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Search and leisure with idiosyncratic endowment shocks in a random-matching model
Journal of Macroeconomics 27 (2005) 385–402 www.elsevier.com/locate/jmacro
Search and leisure with idiosyncratic endowment shocks in a random-matching model J.J. Arias
*
Department of Economics and Finance, Georgia College and State University, Campus Box 014, Milledgeville, GA 31061, United States Received 23 October 2000; accepted 27 February 2004 Available online 5 July 2005
Abstract Agents who realize idiosyncratic endowment shocks (High and Low) choose between leisure and searching for a trading partner. In the non-monetary economy there are two cases: everyone enjoys leisure or everyone enters the market to barter. In the first case, money can generate an equilibrium in which all agents enter the market. In the second, money can generate an outcome impossible without money, in which High agents trade while Low agents rest. Money affects the real side of the economy by driving Low quality goods out of the market, improving the quality of traded goods. Ó 2005 Elsevier Inc. All rights reserved. JEL classification: E40; E42 Keywords: Search; Money; Barter; Leisure
1. Introduction An important characteristic of money as a social convention or mechanism is that it generates behavior that would not occur in its absence. In the present model, *
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0164-0704/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2004.02.004
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money can generate outcomes that are impossible without money. If all agents derive utility outside of the market in a non-monetary economy, money can induce welfareimproving market activity. On the other hand, if all agents engage in barter, money can induce agents with inferior (Low) goods to leave the market. In this second case, the use of money has a real effect by driving Low goods out of the market, improving the quality of traded goods. The theoretical innovation is that agents are heterogeneous in a way other than the usual symmetric specialization in tastes and endowments or production necessary to motivate exchange.1 Agents who receive either a High or Low quality endowment choose between searching for a trading partner in the market or consuming leisure outside the market. The tradeoff between leisure and trade is affected not only by the coincidence of wants and the relative value of consumption to leisure, but also by the presence or absence of High and Low agents in the market. The environment is similar to that found in Ritter (1995) in which non-market activity and an endogenous search decision also play an important role. In the non-monetary economy there are two possible equilibria, depending on the value of leisure. If the utility of leisure is greater than the expected utility of searching when both Low and High agents are present in the market, everyone chooses to enjoy leisure. If the utility of leisure is less, everyone chooses to search for a trading partner. There is no barter equilibrium in which Low agents stay out of the market while High agents enter, which would be the first-best outcome. There are two monetary equilibria corresponding to the two cases in the non-monetary economy, one in which Low good holders join High good holders and enter the market to trade, and another in which only Low good holders enjoy leisure outside the market. The existence of a monetary equilibrium is affected by the difference in consumption value between High and Low goods, the probability of receiving a High endowment, and the discount rate. When a buyer encounters a seller with a Low good, he must decide whether to accept the ‘‘bird-in-the-hand’’, or pass it up in the hopes of finding ‘‘two-in-thebush’’ next period. The first equilibrium requires that buyers accept Low goods instead of waiting to meet a seller with a High good. The second equilibrium requires the opposite, that buyers reject Low goods for the future possibility of meeting a seller with a High good. Both equilibria require that the quantity of money falls within a specific interval. An increase in either the discount rate or the quality difference between High and Low goods causes the interval for the first equilibrium to decrease and the interval for the second to increase. In the second equilibrium money has a real effect by driving Low good holders out of the market. This paper is closely related to the random-matching models pioneered by Kiyotaki and Wright (1991, 1993), especially papers that emphasize an endogenous
1 There is a related literature dealing with heterogeneous agents in the various search and matching models, that Burdett and Coles (1999) summarize and apply to labor markets. The agent heterogeneity in the present model is closer to what they define as ex ante heterogeneity in which the differences between agents are manifest before matches take place.
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search decision.2 For instance Li (1994, 1995, 1997) allows agents to choose their search intensity, which is the argument of a convex cost function. In Burdett et al. (1995) buyers and sellers choose between active search and passive waiting. Although ‘‘movers’’ have a higher probability of exchange than ‘‘stayers’’, they incur a disutility of moving or search.3 The decision to search or rest in this model is similar to the decision to search or wait in Burdett et al. (1995), with two significant differences. One is that in the present model active agents are sampling out of all other trading agents only, so that agents who enjoy leisure never meet another agent. This contrasts with their model in which agents who choose not to search could still meet other agents who are searching. Also, the present model potentially dichotomizes the search decision between High and Low good holders, whereas their model dichotomizes the search decision between buyers and sellers. The second difference is the more important of the two because it gives agents a different decision margin. The paper is organized as follows. Section 2 discusses the environment and assumptions of the model and considers a non-monetary economy. Section 3 introduces fiat money and examines two monetary equilibria. Section 4 addresses robustness of the monetary equilibria to the introduction of lotteries or randomized trading, and Section 5 concludes.
2. The environment and non-monetary economy The environment is similar to the steady-state model of Ritter (1995). There is an endowment economy inhabited by a continuum of infinitely lived agents with unit mass. Time is discrete with a discount factor d 2 (0, 1). At the beginning of each period, agents realize an idiosyncratic endowment shock U 2 {UH, UL}, where UH > UL > 0, and agents receive the High endowment with probability p. Within a particular period, p represents the proportion of the population that has received a High endowment. Endowment goods are perishable across periods, and both goods and money exist as indivisible units. Agents are constrained to holding either one unit of money or one unit of a good, and each agentÕs holdings are observable to others. In addition to varying in consumption value, endowment goods are divided into N P 3 types, with agents specialized in consumption as well as endowments. In addition to being endowed with only one type of good, agents receive utility from only one type of good. Preferences are equally distributed amongst endowment types so that the coincidence of wants is x 1/N. Distributing endowments and preferences in this manner simplifies the model since all goods will be equally acceptable 2
One interesting feature of these models is that the search decision has external effects, relating them to the search and matching models with trading or search externalities (Diamond, 1982; Hosios, 1990; Mortenson, 1982, 1992; Pissarides, 1984; Shi, 1997). 3 Kultti (2000) also develops an interesting model where agents search or wait, and prices are formed through both auctions and bargaining but there is no money in his model.
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in trade. It is also assumed that agents cannot consume their own endowment.4 Agents also receive utility from rest and leisure, denoted by R. Each period agents must choose between entering the market for the possibility of exchange (SEARCH) or staying out of the market and consuming leisure (REST). If they enter the market, agents meet one another according to the arrival rate b, equal to the fraction of agents entering the market.5 This fraction is an endogenous variable and will vary depending on whether no one, everyone or only agents with either a High or Low endowment participate in the market. Consequently, in the nonmonetary economy b 2 {0, p, 1 p, 1}. Since money and endowments are indivisible, agents will either ACCEPT or REJECT their partnerÕs money or good. Finally, assume that x2UH > R > x2UL. Define V as the agentÕs value function after observing their endowment shock. Since agents face an infinitely repeating one-period problem, V is represented by V ¼ maxfbx2 U; Rg þ dE½V ; H
L
ð1Þ H
L
where U 2 {U , EU, U , 0} and EU pU + (1 p)U , so the probability of an exchange opportunity is the arrival rate times the familiar double coincidence of wants and the expected value of exchange U depends on who else decides to enter the market. Equilibrium in this environment depends on the utility of leisure relative to the expected utility of barter when both High and Low agents enter the market.6 If R > x2EU, then the unique sub-game perfect Nash equilibrium is for everyone to REST. If everyone else chooses SEARCH, the best response is REST since R > x2EU. If only agents with a High endowment choose SEARCH, the best response for everyone is REST since R > x2EU > x2pUH. If only agents with a Low endowment choose SEARCH, again the best response is REST since R > x2EU > x2(1 p)UL. However, if everyone chooses REST, the best response is obviously to REST. If x2EU > R, then there is a sub-game perfect Nash equilibrium in which everyone chooses to SEARCH. Given that everyone else has entered the market, the best response is to also enter, so that b = 1 and V = x2EU + dV. There is also a coordination failure equilibrium in which no one enters the market, but the paper hereafter ignores this equilibrium. The first-best outcome would be for High agents to SEARCH and for Low agents to REST. In that case, expected utility for a single period is px2UH + (1 p)R, which is greater than either R or x2EU given the present assumptions. To see this, note that the condition px2UH + (1 p)R > R reduces to x2UH > R, and the condi-
4 This assumption is the analog for an endowment economy to the standard assumption in search theory that agents cannot consume their own output (Kiyotaki and Wright, 1991, 1993; Diamond, 1982, 1984). 5 In an earlier version of the paper, the arrival rate was always equal to one, making the second monetary equilibrium unnecessarily complicated. The present assumption about the arrival rate simplifies the equilibrium considerably without altering any of the central results. It is also consistent with the idea that trade becomes more difficult as the market becomes ‘‘thinner.’’ In any case, the assumption about the arrival rate is one about market structure, which is beyond the scope of this model. 6 All ‘‘ties’’ will go to SEARCH or market activity.
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tion px2UH + (1 p)R > x2EU reduces to R > x2UL. Unfortunately this outcome is not possible in the non-monetary economy. Given that Low agents choose REST, High agents will SEARCH if and only if x2pUH > R, in which case a Low agent would also want to SEARCH. One question is whether the use of money can generate an equilibrium in which High agents SEARCH and Low agents REST.
3. Monetary equilibria We now consider the case in which a constant fraction M of the population holds fiat money. Agents go through the following sequence each period. First, all agents observe the endowment shock. Second, agents with a unit of money decide whether to hold money or hold their endowment. Third, all agents decide whether to SEARCH or REST in that period.7 Fourth, agents who enter the market and find a suitable trading partner decide whether to ACCEPT or REJECT the money or good of their trading partner. From now on the paper will refer to money holders and good holders who enter the market as buyers and sellers, keeping in mind that two sellers can still engage in barter. In a given period, there are possibly two types of sellers depending on their endowment shock: High and Low. Definition. A steady-state, monetary equilibrium is a constant money supply M 2 (0, 1), where given the actions and expectations of other agents, (i) all buyers retain their money until they exchange it for a good, and (ii) there is a positive fraction of money holders who enter the market and accept at least one quality-type (High or Low) of good, and (iii) there is a positive fraction of acceptable good holders who enter the market and accept money. If we restrict our attention to those matches in which trade is possible, there are five possible pairings: Low–Low, Low–High, High–High, Buyer-Low, and BuyerHigh. In the first three pairings, both agents will ACCEPT since foregone leisure is a sunk cost. In the last two pairings a buyer is matched with a seller and the possibility of monetary trade exists. Given that someone has found a match, there are three conditional probabilities that determine the likelihood of trade. Let p denote the conditional probability that a buyerÕs partner will be a suitable seller. Let s denote the conditional probability that a sellerÕs partner will be a suitable buyer. Finally, let b represent the conditional probability that a sellerÕs partner will be another seller suitable for barter. At the beginning of a period the corresponding unconditional probabilities are bp, bs and bb respectively. Let hi 2 {0, 1} indicate whether a buyer will accept the sellerÕs good, where i 2 {H, L}. For example if hL = 0, then buyers will REJECT Low goods, and barter is the only trade available to Low sellers. If money holders enter the market, they choose one of three possible acceptance strategies: ACCEPT All, ACCEPT Low (only), or ACCEPT High (only). 7
Agents with a unit of money actually make the two decisions jointly.
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Denote the state-dependent value functions of agents holding money, High goods, and Low goods at the beginning of a period as VM, V GH , and V GL respectively, and E½V G pV GH þ ð1 pÞV GL . Then V Gi ¼ maxfhi bsdV M þ bbðU þ dE½V G Þ þ ð1 hi bs bbÞdE½V G ; R þ dE½V G g; V
M
M
G
M
ð2:1Þ M
¼ maxfbpðU þ dE½V Þ þ ð1 bpÞdV ; R þ dV g;
ð2:2Þ
where UM is drawn from the same set as U. However, these two variable may differ since U is determined by which good holders enter the market, whereas UM is also determined by the acceptance strategy of buyers. For instance, all good holders may enter the market but buyers may only trade with High sellers. Eq. (2.1) describes the choice faced by agents holding a good at the beginning of a period. If they choose SEARCH and enter the market, there are three possibilities. They could be matched with a buyer who wants their good and engage in monetary exchange. They could be matched with a fellow seller with whom the double coincidence of wants exists, and engage in barter exchange. The final possibility is that they are matched with an agent with whom no trade is mutually beneficial. If good holders choose REST, they enjoy leisure and remain good holders into the next period. Eq. (2.2) describes the choice that money holders face each period. If they choose SEARCH and enter the market, there are two possibilities. Either they meet a seller with a good that they want and buy, or they do not and no trade takes place. If they choose REST, then they enjoy leisure that period and remain money holders into the next period. The optimal action and strategy for an individual agent depends upon their realization of U and the actions and strategies of all other agents. The focus is on monetary equilibria in which the use of money induces agents to alter their behavior relative to the non-monetary economy. 3.1. Money with no leisure If R > x2EU so that agents always choose REST in the non-monetary economy, money can induce all agents to enter the market. Consider a strategy with both High and Low good holders choosing SEARCH and money holders choosing SEARCH and ACCEPT All. Given these strategies, the value functions for High and Low sellers are identical and the arrival and matching rates are b = 1, p = x(1 M), s = xM, and b = x2(1 M). The value functions become V G ¼ bEU þ sd/ þ dV G ;
ð3:1Þ
V M ¼ pEU pd/ þ dV M ;
ð3:2Þ
where / VM VG, which must be non-negative in a monetary equilibrium. Solving for / yields, /ðMÞ ¼
ðp bÞEU ð1 MÞxð1 xÞEU ¼ . 1 dð1 p sÞ 1 dð1 xÞ
ð4Þ
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This function is strictly positive, and linear and decreasing in M. For expositional purposes, define the functions S(M) and B(M) as dMð1 xÞ SðMÞ bEU þ sd/ ¼ ð1 MÞx2 1 þ EU ; 1 dð1 xÞ dð1 MÞxð1 xÞ EU ; BðMÞ pEU pd/ ¼ ð1 MÞx 1 1 dð1 xÞ so that VG = S(M) + dVG, and VM = B(M) + dVM. Both S(M) and B(M) are concave in M. It is also true that S(0) = x2EU, and B(M) > S(M). This equilibrium exists if there is no incentive for any agent to deviate from the strategy of {SEARCH, ACCEPT}. The values of S(M) and B(M) determine whether agents have an incentive to deviate from entering the market. Given the presence of buyers in the market, a good holder will choose SEARCH if S(M) + dVG P R + dVG, which simplifies to S(M) P R. A similar condition holds for money holders. A buyer will always ACCEPT a High endowment but may wish to REJECT a Low endowment depending on the parameter values. Specifically, accepting a Low good requires, UL P d(VM VG). For some parameter values this condition will not hold if M is sufficiently close to zero. Since the function S(M) is concave and always less than B(M), it determines the interval of M in which all agents to enter the market. In fact, if buyers will accept Low goods even as M approaches zero, then the function S(M) determines the interval of M that supports the equilibrium. H
L
U U Proposition 1. Let M* maximize S(M). If (i) S(M*) > R and (ii) 1d d > pxð U L Þ, then * there is an interval ½M; M with 0 < M < M < 1 containing M such that a monetary equilibrium exists in which everyone chooses the strategy {SEARCH, ACCEPT}, if and only if M 6 M 6 M.
Proof. See Appendix A.
h
Note that this equilibrium requires that there is not ‘‘too much’’ or ‘‘too little’’ money in the economy. If M is too small the probability of finding a buyer is too unlikely to justify sacrificing leisure. On the other hand, if M is too large, there are too few goods to make market activity worthwhile. An upper bound for M is a common feature of random-matching models. For example, in a continuous time version of the Kiyotaki-Wright model with production cost C, a monetary equilibrium exists if C + VM VG P 0. Consequently, if M is too high the inequality fails to hold and monetary trade cannot be supported. However, the lower bound on M is new and comes from the fact that agents have a choice between consumption from trade and a non-market activity that gives positive utility. The model of Ritter (1995) also features this non-market alternative but it is assumed that the expected utility from barter trade is higher than non-market utility. In the present scenario, the opposite assumption is made, i.e. R > x2EU = S(0). Therefore, as M increases from zero to one, S(M) initially increases and rises above R, then decreases and falls below
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R. Since B(M) > S(M), condition (i) of Proposition 1 ensures that both money and good holders enter the market. Condition (ii) is a sufficient condition for buyers to ACCEPT Low goods, and is only relevant when a buyer encounters a Low seller with a desirable good. In general a buyer may find it advantageous to REJECT a Low seller in the present period and be a buyer in the next period. In other words, why not forego the inferior ‘‘birdin-the-hand’’ and SEARCH again next period when you may find a High seller? The benefits of this strategy will depend on the discount rate, the coincidence of wants, the proportion of High goods, and the value of UH relative to UL. Specifically, it is better for the buyer to ACCEPT a Low seller when d, x, p, and (UH UL)/UL are sufficiently low. A lower discount rate signifies that the buyer is less patient and more likely to prefer current consumption. When x decreases, the probability of finding a suitable trading partner in the future also decreases. As p decreases, the proportion of High sellers decreases and it is less likely to consume a High good next period. Finally, if the quality of the High good is not much greater than the quality of the Low good in percentage terms, the relative benefit of remaining a buyer into the next period diminishes. Since all these variables are inversely related to the benefit of accepting a Low endowment, they must be sufficiently low for this monetary equilibrium to exist. Given that condition (ii) of Proposition 1 holds, the disparity in consumption values between High and Low goods does not alter the interval that supports this equilibrium. However, since condition (ii) is a sufficient but not a necessary condition, it is possible for the utility differential UH UL to determine the lower bound of M. Suppose that condition (ii) does not hold. In that case, equilibrium requires that dxð1xÞ EU , for some interval of M when all agents are willU L P d/ðMÞ ¼ ð1 MÞ 1dð1xÞ ing to enter the market. Since the right hand side is decreasing in M, the condition holds if M is sufficiently high. This means that one of two constraints can determine the lower bound for M. There is an acceptance constraint that ensures buyers ACCEPT Low goods, and a market participation constraint that ensures good holders choose SEARCH. Either of these two constraints can be binding at the lower bound of M that supports this type of equilibrium. In Proposition 1 it was the participation constraint that determined the lower bound of M, but the acceptance constraint can also determine this lower bound. e such that U L ¼ d/ð M e Þ. If M 6 M e < M, then M e now defines Proposition 2. Define M the lower bound of M that can support this equilibrium. Therefore an increase in the e, utility differential UH UL that leaves EU unchanged will increase the value of M making the interval that can support this equilibrium smaller. The size of the interval e ; M is also decreasing in the discount rate. ½M e is defined by the equality, U L ¼ ð1 M e Þ dxð1xÞ EU , a decrease in UL Proof. Since M 1dð1xÞ when EU remains the same requires a higher value of M to restore the equality. Likewise when there is an increase in d. h
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The relative benefit of accepting a Low good is decreasing in both the discount rate and the quality differential. Consequently, buyers will ACCEPT Low goods over a smaller interval of M when they are more patient and when there is a larger difference in the quality of High and Low goods. This equilibrium demonstrates that money can induce agents to change their behavior from leisure to trade. When R > x2EU all agents choose REST in the non-monetary economy. However, under the conditions of Proposition 1, the introduction of money induces all agents to choose SEARCH and devote all their resources towards market activity. Agents allocate no time to leisure because money has increased the opportunity cost of leisure by facilitating exchange. The essential effect of money in this equilibrium is standard in random-matching models, which is to make exchange more likely by replacing the double coincidence of wants with the single coincidence of wants. Although High and Low good holders behave similarly, the existence of this equilibrium is still sensitive to the heterogeneity of agents since a larger utility differential can increases the minimum value of M that can support this equilibrium. 3.2. Money with some leisure Now assume that x2pUH > R, which implies x2EU > R so agents always choose SEARCH without money. In this case, the use of money can induce agents with Low quality goods to eschew the market, improving the average quality of traded goods. In this equilibrium, only money and High good holders choose SEARCH and buyers ACCEPT High goods but REJECT Low goods. Since Low good holders choose REST, the mass of trading agents is no longer unity but M + p(1 M). Consequently, the arrival rate and matching probabilities in this equilibrium are b ¼ M þ pð1 MÞ; xpð1 MÞ p¼ ; M þ pð1 MÞ xM ; s¼ M þ pð1 MÞ x2 pð1 MÞ b¼ . M þ pð1 MÞ The value functions are now V GH ¼ bbU H þ bsd/ þ dE½V G ; V V
G L M
G
¼ R þ dE½V ; H
ð5:1Þ ð5:2Þ
M
¼ bpU bpd/ þ dV ;
ð5:3Þ
where / VM E[VG]. Solving for the difference in the expected value functions for money and good holders yields /ðMÞ ¼
ð1 MÞpxð1 pxÞU H ð1 pÞR . 1 dð1 pxÞ
ð6Þ
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Unlike the first monetary equilibrium, /(M) is not always positive; however, it is linear and decreasing in M, as in the first monetary equilibrium. A necessary condition for a monetary equilibrium is /(0) > 0, which is true given the assumption x2pUH > R, which implies pxUH > R so that px(1 p)UH > (1 p)R and px(1 px)UH > (1 p)R. This monetary equilibrium exists if, given the strategies of others, no one has an incentive to deviate from their strategy, requiring (i) (ii) (iii) (iv)
V M > maxfV GH ; V GL g, UH P d/ > UL, bbUH + bsd/ P R > bbUH, bpUH bpd/ P R.
The first condition ensures that sellers ACCEPT money, and that money holders retain their money after receiving either a High or Low endowment. Condition (ii) ensures that buyers ACCEPT High goods and REJECT Low goods. Condition (iii) ensures that High good holders choose SEARCH, while Low good holders choose REST. Finally, condition (iv) ensures that money holders choose SEARCH. Proposition 3. There is an interval (M 0 , M00 ) with 0 < M 0 < M00 < 1 such that a monetary equilibrium exists in which money holders and High good holders choose SEARCH, Low good holders choose REST, and buyers ACCEPT High goods and REJECT Low goods: if (i) M 0 < M < M00 and (ii) d/(M 0 ) > UL. Proof. See Appendix A. h Once again the monetary equilibrium involves an upper and lower bound on M. However, in this equilibrium the lower bound discourages Low good holders from barter trade. If M is too low, the probability of barter is high enough to induce Low good holders to deviate from REST and enter the market. Given that Low good holders REST, equilibrium entails a rejection constraint for buyers (d/(M) > UL), a market participation constraint for money holders (bpUH bpd/ P R), and a market participation constraint for High good holders (bbUH + bsd/ P R). The upper bound M00 is defined by one of these three constraints, depending on the parameter values. In contrast to the first equilibrium, the rejection constraint ensures that buyers are now willing to pass up an inferior ‘‘bird-in-the-hand’’ for the prospect of receiving something better in the future. Proposition 4. If the rejection constraint defines M00 , then the size of the interval (M 0 , M00 ) is increasing in both d and the utility differential, UH UL. H
ð1pÞR Proof. When M < M00 , then d½ð1MÞpxð1pxÞU > U L . The left hand side is increas1dð1pxÞ H ing in d and U , and the right hand side is increasing in UL. Therefore, the constraint is binding for a greater value of M when d and UH are larger and UL is smaller. h
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The relative benefit of rejecting a Low good is increasing in both the discount rate and the quality differential. Consequently, buyers will REJECT Low goods over a larger interval of M when they are more patient and when there is a larger difference in the quality of High and Low goods. In this equilibrium, the use of money dichotomizes behavior between High and Low good holders in a way that is not possible in the non-monetary economy. Money makes possible what was previously impossible, inducing High good holders to SEARCH and Low holders to REST. The absence of Low sellers has a positive external effect on buyers by increasing the average value of market goods. 3.3. Multiple equilibria There is the possibility of multiple equilibria in which the intervals associated with each monetary equilibrium overlap. If x2pUH > R then 0 ¼ M < M 0 < M < 1. The key condition for this possibility is d/H(M 0 ) P UL > d/(M 0 ), where /H is defined as VM E[VG] when there are only High sellers in the market, and / is similarly defined when both High and Low sellers are in the market. In this case, a buyer will reject a Low good if all other Low agents are out of the market, but will accept a Low good if all other Low agents are in the market. This can be verified with a formal proof as well as with a numerical example, as in Appendix A.
4. Robustness The existence of one or both of the above monetary equilibria may be sensitive to the indivisibility of goods and money. Under the assumption of indivisibility there is no endogenous price that can affect the ACCEPT/REJECT decision of buyers. One way to introduce an endogenous price is to randomize exchange by incorporating lotteries as in Berentsen et al. (2002). In their model, buyers and sellers bargain over the probability of trading goods and money as a proxy for bargaining directly over the terms of trade. For example, assuming risk neutrality, selling a good in exchange for a 50% probability of receiving a dollar is similar to selling the good for 50 cents. Berentsen et al. (2002) utilize the generalized Nash bargaining solution which allows one to calibrate the relative bargaining power of buyers in a straightforward way. In a related paper Berentsen and Rocheteau (2002) show that some of the standard results of search-theoretic, monetary models are due to the indivisibility of money. In the following analysis, agents make take-it-or-leave-it offers on the probability of trading money as in Berentsen and Rocheteau (2002). In contrast to their model in which only buyers make offers, offers by sellers are also considered. In terms of bargaining power there are two opposite extremes, corresponding to whether buyers or sellers are making the take-it-or-leave-it offer. A priori, there is no reason to assume that either buyers or sellers would have greater bargaining power. The main results are that although both equilibria are sensitive to the introduction of bargaining, monetary trade can take place if sellers make take-it-or-leave-it offers. Conversely, if buyers make the offer, neither monetary equilibrium can be supported.
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4.1. Money with no leisure Consider the first equilibrium when everyone enters the market, and buyers ACCEPT All. If buyers make the offer, they have too much bargaining power to support monetary exchange. This is due to the discrete nature of time and the fact that foregone leisure is a sunk cost for agents in the market. Another factor is the endowment economy with no cost of production for the seller that can provide a lower bound for the terms of trade. When a buyer and seller meet, the buyer offers to give a unit of money with probability s in exchange for the sellerÕs good. A seller will accept any offer that does not make them worse off, i.e. the seller will accept the offer if sdVM + (1 s)dVG P dVG. Since buyers can extract the entire surplus, this condition becomes sd/ = 0. However in this equilibrium / > 0, and the only way the condition holds is if s = 0 in which case there is no benefit for sellers to enter the market because they can never become buyers. Therefore good holders choose REST and monetary exchange will collapse. If sellers make the offer, a rejection by buyers results in foregoing the consumption of the good. Consequently, there could be two prices: s(H) and s(L). When a High seller makes an offer, the condition for indifference becomes UH = s(H)d/, but in this particular equilibrium UH > d/, which implies UH/d/ > 1 = s(H). Sellers extract the entire surplus and money is always traded for High goods. Money is also traded with certainty for Low goods, because condition (ii) of Proposition 1 guarantees UL > d/ for all values of M. Consequently, if sellers make the offer, then s(H) = s(L) = 1 and the equilibrium remains. 4.2. Money with some leisure The second equilibrium also disappears if buyers make the offer. Sellers are indifferent between accepting and rejecting when sd/ = 0. Recall that this equilibrium requires Low good holders to stay out of the market while High good holders enter, that is bbUH + bssd/ P R > bbUH. If sd/ = 0, this condition becomes bbUH P R > bbUH, a contradiction. When sellers make the offer, money is exchanged with certainty with High sellers for reasons similar to the first equilibrium. Indifference for buyers requires UH = s(H)d/ but once again UH > d/ for all values of M and UH/d/ > 1 = s(H). When Low sellers bargain with buyers, there is always a s(L) low enough for buyers to accept the Low good, as seen from the condition for buyer indifference, UL = s(L)d/. If UL 6 d/, then s(L) = UL/d/ < 1; if UL > d/, then s(L) = 1. Consequently, 0 < s(L) 6 1 and buyers no longer REJECT Low goods. Low goods are now acceptable because of attractive terms-of-trade for buyers. In summary, if either buyers or sellers make take-it-or-leave-it offers, the equilibrium cannot exist. Nevertheless, there is a different equilibrium that can also generate an outcome in which Low good holders stay out of the market, while High good holders enter. Low good holders may still want to choose REST even if their goods would be accepted by buyers. In particular, if sellers make take-it-or-leave-it offers, there is a monetary
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equilibrium in which buyers choose SEARCH and ACCEPT All, High good holders choose SEARCH, and Low good holders choose REST. The essential condition is that Low good holders still choose REST while High good holders choose SEARCH when s(H) = 1 and 0 < s(L) 6 1, where s(L) is increasing in M. This requires that bbUH + bsd/ P R > bbUH + bss(L)d/. If s(L) = 1, the conditions becomes a contradiction. Therefore, necessary conditions for this equilibrium are s(L) = UL/d/ < 1 and that only buyers and High good holders choose SEARCH. Consequently, this equilibrium requires that three conditions hold simultaneously: (i) d/ > UL, (ii) bbUH + bsd/ P R > bbUH + bsUL, (iii) bpUH bpd/ P R. These sufficient conditions are similar to those without randomized trading. The only difference is the non-participation constraint for Low good sellers, which is now R > bbUH + bsUL. Recall that this constraint defines the lower bound of M that can support the equilibrium. Assume R > xUL, then the right hand side of the constraint b such that R ¼ x2 pð1 M b ÞU H þ is equal to R at a unique value of M. Define M L b xM U . Proposition 5. Consider the case in which Low sellers offer to give their good with certainty in exchange for money with probability 0 < s(L) < 1. If UL is sufficiently low, b ; M 00 Þ, with 0 < M b < M 00 < 1 such that a monetary then there is an interval ð M equilibrium exists in which buyers choose SEARCH and ACCEPT All, High good b < M < M 00 . holders choose SEARCH, and Low good holders choose REST; if M b , there is an implicit function M b ðU L Þ that is Proof. According to the definition of M L L 0 continuous, increasing in U , and with the limit of M as U approaches zero. Thereb < M 00 and the interval exists. h fore if UL is sufficiently low, M The consumption value of the Low good again plays a role in defining the lower bound of M, but through the non-participation constraint of Low good holders instead of the rejection constraint of buyers. Low goods are traded, but for such a low ‘‘price’’ that the probability of receiving money and becoming a buyer is too small to make entering the market worthwhile.
5. Conclusion This paper develops a model in which the cost of searching for a trading partner is foregone leisure and agents realize idiosyncratic shocks in the consumption value of their endowment. In the non-monetary economy there are two perfect Nash equilibria, one with everyone staying out of the market to enjoy leisure and one with everyone entering the market to seek a trading partner. There is no barter equilibrium
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with only High good holders in the market. When fiat money is introduced, there are two possible equilibria, one with everyone participating in the market and buyers accepting both High and Low goods, and another with buyers rejecting Low sellers and Low agents eschewing the market to enjoy leisure. The behavior of a buyer who meets a Low seller determines which of these two equilibria occur. The first equilibrium occurs when a buyer refuses to forego consuming a Low good in the present for the possibility of consuming a High good in the future. The second equilibrium occurs when a buyer is willing to give up a ‘‘sure thing’’ for the future possibility of something better. Money can affect the real side of the economy through the search-or-rest decision of Low good holders since their presence or absence affects the average quality of traded goods. For instance, when money drives Low good holders out of the market, the value of traded goods increases. Money can generate an outcome that is always impossible without money, namely High good holders entering the market while Low good holders enjoy leisure. If agents bargain over lotteries for money in which sellers make take-it-or-leave-it offers, there is an equilibrium in which buyers enter the market and accept both High and Low goods, High good holders enter to accept money, and Low good holders rest.
Acknowledgements I am thankful to Melanie Baker, Michael Pangia and Zhenhui Xu for helpful comments, and to two anonymous referees who helped substantially improve the content and presentation of the paper.
Appendix A A.1. List of main variables d 2 (0,1) is the discount factor. U 2 {UH, UL} is the utility of consuming the endowment, where UH > UL > 0. N 2 {3, 4, 5, . . .} is the number of different types of endowment goods, representing the level of specialization in the economy. x 1/N is the coincidence of wants. R is the utility of leisure, where x2UH > R > x2UL. p 2 (0, 1) is the probability that an agentÕs endowment is High and the proportion of the population with a High endowment. EU pUH + (1 p)UL is the average endowment size in the economy. M 2 (0, 1) is the proportion of the population that holds a unit of indivisible money. b 2 (0, 1] is the arrival rate, which is equal to the proportion of agents in the market. p 2 (0, x) is the probability that a buyer will find a suitable seller.
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s 2 (0, x) is the probability that a seller will find a suitable buyer. b 2 (0, x2) is the probability that a seller will find another seller suitable for barter. / VM E[VG] s is the probability that money is exchanged. Proof of Proposition 1. We first verify that no one deviates from SEARCH. Given that money traders choose {SEARCH, ACCEPT}, a good holder will enter the market if S(M) + dVG P R + dVG. Since S(M*) > R and S(M) is concave and continuous, there is an interval in the neighborhood of M* where S(M) P R. Turning to the other side of the market, given that good holders choose {SEARCH, ACCEPT}, a money holder will enter the market if B(M) + dVM P R + dVM. Since S(M) P R when M 6 M 6 M, money holders will prefer to retain their money and dx enter the market if B(M) > S(M). This inequality reduces to 1 > 1dð1xÞ , which is true. Consequently, both money and good holders enter the market when M 6 M 6 M. We now verify that no one deviates from ACCEPT. Sellers will accept money as > 0. A buyer will long as / VM VG > 0, which is true since / ¼ ð1MÞxð1xÞEU 1dð1xÞ ACCEPT a High endowment if UH + dVG P dVM. This is equivalently stated as U H P dð1MÞxð1xÞ 1dð1xÞ EU , which is true since the fraction on the RHS is less than one and UH > EU so that U H > EU > dð1MÞxð1xÞ 1dð1xÞ EU . Finally, a buyer will ACCEPT a Low endowment if UL + dVG P dVM, equivalently expressed as U L P ð1 MÞ dx ð1 xÞ 1dð1xÞ EU . A sufficient condition for this inequality to hold is UL >
dx EU . 1 dð1 xÞ
ðA:1Þ
Since (A.1) is algebraically equivalent to condition (ii) of Proposition 1, the sufficient condition holds and buyers will ACCEPT Low endowments. There is thus no incentive for any agent to deviate from {SEARCH, ACCEPT}. h Proof of Proposition 3. It is sufficient to show that four inequalities hold.
h
Lemma. Conditions (iii), (iv), and the second part of condition (ii) are sufficient conditions for this equilibrium. Proof. Condition (i) holds if V M > V GH > V GL . The first equality ðV M > V GH Þ can be expressed as bp(1 x)UH + (1 bx)d/ > 0, which is true if d/ > 0, which is implied by the second part of condition (ii), d/ > UL. The second inequality ðV GH > V GL Þ is true if and only if bbUH + bsd/ > R, which is the first part of condition (iii). The first inequality of condition (ii), UH > d/(M) is true for all values of M if UH > d/(0). dpxð1pxÞ dð1pÞR U H 1dð1pxÞ which holds When M = 0, the inequality becomes U H > 1dð1pxÞ
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dpxð1pxÞ because U H > 1dð1pxÞ U H . The equilibrium therefore exists if one can show that L d/ > U and conditions (iii) and (iv) simultaneously hold for some interval of M. h
These conditions consist of four inequalities, one of which requires that M be sufficiently high. The remaining three inequalities require that M be sufficiently low. These three inequalities can be described as a rejection constraint ensuring buyers REJECT Low goods and two market participation constraints ensuring that both money and High good holders choose SEARCH. The remainder of the proof has two more steps. The first is to show that there is a threshold value of M that determines when M is high enough to cause Low holders to REST, and the second is to show that the remaining three conditions hold at that threshold value. Since bbUH is decreasing in M, define M 0 as the unique value of M where bbUH x2p(1 M)UH = R, so that R > x2p(1 M)UH if and only if M > M 0 . Now define g(M) bbUH + bsd/ and k(M) bpUH bpd/. Taking the second derivative of both functions w.r.t. M gives g00 ðMÞ ¼
2x2 pdð1 pxÞU H <0 1 dð1 pxÞ
and k00 ðMÞ ¼
2x2 p2 dð1 pxÞU H < 0. 1 dð1 pxÞ
Given the strict concavity of these functions, if g(M) > R and k(M) > R when M = M 0 , then there is an interval of M in which Low good holders REST while both money and High good holders SEARCH. By definition, when M = M 0 then bbUH = R so that the inequality g(M) > R can be rewritten as R + dxM/ > R. This is true when / > 0. / is positive if and only if bp(1 px)UH (1 p)R > 0. Using fact that bbUH = R yields (1 M)px(1 px)UH > (1 p)(1 M)x2pUH which reduces to 1 > x, obviously true. Likewise, substituting bbUH = R into the condition k(M) > R gives 2
ð1 MÞpxU H
2
ð1 MÞ p2 x2 dð1 pxÞU H ð1 MÞ p2 x3 dð1 pÞU H þ 1 dð1 pxÞ 1 dð1 pxÞ
> px2 ð1 MÞU H which after dividing by (1 M)pxUH becomes 1
ð1 MÞpxdð1 pxÞ ð1 MÞpx2 dð1 pÞ þ > x. 1 dð1 pxÞ 1 dð1 pxÞ
dpxð1MÞ dpx Rearranging and dividing by (1 x) gives 1 > 1dð1pxÞ . This is true if 1 > 1dð1pxÞ , which reduces to 1 d > 0. Since g(M 0 ) > R > g(1) and k(M 0 ) > R > k(1), and since both functions are continuous and strictly concave in M, then applying the intermediate value theorem each function is equal to R at a unique value of M 2 (M 0 ,1). Let these values be defined as M1 and M2 respectively so that g(M1) = R and k(M2) = R. Finally, since UL is not a part of any other condition, simply assume d/(M 0 ) > UL. Since d/(M) is decreasing in M, there is a threshold value M0 where d/(M0) = UL.
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Depending on the specific parameter values, one of these three constraints is the first to become binding as M increases above M 0 . Therefore one of these constraints defines the upper bound of M where M00 = min{M0, M1, M2}. h A.2. Numerical example of multiple equilibria Since x2pUH > R implies 0 = M < M 0 , there are only two sufficient conditions for multiple equilibria: d/H ðM 0 Þ P U L > d/ðM 0 Þ;
ðA:2Þ
SðM 0 Þ > R.
ðA:3Þ
Eq. (A.2) ensures that buyers will ACCEPT and REJECT Low goods in the respective equilibria. Eq. (A.3) ensures that M 0 < M. When, x = 0.2, p = 0.6, d = 0.9, UH = 100, UL = 26 and R = 1.8 then M 0 = 0.25 and both conditions hold. So there is an interval of M above 0.25 where there are multiple equilibria. For instance, when M = 0.3, d/H = 28.87 > 26 > 25.34 = d/ and S = 3.49 > 1.8. A formal proof is also available.
References Berentsen, A., Molico, M., Wright, R., 2002. Indivisibilities, lotteries, and monetary exchange. Journal of Economic Theory 107, 70–94. Berentsen, A., Rocheteau, G., 2002. On the efficiency of monetary exchange: how divisibility of money matters. Journal of Monetary Economics 49, 1621–1649. Burdett, K., Coles, M.G., 1999. Long-term partnership formation: marriage and employment. Economic Journal 109, F307–F334. Burdett, K., Coles, M., Kiyotaki, N., Wright, R., 1995. Buyers and sellers: should I stay or should I go? American Economic Review 85, 281–286. Diamond, P.A., 1982. Aggregate demand management in search equilibrium. Journal of Political Economy 90, 881–894. Diamond, P.A., 1984. Money in a search equilibrium. Econometrica 52, 1–20. Hosios, A.J., 1990. On the efficiency of matching and related models of search and unemployment. Review of Economic Studies 57, 279–298. Kiyotaki, N., Wright, R., 1991. A contribution to the pure theory of money. Journal of Economic Theory 53, 215–235. Kiyotaki, N., Wright, R., 1993. A search-theoretic approach to monetary economics. American Economic Review 83, 63–77. Kultti, K., 2000. A model of random matching and price formation. European Economic Review 44, 1841–1856. Li, V., 1994. Inventory accumulation in a search-based monetary economy. Journal of Monetary Economics 34, 511–536. Li, V., 1995. The optimal taxation of fiat money in search equilibrium. International Economic Review 36, 927–942. Li, V., 1997. The efficiency of monetary exchange in search equilibrium. Journal of Money, Credit, and Banking 29, 61–72. Mortenson, D.T., 1982. Property rights and efficiency in mating, racing, and related games. American Economic Review 72, 968–979.
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Mortenson, D.T., 1992. Search theory and macroeconomics: a review essay. Journal of Monetary Economics 29, 163–167. Pissarides, C.A., 1984. Search intensity, job advertising, and efficiency. Journal of Labor Economics 2, 123–143. Ritter, J.A., 1995. The transition from barter to fiat money. American Economic Review 85, 134–149. Shi, S., 1997. A divisible search model of fiat money. Econometrica 65, 75–102.
Search and leisure with idiosyncratic endowment shocks in a random-matching model J.J. Arias
*
Department of Economics and Finance, Georgia College and State University, Campus Box 014, Milledgeville, GA 31061, United States Received 23 October 2000; accepted 27 February 2004 Available online 5 July 2005
Abstract Agents who realize idiosyncratic endowment shocks (High and Low) choose between leisure and searching for a trading partner. In the non-monetary economy there are two cases: everyone enjoys leisure or everyone enters the market to barter. In the first case, money can generate an equilibrium in which all agents enter the market. In the second, money can generate an outcome impossible without money, in which High agents trade while Low agents rest. Money affects the real side of the economy by driving Low quality goods out of the market, improving the quality of traded goods. Ó 2005 Elsevier Inc. All rights reserved. JEL classification: E40; E42 Keywords: Search; Money; Barter; Leisure
1. Introduction An important characteristic of money as a social convention or mechanism is that it generates behavior that would not occur in its absence. In the present model, *
Tel.: +1 478 445 2085; fax: +1 478 445 1535. E-mail address: [email protected]
0164-0704/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2004.02.004
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money can generate outcomes that are impossible without money. If all agents derive utility outside of the market in a non-monetary economy, money can induce welfareimproving market activity. On the other hand, if all agents engage in barter, money can induce agents with inferior (Low) goods to leave the market. In this second case, the use of money has a real effect by driving Low goods out of the market, improving the quality of traded goods. The theoretical innovation is that agents are heterogeneous in a way other than the usual symmetric specialization in tastes and endowments or production necessary to motivate exchange.1 Agents who receive either a High or Low quality endowment choose between searching for a trading partner in the market or consuming leisure outside the market. The tradeoff between leisure and trade is affected not only by the coincidence of wants and the relative value of consumption to leisure, but also by the presence or absence of High and Low agents in the market. The environment is similar to that found in Ritter (1995) in which non-market activity and an endogenous search decision also play an important role. In the non-monetary economy there are two possible equilibria, depending on the value of leisure. If the utility of leisure is greater than the expected utility of searching when both Low and High agents are present in the market, everyone chooses to enjoy leisure. If the utility of leisure is less, everyone chooses to search for a trading partner. There is no barter equilibrium in which Low agents stay out of the market while High agents enter, which would be the first-best outcome. There are two monetary equilibria corresponding to the two cases in the non-monetary economy, one in which Low good holders join High good holders and enter the market to trade, and another in which only Low good holders enjoy leisure outside the market. The existence of a monetary equilibrium is affected by the difference in consumption value between High and Low goods, the probability of receiving a High endowment, and the discount rate. When a buyer encounters a seller with a Low good, he must decide whether to accept the ‘‘bird-in-the-hand’’, or pass it up in the hopes of finding ‘‘two-in-thebush’’ next period. The first equilibrium requires that buyers accept Low goods instead of waiting to meet a seller with a High good. The second equilibrium requires the opposite, that buyers reject Low goods for the future possibility of meeting a seller with a High good. Both equilibria require that the quantity of money falls within a specific interval. An increase in either the discount rate or the quality difference between High and Low goods causes the interval for the first equilibrium to decrease and the interval for the second to increase. In the second equilibrium money has a real effect by driving Low good holders out of the market. This paper is closely related to the random-matching models pioneered by Kiyotaki and Wright (1991, 1993), especially papers that emphasize an endogenous
1 There is a related literature dealing with heterogeneous agents in the various search and matching models, that Burdett and Coles (1999) summarize and apply to labor markets. The agent heterogeneity in the present model is closer to what they define as ex ante heterogeneity in which the differences between agents are manifest before matches take place.
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search decision.2 For instance Li (1994, 1995, 1997) allows agents to choose their search intensity, which is the argument of a convex cost function. In Burdett et al. (1995) buyers and sellers choose between active search and passive waiting. Although ‘‘movers’’ have a higher probability of exchange than ‘‘stayers’’, they incur a disutility of moving or search.3 The decision to search or rest in this model is similar to the decision to search or wait in Burdett et al. (1995), with two significant differences. One is that in the present model active agents are sampling out of all other trading agents only, so that agents who enjoy leisure never meet another agent. This contrasts with their model in which agents who choose not to search could still meet other agents who are searching. Also, the present model potentially dichotomizes the search decision between High and Low good holders, whereas their model dichotomizes the search decision between buyers and sellers. The second difference is the more important of the two because it gives agents a different decision margin. The paper is organized as follows. Section 2 discusses the environment and assumptions of the model and considers a non-monetary economy. Section 3 introduces fiat money and examines two monetary equilibria. Section 4 addresses robustness of the monetary equilibria to the introduction of lotteries or randomized trading, and Section 5 concludes.
2. The environment and non-monetary economy The environment is similar to the steady-state model of Ritter (1995). There is an endowment economy inhabited by a continuum of infinitely lived agents with unit mass. Time is discrete with a discount factor d 2 (0, 1). At the beginning of each period, agents realize an idiosyncratic endowment shock U 2 {UH, UL}, where UH > UL > 0, and agents receive the High endowment with probability p. Within a particular period, p represents the proportion of the population that has received a High endowment. Endowment goods are perishable across periods, and both goods and money exist as indivisible units. Agents are constrained to holding either one unit of money or one unit of a good, and each agentÕs holdings are observable to others. In addition to varying in consumption value, endowment goods are divided into N P 3 types, with agents specialized in consumption as well as endowments. In addition to being endowed with only one type of good, agents receive utility from only one type of good. Preferences are equally distributed amongst endowment types so that the coincidence of wants is x 1/N. Distributing endowments and preferences in this manner simplifies the model since all goods will be equally acceptable 2
One interesting feature of these models is that the search decision has external effects, relating them to the search and matching models with trading or search externalities (Diamond, 1982; Hosios, 1990; Mortenson, 1982, 1992; Pissarides, 1984; Shi, 1997). 3 Kultti (2000) also develops an interesting model where agents search or wait, and prices are formed through both auctions and bargaining but there is no money in his model.
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in trade. It is also assumed that agents cannot consume their own endowment.4 Agents also receive utility from rest and leisure, denoted by R. Each period agents must choose between entering the market for the possibility of exchange (SEARCH) or staying out of the market and consuming leisure (REST). If they enter the market, agents meet one another according to the arrival rate b, equal to the fraction of agents entering the market.5 This fraction is an endogenous variable and will vary depending on whether no one, everyone or only agents with either a High or Low endowment participate in the market. Consequently, in the nonmonetary economy b 2 {0, p, 1 p, 1}. Since money and endowments are indivisible, agents will either ACCEPT or REJECT their partnerÕs money or good. Finally, assume that x2UH > R > x2UL. Define V as the agentÕs value function after observing their endowment shock. Since agents face an infinitely repeating one-period problem, V is represented by V ¼ maxfbx2 U; Rg þ dE½V ; H
L
ð1Þ H
L
where U 2 {U , EU, U , 0} and EU pU + (1 p)U , so the probability of an exchange opportunity is the arrival rate times the familiar double coincidence of wants and the expected value of exchange U depends on who else decides to enter the market. Equilibrium in this environment depends on the utility of leisure relative to the expected utility of barter when both High and Low agents enter the market.6 If R > x2EU, then the unique sub-game perfect Nash equilibrium is for everyone to REST. If everyone else chooses SEARCH, the best response is REST since R > x2EU. If only agents with a High endowment choose SEARCH, the best response for everyone is REST since R > x2EU > x2pUH. If only agents with a Low endowment choose SEARCH, again the best response is REST since R > x2EU > x2(1 p)UL. However, if everyone chooses REST, the best response is obviously to REST. If x2EU > R, then there is a sub-game perfect Nash equilibrium in which everyone chooses to SEARCH. Given that everyone else has entered the market, the best response is to also enter, so that b = 1 and V = x2EU + dV. There is also a coordination failure equilibrium in which no one enters the market, but the paper hereafter ignores this equilibrium. The first-best outcome would be for High agents to SEARCH and for Low agents to REST. In that case, expected utility for a single period is px2UH + (1 p)R, which is greater than either R or x2EU given the present assumptions. To see this, note that the condition px2UH + (1 p)R > R reduces to x2UH > R, and the condi-
4 This assumption is the analog for an endowment economy to the standard assumption in search theory that agents cannot consume their own output (Kiyotaki and Wright, 1991, 1993; Diamond, 1982, 1984). 5 In an earlier version of the paper, the arrival rate was always equal to one, making the second monetary equilibrium unnecessarily complicated. The present assumption about the arrival rate simplifies the equilibrium considerably without altering any of the central results. It is also consistent with the idea that trade becomes more difficult as the market becomes ‘‘thinner.’’ In any case, the assumption about the arrival rate is one about market structure, which is beyond the scope of this model. 6 All ‘‘ties’’ will go to SEARCH or market activity.
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tion px2UH + (1 p)R > x2EU reduces to R > x2UL. Unfortunately this outcome is not possible in the non-monetary economy. Given that Low agents choose REST, High agents will SEARCH if and only if x2pUH > R, in which case a Low agent would also want to SEARCH. One question is whether the use of money can generate an equilibrium in which High agents SEARCH and Low agents REST.
3. Monetary equilibria We now consider the case in which a constant fraction M of the population holds fiat money. Agents go through the following sequence each period. First, all agents observe the endowment shock. Second, agents with a unit of money decide whether to hold money or hold their endowment. Third, all agents decide whether to SEARCH or REST in that period.7 Fourth, agents who enter the market and find a suitable trading partner decide whether to ACCEPT or REJECT the money or good of their trading partner. From now on the paper will refer to money holders and good holders who enter the market as buyers and sellers, keeping in mind that two sellers can still engage in barter. In a given period, there are possibly two types of sellers depending on their endowment shock: High and Low. Definition. A steady-state, monetary equilibrium is a constant money supply M 2 (0, 1), where given the actions and expectations of other agents, (i) all buyers retain their money until they exchange it for a good, and (ii) there is a positive fraction of money holders who enter the market and accept at least one quality-type (High or Low) of good, and (iii) there is a positive fraction of acceptable good holders who enter the market and accept money. If we restrict our attention to those matches in which trade is possible, there are five possible pairings: Low–Low, Low–High, High–High, Buyer-Low, and BuyerHigh. In the first three pairings, both agents will ACCEPT since foregone leisure is a sunk cost. In the last two pairings a buyer is matched with a seller and the possibility of monetary trade exists. Given that someone has found a match, there are three conditional probabilities that determine the likelihood of trade. Let p denote the conditional probability that a buyerÕs partner will be a suitable seller. Let s denote the conditional probability that a sellerÕs partner will be a suitable buyer. Finally, let b represent the conditional probability that a sellerÕs partner will be another seller suitable for barter. At the beginning of a period the corresponding unconditional probabilities are bp, bs and bb respectively. Let hi 2 {0, 1} indicate whether a buyer will accept the sellerÕs good, where i 2 {H, L}. For example if hL = 0, then buyers will REJECT Low goods, and barter is the only trade available to Low sellers. If money holders enter the market, they choose one of three possible acceptance strategies: ACCEPT All, ACCEPT Low (only), or ACCEPT High (only). 7
Agents with a unit of money actually make the two decisions jointly.
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Denote the state-dependent value functions of agents holding money, High goods, and Low goods at the beginning of a period as VM, V GH , and V GL respectively, and E½V G pV GH þ ð1 pÞV GL . Then V Gi ¼ maxfhi bsdV M þ bbðU þ dE½V G Þ þ ð1 hi bs bbÞdE½V G ; R þ dE½V G g; V
M
M
G
M
ð2:1Þ M
¼ maxfbpðU þ dE½V Þ þ ð1 bpÞdV ; R þ dV g;
ð2:2Þ
where UM is drawn from the same set as U. However, these two variable may differ since U is determined by which good holders enter the market, whereas UM is also determined by the acceptance strategy of buyers. For instance, all good holders may enter the market but buyers may only trade with High sellers. Eq. (2.1) describes the choice faced by agents holding a good at the beginning of a period. If they choose SEARCH and enter the market, there are three possibilities. They could be matched with a buyer who wants their good and engage in monetary exchange. They could be matched with a fellow seller with whom the double coincidence of wants exists, and engage in barter exchange. The final possibility is that they are matched with an agent with whom no trade is mutually beneficial. If good holders choose REST, they enjoy leisure and remain good holders into the next period. Eq. (2.2) describes the choice that money holders face each period. If they choose SEARCH and enter the market, there are two possibilities. Either they meet a seller with a good that they want and buy, or they do not and no trade takes place. If they choose REST, then they enjoy leisure that period and remain money holders into the next period. The optimal action and strategy for an individual agent depends upon their realization of U and the actions and strategies of all other agents. The focus is on monetary equilibria in which the use of money induces agents to alter their behavior relative to the non-monetary economy. 3.1. Money with no leisure If R > x2EU so that agents always choose REST in the non-monetary economy, money can induce all agents to enter the market. Consider a strategy with both High and Low good holders choosing SEARCH and money holders choosing SEARCH and ACCEPT All. Given these strategies, the value functions for High and Low sellers are identical and the arrival and matching rates are b = 1, p = x(1 M), s = xM, and b = x2(1 M). The value functions become V G ¼ bEU þ sd/ þ dV G ;
ð3:1Þ
V M ¼ pEU pd/ þ dV M ;
ð3:2Þ
where / VM VG, which must be non-negative in a monetary equilibrium. Solving for / yields, /ðMÞ ¼
ðp bÞEU ð1 MÞxð1 xÞEU ¼ . 1 dð1 p sÞ 1 dð1 xÞ
ð4Þ
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This function is strictly positive, and linear and decreasing in M. For expositional purposes, define the functions S(M) and B(M) as dMð1 xÞ SðMÞ bEU þ sd/ ¼ ð1 MÞx2 1 þ EU ; 1 dð1 xÞ dð1 MÞxð1 xÞ EU ; BðMÞ pEU pd/ ¼ ð1 MÞx 1 1 dð1 xÞ so that VG = S(M) + dVG, and VM = B(M) + dVM. Both S(M) and B(M) are concave in M. It is also true that S(0) = x2EU, and B(M) > S(M). This equilibrium exists if there is no incentive for any agent to deviate from the strategy of {SEARCH, ACCEPT}. The values of S(M) and B(M) determine whether agents have an incentive to deviate from entering the market. Given the presence of buyers in the market, a good holder will choose SEARCH if S(M) + dVG P R + dVG, which simplifies to S(M) P R. A similar condition holds for money holders. A buyer will always ACCEPT a High endowment but may wish to REJECT a Low endowment depending on the parameter values. Specifically, accepting a Low good requires, UL P d(VM VG). For some parameter values this condition will not hold if M is sufficiently close to zero. Since the function S(M) is concave and always less than B(M), it determines the interval of M in which all agents to enter the market. In fact, if buyers will accept Low goods even as M approaches zero, then the function S(M) determines the interval of M that supports the equilibrium. H
L
U U Proposition 1. Let M* maximize S(M). If (i) S(M*) > R and (ii) 1d d > pxð U L Þ, then * there is an interval ½M; M with 0 < M < M < 1 containing M such that a monetary equilibrium exists in which everyone chooses the strategy {SEARCH, ACCEPT}, if and only if M 6 M 6 M.
Proof. See Appendix A.
h
Note that this equilibrium requires that there is not ‘‘too much’’ or ‘‘too little’’ money in the economy. If M is too small the probability of finding a buyer is too unlikely to justify sacrificing leisure. On the other hand, if M is too large, there are too few goods to make market activity worthwhile. An upper bound for M is a common feature of random-matching models. For example, in a continuous time version of the Kiyotaki-Wright model with production cost C, a monetary equilibrium exists if C + VM VG P 0. Consequently, if M is too high the inequality fails to hold and monetary trade cannot be supported. However, the lower bound on M is new and comes from the fact that agents have a choice between consumption from trade and a non-market activity that gives positive utility. The model of Ritter (1995) also features this non-market alternative but it is assumed that the expected utility from barter trade is higher than non-market utility. In the present scenario, the opposite assumption is made, i.e. R > x2EU = S(0). Therefore, as M increases from zero to one, S(M) initially increases and rises above R, then decreases and falls below
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R. Since B(M) > S(M), condition (i) of Proposition 1 ensures that both money and good holders enter the market. Condition (ii) is a sufficient condition for buyers to ACCEPT Low goods, and is only relevant when a buyer encounters a Low seller with a desirable good. In general a buyer may find it advantageous to REJECT a Low seller in the present period and be a buyer in the next period. In other words, why not forego the inferior ‘‘birdin-the-hand’’ and SEARCH again next period when you may find a High seller? The benefits of this strategy will depend on the discount rate, the coincidence of wants, the proportion of High goods, and the value of UH relative to UL. Specifically, it is better for the buyer to ACCEPT a Low seller when d, x, p, and (UH UL)/UL are sufficiently low. A lower discount rate signifies that the buyer is less patient and more likely to prefer current consumption. When x decreases, the probability of finding a suitable trading partner in the future also decreases. As p decreases, the proportion of High sellers decreases and it is less likely to consume a High good next period. Finally, if the quality of the High good is not much greater than the quality of the Low good in percentage terms, the relative benefit of remaining a buyer into the next period diminishes. Since all these variables are inversely related to the benefit of accepting a Low endowment, they must be sufficiently low for this monetary equilibrium to exist. Given that condition (ii) of Proposition 1 holds, the disparity in consumption values between High and Low goods does not alter the interval that supports this equilibrium. However, since condition (ii) is a sufficient but not a necessary condition, it is possible for the utility differential UH UL to determine the lower bound of M. Suppose that condition (ii) does not hold. In that case, equilibrium requires that dxð1xÞ EU , for some interval of M when all agents are willU L P d/ðMÞ ¼ ð1 MÞ 1dð1xÞ ing to enter the market. Since the right hand side is decreasing in M, the condition holds if M is sufficiently high. This means that one of two constraints can determine the lower bound for M. There is an acceptance constraint that ensures buyers ACCEPT Low goods, and a market participation constraint that ensures good holders choose SEARCH. Either of these two constraints can be binding at the lower bound of M that supports this type of equilibrium. In Proposition 1 it was the participation constraint that determined the lower bound of M, but the acceptance constraint can also determine this lower bound. e such that U L ¼ d/ð M e Þ. If M 6 M e < M, then M e now defines Proposition 2. Define M the lower bound of M that can support this equilibrium. Therefore an increase in the e, utility differential UH UL that leaves EU unchanged will increase the value of M making the interval that can support this equilibrium smaller. The size of the interval e ; M is also decreasing in the discount rate. ½M e is defined by the equality, U L ¼ ð1 M e Þ dxð1xÞ EU , a decrease in UL Proof. Since M 1dð1xÞ when EU remains the same requires a higher value of M to restore the equality. Likewise when there is an increase in d. h
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The relative benefit of accepting a Low good is decreasing in both the discount rate and the quality differential. Consequently, buyers will ACCEPT Low goods over a smaller interval of M when they are more patient and when there is a larger difference in the quality of High and Low goods. This equilibrium demonstrates that money can induce agents to change their behavior from leisure to trade. When R > x2EU all agents choose REST in the non-monetary economy. However, under the conditions of Proposition 1, the introduction of money induces all agents to choose SEARCH and devote all their resources towards market activity. Agents allocate no time to leisure because money has increased the opportunity cost of leisure by facilitating exchange. The essential effect of money in this equilibrium is standard in random-matching models, which is to make exchange more likely by replacing the double coincidence of wants with the single coincidence of wants. Although High and Low good holders behave similarly, the existence of this equilibrium is still sensitive to the heterogeneity of agents since a larger utility differential can increases the minimum value of M that can support this equilibrium. 3.2. Money with some leisure Now assume that x2pUH > R, which implies x2EU > R so agents always choose SEARCH without money. In this case, the use of money can induce agents with Low quality goods to eschew the market, improving the average quality of traded goods. In this equilibrium, only money and High good holders choose SEARCH and buyers ACCEPT High goods but REJECT Low goods. Since Low good holders choose REST, the mass of trading agents is no longer unity but M + p(1 M). Consequently, the arrival rate and matching probabilities in this equilibrium are b ¼ M þ pð1 MÞ; xpð1 MÞ p¼ ; M þ pð1 MÞ xM ; s¼ M þ pð1 MÞ x2 pð1 MÞ b¼ . M þ pð1 MÞ The value functions are now V GH ¼ bbU H þ bsd/ þ dE½V G ; V V
G L M
G
¼ R þ dE½V ; H
ð5:1Þ ð5:2Þ
M
¼ bpU bpd/ þ dV ;
ð5:3Þ
where / VM E[VG]. Solving for the difference in the expected value functions for money and good holders yields /ðMÞ ¼
ð1 MÞpxð1 pxÞU H ð1 pÞR . 1 dð1 pxÞ
ð6Þ
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Unlike the first monetary equilibrium, /(M) is not always positive; however, it is linear and decreasing in M, as in the first monetary equilibrium. A necessary condition for a monetary equilibrium is /(0) > 0, which is true given the assumption x2pUH > R, which implies pxUH > R so that px(1 p)UH > (1 p)R and px(1 px)UH > (1 p)R. This monetary equilibrium exists if, given the strategies of others, no one has an incentive to deviate from their strategy, requiring (i) (ii) (iii) (iv)
V M > maxfV GH ; V GL g, UH P d/ > UL, bbUH + bsd/ P R > bbUH, bpUH bpd/ P R.
The first condition ensures that sellers ACCEPT money, and that money holders retain their money after receiving either a High or Low endowment. Condition (ii) ensures that buyers ACCEPT High goods and REJECT Low goods. Condition (iii) ensures that High good holders choose SEARCH, while Low good holders choose REST. Finally, condition (iv) ensures that money holders choose SEARCH. Proposition 3. There is an interval (M 0 , M00 ) with 0 < M 0 < M00 < 1 such that a monetary equilibrium exists in which money holders and High good holders choose SEARCH, Low good holders choose REST, and buyers ACCEPT High goods and REJECT Low goods: if (i) M 0 < M < M00 and (ii) d/(M 0 ) > UL. Proof. See Appendix A. h Once again the monetary equilibrium involves an upper and lower bound on M. However, in this equilibrium the lower bound discourages Low good holders from barter trade. If M is too low, the probability of barter is high enough to induce Low good holders to deviate from REST and enter the market. Given that Low good holders REST, equilibrium entails a rejection constraint for buyers (d/(M) > UL), a market participation constraint for money holders (bpUH bpd/ P R), and a market participation constraint for High good holders (bbUH + bsd/ P R). The upper bound M00 is defined by one of these three constraints, depending on the parameter values. In contrast to the first equilibrium, the rejection constraint ensures that buyers are now willing to pass up an inferior ‘‘bird-in-the-hand’’ for the prospect of receiving something better in the future. Proposition 4. If the rejection constraint defines M00 , then the size of the interval (M 0 , M00 ) is increasing in both d and the utility differential, UH UL. H
ð1pÞR Proof. When M < M00 , then d½ð1MÞpxð1pxÞU > U L . The left hand side is increas1dð1pxÞ H ing in d and U , and the right hand side is increasing in UL. Therefore, the constraint is binding for a greater value of M when d and UH are larger and UL is smaller. h
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The relative benefit of rejecting a Low good is increasing in both the discount rate and the quality differential. Consequently, buyers will REJECT Low goods over a larger interval of M when they are more patient and when there is a larger difference in the quality of High and Low goods. In this equilibrium, the use of money dichotomizes behavior between High and Low good holders in a way that is not possible in the non-monetary economy. Money makes possible what was previously impossible, inducing High good holders to SEARCH and Low holders to REST. The absence of Low sellers has a positive external effect on buyers by increasing the average value of market goods. 3.3. Multiple equilibria There is the possibility of multiple equilibria in which the intervals associated with each monetary equilibrium overlap. If x2pUH > R then 0 ¼ M < M 0 < M < 1. The key condition for this possibility is d/H(M 0 ) P UL > d/(M 0 ), where /H is defined as VM E[VG] when there are only High sellers in the market, and / is similarly defined when both High and Low sellers are in the market. In this case, a buyer will reject a Low good if all other Low agents are out of the market, but will accept a Low good if all other Low agents are in the market. This can be verified with a formal proof as well as with a numerical example, as in Appendix A.
4. Robustness The existence of one or both of the above monetary equilibria may be sensitive to the indivisibility of goods and money. Under the assumption of indivisibility there is no endogenous price that can affect the ACCEPT/REJECT decision of buyers. One way to introduce an endogenous price is to randomize exchange by incorporating lotteries as in Berentsen et al. (2002). In their model, buyers and sellers bargain over the probability of trading goods and money as a proxy for bargaining directly over the terms of trade. For example, assuming risk neutrality, selling a good in exchange for a 50% probability of receiving a dollar is similar to selling the good for 50 cents. Berentsen et al. (2002) utilize the generalized Nash bargaining solution which allows one to calibrate the relative bargaining power of buyers in a straightforward way. In a related paper Berentsen and Rocheteau (2002) show that some of the standard results of search-theoretic, monetary models are due to the indivisibility of money. In the following analysis, agents make take-it-or-leave-it offers on the probability of trading money as in Berentsen and Rocheteau (2002). In contrast to their model in which only buyers make offers, offers by sellers are also considered. In terms of bargaining power there are two opposite extremes, corresponding to whether buyers or sellers are making the take-it-or-leave-it offer. A priori, there is no reason to assume that either buyers or sellers would have greater bargaining power. The main results are that although both equilibria are sensitive to the introduction of bargaining, monetary trade can take place if sellers make take-it-or-leave-it offers. Conversely, if buyers make the offer, neither monetary equilibrium can be supported.
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4.1. Money with no leisure Consider the first equilibrium when everyone enters the market, and buyers ACCEPT All. If buyers make the offer, they have too much bargaining power to support monetary exchange. This is due to the discrete nature of time and the fact that foregone leisure is a sunk cost for agents in the market. Another factor is the endowment economy with no cost of production for the seller that can provide a lower bound for the terms of trade. When a buyer and seller meet, the buyer offers to give a unit of money with probability s in exchange for the sellerÕs good. A seller will accept any offer that does not make them worse off, i.e. the seller will accept the offer if sdVM + (1 s)dVG P dVG. Since buyers can extract the entire surplus, this condition becomes sd/ = 0. However in this equilibrium / > 0, and the only way the condition holds is if s = 0 in which case there is no benefit for sellers to enter the market because they can never become buyers. Therefore good holders choose REST and monetary exchange will collapse. If sellers make the offer, a rejection by buyers results in foregoing the consumption of the good. Consequently, there could be two prices: s(H) and s(L). When a High seller makes an offer, the condition for indifference becomes UH = s(H)d/, but in this particular equilibrium UH > d/, which implies UH/d/ > 1 = s(H). Sellers extract the entire surplus and money is always traded for High goods. Money is also traded with certainty for Low goods, because condition (ii) of Proposition 1 guarantees UL > d/ for all values of M. Consequently, if sellers make the offer, then s(H) = s(L) = 1 and the equilibrium remains. 4.2. Money with some leisure The second equilibrium also disappears if buyers make the offer. Sellers are indifferent between accepting and rejecting when sd/ = 0. Recall that this equilibrium requires Low good holders to stay out of the market while High good holders enter, that is bbUH + bssd/ P R > bbUH. If sd/ = 0, this condition becomes bbUH P R > bbUH, a contradiction. When sellers make the offer, money is exchanged with certainty with High sellers for reasons similar to the first equilibrium. Indifference for buyers requires UH = s(H)d/ but once again UH > d/ for all values of M and UH/d/ > 1 = s(H). When Low sellers bargain with buyers, there is always a s(L) low enough for buyers to accept the Low good, as seen from the condition for buyer indifference, UL = s(L)d/. If UL 6 d/, then s(L) = UL/d/ < 1; if UL > d/, then s(L) = 1. Consequently, 0 < s(L) 6 1 and buyers no longer REJECT Low goods. Low goods are now acceptable because of attractive terms-of-trade for buyers. In summary, if either buyers or sellers make take-it-or-leave-it offers, the equilibrium cannot exist. Nevertheless, there is a different equilibrium that can also generate an outcome in which Low good holders stay out of the market, while High good holders enter. Low good holders may still want to choose REST even if their goods would be accepted by buyers. In particular, if sellers make take-it-or-leave-it offers, there is a monetary
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equilibrium in which buyers choose SEARCH and ACCEPT All, High good holders choose SEARCH, and Low good holders choose REST. The essential condition is that Low good holders still choose REST while High good holders choose SEARCH when s(H) = 1 and 0 < s(L) 6 1, where s(L) is increasing in M. This requires that bbUH + bsd/ P R > bbUH + bss(L)d/. If s(L) = 1, the conditions becomes a contradiction. Therefore, necessary conditions for this equilibrium are s(L) = UL/d/ < 1 and that only buyers and High good holders choose SEARCH. Consequently, this equilibrium requires that three conditions hold simultaneously: (i) d/ > UL, (ii) bbUH + bsd/ P R > bbUH + bsUL, (iii) bpUH bpd/ P R. These sufficient conditions are similar to those without randomized trading. The only difference is the non-participation constraint for Low good sellers, which is now R > bbUH + bsUL. Recall that this constraint defines the lower bound of M that can support the equilibrium. Assume R > xUL, then the right hand side of the constraint b such that R ¼ x2 pð1 M b ÞU H þ is equal to R at a unique value of M. Define M L b xM U . Proposition 5. Consider the case in which Low sellers offer to give their good with certainty in exchange for money with probability 0 < s(L) < 1. If UL is sufficiently low, b ; M 00 Þ, with 0 < M b < M 00 < 1 such that a monetary then there is an interval ð M equilibrium exists in which buyers choose SEARCH and ACCEPT All, High good b < M < M 00 . holders choose SEARCH, and Low good holders choose REST; if M b , there is an implicit function M b ðU L Þ that is Proof. According to the definition of M L L 0 continuous, increasing in U , and with the limit of M as U approaches zero. Thereb < M 00 and the interval exists. h fore if UL is sufficiently low, M The consumption value of the Low good again plays a role in defining the lower bound of M, but through the non-participation constraint of Low good holders instead of the rejection constraint of buyers. Low goods are traded, but for such a low ‘‘price’’ that the probability of receiving money and becoming a buyer is too small to make entering the market worthwhile.
5. Conclusion This paper develops a model in which the cost of searching for a trading partner is foregone leisure and agents realize idiosyncratic shocks in the consumption value of their endowment. In the non-monetary economy there are two perfect Nash equilibria, one with everyone staying out of the market to enjoy leisure and one with everyone entering the market to seek a trading partner. There is no barter equilibrium
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with only High good holders in the market. When fiat money is introduced, there are two possible equilibria, one with everyone participating in the market and buyers accepting both High and Low goods, and another with buyers rejecting Low sellers and Low agents eschewing the market to enjoy leisure. The behavior of a buyer who meets a Low seller determines which of these two equilibria occur. The first equilibrium occurs when a buyer refuses to forego consuming a Low good in the present for the possibility of consuming a High good in the future. The second equilibrium occurs when a buyer is willing to give up a ‘‘sure thing’’ for the future possibility of something better. Money can affect the real side of the economy through the search-or-rest decision of Low good holders since their presence or absence affects the average quality of traded goods. For instance, when money drives Low good holders out of the market, the value of traded goods increases. Money can generate an outcome that is always impossible without money, namely High good holders entering the market while Low good holders enjoy leisure. If agents bargain over lotteries for money in which sellers make take-it-or-leave-it offers, there is an equilibrium in which buyers enter the market and accept both High and Low goods, High good holders enter to accept money, and Low good holders rest.
Acknowledgements I am thankful to Melanie Baker, Michael Pangia and Zhenhui Xu for helpful comments, and to two anonymous referees who helped substantially improve the content and presentation of the paper.
Appendix A A.1. List of main variables d 2 (0,1) is the discount factor. U 2 {UH, UL} is the utility of consuming the endowment, where UH > UL > 0. N 2 {3, 4, 5, . . .} is the number of different types of endowment goods, representing the level of specialization in the economy. x 1/N is the coincidence of wants. R is the utility of leisure, where x2UH > R > x2UL. p 2 (0, 1) is the probability that an agentÕs endowment is High and the proportion of the population with a High endowment. EU pUH + (1 p)UL is the average endowment size in the economy. M 2 (0, 1) is the proportion of the population that holds a unit of indivisible money. b 2 (0, 1] is the arrival rate, which is equal to the proportion of agents in the market. p 2 (0, x) is the probability that a buyer will find a suitable seller.
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s 2 (0, x) is the probability that a seller will find a suitable buyer. b 2 (0, x2) is the probability that a seller will find another seller suitable for barter. / VM E[VG] s is the probability that money is exchanged. Proof of Proposition 1. We first verify that no one deviates from SEARCH. Given that money traders choose {SEARCH, ACCEPT}, a good holder will enter the market if S(M) + dVG P R + dVG. Since S(M*) > R and S(M) is concave and continuous, there is an interval in the neighborhood of M* where S(M) P R. Turning to the other side of the market, given that good holders choose {SEARCH, ACCEPT}, a money holder will enter the market if B(M) + dVM P R + dVM. Since S(M) P R when M 6 M 6 M, money holders will prefer to retain their money and dx enter the market if B(M) > S(M). This inequality reduces to 1 > 1dð1xÞ , which is true. Consequently, both money and good holders enter the market when M 6 M 6 M. We now verify that no one deviates from ACCEPT. Sellers will accept money as > 0. A buyer will long as / VM VG > 0, which is true since / ¼ ð1MÞxð1xÞEU 1dð1xÞ ACCEPT a High endowment if UH + dVG P dVM. This is equivalently stated as U H P dð1MÞxð1xÞ 1dð1xÞ EU , which is true since the fraction on the RHS is less than one and UH > EU so that U H > EU > dð1MÞxð1xÞ 1dð1xÞ EU . Finally, a buyer will ACCEPT a Low endowment if UL + dVG P dVM, equivalently expressed as U L P ð1 MÞ dx ð1 xÞ 1dð1xÞ EU . A sufficient condition for this inequality to hold is UL >
dx EU . 1 dð1 xÞ
ðA:1Þ
Since (A.1) is algebraically equivalent to condition (ii) of Proposition 1, the sufficient condition holds and buyers will ACCEPT Low endowments. There is thus no incentive for any agent to deviate from {SEARCH, ACCEPT}. h Proof of Proposition 3. It is sufficient to show that four inequalities hold.
h
Lemma. Conditions (iii), (iv), and the second part of condition (ii) are sufficient conditions for this equilibrium. Proof. Condition (i) holds if V M > V GH > V GL . The first equality ðV M > V GH Þ can be expressed as bp(1 x)UH + (1 bx)d/ > 0, which is true if d/ > 0, which is implied by the second part of condition (ii), d/ > UL. The second inequality ðV GH > V GL Þ is true if and only if bbUH + bsd/ > R, which is the first part of condition (iii). The first inequality of condition (ii), UH > d/(M) is true for all values of M if UH > d/(0). dpxð1pxÞ dð1pÞR U H 1dð1pxÞ which holds When M = 0, the inequality becomes U H > 1dð1pxÞ
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dpxð1pxÞ because U H > 1dð1pxÞ U H . The equilibrium therefore exists if one can show that L d/ > U and conditions (iii) and (iv) simultaneously hold for some interval of M. h
These conditions consist of four inequalities, one of which requires that M be sufficiently high. The remaining three inequalities require that M be sufficiently low. These three inequalities can be described as a rejection constraint ensuring buyers REJECT Low goods and two market participation constraints ensuring that both money and High good holders choose SEARCH. The remainder of the proof has two more steps. The first is to show that there is a threshold value of M that determines when M is high enough to cause Low holders to REST, and the second is to show that the remaining three conditions hold at that threshold value. Since bbUH is decreasing in M, define M 0 as the unique value of M where bbUH x2p(1 M)UH = R, so that R > x2p(1 M)UH if and only if M > M 0 . Now define g(M) bbUH + bsd/ and k(M) bpUH bpd/. Taking the second derivative of both functions w.r.t. M gives g00 ðMÞ ¼
2x2 pdð1 pxÞU H <0 1 dð1 pxÞ
and k00 ðMÞ ¼
2x2 p2 dð1 pxÞU H < 0. 1 dð1 pxÞ
Given the strict concavity of these functions, if g(M) > R and k(M) > R when M = M 0 , then there is an interval of M in which Low good holders REST while both money and High good holders SEARCH. By definition, when M = M 0 then bbUH = R so that the inequality g(M) > R can be rewritten as R + dxM/ > R. This is true when / > 0. / is positive if and only if bp(1 px)UH (1 p)R > 0. Using fact that bbUH = R yields (1 M)px(1 px)UH > (1 p)(1 M)x2pUH which reduces to 1 > x, obviously true. Likewise, substituting bbUH = R into the condition k(M) > R gives 2
ð1 MÞpxU H
2
ð1 MÞ p2 x2 dð1 pxÞU H ð1 MÞ p2 x3 dð1 pÞU H þ 1 dð1 pxÞ 1 dð1 pxÞ
> px2 ð1 MÞU H which after dividing by (1 M)pxUH becomes 1
ð1 MÞpxdð1 pxÞ ð1 MÞpx2 dð1 pÞ þ > x. 1 dð1 pxÞ 1 dð1 pxÞ
dpxð1MÞ dpx Rearranging and dividing by (1 x) gives 1 > 1dð1pxÞ . This is true if 1 > 1dð1pxÞ , which reduces to 1 d > 0. Since g(M 0 ) > R > g(1) and k(M 0 ) > R > k(1), and since both functions are continuous and strictly concave in M, then applying the intermediate value theorem each function is equal to R at a unique value of M 2 (M 0 ,1). Let these values be defined as M1 and M2 respectively so that g(M1) = R and k(M2) = R. Finally, since UL is not a part of any other condition, simply assume d/(M 0 ) > UL. Since d/(M) is decreasing in M, there is a threshold value M0 where d/(M0) = UL.
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Depending on the specific parameter values, one of these three constraints is the first to become binding as M increases above M 0 . Therefore one of these constraints defines the upper bound of M where M00 = min{M0, M1, M2}. h A.2. Numerical example of multiple equilibria Since x2pUH > R implies 0 = M < M 0 , there are only two sufficient conditions for multiple equilibria: d/H ðM 0 Þ P U L > d/ðM 0 Þ;
ðA:2Þ
SðM 0 Þ > R.
ðA:3Þ
Eq. (A.2) ensures that buyers will ACCEPT and REJECT Low goods in the respective equilibria. Eq. (A.3) ensures that M 0 < M. When, x = 0.2, p = 0.6, d = 0.9, UH = 100, UL = 26 and R = 1.8 then M 0 = 0.25 and both conditions hold. So there is an interval of M above 0.25 where there are multiple equilibria. For instance, when M = 0.3, d/H = 28.87 > 26 > 25.34 = d/ and S = 3.49 > 1.8. A formal proof is also available.
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