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Generalized search-theoretic models of monetary exchange
Journal of Monetary Economics 48 (2001) 605–622
Generalized search-theoretic models of monetary exchange$ Peter Ruperta, Martin Schindlerb, Randall Wrightb,* a b
Research Department, Federal Reserve Bank of Cleveland, Cleveland, OH 44101, USA Department of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA 19104, USA
Received 9 May 2000; received in revised form 8 November 2000; accepted 21 November 2000
Abstract Virtually all simple search models of money assume agents with money cannot produce, and so everyone has either 0 or 1 units of money in steady state. We alternatively assume agents can always produce, and simply restrict money inventories to 0 or 1. This seems better for many issues; for example, in the standard model it is difficult to interpret increases in the money supply, because increasing the fraction of agents holding money decreases the economy’s productive capacity. Our model avoids these problems and thus delivers more natural, and simpler, implications in many contexts. We compare results on existence, multiplicity, and welfare across models. r 2001 Elsevier Science B.V. All rights reserved. JEL classification: E00; D83; C78 Keywords: Money; Search; Bargaining; Prices
$ We thank the Research Department at the Federal Reserve Bank of Cleveland for use of their facilities, and the National Science Foundation for financial support. We are also grateful to Yiting Li, Eduardo Siandra, Steve Williamson and an anonymous referee for valuable comments. The usual disclaimer applies.
*Corresponding author. Tel.: +1-215-898-7194; fax: +1-215-573-4217. E-mail address: [email protected] (R. Wright). 0304-3932/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 3 2 ( 0 1 ) 0 0 0 8 8 - 5
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1. Introduction Virtually all of the simple search-theoretic models of money in the literature assume that agents holding money cannot produce. This guarantees that, in steady state, everyone has either 0 or 1 units of money. We explore an alternative where all agents can always produce, and simply restrict money inventories to f0; 1g: This is useful because the standard model has several undesirable implications that our alternative avoids. For example, in the model where agents holding money cannot produce, if two agents with money meet they do not trade even if there is a double coincidence of wants. A related implication of the standard model is that increasing the fraction of agents holding money necessarily decreases the economy’s productive capacity. This makes it difficult, at best, to interpret the effects of changes in the money supply. Our model, where all agents can produce, does not have these unnatural implications. We derive results on existence, the number of equilibria, comparative statics, and welfare in our model, and compare these to results from the standard model. We also present results for all values of the parameter y representing bargaining power, while virtually all previous papers in the monetary search literature consider only the symmetric case y ¼ 12; or the extreme case y ¼ 1: This generality is useful not only for its own sake, but also because it allows us to discuss the relationship between bargaining power and efficiency, as is done in search models of the labor market, for example. Summarizing, this paper analyzes a monetary economy with a general bargaining solution and where all agents can always produce. The latter feature is particularly significant since, as compared to the standard model, the results are often easier to derive and typically more natural in the new model.1
2. The basic model There is a ½0; 1 continuum of agents who live forever and discount at rate r: They meet bilaterally according to a Poisson process with parameter a; and 1
See Kiyotaki and Wright (1989, 1991, 1993), Aiyagari and Wallace (1991, 1992), Trejos and Wright (1995) and Shi (1995) for examples of the standard model. Siandra (1993, 1996) is an example of the alternative model where money holders can produce, although he only considers the case of indivisible goods (see Rupert et al., 2000, for a discussion of this case). Aiyagari et al. (1996) and Kocherlakota and Wallace (1998) also allow agents with money to produce; however, those models assume the probability of a double-coincidence as 0; which renders the main issues discussed here far less interesting because in that case the two models collapse to the same thing (see the discussion preceding Proposition 5). All of the above models in one way or another limit money holdings to f0; 1g: There are also models where money holders can produce and agents can hold more general inventories, but they are an order of magnitude more complicated than what we consider here; see Green and Zhou (1998), Camera and Corbae (1999), Molico (1999), Taber and Wallace (1999) and Zhou (1999).
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trade iff mutually agreeable. There are a variety of nonstorable consumption goods, but each agent produces just one good, at (disutility) cost cðqÞX0 per q units. Given two agents i and j; let iWj indicate ‘‘i wants to consume the good that j produces’’, in the sense that i derives utility uðqÞ from consuming q units of the good that j produces if iWj; and 0 otherwise. We assume u0 ðqÞ > 0 and c0 ðqÞ > 0: Also, c00 ðqÞX0 and u00 ðqÞp0 with at least one of the inequalities strict. Finally, uð0Þ ¼ cð0Þ ¼ 0; u0 ð0Þ > c0 ð0Þ ¼ 0; and there exists q# > 0 such that # ¼ cðqÞ: # We model specialization as follows: For no agent i is it the case uðqÞ that iWi; and for a pair of agents i and j selected at random, prðiWjÞ ¼ x and prð jWijiWjÞ ¼ y: Thus, xy is the probability of a double coincidence of wants.2 There is an exogenous quantity MA½0; 1 of intrinsically valueless fiat money. Initially one unit of money is given to each of M agents chosen at random. The standard version of the model, following Kiyotaki and Wright (1991, 1993), assumes that agents holding money cannot produce. We explore an alternative model, following Siandra (1993, 1996), where they can. Call the first Model-K and the second version Model-S: The Model-K assumption can be interpreted as saying that after producing an agent needs to consume before producing again, which implies that in steady state no one holds more than 1 unit of money. The Model-S assumption does not rule out the accumulation of multiple units of money, so we simply assume agents can store at most 1 unit of money. While obviously it is interesting to study models where agents can accumulate general money inventories, it is also still interesting to study tractable models, and limiting inventories in this way simplifies the analysis significantly. The issue in this paper is to ask, conditional on agents holding 0 or 1 unit of money, what happens if we relax the assumption that agents with money cannot produce. In the standard Model-K; there are only two nontrivial types of meetings: a double coincidence meeting between two agents without money, and a single coincidence meeting between i and j; where iWj and i has money while j does not. The former meeting always leads to a barter trade, and the latter may or may not lead to a monetary trade, but there can be no barter. In Model-S; however, in the latter meeting i might trade goods, cash, or both. Of course, what happens in any meeting depends on the bargaining solution. Since we are most interested in single coincidence meetings, where money plays a role, we explore various alternatives in this case, while for double coincidence meetings we simply impose a particular bargaining solution that gives convenient results. In particular, we show that under our assumptions the unique equilibrium in any double coincidence meeting is for the agents to barter goods for goods, and no money will change hands.
2
This specification nests several models in the literature; e.g., in Kiyotaki and Wright (1993) the events fiWjg and fjWig are independent, so y ¼ x and the double coincidence probability is x2 :
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To pursue this, let Vm denote the value function of an agent with m units of money, mAf0; 1g: First consider a single coincidence meeting. In this case, money must change hands if goods are to change hands, and q is determined by the generalized Nash bargaining solution max ½uðqÞ þ V0 T1 y ½V1 cðqÞ T0 1 y q
ð1Þ
subject to the constraints qX0; D1 ¼ uðqÞ þ V0 V1 X0 and D0 ¼ V1 V0 cðqÞX0: In (1), Tm is the threat point of the agent with m units of money and y represents bargaining power. We will consider below two cases for Tm that have been analyzed in the monetary literature, Tm ¼ Vm and Tm ¼ 0; and generalize previous analyses by allowing y to take on any value in ½0; 1:3 Now consider a double coincidence meeting between an agent with 0 and an agent with 1 unit of money. As said above, we need to determine whether money changes hands, and the two quantities q0 and q1 ; where qm is the quantity of goods that the agent with m units of money receives. The following is proved in the Appendix.4 Lemma 1. In a double coincidence meeting between an agent with and an agent without money; if we assume y ¼ 12 and Tm ¼ Vm ; the unique outcome is that no money changes hands; and q0 ¼ q1 ¼ qn where qn satisfies u0 ðqn Þ ¼ c0 ðqn Þ: There are also double coincidence meetings where the two agents have the same money holdings (m ¼ 0 or m ¼ 1). One can also show in these cases that each agent receives qn and no money changes hands, under the same assumptions used in Lemma 1. Hence, in all double coincidence meetings the net payoff is B ¼ uðqn Þ cðqn Þ: Note that the previous literature typically did not need to worry about what happens in double coincidence meetings, since in Model-K there is only one option (since money holders cannot produce).
3. Monetary equilibrium Having characterized double coincidence meetings, the remainder of the paper is concerned with determining what happens in single coincidence meetings. The first thing we need is the Bellman equations. In Model-S these 3 As is well known, the Nash solution corresponds to the equilibrium of an explicit strategic bargaining model, where y and Tm depend on details of the game; see Osborne and Rubinstein (1990) or, for a presentation in the context of monetary theory, Coles and Wright (1998). 4
For different bargaining powers or threat points, it is possible to have money change hands or to have qm aqn : The point is not that the predictions about double coincidence meetings in Lemma 1 are completely general, but that we can design an environment where they are true, allowing us to focus in more detail on single coincidence meetings.
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are given by rV1 ¼ axyB þ axð1 MÞð1 yÞ½uðqÞ þ V0 V1 ;
ð2Þ
rV0 ¼ axyB þ axMð1 yÞ½V1 V0 cðqÞ:
ð3Þ
For example, a money holder encounters a double coincidence at rate axy; which entails net payoff B; and at rate axð1 MÞð1 yÞ he encounters a single coincidence with an agent without money, which entails net payoff uðqÞ þ V0 V1 : For comparison, the corresponding equations in Model-K are rV1 ¼ axð1 MÞ½uðqÞ þ V0 V1 ;
ð4Þ
rV0 ¼ axð1 MÞyB þ axM½V1 V0 cðqÞ:
ð5Þ
The differences arise because in Model-K money holders cannot produce. In what follows, to reduce notation we normalize ax ¼ 1; with no loss of generality. The first-order condition to the bargaining problem (1) is y½V1 cðqÞ T0 u0 ðqÞ ð1 yÞ ½uðqÞ þ V0 T1 c0 ðqÞ ¼ 0: Choosing Tm and substituting V0 and V1 (which we get from solving the Bellman equations) yields one equation in one unknown, eðqÞ ¼ 0: The different types of equilibria that may occur are best illustrated in a particular case. As a benchmark we start with the standard Model-K; in the case where the threat points are given by Tm ¼ Vm : Then eðqÞ ¼ y½ð1 MÞuðqÞ ðr þ 1 MÞcðqÞu0 ðqÞ ð1 yÞ½ðr þ MÞuðqÞ McðqÞc0 ðqÞ ð1 MÞyB½yu0 ðqÞ þ ð1 yÞc0 ðqÞ:
ð6Þ
A solution to eðqÞ ¼ 0 would be an equilibrium if there were no constraints on the bargaining problem. The constraint D1 X0 can be shown to be redundant if D0 X0 holds, and so we only need to check the latter. This means there are two distinct types of monetary equilibria: an unconstrained equilibrium is a q > 0 such that eðqÞ ¼ 0 and D0 X0; while a constrained equilibrium occurs when D0 X0 binds. If D0 ¼ 0 at some q* then the constraint is violated for any q > q; * and if eðqÞ * > 0 then q* is a constrained solution to the Nash problem. It is easy to see that D0 ¼ 0 iff q ¼ q or q ¼ q; % where q and q% are zeros of the strictly % % concave function f ðqÞ ¼ ð1 MÞuðqÞ ðr þ 1 MÞcðqÞ ð1 MÞyB:
ð7Þ
Note that f is decreasing in r; and for small r the two zeros q and q% exist, while for big r there are no solutions to f ðqÞ ¼ 0: Clearly, qAðq;% qÞ % implies D0 > 0; and so any qAðq; qÞ % such that eðqÞ ¼ 0 is an unconstrained %equilibrium. Also, if eðqÞ > 0 then q% is a constrained equilibrium, and if eðqÞ % > 0 then q% is a % % constrained equilibrium.
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0
q
q
q
-(1-M)y Bθu'(0) -(1-M)yB
f(q)
(a)
e(q) q q
q
(b)
0
θ
1
θ
Fig. 1. Model-K with y > 0; Tm ¼ Vm : (a) Functions e and f : (b) Equilibrium correspondence.
This reduces the search for monetary equilibria to looking for eðqÞ ¼ 0 at qAðq; qÞ; % or for eðqÞ > 0 at q ¼ q or q: % We now provide results on existence and % % present Model-K and then Model-S assuming the number of equilibria. We first y > 0: Later we consider y ¼ 0; where the two models are the same. Beginning with Model-K; the functions eðqÞ and f ðqÞ defined above, as well as the set of equilibrium values for q as a function of y; are depicted in Fig. 1 for the case Tm ¼ Vm and Fig. 2 for the case Tm ¼ 0: Proposition 1. Consider Model-K with y > 0 and Tm ¼ Vm : There exists r% > 0 such that no monetary equilibria exist if r > r%; while if ror% there exists y% Að0; 1Þ ðthat depends on rÞ such that the following is true: for yoy% no monetary equilibria exist; and for y > y% there exists generically an even number of monetary equilibria. All monetary equilibria are unconstrained.
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(1-M)2yBθu'(0)
0
q
q
q
-(1-M)yB
f(q) e(q)
(a)
q q
q
(b)
0
θ
1
θ
Fig. 2. Model-K with y > 0; Tm ¼ 0: (a) functions e and f ; (b) equilibrium correspondence.
Proof. In the limiting case r ¼ 0; clearly there exist q > 0 and q% > q such that % eðqÞp0; f ðqÞ ¼ f ðqÞ % ¼ 0; as seen in Fig. 1a. It is easy to check %that eðqÞp0 and % % % with equality iff y ¼ 1: This implies that no constrained equilibria exist (when y ¼ 1 we actually have an equilibrium where the constraint is satisfied at equality, but since the first order condition also holds with equality the solution is unconstrained). Moreover, if any unconstrained monetary equilibria exist, generically there will be an even number since e will have an even number of zeros in ðq; qÞ: % Setting y ¼ 0 implies eo0 for all q; so no monetary equilibria % that y ¼ 1 implies the existence of monetary equilibria. It can be exist. Recall shown that @e=@yje¼0 > 0 (see Appendix A), so that decreasing y shifts e down in Fig. 1a. This implies the existence of a unique y% such that monetary equilibria exist iff y > y% : See Fig. 1b. All of the above statements are for r ¼ 0; by continuity, the results are the same for small positive r: It is easy to see that e
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is monotonically decreasing in r (for q > 0), and that eo0 for all qAðq; qÞ % when r is sufficiently big. Hence, there is a unique r% such that no monetary %equilibria exist for any y when r > r%: & Proposition 2. Consider Model-K with y > 0 and Tm ¼ 0: There exists r% > 0 such that no monetary equilibria exist if r > r%; while if ror% there exists y% Að0; 1Þ ðthat depends on rÞ such that the following is true: for yoy% no monetary equilibria exist; and for y > y% there are exactly two monetary equilibria. Both equilibria are constrained if y is large; while one equilibrium is constrained and the other unconstrained if y is not so large. Proof. With Tm ¼ 0; the analogue of (6) is eðqÞ ¼ yfð1 MÞðr þ MÞuðqÞ ½rð1 þ rÞ þ Mð1 MÞcðqÞgu0 ðqÞ ð1 yÞf½rð1 þ rÞ þ Mð1 MÞuðqÞ Mðr þ 1 MÞcðqÞgc0 ðqÞ þ ð1 MÞyB½yð1 MÞu0 ðqÞ ð1 yÞðr þ 1 MÞc0 ðqÞ: See Fig. 2a. As before, r ¼ 0 implies that q > 0 and q% > q exist. If y ¼ 0; eðqÞo0 % for any q > 0 and no monetary equilibria exist. If y ¼ 1;%eðqÞ > 0 for all qAð0; q; % and both q and q% are constrained equilibria. One can check @e=@yje¼0 > 0; so there exists% a critical value y% > 0 such that monetary equilibria exist iff yXy% : Also, for yXy% but not too large, eðqÞo0; and so one monetary equilibrium is % unconstrained at qAðq; qÞ % and one equilibrium is constrained at q; while for % very large y; eðqÞ are % > 0% (for arbitrary r), so both monetary equilibria constrained. This can be seen by noting that eðqÞj % y¼1 ¼ ðM þ rÞ½ð1 MÞuðqÞ % ðr þ 1 MÞcðqÞu % 0 ðqÞ % þ ð1 MÞ2 yBu0 ðqÞ % ¼ ð1 þ rÞð1 MÞyBu0 ðqÞ; % which is positive for any r; combined with the fact that e is continuous and increasing in y: These results are qualitatively the same for small positive r: It is shown in Appendix A that, as long as eðqÞ > 0; we have @e=@rje¼0 o0: Note that this is all % exist iff eðqÞ > 0: Setting y ¼ 1; there exists a we need as monetary equilibria % exist when r > r% for any y: For unique r% > 0 such that no monetary equilibria any ror%; by the above reasoning, there exists y% o1 such that monetary equilibria exist for y > y% : To see that there are exactly two monetary equilibria (whenever any exist), it is sufficient to show that any qAð0; qÞ % such that eðqÞ ¼ 0 is unique. This is verified by noting that at e ¼ 0; we have ð1 yÞ
c0 y ¼ fð1 MÞðr þ MÞu u0 D ½rð1 þ rÞ þ Mð1 MÞc þ ð1 MÞ2 yBg;
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where D ¼ ½rð1 þ rÞ þ Mð1 MÞu Mðr þ 1 MÞc þ ðr þ 1 MÞ ð1 MÞyB; and that the left side is increasing and the right side decreasing in q: & We now turn to Model-S: Although the method is the same, the results are quite different: now there exists a unique monetary equilibrium for all y > 0: This is interesting since many policy discussions in this literature focus on multiplicity (for example, Aiyagari and Wallace, 1997; Li and Wright, 1998). Graphically, uniqueness in Model-S can be understood by noting that in this model the functions eðqÞ and f ðqÞ always go through the origin; see Fig. 3 for the case Tm ¼ Vm and Fig. 4 for the case Tm ¼ 0:
0
q
q
f(q) e(q) (a)
q q
(b) 0
1
θ
Fig. 3. Model-S with y > 0; Tm ¼ Vm : (a) functions e and f ; (b) equilibrium correspondence.
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0
q
q
f(q) e(q) (a)
q q
(b)
0
θˆ
1
θ
Fig. 4. Model-S with y > 0; Tm ¼ 0: (a) functions e and f ; (b) equilibrium correspondence.
Proposition 3. Consider Model-S with y > 0 and Tm ¼ Vm : For any r > 0; the following is true: if either y ¼ 0 or y ¼ 1; no monetary equilibrium exists; otherwise; there exists a unique monetary equilibrium and it is unconstrained. Proof. In Model-S; the function f ðqÞ is f ðqÞ ¼ ð1 MÞð1 yÞuðqÞ ½r þ ð1 MÞð1 yÞcðqÞ: With Tm ¼ Vm ; the analogue of (6) is eðqÞ ¼ yf ðqÞu0 ðqÞ ð1 yÞf½r þ Mð1 yÞuðqÞ Mð1 yÞcðqÞgc0 ðqÞ: For y ¼ 1; we have eð0Þ ¼ 0 and eðqÞo0 for q > 0; and so no monetary equilibria exist for any y: Now suppose yo1: It is easy to see that f ðqÞ ¼ 0 at q ¼ 0 and q% > 0: It can also be verified that eð0Þ ¼ 0 and eðqÞp0 for all y; where % %
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the inequality is strict for yo1; implying that there are no constrained equilibria. If y ¼ 0; the only solution to e ¼ 0 is q ¼ 0: For yAð0; 1; e0 ð0Þ > 0; so that there exists a qAð0; q % such that eðqÞ ¼ 0: The monetary equilibrium is unique because at e ¼ 0; we have c0 f ðqÞ ; ð1 yÞ 0 ¼ y ½r þ Mð1 yÞu Mð1 yÞ u where the left-hand side is increasing and the right-hand side decreasing in q: Finally, when y ¼ 1; the (unconstrained) unique monetary equilibrium is at q ¼ q: % See Fig. 3. & Proposition 4. Consider Model-S with y > 0 and Tm ¼ 0: For any r > 0; the following is true: if either y ¼ 0 or y ¼ 1; no monetary equilibrium exists; otherwise; there exists a unique monetary equilibrium; and it is unconstrained for # where 0oyo1: # yoy# and constrained for y > y; Proof. In this case the analogue of (6) is eðqÞ ¼ y½r þ Mð1 yÞf ðqÞu0 þ ð1 þ r yÞyB½yu0 ð1 yÞc0 ð1 yÞ½rð1 þ r yÞu þ Mð1 yÞf ðqÞc0 : The constraint is as before, and so f ðqÞ ¼ 0 at q ¼ 0 and q% > 0: For y ¼ 0; we % equilibrium exists, while at have eð0Þ ¼ 0 and e0 ð0Þo0; and so no monetary y ¼ 1; again, no monetary equilibria exist. Now assume yo1: For yAð0; 1; we have eð0Þ > 0: Note that for y ¼ 0; eðqÞo0, and for y ¼ 1; eðqÞ % % > 0: By # inspection e is monotonically increasing in y: Thus, there exists yAð0; 1Þ such that the monetary equilibrium is unconstrained for yoy# and constrained (at q) % # Uniqueness is shown in the usual way. See Fig. 4. & for y > y: All of the above results are for y > 0: If y ¼ 0 then Model-K and Model-S are identical (to see this, simply compare the Bellman equations). It turns out that the results in the y ¼ 0 case are similar to what we derived above for Model-S with y > 0: for all y > 0 there exists a monetary equilibrium and it is unique. However, note that the equilibrium is always unconstrained when y ¼ 0:5 Proposition 5. Consider either Model-K or Model-S for either Tm ¼ Vm or Tm ¼ 0; where y ¼ 0: For any r > 0; the following is true: if y ¼ 0 no monetary equilibria exist; and if y > 0 there exists a unique monetary equilibrium and it is unconstrained. Proof. For y ¼ 0; we have q ¼ 0 and q% > 0 by inspection of f ðqÞ: For y ¼ 0; the % 0: For yAð0; 1Þ it can be verified that eð0Þ ¼ 0 and only solution to e ¼ 0 is q ¼ This is not inconsistent with Proposition 4, because y# defined in that result goes to 1 as y goes to zero. 5
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eðqÞo0; implying that there are no constrained monetary equilibria. Moreover, % it is easy to see that e0 ð0Þ > 0; which implies that there exists qAðq; qÞ % such that eðqÞ ¼ 0: Finally, when y ¼ 1; the unique nonzero solution to e%¼ 0 is q ¼ q; % this is the unique equilibrium. Function e and equilibria are qualitatively the same as in Model-S with Tm ¼ Vm and y > 0 (see Fig. 3). &
4. Discussion The main elements that distinguish our analysis from the previous literature are: (i) we allow any yA½0; 1; and (ii) we consider Model-S as well as Model-K: Many results from previous analyses continue to hold: for example, for any y; in Model-K with y > 0 the existence of monetary equilibria depends on r being small. Also, as in earlier analyses, the equilibrium quantity q is typically not socially efficient. However, given our generalized model, the interaction between efficiency and bargaining power can be explored in more detail, as is done in search and matching models of the labor market, such as Hosios (1990) or Mortensen and Pissarides (1994). It is easy to see that the efficient (welfare maximizing) quantity is qn ; and that this quantity generically does not satisfy the equilibrium conditions. For example, as remarked in Trejos and Wright (1995), in Model-K with Tm ¼ 0 and y ¼ 12; the equilibrium implies qoqn for all r > 0; although q-qn in the limit as r-0: Once we allow for general bargaining power, however, any quantity qA½q; q % is an equilibrium for some y (in every version of the model). At n least if r is% not too big, we will have qn A½q; q; % and so there exists some y % such that the equilibrium value of money coincides with the efficient qn : Proposition 6. In either model; there exists yn Að0; 1 such that q ¼ qn iff rprn ¼ ð1 MÞð1 yÞB=cðqn Þ: Proof. The proofs of the previous propositions imply that qn can be obtained n as an equilibrium quantity by choosing y appropriately iff qXq : We only need % to determine when this is the case. For r ¼ 0; it is easy to see q% > qn : It is also easy to see that q-0 as r-N: Since f is monotonically decreasing in r; there % n exists rn > 0 such that qXq for rprn : The critical rn is found by setting % n f ðq Þ ¼ 0 and solving for r: & The value of yn that generates q ¼ qn can be derived by setting eðqn Þ ¼ 0 and solving for y (assuming rorn ). When Tm ¼ Vm ; the result is ynK ¼
ruðqn Þ þ ½M þ ð1 MÞyB ; ð1 þ rÞB
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ynS ¼
617
ruðqn Þ þ Mð1 yÞB : ðr þ 1 yÞB
When Tm ¼ 0; the result is slightly more complicated: DK ; ynK ¼ 2DK rðr þ MÞuðqn Þ rðr þ 1 MÞcðqn Þ rð1 MÞyB DS ynS ¼ ; 2 n 2DS r uðq Þ rðr þ 1 yÞcðqn Þ rMð1 yÞB where DK ¼ ½rðr þ 1Þ þ Mð1 MÞuðqn Þ Mðr þ 1 MÞcðqn Þ þ ð1 MÞyBðr þ 1 MÞ; DS ¼ ðr þ 1 yÞyB þ rðr þ 1 yÞuðqn Þ þ Mð1 yÞ½ð1 MÞð1 yÞB rcðqn Þ: In terms of general results, we have the following. If Tm ¼ 0; then yn > 12 in both Model-K and Model-S: Combined with the fact that @e=@y > 0; this implies qoqn whenever y ¼ 12: This result does not hold when Tm ¼ Vm : depending on parameter values, yn may be greater or less than 12; and so whether we have qoqn or q > qn is ambiguous when y ¼ 12: This is related to some results in the mechanism design approach to monetary theory, such as Kocherlakota and Wallace (1998). In the absence of a public record, they show that, for some parameter values, we can support a constrained optimal allocation where qn is exchanged for money every time an agent i with money meets agent j without money and iWj: This allocation is an equilibrium in our model when y ¼ yn :6 Although somewhat complicated in general, the expressions for yn simplify nicely when r-0: On the one hand, when Tm ¼ 0 we have yn ¼ 12 in either Model-K or Model-S: Hence, at least in the limit as frictions vanish, symmetric bargaining power generates the efficient q in either model when Tm ¼ 0: On the other hand, when Tm ¼ Vm we have yn ¼ M þ ð1 MÞy in Model-K and yn ¼ M in Model-S: Notice that yn is greater in Model-K for an exogenously fixed M: However, if we set M to maximize W; conditional on q ¼ qn ; we have M ¼ 12 in Model-S and M ¼ 1 2y=2 2y in Model-K: Inserting these into yn again yields yn ¼ 12 in both cases. Summarizing: Proposition 7. For either Model-S or Model-K; in the limit as r-0; we have the following: when Tm ¼ 0 the efficient bargaining weight is yn ¼ 12 for any M; 6
In a different model, Shi (1997) finds qoqn for any yA½0; 1: A key difference is that his model assumes each household consists of a continuum of agents, each of whom is treated symmetrically in that every one consumes in every period. In our model, a producer in a monetary trade must wait for the next suitable match to be able to consume.
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when Tm ¼ Vm ; yn varies between 0 and 1 depending on the value of M; but if M is set optimally then again yn ¼ 12: In the case Tm ¼ Vm ; the reason that yn is greater in Model-K than in Model-S for an exogenously fixed M can be understood as follows. First, as illustrated in Fig. 5a, the equilibrium q is lower in Model-K for any r; simply because accepting money is more costly, due to the fact that holding money precludes barter. The figure is drawn for uðqÞ ¼ q; cðqÞ ¼ q2 ; y ¼ 0:1; r ¼ 0:01;
W, q WS WK
qS qK
(a)
0
1
M
W, q WS
WK
qS qK
(b)
0
1
M
Fig. 5. W and q as functions of M in Model-S and Model-K: (a) Tm ¼ Vm ; (b) Tm ¼ 0:
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and y ¼ 0:5; but it is true in general that q is lower in Model-K when Tm ¼ Vm ; as we prove in Appendix A. However, when Tm ¼ 0 the equilibrium q may or may not be lower in Model-K than in Model-S: See Fig. 5b, where q is higher in Model-K for low M (note that this is drawn for different parameters than Fig. 5a). Fig. 5 also shows welfare, defined by W ¼ MV1 þ ð1 MÞV0 (when there are multiple equilibria, we depict the one corresponding to the highest q). Note that even though money precludes barter in Model-K; this does not necessarily imply that welfare is lower: in Fig. 5a, we have WS oWK for low M: The intuition is as follows. Given parameters, suppose q is much greater than qn in Model-S: Then welfare can be higher in Model-K; even though money holders cannot produce in this model, because it generates a lower q: However, welfare is lower in Model-K than in Model-S when M is large. In particular, note the discontinuity in WK and qK ; which occurs when M becomes too large and the monetary equilibrium breaks down.
5. Conclusion We have extended existing search-theoretic models of money by generalizing the bargaining solution and by allowing agents with money to produce. We derived results on existence, the number of equilibria, comparative statics and welfare for several versions of the model. We hope that these results will be useful to others who work with search-based monetary theory. In particular, since results are often easier to derive and more natural in Model-S; future applications should consider adopting this version. We emphasize that while this model avoids some of the problems in the standard model, it does not avoid all such problems F e.g., in a single coincidence meeting where both agents have money there is still no trade. These problems ultimately arise from restrictions on money inventories, and so there is reason to consider models that relax these restrictions, such as those mentioned in footnote 1: However, in some contexts it seems better to work with the simpler models, since they deliver analytic results and help develop our intuition. Given one wants to use a model with inventory restrictions for the sake of tractability, it is useful to know that Model-S is available.
Appendix A A.1. Proof of Lemma 1 Since money is indivisible, we allow agents to use lotteries (as in Berentsen et al., 2000), and let d be the probability that money changes hands; this does
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not actually matter as we will show d ¼ 0; but it simplifies the presentation. The bargaining problem is then max ½uðq1 Þ cðq0 Þ þ dðV0 V1 Þ1=2 ½uðq0 Þ cðq1 Þ þ dðV1 V0 Þ1=2
q0 ;q1 ;d
subject to q0 ; q1 X0; uðq1 Þ cðq0 Þ þ dðV0 V1 ÞX0; uðq0 Þ cðq1 Þþ dðV1 V0 Þ X0; and dA½0; 1: It can be shown that the objective function is strictly concave, and so the first-order conditions are necessary and sufficient. If we ignore the constraints, these conditions are given by c0 ðq1 Þ uðq0 Þ cðq1 Þ dðV0 V1 Þ ¼ ; u0 ðq1 Þ uðq1 Þ cðq0 Þ þ dðV0 V1 Þ
ðq1 Þ
u0 ðq0 Þ uðq0 Þ cðq1 Þ dðV0 V1 Þ ¼ ; c0 ðq0 Þ uðq1 Þ cðq0 Þ þ dðV0 V1 Þ
ðq0 Þ
uðq0 Þ cðq1 Þ dðV0 V1 Þ : uðq1 Þ cðq0 Þ þ dðV0 V1 Þ
ðdÞ
1¼
Solving this system yields ðq1 ; q0 ; dÞ ¼ ðqn ; qn ; 0Þ: As no constraints are violated, this is the solution to the constrained problem. & A.2. Claims in the Proofs of Propositions 1 and 2 In Proposition 1, we claim @e=@yje¼0 > 0: Differentiating (6) yields @e ¼ ½ð1 MÞu ðr þ 1 MÞcu0 @y ð1 MÞyB½u0 c0 þ ½ðr þ MÞu Mcc0 : Using e ¼ 0; we have @e ¼ ½ru þ Mðu cÞ þ ð1 MÞyBc0 > 0: @y e¼0 In Proposition 2 we make the same claim. In this case, @e ¼ ½ru þ Mðu cÞ þ ð1 MÞyBðr þ 1 MÞc0 > 0: @y e¼0 We also claim @e=@rje¼0 o0 if eðqÞ > 0: Differentiation yields % @e ¼ y½ð1 MÞu ð1 þ 2rÞcu0 ð1 MÞyBð1 yÞc0 @r ð1 yÞ½ð1 þ 2rÞu Mcc0 ;
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which can be rewritten @e r ¼ e yfð1 MÞMu ½Mð1 MÞ r2 cgu0 @r þ ð1 yÞf½Mð1 MÞ r2 u Mð1 MÞcgc0 ð1 MÞyB½yð1 MÞu0 ð1 yÞð1 MÞc0 : Setting e ¼ 0 and rearranging, we have @e r ¼ r2 ½ycu0 þ ð1 yÞuc0 @r e¼0
½Mðu cÞ þ ð1 MÞyBð1 MÞ½yu0 ð1 yÞc0 : As the first term is strictly negative, a sufficient condition for @e=@rje¼0 o0 is yu0 ð1 yÞc0 > 0: It can be shown easily that eðqÞ > 0 implies yu0 ð1 yÞc0 > % 0; verifying the claim. & A.3. Proof that q is lower in Model-K than Model-S when Tm ¼ Vm It is sufficient to show that eK ¼ 0 implies eS > 0; where ej is the equilibrium condition in model j ¼ K; S: By inspection, eK ¼ eS þ yð1 MÞyðu cÞu0 ð1 yÞyMðu cÞc0 ð1 MÞyB½yu0 þ ð1 yÞc0 and so eK ¼ 0 implies eS ¼ ½ð1 MÞyB þ Myðu cÞð1 yÞc0 þ yð1 MÞyBu0 yð1 MÞyðu cÞu0 ; which is positive since BXuðqÞ cðqÞ for all q:
References Aiyagari, S.R., Wallace, N., 1991. Existence of steady states with positive consumption in the Kiyotaki–Wright model. Review of Economic Studies 58, 901–916. Aiyagari, S.R., Wallace, N., 1992. Fiat money in the Kiyotaki–Wright model. Economic Theory 2, 447–464. Aiyagari, S.R., Wallace, N., 1997. Government transaction policy, the medium of exchange, and welfare. Journal of Economic Theory 74, 1–18. Aiyagari, S.R., Wallace, N., Wright, R., 1996. Coexistence of money and interest-bearing securities. Journal of Monetary Economics 37, 397–419. Berentsen, A., Molico, M., Wright, R., 2000. Indivisibilities, lotteries, and monetary exchange. Journal of Economic Theory, forthcoming. Camera, G., Corbae, D., 1999. Money and price dispersion. International Economic Review 40, 985–1008. Coles, M., Wright, R., 1998. A dynamic equilibrium model of search, bargaining, and money. Journal of Economic Theory 78, 32–54.
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Green, E., Zhou, R., 1998. A rudimentary model of search with divisible money and prices. Journal of Economic Theory 81, 252–271. 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., 1989. On money as a medium of exchange. Journal of Political Economy 97, 927–954. 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. Kocherlakota, N., Wallace, N., 1998. Incomplete record-keeping and optimal payment arrangements. Journal of Economic Theory 81, 272–289. Li, Y., Wright, R., 1998. Government transaction policy, media of exchange, and prices. Journal of Economic Theory 81, 290–313. Molico, M., 1999. The distribution of money and prices in search equilibrium. Unpublished manuscript. Mortensen, D.T., Pissarides, C., 1994. Job creation and job destruction in the theory of unemployment. Review of Economic Studies 61, 397–415. Osborne, M.J., Rubinstein, A., 1990. Bargaining and Markets. Academic Press, San Diego. Rupert, P., Schindler, M., Shevchenko, A., Wright, R., 2000. The search-theoretic approach to monetary economics: a primer. Federal Reserve Bank of Cleveland Economic Review 36, 10–28. Shi, S., 1995. Money and prices: a model of search and bargaining. Journal of Economic Theory 67, 467–496. Shi, S., 1997. A divisible search model of fiat money. Econometrica 65, 75–102. Siandra, E., 1993. A monetary model of market size and specialisation. Economic Theory Discussion Paper No. 182, University of Cambridge. Siandra, E., 1996. Partnerships, search, and money. Department of Economics Discussion Paper No. 96-16, University College London. Taber, A., Wallace, N., 1999. A matching model with bounded holdings of indivisible money. International Economic Review 40, 961–984. Trejos, A., Wright, R., 1995. Search, bargaining, money, and prices. Journal of Political Economy 103, 118–141. Zhou, R., 1999. Individual and aggregate real balances in a random matching model. International Economic Review 40, 1009–1038.
Generalized search-theoretic models of monetary exchange$ Peter Ruperta, Martin Schindlerb, Randall Wrightb,* a b
Research Department, Federal Reserve Bank of Cleveland, Cleveland, OH 44101, USA Department of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA 19104, USA
Received 9 May 2000; received in revised form 8 November 2000; accepted 21 November 2000
Abstract Virtually all simple search models of money assume agents with money cannot produce, and so everyone has either 0 or 1 units of money in steady state. We alternatively assume agents can always produce, and simply restrict money inventories to 0 or 1. This seems better for many issues; for example, in the standard model it is difficult to interpret increases in the money supply, because increasing the fraction of agents holding money decreases the economy’s productive capacity. Our model avoids these problems and thus delivers more natural, and simpler, implications in many contexts. We compare results on existence, multiplicity, and welfare across models. r 2001 Elsevier Science B.V. All rights reserved. JEL classification: E00; D83; C78 Keywords: Money; Search; Bargaining; Prices
$ We thank the Research Department at the Federal Reserve Bank of Cleveland for use of their facilities, and the National Science Foundation for financial support. We are also grateful to Yiting Li, Eduardo Siandra, Steve Williamson and an anonymous referee for valuable comments. The usual disclaimer applies.
*Corresponding author. Tel.: +1-215-898-7194; fax: +1-215-573-4217. E-mail address: [email protected] (R. Wright). 0304-3932/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 3 2 ( 0 1 ) 0 0 0 8 8 - 5
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1. Introduction Virtually all of the simple search-theoretic models of money in the literature assume that agents holding money cannot produce. This guarantees that, in steady state, everyone has either 0 or 1 units of money. We explore an alternative where all agents can always produce, and simply restrict money inventories to f0; 1g: This is useful because the standard model has several undesirable implications that our alternative avoids. For example, in the model where agents holding money cannot produce, if two agents with money meet they do not trade even if there is a double coincidence of wants. A related implication of the standard model is that increasing the fraction of agents holding money necessarily decreases the economy’s productive capacity. This makes it difficult, at best, to interpret the effects of changes in the money supply. Our model, where all agents can produce, does not have these unnatural implications. We derive results on existence, the number of equilibria, comparative statics, and welfare in our model, and compare these to results from the standard model. We also present results for all values of the parameter y representing bargaining power, while virtually all previous papers in the monetary search literature consider only the symmetric case y ¼ 12; or the extreme case y ¼ 1: This generality is useful not only for its own sake, but also because it allows us to discuss the relationship between bargaining power and efficiency, as is done in search models of the labor market, for example. Summarizing, this paper analyzes a monetary economy with a general bargaining solution and where all agents can always produce. The latter feature is particularly significant since, as compared to the standard model, the results are often easier to derive and typically more natural in the new model.1
2. The basic model There is a ½0; 1 continuum of agents who live forever and discount at rate r: They meet bilaterally according to a Poisson process with parameter a; and 1
See Kiyotaki and Wright (1989, 1991, 1993), Aiyagari and Wallace (1991, 1992), Trejos and Wright (1995) and Shi (1995) for examples of the standard model. Siandra (1993, 1996) is an example of the alternative model where money holders can produce, although he only considers the case of indivisible goods (see Rupert et al., 2000, for a discussion of this case). Aiyagari et al. (1996) and Kocherlakota and Wallace (1998) also allow agents with money to produce; however, those models assume the probability of a double-coincidence as 0; which renders the main issues discussed here far less interesting because in that case the two models collapse to the same thing (see the discussion preceding Proposition 5). All of the above models in one way or another limit money holdings to f0; 1g: There are also models where money holders can produce and agents can hold more general inventories, but they are an order of magnitude more complicated than what we consider here; see Green and Zhou (1998), Camera and Corbae (1999), Molico (1999), Taber and Wallace (1999) and Zhou (1999).
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trade iff mutually agreeable. There are a variety of nonstorable consumption goods, but each agent produces just one good, at (disutility) cost cðqÞX0 per q units. Given two agents i and j; let iWj indicate ‘‘i wants to consume the good that j produces’’, in the sense that i derives utility uðqÞ from consuming q units of the good that j produces if iWj; and 0 otherwise. We assume u0 ðqÞ > 0 and c0 ðqÞ > 0: Also, c00 ðqÞX0 and u00 ðqÞp0 with at least one of the inequalities strict. Finally, uð0Þ ¼ cð0Þ ¼ 0; u0 ð0Þ > c0 ð0Þ ¼ 0; and there exists q# > 0 such that # ¼ cðqÞ: # We model specialization as follows: For no agent i is it the case uðqÞ that iWi; and for a pair of agents i and j selected at random, prðiWjÞ ¼ x and prð jWijiWjÞ ¼ y: Thus, xy is the probability of a double coincidence of wants.2 There is an exogenous quantity MA½0; 1 of intrinsically valueless fiat money. Initially one unit of money is given to each of M agents chosen at random. The standard version of the model, following Kiyotaki and Wright (1991, 1993), assumes that agents holding money cannot produce. We explore an alternative model, following Siandra (1993, 1996), where they can. Call the first Model-K and the second version Model-S: The Model-K assumption can be interpreted as saying that after producing an agent needs to consume before producing again, which implies that in steady state no one holds more than 1 unit of money. The Model-S assumption does not rule out the accumulation of multiple units of money, so we simply assume agents can store at most 1 unit of money. While obviously it is interesting to study models where agents can accumulate general money inventories, it is also still interesting to study tractable models, and limiting inventories in this way simplifies the analysis significantly. The issue in this paper is to ask, conditional on agents holding 0 or 1 unit of money, what happens if we relax the assumption that agents with money cannot produce. In the standard Model-K; there are only two nontrivial types of meetings: a double coincidence meeting between two agents without money, and a single coincidence meeting between i and j; where iWj and i has money while j does not. The former meeting always leads to a barter trade, and the latter may or may not lead to a monetary trade, but there can be no barter. In Model-S; however, in the latter meeting i might trade goods, cash, or both. Of course, what happens in any meeting depends on the bargaining solution. Since we are most interested in single coincidence meetings, where money plays a role, we explore various alternatives in this case, while for double coincidence meetings we simply impose a particular bargaining solution that gives convenient results. In particular, we show that under our assumptions the unique equilibrium in any double coincidence meeting is for the agents to barter goods for goods, and no money will change hands.
2
This specification nests several models in the literature; e.g., in Kiyotaki and Wright (1993) the events fiWjg and fjWig are independent, so y ¼ x and the double coincidence probability is x2 :
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To pursue this, let Vm denote the value function of an agent with m units of money, mAf0; 1g: First consider a single coincidence meeting. In this case, money must change hands if goods are to change hands, and q is determined by the generalized Nash bargaining solution max ½uðqÞ þ V0 T1 y ½V1 cðqÞ T0 1 y q
ð1Þ
subject to the constraints qX0; D1 ¼ uðqÞ þ V0 V1 X0 and D0 ¼ V1 V0 cðqÞX0: In (1), Tm is the threat point of the agent with m units of money and y represents bargaining power. We will consider below two cases for Tm that have been analyzed in the monetary literature, Tm ¼ Vm and Tm ¼ 0; and generalize previous analyses by allowing y to take on any value in ½0; 1:3 Now consider a double coincidence meeting between an agent with 0 and an agent with 1 unit of money. As said above, we need to determine whether money changes hands, and the two quantities q0 and q1 ; where qm is the quantity of goods that the agent with m units of money receives. The following is proved in the Appendix.4 Lemma 1. In a double coincidence meeting between an agent with and an agent without money; if we assume y ¼ 12 and Tm ¼ Vm ; the unique outcome is that no money changes hands; and q0 ¼ q1 ¼ qn where qn satisfies u0 ðqn Þ ¼ c0 ðqn Þ: There are also double coincidence meetings where the two agents have the same money holdings (m ¼ 0 or m ¼ 1). One can also show in these cases that each agent receives qn and no money changes hands, under the same assumptions used in Lemma 1. Hence, in all double coincidence meetings the net payoff is B ¼ uðqn Þ cðqn Þ: Note that the previous literature typically did not need to worry about what happens in double coincidence meetings, since in Model-K there is only one option (since money holders cannot produce).
3. Monetary equilibrium Having characterized double coincidence meetings, the remainder of the paper is concerned with determining what happens in single coincidence meetings. The first thing we need is the Bellman equations. In Model-S these 3 As is well known, the Nash solution corresponds to the equilibrium of an explicit strategic bargaining model, where y and Tm depend on details of the game; see Osborne and Rubinstein (1990) or, for a presentation in the context of monetary theory, Coles and Wright (1998). 4
For different bargaining powers or threat points, it is possible to have money change hands or to have qm aqn : The point is not that the predictions about double coincidence meetings in Lemma 1 are completely general, but that we can design an environment where they are true, allowing us to focus in more detail on single coincidence meetings.
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are given by rV1 ¼ axyB þ axð1 MÞð1 yÞ½uðqÞ þ V0 V1 ;
ð2Þ
rV0 ¼ axyB þ axMð1 yÞ½V1 V0 cðqÞ:
ð3Þ
For example, a money holder encounters a double coincidence at rate axy; which entails net payoff B; and at rate axð1 MÞð1 yÞ he encounters a single coincidence with an agent without money, which entails net payoff uðqÞ þ V0 V1 : For comparison, the corresponding equations in Model-K are rV1 ¼ axð1 MÞ½uðqÞ þ V0 V1 ;
ð4Þ
rV0 ¼ axð1 MÞyB þ axM½V1 V0 cðqÞ:
ð5Þ
The differences arise because in Model-K money holders cannot produce. In what follows, to reduce notation we normalize ax ¼ 1; with no loss of generality. The first-order condition to the bargaining problem (1) is y½V1 cðqÞ T0 u0 ðqÞ ð1 yÞ ½uðqÞ þ V0 T1 c0 ðqÞ ¼ 0: Choosing Tm and substituting V0 and V1 (which we get from solving the Bellman equations) yields one equation in one unknown, eðqÞ ¼ 0: The different types of equilibria that may occur are best illustrated in a particular case. As a benchmark we start with the standard Model-K; in the case where the threat points are given by Tm ¼ Vm : Then eðqÞ ¼ y½ð1 MÞuðqÞ ðr þ 1 MÞcðqÞu0 ðqÞ ð1 yÞ½ðr þ MÞuðqÞ McðqÞc0 ðqÞ ð1 MÞyB½yu0 ðqÞ þ ð1 yÞc0 ðqÞ:
ð6Þ
A solution to eðqÞ ¼ 0 would be an equilibrium if there were no constraints on the bargaining problem. The constraint D1 X0 can be shown to be redundant if D0 X0 holds, and so we only need to check the latter. This means there are two distinct types of monetary equilibria: an unconstrained equilibrium is a q > 0 such that eðqÞ ¼ 0 and D0 X0; while a constrained equilibrium occurs when D0 X0 binds. If D0 ¼ 0 at some q* then the constraint is violated for any q > q; * and if eðqÞ * > 0 then q* is a constrained solution to the Nash problem. It is easy to see that D0 ¼ 0 iff q ¼ q or q ¼ q; % where q and q% are zeros of the strictly % % concave function f ðqÞ ¼ ð1 MÞuðqÞ ðr þ 1 MÞcðqÞ ð1 MÞyB:
ð7Þ
Note that f is decreasing in r; and for small r the two zeros q and q% exist, while for big r there are no solutions to f ðqÞ ¼ 0: Clearly, qAðq;% qÞ % implies D0 > 0; and so any qAðq; qÞ % such that eðqÞ ¼ 0 is an unconstrained %equilibrium. Also, if eðqÞ > 0 then q% is a constrained equilibrium, and if eðqÞ % > 0 then q% is a % % constrained equilibrium.
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0
q
q
q
-(1-M)y Bθu'(0) -(1-M)yB
f(q)
(a)
e(q) q q
q
(b)
0
θ
1
θ
Fig. 1. Model-K with y > 0; Tm ¼ Vm : (a) Functions e and f : (b) Equilibrium correspondence.
This reduces the search for monetary equilibria to looking for eðqÞ ¼ 0 at qAðq; qÞ; % or for eðqÞ > 0 at q ¼ q or q: % We now provide results on existence and % % present Model-K and then Model-S assuming the number of equilibria. We first y > 0: Later we consider y ¼ 0; where the two models are the same. Beginning with Model-K; the functions eðqÞ and f ðqÞ defined above, as well as the set of equilibrium values for q as a function of y; are depicted in Fig. 1 for the case Tm ¼ Vm and Fig. 2 for the case Tm ¼ 0: Proposition 1. Consider Model-K with y > 0 and Tm ¼ Vm : There exists r% > 0 such that no monetary equilibria exist if r > r%; while if ror% there exists y% Að0; 1Þ ðthat depends on rÞ such that the following is true: for yoy% no monetary equilibria exist; and for y > y% there exists generically an even number of monetary equilibria. All monetary equilibria are unconstrained.
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611
(1-M)2yBθu'(0)
0
q
q
q
-(1-M)yB
f(q) e(q)
(a)
q q
q
(b)
0
θ
1
θ
Fig. 2. Model-K with y > 0; Tm ¼ 0: (a) functions e and f ; (b) equilibrium correspondence.
Proof. In the limiting case r ¼ 0; clearly there exist q > 0 and q% > q such that % eðqÞp0; f ðqÞ ¼ f ðqÞ % ¼ 0; as seen in Fig. 1a. It is easy to check %that eðqÞp0 and % % % with equality iff y ¼ 1: This implies that no constrained equilibria exist (when y ¼ 1 we actually have an equilibrium where the constraint is satisfied at equality, but since the first order condition also holds with equality the solution is unconstrained). Moreover, if any unconstrained monetary equilibria exist, generically there will be an even number since e will have an even number of zeros in ðq; qÞ: % Setting y ¼ 0 implies eo0 for all q; so no monetary equilibria % that y ¼ 1 implies the existence of monetary equilibria. It can be exist. Recall shown that @e=@yje¼0 > 0 (see Appendix A), so that decreasing y shifts e down in Fig. 1a. This implies the existence of a unique y% such that monetary equilibria exist iff y > y% : See Fig. 1b. All of the above statements are for r ¼ 0; by continuity, the results are the same for small positive r: It is easy to see that e
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is monotonically decreasing in r (for q > 0), and that eo0 for all qAðq; qÞ % when r is sufficiently big. Hence, there is a unique r% such that no monetary %equilibria exist for any y when r > r%: & Proposition 2. Consider Model-K with y > 0 and Tm ¼ 0: There exists r% > 0 such that no monetary equilibria exist if r > r%; while if ror% there exists y% Að0; 1Þ ðthat depends on rÞ such that the following is true: for yoy% no monetary equilibria exist; and for y > y% there are exactly two monetary equilibria. Both equilibria are constrained if y is large; while one equilibrium is constrained and the other unconstrained if y is not so large. Proof. With Tm ¼ 0; the analogue of (6) is eðqÞ ¼ yfð1 MÞðr þ MÞuðqÞ ½rð1 þ rÞ þ Mð1 MÞcðqÞgu0 ðqÞ ð1 yÞf½rð1 þ rÞ þ Mð1 MÞuðqÞ Mðr þ 1 MÞcðqÞgc0 ðqÞ þ ð1 MÞyB½yð1 MÞu0 ðqÞ ð1 yÞðr þ 1 MÞc0 ðqÞ: See Fig. 2a. As before, r ¼ 0 implies that q > 0 and q% > q exist. If y ¼ 0; eðqÞo0 % for any q > 0 and no monetary equilibria exist. If y ¼ 1;%eðqÞ > 0 for all qAð0; q; % and both q and q% are constrained equilibria. One can check @e=@yje¼0 > 0; so there exists% a critical value y% > 0 such that monetary equilibria exist iff yXy% : Also, for yXy% but not too large, eðqÞo0; and so one monetary equilibrium is % unconstrained at qAðq; qÞ % and one equilibrium is constrained at q; while for % very large y; eðqÞ are % > 0% (for arbitrary r), so both monetary equilibria constrained. This can be seen by noting that eðqÞj % y¼1 ¼ ðM þ rÞ½ð1 MÞuðqÞ % ðr þ 1 MÞcðqÞu % 0 ðqÞ % þ ð1 MÞ2 yBu0 ðqÞ % ¼ ð1 þ rÞð1 MÞyBu0 ðqÞ; % which is positive for any r; combined with the fact that e is continuous and increasing in y: These results are qualitatively the same for small positive r: It is shown in Appendix A that, as long as eðqÞ > 0; we have @e=@rje¼0 o0: Note that this is all % exist iff eðqÞ > 0: Setting y ¼ 1; there exists a we need as monetary equilibria % exist when r > r% for any y: For unique r% > 0 such that no monetary equilibria any ror%; by the above reasoning, there exists y% o1 such that monetary equilibria exist for y > y% : To see that there are exactly two monetary equilibria (whenever any exist), it is sufficient to show that any qAð0; qÞ % such that eðqÞ ¼ 0 is unique. This is verified by noting that at e ¼ 0; we have ð1 yÞ
c0 y ¼ fð1 MÞðr þ MÞu u0 D ½rð1 þ rÞ þ Mð1 MÞc þ ð1 MÞ2 yBg;
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where D ¼ ½rð1 þ rÞ þ Mð1 MÞu Mðr þ 1 MÞc þ ðr þ 1 MÞ ð1 MÞyB; and that the left side is increasing and the right side decreasing in q: & We now turn to Model-S: Although the method is the same, the results are quite different: now there exists a unique monetary equilibrium for all y > 0: This is interesting since many policy discussions in this literature focus on multiplicity (for example, Aiyagari and Wallace, 1997; Li and Wright, 1998). Graphically, uniqueness in Model-S can be understood by noting that in this model the functions eðqÞ and f ðqÞ always go through the origin; see Fig. 3 for the case Tm ¼ Vm and Fig. 4 for the case Tm ¼ 0:
0
q
q
f(q) e(q) (a)
q q
(b) 0
1
θ
Fig. 3. Model-S with y > 0; Tm ¼ Vm : (a) functions e and f ; (b) equilibrium correspondence.
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0
q
q
f(q) e(q) (a)
q q
(b)
0
θˆ
1
θ
Fig. 4. Model-S with y > 0; Tm ¼ 0: (a) functions e and f ; (b) equilibrium correspondence.
Proposition 3. Consider Model-S with y > 0 and Tm ¼ Vm : For any r > 0; the following is true: if either y ¼ 0 or y ¼ 1; no monetary equilibrium exists; otherwise; there exists a unique monetary equilibrium and it is unconstrained. Proof. In Model-S; the function f ðqÞ is f ðqÞ ¼ ð1 MÞð1 yÞuðqÞ ½r þ ð1 MÞð1 yÞcðqÞ: With Tm ¼ Vm ; the analogue of (6) is eðqÞ ¼ yf ðqÞu0 ðqÞ ð1 yÞf½r þ Mð1 yÞuðqÞ Mð1 yÞcðqÞgc0 ðqÞ: For y ¼ 1; we have eð0Þ ¼ 0 and eðqÞo0 for q > 0; and so no monetary equilibria exist for any y: Now suppose yo1: It is easy to see that f ðqÞ ¼ 0 at q ¼ 0 and q% > 0: It can also be verified that eð0Þ ¼ 0 and eðqÞp0 for all y; where % %
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the inequality is strict for yo1; implying that there are no constrained equilibria. If y ¼ 0; the only solution to e ¼ 0 is q ¼ 0: For yAð0; 1; e0 ð0Þ > 0; so that there exists a qAð0; q % such that eðqÞ ¼ 0: The monetary equilibrium is unique because at e ¼ 0; we have c0 f ðqÞ ; ð1 yÞ 0 ¼ y ½r þ Mð1 yÞu Mð1 yÞ u where the left-hand side is increasing and the right-hand side decreasing in q: Finally, when y ¼ 1; the (unconstrained) unique monetary equilibrium is at q ¼ q: % See Fig. 3. & Proposition 4. Consider Model-S with y > 0 and Tm ¼ 0: For any r > 0; the following is true: if either y ¼ 0 or y ¼ 1; no monetary equilibrium exists; otherwise; there exists a unique monetary equilibrium; and it is unconstrained for # where 0oyo1: # yoy# and constrained for y > y; Proof. In this case the analogue of (6) is eðqÞ ¼ y½r þ Mð1 yÞf ðqÞu0 þ ð1 þ r yÞyB½yu0 ð1 yÞc0 ð1 yÞ½rð1 þ r yÞu þ Mð1 yÞf ðqÞc0 : The constraint is as before, and so f ðqÞ ¼ 0 at q ¼ 0 and q% > 0: For y ¼ 0; we % equilibrium exists, while at have eð0Þ ¼ 0 and e0 ð0Þo0; and so no monetary y ¼ 1; again, no monetary equilibria exist. Now assume yo1: For yAð0; 1; we have eð0Þ > 0: Note that for y ¼ 0; eðqÞo0, and for y ¼ 1; eðqÞ % % > 0: By # inspection e is monotonically increasing in y: Thus, there exists yAð0; 1Þ such that the monetary equilibrium is unconstrained for yoy# and constrained (at q) % # Uniqueness is shown in the usual way. See Fig. 4. & for y > y: All of the above results are for y > 0: If y ¼ 0 then Model-K and Model-S are identical (to see this, simply compare the Bellman equations). It turns out that the results in the y ¼ 0 case are similar to what we derived above for Model-S with y > 0: for all y > 0 there exists a monetary equilibrium and it is unique. However, note that the equilibrium is always unconstrained when y ¼ 0:5 Proposition 5. Consider either Model-K or Model-S for either Tm ¼ Vm or Tm ¼ 0; where y ¼ 0: For any r > 0; the following is true: if y ¼ 0 no monetary equilibria exist; and if y > 0 there exists a unique monetary equilibrium and it is unconstrained. Proof. For y ¼ 0; we have q ¼ 0 and q% > 0 by inspection of f ðqÞ: For y ¼ 0; the % 0: For yAð0; 1Þ it can be verified that eð0Þ ¼ 0 and only solution to e ¼ 0 is q ¼ This is not inconsistent with Proposition 4, because y# defined in that result goes to 1 as y goes to zero. 5
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eðqÞo0; implying that there are no constrained monetary equilibria. Moreover, % it is easy to see that e0 ð0Þ > 0; which implies that there exists qAðq; qÞ % such that eðqÞ ¼ 0: Finally, when y ¼ 1; the unique nonzero solution to e%¼ 0 is q ¼ q; % this is the unique equilibrium. Function e and equilibria are qualitatively the same as in Model-S with Tm ¼ Vm and y > 0 (see Fig. 3). &
4. Discussion The main elements that distinguish our analysis from the previous literature are: (i) we allow any yA½0; 1; and (ii) we consider Model-S as well as Model-K: Many results from previous analyses continue to hold: for example, for any y; in Model-K with y > 0 the existence of monetary equilibria depends on r being small. Also, as in earlier analyses, the equilibrium quantity q is typically not socially efficient. However, given our generalized model, the interaction between efficiency and bargaining power can be explored in more detail, as is done in search and matching models of the labor market, such as Hosios (1990) or Mortensen and Pissarides (1994). It is easy to see that the efficient (welfare maximizing) quantity is qn ; and that this quantity generically does not satisfy the equilibrium conditions. For example, as remarked in Trejos and Wright (1995), in Model-K with Tm ¼ 0 and y ¼ 12; the equilibrium implies qoqn for all r > 0; although q-qn in the limit as r-0: Once we allow for general bargaining power, however, any quantity qA½q; q % is an equilibrium for some y (in every version of the model). At n least if r is% not too big, we will have qn A½q; q; % and so there exists some y % such that the equilibrium value of money coincides with the efficient qn : Proposition 6. In either model; there exists yn Að0; 1 such that q ¼ qn iff rprn ¼ ð1 MÞð1 yÞB=cðqn Þ: Proof. The proofs of the previous propositions imply that qn can be obtained n as an equilibrium quantity by choosing y appropriately iff qXq : We only need % to determine when this is the case. For r ¼ 0; it is easy to see q% > qn : It is also easy to see that q-0 as r-N: Since f is monotonically decreasing in r; there % n exists rn > 0 such that qXq for rprn : The critical rn is found by setting % n f ðq Þ ¼ 0 and solving for r: & The value of yn that generates q ¼ qn can be derived by setting eðqn Þ ¼ 0 and solving for y (assuming rorn ). When Tm ¼ Vm ; the result is ynK ¼
ruðqn Þ þ ½M þ ð1 MÞyB ; ð1 þ rÞB
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ynS ¼
617
ruðqn Þ þ Mð1 yÞB : ðr þ 1 yÞB
When Tm ¼ 0; the result is slightly more complicated: DK ; ynK ¼ 2DK rðr þ MÞuðqn Þ rðr þ 1 MÞcðqn Þ rð1 MÞyB DS ynS ¼ ; 2 n 2DS r uðq Þ rðr þ 1 yÞcðqn Þ rMð1 yÞB where DK ¼ ½rðr þ 1Þ þ Mð1 MÞuðqn Þ Mðr þ 1 MÞcðqn Þ þ ð1 MÞyBðr þ 1 MÞ; DS ¼ ðr þ 1 yÞyB þ rðr þ 1 yÞuðqn Þ þ Mð1 yÞ½ð1 MÞð1 yÞB rcðqn Þ: In terms of general results, we have the following. If Tm ¼ 0; then yn > 12 in both Model-K and Model-S: Combined with the fact that @e=@y > 0; this implies qoqn whenever y ¼ 12: This result does not hold when Tm ¼ Vm : depending on parameter values, yn may be greater or less than 12; and so whether we have qoqn or q > qn is ambiguous when y ¼ 12: This is related to some results in the mechanism design approach to monetary theory, such as Kocherlakota and Wallace (1998). In the absence of a public record, they show that, for some parameter values, we can support a constrained optimal allocation where qn is exchanged for money every time an agent i with money meets agent j without money and iWj: This allocation is an equilibrium in our model when y ¼ yn :6 Although somewhat complicated in general, the expressions for yn simplify nicely when r-0: On the one hand, when Tm ¼ 0 we have yn ¼ 12 in either Model-K or Model-S: Hence, at least in the limit as frictions vanish, symmetric bargaining power generates the efficient q in either model when Tm ¼ 0: On the other hand, when Tm ¼ Vm we have yn ¼ M þ ð1 MÞy in Model-K and yn ¼ M in Model-S: Notice that yn is greater in Model-K for an exogenously fixed M: However, if we set M to maximize W; conditional on q ¼ qn ; we have M ¼ 12 in Model-S and M ¼ 1 2y=2 2y in Model-K: Inserting these into yn again yields yn ¼ 12 in both cases. Summarizing: Proposition 7. For either Model-S or Model-K; in the limit as r-0; we have the following: when Tm ¼ 0 the efficient bargaining weight is yn ¼ 12 for any M; 6
In a different model, Shi (1997) finds qoqn for any yA½0; 1: A key difference is that his model assumes each household consists of a continuum of agents, each of whom is treated symmetrically in that every one consumes in every period. In our model, a producer in a monetary trade must wait for the next suitable match to be able to consume.
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when Tm ¼ Vm ; yn varies between 0 and 1 depending on the value of M; but if M is set optimally then again yn ¼ 12: In the case Tm ¼ Vm ; the reason that yn is greater in Model-K than in Model-S for an exogenously fixed M can be understood as follows. First, as illustrated in Fig. 5a, the equilibrium q is lower in Model-K for any r; simply because accepting money is more costly, due to the fact that holding money precludes barter. The figure is drawn for uðqÞ ¼ q; cðqÞ ¼ q2 ; y ¼ 0:1; r ¼ 0:01;
W, q WS WK
qS qK
(a)
0
1
M
W, q WS
WK
qS qK
(b)
0
1
M
Fig. 5. W and q as functions of M in Model-S and Model-K: (a) Tm ¼ Vm ; (b) Tm ¼ 0:
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and y ¼ 0:5; but it is true in general that q is lower in Model-K when Tm ¼ Vm ; as we prove in Appendix A. However, when Tm ¼ 0 the equilibrium q may or may not be lower in Model-K than in Model-S: See Fig. 5b, where q is higher in Model-K for low M (note that this is drawn for different parameters than Fig. 5a). Fig. 5 also shows welfare, defined by W ¼ MV1 þ ð1 MÞV0 (when there are multiple equilibria, we depict the one corresponding to the highest q). Note that even though money precludes barter in Model-K; this does not necessarily imply that welfare is lower: in Fig. 5a, we have WS oWK for low M: The intuition is as follows. Given parameters, suppose q is much greater than qn in Model-S: Then welfare can be higher in Model-K; even though money holders cannot produce in this model, because it generates a lower q: However, welfare is lower in Model-K than in Model-S when M is large. In particular, note the discontinuity in WK and qK ; which occurs when M becomes too large and the monetary equilibrium breaks down.
5. Conclusion We have extended existing search-theoretic models of money by generalizing the bargaining solution and by allowing agents with money to produce. We derived results on existence, the number of equilibria, comparative statics and welfare for several versions of the model. We hope that these results will be useful to others who work with search-based monetary theory. In particular, since results are often easier to derive and more natural in Model-S; future applications should consider adopting this version. We emphasize that while this model avoids some of the problems in the standard model, it does not avoid all such problems F e.g., in a single coincidence meeting where both agents have money there is still no trade. These problems ultimately arise from restrictions on money inventories, and so there is reason to consider models that relax these restrictions, such as those mentioned in footnote 1: However, in some contexts it seems better to work with the simpler models, since they deliver analytic results and help develop our intuition. Given one wants to use a model with inventory restrictions for the sake of tractability, it is useful to know that Model-S is available.
Appendix A A.1. Proof of Lemma 1 Since money is indivisible, we allow agents to use lotteries (as in Berentsen et al., 2000), and let d be the probability that money changes hands; this does
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not actually matter as we will show d ¼ 0; but it simplifies the presentation. The bargaining problem is then max ½uðq1 Þ cðq0 Þ þ dðV0 V1 Þ1=2 ½uðq0 Þ cðq1 Þ þ dðV1 V0 Þ1=2
q0 ;q1 ;d
subject to q0 ; q1 X0; uðq1 Þ cðq0 Þ þ dðV0 V1 ÞX0; uðq0 Þ cðq1 Þþ dðV1 V0 Þ X0; and dA½0; 1: It can be shown that the objective function is strictly concave, and so the first-order conditions are necessary and sufficient. If we ignore the constraints, these conditions are given by c0 ðq1 Þ uðq0 Þ cðq1 Þ dðV0 V1 Þ ¼ ; u0 ðq1 Þ uðq1 Þ cðq0 Þ þ dðV0 V1 Þ
ðq1 Þ
u0 ðq0 Þ uðq0 Þ cðq1 Þ dðV0 V1 Þ ¼ ; c0 ðq0 Þ uðq1 Þ cðq0 Þ þ dðV0 V1 Þ
ðq0 Þ
uðq0 Þ cðq1 Þ dðV0 V1 Þ : uðq1 Þ cðq0 Þ þ dðV0 V1 Þ
ðdÞ
1¼
Solving this system yields ðq1 ; q0 ; dÞ ¼ ðqn ; qn ; 0Þ: As no constraints are violated, this is the solution to the constrained problem. & A.2. Claims in the Proofs of Propositions 1 and 2 In Proposition 1, we claim @e=@yje¼0 > 0: Differentiating (6) yields @e ¼ ½ð1 MÞu ðr þ 1 MÞcu0 @y ð1 MÞyB½u0 c0 þ ½ðr þ MÞu Mcc0 : Using e ¼ 0; we have @e ¼ ½ru þ Mðu cÞ þ ð1 MÞyBc0 > 0: @y e¼0 In Proposition 2 we make the same claim. In this case, @e ¼ ½ru þ Mðu cÞ þ ð1 MÞyBðr þ 1 MÞc0 > 0: @y e¼0 We also claim @e=@rje¼0 o0 if eðqÞ > 0: Differentiation yields % @e ¼ y½ð1 MÞu ð1 þ 2rÞcu0 ð1 MÞyBð1 yÞc0 @r ð1 yÞ½ð1 þ 2rÞu Mcc0 ;
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which can be rewritten @e r ¼ e yfð1 MÞMu ½Mð1 MÞ r2 cgu0 @r þ ð1 yÞf½Mð1 MÞ r2 u Mð1 MÞcgc0 ð1 MÞyB½yð1 MÞu0 ð1 yÞð1 MÞc0 : Setting e ¼ 0 and rearranging, we have @e r ¼ r2 ½ycu0 þ ð1 yÞuc0 @r e¼0
½Mðu cÞ þ ð1 MÞyBð1 MÞ½yu0 ð1 yÞc0 : As the first term is strictly negative, a sufficient condition for @e=@rje¼0 o0 is yu0 ð1 yÞc0 > 0: It can be shown easily that eðqÞ > 0 implies yu0 ð1 yÞc0 > % 0; verifying the claim. & A.3. Proof that q is lower in Model-K than Model-S when Tm ¼ Vm It is sufficient to show that eK ¼ 0 implies eS > 0; where ej is the equilibrium condition in model j ¼ K; S: By inspection, eK ¼ eS þ yð1 MÞyðu cÞu0 ð1 yÞyMðu cÞc0 ð1 MÞyB½yu0 þ ð1 yÞc0 and so eK ¼ 0 implies eS ¼ ½ð1 MÞyB þ Myðu cÞð1 yÞc0 þ yð1 MÞyBu0 yð1 MÞyðu cÞu0 ; which is positive since BXuðqÞ cðqÞ for all q:
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