%Paper: ewp-ge/9312001
%From: "Leigh Tesfatsion" <S1.TES@ISUMVS.IASTATE.EDU>
%Date: Sat,  4 Dec 93 22:03:30 CST

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\begin{flushright}
                                                   {\bf May 1993} \\

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\vspace*{15mm} \begin{flushleft} {\large{\bf FURTHER REMARKS ON
WALRAS' LAW \\ AND NONOPTIMAL EQUILIBRIA}}%
     \footnote{ Please address correspondence to L. Tesfatsion,
Department of Economics, Iowa State University, Ames, IA
50011-1070.} \setcounter{footnote}{0}

\vspace*{7 mm} {\large{\bf Mark Pingle}} \\ {\bf Department of
Economics} \\ {\bf University of Nevada, Reno, NV 89557} \\
\vspace*{4mm}

{\large{\bf Leigh Tesfatsion}} \\ {\bf Department of Economics and
Department of Mathematics} \\ {\bf Iowa State University, Ames, IA
50011} \\

\vspace*{10mm}
                 {\bf ABSTRACT}

\vspace{1mm} \end{flushleft}
    The objective of this note is to show that the positively valued
excess supplies which Aiyagari (1992) connects with Pareto
inefficiency for overlapping generations economies represent an
economic opportunity that can potentially be exploited by government
or by a private financial intermediary through the issuance of
unsecured debt.   We demonstrate that, when unsecured debt is
issued, Walras' Law does not fail in the sense described by
Aiyagari.  However, the mere issuance of unsecured debt does not
ensure Pareto efficiency.  We show that Pareto efficiency is
achieved if and only if the opportunity to issue unsecured debt is
optimally exploited, for example, by an {\it earnings-driven\/}
private financial intermediary.

\vspace{7mm}


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\pagebreak

\begin{flushleft} {\bf 1. Introduction} \end{flushleft}

\setlength{\baselineskip}{24pt} In a recent paper, Aiyagari (1992)
demonstrates a connection between the failure of Walras' Law and
nonoptimal equilibria in overlapping generations economies.  The
significant implication of Walras' Law in finite economies, given
all prices are positive and all consumers are locally nonsatiated,
is that an excess supply (in value terms) cannot exist for some
subset of goods without an excess demand (in value terms) existing
for some other subset of goods.  Aiyagari defines the failure of
Walras' Law as a situation in which this implication of Walras' Law
does not hold.   His basic (and interesting) result is to show that
``a competitive equilibrium is nonoptimal if and only if the above
implication of Walras' Law fails in its neighborhood.''

     Aiyagari (1992, section 2) clearly demonstrates that Walras'
Law can fail for an OG economy in this sense.   But shouldn't we be
skeptical of a model in which positively valued excess supplies can
occur in equilibrium?  After all, where do the excess supplies go?
Nonsatiated consumers would not simply throw the excesses away.

    The objective of this note is to show that the positively valued
excess supplies which Aiyagari connects with Pareto inefficiency
represent an unexploited economic opportunity.  Moreover, it is an
opportunity that can potentially be exploited by government or by a
private financial intermediary through the issuance of unsecured
debt.   We demonstrate below that, when unsecured debt is issued,
Walras' Law does not fail in the sense described by Aiyagari.
However, the mere issuance of unsecured debt does not ensure Pareto
efficiency.  We show that Pareto efficiency is achieved if and only
if the opportunity to issue unsecured debt is exploited to its
fullest extent, for example, by an {\it earnings-driven\/} private
financial intermediary.


\begin{flushleft} {\bf 2. An OG Economy With No Unsecured Debt
Issue} \end{flushleft}

     Consider a pure exchange overlapping generations (OG) economy
which begins in period $1$ and extends into the infinite future.
One perishable consumable resource exists, which is distinguished in
period $t$ as ``good $t$.''  At the beginning of each period $t$, a
single two-period lived ``generation $t$ consumer'' is born.  The
generation $t$ consumer is endowed with $w^y > 0$ units of good $t$
and $w^o > 0$ units of good $t+1$.  His preferences over consumption
profiles ($c^y_t$,$c^o_{t+1}$) are represented by a utility function
$U(c^y_t,c^o_{t+1})$ that is twice continuously differentiable,
strictly quasi-concave, strictly increasing, and satisfies
$U(c^y_t,0) = U(0,c^o_{t+1}) = U(0,0)$ and
             \begin{equation}      \label{regcond}
         MRS(w^y,w^o) ~ \equiv ~ U_1(w^y,w^o)/U_2(w^y,w^o) ~ < ~ 1 ~
{}.
            \end{equation} It is also assumed that consumer
preferences satisfy gross substitutability.%
     \footnote{In the model at hand, gross substitutability implies
that an increase in the rate at which the generation $t$ consumer
can trade good $t$ for good $t+1$ results in an increase in his
optimal savings, $t \geq 1$.}

     The population of the economy in period 1 consists of the young
generation $1$ consumer and one old ``generation $0$ consumer.'' The
generation $0$ consumer has an endowment of $w^o > 0$, prefers more
consumption to less, and dies at the end of period $1$.

    Intertemporal trades are facilitated by a price system  ${\bf p}
= (p_1,p_2,\ldots)$, where $p_t$ denotes the price of good $t$ in
terms of a unit of account.  Given this price system, the lifetime
utility maximization problem faced by each generation $t$ consumer,
$t \geq 1$, takes the form:
                       \begin{equation}
                    \max ~ U(c^y_t,c^o_{t+1})
                     \end{equation} with respect to
$(c^y_t,c^o_{t+1})$ subject to the budget and nonnegativity
constraints
               \begin{equation}
    p_tc^y_t + p_{t+1}c^o_{t+1}~ = ~ p_tw^y +p_{t+1}w^o~;
\label{life}
               \end{equation}
                \begin{equation}
        c^y_t \geq  0,~~ c^o_{t+1} \geq 0 ~.  \label{nonneg}
                 \end{equation} Given the stated restrictions on
consumer preferences, any solution to this utility maximization
problem must satisfy
                     \begin{equation} \label{foc}
           MRS(c^y_t,c^o_{t+1}) ~ = ~ p_t/p_{t+1}  ~ .
                     \end{equation} Finally, the consumption level
of the generation 0 consumer is given by
                     \begin{equation} \label{gen0}
                            p_1c^o_1 = p_1w^o.
                       \end{equation}

Let ${\bf c}$ = $(c^0_1,(c^y_1,c^y_2),(c^y_2,c^o_3),\ldots)$ denote
an {\it allocation\/} for the economy.  A nonnegative allocation
${\bf c}$ is {\it feasible\/} if and only if the market for good $t$
clears in each period $ t\geq 1$, in the sense that
                        \begin{equation}   \label{mc}
                     w^y+w^o  ~\geq ~c^y_t + c^o_t ~,~~t\geq 1~.
                           \end{equation} Following Aiyagari (1992,
Section 2.1), an {\it equilibrium\/} for the economy is an
allocation ${\bf c} \geq 0$ and a price system ${\bf p} > 0$ that
satisfy conditions (\ref{life}), (\ref{nonneg}), (\ref{foc}),
(\ref{gen0}), and (\ref{mc}).

      If the market clearing conditions (\ref{mc}) were required to
hold as equalities, then it is straightforward to show that the
unique equilibrium allocation for the economy would be the
``autarkic'' allocation in which each consumer directly consumes his
endowment profile and the price sequence ${\bf p}$ satisfies
$p_t/p_{t+1} = MRS(w^y,w^o)$ for all $t \geq 1$.  However, because
the market clearing conditions (\ref{mc}) allow for excess supply,
the autarkic allocation is {\it not\/} the only equilibrium
allocation for the economy.  In fact, there are an infinite number
of equilibrium allocations.  What is unique about the autarkic
allocation is that it is the only equilibrium allocation for which
the value of excess supply is zero in every market.  For all other
equilibrium allocations, at least one market has a positively valued
excess supply.

     To illustrate, consider Figure 1 where an offer curve is drawn
for the generation $t$ consumer along with three lifetime budget
constraints.  For simplicity, consider only the stationary
equilibria for which the (gross) rate of return $p_t/p_{t+1}$ takes
on a constant value $\rho$ for all $t\geq 1$, with $MRS(w^y,w^o)$
$\leq$ $\rho$ $\leq$ $1$.  The constant rate of return implies that
each generation $t$ consumer consumes the same consumption profile
$(c^y_t,c^o_{t+1})$ = $(c^y,c^o)$.   Moreover, the lifetime budget
constraint (\ref{life}) reduces to $\rho c^y+ c^o=\rho w^y+w^o$ for
all $t \geq 1$, implying that
                      \begin{equation} \label{bcmc}
              w^y+w^o-c^y-c^o = [1-\rho][w^y-c^y] \geq 0.
                           \end{equation} Three possible cases will
now be considered: $\rho = MRS(w^y,w^o)$; $\rho = 1$; and
$MRS(w^y,w^o)$ $<$ $\rho$ $<$ $1$.  These three cases correspond to
the three lifetime budget constraints depicted in Figure 1.

    If $\rho = MRS(w^y,w^o)$, then $(c^y_t,c^o_{t+1}) = (w^y,w^o)$
for each $t \geq 1$, impying that the market clearing conditions
(\ref{mc}) holds as equalities for all $t \geq 2$.  Since $p_1 > 0$,
it follows from (\ref{gen0}) that the generation 0 consumer consumes
his endowment (i.e., $c^o_1 = w^o$), hence market clearing also
holds as an equality for $t=1$.  Consequently, as noted above, no
excess supply exists in this autarkic equilibrium.

     If $\rho = 1$, then $(c^y_t,c^o_{t+1})$ equals the ``golden
rule'' consumption profile $(\bar{c}^y,\bar{c}^o)$ for each $t \geq
1$, and condition (\ref{bcmc}) implies that the market clearing
conditions (\ref{mc}) hold as equalities for all $t \geq 2$.  In
period 1, the generation 0 consumer consumes his endowment (i.e.,
$c^o_1 = w^o$), while the generation 1 consumer consumes less than
his endowment (i.e., $c^y_1 = \bar{c}^y$).  Thus, there is an excess
supply of good 1 in period 1 (i.e., $\bar{c}^y + c^o$ $<$
$w^y+w^o$).  Since $p_1 > 0$, this period 1 excess supply is
positively valued, meaning Walras' Law fails.

    If $\rho = \hat{\rho}$, where $MRS(w^y, w^o) < \hat{\rho} < 1$,
then $(c^y_t,c^o_{t+1}) =(\hat{c}^y,\hat{c}^o)$ for each $t \geq 1$,
where $\hat{c}^y<w^y$.  Together with condition (\ref{bcmc}), this
implies that an excess supply of good $t$ exists in each period $t
\geq 2$.  In period 1, the generation 0 old consumer consumes his
endowment (i.e., $c^o_1 = w^o$) while the generation 1 young
consumer consumes less than his endowment (i.e., $c^y_1 =
\hat{c}^y$).  Thus, an excess supply of good 1 exists in period 1
(i.e., $\hat{c}^y_1 + c^o_1$ $<$ $w^y+w^o$).  Since $p_t > 0$  for
all $t\geq 1$, the excess supply present in each period $t \geq 1$
is positively valued, meaning Walras' Law fails.

    Nonstationary equilibria exist for this model as well.  However,
the results presented here are sufficient to demonstrate the meaning
of Aiyagari's assertion that Walras' Law fails for the OG model.
Considering all equilibria for the economy presented here,
stationary and nonstationary, it can be shown that a positively
valued excess supply exists in every equilibrium except the autarkic
equilibrium.

     To this point, the intermediation process for the economy has
not been explicitly articulated.   The precise form of this
intermediation process would not be significant if positively valued
excess supplies did not occur in equilibrium.  However, positively
valued excess supplies do occur; and because it does not make sense
that nonsatiated consumers would discard valuable resources, it is
important to consider what happens to these excess supplies.  If the
intermediary were a central clearing house, then the clearing house
would hold any excess supplies after the trades had been made.
Recognizing this, it is evident that there is an opportunity
associated with intermediation in this model that is not being
recognized.  What consumer would not want to own the clearing house?

    In the next section, we show that the issuance of unsecured debt
allows this intermediation opportunity to be exploited, and Walras'
Law (in Aiyagari's sense) no longer fails.

\begin{flushleft} {\bf 3. An OG Economy With Unsecured Debt Issue}
\end{flushleft}

    Suppose the economy described in section 2 is now modified by
having the generation 0 old consumer issue unsecured debt in amount
$D_0$.  Let this unsecured debt be taken as the unit of account, so
that the price $p_t$ denotes the number of units of unsecured debt
necessary to buy one unit of good $t$ in period $t$.  Under these
assumptions, the budget constraint of the generation 0 old consumer
becomes
                  \begin{equation} \label{gen0a}
                 c^o_1 = w^o + \frac{D_0}{p_1}~.
                       \end{equation}

     Consumers in generations $t \geq 1$ are not allowed to issue
debt, but they are allowed to purchase old debt and then to resell
it.  Debt can also be sold short, allowing consumers to borrow.  No
other intermediation options are available.   Under these
assumptions, the lifetime utility maximization problem faced by the
generation $t$ consumer, $t \geq 1$, takes the form:
                       \begin{equation}
                    \max ~ U(c^y_t,c^o_{t+1})
                     \end{equation} with respect to
$(c^y_t,c^o_{t+1},D_t)$ subject to the budget and nonnegativity
constraints
           \begin{equation}
    c^y_t ~ = ~w^y - \frac{D_t}{p_t}~;   \label{bcy}
              \end{equation}
           \begin{equation}
    c^o_{t+1}~ = ~ w^o +\frac{D_t}{p_{t+1}}~; \label{bco}
              \end{equation}
            \begin{equation}
        c^y_t \geq  0,~~ c^o_{t+1} \geq 0 ~.  \label{nonneg1}
              \end{equation}

     Let ${\bf D}$ = $(D_0,D_1,D_2,\ldots)$ denote the sequence of
unsecured debt holdings for the economy.  An {\it equilibrium\/} for
the economy is then a triplet $({\bf c},{\bf p},{\bf D})$ consisting
of an allocation ${\bf c} \geq 0$, a price system ${\bf p}$ $>$ $0$,
and a debt sequence ${\bf D}$ that satisfy conditions (\ref{foc}),
(\ref{mc}), (\ref{gen0a}), (\ref{bcy}), (\ref{bco}), and
(\ref{nonneg1}), together with the following market clearing
condition for the unsecured debt:
                       \begin{equation}
             D_{t-1} ~\geq ~ D_t~~ \mbox{for all}~~ t \geq 1~.
\label{mcd}
                        \end{equation} Recall that unsecured debt is
issued only once, in period 1, hence the supply of unsecured debt
available in each period $t\geq 1$ is given by the unsecured debt
$D_{t-1}$ held by generation $t-1$.

    The young age and old age budget constraints (\ref{bcy}) and
(\ref{bco}) together generate the lifetime budget constraint
(\ref{life}). Thus, there are only two essential differences between
the economy presented here and the economy presented in section 2.
First, the generation 0 consumer can here receive a wealth windfall
from the issuance of unsecured debt, whereas no such windfall was
previously possible.  Second, the medium of exchange is here
explicitly identified as being unsecured debt, whereas the medium of
exchange was not previously specified.

     Although Walras' Law was shown to fail for the economy without
unsecured debt, it cannot fail for the present economy.  To
understand this, note that by combining the young age budget
constraint for generation $t$ with the old age budget constraint for
generation $t-1$ one obtains
                      \begin{equation}    \label{wl}
     p_t[w^y+w^o-c^y_t-c^o_t] + [D_{t-1}-D_t]~ = ~ 0 ~,~~t \geq 1~.
                        \end{equation} Using condition (\ref{wl}), a
positively valued excess supply of good $t$ implies an excess demand
for unsecured debt in period $t$, a violation of the market clearing
condition (\ref{mcd}).  Thus, no positively valued excess supply of
any good $t \geq 1$ can exist in equilibrium, meaning Walras Law
cannot fail.

    This restoration of Walras' Law rules out some of the
inefficient equilibria obtained for the section 2 economy.  For
example, because an excess supply is no longer possible in
equilibrium, none of the allocations for the section 2 economy that
were associated with rates of return $\rho$ satisfying $MRS(w^y,w^o)
< \rho < 1$ can now be supported as equilibria.

    Nevertheless, Pareto efficiency is still not ensured.  As is
known from Gale (1973), Pareto efficiency for the present economy
depends upon the real value of the period 1 unsecured debt.  If the
period 1 price $p_1$ is such that the initial real debt level is
given by $D_0/p_1$ $=$ $[\bar{c}^o-w^o]$, then the economy has a
unique stationary Pareto efficient equilibrium allocation in which
each generation $t$ consumer consumes the golden rule consumption
profile $(c^y_t,c^0_{t+1}) = (\bar{c}^y,\bar{c}^o)$, the generation
0 consumer consumes $c^0_1 = \bar{c}^o$, and the rate of return in
each period $t \geq 1$ is given by $\rho = 1$.  Alternatively, if
$p_1 = +\infty$ so that $D_0/p_1 = 0$ (i.e., the unsecured debt is
worthless), then the only possible equilibrium allocation is the
Pareto inefficient autarkic allocation in which each consumer simply
consumes his own endowment in each period $t$.  If $0 < D_0/p_1 <
\bar{c}^o-w^o$, then a nonstationary Pareto inefficient equilibrium
allocation results, with the consumption profile of the generation
$t$ consumer converging to the endowment profile as $t$ becomes
arbitrarily large.  Finally, increasing the initial unsecured debt
level above $[\bar{c}^o-w^o]$ puts the economy on a path to economic
collapse, for the real demand for unsecured debt exceeds the total
endowment of the economy in finite time.

     In summary, in the present economy, Pareto efficiency is
obtained if and only if full advantage is taken of the gain which
can be had from issuing unsecured debt.  In Pingle and Tesfatsion
(1991a,b;1993) it is shown that the opportunity to obtain a wealth
windfall by issuing unsecured debt can be exploited by
earnings-driven private intermediaries (e.g., through the issue of
corporate debentures) as well as by a government (e.g., through the
issue of fiat money).  Indeed, in Pingle and Tesfatsion (1991b) it
is shown that the earnings objective of the private corporate
intermediary is satisfied if and only if price conditions hold which
are analogous to the Cass-Balasko-Shell transversality condition
elaborated in Balasko and Shell (1980), a necessary and sufficient
condition for Pareto efficiency.  But as elaborated in Pingle and
Tesfatsion (1993), these price conditions push to economy to the
very brink of economic collapse, in the sense that any slight
increase in the rate of return in any period $t$ pushes the economy
onto an explosive infeasible path.  Consequently, there is still
much to learn concerning unsecured debt issue (public and private),
efficiency, and economic instability in OG economies.

\pagebreak \begin{flushleft} {\bf REFERENCES} \end{flushleft}

\setlength{\baselineskip}{15pt}

\begin{verse} Aiyagari, S. Rao, (1992), Walras' Law and nonoptimal
equilibria in overlapping generations models, Journal of
Mathematical Economics 21, 343-361. \\[2ex]

Balasko, Y., and Shell, K., (1980), The overlapping generations
Model, I:  The case of pure exchange without money,'' Journal of
Economic Theory 23, 281-306.\\[2ex]

Gale, D., (1973), Pure exchange equilibrium of dynamic economic
models, Journal of Economic Theory 6, 12-36. \\[2ex]

Pingle, M., and L. Tesfatsion, L., (1991a), Overlapping generations,
intermediation, and the first welfare theorem, Journal of Economic
Behavior and Organization 15, 325-345. \\[2ex]

Pingle, M., and L. Tesfatsion, (1991b), Intermediation, bubbles, and
Pareto efficiency in economies with production, Economic Report No.\
24, Iowa State University. \\[2ex]

Pingle, M., and L. Tesfatsion, (1993), Active intermediation in a
monetary overlapping generations economy, Economic Report, Iowa
State University. \\[2ex]

\end{verse}

\pagebreak

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   \put(0,0){\vector(1,0){50}}
       \put(29,-3){$w^y+w^o$}
       \put(52,0){$c^y_t$}
   \put(0,0){\vector(0,1){50}}
       \put(-3,52){$c^o_{t+1}$}
       \put(-11.0,35){$w^y+w^o$}
   \put(0,35){\line(1,-1){35}}
   \put(23,12){\line(-3,2){20}}
   \put(23,12){\line(3,-2){15}}
   \put(23,12){\line(-3,1){15}}
   \put(23,12){\line(3,-1){10}}
       \put(12,25){$\bar{c}$}
   \put(21.5,29){$c^o_{t+1} = w^o + [w^y-c^y_t]$}
      \put(20.5,28){\vector(-1,-2){4.3}}
   \put(43,8){$c^o_{t+1} = w^o + \hat{\rho}[w^y-c^y_t]$}
      \put(42,7){\vector(-4,-1){8}}
   \put(35,16){$c^o_{t+1} = w^o + MRS(w^y,w^o)[w^y-c^y_t]$}
      \put(34,15){\vector(-1,-1){5}}
   \put(5,39){Offer Curve}
   \put(0,0){\dashbox{.5}(10,25)}
   \put(0,0){\dashbox{.5}(23,12)}
   \put(0,0){\dashbox{.5}(15,17)}
   \put(25,13){$w$}
   \put(21.5,-3){$w^y$}
   \put(-4,11.5){$w^o$}
   \put(9.2,-3){$\bar{c}^y$}
   \put(-4,25){$\bar{c}^o$}
   \put(14.0,-3){$\hat{c}^y$}
   \put(-4,16.5){$\hat{c}^o$}
   \put(14.0,-10){\bf Figure 1}

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\end{document}
