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A simple model of bid-ask spread and search.

This paper analyzes a simple search model of market making in which the agents can choose between searching for a trading partner or trading through the market-maker. Explicit solutions for the equilibrium search intensities and the bid-ask spread have been provided. Equilibrium search intensities and the bid-ask spread reflect the strategic interaction between the agents and the market-maker: an increase in the bid-ask spread results in higher search intensities. The bid-ask spread, on the other hand, reacts negatively to an increase in the search intensities (due to lower search costs, higher efficiency of search, or higher gains from a match). It is also shown that the introduction of a market-maker increases the seller's reservation price and decreases the buyer's reservation price, hence narrowing the price dispersion.

I. Introduction

In many markets the process of finding a trading partner requires the agents to expend considerable time and incur search costs. Examples of such markets include financial markets in which buyers and sellers of financial assets search for each other, labor markets in which workers and firms search for each other, and good markets in which buyers and sellers of commodities search for each other. The fact that search is costly, combined with the uncertainty as to whether the search will lead to finding a trading partner, has given rise to intermediaries in most search markets. Some intermediaries are market-makers, who set ask and bid prices at which they buy and sell for their own accounts (e.g., specialists in stock markets, used car dealers, and retailers). Other intermediaries are match-makers, who do not buy or sell but simply match the agents (e.g., employment agencies in labor markets, real estate brokers in housing markets, and travel agents).

This paper studies the role of market-makers in a simple bilateral search model where a buyer and a seller search for each other in order to facilitate a trade. The agents have the choice of searching for each other or trading directly with the market-maker. The objective of the study is to examine the impact of the market-maker on the search and trading behavior of the agents, and the impact that this competition from the agents' search activity has on the bid-ask spread of the market-maker.

This paper is motivated by a gap between the Finance literatue and the Economics literature on market making. These literatures have focused on two different aspects of the problem. The Finance literature (Peck 1990, Garman 1976, O'Hara & Oldfield 1986, Stoll 1978, Ho & Stoll 1981, Glosten & Milgrom 1985) has concentrated on the behavior of market-makers and the determination of the bid-ask spread, without modeling the agents' search behavior explicitly and without allowing the agents to react strategically to the bid-ask spread choice of the market-maker. The Economics literature (Rubinstein & Wolinsky 1987, Gehrig 1990, Moresi 1990, Wooders 1990, Yavas 1990, 1991, 1992), on the other hand, has emphasized the impact of the market-maker on the equilibrium price, trading behavior, and welfare of the agents, without providing an explicit analysis of the impact of the agents' search behavior on the bid-ask spread choice of the market-maker. The main reason why earlier studies have failed to furnish a complete analysis of market-makers in search markets is the complexity of the problem. In particular, having three different sets of agents (buyers, sellers, and market-maker) behaving strategically in a world of uncertainty and incomplete information makes the analysis very difficult. In an attempt to fill the gap, this paper provides a search model that is simple enough to allow us to derive explicit solutions for the equilibrium bid-ask spread and the search intensities, yet rich enough to supply formal support for a number of interesting results.

The results of the paper can be summarized as follows. The market-maker decreases the equilibrium search intensities of the agents. The seller's reservation price becomes higher, while the buyer's reservation price becomes lower; hence, the price dispersion becomes narrower in the presence of a market-maker. The equilibrium bid-ask spread is shown to be positively related to the cost of search, but negatively related to the efficiency of search and to the agents' gains from a trade. It is also shown that the market-maker creates a coordination problem for the agents by generating an equilibrium in which both agents trade directly with her. The problem is one of a coordination because even if it is Pareto optimal for both agents to search for each other, they might get stuck at the (Pareto-inferior) equilibrium in which they both go directly to the market-maker.

The following section presents a bilateral search model, Section III generalizes this model to study the role of market-makers, and Section IV offers some conclusions.

II. The Search Model

First, we present the basic search model without any market-maker. This is a one-period bilateral search model where a risk neutral buyer and a risk neutral seller search for each other in order to trade. The seller has one unit of the commodity and the buyer wants to buy one unit. The seller values the good at [P.sub.s] and the buyer at [P.sub.b], [P.sub.b]>[P.sub.s].(1) The valuations are common knowledge.(2)

Each agent chooses a search intensity to maximize his expected return from search. The choice of the search intensity affects the probability and the cost of meeting the other agent. Let S[element of][0, 1/2] be the search intensity of the seller and B[element of][0, 1/2] be the search intensity of the buyer. The probability that the two agents meet is given by the matching technology 0[less than or equal to][theta](S,B)=[lambda](S+B)[less than or equal to]1, 0[less than or equal to][lambda][less than or equal to]1.(3) The cost of search is given by C(S)=[lambda][S.sup.2] for the seller, and by C(B)=[gamma][B.sup.2] for the buyer. The functions [theta](S,B), C(S) and C(B) are common knowledge. These functional forms have been chosen for two reasons. One is that they allow a simple and clear presentation of the main ideas of the paper, while allowing us to derive a closed-form solution for the equilibrium bid-ask spread. The second reason is that the parameters [lambda] and [gamma] provide an explicit expression of how efficient and costly the search is. An increase in [lambda] increases the probability of a match for any given S and B, hence makes search more efficient, while an increase in [gamma] makes search more costly. This will enable us to examine how the search behavior of the agents and the equilibrium bid-ask spread are related to the efficiency and cost of search.

When the two agents meet each other, they negotiate the price instantaneously. We will not model the bargaining process here. Instead, we simply assume that the negotiated price divides the surplus from trade, R=[P.sub.b]-[P.sub.s], such that the seller receives proportion [omega] of it, 1>[omega]>0, and the buyer gets proportion 1-[omega]. That is, the seller sells his unit to the buyer at a price of [P.sub.s]+[omega]([P.sub.b]-[P.sub.s]). For the current model, the special case of [omega] =.5 is the bargaining solution proposed by Nash (1950), Kalai & Smorodinsky (1975), and Rubinstein (1982). However, we do not assign any specific value for [omega], in order to observe how a change in the bargaining solution affects the search and trading behavior of each side.

Following the Cournot-Nash assumptions, each agent, in choosing his search intensity, treats the search intensity of the other agent as given. Given his conjecture about the search intensity of the buyer, B[degrees], the problem of the seller is to choose a search intensity to maximize his expected return; max [V.sub.s](S,B[degrees])=[lambda](S+B[degrees])[omega]R - [gamma][S.sup.2]. 1/2[greater than or equal]S[greater than or equal]0 Similarly, the buyer solves max [V.sub.b](S[degrees],B)=[lambda](S[degrees]+B)(1-[omega]R - [gamma][B.sub.2], 1/2[greater than or equal]B[greater than or equal]0 where S[degrees] is the buyer's conjecture about the search intensity of the seller.

Note that the search intensities of the agents are strategically independent. That is, a change in the search intensity of one agent does not affect the search intensity choice of the other agent. Given this independence, the equilibrium will consist of a pair of dominant strategies, one for each agent. These strategies are given by the following search intensities: S*=[lambda][omega]R / 2[gamma], and B*=[lambda](1-[omega])R / 2[gamma].

It is possible for [lambda][omega]R/2[gamma] and/or [lambda](1-[omega])R/2[gamma] to exceed 1/2, in which case the solution to problems (1) and (2) will be a corner solution, with S*=1/2 and/or B*=1/2.

It is clear from (3) and (4) that an increase in the buyer's valuation and/or a decrease in the seller's valuation (i.e., an increase in gains from a match, R), an increase in the efficiency of the search, or a decrease in search costs will increase the equilibrium search intensities by both agents. An increase in [omega] will increase the seller's equilibrium search intensity while decreasing the buyer's.

The total welfare (seller's expected surplus + buyer's expected surplus) in this economy is equal to

[MATHEMATICAL EXPRESSION OMITTED]

If the solution to (1) and (2) is a corner solution with S*=B*=1/2, the total surplus, then, becomes TS=[lambda]R-[gamma]/2.

Note that the equilibrium in this search economy involves two sources of inefficiencies. One is the uncertainty as to whether the search efforts of the agents will result in a match, and the other is the positive externalities that exist in the matching process. These inefficiencies could give rise to an institution that could reduce the uncertainty, and could internalize some of the externalities in return for some profits. As a next step, we introduce intermediation into this search economy as such an institution.

III. Market-Maker

The middleman examined here is a market-maker, who sets an ask price and a bid price at which she sells and buys for her own account (e.g., a specialist on a stock exchange or a used-car dealer). In some markets, however, the middleman does not buy or sell, but simply matches the buyer with the seller (e.g., a real-estate broker or an employment agent). Later in the paper, this second type of middleman is briefly discussed and compared with the market-maker.

The market-maker is assumed to be risk neutral, and is assumed to have monopoly power. As explained below, however, the market-maker is subject to some competition from the search market because the agents are allowed to skip the market-maker and search for each other.

The market-maker sets an ask price, [alpha], and a bid price, [beta], at which she sells and buys, respectively. She sets [alpha] and [beta] to maximize her expected profits. She is assumed to have a unit endowment of the good which enables her to sell a unit without buying one, i.e., she is capable of providing the service of immediacy. Her valuation of the unit is given by I, where [P.sub.b]>I>[P.sub.s].(4)

The market-maker posts the ask and bid prices at the beginning of the period, and agents are assumed to know these prices before they choose their search intensities. Agents can trade with the market-maker instantaneously, and at no cost. At the beginning of the period, an agent has two strategies; either he can trade directly with the market-maker, or else he can search for the other agent. The fact that the agents can skip the market-maker and search for each other creates some competition for the market-maker(5), and this competition will be reflected in her choice of the bid-ask spread.

An agent who searches and does not meet the other agent can still go back and trade with the market-maker. However, if an agent trades with the market-maker after having already searched, his surplus is assumed to be discounted by a constant, [delta], 0[less than or equal to][delta][less than or equal to!1. Consider, for example, the seller who has valuation [P.sub.s]<[beta]. If he trades directly with the market-maker, he gets [beta]-[P.sub.s]. If he chooses to search, he has a chance to find the buyer and to receive a surplus greater than [beta]-[P.sub.s]. It is possible that search does not result in a match ([theta](.)<1), in which case the seller can go back to the market-maker and get [delta]([beta]-[P.sub.s]) (similarly [delta]([P.sub.b]-[alpha]) for the buyer). The discount factor [delta] reflects the fact that search is time consuming. Thus, search has the indirect effect of reducing agents' surplus from trading with the market-maker (if their search efforts fail to yield a match) as well as the direct effect of being costly.

Given the sequential nature of the game, we will utilize the concept of subgame perfection (Selten, 1975) in our equilibrium analysis. Note that, given the [alpha] and [beta] choices of the market-maker, if an agent expects the other agent to go directly to the market-maker, then his best response is also to trade directly with the market-maker. The reason is that because if one agent chooses to trade directly with the market-maker, then a search by the other agent will not result in a match. Thus, in equilibrium, either both agents search or they both go directly to the market-maker. Although we have not characterized the equilibrium yet, we can show that the following is true in any equilibrium of this game.

Lemma 1: The market-maker will choose [alpha][less than or equal to] [P.sub.b] and [beta][greater than or equal][P.sub.s].

Proof: Given the assumption that [P.sub.b]>I>[P.sub.s], we can show that it is never optimal for the market-maker just to sell or just to buy a unit. To see this, suppose that [alpha][less than or equal to][P.sub.b] and [beta]<[P.sub.s] so that the market-maker elects only to sell a unit. When the agents go to the market-maker, the market-maker gains a surplus of [alpha]-I by selling her unit. This amount is less than what she can get by setting [beta] at I>[beta][greater than or equal to][P.sub.s] (in which case she sells and buys a unit), ([alpha]-I)+([beta]-I)=[alpha]- [beta], because [beta]<I. A similar argument can be used to show that she also will never choose only to buy a unit. Hence, she will set [alpha][less than or equal to][P.sub.b] and [beta][greater than or equal to][P.sub.s], and will buy and sell a unit.

Because the agents can still trade with the market-maker after having already searched, the surplus that they can get from trading with the market-maker will constitute a threat point in their bargaining process with each other. The bargaining rule divides the surplus in the following way. When the buyer and the seller meet, first each party gets the surplus that he can obtain from trading with the market-maker, so the seller gets [delta]([beta]-[P.sub.s]) and the buyer collects [delta]([P.sub.b]-[alpha]). The remaining part of the surplus is then divided between the two agents according to the [omega] rule of Section II. Hence, the seller gets [delta]([beta]-[P.sub.s])+[omega][([P.sub.b]-[P.sub.s])-[delta]([beta]- [P.sub.s])-[delta]([P.sub.b]-[alpha])] and the buyer recives [delta]([P.sub.b]-[alpha])+(1-[omega])[([P.sub.b]-[P.sub.s])-[delta] ([beta]-[P.sub.s])-[delta]([P.sub.b]-[alpha])]. This also implies that the seller will never sell his unit to the buyer at a price below [P.sub.s]+[delta]([beta]-[P.sub.s]), which constitutes the seller's reservation price under the market-maker. Similarly, the buyer's reservation price becomes [P.sub.b]-[delta]([P.sub.b]-[alpha]). One justification for the choice of the above bargaining rule is that it is equivalent to splitting the difference between the agents' reservation prices. The rule therefore is consistent with the bargaining rule employed in section II, because in section II the valuations are also the reservation prices of the agents. A comparison of the reservation prices in sections II and III leads to the following result:

Proposition 1: The market-maker increases the seller's reservation price and decreases the buyer's reservation price.

Proof: The agents' reservation prices without the market-maker are simply their valuations, [P.sub.s] for the seller and [P.sub.b] for the buyer. Clearly, [P.sub.s][less than or equal to][P.sub.s]+[delta]([beta]-[P.sub.s]) and [P.sub.b][greater than or equal to][P.sub.b]-[delta]([P.sub.b]-[alpha]).

Proposition 1 implies that there is less price dispersion in a market facilitated by a market-maker, in the sense that the range of prices at which a transaction will take place is bounded above by [alpha] and below by [beta]. This result supports the claim that one of the important functions of the specialists in the stock markets is to reduce the volatility of stock prices. It has been shown in Yavas (1992) that Proposition 1 also holds when the valuations of the agents are private information. Note that Proposition 1 is true for any [alpha] and [beta], thus will also be true for the equilibrium values of [alpha] and [beta].

As mentioned earlier, in equilibrium either both agents search or they both go directly to the market-maker. If the agents choose to search, then the seller and the buyer solve the following two problems, respectively: max [v.sub.s](S,B',[alpha],[beta],) = [lambda](S+B')[omega](R-[delta] ([P.sub.b]-[alpha])-[delta]([beta]-[P.sub.s])) + [delta]([beta]-[P.sub.s]) - [gamma][S.sup.2] (6) 1/2[greater than or equal to]S[greater than or equal to]0 max [v.sub.b](B,S',[alpha],[beta],) = [lombda](S'+B)(1-[omega])(R-[delta] ([P.sub.b]-[alpha])-[delta]([beta]-[P.sub.s])) + [delta]([P.sub.b]-[alpha]) - [gamma][S.sup.2] (7) 1/2[greater than or equal to]B[greater than or equal to!0

where B' (S') is the seller's (buyer's) conjecture about the search intensity of the buyer (seller) in the presence of a market-maker. Notice that the threat points, [delta]([beta]-[P.sub.s]) and [delta]([P.sub.b]- [alpha]), are not multiplied by the matching probability because the agents will receive their threat points regardless of whether they trade with each other or with the market-maker.

For a given [alpha] and [beta], we can find the equilibrium search intensity functions (if the agents choose to search) by solving (6) and (7): S**([alpha],[beta]) = [lambda][omega][ (1-[delta])R+[delta]([alpha]-[beta]) ] / 2[gamma] (8) B**([alpha],[beta]) = [lambda](1-[omega])[ (1-[delta]R+[delta]([alpha]-[beta]) ] / 2[gamma]. (9)

Proposition 2: The market-maker decreases the equilibrium search intensities of the agents.

Proof: We know from Lemma 1 that [alpha][less than or equal to][P.sub.b] and [beta][greater than or equal to][P.sub.s], hence [alpha]-[beta][less than or equal to]R. This result yields (1- [delta])R+[delta]([alpha]-[beta][less than or equal to!R, i.e., the market-maker decreases the agents' gains from search. As a result, a comparison of (3) with (8) and (4) with (9) yields smaller search intensities for the agents if there is a market-maker.

It is also clear from (8) and (9) that Proposition 3: S**([alpha],[beta]) and B**([alpha],[beta]) will increase as [alpha]-[beta] increases.

The intuition is that as the bid-ask spread gets larger, agents' net gains from search (net of what they can get from trading with the market-maker in the event that their search efforts do not result in a match) get bigger. This, in turn, increases the agents' equilibrium search intensities.

Next, we use backward induction argument to formalize the market-maker's problem. After solving for the buyer's and the seller's optimum search intensities for given values of her ask and bid prices, S**([alpha],[beta]) and B**([alpha],[beta]) respectively, the market-maker chooses the bid-ask spread that maximizes her expected profits. In choosing [alpha]-[beta] the market-maker takes into consideration the two opposing effects of an increase in [alpha]-[beta]: On one hand, it increases her profits in the event the agents trade with her. On the other hand, it increases the search intensities of the agents (hence increasing the probability that they will meet), which in turn decreases the probability that they will go to her.

As mentioned earlier, there are two pure strategy equilibria in this game. One equilibrium consists of both agents' going directly to the market-maker: if an agent expects the other agent to go directly to the market-maker, then the first agent's best response is also to trade directly with the market-maker. This is so because if one agent chooses to trade directly with the market-maker, then a search by the other agent will not result in a match. In the other equilibrium, both agents choose to search before going to the market-maker.(6) The first equilibrium will be referred to as the "dealer equilibrium" and the second one will be called the "search equilibrium". It is assumed that when there are two equilibria, the search equilibrium will take place with probability q, and the dealer equilibrium will take place with probability 1-q.(7) This multiplicity of equilibria, and the conditions under which one equilibrium Pareto dominates the other, is discussed in the appendix.

The market-maker's problem becomes: max II([alpha],[beta]) = q[1-S**([alpha],[beta])-B**([alpha],[beta])][delta] ([alpha]-[beta]) + (1-q)([alpha]-[beta]). (10) [alpha],[beta]

Substituting (8) and (9) into, and then solving, (10) yields the equilibrium bid-ask spread: [alpha]*-[beta]* = [ 2(1-q)[gamma]+2q[gamma][delta]-q[gamma][delta](1-[delta]) R ] / 2q[gamma][[delta].sup.2]. (11)

It is straightforward from (11) that
Proposition 4:   i. [delta]([alpha]*-[beta]*)/[delta][gamma] > 0,
                ii. [delta]([alpha]*-[beta]*)/[delta][lambda] < 0,
                iii. [delta]([alpha]*-[beta]*)/[delta]R < 0,
                 iv. [delta]([alpha]*-[beta]*)/[delta]q < 0.


The intuition for these results is clear: as search becomes more costly or less efficient, or as the gain from a match becomes smaller, the agents will search less, and this reduced searching activity enables the market-maker to charge a higher spread. The market-maker also charges a higher spread when it is more likely that both agents will go directly to her.

Clearly, we will have [alpha]*>[beta]*. Otherwise, there would be arbitrage opportunities for the agents and negative expected profits for the market-maker. If the profit-maximization yields [alpha]*<[beta]* then the market-maker can simply set [alpha]*> [P.sub.b] and [beta]*<[P.sub.s] (and earn zero profits); in other words, no market-making services will be provided in that market. It is also possible for the First Order Conditions for (10) to yield [alpha]-[beta]>R (if, for example, q or [lambda] are very small), in which case we obtain a corner solution at [alpha]*-[beta]*=R.

Given the parameters of the model, [lambda], [omega], q, [delta], and [gamma], the equilibrium can be characterized as follows:

(i) each agent's choice of whether to search or to trade directly with the market-maker is optimal given the strategy choice of the other agent and the ([alpha]*, [beta]*) choice of the market-maker,

(ii) if the seller chooses to search, his search intensity choice S** ([alpha]*,[beta]*) maximizes his expected return from search given his conjecture about the buyer's search intensity, B**([alpha]*,[beta]*), and given the ([alpha]*,[beta]*) choice of the market-maker, and thus satisfies equation (8),

(iii) if the buyer chooses to search, his search intensity choice B** ([alpha]*,[beta]*) maximizes his expected return from search given his conjecture of the seller's search intensity, S**([alpha]*,[beta]*), and given the ([alpha]*,[beta]*) choice of the market-maker, and thus satisfies equation (9), and

(iv) the market-maker's choice of ([alpha]*,[beta]*) maximizes her expected profits given her conjecture of the optimal reactions of the seller and the buyer, and thus solves equation (10).

Substituting (11) into (8) and (9) gives the closed-form solution for the equilibrium search intensities: S** = [lambda][omega][ 2(1-q)[gamma]+2q[delta][gamma]+q[lambda][delta](1- [delta])R ] / 4[gamma]q[lambda][delta], and (12) B** = [lambda](1-[omega])[ 2(1-q)[gamma]+2q[delta][gamma]+q[lambda][delta](1- [delta])R ] / 4[gamma]q[lambda][delta]. (13)

As in Section II, the equilibrium search intensities increase with R and [lambda], and decrease with [gamma]. It also can be shown that both of the search intensities are negatively related to [delta].

An interesting result here is that the introduction of a market-maker creates a coordination problem for the agents by generating a new equilibrium (dealer equilibrium) in which both agents go directly to the market-maker (see the appendix). It can be shown that when q is small enough, the surplus of the buyer and the seller under the dealer equilibrium is smaller than their surplus when there is no market-maker (provided in equation (5)). Since small q also means higher probability that the dealer equilibrium will be the outcome, it is possible that introduction of the market-maker leaves the agents worse off. A more detailed analysis of the welfare impact of the market-maker can be found in Yavas (1990 and 1992).

The above analysis involves a single profit maximizing market-maker who is subject to an indirect competition from the search market. What happens if we introduce direct competition among market-makers, or the market-maker acts as a social planner and sets [alpha]=[beta]? It can be shown that Lemma 1 still holds; [P.sub.b][greater than or equal to][alpha]=[beta][greater than or equal to] [P.sub.s]. Given that [alpha]=[beta], each agent receives the same surplus from trading with each other as from trading with the market-maker because [P.sub.b]-[P.sub.s]=([P.sub.b]-[alpha])+([beta]- [P.sub.s]). Since search is costly, the result will be a unique (Walrasian) equilibrium where all trades go through the market-maker and the search market collapses.

As mentioned at the beginning of the paper, intermediaries in some markets do not buy or sell, but simply match the agents in return for a commission fee. Examples include real estate brokers, match-makers in marriage markets, and employment agencies. The analysis presented in this paper can easily be applied to these intermediary types by simply interpreting the bid-ask spread as the commission fee. A more detailed analysis of these two intermediary types can be found in Yavas (1992, 1993a, and 1993b).

IV. Concluding Remarks

We have analyzed a simple search model of market making in which the agents can choose between searching for a trading partner or trading through the market-maker. Explicit solutions for the equilibrium search intensities and the bid-ask spread have been provided. Equilibrium search intensities and the bid-ask spread reflect the strategic interaction between the agents and the market-maker: an increase in the bid-ask spread results in higher search intensities. The bid-ask spread, on the other hand, reacts negatively to an increase in the search intensities (due to lower search costs, higher efficiency of search, or higher gains from a match). It is also shown that introduction of a market-maker increases the seller's reservation price and decreases the buyer's reservation price, hence narrowing the price dispersion.

It should be noted that the market-maker modeled in this paper is not a "specialist" per se. The model presented in the paper is probably too simple to describe any real world market. Yet, it is rich enough to yield some interesting results. It is hoped that this study provides a framework for future research directed toward analyzing the market making process in an environment where the traders and the market-maker behave strategically against each other.

In this paper, attention has been confined to search economies that display unique equilibrium. Some search economies, however, might display multiple equilibria and coordination failures (see Diamond 1982). It has been shown in Yavas (1991) that when there are multiple equilibria, the market-maker has an equilibrium selection role, and eliminates some of the Pareto dominated equilibria.

Appendix

Here, we discuss the two equilibria and compare them from the point of view of the agents' welfare.

There is always a dealer equilibrium in which both agents trade directly with the market-maker: If an agent expects the other agent to go directly to the market-maker, then his best response is also to trade directly with the market-maker. What if an agent expects the other agent to search? In that case, he will choose to search if his gains from search exceed his gains from trading directly with the market-maker. Substituting (8) and (9) into (6) and (7), we obtain the conditions under which the seller and the buyer, respectively, prefer search equilibrium over dealer equilibrium: (2[[lambda].sup.2][omega]-[[lambda].sup.2][[omega].sup.2]) [[R(1-[delta])+ [delta]([alpha]-[beta])].sup.2] / 4[gamma] + [delta]([beta]-[P.sub.s]) > ([beta]-[P.sub.s]) (A1) (2[[lambda].sup.2](1-[omega])-[[lambda].sup.2][(1-[omega]).sup.2]) [[R (1-[delta])+[delta]([alpha]-[beta])].sup.2] / 4[gamma] + [delta]([P.sub.b]- [alpha]) > ([P.sub.b]-[alpha]). (A2)

If neither (A1) nor (A2) holds, then one of the agents will always trade directly with the market-maker regardless of what he expects the other agent to do. Since the other agent's best response will be also to trade directly with the market-maker, we will find the dealer equilibrium to be the unique equilibrium (q=0). This implies that when the parameters of the model allow (A1) or (A2) not to hold for some [alpha] and [beta], the market-maker can set [alpha] low enough and/or [beta] high enough so that it is a dominant strategy for at least one of the agents to go directly to her. It is also interesting to note that the market-maker can, for instance, set [alpha] and [beta] low enough so that it is a dominant strategy for the buyer to go directly to the market-maker while the seller prefers to search. Yet the seller's best response to the buyer's strategy is also to go directly to the market-maker even if [beta]-[P.sub.s] is very low. In other words, the market-maker can induce both of the agents to go directly to her by giving enough incentives to only one of the agents, and extracting all the surplus from the other agent. Clearly, the market-maker will follow such a strategy only if it yields more profits than setting [alpha] and [beta] such that the agents will search with probability q and go directly to her with probability 1-q.

However, given that we observe search activity in most intermediated markets, we will confine our attention to the search economies for which (A1) and (A2) hold for any [alpha] and [beta]. For [gamma] small enough or for [delta] close to 1, for example, (A1) and (A2) will hold for any [alpha] and [beta]. (A1) and (A2) are also more likely to hold as [lombda] gets bigger. Note that when (A1) and (A2) hold, both agents prefer the search equilibrium over the dealer equilibrium. Some game theorists (for example, Harsanyi & Selten 1988) argue that the Pareto-dominant equilibrium (i.e., the search equilibrium) is a natural focal point, and hence that the agents will focus on the Pareto-dominant equilibrium (q=1). The experimental results by Cooper et al. (1989), however, indicate that the Pareto-dominant equilibrium is not always the outcome. Based on this finding, we will allow q to be in the open interval (0,1).

Endnotes

(1.)The difference between the two agents' valuations can be explained by their different liquidity needs, information, portfolio considerations, etc.

(2.)This assumption enables us to derive explicit solutions for the equilibrium search intensities and bid-ask spread later in the paper. As has been demonstrated by some previous studies (see, for example, Yavas 1990; and Moresi 1990), it is extremely difficult to derive these explicit solutions when the valuations of the agents are private information. Even though this assumption makes the model too simple to reflect any real world markets, the model remains rich enough to yield a number of interesting and practical results.

(3.)Mortensen (1982a) uses a similar matching technology.

(4.)This assumption is not crucial. All the results of the paper hold when the market-maker's valuation is above that of the buyer or below that of the seller. The reason for this assumption is that it will simplify the profit function of the market-maker later.

(5.)In stock markets, for example, one can interpret the upstairs markets as the competition that the specialists face from the search markets. A greater and greater volume of stock trades are being carried out off the floor of the NYSE. Stoll (1985) reports that about 21% of share volume is prearranged in upstairs markets and executed without specialist participation.

(6.)There might also be mixed strategy Nash equilibria in which the agents randomize between searching and going to the market-maker, and play each strategy with a certain probability. The attention in this paper, though, will be confined to pure strategy equilibria.

(7.)This randomization between the two equilibria has been borrowed from the literature on sunspots (as an example, see Chatterjee, et al. (1993)). The probability q is exogenously given and it is unrelated to the fundamental variables, such as preferences or endowments.

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Author:Yavas, Abdullah
Publication:Review of Financial Economics
Date:Mar 22, 1993
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