18

is among those demanding to withdraw deposits. In addition, the
possibility of being shut out at the bank's window would change the
nature of the consumer's maximization problem. The precautionary
balances held in a bank would no longer be "as good as the mattress."

The introduction of penalty borrowing for the banks makes it
less costly for the intermediary to over-invest ex post in the
illiquid asset than is the case for the consumer. Qualitatively,
this could be interpreted as implying that the consumer has more
difficult access to the short-term capital market than does a
financial intermediary (perhaps due to fixed costs, reputational
effects, etc.) or that a government discount window is available to
the bank. In the next section where numerical estimates of the
increased quantity of investment in the illiquid asset afforded by
the introduction of the financial intermediary are discussed, two
versions of the consumer's problem will be presented. In the first,
consumers will be able to borrow at the same rate Rp as the bank. In
the second consumers will not be permitted to borrow. It will then
be possible to assess the relative importance of deposit pooling vs.
asymmetric liquidity as forces leading to increased investment by the
financial intermediary.

More formally, I will assume that the bank is a risk neutral
firm maximizing expected profits with respect to Kg, the proportion
of initial deposits to invest, holding Rg, R, and for the moment, Rg

constant.


39

TABLE 1
The Percentage Increase in Investment

Due to Deposit Risk Pooling

# of Consumers AI No Consumer AI With Consumer
Borrowing Borrowing

1 8.7 1.7
2 26.0 20.1
3 32.1 25.9
4 35.3 29.0
5 37.3 30.9
7 39.9 33.2
10 41.9 35.2
15 43.8 36.9

44.8 37.7
30 45.9 38.7

45 46.7 39.5


35

au
(i.e., Uy = 3Cy )

TDrake (1967) has an excellent discussion of convolution
theory.

8this condition determines Kg. The bank then invests NW(1-K)Kg
in the irreversible asset. It is this quantity that measures the
social saving on precautionary holdings due to deposit risk pooling.

9the introduction of consumer borrowing will divide the second
stage optimization into three regions. For low values of Y the
consumer will borrow. In an intermediate range he will neither
borrow nor lend. For Y greater than Y* he will save. The region of
borrowing is defined by Y less than Yp, where Yp is defined by

Uy(Cy)

Up(Cp) ~ *B
when Cy = W(I-K) + Yp
Cp = RWK.

For Y between Yp and Y*, the consumer neither borrows nor

lends, This range is defined by
Uy (Cy)
Rs sally § Rw
when Cy = W(I-K) + ¥
Co = RWK.

For Y greater than Y* the analysis is as presented in the text.

For the particular example used in the text augmented by


44

m

So the summation terms become

Sa J

B J-1
] + pcysyNt1-9 .0/BCh- (N+1-g)W(2-K)] -  [inenpw(a-K) =F

(J-1)1

were thus evaluated at J = N +1 the result would be

- Be ome
[ge *] + .0/Bh

Which is identical to the term (**) above.

One can then extend the upper limit on the summation to N + 1 and express the convolution as

(h) = rye) N42 at (N+1)W(1-K)

N+1

ea 3
= wt NAL [8 e *] Lo pqysyNtl-d | .0/8Ch-nel-J]w(a-y]
“ B

J=1 J! (N+1-J)!

for - TM<h < (N+1)W(1-K).


irreversible investment. Using the conditional maximum value
functions for C;, Co, and S developed above, one can rewrite the
consumer's problem as:

Maximize with respect to K

(8) EU =_f U(Cy(K,Y), Co(K,Y))£(Y)aY,

where C1(K,Y) = W(1-K) + ¥-S
Co(K,Y) = RWK + RgS

To choose the optimal K the consumer equates, at the margin,
the benefits of increased period-two consumption and increased
period-one consumption when Y is high, with the costs of increasing K
(due to both the increased probability of being liquidity constrained
and the increased severity of that constraint) at levels of Y less
than Y*,

In section IV below an example of this maximization problem
will illustrate the analysis developed in this section using a
particular parameterization of utility and a specific density for
income. But before doing so there is one more building block to
create from the above analysis, the distribution of individual
deposit withdrawals.

The individual consumer initially deposits W(1-K) at the >»ank
at time t = 1- e. After realizing income Y at t = 1 he can draw
down any or all of that deposit to devote to consumption in period
one or add to the initial deposit if income is unusually high. The
net withdrawal, h, is the difference between initial deposits and
savings held after income is realized.

(9) h=W(1-K) - S$


22

In the next section of the paper I continue to assume that the
deposit premium is exogenous for the purpose of a numerical example.
In that exercise I will seek to measure the increased investment in
the irreversible asset afforded by the introduction of a deposit-
pooling financial intermediary. I will set Rg equal to one so that
the consumer's portfolio allocation will be identical to what would
be the case if he kept his money "in the mattress" (i.e., in the
absence of a financial intermediary).

IV. Numerical Example

In this section I will develop a particular parameterization of
the consumer's problem and then, for a vector of sample parameter
values, compute the bank's optimal portfolio division as a function
of N, the number of consumers in the economy. From these
calculations I can then report the percentage increase in social
investment in the irreversible asset afforded by the introduction of
the deposit risk pooling financial intermediary.

To develop the particular parameterization of the consumer, I
will use an additively separable two-period utility function which
exhibits constant absolute risk aversion in each argument. I will
also use the exponential distribution introduced in the previous

section to model the distribution of consumer income. Assume that:

UC, .Cy) = - Le Mts 9 “82
U wae 21
1
-ac
Usa °

2


THE % INCREASE IN INVESTMENT DUE TO DEPOSIT RISK POOLING

FIGURE 5
7 T I I i T T T oT
= CONSUMERS UNABLE TO BORROW
------- = CONSUMERS ABLE TO BORROW
a
° 4
a

1 | |
on = os i) i) wW w s > on
© 9° n ° n ro) n Oo on °

NUMBER OF DEPOSITORS

36
% INCREASE
IN INVESTMENT

49

42

35

28

21

14


balances, in effect implements a risk sharing arrangement. The
pool:ng of consumers! idiosyncratic withdrawal demands permits the
bank to economize on social resources devoted to liquidity and
therefore to increase the proportion of social wealth which can be
devot.ed to investment for any given rate of return.

In the sections below, I present an analytic representation of
the ideas outlined above. Section II constructs a model of the
individual consumer who derives utility from consumption in each of
two periods and who must divide his initial wealth between
precautionary balances and investment in an irreversible asset prior
to otserving income, which is stochastic. From this, one can derive
the consumer's schedule of optimal deposit withdrawals as a function
of income and then, given the distribution function governing income
uncertainty, the distribution of individual withdrawals.

Section III develops the model of the banking firm. First, the
distribution of aggregate deposit withdrawals is derived from the
distribution of individual withdrawals for any economy of N consumers
with identically and independently distributed income. ‘Then the
bank, with the knowledge of the distribution of aggregate
withdrawals, maximizes its objective function with respect to the
proportion of deposits to invest in the irreversible asset. The
model in this section parallels the decision problem developed in the
literature on stochastic reserve losses and the banking firm.3 The
difference is that the distribution of withdrawals in this paper is
derived from consumer behavior, whereas the previous literature

assumed an exogenous distribution of withdrawals. As a result, I can


33

Obviously, the quantitative importance of this aspect of
financial intermediation depends upon the degree of friction in the
fiaancial markets. Perhaps this is a better model of the nineteenth
century bank, when secondary markets were less efficient (or
nonexistent), than it is of banks in 1983.

In future research several extensions and modifications would
be desirable. First and foremost, the interest rates R and Rg can be
made endogenous as mentioned above in sections III and IV. This will
permit more satisfactory measurement of the efficiencies gained by
deposit risk pooling. On a slightly larger scale, it would be
interesting to model this process in an overlapping generations
context to permit analysis of intertemporal as well as
contemporaneous risk pooling. It would also be desirable to imbed
this model of financial allocation into a general equilibrium model
and to examine the implications of monetary policy and multiple
deposit expansion. Finally, and perhaps most importantly, it would
be nice to dispense with the assumption of complete independence of
stochastic income across consumers in order to model the impact of
aggregate real fluctuations upon the ability of the bank to provide

transformation services and vice versa.


Mathematically, one maximizes with respect to Kg the expression

Dep
(2) E(m) = (R-1)Kg Dep-(Rg-1) f [Dep-h] m(h)dh

Dep
- (Rp-1) sf Ch - (1-Kg)Dep]m(h)dh - F

where (1-Kg)Dep

Dep = N * W(1-K) = initial deposits

m(h) = distribution of aggregate withdrawals
h = aggregate withdrawals

Kp = proportion of deposits invested

Rp = penalty borrowing cost (1+rg)

Rg = return to deposits held at bank (1+rg)
R = return to holding the irreversible asset
F = any fixed costs incurred by the bank

There are four terms in this expression. The first represents
the revenues earned from investing. The second represents the
expected interest payments to depositors. The third term represents
the bank's costs of going to the discount window if reserves do not
cover liquidity. The fourth term, F, represents fixed costs of
operation.

Differentiating the bank's objective function with respect to
Kp yields:

(R-1) . . 8
(13) p1)7 C1 M((1-Kg)Dep)],

where M(h) = the cumulative distribution function of aggregate

withdrawals. For the case developed above when incomes are


25

From the saving schedule it is easy to obtain the net
withdrawal schedule
h = W(1-K) - S
= W1-K) for ¥ < Y*
=a By for Y > Y*
where a = W(1-K) + BY*,
With the savings function and the expressions for C; and Co one
can return to the consumer's portfolio decision and maximize with

respect to K the expression:
(20) EU = 6 - te 21 + e 2 2j9¢ Vay
where Cy; = W(1-K) + Y¥- 5S

Co = RWK + RgS

and S=0 OS Y<« Y¥

1
= —— [y-y* *
Tory CYT Ye y

Differentiating this expression with respect to K, making use of

Leibnitz' Rule, and setting the resultant expression equal to zero

ylelds
ae
+
, O*Rg* (at+e)Y* eY* RS S. 6. (R-Rs) = 0
‘ oes" C1 - - RS *. 6 - (RoRs)
(21) arp E17 e 1+ RLe 1]+ a¥is sg
T+Rs

To insure that the K which solves this condition is a maximum
the second-order condition must be negative at the optimum

32EU

oy* ay*
Sz = (RHI )Weo”” CR - Rge

]


21

greater than zero, one would have to compute the expected profits at
eack. point.)
At the other end of the spectrum, when Kp = 1, if

R-1

1 - M(O) > Spal

all of the initial deposits will be invested in the irreversible
asset. This scenario would develop in the case where N is large
enough to justify the normal approximation and expected savings
exceed W(1-K) for the representative individual. In that case the
distribution of aggregate withdrawals would collapse around N*E(h) <
0 as N increased toward infinity. (Recall also that the mass point
of magnitude F(y*)N goes to zero in the limit as N increases to
infinity.)

Up until this point I have assumed that the bank is a
monopolist and yet it treats Rg, the deposit rate, as exogenous.
Alternatively, one could permit the bank to maximize profits with
respect to both Kg and Rg. As the bank changed Rg it would increase
the cost of deposit payments but it would also induce alterations in
the consumer's portfolio behavior and therefore affect initial
deposits, expected savings and the distribution of withdrawals,
Anotner means to endogenize the deposit return Rg would be via the
assumption that entry is frictionless and drives expected profits to
zero, From this one could establish a schedule relating the deposit
reward Rg to N the number of consumers. This would lead to a "one
bank theorem" where the maximum deposit rate could be offered by one

firm pooling all of the deposit risk in the economy.


34

FOOTNOTES
*vhe author is a staff economist in the International Finance
Division. This paper represents the views of the author and should
not be interpreted as reflecting those of the Board of Governors of
the Federal Reserve System or other members of its staff. I would
like to thank Alan Blinder, William Branson, Stephen Goldfeld, and
Dwight Jaffee for their comments and suggestions. I would also like

to thank Ruby Brooks for the preparation of the manuscript.

1the survey of Baltensperger (1980) is very useful in
organizing the material in this literature.

2this point, and the subsequent developments explaining t-he
absence of complete insurance by relying on asymmetric information,
were first made explicit by Diamond and Dybvig (1983).

3This literature began with Edgeworth (1888). More recent
contributions include Orr and Mellon (1961) and Ratti (1979). The
Baltensperger survey mentioned in footnote 1 also concisely
summarizes the work in this area.

4ror a discussion of irreversibility in assets see the Tobin
manuscript, Chapter 2. For a model of irreversibility which is much
more sophisticated than is needed for the purposes of this paper see
Bernanke (1983). In that paper the author explores the concept of
"option value" when applied to durable irreversible capital
investment.

5see Figure 1.

Ssuberipts of utility denote partial derivatives.


The distribution of hy, call it m(h)y, is of the form

-ey*
C1) my = By Geayra ote}

J J (NW(1-K)-njJ71

n70cs ae oth

¥e (0/3) [n-NW(1-K) J

for - © $ h § NW(1-K)

= F(Y*)N at h = NW(I-K).

The proof of this is contained in the appendix.

As N, the number of independent consumers, becomes large the
distribution of aggregate withdrawals approaches a normal
distribution of mean N-E(h), and standard deviation (NVAR(h))1/2
where

E(h) = W(1-K) - B/ee7OX*

Var(h) = e7OY* Be [2 - e-O¥*),

This result is obtained by the application of the Lindberg and Levy
Central Limit Theorem. In fact, the stochastic reserve loss
literature originated by Edgeworth (1888) and reinvigorated by Orr
and Mellon (1963) begins with the assumption that the distributicn of
aggregate withdrawals is exogenous, normally distributed, and
centered at zero. The present analysis adds to the previous work in
this area in two ways.. First, when N is large enough to justify
normal approximation, as it almost certainly is for any bank, the
mean of the distribution is derived from individual utility
maximizing behavior and is therefore a function of the parameters of

the system, including the return paid on deposits, Rg. Second, an


subject to the liquidity constraint,

(2) Cy, $ W(1-K) + Y,

and the second period budget constraint
(3) Co $ RWK + RgS,
where

s

W

W(1-K) + Y - Cy = savings

W

initial wealth

Y = income

f(Y) = the probability density function of Y

Cy = consumption in period 1

Co = consumption in period 2

K = proportion of initial wealth invested in the irreversible

asset

R = return to investment, (1+r)

Rg = return to savings, (1+rg).

I will assume that the utility function is increasing and
concave in each argument, that utility is additively separable across
periods, and that there is no personal rate of time preference.

Uy, Up > 0,

Ui1» Y22 < 0,

and Uj2 = Uz, . 0.6

The solution to this problem can be approached conveniently by
separating the analysis into two stages. First one can solve for the
values of Cy and Cp which maximize utility conditional on Y and K.
Then, given these conditional maximum value functions, one can

maximize expected utility with respect to K given the density of Y,


31

When N=1, the data in the second column of Table 1 reveal that
AI = 1.7%. This is very small when compared with the magnitude of AI
as N gets large. It appears that in relation to the economies of
risk pooling, the contribution of asymmetric risk aversion to the
increase in investment attributable to the bank is miniscule.

Let me now turn to the examination of Table 1 and its graphical
counterpart in Figure 5. The result plotted there is the relation
between AI, the percentage increase in social investment afforded by
the introduction of the deposit pooling intermediary, and N, the
number of consumers holding precautionary balances at the bank. Two
relazions are plotted in Figure 5. The upper one corresponds to the
case where consumers have equal access to emergency liquidity. As
one would expect, when banks have cheaper access to funds (i.e., less
costly reversibility) than do consumers, the proportionate increase
in investment resulting from the introduction of the bank is greater.

Perhaps more interesting is the trajectory of AI as N increases.
The results indicate that the efficiencies introduced by deposit risk
pooling are quite substantial as N increases from one. But the
marg:.nal increment in AI becomes quite small as N increases beyond
30. This analysis leads one to the conclusion that the increased
allocative efficiency attributable to the existence of the financial
intermediary's pooling of deposits is significant, but, given that
banks have hundreds if not thousands of customers, the marginal
stochastic economies of scale are so small that the determination of
optimal firm size is probably dominated by other considerations.

The calculation displayed in this section represents a rather

crude attempt to investigate the nature of the economies attributable


28

Let me now use the consumer model developed in this secticn in
conjunction with the model of the bank developed in the previous
section and a vector of sample parameter values to compute
numerically the increase in social investment in irreversible
opportunities afforded by the introduction of a deposit risk pocling
financial intermediary. Conceptually I will be analyzing two
economies, both consisting of N identical consumers with
independently distributed incomes, but in one economy I will
introduce a financial intermediary who collects deposits from the N
consumers and invests in the irreversible asset. One can then
compare the two economies to assess the increment in irreversible
investment attributable to the presence of indirect finance.

In what follows I will assume that the environment is
characterized by the following parameter values:

R = 1.10; return to investment in period two

Rg = 1.0; return to saving in period two,

Rg = 1.25; penalty rate

W = 2; initial wealth

6 = .5; reciprocal of expected income

a= .5; the degree of absolute risk aversion
F = 0; fixed cost of banking operation.

In this section I will assume that the supply of irreversible
investment assets is infinitely elastic at R = 1.10. Under this
assumption the efficiencies resulting from the introduction of a
financial intermediary will be reflected solely in the increased
quantity of irreversible investment and not at all by changes in the

price of credit to the ultimate borrower.


maturity and K(= 1+r) in period two. In effect the principal is
"locked-in" until maturity when both the principal and the yield
become available.

The assumption of completely irreversible investment in this
model implies that it is infinitely costly to borrow against the
certain future wealth represented by the asset and/or is infinitely
costly to liquidate early some portion of the rights to second-period
income. This assumption is more severe than simply assuming that the
ult:mate borrower's physical return to investment in an income
generating activity is unavailable prior to the second period.(This
might result from, for example, gestation lags in setting up
productive facilities, i.e., the lumber cannot be delivered until the
tree grows). A secondary market could exist to permit trading in the
rigtits to the future physical output so that when circumstances
change, that which is socially irreversible could be shifted to those
individuals who find it relatively most desirable to hold as a
vehicle to acquire future consumption.

The assumption of complete irreversibility is extreme. It
amplifies the costs associated with ex post over-investment. It
represents a polar case just as the perfectly reversible investment
represents another extreme. In reality there are penalties for early
liquidation, broker's fees, sunk costs result from investment in
asset-specific information which cannot be credibly transferred to a
prospective buyer, and many other tranSactions costs which introduce
an element of costly reversibility into the investment decision.
Surely, the existence of many of the modern financial institutions is

in part due to their ability to economize on these costs and reduce


investigate the impact of changes in exogenous variables upon the
parameters of the distribution in a systematic manner.

Section IV presents the integrated problem of the bank and N
identical consumers assuming particular forms for the utility
function and the distribution of an individual's income. Numerical
calculations of the equilibrium for sample parameter values are
presented. In particular, the increase in the social allocation of
resources to investment in the irreversible asset is plotted as a
function of N, the number of consumers in the economy. This serves
to illustrate both the quantitative importance of deposit risk
pooling and the extent of stochastic economies of scale.

Section V concludes with a discussion of this modelling
strategy and considers directions for future research. The Appendix
contains the proof of the form of the aggregate withdrawal function
for the example used in Section IV.

Il, The Representative Consumer

In this section I develop a model of the individual consumer
under uncertainty in order to provide the microeconomic foundations
of precautionary saving and optimal deposit-withdrawal behavior.

Before turning to the mathematical presentation of the
consumer's problem it is important to discuss the assumption that. the
asset available to the consumer is completely irreversible and to
emphasize the role that this assumption plays in a model of
precautionary savings. A completely irreversible asset is one wrich,
once committed to, cannot be undone prior to maturity, at any cost.4
For example, if an individual invests one unit of wealth in a twc-

period completely irreversible asset he may obtain nothing prior to


explicit form for the distribution of aggregate withdrawals permits
one to examine the extent of stochastic economies of scale in banking
as N increases from 1 to "large enough to use the normal
approximation." I will make use of these features in section IV
below.

With the aggregate withdrawal density m(h)\ one can turn to the
model. of the financial intermediary. The financial institution
collects the precautionary deposits of the N consumers in the economy
and invests some portion of the proceeds in the asset returning R in
period two. The intermediary's liabilities may be withdrawn without
penalty, (Return = 1), at the initiative of the consumer prior to
period two and will pay Rg if left in the bank until the second
period. The intermediary's asset management consists of choosing Kp,
the proportion of initial deposits to invest in the irreversible
asset,, prior to the resolution of consumer uncertainty. The
remaining portion of deposits is held as reserves to meet the demand
for liquidity at the end of period 1. (Timing is shown in Figure 1.)

In order to specify the intermediary's objective function it is
necessary to specify the mechanism for handling those aggregate
states of nature in which the total withdrawals demanded exceed the
intermediary's reserves (i.e., a liquidity crisis). In what follows
I will assume that the financial intermediary has access to an
infinitely elastic supply of liquidity for which it must pay Rp (>>R)
per unit.

Alternatively one could assume that convertibility would be
suspended in the event of liquidity crisis but such a convention

necessitates specifying a mechanism to allocate what liquidity there


The individual consumer has a utility function defined over
consumption in each of two periods. He is endowed with initial
wealth, W, and receives a stochastic realization of income, Y, in
period one, Income is distributed with density f(¥). The price of
consumption goods in terms of wealth or income is constant and set to
one in both periods. The consumer has two means of transferring
purchasing power into the future. He may hold savings balances in a
deposit account paying Rg(= l+rg) in period two and/or invest in an
irreversible asset returning R(>Rg) in the second period. Deposits
are assumed to be available to the consumer at any time, forfeiting
only rg if withdrawn before the second period. The investment asset
is completely irreversible. Once made it is worth R in period two
anc zero prior to that time. The consumer's problem is to choose K,
the proportion of initial wealth to commit to the irreversible asset,
prior to observing the realization of stochastic income.

After the portfolio decision is made the consumer learns of his
particular income realization, Y, and then allocates income and
available precautionary balances, W(1-K), to current consumption, Cy,
anc saving (if any).2 This second stage, or ex post, optimal
allocation is made conditional on the knowledge that RWK, the income
from the irreversible asset, will be available for consumption in
period two.

More formally, the consumer chooses K to maximize expected

utility,

(1) gu =f UCC, .C,)f (x ay


23

= -a2e7@C1
wy ace
= -a2 -aCo
U55 ace
Yip = Yay = 0
and f(¥) = ee 8 os¥se
E(Y) = 1/6

where a is the coefficient of absolute risk aversion and E denotes
the mathematical expectation operator .

As was explained in Section II above, the consumer's problem
cen be separated into two stages. First, one can derive the optimal
choice of C, and Cp as a function of K and Y. Then these conditional
maximum value functions are introduced into the objective function
and the resultant expression is maximized with respect to K. For the
particular specification introduced above, the conditional functions,
Cy (w,k) and Co(w,k) are derived by maximizing the following

expression with respect to C, and Co:
-aly —aCo
L=- [e +e J + ALw(1-k) + Y - Cy] + 6CRWK +RgS - Co]

The first order conditions imply that,

Yr 2 allie) Ay Re
U2

|

where ) is the Lagrange multiplier associated with the liquidity
constraint and 6 is the Lagrange multiplier associated with the

second-period resource constraint. At the point where the


International Finance Discussion Papers
Number 289

July 1986

INCOMPLETE INSURANCE, IRREVERSIBLE INVESTMENT, AND
THE MICROFOUNDATIONS OF FINANCIAL INTERMEDIATION

by

Robert A. Johnson

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publ:.cations to International Finance Discussion Papers (other than an
acknowledgment that the writer has had access to unpublished material)
shoud be cleared with the author or authors.


IFDP
NUMBER

289

288

287

286

285

284

283

282

281

280

279

278

27T

276

46

International Finance Discussion Papers

TITLES

1986
Incomplete Insurance, Irreversible
Investment, and the Microfoundations of
Financial Intermediation

The Yen-Dollar Relationship: A Recent
Historical Perspective

Should Fixed Coefficients be Reestimated
Every Period for Extrapolation?

An Empirical Analysis of Policy
Coordination in the U.S., Japan and
Europe

Comovements in Aggregate and Relative
Prices: Some Evidence on Neutrality

Labor Market Rigidities and Unemployment:
the Case of Severance Costs

A Framework for Analyzing the Process
of Financial Innovation

Is the ECU an Optimal Currency Basket?

Are Foreign Exchange Forecasts Rational?
New Evidence from Survey Data

Taxation of Capital Gains on Foreign

Exchange Transactions and the Non-neutrality

of Changes in Anticipated Inflation

The Prospect of a Depreciating Dollar
and Possible Tension Inside the EMS

The Stock Market and Exchange Rate Dynamics

Can Debtor Countries Service Their Debts?
Income and Price Elasticities for Exports
of Developing Countries

Post-simulation Analysis of Monte Carlo

Experiments: Interpreting Pesaran's (1974)
Study of Non-nested Hypothesis Test Statistics

AUTHOR(s:)_

Robert A. Johnson

Manuel H. Johnson
Bonnie E. Loopesko

P.A.V.B. Swamy
Garry J. Schinasi

Hali J. Edison
Ralph Tryon

B. Dianne Pauls
Michael K. Gavin
Allen B. Frankel
Catherine L. Mann
Hali J. Edison

Kathryn M. Dominguez

Garry J. Schinasi

Jacques Melitz
Michael K. Gavin
Jaime Marquez

Caryl McNeilly

Neil R. Ericsson

Please address requests for copies to International Finance Discussion
Papers, Division of International Finance, Stop 24, Board of Governors of

the Federal Reserve System, Washington, D.C.

20551.


10

consumption, the liquidity constraint ceases to bind and A becomes
zero. The ratio of marginal utilities then equals the return to
saving, Rg. Note also that 6, the shadow value of income in period
two, is never zero because the consumer is never satiated by second-
period consumption. (Uz is greater than zero by assumption). Let us
define Y* as that value of income such that in equilibrium saving is
exactly equal to zero.

Uz(C1)

Up(Cp) ~ Bs

when Cy = W(1-K) + Y*
Co = RWK.
For Y > Y*¥ the liquidity constraint will not bind and

Uy(Cy) Lp,
Up(Cp) ~ Rs

when,
Cy = WI-K) +Y- 5s,
Co = RWK + RgS
andS>0.

For Y < Y* equation (5) holds and

UC)
Upc y ” Rs-

Note also that when utility is additively separable in consumpt:on in
y*

each period, = is increasing

aye UyUpo
(6) g—-C1+ Tue RW > 0


f(Y). This is just a simple "backward induction step" applied to a
two-period dynamic program.

Let me begin with the solution of the second stage problem.
After time t = 1 the individual has chosen K and received income
realization Y. He then allocates liquid balances to consumption C,
and saving to maximize two-period utility subject to the liquidity
constraint and the second period resource constraint. Mathematically
one can use Lagrange's method and maximize the following expression
with respect to C, and C,

(4) L = U(Cy,C,) + ALW(1-K)+¥-Cy] + 6LRWK+R,S - Co]
where \ and 6 are the Lagrange multipliers associated with the
liquidity constraint and the resource constraint respectively.
Differentiating the above with respect to Cy, Co, A and 6 yields

aL
BC, 7 UC) = A+ 6Rs = 0

8L

Ip Yel d-6=0

a. CW(1-K) + Yy - C,] = 0; with complementary slackness
6 . [RWK + RgS - Co] = 0; with complementary slackness

Rearranging these expressions yields

(sy BL) 4A sag

The ratio of the marginal utilities of consumption is equal to the
return to savings plus a term which is the ratio of the shadow value
of liquidity in period one to the shadow value of income in period
two. When, given K, Y is large enough that the consumer desires to

devote some liquidity to saving for the purpose of future


42

nW(1-K)

= J mn), m (Pgh) yay Baya = hy + hy
N+1l-rwW(1-K)
J-1
N NI . fe aoa 73*} . J 0/8 [yg 37 (N-I)W(A-K)] W(1-K) (hwa-x) by thy]
“3, Stora Lee Fiysye b J sear Wt
N+L
N oO Jt wa. J
N! oe. 8B neg _@fhr (N-g)WOL-K)T N+1) W(1-K)-h
(4) “2 Sraeayi * Be ] + FIY*) . B st
=]
Substitution of index d = J+1 into (4) yields
ao_ 4 -
wt NI -fe. 8 riya) (Nel-d). .0/B[h-(N+1-a)W(1-K)] . [(N+2)W(1-K) =f at
dag (2) T(N+I=a)7 B (d-1)1

Also note that if d= 1, the above expression would yield
oe
g eB ecyey® 29/8 [R-Nw(1-K)]

which is identical to (2) above.

One can then express the convolution as



30

model presented above to permit consumers to borrow at Rp if they so
desire.9

The purpose of this exercise is to isolate the effects of
deposit risk pooling, and this modification serves that purpose and
does not alter the consumer problem presented above in any
fundamental way. In fact, for the parameter values listed above, K*,
the consumer's optimal share of initial wealth devoted to the
irreversible asset changes from .663 to .706 and Rp moves from » down
to 1.25.

Before turning to the numerical presentation I must mention one
other asymmetry between the bank and consumers. Consumers in this
model are risk averse, whereas the bank is risk neutral. ‘This
asymmetry tends to amplify the extent to which the risk pooling
depository institution increases the social commitment to
irreversible investment. While I do not make any attempt tio remove
this influence here, I can present a baseline measure of its
importance by examining the case where N=1 and both the consumer and
the bank have access to liquidity at the penalty rate. In such a
case, both asymmetric liquidity and deposit risk pooling cease to be
a factor in promoting investment in the irreversible asset. The
consumer in the absence of the financial intermediary will invest
NWK* directly. After introducing the bank, the quantity of
investment increases to NWK + NW(1-K)Kg. The percentage increase in
irreversible investment afforded by the introduction of the financial

intermediary is therefore;

C1-K*)K; 10
(24) al = ~Qo"B


IF DP
NUMBER

275

274

273

272

271

270

269
268

267

266

265
264

263

47

International Finance Discussion Papers

TITLES

A Method for Solving Systems of First Order
Linear Homogeneous Differential Equations
When the Elements of the Forcing Vector
are Modelled as Step Functions

International Comparisons of Fiscal Policy:
The OECD and the IMF Measures of Fiscal
Impulse

An Analysis of the Welfare Implications of
Alternative Exchange Rate Regimes: An
Intertemporal Model with an Application

1985
(partial listing)

Expected Fiscal Policy and the Recession
of 1982

Elections and Macroeconomic Policy Cycles

Assertion Without Empirical Basis: An
Econometric Appraisal of Monetary Trends
in... the United Kingdom by Milton Friedman

and Anna J. Schwartz

Canadian Financial Markets:
Proposal for Reform

The Government's

Was It Real? The Exchange Rate Interest
Differential Relation, 1973-1984

The U.K. Sector of the Federal Reserve's
Multicountry Model: The Effects of
Monetary and Fiscal Policies

Optimal Currency Basket in a World of
Generalized Floating: An Application to
the Nordic Countries

Money Demand in Open Economies:
Substitution Model for Venezuela

A Currency

Comparing Costs of Note Insurance Facilities
and Eurocredits

Some Implications of the President's Tax
Proposals for U.S. Banks with Claims on
Developing Countries

AUTHOR(s)

Robert A. Johnson

Garry Schinasi

Andrew Feltenstein
David Lebow
Anne Sibert

William H. Branson
Arminio Fraga
Robert A. Johnson

Kenneth Rogoff
Anne Sibert

David F. Hendry
Neil R. Ericsson

Garry J. Schinasi
Richard Meese
Kenneth Rogoff

Hali J. Edison

Hali J. Edison
Erling Vardal

Jaime Marquez
Rodney H. Mills

Allen B, Frankel


29

I also assume, as is implied in this specification, that the
deposit return, ‘Rg, is treated parametrically rather than as a
variable determined by the intermediary. With Rg = 1.0 the consumer
is presented with the same environment in the presence of the
financial intermediary that exists in its absence. I further assume
that when indifferent between the deposit account and holding money
"in the mattress," the consumer will choose to put his money in the
bank. As mentioned above, the introduction of free entry would serve
to endogenize the deposit rate. If the consumer were to respond to
positive interest on deposits by increasing liquid balances, (dK/dRg
< 0) then the introduction of the competitive financial institution
would increase the proportion of total investment done indirectly
when compared with the case displayed below. Furthermore, in utility
terms the existence of positive deposit interest will make the
consumer no worse off and possibly better off. (Revealed
Preference).

In Table 1 and Figure 5 below, I report the results of two
experiments. In the first instance, consumers are unable to borrow,
Rp equals © , whereas banks may borrow at Rp = 1.25. The
implication of this is that the cost of reversing investment is not
the same for the two types of market Participants, and therefore
banks would be likely to undertake a larger share of total investment
than would be the case if both participants were faced with the same
cost of reversibility.

In order to filter out this effect of asymmetric liquidity a

second experiment is undertaken by slightly modifying the consumer


the cost of reversing investments. But even with these institutions
some cost of reversibility remains, if only because of the scarce
real resources required to supply market-making services
competitively.

Whatever its quantitative significance, the qualitative
significance of costly reversibility is that it provides a motive for
holding precautionary balances in the absence of complete insuranze
markets and when 1) some uncertainty exists that is not resolved
until after the portfolio decision is made, and 2) the consumer i3
risk averse. Where portfolio decisions are costless to undo, the
consumer would invest all of his initial wealth in the
interest-bearing asset.(Provided that it offered a return greater
than that earned on bank deposits.) After learning of his particular
income realization he would costlessly retrieve just enough of the
asset to maximize two-period utility. It is the ex ante awareness of
the ex post cost of reallocation in the event of a low-income
realization that serves to restrain the consumer from committing all
of his resources to investment and as a result generates the demand
for precautionary holdings.

In the model developed in this section I will assume complete
irreversibility of investment rather than specify a more elaborate
environment that permits costly reversibility. The model of compl.ete
irreversibility conveys the same qualitative withdrawal-deposit
behavior and has the advantage of relative simplicity. A model of
penalty borrowing to facilitate ex post reallocation of resources

is utilized in the numerical analysis of Section Iv.


20

distributed exponentially the C.D.F. is

- N! ae-oxesg ChenWw(1-K)] Ja
(14) Min) = SE ayciyrae - Cgeroway . yg, Fenn’

. (By beg0/BLE-NW(I-K)}

for -* <$ h $ NW(1-K)
= 1 for h 2 NW(1-K).
The first order condition (13) equates the marginal profit per unit
of investment to the marginal expected cost to penalty financing.

Second order conditions assure that the above is a maximum.

32E(n)

OK

(15) =~ (Ry-1) Dep [(1-k, Dep] < 0

Taking the total differential of the equilibrium condition

reveals the comparative static properties:

aKp 1

(16) $88 = 1g
oR (R,-1)Dep.m(7 K,)Dep)

Oka. <(R=1) KR gg
dRg (Rp-1) «OR

The two extremes, Kg = 0 and Kp = 1 bear some examination
because of the nature of the aggregate withdrawal distribution m(1).
Since Rp > R, the left hand side of the equilibrium condition, (13)
is less than unity. For small enough N it is possible that

R-1

N
1> pot >[1 - F(¥*)°].

(To determine whether Kp = 0 or Kp = €, where ce is infinitesimally


37

Figure 1

Timing

choose K Learns of Y Receives RWK+ RS
knows Chooses C,,S for C,
£(¥) |
UIC) .C,)

tsl-e t=1 th2

Bank Receives
NW (1-K) =Dep
knows m(h)
Chooses Kg

(1-kp) held on reserve

Borrows at
Discount
Window if
Needed

Figure 2
Consumption

Receives RK, “Dep
Pays off any Loans
at Re

Pays RSS



38

gure 3

Saving and Deposit Withdrawals

ye
Figure 4
P.D.F. of Deposit Withdrawals
(a)
walk) h

Mass Point at W(1-K) of Magnitude F(Y*)

Density 2 278th a] -2< h < WO-K)



26

This expression is negative provided that

1 ink
* — ——
(22) K* > TR * SHR)

where K* denotes the value of K that satisfied the first-order
condition.

An additional point is worth raising at this time. To insure
that Y* is within the range of Y, it must be greater than zero. This
in turn implies that:

1, inRg

(23) K* > a * awcreR)

Because R > Rg by assumption, condition (23) is satisfied
provided that the K which satisfied the first order conditions is a
maximum, condition (22) above. The satisfaction of this criterion
assures one that there is some ex ante region of Y in which the
consumer will experience liquidity constraints. The upper limit, on K
is one, so Y* is always less than or equal to (RW-1/alnRg). This
insures that one portion of the range of [0, -] represents ex post
under-commitment to the irreversible asset.

The equilibrium condition above is a trifle cumbersome and in
general numerical computation will be needed to solve for K*, There
is, however, one special case that permits an analytic solution for
K*, This case exists when 6 = a. There is no economically
compelling reason why the degree of absolute risk aversion should be
equal to the reciprocal of expected income, but in such a case one
can solve for K* using the quadratic formula and obtain

1 In(R+(R2-2Rg7)¥,)

Ke 2 4
1+R aW(1+R)


balances of a community of consumers (demand deposits) and invests
some portion of its holdings in interest-bearing assets which are not
redeemable prior to a specified date in the future.

The consumers divide their initial wealth between liquid
balances held in a deposit account at the bank and direct investment
in the interest-bearing security which is completely irreversible
(cannot be redeemed prior to maturity). The consumers derive utility
from holding deposits because they provide self insurance against the
possibility of a "rainy day" (a low stochastic income realization) in
the future.

The consumer's decision to hold precautionary balances is a4
response to the lack of complete insurance markets.2 Were
competitive complete insurance available, the consumer could write a
set of contracts contingent on his income realizations and
precautionary balances would be unnecessary. In the sections that
follow, I assume that the non-availability of complete insurance
derives from the structure of information. I assume that individual
stochastic income realizations are private information and that it is
prohibitively costly for any potential insurer to ascertain the
individual's particular income state.

In an environment characterized by such asymmetries in
information, the individual holds some ex ante optimal level of
precautionary balances and withdraws some of them in response to bad
income realizations. Provided that individual risks are not
perfectly correlated across consumers, the variability of aggregate
bank deposits per consumer is less than individual deposit

variability. The bank, by collecting consumers! precautionary


45

References

Baltensperger, E., 1980, “Alternative Approaches to the Theory of the

Banking Firm," Journal of Monetary Economics 6, pp. 1-37.

, 1972, “Economies of Scale, Firm Size and Concentration
in Banking," Journal of Money, Credit, and Banking 4, pp. 467-489.

Benston, G. J. and C. W. Smith, 1976, “A Transaction Cost Approach to the
Theory of Financial Intermediation," Journal of Finance 31, No. 2,
Pp. 215-32.

Bernanke, B. S., 1983, “Irreversibility, Uncertainty and Cyclical
Investment," Quarterly Journal of Economics #1, pp. 85-106.

Diamond, D. and P. H. Dybvig, 1983, "Bank Runs, Deposit Insurance, and
Liquidity," Journal of Political Economy 91, No. 3, pp. 401-419.

Drake, A. W., Fundamentals of Applied Probability Theory, New York,
McGraw-Hill Co., 1967.

Edgeworth, F. Y., 1888, “The Mathematical Theory of Banking," Journal of
the Royal Statistical Society, v. 51, Part 1, pp. 113-127.

Lavington, F., The English Capital Market, London Methven & Co., Ltd.,

1921.

Orr, D. and W. G. Mellon, 1961, “Stochastic Reserve Losses and Expansion
of Bank Credit," American Economic Review 51, pp. 614-23.

Ratti, R. A., 1979, “Stochastic Reserve Losses and Bank Credit Expansion,"
Journal of Monetary Economics 5, pp. 283-94.

Tobin, J., Unpublished Money Manuscript.


43

= F(y#)N*2 at (en) w(1-K)

mh) an
Nel , 8 4 “2
z : NI —,[8&. >] reyayNtl-d | .0/BCh-[el-d]woi-Kj] | [inenyw(i-K)-A]
a. (1)! Nel-djr* LB (a-1)1
N - @_o J-1
ra Nt . [g e >] recy Ntl-d | .0/BCh- (ntl-gyw(1-Ky] [intiyw(-k)-8
J=1 J!(N-J)! B (J-1)!

-@ <h < (N+1)W(1-K)

The two terms involving summation can then be combined over the range J = 1, N to yield

FcyeyNtl ae (wey w(a-K)

mM) yyy =
a@_ 9 Ore nei. - -
=: Nt) wt BB] pga) Ntl-s ee (mA1-SWO-K)T Cavey wc-wy-n] I?
TE(N-J) 1 (J-1) 1 (N+1-5) 1 * |B . . (J-1)1

Jel

Plus the term from the first of the two summations evaluated at d=N+1

28 .
(2 ee ey - BCA. Ciaywa-Ky-n7TM
B NI
And ut + Nt
J!(N-J)1 (J-1) 1 (N#1-3) 1
= Niwas), wis
JI*(N+1-J)! JI(N+1-J) 1
= (NH) I

J!(N+1-J)!


where S is the savings function developed above. The schedule of
withdrawals as a function of income is depicted in Figure 4, For
realizations of income less than Y* the consumer is liquidity
constrained and withdraws all of his initial deposit. For Y > Y* some
positive deposit balances are maintained and may even constitute a
net deposit (a negative net withdrawal) for large realizations of Y.

In the next section I derive the distribution of aggregate
withdrawals for an N consumer economy under some particular
assumptions. To do that one must develop the distribution of
individual net withdrawals, m(h). Recall that for all Y less than Y*
the consumer withdraws all W(1-K) of his initial deposits. Thus the
distribution of withdrawals will have a probability mass of magnitude
F(Y*) at W(1-K), where F(Y*) is the cumulative distribution function
of f(¥) evaluated Y*. The remaining portion of m(h) will be a region
of density derived from a transformation of f(y).

For Y 2 ¥*, withdrawals are a monotonic function of income.

h = W(1-K) - S(Y).

Inverting this function and transforming variables yields

Y = s7l(W(1-K)-h)

ay

S= a-g7i'
dh s

ay —4 Ke . -
m(h) = ah + f(S7'(W(1-K)-h); for = © $< W(I-K).

The complete distribution of withdrawals is

(10) m(h) = F(Y*) at h = W(1-K)

= Sle(s-1(W(1-K)-h)) for - © $n < W(I-K).


15

seer by the banking firm. Because of the difficulty of aggregation
wher dealing with mixed distributions I am confined to making
particular assumptions about the form of the individual distribution
of income. I will assume that the individual consumer's income is
distributed exponentially.

f(Y) = ee YY ;0s¥se

As was developed in the analysis at the end of the preceding
section, the distribution of individual withdrawals is a

transformation of the distribution of income

9 78 [hea]
mh) = Fe B -= Sh < W(1-K)
= F(Y*) at h = W(1-K)

where F(Y*) = 1 - e76y*

From the distribution of individual withdrawals one can
construct a distribution of aggregate withdrawals under the
additional assumptions that all N consumers are identical and that
the N incomes are independently and identically distributed. Using
the theory of convolution one can then derive the distribution of the

sum of N individual withdrawals, hy, where

hy = a ny .?


Incomplete Insurance, Irreversible Investment, and
the Microfoundations of Financial Intermediation

by
Robert A. Johnson*
i.__Introduction

Economists have long recognized that financial intermediation
is anomalous in a perfectly competitive economy witn complete markets.
Recent efforts to provide an explanation for the existence of
financial intermediaries place great emphasis on imperfections in the
market and proceed to illustrate the role that indirect finance plays
in improving the market allocation in the presence of such
imperfections,

The literature created as a result of this research strategy
can be conceptually divided into two groups corresponding to the
roles: played by the financial intermediary in improving allocative
efficiency.1! The first group models the financial institution as a
broker-dealer which, due to its ability to exploit information or
transactions costs, can lower the cost of borrowing and/or increase
the return to lending when compared with direct interaction between
borrewers and lenders, The second group assumes, often implicitly,
that insurance markets are incomplete and exhibit the economies
achieved by the financial intermediary due to its ability to
consolidate and transform risk. It is this second aspect of
financial intermediation, the transformation of risk, that is the
subject of this paper.

The model developed in the succeeding sections portrays the

financial intermediary as a bank which receives the precautionary


36

borrowing, at Rp, the first order condition evaluated at Rg = 1
becomes:

1 1 _ Ky#Kp
sai t+ warty

= RR
where a T+Rp + 68

/
Ky = R.CRR°’? -1] + 22 ft - Rg 228 J

~ (R-1e

Ke a/2+6

Kg = “ta «yl (CRB)

2

10The percentage increase due to the introduction of the
financial intermediary is,

NWK + NW(1-K)Kp-NWK  _ (1-K)Kg
NWK K



24

equilibrium savings, S, is exactly zero, the liquidity constraint

ceases to bind and Aj goes to zero. At such a point

(17) Cy = Cz -(1/a)1nRg.
At the point where S = 0 in equilibrium,

Cy = W(1-K) + Y*

Cp = RWK

where Y* is the value of income at which, given K, the liquidity
constraint ceases to bind. Using these two expressions for C; and Co
in conjunction with equation (17) reveals that

(18) Y* = [(R+#1)K-1]W - (1/a)1n(Rg)

For values of Y less than Y* savings will be zero, and a1
available liquidity is devoted to consumption in period one. [For Y
greater than Y* some positive savings will be carried over to augment
the return to the irreversible asset.

Cy = WO-K) + ¥ -s

Co = RWK + RgS
Substituting these expressions into (17) and solving for S
reveals

W(1-K) + (1/a)1nRg — RWK + —*

(19) s(x) = 7 Ti

= B(Y-Y*) for Y 2 Y*

where B = FR = marginal property to save out of income.


ABSTRACT

The financial intermediary is shown to result from a market
imperfection related to the costly monitoring of the actions of consumers.
In such an environment complete insurance is not obtainable and consumers
respond by holding some of their wealth as precautionary balances in order
to self-insure. Precautionary balances are those financial vehicles which
permit one to invest and then liquidate with the smallest amount of loss
because of the "sunk costs" associated with the transaction. An economy of
N identical consumers is created and it is shown that a financial
intermediary which collects the precautionary balances of the community can
then implement risk sharing and liberate social resources for greater

investment.


to the presence of a risk pooling financial intermediary. While it
is the author's hope that this exercise provides the reader with a
qualitative feel for the forces at work in this model economy,. the
results presented here are a special case and must be viewed
cautiously. A more sophisticated analysis would, as described atove,
endogenize the deposit rate Rg and perhaps the asset return R. The
author is currently developing a model with free entry in the
financial sector and an asset supply schedule with finite elasticity.
I hope to include presentation of that model in subsequent versicns
of this paper.
V.__Conelusion

In the previous sections I have sought to construct a model of
the interaction between consumers, their holding of precautionary
balances, a financial intermediary, and investment in illiquid assets.
I have not offered a theory of the illiquidity/irreversibility of
assets but have merely assumed that there exists some sunk cost to be
associated with investment. By making extreme assumptions about the
nature of the illiquidity it was possible to develop a model of
consumer deposits based upon individual optimizing behavior. It was
then possible to derive the distribution of the aggregate deposit
pool held by the bank if one assumed that incomes were independently
distributed across individuals. Given the distribution of aggregate
deposits, an expected profit maximizing bank could then choose to
what extent it could invest in the illiquid asset. A particular
parameterization of the problem was solved numerically to illustrate

the significance of the role of the financial intermediary.


41

APPENDIX

Proof by Induction of the Form of the Density
of N Truncated Exponentials

2 go - 8 [h-«-aywea-K
mines E Nt (8 BY. peyay Swany]? | Goowanl]
N JI(N-J)! B (J-1)!
Jel :
N
= F(¥*)" at NW(1-K)

for - © <h < NW(1-K)
@ -de, 8,
m (my, =fe 7B. ,hO/B

cy

< h < W(1-K)

F(Y*) at W(1-K)

at h = W(1-K)

¥
where F(y*) = f{ oe Yay =
0

G- "7

I wish to show that the convolution of M(h) iy and M(h)) equals M(h)

Each density is comprised of an impulse and a region of continuous density. Thus there will be
four components in the convolution.
(1)

N+l
NH, .
F(Y*) impulse at (N+1)W(1-K) (2 impulses)
(2) Fcy*y g e fBenwa-*)-a] 8 (1 impulse & 1 density)
N = 7
Bi -( 8 a) Nti-g
a) Pal TH (N-J) It [pe | F(y*)

— —J-1
~ [(N+1)W(1-K)-h .e

(J-1)!



27

2Rs
where z= R - BS - Bsl*Rg_. [RRs]
2 1 + Rg
T+Ry

Comparative static properties are generally ambiguous, but if
one assumes that interest on the irreversible asset is less than 100
percent (R < 2) and that expected income and initial wealth are of

approximately the same size (i.e., Ow ~ 1), one can evaluate K* when

Rg = 1 and obtain,

se <0
Ss Rg=1

. %

“ >0
Rg=1

aK* _ In(R + (R@-2/3R-1/3)1/2

GEM) 2 = PIGS) do (for R > 1)
sel

ak* _ cin (R+(R2-2/3R-1/3) 1/2)

Ww = WTR) <0 (for R > 1)

Rg=1

For the extremely special case just derived the comparative
statics imply that: raising the deposit rate will increase savings
deposits (K* falls); increasing the return to irreversible investment
attracts more investment; increasing expected income, and therefore
expected liquidity, will increase investment; and finally increasing
the consumer's initial endowment, W, decreases the proportion of
wealth devoted to irreversible investment. Also note that at Rg = 1,

K* is a maximum for R greater than one. Referring to (22) above

. sien + in(R+(R2@= (2/3)R-1/3) 1/2) 4 + Ink
KAT ger 7 1/148 aWC+R) TR” aW(1+R)


11

The value Y* separates tne range of income realizations into
two regions. Realizations of Y greater than Y* constitute the region
of ex post under-allocation to the irreversible asset. Had the
corsumer known the value of Y ex ante he could have invested more
resources in the asset earning R > Rg and improved his allocation.
Similarly, the values of ¥ less than Y* constitute the region of ex
post over-commitment to the irreversible asset. In this region the
cor.sumer would like to transfer wealth back to period one but cannot
because all of his second period wealth is "locked up" until that
time.

From the analysis above one can plot Cj, Co, and S as functions
of income for a given K. This is done in Figures 2 and 3. There one
can see that for low values of Y all additional income is devoted to
first-period consumption. Beginning at Y = Y* some portion of
increasing income will be saved for consumption in the second period.
Optimal saving as a function of Y is continuous and nondifferentiable
at Y*. For values of Y less than Y* all available liquid balances
are devoted to C; and savings are zero. For Y greater than Y* saving

is a positive function of income. At Y = Y* saving is exactly zero.

S=0 -esy¥s¢ y*
(7) = S(Y) Y > Y*
ds
1> a > 0.

Let us now return to the first stage problem of choosing the

optimal portfolio shares to be devoted to precautionary balances and


n

I will illustrate this with an example which will be used in
the next two sections. I will assume that income is distributed
exponentially.

f(¥) = ee 8¥; os Ys @

I will also derive a saving function of the form

S(Y) = Bly-y*); for Y > Y*

So h= W(1-K) + By* - By
Rename W(1-K) + By* = a

Then h = a - By for Y > Y*

ach
So y=
gy 21
dh B

(0/B) [h-a]}

and m(h) = 2 e for - » sh < W(1-k)

= F(Y*) = 1 - e°8Y* at h = W(1-K).
This distribution is pictured in Figure 5. We can now turn to the

derivation of aggregate withdrawal behavior and then to a model of

the bank.

Ill. The Finaneial Intermediary

The previous section developed a model of the precautionary
Savings held by an individual consumer. I now turn to the systemic
interaction between a community of N consumers and the financial
intermediary.

Before directly analyzing the optimizing behavior of the bank
one must derive the functional relationship between individual

deposit-withdrawal behavior and the aggregate withdrawal behavior