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NOISY SIGNALS IN REAL ESTATE AND MONETARY
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BRIAN ARTHUR MCNAMARA

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5/08 K:IProj/Acc&Pres/ClRC/DateOuo.indd

NOISY SIGNALS IN REAL ESTATE AND MONETARY
SEARCH MODELS

By

Brian Arthur McNamara

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Economics

2009

ABSTRACT

NOISY SIGNALS IN REAL ESTATE AND MONETARY SEARCH
MODELS

By

Brian Arthur McNamara

Sellers face much uncertainty when selling a home. If a seller's home does not
sell, it is unclear whether this is due to market conditions or the quality of the real estate
agent. The seller updates her belief on the quality of her agent when the property does
not sell. In the first chapter, I construct a model describing this learning process and test
it empirically with Multiple Listing Service data. The model treats the lack of a sale as a
noisy signal of the agent’s quality and assumes sellers use Bayesian updating when
inferring the quality of the real estate agent. The posterior belief that an agent is a low
quality type is a decreasing function of the current price and the prior price. Assuming
the expected benefit associated with a change in agent increases with respect to this
belief; a seller is more likely to change agents with a lower current and prior price,
conditional on a sale not occurring. My empirical results provide support for these
theoretical implications.

In the second chapter, I investigate the conditions under which endogenously
issued objects are valued, in an economy along the lines of Kiyotaki and Wright (1993)
but with a finite population. In contrast to previous work (Cavalcanti and Wallace

(1999)), we assume that the economy has no exogenous technology that keeps track of

the actions of money issuers. My objective is to identify which additional attributes make
some agents natural candidates to become money issuers. I show that there exists an
equilibrium in which endogenously issued money is valued if the money issuer is
relatively patient as compared to the rest of the economy. Intuitively, patience works as a
commitment device that prevents the overissue of money.

In the third chapter, I analyze the intensive and extensive margins of trade in a
random matching model with divisible money, where productivity differs across agents
and producers can choose whether to enter in the market in every period. The model
exhibits multiple equilibria: one equilibrium in which only high productive sellers enter
and one equilibrium in which both high and low productive sellers enter. The main result
is that the high productive sellers will produce‘more in the equilibrium in which both
types of sellers enter, despite the fact that the average productivity in the economy is
depressed by the presence of the low productive sellers. This result is in contrast to
Camera and Vesley (2006), which consider a similar environment but with indivisible
money. Intuitively, as long as the benefit to the buyer of having a higher probability of
consumption is greater than the average productivity decrease, buyers will choose to
bring more money to the market, thereby encouraging the high productive sellers to work

more.

ACKNOWLEDGEMENTS

I was very fortunate to have two advisors that care greatly about their students. I
will always be gratefiil for their guidance, patience, and support throughout this process.
I would like to thank my co-advisor, Luis Araujo, for always being available to help me
through this process. I would also like to thank my other co-advisor, Mike Conlin, who
had endless patience, encouragement, and guidance. I am forever grateful to both of you.

Many thanks to my fellow graduate students, particularly Deborah, Byung-Cheol,
Nicole, B. Moore, Chris, and Nathan. I would like to thank Don Luidens for giving
conscientious editing. I would like to thank my Mom for her thoughts and faith that it
will all work out. Finally, I would like to thank my father, who I talked constantly to

about real estate, the Jersey City condo market, the frustrations of grad school, and all the

fishing I was missing.

iv

 

TABLE OF CONTENTS

LIST OF TABLES ............................................................................ vi

LIST OF FIGURES ......................................................................... vii

CHAPTER 1: Priced to Change: Dumping Real Estate Agents ........................ 1
Introduction... 1
Institutional Details ................................................................... 3
Literature Review ..................................................................... 6
Model .................................................................................... 7
Data .................................................................................... 13
Empirical Results .................................................................... 15
Conclusion ............................................................................. 20
Data Appendix ........................................................................ 22
Appendix A ............................................................................ 25
Appendix B ........................................................................... 26
References ............................................................................. 37

CHAPTER 2: On the Emergence of Endogenous Money as

Media of Exchange ............................................................................ 39
Introduction ............................................................................. 39
Model ................................................................................... 41
Robustness ............................................................................. 46
Discussion .............................................................................. 48
Conclusion .............................................................................. 51
Appendix ................................................................................ 53
References .............................................................................. 57
CHAPTER 3: The Intensive Margin with Heterogeneous Producers ................... 58
Introduction ............................................................................. 58
Model ................................................................................... 60
Multiple Equilibria .................................................................... 69
Conclusion .............................................................................. 83
References .............................................................................. 86

 

LIST OF TABLES

Table 1.]: Descriptive Statistics .......................................................... 27
Table 1.2 Logit Regression ................................................................. 28
Table 1.3 Logit Regression with Seller fixed effects ................................... 30

Table A1 Distance in months of the end of the previous listing
to the start of linked listing ...................................................... 32

Table 3.1: Comparing the Camera Velsey Model to Rocheteau and Wright. . . . . ....80

vi

LIST OF FIGURES

Figure 1.1: Percentage of Listings that Ended as a Result of a Sale,

Due to Expiration, or an Agent Change33
Figure 1.2: Percentage ofSellers that Changed Their List Price....... .34
Figure 1.3: Average Original List Price and Sales Price ................................... 35
Figure 1.4: Fraction of Sellers That Changed List Price, Changed Agents ............. 36
Figure 2.1: Probability That a Specific Agent Has Seen a Specific Note ................ 52

Figure 3.1: The benefit of Entering the Night Market for Low Productive Sellers. . ...85

vii

Priced to Change: Dumping Real Estate Agents

Sellers face much uncertainty when selling a home. If a home does not sell, the seller is
unclear whether this is due to market conditions or the quality of the real estate agent.
The seller updates her belief on the quality of her agent if the property does not sell. In
this paper, 1 construct a model describing this learning process and test it empirically with
Multiple Listing Service data. The model treats the lack of a sale as a noisy signal of the
agent’s quality and assumes sellers use Bayesian updating when inferring the quality of
the real estate agent. The posterior belief that an agent is a low quality type is a
decreasing function of the current price and the price prior to a price change. Assuming
the expected benefit associated with a change in agent increases with respect to this
belief, a seller is more likely to change agents with a lower current and prior price,
conditional on a sale not occurring. My empirical results provide support for these
theoretical implications.

1. Introduction

In the event a seller's property does not sell, it is unclear whether this is due to
market conditions or to the quality of her real estate agent. The seller may conclude that
she has chosen too high of a list price. If this is the case, she is likely to infer that the
cause is due to unfavorable market conditions, which puts less blame on the agent.
Conversely, if the seller perceives her list price as relatively low, the seller is more likely
to infer that the agent is responsible for the property not selling. Depending on the
seller’s inference, she may decide to change the agent and/or list price. In this paper, I
construct a model describing how the seller updates her belief on the quality of her agent
when the property does not sell. The model treats the lack of a sale as a noisy signal and
assumes sellers use Bayesian updating when accessing the quality of the real estate agent.
The model’s primary implications are that a lower current price and a lower price prior to

a price change increase the belief that the seller has a low quality agent. Assuming that

 

an increase in a seller’s belief of a low quality agent increases the benefit associated with
a change in agents, my empirical results support these implications of the model. This
paper is the first to empirically test how the list price affects the seller’s decision to
change agents. This is done using Multiple Listing Service (MLS) data - a database
containing building characteristics, prices, and listing information on properties for sale.

There are many studies examining the effect of a property’s list price on its sales

price and its time on the market.1 In general, this research concludes that the list price is
positively correlated with the sales price and time on the market. In these studies, the list
price is seen as a signal of the seller’s reservation price which results in a tradeoff
between selling the property quickly versus waiting for a buyer with a higher willingness
to pay. None of these earlier studies, however, considers the relationship between the list
price and the decision to change agents.

In a related study not pertaining to the real estate market, Israel (2005) studies
how consumers learn about their agent in the automobile insurance market. He shows
that “learning events,” such as a non-chargeable claim, cause an interaction between the
customer and their agent. As a result of this interaction, the seller learns about her agent,
which increases the probability of changing agents. In the real estate market, instead of
relying on a single transaction to trigger an update in her beliefs about the quality of her
agent, the seller continuously updates her beliefs based on whether her property sells.
With a higher belief that an agent is low quality, a customer is more likely to change

agents regardless of whether it is an insurance agent or a real estate agent.

 

1
Yavas (1992), Yavas and Yang (1995), Anglin, Rutherford and Springer (2003) for a sample.

 

Section II describes decisions made by the seller while the property is on the
market. Section III summarizes the relevant literature. Section IV presents a model of
how the seller updates her belief about the quality of her agent. Section V introduces the
data, and Section VI presents empirical evidence supporting the model’s comparative

static results. Section VII concludes.

11. Institutional Details

Real estate agents differ in their quality. This quality comes from an agent’s
inherent ability and her experience. Some agents are better at marketing certain
properties, so an agent’s quality may depend on the property. A higher quality agent has
a higher probability of getting the property sold. A seller uses available information
when choosing an agent. This information can be obtained by a friend’s
recommendation, by observing sold signs in her neighborhood, by previous experience
with an agent, and eventually by conducting conversations with potential agents. While
the seller gains information on agent quality fiom these activities, she can not perfectly
infer this quality. After these efforts, the seller chooses an agent, and at this time a listing
is created. A listing is a contract between the seller and the real estate agent, giving the
agent an exclusive right to sell the property. The two most common durations for this
contract are three and six months. With the listing created, the agent is able to go out and
market the property to potential buyers in order to get the property sold. If the property
does not sell, the seller has three actions that she may take: change agents, change the list

price, or withdraw the property from the market.

A seller will change agents in order to obtain a new agent that may be of a higher
quality. Therefore, the benefit of changing agents increases in the seller’s belief that her
current agent is low quality. There is a cost associated with a change in agents. Part of
the process of setting up a new listing is for the seller to have conversations and to visit

with the agent. Spending this time with the new agent is part of the cost when a seller

switches her agent.2 The cost of an agent change is greater if the listing has not reached
the expiration date. When the original listing reaches the expiration date the seller is free
to employ another agent; however, if the seller decides to terminate the relationship prior
to the expiration date, she then needs to persuade the original agent to terminate the
listing. If the original agent does not terminate the listing, the seller is exposed to paying
double the commission because she had signed an exclusive right to sell agreement for
the length of the contract. A seller’s decision to change agents depends on the seller’s
belief about the quality of her current agent and whether the current listing is about to
expire. _

Another option for the seller is to change her list price. To change the price, the

seller needs to sign a form stating the change. The agent then submits the form to the

MLS and the price is updated within two days.3 A seller may lower the list price to
increase the probability of a sale. On the other hand, a seller may choose to increase the
list price. An increase in price could be a result of encouragement from an outside agent.

When a listing is close to expiring, an outside agent could see this as an opportunity to

 

2
This is similar to the cost of a shorter contract length from Miceli (1989). Miceli finds that a shorter
contract duration will increase the effort level of the real estate agent.

The current version of this MLS has since been upgraded so that the agent can submit the price change
online and have the price change immediately. This function was not available during my sample.

pick up a new listing. The outside agent may bring a potential buyer to look at the

property. During this visit, the devious agent4 may insinuate to the seller that the
property should sell quickly and she should be able to get a higher price. As a result of
this conversation, the seller may increase her price. When the listing ends, the seller will
infer the cause of the property not selling is a result of employing a low quality agent and
change agents. An increase in the list price could precede a change in agents.

The final option for a seller is to end the current listing. There are three common
cases when a seller will let the listing expire. These include: (1) when the seller changes
agents; (2) when the property is pulled from the market in order to make home
improvements or until market conditions improve; and (3) when the seller wants to give
the appearance of a new property on the market. In the final case, the seller is trying to
avoid the appearance of a listing that has been passed over by other buyers. By looking
at the listing date, buyers can see that a property has been sitting on the market for a long

time and may infer that other buyers who viewed the property found something wrong
wrth 1t. In order to avord thrs perceptlon, a seller may create a new lrstrng in order to

project the image that the property has not been on the market very long.6

 

4

It is against Article 16-4 of the Code of Ethics and Standards of Practice of the NATIONAL
ASSOCIATION OF REALTORS to solicit a listing which is currently listed exclusively with another
broker.

Taylor (1999) studies conditions when a buyer will be more suspicious of the quality of a house that has
been on the market for a long time.

Obtaining a new MLS number results in a new listing date.

III. Literature Review

While this is the first empirical study of the seller’s decision to change her real
estate agent, as well as the first paper that examines how the list price affects the seller’s
belief of her real estate agent’s quality, the effect of the list price on the sales price and
time on the market has been studied extensively. Yavas (1994) provides an overview of
research on real estate brokerage, including the role of the list price. In line with other
research, Miller and Sklarz (1987) conclude that a lower list price will decrease the time
the property is on the market. An example of later research that also confirms this is
Bjorklund, Dadzie, and Wilhelrnsson (2006) who show that a higher list price is more
likely to result in a higher sales price and increase the time the property is on the market.

Other papers have shown that changing the list price is part of the seller’s process

of learning about the market value of their property.7 Herrin, Knight, and Sirmans
(2004) test different implications of pricing during demand uncertainty. They show that
sellers with more accurate information about the value of their property are less likely to
change their list price. These results are consistent with Lazear’s (1986) theory of price
experimentation under demand uncertainty. Under this theory, the seller enters the
market uncertain about the demand for her good, but she has multiple periods to learn
about the demand. While setting a high initial price decreases the probability of a sale, it
does allow the seller the possibility to receive a high sales price. If the property does not
sell at this high price, the seller will move to the next period with the updated belief that

buyers’ willingness to pay is below the high price. In later periods the seller will lower

 

7
Read (1988), Sass (1988), and Knight (2002) study the property characteristics that results in more
sellers changing their list price.

the price, until the good is sold. In the real estate market, the seller has multiple periods
to get the property sold and learn about her property’s demand through this list price
experimentation.

Current research neglects the role of price experimentation on the seller’s belief
about the quality of her agent. This paper focuses on how the list price affects this belief.
Having a lower list price increases the probability the property will get sold. The seller
will update her belief that the agent is a low quality type more if the property does not
sell at a low, compare to high, list price. The model in the next section illustrates this

point.

IV. Model

Consider a two-period economy populated by a continuum of buyers, sellers, and
real estate agents. Each seller is endowed with one property. At the beginning of the
first period, sellers are randomly and pair wise matched with real estate agents. There are

two types of real estate agents, high and low quality, who differ in the number of

potential buyers that they expose to the seller’s property. B0 is the proportion of real

estate agents that are low quality. I assume that a high quality agent retrieves NG
potential buyers in the first period, while a low quality agent retrieves NB potential
buyers, with NG > NB. Buyers differ in their willingness to pay for the property, and a

buyer’s willingness to pay comes as an independent and random draw from a uniform

X

distribution with support [ _ , x ]. A property is sold if and only if at least one of the

buyers has a willingness to pay above the reservation price of the prOperty.

In the first period, the reservation price of the seller is P1, where f < Pl < x .
Then, depending on the type of the real estate agent, NO or NB buyers will visit the

property. The probability that any given buyer values the property less than P1 is

P11

; _ x . In turn, the probability that the property does not sell in the first period is:

NB NC

_£ x_£ (1)

The seller does not observe the number of buyers who viewed her property. Therefore,
she can not precisely infer the type of agent she has employed, but she can update her
belief about this type by observing whether the property sells. If the property does not

sell in the first period, then all buyers that were exposed to the property valued it less than
P1. Using Bayes’ Rule, the probability of having a low quality agent given that the

property did not sell in the first period is:

 

P—x
30* .1 —
x-&
P N” P N“ 581
30* 33:3 +(1—BO)* :23

Note that since R < X , B 1 > BO . Intuitively, because low quality agents gather

fewer buyers, there is a higher probability that all buyers who viewed the property will

have a valuation below P1. As a result, if the property does not sell in the first period,

the seller increases her belief that the agent is low quality.
In the second period, I assume that a fraction 0 6(0, 1) of first period buyers

remain on the market. This implies that the number of first period buyers who observe

the property again in the second period is equal to UN G or 0N B . Once again, the

seller does not observe this number. New buyers also enter the market in Period 2 and

observe the house for the first time. I assume that the number of new buyers gaining

exposure to the property is N Gf for high quality agents and N Bf for low quality

agents with N Gf > N Bf .
In the second period, the reservation price of the seller is P2. If the seller’s

reservation price decreases (i.e. P2< P1), the probability that a period one buyer values

the property at less than Pg is P _ x . If the property does not sell by the end of the
1 _

second period, the seller’s updated belief that she has employed a low quality agent is:

ONE NBf
P —x
2 _ _
Pl-zE x-x
0N3 NBf
P —x P —x
2 _ 2 _
31(R)* ———- -=—— +
Pl—g x—x
— (3)
O'NG NGf

 

B2.

(1—B.<P.»* 52—39- —:

P11 x-a

Because P2 < x , we have B 2 > B1 , which implies the seller increases her belief
that she has a low quality agent.
Alternatively, if the seller’s reservation price does not decrease (i.e. P2 2P1),

there is zero probability that a first period buyer will purchase the property. With a non-

decrease in price, the updated belief that the seller has a low quality agent after the

second period is:

10

 

 

BI(P1)* -—2 —

 

 

31(B)* 3 ‘ +

(4)

 

(1-B.<P.)>* 3 -
x-zg

For an increase in price, as long as P2 < x , we continue to have B 2 > B 1 .

The above model can be used to describe how the seller’s reservation price affects

the amount the seller updates the belief that an agent is a low quality type.

Lemma 1: The posterior belief that an agent is a low quality type is a
decreasing function of the seller ’s current reservation price.

Proof: See Appendix A.

Lemma 1 states that a lower reservation price increases the sellers’ ex post belief
of a low quality agent. The difference between the two types of agents is the number of
buyers viewing the property. When the seller has a high (low) reservation price, there is
a high (low) probability that a buyer’s willingness to pay is below this price. With a high
reservation price, increasing the number of buyers who view the property will only
increase the probability of a sale by a small amount. Therefore, when the seller’s

reservation price is high, there is a small difference between the probabilities the property

11

is not sold between high and low quality agents; this results in a small increase in the
seller’s belief of a low quality agent. Conversely, with a lower reservation price, the
difference in the number of buyers viewing the property results in a larger difference in

probabilities of the property not selling.

Lemma 2: Conditional on the seller ’s current reservation price ( IDz ), a

decrease in the seller 's reservation price in the prior period ( B ), will
increase the seller '3 belief of a low quality agent. The amount of this

increase diflers depending on whether 1)] is greater than or less than

P2 .
Proof: See Appendix B.

Lemma 2 states that, conditional on not selling and the seller’s current reservation
price, a lower previous reservation price increases the updated belief by the seller that she
has employed a low quality agent. The amount of this increase in the seller’s belief
differs depending on whether the seller’s previous reservation price is greater or less than
her current reservation price.

The expected benefit from an agent change involves increasing the probability the
agent selling your property is a high quality type. The seller will switch agents when this
expected benefit is larger than the cost of obtaining a new agent. Therefore, an increase
in the belief of having a low quality agent will increase the probability of a change in

agents by increasing the expected benefits associated with this change.

12

V. Data

The data used in this analysis come fiom the Hudson County Multiple Listing

Service. The focus is on all condominiums for two neighborhoods in Jersey City (the

Heights and Journal Square) that were listed between January 1998 and December 2006.8
Once a property is listed on the MLS, other real estate agents’ buyers can view the
property. If a seller does not hire an agent, or if the seller hires an agent but wishes to

leave the listing off of the MLS, the property will not reach the MLS. Over 90 percent of

the properties for sale with an agent are listed on the MLS.9

My sample consists of 4,013 listings. Just over half of these listings resulted in a
sale. As mentioned in the Institutional Details section, it may take a seller a couple of
listings to sell her property. By looking at the collection of listings for a property, I
obtain a more accurate measure of time on the market, and can determine whether the
seller changed agents. To accomplish this, I link together listings for the same property,

when the earlier listing does not result in a sale and the later listing starts within six

months of the previous listing expiring. 10 This process created 3,392 sellers. For each
seller, I create an observation for each day the property was on the market which results
in 455,738 property-day observations. The Data Appendix provides a complete

description of how the data were created.

 

Condominiums were chosen for this analysis because they are more homogeneous compared to single
family and multi-family homes.

Dale—Johnson and Hamilton (1998) studied how market conditions affect the decision to hire an agent
and put the property on MLS.

Anglin (2004) links listings together by the same address in order to gain a better perspective of time on
the market.

13

Table 1 shows the descriptive statistics for each variable used in the analysis. The
typical condo in my sample has one or two bedrooms, one bathroom, and square footage
of less than 1,000 square feet. Parking is available in 20 percent of the properties.
Heating by gas and air-conditioning are available in 37 percent and 25 percent of the
properties, respectively. A property is on the market for an average of four and a half
months and the average list price is $204,558. If the property sells, the average sales
price is $172,175.

Figure 1 compares how the percentage of listings that ended as a result of a sale,
the property being taken off the market, or an agent change vary as a fimction of months
on the market. Properties are more likely to be taken off the market after the third and
sixth months (i.e. in the fourth and seventh months), which correspond to the most
common contract lengths. In addition to these spikes, the probability of ending a listing
increases slightly with time on the market. Also shown in Figure 1, listings that changed
agents follow a similar pattern. Figure 1 suggests that sellers are more likely to change
agents when the cost of a change is lower. The pattern for listings that ended because the
property was sold is relatively constant across months but spikes at the second and ninth
months.

The dataset also contains list price changes and dates of the changes from the
MLS property history report. The majority (72 percent) of sellers never changed their list
price. Of the 3,392 sellers, 620 (18 percent) changed their list price once, 218 (six
percent) changed twice, 80 (two percent) changed three times and 47 (one percent)
changed more than three times. Of the 1,494 list price changes, 163 (11 percent) were

increases, with the average increase being $16,373 (10.1 percent). Of the 1,331 that

14

decreased, the average change was $16,368 (6.3 percent). As shown by Figure 2, a
higher percentage of sellers changed their list price in the second month on the market
compared to the first month. This percentage does fluctuate with drops in the fourth and
seventh months and a spike in the ninth month.

Over the course of my sample, the original list price steadily increased until 2002.
There was then a dramatic growth in the original list price between 2003 and 2006. As
Figure 3 shows, the average sales price follows a similar pattern to the original list price
between 1998 and 2004, which resulted in a consistent gap. The gap between the original
list price and sales price increased after 2004. It appears that the market started to slow
down in 2005. With the slowdown, the percentage of sellers who changed their list price
increased. Figure 4 depicts the percentage of sellers who changed their list price and who
changed agents by year. There was a shift in 2005 and 2006; these two years had the
highest percentage of sellers change their list price, perhaps reflecting this change in the
market. There appears to be less of a shift in the percentage of sellers who changed

agents.

VI. Empirical Results

The two comparative statistics that will be tested from the model are a lower
current reservation price or a lower prior reservation price will increase the seller’s belief
that the agent is low quality. With a greater belief of a low quality agent, a seller is more
likely to change agents. In order to test these comparative statistics, I estimate the

following logit model:

15

P P
1 ad + X id ,6 +
D C C
XidflD+ xid ,3 +5121
0 otherwise.

ChangeAgentid = > o

The dependent variable, Chang eAg ent id , is a dummy indicating whether the

. th
seller has changed agents on day (1. The intercept, ad , varies depending on the

P
year and quarter of the observation. The vector X id consists of variables pertaining to

the list price and changes in the list price. This vector includes: current list price, two
dummy variables indicating whether the current list price is less than and greater than the
prior list price, the prior list price when there was a price decrease, and the prior list price
when there was an increase. In the estimation, the list price is a proxy for the seller’s
reservation price, with a change in the list price resulting from a change in the seller’s

reservation price.

D
The vector X id contains duration on the market variables which includes: the

number of days on the market, number of days on the market squared, whether the seller
had previously changed agents, and an interaction term between whether the seller had
previously changed agents and the number of days with the current agent. This duration

vector also includes four dummy variables for whether the day on the market is within

16

the first 15 days of the fourth, seventh, tenth, and thirteenth months. These variables are

included as covariates because the cost of changing agents is lower at the contract

C
expiration date and typical contracts have three or six month durations. The vector X id

contains property characteristics such as square footage, nrunber of bedrooms and

bathrooms, and whether the property has parking, air-conditioning and is heated by gas.

8

The error term, id , has a standard logistic distribution and is assumed to be

uncorrelated with the regressors.

Column I of Table 2 presents estimates from this logit specification.11 Lemma 1
from the model predicts that a lower current list price will increase the probability of a
change in agents. This is a result of the seller increasing the belief that the agent she has
employed is a low quality agent. The coefficient estimate associated with the current list
price is positive and significant at the ten percent level, which is not consistent with the
model’s prediction. A positive coefficient for the current list price suggests that a higher
current list price will increase the probability of an agent change.

Although the decision to change price is not addressed in the model, dummy
variables indicating if the seller has changed her price on a previous day is included in the
estimation. Changes in the list price may capture seller unobserved heterogeneity. A
seller that has changed her price in a previous period may more likely be of an impatient
seller that changes agents. This could explain the positive coefficient estimate associated

with both indicator variables of a prior increase and decrease in list price. The marginal

 

11 . . .
Results were consrstent when mcludmg agent specific characteristics such as: if the agent was an owner

of the agency, if the agent was a member of the board of realtors, and the number of listings the current
agent has open.

17

percentage change in the probability to change agents on a given day is 0.396 for a
previous increase in the list price and 0.126 for a previous decrease in the list price.

The positive coefficient estimate associated with the list price prior to a price
decrease does not support Lemma 2’s prediction that a lower list price prior to a price
decrease will increase the probability of changing agents. The negative coefficient
estimate for a list price prior to a price increase indicates that a seller is more likely to
change her agent with a lower list price prior to an increase in the list price, which does
support Lemma 2. Although, an increase in the list price is outside the scope of the
model, it coincides with the notion that other agents are recommending a higher price to
the seller, which results in the seller increasing her list price and later changing agents.

In terms of the other covariates, duration on the market has a significant role in
determining whether a seller changes her agent. The coefficient estimates suggest that
the probability of a change in agent increases with days on the market, but this increase
diminishes. There are also positive and significant coefficient estimates associated with
the indicator variables denoting that the property has been on the market slightly over
three months and slightly over six months. This estimate is consistent with the theory
that sellers are more likely to change agents when the listing reaches the expiration date.

There is likely significant heterogeneity among sellers in terms of their
motivations to sell and their perceptions of the market. Not adequately controlling for
seller heterogeneity will result in bias estimates if the omitted seller specific variables are
correlated with the seller’s decision to change agents and with her choice of list price.
Omitted variables with respect to seller specific characteristics may explain why the

estimates in Table 2 are not consistent with the model’s comparative static predictions

18

(Lemmas 1 and 2). To address this concern we first expand the set of covariates in an
attempt to address the possible omitted variable bias. Because of the limited set of
covariates available, I also estimate an empirical model that includes seller fixed effects.

Column 11 of Table 2 includes the original list price into the previous estimation.
A more patient seller may set a higher original list price. The patience of a seller will
affect the decision to change agents. By including the original list price in the estimation,
we attempt to control for some of the seller unobserved heterogeneity. The positive
coefficient estimate associated with original list price is economically and statistically
significant and suggests a seller with a higher original list price is more likely to change
her agent. The coefficient estimates for the other variables show little change except that
associated with the current list price which is now close to zero.

The original list price does not completely control for all seller heterogeneity.
There are many aspects to sellers such as: urgency to sell, perceptions of the market, and
prior belief of employing a low quality agent that the original list price will not capture.
In order to better control for seller heterogeneity, I include seller fixed effects in the
initial specification. Seller fixed effects controls for all seller specific characteristics that
do not vary over time. These results are shown in Table 3. Because within seller
variation is used to estimate the coefficients, only observations of the 269 sellers who
change agents are used when estimating this specification. The negative coefficient
estimate associated with the current list price, while not statistically or economically
significant, is consistent with the Lemma 1.

The estimates in Table 3 provide stronger support for the model’s prediction

regarding the effects of the prior price on the seller’s decision to change agents. The

19

coefficient estimates for a previous increase or decrease in price are still consistent with
the premise that sellers are more likely to change agents after a price change. The
marginal percentage change in the probability to change agents on a given day is 0.00845
for a previous increase in the list price and 0.0024 for a previous decrease in the list
price. The estimates also Show that a lower list price prior to a price change increases
the probability of a change in agents. The coefficient estimate for the prior list price
after a list price increase continues to support the second comparative statistic. The prior
list price after a list price decrease now supports the model’s prediction with a negative
and statistically significant coefficient.

The inclusion of seller fixed effects has resulted in stronger evidence to support
the theory that a lower current and prior list price increases the probability of changing
agents. In this estimation both the current list price and prior list price had negative
coefficients. By controlling for all seller specific and property characteristics that do not
vary over time, seller fixed effects pushes the empirical specification closer to the

model’s assumptions.

VII. Conclusion

This paper is the first to address what factors affect the decision to change real
estate agents. The focus of the analysis is on how the list price affects this decision. A
property that stays on the market provides a noisy signal to the seller of her agent’s
quality. Because there is a higher probability of a property not selling with a low quality
agent, the seller will increase her belief that the agent she has employed is low quality.

The strength of this signal changes with the list price. A lower current list price increases

20

the seller’s belief of a low quality agent conditional. Similarly, if the seller has changed
her list price, a lower list price prior to the price change increases the seller’s belief of a
low quality agent. The empirical results support these comparative statics when
including in the estimation seller fixed effects, which accounts for seller specific
characteristics that do not vary over time. By controlling for seller specific
characteristics, the estimation is better able to isolate the updating of agent quality in the
list price.

Because this paper is the first step in this research, there are many areas to
expand. The Bayesian updating model showed how the seller updates her belief in the
quality of her agent. For simplicity, in this paper the list price is assumed to be a proxy
for the seller’s reservation price. Future research could endogenize the list price decision.
Further expansions could include a structural estimate of the model in a manner similar to
Israel (2005).

Implications of the list price’s effect on the strength of the noisy signal to the
seller of her agent’s quality could affect an agent’s level of effort. By varying the
strength of this signal, there are different consequences to the agent from a property that
stays on the market. The inclusion of an agent’s effort level with the seller’s decision to
change agents could be an interesting extension providing further insights into the

process of selling a home.

21

Data Appendix

The number of listings in the sample is 4,176. In order to identify if the seller
changed agents, I must first uniquely identify each property. In order to identify the
property, I need an address and a unit number. The unit number was missing for some

listings. I was able to identify some of these missing unit numbers if the number was in

the address, lot number or the listing matched an entry in tax records.12 There were 68
listings dropped because I was unable to locate a unit number. An additional six listings
were dropped because they had the same address, unit number and listing date as another
listing. A possible explanation for this is clerical error.

There were four listings that had their off market date changed because the
original off market date was earlier than the listing date, causing days on the market to be
negative. In these cases, I looked up each listing’s property history report to determine
an appropriate off market date. There were an additional 18 listings that I updated the off
market date because they overlapped with another listing. These looked like some of the
cases in which the property was on the market and the seller was unhappy with the agent
and moved to another agent before the old listing was completed. An additional 26
listings were dropped because the listing had occurred inside of an existing listing for that
same property. In some cases it could be a clerical error or the seller could be employing
two agents at the same time. Finally, there were 63 listings dropped because the square
footage of the property was set to zero.

The sample is now down to 4,013 listings. Of the current sample, 1,784 listings

did not result in a sale. The majority of the studies using MLS data throw out these failed

 

12
Tax records were found fi'om http://www.njtaxrecords.com/

22

listings. Others use the listings, but see them as a single failed attempt. Anglin (2004) is
one of the few who links different attempts (different listings) to the same property. This
linking is critical to determine if the seller decided to change agents. These different
attempts are linked by identifying whether the seller is the same for the different listings.
I link listings together when they are for the same property, when the earlier listing does
not result in a sale, and when the later listing starts within six months of the ending of the
previous listing. It may take some time for the seller to decide to go back on the market.
The farther from the previous listing, the less validity this previous listing should have
because of changes in market conditions and changes in the seller's personal feelings for
an agent. The seller could decide to put her property back on the market a couple years
later, but that previous attempt should have much less weight on the agent decision then
if the property was back on the market a month later. Cases in which the property went
back on the market more than six months later I treat as a different seller. There are 621
listings tied to a previous listing. Table A1 shows the distance in months from the end of
the previous listing to the start of the later listing.

There are limitations when using MLS data. A seller may get frustrated with
an agent, and instead of going to another agency she may sell the house on her own. I
would not be able to pick up this sale which, in effect, would result in more agent
changes than my data captures. Another potential problem occurs if the property had a
previous listing, was sold without an agent, and then was put back on the market, all
within six months of the original listing going off the market. I would incorrectly treat

this case as the same seller.

23

Considering the above limitations, 3,392 sellers were created in the sample.

Each of these sellers represents the collection of listings needed to sell a particular
property. A dummy variable is set if the linked listings had different agencies and
different agent names. The variable is not set when a seller is switched to a new agent
inside the same office because of vacations, a maternity leave or some other event or if an
agent leaves an office and brings her sellers with her.

For each seller, an observation was created for every day the property was on the
market. After this expansion there are 455,738 observations. I was then able to
determine when a list price was changed and could update the succeeding list prices for

that seller.

24

Appendix A

Proof of Lemma 1: Simplifying equations (2), (3), and (4) to:

Bo

 

 

 

 

 

 

 

C
P —x
B0 +(1_BO)* —1 — (2a)
x-zt
31(3) D E 23,,
B.(R>+(1—B.(R»*[P2‘£] €21 .3.)
P1"! x—)_c
and
B (P)
1 1 P E 532,
BI(B)+(1—B.(P,))* 3‘5 ....
x—g

WhereCZNG-NB, D: ONG- 0N3,andE=NGf-N3f.

In equations (2a) to (4a), an increase in the seller’s current reservation price increases the
denominator, resulting in a lower ex post belief that the seller has employed a low quality
agent.

63] 0 68,
5;; < for equation (2a) and 5].; < 0 for equations (3a) and (4a).

QED

25

Appendix B

Proof of Lemma 2:

ForP1 >P2:

532 _ Bri (Bo — 1)

 

313? Boat-e [[1, -..

andforP1 SPZ:

_a£2__ B0(Bo “—1)

BR (i-zc.) K

QED

(C - D)(

 

 

 

 

 

26

D C
Pz-r Pl-_x_ Pz—ZE
P11 x-J_c 32-29

—] {5—5) [52—5] (l—BO)+BOJ
_ x—)_c x—)_c

 

 

 

 

Table 1.1 Descriptive Statistics
N = 455,738 property-day observations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable Mean Standard Deviation
Square Feet 873 345
Number of Bedrooms 1.50 0.85
Number of Bathrooms 1.15 0.41
Parking Dummy 0.21 0.41
Heating by Gas 0.37 0.48
Air-conditioning 0.25 0.44
Days on the Market 134 120
Current List Price (U .S. Dollars) 204,558 125,855
Sold Price (US Dollars) 172,175 98,339

 

 

 

 

 

27

 

Table 1.2 Logit Regression

 

I II
Original List Price in 100,000 0.227M
(0.081)
Current List Price in 100,000 0178* -0.022
(0.106) (0.113)
Current < Prior [indicator] 0.275 0.317
(0.275) (0.267)
Current > Prior [indicator] 0.764" 0.792"
(0.289) (0.287)
Prior List Price in 100,000* [Current < Prior] 0.016 -0.020
(0.086) (0.083)
Prior List Price in 100,000* [Current > Prior] -0.079* -0.074*
(0.042) (0.040)
Days Since Price Decrease
Days Since Price Increase
Days on the market 0.014" 0.014"
(0.004) (0.004)
Days on the market squared divided by 1,000 -0.026** -0.026**
(0.010) (0.010)
After Seller Changed Agents -1.460** -1.517**
(0.357) (0.367)
After Seller Changed Agents times Days with 0.014" 0.014“
Agent (0.003) (0.003)

 

28

Table 1.2 (Continued)

 

I

II

 

 

The Number of Days on the Market is 1.667" 1.670"
between 90 and 105 (0.162) (0.162)
The Number of Days on the Market is 2.091 ** 2.094”
between 180 and 195 (0.184) (0.184)
The Number of Days on the Market is 0.763" 0.767"
between 270 and 285 (0.342) (0.340)
The Number of Days on the Market is 0.507 0.515
between 360 and 375 (0.619) (0. 620)
Square Footage 0.0002 0.0002
(0.0003) (0.0003)
Number of Bedrooms 0.005 0.013
(0.088) (0.089)
Number of Bathrooms -0.498** -0.503**
(0.181) (0.182)
Parking 0.373" 0.377"
(0.160) (0.160)
Gas -0.340** —0.346**
(0.134) (0.135)
Air-Conditioning -O.307* -0.320**
(0.159) (0.163)
Constant -9.027** -9.009**
(0.638) (0.639)
Year/Quarter YES YES
Observations 417,585 417,585
Pseudo R-squared 0.0976 0.0979

 

Robust Standard errors in parentheses
* significant at 10%; """ significant at 5%

 

29

Table 1.3 Logit Regression with Seller fixed effects

 

dependent variable (day of agent change = l)

 

I

II

 

Current List Price in 100,000

Current < Prior [indicator]

Current > Prior [indicator]

Prior List Price in 100,000* [Current < Prior]

Prior List Price in 100,000* [Current > Prior]

Days Since Price Decrease

Days Since Price Increase

Days on the market

Days on the market squared divided by 1,000

Alter Seller Changed Agents

 

After Seller Changed Agents times Days with
Agent

-0497
(0.995)

1.712*
(0.907)

5.834"
(1.282)

-0731"
(0.296)

-0.520**
(0.158)

0.120"
(0.008)

-0.126**
(0.012)

-15.040**
(1.050)

-0005
(0.004)

-O.632
(0.992)

1.529
(0.944)

4.138"
(1.450)

-0.636**
(0.317)

0523"
(0.140)

-0.00007
(0.00006)

0.00020“
(0.00009)

0.121**
(0.008)

-0124"
(0.012)

-15.278**
(1.056)

-0005
(0.004)

 

30

Table 1.3 (Continued)

 

 

 

I II
The Number of Days on the Market is between 1.452W 1.469“
90 and 105 (0.220) (0.222)
The Number of Days on the Market is between 2.285" 2.245"
180 and 195 (0.263) (0.261)
The Number of Days on the Market is between 0.290 0.286
270 and 285 (0.420) (0.417)
The Number of Days on the Market is between 0.372 0.399
360 and 375 (0.659) (0.664)
Observations 72740 72740

 

Robust Standard errors in parentheses
* significant at 10%; ** significant at 5%

 

31

Table A1 Distance in months of the end of the previous listing to the start
of linked listing.

 

Months from the previous listing to the start of the linked listing

 

One Two Three Four Five Six Tota
l

 

 

No of 380 97 50 39 28 27 621
listings

 

 

 

 

 

 

 

 

 

32

 

Figure 1.1: For each month on the market the percentage of listings that ended as ‘a
result of a sale, due to expiration, or an agent change.

 

 

 

 

 

V. i
<9. - ‘1
W i ‘1
a , I
.9 l “\
—l I, \
“6 or - ;
I: 1' ‘\
% / \ r" “ I \
E // \\\_-—7\_——_\\ It //\\‘/ \\\
LL F _ / ’ \‘ T. ‘I’ \ / “ \
o _
I I I I I I I I I I I
0. 1. 2 3. 4. 5. 6. 7. 8. 9. 10.
Months on the Market
—— Changed Agents ————— Sold
--------- End of Listing Not Resulting in Sale

 

 

33

Figure 1.2: For each month on the market the percentage of sellers that changed their
list price.

.9
N

 

0.184» ~ ~v * , . .
0.16 -...
0.14 — -

 

0.08
0.06 4
0.04
0.02 ---—~-~—-—+--—————- __

 

 

 

 

 

 

 

Percentage of List Price Changes
0

 

 

O
_
_
q
_
_
._
_
_
_

1 2 3 4 5 6 7 8 9 10
Month on the Market

34

Figure 1.3: Average original list price and sales price for listings that resulted in a sale.

I

1

Average Price
1

50000 100000150000 200000 250000 300000
I

 

 

1993. 1099. 2600 2001. 2002 2003. 2004. 2005 2036.
Year

 

 

Original List Prioe(on|y sold)
————— Sales Price

 

35

Figure 1.4: By year, the fiaction of sellers that changed list price, changed agents.

.3
\

.2
1
\

Average that Changed

 

 

 

19'9e 1099. 2600 2601. 2002 2603 2004. 2005
Year

 

-———- Seller Changed Agent
----- Seller Changed List Price

--------- Seller Changed Agent and List Price

 

 

 

36

References

Anglin, Paul M., (2004) “The Selling Process: If, at First, You Don’t Succeed, Try, Try,
Try Again,” working paper, University of Windsor.

Anglin, P., R. Rutherford and T. Springer, (2003), “The trade off between the selling
price and time-on-the-market: The impact of price setting,” Journal of Real Estate
Finance and Economics, 26(1), 95-111.

Bjérklund, Kicki, John Alex Dadzie and Mats Wilhelmsson, (2006), “Offer price,
transaction price and time-on-market,” Property Management, 24(4), 415-426.

Dale-Johnson, David and Stanley Hamilton, (1998), “Housing Market Conditions,
Listing Choice and MLS Market Share,” Real Estate Economics, 26(2), 275-307.

Glower, Michel, Donald R. Haurin and Patric H. Hendershott, (1998), “Selling Time and
Selling Price: The Influence of Seller Motivation,” Real Estate Economics, 26(4), 719-
740.

Herrin, William E, John R. Knight, and C. F. Sirmans, (2004), “Price Cutting Behavior in
Residential Markets,” Journal of Housing Economics, 13(3), 195-207.

Israel, Mark, (2005), “Services as Experience Goods: An Empirical Examination of
Consumer Learning in Automobile Insurance,” American Economic Review, 95(5), 1444-
1463.

Knight, John R., (2002), “Listing Price, Time on Market, and Ultimate Selling Price:
Causes and Effects of Listing Price Changes,” Real Estate Economics, 30(2), 213-237.

Lazear, Edward P., (1986), “Retail Pricing and Clearance Sales,” American Economic
Review, 76(1), 14-32.

Miceli, T., (1989), “The optimal duration of real estate listing contracts,” American Real
Estate and Urban Economics Association Journal, 17(3), 267-77.

Miller, N.G. and MA. Sklarz, (1987), “Pricing Strategies and Residential Property
Selling Prices,” Journal of Real Estate Research, 2(1), 31-40.

Read, Colin, (1988), “Price Strategies for Idiosyncratic Goods — The Case of Housing,”
ARE UEA Journal, 16(4), 379-395.

Sass, Tim R., (1988), “A Note on Optimal Price Cutting Behavior under Demand
Uncertainty,” The Review of Economics and Statistics, 70(2), 336-339.

37

Taylor, Curtis R., (1999), “Time-on-the-Market as a Sign of Quality,” Review of
Economic Studies, 66, 555-578.

Yavas, Abdullah, (1992), “A Simple Search and Bargaining Model of Real Estate
Markets,” ARE UEA Journal, 20(4), 533-548.

Yavas, Abdullah, (1994), “ Economics of Brokerage: An Overview,” Journal of Real
Estate Literature, 2, 169-195.

Yavas, Abdullah and Shiawee Yang, (1995), “The Strategic Role of Listing Price in
Marketing Real Estate: Theory and Evidence,” Real Estate Economics, 23(3), 347-368.

38

.On the Emergence of Money as a Medium of Exchange

The assessment of the physical properties (e.g., homogeneity, storability, divisibility,
durability) that allow some objects to become money and circulate as media of exchange
has been the object of much research. This research usually assumes that the supply of
these objects is exogenous, which greatly simplifies the analysis by circumventing
problems related to the overissue of money. In this paper, by contrast, I investigate the
conditions under which endogenously issued objects are valued, in an economy along the
lines of Kiyotaki and Wright (1993) but with a finite population. In contrast to previous
work, I assume that the economy has no exogenous technology that keeps track of the
actions of money issuers. My objective is to identify which attributes make some agents
natural candidates to become money issuers.

1. Introduction

There is a great deal of research on the attributes of objects that make
them particularly suited to be media of exchange. In contrast, not much has been done on
the study of the attributes of agents that make them natural candidates to be the issuers of
money. Ritter (1995) studies the transition from barter to fiat money and finds that the
money issuer needs to have a large size and patience for this transition to take place. In
turn, starting with Cavalcanti and Wallace (l999a,b) various papers have assumed that a

necessary condition for an agent to become a money issuer is the fact that his behavior is

monitored by the rest of the population.13 Finally, Monnet (2006) shows that agents that

produce a public good have a comparative advantage in the production of money. With

 

13
See also Cavalcanti, Erosa and Temzelides (1999), Mills (2007, 2008) and Ales et alli (2008).

39

the exception of Monnet (2006), all papers have assumed that the behavior of the money

issuer is publicly observable to some extent.14

In what follows, I consider an environment in which there exists no technology
that allows agents to observe behavior in meetings in which they do not participate. In
general, in the absence of such technology, there can be no equilibrium where
endogenously issued money is valued as a medium of exchange. Intuitively, in the
absence of a monitoring technology there is nothing that prevents the money issuer fiom
issuing too much money, as money is costless to produce and he expects that his behavior
will go unnoticed In contrast, I prove that there exists an equilibrium in which
endogenously issued money is valuable, as long as the population is finite. This holds
true, irrespective of the population size. More interestingly, I show that the key attributes
of an agent that make him a natural candidate to be the money issuer are patience,
visibility and “liquidity”.

The basic features on the environment studied in this paper are based on Araujo
and Carnargo (2006; hereafter AC). As in AC, information obtained in pairwise meetings
is important for the sustainability of the monetary equilibrium. The key difference is that
I consider a finite population.

This paper is organized as follows. The next section presents the environment and
describes the equilibrium. Section 3 discusses the robustness of the equilibrium. Section
4 discusses characteristics of the money issuer that makes for a natural candidate to issue

money. Section 5 concludes.

 

Monnet is able to circumvent the need for public observablility because he assumes that notes are costly
and that the matching process is deterministic.

40

2. Model

2.1 Environment

The environment is broadly based on AC. Time is discrete and indexed by t.
There is one large agent that we label the government, which is not able to produce any
good but can issue an intrinsically useless object that we label money. Precisely, at the

beginning of period 1, the government makes a once and for all choice between two

supplies of money, m H and m L with m H > m L >%. The government discounts

the future at a rate 6 per period. The economy is also populated by a finite number N of
agents. This is an important difference with AC, who assumes a continuum of agents.
Agents are able to produce X6 {113?} units of a good that can be stored by the
government. We assume that agents suffer disutility X from producing X units of goods,
and the government (agents) obtains utility X from the storage (consumption) of X units
of goods. At the beginning of every period, agents receive one unit of an indivisible
endowment. This endowment provides utility zero if he is consumed by the agent himself
but provides utility U if consumed by another agent. Agents discount the future at a rate [3
per period. Finally, money is indivisible and agents can hold at most one unit of either
endowment or note at a time.

The timing of events in the economy in every period is as follows. At the start of

every period, the government randomly meets m,- E {m H 5 m L } agents, where m,- is

the government’s choice of money supply made at the beginning of period 1. A key

41

 

assumption is that agents do not observe the value of m. Each agent then makes an offer
of production to the government in exchange for a note. The government can reject this
offer and after incurring a transaction cost A, he can randomly pick another agent in the
pool of agents who did not receive a note. This process continues until In notes are
distributed to the agents. After meetings between agents and the government takes place,
there are r rounds of meetings between agents. For simplicity, agents do not discount the
future in between rounds. In each round, agents are anonymously and pairwise matched
under a uniform random matching technology. In a meeting, if an agent does not have
his endowment and consumes the good of another agent, he receives another unit of
endowment. By assumption, agents cannot barter in a meeting. At the end of the period,
after the last round of meetings in the market, if an agent ends up with a note, he can

redeem this note with the government. In the redemption, the government gives X units

m
_ H
of the good in exchange for the unit of money. Henceforth, we let mh " N and

m1 = N be the fraction of agents in the market with money if the government

chooses m H (respectively, m L ).
2.2 Equilibrium
We first consider the case with perfect information, where agents know the exact

i
number of notes circulating in the economy. Let W1 (7") indicate the value firnction for

an agent holding a note right before the ith round in a period, with i = l,..,r and the

42

. i
fraction of agent wrth money equal to m. Similarly, Wo (m) indicates the value

function for an agent without a note. We have:
(1) Wf (m) = mW1""1(m)+(1- m) {U + WSH}

i+l i+1

(2) WS (m) = mw1 (m) + (l — m)wO
(3a) er+1 (m) = X + flw:

l l
(315) w0+ (m) = flwo

After some computation, we obtain:

(4) W1 (m) = Wi (m) = W?) (m) +(1- m)U

(5) w. (m) = w; (m) = (1— fl)" W + (r — 1>m(1 — m)U}.
Note that
(6) W1 (m) " W0 (m) = (1 — m)U

The additional benefit of entering the first trading period with money is (1-m)U .

Therefore, with the total amount of money injected in the economy equal to Nm, an agent

is willing to produce up to (1 —m)U units of divisible good to enter the market with

money.

In what follows we assume that

43

 

(1—m,)U >X>(1—mh)U >_)_(_>,B(1—m,)U.

This assumption ensures that, if the government chooses m L , each agent who meets

with the government produces X in exchange for a note, while if the government

chooses m H , each agent who meets the government produces i. In fact, as long as the

transaction cost A is small, it is straightforward to show that an agent is going to produce
the minimum amount that ensures acceptance of the offer by the government. If the agent
produces less, the government is going to reject his offer and offer the note to another
agent. Thus, the government receives a total of NmX units of good that he stores during

the period obtaining utilitmeX . Finally, in what follows we also assume that

m L X > m H i . This ensures that, if agents have perfect information with respect

to the government’s choice of money supply, the government’s flow utility is higher
when the government chooses to issue m L o Note that the expected utility of an agent is

also higher when m = m L since this implies a higher fi'equency of trade meetings in

the market. All in all, the discussion above implies that the unique monetary equilibrium
under perfect information involves no overissue of money.

Now consider the scenario where the money supply is not directly observable by
the agents. In this case, because the period 1 flow payoff from issuing more money is
higher than the period 1 flow payoff from not issuing money, the government may have
an incentive to increase the money supply. In fact, if the economy is populated by a

continuum of agents instead of a finite number, this is exactly what is going to happen.

44

 

The intuitive reasoning is as follows. Consider an equilibrium strategy in which there is
no overissue on the equilibrium path. In this case, since agents only face a countable
number of meetings in their lifetime, they always believe that the number of notes in
circulation is consistent with no overissue, irrespective of the behavior of the
government. Thus, the government has an incentive to overissue, a contradiction. This is
the reason why environments with a continuum of agents rely on some exogenous
monitoring technology that allows agents to observe the behavior of the government. In
this paper, as we consider a finite population, it is not necessarily the case that one needs
such technology. This is in itself an interesting approach as it allows the identification of
additional elements (besides exogenous monitoring devices) that may impact the
likelihood of an equilibrium in which no overissue takes place.

In what follows, in order to proceed with our analysis in the case of imperfect

information, we assume that agents are able to uniquely identify the notes issued by the

15 . . .
government. Under thls assumptron, we prove that, as long as the government rs
sufficiently patient, there exists a sequential equilibrium in which no overissue takes
place on the equilibrium path.

Consider the following Strategy profile (Rule 1): The government chooses

m = m L and offers gr: units of goods per note in the redemption process. As long as

an agent has always observed that the number of notes in circulation is below or equal

to m L , he produces (1_ ml )U to the government; otherwise, he produces

 

15
A note is uniquely identified by the serial number for US. currency. Instead of adding the complexity

of given the government a choice to put a serial number on each note, I assume the agent can uniquely
identify each note.

45

 

(1“ m H )U . Finally, after any history, agents always accept money in exchange for

goods in the market.

Proposition 1 describes our main result.

* :1:
Proposition 1:: There exists a discount factor 6 such that, for every 6 2 5 , the
strategy rule Rule 1 is part of a sequential equilibrium. In this equilibrium, the
government does not overissue money.

Proof: See the Appendix.

3. Robustness

With only an agent’s personal trade history as information on the supply of
money, the key assumption fiom the model is that there is a finite population. The
assumption of a once and for all choice of the supply of money simplifies the analysis,
but is not necessary for a monetary equilibrium. The equilibrium could be sustained
when the government is given the option of changing the money supply each period. If

the government found it optimal to deviate one period, the government would find it at

least as optimal to deviate the following period. The benefit of deviating is obtaining X
for additional notes. The only difference between the two periods is that some agents

would have learned that the government has deviated. The probability that the

government would receive X for each note they issue decreases in the following periods.
All agents would understand this and if they caught the government’s deviation they
would believe that the government would continue with the deviation in firture periods.

Similarly, only producing two supplies of money is not necessary, but again. simplifies

46

the equilibrium. All that is needed is a threshold that is considered too much money.
When agents count more than the threshold amount of notes, agents can change their
behavior which punishes the government.

The choice of adding and removing notes each period is designed to ensure that
the government benefits in future periods with a monetary equilibrium. An alternative
environment could be constructed where the government consumes with the entry of each
note, but then benefits from trade with the use of money. By constructing the
environment in Section 2, the calculation of off equilibrium outcomes is greatly
simplified.

The mechanism of trade in rounds also simplifies the analysis, and is not
necessary for a monetary equilibrium. In a new mechanism, such as divisible goods and
divisible money with bargaining, then agents could learn the money supply through their
trade meetings. The monetary equilibrium could be sustained but would cause
complications computing equilibrium and off equilibrium outcomes. In this case, Rule 1
should be modified to use a decision rule similar to Kandori’s (1992) social norm
equilibrium.

The process of distributing notes is constructed in a way so there will be truth
telling by the agent and the government will eventually be punished for over issuing. The
environment could be changed to where there is bargaining between the government and
the agent for a note. Finally, the ability of agents to uniquely identify notes gives the
ability to use their personal trade histories as a mechanism to limit the supply of money.

An alternative approach is for agents to update their beliefs on the money supply using

 

‘6 See Araujo (2004) for an application to monetary theory.

47

Bayesian updating with the modification that there is an epsilon probability that the

government will choose to issue m H notes. A discussion of the behavior of agents in a

similar environment can be found in AC.

4. Discussion

The framework proposed provides an environment where we can study properties
of the money issuer that enable a monetary equilibrium. Our model can thus lend insight
into the types of agents that make good candidates to issue money, and why the
government is a natural candidate. It is clear from Proposition 1 that in order to sustain a
monetary equilibrium, the government needs to be patient. To limit the number of notes,
the government needs to value the opportunity of consumption for many fiiture periods
more than over issuing and being able to consume more today. This result is not new and
has been shown by Ritter (1995).

Another area of exploration is the amount of information available in the
economy. Intuitively, if information about the behavior of an agent is disseminated fast
(say because this agent is more visible than others), he may be a natural candidate to
issue money. To see this, suppose the environment is changed so that or agents, instead
of just one agent, randomly see the note each round. We can see how this change
decreases the benefit of the government from overissuing. With a decrease in the benefit
of deviating, the monetary equilibrium with no overissue can be sustained with fewer
periods. The probability that a specific agent has seen a specific note after 1 rounds in

circulation changes to:

48

.00) = 1 - 1 - £—

09 N—l

As shown in Lemma 1, the case where a = 1 can sustain a monetary equilibrium with a
patient enough government. When a approaches N, this is an environment where the
government is publicly monitored such as in Berentsen (2006), Cavalcanti et a1. (1999),
Cavalcanti and Wallace (1999), Martin and Schreft (2006), Ritter (1995), and Williamson
(1999). As can be seen from Figure 1, the rate of convergence is faster with an increase
ina . It takes fewer rounds for the probability that a specific agent has seen a specific
note to approach one. This probability is related to the probability that a specific agent
has seen all of the notes, with anything that affects the rate of convergence for one will
also affect the other. Holding everything else constant, such as the number of agents in
the economy and the number of rounds in a period, an increase in a will speed up the
probability that an agent has seen all of the notes. The probability that an agent has seen
all of the notes is directly related to the probability that all agents have seen all of the
notes. The value to the government of deviating will thus change depending on the speed
of notes circulating. With fewer periods when money is accepted, there are fewer periods
of future consumption with a deviation. Therefore, an increase in a will decrease the
benefit of deviation, which will increase the sustainability of a monetary equilibrium. A
more visible government is a natural candidate to issue money. This dimension is also not
new as it relates to the idea that the ability to be monitored is an important attribute of the

money issuer.

49

A third dimension, which is novel to the literature, has to do with the redemption

process. If the government is able to commit to redeem a note for X units of goods, the
redemption process now reveals information about a deviation. This is so because on the
equilibrium path, the government is able to firlfill this promise but he cannot do so if he

deviates. If the government deviates, eventually agents will count too many notes and

offer; for a note at the start of the period, leaving the government short when he needs

to redeem the note at}. An agent will now have two ways to spot a deviation: counting

too many notes or receiving less than X during the redemption process. Thus, since a
deviation spreads faster and punishes the government, it prevents deviations from
happening in the first place. We can think of banks who can offer this high quality
redemption as banks that are more liquid in the sense that they can match high levels of
redemption. Thus the ability to be liquid is another dimension that characterizes a note
issuer. To put it in other words, the redemption process can be seen as a signal of the
banks intent. By offering a higher redemption rate, the bank is able to commit to gaining
utility through issuing notes as opposed to a risky investment such as loans. This adds to
the argument that the role of issuing currency should be conducted by an institution that
is solely interested in maintaining a monetary equilibrium instead of individual profits.
The profit seeking bank would be less liquid resulting in a higher probability of default.

This coincides with the nineteen century experience where private profit seeking banks

. . . . . . l7
ovenssued therr notes, resultrng 1n therr notes berng worthless. When a central

government monopolizes currency, overissue is less of a concern.

 

For a discussion of private bankers see Sylla (1976).

50

5. Conclusion

In this paper, an equilibrium with no overissue is sustained without some
technology that allows agents to observe behavior in meetings in which they do not
participate. The money issuer is prevented from over issuing notes by the threat of agents
producing less in exchange for a note. The actions of individual agents will have an
effect on the money issuer because there is a finite population. We study attributes of
agents that make them natural candidates to be the issuers of money, and found that key

attributes are patience, visibility, and liquidity.

51

Figure 2.1: The probability that a specific agent has seen a specific note with varying

as.

 

CorwergencewithmflgerisintheEcoru'ry

1.2

 

 

 

——Alprais1
----A|phai32
------- Alphais3
—----Alphais4

 

 

 

 

 

 

 

 

 

 

52

Appendix

Proof: Note that the assumption that the government offers K in the redemption phase
ensures that the redemption does not transmit any information about a deviation.
Likewise, agent meetings during trade rounds do not transmit any information about a
deviation because an agent will accept and use money during these meetings regardless
on the belief of the money supply because money will increase the probability of
consuming.

First, we need to be careful as to how beliefs are formed after a deviation is
observed in the meeting between the agents and the government. In particular, consider
the case where the government enters a second round in which he offers a note to an
agent. If an agent observes this event, he knows that a deviation took place, as the
government never enters a second round on the equilibrium path. In general, there are
two sources of deviations. First, the government deviated from the equilibrium path; one
agent was able to infer that a deviation took place and punished the bank by producing
less good. The government then entered a second round hoping that another agent would
produce more goods. Second, the agent deviated in the first round with the government
by offering less goods, and the government entered a second round hoping that another
agent would produce more goods.

In what follows we argue that a sensible belief for the agent is to assign a higher
probability that the agent deviated and not the government. The reasoning runs as
follows. If the government deviates from the equilibrium path, it is likely that more than
one agent has observed that there were too many notes in circulation. In this case, if an

agent offers fewer goods in exchange for money in the first round of meetings and the

53

government rejects the offer, he expects that with a positive probability, the agent he will
meet in the second round will also offer less. Thus, as long as the transaction cost A is not
too small, the government prefers to accept the lower offer of the agent in the first round.
Now, if an agent deviates from the equilibrium path and offers fewer goods to the
government, the government will enter a second round as he expects that an agent in the
second round will offer more goods with probability one. All in all, this implies that,
upon observing an offer by the government in the second round, an agent believes that
this offer was the result of an initial deviation by an agent and not a deviation by the
government. Note that, given this belief, an agent has no incentive to deviate and produce
less to the government because the government will reject the offer and enter a second
round as he expects that the agents he will meet will offer more goods. We have already
claimed that the government has no incentive to deviate from the equilibrium path solely
to exploit agent’s beliefs in their meetings with the government because he anticipates
that, even though agents who did not observe a deviation themselves believe that the
deviation was caused by some agent, the government expects that some agents already
have observed a deviation and thus will produce fewer goods. It remains to check the
incentive of the government to deviate from the equilibrium path so as to exploit the fact
that it takes some time for the information of his deviation to spread throughout the

economy.

The value each period to the government if he does not deviate is N ml X . He

obtains this utility each period. With a deviation, the government would obtain

N mh X for a number of periods, which is an increase in consumption compared to the

54

 

value of not deviating. Eventually all agents would have counted m h notes, and

offer 1 in exchange for a note. When this happens, the government would then
continually receive less (mh _)_(_ ). If the increase in consumption with a deviation is

finite, then there exists a 6 * such that, for every 6 2 5 * , the strategy rule Rule 1 is
part of a sequential equilibrium. In order to show that the increase in consumption with a
deviation is finite, I need to show that the probability that all agents has seen all the notes
goes to one as the number of rounds the notes are in circulation goes to infinity.

First consider the probability that a specific agent has seen a specific note after the

note has been circulating 1 rounds:

. _(N-1)"-(N-2)" _ _ __1_"
(9) p(l)_ (N_1)i —1 N-l

 

Two important results can be obtained from this probability. The first is that, as the
number of agents in the economy goes to infinity, the monetary equilibrium breaks down.
When more agents are in the economy, there is a lower probability that an agent has seen

the note. Hence, outside money is essential in large populations.

lim (')——>0
1
N—)oop

The other result is that the probability a specific agent has seen a specific note goes to

one as the number of rounds that note has been circulating goes to infinity.

55

lirn .
. 17(1) -+ 1
l —) 00
With each round a note is circulating, more and more agents are observing this note in

their meetings. Now consider the probability that a specific agent has seen all the notes

m
after i periods where the number of notes is m. This is less than 2111);;
J:

Similarly, the probability that all agents has seen all the notes is less

m

than k2]: ; p— m :1: N . I am partitioning the probability that a specific agent

has seen a specific note for each agent and each note. In essence, I am limiting the

 

number of rounds that a specific agent can see a specific note to , and doing this

1
m * N
for all agents and notes forcing independence which is certainly less than the true

probability that all agents has seen all of the notes.

11111 N m z
. 2 Zp —— -—>1 .. .
l __) 00 k=l m =1: N , therefore the probabrlrty that all agents W111

have seen all the notes goes to one as the number of rounds goes to infinity. QED

56

References

Araujo, L., (2004), “Social Norms and Money,” Journal of Monetary Economics, 51:2,
241-256.

Araujo, L. and B. Camargo, (2006), “Information, Learning and the Stability of Fiat
Money,” Journal of Monetary Economics, 53 :7, 1571-1591.

Berentsen, A., (2006), “Time-consistent private supply of outside paper money,”
European Economic Review, 50, 1683—1698.

R. Cavalcanti, A. Erosa, T. Temzelides, Private money and reserve management in a
random-matching model, J. Polit. Economy 107 (1999) 929—945.

Cavalcanti, R. and N. Wallace, (l999a), “A Model of Private Bank—Note Issue,” Review
of Economic Dynamics, 2:1 January, 104-136.

Cavalcanti, R. and N. Wallace, (1999b), “Inside and Outside Money as Alternative Media
of Exchange,” Journal of Money, Credit and Banking, 31 :3, 443-457.

Kandori, M., (1992), “Social Norms and Community Enforcement,” Review of Economic
Studies, 59, 63-80.

Kiyotaki, N. and R. Wright, (1993), “A Search-Theoretic Approach to Monetary
Economics,” American Economic Review, 83:1, 63-77.

Kiyotaki, N. and R. Wright, (1989), “On Money as a Medium of Exchange,” Journal of
Political Economy, 97, 927-954.

Martin, A. and S. Schreft, (2006), “Currency Competition: A Partial Vindication of
Hayek,” Journal of Monetary Economics, 53:8, 2085-211 1.

Monnet, C., (2006), “Private Versus Public Money,” International Economic Review,
47:3, 951-960.

Ritter, J. (1995), “The Transition from Barter to Fiat-Money,” American Economic
Review, 85, 134-149.

Sylla, R. (1976), “Forgotten Men of Money: Private Bankers in Early US. History,” The
Journal of Economic History, 36:1, 173-188.

Williamson, 8., (1999) ” Private Money”, Journal of Money, Credit, Banking, 31, 469—
491.

Wolinsky, A. (1990), “Information Revelation in a Market with Pairwise Meetings”,
Econometrica, 58, 1-23

57

 

The Intensive Margin with Heterogeneous Producers

This paper analyzes the intensive and extensive margins of trade in a random matching model
with divisible money, where productivity differs across agents and producers can choose whether
to enter in the market in every period. The model exhibits multiple equilibria: one equilibrium in
which only high productive sellers enter and one equilibrium in which both high and low
productive sellers enter. The main result is that the high productive sellers will produce more in
the equilibrium in which both types of sellers enter, despite the fact that the average productivity
in the economy is depressed by the presence of the low productive sellers. This result is in
contrast to Camera and Vesley (2006), which consider a similar environment but with indivisible
money. Intuitively, as long as the benefit to the buyer of having a higher probability of
consumption is greater than the average productivity decrease, buyers will choose to bring more
money to the market, thereby encouraging the high productive sellers to work more.

1. Introduction

This paper analyzes the intensive and extensive margins of trade when the
productivity differs across sellers. The environment in this study assumes that money is
essential as well as divisible. Producers will face a decision to enter the market. As a
result of the different productivities, is it possible to have multiple equilibria, one where
only high productive sellers enter and one where both high and low sellers enter? If both
equilibria exist, how will the amount of consumption by consumers change?

There are other papers that study the affect of heterogeneous producers on the
extensive and intensive margins; these studies use a model with the value of money
endogenous. Camera and Vesely (2006; hereafter CV), develop a model to study the
effects of heterogeneous producers in an environment with money essential and in which
agents have to decide whether or not to enter the market. The focus of their paper is how
the addition of low productive sellers can decrease the value of money, possibly resulting

in an equilibrium in which only high productive sellers being socially preferable. This

58

contrasts with an equilibrium in which both types of sellers are active in the market. In
such a case, there are more buyers consuming, but each buyer will consume less than
they would consume with only high productive sellers in the market. Productivity
decreases during the expansion, and prices will increase.

A limiting factor in the CV model is that money is indivisible. They finish their
paper by stating “[h]owever, we suspect that equilibrium multiplicities should vanish in
models with degenerate distributions on divisible money. In that case, buyers would
benefit by spending a little something — instead of their entire money holdings — even in
matches with inefficient sellers.”

My paper shows that this is not necessarily true. I am able to get multiple
equilibria with divisible money. For the most part there is consistency with CV’s results
and my results; however, my model suggests it is possible for buyers to consume more
with the entry of both types of sellers because the high productive sellers will produce
more in the equilibrium with both types of sellers entering. For this situation to occur,
the benefit to the buyer of having a higher probability of consuming needs to be greater
than the average productivity decrease. In this case the buyer will choose to bring more
money to the economy thereby encouraging the high productive sellers to work more.
This result is not possible in the CV model because of the indivisibility of money.

The rest of the paper is as follows: Section 2 discusses the model. Section 3
shows that multiple equilibria can exist depending on the cost of entry and the number of
different types of producers in the economy. This section is the one in which the
comparison between my model and CV’s model will be made. The final section

concludes with a summary of my findings and future extensions.

59

 

2. Model

The environment is a modification of the model introduced by Rocheteau and
Wright (2005; hereafter RW). There are an infinite number of rounds in which agents
make consumption and production decisions. Each round is divided into two sub-
periods, the day market (centralized) and the night market (decentralized). In these two
markets, there is only one type of good produced and consumed. The good is perishable
and cannot be transferred to other markets or rounds. There is an intrinsically worthless
asset called money that is divisible and costless to store. The amount of money in the
economy stays constant. During the day period, there are no fiictions, all agents are able
to consume and produce. The utility of consuming and the disutility of working are
quasi-linear, which results in no wealth effects. With no wealth effects, agents can enter
the day market with different amounts of money, but they end up leaving the market with
the same amount of money. Setting up the centralized market this way is an innovation
from Lagos and Wright (2005). They show that money holdings are degenerate in

equilibrium, which makes the search model much more tractable.

The night market has fiictions that allow money to be essential”. Agents are
classed into two types: buyers and sellers, and they stay in that class for the life of the
agent. Buyers are unable to produce, and sellers are unable to consume. Buyers can
enter the night market at no cost. As a result of no cost of entry, all buyers enter the night

market. Sellers have a cost, k, to enter the night market, so they have to decide if they

 

18 Essentiality from Kocherlakota (1998), adding money enables a better outcome than
not having money.

60

want to enter. Agents are also anonymous, from Kocherlakota (1998), this is one of the
necessary conditions to produce an essential role for money. The modification to the RW

model that this study introduces is this: it proposes that there are two types of sellers, one

type is more productive than the other. This can be understood as: C 1 (q) < CZ (q)

I r
and C 1 (q) < C 2 (9) where c(q) is the cost of producing q units of the good. For

both types of sellers the cost function satisfies: Ci (q ) > O , Ci (q ) > O and

Ci (0) : Ci (0) = O for i = 1,2. Once again, the difference in productivity occurs
only in the night market. The model allows me to examine the changes in the extensive
and intensive margin effects between the two equilibria, while assuming that money is
divisible.

Only a fraction of buyers and sellers who enter the night market are able to trade;

this is the friction that makes money essential. The number of sellers that participate in

* It
the night market is n, and of those n will find a trade (of course n > n ). There

1|:
are n- n sellers that pay the cost to enter the night market, but due to the frictions in the

market are unable to participate. All buyers enter the night market because there is no

cost to enter. The number of buyers in the economy is normalized to one and of those,

*
B are able to participate. The market is competitive which means that agents who are

able to participate in the night market take the price as given and are able to produce or

consume as much of the good as they would like at that market price.

61

In order to determine the decisions agents will make at night, we first need to
solve decisions made during the day (centralized market). The buyer’s problem during

the day is:

Wb (Zb) = maX{V(X) - y + fldVb (3)}

Zaxay

A
subject to Z + x _ Zb + y , where y is the amount of work in the day market.

Substituting in the constraint yields:

(1)

Wb (2,) = zb + ngax{v(x) —x —2 +fldVb (2)}

The buyer enters the day market with an amount of real balances, Z b . v(x) is

b A
the utility of consuming x, v’(x)>0 and v”(x)<0. V (Z) is the value of entering the

A

night market with real balances, Z . The buyer’s problem is to decide how much to

consume and how much real balances, Z , to bring into the night market.

The seller’s problem is:

6 W<Zs>=maxv(x)—y+fldmaxr:(2>,fl.W(2)]}

25355))

subjectto Z +x = 23 +y.

62

 

The seller enters the day period with the amount of real balances, Z s . The seller

needs to decide how much to consume, should she enter the night market, and if she does,

how much money should she bring.

Lemma 1: from RW holds. For all agents in the centralized market, 2 is independent of
z. The amount of money with which an agent enters the centralized market will not

affect the amount of money the agent will leave with. Also,

Wb(Zb) = Zb +Wb(0)and WS(Zs) : ZS + WS(O) are
linear.

The buyer maximizes (l) with respect to 2 and x.
The first order conditions are:

x: v’(x)=1 and

. —1+,6de(2)$0, :0 if 2>0

(3)Z:

To solve for the amount of money the buyer will bring into the next period we

first need to solve the night market. The value function for the buyer at night is:
Vb (Z b l =
b b b
a(n)ngf,}x{u(q )+ AW (26 - pq )}

+11 — a(n)]fl.W”(z.)

63

b < . . . . .
subject to P4 — Z b where p rs the prrce clearing the competrtrve market. Because

the market is competitive, buyers and sellers take the price as given. The buyer is limited
in the amount she can purchase by the amount of real balances she brought to the market.

The probability the buyer is able to participate in the night market is a(n).
a(n) captures the frictions in the market by limiting the number of agents participating.
If I were using bargaining, a(n) could be thought of as the matching function. The

parameter n is the number of sellers in the night market (the number of sellers that paid

the cost k). The more sellers in the market, the greater the chances are a buyer will trade.
a(n) has the properties a'(n) > 0 , a"(n) < 0
a0) s min{1,n} mm = 0.a'<0) =1,.nd (1(a) = 0.

The next step is to determine if the constraint in (4) will bind. If the constraint in

(4) binds then

Vb(zb) =

b b
Z

ab(n)max u — +flnW" zb—pZ +
(5) 4” P P

[1" ab (n)]flnWb (Zb)

Using Lemma 1 (5) becomes

 

z 1

(6) Vzb (Z) = 050015 :1; P + [1 — a(n)]fln .

64

If the constraint does not bind, then the value of entering the night market with an

additional unit of real money is equal to :8" . Because the extra money will not be used

in the night market, its value comes from its use in the next day market.

1
V2: (Z) = a(n)u" —;— — i i . V2: (Z) = 0 ’If

p 2 , 1f the constrarnt brnds, and

b
the constraint does not bind. V (Zb) is concave, because u”(q) is negative. With

Vb . . .
(Z b ) berng concave, there rs a unrque value of 2 that solves (3).

When the constraint does not bind, (3) becomes:
(7) —1 + ,6 < 0
The buyer does not find it optimal to bring in money that will not be used in the
night market. The added value of bringing money that will not be used in trade is
negative. Given the opportunity to trade, the buyer will spend all of her money.

Therefore, the constraint will bind. Plugging (6) into (3) gives

u'(q" )

1= flda(n)——+fl[1-a(n)l=>

 

1 _ 1 +1_ u___'(qb:>)
060013 0601) .317

__fl_+1-_u'(q”)
‘8) fla(n)+ mp

65

b b
The buyer is going to bring z where Z : pq and q solves (8).

The night value function for the seller of type i is:

 

V’(Z.) = ain)1r;§Xt-C.—(qf)+ fl.W’(Z. + pqm

(9)+ 1_a(n)

 

flnW’(Z.) -k

n

The seller has paid the cost of entering the night market k, and with probability

0601)
, the seller will get to trade in the night market. The seller’s type does not affect

this probability. If the seller is able to trade in the market, the seller’s decision is how
much of the good to produce. Therefore, money brought into the night market will not be
used. The value of bringing money into the night market is the value the money has for

the next day, discounted for the future.

V’(Z.) = flnzs + VS(0)

(10)

V: (Z.) = [3,.

The first order condition for (2) is

—l+,6dVZS(zS)_<_0:>

(11)—1+fl<0

66

All sellers find it optimal to carry no money into the night market. If the seller is
able to participate in the night market, she will decide to produce until the marginal cost

is equal to the marginal benefit. From (9) we get

-C'.-(q.’)+fl..p=0=>

l S _

(.2) C.- (q. ) - A}?
From (12) and having one price in a competitive market, it can be implied that
(13) (910113): 0'2 (615).
Using ( 12), equation (8) gets modified to

l-fl +1: (4'01”)

14 .

( ) 165“”) c'i (q?)

As stated earlier, 11 is the total number of sellers in the night market. This can be
broken apart into the numbers of type 1 and type 2 sellers. (Note: type 1 is high

productive sellers and type 2 is low productive sellers.)

For markets to clear, the quantity demanded by buyers must equal the quantity

supplied by sellers.
0601) 0501) b
”1 n 9: +112 12 q; =a(n)q :

67

n1 5' n2 5 b
(16) ql + q2 = ‘1

Sellers need to decide whether to pay the cost k to enter the night market or avoid
paying and skip the night market. The benefit for the seller of entering the night market

is the probability of being able to trade, times the discounted value of the money earned

from the buyer, minus the cost of production. Using (9), the entry condition is

m) 0“") (flnpqg’ — 6,-(qf)) > k.
n

 

With (12), (17) becomes

(1,) a“) (6'.- (q? )9? - C.- (q? )) > k .
n

 

b s 3
DEFINITION 1: A competitive equilibrium is a list ( q 9 ql 9 q 2 9 n1 9 n 2 ) that

satisfies (12), (l4), (16), (18).

ASS UMPTION 1:

k < 930'2 (CID-62%) < qIC'l (CID-6101?).

Assumption 1 is needed for sellers to enter the night market.

ASSUMPTION 2: There is a limited number of type 1 sellers, ”1, and a limited number

of type 2 sellers, ”2 .

As will be seen later, limiting the number of high productive sellers (type 1),

makes it possible to have an equilibrium where low productive sellers enter.

68

3. Multiple Equilibria"

To describe the circumstances surrounding multiple equilibria, more assumptions
are needed:

ASS UMPT [ON 3:

(.8,“—(——”‘)(q:c (q:)— c.(q:»>k> 0“” ——-’-”(q2‘c c'2(q;)- c2012»
71

1 ”1
The first step is to show that there is an equilibrium with only high productive
sellers. Assumption 3 describes the condition when all high productive sellers enter the

market and no low productive sellers enter the market.

ss _ b SS
q 1 - q so ql solves the buyer’s problem from (15)

1- __fl_+ _ u'(qf’)
“9’ [30601 )2, 6'1 (qu)

 

Some notes on notation: n is the number of sellers in the night market. n1 is the number of high
productive sellers (type 1) and n2 is the number of low productive sellers (type 2).

SS
q 1 is the amount of the good produced by the high productive sellers in an equilibrium with only high
productive sellers.

sb
q 1 is the amount of the good produced by the high productive sellers in an equilibrium with both high

and low productive sellers.

ss
q 2 is the amount of the good that would be produced by the low productive sellers in an equilibrium

with only high productive sellers.

sb
q 2 is the amount of the good produced by the low productive sellers in an equilibrium with both high

and low productive sellers.

bs
q is the amount of the good consumed by the buyer in an equilibrium with only high productive

sellers.

bb
q is the amount of the good consumed by the buyer in an equilibrium with both high and low productive

sellers.

69

Using (1 3)
ss _ 1—1 1 ss
qz ‘62 (61(q1»
SS
q 2 will not be produced in this equilibrium, but would be used to show that if

the type 2 seller entered the market, she would produce q SS , and by plugging 613" into

(17) the value of entry would be less than k.

Using equation (14) there is a unique q, given ”1 sellers in the market, because

the marginal utility of the buyer is decreasing and the cost of production of the seller is
increasing in q, therefore there will be a unique quantity that solves (19). Because the
cost of entry is too high for low productive sellers, this creates equilibrium where only
high productive sellers enter the night market. The change in q, due to a change in n with

one type of seller, is:

QEI.=_ Li “'01) [c'(q)l2
6n .6 r101)2 u"(q)C'(q)-u'(q)0"(q)

With one type of seller in the market, the intensive and extensive margins increase

 

as the number of sellers entering the market increases. As the extensive margin increases
by the properties of the matching function, more sellers in the market will result in more
buyers being able to participate. The intensive margin increases because there is less
consumption risk for the buyer. With less consumption risk, the buyer will be willing to
bring more money, which leads to an increase in the quantity consumed. The final
assumption for multiple equilibria must assure that both types of sellers could enter the

market.

70

ASSUMPTION 4:

 

“(”1 +'1’)(qfc’1(qf)—cl(qf))2
"1 +’72

 

(2°) “(”1 +”Z)(q2c'2(q:)—c2(q:»2k
"1 +n2

Assumption 4 states that the arrival of low productive buyers into the market
leads to an increase in the expected producer surplus. Using Assumption 4 and limiting

the number of type 1 and type 2 sellers, all of the sellers would choose to enter the

sb
market. Equation (14) can be solved as a firnction of q]

Using (13) again
b —1 b
q; = 0'2 (c'1(qi’ ))

sb
q 1 solves the buyer’s problem:

 

 

I n S n I— I S
u —lq1b+_—2'—Cz1 (61(q1b»
l—fl +1: n1+n2 n1+n2
Warm» (2.0.50

. . Sb 1 sb
There exrsts aunrque q1 that solves (14) because C 1 (ql ) is increasing in

n b n —1 b
Sb u' —‘— 5 +46 c' S . Sb
q 1 and "1 +112 ql "1 +712 2 ( 1 (ql )) rs decreasingin q 1

b b Sb 5b
Once q 13 is found, aunique q 3 can be determined; and using ql , qz , "1 and

71

, bb . . . . . . .
n2 , the unrque q can also be calculated. Th1s creates a unrque equrlrbrrum 1n Wthh

both types of sellers enter the night market.

I have shown that there can exist a unique equilibrium with only high productive
sellers and a unique equilibrium with both types of sellers entering the market. But given
the same parameters, can these two equilibria coexist?

With the first equilibrium, the producer surplus for the low productive seller is
less than the cost of entry k. For multiple equilibria, the producer surplus for low
productive sellers needs to rise as the number increases of low productive sellers entering
the market. If this is the case, k and n1 can be set where multiple equilibria can exist. If
the producer surplus increases over a range, k can be set low enough so that a group of
low productive sellers will find it optimal to enter. The condition for multiple equilibria
will have the derivative of the producer surplus positive with respect to low productive

sellers entering the market. Expected producer surplus of a low type seller is

0601)

 

(6'2 (93M; - C2 (615)),

n is the number of sellers currently in the market. All of the high productive
sellers are in the market while no low productive sellers are. The derivative of the

producer surplus positive with respect to low productive sellers is

a '(n)n— a(n)

 

[c2 (q2)qz_ —Cz(q2)]+

(101)

 

 

a s a 2
)_ q2 _CIZ (q2 q2

.. s 861
C 2(q2) 5112q2+02(q2 an an

72

This derivative needs to be positive for existence. We know that
S S S
a'(n)n " “(7’) < 0 because 0"(11) < Oand C'z (q2 )q2 _ cz (qz) > 0

because C. '2 (q) > 0 , for all positive 11 and q, therefore conditions for existence imply

that

a'(—n)n a(n)
[ n2 :|[C 2(q2)q2— 02(q2)1<ann

 

 

H S a S S
C 2 (q2)Eq:—q2

§_q§

The discount rate can be one and still have an > 0. For the buyer, when
another seller enters the market, the probability of consumption increases. Because the
buyer is infinitely patient, this increase does not change the buyer’s behavior. With fewer
productive sellers entering the market, the overall productivity will decrease. The
quantity consumed by the buyer will decrease. Beta equal to one is the hardest case,
because the buyer is not concerned with the probability of finding a seller; this condition
increases when the low productive sellers enter the market.

5615

The driving force for existence is an , and an important component to

 

5615

an is 11. Adding a low productive seller will change the amount of good produced by

 

high and low productive sellers. If there are a lot of high productive sellers already in the
market, the addition of one low productive seller will not dilute the productivity as much.

Therefore it is important to have a low amount of high productive sellers, in order to have

73

a large effect on the change in the amount produced when the low productive sellers
enter.
The existence condition is

a'(n)n - a(n) 0'2 (615)613 - 62 (612’) < 5612’

(*> MO?) 0"2 (613)615 5"

 

 

 

 

 

The first term on the left side is decreasing in n, and the second term on the left side is

increasing in q.

8612’

In order to have existence an needs to be larger than

 

a'(n>n —a(n> 'c'2 <q2>q2 —c2 012*)“
<**) 120601) _ 0"2 (615)615

 

 

 

 

In an attempt to make the calculations easier, I set beta equal to one. The only change in
q is fi'om the change in productivity.

With the simplifying assumption, the buyer’s equilibrium condition (21) simplifies to:

 

n1 _1 n2
'————c' c' 2 +——— 2
1 u[nl+n2 1( 2(q D nl+n2ql
C'2(q2)
n1 n2
0=u' —c"1 c' 2 +——— 2 —' 2
(”1+n2 1( 2(q )) n1+n2qj 62(q)

74

u" _,??L 2 21:32.12
aq2_ (qb)[(- 11)q (n+1n22)2+612[ (n1+n2)2 D

 

2 _ n2
6" 14% ‘1221mb)(ii2“’1 ".(c <q2)>c"2 (q2) + 4 ,2 ]
1+ n2
-c"2 (612)
II q2—q1
" (qb)"li(n1+n2)2j

 

 

- u"(qb)[n1:11150'1 (02 (q2))0"2 (q2)+ 111112112le 6"2 (612)

 

(612-0'1 (62012»)
_ n1+1n 2
(n10 * 1(02(612))C"2 (€12)+ 712)

— if; (n1+n2)

q2 is less than ql, which makes the top term positive. u' ' (qb) is negative and

(***)

_1! l
c"2 (q2) is positive, which makes the bottom term positive, if 0'1 (02 (q2)) is
positive. It is positive because 6'2 (q 2) is increasing in q2 and
0'1 (ql) = 0'2 (q 2) . If q2 is increasing then ql increases, because both c”(q) are

_1' I
positive, which means that 0'1 (C 2 (q 2)) is positive. Certainly the relationship

75

092’

between 01 (q) and 02 (Q) determines the size of an . There needs to be a large

 

difference between ql and q2 for the multiple equilibria to exist.

In the case for existence, n2 equals 0. All high productive sellers are already in
the market. The change in the low productive seller’s quantity should be less because
there is already a large number of high productive sellers in the market. The test for

existence is setting n2 = 0, rearranging (**) and (***) in (*)

a'(nl)nl — a(nl) c'2 (q2)q2 — c2 (q2) I

a(nl) q 2
_l_
u"(qb)

This condition states that for existence, the difference between ql and q2 has to

 

 

< ql — q2
0'11. (0'2 (612)) -

 

be larger than a function of the cost fimctions, utility functions, matching functions at the
equilibrium production value.

With the aforementioned assumptions, this model is able to produce two separate
equilibria, one with only high productive sellers and one with both types of sellers
entering the market. Given that these two equilibria can coexist, what are the intensive
and extensive implications? When both types of sellers are in the market, the total
number of sellers in the market is larger than when only high productive sellers are in the

night market. With more sellers in the market, buyers have a higher probability of

finding a trade, a (”1) < a(n1 + ”2) . Because more buyers are able to trade, the

extensive margin has increased. With everything else being equal, when the probability

76

of finding a trade has increased, the buyer would want to bring more money into that
market, because there is less of a chance that the money would be wasted by not finding a
trade. The opportunity cost of holding money is decreased.

Studying the intensive margin is more difficult due to the changing productivities
of the sellers. But can we study the differences in activity of the agents between these

two equilibria?

S S
Result] q] > qz

This is a result of the assumption that C"l (q) < C'2 (q) and a competitive

market which results in the price equaling the marginal cost. QED

b bb
Result 2 q ls > q

"1 sb n

2 sb bb
_Q1 + q 2 = q
From "I + "2 "2 + ”2 and result 1. QED

Result 3 The high productive sellers will produce more when both types of sellers are in

b
the market. qf‘ < qf

Setting (19) and (21) equal to each other yields

u'(qi”)_ 1-fl =
c.1(q1SS) 780011)
2) —’—1‘——qu+-—172——0'§1(€'1(€1191)))
n1+n2 n1+n2 _ l—fl :2
6101?”) flaw +112)

 

 

 

77

u'(qi”) ___

 

 

 

 

 

c.1(qis)

r "1 3b "2 I- -1

u ”1+nzq1 +nr"'”262(cl @131)» 1,6 1—13
c.1(qib) +28a(n1) _:>,Ba(nr+n2)

 

u'(q1”) > u'(q"”)
611(qf5) C' (qisb) 1fbeta1slessthanone.

Using result 2

 

 

 

u'(q1”) > u'(q””) >u'(q1”’) 3
c'1(q1”) 0'11’(q ) c'1(611"”)

ss sb
Therefore q 1 < ql .QED

From Result 3 and C'i (qzs ) = fin p , we know that the price must be
higher in the equilibrimn with both types of sellers entering the market. This is intuitive
because in order to have both equilibria, there needs to be something that entices the low
productive sellers to enter, and this enticement is a rise in price. With this rise in price,

the high productive seller has more incentive to produce more. This is in direct contrast

ss sb
to the model produced by CV in which q 1 > q 1 . In CV and my models, the entry

of low productive sellers increases the price, but this means different things are
happening in the two models. In CV, because money is indivisible, an increase in price

means a decrease in production, whereas in my model an increase in price means an

78

increase in production. Result 3 is the driving factor in being unable to determine whether

the intensive margin is more or less with both types of sellers entering the market.

ss sb
Result4 q2 < q2

Combining Assumption 3 and 4 implies:

 

a(n‘+n’)(q§bc C'2(q2 )- 02(92 ))>
n1+n,
‘2’) “31’0” '2 (q )— c2(q2 )>

C 22(qb=) Cnnpb>ICps=C2(qSS )jqzb >q2S QED

In order for the lower productive sellers to enter the night market, the price

needs to be high enough, so that the expected benefit of producing is higher than the cost

b . bb
of entry. It is unclear whether qf rs greater than or less than q because of two

opposing forces: the decrease in productivity due to the entry of lower productive
workers, and the increase in effort due to the increase in price. The increasing or
decreasing nature of the intensive margin is determined by how much value the buyer
gains from increasing the chance of trading in the night market versus the decrease in
productivity of the workers that are entering the market. I would expect that an impatient
buyer would have a greater chance of increasing the intensive margin with the entry of
lower productive seller than would a more patient buyer.

The table below compares my results with CV’s results. The main difference is

ss sb
that in my model q 1 < q 1 and CV has the opposite direction. This makes the

79

direction of the intensive margin determined by the production functions, utility firnction,

meeting technology, and the number of agents chosen in the economy.

The following Table smnmarizes the intensive margin effects as found using both

the CV and the RW (with heterogeneous sellers) models. All equilibria are with the same

parameters.

Table 3.1 Comparing the Camera Velsey Model to Rocheteau and Wright

 

 

Camera Velsey model RW with Heterogeneous
Sellers

Equilibrium Equilibrium with Equilibrium Equilibrium

with only High both types of with only with both

Productive Sellers High types of

Sellers Productive Sellers
Sellers

 

Produced

 

 

ss sb ss Sb
> <

by High q] ql q] ql
P d ced

’0 u 0 < q Sb 0 < q sb
by Low 2 2
Consumed q b b
by Buyer q by > (weighted q bs ?? q bb

 

 

sum of consumption

by buyers)

 

 

 

 

 

The following is an example that shows the coexistence of two equilibria with

different intensive margins. The combined discount rate is 0.95. The utility function for

the buyer is: “(q) = (q)

1/2

; the cost fimction for the high productive seller is:

2
C1 (q) = (Q) ; and the cost function for the low productive seller is:

80

 

1
01(q):(q)3/2. Thematching function is: “(’1) :1—[(3O+n)/30]' Iset

the limit of high productive sellers at 10.

Given the two types of cost functions, C '1 (q) < C '2 (q) for the same q will not

be true for all values of q. This is only a problem for large amounts of q (q >
056250000). The highest q from figure 1 is 0.40440000. Figure 1 plots the benefit of

entering the night market for 0 to 30 low productive sellers. The equilibrium values in

my examples do not reach this threshold: C '1 (q) < C '2 (q) continues to hold for the

conditions under which equilibrium could occur, so my results are consistent with the
assumptions of my model.

When there are 10 sellers in the market, the benefit of entry for the low productive
seller is 0.0012638482. Thus, k needs to be greater than 0.0012638482 for no low
productive sellers to enter. As more sellers enter, the benefit increases. At some point
there will be a maximum amount of benefit. In this case it is 0.0013860511, with the
number of low type sellers equal to 6. Thus, k needs to be less than 0.001386051] in
order for low productive sellers to enter. With a limited number of low productive sellers
(in this case fewer than 20), there exists a set of k’s producing multiple equilibria. The
cost of entry needs to be between 0.001263 8482 and 0.001386051] in order to produce
multiple equilibria . On the vertical axis is the benefit of entry for the low productive
seller. On the horizontal axis is the number of low productive sellers in the market.

In order to have multiple equilibria, the price needs to increase enough, so the
lower productive seller now finds it profitable to enter the market. The buyer will only

bring more money if the benefit of having a higher probability of trading is more than the

81

unproductiveness of the additional sellers entering the market. This clears the path
toward an increase in the intensive margin.
Productivity of the sellers is the amount of total production divided by the number

of sellers producing.

 

* S * S * *
nlql +n2q2 _ "1 s ”2 s _ b
a: It — at :0- 1+ :0: *q2_q
n1 +112 n1 +112 n2+n2

bb sb
By comparing q and ‘12 I can tell the change in productivity between the two

bb
equilibria. In my exercises I calculated q by using N1 and N2 because,

N1 , + N2
(N1+N2) ‘11 (N1+N2)q2
5*N1 S 5*N2 S
* ql + * q2 =
a (N1+N2) a (N1+N2)

 

 

 

 

_r_z_l_qs +__n_2_qs _ qb 8 _ a(nl+n2)
at at 1 t u- 2 _ Wh '—
n1 +n2 n1 +n2 ere n1+n2

 

bb bs
q > q can be achieved by changing the matching function to

l
“(’7) =1-[(5 + (”/90))/5] , and changing N1=10, N2=O. This results in

 

bs
k>1.3761085e-005 and q = 0.17480000. If N2=l then k< 1.66503366-005 and q bb =

017506989. This is different from the findings in CV who maintained that the high
productive sellers produce a smaller amount when both types of sellers are in the market.

In their model, the intensive margin for the buyer will be less with both types of sellers in

82

the market. With indivisible money the lower intensive margin translates into higher
prices. My model produces higher prices with both types of sellers in the market, but it
comes to this conclusion because money is divisible, thereby enticing the high productive
seller to produce more.

In a competitive market, the intensive margin is efficient. All sellers are
producing until the marginal cost is equal to the price. With more agents entering the
market, buyers have a higher probability of finding a trade. Therefore, they may bring
more money into the economy. Buyers bringing more money will demand more of the
good, which leads to higher prices. These higher prices could draw in lower productive
sellers. But this will only happen if the buyer prefers the added probability of

consumption versus the increase in prices and consuming less.

4. Conclusion

This paper studied the extensive and intense margin effects of heterogeneous
producers in the RW model and compared the results to the CV model. The main result
is that the high productive seller may produce more when both types of sellers enter the
market. This can not happen in the CV model in which the difference in production by
high productive sellers changes the intensive margin. In the CV model the intensive
margin will decrease with the added low productive sellers; however, in my model, the
intensive margin may either increase or decease.

A path of future research could examine the model with a continuum of different
types of sellers, instead of only two types. In this case, there may be only one

equilibrium, but a demand shock could be added to study business cycle implications of

83

 

having heterogeneous producers. Once the business cycle is implemented, welfare

implications of monetary policy could be made.

84

 

28

24-

20

Figure l
16

12
Number of low productive sellers in the market.

 

. 1 i 1 J n r 1 r r . O
ZHOD'U 1621000 951000 QLLGO'U ULLUU'D

arenas aananpord am am 10} Lima jO triauag

J I

 

 

 

 

85

REFERENCES

Camera, Gabriele and Filip Vesely. “On Market Activity and the Value of Money.”
Journal of Money Credit and Banking, (March 2006), pp. 495-510.

Ennis, Huberto. “Search, Money, and Inflation under Private Information.” Institute for
Empirical Macroeconomics Discussion Paper 142, Federal Reserve Bank of Minneapolis,
August 2004.

Kocherlakota, N. R. (1998). “Money Is Memory.” Journal of Economic Theory, 81,
232—251.

Lagos, Ricardo and Randall Wright. “A Unified Framework for Monetary Theory and
Policy Analysis.” Journal of Political Economy, 113 (3) (June 2005), pp. 463-484.

Rocheteau, Guillaume and Randall Wright. “Money in Search Equilibrium, in
Competitive Equilibrium, and in Competitive Search Equilibrium.” Econometrica, 73 (1)

(January 2005), pp. 175-202.

Trejos, Alberto and Randall Wright. “Search, Bargaining, Money and Prices.” Journal
of Political Economy, 103 (1995), pp. 1 18-141.

86

 

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