Active intermediation in a monetary overlapping generations economy

Journal of Economic Dynamics and Control
22 (1998) 1543—1574
Active intermediation in a monetary overlapping
generations economy
Mark Pingle , Leigh Tesfatsion *
Department of Economics, University of Nevada, Reno, NV 89557, USA
Department of Economics, Iowa State University, Ames, IA 50011-1070, USA
Abstract
This paper establishes that the profit-seeking activities of private intermediaries can
ensure Pareto efficiency in the standard pure-exchange monetary overlapping genera-
tions economy without the need for government monetary or fiscal policy intervention.
Moreover, these profit-seeking activities are shown to rule out all aperiodic and k-
periodic cycles for k greater than 2. Contrary to much recent work on intermediation, the
profit opportunities that arise for intermediaries in this context are not due to assumed
frictions or asymmetric information. Rather, they are due to the dynamic open-ended
structure of the economy, which permits debt roll-over.
1998 Elsevier Science B.V.
All rights reserved.
JEL classification: D61; E44; G23
Keywords: Financial intermediation; Overlapping generations; Pareto efficiency
1. Introduction
The conventional definition of a competitive equilibrium does not ensure
a Pareto efficient outcome for the overlapping generations economy (Gale, 1973;
Samuelson, 1958). This well-known finding has widely been interpreted to
mean that, in the absence of altruistic preferences (Barro, 1974) or a productive
* Corresponding author. E-mail: tesfatsi@iastate.edu
 This paper is a revised abbreviated version of Pingle and Tesfatsion (1996). The authors are
grateful to J. Duffy, H. Quirmbach, the editor, and an anonymous referee for helpful comments.
0165-1889/98/$ — see front matter
1998 Elsevier Science B.V. All rights reserved.
PII S 0 1 6 5 - 1 8 8 9 ( 9 7 ) 0 0 1 0 4 - 8

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nonreproducible asset such as land (McCallum, 1987), some form of government
intervention is essential to ensure Pareto efficiency for overlapping generations
economies. See, for example, Azariadis (1993, pp. 270—271) and Champ and
Freeman (1994, pp. 206—207).
In Pingle and Tesfatsion (1991) we question whether the conventional defini-
tion of a competitive equilibrium, as applied to an overlapping generations (OG)
economy, is truly satisfactory. The conventional definition was developed for
a standard Walrasian economy with a finite number of consumers and goods
whereas the OG economy necessarily contains an infinite number of consumers
and goods. As noted by Shell (1971), the presence of this double infinity in OG
economies introduces a new trading opportunity, namely, the possibility of
incurring and rolling over a debt forever as time proceeds into the infinite future.
The problem with applying the conventional equilibrium definition to the OG
economy is that it does not contain any conditions that recognize this new
trading opportunity. Rather, private agents are assumed to focus narrowly on
consumption and production opportunities, ignoring possible profit opportuni-
ties arising from debt issue and roll over. Is it really surprising, then, that
outcomes in OG economies can fail to be Pareto efficient in the absence of
government intervention?
Using the Samuelson (1958) OG economy as an illustration, we show in
Pingle and Tesfatsion (1991) that the introduction of a private, profit-driven,
price-taking intermediary that is willing and able to arbitrage profit opportuni-
ties associated with debt issue can have a dramatic impact on the efficiency
properties of the economy. For example, regardless of the precise form assumed
for the profit objective of the intermediary, Pareto inefficient outcomes are ruled
out as equilibria since the intermediary necessarily perceives unbounded profit
opportunities.
Extending this prior work, the current paper introduces a private profit-
driven price-setting corporation into the basic monetary OG economy studied
by Grandmont and Laroque (1973), Balasko and Shell (1981), and Grandmont
(1985), among others. The corporation issues unsecured debt in the form of
stock shares. In the initial period the corporation announces its current and
future stock share prices and expected dividend payments in an attempt to
maximize its market value in accordance with the interests of its successive
shareholders. The conventional definition of a monetary equilibrium is general-
ized to include this corporate objective.
We stress our corporate intermediary is a private sector institution, not
a government or government-like institution. As a price setter, the corporation
is fundamentally different from the other private-sector agents in the model, i.e.,
the price-taking consumers. However, the corporation’s price-setting power
does not make it a government. Our corporation uses its price-setting power to
make a market for its corporate stock shares. If the corporation were able to
choose both the price and the quantity of the shares it traded, then it would

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1545
indeed be a government-like agent with extraordinary powers. However, our
corporation’s problem is not unlike that of many real-world intermediaries:
Find a set of prices that will attract customers while generating a positive rate of
return. Competing for the consumers’ dollars are other goods whose prices are
determined by the usual competitive market clearing conditions. In addition,
consumers can choose to hold money for use at a later time.
We find that efficiency is generated as the corporation adjusts the prices it sets
for its shares in pursuit of higher rates of return for its shareholders. More
precisely, we first derive a necessary and sufficient condition for an allocation for
our ‘Corporate Economy’ to be Pareto efficient. We then establish that all
equilibrium allocations for the Corporate Economy are Pareto efficient. More-
over, the equilibrium set is nonempty. In particular, we show that the corpora-
tion plays a meaningful role in the economy — issuing positively priced un-
secured debt and earning a windfall return — whenever the value of the initial
real money balances held by consumers fails to equal the particular value
needed to support the Pareto efficient golden rule equilibrium. We further show
that a first welfare theorem and existence theorem are obtained both in the
presence and in the absence of gross substitutability if the offer curve of each
consumer satisfies certain curvature restrictions of the type studied by Grand-
mont (1985). Without these curvature restrictions, the corporation 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 in
damped cyclical fashion to the golden rule consumption allocation. Conse-
quently, although an endogenous ‘business cycle’ is possible for the Corporate
Economy, the presence of a private profit-driven corporation rules out the
existence of the more complex (and Pareto inefficient) periodic and aperiodic
equilibria that are shown by Grandmont (1985) to arise in the basic monetary
OG economy.
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 firms would proceed to bid up interest rates by
issuing new debt to finance current new expenditures. By explicitly introducing
such firms into the standard OG economy, we obtain results which enable us to
restate Thompson’s important insight in a more general way: namely, like
consumption and production, intermediation should be recognized as a funda-
mental economic activity of the private sector. Moreover, our results enable
us to generalize the information-theoretic view of intermediation expressed
in the research surveyed by Bhattacharya and Thakor (1993) by showing that

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intermediation can play a significant role in dynamic open-ended economies
even in the absence of frictions or asymmetric information.
2. Consumer optimization in the corporate economy
The Corporate Economy is a pure exchange 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 t51 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'0 units of
good t and w'0 units of good t#1. Letting cR and cR> 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 º(cR, cR>)
that is twice continuously differentiable, strictly increasing, strictly quasi-con-
cave, and satisfies º(0, cR>)"º(cR, 0)"º(0,0). Moreover, it will be assumed
that the indifference curves generated by º( ) ) do not come arbitrarily close to
being either kinked or linear, in the sense of Balasko and Shell (1980, Proposi-
tion 5.6, Properties C and C). Finally, letting º and º denote the partial
derivatives of the utility function with respect to cR and cR>, respectively, it is
also assumed that
º
MRS(w, w), (w, w)(1.
(1)
º(w, w)
The implications of these utility function regularity conditions will be clarified
below.
Consumers can hold both government-issued fiat money and stock shares
issued by a private corporation. Let MR denote the quantity of money held by
the generation t consumer from period t to period t#1, and let PR denote the
price of good t in terms of fiat money. The generation t consumer obtains the
MR units of money by trading away MR/PR units of the good t endowment. In
period t#1, the MR units of money can be used to purchase MR/PR> units of
good t#1. Thus, as long as the prices PR and PR> 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 R denote the number of stock shares purchased ( R'0) or sold short
( R(0) in period t by the generation t consumer, and let vR denote the price of
a share in period t, measured in terms of good t. The consumer purchases or sells

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M. Pingle, L. Tesfatsion / Journal of Economic Dynamics and Control 22 (1998) 1543–1574
1547
short the R shares in return for vR R units of good t. In period t#1, the consumer
then receives or pays out [vR>#dR>] R units of good t#1, where dR> 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
max º(cR, cR>)
(2)
with respect to cR, cR>, MR, and R subject to the budget and nonnegativity
constraints
cR"w![MR/PR]!vR R,
cR>"w#[MR/PR>]#[vR>#dR>] R,
04cR, cR>, MR.
The generation t consumer takes as given the positive (possibly infinitely valued)
goods prices PR and PR>, the finite nonnegative share prices vR and vR>, and the
finite nonnegative expected per share dividend dR>.
The regularity conditions on the utility function º( ) ) guarantee that each
consumer in generation t51 will choose cR'0 and cR>'0. Given vR'0, the
first-order conditions for problem (2) require that
MRS(cR,cR>)"qR,
(3)
where
v
q
R>#dR>
R,
(4)
vR
denotes the expected rate of return on holding shares from period t to t#1.
Let sR,w!cR denote the savings of the generation t consumer. For later
purposes, it will now be shown that the budget constraints for problem (2) can
be simply expressed in terms of sR and the share rate of return qR over the range
0(qR(#R whenever vR'0. The proof of the following proposition (and all
subsequent propositions) can be found in an appendix to this paper.
Proposition 2.1. Suppose vR'0 and 0(qR(#R. ¹hen a finite solution exists
for problem (2) if and only if either 04PR/PR>4qR or PR and PR> are both
infinite. In either case the budget constraints for problem (2) can be expressed,
without loss of generality, as
cR"w!sR,
(5)
cR>"w#qRsR,
(6)
04cR, cR>,
(7)

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and the optimal consumption and savings levels of the generation t consumer are
uniquely determined as functions (c(qR), c(qR),s(qR)) of the period t share rate of
return qR, where s(qR)50 if and only if qR5MRS(w, w).
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 units of good 1 and a positive amount M 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 positive quantity of stock shares, . 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 and expects a per share dividend payment d.
The utility of the generation 0 consumer in period 1 is assumed to increase
with increases in his consumption level c. Thus, the generation 0 old consumer
chooses c to satisfy
c"w#[M/P]#[v#d] .
(8)
Note from (8) that the introduction of fiat money and corporate stock shares
gives the generation 0 old consumer a potential wealth windfall.
3. The corporation
A distinguishing feature of the corporate form of business is that a corpora-
tion 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 corpora-
tion 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 pay-out of
positive dividends financed by successive debt accumulation and roll-over,
hence the shares of the corporation need not be valueless.
We assume that the corporation acts in the interests of its shareholders.
Examining the budget constraints (5) and (6), note that the optimized lifetime
utility of the generation t consumer is an increasing function of qR, the expected
rate of return on holding shares from period t to period t#1, over the range
qR5MRS(w, w) where the consumer’s optimal savings level s(qR) is non-
negative. Examining the budget constraint (8), note that the utility of the
generation 0 consumer is an increasing function of [v#d] , his expected
windfall return from stock share ownership. To what extent can the corporation
control these quantities?

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1549
By definition (4), the expected share rate of return qR depends upon the share
prices vR and vR> and the expected dividend dR>. It follows that the corpora-
tion’s control over qR 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
actually 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 corporation announces a se-
quence €"(v, v,2) of finite nonnegative share prices vR together with a se-
quence d "(d,d,2) of finite nonnegative expected dividend payments dR. In
announcing this pair of sequences I"(€, d ), henceforth referred to as a prospec-
tus, the corporation takes nominal goods prices as given. It is assumed that the
announced prospectus is known to all current and potential shareholders.
Although the dividend expectations of shareholders can differ in principle from
the dividend expectations of the corporation as embodied in its prospectus, the
definition of an equilibrium given below in Section 5 will follow the usual
convention in assuming that these expectations coincide. Without loss of gener-
ality, then, we simplify the exposition below by using the same notation to
represent these expectations.
The corporation desires to exist indefinitely, implying that it must be con-
cerned both with the feasibility of its prospectus and with the optimality of its
prospectus as perceived by potential shareholders. We begin by characterizing
the subset of prospectuses perceived by the corporation to be viable. We then
explain how the corporation winnows down this subset lexicographically, in the
interests of its successive shareholders, in order to obtain an optimal choice set
for selection of a prospectus in the initial period 1.
The corporation only considers prospectuses that it expects to be able to
support. 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  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 , must be at least as great as the debt maturing in period 1, v .
More generally, the incremental change in the value of corporate shares out-
standing in any period t51, measured in terms of good t, is given by
R,vR[ R! R\],
(9)
the net earnings of the corporation in period t. It follows by a simple induction
argument that the corporation is solvent in period t only if R50.
If R'0, i.e., if R' R\, 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 earnings in period t equal to R units of good t. Because good t is
perishable, any net earnings held by the corporation become worthless at the

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M. Pingle, L. Tesfatsion / Journal of Economic Dynamics and Control 22 (1998) 1543–1574
end of the period. It is therefore assumed that the corporation pays out all net
earnings in the form of dividends to its shareholders. Letting dR denote the good
t dividend per share paid to the (old aged) shareholder in period t, who holds
R\ shares, it follows that
R"dR R\.
(10)
From an empirical standpoint, the corporation modeled here is an extreme
case in that all of its dividends are financed by incurring new debt. However,
many corporations do occasionally borrow to support a dividend distribution
when earnings are low. In 1989, for example, Kane (1989, p. 36) argued that
‘zombie firms...constitute roughly 25 percent of the FSLIC-insured thrift indus-
try,’ where a zombie thrift is a thrift with zero enterprise-contributed capital that
must rely on FSLIC guarantees to keep attracting new deposits to pay off
previous debts. Our model is an abstraction that allows us to focus on the
efficiency and stability implications of debt roll over. The possible need for
guarantees to ensure the viability of the corporation currently under considera-
tion is taken up in Section 4.
In period 1 the corporation forms an estimate R" R(I, P) for the quantity of
shares R that the corporation expects the generation t consumer to demand,
t51, conditional on a nominal goods price sequence P"(P, P,2) and
a prospectus I"(v, d ). The corporation’s period t expected net earnings are
then given by
R"vR[ R! R\],
(11)
where , . Consequently, the dividend per share d *R that the corporation
privately expects to distribute in period t must satisfy
d *
R R\" R.
(12)
For concreteness, it is henceforth assumed that the corporation sets d *
R "0 if
R\"0.
If the privately-expected dividend payments (12) were to differ from those
announced in the prospectus I, the corporation would be deliberately deceiving
its potential shareholders. We assume that the corporation does not engage in
this behavior. Rather, we assume that the corporation’s publicly announced
prospectus I exhibits dividend consistency in the sense that the dividend sequence
d appearing in I coincides with the dividend sequence d * privately anticipated
by the corporation.
In forming its share demand estimates R, the corporation takes into account
certain general structural implications of the utility maximization problem (2).
In particular, the corporation recognizes that (i) if the share price vR is positive,
then the share demand R for generation t will be bounded above; (ii) if vR is zero
and the expected per-share return [vR>#dR>] is positive, then the share
demand R will be infinitely large; and (iii) if vR is positive and [vR>#dR>] is

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1551
zero, then the share demand
R will be infinitely negative. This corporate
knowledge will be referred to as structural rationality. Finally, the corporation is
also assumed to be aware that the total real resources w#w available in the
economy in each period t constitute a bounded sequence, implying that the
sequence of real share demands, vR R, must also be bounded over time.
A prospectus will be said to be viable from the viewpoint of the corporation
if it satisfies the following three properties: (i) it generates nonnegative
expected net earnings in each period; (ii) it exhibits dividend consistency
relative to the corporation’s structurally rational share demand expectations;
and (iii) it implies a bounded sequence of expected real share demands. The
set of viable prospectuses is nonempty, for it always contains the null prospectus
consisting of zero-valued share prices and zero-valued expected dividend
payments. Moreover, using definition (9) for expected net earnings together
with the assumed structural rationality of the corporation’s share demand
expectations, the following property can be shown to hold for any viable
prospectus I: If vR is zero in any period t51, then I must be the null prospectus.
The corporation is assumed to limit its attention to the subset of viable
prospectuses that are in accordance with the interests of its successive share-
holders. As will now be detailed, this involves the lexicographic construction of
a nested sequence (IR) of subsets of viable prospectuses IR, where each IR is
optimal for generation t conditional on IR\.
The set I is the (possibly empty) subset of viable prospectuses yielding the
highest possible wealth windfall [v#d]  for the generation 0 old consumer.
If I is empty, it is assumed the corporation sets I"I. If I is nonempty and
the wealth windfall entailed by each prospectus in I is positive, it follows by
dividend consistency and the assumed positivity of  that the share price v for
each of the prospectuses in I must also be positive. In this case the share rate of
return q for generation 1 is well-defined, and the corporation is assumed to
restrict its attention further to the (possibly empty) subset I of viable prospec-
tuses in I for which q is as large as possible. If, instead, I is nonempty and
each prospectus in I entails a zero wealth windfall, then it follows by non-
negativity of v and d and positivity of  that the share price v must be zero
for each prospectus in I. As previously explained, viability then implies that
I is a singleton set containing only the null prospectus and hence cannot be
further restricted for the benefit of future generations. In this case it is supposed
that the corporation simply sets I"I.
Suppose, now, that the corporation has constructed IR for some t51.
If IR is the empty set, it is assumed that the corporation simply sets IR>"IR.
If IR is nonempty but vR is zero for some (hence for all) prospectuses in IR,
then, as previously explained, IR must be a singleton set consisting of the
null prospectus. In this case no further winnowing of IR for the benefit of
future generations is possible, and it is assumed that the corporation again
sets IR>"IR.

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To complete the inductive construction of IR>, it remains to show how the
corporation constructs IR> when IR is nonempty and vR is positive for each
prospectus in IR. In this case, by construction of IR, qR is well defined and attains
its largest possible value for each prospectus in IR. This largest possible value
must be positive; for, if not, then vR> must be zero for each prospectus in IR. As
previously explained, IR would then have to be a singleton set containing only
the null prospectus, a contradiction. A positive largest possible value for
qR implies that vR> must be positive, hence the share rate of return qR> for
generation t#1 is well-defined for each prospectus in IR. The corporation is
then assumed to further restrict its attention to the (possibly empty) subset
IR> of viable prospectuses in IR for which qR> is as large as possible.
Let I(P) denote the (possibly empty) set of viable prospectuses that remains
after all winnowing is complete, that is, let I(P) denote the intersection of the
subsets IR for t51. The set I(P) then represents the corporation’s optimal choice
set for selection of a prospectus in period 1. The corporation is assumed to be
indifferent among all prospectuses in I(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. Given the definition (11) for expected net earnings, the definition (12)
for the expected per share dividend, and dividend consistency, if follows that
d "v[ ! ] and hence [v#d] "v . This last relation shows that,
by maximizing the expected windfall return [v#d]  to the generation 0 old
consumer, the corporation also maximizes v , the expected market value of its
outstanding shares at the end of period 1. Furthermore, assuming vR RO0, and
using (12) and dividend consistency to eliminate dR> from the expression (4) for
qR, one obtains
v
q
R> R>
R"
.
(13)
vR R
Consequently, by maximizing the expected share rate of return qR for the
generation t consumer, the corporation also maximizes the expected incremen-
tal increase in the market value of its shares from period t to period t#1.
4. Viability of the corporation
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 vR R 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 qR that the
generation t consumer receives for holding its shares, the corporation also

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