Illiquid Financial Markets and Monetary Policy
aa r X i v : . [ ec on . T H ] S e p Illiquid Financial Marketsand Monetary Policy ∗ Athanasios Geromichalos † , Juan M. Licari ‡ , Jos´e Su´arez-Lled´o § July, 2011
Abstract
This paper analyzes the role of money in markets where financial investmenttakes place in a decentralized fashion. A key methodological contribution is thedevelopment of a dynamic framework that brings together a model for illiquidfinancial assets `a la
Duffie, Gˆarleanu, and Pedersen, and a monetary framework `ala
Lagos and Wright. The presence of decentralized financial markets generates anessential role for money in helping investors re-balance their portfolios. From theequilibrium conditions we are able to derive an asset pricing theory that deliversan explicit connection between monetary policy, the different asset prices, andwelfare. In contrast to the existing monetary literature, we can sustain welfaregains even with positive inflation rates. Also, we obtain a negative relationshipbetween inflation and asset prices, that are always above their fundamental value.This price differential is associated with a liquidity premium.
JEL classification:
E44, E52, G11, G12.
Keywords: monetary policy, asset pricing, decentralized markets, liquidity. ∗ We are very grateful to Darrell Duffie, Ricardo Lagos, and Randall Wright for their guidance andhelpful discussions. We would also like to thank Philipp Kircher, Jan Eeckhout, Aus´ıas Rib´o, andseminar participants at the Summer Workshop on Money, Banking, Payments and Finance at theFederal Reserve Bank of Chicago, the Search and Matching Workshop at University of Pennsylvania,Universidad P´ublica de Navarra, Universitat Aut´onoma de Barcelona, and Universitat de Barcelona.Financial support from the Spanish Ministry of Science and Innovation through grant ECO2009-09847, the Juan de la Cierva Program, and the Barcelona Graduate School Research Network isgratefully acknowledged. All errors are solely our responsibility. † Department of Economics, UC Davis. Contact: [email protected] ‡ Moody’s Analytics. Contact: [email protected] § Departmento de Econom´ıa e Historia Econ´omica. Universidad Aut´onoma de Barcelona. Contact:[email protected]
Introduction
Over the last two decades secondary financial markets have developed considerably,both in size and complexity. New financial products are generated through processesof securitization by which financial assets are derived from the value of an underlyingasset. In their seminal paper Duffie, Gˆarleanu, and Pedersen (2005) document thatmany of these financial products are traded in markets that are characterized bysearch frictions. At the same time, since the influential work of Lagos and Wright(2005) monetary search models have been integrated into main-stream macroeconomictheory. Their model analyzes the role of monetary exchange in economies where tradeis not centralized through some perfect and frictionless (Walrasian) market.Our paper attempts to bring these two strands of the literature together by study-ing the role that money can play, through liquidity provision, in frictional financialmarkets. Duffie et al. analyze Over-the-Counter markets where financial assets tradewithout the use of a liquid asset like fiat money. On the other hand Lagos and Wright(2005) consider a model where money helps overcome certain frictions of decentralizedtrade. However, agents here trade consumption goods for the liquid asset. Our objec-tive is to bring the framework in Duffie et al. (2005) into a dynamic monetary modelwhere financial markets with frictions may generate a role for money. We present amodel with a sequence of markets where agents can store a safe-return real asset andlater on convert part of it into a risky and illiquid financial investment. The modelingof this risky asset follows Duffie et al.: it yields different idiosyncratic returns andagents can re-balance their portfolios in illiquid (decentralized) markets. A relevantinnovative element in our approach is that investors can (and will) use money as amedium of exchange for those transactions. We show that money is not neutral andhas an impact on both the volume of trade and the value of assets.Duffie et al. consider the specific trading structure in decentralized financial mar-kets where assets are traded by procedures like bid-ask pricing, bilateral or multi-lateral bargaining, etc. They build a model that rationalizes standard measures ofliquidity in these markets such as trade volume, bid-ask spreads, and trading delays.However, investors’ ability to re-balance their positions is severely limited by the re-striction that agents can hold either 0 units or 1 unit of the assets. On the otherhand, the search-theoretic literature stemming from Lagos and Wright (2005) devel-ops a framework where centralized and decentralized markets interact. We considerthis an appropriate setup for analyzing these issues that has not been exploited in ourdirection. As it is well known, in models that feature Arrow-Debreu type of markets it is very hard tosupport monetary equilibria other than those where agents are “forced” to hold money, e.g. moneyin the utility function or cash-in-advance models.
The environment that we analyze takes the framework presented in Lagos and Wright(2005), henceforth LW, as a starting point. Time is discrete and there is a [0 ,
1] con-tinuum of agents that live forever and discount future at rate β ∈ (0 , R , at theend of every period, a risky financial asset, and an intrinsically worthless object thatwe call money. In every period agents engage in different activities in three marketsthat open sequentially. At the beginning of a period, t , agents enter an investmentmarket (IM) with a certain amount of the real asset, a t , and money holdings, m t ,that they have previously stored. In this first market agents can decide how muchof the safe asset to transform into a risky financial asset, s t ∈ [0 , a t ], thus giving upany claim on the safe real return by the issued amount. Instead, investing in thisnew asset will yield a high return, y H , or a low return, y L , at the end of the periodwith equal probability. Furthermore, in order to make the analysis more interestingwe are going to assume that 0 ≤ y L < y H , and that y H > R . Following Duffie et al.,we model idiosyncratic returns such that depending on the type of investor I turn outto be I will obtain different return. However, this idiosyncratic uncertainty will onlybe resolved right after the investment decisions have been made and before enteringa decentralized financial market (DFM). These returns are i.i.d. distributed acrossperiods and agents.In the second market, having learned their types, agents can readjust their portfo-lios by re-balancing their holdings of the risky asset. This process happens in bilateralmeetings where agents’ types are public info. In particular, gains from this decentral-ized trade are generated `a la Berentsen and Rocheteau (2003) from low-return agentswanting to sell their investment assets to high-return agents. Anonymity, in the sensethat trading histories are not known among agents, is still present in this environment.Also, there is imperfect enforcement and a double coincidence problem. These aresome of the main elements that render money essential in this type of framework, asdiscussed in Kocherlakota (1998). In particular, we will show that money will allowto reallocate otherwise inefficient investment allocations. Only agents of different types will be willing to trade their risky assets: high types want to takeon larger investment amounts in assets that will give them higher returns than the safe asset; andlow types may want to get ride of low-yield investment. centralized market (CM) with their new holdings of the risky asset, the safeasset, and money. At this last stage agents will choose how much of the safe realasset and money they will carry onto the next period. They also derive net utility, U ( X t ) − H t , from consuming an amount of a general good, X t , and from supplying anamount of labor, H t . Notice that this is the only market in which agents consume andwork. Utility, U ( X t ). is assumed to be twice continuously differentiable with U ′ > U ′′ ≤
0. We also assume there exists X ∗ such that U ′ ( X ∗ ) = 1, with U ′ ( X ∗ ) > X ∗ .At this stage, agents can acquire any quantity of money, ˆ m t , and the real asset, ˆ a t , forthe next period at prices φ and ψ , respectively. The supply of this asset is fixed, A ,and each unit yields dividend, R , in the last market of the next period. Money supplyis controlled by a monetary authority and it evolves according to ˆ M t = (1 + µ ) M t .This paper departs from LW and other related papers in two fundamental aspectsthat will be discussed in detail below: one is the introduction of an investment marketfor securitization, and the other relates to how we model decentralized financial trade.As we mentioned in the introduction, this departure will generate very interesting re-sults, also regarding monetary policy. Some stand in contrast to those in relatedpapers. Finally, it is an important issue in the literature that deals with decentral-ized trade, and particularly in the money-search literature, how different bargainingprotocols may determine the terms of trade. A broad discussion on this is providedby Rocheteau and Wright (2005). In our paper we consider both price taking andbargaining. However, the structure that we endow decentralized trade with turns outspecially relevant. Indeed, we find that Arrow-Debreu equilibria where the price ofthe real asset deviates from its fundamental value can only be supported under pricetaking. Therefore, event though the bargaining version has some interesting aspects,we relegate its analysis and the justification of this result to the Appendix. In orderto save notation, we drop the time subscripts from now on whenever it does not leadto confusion.
In order to derive a more clear intuition of the mechanism of the model, and forconvenience of analysis, we proceed to solve the model backwards, starting from thethird sub-period, a Walrasian centralized market. The value function of entering the Price taking can be regarded as the monetary version of Lucas and Prescott (1974).
5M with money holdings m , real asset holdings b , and financial investment s is V j ( m, b, s ) = max ˆ m, ˆ a,X,H (cid:8) U ( X ) − H + βV ( ˆ m, ˆ a ) (cid:9) s.t. φ ˆ m + ψ ˆ a + X = H + φm + ( ψ + R ) b + ( ψ + y j ) s, where j = L, H , and b = a − s , with a being the amount of the real asset broughtfrom the previous period. Three observations are immediate. First, in every period X = X ∗ and we can write V j ( m, b, s ) = U ( X ∗ ) − X ∗ + φm + ( ψ + R ) b +( ψ + y j ) s + max ˆ m, ˆ a (cid:8) − φ ˆ m − ψ ˆ a + βV ( ˆ m, ˆ a ) (cid:9) . (1)Second, V j is linear in all its arguments, V j ( m, b, s ) = Λ + φm + ( ψ + R ) b + ( ψ + y j ) s, (2)where the definition of Λ is obvious. Finally, it is easy to see from (1) that there arenot any wealth effects: the agent’s choices of ˆ m, ˆ a do not depend on today’s states m, a . This is a consequence of quasi-linearity of preferences. Although we will discussthe solution to the optimization problem above later, we lay out here the first orderconditions for ˆ m and ˆ a , − φ + βV m ( ˆ m, ˆ a ) ≤ , if ˆ m > , (3) − ψ + βV a ( ˆ m, ˆ a ) ≤ , if ˆ a > , (4)where V i stands for the partial derivative of V with respect to its argument i . Fromexpression (2) the envelope conditions would be V jm = φm ; V jb = ψ + R ; V js = ψ + y j . The first order conditions refer to the fact that the cost of carrying the assets acrossperiods must not be negative. Otherwise agents would want to accumulate unboundedamounts of money and the real asset. In a pure Arrow-Debreu world the equilibriumcondition for the real asset would imply its price just reflecting the discounted flow ofreturns. In our model its price will reflect the discounted marginal value of carryingthe asset into the next period, which will differ from its fundamental value.
A large percentage of the financial sector is composed of markets for less liquid assetsthat are more difficult to trade in regular centralized markets. Trade is carried out6n more decentralized procedures in these markets. Over-the-Counter markets are aespecially relevant example (Duffie et al. (2005), Lagos et al. (2010)). However, it isalso the case that in many of these markets agents do not usually bargain. Instead,they feature some type of price taking protocol (bid-ask pricing). Following the logicof these type of markets, we model here the exchange of a risky financial asset, s ,that is less liquid than a standard real asset, a , commonly traded in a centralizedWalrasian market. We also consider a price taking mechanism that resembles thosein the kind of markets we are interested in.When agents enter our DFM they know what return they will obtain for theirinvestment in the risky asset, and they are now allowed to re-balance their positions onthis investment by trading the risky asset. Since 0 < y L < y H , L-types will naturallyarise as the sellers of s , while the H-types will naturally become the buyers. Let p be the dollar price of one unit of investment. Conditional on being an L-type, andhence a seller, an agent who carries an amount s solves the following problem max q s ≤ s V L ( m + pq s , b, s − q s )or alternativelymax q s ≤ s { Λ + φm + ( ψ + R ) b + ( ψ + y L ) s + ( φp − ψ − y L ) q s } , where q s is the supply of securities. The seller’s optimal behavior yields the individualsupply function q ∗ s = , if p < ψ + y L φ , ∈ [0 , s ] , if p = ψ + y L φ ,s, if p > ψ + y L φ . In short, L-types will not sell if the price is low, they will be indifferent if the pricefalls in a medium range, and would want to sell everything if the price exceeds thevalue of the asset. Interestingly, notice that since the price of s is in nominal termsand that of the real asset is in real terms, inflation will have something to say here.Sellers compare p to the nominal value of ψ + y L by dividing it by φ . Therefore, as In principle it makes sense that L-types want to get rid of a bad asset, while H-types want topurchase more of an asset on which they can get a high return. As will be shown, the amount of moneyexchanged in this market will be either such that L-types are at least indifferent, which will maketrade profitable for H-types, or such that H-types are indifferent, in which case trade will be profitablefor L-types. Therefore, the natural arrangement in this market is actually always implemented inany equilibrium. Given the competitive nature of this market, the only variables that matter for the seller’sproblem are s and the price (that she takes as given). Similarly, in the buyer’s problem below therelevant variables are m, p . φ , increases it is more likely that p > ( ψ + y L ) /φ and theywill want to sell everything. Conversely, as inflation is higher, φ →
0, sellers decreasetheir supply, q s → q b ≤ m/p V H ( m − pq b , b, s + q b )or alternativelymax q b ≤ m/p { Λ + φm + ( ψ + R ) b + ( ψ + y H ) s + ( ψ + y H − φp ) q b } . where q b is the demand of securities. The demand function is given by q ∗ b = , if p > ψ + y H φ , ∈ [0 , m/p ] , if p = ψ + y H φ ,m/p, if p < ψ + y H φ . The interpretation for buyers is the opposite as that for sellers, and inflation alsoplays a role; as φ →
0, buyers increase their demand, q b → m/p . Figure 1 depicts theaggregate demand and supply curves. The equilibrium price and the quantity of s that is traded depend on the shape of the demand curve. In particular, inspection ofthe demand and supply functions shows that equilibrium depends on the distributionof the returns on the financial asset. If φm < ( ψ + y L ) s (represented by D in Figure1), we have Q ∗ = ( λ/ φm ) / ( ψ + y L ). If φm ≥ ( ψ + y L ) s (represented by D inFigure 1), we have Q ∗ = λs/
2. Therefore, Q ∗ = ( λ/
2) min { s, ( φm ) / ( ψ + y L ) } . Theparameter λ is the probability of going into the DFM. It can be seen as the degree ofaccess to decentralized financial markets. Thus, this parameter can also be interpretedas a measure of the liquidity of these markets. The equilibrium quantity of s that arepresentative agent (H-type) acquires in the DFM is then q ∗ = min (cid:26) s, φmψ + y L (cid:27) , (5)and the equilibrium price is given by p ∗ = ψ + y L φ , if φm < ( ψ + y L ) s, ms , if φm ∈ [( ψ + y L ) s, ( ψ + y H ) s ] , ψ + y H φ , if φm > ( ψ + y H ) s . (6)Interestingly enough, the equilibrium in this market depends on real money balances, φm . Portfolio re-balancing generates a role for money and takes place at the equilib-rium price, p ∗ , which depends on inflation through φ . Moreover, it is key to noticefrom equation (5) that trade in this market exists, q ∗ >
0, if and only if φm > .3 Investment Market
At the beginning of every period, agents enter a financial market with holdings ofmoney and real asset from previous period. Thus, the vector of state variables is( m, a ), and the agent wishes to maximize her continuation value in the DFM bychoosing optimally what part of a to invest in the risky asset. Therefore, an agentthat enters the IM with portfolio ( m, a ) has a value function V ( m, a ) = max s ∈ [0 ,a ] (cid:26) (cid:2) V L ( m, b, s ) + V H ( m, b, s ) (cid:3)(cid:27) . (7)In order to examine the optimal choice of s for the agent, we need to replace thevalue functions V j with more useful expressions. To that end notice that V L ( m, b, s ) = λV L ( m + pq s , b, s − q s ) + (1 − λ ) V L ( m, b, s ) ,V H ( m, b, s ) = λV H ( m − pq s , b, s + q s ) + (1 − λ ) V H ( m, b, s ) . Equivalently, using the results derived above regarding the terms of trade in the DFMwe can write V L ( m, b, s ) = λV L ( m + p ∗ q ∗ , b, s − q ∗ ) + (1 − λ ) V L ( m, b, s ) ,V H ( m, b, s ) = λV H ( m − p ∗ q ∗ , b, s + q ∗ ) + (1 − λ ) V H ( m, b, s ) . Then, exploiting the linearity of V j , equation (2), and using (5), we obtain12 (cid:2) V L ( m, b, s ) + V H ( m, b, s ) (cid:3) = Λ + φm + ( ψ + R ) a ++ (cid:20)
12 ( y L + y H ) − R (cid:21) s + λ y H − y L ) min (cid:26) s, φmψ + y L (cid:27) . (8)Inserting (8) into (7), we can rewrite the objective function asmax s ∈ [0 ,a ] n (cid:2) ( y L + y H ) − R (cid:3) s + λ ( y H − y L ) min n s, φmψ + y L o o ≡≡ max s ∈ [0 ,a ] n α s + α min { s, α } o . The definitions of the terms α i , i = 1 , ,
3, are obvious and are adopted for analyticalconvenience. The objective function of the agent is very intuitive. The first term, ( y L + y H ) − R , represents that for every unit of a that she turns into s , she forgoesthe return R , and gains y L or y H with equal probability. The second term of theobjective is the expected gain from trade in the DFM, which is equal to the total Since b = a − s it does not matter whether the agent chooses s or b . s that trade in the DFM multiplied by the average gain from thistransaction. For every unit of s that goes from the hands of an L-type to those of anH-type, a surplus equal to y H − y L is generated.Consider now the optimal choice of s , namely, s ∗ . In order to focus on the moreinteresting situations we want to consider the case where the expected return fromthe risky investment is less than the return on the safe asset, i.e. we assume that α ≡ (1 /
2) ( y L + y H ) − R ≤
0. This means that if agents were not able to trade s inthe DFM, they would never choose s ∗ > The objective of the agent is depicted inFigure 2 for various parameter values. The key variable for the determination of s ∗ is α + α = (1 / λ ) y H + (1 − λ ) y L ] − R . This term is the multiplier of s for s ≤ α . When the financial market is fairly liquid, λ big, more weight is put on y H and α + α is also big. If financial markets are illiquid, λ = 0, the agent never getsto trade into the DFM; the gain from holding s coincides with the net return of therisky asset, which is assumed to be non-positive. If λ = 1, α + α = y H − R >
Lemma 1.
For a given state ( m, a ) , the optimal choice of s ∈ [0 , a ] is given by s ∗ = , if α + α < , ∈ [0 , min { α , a } ] , if α + α = 0 , min { α , a } , if α + α > and α < , ∈ [min { α , a } , a ] , if α + α > and α = 0 . (9) Proof.
The result follows from inspection of the objective function and Figure 2.The following lemma will be useful in later analysis and expresses that the netgain of carrying assets across periods is non-positive. The interpretation behind thisis that agents will only be willing to carry assets, money and real asset, if thereare positive gains from re-balancing their investment positions in the DFM. Indeed,we show below that one of the main contributions of the model is that the valuegenerated by this mechanism is reflected in the asset pricing equation. Should thisgain not exist, only the fundamental value of the asset would be reflected by its price,i.e. the discounted flow of expected returns.
Lemma 2.
In any equilibrium ψ ≥ β (cid:16) R + ˆ ψ (cid:17) ,φ ≥ β ˆ φ. Alternatively, the assumption that α ≤ s ∗ > α were positive. roof. This is a standard result in the literature and its proof is omitted. For detailssee for example Geromichalos, Licari, and Su´arez-Lled´o (2007).Having characterized the optimal choice of s we can proceed to analyze the choiceof ˆ m, ˆ a by examining the value function V j . This choice will crucially depend on theparameter values. Once again the key determinant is the term α + α . Case 1: α + α <
0. We know that in the next period the agent will chooseto not invest anything in the risky asset, i.e. ˆ s ∗ = 0 and q ∗ = 0. Hence, we can write V j ( m, b, s ) = U ( X ∗ ) − X ∗ + φm + ( ψ + R ) b + ( ψ + y j ) s + max ˆ m, ˆ a (cid:26) − φ ˆ m − ψ ˆ a + β (cid:2) V L ( ˆ m + µM, ˆ a,
0) + V H ( ˆ m + µM, ˆ a, (cid:3)(cid:27) == Ω + max ˆ m, ˆ a n − φ ˆ m − ψ ˆ a + β h ˆΛ + ˆ φ ( ˆ m + µM ) + ( R + ˆ ψ )ˆ a + λ y H − y L ) min ( , ˆ φ ( ˆ m + µM )ˆ ψ + y L ) == Ω + max ˆ m, ˆ a n − (cid:16) φ − β ˆ φ (cid:17) ˆ m − h ψ − β (cid:16) R + ˆ ψ (cid:17)i ˆ a o , where the first equality follows from (8) and the definition of q ∗ . The terms Ω , Ω ,and ˆΛ do not depend on ˆ m, ˆ a , and their definitions are obvious. Recall from Lemma2 that the multipliers of both ˆ m and ˆ a above are non positive. Hence, it is easy tosee that the optimal choices of these variables satisfyˆ m ∗ = (cid:26) , if φ > β ˆ φ, ∈ ℜ + , if φ = β ˆ φ, ˆ a ∗ = , if ψ > β (cid:16) R + ˆ ψ (cid:17) , ∈ ℜ + , if ψ = β (cid:16) R + ˆ ψ (cid:17) . (10)We will return to describe equilibrium in this case. Before that, we complete ourdescription of the optimal choice of ˆ m, ˆ a . Case 2: α + α ≥
0. Here ˆ s ∗ = min { ˆ a, ˆ α } , where ˆ α = ˆ φ ( ˆ m + µM ) / (cid:16) ˆ ψ + y L (cid:17) . Since ˆ s ∗ = 0, we have ˆ b = ˆ a . Also, the money holdings in the next period of an agent thatchooses ˆ m in the current period’s CM are given by ˆ m + µM . In particular, ˆΛ is the term we get if we lead Λ, defined in (2), by one period. Since Λ isindependent of m, a , ˆΛ is independent of ˆ m, ˆ a . As opposed to α , the terms α , α are constant. Moreover, from Lemma 1 it follows thatˆ s ∗ = min { ˆ a, ˆ α } is always an optimal choice in the forthcoming period, although not the only V j ( m, b, s ) = U ( X ∗ ) − X ∗ + φm + ( ψ + R ) b + ( ψ + y j ) s + max ˆ m, ˆ a n − φ ˆ m − ψ ˆ a + β (cid:2) V L ( ˆ m + µM, ˆ a − min { ˆ a, ˆ α } , min { ˆ a, ˆ α } )+ V H ( ˆ m + µM, ˆ a − min { ˆ a, ˆ α } , min { ˆ a, ˆ α } ) (cid:3) o == Ω + max ˆ m, ˆ a n − φ ˆ m − ψ ˆ a + β h ˆΛ + ˆ φ ˆ m + (cid:16) R + ˆ ψ (cid:17) (ˆ a − min { ˆ a, ˆ α } )+ (cid:18) ˆ ψ + y H + y L (cid:19) min { ˆ a, ˆ α } + λ y H − y L ) ˆ q ∗ (cid:21)(cid:27) . From (5), the quantity of s that changes hands in the DFM is given byˆ q ∗ = min ( ˆ s ∗ , ˆ φ ( ˆ m + µM )ˆ ψ + y L ) = min ( min { ˆ a, ˆ α } , ˆ φ ( ˆ m + µM )ˆ ψ + y L ) == min ( min ( ˆ a, ˆ φ ( ˆ m + µM )ˆ ψ + y L ) , ˆ φ ( ˆ m + µM )ˆ ψ + y L ) = min ( ˆ a, ˆ φ ( ˆ m + µM )ˆ ψ + y L ) . Using this fact to rewrite the the value function V j , we conclude that the objectiveof the agent is to max ˆ m, ˆ a n − (cid:16) φ − β ˆ φ (cid:17) ˆ m − h ψ − β (cid:16) R + ˆ ψ (cid:17)i ˆ a + β λ ) y H + (1 − λ ) y L − R ] min ( ˆ a, ˆ φ ( ˆ m + µM )ˆ ψ + y L )) ≡≡ max ˆ m, ˆ a {− γ ˆ m − γ ˆ a + γ min { ˆ a, γ ˆ m + γ }} . The definitions of the terms γ i , i = 1 , .., γ , γ ≥
0. Also, γ = β ( α + α ) ≥
0. Finally, γ > γ depends on whether the monetary authority is running inflation ordeflation, i.e. the sign of µ . The objective function is linear in ˆ m, ˆ a . The optimalsolution depends on the magnitude of the various gamma’s. Considering that botharguments in the min function depend on decision variables of the agent, the optimalchoice will have both arguments being equal to each other. Thus, it is relatively easy one. This choice is the unique optimal, only if α + α > α <
0. Nevertheless, pluggingˆ s ∗ = min { ˆ a, ˆ α } into the value function V j is always a good idea, since it yields the same result asany other ˆ s ∗ .
12o verify that ( ˆ m ∗ , ˆ a ∗ ) = (0 , , if γ + γ γ > γ γ , (+ ∞ , + ∞ ) , if γ + γ γ < γ γ , ( z, γ z + γ ) , for any z ≥ − γ γ , if γ + γ γ = γ γ . (11) We now proceed to the definition and the analysis of equilibrium.
Definition 1.
An equilibrium for this economy is a set of value functions V ij , i =1 , , and j = L, H that satisfy the Bellman equations, a triplet ( p ∗ , q ∗ , s ∗ ) that satisfy(5), (6), and (9) in every period, and a pair of bounded sequences { φ t M t } ∞ t =0 , { ψ t } ∞ t =0 ,such that the agent behaves optimally under the conditions a t = A and m t = M t , all t . Consider first Case 1, i.e. α + α <
0. From (10), a necessary condition forequilibrium is ψ = β (cid:16) R + ˆ ψ (cid:17) , which implies that the sequence of the asset priceshould follow the difference equation ψ t +1 = − R + (1 /β ) ψ t . Since β <
1, this canonly be true if ψ t = ¯ ψ = βR − β . (12)The asset price given by (12) is the fundamental value of the asset, i.e. the discountedstream of future dividends. The result according to which ψ t = ¯ ψ , reflects the factthat in the case under consideration the asset is only valued for the fruit it yields:agents never buy it in order to invest in the risky asset and possibly trade in theDFM.The role of money in this economy is not essential. Agents have a positive demandfor money only if φ = β ˆ φ . In steady state this implies that µ = 1 − β , i.e the monetaryauthority is following the Friedman rule. Agents carry money only when the costof doing so is zero (equivalently, when the nominal interest rate is zero). Of course,they never get to use that money in the DFM, since s ∗ = 0 and there is nothing totrade money with.The more interesting case is therefore α + α ≥
0. For the rest of this paper In the third line, the condition z ≥ − γ /γ just guarantees that if µ < a satisfies non-negativity. In steady state φM = ˆ φ ˆ M ⇒ φM = ˆ φ (1 + µ ) M . Therefore, 1 + µ = φ/ ˆ φ = β .
13e assume that this condition holds. Optimality (described by (11)) reveals that anecessary condition for the existence of equilibrium is γ + γ γ = γ γ ⇔ φ − β ˆ φ + h ψ − β (cid:16) R + ˆ ψ (cid:17)i ˆ φ ˆ ψ + y L = β ( α + α ) ˆ φ ˆ ψ + y L . (13)As we pointed in the definition above, we focus on steady state equilibria. That is,equilibria in which money balances and asset prices are constant in every period.However, in the appendix we show that bubbles, i.e. equilibria in which the assetprice is above the market price for some periods, can never arise in this environment.To simplify equation (13), divide both sides by ˆ φ and recall that at the steadystate φ/ ˆ φ = 1 + µ . We obtain(1 + µ − β ) (cid:16) ˆ ψ + y L (cid:17) = − ψ + β (cid:16) R + ˆ ψ (cid:17) + β ( α + α ) . Using the definition of α + α and solving with respect to ˆ ψ yields the asset pricingdifference equation followed,ˆ ψ = β [(1 + λ ) y H + (1 − λ ) y L ] − (1 + µ − β ) y L µ − β −
11 + µ − β ψ. (14)As we explained, we drive our attention to the case in which ψ = ˆ ψ = ψ ∗ . Usingthis in (14) and solving with respect to ψ ∗ we obtain ψ ∗ = β [(1 + λ ) y H + (1 − λ ) y L ] − (1 + µ − β ) y L − β ) + µ . (15)It immediately follows from this equation that the price of the asset depends negativelyon the growth rate of money. This result is in contrast with the predictions of relatedpapers, like Geromichalos et al (2007) or Lester, Postlewaite, and Wright (2007). Inthese papers agents bring money and real assets with them in the DFM in order tobuy a specialized good. Higher inflation makes carrying money more costly, and sopeople turn to the relatively cheaper asset, thus increasing its price. Unlike thesepapers, here money and the asset are complements, in the sense that agents needboth objects in order to trade in the DFM. If money is costly to carry, the demand The returns y H and y L are deterministic and with probability 0 .
5. Therefore the only randomnesscomes from the idiosyncratic realizations of those returns. The probability of going to the DFM, λ ,is also constant. Therefore, the intuition behind this is that the gain from decentralized trade is alsoalways going to be the same, so it makes sense that the price of the asset is constant every period aswell. More extensive discussion is provided later. More precisely, agents trade m for s in the DFM, but in order to be in possession of some s ,agents need to bring some a from the previous period. To see why m and s are complements considerthe extreme case in which all agents bring a billion dollars but s ∗ = 0. The surplus generated in theDFM is still zero. λ >
0. Moreover, since the asset price is decreasing in themoney growth rate, it obtains its maximum value when µ = β −
1. The resultingexpression is given by ψ max = β − β
12 [(1 + λ ) y H + (1 − λ ) y L ] ≥ β − β R, where the inequality follows from the fact that α + α ≥
0. Again, unless λ = 0,the asset price is strictly greater than its fundamental value. Furthermore, since theprice, ψ ∗ , can never be smaller than βR/ (1 − β ) because this would violate Lemma2, the upper bound of admissible monetary policies can be uniquely defined by ¯ µ ≡{ µ : ψ ∗ ( µ ) = βR/ (1 − β ) } . A characterization of ¯ µ is provided in the Appendix wherewe also show that equilibria with positive inflation, ¯ µ >
0, can be supported dependingon the distribution of idiosyncratic shocks.The asset pricing equation in this model reflects two dimensions. First, the assetis valued for the dividend it yields, R . Second the asset is also valued for its propertyto be transformed into s , thus allowing agents to generate additional value by re-balancing their portfolios in the DFM. Since access to the DFM is granted withprobability λ , the difference expressed by ψ ∗ − βR/ (1 − β ) reflects the premium ofthe asset over the fundamental value. In particular, this premium is largest at theFriedman rule, in which case the term α + α ≡ (1 /
2) [(1 + λ ) y H + (1 − λ ) y L ] − R would represent the spread. Notice, however, that in general such premium dependson the money growth rate, µ . Then, we consider it appropriate to refer to it asa liquidity premium . After all, additional value in this economy is generated by thedegree of “liquidity” in the DFM. That is, value is generated along with the easiness toreallocate portfolios in the DFM, and this ultimately depends on the value of money. In most other related models welfare is measured in a standard manner by two terms:the excess utility over the cost of producing the general consumption good in the cen-tralized market, U ( X ) − X , and the excess utility of the total amount of consumptionin the decentralized market over the cost of producing that total amount. However, inour economy there is no consumption good being traded in the decentralized market.Instead, the amount of assets traded in the DFM affects the budget constraint in theCM. Therefore, the way we measure welfare in this economy is by computing the totalvalue of trade in the DFM in terms of the CM.Given that α + α >
0, in the IM agents place all their asset holdings into the15isky investment, s ∗ = A . In the DFM the amount of assets that change hands isgiven by Q ∗ = ( λ/
2) min { A, φM/ ( ψ + y L ) } = ( λ/ A . This means that L-types sellall their assets to H-types, which is efficient from a social point of view, and in returnthey receive p ∗ Q ∗ = λA ( ψ ∗ + y L ) / (2 φ ∗ ). The latter accounts for the total value oftrade in the DFM. One result that is as stricking as important is that total welfareis unaffected by inflation as long as µ ∈ [ β − , ¯ µ ]. This result implies, in contrast toother monetary models, that welfare gains can be sustained even by positive rates ofinflation. If µ > ¯ µ carrying money becomes too expensive and the monetary equi-librium collapses, together with any trade in the DFM. A first best is achieved when λ = 1 because decentralized trade (and, therefore, optimal re-alocation of resources)is maximized. The following proposition summarizes the most important results. Proposition 1.
The key variable for the determination of equilibrium is α + α ,which can be interpreted as the net return of the risky asset. (a) If α + α ≤ no trade exists in the DFM, q ∗ = s ∗ = 0 , and the asset is onlyvalued for the dividend it yields, i.e. ψ ∗ = βR/ (1 − β ) . There is no essential role formoney in this economy. Its real allocation coincides with that of an Arrow-Debreueconomy and asset pricing follows the Lucas formula. A monetary equilibrium couldonly be supported by the Friedman rule. (b) If α + α > , ψ t = ψ ∗ ( µ ) given by (15) for all t . The range of policies underwhich monetary equilibria can be supported is [ β − , ¯ µ ] , where ¯ µ is defined by ¯ µ ≡{ µ : ψ ∗ ( µ ) = βR/ (1 − β ) } . For all µ in this range, dψ ∗ /dµ < . The bigger the λ ,the bigger the premium between the actual value and the fundamental value of theasset. This liquidity premium is maximized at the Friedman rule. These results arepresented in Figure 3.Equilibrium welfare depends on λ , i.e. the probability with which agents get totrade in the DFM and re-balance their positions. In the DFM L-types sell all theirrisky assets to H-types, hence all equilibria are (constrained) efficient. The First Bestis achieved when λ = 1 . If the decentralized financial market can generate value ( α + α > φm >
0, exist and can be supported by monetary policies ranging fromthe Friedman rule to potentially positive inflation rates. Risky investments take place Remember that in any equilibrium with positive and finite money and asset holdings, thesevariables are chosen so that the two arguments in the min function are the same. This result is closely connected to the nature of the returns to the risky asset. In an alternativespecification of this model, we consider the case in which investing s ∈ [0 , a ] units of the asset yields ǫ j f ( s ), where j = L, H and f is a strictly increasing and concave function with standard Inadaconditions. Because lim s → f ′ ( s ) = ∞ , it is never optimal for the L-type to give up all her assets.This creates a link between inflation and the amount of s that changes hands in the DFM. However,the analysis becomes very complex and we are only able to solve the model numerically. Since noneof the major results of the model are altered, we prefer the more simple specification presented here,which yields closed form solutions and clear intuition. q ∗ = s ∗ = A , in exchange for p ∗ units of money.Thus, money plays an essential role as a medium of exchange in frictional financialmarkets. In particular, the price at which securities are traded, p ∗ , not only is relatedto the price of the real asset, but will also depend negatively on the money growthrate, µ . The quantity exchanged, s ∗ , is independent of monetary considerations. Moreinterestingly, the price of the real asset, ψ ∗ = ψ ( µ ), is always above its fundamentalvalue and exhibits a liquidity premium that depends crucially on two notions of liq-uidity. On one side, the degree of access to decentralized financial markets, λ . Andon the other side, the degree of liquidity within the financial markets, which dependson money supply, µ . The liquidity premium of the real asset, and therefore its value,is maximized at the lowest feasible inflation rate. Positive money growth rate couldalso generate welfare gains. Certain types of markets have quickly developed to attain an enormous size within thefinancial sector. They have also recently proven to have a considerable impact. Theirstructure deviates from the standard centralization of primary markets that is embed-ded in most macroeconomic models. Markets for securitization and Over-the-Countermarkets stand as a reference of these type of markets. In this context a concern arisesnaturally: does monetary policy have something to say in these environments?We have presented a general equilibrium model that analyzes the role of moneyin decentralized financial markets. We provided an innovative framework that modelsa risky financial asset that can be issued from the value of a safe-return real asset.Agents can re-balance their positions on these relatively illiquid assets with a moreliquid instrument (fiat money) in a decentralized financial market. We show that valuecan be generated from this transfer of assets, and therefore welfare can be increased,by reallocating otherwise inefficient investment.Our new approach yields very interesting results, some in contrast with the exist-ing literature. On the one hand, monetary policy affects the price of the real and thefinancial asset, attaining their maximum values at a policy equivalent to the Friedmanrule. In particular, since money and the real asset are complements, its price is nega-tively correlated with inflation. On the other hand, monetary policy then influencesthe value generated for the economy in the decentralized market. In fact, policy canalways sustain asset prices above their fundamental values and generate welfare gainseven under inflationary money growth rates, as long as those rates are not too large.17 eferences [1] Berentsen, A. and G. Rocheteau, 2003. Money and the Gains from Trade. Inter-national Economic Review 44 (1), 263-297.[2] Duffie, D., Gˆarleanu, N., and L.H. Pedersen, 2005. Over-the-Counter Markets.Econometrica 73, 1815-1847.[3] Ferraris, L., 2010. On the Complementarity of Money and Credit. EuropeanEconomic Review 54 (5), 733-741.[4] Ferraris, L. and M. Watanabe, 2011. Collateral Fluctuations in a Monetary Econ-omy. Journal of Economic Theory, forthcoming.[5] Geromichalos, A., Licari, J.M., and J. Su´arez-Lled´o, 2007. Monetary Policy andAsset Prices. Review of Economic Dynamics 10, no. 4, 761-779.[6] Kocherlakota, N., 1998. Money is Memory. Journal of Economic Theory 81, 232-251.[7] Lagos, R., 2011. Asset Prices, Liquidity, and Monetary Policy in an ExchangeEconomy. Journal of Money, Credit, and Banking, forthcoming.[8] Lagos, R. and G. Rocheteau, 2009. Liquidity in Asset Markets with Search Fric-tions. Econometrica 73 (2), 403-426.[9] Lagos, R., Rocheteau, G., and P.O. Weill, 2011. Crises and Liquidity in Over-the-Counter Markets. Journal of Economic Theory, forthcoming.[10] Lagos, R. and R. Wright, 2005. A Unified Framework for Monetary Theory andPolicy Analysis. Journal of Political Economy 113, no. 3.[11] Rocheteau, G. and R. Wright, 2005. Money in Search Equilibrium, in CompetitiveEquilibrium, and in Competitive Search Equilibrium. Econometrica 73, 175-202.
A Appendix
Non-steady state equilibria.
We prove here that equilibria other than the steady state described above, cannotarise in this model. More precisely, we cannot have sequences { φ t M t } ∞ t =0 , { ψ t } ∞ t =0 thatconverge to a certain limit either monotonically or in an oscillating manner. Therefore,the only way to keep these sequences bounded is to have φ t M t and ψ t constant in allperiods (in order to make this statement one needs to exclude cycles, which is the18ase here). As part of an equilibrium, we are looking for bounded sequences of moneybalances and asset prices such that (13) holds and also φ t M t = ( ψ t + y L ) A, ∀ t. (16)Define real balances as z ≡ φM . If we multiply (13) by ˆ M and use (16) to substitutefor ˆ φ/ ( ˆ ψ + y L ) and ψ , we conclude that real money balances follow a first-order lineardifference equation,ˆ z = − β ( α + α + R ) + (1 − β ) y L β A + 2 + µ β z. However, µ ≥ β − z that (2+ µ ) / (2 β ) > { φ t M t } ∞ t =0 will always be explosive, unless ( φM ) t = ( φM ) ∗ for all t . Since (16) has to hold in every period the same conclusion is true for { ψ t } ∞ t =0 . Optimal monetary policy range.
In the section on equilibrium with the price taking version we discussed the feasiblerange for optimal monetary policies. Here we provide a more detailed characterizationof the upper bound of that range. The whole point of the price taking version is that itsupports asset prices higher than the fundamental value, and such price is decreasingin the money growth rate. Therefore, the upper bound for monetary policy is themoney growth rate that makes the price of the real asset equal to its fundamentalvalue, ¯ µ ≡ { µ : ψ ∗ ( µ ) = βR/ (1 − β ) } . If we take the equation for the equilibriumprice of the asset, (15), and make it equal to its fundamental value, we can solve forthe money growth rate, ¯ µ b . Thus, we write ψ ∗ = β [(1 + λ ) y H + (1 − λ ) y L ] − (1 − µ − β ) y L − β ) + µ = βR − β . After some algebra one arrives at the following expression¯ µ b = β − β (1 − β )( α + α ) y L (1 − β ) + βR . which will be positive as long as α + α > − ββ y L + R > Bargaining in the DFM.
We present here the complete analysis of the version of the model with bargaining inthe DFM. We consider agents bilaterally trading in this market according to a Nashbargaining procedure. It can be easily shown that in any meeting between two agentsof the same type no trade will occur. Thus, we focus here on bilateral meetings ofagents of different type. In this matches the portfolio of a low type agent is ( m, b, s )19nd that of a high type agents is ( ¯ m, ¯ b, ¯ s ). Thus, agents must choose the amount ofmoney, d m , and securities, d s , exchanged in order tomax d m ,d s (cid:2) V L ( m + d m , b, s − d s ) − V L ( m, b, s ) (cid:3) θ ×× (cid:2) V H ( ¯ m − d m , ¯ b, ¯ s + d s ) − V H ( ¯ m, ¯ b, ¯ s ) (cid:3) − θ s.t. − m ≤ d m ≤ ¯ m ; − ¯ s ≤ d s ≤ s In other words, agents maximize the value of trade. In principle, we formulate theproblem in the most general way that allows either agent to sell or buy securities.However, in the equilibrium delivered by this version, in which returns on the riskyasset are linear in the amount of the asset, H-types will always be the buyers andL-types the sellers. Now, using the linearity of the value function, we can rewrite thebargaining problem asmax d m ,d s [ φd m − ( ψ + y L ) d s ] θ [ − φd m + ( ψ + y H ) d s )] − θ (17)s.t. − m ≤ d m ≤ ¯ m ; − ¯ s ≤ d s ≤ s Before proceeding with any further analysis, it is absolutely critical to realize that theproblem we pose here is completely different from most other bargaining problems inthe mainstream literature on money search, at least in those models derived from LW.The key aspect is the following. In those papers the welfare pie in the DFM is givenby u ( q ) − c ( q ), where q is the consumption of the good in that market, and u ( q ), c ( q )are the utility and cost derived from it. Whereas in our model the size of the pie is notpre-determined. In fact, it depends on the size of the portfolio that the parties carrywith them. Therefore, the problem we propose here can only be solved by imposingconstraints, otherwise the agents would optimally choose d m = d s = + ∞ . Thus,the optimal solution to our problem features one of the constraints above binding inequilibrium. We will proceed case by case and then summarize the results.Suppose first that d m = ¯ m and d s ≤ s . The problem has a unique interiorsolution determined by the FOC for d s : θ ( ψ + y L ) φ ¯ m − ( ψ + y L ) d s = (1 − θ )( ψ + y H ) − φ ¯ m + ( ψ + y H ) d s , which yields d s = φ ¯ m [ ψ + (1 − θ ) y H + θy L ]( ψ + y L )( ψ + y H ) ≡ d ∗ s . In all search models where consumption is traded in the DFM, its maximum level is boundedabove by the condition u ′ ( q ∗ ) = c ′ ( q ∗ ).
20o be more precise, d s = min { s, d ∗ s } . Now let us suppose that d s = s and d m ≤ ¯ m .The first order condition for d m can be written as θ [ − φd m + ( ψ + y H ) s ] = (1 − θ )[ φd m − ( ψ + y L ) s ] , which implies that d m = s [ ψ + θy H + (1 − θ ) y L ] φ ≡ d ∗ m . So, in fact, d m = min { ¯ m, d ∗ m } . In more detail, since there can be equilibria whereonly one of the constraint binds or both bind, we have that if ¯ m ≤ d ∗ m then d m = ¯ m .In this case, d s = s only if s ≤ d ∗ s . In other words, both constraints would be binding( d m = ¯ m, d s = s ) in an equilibrium where ¯ m ≤ sφ [ ψ + θy H + (1 − θ ) y L ] ≡ m h and¯ m ≥ sφ ( ψ + y L )( ψ + y H )[ ψ +(1 − θ ) y H + θy L ] ≡ m l . It is easy to check that ψ + θy H + (1 − θ ) y L ≥ ( ψ + y L )( ψ + y H )[ ψ + (1 − θ ) y H + θy L ] , and this condition holds with strict inequality unless y H = y L , or θ = 0, or θ = 1; butnone of these happens in our model. Therefore, for all ¯ m ∈ [ m l , m h ] agents exchange d m = ¯ m and d s = s . The cases where only one constraint binds are described by thepairs ( d m = ¯ m < m l , d s = d ∗ s < s ) and ( m h = d m = d ∗ m < ¯ m, d s = s ). That is, ifmoney holdings of the buyer are enough to purchase all s from the seller, then d ∗ m ishanded over in exchange for all s . Otherwise, the buyer will trade away all her moneyin exchange for d ∗ s . All these possible cases are described by the solution below d m = min (cid:26) ¯ m, s [ ψ + θy H + (1 − θ ) y L ] φ (cid:27) , (18) d s = min (cid:26) s, φ ¯ m [ ψ + (1 − θ ) y H + θy L ]( ψ + y H )( ψ + y L ) (cid:27) . Once we know how the equilibrium in the DFM looks like, we can solve the rest ofthe model proceeding as in the previous section. We use the solution to the DFM inthe optimization of the first market, and then we find the optimal solution for ˆ m andˆ a . We first have to find s so that V ( m, a ) = max s ∈ [0 ,a ] (cid:26) (cid:2) V L ( m, b, s ) + V H ( m, b, s ) (cid:3)(cid:27) . (19)where now V L ( m, b, s ) = λ V L (cid:18) m + Z d m ( ˜ m, s ) dF ( ˜ m ) , b, s − Z d s ( ˜ m, s ) dF ( ˜ m ) (cid:19) + (cid:18) − λ (cid:19) V L ( m, b, s ) , (20)21nd V H ( m, b, s ) = λ V H (cid:18) m − Z d m ( m, ˜ s ) dG (˜ s ) , b, s + Z d s ( m, ˜ s ) dG (˜ s ) (cid:19) + (cid:18) − λ (cid:19) V H ( m, b, s ) . (21)In the previous value functions, F ( ˜ m ) and G (˜ s ) are the distribution of money andsecurity holdings in the economy. Using the solution to the bargaining problem, (19)can be written as V ( m, a ) = max s ∈ [0 ,a ] (cid:8) Λ + φm + ( ψ + R ) a + (cid:0) y L + y H − R (cid:1) s + λφ min n ¯ m, [ ψ + θy H +(1 − θ ) y L ] φ s o − λφ min n m, [ ψ + θy H +(1 − θ ) y L ] φ ¯ s o + λ ( ψ + y H )4 min n ¯ s, φ [ ψ +(1 − θ ) y H + θy L ] φ m o − λ ( ψ + y L )4 min n s, φ [ ψ +(1 − θ ) y H + θy L ] φ ¯ m oo . Remember that at this stage we have to choose s , so we are only interested in the 4th,5th, and 8th terms. Therefore, focusing only on those terms and rearranging them alittle bit we can write the relevant objective function asmax s ∈ [0 ,a ] (cid:26)(cid:18) y L + 12 y H − R (cid:19) s + λ (cid:20) [ ψ + θy H + (1 − θ ) y L ] min (cid:26) φ ¯ mψ + θy H + (1 − θ ) y L , s (cid:27) − ( ψ + y L ) min (cid:26) φ [ ψ + (1 − θ ) y H + θy L ]( ψ + y H )( ψ + y L ) ¯ m, s (cid:27)(cid:21)(cid:27) . or in shortmax s ∈ [0 ,a ] (cid:26)(cid:18) y L + 12 y H − R (cid:19) s + λ h α b min n α b , s o − α b min n α b , s oi(cid:27) , where the definitions of α bi , i = 1 , ..., , are obvious. It is easy to check that α bi ≥ , α b , α b > , and most importantly, α b ≥ α b , α b > α b . Now, let us recall that we arefocusing on the interesting case (1 / y H + y L ) − R ≤
0. The following can easily beverified by visual inspection after straightforward substitution. If we choose s ≤ α b the objective function above is strictly increasing in s as long as ( λθ ) / y H − y L ) > − [(1 / y H + y L ) − R ]. On the other hand, if our choice was s ∈ [ α b , α b ] the objectivewould become ( λ/ φ ¯ m + [(1 / y H + y L ) − R − ( λ/ ψ + y L )] s , which is clearlydecresing in s . Finally, if s ≥ α b the objective would just be decreasing in s with allbut the first term being constant. In a word, the objective function is decreasing forall s beyond α b , which yields the following characterization of the solution s ∗ = , if ( y H + y L ) < R − λθ ( y H − y L )4 , ∈ (cid:2) , min { α b , a } (cid:3) , if ( y H + y L ) = R − λθ ( y H − y L )4 , min { α b , a } , if ( y H + y L ) > R − λθ ( y H − y L )4 . (22)22here α b = φ ¯ mψ + θy H +(1 − θ ) y L . As this solution describes, that the DFM actually opens, s ∗ >
0, depends on whether the expected return on the risky investment plus thevalue generated by re-balancing portfolios with bargaining is high enough. If the ex-pected return of securities is low, ( y H + y L ) ≤ R − λθ ( y H − y L )4 , it is optimal to set s ∗ = 0 and no trade happens in the DFM. All we have to do then is solve for ˆ m andˆ a . In this case people store the safe asset only if its price is constant and behavesaccording to ψ = β ( R + ˆ ψ ), that is ψ = βR/ (1 − β ). Obviously, in this equilibriamoney is not essential and will only circulate as long as φ = β ˆ φ . On the other hand,if the expected return on securities is good enough, ( y H + y L ) ≥ R − λθ ( y H − y L )4 , then s ∗ = min { a, α b } and we would proceed to solve for ˆ m and ˆ a using this choice of s .Surprisingly enough, even though in this case there could be trade in the DFM, theonly equilibrium that can be supported again features the fundamental value of thereal asset, ψ = βR/ (1 − β ).Let us consider first the case of a bad distribution of returns, i.e., (1 / y H + y L ) ≤ R − ( λθ/ y H − y L ). As we said, s ∗ = 0 and we have to choose ˆ m and ˆ a to solve V j ( m, b, s ) = κ ( m, b, s ) + max ˆ m, ˆ a (cid:26) − φ ˆ m − ψ ˆ a + β (cid:20) V L ( ˆ m + µM, ˆ a,
0) + 12 V H ( ˆ m + µM, ˆ a, (cid:21)(cid:27) , for j = H, L , where κ ( m, b, s ) is just a group of terms depending on previous choicesof money, real assets, and securities, but does not depend on the actual choice vari-able. Also, remember that in the next period an amount µM of money is injected(subtracted) in (from) the economy. The details of the rest of the computation areavailable upon request. All that matters is that the optimal decision for next periodreal asset holdings, ˆ a , is given byˆ a ∗ = (cid:26) , if ψ > β ( ˆ ψ + R ) , ∈ R + , if ψ = β ( ˆ ψ + R ) , (23)and the optimal next period money holdings, ˆ m , will only be positive under certainparameterizations, when φ = β ˆ φ . In brief, if the distribution of returns on securitiesis bad, people will optimally choose not to issue any risky assets. As a consequence,there will be no trade in the DFM and money will only circulate under the Friedmanrule. The only feasible path for the asset price is then ψ = βR − β .Finally, let us consider the opposite case of a good enough distribution of returns,(1 / y H + y L ) ≥ R − ( λθ/ y H − y L ). In this scenario, we know that the best decisionon issuing risky assets is s ∗ = min { a, α b } . Using this to solve for ˆ m and ˆ a we can23rite V j ( m, b, s ) = κ ( m, b, s )+ max ˆ m, ˆ a (cid:26) − φ ˆ m − ψ ˆ a + β (cid:20) V L ( ˆ m + µM, ˆ a − min { ˆ a, ˆ α b } , min { ˆ a, ˆ α b } )+ 12 V H ( ˆ m + µM, ˆ a − min { ˆ a, ˆ α b } , min { ˆ a, ˆ α b } ) (cid:21)(cid:27) , for j = H, L , where ˆ α b = ˆ φ ( ˆ¯ m + µM )ˆ ψ + θy H +(1 − θ ) y L . If we are careful with the right expres-sions for d m ( ˆ¯ m + µM, min { a, α b } ), d m ( ˆ m + µM, ˆ¯ s ), d s ( ˆ m + µM, ˆ¯ s ), and d s ( ˆ¯ m + µM, min { a, α b } ), and after some algebra, we can arrive at a more useful objectivefunctionmax ˆ m, ˆ a n − ( φ − β ˆ φ ) ˆ m − [ ψ − β ( ˆ ψ + R )]ˆ a + (cid:20)
12 ( y L + y H ) − R + λ ψ + θy H + (1 − θ ) y L − ( ˆ ψ + y L )] (cid:21) min ( ˆ a, ˆ φ ( ˆ¯ m + µM )ˆ ψ + θy H + (1 − θ ) y L ) + λ ˆ φ ψ + (1 − θ ) y H + θy L ]ˆ ψ + y L min ( ˆ m, ( ˆ ψ + y H )( ˆ ψ + y L )ˆ¯ s [ ˆ ψ + (1 − θ ) y H + θy L ] ˆ φ − µM ) λ ˆ φ ( ˆ m, [ ˆ ψ + θy H + (1 − θ ) y L ]ˆ¯ s ˆ φ − µM )) ≡ max ˆ m {− c m ˆ m + c m min { ˆ m, c m } − c m min { ˆ m, c m }} + max ˆ a {− c a ˆ a + c a min { ˆ a, c a }} , where the definitions of c mi and c ak , i = 1 , ..., k = 1 , ,
3, are obvious. The optimalsolutions for ˆ m and ˆ a are fully described belowˆ m ∗ = , if c m < c m + c m , ∈ [0 , c m ] , if c m = c m + c m ,c m , if c m > c m + c m , and ˆ a ∗ = , if c a < c a ,c a , if c a > c a ∈ [0 , c a ] , if c a = c a , c a = 0 , ∈ [ c a , + ∞ ] , if c a = 0 . (24)However, it is important to notice that with this optimal decisions market clearing inthe DFM fails in the following sense. Within any given period the position of agentson money, m , and the real asset, a , are optimally chosen in the previous period, havingtaken into account the optimal choice for s this period. Thus, within any given periodwe have s = min n a, α b = φmψ + θy H +(1 − θ ) y L o and m = c m = ( ψ + y H )( ψ + y L ) s [ ψ +(1 − θ ) y H + θy L ] φ , which are24learly incompatible since we have shown that ψ + θy H + (1 − θ ) y L > ( ψ + y H )( ψ + y L )[ ψ +(1 − θ ) y H + θy L ] .Therefore, the only possible equilibrium has c a = 0, ψ ∗ = βR − β , ˆ m ∗ = c m , andˆ a ∗ = A = ˆ φ ( ˆ¯ m + µM )[ ψ + (1 − θ ) y H + θy L ]( ˆ ψ + y H )( ˆ ψ + y L ) . Finally, we characterize the range of monetary policies consistent with equilibria underbargaining in the case of a good distribution of return. The optimal choice of moneybalances requires that λ ˆ φ ψ + (1 − θ ) y H + θy L ]( ˆ ψ + y L ) ≥ φ − β ˆ φ + λ ˆ φ . Now, if we use the fact that the only possible path for the price of the asset is ψ = βR/ (1 − β ) and rearrange the expression, we arrive at µ ≤ β − λ − θ )( y H − y L ) βR − β + y L = ¯ µ b , (25)where we have used the fact that in a steady state φ = ˆ φ (1 + µ ). We conclude thatthere are many policies consistent with equilibrium, µ ∈ [ β − , ¯ µ b ]. However, noneof them affect asset prices, since ψψ