GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.90

Asset markets can achieve efficiency
in the directed search framework

Shoko Morimoto

September 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Asset markets can achieve efficiency in the directed
search framework∗
Shoko Morimoto†

Abstract
Using a directed search model, modified from random matching, this paper investigates
how trading frictions in asset markets affect portfolio choices, asset prices, and welfare. Bysolving the model numerically, it is demonstrated that the asset price increases (decreases)in the matching efficiency, if the relative risk aversion is smaller (larger) than unity. Efficient
asset allocation is achieved in the directed search framework, while random matching is
known not to achieve efficient allocation.
JEL Classification: G11, G12, G14, D83
Keywords: directed search, asset market, social welfare, intermediation∗

I am grateful to Yoshiro Tsutsui, Yuichi Fukuta, Masaru Sasaki, Katsuya Takii, and the seminar participants of
the Asset Pricing Workshop and Search Theoretic Workshop at Osaka University for helpful comments. All errors
remain mine.
†
E-mail address:hge012ns@mail2.econ.osaka-u.ac.jp

1. Introduction
If there are no frictions in the asset market, investors can trade immediately. In reality, however,
it is almost impossible for investors to locate trading partners immediately. Therefore, financial
institutions intermediate and conduct transactions for investors in return for payment of intermediation fees. In other words, intermediaries exist to reduce transaction difficulty.1 Of course,intermediaries cannot be perfect and it takes time to find trading partners. Suppose an investor
wants to buy an asset, such as a government bond issued by a developing country whose liquidity is extremely low, and sends an order to an intermediary. If the intermediary happens to havethis asset, the investor will obtain it immediately. Otherwise, it may take several hours or days
to complete the transaction because of the low liquidity of the asset. Therefore, it is interesting
to examine how the ability of the intermediary affects the efficiency of asset markets. That is
the aim of this paper.
Researchers have explored how trading delays affect market efficiency using the search theoretic approach. The search theory assumes that: (a) market participants cannot make transactions without ”searching” for trading partners and (b) they find trading partners stochastically.
The search theory has been applied mainly to explain unemployment and why money is used asa medium of exchange.
Only recently, the search theory was applied to asset markets. Some studies have shown thatsearch friction affects asset prices, market efficiency, risk premiums, etc. Lagos (2006) explains
the equity premium puzzle by analyzing the search premium for risky assets. He assumes thatsearch costs are necessary to trade risky assets, while such costs are not necessary to traderiskfree assets. Since risky assets are subject to a search premium, the premium for risky assets
becomes larger. Voyanos and Wang (2007), Voyanos and Weill (2008), and Weill (2008) built
models in which two assets with identical cash flows were traded at different prices because ofdifferent search frictions. Duffie et al. (2007) investigated the impact of search-and-bargaining
friction on asset prices in over-the-counter markets.2 They found that, under certain conditions,1

Another reason for the existence of intermediaries is to save monitoring costs. See Diamond (1984).
Over-the-counter (OTC) markets provide the opportunity to trade financial instruments such as stocks, bonds,
commodities, and derivatives directly between two parties. This differs from exchange trading, which occurs infacilities such as futures exchanges or stock exchanges which were created especially for the purpose of trading.
21asset prices are highly discounted because of search friction.
Lagos and Rocheteau (2007) is especially important to this paper as it inclusively investigates the effects of trade delays on asset markets in a random matching model framework. The
random matching model assumes two points: (a) that market participants do not have advance
information about their trading partners and (b) they randomly meet their trading partners only
after a search. If the partner is not suitable, market participants cease contact and begin a new
search. Lagos and Rocheteau (2007) found a condition relating to preferences regarding market
participants, i.e., a reduction in trading frictions raises the asset price. They found that, unlessinvestors have all the bargaining power in bilateral negotiations with brokers, asset allocation isinefficient even if the Hosios condition is satisfied.3
Lagos and Rocheteau (2007) show that efficient portfolio allocation is not achieved in the
random matching framework without particular conditions. This raises the question, ”Are there
any market mechanisms that realize more efficient allocations than random matching?” In somelabor market studies, more efficient outcomes were achieved via the directed search framework
than random matching.4 The directed search model differs from the random matching model inthe following: market participants have information about potential trading partners in advanceand choose their partners before they search. However, because of search frictions, they may
not find the chosen partners. Market participants never meet partners who were not chosen.
They meet the chosen partner with some probability, otherwise fail to find any partner. Oncethey meet their partners, transactions are made immediately. In this paper, I follow Lagos andRocheteau (2007)’s framework, but modify it from random matching to a directed search model
to compare the efficiency of asset markets in these two frameworks.
The modification is not merely an intellectual exercise. I believe the directed search model
better describes asset transactions in the real world than the random matching model. First, incontrast to the assumption of the random matching model, investors can actually choose theirbroker. Second, contrary to the assumption of the random matching model that intermediation
3

The Hosios condition indicates that the value of a broker’s bargaining power is equal to the elasticity of thematching function. See Hosios (1990).
4
Moen (1997) used a directed search framework to analyze labor markets. He assumed that firms first determine
their wage rates and then announce them. Workers apply to firms knowing the wage rates and there is no bargaining.

2

fees are determined through bilateral negotiations between investors and brokers, in practicebrokers announce their fees and there are no negotiations. Thus, the directed search model
assumes that investors choose their dealers, and that dealers announce their intermediation feesin advance.
I show that the asset allocation is efficient in the directed search framework. Moreover,by solving the model numerically, I show that the asset price increases (decreases) with thematching efficiency, if the relative risk aversion is smaller (larger) than unity.
The rest of this paper is organized as follows. In section 2, I lay out the model settings.
In section 3, I define the competitive equilibrium of this economy and examine how trading
frictions affect asset price and portfolio choices numerically. In section 4, I show the relationship between trading frictions and social welfare and discuss the efficiency of asset allocation.
Section 5 is the conclusion.

2. Environment
The environment of the model basically follows Lagos and Rocheteau (2007). Time is continuous and infinite. Infinite investors, scaled up to unity, maximize the present value of their
lifetime utility. Homogeneous brokers, indexed by n = 1, 2, ..., maximize their instantaneous
profit. The number of brokers is determined by a free entry condition. There is one durable andperfectly divisible asset. There is one perishable good which we use as the numeraire. Total
supply of the asset is fixed to A. The numeraire goods are produced and consumed by all agents,
who produce in units of numeraire goods per working hour. Each investor’s instantaneous utility function depends on his assets, consumption, and working hours, described as ui (a) + c − w,
where a is the quantity of assets that type i investor holds; i specifies the investor’s preference,
which can be either h(high) or l(low). I assume that u0h (a) > u0l (a) for all a. The utility function is bounded, continuously differentiable, strictly increasing, and strictly concave. c is theinvestor’s consumption of numeraire goods, w is his working hours. Each investor’s preference
type randomly changes from low and high subject to a Poisson process with an arrival rate ofλh (λl ). Each investor’s time preference rate is always ρ.
3There is a competitive asset market that investors cannot access directly but brokers canaccess at any time. Therefore, investors must use brokers when they want to trade in that asset
market. However, they may not be able to meet their broker because of matching frictions
between investors and brokers, as discussed below.
Brokers cannot own assets, so they do not engage in dealing. The only source of revenue forbrokers is the intermediation fee paid by investors. Brokers have an identical operating function, c(q), which is continuously differentiable and nondecreasing in the transaction volume
they intermediate q. Each broker announces the intermediation fee function φn (q) in advanceand commits to it, where index n represents the n-th broker. The function is assumed to becontinuously differentiable and nondecreasing in q. Investors choose brokers based on information about intermediation fees. After deciding to partner with, say, the n-th broker, the investor
must wait to meet the broker. How long the investor waits is determined by a matching function
M(ηn ), where ηn is the measure of investors who choose the n-th broker.The investor meets then)n-th broker according to a Poisson process with an arrival rate of M(η.
ηn

M(ηn )
represents theηnn)implies a shorter waiting time. I assume
frequency of successful meetings, so that a larger M(ηηnthe matching function, M(ηn ), is continuously differentiable, strictly increasing, and strictly
concave. In addition, M(ηn ) = 0 if and only if ηn = 0.

3. Competitive equilibrium
In this section, I characterize the steady-state equilibrium of the model defined in the previous
section. I focus on the steady-state where the problem is solved recursively. Thus, I omit thetime subscript t in this section.

3.1. Broker
I define the n-th broker’s instantaneous profit function πn such that
πn = M(ηn ) φn (qn ) − c(qn ) ,n = 1, 2, · · · ,where M(ηn ) is the intermediation frequency, φ(qn ) is the intermediation fee, and c(qn ) is theoperating cost.
4

Brokers earn no profits, i.e., πn = 0, because of the free entry condition, implying that
M(ηn ) = 0⇔

ηn = 0,φ(qn ) − c(qn ) = 0⇔

φ(qn ) = c(qn ).

orηn = 0 implies that the n-th broker no longer operates in the market, so this case is excluded
from the following analysis and I obtain φ(qn ) = c(qn ). In steady-state equilibrium, the broker’s
fee function is identically characterized asφn (·) = c(·),

n = 1, 2, · · · ,(1)φ0n (·) = c0 (·), n = 1, 2, · · · .

(2)

so thatEq. (1) indicates that every dealer charges the same intermediate fee schedule. All brokers look
the same, resulting in a situation in which investors are uniformly distributed among brokers,
that isη1 = η2 = · · · .

(3)The number of successful matches for all brokers also becomes equal,M(η1 ) = M(η2 ) = · · · .

3.2. Investor
Denoting type i investor’s value function as Vi (·), the Bellman equation of a high-type investor’s
problem is described asρVh (a) = uh (a) + λl Vl (a) − Vh (a)

n 
o 
+ max 0 max
mn Vh (a0 ) − Vh (a) − p(a0 − a) − φn (|a0 − a|) , 0 ,a ∈R,n∈{1,2,··· }5where p is the asset price. The first term, uh (a), represents the instantaneous utility of a hightype investor who owns a units of assets. The second term arises from the possibility that theinvestor will change to a low preference type at the next moment. The third term reflects thechange in asset value stemming from the transaction. Investors must decide in advance whether
or not to trade. If they decide not to trade, the value will not change. If they decide to trade,n)then they need to choose a broker and wait to meet at a Poisson arrival rate of mn ≡ M(η
. The
ηn

investor decides the amount of assets they will own in the next period, a0 ∈ R, and pay theintermediation fee, φn (|a0 − a|), to the n-th broker.
Similarly, the Bellman equation of a low-type investor’s problem is written asρVl (a) = ul (a) + λh Vh (a) − Vl (a)

n 
o 
0
0
0
+ max 0 max
mn Vl (a ) − Vl (a) − p(a − a) − φn (|a − a|) , 0 .
a ∈R,n∈{1,2,··· }The following lemma is necessary for subsequent arguments. All proofs are given in the
Appendix.
Lemma 1. Vi (·) is strictly increasing, strictly concave, and continuously differentiable for i =h, l.
Once investor i decides to trade, his maximization problem is reduced tomax

a0 ∈R,n∈{1,2,··· }mn Vi (a0 ) − Vi (a) − p(a0 − a) − φn (|a0 − a|) .

The first order condition (henceforth FOC) with respect to a0 is
a0 > a,1
Vi0 (a0 ) − p − I(a,a0 ) φ0n (|a0 − a|) = 0, I(a,a0 ) = 
−1 a0 < a,i = h, l,(4)where Vi0 (·) is the first differential of the value function. The existence of Vi0 (·) is warranted by
Lemma 1; φ0n (|a0 − a|) exists by assumption. Eq. (4) indicates that the optimal portfolio depends
on the investor’s type (i) and the portfolio held by the investor before trading (a). Although
Eq. (4) also depends on n, the optimal portfolio does not depend on the broker, because every
broker charges the same fee.
6

Thus, denoting the optimal portfolio for the high-type investor as ah and for the low-type as
al , the Bellman equations are rewritten as:ρVh (ah ) =uh (ah ) + λl Vl (ah ) − Vh (ah ) ,

(5)



ρVh (al ) =uh (al ) + λl Vl (al ) − Vh (al )

(6)



+ mn Vh (ah ) − Vh (al ) − p(ah − al ) − φn (|ah − al |) ,ρVl (ah ) =ul (ah ) + λh Vh (ah ) − Vl (ah )

(7)



+ mn Vl (al ) − Vl (ah ) − p(al − ah ) − φn (|al − ah |) ,ρVl (al ) =ul (al ) + λh Vh (al ) − Vl (al ) .

(8)

I exclude the case of an investor who does not want to trade because that investor’s portfolio
will not change from the beginning, i.e., the steady-state of the economy coincides with theinitial state. In other words, I assume that the intermediation fees are not so large that such acase exists.
Now, I demonstrate the following lemma.
Lemma 2. The optimal portfolio of the high type investor is greater than that of the low type
investor, i.e., ah ≥ al .
Using lemma 2, the following proposition can be proved.
Proposition 1. The demand function for the assets of the high-type investor isp=(ρ + λh + mn )u0h (ah ) + λl u0l (ah )ρ(ρ + λh + λl + mn )hi
2mn λl− 1+
φ0 (ah − al ),ρ(ρ + λh + λl + mn ) n(9)and for that of the low-type investor is(ρ + λl + mn )u0l (al ) + λh u0h (al )p=ρ(ρ + λh + λl + mn )hi
2mn λh+ 1+
φ0n (ah − al ),ρ(ρ + λh + λl + mn )

7

(10)

These demand functions warrant that when trading frictions do not exist and no intermediation
fees are levied, the asset price of this model decays into the Walrasian equilibrium price. To
confirm this, I substitute λh = λl = 0 and φn (ah − al ) = 0 in Eqs. (9) and (10) to get pW ≡u0h (ah )=ρ

u0l (al )ρ

3.3. Law of transition
I classify investors’ states into four classes, hh, hl, lh, ll , whose definitions are given in Table
1.
Using these variables, I can describe the laws of transition among the states of investors,
which should be followed by steady states as follows.
mn ξhl + λh ξlh − λl ξhh = 0,(11)

λh ξll − mn ξhl − λl ξhl = 0,(12)

λl ξhh − mn ξlh − λh ξlh = 0,(13)

mn ξlh + λl ξhl − λh ξll = 0,Xξhl + ξlh =
ηn ,

(14)
(15)

n

ξhh + ξhl + ξlh + ξll = 1.

(16)

Eq. (11) means that inflow and outflow of hh are equal in the steady-state. The first term, mn ξhl ,represents the movement of investors from hl to hh during one period because of transactions.
The second term, λh ξlh , indicates movement of investors from lh to hh because of a change
in type. The third term, λl ξhh , are investors who move from hh to lh because of a change intype. Eqs. (12), (13), and (14), describe the conditions associated with states hl, lh, and ll,
respectively. Eq. (15) shows that investors who tend to trade are identical to investors whosetypes and portfolios are not coincidental. Eq. (16) assumes that investors are scaled up to unity.

3.4. Asset market
The market clearing condition is:(ξhh + ξlh )ah + (ξhl + ξll )al = A.
8

(17)

The left side is demand for the asset. ξhh + ξlh (ξhl + ξll ) is the sum of investors who have ah (al ).
The total supply of the asset is fixed to A by assumption.

3.5. Equilibrium
I define the competitive equilibrium of this economy as follows.

Definition 1. The steady-state competitive equilibrium is characterized by p, ah , al , φ1 (·),
φ2 (·), · · · , η1 , η2 , · · · , ξhh , ξhl , ξlh , ξll that satisfies Eqs. (1), (3), and (9)-(17).

3.6. Numerical Example
3.6.1 Specifications
The equilibrium system defined above is too complicated to solve analytically. Therefore, letme demonstrate a numerical solution instead. To calibrate the system, I specified the matching,utility, and cost functions, and the value of parameters in the functions, as shown in Table 2.
The matching function is specified as continuous, strictly increasing, and strictly convex. A
constant relative risk aversion (CRRA) type of utility function and a linear cost function are
assumed. I determined parameters that are appropriate for fitting quarterly data. The value
adopted for the time preference rate, ρ, is frequently used in macroeconomics studies such as
the real business cycle theory. I set the Poisson arrival rate for the transition of investor type,λh and λl , at approximately every week and parameters of the utility functions, h and l , tomake the marginal utility of low-type investors be half that of high-type. The cost function
parameters, c1 and c2 , were set to be small enough to induce the incentive to trade. With these
specifications, a larger M implies more opportunity for transactions. Thus, we can regard theparameter M as the matching efficiency.
3.6.2. Results
Figure 1 shows how the matching efficiency (M) affects price (p), the high-type investor’s optimal portfolio (ah ), and the low-type investor’s optimal portfolio (al ) when the relative risk
aversion is less than unity. When M equals 10 (20), the investor meets the broker every few
days (several hours). In this case, the asset price increases with the matching efficiency. In
9the reverse case, when relative risk aversion is larger than unity, the opposite result is obtained(Figure 2). In both cases, the optimal portfolio of the high-type investor increases and of the
low-type investor decreases with the matching efficiency. In other words, ah − al increases withthe matching efficiency.
3.6.3. Intuition
Let me explain why the above results were obtained. First, suppose that if the matching efficiency (M) is extremely low, investors know that transactions are hardly possible; they will
have to keep their portfolios as they are even though their portfolios are not optimal. The larger
the gap between ah and al , the more opportunity costs. Thus, when M is low, ah − al has a
low value. Second, when the relative risk aversion is smaller than unity, investors are close toneutral risk and their elasticity of intertemporal substitution is larger than unity. In this case,the substitution effect exceeds the income effect, so that the price elasticity of an asset is larger
than unity. The demand functions corresponding to this case are described in Figure 3. Eqs.
(9) and (10) indicate that an improvement of matching efficiency shifts the high-type investor’s
demand function upward and the low-type investor’s downward. If the asset price remains at p∗ ,expansion of the high-type investor’s demand a0h − ah dominates the reduction of the low-type
investors’ al − a0l , resulting in expansion of the aggregate demand. Since the total asset supply isfixed at A, expansion of the aggregate demand causes excess demand for the asset. As a result,the asset price rises to clear the market. When the relative risk aversion coefficient is larger thanunity, the argument is opposite. These effects of matching efficiency on asset price and optimal
portfolios are consistent with Lagos and Rocheteau (2007).

4. Efficiency
4.1. Efficient allocation
In this section, I analyze a social planner’s problem. I assume that the social planner manipulates
investors’ portfolios. The free entry condition is assumed in this section to compare the social
optimum with the market equilibrium under the same exogenous condition.
For analytical tractability, I consider the case where the time preference rate is close to 0.
10

Under this assumption, the social planner’s problem decays to the maximization of the steadystate instantaneous welfare. The planner solves
maxs s
ah ,als
W =ξhhuh (ahs ) + ξhls uh (als ),+ ξlhs ul (ahs ) + ξlls ul (als ) −(18)
X

M(ηns )c(ahs − als ),nsubject tos+ ξlhs )ahs + (ξhls + ξlls )als = A,
(ξhh

(19)

s
= 0,
mns ξhls + λh ξlhs − λl ξhh(20)

λh ξlls − mns ξhls − λl ξhls = 0,(21)s
λl ξhh− mns ξlhs − λh ξlhs = 0,(22)

mns ξlhs + λl ξhls − λh ξlls = 0,

(23)

s
ξhh
+ ξhls + ξlhs + ξlls = 1,X
ηns = ξhls + ξlhs ,

(24)
(25)

n

η1s = η2s = · · · .
where s denotes the social planner’s problem and mns ≡(26)

M(ηns )
is common for all brokers becauseηnsof Eq. (26). The social planner’s objective function appeared in Eq. (18). It is composed of
the weighted sum of the investors’ utility and the brokers’ operating costs. It does not include
intermediation fees, because the social planner is not interested in these fees that are merelytransferred from investors to brokers.
FOCs with respect to ahs and als ares 0ξhhuh (ahs ) + ξlhs u0l (ahs ) −X

sM(ηns )c0 (ahs − als ) − µ(ξhh+ ξlhs ) = 0,

(27)

n

ξhls u0h (als ) + ξlls u0l (als ) +X

M(ηns )c0 (ahs − als ) − µ(ξhls + ξlls ) = 0,

(28)

n

where µ is a Lagrange multiplier for the constraint (19). Eliminating the Lagrange multiplier in
Eqs. (27) and (28), I obtainPPs 0
uh (ahs ) + ξlhs u0l (ahs ) − n M(ηns )c0 (ahs − als ) ξhls u0h (als ) + ξlls u0l (als ) + n M(ηns )c0 (ahs − als )
ξhh
=
.
s
+ ξlhsξhhξhls + ξlls
(29)
11

From Eqs. (20)-(24), I obtainλh (mns + λh ),(λh + λl )(mns + λh + λl )λh λl
ξhls = ξlhs =,(λh + λl )(mns + λh + λl )λl (mns + λl )ξlls =.
(λh + λl )(mns + λh + λl )
s
=
ξhh

(30)
(31)
(32)

Substituting Eqs. (25), (26), and (30)-(32) into Eq. (29), I obtain(mns + λh )u0h (ahs ) + λl u0l (ahs ) − 2mns λl c0 (ahs − als )= 1.
(mns + λl )u0l (als ) + λh u0h (als ) + 2mns λh c0 (ahs − als )(33)

Eq. (33) is interpreted as follows. If the social planner decreases ahs , the instantaneous utility ofhh and lh investors decreases and als increases to satisfy market clearing conditions. In response,ll and hl investors’ utility increases. The changes in ahs and als also alter the broker’s operating
costs. At the optimum, the weighted sum of these effects is canceled out.
I now define the efficient allocation of this economy.
Definition 2. The steady-state efficient allocation is characterized by ahs , als , η1s , η2s , · · · ,sξhh, ξhls , ξlhs , ξlls that satisfies Eqs. (19)-(26) and (33).

Figure 4 is a numerical example of the relationship between the matching efficiency andsocial welfare in efficient allocations. The vertical axis W is social welfare, defined in Eq. (18).
The horizontal axis represents matching efficiency, parameters of which are shown in Table 2.
Social welfare monotonically increases with matching efficiency, implying that smaller trading
frictions results in higher social welfare, consistent with our intuition.

4.2. Efficiency of competitive equilibrium
In this subsection, I discuss the efficiency of competitive equilibrium. Specifically, I examine
whether competitive equilibrium achieves the social optimum.
It is obvious that Eqs. (19)-(26) of the social optimum are parallel to Eqs. (3) and (11)-(17)in the competitive equilibrium. Thus, what is necessary to show is that Eq. (33) is fulfilled in

12the competitive equilibrium. To confirm this, I incorporate Eqs. (9) and (10) to obtain:i(ρ + λh + mn )u0h (ah ) + λl u0l (ah ) h
2mn λl− 1+
φ0 (ah − al )ρ(ρ + λh + λl + mn )ρ(ρ + λh + λl + mn ) ni(ρ + λl + mn )u0l (al ) + λh u0h (al ) h
2mn λhφ0 (ah − al ).
=+ 1+
ρ(ρ + λh + λl + mn )ρ(ρ + λh + λl + mn ) n(34)

To derive Eq. (33), I substitute Eq. (2) into Eq. (34), multiply both sides of Eq. (34) byρ(ρ + λh + λl + mn ), and take ρ → 0. Thus, the following proposition is established.
Proposition 2. The steady-state competitive equilibrium coincides with the efficient allocation
when each investor’s time preference rate is close to 0.
This proposition ascertains that trading frictions do not distort efficient allocation whenmodeled in a directed search framework. Why is this efficient proposition derived? Close
inspection of the above proof reveals that Eq. (2), which forces marginal intermediation feesto be equal to the broker’s marginal costs, is the key for the competitive equilibrium to besocially optimal. The social planner is concerned not with intermediation fees but with brokers’
operating costs, while investors are concerned with the contrary. Eq. (2) makes these two costs
coincide, so that both the social planner and the investors make the same decision. Since Eq.
(2) comes from competition among brokers, the free entry condition, as well as the obligation
to announce intermediation fees in advance, is inevitable for the intermediaries to be socially
optimal.

5. Conclusion
In this paper, I investigated how trading frictions of asset markets affect portfolio choices, asset
prices, and welfare, essentially following Lagos and Rocheteau (2007)’s framework, but modifying it from random matching to a directed search model. Solving the model numerically, I
show that the asset price increases (decreases) with matching efficiency if the relative risk aversion is smaller (larger) than unity. Lagos and Rocheteau (2007), using the random matchingframework, found that asset allocation in the competitive equilibrium is not efficient. Contrary
to their results and using the directed search framework, I show that asset allocation in the competitive equilibrium is efficient. The key condition for the optimum is that intermediation fees13

be equal to brokers’ operating costs. If the intermediation fees are larger than the brokers’ operating costs, the transaction volume of the asset is smaller than optimal. On the other hand, if
the intermediation fees are smaller than the brokers’ operating costs, the transaction volume islarger than optimal. Note, this condition is not achieved only by Hosios condition but also because the brokers’ bargaining power takes a zero value in the random matching and bargaining
framework.
In this paper, I assumed that investors’ preference type changes according to an identical
Poisson process, and that all brokers have a homogeneous cost function. Thus, a possible
extension of this paper would be introduction of heterogeneity to reflect the real world, where
several types of investors and brokers exist, at least ex-ante.

14

Appendix
Proof of Lemma 1.
I prove Lemma 1 using theorems 4.7 and 4.8 from Stokey and Lucas (1989). It is sufficient tocheck the following three items, which are clearly satisfied in this economy.
1. R is nonempty, compact-valued, continuous, monotone, and convex.
2. u(·) is bounded, continuous, strictly increasing, and strictly concave.
3. ρ > 0.


Proof of Lemma 2.
Suppose ah < al , thenI(al ,ah ) = −1,
I(ah ,al ) = 1,andVh0 (ah ) = p − φ0n (al − ah ) ≤ p + φ0n (al − ah ) = Vl0 (al ),because φ0 (·) is nonnegative by assumption. The strict concavity of value functions indicatesthat
Vl0 (ah ) > Vl0 (al ) ≥ Vh0 (ah ) > Vh0 (al ).
Then noteρVh0 (ah ) = u0h (ah ) + λl Vl0 (ah ) − Vh0 (ah ) ,= u0h (ah ) + λl [sign+],

(35)





ρVl0 (ah ) = u0l (ah ) + λh Vh0 (ah ) − Vl0 (ah ) + mn − Vl0 (ah ) + p + φ0k (al − ah ) ,= u0l (ah ) + λh Vh0 (ah ) − Vl0 (ah ) + mn − Vl0 (ah ) + Vl0 (al ) ,= u0l (ah ) + λh [sign−] + mn [sign−],
15

(36)

where [sign+] and [sign−] mean a positive term and a negative term, respectively. Subtract Eq.
(36) from Eq. (35), then ρ Vh0 (ah ) − Vl0 (ah ) = u0h (ah ) − u0l (ah ) + [sign+],⇒ ρ[sign−] = [sign+],where I assume that u0h (a) ≥ u0l (a) for all a. This is a contradiction because ρ > 0.


Proof of Proposition 1.
Lemma 2 indicates that
I(al ,ah ) = 1,

(37)

I(ah ,al ) = −1.

(38)

Eqs. (37) and (38) indicate that the investors’ FOCs are
Vh0 (ah ) = p + I(al ,ah ) φ0n (ah − al ) = p + φ0n (ah − al ),(39)

Vl0 (al ) = p + I(ah ,al ) φ0n (ah − al ) = p − φ0n (ah − al ).

(40)

First, transforming Eqs. (5) and (8), thenuh (ah ) + λl Vl (ah ),ρ + λlul (al ) + λh Vh (al )Vl (al ) =.
ρ + λhVh (ah ) =

(41)
(42)

Next, differentiating both sides of Eq. (41) with respect to ah and Eq. (42) with respect to al ,thenu0h (ah ) + λl Vl0 (ah ),ρ + λlu0 (al ) + λh Vh0 (al )Vl0 (al ) = l
ρ + λhVh0 (ah ) =

(43)
(44)

Substituting Eqs. (43) and (44) into Eqs. (39) and (40),u0h (ah ) + λl Vl0 (ah )p=− φ0n (ah − al ),ρ + λlu0l (al ) + λh Vh0 (al )p=+ φ0n (ah − al ).
ρ + λh
16

(45)
(46)

Transforming Eqs. (6) and (7),uh (al ) + λl Vl (al ) + αn Vh (ah ) − p(ah − al ) − φn (ah − al )Vh (al ) =,ρ + λl + αnul (ah ) + λh Vh (ah ) + αn Vl (al ) − p(al − ah ) − φn (ah − al )Vl (ah ) =.
ρ + λh + αn

(47)
(48)

Differentiating Eqs. (47) and (48) with respect to al and ah , respectively, and substituting Eqs.
(41) and (42) into them, I obtainu0h (al ) + (λl + αn )p − (λl − αn )φ0n (ah − al ),
ρ + λl + αnu0 (ah ) + (λh + αn )p + (λh − αn )φ0n (ah − al ).
Vl0 (ah ) = l
ρ + λh + αnVh0 (al ) =

(49)
(50)

Substituting Eqs. (49) and (50) for Eqs. (47) and (48), I can derive proposition 1.
Proof of Proposition 2.
See text.
17

References
Diamond, D.W., 1984. Financial intermediation and delegated monitoring. Rev. Econ. Stud.
51, 393-414.
Duffie, D., Gârleanu, N., Pedersen, L.H., 2007. Valuation in over-the-counter markets. Rev.
Finan. Stud. 20, 1865-1900.
Hosios, A.J., 1990. On the efficiency of matching and related models of search and unemployment. Rev. Econ. Stud. 57, 279-298.
Lagos, R., 2006. Asset Prices and Liquidity in an Exchange Economy. Research Department
Staff Report 373, Federal Reserve Bank of Minneapolis.
Lagos, R., Rocheteau, G., 2007. Search in asset markets: Market structure, liquidity, welfare.
Amer. Econ. Rev. 97, 198-202.
Moen, E.R., 1997. Competitive search equilibrium. J. Polit. Econ. 105, 385-411.
Stokey, N.L., Lucas, R.E., 1989. Recursive Methods in Economic Dynamics. Harvard.
Voyanos, D., Wang, T., 2007. Search and endogenous concentration of liquidity in asset markets. J. Econ. Theory 136, 66-104.
Voyanos, D., Weill, P.O., 2008. A search-based theory of the on-the-run phenomenon. J. Finan.
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Weill, P.O., 2008. Liquidity premia in dynamic bargaining markets. J. Econ. Theory 140,
66-96.

18

Table 1: Four classes of investors
state
hh
hl
lh
ll

type portfolio
high
ah
high
al
low
ah
low
al

measure
ξhh
ξhl
ξlh
ξll

Table 2: Values of parameters used in functions
M(ηn ) ui (a)1−γMηβn i a1−γβ
ρ0.5 0.01c(q)c1 + c2 q

λh
12

λl
12

h
20

l
10

A
10

c1
0.01

c2
0.001

316.5

316

p
315.5

315
10

11

12

13

14

15

16

17

18

19

20

11

12

13

14

15

16

17

18

19

20

11

12

13

14

15

16

17

18

19

12.5

ah

12

11.5
10
8.5

al

8

7.5
10

20

M

Figure 1: Asset price and portfolio choices when the relative risk aversion is less than unity (γ = 0.6). This
figure shows how matching efficiency (M) relates to asset price (p), the high-type investor’s optimal portfolio (ah ),and the low-type investor’s optimal portfolio, al .

19

34
33

p

32
31
30
10

11

12

13

14

15

16

17

18

19

20

11

12

13

14

15

16

17

18

19

20

11

12

13

14

15

16

17

18

19

12.5

ah

12

11.5
10
9.6

al 9.4
9.2
10

20

M

Figure 2: Asset price and portfolio choices when the relative risk aversion is larger than unity (γ = 1.5).
This figure shows how matching efficiency (M) relates to asset price (p), the high-type investor’s optimal portfolio(ah ), and the low-type investor’s optimal portfolio, al .

p
ph+
p
h
pl
p l+

p*

a’l
small

al

a’h

ah
large

a

Figure 3: Demand functions (γ = 0.6). ph (pl ) is the demand function of the high-type (low-type) investor. As
the matching efficiency improves, demand functions shift from solid lines (ph , pl ) to dotted lines (p+h , p+l ). If theprice remains at p∗ , the high-type (low-type) investor’s optimal portfolio shifts from ah (al ) to a0h (a0l ).

20

W

79

78.95

78.9

78.85

78.8

78.75

78.7
10

11

12

13

14

15

16

17

18

19

20

M

Figure 4: Welfare. The relationship between matching efficiency (M) as defined in Table 2 and social welfare(W) as defined in Eq. (18).

21