result for τ . It is easy to show τ can be too small – i.e., we can have τ = 0 in equilibrium when the planner wants τ = 1 – but we did not find an example where τ = 1 in equilibrium when the planner wants τ = 0 (although we did not prove it is impossible). We understand the situation in terms of coordination. When τ = 0, M is willing to sell x to C in principle, but not in practice, because inventories take too long to replace. If every M were to change to τ = 1, however, Vp and hence np would rise, and then M would trade x to C since it would take less time to replace. Moreover, when equilibria with τ = 0 and τ = 1 coexist, the latter yields higher welfare. Although the result does not depend on this, it is easiest to see when θpc = θmc = 1, since then welfare is unambiguously measured by Vp = V0 in any equilibrium with nm > 0.10 In this case, note that the two equilibria imply 9 The method involves checking each (τ, n ) pair in Table 1 to determine the parameters for which it solves the planner m

problem. To be a solution (τ, nm ) must satisfy these conditions: a corner solution τ = 0 (τ = 1) requires rW decreasing in τ at τ = 0 (increasing in τ at τ = 1); a corner solution for nm requires something similar; and interior solutions for τ ∈ (0, 1) or nm ∈ (0, 1 − nc ) must satisfy standard FOC’s and SOC’s. After exhausting the candidates in Table 1, we can partition parameter space into regions as follows: for big ρp the efficient outcome is 1T or 1K as ρm is small or big; for small ρp it is 2T or 2K as ρm is small or big. The result is simple enough, and qualitatively the four outcomes are similar to the four classes of equilibria analyzed above, but the regions where they obtain are different. What is slightly harder is this: in some regions there is more than one (τ, nm ) satisfying the FOC’s, so we have to check the SOC’s, and also compare rW across local maximizers to find the global maximizer. 10 Coexistence requires n > 0, which requires V = V , and when θ = θ m p pc mc = 1 all the surplus goes to noncon0 sumers. When θpc < 1 or θmc < 1 consumers get some surplus, too, but the result goes through if we define welfare by rW , which aggregates rVc and rVp .

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rVp (τ = 0) = (1 − nc ) (nc u − γp ) rVp (τ = 1) = (1 − nc ) [nc u − γp + nm (1 − μ)(V1 − V0 )] Hence, rVp (τ = 1) > rVp (τ = 0), consistent with interpreting τ = 0 as a coordination problem. We summarize as follows: Proposition 5. Similar to goods markets, in asset markets the efficient outcome exists and is generically unique, and equilibrium can have nm too big or too small. Different from goods markets, we cannot set the θ ’s so that equilibrium is efficient. Also, when equilibria with τ = 0 and τ = 1 coexist the latter is better. 7. Asset market dynamics It is convenient here to work in (n1 , ) space, where n1 = nm μ is the measure of inventories held by M, which is predetermined at any point in time t , while = V1 − V0 represents (beliefs about) the value of inventories. Then the equilibrium conditions can be collapsed to a two-dimensional dynamical system in (n1 , ). For γj > 0, the Appendix shows the unique ¯ such that steady state is a saddle point: for any initial condition n1 = n¯ 1 there is a unique

¯ the system transits to steady state; and for any other

¯ the system follows an starting at n¯ 1 ,

explosive path. In other words, when γj > 0 equilibrium, not only steady state, is unique. For γj < 0, the dynamics are more complicated.11 To begin, as in the steady state analysis, let us assume type P can at any t become type M, but must start with 0 inventory. Next, since type C agents really make no decisions, we can ignore them and focus on rVp = nc θpc u + n0 θpm − γp + V˙p rV0 = (1 − nc − n1 − n0 )θmp + V˙0

(15)

rV1 = nc τ θmc (u − ) − γm + V˙1 .

(17)

(16)

The interesting case concerns nm > 0, where Vp = V0 . This holds iff n0 = n0 (n1 , ), where we can solve explicitly for γp − nc θpc u + (1 − nc − n1 )θmp .

Next, subtracting (16)-(17), we get n0 (n1 , ) =

˙ = r − nc τ θmc (u − ) + γm + (1 − nc − n1 − n0 )θmp .

(18)

Now τ = τ ( ) is 1, [0, 1] or 0 as − u is positive, 0 or negative, which we can insert along with n0 (n1 , ) into (18) to arrive at ˙ = r − nc τ ( ) θmc (u − ) + γm + [1 − nc − n1 − n0 (n1 , )] θmp .

(19)

Thus we get a differential equation for . The equation for n1 is n˙ 1 = n0 (n1 , ) [1 − nc − n1 − n0 (n1 , )] − n1 nc τ ( ) .

(20)

11 While some previous search models also display interesting dynamics – e.g., Diamond and Fudenberg (1989), Boldrin et al. (1993) or Mortensen (1999) – they hinge crucially on increasing returns in the matching or production technology; that is not the case here.

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Fig. 4. The phase plane with ρj > 0.

Then, as is standard, given an initial n¯ 1 equilibrium defined as a bounded and nonnegative path (n1 , ) satisfying (19)-(20) (boundedness comes from the transversality condition; see, e.g., Rocheteau and Wright, 2013). We know that for some parameters there are 3 steady states, as shown in Fig. 4. Notice in ˙ = 0 curve has a kink at = u, this phase plane the n˙ 1 = 0 curve has a flat spot and the

  L which is where τ switches from 0 to 1. The lower steady state, nL 1 , ,has < u and τ = 1;  H H M has = u the higher one, n1 , , has > u and τ = 0; and the middle one nM 1 ,

 H H  L L and τ ∈ (0, 1). One can check that n1 , and n1 , are saddle points, but the dynamics   M can be complicated. around nM 1 ,

To see this, consider an example with u = 1, r = 0.05, nc = 0.3, θpm = 0.5, θpc = θmc = 1, ρm = 0.36 and ρp = 0. As shown in the upper left panel of Fig. 5, the three steady states are apM H proximately nL 1 =0.05, n1 = 0.07 and n1 = 0.25. The upper right panel zooms in to show local M M dynamics around n1 , . Whether this converges to steady state, or to a small cycle around it, is hard to say from the numerical output, and checking local stability directly is hindered by the n˙ 1 = 0 curve being nondifferentiable at = u. But we can learn a lot numerically. Fig. 6 shows the time path of n1 in the example perturbed by reducing θpc to 0.1, starting with n1 close to nM 1 . Clearly, these equilibrium fluctuations do not settle down, and they not small. of equiMore generally, starting with any n¯ 1 in the neighborhood of nM 1 , there is a continuum  ¯ over some range, because all paths starting at n¯ 1 ,

¯ are bounded libria indexed the choice of

and nonnegative. After a shock to the system –e.g., an unexpected drop in μ for whatever reason  M as in Fig. 5 or 6. Hence, small changes – there are many equilibria that cycle around nM ,

1 in fundamentals can generate very volatile reactions. Also notice that while the fluctuations in

or np are not that big relative to their long-run averages, the fluctuations in n1 and n0 are around 10% and 20%, so while this is not a serious calibration, it is interesting that fluctuations in inventories are much bigger than output, a stylized fact about business cycles. Also, in Fig. 4, the left panel shows the stable manifolds of the two saddle points trapped inside the unstable manifolds. This means the stable manifolds must wrap around the middle steady state, either emanating from it or from a cycle around it. In either case, starting with ¯ and after any n¯ 1 in the neighborhood of nM by , 1 , there is a continuum of equilibria indexed   M , as well as a shock there are many different equilibrium paths that fluctuate around nM ,

1   L or paths starting on one of the stable manifolds and asymptotically approaching either nL 1 ,

 H H n1 , . The right panel is similar, except that for any n¯ 1 in a large range, depending on initial

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Fig. 5. Dynamics in an example with ρm > 0.

Fig. 6. Persistent fluctuations in equilibrium.

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      L or nH , H , or oscillate on its way to nM , M or beliefs, the system can transit to nL 1 ,

1 1 a cycle around it.12 Thus our intermediated asset market not only has multiple steady states, but dynamic indeterminacy and excess volatility (fluctuations in endogenous variables when fundamentals are constant). It is also subject to fragility, where a small change in fundamentals can lead to a structural change in equilibrium. Similar results appear in other models, but the economics is different, arising here from complementarities in trading strategies and endogenous market composition. 8. Robustness The results are robust on several dimensions. First, in the Appendix we extend the uniqueness result for γj > 0 to the case where we give multiplicity a fighting chance by going beyond linear utility and steady state. This is relevant because some search models display multiplicity if one considers nonlinear utility or dynamics, but not if one imposes linearity and looks only at steady state (e.g., Wright and Wong, 2014; Trejos and Wright, 2016); that is not the case here. One commentator conjectured that treating types C and M asymmetrically might be driving some results. We think our setup is natural, but it is true that: (i) C gets payoff u from x while M gets payoff ρm /r from x; and (ii) after acquiring it C can get another x, while M is restricted to {0, 1}. So, consider instead: (i) C gets ρc /r from x; and (ii) C also is restricted to x ∈ {0, 1}. This makes them symmetric, but without further modification, eventually all end users can end up with x and production shuts down. So, suppose that, when C acquires x, he leaves the market to be replaced by a new C (we could also let x depreciate as discussed below). To ease notation, set θpc = θmc = 1 and write rVp = nc ρc /r + nm (1 − μ)θpm (V1 − V0 ) + ρp rV0 = np θmp (V1 − V0 ) rV1 = nc τ (ρc /r + V0 − V1 ) + ρm . Now τ = 1 if ρc /r + V0 − V1 > 0, which reduces to r ρc > ρ ∗ ≡ ρm . r + np θmp Hence, M and C trade when C’s valuation is above ρ ∗ , not surprisingly, but there are two points worth mention: first, np θmp > 0 implies ρ ∗ < ρm , so it is not the case that M trades x to C iff ρc > ρm ; second, ρ ∗ depends on the endogenous np , not merely parameters. Other commentators conjectured that we do not need type M, as similar multiplicities can occur with only P and C. That is more delicate. One approach is to eliminate M and let agents choose to be either C or P , with np + nc = 1. Letting τ now denote the probability P and C trade, we have 12 One can also construct stochastic (sunspot) equilibria using standard methods. Consider a random variable σ that affects nothing fundamental but could affect behavior if agents believe it will. According to a Poisson process σ switches from σ1 to σ2 at rate η1 and switches back at rate η2 , and at each switch the economy jumps from one (bounded) trajectory in Fig. 4 to another. Now given rational expectations about the jumps, Fig. 4 would actually change, but it is qualitatively similar if the η’s are not too big. In particular, when the phase plane looks like the right panel, one equilibrium has the economy jumping between paths approximately given by the stable manifolds of the steady states with τ = 1 and τ = 0, with M’s behavior and the direction of the inventory path switching whenever crosses u. For more discussion see, e.g., Kaplan and Menzio (2016) who study different but related kinds of models.

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Fig. 7. Inventory flows.

  rVc = np τ 1 − θpc u rVp = (1 − np )τ θpc u + ρ. It is easy to check that as long as θpc = 0 the unique equilibrium has τ = 1. For θpc = 0, multiplicity emerges, because then rVc = np τ u and rVp = ρ, so we can have np ∈ (0, 1) as long as np τ u = ρ, and there are many combinations of (np , τ ) satisfying this condition. But that is not interesting, because it only works at θpc = 0, and it is payoff irrelevant. So we do need M in this version. A different conjecture by a referee is that a similar multiplicity may emerge without type M when we give type C a trade off between saving x and consuming it. This is plausible, since C consuming x is similar to M trading it away. To pursue it, suppose there are two types, type C that can save or consume x, and type P that can produce it or be a nonparticipant. We can confirm (in the Supplemental Appendix) this works: if C is more likely to consume x, there will be more C searching for x, which encourages participation by P and makes C more likely to consume. Multiplicity can emerge. This changed our views, given we previously thought middlemen were necessary for this kind of multiplicity; they are not. Yet it does not change a main conclusion: in models with middlemen, it is still true that multiplicity and complicated dynamics emerge only if γm < 0 and only if market composition is endogenous. Another extension is to let M hold inventories i ∈ {0, 1, ...I }, where 1 < I < ∞.13 Let n = (n0 , n1 , ...nI ), where ni is the measure of M with i units of x. As Fig. 7 shows, M moves up the distribution by one unit at rate εU = nP , since he trades whenever he meets P (we maintain that i = n τ + δ, P can only produce one unit per meeting). Also, M moves down one unit at rate εD C i where τi ∈ {0, 1} indicates whether or not M with i units trades with C (we maintain that C only wants one unit per meeting), and each unit depreciates at Poisson rate δ. If Vi is the value function for M with i, then τi = 1 if u > i = Vi − Vi−1 . Now steady state solves n0 np = n1 (δ + nc τ1 ) ... ni np + ni (δ + nc τi ) = ni−1 np + ni+1 (δ + nc τi+1 ) ... nI (δ + nc τI ) = nI −1 np 13 One can consider I = ∞, too, but we think I < ∞ makes sense. The reason is that if there were an increasing marginal cost or decreasing marginal benefit of inventories, and M could choose I , he would choose I < ∞ (see Shevchenko, 2004). Also, for this extension we assume M types never meet each other. With i ∈ {0, 1} this does not matter, since there are no gains from trade between M’s, but here it would matter and might be worth studying (see Afonso and Lagos, 2015 for something along these lines applied to banks in the Fed Funds market).

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The first equation implies n1 = n0 np /(δ + nc τ1 ). Then the second reduces to n1 np = n2 (δ + nc τ2 ), or n2 =

n0 n2p n 1 np = . (δ + nc τ2 ) (δ + nc τ1 ) (δ + nc τ2 )

Continuing in this way ∀i ∈ {1, 2...I } we get ni =

n0 nip i  

δ + nc τ j

j =1



(21)

.

Note the last equation holds automatically when the others hold, so we replace it with

ni =

i=0

1 − nc − np , or, n0 =

I 

1 − nc − n p I  nip 1+ i   i=1

j =1

(22)

.

δ+nc τj



This is the closed-form for n given any τ and np .14 Let ρ (i) = iρM be M’s return from holding i units, with ρm > 0, where ρ (i) is linear only to focus on nonlinearities coming from endogenous behavior. Then rV0 = np θ 1 ... rVi = iρm + nc τi θmc (u − i ) + np θmp i+1 − δ i

(23)

... rVI = Iρm + nc τI θmc (u − I ) − δ I . Thus M with i > 0 gets a flow iρm , gets a share θmc of the surplus when he trades with C, suffers from depreciation at rate δ, and increases i when he meets P as long as i < I . Similarly, we have rVp = ρp + nc θpc u +

I −1

ni θpm i+1 .

(24)

i=0

Note that equilibrium is generically unique when np is exogenous, because while τ affects n it does not affect (24). Also, as in the baseline model, multiplicity obviously requires ρm > 0. It is not hard to solve the model numerically and look for multiplicity with np endogenous and ρm > 0. First note that Vi is nonlinear in i, and to see why this matters, consider a special case that we call reservation equilibria, defined as follows: for some i ∗ ∈ {1, 2, ...I }, τi = 1 iff i ≥ i ∗ . One might think all equilibria have this property, which would be true if Vi were linear, but Vi is not linear. Fig. 8, drawn for I = 5, shows i = Vi − Vi−1 , and note that when i is 14 The process for i is reminiscent of the one for currency holdings in monetary models with m ∈ {0, 1, 2, ...M}, like Green and Zhou (1998), Camera and Corbae (1999) or Berentsen (2002). However, those papers need to justify agents’ decisions to trade 1 unit of money at a time, while here trading 1 unit of x is guaranteed by technology and preferences.

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Fig. 8. Equilibria at ρm = 0.05 and ρm = 0.07.

Fig. 9. All equilibria with I = 100 and reservation equilibria with I = 50. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

increasing (decreasing) Vi is convex (concave); hence Vi starts convex and becomes concave as i increases.15 In the left panel, with ρm = 0.05, τ = (1, 0, 1, 1, 1) is an equilibrium, and in fact the unique equilibrium for these parameters, but it is not a reservation equilibria (M trades when i = 1 but not when i = 2). In the right panel, with ρm = 0.07, τ = (0, 0, 1, 1, 1) is a reservation equilibrium. Also shown are histograms for n and the value of np . It is not hard to get multiplicity. The left panel of Fig. 9 shows the result of checking all 2I candidate equilibria for I = 100 for each point on a grid in (ρm , δ) space.16 The blue region has one equilibrium, the green has two, and yellow three. In this case they all happen to be reservation equilibria, implying multiplicity even within the special class, but one can also find reservation and other equilibria coexisting. The right panel checks for only reservation equilibrium when I = 50. For every i ∗ ∈ {1, 2, ...50} on the horizontal axis, the values of ρm that support a reservation equilibrium with that i ∗ are shown by bars. For low ρm there is a reservation equilibrium with i ∗ = 0, so M trades with C whenever he can. For high ρm there is a reservation equilibrium with i ∗ close to I , so M rarely trades with C. Note that the overlap of the bars again demonstrates 15 Parameters for this example, and for those below, unless indicated otherwise, are u = 1, r = 0.01, δ = 0.09, n = c 0.25, θmp = 0.8, θpc = θmc = 1, ρp = 0 and ρm = 0.05. 16 The parameter values are the same as above except δ = 0.11 in the left panel and δ = 0.10 in the right panel, mainly to make the pictures look nicer.

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multiplicity. Also, note that for intermediate i ∗ no value of ρm supports reservation equilibria, and so reservation equilibria are not pervasive. The intuition is similar to I = 1. First, one can show higher τi lowers ni and increases nj for j = i. Also, setting τi = τ ∀i and increasing τ twists the inventory distribution – i.e., there is an iτ such that ni decreases for i > iτ and increases for i ≤ iτ . So when M is more inclined to trade with C, in the sense that τi = 1 for more values of i, a random middleman is more likely to have lower inventory. This increases profit for P , and to re-establish Vp = V0 we need to raise np , which encourages M to trade with C more often. This strategic effect makes multiplicity possible even when I is big, and shows that insights from our tractable benchmark model go through for I > 1. 9. Conclusion This project continued the development of search-based theories of intermediation, with emphasis on endogenizing the measures of middlemen and producers, nm and np . The analysis delivered clean results on existence, efficiency and dynamics, with some predictions that were surprising, until we understood the theory better, like the possibility of ∂nm/∂γm > 0. A result we find especially interesting is this: absent devices like increasing returns, in models where nm and np are endogenous, equilibrium is unique and simple if holding inventories involves storage costs, as in goods markets; but multiplicity and complicated dynamics emerge if holding inventories involves positive returns, as in asset markets, if we endogenize the composition of the market. There are many extensions and applications one can imagine for future work, but we think the findings here already have taught us a lot about search theory and intermediation. Appendix A. Payment frictions and dynamics To begin let us suppose U  (y) < 0. This is interesting because now we can have U (y) < y, implying the cost to the payer exceeds the value to the payee, which discourages intermediation because it requires two payments, M to P and C to M, rather than one, C to P . Thus we can interpret the model in terms of payment frictions. Given that, we can show uniqueness in goods markets also holds with nonlinear utility. For tractability, let us use Kalai’s (1977) “proportional” bargaining solution, which has several advantages in related models (Aruoba et al., 2007). To reduce notation, set θpc = θmc = 1, which is fairly innocuous since type C really plays a very minor role anyway, and convenient because it implies Vc = 0 and ycp = ycm = u. Also, let z ≡ U (u) and write the dynamic programming equations as rVp = nc z + n0 U (ymp ) − γp + V˙p 1 − θpm rV0 = (1 − nc − n1 − n0 ) U (ymp ) + V˙0 θpm rV1 = nc (z − ) − γm + V˙1 .

(25) (26) (27)

The definition of equilibrium is similar, but the analysis is harder. To begin, let us first analyze steady state, then take up dynamics. Here are generalizations of some basic results for U (y) = y: Lemma 8. A (subgame perfect) class 0 equilibrium exists iff γp ≥ nc z and γp ≥ G(γm ), where G(γm ) ≡ nz + U (y0 )(1 − n), and y0 is given by the bargaining solution for ymp at μ = 0.

(28)

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  Fig. 10. Equilibrium in ymp , μ space.

Lemma 9. A class 1 equilibrium exists iff γp ≤ nc z and γp ≤ F (γm ), where F (γm ) ≡ nc z − U (y)(1 ¯ − nc )

(1 − θpm ) , θpm

(29)

and y¯ is the bargaining solution for ymp at μ = μ. ¯ ˜ (0, y0 ) > 0 > Q ˜ (μ, ˜ is defined in the Lemma 10. A class 2 equilibrium exists iff Q ¯ y), ¯ where Q proof. The main complication comes from the fact that, with a general U (y), we  cannot eliminate ymp from the above conditions. Hence, we work with two curves in ymp , μ space, one representing bargaining and one representing the choice to be type M or P . Setting V0 = Vp implies a quadratic that solves for     

2θpm (1 − nc ) + nc U ymp + θpm nc z − γp − D˜   (30) μ= 2θpm U ymp where D˜ is the discriminant. This defines a function μ = O(ymp ), with O for “occupational choice.” One can check ∂O/∂ymp  −(nc z − γp ), where  a  b means a and b have the same sign. As shown in Fig. 10, this traces a curve in ymp , μ space that slopes up or down, depending ¯ on the sign of nc z − γp , but in any case limymp →∞ O(ymp ) = μˆ ∈ (0, μ). Next, we solve for V1 − V0 and eliminate it from the bargaining solution to get ymp = B(μo), where B is for “bargaining.” The result is θpm (nc z − γm ) − ϒ    , (31) θpm (nc z − γm ) − ϒ + nc 1 − θpm U ymp    

where ϒ ≡ (r + nc ) θpm ymp + 1 − θpm U ymp . This traces a downward-sloping curve. Now we have μ = B −1 (ym ) =

  ¯ iff F (γm ) < γp < G(γm ). Lemma 11. ymp = B (μ) and μ = O ymp intersect in (0, ∞) ×(0, μ) They never intersect more than once in (0, ∞) × (0, μ). ¯ In (γp , γm ) space, F is increasing and concave, G is decreasing and concave, and F (nc z) = G(nc z) = nc z.

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Proposition 6. With γj > 0 and U  < 0 equilibrium exists and is generically unique. Based on this it is not hard to show the results look a lot like Fig. 1 except now the boundaries of the different regions are nonlinear. In any case, the point it that uniqueness with γj > 0 extends holds to nonlinear U (y). Now consider going dynamics in class 2 equilibrium, still allowing U  < 0 and setting θpc = θmc = 1, so again ycp = ycm = u, Vc = 0 and z = U (u). Also, here it is more convenient to work with the n’s, rather than μ, so let the state variable be n1 , the measure of M with x, while n0 = nm − n1 . Then n˙ 1 = n0 (1 − nc − n1 − n0 ) − n1 nc .

(32)

The other state is = V1 − V0 capturing agents beliefs about the value of acquiring x. The bargaining solution is U (ymp ) = θpm [U (ymp ) − ymp + ].

(33)

While this system appears complex, it can be reduced to something manageable as follows: First, notice Vp = V0 ∀t implies V˙p = V˙0 ∀t . Then the dynamic programming equations imply that for nm ∈ (0, 1 − nc ) we have nc z + n0 U (ymp ) − γp − (1 − nc − n0 − n1 )

1 − θpm U (ymp ) = 0. θpm

(34)

They also imply ˙ − (1 − nc − n0 − n1 ) r = nc (z − ) − γm +

1 − θpm U (ymp ) = 0. θpm

Using (34) and simplifying this, we get ˙ = (r + nc ) + γm − γp + n0 U (ymp ).

(35)

Thus (32) and (35) deliver a two-dimensional dynamical system in (n1 , ), with n0 and ymp implicit functions of the state. As is standard, given an initial condition n¯ 1 , an equilibrium is a nonnegative and bounded path for (n1 , ) solving (32) and (35) (boundedness comes from transversality considerations discussed in, e.g., Rocheteau and Wright, 2013). For γj > 0 there is a unique steady state where ˙ = 0. These have slopes the curves solving n˙ 1 = 0 and

 

(n0 + nc ) + (1 − nc − n1 − 2n0 )(1 − θpm ) (1 − θpm )U  + θpm ∂

|n˙ =0 =  ∂n1 1 (1 − nc − n1 − 2n0 ) UU [(1 − θpm )(1 − nc − n1 ) − n0 ]θpm 

U (1 − θpm ) (1 − θpm )U  + θpm ∂



|˙ = . ∂n1 1 =0 (r + nc ) (1 − θpm )U  + θpm + U  (1 − θpm )(1 − nc − n1 )θpm ˙ = 0 is strictly positive, and while the slope of n˙ 1 = 0 can be One can check the slope of

˙ = 0 curve. Also, ∂ n˙ 1 /∂n1 < 0 positive or negative, when it is positive it is steeper than the

˙ and ∂ /∂n1 < 0. Hence the phase portrait looks like Fig. 11, and whether n˙ 1 = 0 slopes up   or ¯ such that starting at n¯ 1 ,

¯ the down, the steady state is a saddle point: ∀n¯ 1 there is a unique

system (n1 , ) transits to steady state, and otherwise the system follows an explosive path. In other words, given any initial n1 equilibrium and not only steady state is unique.

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Fig. 11. The phase plane with γj > 0.

Appendix B. Proofs of nonobvious results Lemma 1: Class 0 and class 2 equilibria coexist in the region where γp ≥ γ¯p and γm < g(γp ), but we claim the former is not subgame perfect. Notice γm < γ¯m in this region, and consider a class 0 candidate equilibrium. Suppose a nonparticipant deviates and produces. When he meets another nonparticipant, which happens with positive probability, that agent has a strict incentive to accept his good and act like type M because γm < γ¯m (i.e., it is not credible to think he would reject it). This constitutes a profitable deviation.  Lemma 3: There are three ways for a convex Q(μ) = 0 to have solutions in (0, μ): ¯ (a) one root with Q(0) < 0 < Q(μ); ¯ (b) one root with Q(0) > 0 > Q(μ); ¯ (c) two-roots, which requires (c1) Q(μ) ¯ > 0, (c2) Q(0) > 0, (c3) Q (μ) ¯ > 0, (c4) Q (0) < 0, and (c5) Q(μ∗ ) < 0, where  ∗ Q (μ ) = 0. Notice that Q(0) = (1 − nc )θpm (γ¯m − γm ) + (r + nc θmc )(γ¯p − γp ) Q(μ) ¯ = nc [r + nc θmc + (1 − nc )θmp ](γ¯p − γp ) − nc (1 − nc )θmp (γ¯m − γm ). In case (a), it is easy to see Q(0) < 0 iff γp > γ¯p + (1 − nc )θpm (γ¯m − γm )/(r + nc θmc ), and Q(μ) ¯ > 0 iff γp < γ¯p − (1 − nc )θmp (γ¯m − γm )/[r + nc θmc + (1 − nc )θmp ]. As these conditions are contradictory, case (a) cannot occur. Turning to case (c), (c1) ⇒ γp < γ¯p while (c2) ⇒ κ3 > 0 ⇒ γm < g(γp ), which is redundant given (c1) and that equilibrium requires that γm ≤ γ¯m . Also, (c3) and (c4) ⇒ r + nc θmc − nc θmp (γ¯p − γp ) nc r + nc θmc − nc θmp (γ¯p − γp ). γm < ψ(γp ) ≡ γ¯m + 2(1 − nc )θpm + nc γm > φ(γp ) ≡ γ¯m +

Finally, (c5) is equivalent to D > 0, where D is the discriminant of Q(μ). We now show r + nc θmc − nc θmp < 0 is necessary for (c3) and (c4). Suppose that r + nc θmc − nc θmp > 0. This implies φ  (γp ) < 0 and ψ  (γp ) < 0, and both of the lines γm = φ(γp ) and γm = ψ(γp ) go through (γ¯p , γ¯m ). Since equilibrium requires γm ≤ γ¯m and γp ≤ γ¯p , condition (c3) is violated, i.e., as illustrated in the left panel of Fig. 12, the intersection of conditions (c3) and (c4) is the empty set when γp ≤ γ¯p . Suppose now that r + nc θmc − nc θmp < 0. It is easy to show (c3) and (c4) are satisfied. The parameter set consistent with the conditions c(1), c(3) and

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Fig. 12. The functions φ(γp ) and ψ(γp ).

  c(4) is given by S1 ≡ {(γp , γm )|0 < γp ≤ γ¯p , φ(γp ) < γm < ψ γp }, shown in the right panel of Fig. 12 Similarly, let S2 be the set consistent with (c5). To characterize S2 , the discriminant of Q(μ), ˆ m |γp ) = κˆ 1 γm2 + κˆ 2 γm +κˆ 3 , where D, can itself be written as a quadratic in γm given γp , Q(γ κˆ 1 = nc 2 + 4nc (1 − nc )θpm θmp κˆ 2 = −2γ¯m [nc 2 + 4nc (1 − nc )θpm θmp ] −2nc (γ¯p − γp )[(r + nc θmc − nc θmp )(1 − 2θpm ) − 2θmp θpm ] κˆ 3 = γ¯m2 [nc 2 + 4nc (1 − nc )θpm θmp ] + (γ¯p − γp )2 (r + nc θmc − nc θmp ) +2nc γ¯m (γ¯p − γp )[(r + nc θmc − nc θmp )(1 − 2θpm ) − 2θmp θpm ]. ˆ is strictly convex. Also, it is straightforward to show that Q( ˆ γ¯m |γp ) < 0 ∀γp ∈ Since κˆ 1 > 0, Q ˆ is strictly convex and Q( ˆ γ¯m |γp ) < 0, S2 = ∅ ⇒ Q(0|γ ˆ [0, γ¯p ). Thus, since Q p ) > 0 ⇒ κˆ 3 > 0, as shown in the left panel of Fig. 13. ˆ m |γ¯p ) > 0 ∀γm ∈ [0, γ¯m ) and Q( ˆ γ¯m |γ¯p ) = 0. Since Q ˆ is conIt can be shown that Q(γ ˆ tinuous, Q(γm |γp ) > 0 for some γm < γ¯m if γ¯p − γp is small. The admissible set of γp ˆ m |γp ) > 0 is pinned down by the lower root of Q(γ ˆ m |γp ) = 0 being positive, for which Q(γ √ 2 − γm (γp ) = (−κˆ 2 − )/2κˆ 1 > 0, where  = κˆ 2 − 4κˆ 1 κˆ 3 > 0. One can show γm− (γp ) > 0 ⇒ κˆ 2 > 0 ⇒ γp > γ p with γ p ≡ γ¯p + γ¯m [nc + 4(1 − nc )θpm θmp ][(r + nc θmc − nc θmp )(1 − 2θpm ) − 2θmp θpm ]. ˆ m |γp ) > 0 is [0, γm− (γp )). Therefore, S2 = {(γp , γm )|γ < Given γp , the set of γm such that Q(γ p

γp < γ¯p , 0 < γm < γm− (γp )}. Suppose for a given γp there exists γm− (γp ) > 0 such that ˆ m− (γp )) = 0. Express the lower root as Q(γ nc [(r γm− (γp ) = λ(γp ) ≡ γ¯m +(γ¯p −γp )

+ nc θmc − nc θmp )(1 − 2θpm ) − 2θmp θpm ] − nc 2 + 4nc (1 − nc )θpm θmp

√ 

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ˆ m |γp ) and λ(γp ). Fig. 13. The functions Q(γ

One can show λ (γp ) > 0. The right panel of Fig. 13 depicts γm = φ(γp ), γm = ψ(γp ) and γm = ˆ ≡ D > 0 ⇒ γm < λ (γh ), a necessary condition for case (c) is λ (γp ) < φ  (γp ), λ(γp ). Since Q as in the right panel of Fig. 13. Hence, (c) requires S1 ∩ S2 = ∅ and λ (γp ) < φ  (γp ). The inequality implies (θmc − θmp )[nc + 4(1 − nc )θpm θmp ] < [nc (θmc − θmp )(1 − 2θpm ) − 2θmp θpm ] − {[nc (θmc − θmp )(1 − 2θpm ) − 2θmp θpm ]2 − (θmc − θmp )[nc + 4(1 − nc )θpm θmp ]}1/2 , ignoring terms with r that strengthen the inequality. This implies −1 + nc (θmc − θmp ) > 4nc θmc θpm + 4(1 − nc )θpm θmp (θmc + θpm ). But the LHS is negative and the RHS positive – a contradiction.  Lemma 5: We derive ∂Q (μ) = nc μ + θpm [2(1 − nc )μ − (1 − nc ) − μ2 ]. ∂γm One can check this vanishes when γp = γ¯p . Moreover,

∂ ∂Q(μ)

= nc + θpm 2(1 − nc ) − 2θpm μ > 0. ∂μ ∂γm μ=μ˜ Hence, ∂μ/∂γm > 0 if γp < γ¯p and ∂μ/∂γm < 0 if γp > γ¯p .  Proposition 2: There are two cases: First, if γm > nc u then nm > 0 cannot be efficient. In this case γp > nc u implies nop = 0 and γp < nc u implies nop = 1 − nc . Second, if γm < nc u then nm > 0 may or may not be efficient. Eliminating np and nm in (10), we reduce the problem to   μ 1 − nc − μ max μnc u − nc γp − μ γm . μ∈[0,μ] ¯ 1−μ 1−μ

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The derivative of this is proportional to Qo (μ) = (1 − μ)2 (nc u − γm ) + nc (γm − γp ). This implies the solution is μo = 0 when γm ≥ g o (γp ), and μo = μ¯ when γm ≥ f o (γp ), with f o and g o defined in (11). If γm < f o (γp ), g o (γp ) the solution is the unique μo ∈ (0, μ) ¯ solving Qo (μo ) = 0.  Lemma 6: As ∂(nm μ)/∂γm = ∂(nm μ)/∂μ × ∂μ/∂γm we need to sign ∂(nm μ)/∂μ. Notice nm μ = μ − nc μ/(1 − μ), which implies ∂(nm μ)  (1 − μ)2 − nc  γp − nc u, ∂μ where a  b means a and b have the same sign. When γp < nc u, ∂μ/∂γm > 0 and ∂(nm μ)/∂μ < 0, so ∂(nm μ)/∂γm < 0; when γp > nc u, ∂μ/∂γm < 0 and ∂(nm μ)/∂μ > 0, so again ∂(nm μ)/∂γm < 0.  Proposition 3: First, it is obvious that the efficient and equilibrium outcomes correspond in general only if θmc = θpc = 1, because that needed for γ¯p = γ¯m = nc u. With θmc = θpc = 1, we have −n2c u + [1 − θpm (1 − nc )]γp (1 − nc ) (1 − θpm ) nc u[nc + θpm (1 − nc )] − nc γp . g(γp ) = (1 − nc ) θpm

f (γp ) =

o = 1 then g(γ ) = g 0 (γ ); so for (γ , γ ) ∈ S o , n = no . If θ o = 0 then f (γ ) = f 0 (γ ); If θpm p p p m p p pm 0 j j o ∈ (0, 1) then γ ≤ f (γ ) implies γ ≤ f o (γ ) and so for (γp , γm ) ∈ S1o , again nj = noj . If θpm m p m p o . It is easy to check γm ≤ g(γp ) implies γm ≤ g o (γp ). Set θmc = θpc = 1 and μ = μo to get θpm o θpm ∈ (0, 1). 

Lemma 10: We need nc z > γm , Vp = V0 , and ymp is the bargaining solution. Now Vp = V0 iff   0 = Q˜ μ, ymp = κ˜ 1 μ2 + κ˜ 2 μ + κ˜ 3 = 0 where   κ˜ 1 = θpm U ymp     

κ˜ 2 = − 2θpm (1 − nc ) + nc U ymp − θpm nc z − γp     κ˜ 3 = θpm nc z − γp + θpm (1 − nc ) U ymp . ˜ ˜ ˜ μ, There are again three cases for Q(μ, ymp ) = 0: (a) one root with Q(0, y0 ) < 0 < Q( ¯ y); ¯ ˜ ˜ ˜ (b) one root with Q(0, y0 ) > 0 > Q(μ, ¯ y); ¯ and (c) two roots, requiring (c1) Q(0, y0 ) > 0, ˜ μ, ˜ ˜ (c2) Q( ¯ y) ¯ > 0, (c3) ∂ Q(μ, ymp |μ = μ)/∂μ ¯ > 0, (c4) ∂ Q(μ, ymp |μ = 0)/∂μ < 0, and (c5) ∗ ∗ ˜ ˜ Q(μ , ymp ) < 0, where ∂ Q(μ , ymp )/∂μ = 0. As in Lemma 3, case (a) is impossible. Also notice ˜ Q(0, y0 ) = θpm [nc z − γp + U (y0 )(1 − nc )] ˜ μ, ¯ − nc )(1 − θpm )]. Q( ¯ y) ¯ = nc [θpm (nc z − γp ) − U (y)(1 In case (c), (c1) implies γp < G(γm ) and (c2) implies γp < F (γm ). From (c3) and (c4), we have

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901

˜ ∂ Q(μ, ymp ) = 2θpm μU (ymp ) − (nc z − γp )θpm − U (ymp )[nc + 2(1 − nc )θpm ]. ∂μ We need this positive at μ = μ, ¯ which means γp > K ≡ nc z − U (y)n ¯ c /θpm , and at μ = 0, which means γp < nc z + U (y0 )[nc + 2(1 − nc )θpm ]/θpm . Given (c2), (c1) and (c4) are not binding. Also, (c2) and (c3) imply γp is between K and F , which holds iff θpm < (1 − 2nc ) / (1 − nc ). Assume this is true and consider (c5). To get μ∗ , solve ∂Q/∂μ = 0 to get ˜ ∗ , ymp )  −(nc z − γp )[(nc z − γp )θpm + 2nc U (ymp )(1 − 2θpm )] Q(μ −U (ymp )2 nc [1 + 4θpm (1 − θpm )(1 − nc )]. ˜ ∗ , ymp ) < 0. With a small abuse of notation, let Q(μ ˜ ∗ , ymp ) ≡ Q(γ ˜ p ) < 0 where We need Q(μ ˜ p ) = −θpm γp2 + 2nc [U (ymp )(1 − 2θpm ) + θpm z]γp − n2c z2 θpm Q(γ −n2c zU (ymp )(1 − 2θpm ) − nc U (ymp )2 [1 + 4θpm (1 − θpm )(1 − nc )]. ˜ p ) < 0. There are three possibilities: (c5.1) one For (c5) we seek the set of γp such that Q(γ ˜ ˜ ˜ ˜ ); (c5.3) two roots, which root with Q(K) < 0 < Q(F ); (c5.2) one root with Q(K) > 0 > Q(F   ∗ ˜ ˜ ˜  (γp∗ ) = 0. Given ˜ ˜ ˜ requires Q(K) < 0 < Q(F ), Q (K) > 0 > Q (F ), and Q(γp ) < 0, where Q γp = K and nc z − γp = U (y)n ¯ c /θpm , n2 ˜ Q(K) = −U (y) ¯ 2 c [1 + 2(1 − 2θpm )] − U (y) ¯ 2 nc [1 + 4θpm (1 − θpm )(1 − nc )] < 0. θpm Given γp = F (γm ) and nc z − γp = U (y)(1 ¯ − θpm )(1 − nc )/θpm , 2 ˜ )  −U (y) ] + nc θpm } < 0, Q(F ¯ 2 {(1 − θpm )(1 − nc )[1 + nc − θpm (1 + 3nc ) + 4nc θpm

for (1 − 2nc )/(1 − nc ) > θpm > 0. This rules out (c5.1) and (c5.2). To check (c5.3), consider Q˜  (γp ) = −2θpm γp + 2nc [U (ymp )(1 − 2θpm ) + θpm z]. ˜  (γp ) > 0 at γp = K, and Q ˜  (γp ) > 0 at γp = F (γm ). As Q ˜  (F ) > 0 violates (c5.3), there Now Q ∗ ∗ ˜ p ) < 0.  is no γp between K and F such that Q(γ Lemma 11: We need B and O to cross in (0, ∞) × (0, μ), ¯ plus γm < nc z. For μ ∈ (0, μ), ¯ ˜ μ, ˜ ˜ ¯ y), ¯ where y0 = B(0) and y¯ = B (μ). ¯ Now Q(0, y0 ) > 0 iff we check Q(0, y0 ) > 0 > Q( γp < G(γm ). At γm = nc z, bargaining implies y0 = 0 and  γp < G(γm ) becomes γp < nc z. As we lower γm , y0 rises, and we need γp < G (γm ). In γp , γm space G (γm ) traces a down˜ ward sloping and concave curve (see below), and Q(0, y0 ) > 0 to the left of γp = G (γm ). Then ˜ μ, Q( ¯ y) ¯ < 0 iff γp > F (γm ). At γm = nc z, bargaining implies y¯ =0, and γp > F (γm ) becomes γp > nc z. As we lower γm , y¯ rises, and we need γp > F (γm ). In γp , γm space, F (γm ) traces   ˜ μ, ymp = 0 iff a upward sloping and concave curve (see below). Hence ∃μ ∈ (0, μ) ¯ solving Q F (γm ) < γp < G (γm ). To check ymp > 0, note that ymp > y¯ ≥ 0. To check Vp = V0 ≥ 0, by construction V0 ≥ 0 if μ ≥ 0. To show B and O cannot cross more than once, write     A − (r + nc ) 1 − θpm U ymp     μ= ≡ μ∗ , A − r 1 − θpm U ymp with A = θpm(nc z − γm ) −  (r +  nc ) θpm ymp > 0, from the bargaining  solution.   Note μ > 0 ⇒ A > (r + nc ) 1 − θpm U ymp , and μ < μ¯ ⇒ A < (1 + r) 1 − θpm U ymp . Then

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  U  θpm nc z − γp ∂O(ymp ) =− (1 − μ) ∂ymp U D˜   nc 1 − θpm ∂B −1 (ymp )   =− [AU  + θpm (r + nc ) U ] ∂ymp [A − r 1 − θpm U ]2 If nc z < γp equilibrium is obviously unique. If nc z > γp , we claim ∂O/∂ymp > ∂B −1 /∂ymp when they cross. To verify this, insert μ = μ∗ to get     U  θpm nc z − γp n 1 − θpm ∂O(ymp )  ,  =− ∂ymp A − r 1 − θpm U D˜ ˜ Using (30) to replace D˜ and μ = μ∗ , we get where D˜ is the discriminant of Q.     U  θpm nc z − γp nc 1 − θpm ∂O(ymp )

   =− , ∂ymp A − r 1 − θpm U     2θ U [A−(r+n )1−θ U ]

where  ≡ 2θpm (1 − nc ) + nc U + θpm nc z − γp − pm A−r 1−θc U pm . Now A = θ (nc z − γm ) − (r + nc ) θy ∗ and U = U (y ∗ ) solves

pm

θ U [A − (r + n) (1 − θ) U ]2 + θ[A − r (1 − θ ) U ]2 [nz − γp + (1 − n)U ]   = [A − (r + n) (1 − θ ) U ][A − r (1 − θ ) U ]{[2θ (1 − n) + n] U + θ nz − γp } Routine algebra implies ∂O(y)/∂y − ∂B −1 (y)/∂y is proportional to   U θ nz − γp [A − r (1 − θ ) U ][U  r (1 − θ ) + (r + n) θ ] +[AU  + θ (r + n) U ]U {n[A − r (1 − θ ) U ] + 2θ n[(1 + r) (1 − θ) U − A]} Since (1 + r) (1 − θ) U > A > r (1 − θ) U , this is positive, establishing the result. Finally, for the properties of F and G, derive G (γm ) =

−θpm (1 − nc )U  (y0 )  

 <0 (r + nc ) θpm + 1 − θpm U  (y0 )

2 (1 − n )U  (y ) y  (γ ) −θpm c 0 0 m  

2 < 0. (r + nc ) θpm + 1 − θpm U  (y0 )     Similarly, F  (γm ) > Thus G (·) is decreasing and concave in γm , γp space or γp , γm space.   0 and F (γm ) > 0. Thus F (·) is increasing and convex in γm , γp space, or increasing and concave in γp , γm space. 

G (γm ) 

Lemma 7: For preliminaries, first solve for rVp =

(r + τ nc θmc + np θmp )(γ p − γp ) + nm (1 − μ)θpm (τ γ m − γm ) r + τ nc θmc + np θmp

np θmp (τ γ m − γm ) r + τ nc θmc + np θmp (r + np θmp )(τ γ m − γm ) , rV1 = r + τ nc θmc + np θmp rV0 =

and notice steady state implies

(36) (37) (38)

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np =

nc τ μ (1 − nc )(1 − μ) − nc τ μ and nm = . 1−μ 1−μ

903

(39)

We now consider each candidate equilibrium. Equilibrium 1K : In a candidate equilibrium with τ = 0 and nm = 0, (36)-(38) reduce to rVp = γ p − γp

(40)

rV0 = (1 − nc )θmp (V1 − V0 )

(41)

rV1 = −γm .

(42)

The best response condition for τ = 0 is V1D ≤ V1 , where V1D is the value to setting τ = 1 then reverting to the candidate equilibrium with V0 given by (41): (r + nc θmc )V1D = γ m − γm −

nc θmc (1 − nc )θmp γm . [r + (1 − nc )θmp ] r

Simplifying, τ = 0 is a best response iff γm ≤ γm where  γm is defined in (12). Similarly, for  nm = 0, Vp ≥ V0 iff γm ≥ f(γp ) where f γp is defined in (13). Hence, class 1K equilibrium exists iff f(γp ) ≤ γm ≤  γm . Equilibrium 1T : Consider next τ = 1 and nm = 0. The  response condition for τ = 1 is  best γm ≥  γm . For nm = 0, Vp ≥ V0 iff γm ≥ f (γp ), where f γp is defined in (7). Hence, class 1T equilibrium exists iff γm ≥ max{ γm , f (γp )}. Equilibrium 2K : Consider τ = 0 and nm ∈ (0, 1 − nc ), where the dynamic programming equations are the same as (40)-(42). The best response condition for τ = 0 is V1D ≤ V1 , where V1D is the value to setting τ = 1 and reverting to the candidate strategy with V0 = Vp . Algebra implies V1D =

r(γ m − γm ) + nc θmc (γ p − γp ) r(r + nc θmc )

.

It is easy to check V1D ≤ V1 iff γm ≤ k(γp ) where k(γp ) is defined in (14). The condition for nm ∈ (0, 1 − nc ), Vp = V0 , now implies np =

r(γ p − γp ) θmp (γm + γ p − γp )

.

Now np < 1 − nc is the binding condition for np ∈ (0, 1 − nc ), and that holds iff γm ≤ f(γp ). So class 2K equilibrium exists iff γm ≤ min{k(γp ), f(γp )}. Equilibrium 2R : Consider next τ ∈ (0, 1) and nm ∈ (0, 1 −nc ). Now τ ∈ (0, 1) iff rV1 = −γm iff γm = −[r + (1 − nc − nm )θmp ]u. Solve this for nm =

γm + [r + (1 − nc )θmp ]u . uθmp

One can check nm ∈ (0, 1 − nc ) implies τ=

(γm + ru)(γm + ru + γ p − γp ) nc u[γm + ru + θmp (γ p − γp ) + θpm θmp (1 − nc )u]

.

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We now obtain the set of parameters such that τ ∈ (0, 1) and nm ∈ (0, 1 − nc ). To see when τ > 0, denote the denominator of τ by D. There are two possibilities, D < 0 and D > 0. The former can be shown to be inconsistent with τ > 0, so we are left with D > 0, which holds iff   γm >  γp ≡ −ru − θmp (γ p − γp ) − θpm θmp (1 − nc )u.   Given this, τ > 0 if γm < k(γp ). So τ > 0 iff  γp < γm < k(γp ). Also, using D > 0, algebra implies τ < 1 iff Q1 (γm ) < 0 where Q1 (γm ) = a1 γm2 + b1 γm + c1 < 0, with a1 = 1, b1 = γ p − γp + 2ru − nc u c1 = ru(γ p − γp + ru) − nc u[ru + θmp (γ p − γp ) + θpm θmp (1 − nc )u]. We cannot sign b or c, but can show Q1 (γm∗ ) < 0, where γm∗ solves Q1 (γm∗ ) = 0. Hence, there are three possibilities for Q1 (γm ) < 0, all of which reduce to γm >  k(γp ) where  k(γp ) is the lower root of Q1 (γm ) = 0: k(γp ); 1. if γm∗ > 0 then Q1 (0) < 0 and so Q1 (γm ) < 0 iff γm >  2. if γm∗ < 0 and Q1 (0) < 0 then Q1 (γm ) < 0 also implies γm >  k(γp ); 3. if γm∗ < 0 and Q1 (0) > 0 then Q1 (γm ) < 0 implies γm+ > γm >  k(γp ), where γm+ is the upper root of Q1 (γm ) = 0, but τ > 0 implies γm+ > γm is not binding. All these  possibilities imply, given the other conditions, that τ < 1 iff Q1 (γ  m ) < 0 iff γm > k(γp ). We claim that when this holds, the earlier condition γm ≥  γp is not binding. To see this, γp , −ru − (1 − nc )θmp u) and (γ p + (1 − nc )θpm u, −ru), and  k(γp ) notice  k(γp ) intersects  at (   k(γp ). In sum, is increasing and concave. Thus  k(γp ) >  γp , so the binding constraint is γm ≥  k(γp ) ≤ γm ≤ k(γp ). class 2R equilibrium exists iff  Equilibrium 2T : Consider τ = 1 and nm ∈ (0, 1 − nc ). We first solve Vp = V0 for μ and check μ ∈ (0, μ), since that is equivalent to nm ∈ (0, 1 − nc ), where μ = 1 − nc . By (36)-(37), Vp = V0 iff (r + nc θmc + np θmp )(γ p − γp ) + [nm θpm (1 − μ) − np θmp ](γ m − γm ) = 0. Using (39) to eliminate np and nm , then simplifying, we get Q2 (μ) = a2 μ2 + b2 μ + c2 = 0 where a2 = θpm (γ m − γm ) b2 = −(r + nc θmc − nc θmp )(γ p − γp ) − [nc + 2(1 − nc )θpm ](γ m − γm ) c2 = (r + nc θmc )(γ p − γp ) + (1 − nc )θpm (γ m − γm ). We need to check when the solution Q2 (μ) = 0 is in (0, μ). Now Q2 (μ) can have one or two roots. Since Q2 (0) > 0 and μ > 0, the one root case corresponds to the lower-root of Q2 (μ), say μ− . To analyze μ− , first, notice μ− > 0 requires γm > γ m + (r + nc θmc )(γ p − γp )/[(1 − nc )θpm ], which is non-binding. Second, μ− < 1 − nc requires γm < f (γp ), which holds iff Q2 (μ) < 0. This last result implies a two-root result is impossible because we have Q2(0) > 0 and Q2 (μ) < 0. Hence only one root of Q2 (μ) = 0 can occur. Finally, we check the best response condition for τ = 1, which reduces to γm ≥ −(r + np θmp )u, or

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μ≤

905

γm + ru . γm + ru − nc θmp u

Substituting μ using μ− and solving for γm yields Q1 (γm ) = a1 γm2 + b1 γm + c1 ≤ 0, as in class k(γp ). Hence, class 2T 2R equilibrium. So the set of γm consistent with Q1 (γm ) ≤ 0 is γm ≥  equilibrium exists iff  k(γp ) ≤ γm ≤ f (γp ).  Proposition 4: Part (i) of the result follows directly from Lemma 7. For part (ii), for fixed np and nm we have rVp = nc θpc u + nm θpm (1 − μ) (V1 − V0 ) − γm rV1 = nc θmc τ (u + V0 − V1 ) − γm rV0 = nc θmp (V1 − V0 ) . Consider a candidate equilibrium with τ = 0. Then μ = 1 and the above equations imply −nc θmp γm γm  and V1 = V0 =  . r r r + nc θnp

  A deviation by M to τ = 1 implies rV1d = nc θmc u + V0 − V1d − γm . After inserting V0 we get   r r + nc θmp (nc θmc u − γm ) − n2c θmc θmp γm   . V1d = r r + nc θmp (r + nc θmc ) The deviation is not profitable, and hence τ = 0 is an equilibrium, iff V1d ≤ V1 . This reduces to  −γm ≥ r + nc θmp u. Now consider a candidate equilibrium with τ = 1. Then we solve in the usual way for   r + nc θmp (nc θmc u − γm )  .  V1 = r r + nc θmc + nc θmp A deviation to τ = 0 implies V1d = −γm /r. This is not profitable, and hence τ = 1 is an equilib  rium, iff V1d ≤ V1 . This reduces to −γm ≤ r + nc θmp u. Equilibrium is generically unique.  Appendix C. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/ j.jet.2019.07.001. References Afonso, G., Lagos, R., 2015. Trade dynamics in the market for federal funds. Econometrica 83, 263–313. Aruoba, S., Rocheteau, G., Waller, C., 2007. Bargaining and the value of money. JME 54, 2636–2655. Berentsen, A., 2002. On the distribution of money holdings in a random-matching model. Int. Econ. Rev. 43, 945–954. Biglaiser, G., 1993. Middlemen as experts. RAND 24, 212–223. Boldrin, M., Kiyotaki, N., Wright, R., 1993. A dynamic equilibrium model of search, production and exchange. JEDC 17, 723–758. Burdett, K., Coles, M., 1997. Marriage and class. QJE 112, 141–168. Camera, G., 2001. Search, dealers, and the terms of trade. RED 4, 680–694. Camera, G., Corbae, D., 1999. Money and price dispersion. IER 40, 985–1008. Cavalcanti, R., Wallace, N., 1999. A model of private banknote issue. RED 2, 104–136. Diamond, P., 1982. Aggregate demand management in search equilibrium. JPE 90, 881–894.

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