OPTIMAL AGENCY RELATIONSHIPS IN SEARCH MARKETS

Christopher Curran and Joel Schrag*

October 2000

Abstract: We compare a seller’s agency regime, in which agents give sellers information about 

buyers’ willingness to pay, with a buyer’s agency regime, in which agents keep buyers’ 

information confidential. Aggregate gains from trade can be higher under either agency 

regime. Aggregate gains from trade are higher under buyer’s agency if traders expect to 

spend less time in the market under buyer’s agency. Equivalently, aggregate gains from 

trade are higher under seller’s agency only if traders expect to spend less time in the 

market under seller’s agency. We use our theoretical results to interpret empirical 

findings on the effects of different agency regimes.

Keywords: search, matching markets, real estate, agency

 

*

 Associate Professor, Emory University and Economist, Federal Trade Commission, respectively. Curran can 

be reached by email at econcc@emory.edu and by phone at (404) 727-6355, and Schrag can be reached by

email at jschrag@ftc.gov and by phone at (202) 326-3588. While all errors remain the authors’ responsibilities, 

they thank two anonymous referees for helpful comments. The views expressed in this paper are the authors’ 

alone and do not necessarily reflect the opinions of the Federal Trade Commission or any individual

Commissioner.
1

1. Introduction

Agents play an important role in many search markets. In the real estate market, for example, 

agents improve the efficiency of the search process by helping buyers and sellers locate trading

partners, identify situations where there are gains from trade, and take care of the technical details 

of a transaction. Agents may also have an important effect on the flow of information between

buyers and sellers, depending on whom the agents work for and the nature of their legal

responsibilities. For instance, if an agent helps a buyer to find trading partners, it is likely that he 

will learn a great deal about the buyer’s preferences. Unless the agent is bound by a duty of

confidentiality, he may reveal what he knows to sellers, who would presumably benefit from

access to this information, since they could use it when formulating their bargaining strategy. It

may be in the agent’s interest to strengthen the seller’s hand if, as is common in the real estate

market, his compensation is proportional to the sales price that the buyer and seller negotiate.

In this paper we investigate how the role that agents play in a search market affects welfare. 

Our principal question is whether agents should be able to transmit information about buyers’ 

willingness-to-pay to sellers, assuming that society’s goal is to maximize the discounted expected 

gains from trade that the market generates. To this end, we analyze the equilibrium that arises in 

a search market under alternative assumptions about the agents’ role. For concreteness, we

develop our results using a model of the real estate market, but our conclusions are applicable to 

any search market in which sellers wish to sell and buyers wish to buy one unit of an indivisible 

good, buyers are privately informed about their valuations, and sellers’ valuations are commonly 

known.

In our analysis, we contrast a seller’s agency regime with a buyer’s agency regime. These are 

the alternative legal regimes that have prevailed in the real estate market in recent years.1 In the 

traditional seller’s agency system, which was the dominant legal regime in the real estate market

 

1

 For a thorough description of these agency relationships, see Curran and Schrag (2000), and especially Lefcoe 

(1993).
2

until relatively recently, all agents represent sellers’ interests. Under this system, an agent in a

transaction has an obligation to share with the seller any information that the buyer has revealed

to the agent. In recent years it has become much more common for buyers to employ a buyer’s 

agent in their search for a house, rather than a traditional seller’s agent. Under buyer’s agency, 

the buyer’s agent works to advance the buyer’s interests, and he is therefore prohibited from

revealing information about his client to sellers.

After establishing that there exists an equilibrium of our theoretical model, we use a pair of

numerical examples to show that it is not possible to conclude that either of the two agency

regimes is generally superior, in the sense of always delivering greater total discounted expected

gains from trade to a representative buyer and seller. Welfare can be higher under either agency 

regime. Nevertheless, we do prove a general result that identifies circumstances in which it is

possible to rank the different agency regimes. Suppose that buyer’s agents are at least as

effective as seller’s agents at introducing buyers to potential trading partners. Then a sufficient

(though not necessary) condition for the buyer’s agency regime to be superior to the seller’s 

agency regime is that the time traders expect to spend in the market before reaching a deal is

lower under buyer’s agency than under seller’s agency. Equivalently, a necessary (though not

sufficient) condition for the seller’s agency regime to be superior to the buyer’s agency regime is 

that the time that traders expect to spend in the market before reaching a deal is lower under

seller’s agency than under buyer’s agency.

These findings are reminiscent of results in the literature on the welfare effects of thirddegree price discrimination by a monopolist. See, e.g., Schmalensee (1981) and Varian (1985). 

In this literature, third-degree price discrimination has a positive effect on social welfare (defined 

as the sum of producer and consumer surplus) only if it leads the monopolist to produce more

units. Equivalently, the elimination of third-degree price discrimination must have a positive

effect on social welfare if it is accompanied by an increase in output. The intuition behind these 

results is straightforward. Price discrimination may lead to an improvement in welfare if it leads 
3

a seller to sell more units and thus reduces the loss from the monopoly restriction on output. But, 

because a shift to price discrimination tends to cause the price of the relevant good to fall for lowvaluation buyers and to rise for high-valuation buyers, units are shifted from high-valuation to

low-valuation buyers, creating a misallocation of units. Thus, a necessary condition for price

discrimination to be welfare improving is that it is accompanied by an output increase that can

offset the negative effect of any misallocation of units. Equivalently, a sufficient condition for

price discrimination to be welfare harming is that it is accompanied by an output reduction.

In our model, a seller’s agency regime allows sellers to acquire information that they can use 

to discriminate between buyers with different valuations, and hence it is like a market in which

price discrimination is possible. Of course, sellers must have some bargaining power in their

meetings with buyers in order to exploit this information; for simplicity we assume that sellers

make take-it-or-leave-it offers. In our model, the probability that traders conclude a deal in any

meeting (which is inversely related to the time that traders expect to spend in the market) is

analogous to the monopolist’s output. Just as a monopolist inefficiently restricts output, sellers

have a tendency to bargain with an inefficiently large number of buyers before concluding a deal. 

Therefore, the discounted expected gains from trade generated by the market tend to increase if

the traders on average reach deals faster.

On the other hand, a seller’s agency regime may lead to a misallocation of units, as occurs

under third-degree price discrimination, because a well-informed seller chooses a reservation

price that is correlated with her estimate of a buyer’s true valuation. This pricing behavior has a 

tendency to reduce the likelihood that a high valuation buyer will purchase a particular house. 

Therefore, even if traders conclude deals faster under seller’s agency, welfare may still be lower

compared to buyer’s agency because of the potential misallocation of properties.

These theoretical findings enable us to interpret recent empirical findings on the effect of the 

recent shift in the real estate market from a seller’s agency regime to a buyer’s agency regime. 

The results that Schrag and Curran (2000) and Elder, Zumpano, and Baryla (1999) present all
4

support a conclusion that traders conclude deals faster under buyer’s agency than under seller’s 

agency. Interpreted through the lens of our theoretical results, these findings are consistent with a 

conclusion that welfare is higher under a buyer’s agency regime than under a seller’s agency

regime.

Our paper contributes to the literature on search models of the real estate market by analyzing 

the role of agents who facilitate trades between buyers and sellers. See, for instance, Wheaton

(1990) and Yavas (1992). Other papers that investigate the matchmaking role that intermediaries 

play in search markets include Yavas (1995), who studies whether agents can help traders avoid

coordination failures, Salant (1991), who studies a seller’s decision about whether or not to use an 

agent to sell a house, and Rubinstein and Wolinsky (1987), who analyze the factors that

determine the extent to which intermediated trade replaces search by the traders themselves. 

None of these papers consider how the institutions that govern the relationship between

intermediaries and traders affect the equilibrium that arises in search markets. This issue is our

main focus. Our paper also contributes to the literature on bargaining and markets (e.g. Binmore 

and Herrero (1988), Samuelson (1992), and Rubinstein and Wolinsky (1990) by showing how

institutions that influence the outcome of bilateral bargaining, such as agency law, have an

important effect on the equilibrium of search markets.

2. A Model of the Real Estate Market

We begin by developing a model of the real estate market. Suppose that, in every period t ∈

{0, 1, 2, ... }, agents introduce prospective buyers to prospective sellers. We assume that there is 

a continuum of ex ante identical buyers and a continuum of ex ante identical sellers. For

simplicity, we assume that the number of traders is the same on both sides of the market.

Furthermore, we assume that, for any given buyer or seller, the probability of being matched in

any given period is qx ∈ (0, 1], x ∈ {BA [Buyer’s Agency], SA [Seller’s Agency]}. We will have 
5

qSA ≠ qBA if the ability of agents to match buyers and sellers is different under the alternative

agency relationships. Suppose, for instance, that buyers are more inclined to reveal information

about their preferences under buyer’s agency, when their agents are required to keep this

information confidential. Or, as Elder, Zumpano, and Baryla (1999) argue, suppose that a

buyer’s agent has higher-powered incentives to find houses for his clients. Then we would expect 

to observe qBA ≥ qSA, i.e. the agent would be a more effective matchmaker under buyer’s agency.

Each buyer wishes to purchase one house, and each seller wishes to sell one house. In each

match between a buyer and a seller, the buyer’s valuation of the seller’s house is a random

variable, and different buyers' valuations for a given seller’s house are independently and

identically distributed. Define vt ∈ V = [0,  ̄v] as a representative buyer’s valuation of a seller’s 

house in a match that occurs in period t. 

It is reasonable to assume that a buyer is privately informed about his valuation for a

particular seller’s house.2 Depending on the nature of the prevailing agency regime, however, a

seller may receive information about a buyer’s valuation for her house. In a seller’s agency

regime, a buyer’s real estate agent transmits information about his valuation for a particular house 

to the appropriate seller. In order to model this flow of information, we suppose that, under

seller’s agency, the seller observes a signal σt ∈ Σ = [σ, σ ̄] of the buyer’s valuation in period t. In 

a buyer’s agency regime, a buyer’s real estate agent does not share information about his or her

client, and the seller cannot observe the realization of σt. 

The buyer’s period t valuation vt and signal σt are jointly distributed according to the

probability density function f(vt, σt). We assume that f(⋅, ⋅) is continuous and has full support on 

the domain V × Σ. Each valuation and signal pair is independently and identically distributed. In 

order to reflect the idea that the seller’s signal is informative about the buyer’s valuation, we

 

2

 For expositional clarity, we use the feminine pronoun to refer to sellers and the masculine pronoun to refer to 

buyers.
6

assume that f(⋅,⋅) satisfies the familiar monotone likelihood ratio property. Under this assumption, 

a higher realization of σt shifts the seller’s posterior beliefs towards higher valuations. Formally, 

for v¢ > v′′, f(v′, σ′)/f(v′′, σ′) > f(v′, σ′′)/f(v′′, σ′′) if and only if σ′ > σ′′.

In order to simplify the model, we assume that sellers make take-it-or-leave-it offers that

buyers either accept or reject.3 Therefore, each seller chooses a pricing rule that identifies the

price that she demands as a function of what, if anything, she knows about the potential buyers

she could meet. Each buyer, meanwhile, chooses the acceptance rule that identifies which

proposed transactions he will accept. In each match, trade occurs— and the traders exit the

market— if the buyer accepts the seller’s proposed price. If, in a particular match, the buyer

rejects the seller’s proposed price, the buyer and the seller part company and reenter the matching 

process in the next period. Both buyers and sellers discount future payoffs according to the

common discount factor δ∈ [0, 1).

The number of participants in the market fluctuates from period to period unless the entry of 

new buyers and sellers in each period exactly offsets the exit of traders who concluded deals in

the previous period. Because the number of traders in the market may influence the probability

of being matched, our assumption that qx is constant may be strong. We do not formally model 

the entry of new traders into the market, but our assumption that qx is constant could be

interpreted as meaning that buyers and sellers believe that the market is in a steady-state 

equilibrium and that the expected number of traders does not fluctuate from period to period. 

Then qx represents the buyers’ and sellers’ beliefs about the probability of being matched in each 

period under the different agency regimes.

 

3

 In an earlier version of this paper, we analyze a strategic bargaining model in which sellers and potential

buyers make alternating offers. Permitting this possibility did not change our main conclusions, because

privately informed buyers did not signal their information in equilibrium. Intuitively, high valuation buyers

always had an incentive to deviate to some offer made by lower valuation buyers in order to elicit a favorable 

counteroffer from the seller. In the present model, the sellers’ offers can be interpreted as counteroffers to

buyers’ initial (uninformative) offers. 
7

Define the conditional probability density and distribution functions f(v|σ) =

∫ σ

σ

v

f x dx

f v

0

( , )

( , )

 

and F(v|σ) = ∫ σ

v

f x dx

0

( | ) . In the following proposition we establish that there exists a symmetric 

Nash equilibrium under both buyer’s agency and seller’s agency if the respective hazard

functions 

∫

∫∫

σ

σ

σ

σ

f x y dy

f z y dydz

v

x

( , )

( , )

 and 

( | )

1 ( | )

s

s

f x

− F x

 are decreasing in x. These assumptions are

sufficient to guarantee that a seller’s expected revenue is a strictly quasi-concave function of her 

reservation price and therefore has a unique maximum. We focus on symmetric equilibria in

which each (ex ante identical) seller chooses the same pricing strategy and each (ex ante

identical) buyer chooses the same offer acceptance strategy.

PROPOSITION 1: Suppose that

∫

∫∫

σ

σ

σ

σ

f x y dy

f z y dzdy

v

x

( , )

( , )

 and 

( | )

1 ( | )

s

s

f x

− F x

 are decreasing in x. Then 

there exists a symmetric Nash equilibrium under both agency regimes.

The proofs of all results are located in Appendix 1.

In general, the symmetric Nash equilibria that arise under the two agency regimes are not

socially efficient, in the sense of maximizing the total discounted expected gains from trade that a 

representative buyer and seller receive. We establish this result in the following proposition.

 PROPOSITION 2: The symmetric Nash equilibria that arise under the alternative agency

relationships do not maximize the total discounted expected gains from trade in the

market. Under buyer’s agency, traders expect to spend more time in the market than is

socially optimal.
8

The intuition for the result in Proposition 2 is straightforward. When sellers and buyers

decide whether or not to conclude a deal, they do not consider the effect of their decisions on

their bargaining partners. So, for instance, when a seller considers whether to reduce her

reservation price and propose a deal that would be acceptable to a lower valuation buyer, she

must weigh the increased probability of concluding a deal against the loss of surplus from

inframarginal buyers. Of course, from a social point of view this surplus is not lost; it is merely 

transferred from the seller to the buyer. Because the seller is concerned with a cost that is

irrelevant from the point of view of aggregate gains from trade, she is, on average, too reluctant to 

conclude a deal in each meeting with a buyer.

The following numerical examples illustrate that it is not possible to conclude that one of the 

two agency regimes always delivers higher welfare. Details of the numerical examples’ solutions 

are located in Appendix 2. Suppose that, in each meeting between a buyer and a seller, the

buyer’s valuation is drawn from one of two possible distributions. If the match is a “high 

valuation” match, the buyer’s valuation is distributed on the unit interval according to the

probability density function fH(v) = 2v and the associated probability distribution function FH(v) = 

v

2

. If the match is a “low valuation” match, the buyer’s valuation is distributed on the unit interval 

according to the probability density function fL(v) = 2 – 2v and the associated probability

distribution function FL(v) = 2v – v

2

. We assume that the probability that any given match is a

“high valuation” match is 0.5. See Figure 1 for an illustration of the two probability distribution 

functions.4

— INSERT FIGURE 1 ABOUT HERE —

 

4

 In terms of the notation of the previous section, the valuation and signal pair (v, s) is distributed according to 

the joint density function 







− =

=

=

v s L

v s H

f v s

1 ,

,

( , ) , v ∈ [0, 1].
9

Under seller's agency, sellers can observe whether a buyer’s valuation is drawn from the

“high valuation” distribution FH(⋅) or the “low valuation” distribution FL(⋅). In this case, the

representative seller’s strategy must specify the price that she proposes as a function of her

information. Thus, the seller’s strategy is a pair of prices (p

H

, p

L

). Under buyer's agency, sellers 

are unable to observe the distribution from which the buyer’s valuation is drawn. In this case a 

seller believes that the distribution of a buyer’s valuation is F(v) = 0.5FH(v) + 0.5FL(v) = 0.5(v

2

) 

+ 0.5(2v – v

2

) = v, v ∈ [0, 1], i.e. the seller believes that the buyer’s valuation is distributed

uniformly on the unit interval. The representative seller’s strategy must now specify a price, say 

p*, which she proposes in every meeting with the buyer. 

The analytic solution to even this simple model is difficult to find and analyze. We

therefore solve for the equilibrium of the model for different values of the parameters δ and q.

5

 In 

Table 1 we describe the symmetric Nash equilibrium that arises under alternative assumptions

about the information available to sellers, and in Table 2 we summarize the properties of these

equilibria, in particular the expected price at which trades occur, the expected time that traders

spend in the market, and the total expected discounted surplus that the representative buyer and

seller together receive. This measure of welfare, which we denote W = B + S, is proportional to 

the total expected discounted surplus generated by the market, provided that the nature of the

prevailing agency relationship does not influence people’s decisions about whether to enter the

market.

The results presented in the two tables are representative of the equilibria that arise under 

all of the different assumptions about the values of δ and q that we examined. Several features of 

the model’s equilibrium are apparent. First, when all else is equal, buyers receive a higher

expected payoff under buyer’s agency, and sellers receive a higher expected payoff under seller’s 

agency. The intuition for this finding is straightforward; when a seller has more information

 

5

 We use Mathematica v. 3.01 in order to solve for the equilibrium of the model.
10

about buyers’ valuations for her property, she is better able to extract the gains from trade in

bargaining. This intuition is also consistent with the results in Table 2, which indicate that, all

else equal, the expected price of housing is higher under seller’s agency.

Second, the results summarized in Table 2 indicate that, all else equal, the expected time that 

traders spend in the market will be lower under seller’s agency. Put differently, the probability

that a transaction will occur in any give match is higher under seller’s agency. Finally, the results 

presented in Table 2 indicate that welfare is higher under seller’s agency, in the sense that the

combined value of the representative seller’s and buyer’s programs is larger. In this version of

the model, welfare is higher under seller’s agency because the expected time that traders spend in 

the market is shorter, compared to buyer’s agency. 

We now analyze a second example that incorporates different assumptions about the

distribution of buyers’ valuations. If the match is a “high valuation” match, the buyer’s valuation 

is uniformly distributed on the unit interval, and the probability distribution function is GH(v) = v, 

v ∈ [0, 1]. If the match is a “low valuation” match, with probability 0.1 the buyer’s valuation is 

zero, and with probability 0.9 the buyer’s valuation is uniformly distributed on the interval [0, .9]. 

In this case the probability distribution function is GL(v) = 0.1 + v, v ∈ [0, 0.9]. We assume that 

the probability that any given match is a “high valuation” match is 0.5. See Figure 2 for an

illustration of the two probability distribution functions.

— INSERT FIGURE 2 ABOUT HERE —

As before, we suppose that, under seller’s agency, sellers can observe whether the buyers’ 

valuations are drawn from the high-valuation or the low-valuation distribution. In Table 3 we

describe the symmetric Nash equilibrium that arises under the alternative assumptions about the

information available to sellers, and in Table 4 we summarize the properties of these equilibria.

The results presented in the two tables are consistent with the results obtained in the first

example, with one prominent exception. In this example, welfare is higher under buyer’s agency, 
11

despite the fact that, as before, traders conclude deals faster under seller’s agency than under

buyer’s agency. Welfare can be lower under seller’s agency because, when a seller can price

discriminate, she proposes a higher price to high valuation buyers and a lower price to low

valuation buyers, compared to a buyer’s agency regime. But then the probability that a seller

trades in a given match with a low valuation buyer rises, and the probability that a seller trades in 

a given match with a high valuation buyer falls, compared to buyer’s agency. Under the

distributional assumptions in this example, this “transfer” of units from high valuation to low

valuation buyers reduces the aggregate gains from trade by enough to overcome the benefit of

faster exit from the market under seller’s agency.

The main conclusion that can be drawn from the two examples is that neither of the two

agency regimes is always superior, in the sense of always generating greater surplus. 

Nevertheless, it is possible to identify some circumstances when we can rank the agency regimes. 

Define Bx as the value of the representative buyer’s program and Sx as the value of the

representative seller’s program at the beginning of a period, before the traders find out if they will 

meet trading partners, under the agency regime x ∈ {BA, SA}. Because our model is stationary, 

the values of the representative buyer's and seller's programs do not depend on the time t ∈ {0, 1, 

2, ... }. Define Tx as the expected time that each trader will spend in the market under agency

regime x. Because we assume that the number of traders is the same on both sides of the market, 

buyers and sellers expect to spend the same amount of time in the market before concluding a

deal. We have the following proposition.

PROPOSITION 3: Suppose that qSA ≤qBA. The representative traders’ total expected discounted 

gains from trade are higher under buyer’s agency than under seller’s agency if the

expected time that traders spend in the market is lower under buyer’s agency. That is, SSA

+ BSA < SBA + BBA if TBA < TSA. Equivalently, SSA + BSA > SBA + BBA only if TBA > TSA.
12

The hypothesis of the proposition means that agents are at least as effective at matching

buyers and sellers under buyer’s agency as they are under seller’s agency. This assumption is

realistic because, as discussed in the introduction, buyers are likely to reveal more information to 

their agents under a buyer’s agency regime. The proposition establishes that discounted expected 

gains from trade that a representative buyer and seller enjoy in this matching market are higher

under a buyer’s agency regime if traders on average conclude deals faster under buyer’s agency. 

That is, a finding that buyers and sellers conclude deals faster under buyer’s agency is sufficient 

to conclude that aggregate gains from trade are higher under buyer’s agency. Equivalently, the

discounted expected gains from trade are higher under the seller’s agency regime only if traders

on average conclude deals faster under seller’s agency.6 

There are two factors at work behind these results. First, a buyer’s agency regime tends to

deliver greater discounted expected gains from trade than seller’s agency because it is not prone

to the misallocation of units from high valuation to low valuation buyers that arises under price

discrimination. Second, sellers tend to inefficiently prolong the search process by setting

reservation prices that are too high, for the familiar reason that they do not want to give up gains 

from trade that they expect to capture from inframarginal buyers. Therefore, unless this

advantage is offset by another factor, the agency regime that produces the shortest search time

tends to yield the largest discounted expected gains from trade. 

Together, these factors enable us to understand the sufficient and necessary conditions

summarized in the Proposition. If search durations are shorter under buyer’s agency, this fact is 

sufficient for a conclusion that welfare is higher under buyer’s agency because that regime is

inherently advantaged by the fact that it does not produce a misallocation of units between

buyers. On the other hand, because seller’s agency does lead to a misallocation of units, it is

necessary for search durations to be shorter under seller’s agency for that regime to be superior.

 

6

 It is important to point out that traders’ search durations depend on both qx and on their bargaining strategies. 

It is possible for traders to exit the market faster under seller’s agency even if qSA < qBA.
13

Our theoretical results provide us with a framework that we can use to interpret recent

empirical findings about the effects of different agency relationships. Curran and Schrag (2000) 

analyze the effect of a change in Georgia real estate law that took place in 1994. They argue that 

this legal change effectively introduced a buyer’s agency regime into Georgia by mandating the

disclosure of the agency relationships available to buyers and by instituting a reform of the

Multiple Listing Service (MLS) information system that eliminated automatic subagency, a

practice by which a buyer’s real estate agent who used the MLS automatically became a subagent 

of the seller’s listing agent. Curran and Schrag find that, after the legal change, both the average 

price of real estate and the average time that houses spent on the market declined. These findings 

are consistent with two conclusions. First, buyers are able to extract more gains from trade under 

buyer’s agency. Second, traders concluded deals and exited the market faster under buyer’s 

agency. 

Elder, Zumpano, and Baryla (1999) use cross-section survey data to assess the impact of the 

different agency relationships. They conclude that real estate agents— including buyer’s agents—

have no independent effect on real estate prices, but they also find that real estate agents in

general, and buyer’s agents in particular, tend to reduce the time that traders spend in the market 

before concluding a deal.

Interpreting these empirical results through the prism of Proposition 3, they constitute

evidence that aggregate, discounted expected gains from trade are higher under a buyer’s agency 

regime than under a seller’s agency regime. Of course, it is important not to overinterpret these 

results, because the data that the researchers use in these two empirical studies are not ideal.7 

Nevertheless, the combination of these empirical results and the theoretical results that we

develop in this paper shed some light on a difficult question. The gains from trade that the real

 

7

 Curran and Schrag (2000) use time on the market data reported to the Multiple Listing Service, giving them a 

measure of how long an individual house was on the market before a sale. An obvious problem with these data 

is that they are biased downward; the time that a house was listed with a realtor who did not sell it is not

reflected in the data. Elder, Zumpano, and Baryla, (1999) meanwhile, rely on survey data provided by buyers 

months after they completed their transactions.
14

estate market generates will never be observable, so any effort to identify the set of legal

institutions that maximize aggregate gains from trade must rely on indirect evidence.

4. Conclusions

In this paper we have analyzed how the flow of information influences the equilibrium in a

particular kind of search market, in which buyers and sellers search for trading partners, meet,

and either strike a deal or reenter the search process. Perhaps unsurprisingly, we found that it is 

not possible to conclude a priori that welfare is helped or harmed when sellers have access to

information about buyers’ valuations. But it is heartening that our results show that empirical

analysis of potentially observable data can help to shed light on this issue.

While our model describes the real estate market reasonably well, certain aspects of this and

other search markets are absent from our formalization. For instance, in our model we have

abstracted away from the need to provide agents with incentives to overcome traditional moral

hazard concerns. It may be worthwhile to investigate how agents’ incentives and the resulting

agency costs depend on the legal rules governing agents’ relationships with traders. A finding

that agency costs are always lower under buyer’s agency would leave the conclusions of

Proposition 3 unchanged, but the opposite finding (which seems unlikely) might change these

conclusions.

Another issue is that, in many search markets, both buyers and sellers are repeat players, and 

they may find it advantageous to establish long-lasting relationships. While this issue is not

present in the real estate market, it is certainly a factor in many retail and labor markets. Clearly, 

the possibility of repeated contact may influence both sellers’ incentives to acquire information

and buyers’ incentives to permit the revelation of information about their personal characteristics. 

An analysis of this issue is an important future task.

While we have focused on the role that (human) agents play in the transmission of

information in search markets, it is clear that information technology both reduces buyers’ costs 
15

of searching for sellers and gives sellers greater opportunities to gather, process, and possible sell 

information about buyers. The recent debate on electronic commerce and privacy highlights the

importance of a careful analysis of the welfare effects of these practices. While it may initially

seem that consumers must suffer if firms can more easily collect information about buyers’ 

preferences, this view neglects the possibility that, in equilibrium, fully informed consumers will

demand compensation in return for revealing personal information. As long as consumers know

what they are giving up, they will know what to demand in return.
16

APPENDIX 1: PROOFS OF PROPOSITIONS

Proof of Proposition 1: We will argue for the seller’s agency case. We must show that there 

exists an acceptance rule for a representative buyer and a pricing rule p(σ) for a representative

seller that are mutual best replies. Consider first the representative buyer’s problem. Define B as 

the value of the buyer’s program at the beginning of a period, before he knows if he will meet a 

trading partner in that period. In a stationary Nash equilibrium the value of the buyer’s program 

does not depend on time. The buyer’s best reply is to purchase a house in period t if and only if vt

– pt ≥ δB, where vt is the buyer’s valuation of the house he sees in period t and pt is the price the 

seller demands. If sellers choose the pricing rule p(σ) in every period, then the value of the

buyer’s program is:

(1.1) 

∫∫

∫∫

∑ ∫∫ ∫∫

σ

σ + δ

σ

σ + δ

∞

=

σ

σ + δ

σ

σ

+ δ

− δ+ δ

−

=

= − ⋅δ − +

v

p y B

SA

v

p y B

SA

t

v

p y B

p y B

t

SA SA

t

SA

q f x y dxdy

q x p y f x y dxdy

B q x p y f x y dxdy q q f x y dxdy

( )

( )

0

( )

( )

0

1 ( , )

( ( )) ( , )

( ( )) ( , ) (1 ( , ) )

.

We now turn to the representative seller’s problem. In each period the seller must choose a 

pricing rule that specifies what price she proposes as a function of the realization of her signal of 

the buyer’s true valuation. In a stationary equilibrium the seller will choose the same pricing rule 

in every period.

The seller’s problem is to choose a vector of pricing rules (p0(σ), p1(σ), p2(σ), ... ) to

maximize the discounted expected payoff:

(1.2) ∑ ∏ ∫∫ ∫∫

∞

=

σ

σ + δ =

σ

σ

+ δ

δ − +

0

( )

0

( )

0

(1 ( , ) ) ( ) ( , )

t

v

p y B

SA t

t

r

p y B

SA SA

t

t

r

q q f x y dxdy q p y f x y dxdy

Denoting the value of the seller’s program in period t by S(t), we can write the seller’s 

problem in period t as:

(1.3) ∫ ∫ ∫

σ

σ

+ δ

+ δ

⋅

+ − δ +







 max ( ) ( , ) + δ( + 1) ( , ) (1 ) ( 1)

( )

( ) 0

( ) q p y f x y dx S t f x y dx dy q S tSA

v p y B

p y B

p SA t

t

t

t

.
17

Appealing to standard optimal control techniques (see Takayama (1993)), the necessary condition 

for an interior maximum is:

(1.4) ( , ) ( ) ( ( ) , ) ( ( ) , ) ( 1) 0, [ , ]

( ) ∫ σ − σ σ + δ σ + σ + δ σ δ + = ∀ σ∈ σ σσ + δ

f x dx p f p B f p B S t

t t t

v

pt B

.

We observe that, under the hypotheses on f(⋅,⋅ ), both (1.3) and (1.4) are continuous functions.

We now show that, whenever (1.4) is satisfied with equality, the second-order condition is

also satisfied. Differentiating (1.4) with respect to pt(σ) and using (1.4) to substitute in for pt(σ) -

δS, we have:

(1.5)

0,

( ( ) , )

( , )

( ( ) , ) ( ( ) , )

( ( ) , )

( , )

2 ( ( ) , ) ( ( ) , )

2 ( ( ) , ) ( ( ) , )( ( ) ( 1))

( )

1

( )

1

1

<

σ + δ σ

σ

− σ + δ σ − σ + δ σ

<

σ + δ σ

σ

− σ + δ σ − σ + δ σ

− σ + δ σ − σ + δ σ σ − δ + =

∫

∫

σ + δ

σ + δ

f p B

f z dz

f p B f p B

f p B

f z dz

f p B f p B

f p B f p B p S t

t

v

p B

t t

t

v

p B

t t

t t t

t

t

where the last inequality follows from the hypothesis that the hazard function is decreasing. The 

finding that the second order condition for a local maximum is strictly satisfied whenever (1.4) is 

satisfied implies that (1.3) has no interior local minima and hence is quasi-concave. But that

implies that there exists a unique optimal pt(σ) ∀ σ ∈ [σ, σ ̄], and we conclude that the

representative seller has a well-defined best reply. It is straightforward to show that this bestreply is always at an interior maximum; we omit the proof to save space.

It remains to show that there exist mutual best replies that form a symmetric stationary

equilibrium. In such an equilibrium the value of the seller’s program is:

(1.6)

∫∫

∫∫

σ

σ + δ

σ

σ + δ

− δ+ δ

=

v

p y B

SA

v

p y B

SA

q f x y dxdy

q p y f x y dxdy

S

( )

( )

1 ( , )

( ) ( , )

.

Define w(y) = p(y) + δB. Adding and subtracting δB from the left-hand side of (1.4), substituting 

(1.6) into (1.4), using the definition of w(y), and dividing by f(w(σ),σ), we rewrite (1.4) as:
18

(1.7) 0, [ , ].

1 ( , )

( , )

( )

( ( ), )

( , )

( )

( ) ( )

= ∀ σ∈ σ σ

− δ+ δ

− σ +

σ σ

σ

∫∫

∫ ∫∫

σ

σ

σ

σ σ

v

w y

SA

v

w y

SA

v

w

q f x y dxdy

q xf x y dxdy

w

f w

f x dx

.

It remains to show that there exists a function w(σ) that satisfies (1.7) for all s. Define w

^

(σ, k) as 

the function that is implicitly defined by:

(1.8) ˆ 0, [0, ].

( ˆ, )

( , )

ˆ

w k k v

f w

f x dx

v

w

− + = ∈

σ

∫ σ

Clearly w

^

(σ, k) is a continuous function, and well-known results about the monotone likelihood 

ratio property guarantee that it is an increasing function of σ. Define the function:

(1.9)

∫∫

∫∫

σ

σ

σ

σ

− δ+ δ

=

v

w y k

SA

v

w y k

SA

q f x y dxdy

q xf x y dxdy

W k

ˆ( , )

ˆ( , )

1 ( , )

( , )

( ) .

Recognizing that W: [0,  ̄v] → [0,  ̄v] and that W(⋅) is continuous, Brouwer’s fixed point theorem 

establishes that ∃k* ∈ [0,  ̄v] such that W(k*) = k*. Define w*(σ) = w

^

(σ, k*). By construction 

w*(σ) satisfies (1.7) for all σ ∈ Σ = [σ, σ ̄]. Using the definitions of w*(σ) and B, it is possible to 

recover B and p(σ), which in turn define the representative buyer’s and seller’s mutual best

replies and the resulting symmetric Nash equilibrium.

Similar arguments establish the existence of a symmetric Nash equilibrium in the buyer’s 

agency case.

QED

Proof of Proposition 2: Define ωt as the valuation of the marginal buyer in period t. To

maximize the aggregate discounted expected gains from trade, choose (ω0, ω1, ω2,... ) in order to 

maximize:

(2.1) ∑ ∏ ∫∫ ∫∫

∞

= =

σ

σ ω

σ

σ ω

δ − ⋅

0 0

(1 ( , ) ) ( , )

t

t

r

v v

t

r r

q f x y dxdy q x f x y dxdy
19

where q ∈ {qSA, qBA} is the probability of being matched in a period. It is straightforward to

differentiate (2.1) and establish that maximizing (2.1) involves choosing the same marginal buyer 

“cutoff” in every period. Writing the aggregate gains from trade W as a function of the cutoff,

say ω, the problem is:

(2.2)

∫∫

∫∫

σ

σ ω

σ

σ ω

ω∈

− δ+ δ

⋅

=

v

v

v

q f x y dxdy

q x f x y dxdy

W

1 ( , )

( , )

max [0, ].

The necessary condition for an interior maximum is:

(2.3) 0

(1 ( , ) )

(1 ( , ) )

( , )

( *, ) *

*

*

*

=

− δ+ δ





















− δ+ δ

δ

ω − ω +

=

ω

∫∫

∫

∫∫

∫∫

σ

σ ω

σ

σ σ

σ ω

σ

σ ω

v

v

v

q f x y dxdy

q f x y dxdy

q xf x y dxdy

q f y dy

d

dW .

Define ω* as the value of ω that satisfies (2.3). Differentiating (2.3), evaluating the result at 

ω*, and using (2.3), we have:

(2.4) 0

1 ( , )

( *, )

*

2

2

<

− δ+ δ

− ω

=

ω

∫∫

∫

σ

σ ω

σ

σ

v

q f x y dxdy

f y dy

d

d W .

Thus, the second order condition is satisfied whenever (2.3) is satisfied, implying that W(ω) 

has no interior minimum and hence is quasiconcave and has a unique maximum. It is

straightforward to show that ω* <  ̄v. Furthermore, W¢ (ω) < 0 for all ω > ω*, and W′ > 0 for all ω

< ω*.

We now consider the buyer’s agency case. Arguments analogous to those in the proof of

Proposition 1 establish that, in equilibrium, the valuation of the marginal buyer in each period,

say ωBA*, satisfies
20

(2.5) 0

1 ( , )

( , )

( , )

( , )

*

* *

*

*

=

− δ+ δ

− ω +

ω σ

σ

∫∫

∫ ∫∫

σ

σ ω

σ

ω σ ω

v

BA

v

BA

BA

BA

v

BA

BA BA

q f x y dxdy

q xf x y dxdy

f

f x dx

Because the hazard function is non-negative, we must have:

(2.6) 0

1 ( , )

( , )

*

*

*

<

− δ+ δ

− ω +

∫∫

∫∫

σ

σ ω

σ

σ ω

v

BA

v

BA

BA

BA

BA

q f x y dxdy

q xf x y dxdy

Comparing (2.6) to (2.3) and using quasiconcavity of W(ω), it immediately follows that ωBA* 

> ω*, establishing that the probability of trade is lower in any given meeting under buyer’s 

agency than if the marginal buyer’s valuation were ω*. Therefore, traders expect to spend more 

time than is optimal in the market under buyer’s agency. A fortiori the Nash equilibrium under 

buyer’s agency does not maximize the aggregate discounted expected gains from trade.

Similar arguments establish that the Nash equilibrium under seller’s agency does not

maximize the aggregate discounted expected gains from trade; we omit the proof to save space.

 QED

Proof of Proposition 3: Define ωSA(σ) = pSA(σ) + δBSA, where pSA(σ) is the seller’s 

equilibrium pricing rule under a seller’s agency regime. Define ωBA = pBA + δBBA, where pBA is 

the seller’s equilibrium price under a buyer’s agency regime. It is straightforward to calculate

that 

∫∫

σ

σ ω

=

v

y

x

x

x

q f x y dxdy

T

( )

( , )

1

, x ∈ {BA, SA}. We must show that

∫∫

∫∫

∫∫

∫∫

σ

σ ω

σ

σ ω

σ

σ ω

σ

σ ω

− δ+ δ

⋅

> + =

− δ+ δ

⋅

+ =

v

BA

v

BA

v BA BA

y

SA

v

y

SA

SA SA

BA

BA

SA

SA

q f x y dxdy

q x f x y dxdy

S B

q f x y dxdy

q x f x y dxdy

S B

1 ( , )

( , )

1 ( , )

( , )

( )

( )

only if ∫∫ ∫∫

σ

σ ω

σ

σ ω

>

v v

y

SA BA

SA BA

q f (x, y)dxdy q f (x, y)dxdy

( )

. 
21

Suppose to the contrary that ∫∫ ∫∫

σ

σ ω

σ

σ ω

<

v v

y

SA BA

SA BA

q f (x, y)dxdy q f (x, y)dxdy

( )

 and SSA + BSA > 

SBA + BBA. Define ω

^

 as satisfying ∫∫ ∫∫

σ

σ ω

σ

σ ω

=

v

y

v

SA

f x y dxdy f x y dxdy

ˆ ( )

( , ) ( , ) . Because ωSA(σ) is 

increasing (see the proof of Proposition 1), it is straightforward to establish that 

∫∫

∫∫

∫∫

∫∫

σ

σ ω σ

σ ω

σ

σ ω σ

σ ω

⋅

<

⋅ v

v

v

y

v

y

dxdy

f x y dxdy

x f x y

dxdy

f x y dxdy

x f x y

SA

SA

ˆ

ˆ

( )

( )

( , )

( , )

( , )

( , )

, implying that 

∫∫

∫∫

∫∫

∫∫

σ

σ ω

σ

σ ω

σ

σ ω

σ

σ ω

− δ+ δ

⋅

<

− δ+ δ

⋅

v

SA

v

SA

v

y

SA

v

y

SA

q f x y dxdy

q x f x y dxdy

q f x y dxdy

q x f x y dxdy

SA

SA

ˆ

ˆ

( )

( )

1 ( , )

( , )

1 ( , )

( , )

. It remains only to show that 

∫∫

∫∫

∫∫

∫∫

σ

σ ω

σ

σ ω

σ

σ ω

σ

σ ω

− δ+ δ

⋅

<

− δ+ δ

⋅

v

BA

v

BA

v

SA

v

SA

BA

BA

q f x y dxdy

q x f x y dxdy

q f x y dxdy

q x f x y dxdy

1 ( , )

( , )

1 ( , )

( , )

ˆ

ˆ

, forcing a contradiction and 

establishing the result.

By hypothesis ∫∫ ∫∫ ∫∫

σ

σ ω

σ

σ ω

σ

σ ω

= <

v v

y

BA

v

SA SA

SA BA

q f (x, y)dxdy q f (x, y)dxdy q f (x, y)dxdy

( ) ˆ

, so 

∫∫

∫∫

∫∫

∫∫

σ

σ ω

σ

σ ω

σ

σ ω

σ

σ ω

− δ+ δ

⋅

>

− δ+ δ

⋅

v

BA

v

BA

v

SA

v

SA

BA

BA

q f x y dxdy

q x f x y dxdy

q f x y dxdy

q x f x y dxdy

1 ( , )

( , )

1 ( , )

( , )

ˆ

ˆ

 only if 

∫∫

∫∫

∫∫

∫∫

σ

σ ω σ

σ ω

σ

σ ω σ

σ ω

⋅

>

⋅ v

v

v

v

BA

BA

dxdy

f x y dxdy

x f x y

dxdy

f x y dxdy

x f x y

( , )

( , )

( , )

( , )

ˆ

ˆ

, i.e. only if ω

^

 > ωBA. Define W(ω, q) 

= 

∫∫

∫∫

σ

σ ω

σ

σ ω

− δ+ δ

⋅

v

v

q f x y dxdy

q x f x y dxdy

1 ( , )

( , )

. Differentiation establishes that ∂W/∂q > 0, so the hypothesis that 

qSA ≤qBA implies that W(ω, qSA) ≤W(ω, qBA). Arguments given in the proof of Proposition 2 

establish that W(ω,q) is a quasiconcave function of ω, and Proposition 2 establishes that ωBA > 

argmaxω ∈ [0, v ̄] W(ω, qBA). Because ω

^

 > ωBA, we then have W(ωBA, qBA) > W(ω

^

, qBA) ≥ W(ω

^

, qSA), 
22

from which it follows that 

∫∫

∫∫

∫∫

∫∫

σ

σ ω

σ

σ ω

σ

σ ω

σ

σ ω

− δ+ δ

⋅

<

− δ+ δ

⋅

v

BA

v

BA

v

y

SA

v

y

SA

BA

BA

SA

SA

q f x y dxdy

q x f x y dxdy

q f x y dxdy

q x f x y dxdy

1 ( , )

( , )

1 ( , )

( , )

( )

( )

, a 

contradiction.

QED
23

APPENDIX 2: NUMERICAL EXAMPLES

Example 1:

We first consider the seller’s agency case. The representative seller’s strategy (p

H

, p

L

) must 

satisfy the following equilibrium conditions, corresponding to equation (1.4) in the proof of

Proposition 1.

(1) 0

1 (0.5(1 ( ) ) 0.5(1 2( ) ( ) )

(0.5 2 0.5 (2 2 ) )

2( )

1 ( )

2 2

1 1

2

=

− δ+ δ − + δ + − + δ + + δ

δ + −

− +

+ δ

− + δ ∫ + δ ∫ + δ

q p B p B p B

q xp dx x p dx

p

p B

p B

H L L

p B

L

p B

H

H

H

H H L

(2) 

0

1 (0.5(1 ( ) ) 0.5(1 2( ) ( ) )

(0.5 2 0.5 (2 2 ) )

2 2( )

1 2( ) ( )

2 2

1 1

2

=

− δ+ δ − + δ + − + δ + + δ

δ + −

− +

− + δ

− + δ + + δ

∫ + δ ∫ + δ

q p B p B p B

q xp dx x p dx

p

p B

p B p B

H L L

p B

L

p B

H

L

L

L L

H L

Meanwhile, the value of the representative buyer’s program is given by:

(3) 0

1 (0.5(1 ( ) ) 0.5(1 2( ) ( ) )

(0.5 2 ( ) 0.5 (2 2 )( )

2 2

1 1

=

− δ+ δ − + δ + − + δ + + δ

δ − + − −

=

∫ + δ ∫ + δ

q p B p B p B

q x x p dx x x p dx

B H L L

p B

L

p B

H

H L

.

In order to solve for an equilibrium, it is helpful to define the variables w

H

 = p

H

 + δB and w

L

= p

L

 + δB. Adding and subtracting δB to the left-hand side of (1) and (2) and using the definition 

of B, we have the following equilibrium conditions.

(4) 0

1 (0.5(1 ( ) ) 0.5(1 2 ( ) )

(0.5 2 0.5 (2 2 ) )

2

1 ( )

2 2

1 1

2

2

=

− δ+ δ − + − +

δ + −

− +

− ∫ ∫

H L L

H w w

H

H

q w w w

q x dx x xdx

w

w

w H L

(5) 0

1 (0.5(1 ( ) ) 0.5(1 2 ( ) )

(0.5 2 0.5 (2 2 ) )

2 2

1 2 ( )

2 2

1 1

2

2

=

− δ+ δ − + − +

δ + −

− +

−

− + ∫ ∫

H L L

L w w

L

L L

q w w w

q x dx x xdx

w

w

w w H L

Equations (4) and (5) represent a system of two equations in two unknowns, w

H

 and w

L

. In 

principle, it is possible to solve this system of equations and then use the definitions of B, w

H

, and 

w

L

 to solve for B, p

H

, and p

L

. 
24

We now consider the buyer’s agency case. The seller’s strategy p* must satisfy the following 

equilibrium condition, which is analogous to equations (1) and (2).

(6) 0

1 (1 * )

1 ( * ) *

1

*

=

− δ+ δ − − δ

δ

− + δ − +

∫+ δ

q p B

q pdx

p B p

p B

Meanwhile, the value of the representative buyer’s program is given by:

(7)

1 (1 * )

( *)

1

*

q p B

q x p dx

B

p B

− δ+ δ − − δ

−

=

∫+ δ

.

In order to solve for an equilibrium, it is helpful to define the variable w* = p* + δB. Adding 

and subtracting δB to the left-hand side of (6) and using the definition of B, we have the

following equilibrium condition.

(8) 0

1 (1 *)

1 2 *

1

*

=

− δ+ δ −

δ

− +

∫

q w

q xdx

w

w

In principle, it is possible to solve equation (8) for w* and then use the definitions of B and 

w* to solve for B and p.

Example 2:

We first consider the seller’s agency case. Assuming that the seller trades with positive

probability in every match with a buyer, meaning that p

L ≤ 0.9, the representative seller’s strategy 

must satisfy the following equilibrium conditions.

(9) 0

1 (0.5(1 ) 0.5(.9 ))

(0.5 0.5 )

1 ( )

1 .9

=

− δ+ δ − − δ + − − δ

δ +

− + δ − +

∫ + δ ∫ + δ

q p B p B

q p dx p dx

p B p H L

p B

L

p B

H

H H

H L

(10) 0

1 (0.5(1 ) 0.5(.9 ))

(0.5 0.5 )

.9 ( )

1 .9

=

− δ+ δ − − δ + − − δ

δ +

− + δ − +

∫ + δ ∫ + δ

q p B p B

q p dx p dx

p B p H L

p B

L

p B

H

L L

H L

.

Meanwhile, the value of the representative buyer’s program is given by:
25

(11)

1 (0.5(1 ) 0.5(.9 ))

(0.5 ( ) 0.5 ( ) )

1 .9

q p B p B

q x p dx x p dx

B H L

p B

L

p B

H

H L

− δ+ δ − − δ + − − δ

− + −

=

∫ + δ ∫ + δ

.

It is helpful to define the variables w

H

 = p

H

 + δB and w

L

 = p

L

 + δB. Adding and subtracting 

δB to the left-hand side of (9) and (10) and using the definition of B, we have the following

equilibrium conditions.

(12) 0

1 (0.5(1 ) 0.5(.9 ))

(0.5 0.5 )

1 2

1 .9

=

− δ+ δ − + −

δ +

− +

∫ ∫

H L

H w w

q w w

q xdx xdx

w

H L

(13) 0

1 (0.5(1 ) 0.5(.9 ))

(0.5 0.5 )

.9 2

1 .9

=

− δ+ δ − + −

δ +

− +

∫ ∫

H L

L w w

q w w

q xdx xdx

w

H L

Equations (12) and (13) represent a system of two equations in two unknowns, w

H

 and w

L

. In 

principle, it is possible to solve this system of equations and then use the definitions of B, w

H

, and 

w

L

 to solve for B, p

H

, and p

L

.

We now consider the buyer's agency case. Assuming that the seller trades with positive

probability in every match, i.e. that p < 0.9, the seller’s strategy p* must satisfy the following

equilibrium condition.

(14) 0

1 (0.5(1 * ) 0.5(.9 * ))

(0.5 0.5 )

.95 ( * ) *

.9

*

1

*

=

− δ+ δ − − δ + − − δ

δ +

− + δ − +

∫+ δ ∫+ δ

q p B p B

q pdx pdx

p B p

p B p B

Meanwhile, the value of the representative buyer’s program is given by:

(15)

1 (0.5(1 * ) 0.5(.9 * ))

(0.5 ( *) 0.5 ( *) )

.9

*

1

*

q p B p B

q x p dx x p dx

B

p B p B

− δ+ δ − − δ + − − δ

− + −

=

∫+ δ ∫+ δ

.

In order to solve for an equilibrium, it is helpful to define the variable w* = p* + δB. Adding 

and subtracting δB to the left-hand side of (14) and using the definition of B, we have the

following equilibrium condition.
26

(16) 0

1 (0.5(1 *) 0.5(.9 *))

(0.5 0.5 )

.95 2 *

.9

*

1

*

=

− δ+ δ − + −

δ +

− +

∫ ∫

q w w

q xdx xdx

w

w w

In principle, it is possible to solve equation (16) for w* and then use the definitions of B and 

w* to solve for B and p*.
27

Seller’s Agency Buyer’s Agency

Seller’s Strategy p

H

 = .6601, p

L

 = .5793 p* = .6444

Seller’s Expected Payoff .4903 .4849 δ = .9, q = 1

Buyer’s Expected Payoff .1608 .1640

Seller’s Strategy p

H

 = .6482, p

L

 = .5570 p* = .6295

Seller’s Expected Payoff .4489 .4432 δ = .9, q = .75

Buyer’s Expected Payoff .1523 .1554

Seller’s Strategy p

H

 = .6642, p

L

 = .6061 p* = .6544

Seller’s Expected Payoff .5294 .5252 δ = .95, q = 1

Buyer’s Expected Payoff .1972 .2001

Seller’s Strategy p

H

 = .6540, p

L

 = .5878 p* = .6421

Seller’s Expected Payoff .4968 .4922 δ = .95, q = .75

Buyer’s Expected Payoff .1899 .1930

TABLE 1: NASH EQUILIBRIUM IN EXAMPLE ONE
28

Seller’s Agency Buyer’s Agency

Expected Price (P ̄*) .6457 .6444

Expected Time in Market (T ̄*) 4.67 4.81 δ = .9, q = 1

Welfare (B + S) .6511 .6489

Expected Price (P ̄*) .6303 .6295

Expected Time in Market (T ̄*) 5.59 5.78 δ = .9, q = .75

Welfare (B + S) .6012 .5985

Expected Price (P ̄*) .6564 .6544

Expected Time in Market (T ̄*) 6.30 6.43 δ = .95, q = 1

Welfare (B + S) .7265 .7253

Expected Price (P ̄*) .6440 .6421

Expected Time in Market (T ̄*) 7.46 7.64 δ = .95, q = .75

Welfare (B + S) .6867 .6852

TABLE 2: NUMERICAL RESULTS FROM EXAMPLE ONE
29

Seller’s Agency Buyer’s Agency

Seller’s Strategy p

H

 = .5916, p

L

 = .5416 p* = .5592

Seller’s Expected Payoff .4071 .3999 δ = .9, q = 1

Buyer’s Expected Payoff .2035 .2122

Seller’s Strategy p

H

 = .5843, p

L

 = .5343 p* = .5538

Seller’s Expected Payoff .3747 .3689 δ = .9, q = .75

Buyer’s Expected Payoff .1873 .1938

Seller’s Strategy p

H

 = .6086, p

L

 = .5586 p* = .5680

Seller’s Expected Payoff .4573 .4415 δ = .95, q = 1

Buyer’s Expected Payoff .2287 .2457

Seller’s Strategy p

H

 = .6024, p

L

 = .5524 p* = .5657

Seller’s Expected Payoff .4313 .4195 δ = .95, q = .75

Buyer’s Expected Payoff .2156 .2285

TABLE 3: NASH EQUILIBRIUM IN EXAMPLE TWO
30

Seller’s Agency Buyer’s Agency

Expected Price (P ̄*) .5697 .5592

Expected Time in Market (T ̄*) 4.99 5.00 δ = .9, q = 1

Welfare (B + S) .6106 .6115

Expected Price (P ̄*) .5621 .5538

Expected Time in Market (T ̄*) 6.00 6.01 δ = .9, q = .75

Welfare (B + S) .5620 .5627

Expected Price (P ̄*) .5878 .5680

Expected Time in Market (T ̄*) 6.71 6.73 δ = .95, q = 1

Welfare (B + S) .6860 .6872

Expected Price (P ̄*) .5812 .5657

Expected Time in Market (T ̄*) 7.95 7.97 δ = .95, q = .75

Welfare (B + S) .6469 .6479

TABLE 4: NUMERICAL RESULTS FROM EXAMPLE TWO
31

0 1

1

FH(v)

FL(v)

v

F(v)

FIGURE 1: DISTRIBUTIONS IN EXAMPLE ONE
32

0 1

1

GH(v)

GL(v)

v

FIGURE 2: DISTRIBUTIONS IN EXAMPLE TWO
33

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