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%From: "Leigh Tesfatsion" <S1.TES@ISUMVS.IASTATE.EDU>
%Date: Sat,  4 Dec 93 19:49:21 CST
%Date (revised): Sat, 4 Dec 93 20:19:02 CST

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\begin{titlepage} \setlength{\baselineskip}{15pt}
            \begin{flushright} {\bf Revised November 1993} \\
             \end{flushright} \vspace*{6mm}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
                                \begin{center} {\large{\bf ACTIVE
INTERMEDIATION IN A MONETARY \\ OVERLAPPING GENERATIONS
                      ECONOMY}}%
     \footnote{Previous versions of this paper have been presented
at the University of Iowa and at the American Mathematical Society
Meeting.  The authors are grateful to seminar participants for
helpful comments.  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}
                                 \end{center}

\vspace{3mm} It is now widely believed that government intervention
is essential to ensure Pareto efficiency in the standard overlapping
generations economy with nonaltruistic agents.  This paper argues
that the normal profit-seeking activities of {\it private
intermediaries\/}---missing from the standard overlapping
generations economy---would tend to eliminate the need for such
government intervention.  A private earnings-driven corporate
intermediary is introduced into a standard monetary overlapping
generations economy which, in the absence of the intermediary,
generates Pareto inefficient equilibria.  The intermediary issues
unsecured corporate debt and maximizes its market value in direct
accordance with the interests of its successive shareholders.  The
significant impact of private intermediation is demonstrated by
establishing that all equilibria for the resulting ``Corporate
Economy'' are Pareto efficient.  Dynamic properties of equilibrium
paths are also determined.  It is shown, for example, that
endogenous cyclic equilibria are possible for the Corporate Economy
in the absence of gross substitutability.

\vspace{7mm}


{\large {\bf Keywords:}}  Financial Intermediation, Overlapping
Generations, Pareto Efficiency, Endogenous Cycles

\vspace{5mm} {\large {\bf J.E.L. Number:}} E44

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%HERE IS RRACT1.TEX


\end{titlepage}

\begin{center}
                            {\bf 1. INTRODUCTION} \end{center}

     The conventional definition of a competitive equilibrium,
originally developed for economies with finitely many consumers and
goods, does not ensure a Pareto efficient outcome for the
overlapping generations economy [Samuelson (1958), Gale (1973)].
This well-known finding has widely been interpreted to mean that
some form of government intervention is essential to ensure
efficiency in dynamic open-ended economies with nonaltruistic
finite-lived agents.  Recently this ``folk theorem'' has even begun
to make its way into graduate and undergraduate textbooks; see, for
example, Azariadis (1993, pp. 270-271) and Champ and Freeman (1994,
pp. 206-207).

     It is therefore important to ask whether the conventional
definition of a competitive equilibrium, as applied to an
overlapping generations economy, is truly satisfactory in the sense
that private agents under this definition are permitted to exploit
all available profit opportunities.  In particular, even if
government intervention can eliminate inefficiency by an appropriate
manipulation of monetary or fiscal policy instruments, is it
possible that the normal profit-seeking activities of {\it private
intermediaries\/}---missing from the standard overlapping
generations economy---would tend to eliminate the need for any such
government intervention?

     Preliminary support for this conjecture is provided in Pingle
and Tesfatsion (1991a,b).  The basic argument is that the profit
opportunities associated with intermediation per se are not
exploited by private agents under the conventional definition of a
competitive equilibrium because private agents are assumed to be
narrowly focused on consumption and production opportunities, taking
prices (terms of trade) as given.  Using Samuelson's (1958)
overlapping generations economy as an illustration, it is shown that
the introduction of an earnings-driven private intermediary willing
and able to arbitrage profit opportunities on terms of trade has a
dramatic impact on the efficiency properties of the economy.  For
example, regardless of the precise form assumed for the earnings
objective of the intermediary, Pareto inefficient outcomes are ruled
out as equilibria since the intermediary necessarily perceives
unbounded earnings opportunities.

     Extending this prior work, the present paper introduces an
earnings-driven corporate intermediary into the basic monetary
overlapping generations economy studied by Grandmont and Laroque
(1973), Balasko and Shell (1981), and Grandmont (1985), among
others.  The corporate intermediary issues unsecured corporate debt
(i.e., debentures) and strives to maximize its market value in
direct accordance with the interests of its successive shareholders.
The conventional definition of a monetary equilibrium is generalized
to include this corporate objective.

     We first derive a necessary and sufficient condition for a
consumption allocation for this modified ``Corporate Economy'' to be
Pareto efficient.  We then establish that all equilibria for the
Corporate Economy are Pareto efficient, yielding the golden rule
consumption allocation, if consumer preferences satisfy gross
substitutability; and the set of such equilibria is not empty.  In
particular, the corporate intermediary has a viable way to enter the
economy and earn a windfall return by issuing unsecured debt
whenever the value of the initial real money balances held by
consumers is ``too low.'' We further show that a similar first
welfare theorem and existence theorem are obtained in the absence of
gross substitutability if the offer curve of each consumer satisfies
certain curvature restrictions of the type studied by Grandmont
(1985).  Without these curvature restrictions, it is shown that the
corporate intermediary faces an interesting time inconsistency
problem.

     We also investigate the dynamic properties of Corporate Economy
equilibria.  Given gross substitutability, the golden rule
consumption allocation is immediately obtained.  In the absence of
gross substitutability, only three types of dynamic behavior are
possible in equilibrium: either (a) the economy enters immediately
into a cycle with a period 2 orbit; or (b) the economy converges to
a limit cycle with a period 2 orbit; or (c) the economy converges
cyclically to the golden rule consumption allocation.  In contrast,
Grandmont (1985) establishes for the basic monetary overlapping
generations economy that, in the absence of gross substitutability,
$k$ periodic ($k > 2$) or even aperiodic equilibria can exist.  The
introduction of an earnings-driven corporate intermediary thus
eliminates these more complex (and Pareto inefficient) equilibria,
although an endogenous ``business cycle'' is still possible.

     The work closest in spirit to our own is the seminal paper by
E.  Thompson (1967).  Thompson asserts (p.\ 1205) that if interest
rates were forever too low for Pareto efficiency, then private
corporate firms would proceed to bid up interest rates by issuing
new debts to finance current new expenditures.  Despite Thompson's
important insight, most researchers using the overlapping
generations framework continue to exclude private corporate
intermediaries.  The current paper supports Thompson's claim that,
to be a fully articulated competitive equilibrium model, the
standard overlapping generations economy must be extended to permit
the entry of private corporate intermediaries willing and able to
take advantage of unexploited profit opportunities through private
debt issue.

     The basic structure of the Corporate Economy is described and
motivated in sections 2 through 4.  Section 5 sets out a definition
of equilibrium for the Corporate Economy, and section 6 investigates
the efficiency and dynamic properties of these equilibria.
Concluding comments are given in section 7.  Proofs of propositions
are provided in an appendix.

%HERE IS RRACT2.TEX

\vspace{2mm} \begin{center}
           {\bf 2. CONSUMER OPTIMIZATION IN THE CORPORATE ECONOMY}
\end{center}

    Consider a pure exchange overlapping generations (OG) economy
that begins in period 1 and extends into the infinite future.  The
economy's population growth rate is equal to zero, and each
generation consists of one two-period lived consumer.  The economy
contains a single perishable consumable resource that provides
consumers with utility.  The resource available during period $t$
will be referred to as good $t$.

     Consumers born in periods $t \geq 1$ are identical aside from
time of birth.  The ``generation $t$'' consumer is born at the
beginning of period $t$ and lives through the end of period $t+1$.
Each generation $t$ consumer is endowed with $w^y>0$ units of good
$t$ and $w^o>0$ units of good $t+1$.  Letting $c^y_t$ and
$c^o_{t+1}$ denote the young and old age consumption levels of the
generation $t$ consumer, it is assumed that his lifetime consumption
preferences are measured by a utility function $U(c^y_t,c^o_{t+1})$
that is twice continuously differentiable, strictly increasing,
strictly quasi-concave, and satisfies $U(0,c^o_{t+1})$ =
$U(c^y_t,0)$ = $U(0,0)$.  Letting $U_1$ and $U_2$ denote the partial
derivatives of the utility function with respect to $c^y_t$ and
$c^o_{t+1}$, respectively, it is also assumed that
                           \begin{equation}  \label{sam}
    MRS(w^y,w^o)~ \equiv ~ \frac{U_1(w^y,w^o)}{U_2(w^y,w^o)} ~ < ~ 1
{}~.
                            \end{equation} The implications of these
utility function regularity conditions will be clarified below.

     Consumers in the Corporate Economy can hold both
government-issued fiat money and stock shares issued by a private
corporation.  Let $M_t$ denote the quantity of money held by the
generation $t$ consumer from period $t$ to period $t+1$, and let
$P_t$ denote the price of good $t$ in terms of fiat money.  The
generation $t$ consumer obtains the $M_t$ units of money by trading
away $M_t/P_t$ units of the good $t$ endowment.  In period $t+1$,
the $M_t$ units of money can be used to purchase $M_t/P_{t+1}$ units
of good $t+1$.  Thus, as long as the prices $P_t$ and $P_{t+1}$ are
not infinite, the generation $t$ consumer is able to transfer wealth
from period $t$ to period $t+1$ by choosing to obtain and hold fiat
money.

     Let $\theta_t$ denote the number of stock shares purchased or
sold short in period $t$ by the generation $t$ consumer, and let
$v_t \geq 0$ denote the price of a share in period $t$, measured in
terms of good $t$.  The generation $t$ consumer purchases or sells
short the $\theta_t$ shares in return for $v_t\theta_t$ units of
good $t$.  In period $t+1$, the consumer then receives or pays out
$[v_{t+1} +d^e_{t+1}]\theta_t$ units of good $t+1$, where
$d^e_{t+1}$ denotes the expected per share dividend.  Thus, in
addition to saving through money holding, the generation $t$
consumer is able to save or borrow from period $t$ to period $t+1$
through share transactions.

     Given these specifications, the lifetime utility maximization
problem facing the generation $t$ consumer can be represented as
                            \begin{equation}   \label{umax}
                         \max U(c^y_t,c^o_{t+1})
                              \end{equation} with respect to
$c^y_t$, $c^o_{t+1}$, $M_t$, and $\theta_t$ subject to the budget
and nonnegativity constraints
                                  \begin{eqnarray*}
         c^y_t  & ~=~ & w^y - [M_t/P_t] - v_t\theta_t ~;  \\
         c^o_{t+1} & ~=~ & w^o + [M_t/P_{t+1}] +
                  [v_{t+1} + d^e_{t+1}]\theta_t ~;  \\
             0 ~ & \leq & ~ c^y_t, c^o_{t+1},M_t~ .
                       \end{eqnarray*} The generation $t$ consumer
takes as given the (possibly infinite) positive goods prices $P_t$
and $P_{t+1}$, the finite nonnegative share prices $v_t$ and
$v_{t+1}$, and the finite nonnegative expected per share dividend
$d^e_{t+1}$.

     The regularity conditions on the utility function $U(\cdot)$
guarantee that each consumer in generation $t \geq 1$ will choose
$c^y_t > 0$ and $c^o_{t+1} > 0$.  Since $\theta_t$ is unrestricted
in sign, the first-order conditions for problem (\ref{umax}) require
that
                  \begin{equation}   \label{mrs}
                       MRS(c^y_t,c^o_{t+1}) = q_t~,
                         \end{equation} where
                \begin{equation} \label{sharerate}
            q_t ~ \equiv ~ \frac{v_{t+1}+d^e_{t+1}}{v_t}
                         \end{equation} denotes the expected rate of
return on holding shares from period $t$ to $t+1$.

     Let $s_t$ $\equiv$ $w^y - c^y_t$ denote the savings of the
generation t consumer.  For later purposes, it will now be shown
that the budget constraints for problem (\ref{umax}) can be simply
expressed in terms of $s_t$ and the share rate of return $q_t$ over
the range $0$ $<$ $q_t$ $<$ $+\infty$.  The proof of the following
proposition (and all subsequent propositions) can be found in an
appendix to this paper.

\vspace{2mm} {\it \noindent {\bf Proposition 2.1.} Suppose $0 < q_t
< +\infty$.  Then a finite solution exists for problem (\ref{umax})
if and only if either $0$ $\leq$ $P_t/P_{t+1}$ $\leq$ $q_t$ or $P_t$
and $P_{t+1}$ are both infinite.  In either case the budget
constraints for problem (\ref{umax}) can be expressed without loss
of generality as
                   \begin{eqnarray}
          c^y_t ~ & = & ~ w^y - s_t~; \label{cy} \\
          c^o_{t+1} ~ & = & ~ w^o + q_ts_t~;  \label{co} \\
          0 ~& \leq & ~ c^y_t, c^o_{t+1}~;
                        \end{eqnarray} and the optimal consumption
and savings levels of the generation $t$ consumer are uniquely
determined as functions $(c^y(q_t),c^o(q_t),s(q_t))$ of the period
$t$ share rate of return $q_t$, where $s(q_t) \geq 0$ if and only if
$q_t$ $\geq$ $MRS(w^y,w^o)$.}
    \vspace{2mm}

     In the initial period 1 the Corporate Economy consists of one
generation $1$ young consumer and one generation $0$ old consumer.
The generation $0$ old consumer is endowed with $w^o$ units of good
1 and a positive amount $M_0$ of fiat money issued (once and for all
time) by government.  The generation 0 old consumer is the
entrepreneur who starts the corporation, hence he is also endowed
with an initial quantity of stock shares, $\theta_0 > 0$.  To retain
symmetry with other consumers, it is assumed that the generation 0
old consumer plans to sell these shares at the unit price $v_1$ and
expects a per share dividend payment $d^e_1$.

     The utility of the generation $0$ consumer in period $1$ is
assumed to increase with increases in his consumption level $c^o_1$.
Thus, the generation 0 old consumer chooses $c^o_1$ to satisfy
                     \begin{equation}  \label{bc0}
             c^o_1 ~= ~w^o + [M_0/P_1] + [v_1 + d^e_1]\theta_0~.
                        \end{equation} Note from (\ref{bc0}) that
the introduction of fiat money and corporate stock shares gives the
generation 0 old consumer a potential wealth windfall.

%HERE IS RRACT3.TEX

\vspace{2mm} \begin{center}
                          {\bf 3.  THE CORPORATION} \end{center}

     A distinguishing feature of the corporate form of business is
that a corporation can outlive any particular shareholder and
generally has no foreseeable date of termination.  As an
approximation to this reality, we suppose that the corporation has
an infinite planning horizon spanning all successive generations of
its shareholders.  Moreover, since the focus of the present study is
on the ability of corporations to incur and roll over debt, we
simplify the analysis by abstracting from the production process.
That is, we assume that the corporation has no capital assets and
employs no labor, and hence produces no physical output by which to
generate earnings.  Nonetheless, the corporation can borrow.  As
will be clarified, below, this permits in principle the continual
payout of positive dividends financed by successive debt
accumulation and roll over, hence the shares of the corporation need
not be valueless.

     It is also assumed that the corporation acts in the interests
of its shareholders.  Examining the budget constraints (\ref{cy})
and (\ref{co}), it is seen that the optimized lifetime utility of
the generation $t$ consumer is an increasing function of $q_t$, the
expected rate of return on holding shares from period $t$ to period
$t+1$, over the range $q_t \geq MRS(w^y,w^o)$ where the consumer's
optimal savings level $s(q_t)$ is nonnegative.  Examining the budget
constraint (\ref{bc0}), it is seen that the utility of the
generation 0 consumer is an increasing function of
$[v_1+d^e_1]\theta_0$, his expected windfall return from stock share
ownership.  To what extent can the corporation control these
quantities?

      By definition (\ref{sharerate}), the expected share rate of
return $q_t$ depends upon the share prices $v_t$ and $v_{t+1}$ and
the expected dividend $d^e_{t+1}$.  It follows that the
corporation's control over $q_t$ depends upon its control over share
prices and expected dividends.  In reality, corporations can and do
influence their stock share prices by buying and selling their own
shares.  Here it is assumed that the corporation sets its own share
prices by agreeing to buy or sell any quantity of shares at the
share prices it desires to support.

     In particular, at the beginning of period 1 the corporate
intermediary announces a sequence ${\bf v}$ = $(v_1,v_2,\ldots )$ of
finite nonnegative share prices $v_t$ together with a sequence ${\bf
d^e}$ = $(d^e_1,d^e_2,\ldots )$ of finite nonnegative expected
dividend payments $d^e_t$.  In announcing this pair of sequences
${\bf I}$ = $({\bf v},{\bf d^e})$, henceforth referred to as a {\it
corporate plan\/}, the corporation takes goods prices as given.  It
is assumed that the announced corporate plan is known to all current
and prospective shareholders.  Also, while actual dividends can
differ from expected, it is assumed that the dividend expectations
of current and prospective shareholders coincide with the dividend
expectations of the corporation as reflected in its announced
corporate plan.

     The corporation must of course be concerned about the
feasibility of supporting any particular corporate plan.  Since the
corporation has no physical assets and no earnings capacity from
physical production, the shares that it issues represent unsecured
debt.  The quantity $v_1\theta_0$ measures the value of corporate
debt which matures in period $1$.  The only way that the corporation
can repay this debt is by rolling it over.  In order for the
corporation to remain solvent, the value of debt that it issues in
period $1$, $v_1\theta_1$, must be at least as great as the debt
maturing in period $1$, $v_1\theta_{0}$.  More generally, the
incremental change in the value of corporate shares outstanding in
any period $t \geq 1$, measured in terms of good $t$, is given by
                  \begin{equation} \label{cf}
    \pi_t ~ \equiv ~ v_t[\theta_t - \theta_{t-1}]~;
                        \end{equation} and it follows by a simple
induction argument that the corporation is viable in period $t$ only
if $\pi_t$ $\geq$ $0$.

     If $\pi_t > 0$, i.e., if $\theta_t > \theta_{t-1}$, the
corporation is issuing new shares in period $t$ to cover an increase
in the demand for its shares.  In this case the corporation has
positive net receipts equal to $\pi_t$ units of good $t$.  Because
good $t$ is perishable, any net receipts held by the corporation
become worthless at the end of the period.  It is therefore assumed
that the corporation pays out all net receipts in the form of
dividends to its shareholders.  Letting $d_t$ denote the good $t$
dividend per share paid to the (old aged) shareholder in period $t$,
who holds $\theta_{t-1}$ shares, it follows that
                       \begin{equation}   \label{div}
                   \pi_t ~ = ~ d_t\theta_{t-1} ~.
                           \end{equation} Note that all of the
corporation's dividends are financed by incurring new debt.  From an
empirical standpoint, this is an extreme case.  However, many actual
corporations do occasionally borrow to support a dividend
distribution when earnings are low.%
            \footnote{Interestingly, Kane (1989, p.\ 36) argues that
``zombie firms now constitute roughly 25 percent of the
FSLIC-insured thrift industry,'' where a zombie thrift is
characterized as a thrift with zero enterprise-contributed capital
that must rely on FSLIC guarantees to keep attracting new deposits
to pay off previously accumulated debts.  The possible need for
guarantees to ensure the viability of the corporation currently
under consideration is taken up in section 4.}
     Our model is an abstraction that allows us to focus on the
efficiency and stability implications of debt roll over.

     By assumption, the corporation announces share prices  and
expected dividend payments in the form of a corporate plan and then
agrees to buy or sell shares in accordance with consumer
preferences.  As seen in  (\ref{cf}), actual  period  $t$ net
receipts $\pi_t$, and hence actual dividend payments, depend  on
the  actual  consumer  share demands  $\theta_t$ and $\theta_{t-1}$.
These share demands in turn depend on the share prices and expected
dividend payments  announced in the corporate plan.  It  is  assumed
that  the  corporate  plan  truthfully  reveals  to potential
shareholders the dividend payments  that  the  corporation actually
expects  to make.  This ``dividend consistency'' condition will now
be formalized.

     Suppose the corporation in period 1 is able to  form  estimates
for the share  demands  $\theta_t$,  $t \geq 1$, conditional on a
goods price sequence ${\bf P}$ and a corporate plan  ${\bf  I}$  =
$({\bf v},{\bf d^e})$.  Letting $\theta^e_t$ = $\theta_t({\bf
I},{\bf P})$ denote the quantity of  shares  that  the  corporation
expects  the generation  $t$ consumer  to purchase, conditional on
${\bf P}$ and ${\bf I}$, the corporation's period $t$ expected  net
receipts  are given   by
               \begin{equation}\label{nrexpected}
    \pi^e_t   ~   =  ~ v_t[\theta^e_t-\theta^e_{t-1}]~,
                \end{equation} where  $\theta^e_0$ $\equiv$
$\theta_0$.   The  period  $t$ expected dividend per share implied
by (\ref{nrexpected})  is  then  given  by
                      \begin{equation} \label{divexpected}
          d^{e*}_t   =  \frac{\pi^e_t}{\theta^e_{t-1}}~.
                               \end{equation}

     If the expected dividend payments (\ref{divexpected}) were to
differ from the expected dividend payments announced in the
corporate  plan ${\bf I}$,  the  corporation  could  be  accused  of
deliberately deceiving its potential  shareholders.   It  is
assumed  that  the corporation  does  not engage in this behavior.
Specifically, it is assumed  that  the  corporate plan  ${\bf  I}$
announced  by   the corporation  exhibits {\it dividend
consistency\/} in the sense that the expected dividend sequence
${\bf d^e}$ appearing  in  ${\bf  I}$ coincides with the expected
dividend sequence ${\bf d^{e*}}$ implied by relation
(\ref{divexpected}).

     Finally, the corporation is assumed to be aware that the total
economy resources $w^y+w^o$ available in each period $t$ constitute
a bounded sequence, implying that the sequence of real share
demands, $v_t\theta_t$, must also be bounded over time.
Consequently, the corporation is assumed to limit its choice of a
corporate plan to those plans entailing a {\it bounded\/} sequence
of expected real share demands $(v_t\theta_t^e)$.

    The behavior of the corporation can now be summarized.  The
corporation in period 1 takes the sequence ${\bf P}$ = $(P_1,P_2,
\ldots )$ of current and future goods prices as given, as do its
consumer-shareholders.  Conditional on this information, the
corporation chooses a corporate plan ${\bf I}$ = (${\bf v},{\bf
d^e})$ for periods $t \geq 1$ consisting of a finite-valued
nonnegative share price sequence ${\bf v}$ and a finite-valued
nonnegative expected dividend sequence ${\bf d^e}$.

     The corporation desires to exist indefinitely, implying that it
only considers corporate plans that are perceived to be viable. In
particular, then, the corporation only considers corporate plans
that generate nonnegative expected net receipts in each period, that
exhibit dividend consistency, and that imply a bounded sequence of
expected real share demands.  The corporation is only interested in
the subset of viable corporate plans that are in accordance with the
interests of its successive shareholders.  In particular, the
corporation first identifies the subset of viable corporate plans
yielding the highest expected wealth windfall for the generation 0
old consumer.  This subset is then further winnowed down by
requiring the successive maximization of the expected share rates of
return $q_t$, $t \geq 1$.  The (possibly empty) set of corporate
plans that remains will be referred to as the corporation's {\it
optimal choice set\/}, denoted by $\cal{I}({\bf P})$.  It is assumed
that the corporation is indifferent among all corporate plans in
$\cal{I}({\bf P})$.

     The success of a modern corporation is often judged by the
market value of its outstanding shares.  The behavior of our
corporation is consistent with this viewpoint.  To see this, note
that it follows from the definition (\ref{nrexpected}) for expected
net receipts, the definition (\ref{divexpected}) for the expected
per share dividend, and dividend consistency that $d^e_1\theta_0 =
v_1[\theta^e_1-\theta_0]$, hence $[v_1 + d^e_1]\theta_0$ =
$v_1\theta^e_1$.  This last relation shows that, by maximizing the
expected windfall return $[v_1 + d^e_1]\theta_0$ to the generation 0
old consumer, the corporation also maximizes $v_1\theta^e_1$, the
expected market value of its outstanding shares at the end of period
1.  Furthermore, using the expected dividend definition
(\ref{divexpected}) to eliminate $d^e_{t+1}$ from the expression
(\ref{sharerate}) for $q_t$, it follows that
                   \begin{equation}  \label{imv}
     q_t ~ \equiv ~ \frac{v_{t+1}\theta^e_{t+1}}{v_t\theta^e_t}~.
                     \end{equation} Consequently, by maximizing the
expected share rate of return $q_t$ for the generation $t$ consumer,
the corporation also maximizes the expected incremental increase in
the market value of its shares from period $t$ to period $t+1$.

%HERE IS RRACT4.TEX

\begin{center}
                   {\bf 4.  VIABILITY OF THE CORPORATION}
\end{center}

     The only way the corporation described in section 3 can viably
enhance the welfare of its shareholders is by incurring debt and
rolling it over forever.  Although $v_t\theta_t$ measures the market
value of the corporation's stock shares during period $t$, it also
measures the market value of the corporation's debt during period
$t$.  Thus, in attempting to increase the rate of return $q_t$ that
the generation $t$ consumer receives for holding its shares, the
corporation also increases the rate at which it assumes unsecured
debt.   This has led some to conclude that a privately owned firm
such as the one described here is not viable.  There are three basic
arguments.

     One argument, discussed by Lerner (1959; p.\ 523), is that such
a firm operates as an illegal ``chain letter'' or ``Ponzi scheme.''
However, as Lerner also recognized, if the economy does extend into
the infinite future, then in fact no one need be hurt by the chain
letter aspect of the operation because correspondents need not run
out.   More precisely, our corporation can viably operate in an
economy with an infinite time horizon as long as it attracts enough
shareholders in each period $t$ to fund the obligations it incurs in
period $t-1$.

     The second argument is related to the corporation's net worth
position.  Note that the net worth of the corporation is negative at
the end of each period $t$ because the corporation has no assets
(all receipts are paid out in the form of either share redemptions
or dividends) but still has a liability equal to $v_t\theta_t$.
Recognizing this situation, Cass and Yaari (1966; p. 360) conclude
that ``it is certainly the case that a privately owned financial
intermediary will not rescue the economy from inefficiency.'' They
arrive at this conclusion by noting that, while it has a negative
net worth whenever it operates, the firm can guarantee itself a zero
net worth by not operating.  They argue that a privately-owned firm
would therefore choose not to operate.

      This argument assumes that a corporation will use net worth
per se to gauge its proper course of action.   However, the owners
of the corporation---its shareholders---only care about net worth to
the extent that it affects the total net returns (capital gains or
losses plus dividend distributions) that are associated with holding
shares in the corporation.  Hence, a negative net worth per se will
not necessarily convince a corporation to cease its operations.
Indeed, in our context, a negative net worth for the corporation is
{\em necessary} in order for the corporation to distribute positive
dividends.

     The third argument against the existence of the type of
corporation we are considering is that the corporation's survival
might be threatened by the entry of other organizations which also
seek to incur and roll over debt.  In our context, the generation 0
old consumer who starts the corporation receives a wealth windfall.
However, this windfall can only be obtained if the generation 1
consumer can be persuaded to buy shares in the corporation.  Suppose
that the generation 1 consumer ``gets smart'' and refuses to
purchase shares in the corporation started by the generation 0
consumer, opting instead to start a similar corporation at the
beginning of period 2.  If successful, the generation 1 consumer
would receive the wealth windfall rather than the generation 0
consumer.   Of course, the generation 1 consumer would face the same
problem that the generation 0 consumer faced.  Thus, there is a real
possibility that the corporation could never get started.  Moreover,
even if the corporation manages to stay in existence for some number
of periods $t$, there is always the possibility that it will be
bankrupted in period $t+1$ if the generation $t+1$ consumer refuses
to buy the corporation's shares, opting instead to start his own
corporation.

     Samuelson (1958; pp.\ 480-482) discusses at some length the
potential usefulness of social compacts for enhancing social welfare
in situations characterized by a ``prisoner's dilemma.''  Roughly
described, a prisoner's dilemma occurs when all agents are better
off if all agents cooperate rather than mutually defect, yet it is
always privately rational for individual agents to defect.  This is
the type of dilemma that undermines the viability of the private
corporation introduced in the current context.  Even if all
consumers are potentially better off with the corporation in
existence, each individual consumer always perceives a private
advantage to be gained by ``defecting'' from the existing
corporation and starting a new one.

      Note, however, that this same dilemma undermines the viability
of fiat money.  An agent might anticipate that a wealth windfall
could be obtained by refusing to accept the fiat money currently in
use, and by instead issuing a new form of fiat money.  It was this
fragility of fiat money that motivated the discussion in Samuelson
(1958) regarding the significance of social compacts.   By agreeing
to accept fiat money issued by government as ``legal tender'' in
exchange for goods, private agents make it possible for this fiat
money to act as a store of value and hence also as a medium of
exchange.  A violation of this social compact could lead to the
demise of fiat money in any form [Bryant (1981)].

     This discussion makes a simple but important point:  An
organization will only be able to incur and roll over debt from one
period to the next if it obtains and maintains the confidence of
savers.  In the model presented here, the confidence that savers
have in fiat money allows a government to incur and roll over a
debt, while the confidence that savers have in share values allows a
corporation to incur and roll over a debt.  If confidence is lost in
either financial asset for whatever reason, then the ability to roll
over debt is lost.

     In reality, various regulations protect government's monopoly
over the issue of fiat money, just as chartering and other types of
regulations restrict entry into the private intermediation sector.
In this paper we abstract from the issue of entry.  We are concerned
with the efficiency implications of government intermediation
through fiat money versus private intermediation through corporate
shares, assuming both assets are protected from threat of entry.
Understanding whether or not a private earnings-driven intermediary
can even {\it in principle\/} enhance the efficiency of a
monetarized economy would seem to be a necessary prerequisite for
determining the extent to which social welfare might be improved by
the regulation of private intermediaries.

%HERE IS RRACT5.TEX

\vspace{2mm}
                               \begin{center}
               {\bf 5.  EQUILIBRIUM IN THE CORPORATE ECONOMY}
                                \end{center}

      This section sets out a definition of equilibrium for the
Corporate Economy.  It is assumed that the private intermediation
sector consists of a single corporation issuing unsecured debt, with
no threat of entry.  As will be clarified below (Propositions 6.5
and 6.8), the share prices announced by the optimizing private
intermediary will be positive whenever the initial real money supply
is lower than the particular level needed to achieve the golden rule
Pareto efficient outcome in the absence of the intermediary.  For
expositional simplicity, our definition of a Corporate Economy
equilibrium rules out the special case in which the initial real
money supply is set ``just right'' and focuses on those outcomes in
which the optimizing private intermediary plays a nontrivial role in
the economy through positive share price announcements.

\vspace{2mm} \noindent {\bf Definition 5.1.} Given an initial level
of money balances $M_0 > 0$ and an initial level of share holdings
$\theta_0 > 0$, a vector
                       \begin{equation} \label{bme} {\bf
e}(M_0,\theta_0) = ({\bf c},{\bf M},{\bf \theta},{\bf \theta^e},{\bf
      I},{\bf P})
                         \end{equation} consisting of a consumption
allocation ${\bf c}$ = $(c^o_1,(c^y_1,c^o_2),(c^y_1,c^o_2),\ldots
)$, a nominal money demand sequence ${\bf M}$ = $(M_1,M_2,\ldots )$,
a share demand sequence ${\bf \theta}$ = $(\theta_1,\theta_2,\ldots
)$, a corporate expected share demand sequence ${\bf \theta^e }$ =
$(\theta^e_1,\theta^e_2,\ldots )$, a corporate plan ${\bf I}$ =
$({\bf v},{\bf d^e})$ with ${\bf v} > 0$, and a nominal goods price
sequence ${\bf P}$ = $(P_1,P_2,\ldots )$ is an {\it equilibrium\/}
for the Corporate Economy, conditional on $M_0$ and $\theta_0$, if
it satisfies the following five conditions:
                            \begin{itemize}
   \item {\bf Positive Nominal Goods Prices:} $0 < P_t$ for each
$t\geq 1$.
   \item {\bf Consumer Optimization:} Consumer demands for goods,
money, and shares constitute finite-valued solutions to the
generation 0 consumer's budget constraint (\ref{bc0}) and the
lifetime utility maximization problem (\ref{umax}) for each $t \geq
1$.
    \item {\bf Corporate Optimization:} The corporate plan $I$ is an
element of the corporation's optimal choice set $\cal{I}({\bf P })$.
     \item {\bf Market Clearing for Goods and Money:} For each $t
\geq 1$,
                   \begin{equation} \label{cgmc}
                   w^y + w^o \geq c^y_t + c^o_t~;
                           \end{equation}
                   \begin{equation}\label{cmmc}
                       M_{t-1} \geq M_t~.
                          \end{equation}
    \item {\bf Fulfilled Share and Dividend Expectations:} For each
$t \geq 1$, $\theta^e_t$ = $\theta_t$ and $d^e_t$ =
$v_t[\theta_t-\theta_{t-1}]/\theta_{t-1}$.
                           \end{itemize}

     While the first condition is a non-primitive restriction that
excludes equilibria with zero nominal goods prices, it does permit
the price of money, $1/P_t$, to be zero in finite time.  In the
latter case, only shares can be used to store value.  The consumer
optimization condition ensures that each consumer is maximizing his
utility, conditional on expected prices and dividends, and the
corporate optimization condition ensures that the corporation is
acting in accordance with the interests of its shareholders.

     The goods market clearing condition (\ref{cgmc}) ensures the
feasibility of the equilibrium consumption allocation.  The money
market clearing condition (\ref{cmmc}) ensures that the demand for
money in period $t$ does not exceed the supply, which is given by
the amount of money held by the generation $t-1$ old agent.  Recall
that government only issues money in period 1.  The absence of a
market clearing condition for shares implies that the corporation is
free to issue new shares in each period $t$.  Consequently, the
number of shares $\theta_{t-1}$ which the generation $t-1$ old
consumer redeems in period $t$ places no direct restriction on the
number of shares $\theta_t$ which can be issued to the generation
$t$ young consumer.

     The final condition ensures that the corporation's share demand
and dividend expectations are correct.

%HERE IS RRACT6.TEX

\vspace{2mm} \begin{center}
                 {\bf 6. EFFICIENCY AND DYNAMIC PROPERTIES}
\end{center}

     If the corporate intermediary is eliminated from the Corporate
Economy, i.e., if share prices and expected dividend distributions
are simply constrained to be zero, then the Corporate Economy
reduces to the basic monetary OG economy.  The no-trade outcome in
which money is without value and each consumer in each period $t$
simply consumes his own endowment can be supported as a competitive
equilibrium for the latter economy; and, given the regularity
condition (\ref{sam}), this no-trade outcome is {\it not\/} Pareto
efficient [Gale (1973)].  Does the presence of an earnings-driven
corporate intermediary engaging in debt issue and rollover
necessarily eliminate the possibility of Pareto inefficient
equilibria for the Corporate Economy?  Can such an economy exhibit
an endogenously generated ``business cycle''?

     To begin the analysis of these questions, consider the
cross-sectional summation of the young age budget constraint of the
generation $t$ consumer and the old age budget constraint of the
generation $t-1$ consumer.  Assuming that dividend expectations are
fulfilled, one obtains
                     \begin{equation}\label{wl}
     0 ~ =  ~ [w^y+w^o-c^y_t-c^o_t] + [M_{t-1} - M_t]/P_t~,~ t \geq
1.
                          \end{equation} The market clearing
conditions (\ref{cgmc}) and (\ref{cmmc}) for goods and money then
imply that
                         \begin{eqnarray}
    0 ~&  = & ~ w^y + w^o - c^y_t -c^o_t ~,~ t \geq 1; \label{mc1}
\\
    0 ~ & = & ~ [M_{t-1} - M_t]/P_t~, ~t \geq 1.  \label{mc2}
                              \end{eqnarray}

     Recalling that the generation $t$ savings level is defined by
$s_t$ $\equiv$ $w^y - c^y_t$ for each $t \geq 1$, it follows from
the budget constraint (\ref{co}) and the goods market clearing
condition (\ref{mc1}) that $c^o_1 = w^o + s_1$ and
                     \begin{equation} \label{diff}
                     s_{t+1} ~= ~ q_ts_t~ , ~t \geq 1~.
                          \end{equation} Moreover, the first order
condition (\ref{mrs}) can be expressed as
                        \begin{equation}  \label{mrs1}
               MRS(w^y - s_t, w^o + q_ts_t) = q_t~,~t \geq 1~.
                       \end{equation} Given a value for the initial
share rate of return $q_1$, relations (\ref{diff}) and (\ref{mrs1})
determine all possible equilibrium paths for $s_t$ and $q_t$.

     In the following subsections the efficiency and dynamic
properties of Corporate Economy equilibria are examined, first under
the assumption that consumer preferences exhibit gross
substitutability, and then without this assumption.  The next two
propositions will be useful for this purpose.  The first proposition
establishes a uniform positive lower bound on the share rates of
return in any Corporate Economy equilibrium.

\vspace{2mm} {\bf PROPOSITION 6.1.} {\it In any Corporate Economy
equilibrium, the share rates of return $q_t$ satisfy $q_t \geq
MRS(w^y,w^o)$ for all $t \geq 1$.} \vspace{2mm}

     Recalling Proposition 2.1, it follows from Proposition 6.1 that
all young consumers in a Corporate Economy equilibrium choose
nonnegative savings levels $s_t$.  The next proposition relies on
the well-known Cass-Balasko-Shell ``transversality condition''
elaborated in Balasko and Shell (1980, Prop.\ 5.6, p.\ 296) to
obtain a general characterization of Pareto inefficient consumption
allocations for the Corporate Economy in terms of the long-run
behavior of these savings levels.%
     \footnote{Balasko and Shell impose certain technical regularity
conditions (referred to as Properties C, C', and G) on the curvature
of the utility function $U(\cdot )$ to rule out the possibility that
indifference surfaces come arbitrarily close to being kinked,
linear, or infinitely steep or flat along the equilibrium path,
respectively.  The uniform positive lower bound on the rates of
return $q_t$ established in Proposition 6.1 ensures that Property G
holds.  Hereafter it will be assumed that Properties C and C' hold
for $U(\cdot )$ as well, as additional strengthenings of our
previous restrictions on consumer preferences.  The reader is
referred to Balasko and Shell (1980) for a detailed discussion of
these regularity conditions.}

\vspace{2mm} {\bf PROPOSITION 6.2.} {\it Let ${\bf e}(M_0,\theta_0)$
= $({\bf c},{\bf M},{\bf \theta },{\bf \theta^e },{\bf I},{\bf P})$
denote an equilibrium for the Corporate Economy.  Suppose the
equilibrium consumption profiles $(c^y_t,c^o_{t+1})$ are uniformly
bounded above and below by strictly positive vectors.  Then ${\bf
c}$ is a Pareto inefficient consumption allocation if and only if\/
$\lim_{t \rightarrow \infty} s_t = 0$, where $s_t \equiv
[w^y-c^y_t]$.}

\vspace{4mm} \begin{center}
   6.A {\it Efficiency and Dynamic Properties Under Gross
Substitutability} \end{center}

        As established in Proposition 2.1, the optimal savings level
of each consumer is uniquely determined as a function $s(q)$ of the
share rate of return $q$ for each $q > 0$.  Suppose the preferences
of consumers in the Corporate Economy satisfy gross
substitutability, in the sense that $s(q)$ is a strictly increasing%
     \footnote{If the nonnegative bordered Hessian determinant
associated with the strictly quasi-concave utility function $U(\cdot
)$ is strictly positive, it follows from a comparative static
analysis of relation (\ref{mrs1}) that $\mbox{sign}(ds(q)/dq)$ =
$\mbox{sign}(U_2+s(q)[qU_{22} - U_{12}])$, where all partials of
$U(\cdot )$ are evaluated at $(w^y-s(q),w^o+qs(q))$.}
     function of $q$.  In this case each optimal savings level is
supported by a unique rate of return, and at most one equilibrium
path for $q_t$ and $s_t$ is associated with each initial rate of
return $q_1$.  As shown in Figure 1, the offer curve for the
generation $t$ consumer is everywhere negatively sloped.

     In any stationary-structured OG economy such as the Corporate
Economy, the ``golden rule'' rate of return is defined to be the
stationary rate of return $\bar{q}$ that supports the highest
possible lifetime utility for a representative consumer.  As shown
by Samuelson (1958), this rate of return (in gross terms) coincides
with $1$ plus the population growth rate $g$, where $g$ is assumed
to be zero for the Corporate Economy.  The golden rule consumption
profile and savings level supported by the Corporate Economy golden
rule rate of return $\bar{q}$ $=$ $1$ are depicted in Figure 1 as
$\bar{c}$ = $(\bar{c}^y,\bar{c}^o)$ and $\bar{s}$ $\equiv$
$w^y-\bar{c}^y$, respectively.  Given the regularity conditions
imposed on preferences in section 2, $\bar{q}=1$ is strictly greater
than $MRS(w^y,w^o)$, and $\bar{c}^y$, $\bar{c}^o$, and $\bar{s}$ are
necessarily positive.  Consequently, it follows from Proposition 6.2
that any Corporate Economy equilibrium supported by the stationary
golden rule rate of return $\bar{q}$ = $1$ is Pareto efficient.

     Do any such equilibria exist?  The next proposition provides a
first step toward answering this question.

\vspace{2mm} {\bf PROPOSITION 6.3.} {\it Given gross
substitutability, in any Corporate Economy equilibrium the
optimizing corporation must set $q_1 = 1$.} \vspace{2mm}

     As seen in the appendix proof of this proposition, $q_1$ $>$
$1$ violates the assumption that the corporation chooses a corporate
plan it perceives to be viable, and $q_1$ $<$ $1$ violates the
assumption that the corporation acts in the best interests of its
shareholders.  But this raises two new questions.  Do any Corporate
Economy equilibria exist that support the initial share rate of
return $q_1=1$?  If so, what are their efficiency properties?  The
following proposition answers both of these questions.

\vspace{2mm} {\bf PROPOSITION 6.4.} {\it Given gross
substitutability, the only possible Corporate Economy equilibria are
the Pareto efficient golden rule equilibria supported by $q_t$ = $1$
for all $t\geq 1$; and the set of such equilibria is not empty.}
\vspace{2mm}

     Proposition 6.4 shows that the corporation has a dramatic
impact on the efficiency and dynamic properties of the Corporate
Economy.  Assuming gross substitutability, the economy has a unique
equilibrium consumption allocation: namely, the Pareto efficient
golden rule consumption allocation characterized by the stationary
consumption profile $\bar{c}$, as depicted in Figure 1.  As seen in
the appendix proof of Proposition 6.4, this unique equilibrium
consumption allocation can be supported by Corporate Economy
equilibria entailing different corporate plans ${\bf I}$, money
demand sequences ${\bf M}$, share demand sequences ${\bf \theta}$,
expected share demand sequences ${\bf \theta^e}$, and price
sequences ${\bf P}$.

     In contrast, Gale (1973) establishes that the basic monetary OG
economy with preferences satisfying gross substitutability and
regularity condition (\ref{sam}) has {\it infinitely\/} many
equilibrium consumption allocations.  Only one of these consumption
allocations---the golden rule consumption allocation supported by
the stationary rate of return $\bar{q}=1$---is Pareto efficient.
All other equilibrium consumption allocations are Pareto
inefficient, and are associated with initial rates of return that
are less than one.  In the Corporate Economy an initial rate of
return less than one on both money and shares gives the corporation
an opportunity to increase the welfare of each of its shareholders
by raising the initial rate of return on shares and by increasing
its issue of unsecured debt.  But active private financial
intermediaries are missing in the basic monetary OG economy, hence
the potential profit opportunities arising from intermediation can
remain unexploited.

     Proposition 6.4 also raises another interesting point: the
fundamental trade-off between efficiency and stability in both the
Corporate Economy and the basic monetary OG economy.  Assuming gross
substitutability, the setting of an initial rate of return exceeding
one ultimately results in insolvency for either economy.  For
example, making repeated use of relation (\ref{diff}) and gross
substitutability, a simple induction argument establishes that the
optimal savings sequence ($s_t$) for the Corporate Economy must
diverge to $+\infty$ if $q_1>1$, implying that the finite resources
$w^y+w^o$ available in the economy in each period $t$ are exceeded
in finite time.  The drive of an active earnings-driven corporation
to enhance the welfare of its shareholders pushes the initial rate
of return to $q_1=1$, and hence increases efficiency; but it also
pushes the economy to the brink of collapse.  Similarly, for the
basic monetary OG economy, a government monetary policy designed to
achieve efficiency would (if possible) set the initial rate of
return on money to one, and hence bring the economy to the brink of
collapse as well.

    Although the set of possible Corporate Economy equilibria
exhibits a considerable degree of indeterminacy with regard to the
setting of nominal variables, some interesting inferences can be
drawn from the finding in Proposition 6.4 that the equilibrium
wealth windfall of the generation 0 old consumer, $M_0/P_1 +
[v_1+d_1]\theta_0$, must equal the golden rule savings level,
$\bar{s}$.  Since $[v_1+d_1]\theta_0$ $\geq$ $0$, this relation
implies that the initial real money supply, $M_0/P_1$, cannot exceed
$\bar{s}$ in equilibrium.  As the next proposition shows, the
corporation has a viable way to enter the economy if and only if
$M_0/P_1 < \bar{s}$.

\vspace{2mm} {\bf PROPOSITION 6.5.} {\it Let $M_0 > 0$ and $\theta_0
> 0$ be given, and suppose consumer preferences satisfy gross
substitutability.  Then in order for a Corporate Economy equilibrium
${\bf e}(M_0,\theta_0 )$ to exist, the nominal goods price sequence
${\bf P}$ = $(P_1,P_2,\ldots )$ must satisfy $M_0/P_1$ $<$ $\bar{s}$
and $P_t = P_{t+1}$ for all $t \geq 1$.  Conversely, given any such
price sequence, there exists a Corporate Economy equilibrium ${\bf
e}(M_0,\theta_0 )$ that generates this price sequence; and the
market value of the corporation in this equilibrium is given by
$v_t\theta_t$ = $[\bar{s}-M_0/P_1]$ $>$ $0$ in each period $t \geq
1$. } \vspace{2mm}

     Consequently, if preferences exhibit gross substitutability,
then government can prevent the viable entry of a private corporate
intermediary issuing unsecured debt if it can set the initial real
money supply equal to the golden rule level $M_0/P_1$ = $\bar{s}$.
In this special case the Pareto efficient golden rule outcome is
achieved without the need of a private intermediary.  On the other
hand, if the initial real money supply is below this golden rule
level for any reason, then a profit opportunity exists in the
economy that can be exploited by the entry of a private intermediary
willing to issue and forever roll over unsecured debt.

\vspace{4mm} \begin{center}
 6.B {\it Efficiency and Dynamic Properties Without Gross
Substitutability} \end{center}

     Suppose, instead, that consumer preferences fail to satisfy
gross substitutability.  The optimal consumption and savings levels
of each consumer are still uniquely determined as functions
$(c^y(q),c^o(q),s(q))$ of the rate of return $q$ for each $q>0$.
However, the absence of gross substitutability implies that there
exist savings levels $s_t$ for which the first order condition
(\ref{mrs1}) determines at least two possible supporting rates of
return $q_t$.  Consequently, the mapping from $q$ to $s(q)$ is not
one-to-one and the offer curve of each consumer bends back upon
itself at least once, as illustrated in Figure 2.  It follows that
more than one equilibrium path can be associated with an initial
rate of return $q_1$.

     In view of the potential complications caused by this
indeterminacy of equilibrium paths, most researchers using the OG
framework have assumed gross substitutability.%
     \footnote{An important exception is Grandmont (1985), who
relies on the absence of gross substitutability to obtain his
endogenous competitive business cycle.}
     Nevertheless, some efficiency properties can still be
established in its absence.

     By assumption (\ref{sam}), the offer curve of each consumer has
a slope $-MRS(w^y,w^o)$ $>$ $-1$ at the endowment point $(w^y,w^o)$,
implying that the optimal savings level $s(q)$ is a positive and
increasing function of $q$ for sufficiently small $q$ $>$
$MRS(w^y,w^o)$.  Suppose that the offer curve of each consumer is as
depicted in Figure 2.  More precisely, suppose that the offer curve
bends back upon itself only once, that the bend point is supported
by a rate of return $q^*$ that is strictly less than one and
strictly greater than $MRS(w^y,w^o)$,%
     \footnote{Another alternative would be for the offer curve
shown in Figure 2 to bend back upon itself at a supporting rate of
return $q^* \geq 1$.  However, using the geometric analysis
described below, one can deduce for this case that no equilibrium
could exist with $q_t$ $>$ $1$ in any period $t \geq 1$.  Thus, this
case is essentially the same as the case examined in section 6.A for
gross substitutability.}
      and that the optimal old-age consumption level $c^o(q)$
implied by this offer curve is a strictly increasing function of $q$
for all $q \geq MRS(w^y,w^o)$ and satisfies $\lim_{q \rightarrow
+\infty}c^o(q)$ = $+\infty$.%
   \footnote{Recalling that $c^o(q)$ = $w^o+qs(q)$, it can be shown
using footnote $3$ that a sufficient condition for $dc^o(q)/dq$ $>$
$0$ for all $q > 0$ is that the utility function be separable; i.e.,
$U_{12}$ $\equiv$ $0$.  In addition, separability of the utility
function also guarantees that $\lim_{q \rightarrow +\infty}c^o(q) =
+\infty$; for it follows from $c^o(q) = w^o + qs(q)$ that $s(q)$ =
$[w^y - c^y(q)]$ must tend to $0$ as $q$ approaches $+\infty$ if
$c^o(q)$ remains bounded.  But it then follows from separability
that $U_1(c^y(q),c^o(q))$ remains bounded while
$qU_2(c^y(q),c^o(q))$ diverges to $+\infty$, a violation of the
first-order condition (\ref{mrs}).}
     Given these restrictions on the offer curve of each consumer,
the Corporate Economy in the {\it absence} of the corporation
essentially reduces to the basic monetary OG economy studied by
Grandmont (1985).

     To understand more clearly what is depicted in Figure 2,
suppose ${\bf q} = (q_1,q_2,\ldots)$ is an equilibrium sequence of
return rates.  Given any point $A$ on the offer curve in Figure 2
associated with the equilibrium savings level $s(q_t)$ = $w^y -
c^y(q_t)$ for the young consumer in period $t$, note that the
projection of $A$ onto the vertical axis gives the equilibrium
consumption level $c^o_{t+1}$ = $c^o(q_{t})$ = $w^o + q_ts(q_t)$ for
the old consumer in period $t+1$.  Thus, moving horizontally from
$A$ to the point $B$ on the budget constraint associated with the
rate of return $\bar{q} = 1$, it follows from the goods market
clearing condition (\ref{mc1}) that the projection of $B$ onto the
horizontal axis must yield the equilibrium consumption level
$c^y_{t+1}$ = $c^y(q_{t+1})$ = $w^y + w^o - c^o(q_t)$ for the young
consumer in period $t+1$.  The equilibrium savings level
$s(q_{t+1})$ for the young consumer in period $t+1$ is then given by
the distance $w^y-c^y(q_{t+1})$, which by construction equals
$q_ts(q_t)$.  This is simply a geometric illustration of condition
(\ref{diff}), shown earlier to be a necessary condition for
equilibrium.

     Let $s_{max}$ denote the maximum possible level for the
consumer's optimal savings, achieved at the first and only bend
point in the offer curve.  By assumption, this maximum savings level
is supported by the rate of return $q^*$, i.e., $s_{max}$ =
$s(q^*)$.  Note that $s_{max}$ in Figure 2 is associated with a rate
of return $q_{max}$ through the relation $s_{max}$ =
$q_{max}s(q_{max})$.  Recalling (\ref{diff}), this relation has the
following interpretation:  given the rate of return $q_{max}$ in
some period $t$, the equilibrium savings level for period $t+1$ is
$s_{max}$.  For later purposes, various properties of $s_{max}$ and
$q_{max}$ depicted in Figure 2 will now be established analytically.

\vspace{2mm} {\bf PROPOSITION 6.6.} {\it Suppose the offer curve of
the consumer bends back upon itself exactly once, at a point
supported by a rate of return $q^*$ satisfying $MRS(w^y,w^o)<q^*<1$,
and that the optimal old-age consumption level $c^o(q)$ is a
strictly increasing function of $q$ for all $q \geq MRS(w^y,w^o)$
and satisfies $\lim_{q \rightarrow +\infty}c^o(q)$ = $+\infty$.
Then $\bar{s}$ $<$ $s_{max}$ $<$ $w^y$, and there exists a unique
solution $q_{max}$ to the equation $s_{max}$ = $qs(q)$, where
$q_{max}$ $>$ $1$.  Moreover, in any Corporate Economy equilibrium
the share rates of return $q_t$ lie in the interval
$[MRS(w^y,w^o),q_{max}]$, and the consumption profiles
$(c^y_t,c^o_{t+1})$ are uniformly bounded above and below by
strictly positive vectors.} \vspace{2mm}

     Propositions 6.2 and 6.6 can now be used to establish a first
welfare theorem for the Corporate Economy in the absence of gross
substitutability.  The proof of this result, given in the appendix,
proceeds by showing that the existence of a Pareto inefficient
equilibrium contradicts the assumption that the corporation acts in
the best interests of its shareholders.

\vspace{2mm} {\bf PROPOSITION 6.7.} {\it Under the assumptions of
Proposition 6.6, all Corporate Economy equilibria are Pareto
efficient.}
   \vspace{2mm}

     Several important questions still need to be addressed:  Do
equilibria necessarily exist for the Corporate Economy in the
absence of gross substitutability?  And, if so, what dynamic
properties do they exhibit?  The next proposition establishes that
equilibria do exist, given the assumptions of Proposition 6.6 and an
additional restriction on the curvature of the upper branch of the
offer curve.  Interestingly, in the absence of the latter curvature
restriction the optimizing corporation may exhibit time inconsistent
behavior---i.e., a desire in some period $t > 1$ to veer away from
the corporate plan announced in period $1$ in order to improve the
welfare of its shareholders.  We shall return to this point below,
after a discussion of the implications of Proposition 6.8.

\vspace{2mm} {\bf PROPOSITION 6.8.} {\it  Suppose the assumptions of
Proposition 6.6 hold.  In addition, suppose the curvature of the
upper branch of the offer curve is such that
                     \begin{equation} \label{regoffercurve}
                       q_{max} ~ \geq~ 1/q^* ~.
                               \end{equation} Then there exist
infinitely many Corporate Economy equilibria; but all of these
equilibria entail the same real equilibrium solution values for
consumption levels, savings levels, and share rates of return.  In
particular, in any Corporate Economy equilibrium the optimizing
corporation sets $q_1=q^*$, and all equilibrium share rates of
return $q_t$ lie in the interval $[q^*,q_{max}]$.}
   \vspace{2mm}

     It is informative to sketch here a heuristic proof for
Proposition 6.8.  A detailed proof can be found in the appendix.

     Let $M_0$ and $\theta_0$ be any given positive values for the
money and share holdings of the generation 0 consumer.  In any
Corporate Economy equilibrium the windfall return of the generation
0 consumer, $M_0/P_1 + [v_1 + d^e_1]\theta_0$, must equal the
savings level $s_1$ of the generation 1 consumer.  Given the
assumptions of Proposition 6.6, the maximum possible value for $s_1$
is $s_{max}$.

     Let $P^*_1$ denote any positive (possibly infinite) nominal
price level for period 1 that satisfies $0 \leq M_0/P^*_1$ $<$
$s_{max}$.  Then, taking $M_0/P^*_1$ and $\theta_0 $ as given, it
follows from the definition of a Corporate Economy equilibrium that
the optimizing corporation sets $[v_1 + d^e_1] > 0$ in period 1 so
that the windfall return of the generation 0 consumer is equal to
$s_{max}$.  But, given such a promised windfall to the generation 0
consumer, the highest feasible rate of return that the corporation
can offer to the generation 1 consumer is $q^*_1 \equiv q^*$,
resulting in the savings level $s^*_1 = s(q^*) \equiv s_{max}$; for
any higher rate of return would result in a lower savings level, and
the corporation would not be able to fulfill its contractual
obligations to the generation 0 consumer.

     By relation (\ref{diff}), the equilibrium savings level for
period $2$ must then be given by $s^*_{2}$ = $q^*_1s(q^*_1)$  $<$
$s_{max}$.  The restriction (\ref{regoffercurve}) implies that
$s^*_2 \geq s(q_{max})$.  Consequently, as depicted in Figure 2, two
possible rates of return $q_2$ support the savings level
$s^*_2$---that is, two possible rates of return $q_2$ satisfy
relation (\ref{mrs1}) for the given $s^*_2$---and neither of these
rates of return exceeds $q_{max}$.  But it follows from the
definition of a Corporate Economy equilibrium that the optimizing
corporation always chooses the highest possible rate of return for
each successive generation.  The economy thus moves along the upper
branch of the offer curve, the branch that passes through the golden
rule consumption point $\bar{c}$.

     Continuing this line of reasoning, the equilibrium savings
levels $s^*_t$ and rates of return $q^*_t$, and hence also the
equilibrium consumption levels $c^{y*}_t$ and $c^{o*}_{t+1}$, are
uniquely determined by successive applications of relations
(\ref{diff}) and (\ref{mrs1}), together with the ``selection
principle'' that the highest possible rate of return is always to be
chosen.  The regularity condition (\ref{regoffercurve}) guarantees
that the rates of return $q^*_t$ all lie in the interval
$[q^*,q_{max}]$, hence the economy remains on the upper branch of
the offer curve for all periods $t \geq 1$.

     As shown in the detailed appendix proof of Proposition 6.8 ,
infinitely many Corporate Economy equilibria can be constructed to
support this unique real equilibrium solution for consumption,
savings, and share rates of return.  One source of indeterminacy is
the degree of freedom the corporation has to realize the rates of
return $q^*_t$ either in the form of share price appreciation or in
the form of dividend distributions.  Another source of indeterminacy
is the setting of the initial real money balances $M_0/P^*_1$.  In
analogy to the case of gross substitutability, the corporation is
viable as long as these real balances are strictly less than
$s_{max}$, the maximum possible level for consumer savings; for this
permits the corporation to issue and roll over positively valued
shares in each period $t \geq 1$.

     What dynamic properties are exhibited by the unique real
equilibrium solution for the Corporate Economy?  The next
proposition establishes that only three relatively simple types of
dynamic behavior are possible.

\vspace{2mm} {\bf PROPOSITION 6.9.} {\it Under the hypotheses of
Proposition 6.8, only three types of dynamic behavior are possible
for the rate of return sequence $(q^*_t)$ in any Corporate Economy
equilibrium: either (a) condition (\ref{regoffercurve}) holds as an
equality and the sequence $(q^*_t)$ cycles back and forth between
$q^*$ and $q_{max}$ in a period $2$ orbit, starting with $q^*_1$ =
$q^*$; or (b) condition (\ref{regoffercurve}) holds as a strict
inequality and the sequence $(q^*_t)$ converges to a limit cycle
supported by a period $2$ orbit $\{q_L,q_U\}$ with $q^*$ $<$ $q_L$
$<$ $1$ $<$ $q_U$ $<$ $q_{max}$ and $q_Lq_U$ = $1$; or (c) condition
(\ref{regoffercurve}) holds as a strict inequality and the sequence
$(q^*)$ converges cyclically to the golden rule rate of return
$\bar{q}=1$.} \vspace{2mm}

    In summary, under the hypotheses of Proposition 6.8 the
Corporate Economy in equilibrium either exhibits a deterministic and
endogenously generated period $2$ ``business cycle'' or converges
cyclically to the golden rule consumption allocation.  In either
case, however, it follows from Proposition 6.7 that the equilibrium
is Pareto efficient.

     If condition (\ref{regoffercurve}) does not hold, implying that
$q^*s(q^*)$ $<$ $s(q_{max})$, then no equilibrium exists for the
Corporate Economy.  To understand why, note that it is always in the
best interest of the generation 0 shareholder if the corporation
sets $q_1=q^*$ in period 1 to ensure that the generation 1 consumer
saves the maximum amount $s(q^*) = s_{max}$.  The consumer budget
constraints and the product market clearing condition embodied in
condition (\ref{diff}) then imply that the savings level $s_2$ of
the generation $2$ consumer must equal $q^*s(q^*)$.

     From the consumer's perspective, two different rates of return
$q^{'}_2$ and $q^{''}_2$ support the savings level $s_2=q^*s(q^*)$
in period $2$, with $q^{'}_2 < 1 < q^{''}_2$.  Given $q^*s(q^*)$ $<$
$s(q_{max})$, however, neither represents a feasible equilibrium
rate of return.  The higher rate of return $q^{''}_2$ exceeds
$q_{max}$, and is thus not feasible by Proposition 6.6.  The lower
rate of return $q^{'}_2$ leads the corporation into a time
inconsistency problem.  The corporation in period $2$ would desire
to deviate from the corporate plan it announced in period $1$ by
giving the generation $1$ old consumer an unexpected windfall
return, but such reoptimization is not allowed under the current
definition of a Corporate Economy equilibrium.

     More precisely, the following scenario might occur when
condition (\ref{regoffercurve}) fails to hold.  To attract and
satisfy its potential shareholder in period 1, the corporation in
period 1 announces a corporate plan with $q_1 = q^*$ and $q_2$ =
$q^{'}_2$.  The period 1 rate of return ensures that the generation
1 consumer chooses the maximum possible savings level and hence that
the generation 0 old consumer receives the maximum possible windfall
return.  Once period $2$ is actually reached, however, the rate of
return $q^{'}_2$ is not in the best interest of either the
generation 1 old shareholder or the generation $2$ young
shareholder.  Rather, these shareholders would best be served by the
corporation changing its corporate plan and setting $q_2=q^*$ to
generate the maximum savings level from the generation $2$ young
consumer and hence an unexpected windfall return for the generation
1 old consumer.

    The difficulty is that if the generation 1 young consumer
perceives a positive probability that he will receive a windfall
return from the corporation in period $2$---i.e., a return above and
beyond the return generated by the dividend and/or share price
appreciation promised in the original corporate plan---then he might
lower his savings level in period 1 below $s_{max}$, making the
generation 0 old consumer worse off.  Consequently, to obtain a
deterministic outcome for this case, more detail would have to be
added to the model concerning how the consumers and the corporation
behave in the uncertain environment created by the possibility of
time inconsistent reoptimization.

%HERE IS RRACT7.TEX

\vspace{2mm} \begin{center}
                            {\bf 7. CONCLUSION} \end{center}

     Trade and credit arrangements in modern market economies are
primarily accomplished through earnings-driven private
intermediaries such as retail stores, banks, and brokerage firms.
Understanding how private intermediaries affect the allocation of
resources is therefore of considerable importance.  The findings of
this paper suggest that the inclusion of private intermediaries is
essential for the study of efficiency in OG economies, even in the
absence of transactions costs and asymmetric information problems.
In particular, it is shown that a first welfare theorem can be
recovered for the basic monetary OG economy if the economy is
generalized to encompass a corporate intermediary that maximizes its
market value in direct accordance with the interests of its
successive shareholders.

     Further work is of course needed to check the robustness of
these findings.  One important issue is the degree of market power
exercised by private intermediaries.  In this paper it is assumed
that the corporate intermediary exercises control over both its
dividend distributions and its share prices, which in turn determine
the equilibrium interest rates for the economy.  Because of product
differentiation, chartering restrictions, and so forth, actual
financial intermediaries often do exercise some market power in
setting rates of return for local deposits and loans [Hannan and
Berger (1992)], but the intermediary may be too small relative to
the securities market to influence its marginal funding costs and
its marginal earnings on other financial assets such as
large-denomination CD's [Fama (1985)].

     The issue of market power is closely tied to the issue of
entry.  Empirical findings suggest that chartering and other
restrictions currently in force do reduce initial entry into
financial markets [Amel and Liang (1992)], but the special features
of financial intermediaries that might warrant such particular forms
of supervision are still under debate.  Nearly all arguments for the
regulation of private financial intermediaries to date have been
based on the belief that information problems are particularly
severe in financial markets [Gertler (1988), Williamson (1992)].
This paper suggests that some form of initial entry deterrence may
be necessary to ensure the long-run viability of private financial
intermediaries, even in the absence of information problems; but
complete deterrence resulting in a sole reliance on financial assets
passively supplied by government to coordinate trade and credit will
generally be inefficient.

     Another interesting area for future research concerns the
relationship between efficiency and private intermediation in
economies with capital accumulation.  In Pingle and Tesfatsion
(1991c) it is shown that the inclusion of a private earnings-driven
corporate intermediary can eliminate the inefficiency that arises in
the one-sector growth model studied by Diamond (1965) and Tirole
(1985).  Still unresolved, however, is whether private
intermediation alleviates the ``inadequate distribution of capital
among firms'' that can arise in multi-sector growth models when the
production decisions of individual firms result in aggregate capital
overaccumulation [Malinvaud (1953)].

     Finally, Grandmont (1985) stresses the importance of taking due
account of learning when formulating the dynamics of an economy.  In
particular, he establishes for the basic monetary OG economy that
the stability properties of equilibria depend on the processes that
agents use to form their price expectations; and this point is
surely relevant for the Corporate Economy as well.  Interestingly,
as seen in section 6, another type of learning problem also arises
for the Corporate Economy: namely, in the absence of gross
substitutability, it can happen that the corporate intermediary will
face a time inconsistency problem.  That is, in some period $t > 1$
the corporate intermediary might desire to deviate from the
corporate plan it announced in period 1 in order to increase the
welfare of its current and future shareholders.  In this case the
dynamic path of the economy cannot be determined without a more
fully articulated modelling of consumer decision making in the face
of uncertain corporate behavior.

%HERE IS RRACTR.TEX

\noindent {\bf REFERENCES} \vspace*{2mm}

\setlength{\baselineskip}{15pt}

\begin{verse}

AMEL, D. F., and LIANG, J. N. (1992), ``A Dynamic Model of Entry and
Performance in the U.S.\ Banking Industry'' (Federal Reserve Board,
Washington D. C., Discussion Paper 210, Finance and Economics
Discussion Series) \\[2ex]

AZARIADIS, C. (1993), {\it Intertemporal Macroeconomics\/}
(Cambridge, MA:  Blackwell). \\[2ex]

BALASKO, Y., and SHELL, K. (1980), ``The Overlapping Generations
Model, I:  The Case of Pure Exchange Without Money,'' {\em Journal
of Economic Theory}, 23, 281-306.\\[2ex]

BALASKO, Y. and SHELL, K. (1981), ``The Overlapping Generations
Model, II:  The Case of Pure Exchange with Money,'' {\em Journal of
Economic Theory\/}, 24, 112-142. \\[2ex]

BRYANT, J. (1981), ``Bank Collapse and Depression,'' {\em Journal of
Money, Credit, and Banking\/}, 13, 454-464.\\[2ex]

CASS, D., and YAARI, M. (1966), ``A Re-examination of the Pure
Consumption Loan Model,'' {\em Journal of Political Economy}, 74,
353-367. \\[2ex]

CHAMP, B. and FREEMAN, S. (1994), {\em Modelling Monetary
Economies\/} (New York: John Wiley and Sons, Inc.). \\[2ex]

DIAMOND, P. (1965), ``National Debt in a Neoclassical Growth
Model,'' {\it American Economic Review\/}, 55, 1126-1150.\\[2ex]

FAMA, E. (1985), ``What's Different About Banks?,'' {\em Journal of
Monetary Economics\/}, 15, 29-39.\\[2ex]

GALE, D. (1973), ``Pure Exchange Equilibrium of Dynamic Economic
Models,'' {\em Journal of Economic Theory}, 6, 12-36. \\[2ex]

GERTLER, M. (1988), ``Financial Structure and Aggregate Economic
Activity: An Overview,'' {\em Journal of Money, Credit, and
Banking\/}, 20, 559-596.\\[2ex]

GRANDMONT, J.M., and LAROQUE, G. (1973), ``Money in the Pure
Consumption Loan Model,'' {\em Journal of Economic Theory\/}, 6,
382-395. \\[2ex]

GRANDMONT, J.M. (1985), ``On Endogenous Competitive Business
Cycles,'' {\it Econometrica\/}, 53, 995-1045.\\[2ex]

HANNAN, T. H., and BERGER, A. N. (1991), ``The Rigidity of Prices:
Evidence from the Banking Industry,'' {\em American Economic
Review\/}, 81, 938-945.\\[2ex]

KANE, E. (1989), ``The High Cost of Incompletely Funding the FSLIC
Shortage of Explicit Capital,'' {\it Journal of Economic
Perspectives\/}, 3, 31-47. \\[2ex]

LERNER, A. (1959), ``Consumption-Loan Interest and Money,'' {\it
Journal of Political Economy\/}, 67, 512-518, plus ``Rejoinder,''
{\it Ibid.\/}:523-525. \\[2ex]

MALINVAUD, E. (1953), ``Capital Accumulation and the Efficient
Allocation of Resources,'' {\em Econometrica\/}, 21, 233-268, plus
``Efficient Capital Production: A Corrigendum,'' {\em Ibid.\/}, 30
(1962), 570-573.\\[2ex]

PINGLE, M., and TESFATSION, L. (1991a), ``Overlapping Generations,
Intermediation, and the First Welfare Theorem,'' {\em Journal of
Economic Behavior and Organization\/}, 15, 325-345. \\[2ex]

\underline{\hspace{2in}} (1991b), ``Intermediation in Overlapping
Generations Economies'' (Iowa State University, Economic Report No.\
29). \\[2ex]

\underline{\hspace{2in}} (1991c), ``Intermediation, Bubbles, and
Pareto Efficiency in Economies with Production'' (Iowa State
University, Economic Report No.\ 24). \\[2ex]

SAMUELSON, P. (1958), ``An Exact Consumption-Loan Model of Interest
With or Without the Social Contrivance of Money,'' {\em Journal of
Political Economy}, 66, 467-482.\\[2ex]

TIROLE, J. (1985), ``Asset Bubbles and Overlapping Generations,''
{\it Econometrica\/}, 53, 1499-1528. \\[2ex]

WILLIAMSON, S. (1992), ``Laissez-Faire Banking and Circulating Media
of Exchange'' (University of Western Ontario, Department of
Economics Working Paper). \\[2ex]

THOMPSON, E. (1967),``Debt Instruments in both Macroeconomic Theory
and Capital Theory,'' {\it American Economic Review\/}, 57,
1196-1210. \\[2ex]

\end{verse}

%HERE IS RRACAP.TEX

                             \begin{center}
                 {\bf  APPENDIX: PROPOSITION PROOFS}
                              \end{center}

{\bf PROOF OF PROPOSITION 2.1.} Suppose $P_t$ $<$ $+\infty$.  The
budget constraints for problem (\ref{umax}) can then be written as
                       \begin{eqnarray}
         c^y_t  & ~=~ & w^y - s_t ~; \label{sbcy} \\ c^o_{t+1} & ~=~
& w^o + q_ts_t + \frac{M_t}{P_t}[r_t-q_t]~,\label{life1}
             \end{eqnarray} where $r_t$ $\equiv$ $[P_t/P_{t+1}]$
denotes the gross rate of return on holding money from period $t$ to
$t+1$.  If $r_t$ $>$ $q_t$, it is apparent from (\ref{life1}) that
the generation $t$ consumer would desire to sell shares short
without bound in his youth in order to finance the acquisition of an
arbitrarily large holding of money.  Consequently, in order for
problem (\ref{umax}) to have a finite solution when $P_t < +\infty$,
it is necessary that $r_t \leq q_t$.

     If $P_t$ $<$ $+\infty$ and $r_t < q_t$, the young consumer in
generation $t$ will choose not to hold money (i.e., $M_t = 0$), and
the optimal consumption and savings levels of the generation $t$
consumer will be uniquely determined as functions of $q_t$ by the
relations (\ref{mrs}), (\ref{cy}), and (\ref{co}).  If $P_t$ $<$
$+\infty$ and $r_t = q_t$, the generation $t$ young consumer will be
indifferent between holding shares and holding money.  In this case
the particular levels of $M_t$ and $\theta_t$ will be indeterminate,
but the optimal consumption and savings levels of the generation $t$
consumer will again be uniquely determined as functions of $q_t$ by
relations (\ref{mrs}), (\ref{cy}), and (\ref{co}).

     In summary, if $P_t < \infty$, then a finite solution exists
for problem (\ref{umax}) if and only if either $r_t < q_t$ (implying
$M_t = 0$) or $r_t = q_t$.  In either case the budget constraints
(\ref{sbcy}) and (\ref{life1}) reduce to (\ref{cy}) and (\ref{co})
and the optimal consumption and savings levels are uniquely
determined as functions of $q_t$.  Moreover, it follows from the
regularity condition (\ref{sam}) that $s(q_t) \geq 0$ if and only if
$q_t \geq MRS(w^y,w^o)$.

     Suppose, instead, that $P_t$ and $P_{t+1}$ are both infinite.
Then the real money balances of the generation $t$ consumer are
zero-valued regardless of his choice of nominal money holdings
$M_t$. Consequently, the budget constraints (\ref{sbcy}) and
(\ref{life1}) once again reduce to (\ref{cy}) and (\ref{co}), and
the remaining assertions of Proposition 2.1 follow as for the case
$P_t < \infty$.  Finally, if $P_t$ = $+\infty$ and $P_{t+1}$ $<$
$+\infty$, then problem (\ref{umax}) has no solution since the
generation $t$ consumer will desire to hold an arbitrarily large
amount of (costless) money in period $t$. ~~~Q.E.D. \\

{\bf PROOF OF PROPOSITION 6.1.} By Proposition 2.1, $s(q) \geq 0$ if
and only if $q \geq MRS(w^y,w^o)$.  Suppose there exists a Corporate
Economy equilibrium in which the share rate of return $q_t$ first
falls below $MRS(w^y,w^o)$ in some period $t^* \geq 1$.  The
generation $t^*$ young consumer would then plan to borrow rather
than to save, i.e., $c^y_{t^*} > w^y$.  But the planned optimal
savings of the generation $t^*-1$ old consumer are nonnegative,
implying that $c^o_{t^*} \geq w^o$, since either this consumer is
the initial old consumer or he faced a share rate of return
$q_{t^*-1} \geq MRS(w^y,w^o)$ in period $t^*-1$.  The goods market
clearing condition (\ref{mc1}) thus fails for period $t^*$.  It must
therefore hold that $q_t \geq MRS(w^y,w^o)$ for all $t \geq 1$.
{}~~~Q.E.D. \\

{\bf PROOF OF PROPOSITION 6.2.} Let $p_t$ $\equiv $ $[q_{t-1}\cdots
q_1]^{-1}$ $>$ $0$ denote the real price of good $1$ measured in
units of good $t$, $t \geq 2$, with $p_1 \equiv 1$.  Then $p_{t+1}$
= $[1/q_t]p_t$, where it follows from Proposition 6.1 that $q_t$ is
uniformly bounded from below by the positive constant $MRS(w^y,w^o)$
for all $t\geq 1$.  It is then straightforward to show that Property
G in the Balasko-Shell Proposition 5.6 is satisfied.  Applying the
latter proposition to the Corporate Economy---a special case of the
Balasko-Shell pure exchange model in terms of the structural
specifications for preferences, real endowments, and technology
(nonstorable resource)---the equilibrium consumption allocation
${\bf c}$ is Pareto inefficient if and only if
$\sum_{t=1}^{t=+\infty} 1/p_t < +\infty$.

     Since the optimal savings levels $s_t$ are bounded between $0$
and $w^y$ for $q_t \geq MRS(w^y,w^o)$, either $\limsup s_t$ $=$ $0$
or $\limsup s_t$ $>$ $0$.  Suppose $\limsup s_t > 0$.  From
condition (\ref{diff}), $s_{t+1}$ = $[p_t/p_{t+1}]s_t$ for all $t
\geq 1$, implying that $s_t = [p_1/p_t]s_1$ for all $t\geq 1$.  If
$s_1 = 0$, then $s_t = 0$ for all $t \geq 1$, contradicting the
supposition.  Therefore, $s_1 > 0$.  If $\lim_{t \rightarrow +
\infty} [1/p_t] = 0$, then $\lim_{t \rightarrow + \infty} s_t = 0$
again contradicting the supposition.  Therefore $\sum^{t=T}_{t = 1}
[1/p_t]$ does not converge as $T \rightarrow +\infty$.  Because $p_t
> 0$ for all $t \geq 1$, it follows that $\sum^{t=+\infty}_{t = 1}
[1/p_t] = + \infty$.

     Conversely, suppose $\limsup s_t = 0$, which in turn implies
that $\limsup c^y_t = w^y$ and $\limsup c^o_{t+1} = w^o$.  It
follows from the regularity condition (\ref{sam}) that, for
sufficiently small $\epsilon$ in $(0,1)$, there exists a period $k$
such that $p^t/p^{t+1}$ = $MRS(c^y_t,c^o_{t+1})$ $\leq$ $[1-\epsilon
]$ for all $t \geq k$.  Defining $\rho_t$ $\equiv$ $1/p_t$ $=$
$[\frac{1}{p_2} \frac{p_2}{p_3} \cdots \frac{p_{t-1}}{p_t}]$, one
has $\rho_t$ $\leq$ $\rho_{k} [1-\epsilon]^{t-k}$ for all $t \geq
k$.  Thus, $\sum_{t = k}^{t = + \infty} 1/p_t$ $\leq$
$\rho_{k}/\epsilon$ $<$ $+ \infty$.  Finally, since $\sum_{t = 1}^{t
= k} 1/p_t < + \infty$, it follows that $\sum_{t = 1}^{t =+\infty}
1/p_t < + \infty$.

     In summary, the equilibrium consumption allocation ${\bf c}$ is
Pareto inefficient if and only if $\limsup s_t$ $=$ $0$.  However,
$s_t \geq 0$ for all $t \geq 0$ implies that $\limsup s_t = 0$ if
and only if $\lim_{t \rightarrow \infty} s_t = 0$. ~~~Q.E.D. \\

{\bf PROOF OF PROPOSITION 6.3.} Let a Corporate Economy equilibrium
be given, characterized by a rate of return sequence $(q_t)$.  By
Propositions 2.1 and 6.1, $q_t$ $\geq$ $MRS(w^y,w^o)$ and $s_t \geq
0$ for each $t \geq 1$.

     Suppose $q_1$ $\geq$ $k$ for some $k > 1$.  Then gross
substitutability implies $s_1$ $>$ $\bar{s}$ and relation
(\ref{diff}) implies $s_2$ $>$ $s_1$.  Knowing $s_2$ $>$ $s_1$,
gross substitutability implies $q_2$ $>$ $q_1$.  Making repeated use
of relation (\ref{diff}) and gross substitutability, a simple
induction argument then establishes that $q_{t+1}$ $>$ $q_t$ for all
$t\geq 1$, hence $q_t$ $\geq$ $k$ for all $t\geq 1$.  It follows
from the expression (\ref{imv}) for $q_t$ that
$v_{t+1}\theta^e_{t+1}$ $\geq$ $kv_t\theta^e_t$ for all $t \geq 1$,
which implies that the sequence of expected real share demands,
$v_t\theta^e_t$, diverges to $+\infty$.  But this violates the
section 3 assumption that the optimizing corporation only chooses
from among those corporate plans it perceives to be
viable---implying, in particular, that the corporate plan must
entail a bounded sequence of expected real share demands.
Consequently, in any Corporate Economy equilibrium, it cannot be
true that $q_1 > 1$.

    Suppose, instead, that $MRS(w^y,w^o)$ $\leq$ $q_1$ $<$ $1$.  It
will be shown that the corporation could then increase the welfare
of each of its shareholders by making a viable change in its
corporate plan, a violation of the assumption that the corporation
in any Corporate Economy equilibrium chooses a viable corporate plan
in accordance with the best interests of its shareholders.

     Suppose the corporation increases the share rate of return
$q_1$ to $\hat{q_1} \leq 1$ by increasing the share price $v_2$.
The gross substitutability assumption implies that this increase in
$q_1$ leads the generation 1 consumer to increase his savings level;
i.e., $\hat{s}_1 > s_1$.  Also, taking the price sequence $(P_t)$ as
given, the increase in $q_1$ would lead the generation $t$ consumer
to reduce his money demand to zero (if it were not already zero);
i.e., $\hat{M}_1 = 0$.  This implies that the real demand for shares
would become $v_1\hat{\theta}_1 = \hat{s}_1$.  Since $\hat{s}_1 >
s_1$ and $\hat{M}_1/P_1 \leq M_1/P_1$, it follows from the
generation 1 consumer's young age budget constraint that
$\hat{\theta_1} > \theta_1$.  This implies that the dividend of the
generation 0 consumer would increase from $d_1 = v_1[\theta_1
-\theta_0]/\theta_0$ to $\hat{d}_1 = v_1[\hat{\theta}_1
-\theta_0]/\theta_0$.  Thus, the welfare of the generation 0
consumer is enhanced due to a ``windfall'' dividend.

    Because $\hat{q}_1 > q_1$ and $\hat{s}_1 > s_1$ as determined
above, condition (\ref{diff}) implies that $\hat{s}_2 > s_2$ would
also have to hold in order for the sequences $(\hat{q}_t)$ and
$(\hat{s}_t)$ to be part of an equilibrium.  By gross
substitutability, $\hat{s}_2 > s_2$ implies $\hat{q}_2 > q_2$.
Continuing this line of reasoning, it is seen that for all $t \geq
1$, $\hat{q}_t$ $>$ $q_t$ $\geq$ $MRS(w^y,w^o)$.  Suppose, then,
that the corporation adjusts its corporate plan to generate these
higher share rates of return $\hat{q}_t$, $t \geq 1$.  Since the
optimal savings level of each consumer in the original equilibrium
was nonnegative, it follows from the budget constraints (\ref{cy})
and (\ref{co}) that each consumer is strictly better off. ~~~Q.E.D.
\\

{\bf PROOF OF PROPOSITION 6.4.} Recall that $\bar{q}=1$ is the
golden-rule rate of return for the Corporate Economy.  By
Proposition 6.3, $q_1$ must equal 1 in any Corporate Economy
equilibrium.  Given $q_1=1$, it follows by gross substitutability
that $s_1$ must equal the golden-rule savings level $\bar{s}$.
Also, relation (\ref{diff}) implies that $s_2=s_1$.  Hence, applying
gross substitutability once more, $q_2=q_1=1$.  A simple induction
argument then gives $q_t=1$ and $s_t$ = $\bar{s}$ for all $t \geq
1$.  Consequently, in any Corporate Economy equilibrium, the only
possible real outcome is the Pareto efficient golden-rule outcome
characterized by the stationary share rate of return $\bar{q}=1$,
the stationary savings level $\bar{s}$ for each young consumer, and
the stationary consumption levels $\bar{c}^y$ and $\bar{c}^o$ for
each young and old consumer, respectively.

     It will now be shown that there exists at least one Corporate
Economy equilibrium that supports this golden-rule outcome.  Suppose
that the sequence ${\bf P}$ = $(P_1,P_2,\ldots )$ of nominal goods
prices is a positive sequence satisfying
             \begin{equation}
 0 ~ \leq ~  M_0/P_1  ~ < ~ \bar{s}~;
   ~~  P_t ~ = ~ P_{t+1} ~,~ t \geq 1~. \label{gpc}
              \end{equation} Taking this sequence of nominal goods
prices as given, suppose the corporation selects the corporate plan
${\bf I}^*$ defined as follows: the corporation sets $d^e_t$ = $0$
for all $t\geq 1$, sets $v_1 > 0$ so that $v_1\theta_0$ =
$\bar{s}-M_0/P_1$, and sets $v_t=v_1$ for all $t \geq 2$.  Note that
the share rates of return implied by this corporate plan satisfy
$q_t = 1$ for all $t \geq 1$ by definition (\ref{sharerate}).

     Consider, first, the perceived viability of the plan ${\bf
I}^*$.  Suppose the share demands expected by the corporation under
this plan are $\theta^e_t$ = $\theta_0$ for all $t\geq 1$.  The real
share demands expected by the corporation over time then satisfy
$v_t\theta^e_t$ = $v_1\theta_0$ for all $t\geq 1$, a bounded
sequence.  Also, by relation (\ref{nrexpected}), the net receipts
expected by the corporation in any period $t$ are zero.  Finally,
zero expected net receipts in any period $t$ implies a zero expected
dividend distribution, hence the plan ${\bf I}^*$ exhibits dividend
consistency.  It follows that there is a set of corporate
expectations for share demands under which the corporation will
perceive the plan ${\bf I}^*$ to be viable.

     What about the optimality of the plan ${\bf I}^*$ from the
viewpoint of the corporation?  Under this plan, $q_t = 1$ for all
$t\geq 1$.  If the corporation were to attempt to increase the share
rate of return $q_t$ above $1$ in any period $t\geq 1$, relation
(\ref{imv}) implies that the sequence of real share demands expected
by the corporation would become unbounded, a violation of the
viability of the corporate plan.  Consequently, by gross
substitutability, the corporation cannot increase the savings level
$s_1$ of the generation 1 young consumer above $\bar{s}$, implying
that the corporation is maximizing the windfall return it can
provide to the generation 0 old consumer in period 1.  In addition,
the corporation is maximizing the share rate of return $q_t$ it can
provide to the generation $t$ young consumer in each period $t \geq
1$.  Hence, the corporation perceives the plan ${\bf I}^*$ to be
optimal.

     What about consumer optimization?  Given $q_t=1$ for all $t\geq
1$, it follows from (\ref{cy}) and (\ref{co}) that each young
consumer will choose to consume the golden-rule consumption profile
$(c^y_t,c^o_{t+1})$ = $(w^y - \bar{s}, w^o+\bar{s})$ =
$(\bar{c}^y,\bar{c}^o)$.  Moreover, under the plan ${\bf I}^*$, the
value of the shares held by the generation 0 old consumer is
$v_1\theta_0$ = $\bar{s} - M_0/P_1$, and his expected dividend is
$0$.  It follows from the budget constraint (\ref{bc0}) that the
generation 0 old consumer will choose to consume $c^o_1$ = $w^o +
\bar{s}$ = $\bar{c}^o$.

     The above discussion establishes that, for positive nominal
goods prices satisfying (\ref{gpc}), the consumer and corporate
optimization conditions in the definition of a Corporate Economy
equilibrium are satisfied if the corporation sets the plan ${\bf
I}^*$ and has share demand expectations $\theta^e_t$ = $\theta_0$
for all $t\geq 1$.  But what about the remaining conditions
requiring market clearing for goods and money and fulfilled share
and dividend expectations?

     Given ${\bf I}^*$, real outcomes must coincide with the
golden-rule solution, and the goods market clearing condition
(\ref{cgmc}) holds by construction for this golden-rule solution.
If $P_1 < +\infty$, then it follows from (\ref{gpc}) that the money
rate of return $r_t$ equals the share rate of return $q_t = 1$ in
each period $t$, implying that the consumer is indifferent between
holding his optimal real savings $\bar{s}$ in the form of money or
shares.  Without loss of generality, it can then be assumed that
$M_t$ = $M_0$ for all $t\geq 1$, implying that the money market
clearing condition (\ref{cmmc}) is satisfied.  Alternatively, if
$P_1$ = $+\infty$, then it follows from (\ref{gpc}) that $P_t$ =
$+\infty$ for all $t \geq 1$.  In this case money has no value and
$M_t$ = $0$ for all $t\geq 1$; but the money market clearing
condition (\ref{cmmc}) is clearly satisfied for all $t \geq 1$.  In
either case, $v_t\theta_t$ = $v_1\theta_0$ = $\bar{s} - M_0/P_1$ for
all $t\geq 1$.  But $v_t$ = $v_1$ $>$ $0$ for all $t\geq 2$ under
plan ${\bf I}^*$, which forces $\theta_t$ = $\theta_0$ for all
$t\geq 1$.  It follows that the actual net receipts (dividend
payout) of the corporation in each period $t$ are zero, exactly what
the corporation expects under plan ${\bf I}^*$.  Also, the
corporation's expected share demands are fulfilled.

      In summary, given gross substitutability, the only possible
Corporate Economy equilibria are the Pareto efficient golden-rule
equilibria supported by $q_t = 1$ for all $t\geq 1$.  And it has
been shown that at least one Corporate Economy equilibrium exists
for each nominal goods price sequence satisfying (\ref{gpc}).
{}~~~Q.E.D. \\

{\bf PROOF OF PROPOSITION 6.5.} The necessity of having $M_0/P_1
\leq \bar{s}$ in equilibrium follows from the fact that $M_0/P_1 +
[v_1+d_1]/\theta$ = $\bar{s}$ in equilibrium, as established in
Proposition 6.4.  But, given $\theta_0 > 0$, $M_0/P_1$ = $\bar{s}$
would force $[v_1+d^e_1]$ = 0, a violation of the requirement that
$v_1$ $>$ $0$ and $d^e_1$ $\geq$ $0$.  If $P_1$ $<$ $+\infty$, it
follows from the money market clearing condition (\ref{mc2}) that
$M_1$ = $M_0$ $>$ $0$; but the period 1 young consumer will only be
willing to hold a positive amount of money in lieu of shares if
$r_1=q_1=1$, implying $P_1 = P_2$.  A simple induction argument then
yields the necessity of having $P_t=P_{t+1}$ for every $t\geq 1$ in
any equilibrium.  Conversely, if $P_1$ = $+\infty$, no solution
exists for the utility maximization problem (\ref{umax}) in period 1
unless $P_2$ = $+\infty$; and a simple induction argument again
yields the necessity of having $P_t$ = $+\infty$ for all $t \geq 1$
in any equilibrium.

     Given a nominal goods price sequence satisfying $M_0/P_1 <
\bar{s}$ and $P_t$ = $P_{t+1}$ for each $t \geq 1$, together with
gross substitutability, the proof of Proposition 6.4 establishes the
existence of a Corporate Economy equilibrium with the market value
of the firm given by $v_t\theta_t$ = $\bar{s}-M_0/P_1$ $>$ $0$ for
each $t\geq 1$.  ~~~Q.E.D. \\

{\bf PROOF OF PROPOSITION 6.6.} The maximum optimal savings level
$s_{max}$ must be strictly less than $w^y$; for otherwise a consumer
would choose to consume nothing when young when faced with the rate
of return $q^*$, an impossibility given the restrictions on $U(\cdot
)$ imposed in section 2.  Moreover, the regularity condition
(\ref{sam}), together with the assumption of a unique backward bend
in the offer curve at $q^* < 1$, implies that $s_{max}$ $\equiv$
$s(q^*)$ is greater than the golden rule savings level $\bar{s}$ =
$s(1)$; see Figure 2.

     Since $qs(q)$ = $c^o(q) - w^o$ is a strictly increasing
function of $q$ that takes on the value $0$ at $q = MRS(w^y,w^o)$
and diverges to $+\infty$ as $q \rightarrow +\infty$, the equation
$s_{max} = qs(q)$ has a unique solution $q_{max} > 0$ for every
$s_{max} > 0$.  By definition of $s_{max}$, the rate of return
$q_{max}$ must be greater than $1$; for $q_{max}$ $<$ $1$ would
imply that $0$ $<$ $s_{max}$ = $q_{max}s(q_{max})$ $<$ $s(q_{max})$
$\leq$ $s_{max}$, a contradiction, and $q_{max}=1$ would imply that
$s_{max}$ = $\bar{s}$, another contradiction.  Also, no share rate
of return $q_t$ $>$ $q_{max}$ can occur in any Corporate Economy
equilibrium; for this would imply that the equilibrium savings rate
$s_{t+1}$ = $q_ts(q_t)$ for period $t+1$ exceeds $s_{max} =
q_{max}s(q_{max})$, a contradiction of the definition of $s_{max}$.

    Since by assumption $c^o(q)$ is an increasing function of $q$,
the finding $q_t \leq q_{max}$, together with the goods market
clearing condition (\ref{mc1}) and Proposition 6.1, imply that $0$
$<$ $w^o$ $=$ $c^o(MRS(w^y,w^o))$ $\leq$ $c^o(q_t)$ $\leq$
$c^o(q_{max})$ $\leq$ $w^o + w^y$, $t\geq 1$, in any Corporate
Economy equilibrium; that is, the equilibrium old-age consumption
levels are uniformly bounded above and below by positive finite
quantities.  Finally, it follows from the goods market clearing
condition (\ref{mc1}) and the finding $s_{max}$ $<$ $w^y$ that
$c^y(q_t)$ must satisfy $0$ $<$ $w^y - s_{max}$ $\leq$ $w^y -
s(q_t)$ = $c^y(q_t)$ $\leq$ $w^y + w^o$, $t\geq 1$; that is, the
equilibrium young-age consumption levels are also uniformly bounded
above and below by positive finite quantities. ~~~Q.E.D. \\

{\bf PROOF OF PROPOSITION 6.7.} Suppose there exists a Corporate
Economy equilibrium for which the consumption allocation is not
Pareto efficient.  By Propositions 6.2 and 6.6, the sequence $(s_t:t
= 1,2,\ldots )$ of (nonnegative) equilibrium savings levels $s_t$
must then converge to zero.  In particular, given any $\epsilon >
0$, there exist at most finitely many periods $t$ such that $s_t$
$>$ $\epsilon $.

     Let $\epsilon $ $\equiv$ $s(q_{max})/2$ $>$ $0$.  Then for each
$b \leq \epsilon$ the equation $s(q) = b$ has only one solution
$q(b)$ satisfying $q(b) \leq q_{max}$, and this solution satisfies
$0$ $<$ $q(b)$ $\leq$ $q(\epsilon )$ $<$ $q^*$ $<$ $1$; see Figure
2.  Let $k \geq 1$ be such that the equilibrium savings levels
satisfy $s_t$ = $s(q_t)$ $\leq$ $\epsilon /2$ for all $t \geq k$.
By Proposition 6.6, the equilibrium rates of return are uniformly
bounded above by $q_{max}$ and below by $MRS(w^y,w^o)$.  Hence, by
choice of $\epsilon $, the equilibrium rate of return $q_t$ for any
period $t \geq k$ must satisfy $MRS(w^y,w^o)$ $\leq$ $q_t$ $<$
$q(\epsilon )$ $<$ $q^*$ $<$ $1$.  It follows from (\ref{diff}) that
the (nonnegative) equilibrium savings rates $s_t$ converge to $0$
and the equilibrium rates of return $q_t$ converge to $MRS(w^y,w^o)$
as $t \rightarrow \infty$.

     Suppose the corporation increases the rate of return from $q_t$
to $q(\epsilon )$ $<$ $q^*$ in each period $t \geq k$ by an
announcement of new higher share prices ($\hat{v}_{t+1}: t \geq k)$
satisfying $\hat{v}_{t+1}$ = $q(\epsilon )\hat{v}_{t}$ for all $t
\geq k$, with $\hat{v}_k$ $\equiv$ $v_k$, and an announcement of an
increased expected dividend $\hat{d}^e_k$ for period $k$ (derived
below) and zero expected dividends for all periods $t+1$ with $t\geq
k$.  It will now be shown that these changes in its corporate plan
would be perceived by the corporation to be viable changes that
would benefit each of its shareholders, contradicting the assumption
that the economy was originally in a Corporate Economy equilibrium.

     By assumption, the optimal savings level $s(q)$ is a strictly
increasing function of $q$ for all $q$ $<$ $q^*$.  The increase from
$q_k$ to $q(\epsilon ) < q^*$ in period $k$ thus leads the
generation $k$ young consumer to increase his savings level; i.e.,
$\hat{s}_k > s_k$.  In particular, since the price sequence ${\bf
P}$ is taken as given, the increase in $q_k$ leads the generation
$k$ young consumer in period $k$ to reduce his money demand to zero
(if it is not already zero) and to increase his real demand for
shares.  Recalling that $\hat{v_k} \equiv v_k$, this implies that
$\hat{s}_k$ = $v_k\hat{\theta}_k$ $>$ $s_k$ = $v_k\theta_k +
M_k/P_k$ $\geq$ $v_k\theta_k$, hence $\hat{\theta_k}$ $>$
$\theta_k$.  It follows that the expected dividend of the generation
$k-1$ consumer increases from $d^e_k$ = $v_k[\theta_k
-\theta_{k-1}]/\theta_{k-1}$ to $\hat{d}^e_k$ = $v_k[\hat{\theta}_k
-\theta_{k-1}]/\theta_{k-1}$.  Thus, the welfare of the generation
$k-1$ consumer is enhanced due to a ``windfall'' dividend.
Moreover, the optimized lifetime utility of each generation $t$
consumer is an increasing function of the share rate of return $q_t$
over the range $q_t \geq MRS(w^y,w^o)$ where the consumer desires to
save, hence the change in corporate plan benefits all consumers in
generations $t \geq k$ as well.  The welfare of shareholders in any
generations previous to $k-1$ is unaffected.

     Setting the share rate of return equal to the constant level
$q(\epsilon )$ $<$ $q^*$ $<$ $1$ for all periods $t \geq k$ ensures
by (\ref{imv}) that the real share demands expected by the
corporation under its modified corporate plan converge to zero over
time, and hence constitute a bounded sequence.  Moreover, expected
profits $\hat{\pi}^e_{t}$ for periods $t \geq k$ are nonnegative.
More precisely, as explained above, the expected dividend (hence the
expected amount of profit) for period $k$ is strictly increased from
its original (nonnegative) equilibrium level; and, for any period
$t+1$ with $t \geq k$, it follows from (\ref{nrexpected}),
(\ref{imv}), and the specification of the new share prices that
expected profits take the form
    \begin{equation} \label{newprofits}
 \hat{\pi}^e_{t+1}  = \hat{v}_{t+1}[\theta^e_{t+1} - \theta^e_{t}]
                =  \theta^e_{t}[\hat{v}_{t}q(\epsilon
)-\hat{v}_{t+1}] =  0~.
                \end{equation} Finally, it follows from
(\ref{divexpected}) that dividend consistency holds as well.

    In summary, the assumption that there exists a Corporate Economy
equilibrium that supports a Pareto inefficient consumption
allocation has led to the contradictory conclusion that the
criterion for corporate optimization appearing in the definition of
a Corporate Economy equilibrium---namely, ${\bf I}$ $\in$ ${\cal
I}({\bf P})$---is not satisfied by this equilibrium.  It follows
that any Corporate Economy equilibrium must support a Pareto
efficient consumption allocation. ~~~Q.E.D. \\

{\bf PROOF OF PROPOSITION 6.8.} Consider the sequential generation
of savings and rate of return sequences $(s^*_t)$ and $(q^*_t)$ by
the relations (\ref{diff}) and (\ref{mrs1}), starting from the
initial condition $q^*_1$ = $q^*$, under the ``selection principle''
that the highest rate of return is always to be chosen when multiple
solutions are possible.  It will first be shown (Lemma 2, below)
that, under the regularity condition (\ref{regoffercurve}), such
sequences can be generated because the rates of return $q^*_t$ all
lie in the interval $[q^*,q_{max}]$, implying in turn that the
savings levels $s^*_t$ all lie in the feasibly supported interval
$[s(q_{max}),s(q^*)]$, where $s(q^*)$ $\equiv$ $s_{max}$.

\vspace{2mm} {\bf LEMMA 1.}  {\it Condition (\ref{regoffercurve})
holds if and only if $q^*s(q^*) \geq s(q_{max})$.} \vspace{2mm}

{\bf PROOF OF LEMMA 1.} By construction, $s_{max}$ $\equiv$ $s(q^*)$
= $q_{max}s(q_{max})$, where $s(q^*)$ and $s(q_{max})$ are both
strictly positive.  Multiplying each side by $q^*$, the proof is
immediate.  Q.E.D.

\vspace{2mm} {\bf LEMMA 2.} {\it Suppose condition
(\ref{regoffercurve}) holds as a strict inequality.  Then, for all
periods $m \geq 1$ and $k \geq 1$, one has:
    (a) if $q^*$  $\leq$  $q^*_m$  $<$  $q^*_k$  $<$  $1$, then
                $1$ $<$ $q^*_{k+1}$ $<$ $q^*_{m+1}$ $<$ $q_{max}$;
and
   (b) if $1$ $<$ $q^*_k$ $<$ $q^*_m$ $<$ $q_{max}$, then $q^*$ $<$
                     $q^*_{m+1}$ $<$ $q^*_{k+1}$ $<$ $1$.}
\vspace{2mm}

{\bf PROOF OF LEMMA 2.} By assumption, the optimal old-age
consumption level $c^o(q)$ = $w^o + qs(q)$ is a strictly increasing
function of $q$ over the interval $q \geq q^*$, and the optimal
savings level $s(q)$ is a strictly decreasing function of $q$ over
$q \geq q^*$.  Moreover, by (\ref{diff}), $s(q^*_{t+1}) =
q^*_ts(q^*_t)$ for all $t\geq 1$.   Hence the conditional statement
in (a) implies that $c^o(q^*)$ $\leq$ $c^o(q^*_m)$ $<$ $c^o(q^*_k)$
$<$ $c^o(1)$, which in turn implies---using (\ref{diff}),
(\ref{regoffercurve}), and Lemma 1---that $s(q_{max})$ $<$
$s(q^*_{m+1})$ $<$ $s(q^*_{k+1})$ $<$ $s(1)$.  The selection
principle then guarantees that the desired conclusion in part (a)
holds.  Similarly, the conditional statement in part (b) implies
that $s(1)$ $<$ $s(q^*_{k+1})$ $<$ $s(q^*_{m+1})$ $<$ $s_{max}$,
thus the selection principle guarantees that the desired conclusion
in part (b) holds. Q.E.D.

\vspace{2mm} {\bf LEMMA 3.} {\it The rate of return $q^*_t$ lies in
the interval $[q^*,q_{max}]$ for each $t \geq 1$.  In particular, if
(\ref{regoffercurve}) holds as an equality, then $q^*_t \in
\{q^*,q_{max}\}$ for all $t\geq 1$; and if (\ref{regoffercurve})
holds as a strict inequality, then $q^*$ $\leq$ $q^*_{2t-1}$ $<$
$q^*_{2t+1}$ $<$ $1$ $<$ $q^*_{2t+2}$ $<$ $q^*_{2t}$ $<$ $q_{max}$
for all $t \geq 1$.} \vspace{2mm}

{\bf PROOF OF LEMMA 3.}  If $q^*q_{max}$ = $1$, implying that
$q^*s(q^*)$ = $s(q_{max})$, then the sequence $(q^*_t)$ cycles back
and forth between $q^*$ and $q_{max}$ in a simple period 2 orbit
starting with $q^*_1$ = $q^*$.  To see this, note by (\ref{diff})
that, if $q_t$ = $q_{max}$ in some period $t$, then the equilibrium
savings level for period $t+1$ is $q_{max}s(q_{max}) = s_{max}$, a
savings level that is uniquely supported by the share rate of return
$q^*$.  Conversely, if $q^*_t$ = $q^*$ in some period $t$, then the
equilibrium savings level for period $t+1$ is $q^*s(q^*)$ =
$s(q_{max})$.  The rates of return $q_t$ thus cycle back and forth
between $q^*$ and $q_{max}$.

     Suppose, instead, that $q^*q_{max}$ $ > 1$, implying that
$q^*s(q^*)$ $>$ $s(q_{max})$.  By construction, $c^o(q^*_1)$ $<$
$\bar{c}^o$; and all savings levels greater than $s(q)$ are
supported by rates of return that are strictly smaller than $q$, for
all $s(q)$ $\geq$ $s(q_{max})$.  Also, from (\ref{diff}), one has
the goods market clearing condition $c^o(q^*_t) + c^y(q^*_{t+1})$ =
$w^y + w^o$ for every $t\geq 1$.  In particular, the golden rule
consumption profile satisfies this condition, implying that
$\bar{c}^y + \bar{c}^o$ = $w^y+w^o$.  It then follows from
$s(q_{max})$ $<$ $q^*_1s(q^*_1)$ = $s(q^*_2)$ that $q^*_2$ $<$
$q_{max}$.  And $c^o(q^*_1)+c^y(q^*_2)$ = $w^y+w^o$, with
$c^o(q^*_1)$ $<$ $\bar{c}^o$, implies that $c^y(q^*_2)$ $>$
$\bar{c}^y$ = $c^y(1)$; hence $q^*_2$ $>$ $1$, for otherwise the
selection principle used in the construction of $(q^*_t)$ is
violated.  Thus, $q^*_1$ $<$ $1$ $<$ $q^*_2$ $<$ $q_{max}$.  To
complete the proof, it suffices to show that $q^*_1$ $<$ $q^*_3$ $<$
$1$; for then, using Lemma 2, it follows by a simple induction proof
that $q^*$ $\leq$ $q^*_{2t-1}$ $<$ $q^*_{2t+1}$ $<$ $1$ $<$
$q^*_{2t+2}$ $<$ $q^*_{2t}$ $<$ $q_{max}$ for all $t \geq 1$.

     By assumption, $c^o(q)$ = $w^o+qs(q)$ is a strictly increasing
function of $q$ for all $q \geq MRS(w^y,w^o)$.  Together with
relation (\ref{diff}), and the earlier finding that $q^*_2 > 1$,
this implies that $s(q^*_3)$ = $q^*_2s(q^*_2)$ $>$ $s(1)$.  But, by
the restrictions on the offer curve, all savings levels greater than
$s(1)$ are supported by rates of return less than $1$; cf.\ Figure
2.  In particular, then, $q^*_3$ $<$ $1$.  Moreover, the selection
principle guarantees that $q^*_3$ $\geq$ $q^*_1$ $\equiv$ $q^*$.  If
$q^*_3$ = $q^*_1$, then $s(q^*_3)$ = $s(q^*)$ = $s_{max}$ =
$q_{max}s(q_{max})$.  But this leads to a contradiction; for it
holds by (\ref{diff}) that $s(q^*_3)$ = $q^*_2s(q^*_2)$, and it has
been established above that $q^*_2$ $<$ $q_{max}$, hence $s(q^*_3)$
$<$ $s_{max}$.  It follows that $q^*_3$ must be strictly greater
than $q^*_1$. Q.E.D. \vspace{2mm}

     By Lemma 3, the savings and rate of return sequences $(s^*_t)$
and $(q^*_t)$ constitute a feasible real outcome for the Corporate
Economy.  For the reasons explained in the main text, under the
hypotheses of Proposition 6.6 any Corporate Economy equilibrium must
support these sequences.  It will now be shown that there exist
infinitely many such equilibria.

     Let $M_0$ and $\theta_0$ be any given positive values for the
money and share holdings of the generation 0 consumer, and let
$P^*_1$ denote any positive (possibly infinite) nominal price level
for period 1 that satisfies $0 \leq M_0/P^*_1$ $<$ $s^*_1$, where by
construction $s^*_1$ = $s(q^*_1)$ = $s_{max}$.  It follows from
(\ref{mc2}) that the nominal price sequence must then be given by
                     \begin{equation} \label{price} P^*_t ~ = ~
\left[ \frac{1}{q^*_{t-1}\cdots q^*_1} \right] P^*_1~,~t \geq 2~,
                    \end{equation} in order for the money market to
clear in each period $t \geq 1$.

     Taking the price sequence $(P^*_t)$ as given, suppose the
corporation announces a corporate plan ${\bf I}$ = $({\bf v},{\bf
d}^e)$ that is consistent with the following requirements.  The
corporation announces a period 1 share price $v_1 > 0$ and expected
dividend $d^e_1 \geq 0$ that satisfy
                    \begin{equation} \label{vone}
     [v_1 + d^e_1]\theta_0 ~ = ~ [s^*_1 - M_0/P^*_1] ~ > ~ 0 ~.
                     \end{equation} This setting supports the
maximum possible windfall return for the generation 0 consumer in
period 1.  All remaining share prices $v_{t+1} > 0$ and expected
dividends $d^e_{t+1} \geq 0$ are then set so that they are
consistent with the requirement
                      \begin{equation}  \label{vrest}
         \frac{v_{t+1} + d^e_{t+1}}{v_t} ~ = ~ q^*_t~, ~ t \geq 1~.
                        \end{equation} By construction, these
settings yield the highest feasible share rate of return $q^*_t$ in
each period $t \geq 1$.  Finally, the real share demand $\theta^e_t$
expected by the corporation in each period $t$ is determined by
                      \begin{equation}  \label{theta}
        s^*_t ~ = ~ M_0/P^*_t + v_t\theta^e_t ~, ~ t \geq 1~.
                        \end{equation} By construction, given
$(P^*_t)$, these real share demand expectations are fulfilled along
any path for which the savings sequence $(s^*_t)$ is fulfilled.

      Finally, what about dividend consistency, nonnegative net
receipts, and fulfilled dividend expectations in each period $t$?
Combining (\ref{vone}) with (\ref{theta}) for $t=1$, one obtains
                    \begin{equation} \label{saving1}
             v_1[\theta^e_1 - \theta_0] ~= ~d^e_1\theta_0 ~.
                      \end{equation} And noting that (\ref{price})
implies that $M_0/P^*_t$ = $q^*_{t-1}M_0/P^*_{t-1}$ for all $t\geq
2$, it follows from (\ref{diff}), (\ref{vrest}), and (\ref{theta})
that
            \begin{equation} \label{coherency}
 v_t\theta^e_t ~ = ~ q^*_{t-1}v_{t-1}\theta^e_{t-1} ~ = ~
                             [v_t+d^e_t]\theta^e_{t-1}~,~ t \geq 2~.
             \end{equation} Consequently, expected net receipts
$\pi^e_t$ in each period $t \geq 1$ satisfy
                    \begin{equation}     \label{pi} \pi^e_t ~\equiv~
v_t[\theta^e_t - \theta^e_{t-1}] ~=~ d^e_t\theta^e_{t-1}~,
                       \end{equation} where $\theta^e_0 $ $\equiv $
$\theta_0 $ $>$ $0$.  Thus, in period 1, dividend consistency holds
and expected net receipts are nonnegative.  A simple induction
argument then establishes that $\theta^e_{t} $ $\geq $
$\theta^e_{t-1} $ $>$ $0$ for all $t\geq 1$, hence dividend
consistency holds and expected net earnings are nonnegative in each
period $t\geq 1$.  Finally, given the fulfillment of the real share
demand expectations in each period $t \geq 1$, it follows from
(\ref{pi}) that all dividend expectations are fulfilled as well.

        In summary, infinitely many Corporate Economy equilibria can
be constructed to support the savings and rate of return sequences
$(s^*_t)$ and $(q^*_t)$.  ~~~Q.E.D. \\

\vspace{3mm} {\bf PROOF OF PROPOSITION 6.9.} The proof of case (a)
follows from Lemma 2 and Lemma 3, established in the course of
proving Proposition 6.8.

     Suppose (\ref{regoffercurve}) holds as a strict inequality. It
then follows from Lemma 3 that the subsequence $(q^*_{2t-1}: t \geq
1)$ is strictly increasing in $t$ over the subinterval $[q^*,1]$ and
the subsequence $(q^*_{2t}: t\geq 1)$ is strictly decreasing in $t$
over the subinterval $[1,q_{max}]$.  It follows by the monotone
convergence theorem that these subsequences must converge to
elements in these subintervals, say $q_L$ $\in$ $[q^*,1]$ and $q_U$
$\in$ $[1,q_{max}]$, respectively.  If $q_L = q_U= 1$, then $q^*_t$
converges cyclically to the golden rule rate of return $\bar{q}=1$.
Suppose $q_L < q_U$, implying that the full sequence ($q^*_t)$
converges to a limit cycle supported by the period $2$ orbit
$\{q_L,q_U\}$ with either $q_L < 1$ or $1 < q_U$.  In order for such
a limit cycle to exist, it must hold by (\ref{diff}) that
$s(q_L)=q_Us(q_U)$ and $s(q_U) = q_Ls(q_L)$.   But this implies that
$s(q_L)$ = $q_Lq_Us(q_L)$, which is only possible if $q_Lq_U$ = $1$
with $q_L < 1 < q_U$. ~~~Q.E.D.


%HERE IS RRACTFIG.TEX


%Here is FIGURE 1.

\vspace*{3in}

\begin{center} \setlength{\unitlength}{.05in} \begin{picture}(39,39)
   \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(-12.0,35){$w^y+w^o$}
   \put(0,35){\line(1,-1){35}}
       \put(12,25){$\bar{c}$}
   \put(34,16){$c^o_{t+1} = w^o + [w^y-c^y_t]$}
      \put(33,15){\vector(-1,-2){4.3}}
   \put(8,38){Offer Curve}
   \put(0,0){\dashbox{.5}(10,25)}
   \put(0,0){\dashbox{.5}(23,12)}
   \put(25,13){$w$}
   \put(21.5,-3){$w^y$}
   \put(-4,11.5){$w^o$}
   \put(9.5,-3){$\bar{c}^y$}
   \put(-4,25){$\bar{c}^o$}
   \put(9.5,-4){$\underbrace{\mbox{\hspace{.7in}}}$}
   \put(15.5,-8){$\bar{s}$}

\end{picture}

\vspace{.75in}
    {\bf Figure 1} \end{center}

%Here is FIGURE 2.

\pagebreak \vspace*{2.5in}

\begin{center} \setlength{\unitlength}{.06in} \begin{picture}(39,39)
   \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(-2,1){20}}
   \put(23,12){\line(2,-1){15}}
   \put(23,12){\line(-1,3){10}}
   \put(23,12){\line(1,-3){3}}
       \put(12,25){$\bar{c}$}
   \put(34,16){$c^o_{t+1} = w^o + [w^y-c^y_t]$}
      \put(33,15){\vector(-1,-2){4.3}}
   \put(43,8){$c^o_{t+1} = w^o + q^*[w^y-c^y_t]$}
      \put(42,7){\vector(-4,-1){5}}
   \put(22,41){$c^o_{t+1} = w^o + q_{max}[w^y-c^y_t]$}
      \put(21,40){\vector(-1,-1){5}}
   \put(25,30){Offer Curve}
   \put(0,0){\dashbox{.5}(10,25)}
   \put(0,0){\dashbox{.5}(23,12)}
   \put(4.5,0){\dashbox{.5}(12,30)}
   \put(25,13){$w$}
   \put(21.5,-3){$w^y$}
   \put(-4,11.5){$w^o$}
   \put(4,-3){$c^y_{min}$}
   \put(9.5,-3){$\bar{c}^y$}
   \put(-4,25){$\bar{c}^o$}
   \put(16.5,-4){$\underbrace{\mbox{\hspace{.4in}}}$}
   \put(16.5,-8){$s(q_{max})$}
   \put(9.5,-9){$\underbrace{\mbox{\hspace{.85in}}}$}
   \put(15.5,-12){$\bar{s}$}
   \put(4.5,-13){$\underbrace{\mbox{\hspace{1.15in}}}$}
   \put(14,-17){${s_{max}}$}

\end{picture}

\vspace{1.75in}
    {\bf Figure 2} \end{center}
