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Pairwise Trade, Asset Prices, and Monetary Policy
Ed Nosal
Guillaume Rocheteau
Federal Reserve Bank of Chicago
University of California, Irvine
December 9, 2009
Abstract
We construct a search-theoretic model in which …at money coexists with real assets, and no restrictions
are placed on the use of assets as means of payment. The terms of trade in bilateral matches are
determined by a pairwise Pareto-e¢ cient pricing mechanism. From the view point of agents with a
…nancing need, this mechanism replicates the explicit liquidity constraints found in Kiyotaki and Moore
(2005) or Lagos (2006) that are needed to generate asset pricing facts found in the data. A critical
di¤erence, however, is that we do not impose any such constraints in our environment. We show that …at
money can be valued despite being dominated in its rate of return. Moreover, real assets can generate
di¤erent rates of returns even if agents are risk-neutral. An increase in in‡ation raises assets’ prices,
lowers their returns, and widens the rate-of-return di¤erences between real assets. Finally, there is a
range of in‡ation rates that implement the …rst-best allocation.
J.E.L. Classi…cation: E40, E50
Keywords: Search, money, bilateral trades, in‡ation, asset prices.
The views expressed here are those of the authors and not necessarily those of the Federal Reserve Bank of Chicago, or the
Federal Reserve System. We thank Veronica Guerrieri and Jonathon Chiu for insightful discussions, participants at the 2008
meeting of the Society of Economic Dynamics and 2009 meetings of the Canadian Macro Study Group, and seminar participants
at the Bank of Canada, the University of California at Irvine, and the Federal Reserve Banks of Chicago, New York, and St.
Louis.
1
1
Introduction
What are the determinants of an asset’s liquidity? Twenty years ago, Kiyotaki and Wright (1989)— KW
hereafter— provided an answer to this question in the context of a monetary model with bilateral trades.
They found that the moneyness of an asset depends on its physical properties (e.g., storage costs), the
fundamentals (e.g., the pattern of specialization), and on conventions (e.g., the coordination on one among
multiple equilibria). These insights were derived under extreme portfolio restrictions— agents cannot hold
more than one unit of an asset— and stark assumptions, such as indivisible assets and goods. More recent
developments in the search-theoretic approach to monetary economics that allow for unrestricted portfolios
and divisible assets— e.g., Shi (1997) and Lagos and Wright (2005)— have led to a renewed interest for the
initial question that prompted this literature: What makes money money?
According to the original insights of KW, the liquidity di¤erences among assets can stem from di¤erences
in terms of assets’physical properties. Lester, Postlewaite, and Wright (2007) focus on the recognizability
property of assets, and show that …at money is a superior means of payment because it is harder to counterfeit
and easier to authenticate than other assets.1 Fundamentals can also help explain the moneyness of assets.
Rocheteau (2008) shows that in the presence of informational asymmetries the liquidity of assets depends
on the properties of their dividend processes.
In this paper, we take yet a di¤erent approach. We pursue the idea that when agents interact in small
groups, the possible set of equilibrium outcomes can be a non-degenerate core, and that agents use bargaining
conventions to determine prices. The non-degeneracy gives the modeler some freedom, in terms of choosing
the conventions that agents use, for determining the actual terms of trade that will prevail. Our approach
is inspired by the work of Zhu and Wallace (2007). We are not alone in believing that such an approach
can yield useful insights. Kocherlakota (2005, pp. 726-727) points out that “we are likely to continue to
learn a lot about monetary economics, and economics more generally, by studying the implications of ...
trade tak[ing] place in small groups.” So, whereas in KW, network-like externalities can generate multiple
equilibria with di¤erent payment arrangements, and a convention will dictate which equilibrium is played,2
1 The recognizability of money is an old theme in monetary theory. Recent formalizations include Williamson and Wright
(1994) and Banarjee and Maskin (1996).
2 Kiyotaki and Wright (1989) consider two versions of their model, models A and B, that di¤er in terms of the pattern
of specialization. While the equilibrium of model A is unique, model B delivers multiple equilibrium for a nonempty set of
parameter values. For other examples of search-theoretic models of payments with multiple equilibria, see, e.g., Matsuyama,
Kiyotaki and Matsui (1993) and Ayagari, Wallace and Wright (1996).
2
in our environment, the possibility of multiple equilibria presents itself in a very natural way, thanks to a
large set of Pareto-e¢ cient allocations in core, and the convention determines which allocation is chosen.
To develop our argument, we adopt the search-theoretic model with divisible money of Lagos and Wright
(2005), in which a lack of double coincidence of wants and the absence of a record-keeping technology in
decentralized markets generate an explicit need for a tangible medium of exchange. Besides money, we
introduce real assets, “Lucas’trees,” that yield a ‡ow of real dividends.3 We adhere to the Wallace (1996)
dictum and place no restrictions on the use of assets as means of payment. Agents are free to use any quantity
of their real assets and money holdings for transactions purposes. A crucial aspect of the environment is that
some trades take place in bilateral meetings. Because the Pareto-frontier of the bargaining set in a bilateral
match is non-degenerate, a large set of pairwise Pareto-e¢ cient allocations are potentially implementable.
We will exploit this feature of the model to select a mechanism that generates outcomes that are qualitatively
consistent with some observations in the data.4
While most of the recent search literature assumes an axiomatic bargaining solution to determine terms
of trades in bilateral meetings, we follow Zhu and Wallace (2007) and only retain the axiom of Pareto
e¢ ciency.5 The axiom of Pareto-e¢ ciency, however, is not enough to get a unique outcome. Instead of
imposing other axioms, such as independence of irrelevant alternatives or monotonicity, we take a di¤erent
route and let some qualitative aspects of the data guide our selection of a trading mechanism. We demand
that the trading mechanism accounts for the rate-of-return-dominance puzzle, according to which …at money
can coexist with interest-bearing assets, and various other asset pricing anomalies. We also want the model
to generate the observed negative relationship between in‡ation and assets’returns.
Kiyotaki and Moore (2005) are able to generate the asset pricing anomalies and the e¤ects of monetary
3 The …rst attempt to introduce capital goods into the Lagos-Wright model was by Aruoba and Wright (2003) but capital
goods were not allowed to be used as means of payment in bilateral matches. Lagos and Rocheteau (2008) relax the restriction
on the use of capital as a competing means of payment and show that …at money and capital can coexist provided that there is
a shortage of capital to be used as means of payment. Geromichalos, Licari and Suarez-Lledo (2007) follow a similar approach
but assume that capital is in …xed supply. Lagos (2006) calibrates the model where money is replaced by risk-free bonds and
capital is a risky asset, and shows that it can account for the risk-free rate and equity premium puzzles under a mild restriction
on the use of capital as means of payments.
4 Interestingly, the same feature of the labor search model is used by Hall (2005) to explain wage rigidity. See the discussion
in Kocherlakota (2005). In an earlier version of their paper Zhu and Wallace (2007) called their pricing mechanism “cash-inadvance with a twist” because from the view point of the buyer, it is as if he faces a cash-in-advance constraint. To mirror this
terminology, our pricing protocol could be called “Kiyotaki-Moore with a twist.”
5 Shi (1997) and Lagos and Wright (2005) assume that terms of trade in bilateral matches are determined by the generalized
Nash solution. Aruoba, Waller, and Rocheteau (2007) investigate the robustness of the results to alternative bargaining
solutions. Other mechanisms have been studied such as auctions and competitive posting. For a mechanism design approach
where the mechanism is chosen by normative considerations, see Hu, Kennan and Wallace (2008).
3
policy we want to explain.6 They do so, however, by imposing liquidity constraints on agents, where the
constraint takes the form of a restriction on the fraction of real asset holdings that agents can use to …nance
investment opportunities. We design a family of trading mechanisms that replicate the same asset pricing
patterns as the ones in the economy with liquidity constraints of Kiyotaki and Moore (2005) but we do not
impose any exogenous liquidity constraints in our environment. Unless a Pareto-e¢ cient trade is achieved,
our mechanism does not leave any asset unused as a means of payment. The family of trading mechanisms
we consider is parametrized by a single parameter— just like the generalized Nash solution— and it admits as
particular cases the pricing mechanisms considered in Geromichalos, Licari and Suarez-Lledo (2007), Lagos
(2006), Lagos and Rocheteau (2008), Zhu and Wallace (2007).
The main insights of our theory are as follows. First, …at money can be held and valued despite being
dominated in its rate of return by competing assets. In contrast to earlier works, we do not need to impose
trading restrictions to generate a rate of return dominance pattern: a plain-vanilla Lagos-Wright model
can account for this puzzle. Second, the model is capable of generating a liquidity-based structure of assets’
yields, where real assets can exhibit di¤erent rates of returns, even if they share the same risk characteristics,
or if agents are risk-neutral.7 Those rate of return di¤erences, which are all anomalies from the standpoint of
consumption-based asset pricing theories, re‡ect di¤erences in the liquidity of the assets; some assets can be
liquidated at better terms of trade than others. Third, our model also has implications for the transmission
mechanism of monetary policy to asset prices. Under our pricing protocol, real assets have a liquidity value—
i.e., a buyer is able to capture some surplus in a bilateral meeting when paying with real assets— and, hence,
an increase in in‡ation will raise their prices and lower their returns. Moreover, in‡ation widens the rate of
return di¤erences between real assets. Finally, from a normative standpoint, there is a range of in‡ation rates
that implement the …rst-best allocation, including the Friedman rule. As a consequence, a small in‡ation
above the Friedman rule does not impose a welfare cost on society.
6 More
generally, our paper is related to the recent literature in macroeconomics that takes into account the transaction role
of assets in order to explain asset pricing anomalies and the e¤ects of monetary policy on assets’returns. For instance, Bansal
and Coleman (1996) explain the risk-free rate and the equity premium puzzles in a pure exchange economy in which there
are di¤erent transactions costs associated with the use of di¤erent means of payment, e.g., …at money, government bonds and
credit.
7 Wallace (2000) provides a theory of the liquidity structure of asset yields based on the indivisibility of assets. Weill (2008)
explains di¤erences in liquidity across seemingly identical assets by the presence of thick market externalities in decentralized
asset markets.
4
2
The environment
Time is discrete and continues forever. The economy is populated with a [0; 1] continuum of in…nitely-lived
agents. As in Lagos and Wright (2005), each period is divided into two subperiods, called AM and PM.
In the AM, trade takes place in decentralized markets, where agents are bilaterally matched in a random
fashion. In the PM, trade takes place in competitive markets.
In the AM decentralized market, agents produce and consume perishable goods that come in di¤erent
varieties. The probability that an agent is matched with someone who produces a good he wishes to consume
is
1=2. Symmetrically, the probability that an agent meets someone who consumes the good he produces
is
1=2. For convenience, and without loss of generality, we rule out double-coincidence-of-wants meetings.
In the PM subperiod, all agents are able to consume and produce a perishable (general) good.8
An agent’s utility function is
E
"
1
X
t
u
ytb
c (yts )
+ xt
t=0
ht
#
where y b is consumption and y s is production of the AM good, x is consumption of the general good, h is
hours of work to produce the general good, and
We assume that u(y)
= (1 + r)
1
2 (0; 1) is the discount factor across periods.
c(y) is continuously di¤erentiable, strictly increasing and concave. In addition,
c(0) = u(0) = 0, u0 (0) = +1, u0 (+1) = 0, and there exists a y < +1 such that u0 (y ) = c0 (y ). The
technology to produce general good is linear and one-to-one in hours, i.e., h hours of work produce h units
of the general good in the PM.9
Agents are unable to commit, and their trading histories are private information. This implies that
credit arrangements are infeasible. The infeasibility of credit, in conjunction with the specialization of
agents’ consumption and production in the AM decentralized markets, generates a role for a medium of
exchange. There are two storable and perfectly divisible assets in the economy, and both can serve as media
of exchange without restrictions. There is a real asset that is in …xed supply, A > 0. In each PM subperiod,
one unit of the real asset generates a dividend equal to
8 We
> 0 units of the general good. There also exists an
could assume that the same goods which are traded in the AM decentralized market are also traded in the PM
competitive market. However, the specialization in terms of preferences and technologies is irrelevant in a complete information,
competitive environment.
9 Following Lagos and Wright (2005), we could adopt a more general utility function in the PM, U (x)
h with U 00 < 0.
Our results would not be a¤ected provided that the nonnegativity constraint for the number of hours, h 0, is not binding in
equilibrium.
5
intrinsically useless asset called …at money. The quantity of money at the beginning of period t is denoted
Mt . The money supply grows at the gross rate
Mt+1
Mt ,
> , where
via lump-sum transfers or taxes in
the PM subperiod.10 In the AM subperiod, producers and consumers in a bilateral match can exchange the
assets for one another or for the consumption good. In the PM subperiod, assets are traded with the general
good in competitive markets.
The asset prices at date t are measured in terms of the general good in the date t PM subperiod. The
price of money is denoted by
t
and the price of the real asset is denoted by qt . In what follows, we will
focus our attention on stationary equilibria, where
3
t Mt
and qt are constant.
Pricing
In this section, we describe the determination of the terms of trade in bilateral meetings in the AM decentralized market. Before we do this, however, it will be useful to show some properties of an agent’s value
function in the PM subperiod, W , since it tells us how agents will value the assets they give up or receive in
the AM decentralized market. The value function of an agent entering the PM competitive markets holding
a portfolio of a units of real asset and z units of real balances is,
W (a; z) = max
fx
0 0
x;h;a ;z
s.t.
where T
t (Mt+1
h + V (a0 ; z 0 )g
(1)
z 0 + qa0 + x = z + h + a(q + ) + T;
(2)
Mt ), which is measured in terms of the general good, is the lump-sum transfer
associated with money injection. At the start of the PM subperiod, each unit of the real asset can be bought
or sold in a competitive market at the price q, and it generates
his net consumption, x
units of the general good. The agent chooses
h, and the portfolio, (a0 ; z 0 ), that he will bring into the subsequent decentralized
market. Each unit of real balances acquired in the PM subperiod of date t will turn into
t+1
t
=
1
units
of real balances in date t + 1. Hence, to have z 0 units of real balances next period, an agent must acquire
z 0 units in the current period.
1 0 If
< 1 the government can force agents to pay taxes in the PM. In a related model, Andolfatto (2007) considers the
case where the government has limited coercion power— it cannot con…scate output and cannot force agents to work— and the
payment of lump-sum taxes is voluntary: agents can avoid paying taxes by not accumulating money balances. He shows that
if agents are su¢ ciently impatient, then the Friedman rule is not incentive-feasible, i.e., there is an induced lower bound on
de‡ation.
6
Substituting x
h from (2) into (1) gives
W (a; z) = z + a(q + ) + T + max
f
0 0
a ;z
z0
qa0 + V (a0 ; z 0 )g :
(3)
From (3), the PM value function is linear in the agent’s wealth: this property will prove especially convenient
in terms of simplifying the pricing problem in the AM decentralized market. Note also that the choice of the
agent’s new portfolio, (a0 ; z 0 ), is independent of the portfolio that he brought into the PM subperiod, (a; z),
as a consequence of quasi-linear preferences.
Consider now a match in the AM decentralized market between a buyer holding portfolio (a; z) and a
seller holding portfolio (as ; z s ). The terms of trade are given by the output y
and the transfer of assets (
of real balances and
a
m;
a)
2 [ z s ; z]
0 produced by the seller
[ as ; a] from the buyer to the seller, where
m
is the transfer
is the transfer of real assets. (If the transfer is negative, then the seller is delivering
assets to the buyer.) The procedure that determines the terms of trade in the AM decentralized market
generalizes the one suggested by Zhu and Wallace (2007). The procedure has two steps. The …rst step
^ b , which is equal to what he would obtain in a
generates a payo¤ or surplus for the buyer, denoted as U
bargaining game if he had all the bargaining power, but was facing liquidity constraints. Speci…cally, in this
“virtual game” it is assumed that the buyer can at most transfer a fraction
a
a. Zhu and Wallace (2007) assume that
and Suarez-Lledo (2007) assume
than he holds, i.e.,
m
of his real asset holdings, i.e.,
= 0; Lagos and Rocheteau (2008) and Geromichalos, Licari
= 1. In terms of real balance transfers, the buyer cannot transfer more
z.11 The liquidity constraint on real asset holdings in the virtual game is chosen
purposely to be reminiscent of the one used in Kiyotaki and Moore (2005), where individuals can only use a
fraction of their capital goods to …nance investment opportunities.12 From the buyer’s standpoint, it is as if
he was trading in the Kiyotaki-Moore economy. The liquidity constraint is also reminiscent of the constraint
in Lagos (2006), where
= 0 in a fraction of the matches and
= 1 in the remaining matches. Note that the
output and wealth transfers that the …rst step generates are virtual in the sense that they are simply used to
^ b . The actual output and wealth transfer are determined in the second step
determine the buyer’s surplus, U
so as to generate a pairwise Pareto-e¢ cient trade. The actual trade maximizes the seller’s surplus subject
^ b , and the only restrictions placed
to the constraint that the buyer receives a surplus at least equal to U
1 1 We
do not constraint the transfer of asset holdings of the seller, but this is with no loss in generality.
and Moore (2005) consider an economy with two assets, capital and land. Land is in …xed supply while capital is
accumulated. Both assets are inputs in the production of the …nal good. Individuals receive random opportunities to invest.
In order to …nance investment, they can use all their land— land is “completely liquid”— but only a fraction of their capital
holdings. So land is analogous to money in our formulation, while capital is similar to our real asset.
1 2 Kiyotaki
7
on the transfer of either asset is that an agent cannot transfer more than he has, i.e.,
zs
m
as
a
a and
z.
^ b , solves the following
The …rst step of our pricing protocol, which determines the buyer’s surplus U
problem,
^ b (a; z) = max [u(y) + W (a
U
y;
s.t.
a; z
m; a
c(y) + W (as +
a; z
2 [ z s ; z];
a
m
s
+
z)
W (a; z)]
W (as ; z s )
m)
2 [ as ; a]
The buyer in this virtual game maximizes his surplus, subject to the participation constraint of the seller
and the constraints on the transfer of his asset holdings: while the buyer can transfer all his money balances,
of his real asset.13 Using the linearity of W (a; z), the above problem can
he can only hand over a fraction
be rewritten as
^ b (a; z) = max [u(y)
U
y;
s.t.
zs
as (q + )
m; a
c(y) +
m
+
a (q
a (q
+ )
m
+
a (q
m
+ )
+ )]
(4)
0
(5)
z + a(q + )
From this formulation, note that what matters is the total value of the transfer of assets,
(6)
m + a (q +
), and
not its composition in terms of money and real asset. Moreover, from the seller’s participation constraint,
y
0 requires
m
+
a (q +
)
0. Thus, the constraint that the seller cannot transfer more than his wealth
is irrelevant, and the buyer’s payo¤ is independent of (as ; z s ). From this, it is easy to see that the buyer’s
payo¤ is a function of only his liquid wealth, z + a (q + ), the wealth he can use in the virtual game to
^ b (a; z).
maximize his payo¤. We now describe some of the properties of buyer’s surplus function, U
Lemma 1 The buyer’s payo¤ is uniquely determined and satis…es,
^ b (a; z) =
U
u(y ) c(y ) if z + a(q + ) c(y )
u c 1 [z + a(q + )] z
a(q + ) otherwise
(7)
^ b (a; z) is strictly increasing and strictly concave with respect to each of its
If z + a(q + ) < c(y ), then U
^ b (a; z) is jointly concave (but not strictly) with respect to (a; z).
arguments. Moreover, U
1 3 In principle, U
^ b should also have z s and as as arguments. Here we anticipate the result that the terms of trade in this
virtual game are independent of the seller’s portfolio.
8
^ b = u(y )
Proof. The solution to (4)-(6) is y = y and U
y=c
1
[z + a(q + )] and (
m;
a)
c(y ) i¤ z + a(q + )
c(y ); otherwise,
= (z; a).
If z + a(q + ) < c(y ), then
where ! = c
1
^ab
U
=
^zb
U
=
^b
@U
u0 (!)
= (q + ) 0
@a
c (!)
b
0
^
@U
u (!)
= 0
1 > 0;
@z
c (!)
1 >0
(8)
(9)
^ b (a; z) is increasing with respect to each of its arguments. As well,
[z + a(q + )]; U
^b
U
zz
^b
U
za
b
^aa
U
=
u00 (!)c0 (!)
<0
3
=
=
u0 (!)c00 (!)
(q + )
[c0 (!)]
"
u00 (!)c0 (!)
"
2
[ (q + )]
u0 (!)c00 (!)
3
[c0 (!)]
u00 (!)c0 (!)
#
<0
u0 (!)c00 (!)
3
[c0 (!)]
#
< 0;
b ^b
^ b (a; z) is strictly concave with respect to each of its arguments, and U
^aa
Uzz
U
b
^za
U
2
^ b (a; z)
= 0. Hence, U
is jointly concave, but not strictly jointly concave.
^ s (a; z), and the actual terms
The second step of the pricing protocol determines the seller’s surplus, U
of trade, (y;
m;
a ),
as functions of the buyer’s portfolio in the match, (a; z). By construction, the terms
of trade are chosen so that the allocation is pairwise Pareto-e¢ cient. The allocation solves the following
problem,
^ s (a; z) = max [ c(y) +
U
y;
s.t. u(y)
zs
m; a
a (q
m
+ )
as
z;
m
m
+
a (q
+ )]
(10)
^ b (a; z)
U
(11)
a
(12)
a
Notice that in this problem, the use of the real asset as means of payment is not restricted. Moreover,
^ s (a; z)
U
0 since the allocation determined in the …rst step of the pricing protocol is still feasible in the
second step. It is straightforward to characterize the solution to the seller’s problem.
Lemma 2 If z + a(q + )
^ b (a; z), then the terms of trade in bilateral meetings satisfy
U
u(y )
y
m
+
a (q
+ )
= y
= u(y )
9
(13)
^ b (a; z);
U
(14)
otherwise,
y
(
a;
m)
= u
=
1
h
i
^ b (a; z)
z + a(q + ) + U
(15)
(a; z):
(16)
The seller’s payo¤ and output are uniquely determined. The composition of the payment between money
and the real asset is unique if the output is strictly less than the e¢ cient level, y . If z + a(q + ) >
u(y )
^ b (a; z), then there are a continuum of transfers (
U
a;
m)
that achieve (14).
^ b (a; z). Here, output, y, depends on the buyer’s portfolio.
Consider the case where z +a(q + ) < u(y ) U
From (15)-(16) one can compute the quantity of output a buyer can acquire with an additional unit of wealth.
If an agent accumulates an additional unit of real balances, his consumption in the AM increases by
@y
u0 (!)
= 0
;
@z
c (!)u0 (y)
where ! = c
q+
1
[z + a(q + )]. If the agent accumulates an additional unit of the real asset, which promises
units of output in the next PM, then
1+
@y
(q + ) 1
=
@a
If
h
u0 (!)
c0 (!)
u0 (y)
i
1
:
< 1, then a claim on one unit of PM output buys more output in the AM if it takes the form of …at
money instead of the real asset. It can also be checked from Lemma 1 that
^ b (a; z)
^ b (a; z)
@U
@U
= (q + )
:
@a
@z
One unit of the real asset generates an increase of the buyer’s surplus that is
q+
units of real balances. In this sense,
times the one associated with
is a measure of the liquidity of the real asset, i.e., its ability to
buy AM output at favorable terms of trade.
In Figure 1 the determination of the terms of trade is illustrated. The surpluses of the buyer and the seller
^ b and U
^ s , respectively. There are two Pareto frontiers: the lower (dashed) frontier corresponds
are denoted U
to the pair of utility levels in the …rst step of the pricing protocol, where the buyer cannot spend more than
a fraction
of his real asset, and the upper frontier corresponds to the pair of utility levels in the second
step of the procedure, where payments are unconstrained.14 In Figure 1, the upper frontier is constructed
1 4 If
z + a (q + ) > c(y ) then both Pareto frontiers have a part in common with the e¢ cient line U b + U s = u(y )
^ b = u(y ) c(y ) and U
^ s = 0.
In that case, U
10
c(y ).
Us
Constrained payments (τa<θa)
Unconstrained payments
U b + U s = u ( y*) − c( y*)
Uˆs
Ub
Uˆb
Figure 1: Pricing mechanism
in the case where the …rst-best level of output is incentive-feasible, i.e., z + a (q + ) > c (y ). Along the
linear portion of the upper frontier— moving in a north-west direction— output remains at y but wealth
transferred to the seller increases. The linear portion ends when the wealth transference to the seller equals
z + a (q + ); beyond that point on the upper frontier, the seller’s surplus increases by having him receive all
of the buyer’s wealth in exchange for producing successively smaller amounts of the consumption good. The
^ b < u (y )
lower frontier is constructed under the assumption that z + a (q + ) < c (y ), i.e., U
c (y ). In
Figure 1, in the …rst step of the pricing protocol, the buyer’s surplus is the one he obtains in a virtual game
where he o¤ers his entire “liquid”wealth in exchange for the maximum level of output the seller is willing to
^ b; U
^ s ) lies on the upper
produce. In the second step, the seller’s surplus is chosen so that the agreement (U
frontier in Figure 1 so that the trade is Pareto e¢ cient. Given the con…guration of Figure 1, the seller will
receive all of the buyer’s wealth and will produce a level of output y < y that provides the buyer with a
^ b.
surplus equal to U
11
4
Equilibrium
We incorporate the pricing mechanism, described in Section 3, in our general equilibrium model. Let y (a; z),
a
(a; z) and
m
(a; z), represent the output and transfer outcomes from the pricing mechanism, when the
buyer in the match has portfolio (a; z). The value to the agent of holding portfolio (a; z) at the beginning of
the AM subperiod, V (a; z), is given by
V (a; z)
=
fu[y(a; z)] + W [a
a (a; z); z
+ E f c[y(~
a; z~)] + W [a +
+(1
With probability
[
a (a; z);
m (a; z)]
m (a; z)]g
a; z~); z
a (~
+
a; z~)]g
m (~
2 )W (a; z):
, the agent is the buyer in a match.
He consumes y(a; z) and delivers the assets
to the seller.15 As established in Lemmas 1 and 2, the terms of trade (y;
depend on the portfolio of the buyer in the match. With probability
He produces y and receives (
a;
m)
(17)
from the buyer where (y;
a;
m)
a;
m)
only
the agent is the seller in the match.
is a function of the buyer’s portfolio
(~
a; z~). The expectation is taken with respect to (~
a; z~), since the distribution of asset holdings might be nondegenerate, assuming that the buyer partner is chosen at random from the whole population of potential
buyers. Finally, with probability 1
2
the agent is neither a buyer nor a seller. Using the linearity of
W (a; z) and the expressions for the buyer’s and the seller’s surpluses, (17) can be rewritten more compactly
as
^ b (a; z) + EU
^ s (~
V (a; z) = U
a; z~) + W (a; z):
(18)
If the agent was living in an economy with exogenous liquidity constraints where he can only transfer a
fraction of his real asset holdings, as in Kiyotaki and Moore (2005) or Lagos (2006), then the main di¤erence
^ s (~
would be that EU
a; z~) = 0 (assuming that the buyer makes take-it-or-leave-it-o¤ers). Consequently, the
buyer’s choice of portfolios in both economies would be similar.
Substituting V (a; z), as given by (18), into (3), and simplifying, an agent’s portfolio solves
(a; z) 2 arg max
a;z
1 5 Recall
m
+
a(
n
z
qa +
from Lemma 2 that even though the terms of trade (
+ q), is unique.
h
a;
12
io
^ b (a; z) + z + a(q + ) :
U
m)
(19)
may not be uniquely determined, the transfer of wealth,
The portfolio is chosen so as to maximize the expected discounted utility of the agent if he happens to be
a buyer in the next AM market minus the e¤ective cost of the portfolio. The cost of holding an asset is its
purchase price minus its discounted resale price and its dividend. Rearranging (19), it simpli…es further to
(a; z) 2 arg max
a;z
where i
where q
n
iz
o
^ b (a; z) ;
q )+ U
ar (q
represents the cost of holding real balances, r (q
r
(20)
q ) is the cost of holding the real asset
is the discounted sum of the real asset’s dividends. The …rst-order (necessary and su¢ cient)
conditions for this (concave) problem are:
r (q
+
i+
u0 (!)
c0 (!)
1
q ) + (q + )
u0 (!)
c0 (!)
1
where [x]+ = max(x; 0) and ! = c
1
0;
“ = ” if z > 0
(21)
0;
“ = ” if a > 0
(22)
+
[z + a(q + )]. The term
h
u0 (!)
c0 (!)
1
i+
in (21) represents the liquidity
return of real balances, i.e., the increase in the buyer’s surplus from holding an additional unit of money.
From (22), the liquidity return of
1
q+
units of the real asset is
times the liquidity return of real balances.
Finally, the asset price is determined by the market clearing condition
Z
a(j)dj = A;
(23)
[0;1]
where a(j) is the asset choice of agent j 2 [0; 1].
De…nition 1 An equilibrium is a list f[a(j); z(j)]j2[0;1] ; [y(a; z); a (a; z);
R
(16), (20) and (23). The equilibrium is monetary if [0;1] z(j)dj > 0.
m (a; z)]; qg
that satis…es (13)-
Consider …rst nonmonetary equilibria, where the real asset is the only means of payment in the AM
market. In this case, z(j) = 0 for all j.
Proposition 1 There is a nonmonetary equilibrium, and it is such that q 2 [q ; +1).
(i) If
(ii) If
= 0, then q = q .
> 0, then q is the unique solution to
q +
q+
If A(q + )
+
r
u0 c
c0 c
+
1
[ A(q + )]
1 [ A(q + )]
c(y ), then q = q ; otherwise q > q .
13
1
= 1:
(24)
Proof. De…ne the demand correspondence for the real asset as
(Z
)
n
o
d
b
^ (a; 0)
A (q) =
a(j)dj : a(j) 2 arg max
ar (q q ) + U
:
[0;1]
The market clearing condition, (23), can then be re-expressed as A 2 Ad (q). First, suppose
^ b (a; 0) = 0 for all a
(7), U
= 0. From
0. Then, Ad (q) = f0g for all q > q and Ad (q ) = [0; +1). Consequently, the
unique solution to A 2 Ad (q) is q = q .
Next, suppose
> 0. In order to characterize Ad (q) we distinguish three cases:
1. If q > q then Ad (q) = fag where a <
c(y )
(q+ )
is the unique solution to
r (q
^ab (a; 0) :
q )= U
(25)
^ b (a; 0) is strictly concave with respect to its …rst argument
To see this, recall from Lemma 1 that U
h
i
c(y )
)
^b
^b
^b
over the domain 0; c(y
(q+ ) , Ua (0; 0) = 1 and Ua (a; 0) = 0 for all a
(q+ ) . Substituting Ua by its
expression given by (8) into (25) and rearranging, one obtains
q +
q+
+
r
u0 c
c0 c
1
[ a(q + )]
a(q + )]
1[
1 = 1:
(26)
Note that the left-hand side is strictly decreasing in both q and a; hence, a is strictly decreasing in q.
So, for all q > q , Ad (q) is single-valued and strictly decreasing. Moreover, as q ! q , a !
as q ! 1, a ! 0.
2. If q = q , then Ad (q ) = argmaxa
0
n
o
^ab (a; 0) = [
U
c(y )
(q + )
and
c(y )
(q + ) ; +1).
3. If q < q , then the agent’s problem has no solution.
In summary, Ad (q) is upper hemi-continuous over [q ; 1) and its range is [0; 1). Hence, a solution
A 2 Ad (q) exists. Furthermore, any selection from Ad (q) is strictly decreasing in q 2 [q ; 1), so there is
a unique q such that A 2 Ad (q). If A
c(y )
(q + ) ,
then A 2 Ad (q) implies q = q . If A <
c(y )
(q + ) ,
then
A 2 Ad (q) implies that q solves (26) with a = A, i.e., (24).
If
= 0, as in Zhu and Wallace (2007), then the real asset is fully illiquid in the sense that holding the asset
does not allow the buyer to extract a surplus from his trade in the decentralized AM. The allocation coincides
with the one where the seller makes a take-it-or-leave-it o¤er, and the asset is priced at its fundamental value,
q=q .
14
If
> 0, then the buyer can obtain a positive surplus from holding the asset in the decentralized market.
If the intrinsic value of the stock of the real asset, A (q + ), is su¢ ciently high, and if it is not too illiquid,
i.e.,
is not too low, then the buyer can extract the entire surplus of the match. An additional unit of
asset does not a¤ect the buyer’s trade surplus in the AM so that the asset has no liquidity value and its
price corresponds to its fundamental price, q = q = =r. The distribution of asset holdings is not uniquely
determined, but this indeterminacy is payo¤ irrelevant since the output traded in all matches in the AM
is y . In contrast, if the intrinsic value of the asset is low, or if the asset is very illiquid ( is low but
positive), then the price of the asset raises above its fundamental value because it is useful to the buyer to
increase his surplus in the AM. In this case, the equilibrium and the distribution of asset holdings— which
is degenerate— are unique.
It is rather interesting to note from (24) that the allocation can be socially e¢ cient, y = y , even when
q > q . It can be the case that the value of the buyer’s liquid portfolio, a (q + ), is insu¢ cient to purchase
y in the decentralized AM subperiod— i.e., a (q + ) < c (y )— but the total value of the buyer’s portfolio
can be su¢ cient to support an output of y , i.e., a(q + )
u(y )
^ b (a; z).
U
Let’s now turn to monetary equilibria.
Proposition 2 There exists a monetary equilibrium i¤
A<
(r i )
`(i);
(1 + r)
(27)
where `(i) is unique and implicitly de…ned by
1+
i
=
u0 c
c0 c
1
(`)
1 (`)
:
(28)
In a monetary equilibrium, the asset price is uniquely determined by
q=
(1 + i )
r i
q :
(29)
^ b (a; z) where ` = z + a(q + ). (Notice from
^ b (`) = U
Proof. With a slight abuse of notation, de…ne U
Lemma 1 that (a; z) matters for the buyer’s payo¤ only through z + a(q + ).) Then, the agent’s portfolio
problem (20) can be re-expressed as
(a; `) 2 arg max
a;`
n
i`
a [(r
o
^ b (`) s.t. a(q + )
(1 + i )] + U
i )q
15
`:
(30)
De…ne the demand correspondence for the real asset as
(Z
)
d
A (q) =
a(j)dj : 9` s.t. [a(j); `] is solution to (30) :
[0;1]
The market-clearing condition (23) requires A 2 Ad (q) for some q. In order to characterize Ad (q) we
distinguish three cases:
1. If (r
i )q >
(1 + i ) then, from (30), Ad (q) = f0g.
2. If (r
i )q <
(1 + i ), then constraint a(q + )
` must bind and z = 0, i.e., the equilibrium is non-
monetary. Hence, from the proof of Proposition 1, Ad (q ) = [
c(y )
(q + ) ; 1)
and, for all q 2 q ;
(1+i )
(r i )
,
Ad (q) = fag where a solves
r (q
3. If (r
i )q =
q )=
u0 c
c0 c
(q + )
(1 + i ), then Ad (q) = [0;
`
(q+ ) ]
1
1
[ a(q + )]
[ a(q + )]
1 :
(31)
where, from (30), ` solves (28).
In conjunction with (28), it can be checked that the solution to (31) for a approaches
`
(q+ )
as q %
(1+i )
(r i ) .
Hence, Ad (q) is upper hemi-continuous on [q ; +1) with range (0; +1). Moreover, any selection from Ad (q)
is strictly decreasing. Therefore, there is a unique q 2 [q ; +1), such that A 2 Ad (q). Moreover, if r > i
then q 2 [q ;
(1+i )
].
r i
See Figure 2.
A monetary equilibrium exists if z(j) > 0 for a positive measure of agents. From the discussion above,
`(i)
(q+ ) ,
a monetary exists if and only if r > i and A <
where `(i) is de…ned by (28) and q =
(1+i )
,
r i
i.e.,
equation (29). This gives (27).
Since the right-hand side of (27) is decreasing in i, by taking the limit as i approaches 0 we obtain the
following necessary condition for the existence of a monetary equilibrium: A <
c(y )
(q + ) .
If this condition
holds, (27) can be restated as i < i0 , where i0 is the unique solution to16
(r
1 6 In
a monetary equilibrium, ` (i) > A
(1+r)
r i
i0 )
`(i0 ) = A:
(1 + r)
(33)
or, from (28),
1+
i
<
u0
c
1
A
(1+r)
r i
c0
c
1
A
(1+r)
r i
:
(32)
Since the left-hand side is increasing and continuous in i and the right-hand side is decreasing and continuous in i, there exists
a unique i0 such that (32) holds at equality, which implies that for any i < i0 , condition (32) holds and, hence, a monetary
equilibrium exists.
16
A d (q)
Non-monetary
equilibrium
c( y* )
θ (q* + κ )
A'
r −iθ
θκ(1+r)
Monetary
equilibrium
l(i)
A
q*
κ(1 + iθ)
r − iθ
Figure 2: Asset demand correspondence
Thus, a monetary equilibrium exists whenever the asset price in the nonmonetary equilibrium is greater than
its fundamental value, provided that the in‡ation rate is not too large. If money is valued, then the asset
price is still greater than its fundamental value if i > 0, the real asset is not fully illiquid and the cost of
holding real balances is positive.
The determination of the equilibrium is characterized in Figure 2. The price q is at the intersection of the
constant supply, A, and the downward-sloping demand, Ad (q). It is uniquely determined. The equilibrium
is monetary whenever the supply intersects the demand in its vertical portion.
A rather important result from above is that, if A <
c(y )
(q + ) ,
then there always exists a i > 0, such
that a monetary equilibrium exists. Diagrammatically speaking, it is easy to see this result. Suppose that
A<
c(y )
(q + ) ,
but A >
r i
(1+r) ` (i)
for a given value of i; in …gure 2, let A = A0 . This implies that at the
current in‡ation rate, the equilibrium is non-monetary, as A0 intersects the strictly downward portion of
Ad (q). However, since
(1+ i)
r i
is increasing in i, by decreasing i from its current level, the vertical portion
of Ad (q) will “move”to the left. If i is decreased su¢ ciently, A0 will intersect the vertical portion of Ad (q).
Since in a monetary equilibrium the “liquid wealth”of the buyer, `(i), is uniquely determined, the buyer’s
payo¤ in the decentralized market is unique. However, there are in…nitely many ways to to combine a and
17
z to obtain a given `(i). As a consequence, the terms of trade (y;
a;
z)
and a seller’s payo¤ need not
be unique. So, even though the asset price is unique, the real allocation may be indeterminate. In the
following we restrict our attention to symmetric steady-state equilibria. All agents hold A of the real asset
and z(i) = `(i)
5
A(q + ) real balances.
Asset prices and monetary policy
Let us turn to asset pricing considerations and the implications for monetary policy. We have established in
Proposition 2 that in any monetary equilibrium the asset price satis…es q =
(1+ i)
r i ,
which can be rewritten
as
q=q + i
If
q +
:
r
i
(34)
= 0— the pricing mechanism corresponds to the one in Zhu and Wallace (2007)— then the asset price is
equal to its fundamental value, i.e., the discounted sum of its dividends, and it is una¤ected by the money
growth rate. In contrast, if the real asset is at least partially liquid, i.e., 0 <
1, then the asset price is
above its fundamental value.
We de…ne the liquidity premium of the asset as the di¤erence between q and q . From (34), it is equal to
L= i
q +
:
r
i
(35)
This liquidity premium arises because the real asset allows the buyer in a bilateral match in the AM to capture
some of the gains from trade. Monetary policy a¤ects the asset price through this liquidity premium. The
asset price increases with in‡ation, i.e., @q=@i > 0. As the cost of holding money gets higher, agents will
attempt to reduce their real balance holdings in favor of the real asset. The price of the real asset will,
therefore, increase. As the cost of holding real balances is driven to zero, i ! 0, then the liquidity premium
vanishes and the asset price approaches its fundamental value. In this limiting case, agents can use …at
money to extract all the gain from trade in the decentralized AM market, and hence the real asset has no
extra value beyond the one generated by its dividend stream.
In a monetary equilibrium, the gross rate of return of the real asset is R =
R=
1+r
:
1+ i
q+
q
or, from (34),
(36)
From (36), the rate of return of the asset depends on preferences, r, monetary policy, i, and the characteristics
of the pricing mechanism, .
18
Proposition 3 In any monetary equilibrium, R
1 + r. If i > 0, then @R=@i < 0 and @R=@ < 0.
Proof. Immediate from equation (36).
If i = 0, then the liquidity premium of the asset is 0 and hence its rate of return is equal to the rate of
time preference. In contrast, if i > 0, then the asset price exhibits a liquidity premium and its rate of return
is smaller than the rate of time preference. The model predicts a negative correlation between the rate of
return of the real asset and in‡ation. As in‡ation increases, agents substitute the real asset for real money
balances and, as a consequence, the asset price increases and its return decreases.17 The rate of return of
the asset also decreases with its liquidity as captured by
for much the same reason: as the liquidity of the
asset increases, the value of the asset for transactions purposes increases. Hence, the asset price increases
and its return decreases.
The absolute value of the elasticity of the asset rate of return with respect to i is given by
R=i
=
i
@R=R
=
:
@i=i
1+ i
This elasticity is less than one and is increasing with . Hence, if the asset becomes more liquid, as measured
by an increase in , its return becomes more sensitive to in‡ation. Also, in high-in‡ation environments the
rate of return of the asset is more sensitive to changes in monetary policy.
Let’s now turn to the rate of return di¤erential between the real asset and money. Since the gross rate
of return of …at money is
1
, the rate of return di¤erential is
R
1
=
1
1+i
1+ i
1 :
(37)
Proposition 4 In any monetary equilibrium, the real asset dominates money in its rate of return i¤ i > 0
and
< 1.
Proof. Immediate from equation (37).
If
= 1, as in Lagos and Rocheteau (2008) or Geromichalos, Licari and Suarez-Lledo (2007), the model
is unable to explain the rate of return di¤erential between money and the real asset. Since both capital and
money are “equally liquid,”in order for the two media of exchange to coexist, they must have the same rate
of return. As well, since R
1, a monetary equilibrium cannot exist if in‡ation is positive, i.e., if
Both of these results are counterfactual.
1 7 This
…nding is in accordance with the empirical evidence. See, e.g., Marshall (1992).
19
> 1.
In our model, if i = 0, then …at money and the real asset will have the same rate of return. By running
the Friedman rule, the monetary authority can satiate agents’need for liquidity, in which case the rate of
return of the real asset is equal to the rate of time preference— since the asset has no value as a medium of
exchange— which is also the rate of return of …at money.
However, if
< 1 and i > 0, then our model delivers a rate of return di¤erential between the real asset
and money. For given i, this di¤erential decreases with : as
increases, the value of the real asset increases
owing to its increased bene…t as a medium of exchange. As a result, its rate of return declines. One can
relate the rate of return di¤erential and the elasticity of the asset return with respect to in‡ation, i.e.,
1
R
(1 + i) 1
R=i
=
1
:
There is a negative relationship between the rate of return di¤erential and the elasticity of the asset rate of
return with respect to in‡ation.
We conclude this section by investigating the optimal monetary policy.
Proposition 5 Assume A(q + ) < c(y ) and
< 1. Then there is ^{ > 0 such that for all i 2 [0; ^{], y = y
at the symmetric monetary equilibrium.
Proof. From Proposition 2, since A(q + ) < c(y ), there is a symmetric monetary equilibrium,
provided that i < i0 (where i0 is de…ned in (33)). Moreover, q(i) =
(1+ i)
(r i) ,
`(i) solution to 1 +
i
=
u0 c
c0 c
1
(`)
1 (`)
and z(i) solution to
z(i) = `(i)
A
(1 + r)
(r
i)
(38)
are all continuous in i. De…ne
(i)
^ b [`(i)]
A(q(i) + ) + z(i) + U
u(y ):
The function (i) is continuous over [0; i0 ) and, from Lemma 2, y = y whenever (i) > 0. Substitute z(i)
by its expression given by (38) to get
(i)
(1
)A
^ b [`(i)] ! u(y )
As i ! 0, `(i) ! c(y ) and U
(1 + r)
(r
i)
c(y ). Since
lim (i) = (1
i!0
^ b [`(i)]
+ `(i) + U
)A
20
< 1,
(1 + r)
>0
r
u(y ):
By continuity, there exists a nonempty interval [0; ^{] such that (i) > 0.
In most monetary models with a single asset, the Friedman rule is optimal and it achieves the …rst best
(provided that there are no externalities, no distortionary taxes, and the pricing is well-behaved); we have
that result too.18 However, in contrast to standard monetary models, a small deviation from the Friedman
rule is neutral in terms of welfare in our model. Hence, a small in‡ation is (weakly) optimal. The only e¤ect
of increasing the in‡ation above the Friedman rule is to increase asset prices.
This …nding has the following implications. First, a small in‡ation will have no welfare cost. Hence,
we conjecture that moderate in‡ation (let say, 10 percent) will have a lower welfare cost than in standard
models. Second, even though asset prices respond to monetary policy, these movements do not correspond
to changes in society’s welfare. Hence, asset prices may not be a very good indicator of society’s welfare or
monetary policy e¤ectiveness. Third, since there is a range of in‡ation rates that generates the …rst-best
allocation, the optimal monetary policy is consistent with a rate of return di¤erential between …at money
and the real asset.
6
Liquidity structure of asset yields
In this section, we extend the model to allow for multiple real assets. We will show that the same model that
can explain the rate of return dominance puzzle— that …at money has a lower rate of return than risk-free
bonds— can also deliver a non-degenerate distribution of assets’yields despite agents being risk neutral. We
will investigate how this structure of yields is a¤ected by monetary policy.
Suppose that there are a …nite number K
1 of in…nitely-lived real assets indexed by k 2 f1; :::; Kg.
Denote Ak > 0 as the …xed stock of the asset k 2 f1; :::; Kg,
k
its expected dividend, and qk its price.
Agents learn the realization of the dividend of an asset at the beginning of the PM centralized market.
Consequently, the terms at which the asset is traded in the AM decentralized market only depend on the
expected dividend
k.
Moreover, since agents are risk-neutral with respect to their consumption in the PM
centralized market, the risk of an asset has no consequence for its price.19
th
Consider a buyer in the AM with a portfolio (fak gK
real asset.
k=1 ; z), where ak is the quantity of the k
1 8 In search monetary models, the Friedman rule can be suboptimal because of search externalities (Rocheteau and Wright,
2005) or distortionary taxes (Aruoba and Chugh, 2008). Also, if the coercive power of the government is limited, then the
Friedman rule might not be incentive-feasible (Andolfatto, 2007).
1 9 The result that the price of an asset does not depend on its risk would no longer be true if the realization of the dividend
was known in the AM when agents trade in bilateral matches. See Lagos (2006) and Rocheteau (2008).
21
The pricing mechanism is a straightforward generalization of the one studied in the previous sections. The
buyer’s payo¤ is given by
^b =
U
max
m ;f k g
y;
s.t.
"
u(y)
K
X
m
k (qk
#
+
k)
(39)
0
(40)
k=1
c(y) +
m
+
K
X
k (qk
+
k)
k=1
m
+
K
X
k (qk
+
k)
z+
k=1
where
k
K
X
k ak (qk
+
k );
(41)
k=1
2 [0; 1] for all k. According to (39)-(41), the buyer’s payo¤ is the same as the one he would get
in an economy where he can make a take-it-or-leave-if-o¤er to the seller, but where he is constrained not to
spend more than a fraction
of the real asset k.
k
One can generalize Lemma 1 in the obvious way; in particular,
u(y ) c(y ) if ` c(y )
;
u c 1 (`) ` otherwise
^ b (`) =
U
where ` = z +
PK
k=1
k ak (qk
+
k)
is the buyer’s liquid portfolio. Assume that
^b
(qk +
So 1=(qk +
k)
(42)
1 @U
k)
@ak
=
k
1
2
:::
K.
Then,
^b
@U
:
@z
units of the k th asset, which yields one unit of PM output, allows the buyer to raise his surplus
in the AM decentralized market by a fraction
k
of what he would obtain by accumulating one additional
unit of real balances instead. The parameter
k
can then be interpreted as a measure of the liquidity of
the asset k, that is, the extent to which it can be used to …nanced consumption opportunities in the AM at
favorable terms of trade. Given our ranking, the asset 1 is the most liquid one and the asset K is the least.
The second step of the pricing procedure is a generalization of (10)-(12). The seller’s payo¤ and the
actual terms of trade are determined by
^s =
U
y;
max
m ;f k g
s.t. u(y)
"
c(y) +
m
+
K
X
k (qk
+
k)
^b
U
k=1
m
K
X
k (qk
+
#
k)
k=1
zs
m
z;
ask
k
22
ak ;
k = 1; :::; K
In the PM agents choose the portfolio, (fak g; z), that they will bring into the decentralized market. The
portfolio problem becomes
(fak g; z) 2 arg max
fak g;z
and qk =
k =r.
(
iz
r
K
X
ak (qk
^b
qk ) + U
z+
k=1
K
X
k ak (qk
k=1
+
!)
k)
;
(43)
According to (43), the agent maximizes his expected utility of being a buyer in the AM
decentralized market, net of the cost of the portfolio. The cost of holding asset k is the di¤erence between
the price of the asset and its fundamental value (expressed in ‡ow terms), while the cost of holding real
balances is i =
, approximately the sum of the in‡ation rate and the rate of time preference. An agent’s
portfolio choice problem, (43), can be rewritten as
(
K
X
ak [i k (qk +
i` +
max
fak g;`
k)
)
^ b (`)
qk )] + U
r (qk
k=1
s.t.
K
X
k ak (qk
+
k)
(44)
`:
(45)
k=1
As in Section 4, in a monetary equilibrium, constraint (45) does not bind, ` solves (28), and the asset prices
must satisfy i k (qk +
k)
r (qk
qk ) = 0 or
qk =
1+i
r i
k
k;
8k 2 f1; :::; Kg
k
for all k 2 f1; :::; Kg.20 Note that the price of the real asset k increases with in‡ation, provided that
From (45) and (46), a monetary equilibrium exists if r > i
K
X
k Ak
k=1
1+r
r i k
k
(46)
k
> 0.
for all k and
k
< `(i):
(47)
For money to be valued, the total stock of real assets, adjusted by their liquidity factors, must not be too
large. The rate of return of asset k is given by
Rk =
k
1+r
+ qk
=
;
qk
1+i k
8k 2 f1; :::; Kg
(48)
Provided that the nominal interest rate is strictly positive, the model is able to generate di¤erences in the
rates of return of the real assets, where the ordering depends on the liquidity coe¢ cients f k g.
2 0 The
proof to these claims are almost identical to the proof of Proposition 2. Speci…cally, if r qk qk > i k (qk + k ), then
(qk ) = f0g, which cannot be an equilibrium; if r qk qk < i k (qk + k ), then constraint (45) binds and z = 0, i.e., the
P
d
equilibrium is non-monetary; and if r qk qk = i k (qk + k ), then
k Ak (qk ) k (qk + k ) 2 [0; `], where ` solves (28).
Ad
23
Proposition 6 In any monetary equilibrium, RK
Rk 0
RK
Rk = (1 + r)
1
:::
i( k
k0 )
(1 + i k )(1 + i
1
R1
k0 )
. Moreover,
> 0:
(49)
Proof. Direct from (48).
The di¤erences between the rates of return across assets emerge even if the assets are risk-free, or agents
are risk-neutral. They result from the pricing mechanism according to which di¤erent assets are traded at
di¤erent terms of trade in the decentralized market. These rate-of-return di¤erences would be viewed as
anomalies by standard consumption-based asset pricing theory.
Let us turn to monetary policy. A change in in‡ation a¤ects the entire structure of asset returns. Denote
fRki gK
k=1 the structure of asset yields when the cost of holding …at money is equal to i.
0
i K
Proposition 7 In any monetary equilibrium, fRki gK
k=1 dominates fRk gk=1 in a …rst-order stochastic sense
whenever i0 > i. Moreover,
@(Rk0 Rk )
@i
> 0 if and only if (
k
k0 )
1
i2
k k0
Proof. The …rst part of the proposition is direct from (48). From (49), if
which is independent of i. With no loss, assume
di¤erentiate ln (Rk0
k
>
k0
> 0.
k
=
and i > 0 so that Rk0
k0
then Rk0
Rk = 0
Rk > 0. From (49),
Rk ) to get
@ ln (Rk0
@i
Rk )
=
1 i2 k k0
i (1 + i k ) (1 + i
k0 )
:
An increase in in‡ation raises the rates of return of all real assets because agents substitute the real
assets for real balances which are more costly to hold. Moreover, the premia paid to the less liquid assets,
Rk 0
Rk with
k
>
k0 ,
increase provided that i is not too large. So if one interpret the less liquid asset as
risky equity and the most liquid one as risk-free bonds, then the equity premium increases with in‡ation.
7
Conclusion
The main contribution of this paper is to show that the rate-of-return-dominance and other asset pricing
puzzles need not be puzzles when viewed through the lens of monetary models with bilateral trades. These
seemingly-anomalous asset pricing patterns can be generated by trading mechanisms that are pairwise Paretoe¢ cient and that do not impose any restriction on the use of assets as means of payment, i.e., the Wallace
24
dictum is satis…ed. Liquidity di¤erences across assets can arise because agents coordinate on a mechanism
that makes it cheaper for buyers to trade some assets relative to others. For instance, sellers can agree to
o¤er better terms of trade to buyers trading with money instead of bonds or equity. Lagos (2006) shows that
a calibrated version of a search-theoretic monetary model can account for both the risk-free rate and the
equity premium puzzles once a small restriction is introduced on the use of equity as means of payment. Our
analysis indicates that such a restriction is, in fact, not needed once one allows for a broader class of trading
mechanisms. The trading mechanisms we have considered have no axiomatic or strategic foundations, where
the only property that we impose on the mechanism is that it be pairwise Pareto-e¢ cient. Ideally, one
would like to reconcile the existing approaches by providing deeper foundations for the mechanisms we have
considered. We leave this for future research.
25
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@FedFRASER