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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
1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan Asset markets can achieve efficiency in the directed 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. Efficientasset allocation is achieved in the directed search framework, while random matching isknown not to achieve efficient allocation.
JEL Classification: G11, G12, G14, D83Keywords: 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 errorsremain mine.
†E-mail address:[email protected] 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 inter- mediation 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 liquid- ity is extremely low, and sends an order to an intermediary. If the intermediary happens to have this 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 Researchers have explored how trading delays affect market efficiency using the search the- oretic approach. The search theory assumes that: (a) market participants cannot make transac- tions 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 as Only recently, the search theory was applied to asset markets. Some studies have shown that search 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 that search costs are necessary to trade risky assets, while such costs are not necessary to trade riskfree 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 of different 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, 1Another reason for the existence of intermediaries is to save monitoring costs. See Diamond (1984).
2Over-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.
asset prices are highly discounted because of search friction.
Lagos and Rocheteau (2007) is especially important to this paper as it inclusively investi- gates 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, unless investors have all the bargaining power in bilateral negotiations with brokers, asset allocation is inefficient 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 some labor 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 in the following: market participants have information about potential trading partners in advance and 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. Once they meet their partners, transactions are made immediately. In this paper, I follow Lagos and Rocheteau (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, in contrast to the assumption of the random matching model, investors can actually choose their broker. Second, contrary to the assumption of the random matching model that intermediation 3The Hosios condition indicates that the value of a broker’s bargaining power is equal to the elasticity of the matching function. See Hosios (1990).
4Moen (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.
fees are determined through bilateral negotiations between investors and brokers, in practice brokers 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 fees 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 the matching 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 relation- ship between trading frictions and social welfare and discuss the efficiency of asset allocation.
2. Environment
The environment of the model basically follows Lagos and Rocheteau (2007). Time is con- tinuous 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 and perfectly 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 util- ity 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 u (a) > u (a) for all a. The utility func- tion is bounded, continuously differentiable, strictly increasing, and strictly concave. c is the investor’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 of λh (λl). Each investor’s time preference rate is always ρ.
There is a competitive asset market that investors cannot access directly but brokers can access 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 for brokers is the intermediation fee paid by investors. Brokers have an identical operating func- tion, 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 advance and commits to it, where index n represents the n-th broker. The function is assumed to be continuously differentiable and nondecreasing in q. Investors choose brokers based on informa- tion 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 the n-th broker according to a Poisson process with an arrival rate of M(ηn) . M(ηn) represents the frequency of successful meetings, so that a larger M(ηn) implies a shorter waiting time. I assume the matching function, M(ηn), is continuously differentiable, strictly increasing, and strictly 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 the time subscript t in this section.
3.1. Broker
I define the n-th broker’s instantaneous profit function πn such that M(ηn) φn(qn) − c(qn) , where M(ηn) is the intermediation frequency, φ(qn) is the intermediation fee, and c(qn) is the 0, because of the free entry condition, implying that φ(qn) − c(qn) = 0 φ(qn) = c(qn).
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 as φn(·) = c(·), n = 1, 2, · · · , φ (·) = c (·), n = 1, 2, · · · .
Eq. (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, 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 ρVh(a) = uh(a) + λl Vl(a) − Vh(a) mn Vh(a ) − Vh(a) − p(a − a) − φn(|a − a|) , 0 , a ∈R,n∈{1,2,··· }
where p is the asset price. The first term, uh(a), represents the instantaneous utility of a high- type investor who owns a units of assets. The second term arises from the possibility that the investor will change to a low preference type at the next moment. The third term reflects the change 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, then they need to choose a broker and wait to meet at a Poisson arrival rate of m ≡ M(ηn) investor decides the amount of assets they will own in the next period, a ∈ R, and pay the
intermediation fee, φn(|a − 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) 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 Lemma 1. Vi(·) is strictly increasing, strictly concave, and continuously differentiable for i =
Once investor i decides to trade, his maximization problem is reduced to mn Vi(a ) − Vi(a) − p(a − a) − φn(|a − a|) .
a ∈R,n∈{1,2,··· }
The first order condition (henceforth FOC) with respect to a is V (a ) − p − I φ (|a − a|) = 0, I where V (·) is the first differential of the value function. The existence of V (·) is warranted by Lemma 1; φ (|a − 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 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) , ρVh(al) =uh(al) + λl Vl(al) − Vh(al) n Vh(ah) − Vh(al) − p(ah ρVl(ah) =ul(ah) + λh Vh(ah) − Vl(ah) n Vl(al) − Vl(ah) − p(al ρVl(al) =ul(al) + λh Vh(al) − Vl(al) .
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 the initial state. In other words, I assume that the intermediation fees are not so large that such a Now, I demonstrate the following lemma.
Lemma 2. The optimal portfolio of the high type investor is greater than that of the low type
Using lemma 2, the following proposition can be proved.
Proposition 1. The demand function for the assets of the high-type investor is
mn)u (ah) + λlu (ah) and for that of the low-type investor is mn)u (al) + λhu (al) 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 0 and φn(ah al) = 0 in Eqs. (9) and (10) to get pW ≡ uhρ 3.3. Law of transition
I classify investors’ states into four classes, hh, hl, lh, ll , whose definitions are given in Table Using these variables, I can describe the laws of transition among the states of investors, which should be followed by steady states as follows.
Eq. (11) means that inflow and outflow of hh are equal in the steady-state. The first term, m ξ represents the movement of investors from hl to hh during one period because of transactions.
h lh, indicates movement of investors from lh to hh because of a change l hh, are investors who move from hh to lh because of a change in type. 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 whose types and portfolios are not coincidental. Eq. (16) assumes that investors are scaled up to unity.
3.4. Asset market
The left side is demand for the asset. ξ + ξ ll) 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, a , , φ
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, let me 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, to make 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 the parameter 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 op- timal 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 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, a − 3.6.3. Intuition
Let me explain why the above results were obtained. First, suppose that if the matching ef- ficiency (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 h and al, the more opportunity costs. Thus, when M is low, ah low value. Second, when the relative risk aversion is smaller than unity, investors are close to neutral 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 a − a h dominates the reduction of the low-type a , resulting in expansion of the aggregate demand. Since the total asset supply is fixed 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 than unity, 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.
Under this assumption, the social planner’s problem decays to the maximization of the steady- state instantaneous welfare. The planner solves M(ηs)c(as − as), (ξs + ξs )as + (ξs + ξs )as = A, msξs + λ ξs − λ ξs = 0, msξs + λ ξs − λ ξs = 0, ξs + ξs + ξs + ξs = 1, ηs = ηs = · · · .
where s denotes the social planner’s problem and ms ≡ M(ηsn) is common for all brokers because of 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 merely transferred from investors to brokers.
FOCs with respect to as and as are ξs u (as) + ξs u (as) − M(ηs)c (as − as) − µ(ξs + ξs ) = 0, ξs u (as) + ξsu (as) + M(ηs)c (as − as) − µ(ξs + ξs ) = 0, where µ is a Lagrange multiplier for the constraint (19). Eliminating the Lagrange multiplier in ξs u (as) + ξs u (as) − ξs u (as) + ξsu (as) + Substituting Eqs. (25), (26), and (30)-(32) into Eq. (29), I obtain Eq. (33) is interpreted as follows. If the social planner decreases as, the instantaneous utility of hh and lh investors decreases and as increases to satisfy market clearing conditions. In response, ll and hl investors’ utility increases. The changes in as and as 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 as, as, ηs, ηs, · · · ,
ξs , ξs , ξs , ξs that satisfies Eqs. (19)-(26) and (33). Figure 4 is a numerical example of the relationship between the matching efficiency and social 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 the competitive equilibrium. To confirm this, I incorporate Eqs. (9) and (10) to obtain: To derive Eq. (33), I substitute Eq. (2) into Eq. (34), multiply both sides of Eq. (34) by 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 when modeled 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 fees to be equal to the broker’s marginal costs, is the key for the competitive equilibrium to be socially 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 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 mod- ifying 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 aver- sion is smaller (larger) than unity. Lagos and Rocheteau (2007), using the random matching framework, 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 com- petitive equilibrium is efficient. The key condition for the optimum is that intermediation fees be equal to brokers’ operating costs. If the intermediation fees are larger than the brokers’ op- erating 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 is larger than optimal. Note, this condition is not achieved only by Hosios condition but also be- cause the brokers’ bargaining power takes a zero value in the random matching and bargaining 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.
Appendix
Proof of Lemma 1.
I prove Lemma 1 using theorems 4.7 and 4.8 from Stokey and Lucas (1989). It is sufficient to check 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.
Proof of Lemma 2.
because φ (·) is nonnegative by assumption. The strict concavity of value functions indicates h) + λh[sign−] + mn[sign−], where [sign+] and [sign−] mean a positive term and a negative term, respectively. Subtract Eq.
= uh h) − ul h) + [sign+], where I assume that u (a) ≥ u (a) for all a. This is a contradiction because ρ > 0.
Proof of Proposition 1.
Eqs. (37) and (38) indicate that the investors’ FOCs are h) = p + I(al,ah) n l) = p + I(ah,al) n First, transforming Eqs. (5) and (8), then h(ah) + λlVl(ah) , l(al) + λhVh(al) .
Next, differentiating both sides of Eq. (41) with respect to ah and Eq. (42) with respect to al, u (ah) + λlV (ah) u (al) + λhV (al) Substituting Eqs. (43) and (44) into Eqs. (39) and (40), u (ah) + λlV (ah) u (al) + λhV (al) h(al) + λlVl(al) + αn Vh(ah) − p(ah l(ah) + λhVh(ah) + αn Vl(al) − p(al Differentiating Eqs. (47) and (48) with respect to al and ah, respectively, and substituting Eqs.
Substituting Eqs. (49) and (50) for Eqs. (47) and (48), I can derive proposition 1.
Proof of Proposition 2.
References
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Hosios, A.J., 1990. On the efficiency of matching and related models of search and unemploy- 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.
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 mar- Voyanos, D., Weill, P.O., 2008. A search-based theory of the on-the-run phenomenon. J. Finan.
Weill, P.O., 2008. Liquidity premia in dynamic bargaining markets. J. Econ. Theory 140, Table 2: Values of parameters used in functions 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.
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.
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+, p+). If the
price remains at p∗, the high-type (low-type) investor’s optimal portfolio shifts from ah (al) to a (a ).
Figure 4: Welfare. The relationship between matching efficiency (M) as defined in Table 2 and social welfare
(W) as defined in Eq. (18).

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