%Paper: ewp-mac/9609002
%From: "Edward J. Green" <ejg@sup.mpls.frb.fed.us>
%Date: Mon, 9 Sep 1996 21:46:36 -0500
%Date (revised): Mon, 9 Sep 1996 21:54:16 -0500

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\begin{document}

\title{Money and Debt in the \\ Structure of Payments} \author{Edward
  J.~Green\thanks{This paper is based on research conducted when the
    author was a Visiting Scholar in the Institute for Monetary and
    Economic Studies of the Bank of Japan. He would like to thank
    members of the Institute's Division I, including especially
    Dr.~Hiroshi Fujiki, for comments and suggestions. However, views
    expressed in the paper are solely those of the author, and do not
    necessarily represent those of the Bank of Japan, the Federal
    Reserve Bank of Minneapolis, or the Federal Reserve System.} \\ 
  \strut \\ Federal Reserve Bank of Minneapolis \\ 
  ejg@res.mpls.frb.fed.us} \date{August, 1996 \\ ewp-mac/9609002 \\
http://econwpa.wustl.edu} \maketitle

\begin{abstract}
Freeman (1996) formulates a model in which payment arrangements based
  on intermediated debt that is settled using money can achieve higher
  welfare than direct money payment achieves.  Freeman finds that a
  monetary authority can sometimes further improve welfare, and
  achieve efficiency, by participating in a secondary market for
  debt. The main result of this paper is that a private intermediary
  can also achieve efficiency by means of novation and substitution, a
  contractual device widely used by clearinghouses. The features of
  institutional governance required for either a central bank or a
  clearinghouse to achieve efficiency, particularly features related
  to ``central-bank independence,'' are discussed informally.
\end{abstract}

\setlength{\parskip}{4pt}

\Section a{Introduction}

A famous aphorism in economics is that money exchanges for goods, and
goods for money, but goods do not exchange for goods. However, if one
interprets `money' to mean base money or other outside money (such as
balances held at a central bank), then the aphorism's simple pattern
of money-for-goods exchange hardly captures the structure of actual
transactions. The goal of this paper is to understand the structure of
transactions more closely, and to begin to address two major issues
regarding it.

Notwithstanding the dissimilarity among various payment arrangements
at a fine-grained level, most such arrangements have two main
structural features in common.  First, with few exceptions (such as
cashier's checks and some wire-transfer networks based on real-time
gross settlement), payment arrangements involve the creation of
short-term debt of the payor to the payee that is settled through
intermediaries.  Second, although incurring short-term indebtedness is
a substitute for using money for the purchase of a good, these
debt-based arrangements do not wholly replace money because money is
used to settle the debt.\footnote{Throughout this paper, the term
  `money' refers to outside money.}

Specifically, then, this paper concerns payment arrangements based on
intermediated debt that is settled using money. These arrangements
include checks, wire-transfer systems with netting arrangements,
credit cards, and the like. These two features lie at the root of
current discussions regarding welfare and policy aspects of the
``payments system.'' To begin, regarding large-value payments
especially, there is controversy over whether or not the creation of
debt is a desirable feature of a payment system. Given that there is a
feasible way to make a cash transaction or to achieve gross settlement
of an electronic transaction in real time, it is not obvious what is
the gain from making payments in a way that involves creation of debt
at an interim stage. In practice the creation of debt carries at least
a small risk of inability to settle, so one would not choose
arrangements involving netting or other forms of debt creation if cash
or gross-settlement arrangements were equally good in other respects.
To the extent that the concentration of this debt in the possession of
an intermediary should be cause for additional concern, this argument
becomes even more persuasive. In order to make any case for
payment-system arrangements involving intermediated debt, therefore,
some specific benefit must be found. Particularly in the case of
electronic payments where the real cost of making a transaction is
extremely small, the mere fact that netting economizes on the number
of transactions is unlikely to be a sufficient consideration. Thus it
is important to understand whether or not there is some additional
benefit from using intermediated debt as a means of payment. The
theoretical basis for such understanding is provided by Scott Freeman
(1996), who shows that such a benefit does exist in some model
environments.\footnote{Although I do not explicitly consider risk of
  the payor's inability to settle in this paper, Freeman does consider
  it.  He finds that the benefit to using intermediated debt is robust
  to the existence of some level of settlement risk.}

The use of cash settlement for the debt created in the payment system
raises a further issue regarding the appropriate role of the public
sector, and especially of a central bank. As of today, different
countries are taking various stands on this issue. In some countries
the government is solely a regulator of the payment system, while in
others the government is an active participant. In either case, there
is a subordinate issue of how to apportion the responsibility for
public-sector involvement among the treasury, the bank-supervisory
agency, and the central bank; and countries differ in their approach
to this as well.\footnote{An issue that is related, although beyond
  the scope of this paper, concerns the scope of private-sector
  participation. By their regulatory policies, some governments are
  encouraging non-bank firms to enter the payments industry, while
  others are inclined to erect legal barriers to such entry.}

Most current discussion of these issues considers the extent to which
profit-maximizing operation of the payment system might potentially
interfere with the conduct of monetary policy. There is consensus,
although not unanimity, that this is not an urgent problem. However,
there is another relevant issue that has not been much discussed:
whether participation by the monetary authority can potentially
enhance the economic efficiency of the payment system.\footnote{It is
  sometimes suggested that the central bank can enhance payment-system
  efficiency due to its ability to guarantee immediate and final
  payment, which private-sector intermediary cannot do. This
  suggestion seems to reflect a view that a private intermediary would
  face potential liquidity crises analogous to bank runs, which the
  central bank would not face because of its ability to issue new fiat
  money.  However, if a central bank is empowered to serve as a lender
  of last resort to a private payment-system intermediary, then this
  observation is not sufficient to show that it must also participate
  in the payment system on a day-to-day basis, any more than the
  possibility of bank runs shows that the central bank must do a
  day-to-day business as a commercial bank.} In this paper, I adress
this efficiency issue the context of Freeman's 
model.\footnote{As a model of a central bank, Freeman's model is
  clearly a partial-equilibrium model. An overall judgment about
  whether a central bank should participate in the payment system
  should take into account the opportunity cost of such participation
  with respect to its other objectives. However, if the participation
  of the monetary authority in the payment system can enhance its
  economic efficiency, then there is at least a prima facie case for
  that participation.}

Freeman  shows that the potential of a central bank to enhance
payment-system efficiency can only be evlauated by close study of the
economy concerned. For some parametrizations of the model economy, a  
laissez-faire market in intermediated debt is efficient. For others, 
restrictions on private agents' market access entail that the 
monetary authority can improve welfare relative to some baseline by
participating in a secondary market for debt that has not yet settled.

The baseline to which I refer is the payment system that would be
efficient if only a subset of the restrictions on market access were
in force. Of course, to make a strong case for the need for the
central bank to be a payment-system operator, it would have to be
shown that its participation can improve welfare relative to the best
payment system that a purely private system could implement in
precisely the economy where its participation is being envisioned.
Freeman's model is not formulated at a sufficiently fundamental level
to answer this question in a fully convincing way, but it comes close
to doing so. I will show that efficiency requires an asset that is a
perfect substitute for currency, in a sense that I will make precise.
I extend the model economy to permit a private-sector intermediary to
trade its own debt obligation for the debt issued by the initial
payor, thus providing such a perfect-substitute asset in the model
environment. Since the original debt claim is transferred from the
payee to the intermediary, this trade of debt claims is tantamount to
novation and substitution, a contractual device widely used by
clearinghouses. Direct participation of the monetary authority is not
essential to achieve efficiency in this model. This result can even
hold in the extended version of a model environment that Freeman
studies where intermediaries are unable to settle some of the debt
that they issue.

Both the version of the model with central-bank participation and the
version with novation and substitution implicitly prejudge the issue
of asset substitutability, since they abstract from aspects of the
economy such as privacy of information and limited or costly
enforceability of commitments, which might or might not give agents
reason to regard a central bank as a more (or possibly less)
trustworthy institution than a private clearinghouse.  Although such
issues related to ``credibility'' and institutional governance lie
beyond the scope of the formal model, it is clear that they are
inseparable from the market-equilibrium issues that are formalized in
the model. In particular, issues that determine the effectiveness of
participation by a central bank in the payments system appear closely
related to those that arise with respect to ``political independence''
of a central bank.

\Section b{Modelling strategy}

To address the welfare questions discussed in the introduction
requires a model in which the following three means of payment, which
are observed in actual economies, emerge endogenously in an
equilibrium. 

\begin{itemize}

\item Money is used directly as a medium of payment for goods.

\item Some purchases of goods are also financed by the issuance of
private debt, 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 the equilibrium, one should be able to
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 debt that has not yet been settled.\footnote{This secondary-market
transaction can be structured in various ways. The debt can be in the
form of a security payable to the bearer, or it can be assignable, or
novation can occur.}

\end{itemize}

To formulate such a model, I follow the general strategy that was
introduced by Sargent and Wallace (1982), who exhibited an equilibrium
that has the first two attributes. The idea is to use an
overlapping-generations model, so that money can have value in
equilibrium and its use can be essential for efficiency, and to posit
some heterogeneity among agents within each generational cohort in
order to provide incentive and efficiency rationale for other types of
transaction to occur. I proceed by first constructing two simpler
model economies, in order to make clear how subsystems of the main
model work. To begin, I specify the population and endowment structure
that are common to all of the models.

Before beginning the technical exposition, let me emphasize that the
overlapping generations structure of the model is a technical
convenience. The aim is to formulate the simplest possible model in
which the various kinds of transaction observed in actual economies
can all play a role, and in which welfare questions regarding those
transactions can be framed and analyzed. The spirit of the modelling
exercise is that this model is exemplary of models with lack of double
coincidence of wants, and with restrictions on agents' access to
markets. These fundamental economic features of the model are what
lead to the results; consequently one would confidently expect
parallel results from the analysis of more ``realistic'' models with
the same features. From this perspective, the specific demographic
structure of the model formulated here is a matter of convenience,
although it might be of great significance in the case of other
applications.

\Section c{The model}

\Subsection d{The population}

At each date $1,2,3,\ldots$, a set $A_t = C_t \cup D_t$ of agents is
born.  $C_t$ and $D_t$ each consist of a continuum of agents, of
measure 1. I will sometimes refer to the agents in $C_t$ and $D_t$ as
{\it creditor} and {\it debtor} agents, respectively, since the
debtors will borrow from the creditors in the equilibrium trading
pattern of the model. Each agent lives for two periods (dates $t$ and
$t+1$). Furthermore there is a set $C_0$ of agents, the ``initial
old'' (also a continuum of measure 1) who live only at date 1. Define
$C = C_1 \cup C_2 \cup \ldots$ and $D = D_1 \cup D_2 \cup \ldots$. 

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 $C_0$ is endowed with one unit of fiat money but with no
consumption good. 

Let $x_{1t}$ (resp.~$x_{2t}$) be an agent's consumption of the
endowment good of agents in $C$ (resp.~$D$) at date $t$. 

An agent must consume a nonnegative quantity of each good at each
date. Let the utility function of an agent be
%
\display a{w_i(x_{1t}, x_{2t}, x_{1(t+1)}, x_{2(t+1)}) = 
\cases{u(x_{1t}) + v(x_{2(t+1)}) & if agent $i$ is in $C$;\cr
&\cr
u_*(x_{1t}) + v_*(x_{2t}) & if agent $i$ is in $D$;\cr
&\cr
v(x_{21}) & if agent $i$ is in $C_0$.\cr
}}
%
Assume that all of the functions on the right side are strictly
increasing and strictly concave, and satisfy the Inada condition that
the limit of the derivative as the argument tends to 0 from the right
is infinite.

Given this specification of utility functions, and given the focus on
stationary allocations in this paper, the following notation that
supresses time subscripts will be convenient.
%
\begin{eqnarray*}
x_1 && \hbox{consumption of $x_{1t}$ by an agent in $C_t$;} \\
\strut \\
x'_2 && \hbox{consumption of $x_{2(t+1)}$ by an agent in
$C_t$;} \\
\strut \\
x^*_1 && \hbox{consumption of $x_{1t}$ by an agent in $D_t$;} \\
\strut \\
x^*_2 && \hbox{consumption of $x_{2t}$ by an agent in $D_t$.} \\
%
\end{eqnarray*}

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 it will be seen that, 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.

\Subsection e{Efficiency}

I concentrate on stationary allocations, that is, those in which
corresponding agents in distinct age cohorts receive identical
lifetime-consumption bundles, except for the dating of their goods.
(The consumption of an agent in $C_0$ is identical to the consumption
of an agent in $C_t$ at date $t+1$.) 

An {\it efficient stationary allocation} is a stationary allocation
problem that solves the problem of maximizing a weighted sum of
utilities of the members of $C$ and $D$ in each age cohort. That is,
$(\hat x_1, \hat x'_2, \hat x^*_1, \hat x^*_2)$ is efficient if, for
some $\weight > 0$, it solves the problem of maximizing
%
\display c{\hbox{Maximize}  \left[ u(x_1) + v(x_2') \right]
+ \weight \left[ u_*(x^*_1) + v_*(x^*_2) \right] }
%
subject to the feasibility constraints that 
%
\display d{x_1+x^*_1=1 \hbox{\ and\ } x_2'+x^*_2=1.}

A necessary and sufficient condition for a feasible stationary
allocation to be efficient is that
%
\display e{{v'(x'_2) \over u'(x_1)} = {v_*'(x^*_2) \over u_*'(x^*_1)}.}

I study this criterion of efficiency because of its technical
simplicity, and because it implies the standard Pareto-efficiency
criterion.  An efficient stationary allocation is Pareto efficient in
the set of all feasible allocations of the infinite-horizon economy,
by a result of Okuno and Zilcha (1980). 

There is a specific allocation that I will be concerned with
implementing under various constraints on market access. To define it,
consider a two-agent exchange economy. The first agent is endowed with
one unit of good 1, and has utility function $w(x) = u(x_1) +
v(x'_2)$. The second agent is endowed with one unit of good 2, and has
utility function $w_*(x^*) = u_*(x^*_1) + v_*(x^*_2)$.  Define the
stationary allocation $(\tilde x_1, \tilde x'_2, \tilde x^*_1, \tilde
x^*_2)$ by stipulating $(\tilde x_1, \tilde x'_2)$ and $(\tilde x^*_1,
\tilde x^*_2)$ to be the Walras consumption bundles of these two
agents. Note that equation \eqn e is a necessary condition for a
Walras equilibrium of the two-agent economy, so the corresponding
stationary equilibrium of the infinite-horizon economy is efficient.

Clearly the Walrasian price that supports this equibrium is 

\display f{\tilde p = \left( {1 \over \tilde x^*_1}, {1 \over
\tilde x'_2} \right).}

\Subsection f{Market access, securities, and equilibrium}

I complete the specification of the economy by imposing explicit
constraints on agents' access to markets in each
periods.\footnote{This access is called `market participation'
  elsewhere, but I have already used `participation' in a different
  sense in the introduction. In a formal sense, of course, the fact
  that each agent has access to markets in only two periods is already
  a constraint. The constraints to be introduced here will impede
  trade within an age cohort.} In each period, there will be a
sequence of sub-periods. In each sub-period, only a subset of the
agents currently alive will be able to trade or settle debts. In order
for trade or debt settlement to be transacted between agents who do
not have direct access to one another, money or another security must
be accepted by a third agent or even by several intermediate agents.

Equilibrium is defined in terms of two features: that agents are price
takers who make optimal trading plans, given prices in the markets to
which they have access (including correctly anticipated prices in
markets to which they will have future access); and that markets
clear.\footnote{That is, the definition of equilibrium is in the
  spirit of Radner (1972). A fully adequate equilibrium concept for
  this environment would allow for the endogenous introduction of
  securities, as in Allen and Gale (1988) Instead, for each market, I
  specify an exogenous set of securities to be traded. In principle
  this is a shortcoming but, particularly since the equilibria to be
  studied here support efficient stationary allocations, apparently
  there would be no scope for the introduction of further securities.
  That is, I believe that these equilibria would continue to be
  equilibria if a robustness-to-innovation requirement were explicitly
  imposed.}

For clarity, I will consider three different access-constraint
specifications below. In the next section, I will specify the
constraints in such a way that only the use of fiat money is required
to support an efficient equilibrium. Following that, I will specify
them in such a way that debt needs to be used, as well as fiat money. 
Finally I will specify the constraints in such a way that the debt
must be intermediated in order to be settled. Also in this final
specification, either the stock of money must fluctuate within each
period, or else the debt must be exchanged for debt issued by the
intermediary (that is, novation must occur), in order for an efficient
stationary equilibrium to be supported.

\Section g{Modelling money, debt, and intermediation}

\Subsection h{A basic overlapping-generations structure: valued fiat money}

Suppose that, at each date $t = 1,2,\ldots$, all of the traders
currently alive are able to trade among themselves in the following
pattern. First, the agents in $C_{t-1}$ trade with those in $D_t$.
Subsequently the agents in $D_t$ trade with those in $C_t$.

I will show that, because each agent in $C_0$ holds a unit of money,
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. The entire money stock is passed in
the opposite direction to goods at each stage, so that the old $C$
agents continue to 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.

To formalize this idea, let each period $t$ be divided into two
sub-periods, $t.1$ and $t.2$. Market participation is described by the
following table, which lists the traders who have access and the goods
that are traded within each sub-period. Money is also traded in each
sub-period, and it is the numeraire.\footnote{The only equilibrium in
which the price of money is zero is autarky.} It will be represented
as the last coordinate of a price vector.

\display g{\begin{array}{l c c c c}
\hbox{sub-period} & \qquad & t.1 & \quad & t.2 \\
\strut \\
\hbox{access} && C_{t-1}, D_t && C_t, D_t \\
\strut \\
\hbox{traded} && 2, m && 1, 2, m
\end{array}} 

That is, in the market at $t.1$, there is a
price $p^1 = (p^1_2, 1)$, which has only two coordinates since good 2
(that is, the debtors' endowment good) is the only good available to
be traded. In the market at $t.2$, there is a price vector $p^2 =
(p^2_1, p^2_2, 1)$, since both goods 1 and 2 are available in the
market. (By the Inada condition on $v_*$, debtors will not trade away
their entire endowments at $t.1$.)

I adopt the following notation to represent net trades. A net trade is
always represented by the variable $z$, which can have the following
superscripts and subscripts. 

\begin{itemize}
%
\item[-] An asterisk superscript immediately following $z$
indicates that the net trade belongs to an agent in $D$.
%
\item[-] A numerical superscript indicates the subperiod in which the
net trade is made.
%
\item[-] If the numerical superscript is primed, it indicates that the
net trade is made by an agent in the second period of life (that is,
by an agent in $A_{t-1}$ in period $t$).
%
\item[-] A subscript indicates a coordinate of $z$. A numerical
subscript 1 or 2 refers to one of the two goods available in the
period of the market, and a letter subscript $m$ (money), $d$ (debt
issued in the current period), $d'$ (debt issued in the preceding
period). or $n$ (debt arising from novation, which will be introduced
later in the paper) may also occur.
%
\item[-] The letter $p$ denotes a price vector. A numerical
  superscript indicates the sub-period of the market to which this
  price corresponds. A subscript, which can take the values just
  defined, indicates a coordinate.
%
\end{itemize}

An agent in $C_t$ has access at $t.2$ (for $t>0$) and at $(t+1).1$. At
$t.2$, the agent makes a net trade $z^2 = (z^2_1, z^2_2, z_m^2)$
and at $(t+1).1$ he makes a net trade $z^{1'} = (z^{1'}_2, 
z_m^{1'})$. Thus, the market-constrained optimization problem of an
agent in $C_t$ is to maximize
%
\display h{u(x_1) + v(x'_2)}
%
subject to 
%
\begin{displaymath} 
\begin{array}{r c l c r c l}
(x_1, x'_2) & \in & \Re^2_+; & \qquad & z_m^2 & \ge & 0; \\
\strut \\
x_1 & = & 1 + z^2_1; && z_m^{1'} & \ge & -z_m^2; \\
\strut \\
x'_2 & = & z^{1'}_2; && p^2 \cdot z^2 & \le & 0; \\
\strut \\
z^2_2 & \ge & 0; && p^1 \cdot z^{1'} & \le & 0.
\end{array} 
\end{displaymath}
%
(Note that, by the specification of the trader's endowment and utility
function, utility maximization will clearly imply that $z^2_2 = 
z^{1'}_1 = 0$ and $z_m^{1'} = -z_m^2$. A trader in $C_0$ only makes net
trade $z'$, and utility maximization clearly implies that $z_m^{1'}=-1$,
that is, an old creditor disposes of his entire money stock.)

An agent in $D_t$ has access at $t.1$ and $t.2$, and makes net trades 
$z^{*1} = (z^{*1}_2, z_m^{*1})$ and $z^{*2} = (z^{*2}_1, z^{*2}_2,
z_m^{*2})$ at these dates respectively. His market-constrained 
optimization problem is to maximize
%
\display i{u_*(x^*_1) + v_*(x^*_2)}
%
subject to 
%
\begin{displaymath}
\begin{array}{r c l c r c l}
(x^*_1,x^*_2) & \in & \Re^2_+; & \qquad & z_m^{*1} & \ge & 0; \\
\strut \\
x^*_1 & = & z^{*2}_1; && z_m^{*2} & \ge & -z_m^{*1}; \\
\strut \\
x^*_2 & = & 1 + z^{*1}_2 + z^{*2}_2; && p^1 \cdot z^{*1} & \le & 0; \\
\strut \\
z^{*1}_2 & \ge & -1; && p^2 \cdot z^{*2} & \le & 0.
\end{array}
\end{displaymath}

Since $C_{t-1}$, $C_t$, and $D_t$ all have the same number of agents,
the market-clearing conditions for this economy are that 
%
\display j{z^{1'} = -z^{*1} \hbox{\ and\ } z^2 = -z^{*2}.}

Now it is straightforward to verify, using equation \eqn e, that the
Walrasian stationary allocation $(\tilde x_1, \tilde x'_2, \tilde
x^*_1, \tilde x^*_2)$ is an equilibrium allocation of this market
structure. Equilibrium is supported by the following net trades and
prices.
%
\display k{\begin{array}{r c l c r c c c l}
p^1 & = & \left( {1 \over \tilde x'_2}, 1 \right); &
\qquad & z^{1'} & = & -z^{*1} & = & (\tilde x'_2, -1); \\
\strut \\
p^2 & = & \left( {1 \over \tilde x^*_1}, {1 \over \tilde x'_2}, 1 \right);
&& z^2 & = & -z^{*2} & = & (-\tilde x^*_1, 0, 1). 
\end{array}}

Because the $C$ agents closely resemble the agents in the standard
overlapping-generations 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 overlapping-generations equilibrium. In particular,
money has value but there is no credit and there is no role for a
monetary authority.

\Subsection i{Reversing the order of transactions within a period: debt
securities} 

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 $C_{t-1}$ 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. 

If it is possible for young agents to issue debt securities that they
pay in money when they are old, then there is a solution. The young
$D$ agents can use these securities to finance their consumption of
goods purchased from young $C$ agents, then give some of their
endowments to old $C$ agents in return for their money, and finally
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 debt
securities are issued). This repayment of debt requires an additional
sub-period in each period, which I will assume to occur between the
two sub-periods where markets occur. Although from an ex-post
perspective repayment of debt is a mandatory transfer, not a voluntary
exchange, it will be treated formally as an exchange. That is, after
repaying his debt, a debtor holds a zero quantity of debt in his
portfolio.

The debt security traded in this economy is a commitment to pay one
unit of money to the bearer, at some time during the period following
the period in which it is issued. The quantity of this security that
an agent acquires will be denoted by $d$. That is, issuing a unit of
debt corresponds to choosing $d=-1$.

The following table shows the order of transactions within each period
$t$. The bottom row shows, for each sub-period, which goods (1 and 2)
and assets ($d$ and $m$) are traded. These are listed in the order
that they appear in the price vector. The numeraire is last.
%
\display l{\begin{array}{l c c c c c c}
\hbox{sub-period} & \qquad & t.1 & \quad & t.2 & \quad & t.3 \\
\strut \\
\hbox{access} && C_t, D_t && C_{t-1}, D_{t-1} && C_{t-1}, D_t \\
\strut \\
\hbox{traded} && 1, 2, d && d, m && 2, m
\end{array}} 

The market-constrained optimization problem of an agent in $C_t$ is to
make net trades $z^1$, $z^{2'}$, and $z^{3'}$ that maximize
%
\display m{u(x_1) + v(x'_2)}
%
subject to 
%
\begin{displaymath} 
\begin{array}{r c l c r c l}
(x_1, x'_2) & \in & \Re^2_+; & \qquad & \\
\strut \\
x_1 & = & 1 + z^1_1; && p^1 \cdot z^1 & \le & 0; \\
\strut \\
x'_2 & = & z^{3'}_2; && p^3 \cdot z^{3'} & \le & 0; \\
\strut \\
z^1_2 & \ge & 0; && z_m^{3'} & \ge & -z_m^{2'}; \\
\strut \\
z^{2'}_m & \ge & 0 && z^{2'}_m & = & z^1_d.
\end{array} 
\end{displaymath}

The market-constrained optimization problem of an agent in $D_t$ is to
make net trades $z^{*1}$, $z^{*3}$, and $z^{*2'}$ that maximize 
%
\display n{u_*(x^*_1) + v_*(x^*_2)}
%
subject to 
%
\begin{displaymath} 
\begin{array}{r c l c r c l}
(x^*_1, x^*_2) & \in & \Re^2_+; & \qquad & \\
\strut \\
x^*_1 & = & z^{*1}_1; && p^1 \cdot z^{*1} & \le & 0; \\
\strut \\
x^*_2 & = & 1 + z^{*1}_2 + z^{*3}_2; && p^3 \cdot z^{*3} & \le & 0; \\
\strut \\
z^{*1}_2 & \ge & -1; && z^{*2'}_m & \ge & -z^{*3}_m; \\
\strut \\
z^{*3}_m & \ge & 0 && z^{*2'}_m & = & z^1_d.
\end{array} 
\end{displaymath}

The market-clearing conditions for this economy are that 
%
\display o{z^1 = -z^{*1}; \hbox{\ } z^{2'} = -z^{*2'}; \hbox{\ and\ } 
z^{3'} = -z^{*3}.}

Again, it is straightforward to verify that the
Walrasian stationary allocation $(\tilde x_1, \tilde x'_2, \tilde
x^*_1, \tilde x^*_2)$ is an equilibrium allocation of this market
structure. Equilibrium is supported by the following net trades and
prices.
%
\display p{\begin{array}{r c l c r c c c l}
p^1 & = & \left( {1 \over \tilde x^*_1}, {1 \over \tilde x'_2}, 1 \right);
& \qquad & z^1 & = & -z^{*1} & = & (-\tilde x^*_1, 0, 1); \\
\strut \\
& & && z^{2'} & = & -z^{*2'} & = & (-1, 1); \\
\strut \\
p^3 & = & \left( {1 \over \tilde x'_2}, 1 \right); &&
z^{3'} & = & -z^{*3} & = & (\tilde x'_2, -1).
\end{array}}

The efficient equilibrium in this transactions structure involves use
of both valued fiat money and debt securities, but the debt securities
are not intermediated and there is no role for a monetary authority.

\Subsection j{Separation within a cohort: intermediated debt securities}

Now I come to one of the two main market structures to be studied in
this paper. In this structure, not all agents of the same cohort can
communicate directly with one another in the second period of their
lives.  Specifically, some debtors are not able to repay creditors to
whom they have issued debt. Those creditors therefore need to sell
their debt to other agents with whom the debtors can communicate.
These purchasers of debt thus serve as intermediaries in the
settlement of the original transactions.

To formalize this environment, define the partitions $C_t = C_t' \cup
C_t''$ and $D_t = D_t' \cup D_t''$, for each $t \ge 1$. Define $C_0''
= C_0$. For each $t \ge 1$, let there be $\gamma \in (0, 1)$ traders in
$C_t'$ and $\delta \in (0, 1)$ traders in $D_t'$. 

The market structure will be specified in such a way that
creditors in $C_t'$ cannot be repaid at $t+1$ by debtors in $D_t''$. 
To do so, consider the following sequence of trading-opportunity
stages within each period $t > 1$. (Only the first and last stages
occur for $t=1$.)

\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 debt securities to their creditors.
%
\item All agents in $C_{t-1}$ can trade money for outstanding debt
securities that have not been settled. For now, assume that no new
debt can be issued in this sub-period.
%
\item Agents in $C_{t-1}'$ trade with agents in
$D_t$, and then leave the market.\footnote{Alternatively it could be
specified that all agents in $C_{t-1}$ trade with agents in $D_t$ in
this sub-period.  In equilibrium, every agent in $C_{t-1}''$ would
make a zero net trade in this market.}
%
\item Agents in $D_{t-1}''$ enter and have the
opportunity to pay their debt securities to anyone in $C_{t-1}''$ who
is holding them.
%
\item Agents in $C_{t-1}''$ trade with agents in $D_t$.
%
\end{enumerate}

This structure can be represented in tabular form as follows.
%
\display q{\begin{array}{l c c c c c c c c c c c c}
\hbox{sub-period} & \quad & t.1 & \enspace & t.2 & \enspace & t.3 &
\enspace & t.4 & \enspace & t.5 & \enspace & t.6 \\
\strut \\
\hbox{access} && C_t && C_{t-1} && C_{t-1} &&
C_{t-1}' && C_{t-1}'' && C_{t-1}'' \\
&& D_t && D_{t-1}' && &&
D_t && D_{t-1}'' && D_t \\
\strut \\
\hbox{traded} && 1, 2, d && d', m && d', m && 2, m && d', m && 2, m
\end{array}} 

There is an important distinction between sub-periods $t.2$ and $t.5$,
on the one hand, and sub-period $t.3$, on the other. In $t.2$ and
$t.5$, debt is being settled at face value. In contrast, in $t.3$,
debt is being purchased at market terms prior to settlement. As in the
other markets where voluntary exchange occurs, the price must be
determined endogenously by agents' optimization together with
market-clearing.

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. I will assume that no information about
these matters is available until the second period of agents' lives.
Later I will discuss an implication of this assumption for welfare
analysis. 

Another question concerns the structure of debt security issuance. Is
trade bilateral, so that each young $D$ agent issues one debt security
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 debt securities
afterwards?  Risk-diversification considerations would seemingly lead
the $C$ agents to prefer the latter arrangement, if it is
feasible.\footnote{Moreover, if a bilateral arrangement is what one
intends to have emerge as an equilibrium trading pattern, there must
be some constraint on (or cost of) debt security 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.} The
diversified, non-strategic trading arrangement will be modelled here.

This arrangement implies an asymmetry in the interpretation of the
quantity of debt securities held by an agent. If an agent holds a
positive quantity of these, then that quantity represents a
diversified portfolio of securities payable by all issuers in the
economy, in proportion to those issuers' amounts of debt outstanding.
If the quantity of debt is negative (that is, if the agent is an issuer
of debt), then it represents debt issued by the agent himself.

As in the market structures studied above, equilibrium is defined in
terms of agents' optimization together with market clearing. The
objective function of an agent in $C_t$ is slightly different here than
above, since his consumption can depend on whether he is in $C_t'$ or
$C_t''$. I assume that such an agent maximizes expected utility,
assigning probability $\gamma$ to the event that he is in $C_t'$  and 
consumes bundle $x'$, and $1 - \gamma$ to the event that he is in
$C_t''$ and consumes bundle $x''$, in period $t+1$.

The optimization problem of an agent in $C$, then, is to choose net
trades  $z^1$, $z^{2'}$, $z^{2''}$, $z^{3'}$, $z^{3''}$, $z^{4'}$, 
$z^{5''}$, and $z^{6''}$, to maximize
%
\display r{u(x_1) + \gamma v(x_2') + (1 - \gamma) v(x_2'')}
%
subject to constraints. The constraints and market conditions are
conceptually straightforward but they are numerous, because the
environment is so complex. They are presented in the appendix.

The structure of trading in this environment is indicated in Figure 1.
Time is on the horizontal axis. A trader's lifespan is depicted by a
horizontal bar of two periods' length. Within each period, the
sub-periods in which a trader has market access are shown by a
thickening of the bar. The top bar (extending only through period 1)
is $C_0$. After that, there are three generations having four bars
each, representing $D_t'$, $D_t''$, $C_t'$, and $C_t''$ in descending
order. $D_4$ is also shown, since those agents have transactions with
$A_3$. Thin vertical bars indicate market or settlement access during
sub-periods.

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\vfil \break

\Section k{Inefficiency of equilibrium}

The market structure just described permits trading of goods and three
financial assets: money ($m\/$), new debt ($d\/$), and seasoned debt
($d'\/$). It is clear that there exists a pattern of trade, involving
goods-for-new-debt, goods-for-money, and seasoned-debt-for-money
market transactions, as well as settlement of seasoned debt, that
achieves the stationary efficient allocation. That pattern of trade
requires goods and assets to be exchanged in particular ratios. If
those ratios are not the same as the price ratios in a competitive
equilibrium, though, then the stationary efficient allocation will not
be a competitive equilibrium allocation of the economy. Following
Freeman, I show that equilibrium is inefficient in an economy where
$\gamma > \delta$.

The argument begins by supposing that, in sub-period 1
at date $t-1$, each agent in $C_{t-1}$ has acquired debt securities
for 1 unit of money to be delivered at date $t$. (It is easy to see
that, except in autarky equilibrium, the entire money stock of 1 unit
must be passed from cohort to cohort in a stationary equilibrium.)
Note that market clearing in that sub-period implies that each agent
in $D_{t-1}$ owes 1 unit of money at date $t$. By diversification, in
sub-period 2 at date $t$, each agent in $C_{t-1}$ receives a total of
$\delta$ units of fiat money from the traders in $D_{t-1}'$, and is
still owed $(1-\delta)$ 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 sub-period 4, though, so in sub-period 3 they will
sell the debt securities still in their possession to other creditors
who will participate in sub-period 4.

Agents in $C_{t-1}'$ regard debt as worthless except in trade at
Stage 3. They will trade away their full inventories at any
positive price.

Debt is certain to be paid by Stage 5, and agents in $C_{t-1}''$ do
not need to use fiat money until Stage 6, so these agents will be
willing to pay up to the face value of debt to obtain it at Stage 3.

Thus all money held by agents in $C_{t-1}''$, up to the face value of
the debt held by agents in $D_{t-1}''$, will be exchanged for that
debt. This determines the equilibrium price in the secondary market.

At the beginning of Stage 3, the aggregate amount of money that will
be provided in settlement of the debt in the possession of agents in
$C_{t-1}'$ is $\gamma (1-\delta)$. The total amount of money in the
possession of agents in $C_{t-1}''$ is $(1-\gamma) \delta$. Thus the
competitive price at Stage 3 of a debt claim for one unit of money is

\display b{p^3_{d'} = \min \left[ 1, {(1-\gamma) \delta \over 
\gamma (1-\delta)} \right] .}

If $\delta < \gamma$, then $p < 1$.

Thus, if $\delta < \gamma$, then availability of money to
intermediaries is a bottleneck in some sense. It remains to be shown
that this bottleneck causes Pareto inefficiency. Freeman's argument
continues by comparing the amount of consumption enjoyed by an agent
in $C'$ with the amount enjoyed by an agent in $C''$ in equilibrium.
The following allocation shows that the consumption of an agent in
$C'$ is lower, so the fact that ``too few'' debtors have market access
in sub-period 2 induces consumption inequality among agents who are
identical except for market access. This inequality is risk from an
ex-ante perspective, so from that perspective it is a
Pareto-inefficient allocation among risk-averse agents.\footnote{The
  specification that all agents in $C$ are identical ex ante is
  inessential to producing consumption inequality, although it
  simplifies the calculation of equilibrium by making all young
  creditors' decisions identical. Its significance is to make an
  allocation with consumption inequality, which would be
  Pareto-incomparable to the equal-consumption allocation if agents
  were distinguishable ex ante, into a Pareto-inefficient allocation.}

Specifically, an agent in $C_{t-1}'$ receives $\delta$ units of money
in settlement of debt in Stage 2, and $p^3_{d'}(1-\delta)$ units of
money from sale in Stage 3 of debt not yet settled. Thus an agent in
$C_{t-1}'$ holds less than 1 unit of fiat money to trade in Stage 4.

At Stage 3, an agent in $C_{t-1}''$ spends all of his money received
in settlement of debt in Stage 2 to purchase debt at price $p < 1$,
which will be settled at par at Stage 5.  Thus he will hold more fiat
money in Stage 6 than if he had not traded in the secondary market.
That is, he will hold more than 1 unit of fiat money to trade in Stage
6.

In equilibrium, agents in $D_t$ must sell their endowment good for the
same price in Stage 4 as in Stage 6. Therefore an agent in $C_{t-1}''$
consumes more of that good than does an agent in $C_{t-1}'$, since he
has more money to spend at the identical price for goods.

\Section l{Chartering a monetary authority to achieve efficiency}

\Subsection m{Representing a monetary authority within the model}

Before presenting a result of Freeman regarding the
potential role of a monetary authority in achieving efficiency, it is
worthwhile to reflect on what a monetary authority is, and on how it
ought to be modelled in this formal environment. First, consider what
Freeman (1996, pp.~6, 14) assumes about the monetary authority, and
how he characterizes its optimal policy.

{\narrower \noindent There exists\dots a monetary authority able to
  issue fiat money\dots . This authority issues an initial stock of
  [money] to each initial old creditor. \dots\ Suppose that the\dots
  monetary authority (or ``central bank'') is now authorized to issue
  and lend fiat money equal to the nominal amount of debt presented by
  any of the late-leaving creditors.  This central bank loan must be
  repaid with fiat money upon the arrival\dots of the late-arriving
  borrowers.\par }

Because the monetary authority is described as dealing with the
creditors in every cohort, superficially it might seem that the
authority must be an infinite-lived agent. In that case, it would be
in a position to provide intermediation services that no private agent
could provide.

There is a convincing argument that this is an inadvisable way to
think about the role of a monetary authority or, in general, an agent
that carries out public policy.\footnote{A very clear development of
  this argument is by Neil Wallace (1988), in a discussion of an
  analogous issue regarding the Diamond-Dybvig model of
  intermediation.} The criticism has to do with a dilemma regarding
how to interpret the restrictions on market access in the model
economy. These restrictions could be interpreted as reflecting
technological restrictions, but then it would be inexplicable why the
monetary authority is not bound by the same constraints that private
agents face. Alternatively the restrictions could be interpreted as
reflecting institutional or legal constraints from which the monetary
authority is exempt, but then the most natural welfare conclusion to
draw from the inefficency of competitive equilbrium would be that
those constraints on private agents ought to be relaxed in general,
not that there is a rationale for a distinguished agent to be granted
a special exemption.  These seem to be the only tenable
interpretations of the market-access restrictions, and neither
provides a good basis for understanding the role of a monetary
authority.

On closer inspection, though, the monetary authority does not
intermediate between agents who do not meet one another. At every
date, it issues money in sub-period 3, which it uses to purchase
seasoned debt. Then, in sub-period 5 at the same date, it absorbs the
money that it receives in settlement of this debt. Thus, rather than
specifying that there is a special, infinite-lived, agent in the
model, one can equally well specify that, in sub-period 2 at each date
$t$, one of the agents in $C_{t-1}''$ is designated to be the monetary
authority.

\Subsection n{How a monetary authority can achieve efficiency}

Consider what can be accomplished by such a monetary authority,
consisting of one agent in each cohort (specifically in $C_{t-1}''$,
at each date $t$) who is authorized to behave differently in one
respect, and constrained to behave differently in another respect,
from the other agents. This distinguished agent is authorized to
create money in sub-period 3, and is required to destroy in sub-period
5 an amount of money equal to what he has created.  Specifically the
agent is authorized to create $\gamma(1-\delta) - (1-\gamma)\delta)$
units of money for purchase of seasoned debt in sub-period 2, so that
(by the argument leading to \eqn b in the laissez-faire case)
$p^3_{d'} = 1$. This intervention eliminates the inequality of
consumption between agents in $C_{t-1}'$ and those in $C_{t-1}''$ Thus
it attains efficiency from the ex-ante perspective.

To interpret the monetary authority in this way, as being one of the
private agents in the population who is selected to carry out a
special responsibility, avoids making the suspect assumption that it
has a mysterious technological superiority over the private agents.
This interpretion also has a clear implication regarding the nature of
the contract to which the monetary authority is subject. It is exempt
from the prohibition that other agents face against creating money
(that is, against counterfeiting). However, it is expected to absorb
the money received in settlement in sub-period 5 (with the exception
of money received in settlement of debt in its private portfolio, as
opposed to the debt initially purchased with newly created money),
rather than to spend that money in sub-period 6 to finance consumption
for itself. For such an expectation to be fulfilled, the monetary
authority must be constrained in some way, or its incentives must be
modified in some way, that is not represented explicitly in the model.
This implicit assumption is analogous the implicit assumption of some
enforcement technology to compel repayment of debt. Subject to this
assumption, the present analysis shows that the difference between a
monetary authority and an ordinary private agent is simply one of
incentives, and not of intrinsic opportunities or capabilities. (The
one obvious advantage that a monetary authority typically enjoys with
respect to private banks---a monopoly or at least a competitive
advantage in note issuance---is an artifact of government policy
rather than being intrinsic.)

Nothing in the formal model requires that this special incentive
arrangement should be offered only to a single agent. It could be
supposed instead that all agents in $C_{t-1}''$ would be subject to
the arrangement. However, the implicitly assumed monitoring and
enforcement functions are presumably costly to carry out. It would be
inefficient to exercise them over all agents in $C_{t-1}''$, or even
over several of them, if one agent can make all of the transactions
that are required for efficiency. This consideration suggests that the
activity of central banking is probably a natural monopoly.

\Subsection p{Relationship to central-bank independence}

This interpretation of the nature of a monetary authority is
different from the social-planner interpretation that
economists often make. Nevertheless, it is consonant with the
views expressed by distinguished scholars of central banking, such as
Cairncross (1988), Cuikerman (1995), Goodhart (1988), and Sayers (1967).
Numerous central banks, including the Bank of England, were initially
chartered as private joint-stock companies and continued to operate
under that form of ownership long after their public-policy roles were
firmly established. In many countries today, including the United
States, payment-system activities of the central bank continue to be
conducted under a corporate charter, and the government is at most a
minority owner.  Thus it very appropriate to model the monetary
authority as being identical to a private agent in most respects.

However, despite their corporate form, central banks are organized in
a way that induces a markedly different outcome from the operation of
an ordinary corporation. ``Ownership'' of a central bank is typically
an entitlement to a fixed income stream (analogous to ownership of
preferred stock, rather than common stock), with residual profits
actually accruing to the government. From a perspective such as that
taken by Jensen and Meckling (1976), the government is the true owner of
the central bank (as the residual claimant of its profit stream), and
thus control of it by the nominal owners is really a means of
separating ownership and control in economic terms.

To the extent that the nominal owners of the central bank have the
primary influence in the appointment and retention of its governor and
other senior executives, it is foreseeable that the executives will 
have relatively small incentive to maximize profit. Other charter
provisions, such as restrictions on the types of asset that can be
held in the portfolio, complement the ownership structure by
constraining the central bank from emulating the decisions that
private agents would take to maximizing profits.

The fact that central-bank charters have such striking and
idiosyncratic provisions, which are recognized to safeguard
``central-bank independence'' from the residual claimant of its
profit, constitutes evidence in favor of the modelling approach taken
here: to represent a monetary authority as an agent with the same
intrinsic opportunities and capabilities as other agents, but with
different induced incentives or legal constraints. Conversely, if the
market structure specified in \sec j is the one that would exist under
laissez-faire, then the fact that an efficient allocation can be
achieved by a departure from profit maximization on the part of the
monetary authority provides a normative argument in favor of
central-bank independence.

\Section o{Institutional and contractual alternatives
  to central-bank participation}

A careful statement of the conclusion reached in \sec n is that, if the
market structure defined in \sec j would be in effect were there no
participation by a monetary authority, then the participation described
in \sec m (that is, open-market operations or equivalent intervention
to support the secondary-market price of debt in sub-period 3) will
support an efficient allocation that Pareto-dominates the
laissez-faire equilibrium allocation from an ex-ante perspective. 

The applicability of this analysis to actual markets is an open
question, because it is not certain that the market structure of \sec
j is the one that would emerge under laissez-faire. That market
structure abstracts from private-sector agents that provide payment
services, such as escrow agents and clearinghouses. It also abstracts
from contractual features of payment, such as contract netting and
novation. 

In this section, I discuss one such private-sector arrangement that
can achieve efficiency in the environment described in \sec j. This
arrangment resembles a clearinghouse that uses \textit{novation and
substitution} (that is, substitution of debt payable by itself for debt
payable by the original issuer) to settle contracts.

\Subsection q{A market structure with novation securities}

An alternative to having a monetary authority is for traders in
$C_{t-1}''$ to issue debt securities---call them \textit{novation
securities} at sub-period $t.3$ in return for the debt securities of
traders in $C_{t-1}'$ that have not yet been settled. The traders in
$C_{t-1}'$ will exchange these novation securites for good 1 in
sub-period $t.4$.  The novation securities will be paid in sub-period
$t.6$, when the agents in $D_t$ who have acquired them will meet the
agents in $C_{t-1}''$ who issued them. In equilibrium, both the
initial securities and the novation securities will trade at face
value. Thus, again, the risk induced by trading-opportunity
uncertainty will be fully insured, and efficiency will be attained.

The asset structure of this economy is described by adding novation
debt (denoted by $n\/$) to the trading structure described in table
\eqn q as follows.
%
\display s{\begin{array}{l c c c c c c c c c c c c}
\hbox{sub-period} & \quad & t.1 & \enspace & t.2 & \enspace & t.3 &
\enspace & t.4 & \enspace & t.5 & \enspace & t.6 \\
\strut \\
\hbox{access} && C_t && C_{t-1} && C_{t-1} &&
C_{t-1}' && C_{t-1}'' && C_{t-1}'' \\
&& D_t && D_{t-1}' && &&
D_t && D_{t-1}'' && D_t \\
\strut \\
\hbox{traded} && 1, 2, d && d', m && d', n, m && 2, n, m && d', m &&
2, n, m
\end{array}} 
%
The budget constraints and market-clearing conditions for this market
structure are straightforward modifications of those for the 
market structure specified in \sec j.

With respect to the characteristics of securities that are represented
explicitly in this model, there is hardly any difference between this
novation security and the money issued and reabsorbed by the central
bank in \sec n. Both money and the novation security are issued by
agents in $C_{t-1}''$ in sub-period $t.3$ to agents in $C_{t-1}'$ in
return for the debt held by those agents. The agents in $C_{t-1}'$
trade the newly-issued security (money or the novation security,
depending on the payment arrangement) in sub-period $t.4$ to agents in
$D_t$ for those agents' consumption good. The security, or another
security of the same type, is thereafter removed from circulation by
the issuer. In the case of money, this happens at in sub-period $t.5$
when the seasoned debt that was purchased with newly issued money is
settled. In the case of novation debt, the money received settlement
of seasoned debt in sub-period $t.5$ is used to settle the novation
debt in sub-period $t.6$. Only with respect to the specifics of how
removal from circulation is accomplished, does novation debt differ
from money in more than name.

Implicitly, though, money and novation securities differ in much more
than name. What differs between the two asset structures is the
institutional framework of ownership, monitoring, and enforcement that
must exist to support them. In contrast to the distinctive features of
a central bank that have been mentioned in \sec p, a clearinghouse
that operates by novation and substitution is subject to roughly the
same framework of contract and enforcement as is a private debtor.
Although it would be an exaggeration to claim that a central bank is
totally unlike a clearinghouse (especially since a clearinghouse is
typically chartered as a nonprofit corporation, jointly owned by a
group of the banks that it serves, and with restricted powers that
prevent it from competing directly with them), in practice the
distinction between them is substantial and easy to recognize.

Historically, clearinghouses preceded central banks in most
industrialized countries, and central banks were chartered in part to
address perceived inefficiencies in the payment systems where those
clearinghouses were already operating. Despite the presence of central
banks, which have tended to be advantaged relative to clearinghouses
in point of their legal powers, clearinghouses continue to exist and
to play a major role. These facts suggest that probably neither
institutional form has an absolute advantage over the other. The basic
model of intermediated debt and its extension studied in this section
can perhaps provide a basis for thinking systematically about the
relative advantages of each type of institution in various
circumstances.

\Subsection r{An economic definition of \textit{novation and substitution}}

A noteworthy feature of the extended model just discussed is that it
permits an economic definition of novation and substitution. This
operating procedure of a clearinghouse is typically described in
institutional terms related to contract law, as in the following
quotation from the Group of Experts on Payment Systems of the Central
Banks of the Group of Ten Countries (1989).\footnote{This document is
  widely known as the ``Angell Report.''} 

{\narrower \noindent ``One type of arrangement would establish a
  clearing house that would be substituted as the central counterparty
  in deals submitted for netting by participants in the arrangement,
  in order to effect a binding multilateral netting among those
  participants (``multilateral netting by novation and
  substitution'').\par }

Such substitution is exactly what takes place, in the equilibrium of
the asset structure discussed in \sec p, when agents in $C_{t-1}'$
swap debt issued by agents in $D_{t-1}$ for novation securities (also
debt) issued by agents in $C_{t-1}''$. Each agent in $C_{t-1}$ has a
different portfolio of specific debt securities after this swap than
beforehand. However, each of these agents has the same net credit
position afterwards as beforehand. Agents in $C_{t-1}'$ hold debt
securities both before and after the swap, and in equilibrium the face
value of the securities (as well as the market value) is the same.
Agents in $C_{t-1}''$ change from being only holders of debt before
the swap to being both holders and also issuers afterwards, but again
there is no change in their net credit position. Thus it is clear that
novation and substitution can be defined in economic terms to be
\textit{an issuance and exchange of debt that leaves the net credit
  position of all agents unaffected.}

The economic role of novation and substition is to transfer debt from
agents who do not have an opportunity to receive settlement of it to
other agents who do have that opportunity, without affecting anyone's
wealth position, and in such a way that the initial debt-holders have
equivalent trading opportunities (that is, ``liquidity'') to what they
would have had if their initial debt holdings had been settled rather
than being traded.

\Section s{Failure of a clearinghouse to settle}

There is a consensus among payment-system experts that the failure of
a clearinghouse to settle its obligations created by novation and
substitution is an especially worrisome systemic risk. This view is
clearly expressed by the Group of Experts on Payment Systems of the
Central Banks of the Group of Ten Countries (1989).

{\narrower \noindent 
[M]ultilateral netting by novation and substitution has the potential to
reduce liquidity risks more than any other institutional form, but this
depends critically on the financial condition of any central counterparty to
the netting; if the liquidity of a central counterparty is weak, the
liquidity risks of this institutional form may be greater than in the case
of bilateral netting by novation; the credit risks of this institutional
form are generally less than in other forms that have been considered,
subject again to the identity and condition of any central counterparty.
\par }

Although Freeman  does not make such a claim, one tempting way
to interpret his result is that the inability of agents in $C''$ to
settle novation securities makes the involvement of a monetary
authority indispensible to attain efficiency in his model economy.
Such an interpretation would be mistaken for two reasons.

Before I discuss these reasons, let me mention that Freeman's model
has a feature that I have omitted from the version developed in 
\sec n. Freeman posits that, before the beginning of sub-period $t.6$,
the agents in $C_{t-1}$ and in $D_t$ are exogenously and randomly
dispersed among several \textit{islands}. (This sequestration lasts
only for the duration of the sub-period, so the debtor agents are able
to trade in period $t+1$ exactly as specified in \sec n, or in \sec q
if novation securities are traded.) If agent $\alpha \in
D_t$ has accepted a novation security issued by agent $\alpha'' \in
C_{t-1}''$, and $\alpha$ is on island $\iota$ in sub-period $t.6$,
and $\alpha''$ is on island $\iota'' \neq \iota$ in sub-period $t.6$,
then $\alpha''$ cannot settle the novation security that $\alpha$
holds. 

Despite this inability of prospective intermediaries in Freeman's
model economy to settle all (or even most) of the novation securities
that they issue, the market structure involving those securities will
still be efficient. To see this, suppose that there are $I$ distinct
islands. If the face value of novation securities issued by an agent
in $\alpha'' \in C_{t-1}''$ is $\phi$, and if those securities are
traded to agents in $D_t$ who are dispersed equally among the islands,
then only a subset of the securities having value $\phi / I$ will be
able to be settled. In sharp contrast, $\alpha''$ will receive
settlement on all of the seasoned debt $d'$ for which he trades
novation securities that he issues. Consequently traders in
$C_{t-1}''$ will bid the novation-security price of seasoned debt
(that is, $p^3_{d'} / p^3_n\/$) up to $I$, rather than only up to par.
Subsequently, in sub-period $t.4$, agents in $D_t$ will recognize that
only $1/I$ of the novation debt will be settled, so as a result of
their optimization, the money price $p^4_n$ of a a unit of the
novation security (specified to be settled in sub-period $t.6$ for one
unit of money) in that sub-period will be only $1/I$. Thus, the amount
of good 2 that an agent in $C_{t-1}'$ can obtain by exchanging a unit
of seasoned debt for novation securities and then exchanging those for
consumption is $i \cdot (p^4_n / p^4_2) = 1/p^4_2$, which is the same
amount that the agent could obtain in the model economy studied in
\sec q. That is, equilibrium in a version of that model economy with
islands would still be efficient, because agents with rational
expectations would fully adjust in market equilibrium for the
occurrence of settlement failure on the part of intermediaries. 

The efficiency of this equilibrium is the first reason why it would be
mistaken to suppose that participation of a monetary authority is
necessary to attain efficiency in Freeman's model. Of course, the
argument in the preceding paragraph makes it clear that the
intermediary's inability to settle in the model economy differs
radically in its foreseeability from the type of settlement failure on
the part of an actual intermediary that concerns policymakers so much.
This is not to say that policymakers' concerns are necessarily
warranted, but rather that models of settlement do not yet reflect
some of the features of the actual economy that are crucial to
reasoning conclusively about those concerns.

The other reason why it would be mistaken to interpret Freeman's model
as justifying a necessary role for a monetary authority is directly
related to the considerations in \sec m and \sec q, regarding the
constraints facing a central bank and their relationship to the 
constraints that face a clearinghouse. The import of my arguments in
those sections is that a central bank cannot be regarded as being
intrinsically a better type of institution than a clearinghouse.
Certainly, given the potential for the payment system to be abused for
political ends, few people would be enthusiastic about transferring
the main settlement responsibilities from a smoothly functioning
clearinghouse to a central bank that lacked independence. On the other
hand, as policymakers recognize, if the structure of a clearinghouse
raises prudential concerns, one needs to examine whether the structure
can be strengthened before concluding that the only solution is for
the central bank to take over its function.

\Section t{Conclusions}

This paper has been concerned with the welfare analysis of
central-bank and clearinghouse intervention in payment arrangements.
At a formal level, this is done by extending a model of the use of
intermediated debt for payment, so that private-sector intermediaries
can issue debt that corresponds to the clearinghouse practice of
novation and substitution. If such debt can be issued, then the
resulting market equilibrium under laissez-faire is efficient, so
there is no need for direct participation by a monetary authority.
This result can even hold in the extended version of a model
environment (which is seen to be very special, however) where
intermediaries are unable to settle some of the debt that they issue.

Although issues of institutional governance lie beyond the scope of
the formal model, the analysis makes it clear that they are
inseparable from the market-equilibrium issues that are treated
explicitly. Whether or not efficiency might require a central bank to
participate in the payments system depends on the degree to which a
central bank can promise reliably and credibly to reabsorb money that
it issues to facilitate payments, and also on whether the
commercial-law framework governing the operation of a private-sector
payments intermediary is sufficient to warrant agents' use of debt
issued by the intermediary as a money-like medium of exchange.

The credibility of a central bank's promise about reabsorption
evidently depends, in turn, on its governance structure. It is likely
that the institutions of central-bank governance necessary for
credible participation in the payments system are essentially
identical to those that are necessary for effective conduct of
monetary policy in a narrow sense. Thus, to whatever extent that there
is a need for a central bank to participate directly in the payments
system, this need reinforces the considerations in favor of chartering
a ``politically independent'' central bank. Moreover, the need for
political independence suggests that the central bank would typically
be a more appropriate public-sector participant in the payments system
than would the treasury or another agency under the immediate control
of the government.

\vfill \eject

\begin{center}
{\Large Appendix: Optimization and Market Clearing}
\end{center}


The optimization problem of an agent in $C$ is to choose net
trades  $z^1$, $z^{2'}$, $z^{2''}$, $z^{3'}$, $z^{3''}$, $z^{4'}$, 
$z^{5''}$, and $z^{6''}$, to maximize
%
\begin{displaymath}
u(x_1) + \gamma v(x_2') + (1 - \gamma) v(x_2'')
\end{displaymath}
%
subject to
%
\begin{displaymath} 
\begin{array}{r c l c r c l}
(x_1, x_2', x_2'') & \in & \Re^3_+; & \qquad & \\
\strut \\
x_1 & = & 1 + z^1_1; && p^1 \cdot z^1 & \le & 0; \\
\strut \\
x_2' & = & z^{4'}_2; && p^4 \cdot z^{4'} & \le & 0; \\
\strut \\
x_2'' & = & z^{6''}_2; && p^6 \cdot z^{6''} & \le & 0; \\
\strut \\
z^{2'}_m & = & \delta z^1_d; && z^{2'}_m & \ge & 0; \\
\strut \\
z^{2''}_m & = & \delta z^1_d; && z^{2''}_m & \ge & 0; \\
\strut \\
z_d^{3'} & \ge & (1-\delta) z^1_d && z_d^{3''} & \ge & (1-\delta) z^1_d \\
\strut \\
z_m^{3'} & \ge & -z_m^{2'}; && p^3 \cdot z^{3'} & \le & 0; \\
\strut \\
z_m^{3''} & \ge & -z_m^{2''}; && p^3 \cdot z^{3''} & \le & 0; \\
\strut \\
z_m^{4'} & \ge &  -(z_m^{2'} + z_m^{3'}); && 
p^4 \cdot\ z^{4'} & \le & 0; \\
\strut \\
z^{5''}_m & = & (1 - \delta) z^1_d + z_d^{3''}; && 
z^{5''}_m & \ge & -(z_m^{2''} + z_m^{3''}); \\
\strut \\
z^1_2 & \ge & 0; && 
z^{6''}_m & \ge & -(z_m^{5''} + z_m^{2''} + z_m^{3''}); \\
\end{array} 
\end{displaymath}

The optimization problem of an agent in $D$, is to choose net trades 
$z^{*1}$, $z^{*4}$, $z^{*6}$, $z^{*2'}$, and $z^{*5''}$, to maximize
%
\begin{displaymath}
u_*(x^*_1) + v_*(x^*_2)
\end{displaymath}
%
subject to
%
\begin{displaymath} 
\begin{array}{r c l c r c l}
(x^*_1, x^*_2) & \in & \Re^2_+; & \qquad & \\
\strut \\
x^*_1 & = & z^{*1}_1; && x^*_2 & = & 1 + z^{*1}_2 + z^{*4}_2
 + z^{*6}_2; \\
\strut \\
p^1 \cdot z^{*1} & \le & 0; \\
\strut \\
z^{*4}_m & \ge & 0; && z^{*6}_m & \ge & -z^{*4}_m; \\
\strut \\
z^{*2'}_m & = & z^1_d; && z^{*2'}_m & \ge & -(z^{*4}_m + z^{*6}_m); \\
\strut \\
z^{*5''}_m & = & z^1_d; && z^{*5''}_m & \ge & -(z^{*4}_m + z^{*6}_m).
\end{array} 
\end{displaymath}

The market-clearing conditions in this economy are 
%
\begin{displaymath}
\begin{array}{r c l c r c l}
z^1 & = & -z^{*1}; & \quad &  \gamma z^{2'} + (1-\gamma) z^{2''}
& = & -\delta z^{*2'}; \\
\strut \\
\gamma z^{3'} & = & -(1-\gamma) z^{3''}; && 
\gamma z^{4'} & = & -z^{*4}; \\ 
\strut \\
(1-\gamma) z^{5''} & = & -(1-\delta) z^{*5''}; 
&& (1-\gamma) z^{6''} & = & -z^{*6}.
\end{array}
\end{displaymath}

\vfill\eject

\begin{center}
{\Large \bf References}
\end{center}

\parindent=0pt

\bigskip

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\bib Cairncross, Alec, ``The Bank of England: Relationships with
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\bib Cukierman, Alex, {\sl Central Bank Strategy, Credibility, and
  Independence: Theory and Evidence}, Cambridge, MIT Press, 1995.

\bib Freeman, Scott, ``The Payments System, Liquidity, and
Rediscounting'' University of Texas working paper, 1996 (forthcoming
in {\sl American Economic Review\/}). 

\bib Goodhart, Charles A.~E., {\sl The Evolution of Central Banks},
Cambridge, MIT Press, 1988.

\bib Group of Experts on Payment Systems of the Central Banks of
the Group of Ten Countries, ``Report on Netting Schemes,'' Bank for
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\bib Jensen, Michael C.~and William H.~Meckling, ``Theory of the Firm:
Managerial Behavior, Agency Costs and Ownership Structure, {\sl
  Journal of Financial Economics,} 3, 1976, pp.~305--360.

\bib Okuno, Masahiro and Itzhak Zilcha, ``On the Efficiency of a
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\bib Radner, Roy, ``Existence of Equilibrium of Plans, Prices, and
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\end{document}
