%Paper: ewp-meet/9606001
%From: ejg@sup.mpls.frb.fed.us (Edward J. Green)
%Date: Sat, 1 Jun 1996 09:10:54 -0500

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\begin{document}

\title{Notes on an Environment with \\ Debt-Settlement
  Intermediation}

\author{Edward J.~Green\thanks{A first draft of these notes was
    presented at the Federal Reserve Bank of Minneapolis in November,
    1995. This revision is prepared for the 1996 TCER conference.
    I would like to thank the conference organizers for inviting me to
    present this work, and the Institute for Monetary and Economic
    Research of the Bank of Japan for making it possible for me to
    accept the invitation during the period while I will be a visitor
    there. The views expressed in these notes are solely mine, and do
    not necessarily represent those of the Federal Reserve Bank of
    Minneapolis, the Federal Reserve System, the IMES, or the Bank of
    Japan.} \\ Federal Reserve Bank of Minneapolis \\ 
  ejg@res.mpls.frb.fed.us}

\date{May, 1996}

\maketitle

\bigskip

\section{Introduction}

These notes concern a model of an economy where the following four
things happen in an equilibrium.

\begin{itemize}

\item Fiat money is useful as a medium of exchange. Consequently it 
has value.

\item Some trades are also financed by the issuance of private IOUs,
and money must be used to pay these off. The use of money for settling
debt is conceptually distinct from its direct use as a medium of
exchange. In this equilibrium, one can identify separate transactions
where the two types of use occur.

\item Besides there being transactions in which money is exchanged
for a good, there are also transactions in which money is exchanged
for an IOU that has not yet matured. That is, the IOUs are a form of
{\it circulating} debt---a privately issued analogue of government
bonds. 

\item Relative to maintaining a fixed supply of fiat currency,
efficiency can be increased by having a ``monetary authority'' that
engages in some pattern of transactions that causes the quantity of
fiat money to fluctuate. 

\end{itemize}

The present model owes a great deal to recent work of Scott Freeman
(1995). The equilibrium that Freeman studies has all four of the
attributes that I have just listed. The present model is simpler than
Freeman's, making some aspects of the definition of equilibrium more
straightforward. It is easier here to characterize the feasible
transactions in the economy, and also to see clearly that the
equilibrium concept for the economy involves parametric price-taking
on the part of all agents rather than involving any strategic
interaction among them.

Freeman concentrates on transactions that
might best be regarded as representing discount-window policy, but it
is clear that there are others (such as open-market operations---which
I will outline in this note---and provision of float in the checking
system) that would have identical implications in the context of this
model. 

The aim is to study how allocations that are Pareto-efficient in the
set of all allocations satisfying materials-balance constraints can be
supported as equilibria of trading institutions by the use of money
and credit, and specifically by the nontrivial involvement of the
monetary authority. This question can be divided into two parts, by
considering the market system as a kind of communication network in
which both goods and messages are passed among traders. If traders
meet one another only in a specified sequence of combinations, then a
paucity of trading opportunities can impose a binding logistical
constraint on moving goods from the traders who are endowed with them
to the traders who can best use them. So the first question is, for
which networks can this logistical problem of distributing goods
efficiently be solved? If a network does solve this problem then the
second question is whether rules for message-passing in the network
can be devised so that agents have incentive to transfer goods among
themselves in the efficient pattern rather than in some other,
inefficient one. Fiat money and IOUs are examples of communication in
this sense. To give a dollar to a merchant is to announce to him, ``I
have provided a dollar's worth of goods or services to someone
already, which has given me a right to take a a dollar's worth of them
from someone else, and now I am exercising that right.''

\section{The population}

At each date $0,1,2,\ldots$, a set $A_t = C_t \cup D_t$ of agents is
born.  $C_t$ and $D_t$ each have $N$ agents. Each of these agents
lives for two periods (dates $t$ and $t+1$). Furthermore there is a
set $C^*$ of agents, the ``initial old,'' who live only at date 0. 
Define $C = C_0 \cup C_1 \cup \ldots$ and $D = D_0 \cup D_1 \cup \ldots$

Each agent in $A_t$ is endowed with one unit of a perishable good at
date $t$, and with nothing at date $t+1$. Agents in $C$ and $D$ are
endowed different goods.

Each agent in $C^*$ is endowed with one unit of fiat money but with no
consumption good. 

Let $x_{c\tau}$ (resp.~$x_{d\tau}$ be an agent's consumption of the
endowment good of agents in $C$ (resp.~$D$) at date $\tau$.

An agent must consume a nonnegative quantity of each good at each
date. Let the utility function of an agent be
%
\display a{W(x_{ct}, x_{dt}, x_{c(t+1)}, x_{d(t+1)}) = 
\cases{u^c(x_{ct}) + v^c(x_{d(t+1)}) & if the agent is in $C$;\cr
u^d(x_{ct}) + v^d(x_{dt}) & if the agent is in $D$;\cr
v^*(x_{d0}) & if the agent is in $C^*$.\cr
}}
%
Assume that all of the functions on the right side are strictly
increasing and strictly concave, and that no agent can consume his
endowment in an efficient and individually rational
allocation.\footnote{This last condition will be satisfied if all of
  the functions on the right side satisfy the Inada condition that the
  limit of the derivative as the argument tends to 0 from the right is
  infinite.}

Note that agents in $D$ wish to trade with members of their own age
cohort in $C$, while agents in $C$ wish to trade with members of the
next age cohort in $D$. Thus, as in the standard
overlapping-generations model of money (as well as most other models
in which fiat money is endogenously valued in equilibrium), there can
be no mutually advantageous trades unless fiat money has value.

\section{A variant of the standard OG structure: valued fiat money}

Suppose that, at each date $t$, all of the traders currently alive are
able to trade among themselves in the following pattern. First, the
agents in $C_{t-1}$ (or $C^*$ if $t=0$) trade with those in $D_t$.
Subsequently the agents in $D_t$ trade with those in $C_t$. 

There is a trading pattern for goods that can achieve efficiency in
this market structure. Young $D$ agents give some of their endowment to
old $C$ agents, and subsequently they receive some of the endowment of
the young $C$ agents. If the entire money stock is passed in the
opposite direction to goods at each stage, then the old $C$ agents
will be the money holders at the beginning of each period. If prices
are set appropriately, markets clear and all agents have incentive to
make the efficient trades.

Because the $C$ agents closely resemble the agents in the standard OG
model, and the $D$ agents want only to trade their endowment good for
a contemporaneous good, it is not surprising that the efficient
equilibrium here bears very close resemblance to the efficient OG
equilibrium. In particular, money has value but there is no credit and
there is no role for a monetary authority.

\section{Reversing the order of transactions within a period: IOUs}

Now consider the opposite order of transactions. That is, suppose that
first the agents in $D_t$ trade with those in $C_t$ and subsequently
the agents in $C_{t-1}$ (or $C^*$ if $t=0$) trade with those in $D_t$.

For fiat money to be passed from the old $C$ agents to the young ones,
it would have to pass through the hands of the young $D$ agents. But
since those agents do not meet the old $C$ agents until it is too late
to deal with the young ones, that cannot happen. The solution is for
the $D$ agents to use IOUs to finance their consumption of goods
purchased from young $C$ agents, to give some of their endowments to
old $C$ agents in return for their fiat money, and to carry the money
into the next period and then use it to repay the holders (who will 
still be alive since they are young when the IOUs are issued). 

The efficient equilibrium in this transactions structure involves use
of both valued fiat money and IOUs, but the IOUs still do not
circulate and there is no role for a monetary authority.

\section{Separation within a cohort: circulating IOUs}

In the preceding market structures, the young $D$ agents have to trade
separately with the young and old $C$ agents. Now, in order to provide
a reason for IOUs to circulate, consider a structure in which not all
agents of the same cohort can communicate directly with one another in
the second period of their lives. Specifically, let $C_t' \subseteq
C_t$, $D_t' \subseteq D_t$, and consider the following sequence of
trading-opportunity stages within each period $t > 0$. (Only the first
and last stages occur for $t=0$.)

\begin{enumerate}

\item All agents in $A_t$ trade with one another.

\item All agents in $C_{t-1}$ enter the market. Agents in
  $D_{t-1}'$ also enter the market, and have the opportunity to pay
  the IOUs to their creditors.


\item Agents in $C_{t-1}$ can trade IOUs for money with one
  another. 

\item Agents in $C_{t-1}'$ leave the market. Agents in
  $D_{t-1}/D_{t-1}'$ enter and have the opportunity to pay their IOUs
  to anyone in $C_{t-1}/C_{t-1}'$ who is holding them.

\item Agents in $C_{t-1}$ trade with agents in $D_t$.

\end{enumerate}

When he is young, an agent's incentive to trade with another member of
his cohort is evidently affected by what he knows or believes about
both his own subgroup and his trading partner's subgroup in the market
structure when they are old. These considerations are made as simple
as possible by assuming that no information about these matters is
available until the second period of agents' lives.

Another question that comes to the fore now concerns the structure of
IOU issuance. Is trade bilateral, so that each young $D$ agent issues
one IOU to a single young $C$ agent, or does each young $D$ agent make
small purchases from many young $C$ agents, so that each $C$ agent
holds a diversified portfolio of small-denomination IOUs afterwards?
Risk-diversification considerations would seemingly lead the $C$
agents to prefer the latter arrangement, if it is feasible. Thus, if
the former, bilateral arrangement is what one intends to have emerge as
an equilibrium trading pattern, there must be some constraint on (or
cost of) IOU issuance to induce it. In that case, the terms of trade
would be negotiated by bargaining within each two-member trading
coalition, rather than taken by agents as parametrically determined by
an economy-wide price (to which Freeman refers). Thus the diversified,
non-strategic trading arrangement, which seems to be the one that
Freeman has in mind, will be adopted here.

Now consider what sort of trading pattern will emerge in the
transaction structure posited here. Let there be $\Gamma$ traders in
$C_{t-1}'$ and $\Delta$ traders in $D_{t-1}'$. Suppose that, in Stage
1 at date $t-1$, each agent in $C_{t-1}$ has acquired IOUs for $f$
units of fiat money to be delivered at date $t$. Note that market
clearing in that stage implies that each agent in $D_{t-1}$ owes $f$
units of fiat money at date $t$. By diversification, in Stage 2 at
date $t$, each agent in $C_{t-1}$ receives a total of $\Delta f / N$
units of fiat money from the traders in $D_{t-1}'$, and is still owed
$(N-\Delta) f / N$ from the remaining traders in $D_{t-1}$. Traders in
$C_{t-1}'$ will not be able to collect their payments from those
debtors in Stage 4, though, so in Stage 3 they will sell the IOUs
still in their possession to other creditors who will participate in
Stage 4. 

Agents in $C_{t-1}'$ regard their IOUs as worthless except in trade at
Stage 3, so they will trade away their full inventories at any
positive price.  The IOUs are not subject to any payment risk in this
model, and creditors do not need to use fiat money until Stage 5, so
creditors who will be market participants at stage 4 regard an IOU and
the amount of fiat money specified for its repayment as being equally
valuable. so they will be willing to pay up to the face value of an
IOU to obtain it. At the beginning of Stage 3, the total face value of
the IOUs in the possession of agents in $C_{t-1}'$ is $\Gamma
(N-\Delta) f$. The total amount of fiat money in the possession of
agents in $C_{t-1}/C_{t-1}'$ is $(N-\Gamma) \Delta f$. Thus the
competitive price at Stage 3 of  an IOU with face value 1 is 
%
\display b{\min \left[ 1, {(N-\Gamma) \Delta f \over \Gamma (N-\Delta)
      f} \right] = \min \left[ 1, {(N-\Gamma) \Delta \over \Gamma
      (N-\Delta)} \right] .}
%
If $\Gamma > \Delta$, then this minimum is strictly less than 1. and
in this case the random timing of trading opportunities induces
consumption risk. Specifically, an agent in $C_{t-1}$ will be
advantaged at date $t$ if he turns out not to be in $C'_{t-1}$,
because he will be able to purchase IOUs below par that will be fully
repaid before he needs to spend money, but he will be disadvantaged if
he does turn out to be in $C'_{t-1}$, because he will have to sell his
IOUs below par before settlement.  ``Open-market operations'' at Stage
3--4, involving the monetary authority purchasing securities at par
from agents in $C'_{t-1}$ and then re-absorbing this money when the
agents in $D_{t-1} / D'_{t-1}$ settle their debts, can eliminate this
risk and thus can increase welfare. Equivalently in this environment,
the monetary authority could issue new money at Stage 3 to agents in
$C_{t-1} / C'_{t-1}$ in return for IOUs that would be settled at Stage
4. This is the analogue of discount-window policy that Freeman
considers.

\bigskip \bigskip

\begin{center}
{\bf Reference}
\end{center}

\bigskip

\noindent Freeman, Scott, ``Currency Shortages and the Discount
Window,'' University of Texas working paper, 1995.

\end{document}