a uniqueness proof for monetary steady state

Journal of Economic Theory 145 (2010) 382–391

www.elsevier.com/locate/jet

Note

A uniqueness proof for monetary steady state ✩

Randall Wright a,b,∗

a University of Wisconsin-Madison, Madison, WI, United Statesb Federal Reserve Bank of Minneapolis, United States

Received 12 April 2008; final version received 28 September 2008; accepted 6 April 2009

Available online 3 December 2009

Abstract

The framework in Lagos and Wright (2005) [20] combining decentralized and centralized markets is usedextensively in monetary economics. Much is known about that model, but there is a loose end: only underspecial assumptions about bargaining power or decentralized market preferences has it been shown that themonetary steady state is unique. For general decentralized market utility and bargaining, I prove uniquenessfor generic parameters with fiat money, and for all parameters with commodity money. As a corollary, I getmonotone comparative statics.© 2009 Elsevier Inc. All rights reserved.

JEL classification: D51; E40

Keywords: Monetary equilibrium; Uniqueness

1. Introduction

The model in Lagos and Wright [20], hereafter LW, combining some decentralized and somecentralized trade, has been used extensively in the recent monetary economics literature. Onereason is that the LW model has relatively solid microfoundations for the role of money (due tothe decentralized markets) and yet is extremely tractable (due to the centralized markets). Otherreasons include the fact that it is easy to quantify, using either calibration or estimation, and that

✩ I thank Ricardo Lagos, Christian Hellwig, Robert Shimer, Philipp Kircher, Rafael Silveira, Guido Menzio, StanRabinovich, participants in several seminars and workshops, and two anonymous referees for comments and discussionson this project. The NSF provided research support. The usual disclaimer applies.

* Address for correspondence: University of Wisconsin-Madison, Granger Hall, 975 University Ave., Madison, WI53706, United States.

E-mail address: [email protected].

0022-0531/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2009.11.004

R. Wright / Journal of Economic Theory 145 (2010) 382–391 383

it integrates nicely with other models, including standard theories of growth, unemployment,taxation, banking, and so on.1

Much is known about the LW model, but there is a loose end: only under severe conditions onbargaining (take-it-or-leave-it offers) or restrictions on preferences (log-concave marginal utility,or decreasing absolute risk aversion) in the decentralized market has it been shown that monetarysteady state is unique. Hence, for all the papers using this model, only an incomplete character-ization of steady state equilibria is available. In this note, for any decentralized market utilityfunction, and general Nash bargaining, I prove the following: with fiat money, we have unique-ness of the monetary steady state for generic parameter values; and with commodity money,we have uniqueness for all parameter values. The key step is to characterize individuals’ opti-mization problem globally, rather than focusing on first order conditions. As a corollary, I getmonotone comparative statics – e.g. the value of fiat money is unambiguously decreasing withmoney growth.

2. The model

A [0,1] continuum of agents live forever. Alternating in discrete time are two types of mar-kets: a frictionless centralized market CM, and a decentralized market DM with anonymous tradeand a double-coincidence problem detailed below. These two frictions in the DM make bothbarter and credit impossible, which makes some medium of exchange essential. See Kocher-lakota [17], Wallace [34], Corbae et al. [11], Araujo [3], or Aliprantis et al. [1,2] for more onthe role of anonymity and the essentiality of money. In the CM, good X is produced using la-bor H ; in the DM, a different good x is produced using h. One can use general technologies, butsince this adds little other than notation, I consider only the case where X and x are producedone-for-one using H and h.

Utility over a period encompassing one CM and one DM is given by U = U (X,H,x,h). Whatkeeps the framework tractable is the assumption that U is linear in one CM good, either X or H .Here we assume U = V (X,x,h) − H . Further, although it is not really necessary, it simplifiesthe presentation to assume V is additively separable and write

U = U(X) − H + u(x) − c(h), (1)

where U , u and c satisfy the usual monotonicity and curvature properties.2 Also, without lossof generality, I assume agents discount between the CM and the next DM at β ∈ (0,1) but notbetween the DM and CM.

The DM good is not storable: it must be produced and consumed simultaneously. The CMgood may be storable but is not portable: if one invests k units of X in a storage technology in

1 Hence the results presented here should be of wide interest, because this is not an arbitrary choice from the set ofmodels, but one that is being used extensively in applications. Rather than list individual papers, see the bibliography athttp://www.ssc.upenn.edu/~rwright/courses/LW-bib.pdf, with entries showing how the model can be applied to substan-tive issues and substantiating the claim that it can be easily quantified, extended to incorporate growth or unemployment,etc. A few of these applications are mentioned in the conclusion.

2 Quasi-linear utility together with alternating centralized and decentralized trade are the defining characteristics ofthe LW framework; other assumptions, like separable utility, linear technologies, a single good in each market, and soon, are merely for convenience. One could also proceed without quasi-linearity, but then the analysis becomes a lotmore complicated, and usually requires numerical methods (see Molico [25], Chiu and Molico [10], or Dressler [13]).Actually, one can replace quasi-linearity with nonconvexities, such as indivisible labor, and lotteries, as in Rogerson [28],and everything goes through as stated (see Rocheteau et al. [26]). A different approach is described in Shi [30].

384 R. Wright / Journal of Economic Theory 145 (2010) 382–391

this CM, it yields f (k) in the next CM, but in between it is illiquid in the sense that it cannotbe brought into the DM. The standard LW model has f (k) ≡ 0, but it is easily extended to haveX storable as long as it is not portable. I include storage here merely as a way to pin down thereturn on illiquid assets, which aids in the economic discussion.3 By contrast, there is anotherasset m that is liquid, in that it can be brought into the DM and used there in trade. Let φ be therelative price of the liquid asset in terms of the CM numeraire X, and let z = φm denote liquidityholdings in real terms.

In general one should think of the liquid asset m as the standard “tree” in the Lucas [23] asset-pricing model, bearing “fruit” γ � 0, in units of X, as a dividend in each CM. A special caseoften studied is the one where m is fiat money, which by definition is an asset that in intrinsicallyuseless, meaning γ = 0 (Wallace [33]). Given this, I refer to the case γ > 0 as commodity money.In general, the total supply of this asset at date t is Mt , and it grows according to Mt+1 =(1 + π)Mt . When γ = 0, and M is fiat money, in steady state π is the inflation rate, and one candefine a nominal interest rate i via the Fisher equation 1 + i = (1 + π)/β . Note either π or i canequivalently be set as an exogenous policy instrument, and I assume π > β − 1, or i > 0, butallow π → β − 1, or i → 0 (the Friedman Rule). When γ > 0, and M is commodity money, forsteady state I want real resources to be constant over time, and therefore assume π = 0.

The double-coincidence problem in the DM is modeled as follows. With probability σ anagent wants to consume x but cannot produce, in which case he is called a buyer; with probabilityσ he can produce but does not want to consume, in which case he is called a seller; and withprobability 1 − 2σ he neither producers nor consumes. Buyers and sellers are paired off in orderto trade.4 Trade involves a seller giving x to a buyer in exchange for p units of the asset, subjectto the constraint p � m where m is the amount of liquid assets the buyer brought to the DM. Theagents bargain over the terms of trade (p, x) according to the generalized Nash solution, withthe bargaining power of the buyer given by θ and threat points given by the continuation valuesfrom going to the next CM without trading in that meeting.

Let Wt(m,k) and Vt(m, k) be the value functions at t for an agent in the CM and DM withportfolio (m, k) (notice that these may in general depend on the date t). The CM problem is

Wt(m,k) = maxX,H,m̂,k̂

{U(X) − H + βVt+1(m̂, k̂)

},

s.t.

X = H + φtm + γm + f (k) − T − φt m̂ − k̂

where (m̂, k̂) is the portfolio taken into the next period, and T is a lump-sum tax that adjusts tobalance the government budget when M varies over time (so that π > 0 or π < 0 implies lump

3 A similar notion of liquidity underlies much work in economics, including e.g. banking research using Diamondand Dybvig [12], where resources invested in projects cannot be accessed until projects come to fruition. Also noteagents in the DM cannot trade claims on f (k), to be paid off in a future CM, any more than they can trade other IOU’s,because anonymity implies they can renege on such claims. Again, storage is not critical and is used merely to define anaccounting return on an illiquid asset.

4 This setup is equivalent to the usual one with random matching and specialized goods, as in LW. A minor detail isthat, in LW, a DM meeting occurs with probability α, and given this there is some probability δ of a double coincidence,where both agents consume and produce. Effectively, this paper sets α = 1 and δ = 0 to ease the presentation, but noneof the results depend on it. The results also hold for some other specifications in the literature, e.g. when agents knowbefore they leave the CM whether they will be buyers or sellers in the DM (Rocheteau and Wright [27]).


sum transfers or taxes). Notice that, since we can convert H into X one-for-one, the equilibriumreal wage is 1.

Given an interior solution for H , sufficient conditions for which are discussed in Lagos andWright [20], one can rewrite the above problem as

Wt(m,k) = φtm + γm + f (k) − T + maxX

{U(X) − X

}+ max

m̂,k̂

{−φt m̂ − k̂ + βVt+1(m̂, k̂)}. (2)

This immediately yields several standard results for this type of model: X = X∗ whereU ′(X∗) = 1; the choice (m̂, k̂) is independent of (m, k); ∂Wt/∂m = φt +γ ; and ∂Wt/∂k = f ′(k).In particular, the fact that (m̂, k̂) is independent of (m, k) suggests that we might expect a degen-erate distribution of (m̂, k̂) across agents, although we do not need to impose this as of yet.

Consider now the DM at t + 1. When a buyer with a particular portfolio (m, k) meets a sellerwith some arbitrary portfolio (M,K), the terms of trade are determined by generalized Nashbargaining. The payoff of the buyer is u(x)+Wt+1(m−p,k) and his threat point is Wt+1(m, k).Using ∂Wt+1/∂m = φt+1 + γ , his surplus from trade is u(x) − (φt+1 + γ )p. Similarly, thesurplus of the seller is (φt+1 + γ )p − c(x). Therefore the generalized Nash solution is

max[u(x) − (φt+1 + γ )p

]θ [(φt+1 + γ )p − c(x)

]1−θ (3)

where the maximization is of course subject to the constraint p � m since the buyer cannot paymore than the liquidity he has on hand. It is immediate that the outcome depends on m if andonly if the constraint p � m binds, and does not depend at all on k, or on the portfolio of theseller (M,K).

It is easy to show that in equilibrium p = m.5 Inserting this into (3) and maximizing withrespect to x, the first order condition can be rearranged to give x as the solution to (φt+1 +γ )m =g(x), where

g(x) ≡ θc(x)u′(x) + (1 − θ)u(x)c′(x)

θu′(x) + (1 − θ)c′(x).

Notice ∂x/∂m = (φt+1 + γ )/g′(x) > 0. Also, the DM value function can be written

Vt+1(m, k) = σ[u(x) + Wt+1(0, k)

] + (1 − 2σ)Wt+1(m, k)

+ σE[Wt+1(m + p̃, k) − c(x̃)

]where the expectation E is over (p̃, x̃), the terms of trade when the agent makes a sale, whichdepend on the m̃ held by a random buyer but, as we saw above, do not depend not on his own(m, k). Simplification yields:

Vt+1(m, k) = σu(x) + (1 − σ)(φt+1 + γ )m + Wt+1(0, k) + σE[(φt+1 + γ )p̃ − c(x̃)

].

Inserting Vt+1 into (2), and ignoring constants, we can summarize the portfolio choice prob-lem in the CM by

maxm̂

{−φtm̂ + βσu(x) + β(1 − σ)(φt+1 + γ )m̂} + max

k̂

{−k̂ + βWt+1(0, k̂)}, (4)

5 Is the case γ = 0, the argument is identical to that in LW. Intuitively, there is no reason to bring fiat money to the DMif you are not going to spend it. With γ > 0, things are slightly different, but buyers still spend everything they bring tothe DM, as discussed in detail in Section 4.


subject to x satisfying the bargaining solution, g(x) = (φt+1 + γ )m̂. Since ∂Wt+1/∂k̂ = f ′(k̂),the first order condition for k̂ implies 1 � βf ′(k̂), with equality if k̂ > 0. Let k∗ be the solutionto this condition. Given we know k̂ = k∗, it will be dropped as a state variable from now on –again, k is only in the model to calculate the return on illiquid assets.6

Now consider the more interesting choice of the liquid asset m̂. Given the bargaining solutiong(x) = (φt+1 + γ )m̂, it is obviously equivalent to consider the direct choice of x, which can bewritten

maxx

{− φt

φt+1 + γg(x) + βσu(x) + β(1 − σ)g(x)

}. (5)

Assuming for now an interior solution, the first order condition reduces to

u′(x)

g′(x)=

φt

φt+1+γ− β(1 − σ)

βσ. (6)

Below we study the choice of x, and hence the implied choice of m̂, in detail. Note in particularthat if there is a unique maximizing choice of x then the distribution of money holdings in theCM must be degenerate.

3. Fiat money

Consider first the case of fiat money, γ = 0. Then in steady state φt/φt+1 = 1 + π , and theFisher equation (1 + π)/β = 1 + i implies (6) can be written

�(x) = i, (7)

where �(x) ≡ σ [u′(x)/g′(x)−1] can be interpreted as the marginal value of liquidity. Discussingthe existence of a solution to (7) is routine. In terms of uniqueness, however, there is no way toguarantee �(x) is monotone without special assumptions, because it depends on third derivativesof u and c.7

Note, however, that (7) is a necessary but not a sufficient condition for a solution to the (5).What I do now is characterize the solution globally, rather than focusing on first order conditions.In the fiat money case γ = 0, (5) can be rewritten

maxx

{−(i + σ)g(x) + σu(x)}, (8)

which looks like a simple decision problem. At first blush, this does not seem to help much,since the first order condition from (8) is of course exactly (7). Still, interpreting (8) as a sim-ple decision problem, as opposed to thinking of (7) as an equilibrium condition, allows one toimpose structure that can be exploited (as Cavalcanti and Puzzello [8] discuss in a related analy-sis).

6 In particular, if there is positive storage, k∗ > 0, the gross return is 1 + r = f ′(k∗) = 1/β . This is why 1/β is used asthe gross real interest rate when writing the Fisher equation as 1 + i = (1 + π)/β .

7 Indeed there are versions of the model where is �(x) is definitely not monotone (see e.g. Silveira and Wright [31]).As mentioned, there are results for special cases. If θ = 1 then third derivative terms vanish, and �′(x) < 0; θ = 1 is veryspecial, however, and precludes applications where sellers have to make ex ante investments. Also, if c(x) is linear andlogu′(x) is concave (i.e. u displays decreasing absolute risk aversion) then one can show �′(x) < 0 for any θ . The goalhere is to get uniqueness for any u(x) and c(x).


Fig. 1. At i = 0 global max occurs at x2; at ı̂ it jumps from x2(ı̂) to x1(ı̂).

Consider Fig. 1, where to make things interesting �(x) is non-monotone. Look first at the lim-iting case i ≈ 0, where, as shown in the left panel, I suppose there are three solutions to �(x) = 0that are candidate equilibria (the argument obviously generalizes to any number of candidates).I now show that generically only one of these actually solves the problem (8) and hence onlyone is an equilibrium. First, it is easy to see that the second order condition from (8), which is−(i + σ)g′′(x) + σu′′(x) � 0, holds at �(x) = 0 if and only if �′(x) � 0. This means candidatex0 is a local minimizer, while x1 and x2 are both local maximizers. For generic parameters (e.g.for almost all θ ) only one of the two is a global maximizer. It is easy to determine which one:moving from x1 to x2 entails a loss and a gain, indicated by the areas in Fig. 1. In the case shown,at i ≈ 0 the global maximizer is x2.

As i increases, the following is clear: there are still two local maximizers and a local mini-mizer, say x1(i), x2(i) and x0(i), with the maximizers decreasing and the minimizer increasingin i, and the global maximizer is x2(i) when i is close to 0. As i increases to ı̂, as shown, thevalue of the objective function is the same at x2(ı̂) and x2(ı̂), because the loss just equals thegain. And as we increase i from slightly below to slightly above ı̂ the global maximizer jumpsfrom x2(i) to x1(i). It should be clear from the diagram that any jump must be to the left, and notto the right. So the global maximizer, call it x(i), decreases with i, and may jump to the left. Suchjumps occur, if at all, on a set of values for i with measure 0. At a high enough value for i it isalso possible that x(i) = 0 – i.e., the global maximizer may be a corner solution, correspondingto nonmonetary equilibrium.

The function x(i) can be interpreted as an individual demand for x. We have just establishedthat it is single valued for almost all i; or, for any i, it is single valued for almost all values ofthe other parameters. Hence for generic parameters all agents in the CM choose the same m̂.Moreover, aggregate demand X(i) looks similar to individual demand x(i), except I can fillin any gaps between jumps, since if there are multiple global maximizers any fraction of thepopulation can be assigned to each.8 This is shown in Fig. 2, where the dashed curve is �(x)

and the solid curve is X(i), the aggregate demand curve, which is continuous and downwardsloping, with possibly some horizontal segments, and may or may not hit the vertical axis forbig i.

Steady state is characterized by the intersection of X(i) with the relevant supply curve. Butwith fiat money, the supply curve is trivially horizontal at the exogenous value of i set by policy.The nominal interest rate is the relevant price, and it is a policy choice. The equilibrium quantityis whatever x is given by demand, since real balances φM = g(x) can adjust through the price

8 This means that, for non-generic parameters, there may have to be a two-point distribution of money holdings in theCM, but of course both entail the same expected payoff. This feature has not been discussed in existing analyses of themodel.


Fig. 2. Aggregate demand curve X = X(i).

φ for any given nominal money supply M . It is now evident that there cannot be multiple steadystates with x > 0, except for a measure 0 set of values for i or, for any fixed i, for a measure 0set of parameters. This completes the proof. Given this, comparative statics are simple and leftas exercises. We summarize as follows:

Theorem 1. When γ = 0, for generic values of i, or for any value of i and generic values of theother parameters, aggregate demand X(i) is single valued, and hence there cannot be multiplesteady state monetary equilibria. Given x > 0, x and φM = g(x) are monotone decreasing in i,and monotone increasing in θ or σ .

4. Commodity money

We now study the model with γ > 0, and a fixed M , so that it makes sense to consider steadystates. This means φ is constant over time. One slight complication is that, when γ > 0, as iswell known you may want to hold more m̂ than you bring to the DM, leaving some in the CM.This is because in bargaining models the terms of trade can turn against you when you have toomuch liquidity at hand.9 To model this, let agents in the CM now choose (m̂1, m̂2), where m̂1 isbrought to the DM and m̂2 is left behind. Then the relevant version of (4) is

maxm̂1,m̂2

{−φt (m̂1 + m̂2) + βσu(x) + β(1 − σ)(φt+1 + γ )m̂1 + β(φt+1 + γ )m̂2}

again subject to the bargaining constraint g(x) = (φt+1 + γ )m̂1.In steady state with M constant, φt = φ for all t . Let φ∗ = βγ/(1−β) denote the fundamental

price of the asset. We cannot have φ < φ∗, since it opens up an arbitrage opportunity. Hence, thereare two possibilities: either φ > φ∗ and then m̂2 = 0, so no one leaves any assets in the CM; orφ = φ∗ and everyone is indifferent about how much they leave in the CM. Given this, the choiceof m̂1, or equivalently x, reduces to (5) with φ constant. The first order condition reduces to (6)with φ constant, which we simplify using 1 + r = 1/β to

�(x) = φ(1 + r)

φ + γ− 1 = r − γ /φ

1 + γ /φ. (9)

To interpret this, let i1 and i2 be the interest rates on liquid and illiquid assets – i.e. the returnon m̂1 and on either k̂ or m̂2 – and define the spread by

9 See Lagos and Rocheteau [18], Geromichalos et al. [14], and Lester et al. [21]. This is a non-issue if γ = 0: no oneacquires fiat money to leave in the CM.


Fig. 3. The commodity money model.

s ≡ i2 − i1

1 + i1.

In equilibrium 1 + i2 = 1/β = 1 + r and 1 + i1 = (φ + γ )/φ. Hence, s = (r − γ /φ)/(1 + γ /φ).Hence the first order condition (9) can be rewritten

�(x) = s, (10)

which equates the marginal value of liquidity to the cost of liquidity as measured by thespread.

The logic in the previous section allows us to characterize the demand for x as a function of s.Here it is actually slightly more convenient, if logically equivalent, to use the demand for liquiditymeasured by z = (φ + γ )m̂1. Note from the bargaining solution that z = g(x). Exactly as before,aggregate demand ZD(s) is continuous, and downward sloping with horizontal segments on atmost a measure 0 set of values for s. Also, ZD(s) hits the horizontal axis at z̄ = g(x̄) where x̄

solves �(x̄) = 1. At z̄ agents are satiated in liquidity. Since s < 0 violates no-arbitrage, if s = 0then agents bring m̄1 to the DM where (φ + γ )m̄1 = z̄, and are willing to keep any amount ofthe asset in the CM, while if s > 0 then they bring m̂1 < m̄1 to the DM and keep nothing inthe CM.

These results on demand are straightforward given the previous analysis of the case γ = 0.What is different from that case is that now the real supply of liquidity ZS = (φ + γ )M isendogenous, since it involves the price φ. Simple algebra implies this price can be expressed interms of the spread as φ = γ (1 + s)/(r − s), and so

ZS = (1 + r)γM

r − s.

The aggregate real supply of liquidity is therefore increasing in s (obviously, since higher s

means higher φ). As shown in Fig. 3, with s on the vertical axis, supply is strictly increasingand concave. It hits the horizontal axis at z = (1 + r)γM/r , and s → r as ZS → ∞. Notice thatwhen γ → 0, ZS becomes a horizontal line at s = r , capturing as a limiting case the fiat moneymodel with π = 0.

A steady state equilibrium solves ZS(s) = ZD(s). In Fig. 3, drawn for γM small enough thatz < z̄, this occurs at s > 0, which means agents take all of their assets to the DM and leave nonein the CM. If γM is larger, then z > z̄ and s = 0 in equilibrium, whence agents take enough to theDM to be satiated in liquidity, and keep the rest of their assets in the CM. The main point fromthe perspective of this paper is the following: for all parameters, with commodity money, there


exists a unique intersection between the decreasing function ZD(s) and the strictly increasingfunction ZS(s). Summarizing:

Theorem 2. When γ > 0 there exists a unique steady state equilibrium. If γM is small thenequilibrium entails a liquidity premium, s > 0; if γM is big the asset is priced by fundamentalsand s = 0.

5. Conclusion

In this paper I have established the uniqueness of monetary steady state in the LW model.There are a few steps to the argument. First, I reduced the equilibrium problem to somethingthat looks like a simple demand problem. Second, I characterized the solution to this problemglobally, instead of focusing on first order conditions, and showed the implied aggregate de-mand curve is continuous and weakly monotone. Finally, given a supply curve that simply setsthe relevant price with fiat money and is strictly monotone with commodity money, the resultsfollowed.10

Perhaps one might not be surprised the demand curve is monotone, and hence we get unique-ness, since the model has quasi-linear utility, and at least in elementary demand theory this oftenimplies strong results. But first one has to reduce the equilibrium model to a demand-like prob-lem, and moreover it is not at all clear that results from elementary demand theory should apply,for two reasons. First, agents here bargain rather than taking prices parametrically. Second, thisafter all is a genuine monetary model, which is not covered by elementary demand (or generalequilibrium) theory.11

I cannot claim the results are completely general, in two senses. First, I focused on steadystates, and it is well known that LW, like most monetary models, has many dynamic equi-libria, including sunspot, cyclic, and chaotic equilibria (Lagos and Wright [19]). Second, themodel is special, and I am not asserting that the uniqueness argument generalizes to all mon-etary theory. Still, many interesting generalizations within the LW framework are available,and the proof here applies directly to recent extensions to include multiple assets (Lagos andRocheteau [18], Geromichalos et al. [14]), physical capital (Aruoba et al. [5]), unemployment(Berentsen et al. [7], Liu [22]), banking (Berentsen et al. [6], He et al. [15], Chiu and Meh [9]),optimal taxation (Aruoba and Chugh [4]), and so on.

To be clear, my uniqueness result applies to those models in the following sense: they may ormay not have multiple equilibria due to, say, something going on in the capital or labor markets,but not because of the money market. Thus, given i and other variables that agents take paramet-rically, equilibrium real balances are pinned down uniquely. For example, in Berentsen et al. [7],the argument in this paper implies there is a unique steady state value for real balances and for

10 A referee suggested that this method of reducing the equilibrium problem to a one-dimensional demand-like prob-lem is also useful because it makes the model easier to compute in otherwise relatively complicated applications andextensions.11 Note that many ostensibly similar models in the literature, including those in Shi [29], Trejos and Wright [32], orJohri [16], yield a unique monetary steady state only in very special cases. In the model in Shi or Trejos–Wright, e.g.,uniqueness requires a zero probability of double-coincidence meetings. It should be emphasized that this is not whatis going on here: the proof of uniqueness goes through exactly as stated with some double-coincidence meetings. Asanother technical point, a corollary to uniqueness is the monotone comparative static results. There is an establishedliterature on this topic, including Milgrom and Shannon [24], although again note that the model here uses bargainingrather than price taking.


any given unemployment rate, but there could be still be multiple combinations of real balancesand unemployment consistent with steady state equilibrium. In the language of that paper, theresults here guarantee that the LW curve (mapping unemployment into real balances) is singlevalued, which is good to know, even if this does not imply it cannot cross the MP curve (mappingreal balances into unemployment) more than once.

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a uniqueness proof for monetary steady state

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