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THREE ESSAYS ON POLICIES TOWARDS RISK

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THREE ESSAYS ON POLICIES TOWARDS RISK
By

Cheong-Seok Chang

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Economics

2006

ABSTRACT

THREE ESSAYS ON POLICIES TOWARDS RISK
By

Cheong-Seok Chang

Payment systems facilitate all types of trade in monetary economies by providing a range
of mechanisms including instruments and operating procedure through which
transactions are settled. A break down or malfunction of the system can be seriously
costly. In this regard. governments may implement policies to keep the payment system
stable. AS the former Federal Reserve Board Chairman Greenspan (2004) suggests, IT IS
IMPORTANT TO UNDERSTAND THE MANY SOURCES OF RISK AND (.'.'\'(‘ER7}II.\'T)' 'I'll.l'l‘
P()1.l( 'l'.ll. lKIiRS FACE.

Chapter 1: “lntraday credit risks in a real time gross settlement system“ identifies credit
risks in the market and analyzes intraday credit policy in a large-value real time gross
settlement payment system. A large-value interbank payment (LVIP) system such as
Fedwire is the backbone of the payment systems, since it provides finality of settlements.
Usually the LVlP system depends on the central bank for intraday credit (or daylight
overdrafts). Credit risks in the market induce the central bank to implement several
policies to manage risks on intraday credit. The apprehension of an economy about
potential loss from credit risks is a determinant of the policy choice. lmpediments to
conducting monetary policy toward shocks from payment system credit risks can be a
source of this apprehension. Quantitative limits (or caps) can be used when the economy
fails to take into account potential loss from credit risks in the payment system. Price can
be used when the economy puts greater weight on this potential loss. Since there are two

types of credit risk in the market. it is enough to employ two tools. The model illustrates

that a price and collateral policy achieves the highest social welfare among other sets of
polices.

Chapter 2: “Payment instruments and the central bank” explores the sensitivity of
payment system stability to macroeconomic shocks through the assets (from lenders‘
viewpoint) used as media of exchange. Specifically, this chapter studies a “payment
instrument policy,” such as the one implemented by the Bank of Korea, to promote usage
of a certain type of payment instrument to stabilize the payment system. This chapter
analyzes two payment instruments, promissory notes (non-intermediated credit) and bills
of exchange (intermediated credit) using a simple partial equilibrium model of the credit
market with moral hazard. The model shows that bills of exchange have a lower market
risk than promissory notes under negative macroeconomic shocks. If, therefore, both
promissory notes and bills of exchange are used in addition to fiat money, systems with a
greater proportion of promissory notes in circulation will be riskier and more prone to
collapse than systems with fewer promissory notes. The model also shows that a subsidy
for bills of exchange is better than a tax on promissory notes.

Chapter 3: “Production uncertainty and private information in a search theoretic model”
is a methodological exercise looking at production uncertainty and private information in
a search theoretic model based on Williamson and Wright (1994). Different types of
uncertainty generate different effects: the model Shows that there exist more equilibria in
the production uncertainty case than under the no production uncertainty case.
Comparative statics Show that an increase of production uncertainty raises possible
returns from choosing the bad technology. These findings support agents’ choice under a
macroeconomic Shock and generate important implications for designing policies to

alleviate risk as in Chapter 2.

To my parents
And to my wife

Jeongiin

iv

ACKNOWLEDGEMENTS

I am indebted to several groups. First of all, 1 would like to thank members of my
dissertation committee: Professor Rowena A. Pecchenino, Assistant Professor Andrei
Shevchenko, and Assistant Professor Luis Araujo. Especially. I am grateful to Professor
Pecchenino for her invaluable guidance.

I would also like to thank the Bank of Korea for its financial support during the
first two years of the program.

And my final thanks go to my family for their constant encouragement. There
have been series of challenges during the program. My Wife, Jeongiin, always supports
me; my son, Joon Nyong, and my daughter, Soo Yon, cheer me up with their lovely

smiles. It would be impossible to complete the degree without their love and support.

TABLE OF CONTENTS

LIST OF TABLES ...................................................................................................... ix

LIST OF FIGURES ...................................................................................................... x
CHAPTER I

lntraday credit risks in a real time gross settlement system

1. Introduction ......................................................................................................... 1

1.1. Overview of payment system .................................................................... 6

2. The environment ................................................................................................. 8

3. The model ........................................................................................................ 12

3.1. Debtors‘ problem ..................................................................................... 12

3.2. Banks” problem ........................................................................................ 14

3.3. Equilibrium .............................................................................................. 17

4. lntraday credit policy ........................................................................................ 19

4.1. The central bank and pricing rule ............................................................ 20

4.2. Optimization and equilibrium .................................................................. 23

4.3. Quantitative restrictions ........................................................................... 26

4.3.1. Caps ................................................................................................. 27

4.3.2. Collateral ......................................................................................... 28

4.4. Calibration of social welfare and policy analysis .................................... 29

5. Concluding remarks .......................................................................................... 32
Appendix

A.1. Proof of Lemma 1 ................................................................................... 35

A2. Proof of Lemma 2 ................................................................................... 35

A3. Proof of Lemma 3 ................................................................................... 36

A4 Equilibrium under collateral ................................................................... 37

References ............................................................................................................. 39

vi

CHAPTER 2

Payment instruments and the central bank

1. Introduction ....................................................................................................... 43
2. The environment ............................................................................................... 47
3. Simple model for payment instruments ............................................................ 49
3.1. Promissory notes ...................................................................................... 49
3.2. Bills of exchange ...................................................................................... 50
4. Macroeconomic shock and payment instrument policy .................................... 52
4.1. Pure promissory notes economy .............................................................. 52
4.2. Pure bills of exchange economy .............................................................. 55
4.3. Preference of central bank ........................................................................ 56
4.4. Payment instrument policy ...................................................................... 58
5. Concluding remarks .......................................................................................... 61
Appendix
A.l. Proof of Lemma 3 ................................................................................... 64
A2. Proof of Lemma 4 ................................................................................... 64
A3. Proof of Lemma 7 ................................................................................... 67
References ............................................................................................................. 69
CHAPTER 3
Production technology and private information in a search theoretic model
1. Introduction ....................................................................................................... 72
2. Basic Model ...................................................................................................... 75
3. Equilibrium ....................................................................................................... 77
4. Welfare Analysis ............................................................................................... 84
5. Comparative statics ........................................................................................... 86
5.1. Production technology and type A, B and E equilibria ............................ 86
5.2. Production technology and type D equilibrium ....................................... 87
5.3. Discount rate and equilibria ..................................................................... 89
6. Concluding Remarks ......................................................................................... 91
Appendix

vii

A]. An infinitely repeated game ................................................................... 92

-

A.2. Proof of Proposition 1 ............................................................................. 93
A.3. Proof of Proposition 2 ............................................................................. 96
A4 Pareto rank of social welfares ZA, 23. 20 and ZE ................................... 97
A5. Production technology and existence region for equilibrium
type A, B and E ....................................................................................... 98
A6. Proof of Proposition 3 ........................................................................... 100
A7 Proof of Proposition 4 ........................................................................... 101
References ........................................................................................................... 103

viii

LIST OF TABLES

Table 1.1: Timing of arrival and departure with fraud risks (banks’ action) .............. 11
Table 1.2: Consumption bundles and utility in equilibrium ...................................... 32
Table 3.1: Types of equilibria ..................................................................................... 80
Table 3.2: Production technology interval for type D equilibrium when

production technology improves ................................................................ 87
Table 3.3: Discount rate and equilibria ....................................................................... 90
Table 3.A. l: A table of payoffs .................................................................................. 92

ix

LIST OF FIGURES

Figure 1.1 : Modeling timing ...................................................................................... 1 1

 

. A
Figure 1.A.1: Equilibrium range it 6 [A4, 1 — Ad) or X e [l d . 1)

“ d

under intervention by the CB ................................................................ 37
Figure 2.1: Timing of events (debtor’s viewpoint) ..................................................... 48
Figure 2.2: Critical levels of wealth for private monies ............................................. 51
Figure 2.3: Timing of events (debtor’s viewpoint) ..................................................... 52
Figure 2.4: Critical level of wealth and macro shock (PN case) ................................. 56
Figure 2.5: Payment instrument policies .................................................................... 59
Figure 2.6: Macro Shock and anticipation .................................................................. 61
Figure 3.1: Production technology and critical values of 0 ........................................ 81
Figure 3.2: A graph oft). - 04 when 0,, = 1 (U = 2.5, r = 0.05) ................................... 84

Figure 3.3: Social welfare by types of equilibria
(U = 2.5. y = 0.75, 0,, = 0.75 and r = 0.05) ............................................... 85
Figure 3.4: Derivatives of type D equilibrium with respect to 0p ............................... 88
Figure 3.A.1: Negative determinant ............................................................................ 95

Figure 3.A.2: Private information and equilibria type D and E
(U = 2.5, y = 0.75, 0p = 0.75, r = 0.05) ................................................... 98
Figure 3.A.3: Derivative of critical values of 0 for type D equilibrium

with respect to r .................................................................................. 102

Chapter 1
lntraday credit risks in a real time gross

settlement system

1. Introduction

Payment systems are the instruments, organizations, operating procedures. and
information and communications systems used to initiate and transmit payment messages
from payer to payee and to settle payments (Balino et al. 1994). The payment system
often refers only to a large-value interbank payment system such as Fedwire of the
Federal Reserve System or TARGET of the European Central Bank. Payment systems
facilitate all types of trade in monetary economies, and accordingly 3 breakdownl of a
system can be seriously costly. For example, if there is a severe shock such as settlement
failure from one or more banks which makes a payment system fail, then the ability to
make payments would be impaired. This would cause serious disruptions in goods
markets and instability in financial markets. In order to remove or alleviate this
possibility or “risks”. the central bank implements payment polices. In this paper. I
analyze the intraday credit policy tools implemented by the central bank under a large-

value real time gross settlement (hereafter RTGS) system with several types of

exogenous risks.

 

' Bank failures occur when banks are unable to meet the demands of their creditors (in earlier times these
were note holders; later on, they were more oflen depositors) [Grossman 2003]. During the US. National
Banking Era (1863-1913). one method of dealing with banking panics in light of the payment system was
for the bankers of a city to pool their resources, through the local bankers‘ clearinghouse and to jointly
guarantee the payment of every member banks' liabilities (Gorton 19853. b).

As Greenspan (1996) suggests, the central bank needs to improve risk
management within payment systems themselves. This rationale shows the importance of
a careful study of how to implement intraday credit policy more effectively. Though
many combinations of policy exist in payment systems. I consider the three basic
elements employed by the central banks: price, quantitative limits (also known as ‘caps’)
and collateral. Among 15 RTGS systems to which the central banks provide intraday
credit, 12 systems employ a price2 and collateral policy and the other three3 employ a
price, caps and collateral policy (Bank for International Settlements [BIS] 2005). In this
regard, the questions that I am going to study are why every central bank employs
multiple tools for managing the risks within a seemingly Simple interbank payment
system and which set of tools is efficient to the economy.

Existing studies do not pay much attention to these questions4 but mainly focus on
choosing efficient policy tools under limited types of credit risk. For price, Mills (2004b)
and Lacker (1997) are typical. Mills (2004b) considers endogenized credit risk committed
by debtors and justifies charging interest on intraday credit by arguing monitoring is
costly. Lacker (1997) supports the interest charge policy not only for overnight but also
for intraday credit. But in his model, the interest charge policy is only an outcome of the
non-interest bearing overnight reserve requirement. Some other economists such as
Faulhaber et al. (1990) and Angell (1991) also argue for charging interest on intraday

credit. There are also arguments for quantitative constraints in Rochet and Tirole (1996).

 

2 Only the Federal Reserve charges it in the form of explicit interest. other central banks charge it in the
form of a haircut on collateral.

3 These three systems employ all three tools: the Fedwire system of US. (partial collateralization), CHAPS
Euro system of UK. and E-RIX system of Sweden.

4 Furfine and Stehm (1998) analyze G-lO countries’ intraday credit policy tools as of 1996. They identify
private and social costs regarding RTGS system. They demonstrate the cost structure of a central bank
determines the choice of intraday credit policy tools.

Kahn and Roberds (2001), and Martin (2004). Rochet and Tirole (1996) Show the same
rationale for the price of intraday credit as Mills (2004b), but they implicitly consider
credit risk committed by borrowers of intraday credit and argue that a cap (or collateral)
is a better means to control the overuse of intraday credit than price because of
incomplete information. Kahn and Roberds (2001) implicitly consider credit risk
committed by borrowers of intraday credit from the central bank (CB) Since they do not
consider private IOUs. They argue that collateral with free intraday credit can eliminate
the liquidity Shortage and restore the first-best consumption allocation. Martin (2004)
considers an endogenous choice of risk committed by borrowers of intraday credit. He
supports a zero intraday interest rate. He argues collateral is the effective means of
controlling risk.

In the real world the credit provided by the central bank (or daylight overdraft) is
used to measure the central bank’s potential exposure to an end-of-day settlement failure
(Milano 1991). Recent huge increases in the volume and the value of payments put
central banks at greater potential loss as they provide intraday credit to the system. For
instance. the average daily volume of funds transfer through Fedwire 5 was 296.4
thousand transactions, and average daily value of transfers was $857.5 billion as of 4lh
quarter of 1994. The figure became 513.2 thousand transactions and $2,007.2 billion as
of 4lh quarter of 2004. During the same period, the amount of intraday credit (daylight
overdraft) provided by the Federal Reserve has grown at a slower pace". The amount of

peak daily intraday credit increased7 from $66.4 billion as of 4th quarter of 1994 to $11 1.5

 

5 http://www.federalreserve.gov/pavmentsvstems/fedwire/fedwirefundstrtQtrhtm

° The fee for the use of intraday credit has been changed: 24 basis points in 1994. 36 basis points since
1995.

7 http://www.federalreserve.gov/PaymentSystems/psr/dlodpeakqtrhtm.

billion as of 4lh quarter of 2004. These dramatic increases also highlight the importance
of the RTGS system and accordingly intraday credit policy to the financial systems.

I modify F reeman's isolated island overlapping generation (OLG) model (1996b.
1999) to analyze intraday credit policy. Freeman's model is useful because 1) as agents
are spatially separated, money works as a medium of exchange; 2) private debt is
incurred between two parties as a payment instrument; 3) private money (IOUS) is repaid
by fiat money; and 4) there can arise liquidity problems which lead to a role for a C88.
Because his model was originally designed to study the use of open-market operations
and the discount window in conducting monetary policy in environments with a liquidity
Shortage, I adjust several of its components. I adjust the timing of events so that an
explicit private intraday credit market emerges. This is important because the price in the
private intraday credit market works as a benchmark for the CB’s pricing rule. The risk
premium for intraday credit is a major ingredient for this benchmark. I substitute ‘one
settlement stage and one consumption stage’ with two groups of stages each consisting of
one consumption stage followed by one settlement stage. During each settlement stage
the intraday credit market is open: one for financing intraday credit and the other for
repaying it. Then, following Zhou (2000), I reinterpret the debt settlement market as a
payment clearing market. resale of debt as private intraday borrowing/lending of reserves.
and the CBS injection of liquidity as intraday credit lending rather than an open market
operation or discount window lending. Finally, I introduce an additional type of credit

risk. Creditors provide overnight consumption credit (in other words, consumption debt)

 

8 As Mills (2004b) noted, one can interpret a central bank as a private clearinghouse that is separate from
the other agents. As noted in Green (1997), the liquidity—providing institution in the model can be either a
public or private one. This is a payment system version of the inside and outside money debate which

remains an open question. Refer to Williamson (1999) for a further review on historical debates about
inside and outside money.

to debtors. Creditors face the risk that debtors will not repay this loan (1 will call this type
I risk). Users of intraday credit (in other words, payment debt, which arises only in the
process of payment and settlement) may not repay the intraday credit (I will call this type
II credit risk). AS a payment model without explicit production processes, creditors have
no opportunity to invest. The type 11 credit risk, however, can be a proxy for creditors'
moral hazard in the model. I analyze the intraday credit policy tools under these two
types of credit risks.

One contribution of the essay is to identify credit risks in the market which
explain the combined usage of intraday credit policy tools in the real world. The model
considers two types of credit risk. This consideration has more meaning than a simple
addition of two credit risks. It serves as a qualitatively different approach to modeling the
RTGS system. Removing or alleviating type 1 credit risk and type II credit risk are the
goals of intraday credit policy. In order to manage a couple of credit risks. a combination
of tools is implemented. Price deals with both type I and II credit risks. A cap works for
type 1 credit risk. Collateral is an effective tool to avoid type II credit risk.

A practical contribution is a direct calculation and comparison of welfare under
possible sets of three tools. Under credit risk free9 circumstances, the Friedman rule
works in the intraday credit market. Different from that implication, however, under
credit risks the model Shows that a positive interest rate on intraday credit set by the CB
_ could achieve higher social welfare than a zero interest rate. Free intraday credit with
caps policy can be implemented by the CB, when the CB does not take serious

consideration of potential loss from credit risks. Even though there may be some

 

9 Strictly speaking. we need not analyze the RTGS system, since the DNS system is better under credit risk
free circumstances.

justification for caps, the model illustrates that collateral is better than caps in terms of
social welfare. A price policy is implemented when the CB places Significant weight on
potential loss. Finally, one of the policy implications from the illustration of the model is

that it is desirable to employ a price and collateral policy to achieve higher social welfare.

1.1. Overview of payment system

There are two alternative systems”): RTGS and delayed net settlement (DNS) system. An
RTGS system is one in which payment messages are processed in a timely manner and
settlement occurs immediately on bilateral and gross bases. The RTGS system has
advantages in finality of payment, security and timeliness. A DNS system is one in which
payment messages are processed continuously but are settled at a given time (usually end
of the day) at which incoming and outgoing messages are netted. Under the DNS system,
synchronization of payment messages does not matter. The DNS system is efficient but
vulnerable to systemic risk. For example, when a participant of the DNS system commits
settlement failure, then all messages from him should be unwound and all net amounts
should be recalculated. Some other participants may be forced to commit settlement
failure which otherwise would not happen. This leads the system to be shut down.

There are five payment system risks: credit risk, liquidity risk, systemic risk,
operational risk and legal risk. According to the BIS (2001), credit risk is the risk that a
party within the system will be unable fully to meet its financial obligations within the
system either when due or at any time in the future. Liquidity risk is the risk that a party
within the system will have insufficient funds to meet financial obligations within the

system as and when expected, although it may be able to do so at some time in the future.

 

") There is a hybrid of the two systems. This is, however, beyond our scope of discussion.

Systemic risk iS the risk that the inability of one of the participants to meet its obligations.
or a disruption in the system itself, could result in the inability of other system
participants or of financial institutions in other parts of the financial system to meet their
obligations as they become due. Such a failure could cause widespread liquidity or credit
problems and, as a result, could threaten the stability of the system or of financial markets.
Operational risk is related to operational factors such as technical malfunctions or
operational mistakes. and legal risk is related to a poor legal framework or legal
uncertainties.

In order I) to remove or alleviate the systemic risk; or 2) to enhance the efficiency
of the payment system; or, more fundamentally, 3) to maintain the payment system (or
monetary economy), payment (system) policy is implemented. If there are no credit risks.
the DNS system is better, and we need no further payment policy. Once there are credit
risks, however, the DNS system is inferior. A settlement failure may need a huge amount
of money from the CB as a lender of last resort to restore the system, Since the DNS
system uses multilateral netting at the end of the day. Furthermore, because of the netting
procedure, the DNS system may have much higher probability of failure. In the RTGS
system, however, settlement failure can be resolved at ‘relatively’ low cost. AS a resultll
of the payment policy, most central banks choose the RTGS system. The RTGS system
has one drawback: liquidity shortages owing to imperfect synchronization of ordering of
payment messages. If the intraday balance available for payments is too small relative to

the value of payments to be made in a given time. it could bring about gridlock which

 

" Finality of the payment is major component to the alleviation of the systemic risk. Even a privately
operated, a used-to-be typical DNS system. CHIPS (Clearing House Interbank Payment System) has

employed a hybrid system by borrowing the finality of payment from the RTGS system since January 2001
(Coleman 2002).

prevents payments from being executed, and therefore the system becomes less efficient.
Central banks. therefore, implement another policy: provision of intraday liquidity in
order to enhance the efficiency of the payment system. If a bank cannot repay the
intraday credit until the end of the day, the CB suffers a loss. Accordingly, central banks
are exposed to credit risks under the RTGS system.

The paper is organized as follows. Section 2 provides the environment of the
model. Section 3 solves agents’ decisions and presents optimal allocation without
intervention. lntraday payment policies will be analyzed in section 4. Section 5

summarizes findings and concludes. Lengthy proofs are in the appendix.

2. The environment

A large number of outer islands are arranged in I pairs around central islands. Each pair
contains both of two types of islands, which are called ‘bank’ and ‘debtor’ islands. Time
is discrete and infinite. There are two types of consumption goods C and D in each period
t 2 1. On each island, N two-period-lived agents are born in each period t 2 1. In the first
period each island also has N agents (the initial old) who live only in the first period. For
simplicity, N is normalized to 1. The central islands consist of three islands: a redemption
island where normal repayments of IOUS occur; and an intraday island where an intraday
credit market opens, and a payment policy is implemented; and a fraud island where
agents who are deviated from their given route arrive before they travel to debtor islands
to trade with young debtors. There exists on the intraday island a monetary authority able

to issue fiat money, which is noncounterfeitable, unbacked. intrinsically useless. and

costlessly exchanged. This authority issues an initial stock of M dollars at period t = 0 to
initial old banks.

Each agent born on a bank island (“bank“) is endowed at birth with 1 unit of a
non-storable good C Specific to this island (and with nothing when old). He wishes to
consume the good C when young and good D when old. No other consumption is desired.
I assume that no agents want to leave a bequest in any form. Because of imperfect
synchronization in receipt and consumption, some banks can borrow intraday credit and
become ‘debtors in the intraday credit market (d.i. in short),' and others can lend intraday
credit and become ‘creditors in the intraday credit market (c.i. in short).’ The utility of a

bank is given by the function U = u(Cc,)+u(Dc,, +,) where Ca and Dam represent his

consumption of good C when young and good D when old respectively.

Each agent born on a debtor island (each “debtor”) is endowed at birth with 1 unit
of a non-storable good D specific to his island (and with nothing when old). Agents wish
to consume the goods D and C when young without uncertainty, good D when old with
some probability.

There is a credit risk in the form of a fraud shock. A fraud is defined as a Situation
in which the ships of old debtor or of old d.i. banks deviate from the given route on the
way to repay their debts (in case of old debtors) or their intraday credit (in case of old d.i.
banks); they arrive at the fraud island instead; after that they proceed to trade with young
debtors. The agents who arrive at the fraud island are called to be dishonest. Every old
debtor has a probability Ad 6 (0,1 ), with which the ships deviate from the given route and
therefore old debtors cannot arrive at the redemption island (I call this type I credit risk).

Each debtor has an equal chance to have a fraud Shock and it is not known until a debtor

departs his island. Old d.i. banks have a probability Ac 6 (0,1). with which the d.i. banks
do not repay intraday credit at the intraday island due to the deviation of the ships from
the given route on the way to the intraday island (I call this type 11 credit risk). Each d.i.
bank has an equal chance to have a fraud Shock, and it is not known until a d.i. bank
departs from the redemption island. Each probability is independent of the other.
Therefore, the intraday credit risk is union of two types of risks: the credit risk (type I)
committed by debtors for overnight credit as in Freeman (1999), Mills (2004a) and
Martin (2004), and the credit risk (type II) committed by intraday credit users. The agents
in the fraud island have additional chance to trade with young debtors. Notice that the
number of borrowers who experience fraud shocks are equal to that of lenders who are
not repaid; therefore. young debtors do not suffer from loss of trading partners from the
fraud Shock.

The expected utility of a debtor is given by the function
V =v(Dd,)+v(("d,)+Adv(Ddg, +1)» where Dd, , Cd, and DC”, +1 represent his consump-
tion of good D and of good C at period t, and of good D at period t+l, respectively. I use
logarithmic utility functions for both debtor and bank.

As in the timing of meeting, the arrivals and departures are assumed not to be
fully synchronized. At the settlement stage, all banks arrive in the central island, but only
a fraction of debtors arrive. Let it. 6 [0,1) be the intrinsic fraction of early arrival debtors.
Since this probability is independent of the probability of a fraud shock, the actual
fraction of early arrival debtors under credit risk is It. s A. (1 — Ad) 6 [0, 1 — Ad). Now I
describe the timing of banks’ departure. Before the remaining 1 — A. debtors arrive. I — or

old banks leave the central island, where on e [0.1]. For an individual agent, the timing of

10

his or her arrival or departure (early or late) is completely random and is learned only
before the early settlement. Table 1.1 summarizes the timing of arrival and departure. Old
banks may fall into one of the three groups: Y, banks who benefit from leaving early; Y'.
banks who benefit from leaving early and not repaying the intraday credit; and Z. banks

who benefit from leaving late.

Table 1.1: Timing of arrival and departure with fraud risks (banks’ action)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

debtor it. (early-arriving) l - A. (late-arriving)
bank EMI—Ad) (l-A)(l—Ad)=l—}t'—Ad Ad(fraud)
_ Borrows from late leaving i bank and consumes
- (I A.)
(I — or) f d Consume. , TE
(eafly- (“0t rau ) I Does repay intraday credit Cannot repay
leaving: ------------------ 4 --------------------------- — _- Does not repay his
group Y) A; d intraday credit and 5:32;: and does
( rau ) proceeds to consume (VJ p y
Lend to an early leaving
a . bank. Sometimes '8 Consume Can not consume.
(late-leavmg: group Z) repaid and consume;
sometimes is not repaid.
Figure 1.1: Modeling timing
Early settlement . Late settlement
A if A \
Old agents meet First intraday 5 Old agents meet Second intraday
with each other. market is open. : with each other. market is open.
L l .
I . 1
Young debtors Early leaving old banks Late leaving old banks. dishonest
meet with young and dishonest debtors debtors and dishonest d.i. banks
banks meet with young debtors meet with young debtors

 

 

 

 

 

 

 

 

 

11

3. The model

This section studies the intraday market under credit risks. Specifically, I start with
intraday credit traded in the market without intervention into the intraday credit market

by the CB.

3.1. Debtors’ problem

Debtors want to consume both good C and good D when they are young and have an
additional chance to consume good D with a probability of Ad when old. Let p0. denote
the price of good D and pg denote the price of good C at period t. Because of a fraud
shock, when a debtor buys good C, he borrows overnight credit at a discount factor 6‘. <
1 from the trading partner bank. More specifically, at period t, a young debtor visits his
companion bank’s island and buys Cd! of good C at the price of pa and pays S‘toh‘, dollars
with his newly-issued IOU hit discounted by 8“. After consumption, he returns to his
home island. The possible trading partner of a young debtor iS either an old bank or an
old debtor. The young debtor sells (1 — Dd.) units of his endowment at the price of pm.
and consumes Ddt. Therefore the young debtor receives mi, units of fiat money from the
transaction. Here mi. is the debtor’s nominal acquisition of fiat money or is the debtor‘s
nominal demand for fiat money. The next morning, the debtor visits the redemption
island, where he pays back the debt, hit with his fiat money m', with probability 1 — Ad. If
he has a fraud Shock, he arrives at the fraud island instead. He can use the fiat money to
buy good D. The representative debtor maximizes his utility by choosing the
consumption bundle (Ddi, Cm, Dam) to maximize his utility subject to his budget

constraints. The following are optimization problem and budget constraints.

12

-. max log Dd: + log Cd, + Ad log 0d,: +1 (1.1)
C (ll’Ddt’DdJ—I-I

subject to:

pDder + [7(‘ert + (1 “5: )hr + Adpl).t+lDd,r+l = P01 + Adht - (1-2)

A debtor’s income (right hand side of budget constraint (1.2)) consists of one endowment

(pm-1) and one conditional income. When the debtor has a fraud shock, he gets hit ,

therefore his expected income from the fraud shock is Adh: . From the trade itinerary.

one can find the following feasibility relations for the debtors’ budget constraint:

PCtht = 51*}; (1-3)
PDtU-Ddt) = m? (1.4)
PD,r+lDd,r+l = m! - (15)

From the budget constraint (1.2) and the relations of (1.3) through (1.5), the following is

derived

m, =h,. (1.6)

Using (1.3) through (1.6), the optimization problem becomes the choice of money

demand mi, from the choice of consumption bundle:

 

maxlog{1—-’£’— +log§fl’—+Adlog m, . (1.7)
P1): P0 pDJ+I

The resulting first order condition for the debtor’s problem is

13

—————,—+—;+ ,=0. (1.8)
th—mt m, mt

 

3.2. Banks' problem

At period t. a young bank sells (1 — Cm) units of his endowment to a young debtor in
exchange for 1‘! dollars of IOU in the morning. He consumes remaining Cc! units of C
good. At period t + 1. he goes to the redemption island to get his I‘. paid back. When the
debtors repay their overnight loans, the bank can consume. The bank proceeds to trade

I

with young debtors. Let Di, +1 denote the amount of good D consumed by the old banks

in group G = Y. Y. and Z. An old bank in group Y can borrow liquidity (payment debt) at
discount factor pi}... < 1 at the intraday market from banks in group Z. Let q‘m denote
the amount of intraday credit transacted in the intraday market and valued at the end of
the market. The total amount of intraday credit in the market. pifilqnu. cannot exceed
cash balances available to a bank at the settlement stage, flit. Every early-leaving old
bank retums to the redemption island to be repaid by the late-arriving old debtors who do
not have a fraud Shock.

AS I assumed, there is a probability Ac with which the d.i. bank does not repay its
intraday credit in the second settlement stage. When a d.i. bank has a fraud shock, he can

have one more chance to visit a debtor island to consume good D again (group Y'). Let

1— A. A
A, 2 A. + (——1——‘%—" denote the overall fraud rate in the intraday market.

When banks become old, they can participate in the intraday credit market and

consume good D. If they are revealed to be early-leaving. then they have utility

l4

log( DZ, +1 ). In addition to that, if they have fraud shock after they are repaid by late

=- . . . A .A
arrival debtors. they can enjoy utility log(Dc): , +1) With a probabllity AC ”1;?“ If they

are revealed to be late-leaving. then they can enjoy utility log(DfJ+l); but they can only

ACA‘I J]. Because of the
I—xi.

 

consume when their intraday credit is repaid [1 — Ad —- (Ac —

risk premium. all consumption after second settlement stage is discounted with discount

factor pil+|e(0. 1]. Now the bank’s problem is to choose a consumption bundle (Cu.

no:

DZ,“ . 031+. . DCZJH ) to maximize his utility subject to budget and liquidity constraints.

 

 

 

 

l , A .A .. ,1. ‘
(I ‘0’) log Dim + (Ac — 1 L I: )Pm log Dam
m}ax log CC, +< } (1.9)
C. ,D .V , ALA * z
()2, (”HZI +a [1 — Ad —[Ac — $1”le log Du,“
Dc'.t+l' c'.l+l l IT’I' 1
subject to:
PC: = paCc. + 5:1; (when young) (1.10)
(1—A*)p,:]l: +231; = p0,,+ng:,+l (early-leaving old) (1.11)

- When his IOU is repaid late, he needs to borrow 1‘, units of money with
discount factor pig]. When his IOU is repaid early. he just proceeds to
consume.

(1 — Ad )1: = pD,,+chY;+l (early-leaving old: additional dishonest) (1.12)

- When he uses intraday credit as he leaves the intraday island early before he is
repaid by the debtor in the redemption island, he needs to repay intraday credit

with his own non-fraud overnight loan 1... But with a probability Ac. he has

15

fraud shock on the intraday credit market after he is repaid by the
corresponding late arrival debtor. If he has a fraud Shock. he has another

chance to consume good D.

(1 —Ad )1: +[l — A,—p:+1]q:+l = pquDci/‘JH (late-leaving old) (1.13)
When a corresponding debtor does not have a fraud Shock, a late-leaving old
bank can be repaid 1“, some of the late-leaving banks lend this money as
intraday credit to early-leaving banks. This intraday credit can earn (1 — p2. .)

q}. when the d.i. bank repays safely, but loses q‘t... if the corresponding d.i.

bank cannot repay.

#* * it

x1 I, — ,0, Hq, +1 2 0 (intraday liquidity constraint) (1.14)
The total amount of the intraday credit in the market, p'm q.t+], cannot exceed

cash balances available to a bank at the first settlement stage. it. 1“.

Using the binding budget constraints the optimization problem can be adjusted from the

choice of consumption bundle to the choice of financial asset bundle, I“ and q'm. Now

Lagrangian function of the optimization problem is:

(1— [mfg]: + 2’1,’

 

,tt
”1!.

I J
+<

I ’(_ ‘1

PD,t+1
*mazc L6. = log
1! “0+1

[I (.-a)[...[

A.A
Mlle— ..
I

l

+lul’1 [I " pr+|qr+li

.11

t

pr+l

 

a

 

where u is a Lagrangian multiplier.

First order conditions for the bank’s problem are

16

log

is.

l

A.A

(

d

 

.

xi. )pul IOg[

pDJ+l

pl).l+l

(1 “Ad )1: +[1“AI "Pinyin

(l - Ad )1,’

l

1,

 

(1.15)

 

l 1 A A 1 W
. (I’d) 7+(Ac‘ dijl— It
(5, 1, 1—1. I,
A

 

 

 

 

 

— **+< L+Ju/i‘=0
PU 0th a|:l— _(AC_AC dual] (I_Ad) .
l 1‘3 (l-Adfl, +[1—A,-p,+l]q,+l .

(1.16)

and
[I‘AI‘P*1]
a[1-Ad—[Ac— A—LA—dfl H J .. —,u=0. (1.17)
1“)“ (I—Ad)[: +[1 AI pt+l]qt+l

3.3. Equilibrium

In this section. I consider a symmetric competitive equilibrium.

0 0 V0 ‘ a c t t 0 o a ‘ o
Defimtlon Given a. it . and M. a symmetric competitive equilibrium IS a set of prices (pp).
3 :1: _ . Z ‘
pm). pa. 0 ,. p,./) and a set ofallocations (0d,, DdJH, D( J“. .D 1+1, D (+1, Cd, (a)

C.

such that

(i) Gitert prices, a set ofallocations (Dan, Dd 1+1 D)J,+l D)J+l, DZJH, C4,. Cu) solves
maximization problems of(1.1) and (1.9).
(ii) Markets clear:
For overnight credit market.

h, =1,. (1.18)

For good D and good C markets

A Ad A A 7
DdI+A(/D(.—il+l+(l a)[D ci,c_t+l+[A 1 A Wch;t+l]+a[l"Ad‘—[Ac* c 1]]Dc.;,+l=l

 

 

1—2
(1.19)

17

Cc, +Cd, =1. (1.20)

For money supply and demand,

mt =M. (1.21)

If liquidity constraint (1.14) is not binding (or u = 0), intraday credit will be
provided at a price p.) which reflects only the intraday credit’s risk of fraud. Banks who
leave early will inelastically demand intraday credit at any price so the price of intraday

credit will be determined by the supply of that by late-leaving banks (1.17). which yields

a:

A
Pi d

l—A,=(l—Ac)(1— *). (1.22)
1 ,1

 

In order to find p}. when liquidity constraint (1.14) is binding (or u > 0), notice

that the clearin condition'2 of the intrada credit market re uires
g y Cl

<2qu = (1 —a)(1 -131; - (1.23)

If liquidity constraint (1 . 14) is binding (or u > 0). from (1.14) and (1.23) we have

*

at 01/1
pt+1= * '
(l-a)(1-/1 )

 

(1.24)

This means the proportion of intraday suppliers [0L )3] is smaller than the proportion of
intraday credit demanders among agents adjusted by the p2: [(1 — a) (l — 1.)] p... This
result can be extended to the following equilibrium results of intraday credit market price

a.
pH]:

 

'2 The intraday credit is traded at a certain price. In the second opening of the intraday market. the intraday
credit should be repaid by the d.i. banks to the c.i. banks. The potential amount of intraday credit repaid is

(I — a) (l — A.) I'.; therefore at the end of the second opening of the intraday credit market, demand side of
the intraday market implies a q‘... = (1 - a) (1 — A.) I".

18

t

“A , <p}' ifandonlyifa< “"AC)(1’A"““,
(l-O'Xl—Ii) l—Ad—ACU—Ad—Ii.)

 

 

pf“ = (1.25)

(l-Ac.)(1-Ad—,t‘)
l-Ad-Aca-Ad-A‘)

 

p‘w = p‘i if and only if a Z (1.26)

4. lntraday credit policy

This section analyzes intervention into the intraday credit market by the CB and intraday
credit policy including pricing rules under credit risks. I have analyzed the payment
system in which liquidity needs arise due to the imperfect synchronization at the
settlement stage. When the liquidity constraint is binding (liquidity shortages). there is a
role for the CB to improve social welfare by providing liquidity to the system. I restrict
attention to two different objectives of the CB. According to the Fed charter, for example.
the Fed is supposed to supply an elastic currency and insure financial stability. In this
regard. firstly I consider the case when the CB simply provides sufficient liquidity, or
liquidity-only objective. If there is no credit risk, then the intraday credit is repaid on the
same day. A liquidity shortage can easily be corrected by the simple intraday credit
policy by the CB: free lending at zero interest on intraday credit. This implies the
liquidity-only objective is sufficient for the CB to resolve the problem. Under credit risks.
however, this may not be an appropriate policy, since this objective may cause several
problems including possibility of losses from credit risk which may make the financial
system unstable. Therefore. the CB needs to provide the necessary amount of liquidity

under some constraint considering risks. This is the second objective of the CB. The

19

objectives of the CB are characterized by a pricing rule of intraday credit in the next
subsection.

The liquidity provision by the CB is characterized by endogenization of on in the
model. When there are liquidity shortages at the settlement stage, receipts and payments
are not synchronized. As in (1.24), liquidity shortages (or binding liquidity constraint) is
expressed by the synchronization parameters (1’, a) in the model. Since the CB provides
liquidity to the banks and relaxes the binding liquidity constraint on them, the provision
by the CB implies synchronization of receipts and payments. In other words, the CB
modifies the payment timing of banks [a (1')] under given receipt (debtors’ repayment)
timing. Once the CB relaxes the binding liquidity constraint by providing intraday
liquidity to the system this liquidity can enable the economy to obtain an equilibrium

which would otherwise not be achieved. One may consider this as one rationale of

intervention by the CB.

4.1. The central bank and pricing rule

A safe and efficient payment system is critical for the effective functioning of financial
systems [BIS, 2001]. Central banks put safety'3 as the first criterion for designing
payment systems; therefore most central banks choose RTGS systems. However, the
RTGS system is inefficient. Shortages of liquidity owing to imperfect synchronization of
the ordering of payment messages are typical. Central banks provide intraday liquidity to

enhance the efficiency of the payment system as well as social welfare.

 

'3 Finality of the payment is major component to the alleviation of the systemic risk.

20

Since intraday credit policy implies the CB supplying fiat money during the day
and absorbing it at the end of the day, if there are credit risks”, an intervention into the
intraday credit market by the CB affects the money stock. If there is a fraud shock on the
intraday credit market, then the amount extracted by fraud is not repaid to the CB at the
end of the day and therefore the money stock as of the end of day increases by that
amount; therefore there is a positive change in money stock over all. If the CB provides
intraday credit at a discount factor p.03, then the CB earns (1 — pica) and therefore
money stock decreases by that proportion; therefore there is a negative change in the
money stock over all. During the day, besides these changes there can be additional
changes in the money stock by the CB as a result of monetary policy. Now we have

following law of motion in money supply

ML. = M: + [[3, —(1 — [2213)] N: +change in M: from monetary policy

where N ,* denotes the amount of intraday credit provided by the CB at period t.

Following Greenspan’s (1996) suggestion that the central bank needs to improve risk
management within payment systems themselves, I consider only a change in money

stock during the payment procedure. Therefore, I assume that there is no smoothing
operation by the CB. or the change in M I from monetary policy equals zero. Now we

have

Mg. = M,’ +[A, —(l-—pz~3)]N:. (1.27)

 

'4 I consider end of day money stock. In the credit risk free case, the provision of intraday credit by the CB
can affect the money supply. Since the price of intraday credit is p = I (or zero interest), however. and the
intraday credit is repaid at the end of the day for certain, the money supply does not change.

21

I consider two cases of CBs whose objectives are presumably different from each
other: a CB with a liquidity-only objective (CB_L) and a CB providing necessary
liquidity under risk (more precisely, loss from risk) management'5 objective (CB_R).
Necessary liquidity means that the amount of liquidity the CB provides does not decrease
the overall money stock in (1.27). Since the only concern of the CB_L is liquidity. I

restrict the case in which the CB_L ignores the risk. therefore the CB_L provides
liquidity free (or p21,} = 1). Since CB_R provides at least necessary liquidity and
minimizes its loss from the provision of intraday credit at the same time. CB_R sets the
price at l — A. or ,0sz 21— A I . Notice that the price of intraday credit set by the CB_R is

equivalent to the price of intraday credit in the private market when the liquidity

constraint is not binding. In other words, the CB_R comes to set its price by taking the

risk premium into consideration. Since the CB’s loss'6 is [A, —(l - p279] N: as in (1.27).

when the CB_R sets the price as p279 =l—A I (= p; ), the CB_R does not experience

expected losses from the intraday credit provision.

The CB_L makes a private intraday credit market collapse; the price of the
intraday credit the CB sets is lower than that in the competitive market, therefore the CB
crowds out private lenders (no late-leaving banks lend their money to the market). As a

result, the demand for the intraday liquidity is solely supplied by the CB.

 

'5 The loss from the risk in the RTGS system becomes the burden of the general public either in the form of
increased tax or higher prices (Clair I991)

'6 l implicitly take this as a policy loss function.

22

4.2. Optimization and equilibrium
I keep all the assumptions in the previous section. The budget constraint of the debtor‘s

problem is the same as in the previous section,

. ,1! t t
.3de1 + PnC-dz ”1“": )hr +AdPl).1+IDd.z+1 = I’D! +Adh1 °

We have an identical problem to the previous section for a debtor.
In the banks‘ case, we need to adjust the objective of late-leaving banks. Under the

CB_L. since late-leaving banks do not provide intraday credit, they can enjoy utility

log( DZ, +1) once the relevant IOU is repaid; but under the CB_R, late-leaving banks can

. . A .A
only consume when their intraday credit 18 safely repa1dl7 [or 1 — Ad — (Ac ———‘——‘:—]].

The bank's problem is

’1

. A .A t
(1— a)[log D3,,” + (Ac - 1 L at: JPCB log DAM

 

j (1.28)

max logC'C, +
( ct~Dc,t+l‘
1)" DZ

c.I+l c.t+l

+a(l - Ad - A)pz~B log DEM

0, under the CB_L

Where A 2 {AC — L'A‘é). under the CB_R
l— 2.
subject to:
p0 = p(‘,Cc, +6,*1,* (when young) (1.29)
(l — 1* )pz‘Bl: +217: = pmHDL... (early-leaving old) (1.30)

(l — Ad )1: = PDJ+ID£+I (early-leaving old: additional dishonest) (1.31)

 

‘7 l implicitly consider that the C B_R employs a cap. If the C B_R does not employ a cap, this probability
becomes 1 - Ad.

23

(l—Ad )1: = p,)J+lDZ ,z+1 (late-leaving old). (1.32)

Notice that the provision of the intraday credit by the CB relaxes the liquidity constraint.
In other words. the CB can adjust a synchronization parameter or given 1' to relax the
constraint. The optimization problem can be adjusted from the choice of consumption

bundle to the choice of financial asset, lit. Now the maximum problem becomes:

 

 

 

 

 

'- at: :1: :1: ta: 7)
I log[(1—/1 )pCBl, +2.1, ]
Pom
:1: :1: (]_a) at:
mach =log 1—(S 1’ +< +[Ac—ACAdeCBlo 0g m i. (1.33)
p(‘( _ l—Il pD,t+l -
+a(1—Ad—A)pzv3 log W]
pD,t+l J

 

 

The first order condition for the bank’s problem is

     

—%T=é{<1-a)[1+zsc[ Ad.)p23]+aa—Ad—A)p2~3} (1.34)
p(~,—o,l, 1, 1—11.

Since now the money stock is not constant and the timing of the banks (a) is

determined under the equilibrium condition, I modify the definition of equilibrium.

Definition Given 1‘, M‘, and pin), a symmetric intervention equilibrium is a set of prices
(pm. pm I. pm, 53) and a set of allocations (0d,, DdJ +1, DC J+l, DC 1+1 DCZJH, (11,, ('a)
such that

(i) Given prices, a set of allocations (0d,, Dd,r+1v Dc , H D), +1 DC. , +| Cm. Cu)

solves maximization problems of a representative debtor and a representative bank.

24

(ii) Markets clear.“
For overnight credit market.
h, =1, . (1.35)

For good D and good C markets

Dd! +Ach/J+l +(1—Q’)[D3J+| +(AC -—g_—i)DgJ+l:l+a[l _Ad _A]D(fl+l =1 (1.36)

1— 2"
Cc,+Cd,=l. (1.37)
For money supply and demand,
m," = M,’ . (1.38)

From the clearing condition for the good D market, we have the following after some

algebra

TCB J," = mg] — Adm: (1.39)

 

where 72.” :(|_a)|:l:(l_/1*)p:,B+/l*]+AC[I—IAir)(l-Ad):|+a(l-Ad-A)(l-Ad).

Lemma 1. When the ('13 sets price of intraday credit at one, or pg], = l (orfree intraday

Ac[/1*(1 —A*)-Ad(1—Ad)]
credit), there exists an equilibrium if a = q t , j, E a]. ).
Ac(1—/1)/l +Ad(l—Ad)(l—Ac —-/1)

 

Proof: See the Appendix (A.1). 1

Lemma 2. When the CB sets price of intraday credit at p; . or pz~3 = l —A,, there exists

an equilibrium if

25

AL.[A*(l—/l*)—Ad(l—Ad)]
a: t * *
(l—Ad)Ad(l—/i )+Ac[(l—2Ad)(1—Ad)—/l (,1 —Ad)

 

:|(E aR|with caps ) (1 ~40)

Ol'

AC[in]—/i*)—Ad(1-Ad)1

 

a:

 

* * ‘ . Ear 'h - (1.41)
Ac(1—/l )li +Ad(1’Ad)(1—Ac-/l )( IW1t out caps)

Proof: See the Appendix (A.2). L;

Lemmas l and 2 describe equilibria which are the result of the CB’s intraday credit

policy.

Lemma 3. There is an equilibrium under the intervention by the CB when A. E [214. l —

Ad

#16
an [Md

 

.1).

Proof: See the Appendix (A3). 3

Lemma 3 provides a limitation of the intervention by the CB. When the proportion of

early-arriving debtors is too low, the CB cannot achieve an equilibrium.

4.3. Quantitative restrictions
This subsection studies two quantitative policy tools: caps and collateral. In addition to
the pricing rules above, these are two possible policy tools for risk management.

Different central banks use different policy tools. For example. the Federal Reserve

26

mainly depends on caps and (partial) collateral, whereas the European Central Bank
(ECB) uses collateral.

Until now, the two policy tools, price and quantity, have been considered to be
substitutes; therefore, the arguments on price vs. quantity are dichotomous. For example.
Faulhaber et al. (1990) argue against a cap because a cap is distortionary. Rochet and
Tirole (1996), on the contrary, argue for the quantitative restrictions. Quantitative
restrictions such as caps or collateral may be better means to control the overuse of
intraday credit than price. They argue that price may induce moral hazard or adverse
selection in situations where information is incomplete. Furthermore, Kahn and Roberds
(2001) argue optimally set quantitative restrictions at zero interest rate can eliminate the

liquidity shortages and restore the first-best consumption allocation.

4.3.1. Caps

The CB relaxes the liquidity constraint by providing intraday liquidity. Banks can shift
their potential credit risks onto the CB by using as much intraday credit from the CB as
possible; therefore. imposing quantitative restrictions helps to control potential loss. In
the model, the caps are set as (l — 7C) (1 — or) 1.1, which is sufficient to relax the liquidity
shortage. If the size of caps is larger than this, all other banks may try to borrow intraday
credit from the CB in order to shift their potential type I credit risk to the CB. If there are
more early arriving debtors in an economy, then caps are low. Notice that the CB needs
to know the information about 1* which consists of it and (l — Ad). This implies the CB

has information about the assets of commercial banks in real world. If the CB is unable to

27

access the information on the assets of commercial banks, then the CB can not manage

risks effectively through caps.

4.3.2. Collateral

I consider full collateralization in this subsection. lnstitutionally, banks put up collateral
before their types are revealed in order to use the facilities provided by the central bank;
specifically, for access to the discount window. According to F urfine and Stehm (1998),
the nature of these collateral arrangements typically involve either pledging collateral to
the central bank or entering into an intraday repurchase agreement (repo) with the central
bank. In the model, since the commodities are perishable and there are no other assets

except IOUs, only the lOUs can be used as collateral'8

. Therefore, I focus on repo type
collateral in this section. Specifically, the banks sell their IOUs to the CB in the form of a
repo; therefore they do not put up collateral before their types are revealed which is
different from usual institutional set up of collateral. In this repo set up, collateral has the
value of repayment from the debtor just as does an IOU. I assume that the “haircut” on
collateral is calculated by the pricing rule in the previous section. In the model, therefore.
a collateral policy always goes with a price policy. Notice that I restrict attention to
collateral for the CB; that is I do not consider a case where collateral is allowed in the
private intraday market.

Collateral changes the market timing so the event of fraud shock by the d.i. bank

no longer occurs. As I consider a repo type collateral, a d.i. bank hands over the IOU to

the CB. Since the CB holds the IOU now, according to the sequence of events, borrowers

 

'3 . . . . .
Slnce there were concerns about defic1ency of collateral, the EC B entitled a w1de range of assets
including commercial loans for collateral.

28

of intraday credit have no possibility to have another chance to consume. The CB can be
repaid directly from the issuer of the IOU in this case. Notice that there still remains a
risk that the consumption debt will not be repaid by the debtor. Therefore, this policy can
not remove all fraud shocks in the economy. This means collateral cannot control shifts
of potential risks from banks to the CB. This is an effective policy, however. to improve
social welfare as well as to limit losses from credit risk.

By imposing collateral, the equilibrium has the following properties. First, since
now Ac is zero in the model, collateral makes A] decrease which results in p‘i increasing.
As a result, the CB_R can set higher p‘CB (or lower intraday interest) according to the
pricing rule. The economy can sustain higher social welfare than before. In the case of
C B_L_. the decreased Al makes the price of good C higher than before. The bank is better
off from the price change. but the debtor is worse off. This has unclear effects on social
welfare. Notice that collateral perfectly removes type II credit risk.

Second, the equilibrium under the intervention of the CB with charging interest

on intraday credit exists]9 only when or (if) = 0.

4.4. Calibration of social welfare and policy analysis

This subsection calibrates social welfare of the combinations of the policy tools. I
compare social welfares under equilibrium. I use Ad = 0.03, Ac = 0.01, Mn = 1. Using
Lemma 1 and 2, I calculate CL (71:) given the synchronization parameter of debtors (if =

0.814). Consumption bundles under each policy set are calculated using above parameter

values.

 

'9 See the Appendix (A.4).

29

The result of calibration is summarized in Table 1.2. Under competitive
equilibrium social welfare is 10.57 with a debtor’s being 5.42 and a bank's being 5.15.
When the CB provides intraday credit at zero interest rate without caps, social welfare
becomes 10.60 with a debtor’s being 5.43 and a bank’s being 5.17. Once the economy (or
the CB as a representative of the economy) takes into account, however, its loss from this
policy, social welfare is reduced according to the weight the CB places on the loss. For
example, if the CB takes the loss into full consideration, social welfare becomes 9.10 (=

10.60 — 1.50).

Property 1. When the ('8 places a low weight on the loss from free intraday credit, the

('B can make the economy better off just by providing intraday liquidity at zero interest.

As the CB considers the loss more, the CB may use quantitative restrictions. caps or
collateral. According to Table 1.2, when the CB puts less than 2% weight on its loss from

the risks. social welfare under zero interest is higher than competitive equilibrium (10.57).

Property 2. There is a critical weight w, to the CB '5 loss such that if a weight is less than

w). caps with. free intraday credit can improve social welfare.

According to Table 1.2, when the society puts less than 2.1% of weight to its loss, social

welfare under caps with free intraday credit is higher than competitive equilibrium.

Property 3. Price tool should be applied with quantitative restriction, specifically

collateral. in order to improve social welfare.

30

The economy becomes worse off under a price only policy whose social welfare is 10.54.
This is not a good policy. because this does not improve social welfare compared to that
of no intervention. Under priced intraday credit with caps, however, social welfare
becomes 10.58 with a debtor’s being 5.42 and a bank’s being 5.16. This is one

justification of caps along with positive interest rate.

Property 4. There is a critical weight w; on the CB 's loss such that if a weight is greater
than W3, a price and caps policy dominates a caps with free intraday credit policy in

terms ofsocial welfare.

According to Table 1.2. when the CB puts more than 1.6% of weight to its loss from the

risks. social welfare under a price and caps policy (10.58) is higher than a caps policy at

zero interest.

Property 5. When intraday credit is costly, collateral is a better tool than caps in terms

ofsocial welfare.

Property 6. Social welfare from a price and collateral policy is the same as that from a

price. caps and collateral policy.

Price considers both types I and 11 credit risks, and caps deal with type I credit risk, and

collateral controls type 11 credit risk. As long as the CB places sufficient weight on

31

potential loss from the risks, Property 6 implies that using a price and collateral policy is

more efficient than any other policies in terms of social welfare.

Table 1.2: Consumption bundles and utility in equilibrium

 

without intervention : intervention
1 (Priced+ (Priced+ (Priced+caps
LRiskfree) (Risky)i Free Free+Caps: Priced Caps) Collateral) Collateral)
Debtors’ Utility 5.4071 5.41581 5.4269 5.426L54202 5.4216 5.4269 5.4269

Banks‘ Utility 5.2407 5.15455 5.1734 5.1734‘ 5.1178 5.1567 5.3911 5.3911

 

 

1119983999 ......................... l--.Q-9.3.1.§_---_9.-9.2§Z ...............................................
Disutility 1 1.4991 1.4108

Social 10.6479 10.5703} 10.6003 10.6003: 10.5380 10.5783 10.8180 10.8180

Welfare-20 5 (9.1012) (9.1897)

 

Figures in ( ) are the utility after full consideration of the loss.

5. Concluding remarks

This paper has analyzed the intraday credit policy tools implemented by the central bank
under a large-value RTGS system with several types of exogenous risks.

This paper identifies credit risks in the market which explain joint usage of
elements of intraday credit policy under RTGS system in the real world. Price deals with
both type I and II credit risks; a cap works for managing type 1 credit risk; even though
this chapter does not consider moral hazard problem, collateral removes type 11 credit risk
just as in Martin“ (2004).

From the analysis of intraday credit policy, the model suggests the following.
When the economy (or the CB as a representative of the economy) does not place

significant weight on the potential loss from credit risks, the CB can make the economy

 

2” Because of logarithmic functional form, I multiply consumption amount by 1000 in order to make
positive utility.

" More specifically, Martin argues that collateral lending with zero interest can prevent moral hazard
committed by banks.

32

better off simply by providing intraday liquidity at zero interest. Caps may be used only
when the economy takes little consideration of the potential loss from the credit risks in
the RTGS system. The CB starts to use the price tool when there is an increase in the
economy's consideration of the potential loss from credit risks. The price tool, however,
should be used with other quantitative tools specifically with collateral. When intraday
credit is costly, collateral is a better tool than caps in terms of social welfare. Since there
are two types of credit risk in the market, it is enough to employ two tools. The model
illustrates that a price and collateral policy achieves the highest social welfare among
other sets of polices.

The model demonstrates that intraday credit policy implemented by the CB may
fully displace a private market for intraday credit. Specifically, collateral makes the
private market collapse. This is why an active private market for intraday credit is not
common. According to BIS (2005), there are only 5 RTGS systems (Germany, Japan.
Singapore, and two systems in Sweden) out of 16 systems (including TARGET of the
EC B) to which the CB provides intraday credit that have additional private market.

Now I discuss sources of apprehension of the society to the potential loss from the
credit risks in the payment systems. The first source can be limitations in the conduct of
monetary policy. Potential loss from the payment policy is revealed in the form of change
in fiat money stock. If the CB faces any impediment to implementing monetary policy for
this purpose, then the society would pay much attention to the potential loss. This can be
an explanation of the payment system policy tools of the ECB. According to Pollard
(2003), the Maastricht Treaty, under which the ECB is established, does not mention a

lender of last resort function for the ECB and the ECB has been criticized for lacking this

33

function. Another possible limitation of the ability originates from the procedure22 of
open market operations. The ECB conducts its main open market operations only once
per week. Additionally, open market operations are decentralized in the euro area. From
these environments. the ECB should pay a lot more attention to the potential loss from
the payment system rather than Federal Reserve.

The second source can be the size of potential loss (daylight overdraft). As the
size of the potential loss increased, for example, the Federal Reserve started to employ
caps and to price intraday credit.

As the model separates senders (debtors) and receivers (banks), a price policy
does not affect the synchronization parameter in this paper. Some studies”, however.
show that a price policy may cause a network extemality which may induce banks to
postpone payment activity.

Since there are too many parameters, one disadvantage of the paper is that it does
not analyze welfare in a clear analytical form, but in calibrated ones. Another limitation is
the model simplifies settlements procedure too much. One more disadvantage is that this
paper considers exogenous risks. One possible extension of research, therefore, is to
endogenize the risks. In this case there may be a cost for committing fraud. If the CB has
full information of the payment timing and ability to supervise the banks, the cost for
committing fraud by the bank can be sufficiently large enough to prevent commitment of

fraud by the intraday users. This gives the same result as collateral.

 

33 The Federal Reserve deals exclusively in US. government securities, whereas the ECB has a broader
range of assets including commercial loans. Since the policy makers of the Fed concern that collateralized
intraday loan may impair the ability to conduct monetary policy through open market operations. the Fed
does not heavily depend on collateral among payment policy tools.

1‘ See Angelini (1998). Kobayakawa (1997).

34

Appendix
A.1. Proof of Lemma 1

In case of the CB_L, 7(‘3 =(I—a)|:1+Ac[l— A2,](l—Ad)]+a(l—Ad)2. From (1.27).

 

(1.39) and money supply and demand equilibrium condition, we have

Tag-l: =(1-Ad)M,‘+A,N,* (1.A.1)

Under the C B__L. the only provider of the intraday credit is the CB. This implies

N; = (1 — a)(1— 2* )1: . Applying this into (1.A.1) and rearranging gives

{7”, —[Ac +tfi%éi](1—a)(1—2*)}If =(1—Ad)M;‘. (1.A.2)
1-
From (1.A.2). if T”, —[Ac +“—’9-C—¥‘—d](1—a)(1—2*)=1—Ad, we have 1‘. = h‘. = m‘.=

t . I“ :1:
M 1. Solvmg T”; —l:AL. +L—éfifl](l—a)(l—2 )=1—Ad for a, we get

Ac[’1‘(1-1*)—Ad(l’Ad)] _
a: 4 q * (EaL).lj (1.A.3)
8,.(1—2 )2 +Ad(1—Ad)(1—A,.—2 )

 

A.2. Proof of Lemma 2

We have two different forms of Tc-B in the CB_R:

 

11:

T”, :a(l-Ad)2 +(1—a)[(1—Ac)(1—2* —Ad)+2*]+(1-2a)Ac[1-IA: ](l—Ad)(in

case of the CB_R with caps) or

35

 

5 e A .
72.B=(1—a)[(1—AC)(1—/1 —Ad)+2 +Ac(l— d, )(l—Ad)]+ar(1—Ad)2 (1n case of
—2
the CB_R without caps)

In either case of Tu}. we have TCB J: = (l—Ad)M: from (1.39) and money supply and

demand equilibrium condition. When Tea = (l - Ad). we have 1‘. = h‘, = ml, = M‘..

Solving Ta; = (1 — Ad) for a. we get

(.

a: * * ,.. .. *(Ealeithcaps) (1'A°4)
(l—Ad)Ad(1—/l )+Ac[(l—2Ad)(1-Ad)—/1 (2 —Ad)‘

A . L2*(1—2*)—Ad(l—Ad)q

 

 

 

 

 

0r
A.-[,1*(1—,(*)-Ad(1-Ad)] ( )
A.3. Proof of Lemma 3

Since (1.A.5) is identical to (1.A.3), I look at (1.A.3) and (1.A.4). From the property of or.

we need to have 0 s a s 1 . Applying (l .A.3) and (1 .A.4) to this property. we have

Ac[2*(1—2*)-Ad(1—Ad)]
OS :0: It: * -
8,.(1—2 )2 +Ad(1-Ad)2(1—Ac—2 )

 

and

0< Ac[2*(1_,1*)_Ad(1—Ad)] <1.

_ (1—A,,)Ad(1—2*)+Ac[(1-2A,,)(1—Ad)—2‘(2" 41(1)] _

 

36

The numerators of both inequalities are identical and are smaller than denominators.
Therefore. these inequalities hold once the numerator is non-negative. The numerator is a
quadratic equation in if. We have 01 = 0 when if = 1 — Ad or 7.. = Ad. Therefore, the

equilibrium under intervention by the CB exists in the region of if 6 [Ad, 1 — Ad) or

 

 

A . . . .
,1 e[ d ,1). The lower bound Ad should be less than 1. This 1mplles Ad < —]—. See
l—Ad l—Ad 2
Figure LA]. ‘1‘ ‘

Figure l.A.l: Equilibrium range A. 6 IA... 1 — Ad) or A e | Ad , I) under intervention by the CB
l—Ad

 

 

Eli:

 

 

0.5 1 4.:

A.4. Equilibrium under collateral

In the equilibrium where M is changing (The CB_L: the CB sets pz~3=l). we have Ill =

t

h ( = mt, = M2. This implies that

at: at A .1.
TL-l, =(1—Ad)M, +I—‘i—;N, (1.A.6)

where TL=(l—a)+a(1—Ad)2.

37

When the CB sets pz~3=l, then the only provider of the intraday credit is the CB.
This implies24 N : =(1— a)(1 — 231: . Applying this and rearranging gives us

[TL —Ad(1—a)]1,* =(1—Ad)M,* (1.A.7)

Only when TL —Ad(1—a) =l—Ad, does there exist an equilibrium. Solving this for 01,

weget (120.

In the equilibrium where M is fixed (the CB_R). we have 1‘l = h“. = m‘[ = M“..
This implies that TR 7: = (l —Ad)M; where TR =(l—aAd)(1—Ad). Only when TR = 1 —

Ad. does there exist an equilibrium. Solving this for 01, we get a = 0 .

 

2‘ I assume that the CB imposes a cap. If there is no cap, we have x: =(1 _ 231:. The result. however. does
notchange.

38

REFERENCES

Angelini. P. (1997) “An analysis of competitive extemalities in gross settlement
systems," Journal of Banking and Finance, Vol. 22, pp.1-18.

Angell, W. (1991) “A proposal to rely on market interest rates on intraday funds to
reduce payment system risk.” Governing Banking 's Future, edited by C. England.
ch.10, Boston: Kluwer.

Bank for International Settlements (1997) “Real-time gross settlement systems,” report
prepared by the Committee on Payment and Settlement Systems, CPSS Publication No.
22. (available at httLL/i’xmwbis.org/publ/cpssZZ.htm)

__ (2001) “Core Principles for systematically important payment systems", report
prepared by the Committee on Payment and Settlement Systems. CPSS Publication No.
43. (available at http://www.bis.org/publ/cpss43.htm)

__ (2003) “The role of central bank money in payment systems”, report prepared by
the Committee on Payment and Settlement Systems. CPSS Publication No. 55.
(available at http://www.bis.org/publ/cpssS5.htm)

__ (2005) "New developments in large-value payment systems”, report prepared by the
Committee on Payment and Settlement Systems, CPSS Publication No. 67. (available
at http://www.bis.org/g1bl/cpss67.htm)

Balino, T.; J. Dhawan; V. Sundararajan (I994) "Payments Systems Reforms and
Monetary Policy in Emerging Market Economics in Central and Eastern Europe," IMF
Staff Papers, Vol. 41 (September), pp. 383-410. (an abridged version available at

http://www.worldbank.org/fandd/english/0396/articles/080396.htm)

39

Calomiris. C. (1993) "Regulation, industrial structure, and instability in US. Banking: an
historical perspective". in Structural Change in Banking edited by M. Klausner and L.
White. pp.l9-l 16. .

Champ, B.: B. Smith; S. Williamson (1996) “Currency Elasticity and Banking Panics:
Theory and Evidence", Canadian Journal of Economics, Vol.29, pp.828-64.

C lair, R. (1991). "Daylight overdraft: who really bears the risk?” in Governing banking '5
future: Markets vs. regulation, edited by C. England. p.117-140. Boston: Kluwer

Coleman, S. (2002) “The evolution of the Federal Reserve’s intraday credit policies",
Federal Reserve Bulletin, February, pp.67-84
(available: http://www.federalreservegov/pubs/bulletin/2002/0202lead.pdf )

Faulhaber, G.; A. Phillips; A. Santomero (1990) “Payment risk, network risk, and the role
of the Fed” in The US. payment system: efficiency, risk and the role of the Federal
Reserve, edited by D. Humphrey, p.197-213. Boston: Kluwer

Freeman, S. (1996a) “Clearinghouse banks and banknote over-issue”, Journal of
Monetary Economics Vol.38, pp.101-15.

Freeman. S. (1996b) “The Payment System, Liquidity. and Rediscounting". American
Economic Review Vol.86 (5), pp.1126-38.

Freeman, S. (1999) “Rediscounting under Aggregate Risk”, Journal of Monetary
Economics Vol.43, pp. 1 97-216.

F urfine, C .; J. Stehm (1998), “Analyzing alternative intraday credit policies in real-time

gross settlement systems". Journal of Money. Credit and Banking Vol.30 (4). pp.832-

48.

4O

Gorton, G. (19853) "Bank Suspensions of Convertibility." Journal of Monetary
Economics, Vol. 15 pp.177-93.

__ (1985b) "Clearing Houses and the Origin of Central Banking in the United States."
Journal of Economic History. Vol. 45, pp.277-83.

Green. E. (1997) “Money and Debt in the Structure of Payments” Monetary and
Economic Studies. Vol.15 (1), pp.63-87.

__ ; R. Todd (2001) “Thoughts on the Fed’s role in the payments system”, Federal
Reserve Bank of Minneapolis Quarterly Review, Winter, Vol.25(l), pp.12-27.

Greenspan, A. (1996) “Remarks on evolving payment system issues,” Journal of Money,
Credit and Banking, Vol.28 (4), Part 2, pp.689-695.

Grossman. R. (2003) "US Banking History, Civil War to World War II". EH.Net
Encyclopedia. edited by R. Whaples. March 17, 2003. (Available at
http://eh.net/encyclopedia/article/grossmanbanking.history.us.civil.war.wwii)

Kahn, C.; W. Roberds (2001) “Real-time gross settlement and the costs of immediacy”,
Journal of Monetary Economics, Vol.47, pp.299-319

Kobayakawa, S. (1997) “The comparative analysis of settlement systems,” Centre for
Economic Policy Research Discussion Paper No.1667.

Lacker, J. (1997) “Clearing. settlement and monetary policy”, Journal of Monetary
Economics, Vol.40, pp.347-81.

_, J. Weinberg (2003) “Payment economics: studying the mechanics of exchange”.
Journal ofMonetary Economics, Vol.50, pp.38 l -7.

Martin, A. (2004) “Optimal pricing of intraday liquidity”, Journal of Monetary

Economics, Vol .51 pp.401-24

41

Martin, A. (2005) “Recent evolution of large-value payment systems: balancing liquidity
and risk”, Economic Review FRB Kansas City, First Quarter, Vol.90, pp.33-57

Mengle, D.; D. Humphrey; B. Summers (1987) “lntraday credit: risk, value, and pricing”.
Federal Reserve Bank of Richmond Economic Review, Vol.73, pp.3-14

Milano. G. (1991) “Payment system risk: a private-sector view” in Governing banking ‘s
future: Markets vs. regulation. edited by C. England, p.141-160. Homewood: Business
One Irwin

Mills, D. C. (2004a) “Mechanism design and the role of enforcement in Freeman’s model
of payments”, Review ofEconomic Dynamics, Vol.7, pp.219-36.

(2004b) “Alternative central bank credit policies for liquidity provision in a

model of payments”. mimeo

Pollard, P. (2003) “A look inside two central banks: the European Central Bank and the
Federal Reserve.” Federal Reserve Bank of St. Louis Review, January/February,
Vol.85(1), pp. 11-30.

Rochet. J.; J. Tirole (1996) “Controlling risk in payment systems”, Journal of Money.
Credit, and Banking, Vol.28 (4), pp.832-62

Williamson, S (1999) “Private money”, Journal ofMoney. Credit, and Banking,
Vol.3l(3) Part 2, pp.469-91

Zhou. R. (2000) “Understanding intraday credit in large-value payment systems”, Federal

Reserve Bank ofChicago Economic Perspectives, 3rd quarter, pp.29-44.

42

Chapter 2

Payment instruments and the central bank

1. Introduction

This chapter explores the sensitivity of payment system stability to macroeconomic
shocks through the assets (from lenders’ viewpoint) or liabilities (from borrowers‘
viewpoint) used as media of exchange. A safe and efficient payment system is a gateway
to the effective functioning of the financial system in monetary economies. In this regard.
governments may implement policies to keep the payment system stable. Specifically,
this chapter studies a “payment instrument policy,” such as the one implemented by the
Bank of Korea, to promote usage of a certain type of payment instrument to stabilize the
payment system.

In addition to fiat money, there are several types of credit which circulate as
payment instruments (or “private money ' ”). Examples of these instruments are
promissory notes, which are simple credits in the sense that sellers provide credit2
directly to buyers; or bills of exchange, which are interrnediated credits in the sense that a
commercial bank is necessary to provide credit and repay the bill between buyers and
sellers. These two payment instruments coexist and are used by different agents. In

practice direct credits are used by relatively wealthy agents about whom much is publicly

 

' In the real world. these payment instruments are circulated when they are endorsed.

7 . . . . .

“ There are a group of papers regardmg “trade credit” whtch means goods lent by suppliers to their
customers. I consider a transaction under “quid pro quo” principle in this chapter: therefore no transaction

occurs without any (private) money. [Refer Petersen and Rajan (1997) for comprehensive review of
theories on trade credit].

43

known, whereas others demand more information-intensive intermediated credits
(Holmstrom and Tirole. 1997), relying on the financial intermediary’s reputation to
secure funds.

In contrast to fiat money, these private monies are risky. Specifically unlike fiat
money the market value of these payment instruments is affected by moral hazard and
macroeconomic shocks such as recessions or oil shocks. The moral hazard may be
accurately priced, but macroeconomic shocks may not. In case of macroeconomic shocks.
expectations of them are a major determining factor of the price. If there are fewer agents
who anticipate macroeconomic shocks, then the shocks would not be accurately priced
and. consequently. the distribution of product return would be wide. Thus, the shocks
alter production choices of agents. As a result, the economy has a higher risk when there
are unanticipated macroeconomic shocks. For example, the Korean currency crisis3 of
November 1997 is considered an unanticipated shock (Radelet and Sachs 1998). As a
result, the bankruptcy rate of firms skyrocketed as represented by the historically high
dishonored bill ratio4 at the end of 1997 (Hahm and Mishkin 2000). I call this a market
risk. A market risk can be considered as systemic risks, since all agents are affected by
the risk and cannot fully control it at the individual level.

In order to alleviate overall market risk from private monies, the government may
implement a “payment instrument policy.” In this chapter, the government may

implement a payment instrument policy by subsidizing or taxing a certain type of

 

3 Refer to Hahm and Mishkin (2000) for details.

4 The dishonored bill ratio is the percentage of corporate promissory notes, checks and bills in default out
of total promissory notes. checks and bills cleared at the clearing house during the day.

5 Systemic risk is the risk that the inability of one of the participants to meet its obligations, or a disruption
in the system itself. could result in the inability of other system participants or of financial institutions in
other parts of the financial system to meet their obligations as they become due. Such a failure could cause
widespread liquidity or credit problems and, as a result, could threaten the stability of the system or of
financial markets (Bank for International Settlements [BIS] 2001).

44

payment instrument in order to achieve a safe and effectively functioning payments
system. For example. in 2000 after the currency crisis, the Bank of Korea (BOK)
introduced a new facility, the “Corporation Procurement Loan,” for subsidizing bills of
exchange in order to promote the usage of bills of exchange and reduce systemic risk
caused by the market risk from promissory notes.

The main question of this chapter is why the BOK chose to promote the usage of
bills of exchange instead of promissory notes as a part of its “payment instrument
policy.” Usually negative macroeconomic shocks hit small size debtors the hardest
(Holmstrom and Tirole, 1997). As stated above, promissory notes (or direct credit) are
used by relatively wealthy agents. One may think from this fact that the government
intervention can be justified by supporting the economically relatively weak. This,
however, does not answer the question sufficiently from the viewpoint of a “payment
instrument policy.” I look at market risk of each payment instrument in order to answer
this question. The next question is an extension of the first one: why these two payment
instruments incur different market risk; and which one is better in terms of risk
management.

This chapter studies the role of the government regarding private money in the
presence of macroeconomic shocks. Two types of private money are studied: promissory
notes (in short PN) and bills of exchange (in short BE). I modify a simple partial
equilibrium moral hazard model of wealth constrained agents. The model features (i) two
types of consumption credit (IOUS); (ii) two different types of lenders: creditors who

provide a direct consumption credit and financial intermediaries which provide

45

interrnediated credit and monitoring services; (iii) the central bank 6 which may
implement a “payment instrument policy” with a tax or subsidy; (iv) a macroeconomic
shock which affects incentives of debtors and lenders. I first consider an economy where
PN and BE coexist and work as consumption credit and study the critical level of
debtors” wealth. I introduce a macroeconomic shock and assume that agents do not
anticipate the shock; I then examine a pure PN economy and a pure BE economy to study
market risk of each payment instrument. Then, I look at two different payment instrument
policy tools: tax and subsidy. Finally, I relax the non-anticipation assumption of the
macroeconomic shock.

This chapter provides a different perspective on intervention for macroeconomic
risk management by the government. The model shows that bills of exchange have a
lower market risk than promissory notes under negative macroeconomic shocks. If,
therefore, both promissory notes and bills of exchange are used in addition to fiat money,
systems with a greater proportion of promissory notes in circulation will be riskier and
more prone to collapse than systems with fewer promissory notes. Payment instruments
play an important role in payment system stability in the presence of moral hazard. This
is a justification for a policy which supports usage of bills of exchange. The model also
shows that a subsidy for bills of exchange is better than a tax on promissory notes.
Another contribution is the identification of types of risk: credit risk from the production

technology (moral hazard) and market risk caused by a macroeconomic shock. The

 

6 As Mills (2004b) noted, one can interpret a central bank as a private clearinghouse that is separate from
the other agents. As noted in Green (1997), the liquidity-providing institution in the model can be either a
public or private one. This is a payment system version of inside and outside money debate which remains

an open question. Refer to Williamson (1999) for a further review on historical debates about inside and
outside money.

46

model demonstrates that unanticipated macroeconomic shocks bring higher market risk
than anticipated shocks.

In contrast to this chapter, existing studies on payment policy mainly pay
attention to intraday credit policy. See, for example, Mills (2004b), Lacker (1997),
Faulhaber et a1. (1990), Angell (1991), Rochet and Tirole (1996), Kahn and Roberds
(2001), and Martin (2004). However, none lend insight into why governments implement
a policy which encourages certain types of payment instrument in order to achieve
payment system stability.

This chapter is organized as follows. Section 2 describes the environment of the
model; section 3 analyzes an economy with two types of private money. Section 4
analyzes an economy with a macroeconomic shock and discusses the government‘s role,

and section 5 summarizes our findings and concludes. Lengthy proofs are in the appendix.

2. The environment

The model shares environments of Holmstrom and Tirole (1997, 1998). Consider a two-
period economy with three types of risk neutral agents: debtors, creditors and banks.
Debtors have an initial endowment of W, which comes as a random draw from a uniform
distribution with support [0, sup (W)]. The amount of wealth is assumed to be public

information. The utility of a debtor is given by U = u(E)+ u(Rd) where E is an amount

of consumption at date 0, exogenously set such that sup (W) < E; where Rd 2 0 is an
amount of consumption at date 1 when production succeed.
Hence, debtors need to borrow resources from other agents. These resources are

repaid at date 1. but repayment is subject to a moral hazard problem. Precisely. at date I

47

each debtor has the option between a good and a bad technology of production and this
choice is unverifiable. The good (bad) technology leads to a probability p (p — Ap. where
Ap > O) of a successful production but the bad technology provides a private benefit to
the debtor. Returns from production are verifiable and success (failure) means that an
output of R (0) is produced where R > 0. Creditors and banks only consume at date 1.
Moreover, they start with an amount of endowment sufficient to meet the debtors“
demands. Banks have a more costly but more efficient monitoring technology. In
particular, banks (creditors) face a monitoring cost of C (0) and induce a private benefit
of b (B) to the debtor with a bad technology, with B > b. If a creditor lends to a debtor,
the payment instrument that underlies this transaction is denoted as a PN. Alternatively,
when the debtor borrows from the bank, the corresponding payment instrument is a BE.
These are the only payment instruments available in the economy. Attention is focused
on debt contracts where (i) neither party is paid anything if the production fails; (ii) if the
production succeeds and there is no systemic risk, Rd 2 0. and the creditor (bank) is

repaid Rc > 0 or Rb > 0. where

Rd+RC=R (Rd+Rb=R).

Figure 2.1: Timing of events (debtor’s viewpoint)

 

 

 

 

 

 

 

t=0 5 1:1
Consumption :
using payment ; Moral hazard Outcome (R or 0)
instrument (Production) and repayment

Creditors and banks are assumed to have sufficient initial wealth to be able to provide

consumption credit to debtors at date 0 which is repaid at date I. The utility of a creditor

48

(bank) is V = u(RC )( V = u(Rb -C) ). There is an authority which enforces a debt contract.

The risk free (gross) interest rate is normalized to 1. Let y be the (endogenous) expected
rate of return on promissory notes; and B be the (endogenous) expected rate of return on
bills of exchange. I also assume that consumption credit is small relative to the overall

size of a bank‘s asset portfolio. This implies that a default by an individual debtor will

not compel the bank into a default.

3. Simple model for payment instruments

There are two different types of payment instruments, promissory notes and bills of
exchange. PN is a non—intermediated credit provided by individual creditors, whereas BE
is an intermediated credit provided by banks. Payment instruments are issued at date 0

and settled at date 1 after production of output R.

3.1. Promissory notes

In this subsection, debtors use PN for consumption. Incentive compatibility constraint for

debtors to choose a good production technology is

p'HRdZ(p-Ap)Rd+B (ICd)

Individual rationality constraint for a creditor is

P'Rc 2}"(l’N) (IRe)

Lemma 1. There is a critical level of debtor’s wealth W such that creditors provide

consumption credit only to debtors whose wealth is above the critical level W .

Proof: From (IRC) we have an upper bound on PN

49

 

and from (ICd) this becomes

 

(PN)sp'RC sf[R——li:l.

In order to consume debtors have
W + (PN ) 2 E .
Rearranging this for W and using the upper bound on PN above. we have

W_>_W

where W :— E—£[R-fp:]; therefore, debtors who are sufficiently wealthy can use
7

promissory notes. CT

3.2. Bills of exchange

In this subsection, I study debtors who use BE for consumption. Banks can help a wealth
constrained debtor to consume. Now the banks monitor debtors. Since monitoring

reduces private benefit of a debtor from B to b, the incentive constraint of a debtor is

P'RdZ(P-AP)'Rd+b- (IC'd)

Participation constraint for the bank is:

p.12, ZB-(BE) (1a.)

Lemma 2. There is a critical level of debtor’s wealth W such that banks provide

consumption credit only to debtors whose wealth is above the critical level W .

50

Proof: Similarly as in Lemma 1. we have W 2 W
where W a E —%[R ”Ab—p]; therefore, debtors who are wealthy enough can use bills of

exchangei)

Lemma 3. When 7 < B. there is a critical output level R such that if R e (0,R) , we have
W < W .

Proof: See the Appendix. I?

Figure 2.2: Critical levels of wealth for private monies

No debtors can Debtors can use BE Debtors can use PN
access to payment
instruments

 

 

 

W W

Lemma 3 implies that the level of output R is in a reasonable range. If the output is too
abundant such as air or water, then size of wealth does not matter. If W > W , there is no
demand for BE. This implies that the monitoring technology is too costly to be socially
useful. Since I consider an economy with coexisting PN and BE, I assume that R e (0, R) .
Since PN is cheaper than BE7, debtors whose wealth is greater than W have no incentive

to use BE; debtors whose wealth is between W and W can only use BE and be

monitored by the banks.

 

7 See Lemma 8.

51

4. Macroeconomic shock and payment instrument policy

This section studies market risk from macroeconomic shocks and payment instrument
policy of the central bank (CB). In order I) to remove or alleviate the systemic risk; or 2)
to enhance the efficiency of the payment system; or, more fundamentally, 3) to maintain

the payment system (or monetary economy), payment (system) policy is implemented.

Figure 2.3: Timing of events (debtor’s viewpoint)

 

 

 

 

 

 

 

t = 0 : t = 1/2 : t = 1
Consumption ' Unanticipated i Moral hazard Outcome (R5 or 0)
using payment : Macro shock 1 (Production) and repayment
instrument j

Following Holmstrom and Tirole (1998), a macroeconomic shock is introduced
by assuming that at an interim date (t = 1/2), each debtor suffers from an output shock.
For simplicity, the economy is assumed to consist of homogenous agents who do not
anticipate macro shocks. The output R decreases to Rs. This shock is non-diversifiable
event like oil shock or a sharp economic contraction after a currency crisis. In order to
verify market risks of each payment instrument, I study a PN only economy and a BE

only economy under macroeconomic shock.

4.1. Pure promissory notes economy

There are two types of market risk from the macro shock in the payment and settlement
system: let mm and n3}; be the endogenous market risk of PN and BE from the macro
shock respectively (these are calculated in next subsection). If the probability of success

from the bad technology (p — Ap) is lower than the market risk from the macro shock.

52

then the expected amount of repayment of consumption debt is less than zero. This
implies that the monetary economy cannot exist, therefore I rule out this case and restrict
analysis to the case8 where the market risk from the macro shock is lower than the
probability of success from the bad technology (p — Ap) or

P — AP > ’7P.\'
and

P“AP>’73£-

Lemma 4. There is an upper limit B for private benefit B such that if B S B , the market
risk rim is negatively related to the macro shock.

Proof: See the Appendix. 1 1

Since the same logic is applied, there is an upper limit b for private benefit b such that

b313, the market risk 1135 is negatively related to the macro shock. From now on. I

assume that the private benefit B and b are not too big.

Because of the macro shock we have the following adjusted (ICSd) and (IRSC)

constraints:
P'de 2(P—AP)'RSd+B (ICSd)

(p—nerRs. 2 r-(PN) HRS.)

From Lemma 1. a new critical wealth for PN is

 

8 A sufficient condition for this restriction is
p > 2 . Ap .

53

WSEE—W[RS——li]. (2.1)
7 AP

There are two criteria of wealth for decisions of agents in this model: a lending
criterion for lenders at date 0 and a technology choice criterion for debtors at date 1. A
lending criterion determines the type of payment instrument, PN or BE as in Lemma 1
and 2. Lenders (creditors or banks) provide either direct credit or interrnediated credit

according to the critical level of wealth at date 0. As a result, for example, debtors for
whom W > W can use PN for their consumption credit. A technology choice criterion
determines the type of production technology, p or (p — Ap). For example, debtors with

W > W who uses PN choose a high technology (p) consistent with the (IQ) constraint at
date 1. When there is no macro shock, these two criteria are identical. The macro shock.
however, makes these two criteria differ from each other. As the economy consists of
homogeneous agents, the lending criterion at date 0 is the same as before: debtors with
W > W can use PN. A technology choice criterion, however, differs from the case of no

macro shock. Since the agents do not anticipate the macro shock, the critical level of
wealth after the macro shock increases to W S as in (2.1); therefore now debtors with

W > W 3 choose a high technology (p) consistent with the new (ICSd) constraint at date 1.

To sum up, the macro shock separates the two criteria of wealth for decisions of agents.

Lemma 5. When a negative macro shock occurs, the critical wealth W S for promissory

notes increases.

54

Proof: Derivative of W3 with respect to RS has negative sign:

 

2c-___.<p—w.eanp~.(p—nminila...

air" 7 7 6R3 72 41) 6R5
W" 7“

4.2. Pure bills of exchange economy

As in PN case. because of the macro shock we have following adjusted (IRsb) constraint:
p-de2(p—Ap)-de+b. (IC'Sd)

(P-03£)'Rsc 24(35) (IRSh)

From Lemma 2. a new critical wealth for BE is

W5 _=_ amps—i]. (2.2)
.6 AP

Lemma 6. When a negative macro shock occurs, the critical wealth W s for bills of

exchange increases.

Proof: Derivative of W s with respect to R5 has negative sign:

£=_W+R_56035 +(p_ZBE)|:RS _i]_ag_<0.;_1
6R5 ,6 fl ates ,6 41) gig:

55

4.3. Preference ofcentral bank
First I consider credit risk from the production technology p, and then consider market
risk from the macro shock. From the lender’s point of view we have the following default

rate 5 for each payment instrument without a macro shock:

WZW' fPN
5=l-p{ 1ncaseo (2.3)

W 2 W in case ofBE

Notice that without a macro shock, both payment instruments incur identical default rates.
even though the range of wealth is different from each other.
One corollary from the default rate from the production technology (2.3) is that if

the wealth level is lower than the critical levels, the default rate becomes higher such as

1-(p-Ap).

This corollary helps us to consider market risk after a macro shock, since debtors have a

new critical level of wealth after the macro shock which determines debtors' production

technology choice.

Figure 2.4: Critical level of wealth and macro shock (PN case)

No debtors can Because of macro shock, The debtors keep
access to PN the debtors are choosing choosing p.

(p — Ap) instead of p.

 

 

 

56

From Lemma 5 and Lemma 6, we have two “grey areas” of wealth level for
debtors. W ~ W s for PN and W ~ W5 for BE. Since there is a macro shock in the

economy, debtors whose wealth is in the “grey area” change their action from choosing a
good technology (p) to choosing a bad technology (p — Ap) as in the above corollary (See

Figure 2.4).

Lemma 7. When y < B. there is a production technology 2 such that if p > p , the critical

wealth W3 for promissory notes increases more than W s for bills of exchange when a
negative macro shock occurs.

Proof: See the Appendix. 1‘

Proposition. When R < R and p > p, market risk from bills of exchange is lower than
promissory notes in the case of a macro shock.
Proof: From Lemma 7. the critical wealth W s for promissory notes increases more than

W s for bills of exchange when a negative macro shock occurs. This means that the “grey

area” for promissory notes (W ~ W s ) is wider than that of bills of exchange (W ~ W s ). 13

Proposition provides a rationale for the “payment instrument policy” of the BOK to

support bills of exchange.

57

4.4. Payment instrument policy

I start by studying the equilibrium prices of promissory notes (y) and bills of exchange
(B). In the absence of monitoring and settlement cost, the gross interest rate on PN at
equilibrium is obtained such that

(1—6)-7=l.

where the risk free gross interest rate is 1. I introduce a settlement cost which occurs in
the process of payment and settlement. This cost contains all costs related to payment and
settlement such as costs from reserve requirements, from non-synchronization of receipts
and payments, and admission or membership fees for payment systems. Let 0 be a
settlement cost. Since the expected return on the promissory notes fully covers the cost of

the credit as above, the expected rate of return on PN at equilibrium becomes
(1_6PN)°7=1+0-P1\'° (2.4)
Assuming perfect competition between banks, similarly, the rate of return on bills of

exchange at equilibrium is determined by the break-even condition

(l—6BE)',B=1+C+O'BE (2.5)

For the time being. I assume that the settlement costs of PN and BE are identical. This

assumption will be relaxed when the payment instrument policy is implemented.

Lemma 8. When agents do not anticipate a macro shock, y < B.

Proof: When agents do not anticipate macro shock, both payment instruments are priced

by the default rates which are determined by debtors’ moral hazard. According to (2.3).

58

both payment instruments have identical default rates. The remaining is straightforward.

Now let me discuss the payment instrument policy tool of the BOK. I argue that
subsidy for BE is better than tax to PN in terms of credit risk control. The critical level of

wealth changes as the payment instrument policy is implemented. Taxing PN makes the

. . —. . 6W . . . - .
cr1t1cal wealth W 1ncrease, srnce .6— >0. Thrs makes the crrtrcal wealth W S 1ncreases
7

from the level which would otherwise reach after the macro shock. This makes the grey
area (W ~ W S ) widen and therefore credit risk increase.

Figure 2.5: Payment instrument policies

A. Tax to promissory notes

 

 

l i

I I
I .............. " ‘73

Macro shock
I l l
T I r
it" """ '5 _’ """"""" '5 is
Tax to PN” Macro shock 11

B. Subsidy to bills of exchange

 

 

I l
I I
I1 """"""" '> s
’ ...... Macro shock
" ‘a
l 1 r
I I T
l! "“’Ii’ 11.5
Subsidy to BE

59

. W .
Subsidizing BE makes the critical wealth W decrease, srnce 53:— > 0. Thrs makes

the critical wealth W5 decrease from the level which would otherwise reach after the

macro shock. This makes the grey area (W ~ W s ) narrow and therefore credit risk does
not increase.

Until now the economy is assumed to consist of homogenous agents who do not
anticipate macroeconomic shocks. If this assumption is relaxed so that a certain
proportion of the agents anticipate this shock at date 0, then the model has a different

implication on market risks: the greater the proportion of agents that anticipate the shock,
the lower market risk. Let q 6 [0,1] be the portion of the agents that anticipate the shock.

The economy has weighted critical wealth for lending standard for each types of

consumption credit:
WW =(1—q)-W+q-W*"

and
VZW=(1-q)-PZ+9°VKS

where W w and W "' are weighted critical wealth of debtors for PN and weighted critical

wealth for BE. respectively. As agents anticipate the macro shock, the economy has new
critical levels of wealth for payment instruments. Now debtors of W > W” can access PN.

and debtors of W > W W can use BE. This change makes the grey area (W "'~ W S for PN

or W W ~ W S for BE) narrow and therefore lowers the market risk.

60

Figure 2.6: Macro shock and anticipation

 

, " " Macro shock N“
I 1 1
K 4 '
............. .’
_ \ _-—- /_ _
I" . , , 11"“ 11"
Antrcrpatron

5. Concluding remarks

This chapter has studied the role of the government on private money without legal
restriction concerning usage of certain types of private money under macroeconomic
shocks.

Payment systems are defined as organizations, Operating procedures, and
information and communications systems used to initiate and transmit payment messages
from payer to payee and to settle payments (Balino et al. 1994). Since there occurs some
cost in the process of payment and settlement of private monies, governments can
manage market risk of private monies by adjusting their cost in payment and settlement.
Payment instruments connect payer and payee from the beginning of a transaction.
Different from other central bank policesg, the effect of the “payment instrument policy”
falls directly on the agents in a sense that it alters the initial asset choice problem of the
individual.

By assumption, the private benefit of choosing a bad technology is higher under
promissory notes as compared to bills of exchange; therefore, given a set of parameters,
the range of values of wealth such that lending and good technology choice occurs is

higher under bills of exchange than promissory notes. If we introduce a negative macro

 

9 For example, monetary policy (open market operations) changes distribution of assets in the economy,
and therefore it affects the economy as a whole not the individual agent.

61

shock such that the incentive to choose the good technology goes down, this shock is
going to have a bigger impact in an economy with promissory notes.

Since promissory notes have a direct relationship between debtors and creditors.
one may think that the use of promissory notes has a contagion effect”). However, the
bottom line of the problem with promissory notes is shown to be moral hazard. Intrinsic
market risk of promissory notes is higher than bills of exchange in the presence of
macroeconomic shocks. and this property of promissory notes may cause collapse of
payment system. This is a justification for a policy which supports usage of bills of
exchange. The model also shows that a subsidy for bills of exchange is better than a tax
on promissory notes. The “payment instrument policy” of the BOK can be justified by
these results.

Another contribution is that this chapter provides a simple model for policy
analysis. This partial equilibrium model is generally said to be too primitive to be used
for policy analysis (Holmstrom and Tirole 1997); but if we care about risks alone, the
model provides a framework which is sufficient for the analysis of risk management.

There are too many exogenous variables in the model and too few choices
available to the agents. If we relax some of these relationships, one can find an interesting
implication: what would happen if C and b, instead of being independent from each other.
were negatively correlated? This seems like a natural relation if one considers a scenario
where monitoring is an endogenous variable, i.e., we would expect that an increase in the

monitoring effort would lead to both an increase in the cost of monitoring and a reduction

 

10 Since trade credit also has a direct relationship between debtors and creditors, the use of trade credit
causes shocks to propagate in the economy [Refer to Kiyotaki and Moore (1997) or Boissay (2006)]. In
addition to that, the BOK mentioned this contagion effect (or in their term ‘chain of defaults”) of
promissory notes as one reason to introduce the “Corporate Procurement Loan” (BOK, 2001).

62

in the private benefit of the bad technology. As a result, the economy has a lower critical
level of wealth W , and therefore more agents who are relatively poorer can use payment
instruments.

One can extend this model in several ways: one can consider the moral hazard
problem committed by the banks. In relation to this, one may extend the model by

considering the size of capital of the banks which may limit the size of lending.

63

Appendix

A.I. Proof of Lemma 3
— p B p b . — . .
We have W .=_ —— -— and W a E —— -— . Subtracting W from W gives.
7 AP 16 Ar

W—W = E—£[R-—§-]—E+£[ -—b—].
‘ 7 Ap [3 417

which is simplified

=fi{wR+(B-fl—b-y)j;}.

- +

If (y—B)R+(B-B—b~y)A—lp->0 or

(B-B-b-y)_1_
(fl-r) Ap

 

R< R.

wehave W—W>0.Li

A.2. Proof of Lemma 4
I calculate the default rate after a macro shock, then prove Lemma 4. Let w be the weight
of the “grey area” with respect to the “accessible area” in which the agent can use a

payment instrument. We have

 

Gum—01W)
WPN = — -
l-G(W)
Total default rate is
[1 —(p—Ap)]-WPN +(1- p)-(I — pr) (2.A.6)

64

The first term is default rate from “grey area”, and the second term is default rate from

remaining area (= “accessible area” — “grey area”). Rearranging (2.A.6), we have

(I —p)+Ap'WPJ\' (2A.7)

(l — p) is default rate from production technology (or moral hazard), and Ap-wPN is

default rate from the macro shock. Therefore, market risk from the macro shock is

nP/V = AP ' WPN (2"A8)

Derivative of market risk with respect to the macro shock R5 is

50px ___ A .all’riv .5WS

 

 

 

 

 

 

 

 

an“ 51173 6R3
:APPWPN . _(P-UPN)+§:5’7PN +(P-UPN)[Rs__{3_] 57
6W3 7 7 6R3 72 AP 5R3
7s . _ J . s _
since .:_(P ’ll.\)+_li_a’lP/’V+(P ZPN)|:Rs__B:_]_a_7_.
6R" 7 7 6R5 7 AP 6R5
- - 577M
Solvrng the above equation for —;—- , we have
.flyflll
arm z are , _(P—UPN)+(P—77P.~')[Rs_£]5_7
. 2
8R? ]_ .yfl.£ lyl . 7 v A1726:
(3W5 7 + + ‘

6w » . . . . . .
We know that 3% has posrtrve s1gn from the definition of weight wPN. The

677px:

only thing that decides the sign of ‘
6R5

is the sign of the

65

al_L_p\ .‘R_S If] ——--—-—Ap (2pr RS

BWS 7 6W3 7

denominator, l—Ap—— —>0, then it will suffice. Let me

specify the distribution of wealth. Wealth is unifome distributed with lower limit of

zero and upper limit of N. The cumulative distribution function of wealth, G(-) is defined

as

W
G(W) _ W

for 0 S W S N. The weight, therefore, wPN is

w '_ G(W5)—G(W) _ W5 —W
P” 1—G(W) N—W ’

 

 

 

 

and
6W5 N —W '
The denominator is
,) r S S . S
]_ .auf"..8__1_Ap.l._R_=]_ AP R (2.A.9)
5W8 7 -W 7

B
(N—E)+ 11-—w
y pl Apl

If the private benefit B is not too big, or

Ap-l:(l+0')(N—E)+p2-R—p-Ap°RS]

B<
p. 1

 

5)
we have positive sign. This is straightforward from (2.A.9). We have

7(N-E)+p[R-—€-]-Ap-RS
Ap >0

 

B
N—E R———e—
7( )+p[ , l

01'

66

S
B<ll(N—E)+pR ApRAp.
p

 

. . . . 1+ 0 . . . .
Smce the mrnrmum of y 18 , we have posrtrve srgn of the denomrnator when

-1—:g-(N—E)+p-R—Ap-Rs

B< p p -Ap..'1

 

A.3. Proof of Lemma 7
Since we consider a negative macro shock, both of the derivatives of critical values of
wealth are negative. I compare absolute value of derivatives of wealth with respect to

macro shock.

 

 

 

 

 

I‘m—[s _la’i/sl:(p—llPN)_R_Sa”PN _(p_7lP.\')—Rs__5__ 57
laRSI MRI 7 7 6R3 72 L Ap_ 6R"

 

 

_(P’085)+R_Sa’73£+(P-773£) Rs__b_- 513
s 2 s
fl ,6 6R fl AP‘BR

 

 

Rearranging this and solving for p, we have

(p—rgpsipfl,‘ 57 ]+<p—ZBE)[_J,+A2 6.6 ]+R.[_lamw +i5'735]
y- airs p 6R3 7 6R5 4 6R3

 

 

where Al = R“ —£ and A2 = R5 --:;.

AP

. 6 . 1 6 .
Since BIZ—5‘ < 0 and 7:133:5— < 0 are very small, and we have —1— > —1— from Lemma 8; the
‘ 7

6W5 6W5
ldRs i—I 6R5 l>0. when

 

last term has positive sign. We have

67

 

 

‘ 2 ”l 2' I ”ll
{:8 ([7 ’lP.\)|:7 laRs 7 (P ”35 fl zaRs

2 2

 

7

01'
a a “
flzl:7-A15%:IUPN’72[,B’A2 a": 77313
a - 04: ( ). (2.A.10)
2 7 2
—A __ _ —A .._.._
16 [7 16”] 7 [.3 26R,

 

111
1‘3

p>

 

d

Since the maximum weights of PN and BE are one from (2.A.8) and its corollary for 773,;

a sufficient probability p is Ap , or

 

 

 

 

 

9 6 0
{fl”[7-Ara—;I;:|—72[fl’42$]}419
p> . 67? 2- 6,6 Zp.D (2.A.ll)
2 —A — —A.
16 [7 'aRK 7 -fl “5193]

68

REFERENCES

Aiyagari, S.; N. Wallace (1997) “Government Transaction Policy, the Medium of
exchange, and Welfare,” Journal of Economic Theory Vol.74, pp.1-18

Angell, W. (1991) “A proposal to rely on market interest rates on intraday funds to
reduce payment system risk,” Governing Banking ’s Future, edited by C. England.
ch. 1 0. Boston: Kluwer.

Balino, T.; J. Dhawan; V. Sundararajan (1994) "Payments Systems Reforms and
Monetary Policy in Emerging Market Economics in Central and Eastern Europe," IMF
Staff Papers, Vol. 41 (September), pp. 383-410. (an abridged version available at
http://www.worldbank.org/fandd/english/0396/articles/080396.htm)

The Bank of Korea (2001), Annual Report 2000, pp. 26-7 (available at
hm)://www.bok.or.kr/content/old/attach/OOOOOl 79/0308522000annual_report.pdf )

Boissay, F. (2006) “Credit chains and propagation of financial distress”. European
Central Bank Working Paper No. 573.

Burkart, M.; T. Ellingsen (2004) “In-kind finance: a theory of trade credit”. American
Economic Review. Vol. 94 (3), pp. 569-90.

Faulhaber, G.; A. Phillips; A. Santomero (1990) “Payment risk, network risk, and the role
of the Fed” in The US. payment system: efficiency. risk and the role of the Federal
Reserve, edited by D. Humphrey, p.197-213. Boston: Kluwer

Friedman, M. (1960) A Programfor Monetary Stability, Fordham University Press. New

York.

69

Green, E. (2004) “Challenges for research in payments,” Conference: The economics of
payments, Federal Reserve Bank of Atlanta (available at
http://www.frbatlanta.org/news/CONFEREN/epfcontZ004/green.pdf)

Hahm. J .; F. Mishkin (2000) “Causes of Korean financial crisis: lessons for policy”,
NBER Working Paper No. 7483 (available at http://www.nber.org/paper/w7483)

Holmstrom, B.; J. Tirole (1997) “Financial intermediation, loanable funds, and the real
sector”, Quarterly Journal of Economics, Vol. 112 (3), pp.663-91.

Holmstrom, B.; J. Tirole (1998) “Private and public supply of liquidity”, Journal of
Political Economy, Vol. 106 (I), pp.1-40.

Kahn. C.; W. Roberds (2001) “Real-time gross settlement and the costs of immediacy”,
Journal ofMonetary Economics, Vol.47, pp.299-319

Kiyotaki, N.; J. Moore (1997): “Credit Chains", Mimeo, London School of Economics.

Lacker, J. (1997) “Clearing. settlement and monetary policy”, Journal of Monetary
Economics, Vol.40, pp.347-8 1.

Martin, A. (2004) “Optimal pricing of intraday liquidity”, Journal of Monetary
Economics. Vol.51 pp.401-24

Mills, D. C. (2004a) “Mechanism design and the role of enforcement in Freeman’s model
of payments”. Review of Economic Dynamics, Vol.7, pp.219-36.

(2004b) “Alternative central bank credit policies for liquidity provision in a
model of payments”, mimeo

Peterson, M.; R. Rajan (1997) “Trade credit: theories and evidence”, The Review of

Financial Studies, Vol. 10 (3), pp. 661-91.

70

Prescott. E.; J. Weinberg (2003) “Incentives, communication, and payment instruments”.
Journal of Monetary Economics, Vol. 50, pp.433—54

Radelet, S.; J. Sachs (1998) “The onset of East Asian financial crisis”, NBER Working
paper No. 6680 (available at http://www.nber.org/papers/w6680)

Rochet, J. (2004) “Macroeconomic shocks and banking supervision”, Mimeo (available at
http://idei.fr/doc/wp/2004/macroshocks.pdi)

Rochet, J.; J. Tirole (1996) “Controlling risk in payment system”, Journal of Money.
Credit. and Banking, Vol.28(4) Part 2, pp.832-62

Sargent, T.; N. Wallace (1982) “The Real-Bills Doctrine versus the Quantity Theory: A
Reconsideration”, Journal of Political Economy, Vol.90, pp. 1 212-36.

Williamson, S. (1999) “Private money”, Journal of Money, Credit, and Banking,
Vol.31(3) Part 2, pp.469-91

__ (2002) “Private money and counterfeiting”, Federal Reserve Bank of Richmond
Economic Quarterly, Vol.88/3 Summer, pp.37-75

__ (2003) “Payments systems and monetary policy”, Journal of Monetary Economics,

Vol.50, pp.475-95

71

Chapter 3
Production technology and private

information in a search theoretic model

1. Introduction

Risk management is a core element of payment policy. In order to implement an effectual
policy. it is necessary to understand properties of the risks. As former Federal Reserve
Chairman Greenspan mentioned in his 2004 AEA meeting address', it is important to
understand the many sources of risk and uncertainty. In Chapter 2, market risk from
moral hazard of debtors” choice of production technology plays an important role in the
sensitivity of the payment system when there is a macroeconomic shock.

Since Akerlof (1970), qualitative uncertainty in the trading sector has been
studied in repeated game setups. With a search theoretic approach, Williamson and
Wright (1994, hereafter WW) develop a model on the same topic. This has inspired other
research using the search theoretic approach to an economy with private informationz.
They all focus on informational frictions which arise in the trading sector. There are also

various types of uncertainty in production processes. For example, farmers sow good

 

' “As a consequence. the conduct of monetary policy has come to involve, at its core, crucial elements of
risk management. This conceptual framework emphasizes understanding as much as possible the many
sources of risk and uncertainty that policymakers face, quantifying those risks when possible, and assessing
the costs associated with each of the risks.”

3 Kim (1996) studies the endogenous information structure with fixed acquisition cost; Trejos (1997)
studies the effects of monetary matters on the incentives to produce high-quality output; and Trejos (1999)
studies informational friction economy with divisible goods. Some other related research includes Cuadras-
Morato (1994) and Li (1995, 1998).

72

quality seeds, and grow them with care, but unanticipated events, such as drought, flood,
or disease affects farm production. Or, because there are so many constituent parts, there
is a multitude of ways in which a car can be defective. The first production uncertainty
can be characterized as bad luck, and the latter as a limitation of technology. Both have
the common feature that agents cannot completely control the outcome. This gives rise to
quality control to reduce the number of defective products produced, and warranties to
protect consumers from suffering should they purchase, unwittingly, a defective product.

I study the role of production uncertainty in a search theoretic model with private
information. A search theoretic model impedes any possibility to compensate or punish a
willful cheat since one will never meet the same trading partner twice. As a result of this
property, the equilibrium trading pattern differs from that of a Walrasian market. Search
models amplify the effect of frictions and uncertainty in the sense that they return a
narrower and smaller range of parameter values in which an equilibrium exists than in a
repeated game setup3.

There are several search-theoretic papers dealing with the production sector.
although not with private information. Diamond (1982) designs a search model with a
production sector in which production costs are randomly drawn from a given
distribution. Since his model generates multiple steady state equilibria, even without lags
in the ability of the government to affect private decisions, an economic policy for
stabilization may not have the power to achieve instantaneously its goal. Wallace and
Zhou (1997) and Boyarchenko (2000) deal with heterogeneous agents who have different
production costs or productivity. Both papers show that higher productivity agents

dominate trade. The findings in Diamond (1982) depend on the assumption of an

 

3 See the Appendix (A. I)

73

increasing returns to scale meeting technology. Johri (1999) relaxes this assumption and
generates multiple equilibria as Diamond did. Johri uses the production sector to show
the different equilibrium price levels are associated with different levels of the
reservation production cost.

I extend the WW model by introducing exogenous uncertainty in the production
process. The production uncertainty implies that a product or process will not meet the
quality requirements, therefore it can be referred as probability of defect. Agents in the
model choose between good and bad production technologies. Because of production
uncertainty, agents may get a bad product even under a good production technology.
Different types of uncertainty generate different effects: the model shows that there exist
more equilibria in the production uncertainty case than in the no production uncertainty
case. Comparative statics show that an increase of production uncertainty increases
possible returns from choosing the bad technology. These findings generate important
implications for designing policies to alleviate risk. When the economy experiences a
deterioration of technology such as a macroeconomic shock, it amplifies the tendency of
the economy to choose the bad technology. This may lead the economy to autarchy.

This chapter is organized as follows. Section 2 describes the basic model, section
3 analyzes equilibria, section 4 analyzes welfare and section 5 discusses comparative

statics. Section 6 summarizes our findings and concludes. Lengthy proofs are in the

appendix.

74

2. Basic Model

This model is essentially the same as in WW. The economy is populated by a continuum
of homogeneous, infinitely-lived agents distributed on the unit interval. There are two
objects: a good quality commodity and a bad quality commodity. The good commodity
and the bad commodity can be produced by all agents. All commodities are indivisible.
freely disposable and storable at zero cost, but only one unit at a time. There are two
types of production processes: a good process which generates a good quality commodity.
and a bad process which generates a bad quality commodity. Because of an imperfection
in the production technology there is a defect in the good process so agents get the quality

that they intended to produce with probability 6,, e [0. 1]. This probability is independent

across agents. The disutility of producing one unit of a good under the good process
equals y > 0. Given the production technology, this implies that the average unit cost of
production4 is y / Op. The cost of using the bad process is equal to zero. I assume that an
agent cannot consume his own produced commodity and has full information on the
quality of his commodity after production. Consumption of a good commodity yields
utility U > 0 and consumption of a bad commodity yields zero utility. All agents are risk
neutral.

Each period, agents are pairwise matched under a uniform random matching
technology. Trade takes place if and only if mutually beneficial. Moreover, trade always
involves a one-for-one swap of inventories since all types of objects are indivisible. So I

do not consider the prices of the objects in this paper. An agent recognizes the quality of

 

4 Because of the change of production cost, one may say the distribution of production cost can return the
same result as the present paper. This is true when we consider type A, B and D equilibria. Under the
distribution of the production cost y, we cannot get type E equilibrium as in the present paper.

75

a good produced by another agent with probability 66m, 1]. This probability is

independent across commodity traders when they meet. Agents do not know whether
agents recognize their commodities and do not know anything about other agents”
histories. Once a trade decision is made, it is irreversible. There are neither private credit
arrangements nor promises, since agents who meet in one period will meet again in
another period with probability zero. Let p denote the proportion of traders holding good

output. The following summarizes the sequence of events within a period:

1. Agents meet pairwise at random. They check the quality of the other trader’s
commodity. Traders swap their inventories only if mutually agreeable.

2. After trade, agents separate and the types of commodities are revealed. Agents
decide to consume, dispose or store the commodity. If they consume or dispose of
the commodity, they choose the production process for the next period.

3. If the good process is chosen, a good output obtains with probability 0,. Under a

bad process. only bad commodities are produced.

I look for stationary Nash equilibria in which the meeting probabilities are
summarized by p, the fraction of goods holders with a good output. Moreover, when
making his decision, an agent takes as given the probabilities with which he believes

other traders with good commodities will accept commodities of unrecognized quality.

76

3. Equilibrium

In this chapter I restrict attention to non-monetary (or barter) equilibria where at least
some good commodities are produced and consumed. 1 denote these equilibria as active
equilibria. I assume that 0 < 1, and 0p < 1, so that agents are not able to recognize the
quality of the other potential trader’s commodity in some meetings and get a quality of
commodity different from their original intention in good production processes. An agent
with a bad commodity is willing to trade with any agent at every possibility because at
worst he gets another bad commodity, however an agent with a good commodity is not
willing to do so. Agents decide whether to choose the good or bad production process.
and whether to accept or reject a commodity of unrecognized quality when they are
currently holding a good commodity themselves.

Let 2 denote the probability with which an agent believes that other traders with
good commodities will accept commodities of unrecognized quality, and let 0 be an

individual’s best response.

Lemma 1. 0 < p S 19,, < 1 in any active equilibrium.

Proof. Given the production technology 0p, we cannot have p > 0p. When every agent

tries to produce good commodities then we have p = 0,,. Moreover, p = 0 is not an active

equilibrium. i

Let V,- denote the value function at the end of period for an agent holding object j,

where j = g. b denotes a good commodity and a bad commodity respectively. Let r > 0 be

77

the discount rate. Let W = max{—y + HpVg +(1—6p)V,,, Vb} represent the value function for a

producer with nothing in inventory, who is deciding which production process to use.
The first element of W is when agents choose a good production process. Usually this
effort leads agents to a good outcome, but there is some noise in the process. So, agents
would get low quality output with probability 1 — 0... even though they made an effort in

terms of y units of disutility.
An individual’s best response problem is described as follows:

rVg =6pAIV(U+W —-Vg)+(1—6)maxo*[pAN(U+W—Vg)+(1—p)(W—Vg)] (3.1)
0’

er = p(l — e)z(U + W — Vb) (3.2)

where A.- EB+(l-19)E is the probability that an agent with a good commodity is

willing to trade. The first term on the right hand side of (3.1) is the probability that the
individual meets someone with good output he can identify (0p) multiplied by the

probability that his partner is willing to trade (AN) times the gain from trading.
U + W — Vg. The second term is the probability that the individual meets someone he can

not identify, 1—19 . multiplied by the net expected gain from choosing the acceptance

probability 0. Expression (3.2) can be explained in a similar way. By Lemma 1, p > 0 and

W = —y + BpVg +(1— 6pm, 2 Vb in any active equilibrium. Hence, the difference between

the value functions from the good process and the bad process is greater than or equal to

the expected cost of a unit of good quality good. We have:

Vg —V,, 257—. (3.3)
P

78

Axiom Vg ~- I"), > 0.

Definition. An active symmetric Nash equilibrium is a set (Vg. V1,, p, X) such that:
(i) —y +0pl’g + (1 —6p)V,, > Vb impliesp = 6,,

andp < 19,, implies —y +9pVg +(1—19p)Vb = Vb

(ii) given p. o = 2. must solve the maximization problem in (3.1).

According to the range of p and 22, there are potentially five types of equilibria that may
occur. Table 3.1 shows types of equilibria by p and Z. Plugging
W = —y + BPI/g + (l —6P)V,, into (3.1) and (3.2), we have

r71. =9PA.\'lU—7—(1-9pXVg —Vb)1

+(l—19)rmxo-{pAN[U—y—(1-19p)(Vg _I/D)]+(1—p)[—7_(l_ngVg _lbll} (3,1')
0

and

er=p(1—6)2[U—7+6p(Vg—Vb)]. (3.2')

Now I will check the existence of equilibria. The procedure for checking the
existence of each equilibrium is as follows: first I plug the parameters of Table 3.1. into
the value functions (31') and (3.2'), and then solve the value functions for parameters.
and then check restrictions for parameters in a way that conditions for each equilibrium
are satisfied. Consider type A equilibrium. Type A is the case of every agent choosing the
good production process, and every agent accepts unrecognized commodity: p = 0p and Z
= 1. After plugging these two parameters into the value functions, I check the condition

under which choosing a good production process is a best response, or i.e. choosing a

79

good production process is unimprovable by a one-shot deviation. One can show that

 

condition (3.3) holds if and only if 0 Z 0. where 6] = 19 (1552:”), )+ . Type B, D and E
‘7 7
P P

equilibria can be checked with the same procedure as described above for type A

equrlrbrrum. Proposrt1on 1 summarizes the exrstence condrtron for each equrlrbrrum .

Table 3.1: Types of equilibria

 

 

 

 

 

p=9p p6(0.0,.)
2:1 A B
26(O,1) C D
2:0 E -

 

 

 

 

Proposition 1. From the existence condition for each type of equilibrium, each type of
equilibrium has following critical values of 6:

1. Type A equilibriumfor 191(r,6p) S 6
2. Type B equilibriumfor 61(r,6p) < 6 S 612(r,t9p)
3. Type D equilibriumfor 64 (r,t9p) < 19 < 02 (rflp)
4. Type E equilibriumfor 95 (r,t9p) S 6 <1 when 61,, is low,
and for 662(r,6p) S 6 <1 or 650,610) S 6 < 661(r,6p) when 6,, is

high.

Proof. See the Appendix. [I

 

" One may consider type C equilibrium: every agent chooses a good production process, but some agents
would not accept to trade with unrecognized commodity. Unfortunately, however, it is impossible to check
the existence of the equilibrium.

8O

Figure 3.1: Production technology and critical values of 0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(3) Type D equilibrium (4) Type E equilibrium

Proposition 1 shows that types of equilibria depend on critical values of 0, and
those critical values are functions of production technology 0p and discount rate r. Figure

3.1 shows that the relationship between 0 and 0,, with arbitrary values6 of U, y and r. Type
E equilibrium exists 95 (nap) S 6 when production technology is lower than 9; , the

solution to

DEl a r26p2(U — y)2 — 4(1—op)3ropUy,

and exists 662(r,6p)S19 or 65(r,6p)S6<661(r,0p) when production technology is

higher than 6;. Notice that if the production technology is higher than 5,, , then the type

 

‘1useU=2.5,y=0.75 andr=0.05.

81

E equilibrium does not exist any more. WW is a special case of this. Lower boundaries of
each equilibrium have negative slope (except 952 for type E), so one can find a substitute

relation between inspection technology 0 and production technology 0p.

Lemma 2. When the equilibria exist, traders in type A equilibrium hold more good
products than traders in type B. Also, traders in type B equilibrium hold more good

products than traders in type D equilibrium; i. e. p A(= 6,) > p B > p D

Proof. Notice that pJ = 6,, is the highest proportion which the economy can achieve,

therefore pJ(= 6p) > [)3 is trivial. Next, I will show that p3 > pD.

 

We have

(r +1— t9)y

P3 = _—

6l(6lpU — 7)

and
1) = PHI-@2017
{16+ (1 -6>201’ —(1-6)>:D}(6,,U —7>
19—06 9 U - -r

where 20 = ( ( p 7) 7 [See the Appendix (A.2)]. The ratio p3 to pr)

 

(1— 6110 — t9 +1)(¢9,,U - 7) + 71
is

Q = (r +1 —o)[o(opu —- y)2 + 719,11]
p0 o(o,,u — 7)[y(t9—r) + 0,110 +r -19)] '

 

f £13— minus the denominator of £5- gives
P0 P0

The numerator 0

70/6 + BpU + r6pU - 290PU).

82

From 2. and 3. of Proposition 1 and y > 0 , one can show 761+6pU +rt9pU —266lpU > 0.

When the private information problem is severe under perfect production
technology (0p = 1; WW), WW argue that type D has the greatest chance of surviving;
because 2 < 1 imposes the greatest discipline on producers of the bad commodity. WW
explain that the existence of the type D equilibrium depends on the willingness of agents
with good output sometimes to give up a trade for something that they cannot recognize
and wait to trade later for something that they can recognize. They emphasize trade
process and the role of discount rate r for this willingness to wait. I argue that production

cost 7 also be considered as one factor for this willingness to wait. When comparing the

critical regions for each type equilibrium, one can find HIIH -l < 614|6 -1 . which means
I)“ I)“

type A or B equilibria exist under more severe private information problem than type D.
Figure 3.2 shows this possibility. For example, there are two types of water in the
economy; filtered water (good output) and tap water (bad output). The production cost of
filtered water is very small compared with its utility. In this economy, if there are no
other frictions, agents would accept the trade even though they cannot recognize the
quality. In this example type A or type B equilibrium survives under more severe private
information conditions. For another example, there are two types of cars in the economy;
plums (good output) and lemons (bad output). I define the production cost of plums is
sum of maintenance cost for a car, such as timely change of lube, brake and transmission,

etc.. and careful driving; In this case production cost is high compared with its utility. In

83

this economy more agents would not accept the trade when they cannot recognize the

quality. In this example type D has more possibility to survive.

Figure 3.2: A graph of 0. - 0,, when 0,, = l (U = 2.5, r = 0.05)

 

 

9. — 9,
0 1 1 y
0.4 0.6
4. Welfare Analysis

Let Z. denote social welfare in type i equilibrium, defined as the expected utility of the

representative agent in the initial period. Then social welfare Z. is measured by

z, = (1-p,)V,, +p,-(—:97—+Vg). (3.4)

P
As Li (1998) argues welfare is determined by the frequency of trade and the probability
of getting high-quality commodities from those transactions. The frequency of trade
depends on, among other things, 2. When we plug in the values in Table 3.1 to equation

(3.4), we have following social welfares

rZA = 6p(1—6)(U—7)—r7+t9p[(1—9)6lp +r]-K14

84

 

a eu- - 1—9 1—e
whereKJ(EV‘l—V‘4)= P( 7)( X ””2 [EA/Kt], (3.5)
g " r+(1-e)(1—ep)+epe(1-op)+op (r—e) DKA

= 7(1+r—6)(l—19)U

rZB
6(6pU —7)

 

yU[-r7 + r76 — 792 — r919pU + 920,11]

rZ
D 726 + yopu - 2yoopu + 6119,,2U2

 

rozagw—y)
2 .
r +6 19p(1 —6lp)

 

rZE =—r}/+

Proposition 2. When utility U is sufficiently large relative to the average production cost
()1 / 61p). and the discount rate r is not too high, then welfare Z4 and welfare Z3 are

decreasing in t9. welfare Z1) and welfare Z5, however, are increasing in 0.

Proof. See the Appendix (A.3). Ll

Figure 3.3: Social welfare by types of equilibria (U = 2.5, y = 0.75, 0,, = 0.75 and r = 0.05)

rZ

r Z 8 \

 

 

85

Social welfare at type A equilibrium is decreasing in 0. This comes from the fact that
payoffs of agents who have bad output decreases because it takes more time to conduct a
successful trade. Social welfare at type B equilibrium is decreasing in 0. As WW point
out, this results from the fact that the fraction of high quality goods, p3, is decreasing in 0.

as can be seen in rZ 13- Figure 3.3 represents the properties of social welfare Z, using

given but arbitrary values of U, y, 0p, and r. One can verify ZA > Z3 > ZD > ZE in the
coexistence region (shaded area in Figure 3.3). I will show this in the Appendix (A.4).
Type D and type E equilibria have a couple of advantages. First, as inspection
technology improves social welfare at these equilibria increases, whereas welfare at other
types of equilibria decreases. Second, type D equilibrium is the most feasible choice
when agents choose their trading strategy. 22 < 1 imposes a discipline on those who
choose a bad production process in a way that the agent takes more time to have a
successful trade. In type A and type B equilibria, by accepting every unrecognized

commodity can give society higher welfare, but it makes the lemons problem more severe.

5. Comparative statics

In this section I analyze comparative statics with a production technology shock and a

change in agents’ patience.

I start with the change of production technology. A type D equilibrium has some

interesting properties. so I discuss type D separately.

5.1. Production technology and type A, B and E equilibria. When the utility U is

sufficiently large relative to the average production cost (y / 0p), an improvement in

86

production technology extends the critical regions for equilibrium A, B and E. This is
straightforward from the result of derivatives on critical values of 0 in Proposition 1 with

respect to 0,,. See the Appendix (A5) for detail.

Proposition 3. When the utility U is sufficiently large relative to the average production
cost ()1 / 6p), an improvement in the production technology increases the social welfare at

type A, B. E equilibria.

Proof. See the Appendix (A6). I]

5.2. Production technology and type D equilibrium. I consider the critical region first. I
keep the condition that the utility U is sufficiently large relative to the average production
cost (7 / 0p). We have following relation between the derivatives of the critical values of 0

for type D equilibrium with respect to 0p,

 

662(r,19p) 664(r,6,_,) _ 7U [r(26pU - 7)2 — (l + r)(6,,U - 7)Jr(6pU - 7)[47 + r(6’,,U — 7H]
69 66 (MPH—yizwpu—7)./r16,.U—7)147+r09pu-7)1

 

 

P P

Table 3.2: Production technology interval for type D equilibrium when production technology
improves

 

 

 

 

 

 

 

r=0.01 r=0.05 r=0.1
QP 9p 9p 977 QP 9p
y = 0.1 0.0488 1( 1.0415) 0.0721 0.1687 n.a. n.a.
y = 0.5 0.2437 l(5.2076) 0.3606 0.8436 n.a. n.a.
y = 0.75 0.3656 H78] 14) 0.5408 l(l.265) n.a. n.a.
y = 1.0 0.4875 1( 10.415) 0.7211 1(1.687) n.a. n.a.
y = 1.5 0.7312 1( 15.623) n.a. n.a. n.a. n.a.

 

 

 

 

 

 

 

n.a. means there is no production technology between 0 and l to reduce the existence region of type D.

87

We can verify from Table 3.2 the value depends on the initial production technology
level [Figure 3.4 (a) uses U = 2.5, y = 0.5 and r = 0.05, and show that when the initial
production technology is in (0.36, 0.84), the derivative of critical values of 0 for type D
has a negative sign; therefore the existence region for type D shrinks as the production
technology improves].

Now I consider social welfare at type D. The derivative of welfare at type D
equilibrium is

agjp) = yU2[r72 —3r726+ 72192 ”7292 _7293 +206ng _2r71926pU +2793ng

p
”cleft/2 —6319p2U2]+[;/26+ repo-2709M weft/2 ]2

Look at Figure 3.4 (b) (U = 2.5, y = 0.5, r = 0.05 and 0 = 0.75). It shows that ifthe initial
production technology is high enough, the derivative of welfare at type D has negative
sign; therefore the welfare of type D equilibrium decreases as the production technology

improves.

Figure 3.4: Derivatives of type D equilibrium with respect to 0,.

 

 

 

 

 

6(rZ
6(02 ‘ 94) “552
p
an,
0 r
0.4
0
0.3 0.4 0. 0.9
90
(a) Derivative of critical values of existence region (b) Derivative of social welfare

88

5.3. Discount rate and equilibria. Proposition 4 summarizes the relationship between the
discount rate and the existence of each equilibrium and Proposition 5 deals with the

relationship between the discount rate and social welfare at each equilibrium.

Proposition 4. When the utility U is sufficiently large relative to the average production
cost ()1 / 0p), if the discount rate r increases then the existing regionsfor equilibria shrink

exceptfor the case of type B.

Proof. See the Appendix (A.7). C1

Proposition 5. When the utility U is sufficiently large relative to the average production
cost ()1 / 0p), the social welfares of type B, D and E equilibria decrease in the discount

rate r. The welfare of type A, however, increases in r.

Proof: We will check the sign of derivatives of social welfare with respect to r. In case of

 

 

 

 

 

type A equilibrium.
62:1 _ 9p(1—6)(U—7) _ 6p2(1—6) . 6K4
6r r2 r2 6r
-— + + —2 6 +019 U
where 6K‘4 = ( 7 76’) 76 76 p p2 ) 2 . Since 6K" has negative
or [r+(1—o)(1—op)+opo(1—ep)+ep (1—6)] 67

. az . . . . . .
srgn from (3.5. 6—4 has posrtrve srgn. Other derrvatrves are straight forward;
r

gs _ —7(1—6)2U <
or r20(6pU—7)

 

0.

89

82,, _ -76’U<6,.U—7>
6r 9939+ repo -2y66pU weft/3)

 

 

<0,

82,; _ —626p’(U-7)
6r (Haze, —eze,,2)’-

 

<0.C1

Welfare ZA increases when agents become less patient. This is because type A
equilibrium has no lemons problem, so less patient agents trade more frequently which
helps welfare increase. Intuition for Proposition 5 is that when there is a private
information problem, an economy consisting of less patient agents has lower welfare than
that consists of patient agents. Table 3.3 describes Proposition 4 and 5 when discount rate
r decreases. When agents become patient without a private information problem, critical
regions for equilibria become larger. From the social welfare point of view, not accepting
unrecognized commodity works in this case, therefore the social welfare at the type E
equilibrium increases under this circumstance. When agents become patient with a
private information problem, social welfare increases. The critical region for the type D
equilibrium, however, expands, because there are agents who deny the trade when they

cannot recognize the quality.

Table 3.3: Discount rate and equilibria

 

 

 

 

 

 

 

Welfare
rdecreases
Increase Decrease
Critical Expand D,E A
Region Shrink B -

 

 

 

90

6. Concluding Remarks

This chapter has studied the role of production uncertainty in a search model with private
information. In active equilibria, there exist multiple equilibria by the degree of
production technology. Furthermore, the introduction of another type of noise, production
technology, brings two additional types of equilibria which do not exist in the original
model; type C — an equilibrium where all agents choose good production processes, but
some of them will not accept unrecognized commodity — and type E — an equilibrium
where all agents choose good production processes, but all of them do not accept
unrecognized commodities. These equilibria can be Pareto-ranked. An increase of
production uncertainty makes social welfare decrease and existing regions for type A, B
and E equilibria shrink. Type D — some agents choose bad production processes, and
some others will not accept unrecognized commodity — has a different property. The
model has shown that both welfare and existing region of type D equilibrium depends on
initial production technology. Considering that type D equilibrium is most feasible
equilibrium under private information problem, a macroeconomic shock which would
increase production uncertainty makes agents choose a bad process. When agents become

patient, not accepting an unrecognized commodity is a helpful means for improving

social welfare.

91

APPENDIX

A.l. An infinitely repeated game
In this subsection, 1 study an infinitely repeated game with production uncertainty. In this
game I show that punishment affects current behavior. There are two agents in the
economy; agent A and B. They meet each other to trade their product. I hold the same
cost and utility structure as in the search model, and I assume that they cannot consume
their own product. They can choose either a good production process (GP) in which they
can get a good product with probability of 0,, or a bad production (BP) process in which
they only get a bad output. I suppose that for each time t, the outcomes of the t — 1
preceding choice are observed before the choice at time t. Let B be discount factor which
defined as 1 / (l + r) where r is discount rate. Table 3.A.1 shows the payoffs of each
agent

Let agent A‘s strategy be as follows: (a) He starts with GP, (b) From t =2 and on
if agent B chose GP on the previous time then choose GP, otherwise choose BP. In the
same manner, agent B’s strategy is as follows: (a) He starts with GP, (b) From t =2 and

on if agent A chose GP on the previous time then choose GP, otherwise choose BP.

Table 3.A.1: A table of payoffs

 

 

 

 

 

A \ B GP BP
GP (— Y + 9pU1 “ Y + 0pU) (— 'Ya 9pU)
BP (BpU. — y) (0, O)

 

 

 

Now I consider the present value of the payoff. When an agent chooses to stay GP,
his present value of the payoff is

—y+6pU

(—7+6pU)(1+r+rz+e--i= 1_fl

(3.A.l)

92

Present value of payoff when agent exercises one time deviation from GP is

BPU. (3.A.2)

The given strategies are Nash equilibrium if and only if (3.A.1) Z (3.A.2)
(unimprovability criteria). As a result one can get

HpU—y
7

 

2 r. (3.A.3)

I look at the critical value of type A equilibrium in the search model. Rearranging the

critical value. we have

619p(9pU-7)+7l-7 > r
7

 

When there is no private information problem (0 = l) in the search model, the

equilibrium condition in an infinitely repeated game (3.A.3) is weaker than that in the

search model.

A.2. Proof of Proposition 1
The value functions of type B (Z = 1, 0 < p < 0p) equilibrium are;

,Vg sz_(g.p+1—o)[y+(1-6p)(Vg 'Vb)]»

er = p(l—6)U.

Solving for p with the above value functions, we have p3. From the condition of

0 < p B < 19p. and the condition of o = 2 = 1 being a best response or i.e.

pB[U —7—(1—t9p)(Vg — ’b)]+(1—pB)[-—7—(l—-6lp)(Vg —V,,)] 2 0, we have

93

19pU(6pU—y)+7 ‘ 219 U-7 ‘ '

 

The value functions of type D (0 < 2 < 1, 0 < p < 0,.) equilibrium are
r 2 /
rtg = [6+(1—6)2] p(U —K)—(1—6)2(1—p)(Vg —1b) ,

er =p(1—o)zu.

Solving for p using these value functions, we have pD. We can get the ED from the

condition of o = ED 6 (0.1) being a best response or i.e.

P019+(1‘9)£DI(U - Vg +Vb)=(1-PD)(Vg —Vb) , and then solving 0 < ZD< I. we have

 

(opa — 7)r + \kopu — 7)2r2 +4r7(6pU - 7)
2(6pU — 7)

(r +1)6pU
26pU - 7 '

 

(564)<9<

The value functions of Type E (Z = 0, p = 0,.) equilibrium becomes

1929,,(11—7)
g r+926p(1—op)’

 

Vb=0.

Choosing the good production process (p = 0,.) is a best response if and only if condition

(3.3) holds. When solving this, we have

 

 

r7 =
\/6p(t9pU—7) (_ 65)se. (3.A.4)

From the condition for o = Z = 0 being a best response, or i.e.

196,,[U —7—(1—6p)(Vg —V,,)]+(1—t9p)[—7—(1—6p)(Vg —V,,)] < 0, we have

94

(1—19 )26 UBZ—rB (U—7)t9+ry(l—t9 )>0. (3.A.5)
p p p 7

Before solving the inequality for 0, we need to check its determinant:

DE, = r36p2(u — 7)2 —4(1—6lp)3r6pUy = r6p[r9p(U — y)2 — 4(1—6p)3U;/] .

If DE. < 0 or r9,,(U — 7)2 <4(1—t9p)3 U y (in case the production technology 0p is low.

look Figure 3.A.1), then (3.A.5) always holds, and therefore we need to consider

condition (3.A.4) only. If

DEI > 0 or r6p(U — y)2 > 4(1—6p)3U7 (3.A.6)

(in case the production technology 0p is high), then we have to consider second condition
at the same time. Figure 3.A.l (U = 2.5, y = 1) shows that production technology affects

the sign of determinant.

Figure 3.A.1: Negative determinant

Dre)

 

 

Now, let us solve (3.A.5) for 0 when DE, >1 ; we have

 

19 < r6p(U - 7) — Jr26p2(U - 7)2 -4(1 —19p)3rt9pU7
2(1—op)zopu

 

(566,) (3.A.7)

95

and

 

2 2 2 3
> r6,,(U—y)+\/r 19,, (U—y) 4(1—op) repay
2(1-6p)219pU

 

(E 662) (3.A.8)

When (3.A.6) holds. we have a condition (3.A.4) fl [(3.A.7) U (3.A.8)]; therefore, we
need to compare 05, 0... and 952. It is hard to show analytically, but we can verify 195 < 962 .
We consider U = 2.5, y = 0.5, I, 1.5, r = 0.05. 0.10, 0.15, 0.20 and 0p = 0.5, 0.6, 0.7, 0.8,
0.9. We have 66' < 65 in all cases but when U = 2.5, y = 0.5, r = 0.05, 0p = 0.7. Therefore,

we have 65(r.t9p)<6 when 0,, is low (or DE. < 0), and for 962(r,6lp)S0 or

65(r,6p) S 6 < 96' (rflp) when 0p is high (or DE) > 0). l1

A.3. Proof of Proposition 2

6(rZi)

We need to check sign of . First, differentiating rZA with respect to 0, we have

 

 

 

1 a 1—66 +r NK 'rDK —NK ~DK,'
W24) ._._ep(U—7)—ep2.1<,+ ”K ) P H A A A "1 (3.A.9)
where NKJ' =19J,(U—7)+(1—6p)7>0, DKA'=—1+219p —2op2 <0.
Dividing (3.A.9) by 0,, and rearranging it, we have
-1—t9 r,6,6 U- r,t9,6l
( p)[gU( . p) gy( p)7] (3HAIO)

DKAZ

where g(/.r(r,9.9p) =1—ep +9,"- + 2r —rt9p we," + r2 — 219+4oop 41919,}
3 2 2 2 2 2 3 ’
+209], —2rt9+2r96p+t9 —39 6p+46 6p —26 6p

96

g),(r,0.6’p) = 1+9],2 +2r+2r6p +2r2 -2¢9+209p — 266p2 —2r6+t92 49249,, +29%}.
It is hard to show (3.A.10) has negative sign analytically, but one can show that if
gU(r.6.6p)—g},(r.t9,0p)>0, then (3.A.10) has negative sign. Differentiating rZB with
respect to 0 is straightforward;

6023) _ —yU(1+r-—62)
06 ezwth-y)

 

 

<0

Differentiating rZD with respect to 0, we have

5021)) _ —7U(9,,U - y)(72r - 72.92 — zygpau + zygpazu —6p262U2)
00 0'26 + my - 27‘9ng + 6,30%?

 

It is obvious that the denominator of has positive sign. Rearranging the

6(7'ZD)
66
numerator, we have

—-yU(6pU — y)[—7(296pU — 7r) —(6pU — “292] > 0.
+

Differentiating rZL: with respect to 0, we have

.9025) _ zepzrzaw - y)
66 (r + 9,,92 —9p292)

 

2>0.D (3.A.ll)

A.4. Pareto rank of social welfares ZA, Z3, Zn and la
This part shows Pareto-ranked social welfares in analytic and graphical way. First, I

compare rZA with r23;

97

(—}/— 7r+ 76— y6p6+6p26U)x[6p(1-6)+r]6(6pU— y)+U(1—6)-DK‘4

rZ -rZ =
'4 B [DKA-6(6pU-7)]

 

From 0< p3 <6], . I can show the first part of the first term of the numerator in

rZA — rZB has positive sign, and the second part of the first term and denominator have

positive sign. I can show rZB — rZD > 0 by Lemma 2. Finally. I compare rZD with rZE;

 

 

7U[—r7 + r76 —r92 - r69pU + 929,21] razapzw - 7)
+r7—

rZ —rZ ~ 2 .
D b 726+76pU—2766pU+66p2U2 r+626p(1—6p)

It is hard to show analytically, but I can show this with graph with U = 2.5, r = 0.], 0,, =

0.6 (Figure 3.A.2).

Figure 3.A.2: Private information and equilibria type D and E (U = 2.5, y = 0.75, 0,, = 0.75, r = 0.05)

I'ZD — I'ZE

 

 

A.5. Production technology and existence region for equilibrium type A, B and E

In order to check the relationship between the production technology and the existence
region for each equilibrium, we get derivatives of critical values of 0 for each equilibrium

with respect to 9p. We have following derivatives of critical values of 0;

98

6910.6,» = —(r +1)7(26pU — y) < O
66,, [9,(6,U - 7) + y]2

 

9

6620.6,» __ —7(r+1)U

2 <0.
69p (29pU-7)

 

884(r,9p)= —r}/U <0
(36,, (QPU — 7)Jr(6pU - 7)[4}’ + YWPU - 7)]

 

 

665 (r,6?,,) _ —r7(2(9pU -}’)

69” 26 2(19,,U—y)2 __IL_--,__
P 6,,(6,,U — 7)

<0,

 

 

 

2r(l —t9,,)29,,U(U — y) +[r6,,(U — 7)+ ./DEl 12(1 —6,,)(—1 + 36,,)U
6662(rflp) _ +2r(1—6p)20pU[r6p(U — 7)2 + 2(1—6,,)2U7(—1+46,,)]/./DE,

69,, [2(1-6p)26pU]2

 

Type A is trivial. Let us look at type B equilibrium. We need to check the sign of

562 (r, 0,,) 6610,67,»

56p p

 

for type B. We have,

 

662(r,6,,) _ 69, (r,6,,) _ 7(r +1)[(26,,U — n3 — U(6,,2U —9,,y + 7)2]
66,, 66,, (20,,U - ”2(9sz —6,,7+7)2

aazoflp) _ 66,(r,6p)

66,, 66,,

has positive sign; therefore it is enough to

 

The denominator of

show the numerator is positive, i.e. (ZQPU — y)3 -U(6p2U —t9,,y+ 7)2 > 0. Letting J =
0,,U — y > 0 and rearranging the numerator, we have

,3. i
J3 +U[J26,,(3—6,,)+6’pJ(36pU—27)+(6p3U —y)(6,,2u +7)] > o.

99

When the utility U is sufficiently large relative to the average production cost (7 / 0p), we

662 (r,t9,,) _ 66. (r,6l,,)

p 56p

have a positive sign for the numerator; therefore > 0. In case of

 

type E, the critical region is mainly related to 05 and 062. 05 has a negative derivative.
whereas 062 has positive derivative. When the initial production technology is low, only
05 is related to the existence of type B. When the initial production technology is high. 062
is related; however the equilibrium disappears quickly as the production technology
improves. Therefore, when the initial production technology is low, the type E

equilibrium also expands as production technology improves. E

A.6. Proof of Proposition 3

Derivatives of welfare of type A. B and E with respect to 0,, are straightforward;

a(rZ.~1)_ 2 5K4
Rp— — (1-6)(U—7)+[2(1—6)0p +I‘]KA +[(I—6)6p +r6p]—a-67 >0

ex, = [6(U—y)+(l-6)y][r+(l—6)(l—t9,,)+66,,(l—6,,)+(9,,2(I—6)]
60,, [r +(1-6)(I-6,,)+96,(1 -9,,)+0,,2(1—49)]2
+ {66pm — y) — (I —0)<I - 6,.)210- 2600 —29,,) > 0
[r+(l —6)(l—6,,)+06,,(1—6,,)+6,,2(I-6)]2

where

 

 

(3.A.12)

 

K . . . . . .
The first term of 52—6—4 18 posrtive. and the second term 13 posrtlve when 9 and 0,, are both
p

greater than 1/2.

 

5023) = y(1+r—6)(1—6)6U2 >

66 2 2 0’
P 0 (apU-y)

100

(30.2 1:) __ r626p(2r + 626p)(U - 7)
66,, (—r — 626,, + 626,2)2

”1

 

 

A.7. Proof of Proposition 4

The derivatives of critical values for each equilibrium from Proposition 1 with respect to
discount rate r have positive values, therefore it is trivial that existing regions for type A
equilibrium shrinks by the increase of discount rate r. Critical region for type E
equilibrium is mainly related to 05 and 062. They have positive derivatives; therefore the

existing region for type B equilibrium also shrinks as r increases.

a61(r96p) _ 7 > 0
6r 6p(6pU—y)+y

 

 

56 ,9 9 U
2(r p): p >0,
6r 26 U—y

 

 

694(r,6,,) _ 27+ r(6,U — 7)+\/r(6lpU — y)[4y +r(6,,U — 7)] > o

 

9

 

 

 

 

5r 2 r(t9,,U —y)[4y+r(6pU -}’)l

665 (7,6p) _ y > 0
6r yr
26 (6 U—y) ~—-—--—----—

p p (9,,(6pU —-7) ’
r6 (U—y)2-2(l—0 )3U7

(ll—7) + 2 l2 2 p 3

6662(r96p) : \/r 6,, (U—y) —4(1—6,) r6,,Uy >0
6r 2(1—6,)2U

Type B equilibrium case is:

662(r,6p) _ 66' (rsgp) : 6pU[6p(6pU-7)+7]_7(26pU_}/)
6r 6r (26pU—y)[6,,(6pU —r)+7l °

 

(3.A.13)

101

We have positive denominator in (3.A.13), and we can show numerator > 0. Rearranging

the numerator of (3.14 gives (6,,ZU — 7)(6l,,U - 7) .. and this is true when U is sufficiently

large.

Figure 3.A.3: Derivative of critical values of 6 for type D equilibrium with respect to r

 

0

 

 

In case of type D. I am going to use a graph to show its shrinkage. Type D equilibrium

case is;

66206,) 664(r.6,,)

6r 6r
_ rr(0,,U — 7H4? + r(0,,U - 2)] - (20,,U - 7)[27 + r(0,,U -7)]./r(¢9pU - 7)[47 + r(¢9pU - 7)
_ 2(26pU - 7)r(t9,,U — 7)[47 + r(6,,U - 7)]

 

 

662(r,6,,) _ 664(r,6,,)

6r 6r

 

When discount rate r is not too big (r = 0.1), One can verify that

has negative sign as in Figure 3.A.3 (U = 2.5). D

102

REFERENCES

Akerlof, G. (1970), “The Market of ‘Lemons’: Qualitative Uncertainty and the Market
Mechanism.” Quarterly Journal of Economics. Vol.84 (3), pp.488-500

Boyarchenko. S. (2000), “A Monetary Search Model with Heterogeneous Agents,”
Mimeo

C uadras-Morato, X. (1994), “Commodity money in the presence of goods of
heterogeneous quality”, Economic Theory, Vol. 4, pp.579-591.

Diamond. P. (1982), “Aggregate Demand Management in Search Equilibrium,” Journal
of Political Economy, Vol.90 (5), pp.881-94

Johri, A. (1999), “Search, Money, and Prices,” International Economic Review, Vol.40
(2), pp.439-54

Kim, Y. S. (1996), “Money, Barter, and Costly Information Acquisition,” Journal of
Monetary Economics, Vol. 37 (I), pp.119-42

Li, Y. (1995), “Commodity Money under Private Information,” Journal ofMonetary
Economics, Vol.36 (3). pp.573-92

. (1998), “Middlemen and Private Information,” Journal of Monetary

Economics, Vol.42 (1), pp. 1 3 1-59

Trejos. A. (1997), “Incentives to Produce Quality and the Liquidity of Money,”
Economic Theory, Vol.9 (2), pp.355-65

. (1999) “Search, Bargaining, Money and Prices Under Private Information.”

International Economic Review, Vol.40 (3), pp.679-95

Wallace, N.; R. Zhou (1997), “A Model of a Currency Shortage,” Journal ofMonetary
Economics, Vol.40, pp.555-72

Williamson. 8.; R. Wright (1994), “Barter and Monetary Exchange Under Private

Information.” American Economic Review, Vol. 84 (1), pp.104-23

103

 

 

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