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"'95 I:

This is to certify that the

thesis entitled
EXPECTED UTILITY MAXIMIZATION AND THE
THEORY OF MULTI PRODUCT BANKING FIRM
UNDER UNCERTAINTY
presented by
Elyas Elyasiani

has been accepted towards fulfillment
of the requirements for

Ph 0 D 0 degree in Economics

 

LEBAQXIEILEX

Michigan State
Univeww

 

 

Date November 5, 1979

 

0—7639

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PER ITEM

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EXPECTED UTILITY MAXIMIZATION AND THE
THEORY OF MULTI PRODUCT BANKING FIRM
UNDER UNCERTAINTY
by
Elyas Elyasiani

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1979

ABSTRACT

EXPECTED UTILITY MAXIMIZATION AND THE
THEORY OF MULTI PRODUCT BANKING FIRM
UNDER UNCERTAINTY

BY

Elyas Elyasiani

An attempt is made in this dissertation to integrate
some different aspects of bank behavior such as physical
production, portfolio selection, costs, liquidity management,
risk aversion, and institutional constraints into a unified
banking model.

The existing banking models may be categorized as
theories of costless intermediation. The banking literature
has undersold the neoclassical theory of the firm for
portfolio theory. Banking activity is summarized in
portfolio management, operating costs are overlooked, and
the role of banks as the administrators of the national
payment mechanism is ignored.

In this study microeconomic theory is applied to
banks as multiproduct firms. The banking firm is not a
pure financial intermediary. In addition to loans and
investments which it produces as an intermediary, it offers
clearance services as a direct supplier. Banking outputs
are joint and produced under uncertainty. Uncertainty is
present due to randomness in deposits, interest rates, and

output quantities. The bank maximizes the expected value of

an exponential utility function as the objective function.

The effects of changes in exogenous variables on bank
behavior are investigated through comparative static results.
The basic model indicates that portfolio composition and
the price charged by the bank on clearance output are not
independent of the rate paid on demand deposits. With an
increase in the latter rate, liquidity declines and the
percentage of the portfolio held in the risky asset
increases. This means that regulation Q, when effective,
serves the purpose of keeping banks liquid and safe.
Regulation Q, however, is also found to lower the price paid
for the clearance output. This artificially low price leads
to overutilization of the productknrconsumers andaaresource
allocation which is not socially optimal. A trade-off may
therefore be said to exist between bank liquidity and
allocative optimality. It is desirable that the policy
maker frequently chooses the optimal regulation Q ceiling
on the trade-off possibility curve according to prevailing
priorities. The existing zero ceiling is not necessarily'
apprOpriate over time.

Comparative static results on variances indicate that
uncertainty increases liquidity, reduces the loan output,
and lowers the certainty equivalent return on unencumbered
funds. This feature should be considered in the making of
monetary policy. The effect of uncertainty created by the
policy should be taken into account, as well as the effect

of the change in the policy instrument. The former effect

may even overshadow the latter.

Asset reserve credit and payment of interest on
excess reserves are investigated as potential monetary
policy instruments. It is found that these instruments may
be used to broaden the Fed's control over credit. The
effects of an increase in the reserve requirement and the
discount rate are found to be in general indeterminate.
Only under specific conditions are these effects restrictive.

The basic model is extended to include more outputs
and to incorporate liability management. The extended
model indicates the possibility of both complementarity and
substitution between outputs depending on the output mix.
Results in the basic model do not all carry over to the
extended model. In particular, fewer results have
determinate signs.

Monthly data on the large weekly reporting banks in
New York City are used to estimate the reduced form of the
linearized version of the model. The sample covers the
period June l969-December 1978. To measure expectations
on demand deposits and interest rates Box-Jenkins time
series techniques are invoked to develop ARIMA models and
make forecasts on these variables. The forecasts are used
as data in estimation. The empirical results indicate that
the hypotheses of risk aversion and dependence of bank

decisions on operating costs can not be rejected.

To my Father,
and to the souls of

Mother, Moshe, and Avner

ACKNOWLEDGMENTS

I wish to thank the members of my dissertation
committee, Professors Robert Rasche, James Johannes, Norman
Obst, and Bruce Allen.

My special thanks go to the committee chairman,
Professor Robert Rasche. He was a rich source of inspiration
and encouragement. Professor Johannes helped me with time
series forecasting techniques. Professors Obst and Allen
read the manuscript and made helpful comments. Professor
Allen also provided editorial help.

Terie Snyder typed both the preliminary and final
drafts. I would like to thank her for her efforts and

patience.

ii

TABLE OF CONTENTS

List of Tables
List of Figures
Introduction

Chapter
I. REVIEW OF LITERATURE

II. A TWO PRODUCT BANKING FIRM UNDER PRICE UNCERTAINTY

2-1 Introduction
2-1-1 The Partial Nature of Banking Models
2-1- 2 The Bank as a Pure Financial Intermediary
2- l- 3 The Intermediary-Direct Supplier Theory
2-1-4 The Multi Product Production Frontier
2- 1- 5 Jointness in Outputs and Separability
2- l- 6 Banking Costs
2- l- 7 The Market Structure
2-1- 8 Sources of Uncertainty
2-1- 9 The Precautionary Demand for Cash
2- l- 10 The Objective Function
2- 1- 11 The Constraints
2- l- 12 Output and Input Measures

2—2 The Structure of the Model and the Optimality
Conditions

2-3 Displacement of Equilibria
III. THE BANKING FIRM UNDER PRICE AND OUTPUT UNCERTAINTY

3-l Introduction ‘

3-2 The Optimality Conditions

3-3 Displacement of Equilibria

IV. THE MULTI PRODUCT BANK AND LIABILITY MANAGEMENT
4-1 Introduction ‘
4-2 The Optimality Conditions

4-3 Displacement of Equilibria

iii

Chapter

V. THE ESTIMATION PROCEDURE
5-1 A System of Reduced Form Equations
5-2 Planned and Expected Values
5-3 The Disturbance Term PrOperties
5-4 The Data
VI. TIME SERIES FORECASTING
6-1 Introduction
6-2 Time Series Model Building
6-2-1 Identification
6-2-2 Estimation
6-2-3 Diagnostic Checking
6-2-4 Forecasting
6-3 Accuracy Analysis of Post Sample Forecasts
VII. THE EMPIRICAL RESULTS
VIII. CONCLUSION
8-1 Results and Policy Implications
8-2 Limitations
Appendix
1. Moments of an Incomplete Distribution
2. Second Order Conditions
3. Total Differentiation of the First Order Conditions
4. Comparative Static Results
5. Moments of a Product
6. Graphs of Actual Versus Forecast Values of Demand
Deposits and Interest Rates
Bibliography

iv

Table

Table
Table

Table

Table

Table

Table
Table

Table

TablelO:
Tablell:

TablelZ:

7:
8:

9:

LIST OF TABLES
The Differentiated Form of the First Order 43-45
Conditions
Cofactors Signs 46
The Comparative Static Results 47-48

The Differentiated Form of the First Order 65-67
Conditions

The Comparative Static Results 68-69

The Differentiated Form of the First Order 81-82
Conditions

The Signs of the Cofactors 83
The Comparative Static Results 84-85
Box-Jenkins Models 106
Accuracy Measures For Forecasting Models 110
Theil's Accuracy Measures 112
The Empirical Results 117-120

Figure 1:

Figure 2:

Figure 3

Figure 4:

Figure 5:

Figure 6:

Figure 7:

Figure 8:

Figure 9:

FigureJO:

LIST OF FIGURES

Product Transformation on An Aggregate
Production Function

Indifference Curves Between Mean and
Variance

Indifference Curves Between Mean and
Standard Deviation

Dependence of the Planned Output on the
Expected Price

The Distribution of Free Reserves

Graph of Actual Versus Forecasted
Demand Deposits

Graph of Actual Versus Forecasted Prime
Loan Rate

Graph of Actual Versus Forecasted
Treasury Bill Rate

Graph of Actual Versus Forecasted Federal
Funds Rate

Graph of Actual Versus Forecasted CD Rate

vi

18

33

33

60

128

147

148

149

150

151

INTRODUCTION

Commercial banks are one of the most vital institu-
tions in today's economies. They constitute an important
link in the monetary transmission process, play a key role
in the capital market, and strongly influence the money
supply, business activity, and the price level.

In spite of such prominence, a definitive model of
bank behavior has not yet appeared. The existing models,
greatly overshadowed by portfolio theory, treat banks as
pure intermediaries that allocate a sum of funds among
competing assets so as to optimize some objective function.
These models are often unintegrated, lack a satisfactory
theoretical framework, and overlook some of the relevant
functions of banks. The marginalist micro theory has
rarely been utilized to analyze the banking enterprise.

The complex and diverse nature of commercial bank activities
has made the establishment of a workable theory of produc-
tion and cost prohibitively difficult.

In this study an attempt is made to develop an inte-
grated model of bank behavior in a microeconomic firm
theoretic context. The bank is considered as a multi
product firm which transforms its inputs of capital, labor,
and deposits into some outputs. The clearance activity which
makes banks the administrators of the nation's payment
mechanism is considered as a non-intermediary output. Loans

and investments constitute intermediary outputs. The

vii

transformation of inputs into outputs is, like in any other
firm, subject to a production function. It is argued that
banking outputs are technically interdependent and therefore
joint.

The study may be summarized as follows: The first
chapter provides a brief review and a critique of the
existing literature. In Chapter Two a theoretical framework
is established and utilized to develop a model of multi
product firm under uncertainty. In this chapter only two
outputs are introduced, the clearance output and loans.

The sources of funds are the exogenously determined demand
deposits and the net worth. Uncertainty, is due to random-
ness in demand deposits, the loan rate, and the demand for
the clearance output. The quantity of loans is determinis-
tically set and achieved. The bank holds excess reserves
to satisfy its liquidity needs and borrows at the discount
window in case of a deficiency. The comparative static
results are derived and interpreted. Randomness in loans
is incorporated into the model in Chapter Three. This
makes the quantity of loans and the loan rate simultaneously
random. The comparative static results are found to remain
essentially unchanged.

Commercial banks may not, in our time, be considered
as passive intermediaries. The prevalence of liability
management has transformed banks to aggressive enterprises.
Chapter Four extends the model to introduce liability

management. The number of outputs is also increased. With

viii

liability management the bank has access to new sources of
funds which are under its control. It can sell CD's or buy
funds in the federal funds market to increase the scale of
its operation. The comparative static results for the
extended model are derived, interpreted, and contrasted to
previous models.

In Chapter Five the model is linearized, the adjust-
ment mechanism between actual and desired values is hypothe-
sized, and a system of reduced form equations is derived
which is adaptable to estimation. To measure expectations
on demand deposits and interest rates, which appear as
regressors, in Chapter Six Box-Jenkins techniques are used
to develop time series models and make forecasts on these
variables.

In Chapter Seven the estimation techniques are
discussed. The empirical results are displayed and con-
trasted to theoretical comparative static results.

Chapter Eight derives the policy implications of the
results and provides some of the limiting features of the
model. The mathematical derivations of moments of an
incomplete distribution - discount window borrowing,-—the
second order conditions, total differentiation of the first
order conditions, comparative static results, and the
moments of products of random variables are provided in
Appendices one through five respectively. Graphs of actual
versus expected values of demand deposits and interest rates

are given in Appendix Six.

ix

CHAPTER ONE
A REVIEW OF THE BANK BEHAVIOR MODELS

Even though there exists an extensive literature on
banking, a satisfactory model of bank behavior is still
lacking. It is surprising that the nee-classical micro-
economic theory did not find its way to banking literature
until the 19605, and even since then this apparatus has not
been extensively used. The nature of the bank as a
productive firm has not received adequate attention. In
1961 Porter rightly complained: "Over the course of the
last century, the implications of the assumption of profit
maximization for the behavior of the firm have been tracked
down in ever greater detail, curiously however, this firm
has almost always been a seller of non-financial goods;
banking has been studiously exempted from the application of
such theory. The exemption is curious because the
commercial bank seems in many respectes more likely to fit
the conditions of such static theory than the product
manufacturer" [33 , p. 12] .

To bridge the gap between the bank behavior models
and the microtheory, Porter set up a model of banking firm
which produces loans, securities, and cash under uncertainty.
Porter recognized uncertainty as the crux of bank Operation
and explicitly introduced deposit variability into the asset
selection process. This introduction came just about seventy
years after Edgeworth had indicated the importance of such

1

2

uncertainty in banking [ 9 ]. Porter applied an inventory-
theoretic approach to the banking firm, considering cash
and assets readily convertible to cash as inventories. The
carrying costs of these inventories are the interest income
foregone. Insufficient inventories are subject to penalty.

In Porter's model, the bank's sole concern is the
deposit lowpoint. This is the lowest point to which deposits
fall during the period and consequently, the point which
necessitates the most radical adjustments in the bank's
portfolio. In case of deficiency, at the deposit low point,
the bank will sell securities at prevailing market prices,
and when securities are exhausted, it will borrow from the
discount window. Both competitive and imperfect loan
markets are introduced and the firm is assumed to choose its
portfolio so as to maximize the expected value of its profits.

Porter's model can explain the observed diversifica-
tion behavior, the demand for excess reserves, and the
dependence of the Optimal asset-mix on the asset-returns,
the discount rate, and the distribution of the deposit low.

In the Porter model, the bank's demand for reserves
is not determined by the optimization process. It is set
to be a fixed percentage of its deposits. The Porter
banker always holds some interest-free excess reserves for
no purpose, except to maintain the fixed desired excess
reserve ratio. Porter's liquidity adjustment pattern is
equally mechanical. In case of a deficiency, securities are

completely exhausted before any other liquidity source is

3

considered. This is a hierarchical pattern imposed on the
bank and one that a rational decision making unit would not
necessarily follow. Additionally, Porter's results are
derived under specific simple distributions for the model's
random variables and wduld not hold in general. Finally,
Porter's model has not undergone any empirical testing;
theory has not simply been made Operational. To make the
model testable, at least an expectation formation mechanism
has to be introduced.

Another approach to bank behavior is the portfolio
approach introduced by Markowitz [ 25 ]. In this approach
returns on assets are taken to be random, and the investor
minimizes the risk for any given expected return. The
portfolio approach can explain the observed diversification
phenomenon; when the returns are not perfectly correlated,
diversification reduces the risk. This approach, however,
ignores all the dimensions of the asset selection which are
not translated into the two first moments of the return on
portfolio. Specifically, if bank behavior can be divided
into a portfolio decision and a liquidity decision, the
portfolio approach certainly leaves the latter out.

Kane and Malkiel’s (1965) model is based on the
introduction of deposit variability into such a framework
[ 20 ]. The model includes loans and securities as assets
with equal maturities and random returns. No reserves are
held and no borrowing is possible, so that deposit flows

are supposed to be totally reflected in the securities.

4

Kane and Malkiel enrich this apparatus by introducing
the idea of customer relationship and its implications on
short and long-run earning and risk. They argue that there
exists a class of loan requests L*, distinguished primarily
by a continuing relationship between borrower and the bank,
where the very failure to grant the loan itself would change
the banker's opportunity set. It would reduce the strength
of the relationship between the borrower and the bank,
leading to a decline in the expected value of both short
and long-run profits and an increase in aggregate risk
[ 20, p. 119]. In such a world, Kane-Malkiel claim that,
in a tight money condition banks stretch the liquidity
limits, sell their securities with capital losses, and
ration credit on the basis of the strength of the customer
relationship to satisfy the L* customers. This behavior,
they conclude, is in contrast with the three basic points
of the availability doctrine which ignores the bank's
reaction to monetary policy.

One criticism of the model is that if the bank adjusts
its portfolio of loans and securities with the possible
arrival of the L* customers, the quantities of loans and
securities may not be set and achieved. They must be
stochastic. In addition, Kane and Malkiel do not provide an
empirical test of their model.

Shull's (1963) banking model is based on the Clemens
(1951) model of multi product price-discriminating firm

[38,71- The latter views the firm as a combination of Joan

5

Robinson's price-discriminating monOpolist and Chamberlin's
product-differentiating monOpolistic competitor.

The firm uses its mobile resources to produce m
distinct outputs which are cost-wise homogenous. It sells
each output i to ni separable groups of customers with
different demand gIasticities. Each group of customers
demanding one output constitutes a market and is represented
by a demand curve. The demand curves are assumed to be
independent both across customer groups and across products.

The firm starts off producing for the market with the
least elastic demand. It then diversifies into new markets
with successively more elastic demand curves. It carries
its production to the point at which marginal cost equals
the marginal revenue in the least profitable (marginal)
market. The quantities and prices are then adjusted in
other markets so that all markets have an equal marginal
revenue. This adjustment maximizes the firm's profits.

Such a firm will have a barely profitable marginal
market which might have a quite elastic demand curve. It
increases its prices and restricts its output in less elastic
markets as it moves to new ones and tries to increase the
numberof its markets as much as possible, because this leads
to greater profits. The firm, however, may end up with
normal profits if the number of competitors is large.

Shull claims that such a model fits banking activity.
He argues that: a) the bank's main resource (funds) is mobile,

b) outputs can be considered cost-wise homogeneous since the

6

main cost is the cost Of funds, c) the bank customers can be
divided into groups with different demand elasticities, d)
banks do seem to diversify into new markets (new types Of
loans) instead Of lowering the price in the Old markets, and
e) they do seem to discriminate in pricing by negotiating
with each customer.

Shull's model may be criticized for its unrelated
demand assumption. [See I[37]] and cost homogeneity. Besides,
Shull fails tO provide a mathematical formulation Of the
model.

Klein (1971) introduced a neoclassical microeconomic
analysis Of the banking firm [ 22 ]. .In this analysis the
bank is assumed to set the deposit rates in the deposit
market and to use the funds to acquire loans, securities, and
cash. The security market is assumed competitive and the
loan market imperfect. Uncertainty is present due to random-
ness in deposits and asset returns. Cash holding represents
a precautionary demand. The bank's objective is to maximize
the rate Of return on equity.

In this study Klein finds that: a) the Optimal asset
mix is independent Of rates paid-on deposits [ 22 , p. 215],
so that the riskiness Of the bank portfolio is not altered
by variations in rates paid on deposits and the justification
for regulation Q is unsound, b) loans are held up to the
point at which their marginal return is the same as the
exogenous expected return on riskless government securities

E(g) [ 22 , p. 213], in other words, E(g) is the cut-Off

7
rate, c) the rate on each kind Of deposit is set at a point
which is below the portfolio yield by a factor which depends
on the parameters Of the deposit supply function (p. 214).
Klein also estimates a production function for the bank's
clearance output.

Pringle (1973) challenged the generality Of these
results [ 34 ]. He showed that result (a) "follows from
his [Klein's] assumption that all controllable sources Of
funds have rising cost curves (all are endogenous) and that
something other than deposit rates, namely the exogenous
expected return on riskless government security E(g) consti-
tutes the exogenous rate that pegs the system." In the same
way Pringle showed that result (b) "follows from the assump-
tion that E(g) alone is exogenous, and that investors are
risk neutral" [ 34 I p. 992]. Pringle introduced a set of
assumptions under which these results did not follow.

Hester and Pierce (1975) developed an econometric
micro model Of bank behavior in the framework Of a profit
maximizing firm subject to legal and institutional constraints,
and deposit and loan uncertainty [ 17 ]. In this study, the
exogenously determined bank liabilities--demand and time
deposits-~are assumed to be generated by auto-regressive
stochastic processes. The processes are taken to be indepen-
dent and uninfluenced by the portfolio composition [ jr7, p.
36]. Hester and Pierce banks consider the deposit inflows
as largely transitory. They ultimately expect to retain

only a small fraction Of each inflow. Therefore, they try

8
to lend only this small permanent component, holding the
remaining in securities and cash, depending on the dates the
deposits are expected tO be withdrawn (p. 69). More
specifically Hester and Pierce hypothesize a lagged portfolio
adjustment mechanism by which deposit inflows are initially
held in the form Of cash, but are transferred gradually to
less and less liquid assets. In this dynamic specification
the asset structure is strongly dependent on the time pattern
Of changes in liability items. Hester and Pierce test their
lagged adjustment hypothesis in an input-output version and
an adaptive-expectation version. Their results for commercial
banks are found tO be satisfactory. The extent Of their
empirical work and the huge set of information which they
use are impressive. Their empirical model and their empirical
results, however, do not follow the theoretical framework
they establish.

Scott (1977) developed a model Of a multi product
banking firm facing interrelated demand curves [ 117 ]. The
idea is that prices charged to various customers for one
output (e.g., loans) will influence their demand for other
outputs (like deposits). Scott argues that with related
demands "intuitively we might expect that the firm under
certain conditions would set Pl < MCl (marginal cost Of good
1) to stimulate the demand for good 2. Good 1 would be a
loss-leader in business terminology. The simple tactic Of
lowering prices on some goods tO increase demand for others,

becomes increasingly likely in a world of imperfect

9

information where the seller wants to "get the customer's
attention" and win his business [ 37, r» 16].

In this model deposits are considered as both outputs
and inputs or as outputs sold at ostensibly negative prices
because they can support the sale the income-earning assets.
"Formally the marginal revenue from an additional dollar of
liabilities 'sold' includes a 'multiplier' fromthe relation-
ship constraining assets given liabilities" (p. 22), so that
if we incorporate the effect Of this multiplier, the
"effective marginal revenue" will be positive and equal to
marginal cost Of deposits production. Scott's Objective is
to examine the impact Of concentration on prices, but his
model can be used to examine the other aspects Of bank
behavior as well. He claims that his model includes Hodgman's
model as a special casell9].

It is Observed that banking models consider banks
either as rational investors or productive firms. The rational
investor models summarize banking activities to portfolio
management and leave out the other dimensions. It may be
noticed that the models developed by Porter, Kane and Malkiel,
and Hester and Pierce are in this category. Among the models
that consider banks as firms there is no general agreement
on what constitute outputs, or what the technological and
demand interdependence patterns between inputs or outputs are.

One output Of the bank which is not Often recognized
is the clearance output. Generally, this output is over-

shadowed by portfolio management and taken as incidental to

10

it. The clearance service is the distinctive feature between
banks and other financial intermediaries; a service for which
Pesek claims that portfolio management is maintained akin
to the maintenance Of cables for the phone services [ 31 ].

Another feature which is Often ignored is the banker's
porverbial conservatism. Models presented by Porter, Shull,
Klein, and Hester and Pierce all display risk neutrality.
Under uncertainty stochastic profits may deviate from their
expected levels tOO much tO be acceptable to the banker.
The banker is therefore expected to consider the risk and tO
act accordingly. This point has been explained by Drhymes
by using the Tchebychev inequality [ 3 , p. 245]. The
inequality as applied to profit may be expressed as:

P [1n-E(n)|>xon1<-l—2 K>0
r K

where E(w), on are the mean and the standard deviation Of
profits (n), and K is a constant. In this inequality for
large K the deviation [n - E(n)] may be beyond the maximum
level acceptable to the bank.

Exogeniety Of deposits is another problem with some
of the models. This feature might be adequate for pre 19605,
but no longer prevails. The banking world has undergone
great changes over the last decade. Banks nO longer leave
the level and the composition Of their liabilitieSotO the
public. Rather, they actively influence the size and the
distribution Of these liabilities by aggressively seeking

CD's, federal funds, Eurodollars, etc.

11

The emergence Of these new sources Of funds discredits
some Of the conclusions which can be derived from the models
based on exogenous liabilities. For example, in Hester and
Pierce model satisfactory results were Obtained for pre 1963
period on the assumption that demand and time deposits are
independent. However, the existence Of the CD market, which
allows banks tO Optimize their CD sales in response to
unexpected variations in deposits, falsifies such an assump-
tion and restricts the cases in which the model can give
satisfactory results.

In introduction to the next chapter, some other

features Of the existing models will be discussed.

CHAPTER TWO

A TWO PRODUCT BANKING FIRM UNDER UNCERTAINTY

2-1 Introduction

 

2-l-J.The Partial Nature Of the Banking Models and the
Necessity Of a Theoretical Framework

A major problem with the banking models that have
appeared in the literature is that they typically abstract
from the interdependent nature Of banking activities. Each
function Of the bank - portfolio selection, reserve manage-
ment, or borrowing at the discount window - is analyzed in
isolation and separated from other functions. [See for
example [20], [26], [12]]. For unrelated functions such an
approach is justified, but as soon as we realize that each
function Of the bank is an integral part Of the whole and
dependent upon others, such an isolation becomes equivalent
to ignoring the linkages which relate these functions to one
another tO form an interdependent and unified complex. It
is the conglomeration Of all Of the bank's functions which
makes the bank a unique financial intermediary. Hence, the
integration and unification Of various dimensions Of the
bank behavior is essential to the better understanding of
banking activity.

Another problem in the literature is that issues are
Often analyzed outside any explicitly introduced framework.
As Klein [22] has pointed out, questions are Often asked
and answered about the effect on the bank behavior Of

12

13
changes in some exogenous factors - market structure,
regulations, etc. - without bothering first to develOp a
theory which describes the bank behavior under any specific
conditions. [See [22]]. DevelOping a model within which
issues can be apprOpriately analyzed seems tO be a
prerequisite tO a sensible analysis of those issues.

The model developed in this study integrates such
different aspects Of bank behavior as physical production,
portfolio selection, liquidity, costs, risk attitude, and
institutional constraints into a unified banking model.
Then, on the basis Of this model, the effect Of variations

in the exogenous factors on banking activities are analyzed.
2-1-2The Nature of the Bank as a Pure Financial Intermediary

The banking literature treats the bank as a pure
financial intermediary, which borrows short and lends long,
expediting the transfer Of funds from surplus spending units
tO deficit spending units. As a borrower, it issues deposit
liabilities against itself. As a lender, it allocates the
secured funds among competing assets so as to Optimize some
Objective function. Such a view treats banks as rational
investors or portfolio holders rather than productive firms.
Consequently, banks have typically been analyzed in a
portfolio-choice context, as Opposed to microeconomic firm-

theoretical context. [See [20] and [3 1].

l4

2iL<3The Banking Firm, the Clearance Output, and the Inter-
mediary-Direct-Supplier Theory

In spite Of the traditional theory, it may be argued
that the demand deposit holder is not a surplus spending unit
who is trying tO lend to the bank. Quite the contrary, it
might be a deficit spending unit who has borrowed the.
deposited funds from the banker. A demand deposit holder
does not necessarioy prefer tO spend on current consumption
and/or investment goods less than his current income as the
definition Of surplus-spending unit requires [See [21], p.
49]. Instead, the depositor may plan to only use the bank's
clearance service, one prerequisite Of which is holding
deposits.

The pure intermediary theory actually overlooks the
non-intermediary portion Of the banking activity. A
distinct feature of banks as Opposed to other financial
intermediaries is that banks are the administrators Of the
nation's payment mechanism. As indicated by Klein [p. 206],
such administration utilizes scarce resources and constitutes
a service provided by the bank tO the non-bank public; a
service that may not be considered intermediation. Klein
has estimated a linear homogenous Cobb-Douglas production
function for the clearance output based on the Federal
Reserve functional cost data. He imputes an implicit return
Of 1.6 percent on demand deposits through the clearance
activity. The Klein model, however, isolates the clearance

output from the rest Of the banking activity and implies

15

their independence.

Pesek [31] has also emphasized the non-intermediary
nature of at least a portion Of banking activity. He has
prOposed that "one part of our banking business does not
consist Of intermediation at all and does consist only Of
direct supply of a good (medium Of exchange) and a service
(accounting) to the demanders" (p. 876). Pesek Offers an
intermediary-direct-supplier view of banks. In this View a
bank becomes, "a conglomerate Of a pure bank....and a
productive enterprise indistinguishable from say a C.P.A.
firm or from H & R Block's tax advisory firm." (p. 875). It
acts as an intermediary when it borrows from surplus spending
units and lends to deficit spending units, but as a direct
supplier when it produces the clearance output as a service
for its depositors.

Following Klein [22], Pesek [31], and others, the
present study views banks as multi product firms which are
both a financial intermediary and a direct-supplier. In
this view the bank becomes a technical unit which transforms
some inputs into outputs subject to a production function.
It employs capital and labor inputs to complement deposits
in producing the clearance output (X1) and the loan output
(X2). These outputs are claimed tO be produced jointly.
2-l-4The Technological Constraint and a Critique of

Aggregation
The multiproduct banking firm has a technological

constraint specified by its production function. Commonly

16
two methods have been adopted in adapting single output
production functions to multiple outputs. One method,
adopted for example by Bell and Murphy [44], is to assume
that outputs are independent so that a separate production
function may be constructed for each individual product.
This method ignores the interdependence bewteen different
outputs. The other method is to construct a weighted index
or an aggregate measure, which may be considered as a
composite good. This method, which was adopted for example

by Greenbaum [13], is further analyzed below.
2-1.41LA.Critique of Aggregation

The most common procedure of aggregation is the
weighted-average approach. In this approach output prices
are generally used as weights to obtain total revenue as an
aggregate measure. The problem with this approach is that
it violates the neoclassical requirement that the production
function should be single-valued [11), p. 7]. It also leads
to non-positive marginal products. The argument (due to
Mundlak [27]) is as follows:

Consider the product transformation lOcus between
outputs X1 and X2, on which a and a* are two Optimal output
combinations, selected under different price situations Pa
and Pa*’ where each combination corresponds to a vector of

l 2

outputs; e.g., a = (Xa , Xa ). Now if K is the aggregate

output measure, the outputs atcxand a* will be measured as:

The problem is that although all the output combinations on
the product transformation curve, including a and a* are by
definition produced by the same bundle of inputs, theyék>not
necessarily give rise to the same monetary value output
measure (X). In fact, depending on the choice of prices,

X can be larger than, equal to, or smaller than Xa*’

Q.

X * > X . This result indicates that the single output

0. < C!

production function, with total revenue as the output
measure, will not be single-valued[Figure 1].

Further, consider an increase in the bundle of inputs
under the conditions that the relative output prices remain
unchanged, i.e., assume the tangent isorevenue lines shift

parallel to themselves so that we have:

pl Pl 9*1 p*1
__=__B_ 0‘

2 2 ’
Pa pB Pa 8

 

 

Now consider that from a to B and from a* to 8*, with
outputs having increased and relative prices held constant,
the aggregate output measure - total revenue - has increased,

that is:

18

 
 

 

  

><
Q
*l—‘
><
Q
...:
X
....-

FIGURE 1

PRODUCT TRANSFORMATION ON AN

AGGREGATE PRODUCTION FUNCTION

19

the following were also shown to hold:

>
X* -, X
a x a
>
X* - X
B < 8

Putting these results together, it may be concluded that:

> >
* — — *
X < XB , X < X .

These relations indicate that an increase in the bundle of
inputs may correspond to a larger, smaller, or equal value
of aggregate output measure. In other words, these relations
indicate that marginal products may be positive, negative,
or zero~[Figure 1] .

The non-single-valuedness, and the non-positive
marginal product problems, will be removed only in the
special case of fixed proportion production. This is
because in this case no transformation is possible between
the two outputs and the product transformation curve reduces
to a single point. It may be concluded that the use of the
monetary value measure is equivalent to ignoring the

transformation possibilities [28 ].
2-l-4-2.The Multi Product Frontier

With the problems discussed above, it seems necessary
that a multi product production frontier be used whenever
multiple outputs are present. A multi product production

frontier can be represented by an implicit function

20
F(X,V) = y, where X and V denote vectors of n outputs and m

inputs respectively and y is a positive scalar.

><
ll

(x1....xn)

V

(V1....Vm)

The frontier can then be defined as a function which (i) for
given values of all inputs and all but one output Xi,
specifies the maximum amount of Xi producible, (ii) for
given levels of all outputs and all but one input Vj’
determines the minimum required Vj.

The production frontier is usually normalized so that
its value varies directly with outputs and inversely with
input levels. When the frontier is differentiable, normali-
zation implies the following restrictions:

8F(X,V)
BXi

8F(X,V)

V.
3

> 0 i = 1,....n

After the normalization, the positive scalar Y may be inter-
preted as the efficiency parameter, since for given inputs,

larger output levels will correspond to larger y values.

Mathematically:
F(X,V) = Y
F(X*,V) = y*
Y* > Y

imply X* > X .

21

Parameter y can be absorbed in the frontier, so that the
frOntier may be written as F(X,V) = 0.

In this study the logarithmic version Of the Mundlak
trnascendental production frontier is adopted. This frontier
is non-homogenous, has a variable elasticity of transforma-
tion, and is separable between outputs and inputs. The
frontier can mathematically be represented as:

a a B X + B X a a a y
F(X,V) = X 1X 2 e l l 2 2 - DD DL LK Ke = 0

or equivalently in logarithms as:

F(X,V) = a1 log (x1) + a2 log (x2) + 81x1 + 32x2
- GD log (DD) - aL log (L) - aK log (K) - Y = 0
where Xi's are the clearance and loan outputs, K, L, and DD
are capital, labor, and demand deposit inputs, and Y is the
efficiency parameter. The following restrictions are
required by normalization:
Fi > 0 Fj < 0 where Fi = aF/BXi Fj == aF/avj

2-1-5.Lointness in Outputs and Separability in the Production
Function

2-1-5-1,Theoretical Background

A multi product production process may correspond to
a non-joint or a joint technology. If the technology is
non-joint, multiple outputs are each produced under a

separate process which is independent of other processes.

22
Hall [14] has defined non-jointness in the following manner.
"A technology with transformation function
t(X,V) is non-joint if there exist functions
fl(Vl)....fn(Vn) (interpreted as individual
production functions) with the prOperties: (i)
There are no economies of jointness: if V can

produce S, there is a factor allocation

v1 + v2 + vn = v such that fi(vi) > xi

i = l,....n. (ii) There are no diseconomies
of jointness: if Xi = fi(Vi), all i. Then
V = V1 + + Vn can produce X."

This definition implies that if there are any kind of inter-
dependence between outputs, so that their co-production
results either in some economies or diseconomies, the process
is joint. In a joint technology all the outputs are
produced through a single production process and are
technically interdependent.

Separability is a feature related to jointness. The
frontier F(X,V) = 0, is said to be separable between
outputs and inputs if it can be written as t(X) - g(V) = 0.
t(X) is called the output or transformation function, and
g(V), the input or production function. Separability almost
always implies jointness in outputs. In fact, Hall [14] has
shown that if a multiple output technology which is separable
*is non-joint, it must be a fixed prOportion process. In
this case a single output production frontier may satisfac-

torily be used as explained before. The jointness

23

implication of separability makes it practically equivalent
to jointness. This equivalence forecloses the investigation

of jointness hypothesis for any separable production frontier.
2-1-5 Jointness in Banking Outputs

Here it is claimed that the bank outputs are
produced jointly. In financial intermediary-direct-supplier
theory, the bank has at least two outputs Of clearance (X1)
and intermediation (X2). The production of X1 provides a
source of funds for the bank which it uses an an input to
produce X2. The two outputs are produced in a technically
interdependent manner. It is possible, of course, for the
two outputs to be produced separately, but in that case they
will not be produced as efficiently. We can think of
institutions which would accept deposits and clear checks
for some fee without engaging in intermediation, where some
others would borrow through time deposits and would lend
without producing the clearance output. The aggregate cost
of these two separate production processes would exceed their
counterpart in the joint production case, because in the
former case the demand deposits would not be as efficiently
utilized.

It is noteworthy that these outputs have not always
been produced jointly. Sixteenth centurygoldsmiths held
gold coins and issued transferable liability notes which
eased the clearance mechanism to a large extent, but for a

long time did not engage in lending. However, as time

24

progressed they recognized the profit Opportunities inherent
in the co-production of the two outputs and adopted a joint
process.

The existence of profit Opportunities in the co-
production of different outputs indicates some economies of
jointness. The interdependent nature of banking outputs
alongside the presence of these economies in turn indicates
that the production process has a joint nature. Bell and
Murphy [4 ]have claimed that the bank's functions and
services are not joint products, since they can be produced
in varying prOportions. There seems to be some misconcep-
tion in such an argument. It is true that the bank outputs
can be produced in varying prOportions, but this does not
rule out their jointness, since fixed prOportion production
is only a special case of joint production. Jointness exists
whenever the quantities Of outputs are technically inter-
dependent [15]. The best way to resolve the controversy Of
course is to empirically test the jointness hypothesis. One
such test is provided by Mundlak and Razin [28]. Another
test is suggested by Samuelson [35] and adjusted to banking
by Adar, Agmon, and Orgler [1 ]. These tests however, are
based on functional cost data which are not available to
the general researcher. TO incorporate jointness, a

separable production frontier is adOpted in this study.

—{"_

25

2-1-6Banking Costs

It is customary in the banking literature, to ignore
the bank's Operating costs together with the production
function constraint. These costs are considered to be
irrelevant in the bank's decision making process. As a
consequence of this tradition and as Opposed to the neo-
classical theory Of the firm, the bank's output supply
functions are not based on the production costs. The
banking theory, as Pesek [31] has pointed out, is a theory
Of costless banking. Pesek [31] has argued that money supply
theory should be based on the production costs, rather than
the mechanics of money multiplier. In this study Operating
costs are incorporated into the decision making process.
This will allow a test of significance of costs on output

supply functions.
2-1-7The Market Structure

In the clearance output market the proximity of the
depositor to the bank is of prime importance. This market
has therefore a local nature [22]. It seems unlikely that
non-local competition can induce the depositors to transfer
their transaction funds to non-local banks, or at least it
is so in the presence of requlation Q which prohibits explicit
interest payment on demand deposits.

Under this condition, the clearance output of
different banks are spatially differentiated and the bank

has a degree of monopoly power which enables it to set the

26

price in the local clearance output market. The demand for
clearance output is assumed to be represented by a linear
function of the stock of demand deposits, the price charged
by the bank, and a random term. The form of the demand
function is expressed by equation (1) given in the next
subsection. The loan market is assumed to be apprOpriately
approximated by perfect competition, so that the demand
function for loans is perfectly elastic. Capital and labor

are also assumed to be hired competitively.
2-1-8 Sources of Uncertainty

One inherent characteristic Of bank liabilities is
deposit variability. Although this feature was recognized
by Edgeworth (1888) [9 ], it was first incorporated into the
banking models by Porter (1961) [33], and Mellon and Orr
(1961) [29]. The stochastic nature of the deposits makes
banks live under uncertainty created by random deposit
flows and unexpected cash needs.

The demand for clearance output is assumed to be
stochastic for two reasons besides deposit uncertainty.
First, the transaction pattern of the demand deposit holders
which is the source Of their demand for clearance output is
indeterminate. Second, technological changes in the payment
mechanism may cause the clearance output to vary widely for
given deposit levels and the clearance output prices.

A third source of uncertainty is randomness in the

loan rate. To analyze the effect of default, assume the

27

banker adds a premium to the contract rate to compensate for
default [1x3]. With this additive default rate, the pure

(net) loan rate P may be written as the difference between

2

the contract rate C2 and the default rate d2. It is assumed

that both the contract rate and the default rate are random.

Following these explanations we have:
(1) X1 = alDD + blPl + El al > 0 131 < o
(2) DD ~ (56, V(D))
(3) E .. (O, V(El))

(4) C2 ~ (C2, V(C2))

(5) d2 ~ (32, V(d2))

(6) P2 ~ (P2, V(P2))
Where DD indicates demand deposits, E1 is a disturbance
term, and C2, d2, and P2 were defined before.

The quantity of loans (X2) is assumed to be set and
achieved by the bank. Hence, any unexpected deposit flows
will be reflected in reserve holdings. .The distributions
of the random variables are assumed known, and therefore
bank behavior can be said to be under risk rather than under

uncertainty.

28

2-1-9 The Precautionary Demand for Cash and the Recourse
to the Discount Window

In the spirit of Mellon and Orr [ 29] and Klein
[ 22] it is assumed that the bank holds a cash inventory
as a precaution against the uncertainty in deposit flows.

It plans to hold some optimum level of free reserves at the
beginning of the period, but unexpected deposit shocks may
make its actual reserve holdings deviate from the planned
level. Unexpected deposit flows will lead to excess reserve
holdings larger than the planned level and unexpected
deposit drains may lead to insufficient reserves, in which
case the bank will have to borrow at the discount window.

Since the level of free reserves is a random variable,
its value can not be Optimized. It is instead assumed that
the bank chooses the mean value of this variable which will
be its planned level of reserve holdings.

Excess reserves are assumed to have no explicit return
while cash deficiencies are subject to a fixed penalty rate.
It can be assumed that deficiencies will be eliminated by
borrowing from the discount window or that they will occur
at the end of the period and are covered by the sale of
loans. The penalty rate in the first case will be the
discount rate, and in the latter case the transaction cost
of asset sale. The return on cash holding in this framework
can be examined by evaluating its effect on the distribution

of cash deficiencies and hence on the penalty costs incurred.

29

2-1-10'Fhe Objective Function

The bank is assumed to be risk-averse and to maximize
the expected value of an exponential utility function, where
utility may be expressed as a function of the end Of the

period wealth (W).

-2aW

U = a - be a a, b > 0

> 0
'<0
where (W) is the sum of initial net worth (W0) and the
period's profit (n). a represents the risk-aversion co-
efficient.

The exponential utility function exhibits constant

absolute and increasing relative risk-aversion. This may be

illustrated by noting:

U' = zome'zo‘W > o

2 -2aW

U" -4a be < 0

_u
The degree of absolute risk-aversion as measured by (7%7

has the constant value of 2a. The degree of relative risk-
aversion as measured by (3%;E) has the value of 2dW which
increases with wealth. The latter can be shown to be the
wealth elasticity of marginal utility.

The choice of a utility function with constant
absolute risk-aversion, though generally assumed to be
inadequate for long-run and large decisions, seems reasonable

for short-run and intermediate decisions. When decisions

are revised in short intervals Of time, no single decision

30

can cause large changes in wealth [24 ].

To simplify matters, we assume that wealth or
equivalently profits are distributed normally. If profits
are normal it can be shown that maximization of the
expected value of any utility function of wealth leads to
maximization of a function of expected profits, variance of
profits, and initial net worth. [See [41 ]]. With
exponential utility and normally distributed profits, the

expected utility function can be simplified as follows:

 

 

 

E(U) = a - bE (e-2aW)
where W~(W, V(W)). E indicates the expectation Operator
and V indicates the variance.
+ -(W-W)2
E(e-ZOW) = e 2v(W) * e-ZaW
- °° (21TV(W))
H °° 73% (w2 + W2 - zwfi + 40LWV(W))
= e
dW
_ m (2nv(W))l/2
r+ m
fig {[w - (W - 2 v<wn12— 4a2(V(W))2+4dWV(W)}

= e dw
(2‘rrv(W))l/2

 

 

31

r+ m
-2a(W - (in) Tim {W ‘ [W - ZOMWH}2
e av * e

 

dW
(zmm ) ”2

 

e-Za (W'- av(W))

-2dW
.4.

Consequently, max E(U) ++ max - Ee +

max _ e-2a(W - av(W)) ++ max W’- av(W) +-

max E(n) - av(n) as W = W0 + v

To maximize expected utility, therefore, the bank has

to maximize an operational objective function G where:
G = E (w) - av(n) (7)

If we are maximizing expected utility, it can be
shown that a Taylor-series expansion of a logarithmic utility
function of wealth, and also a quadratic utility function
lead to a similar but not exactly the same objective function.
The latter functions are quadratic in mean and linear in
variance of profits. An affine transformation, however,
would make these functions linear in both terms [ 8 ].

With adoption of the above objective function a few
points should be noted. First, under risk-aversion, risk
as measured by profit variability becomes a utility cost
av(n) which is subtracted from the expected profits. Second,
with this measure, G as defined by equation (7), can be

termed the certainty-equivalent profits. [See [24 ]].

32
Third, the indifference curves which are.the loci of equal
expected utility, can be equivalently defined as the loci
of points on which G remains unchanged, so that on each
indifference curve we have dG = dE - adv(n) = 0. For the
measure of risk, we can take either the variance or the
standard deviation of the profits. In the former case we
can draw the indifference curves between the expected value
and variance of profits. The trade-off between these two
variables and the lepe of the indifference curves are the
constant a which merely depends on the degree Of risk-
aversion. In the latter case, however the indifference
curves are drawn between the expected value and standard
deviation Of profits; the trade-off and the slope of the
indifference curves are measured by 20L(V(1T))l/2 which is
increasing with the level of risk as well as with the risk-
aversion coefficient a. In the former case, the loci are
straight lines, while in the latter case, each indifference
curve is generated by the positive half of the parabola
E(w) = 20L(V(rr))]'/2 + C, where C is a constant. On each of
the latter indifference curves, successive units of risk

have to be compensated by larger and larger gains in

expected profits [Figures 2 and 3].
2-l-llThe Constraints

The production function, the demand function for
clearance output, the balance sheet identity, and reserve

requirement equation constitute the constraints on the model.

)

E(n)

z/:::::::::::::::::::::::://’E(n) == aVLN) + C

.fi

V(n)

 

 

FIGURE 2 : Indifference Curves Between Mean and

 

 

Variance
+
E(w)
2
l 2
J W = 0W) ” +C
I12
(V(nll /
FIGURE 3 : Indifference Curves Between Mean and

Standard Deviation

34

F = o1 log (alDD + blPl) + a2 log (x2) + 81(alDD

+ blPl) + 82x2 - OD 10g (DD) - aL log (L)

II
c:

- GK log (K) - Y (8)

Now consider the other two constraints. We define
free reserves as FR = ER - BR and substitute constraint (d)
into (c) to get the substituted version of the balance

sheet:

x<1-s2)+fii= (l-rd)—DT5+W (9)

2 0

This form of the balance sheet implies that the sum

of the planned values Of assets and the sum of the expected
liabilities must be the same. This formulation of the
balance sheet is similar to the concept of "desired balance
sheet" introduced by Tobin and Brainard [43 ] and has been
adopted as the balance sheet constraint by Parkin, Gray and
Barrett [30 1.

The equivalence of asset credit to compensating
balances may be seen by considering the balance sheet and

reserve requirement constraints. If r percent of loans are

2
to be held as compensating balances, these two constraints

will be written as:

_=_ +—
X2 +R DD+r2X2 BR+WO

‘E

rd(DD + rZXZ) + ER

where r2X2 is the amount of deposits created through

 

 

35
compensating balances. Now substitution of the second

constraint into the first will result in:

x [(l-r2(l-rd)] + FR = (l-rd)fif5 + Eff + W

2 0

This relation is the same as the substituted version of the
balance sheet given before except that 82 is replaced by r2

(l-rd).
2-1-12 Output and Input Measures
Outputs and inputs are measured as follows:

X number of dollars debited per period

1!

X number of dollars of loans

2!
DD, number of dollars of demand deposits held at

bank (stock)

L, man-hours labor hired
K, units of machines rented
P service charges per dollar of debits

P , pure loan rate, contract rate adjusted for

default
RD' rate paid on demand deposits
wage rate per man-hour

capital input rental price, dollars per machine

36
2-1 The Structure of the Model and the Optimalitng0nditions

 

The bank chooses its policy variables P FR, L,

ll x2!
K, so as to maximize its Objective function (7) subject to

the constraints (8) and (9). The Objective function (7) can

be measured as follows:

E = Ple + (C2 - d2)X2 - PLL - PKK - RDDD - RBBR

where X1, DD, E C d were given by (1), (2), (3), (4),

2' 2

and BR is the borrowing from the discount

1'
(5). C2 - d2 = P2,

window. We consequently have:

2

17 = (alpl- RD) DD+ blPl + P131 + (CZ-d2)X2-PLL-PKK-RBBR
E(Tr) = (alpl- RD) 5+ blP12+ (CZ-dz)X2-PLL-PKK - REE-R
mm = (alPl-RD)2V(DD) + p12 V(El) + x22 V(c:2 -d2) + RB2V(BR)
+ 2 (alPlZ-leD) covwo, El) + 2x2(alpl - RD)
COV(DD, (c2 - d2)) - 2RB (alpl - RD) COV(DD, BR)

- 21:sz COV(BR, (CZ-d2” + 2Pl x2 COV(E1, (CZ-dzn

- ZPlRB Cov<Elr BR>

It is assumed that the exogenous loan rate is uncorrelated
with the individual bank's clearance output, and that
borrowing is uncorrelated with the disturbance term El.

It follows that:

 

37

c0V ((c2 - d2), DD) = 0
COV ((C2 - d2), E1) = 0
COv (BR, El) = 0

These values are substituted in the objective function to

Obtain the Lagragian function G* to be maximized.

__ 2 _
'k —_- _
G (alPl RD) DD + b P + (C

11 -d2)X2-PL-PK

2 L K

2 2

- RBER - a{(alPl - RD)2V(DD) + Pl V(El) + x2

V(C2-d2) + RBZ V(BR) + (2alPlz-2P1RD) COV(DD, El)
- 2R3 <aiPl - RD) Cov (DD: El)

- zszB COV((c2-d2), BR)} — AF-—p[x2
(1 - 52) + FE - (1 - rd)DD - W0]

where F is given by (8).

The first order conditions (F.O.C.) for maximization

are:

* -_- _ _
3G /3Pl a DD + 2b P 1 RD) V(DD)

1 1 1 ' O‘mai‘a‘f’

+ 2P1V(El) + (4alP1 - 2RD) COV(DD, El)

- 2alRB COV(DD, BR)} - AFPl = 0 (10)

* —..""_ _ ..
8G /3x2 C2 d2 0L{2X2V(C2 d2) ZRB C0v

((C2 - d2), BR)} - AFZ - u(1 - 52)‘=°
(11)

38

3G*/3FR = - RB (aER/afii) - QRZB (3V(BR)/a§§)-p = o

(12)
ac*/aL = - PL - AFL = o (13)
3G*/3K = - PK - AFK = o (14)
aG*/aA = F = o (15)
ac*/au = - [x2(1 - 52) + FR - (1 - rd) BB'- woi = o

where F1, F2

frontier,

and F
P1

If we
constraint's

the following

Since‘y is th
the marginal

equivalent pr
price of effi
u can be inte
each dollar O

equivalent pr

(16)

F < 0 by normalization of the

>
0 FL’ K

= (SF/3X1) (3Xl/8Pl) = blFl < 0

consider 7 and (l - rd) DD + W0 as our
constants, the Lagrangian multipliers Satisfy

relationships:
aG/By = l,
ac/a [(l-rd)DD - W0] = u.

e efficiency parameter,)(can be interpreted as
effect of efficiency on the certainty-

ofit (G). It has also been termed the shadow
ciency by Hasenkamp [15 ]. In the same way
rpreted as the equilibrium contribution Of

f unencumbered funds to the certainty-

Ofit (G); in other words, n is the Opportunity

39

cost of each dollar of funds allocated in equilibrium.

It may be noted that the variable "borrowing" does
not have a well-defined distribution. It represents the
negative portion Of the distribution of free reserves (FR),
so that its moments can be considered as the incomplete
moments Of the latter variable. These moments and their
partial derivatives with respect to FR are derived in
Appendix 1. There it is shown that holding free reserves
reduces both the mean and variance of the reserve deficiency

or borrowing. Mathematically we have:

BER/317R = -Pr < 0

3V(BR)/3FR

- Z—R (l - P ) < 0
r

where Pr represents the probability of reserve deficiency.
Using these relations and the third of the first order
conditions, the sign of u is determined to be positive. The
sign of A is also positive due to the fifth of the first-
Order conditions.

The first order conditions implicitly define the
asset demand functions for loans and free reserves and also
the input demand functions for capital and labor. The
optimal value of each endogenous variable is a function of
all the exogenous variables in the model so that the
decisions about production, portfolio selection, liquidity,
etc. are simultaneous and interrelated.

The bank's demand for assets will generally depend

on:

40

(i) the expected return on both assets
(ii) variances and covariances of returns
(iii) the distribution of deposits

(iv) the return on deposits (RD)

(v) the risk attitude of the bank (a)

(vi) reserve requirement (rd), discount rate (RB)

input prices (PL, P ), and net worth (W0). Therefore one

R
asset can be attractive to some banks and not to others,
because of the differences in their distribution of deposits,
their estimates of variances, and risk attitudes. The
presence of the taste element (a) also rules out the
separation feature in the portfolio.

The diversification can be explained by both risk-
aversion and precautionary motive. Since cash holding
reduces the mean and variance of borrowing, it will be held
so long as its contribution exceeds its Opportunity cost
(u),[see equation (12fl. One feature which makes the present
model distinct from, for example, the one develOped by Klein
[22 ], is that in the latter, all the assets are pegged to a
rate imposed on the model from outside. Each asset is held
up to the point at which its marginal return, regardless Of
its riskiness, is the same as the expected rate of return on
government securities. In such a model, riskiness of
different assets has no effect on the asset mix. 'On the
contrary, in the present model, assets are pegged to the
equilibrium Opportunity cost n, which is endogenously

determined by the system. Each asset is held up to the

41

point at which its return, adjusted for its riskiness, is
the same as the common equilibrium contribution n. In other‘
words n includes both risk and expected return elements.

One last point to consider is the dependence of the
Optimal values on the rate paid on deposits (RD). This
dependence has implications on the establishment Of the
regulation O, which will be further explained in interpre-
tations of the comparative static results.

The second order conditions for a maximum are derived
in Appendix 2. The following restrictions are shown to be

sufficient for these conditions to be satisfied.

F > 0

F LL' KK

F F

11' 22’

32V(BR)/ F132 3 o

It is also shown that the first two conditions imply
increasing marginal rate of product transformation between

X1 and X and the latter two imply decreasing marginal rate

2!
of substitution between K and L. The last term is shown to

‘have an indeterminate sign in Appendix 1.

2-3 :Displacement of Equilibria

 

Here the first order conditions (10) - (16) are
totally differentiated to derive the comparative static
results. One additional variable is also added to the model.
To analyze the effect of improvement in the clearance

technology on the system's variables, such improvement is

42

introduced as an increase in the productivity of inputs in
producing output X1. The marginal products of inputs for

X can be measured as follows:

 

l
_ 3" PV. O‘j/vj + 33. _
MPV.' X1 ‘-" 5"?— = - 4F = a __ Vj = K, L, DD.
3 j l 1/X1'+B
1
For these marginal products to increase either o1 and/or 81
has to decline with technological progress. Here a is taken

1
to be a decreasing function of technology, so that Eel/3 is
t

negative. This prOperty will be used to determine the
response directionscflfthe endogenous variables to variations
in technology.

Total differentiation of the first order conditions
is given in Appendix 3, and summarized in Table l. The
cofactors Hij Of the hessian matrix H, are needed for
comparative static results. The calculations are omitted
here, their signs are displayed in Table 2. The comparative
static results are derived in Appendix 4. The signs are
displayed in Table 3.

The conditions under which some of the comparative
static results are determined require some explanations.

The terms 5%I—ERT and 335%. refer to the effect OfV(FR) on
the mean borrowing. These terms will vanish in the special
case that ER is affected only by variations in FR and not
those of V(FR). The condition RD < 2alPl imposes a ceiling
on RD. The zero ceiling imposed by regulation Q is

sufficient for this conditions to be satisfied. Since

borrowing from the discount window results from unexpected

43
TABLE 1

THE DIFFERENTIATED FORM OF THE FIRST ORDER CONDITIONS

 

 

r o o FPl F2 0 FL FK ‘ r-dl
o o o 1--s2 1 o o -du
FPl o All 0 o o o dPl
F2 1-s2 0 A22 0 o o ax2
o 1 o 0 A33 0 o dFR
FL 0 o o o -1FLL o dL
PK 0 o o o o -AFKK dK
l . l

 

 

44

TABLE 1 (cont'd)

r 3d
_ _ l
dDD[alFl dD/DD] + dy + log (alDD + blPl) in: dt
(l-rd)dDD - DDdrd + dWO + X2d82
2
(20La1Pl - Zoral RD )dV(DD) + 20LP1dV(El ) + 2a(2al Pl -RD)dC0v

(DD,E1) - ZualRB dCOV(DD,BR) - 2aa COV(DD,BR)dRB

1
- [2da1V(DD) + 201CO (DD, E l)]dRD + [lalb ]-F 11 - alldDD
+ [(ZaiPl - alRD)V(DD) + 2Pl V(El) + (4alPl-2RD)COV
(DD,El) - 2alRB COV(DD,BR)] dd
-dE2 + daz -‘uds2 + 2ax2dv(c2) + 2aX2dV(d2)
- 40cX2dCOV(C2,d2 - 2aRBdCOV( C2,BR) + 20LRBdCO v(d2,BR)
- 2aC0v(C2-d2,BR)dRB + [2X2V(C2-d2) - ZRBCOV(C2,BR)

+ 2RB COV(d2,BR)] da

3P '——
2-—- r 2 2 23BR
- { [R - ZQRBBR] W (l’rd) + 20RB(1‘Pr) (l-rd) TV ('FR)}

__ 2 __
dV(DD) - (Pr + 2aBR RB(1-Pr)]dRB-[2RB BR(l-Pr)]

3P
r

2 —— 2
(10. + {[RB " ZGRB BR] [2(1-rd)V(DD) m] + ZGRB

afifi

FVTFRT} drd

(l-Pr)(1-rd)V(DD)
dP

dP

 

 

11

22

33

ii

45

2
Zbl 2aV(Xl) - Abl Fll

- 20LV(C2 - d2) - AFZZ
__. 2 __ .__
RB aPr/aFR - 2aRB [(l-Pr) Pr + 2BR apr/aFR]

O i = 1,2,3.

 

46
TABLE 2

COFACTORS SIGNS

H0,0 ’ H01 + H02 ‘ Ho3 + Ho4 + H05
H0,00 + H001 ’ H002 ’ H003 ’ H004 ' H005
H00,00 ‘ H11 + H12 + H13 " H14 ’ H15
H22 + H23 ' H24 + H25

H33 + H34 ‘ H35

H44 + H45

H55

47

TABLE 3

THE COMPARATIVE STATIC RESULTS

 

Endogenous Variables

 

 

Exogenous Conditions __

Variables Imposed Pl X2 FR L K u A
52 + + - + + + +
d2 - - + - - - -
52 + + ? + + ? +

< - - + '2 ? '2 ?
RD COV(DD'X1) > O + + — - — + _
+ - + - + - +
PL
+ - + .+ - - +
PK
v(C2) - - + - - - -
- _ + .. _ .. -
V(dz)
-—- 8P
BBR r
—————— = —————— = - — -+ + + - +
V(DD) av(RF) 3V(FR) 0'
RD < 2a1P1
V(El) — - + ‘+ + — +
- ' + +
C (C2,BR) + + - ‘+ + + +
.. .. + .. .. .. _

48
TABLE 3 (cont'd)

 

COV(DD,BR)

COV(DD,E1)

3V(FR)

RD < 2alPl

BPr' = 3BR _
3V(FR) ’

RD < 2alPl,
COV(DD,BR) < 0,

COV(C2-d2,BR) = 0

'0

'0

'0

'0

'0

'0

‘0

 

\3 .

 

49

deposit drains and occurs only in emergencies, it may be
argued that BR varies indirectly with deposits and is
uncorrelated with the loan rate. Following these arguments
the conditions COV(BR,DD) < 0 COV(C2-d2, BR) = 0 seem
reasonable.

Some of the comparative static results are worth
noting: The model shows that as 62 rises the bank switches
from Xl to X2. It charges a higher P1 to reduce X1, and

holds less reserves using the funds together with additional

capital and labor inputs to produce more X The bank

2.
achieves a larger certainty-equivalent return per dollar (u).
The negativity of aFR/aCZ indicates that the bank's cash
holding is interest elastic. (Row 1)

The effect of mean default rate dz is in the Opposite
direction to mean contract rate CZ. If expected mean default
rate increases fewer loans will be planned, and more funds
will be held as cash. The equilibrium contributioncflfeach
dollar of unencumbered funds (u), will be smaller due to
larger default. It may becnfinterest to consider 5? as a
per unit tax (subsidy) on loans. The effect of such tax (subsidy)
would be the same as (opposite to) that of default. The
effect of collateral could also be analyzed in the same way.
Collateral reduces the probability of default, and conse-
quently the mean default rate dz. The effect of collateral
therefore would be Opposite to that of default.

The equivalence of asset reserve credit to compensating

balances was shown before, both features may be considered

 

 

 

50

as additional compensation on loans, since they raise the

P
effective loan rate as measured by P = ——Z—-, where S is
2e l-S2 2

the percentage of loans credited to the bank's reserves.

The comparative static effects of these variables are similar
to those of the loan rate whenever the former have deter-
minate signs.

The effect of asset reserve credit or compensating
balances on the bank's liquidity-cash holding-is indeter-
minate. This is due to the fact that these variables - on
one hand raise the effective loan rate, which increases the
loans and reduces free reserves, and on the other hand
provide a source of funds. The over all effect on cash can
therefore be of either sign. The conditional supply of
loanable funds, and demand for endogenous inputs have
conventional signs, that is, the former is positively and
the latter are negatively sloped (Columns 2, 4, 5).

Increased uncertainty represented by increases in the
variances acts as an increase in the default rate; it lowers

the clearance price P1 and the loan output X while it

2)
increases the mean cash holding FR. These results on loans
and free reserve holdings are consistent with Hicks [ 18].
The result about the loan output is also consistent with
Sandmo [1M5]. All variances lower the equilibrium certainty-
equivalent return n. This result justifies Hester and
Pierce's claim that realized profits under certainty fall

short of their counterpart under perfect foresight [17 ].

The monetary policy effect of the discount window

51
Operation is shown to be in the intended direction (Row 4).
The monetary policy effect of the reserve requirement (rd)
on the level of credit (X2) is restrictive. It is also found
that the reserve requirement and the mean free reserves move
in opposite directions, so that in the absence of reserve
requirement, a voluntary reserve holding would prevail (as
aFR/ard < 0).

The portfolio composition and the risk undertaken by
the bank are not independent of the rate paid on deposits
(RD). This result contradicts Benston [ 5 ] and Klein [22 ],
but is consistent with Gambs []J.]. The model indicates
that the percentage of the portfolio held in the risky asset

(X2) does rise with R Therefore regulation Q, when

D'
effective, does reduce the percentage of the risky asset in
the portfolio, and does raise the liquidity as measured
by FR/X2 or FR/DD. It can therefore be concluded that
regulation Q serves its purpose.

Yet another conclusion can be drawn about regulation
Q. The positive sign of aPl/BRD ‘ gives a positive answer to
the question posed by Klein [22 ] about whether a relation-
ship exists between pricing policy Of the clearance output
and the rate of interest offered by the bank on the stock
of demand deposits. By limiting the rate paid on demand
deposits, regulation Q has led to a reduction of the service
charges, so that the costs of the clearance output is at

least partially Offset by interest free demand deposits. In

addition to this, the reduction in service charges leads to

52

over utilization of the clearance output by its demanders
and hence to a resource allocation which is not socially
optimal. These results about regulation Q, however, cease
to hold true if the model is based on risk neutrality. This
is one explanation why the Klein and the Benston models do
not produce analogous results.

Inputs involved in a production process may be either
substitutes or complements. A rise in an input price always
lowers the demand for its complements, but it may increase
or decrease demand for its substitutes depending on the
magnitudes of substitution and output effects. Comparative

static responses of capital and labor to input prices P P

K’ L’

and RD indicate that capital and labor are substitutes
between themselves, but the relations between capital and
labor on one hand and demand deposits on the other may be
either one of substitution or complementarity.

An inferior input is defined as one an increase in
whose price leads to an increase in the equilibrium output
Of the firm [ 10, p.139]. In this model demand deposits,
when considered as an input in producing loans, satisfy the
inferiority condition [BXZ/BRD > 0]. As RD increases, the
bank switches some of its cash funds to loans to increase
its earning. The higher earning will Offset the increment
in costs due to the rise in RD. Demand deposits are a normal
(or superior) input for the clearance output. Capital and

labor are normal (or superior) inputs for both outputs.

Increased risk aversion as measured by an increase

 

 

53

in the risk aversion coefficients a is shown to reduce the
service charges P1 and loans X2. These features both lead
to a higher Xl/X2 ratio. Lower service charges lead to a
larger clearance output level and increase the output ratio
_Xl/X2, this increase is further reinforced by reduction in
loans. Some of the loan funds are shown to be transferred
to free reserves to increase liquidity as measured by the
ratios FR/X2 and FR/DD. The comparative static effects of
changes in demand deposits are indeterminate. An increase
in wealth will not be totally allocated to loans as Porter
claims [ 33]. It will be divided between loans and reserves.

The efficiency parameter y may be considered as an
indicator of technological progress. The model shows that
as y increases, the clearance output price Pl’ the planned
free reserves FR, and capital and labor decline, while the
loan output X2 increases. The decline in the clearance
output price Pl leads to a higher demand for X1. The bank
will employ less Of capital and labor inputs, will hold less
reserves on the average, and will produce more of both
clearance and loan outputs. The shadow price of efficiency
A (or the shadow price of technology) declines as efficiency.
increases. The equilibrium return or the Opportunity cost
of funds to the bank increases with the technology. [ROWZH].
The rise in the clearance output Xl for given level of
deposits indicates an increase in the velocity Of deposits
as defined by the ratio Xl/DD.

The variable t represents the technological

54
improvement in X1 while the technology in production of X2

is unchanged. From the comparative static results, it can
only be said that such improvement reduces the service

charge Pl' This result indicates that technological progress
in the clearance mechanism - an electronic fund transfer
system, VISA etc., - is beneficial to the consumer and
increases the demand for clearance output.

It might be of interest to contrast these comparative
static results to their counterparts when the model is
adjusted to include only one of the two outputs. If the
clearance output is excluded from the model, the results
remain essentially unchanged - except that the rate paid on
demand deposits (RD) no longer appears bathe first order
conditions. The system is consequently independent of this
rate and the establishment of regulation Q is found to have
no theoretical basis. It may be argued that the results
obtained on regulation Q by Klein [22 ], which contradict
the results in the present study, may have followed his
failure to include the clearance output.

A model without the loan output is equivalent to
imposition of a 100 percent reserve requirement on demand
deposits. Demand deposits receive no interest in this case,
since they can not be used to make loans. In this model,
few exogenous variables appear. The comparative static
results indicate that the bank capital input rises with the
expected demand deposit flow. The bank buys additional

machinery to produce a larger clearance output. The

55

remaining results are not surprising.

In this chapter an attempt was made to integrate such
different aspects of bank behavior as physical production,
portfolio selection, costs, liquidity, risk aversion, and
institutional constraints into a unified banking model, but
much more can be added. Randomness in outputs, simultaneous
randomness in outputs and prices, and liability management

will be incorporated in the following chapters.

CHAPTER THREE

THE BANKING FIRM UNDER PRICE AND OUTPUT UNCERTAINTY

3-1 Introduction

 

In the last chapter the bank was considered as a two
product firm under deposit variability and price uncertainty,
where the quantity of the loan output was deterministically
set and achieved. In the real world, however, banks face
quantity uncertainty as well as price uncertainty. If the
bank's anticipations are not realized, the planned outputs
will not necessarily prevail. When deposits unexpectedly
run off, or the demand for loanable funds drops, the planned
loans will be curtailed. On the other hand, when the bank
confronts unexpected deposit inflows or loan demands which
it is unwilling to turn down, it may end up lending beyond
the planned level. Since decisions on loans have implications
on the strength of customer relationship and long-run profits,
loan uncertainty is critical to banks.

In this chapter output uncertainty is incorporated
into the model. It is assumed that the bank plans the loan
output, but the planned level is not necessarily achieved.

The actual output is assumed to be stochastic and to deviate
from the planned level by a random term with zero mean and a
constant variance. The mean of the actual output Will
therefore be the planned output, and its variance the variance
of the random term.

The incorporation of default and asset reserve credit

56

57
instrument are abstracted in this chapter. The default
rated2 is subtracted from the contract rate c2, and the
net loan rate P2 will be used throughout. Loan rate

uncertainty and perfect competition in the loan market are

retained.

3-2 The Structure of the Model and the Optimality1Conditions

3-2—1The Objective Function

The bank chooses the service charges Pl’ the planned
loans (R2), the planned level of free reserves (FR), and
capital and labor inputs so as to maximize its objective
function. The objective function may be expressed as
G = E(fl) - dv(w), where the profit (w), and its mean and

variance can be measured as follows:

 

 

 

 

 

 

n = Plxl + pzx2 — PLL - PKK - RBBR — RDDD
Where: X1 = alDD + blPl + El
1 _
f 1 __ f 1
DD DD V(DD) COV(E1,DD) COV(DD,X2)
El 0 COV(E1,DD) V(El) COV(El,x2)
— )
sz x2 LCOV(DD,X2 COV(El,x2) V(Xz)
2
n - (alPl-RD)DD + blpl +-PlEl+-pzx2-PLL - PKK - RBBR
. __ 2 __
E(n) — (alPl - RD)DD+blPl +-E(P2X2) - PLL - PKK - RBBR

_ _ 2 2 2
V(n) — (alPl RD) V(DD) + P1 V(El) + V(P2X2) + R BV(BR)

58

+2 Pl(alPl - RD) COV(DD,E1) + 2(alPl — RD) COV(DD,P2X2)

-2(alPl - RD) RB COV(DD,BR) + 2 Pl COV(E1,P2X2)

The objective function includes moments and cross—
moments of sz2 which is the product of two random variables.
To measure these moments some simplifications are necessary.
Here each random price is written as P = 5 + U and each
random quantity as Z = E + V where 5 and 5 represent means
(expected price or planned output) and U and V are random
terms. It is assumed that random terms in prices are

independent from those in quantities and each term has a

zero mean and a constant variance.

-

U 0 V (U) 0
I where: V(U)
V O 0 ‘V(U)

V(P)

V(V)

V(Z)

 

 

In consequence to this treatment of the random variables,

the following results are shown to hold in Appendix 5.

E (PZXZ) = szz
_2 _2
v (P2X2) = V(P2)V(X2) + PV(X2) + x V(PZ)
COV(BR,P2X2) = P2(%N/(BR,X2)
COV(DD,P2X2) = P2C0v(DD,X2)

Besides, it is assumed that the random term in the demand

function for clearance output (E1) is uncorrelated with

59

borrowing (BR) and total loan income P2X2 so that the

following covariances vanish.

I
O

COV(E1,BR)

l
0

3—2-2 The Relation Between Loans and the Loan Rate

It should be explained that the assumption of
independence between the error terms in the quantity of loans
and in the loan rate does not imply the independence of
loans from the loan rate. The assumption indicates that
deviations of loans from the planned level do not result
from deviations of the loan rate from its expected value.

For example, a situation is possible in which the loan rate
is higher than expected and still the quantity of loans falls
short of its planned value. The assumption can be justified
on the ground that deviations in the loan output often
result from unexpected deposit shocks and loan demand rather
than price variations.

The relationship between loans and the loan rate may
be explained better in a hypothetical one output model. As
the output decisions occur prior to production period, the
planned output level depends on the expected rather than
actual price. For a given distribution of the loan rate the
hypothetical marginal cost curve intersects the expected

price P2 at point A which determines the planned output i2.

The actual output will have a distribution whose mean is the

60

MC

 

 

 

4r

.3“ X 3»

FIGURE 4
DEPENDENCE OF II‘HE PLANNED OUTPUT ON THE

EXPECTED PRICE

61
planned output i:. It may take an infinite number of values,
however, when the production occurs only one of these values
will materialize and will become the actual output level.
The fact that when 52 shifts up, a larger output will be
planned explains the dependence of the planned output on the
expected price, in other words it shows how the location of
the distribution of the loan rate determines the location
of the loan distribution. However, since loans and the loan
rate can, regardless of each other, take on any of the values
described by respective distributions, their deviations

from respective means may be considered independent [Figure4].
3-2-3 The Constraints

As in the preceding chapter, the production function,
the balance sheet, the reserve requirement equation and the
demand function for clearance output must be imposed on the
model. For simplicity the demand function for clearance is
substituted in the production function, and the reserve
requirement constraint is substituted in the balance sheet.

The substituted version of the constraints may be expressed

as:
F = 0L1 109 (alDD + blPl) + a2 log (x2) + elm-1100+ blPl)
+ 82X2 - OLD log (DD) - OLL log (L) - ak log (_K') “Y = 0
X" +T-(1-rd)fi-w = o

2 O

62

3-2-4 The Lagrangian Function

After substitution for moments of products the

Lagrangian function may be written as G*:

__ 2 _._
* = - - -
max G (alPl RD)DD + blPl + P2X2 PLL PKK
- fiE - a{(a p - R )2 V(DD) + p 2V(E )4-V(p )
RB 1 1 D 1 1 2
2

-2
V(Xz) + P2 V(Xz) + £2 V(PZ) + R32 V(BR) + 2Pl

(alPl - RD) COV(DD'E1) + 2(alPl - RD) P2 COV(DD,

x2) - 2(alPl — RD) RB COV(DD,BR) - ZRB P2
COV(BR,X2)} - AF - u[X2 + FR - (l-rd) DD - W0]
3—2—5The First Order Conditions (F.O.C.)

The F.O.C. require that the partial derivatives of
the Lagrangian function with respect to the endogenous

variables be set to zero.

3g*/3Pl = a 65 + 2b P

1 1 1 - a{2al(alPl - RD) V(DD) +

2Pl V(El) + (4alPl - 2RD) COV(DD'E1) +

2a 5 c (DD,X

1 2 ov - ZalRB COV(DD,BR)} - AF

)
2 P1

P2 - ZOLX2 V(P

3G*/3X2 ) - XFZ - u = O

2

- RB BEE/3513 - aRBZ aV(BR)/afi = 0

aG*/3F§

aG*/3L = ‘PL - AFL = o

aG*/aK = “PK - AFK = o

aG*/ax = - F = o

8G*/8u = - [322 + Ffi - (1-rd)BE — W0] = o

§§ and V (BR) are the incomplete moments of the
distribution of FR. The derivation of their partial deri-
vatives with respect to FF follows the same lines as in the
preceding chapter. Although double integrals should be
used due to randomness in loans, the partial derivatives
are the same as in the last chapter.

Notice the first order conditions are essentially
the same as in the last chapter, where now i2 appears
instead of X . The same interpretation of the optimality

2
conditions is still applicable.

3-2-6 The Second Order Conditions (S.O.C.)

The Hessian matrix and the S.O.C. are similar to

their counterparts in the last chapter, where now X2 is

substituted for X2 and S2 is set to zero. The S.O.C. lead

to the following restrictions:

F > O

F LL ' KK

F F

11’ 22’

82V(BR)
affi

2 o

64

323 Displacement of Equilibria

The first order conditions are totally differentiated
to derive the comparative static results. The differentiated
form of the first order conditions appears in Table 4. The
cofactors Hij of the Hessian matrix have similar signs to
their counterparts in the last chapter. These signs were
displayed in Table 2. .The comparative static results are
presented in Table 5. Notice most results are the same as
in the last chapter. The covariation between loans and
deposits has led to indeterminacy of some previously
determinate results (Row 1). Due to randomness in loans two
new variables, V(XZ) and COV(DD,X2), are now added to the
set of the model's exoqenous variables, but they do not have
a determinate effect. Only under the special condition,
that variability in free reserves does not affect the
probability of borrowing, can we say that with uncertainty
in loans the bank prudently plans smaller loans and holds

more free reserves.

65

TABLE 4

THE DIFFERENTIATED FORM OF THE FIRST ORDER CONDITIONS

 

_AFKK

 

 

-dA

 

66

TABLE 4 (cont'd)

- (a - aD/BE) dBE + dy

1F1

(l-rd) d65'- BB drd + dwO

2aal COV(DD,X2) dP2 + Zdal(alPl - RD) dV(DD) + 2aPldV(El)
+ (4aalPl - 2aRD) dCOV (DD,El) + 2aalP2 dCOV (DD,X2)
- ZdalRB dCOV(DD,BR) - 20!.[alV(DD) + COV(DD'E1)] dRD

- D _ _ _—
2aal COV( D,BR) dRB (al AalblFll) dDD

 

+ [2al(alPl-RD)V(DD) + 2PlV(El) + (4alPl-2RD) COV
(DD,Bl) + 2alP2 COV(DD,X2) - 2alRB COV(DD,BR)] dd
sz + 2aX2 dV(PZ) + 2X2 V(P2) dd
-Q(l-rd)2 dV(DD) ~ QdV(X2) - 2Q[COV(DD,X2)-(l-rd)V(DD)]drd
__ 2__
+ 2Q(l-rd)dCOV(DD,X2) - dRB[Pr+ 4aRBBR(l-Pr)] - ZRBBR
(l-P )da
r
dPL
dPK
l
A = 2b — ZaV(X ) - AbZF
ll 1 l 1 11
A22 = -2aV(P2) - AFZZ
_ 2 -— __
A33 - RB BPr/Bfi - 2(1RB [(l—Pr)Pr + ZBR SPr/BFR]

 

67
TABLE 4 (cont'd)

A.. < O i = 1,2,3
11

Q _ RB aPr/3V(FR) + ZaRB(l-Pr) aBR/aV(FR)—2aRBBR apr/3V(FR)

68
TABLE 5

THE COMPARATIVE STATIC RESULTS

 

Endogenous Variables

 

 

Exogenous Conditions _ __
Variables Imposed Pl X2 FR L K n
— 9 9 9 9 9
P2 . + + .
.. .. 'D 9 9

RB +

RD + + - - - +

PL + - + - + -

PK + - + -+ - -
V(PZ) - - + - — -
V(DD) BER/3V(FR) = - _ .+ + + _

3Pr/3V(FR) = 0,
RD < ZalPl
V(XZ) 8BR/aV(FR) > 0 ' ' + ' ‘ +
8Pr/ava) = 0
9 9 ? + + -
Cov(DD'X2) aPr/3V(FR) = o
c DD,E - - + -+ + -
ov( l) '

+ + - - - +
COV(DD,BR)

69

TABLE 5 (cont'd)

 

aBR/3V(FR =

3P
r/3V(FR)

COV(DD,X2) > 0
COV(DD,BR) < 0
R

D < ZalPl

'0

'0

'0

 

CHAPTER FOUR
THE MULTIPRODUCT BANK AND LIABILITY MANAGEMENT

4—1 Introduction

 

In this chapter we introduce a third output and
liability management into the model. The bank now has a
new source of funds which is under its control. It can
choose the level of time deposit liabilities. This can be
done through the sale of large negotiable certificates of
deposits, CD's. It is also assumed that the bank has access
to the federal funds market. It sells its excess reserves
to and borrows its reserve deficiencies from other banks
through this market. The funds secured through sale of
CD's are used alongside funds acquired through clearance
X and X . X is

l' 2 3 l

the clearance output. X2 and X3 may be considered as two

categories of loans, or as loans and investments respect-

activity to produce three outputs X

ively.
4-LfiLSources of Uncertainty

The bank faces uncertainty in both quantities and
prices. In regard to quantity uncertainty, the quantity of
demand deposits, DD, time deposits, CD, federal funds, FF,

1’ X2 and X3 are assumed to be random.

In regard to price uncertainty, the CD rate, R

and the three outputs X
CD’ the

federal funds rate RF, the loan rate, P2, and the security

70

71

rate, P are assumed to be stochastic.

3:
Since X1, X2, X3, CD and FF are random variables,
their values cannot be optimized. It is therefore assumed
that the bank optimizes their mean values 2', Y2, Y3, CD,
and FF. In the decision period the bank can be considered
to be facing a series of distributions for each variable
with different means. By making arrangements, the bank
chooses the distribution with the optimal mean. When the
mean is chosen, it will constitute the planned or the
desired quantity of the corresponding variable (loans for
example). This quantity will not necessarily prevail
however. When the production period arrives, the variable
will take on one of the possible values on the corresponding
distribution. This actual value will deviate from the
planned or desired level by a random term.

In the same way, demand deposits, the loan rate,
the security rate, the CD rate and the federal funds rate
are random variables whose distributions are exogeneous to
the bank. It is assumed that the bank can forecast and
somehow measure the first two moments of these distributions.
The forecasted mean will be considered as the expected
deposit or expected price by the bank. The actual deposits
or actual price will deviate from their expected levels by
a random factor. Mathematically, this argument may be
introduced by writing each variable Z as the sum of the
corresponding mean 5 and an error term Ez. Z = E + Ez.

This formulation will later be used to measure and simplify

72

the objective function.

4-1-2 Market Structure

The market structure for the clearance output is
assumed imperfect, while the market for demand deposits,
CD's, federal funds, loans, securities and capital and labor
inputs are competitive. The demand for clearance output is
consequently less than perfectly elastic; the bank sets the
clearance output price or equivalently the mean output
level; while CD rate, loan rate, security rate and federal

funds rate are exogenous to the individual bank.

4-2 The Structure of the Model and the Optimality Conditions

 

4-2-1 The Objective Function

The bank's objective is to maximize the certainty-
equivalent profit G where G = E(w) - V(fl). Here the profit
equation n is used to derive its two first moments in order

to measure the objective function.

= + -— _ _ ..
Tr Ple + P2X2 + P3X3 RFFF RDDD RCCD PLL PKK

where FF is the net federal funds sold.

E = P i + E P x -+E P x + E FF - R 55
(w) 1 1 ( 2 2) ( 3 3) (RF ) D
- E(RCDCD) - PLL - PKK
_ 2 2
V(w) - Pl V(Xl) + V(PZXZ) + V(P3X3) + V(RFFF) + RD V(DD)

73
+ V(RCDCD) + 2P1[Cov(xl,P2x2) + COV(xl,P3X3)

+ COV(xl'RFFF) - R C (Xl,DD) - COV(X

D OV CD)]

l'RCD
+ 2Cov(P2X2,P3X3) + 2C0V(P2X2,RFFF) - 2RDCOV

(P2X DD) - 2Cov(P X R CD) + 2C0V(P3X3,RFFF)

2' 2 2' CD

- 2RDCOV(P3X3,DD) - 2C0V(P3X3,RCDCD) - 2RDCOV

(RFFF,DD) - 2C0V(RFFF,R CD) + 2RDCOV(DD,R CD)

CD CD

As explained before, some simplifications are
required to derive the desired product moments. Random
prices and quantities are written as the sum of the
respective means and random terms which represent the
deviations of the actual values from the corresponding
expected or planned levels, P = R + U, Z = §'+ V. Then, it
is assumed that the random terms in prices (U) are indepen-
dent from those in quantities (V). Following these
assumptions it is shown in Appendix 5 that the desired

product moments may be written as follows:

E(PiXi) = ixi i = 2,3
_ — 2 — 2 . _
V(PiXi) — xi V(Pi) + Pi V(Xi) + V(Pi)V(Xi) 1 — 2,3
_ — 2 -——2
V(RCDCD) - RCD V(CD) + CD V(RCD) + V(CD)V(RCD)
V(RFFF) = ‘RF2V(FF) + FF2V(RF) + V(RF)V(FF)

COV(PiXi,DD) = PiCOV(Xi,DD) 1 = 2,3

c V(Pixi'xl) = Pi Hov(x ,xl ) i = 2,3
c V(RFFF,X1) = RF WOV(X ,FF)
c V(RFFF,DD) = RF (DD,FF)
COV(RCDCD, DD) = RCD COV(DD, CD)
“(szz 3X3) = Cov(P2'P3)Cov(X2'X3) + Y2§3CDV(P2'P3)
+ PiP3c0V(x2,x3)
c V(Pixi,RFFF) = COV(Pi,RF)COV(Xi,FF) + Riff COV(Pi,RF)
+ RF P1C0V(X ,FF) 1 = 2,3
0 V(PiXi,RCDCD) = Vi(P ,RCD) HOV(X ,FF) + x. lCD COV(Pi’
RCD) + Pi RCD HOV(X ,CD) 1 = 2,3
C V(RFFF,RCDCD) = COV(RF,RCD) COV(FF,CD) + RCDRF COV

(FF,CD) + FF CD COV(RF.RCD)

These relations will be used to set up the Lagrangian

function.
4—1-3The Constraints

The following Constraints will be imposed on the
model

a — Production Function

F = on 109021) + CL

1 lOg(X2) + a310g(X3) + Ble + 82x2

2

7S

+~B X - aDDlog‘DD) - aCDlog(CD) - aLlog(L) - aLlog(K)

b - Balance sheet identity

+ + =
x2 x3 FF DD + CD + W0

c - Demand function for clearance output

X1 = alDD + blpl + El

d - Reserve requirement
RR = rd * DD + rc * CD

where rd and rc are reserve requirement ratios for demand
deposits and CD's respectively.

Due to the accessibility of the federal funds market,
the bank is assumed not to hold any interest—free excess
reserves. Since the optimization occurs prior to the
production period, the Constraints will be imposed in their
expected value form. To simplify, we substitute the last
constraint into the balance sheet constraint. We also use

the mean demand function for the clearance output to derive

 

P1 in terms of X1 and substitute that in the objective
function.
Xi = alDD + blpl
or
P = Xl - aIDD
l b

76
Consequently there are only two constraints which

must be imposed on the objective function.

4-1-4The Langrangian Function

If the mean demand function for clearance output is

used to substitute for P and the simplifications given in

1!
Appendix 5 are used to substitute for the means, variances,
and covariances in the objective function, the Lagrangian

function can then be written as follows:

 

 

 

i1 - alDD _ ._ _
*= ..
G ( b1 ) x1 + 92x2 + 93x3 + RFFF RDDD
__ ail—aim?
- RCDCD - PLL - PKK - a{( bl‘ ) V(Xl) + V(PZ)
V(X)+332V(P)+§2V(X)+V(P)V(X)+SE2
2 2 2 2 2 3 3 3
V(P ) + 5'2 V(X ) + V(R ) V(FF) + EFZ V( ) + ‘ 2
3 3 3 F RF RF
V(FF) + 2 V(DD) + V(R ) V(CD) + i 2 V(CD)
RD CD CD
__2 i1 - alfifi _ _
+ CD V(RCD) + 2 ( bl )[P2 COV(X1,X2) + P3
c V(X1,X3) + RF COV(X1,FF) - RD COV(DD,X1) - RCD

C V(Xl,CD)] + 2C0V(P2,P3) COV(X2,X3) + 2P2P3

C V(X2 X) + 2X (P2 ,}?3) + 2C0V(P2'RF)COV

2— X3 COV

(X2,FF) + ZRFP C (X2,FF) + ZFF x COV(P2,RF)

2 0V 2
- 2RDP2 COV(X2,DD) - 2C0V(P2,RCD) COV(X2,CD)
- — - —'__ +
2P2R CD CZ'OV(X ,CD) 2X2CD COV (P2, RCD)

77

C V(P3,RF) COV(X3,FF) + 2 P3 RF COV(X3,FF) + 2X

3
FE COV (P3,RF) - 2RD§3 COV(DD,X3) - 2RD§f COV(DD,FF)
’ Cov(P3' 'RCD) Cov(x3'DD) ' 21D-3R CD Cov(x3’CD)
- 223 ED COV (P3,RCD) — 2C0v (RF,RCD) COV(FF,CD)

- ZRF RCD COV (FF,CD) - ZFF CD COV(RF'RCD) + 2RD

(DD, CD)}- AF - p[X2 + x + FF - (1-rd)BB'

RCD COV 3

- (l-rc) CD — W0]
4-1-5First Order Conditions For Optimality (F.O.C.)

The F.O.C. require that partial derivatives of the
lagrangian function with respect to endogenous variables
be set to zero, that is:

2? - a 55 I? - a 55

 

 

aG*/8§l = l b l - 2a {( l b l ) V(Xl) + g;
1 1 1
[E2 C V(xl’XZ) + 53 Cov(xi'x3) + EF Cov(X1'FF)
- RD Wov<x ,DD) — ECD COV(X1,CD)]} - lFl = o
ac*/a§2 = 52 - 2a {22 V(Pz) + 23 cOV (P2,P3) + FE
cOV (P2,RF) - EB cOV (P2,RCD)} - AFZ ‘11 = o
ac*/a§3 = 53 - 2a {23 V(P3) + Y2 COV(P2,P3) +f§§

c V(P3,RF) - CD COV(P3,RCD)} - AF3 - u = (3

78

aG*/3FF = EF - 2a {FF V(RF) + £2 c0V (P2,RF)
4' X3 Cov (P3'RF) ’ EB (RF’RCD)}
- u = 0

aG*/aEB = - ECU - 2a {55 V(RCD) - i2 C0V (P2,RCD) - ff
C0v (BF'RCD) " i3 C0v (PB'RCD)} ' AFCD
+ u(l-rC) = 0

3G*/3L = -PL - AFL = o

aG*/aK = -PK __AFK = o

SG*/3A = -F = o

‘aG*/ap = -[Yé + 23 + if - (1-rd)Bfi - (1-rC)ED - wO] =
where Fz = BF/Bz

The interpretation of the F.O.C. follows the same

lines as in the previous chapters.

These conditions define

the implicit demand function for CD's, Capital, and labor

inputs.

loanable funds, investment funds,

They also define the implicit supply functions for

and federal funds. The

model explains the observed diversification behavior and

the dependence of the loan output

demand deposits.

on the rate paid on

0

79

4-1-6Second Order Conditions (S.O.C.)

The S.O.C. require the Hessian matrix H to be negative
definite. This, in turn, implies that the principal minors
alternate in sign with the first being positive. In the
present form of the model, it can be shown that the S.O.C.
become very complicated and will not necessarily be satis-
fied. The comparative static results will also be indeter-
minate in large part. However, if we assume that the
relationships between different interest rates involved are
highly nonlinear, so that the degree of their linear
relationship as measured by their correlation coefficient is
zero, then the covariances in the Hessian matrix disappear,
the S.O.C. can be satisfied by certain requirements and
some comparative static results will become determinate at
least under certain conditions. Having made these assumptions,

the S.O.C. require:

 

32F
F.. > 0 i = 1,2,3 where F.. =

1]. ll 32"2
l

2
. 3 F
F-o > 0 = L’K F.. = _
33 3 33 3j2

4-3 Displacement of Equilibria

 

To get the comparative static results, the first
order conditions are totally differentiated. The differen-
tiated form provides a simultaneous equation system. The

matrix of the coefficients in the system is the Hessian

80

matrix. If a consistent solution exists, the system may be
solved for the endogenous variables by using Cramer's rule.

The total differentiation and the cofactors of the
Hessian matrix H will not be given here. The differentiated
form of the first order conditions is given in Table 6. The
signs of the cofactors Hij are given in Table 7. The
comparative static results are given in Table 8. The
cofactors and the comparative static results have the given
signs under the conditions specified in respective tables.

The cofactors which are marked by questions marks (?)
have indeterminate signs. The conditions given beneath
Table 7 can be translated into meaningful concepts. The

conditions l and a can be written as F2/F > 1, where,

_ 3 <
Fz/FB = - dx3 represents the marginal rate of product
d'iz

transformation between outputs X2 and X . Conditions 3 and

4 can be written as 392 / F > 1 £92 3/ F re resents the
_ rC-l 2< ° (rC-l) 2 p

marginal product of the unencumbered CD input in producing

output X Analogously, conditions i and g have implica-

2.
tions on the level of marginal product of unencumbered CD‘s

in producing output X Some of the comparative static

3.
results are especially noteworthy. In the profit
maximizing competitive multiproduct firm, the conditional
supply and demand functions for outputs and inputs are upward
and downward sloping respectively [ 15,p. 98]. Such a

determinacy is not present in our model. The conditional

supply functions of loanable and investment funds are upward

 

81

TABLE 6

THE DIFFERENTIATED FORM OF F.O.C.

F1

F2

F3

0

 

 

-dl

-du

dFF‘

d‘éi

dL

dK

 

 

L

82

TABLE 6 (cont'd)

f _
-F dDD

D

(l-rd) difi - BE drd - CD drc + dWO

 

 

a __ Zaa V(X ) Y1-a 56
J:- dDD - l 1 dD—D + 2a{ 1 dV(X )+—— 1 (x1)
b 2 1 b C0v x2
1 b b l
l l
dP +i— dC (x X)+—l—C (x1,X)d'F'+—l—'§dc
2 bl 2 OV l’ 2 bl C:OV 3 3 bl OV
1 1 — RD
(X1, ,X 3)+-bl RFdCO V(X1,FF)+blCO V(X1,FF)dRF-EI dCOV
C (DD x ) R C
0v ' 1 CD OV
(DD,X1)- b T?’ dCOV(Xl,CD)-T;— (X1,CD)
_1 _ 1 1
Xl-alDD l _ _
dRCD}4-2da{( )V(Xl), + B‘ [P2C0V(X1,X2) + P3
b1 1
C V(Xl ,x 3)+-RFCOV(X1,FF)-R DC (DD, xl )- RCD COV(xl ,CDH
-dF2 + 2ax2 dV(PZ) + 2x2 V(Pz) dd
-dP3 + Zax3 dV(P3) + 2X3 V(P3) dd
-dRF + 2aFF dV(RF) + 2FF V(RF) dd
+dRCD + ZaCD dV(RCD) + 2CD V(RCD) da + pdrc
dPL
dPK
2 2aV(Xl)
where All = B— - 2 - AFll A44 - —2aV(RF)
1 b1 :
_ _ _ A = -2dV(R )-AF-—
A22 — 2aV(P2) AF22 55 CD CD
A33 = -2aVLP3) - AF33

83

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86
sloping only if these outputs each has a positive correlation
with clearance output. (Rows lanni3). the conditional supply
of federal funds is upward sloping and the conditional
demand for CD's is downward $10ping only if each one is
uncorrelated with clearance output. (Rows 5 and 7). Only the
conditional demand for capital and labor are negatively
sloped in general (Rows 9 and 10).

. In the profit maximizing model with a general form
production function, the comparative static effects of a
change in price of a given output or input on other outputs
and inputs are indeterminate [16 ,p. 98]. With our model,
however, it was shown in the two product case (Chapter Two)
that vfluni the loan rate rises, the bank will charge a
higher price on the clearance output to reduce its quantity
demanded. In other words it switches from X1 to X2. This
result does not carry over to the present three output case.
Here?l rises with i2 (or i3) as the latter's price is
expected to increase (Rows l and l)° It is not even clear

whether the bank will switch between X and X3 when one of

2
their prices change (Rows l and 3). It is possible for X2

(X3) to rise with 5 (52) at a given output mix and to fall

3
at another. This allows the possibility of both complemen-
tarity and substitution between the two outputs.

The relation between demand-deposits and CDfs is
shown to be one of substitutability. The substitution effect,

which results from the increase in the rate paid on demand

deposits, outweighs the output effect so that CD'S rise

87
with an increase in the rate paid on demand deposits. (Row
8). The same argument holds about the relationship between
capital and labor inputs (Rows g and l9)- The nature of
the relationship between capital and labor in one hand and
CD's on the other is indeterminate (Rows g and $9).

The comparative static effects of input prices on
outputs are in general indeterminate [ l6, pu 98]. In our
model a rise in expected input prices EED’ K
and X3 (Rows 6 through 10).

RD! PL and P
reduces the planned outputs of X2
These inputs are therefore normal (or superior) in production
of these two outputs.

Demand deposits are normal (or superior) inputs for

X The larger the rate paid on demand deposits, the lower

1'
will be the planned clearance output (Row 8). This feature
has implications on regulation Q. Since the clearance

output Y and its price (service charges) move in the

l
Opposite directions, the above result indicates that service
charges and the rate paid on demand deposits have a positive
relationship, i.e., BPl/BRD > 0. As explained in Chapter
Two, this means that regulation Q, which lowers the rate
paid on demand deposits to zero, artificially reduces the
service charges, leads to over utilization of the clearance
output by consumers, and an allocation of resources which

is not socially optimal. Capital and labor are shown to be
inferior inputs of X1 (Rows 9 through 10).

The federal funds market in the model can be

considered as a discount window at which the bank can borrow

88
and lend at the same rate. The federal funds rate plays the
role of the discount rate, or equivalently, the interest rate
paid on cash reserves. Such a mechanism has actually been
suggested by Tobin [42:]. Tobin suggests that, this
mechanism can be a strong monetary policy instrument for the
Fed. An increase in the rate paid on reserves would be a
restrictive monetary policy and a decrease would be expan-
sionary. Our model confirms Tobin's View. With an increase
in the federal funds rate the bank lowers its scale of
operation. It plans to buy less CD'S, to lend less, and to
invest less funds. It would rechannel the funds to interest
bearing reserves (federal funds) (Row 5).

The effect of uncertainty can be evaluated through
the comparative static results of variances and covariances.
Increased uncertainty in price of each output X2 or X3, as
measured by price variabilities, makes the bank switch to
the other output (Rows 11 and 12). The bank will plan to
buy less CD's and hire fewer units of capital and labor
(Rows 11 and 12). In brief, uncertainty reduces the bank's
scale of operation.

The effect of federal funds rate variability depends
on whether the bank is a seller or a buyer in the federal
funds market. We define a seller (buyer) bank as one whose
average federal funds holding is positive (negative), i.e.
although it may both buy and sell funds its sales exceed
(fall short of) its purchases. The model indicates that

when federal funds rate becomes more uncertain, a seller

89
plans to sell less federal funds and allocate more funds to
production of its outputs (Row 13). A buyer on the other
hand will plan to buy less federal funds and to reduce its
outputs (Row 14). In other words, uncertainty in the federal
funds rate redistributes outputs from buyer to seller banks.
[Note: the model was developed in terms of a seller bank,
so that F? is the average federal funds sold. Then for a
buyer bank afi/awm") > 0 indicates that a larger sale (and
therefore a smaller purchase) will correspond to greater
variability in the federal funds rate].

Variability in the CD rate reduces the bank's demand
for CD's. The bank will also restrict its loans and invest-
ments (Row 15). Variability in the clearance output V(Xl)
includes both variations in demand deposits and their
turn-over rate. This is evidenced by the demand function
for the clearance output X1 = alDD + blpl + El' More
uncertainty in either of the two components lowers all the
three outputs. When the availability of funds grows more
uncertain, that is variance of demand deposits rises, the
banker becomes cautious in order to avoid fund insufficien-
cies. The bank plans to buy more CD's, produce less, and
hold more of its funds in federal funds market (Row 16).

As explained in Chapter Two, u is the equilibrium
contribution of each dollar of unencumbered funds to
certainty equivalent profit. The model indicates that this
contribution declines as uncertainty grows. (Rows 11 through

14, and 16).

 

90

Increased covariation between the clearance output in
one hand and loans and investments on the other induces the
bank to plan larger quantities of all outputs. As the
clearance output X1 is related to demand deposits through
the demand function X1 = alDD + blpl + E1, the covariances
between X]- and xi (i = 2,3) can be written as COV(X1,Xi)

al COV(DD,Xi) + COV(E1,Xi) i = 2,3 so that COV(xl,xi)

mainly represents COV(DD,Xi). This relation can be used to
explain the above comparative static result. As covariation
between demand deposits and loans increases, the bank becomes
more confident that deviations of loans above their planned
level can be covered by simultaneous covariations in demand
deposits, therefore it plans to buy less CD'S and make more
loans in the same time. The same argument holds about
investments.

The reserve requirement instrument of monetary policy
is shown to work in the intended direction. Larger reserve
requirements on demand deposits or CD's will lower the

planned outputs of loans and investments. This effect is

similar to that of a decline in the net worth. The effect of

changes in reserve requirements on CD transactions is not
clear. The comparative static results for expected demand
deposits and risk aversion coefficient are indeterminate.
Contrasting these results with those of the,preceding
chapters, it may be concluded that as the model extends to
include more inputs and outputs, the results become less

determinate. This is because when more alternatives are

91

available the choice of each alternative is less likely.
Notice in the extended model results lose their generality
and hold only over certain ranges of variations as specified
in Table 7. This indicates that it is quite possible for
two variables to have a certain relationship over one range
of variations and a different relationship over another.

In the next chapter, the model will be adapted to

estimation and the sources of data will be described.

CHAPTER FIVE

THE ESTIMATION PROCEDURES

591 System of Reduced Form Equations

 

To empirically test the model, either the first order
conditions or the system of equations displayed in Table 6
may be used. One problem with estimating the model is that
the data on all the variables are not available. The best
that can be done is to proxy some of these variables by
other variables or, if they are endogenous, eliminate the
corresponding equations.

If the first order conditions are to be estimated,

a linearization of the system is necessary to allow the use
of the ordinary linear techniques. Here, the first order
conditions are approximated by a Taylor series up to the
first component. The resulting linear system may now be
used to obtain a reduced form in terms of the level of

the variables. The reduced form will in general represent
each of the endogenous variables as a function of all the
exogenous variables plus an error term. The error term in
each equation is a linear function of all the remainder
terms in the Taylor series approximation. Since the
Lagrangian multipliers cannot be measured and the data on
capital and labor inputs are not available, the equations
describing these variables are eliminated from the system.

The system of equations displayed in Table 6 may
also be used to derive a set of reduced form equations in

92

93
terms of first differences. The system may be written as
Adz = b, where dz is the vector of first differences of
endogenous variables, A, when evaluated around some initial
condition, is the matrix of the coefficients, and b, when
evaluated around some initial condition, is a vector of
constants. The rows in b are linear functions of the first
differences of the exogenous variables. To get the reduced
form, the system is alternatively written as dz = A-lb = C

and partitioned to two parts:

r ‘ (T

dzl = Cl

dz2 = C2

L J L J

 

 

 

 

dz includes the endogeneous variables on which the data is

1

available and dz includes the remaining variables. dzl==Cl

2
is a system of linear equations which can be adapted to

estimation.

5-1 Planned and Expected Values

 

The dependent variables in the reduced form equations
are the planned or desired values of the endogenous varia—
bles. These values cannot be directly measured. To get
the system in terms of actual values, one way is to assume
that the planned values (§)differ from the actual values
(2) by a random term (E). z = E + E. In this case the
random term in each equation will be added to the distur-

bance term. A second way is to assume that the actual

94

values adjust to the desired values by a partial adjustment

mechanism [23 , p. 476].

z - z = y(zt - zt_l) + U o < y 5 1

or

- _ 1 1:1 _ l
2 — Y zt + zt_l Y U

In this case, after substitution in each equation for the
desired value in terms of the actual value, the lagged value
of the dependent variable will be added to the set of
regressors and the error term U will be added to the distur-
bance term. The presence of the own lagged value allows the
own adjustment coefficient (the coefficient of the lagged
value) to be estimated. The partial adjustment hypothesis
requires this coefficient to be between zero and unity.

O < l -'y < l.

The set of regressors in the system includes the
expecged values of interest rates and demand deposits. Here
it is assumed that the bank uses Box-Jenkins time series
models to forecast or measure these expected values. In the
next chapter, such models are develOped for demand deposits,
the prime loan rate, the treasury bill rate, the CD rate,
and the federal funds rate. These models are used to
generate data on the corresponding variables over the sample

period.

95

5-3 The Disturbance Term PrOperties

 

Since all the non-linearities in the model (remainder
terms in the Taylor series approximation), and also the
random terms in adjustment of actual to desired values, are
relegated to the disturbance terms, there is a high probab-
ility that the effect of the omitted variables (variables
relegated to the disturbance term) will carry over into
subsequent periods and lead to serial correlation. The use
of monthly time series data increases this probability. The
disturbance terms therefore may not be spherical.

Here, it is assumed that the disturbance term for
each equation in the reduced form system has the following
prOperties:

(a) it is normally distributed

(b) it has a zero mean

2

(c) it is homoskedastic Ui ~ NCO, oi)

t
(d) if the data show that serial correlation is
present, it is assumed that the disturbance

term follows a first order autoregressive

scheme.
= <
Uit piUit-l + Vit lo! 1
where Vit is spherical,that is:

2
Vit ~ N(0. oiV)

E(Vit'vit—l) = 0

96
Besides, after removing the serial correlation the distur-
bance terms might be correlated across the equations so that

E(Vi ) = 0.. is non-zero. If this condition prevails,

t'Vjt 1]
we have a case of "seemingly uncorrelated" regressions.

It is well known [23 , p. 520] that in this case
the generalized least squares approach, which takes into
account the correlations across equations, is more efficient
than the ordinary least squares. The gain in efficiency,

however, is zero for the present system since the regressors

of all the equations are the same [40 , p. 309].

5-4 The Data

 

A combination of cross-section and time series
observations on a group of homogenous individual banks may
be considered as an ideal data set for the model. Since
the model includes four simultaneous interest rates as
regressors, a reasonably large number of observations might
be necessary to reduce the degree of multicollinearity.
Measurement of expectations, variances, and covariances, as
conceived by the bankers, require formulation of moment
generating mechanisms which generate these moments over time.
Developing such mechanisms is an art rather than a science.

The data used in this study were drawn from a survey
of weekly reporting large commercial banks in New york City.
The sample period for the estimation starts at June, 1969.
For loans, investments, federal funds, and CD's the sample

ends at December, 1976. Since some of the items were

97

available only in monthly figures, the weekly figures during

each month were averaged to construct monthly series for the

rest of the items.
The variables in the model were measured as follow:

demand deposits (DD): total demand deposits adjusted to
exclude U.S. government demand deposits, domestic
commercial banks demand deposits, and cash items in
process of collection.

loans (X2): total gross loans, adjusted to exclude loans
and federal funds transactions with domestic
commercial banks. The adjective "gross" indicates
that reserves for loans losses have not been deducted
from total loans.

investments (X3): total security investments

CD's: large negotiable time certificate of deposits. These
are the certificates of deposits issues at demonina-
tions of $100,000 or more.

federal funds (FF): net federal funds sold = federal funds
sold - federal funds purchased

debits (X1): data on debits are not available for the banks
under consideration, but can be measured as the
product of demand deposits and their turn over rate.
The turn over rate for the banks were proxied by
monthly turn over rate of adjusted demand deposits
in New York City.

Net Worth (W0): total capital account.

98

reserve requirement ratio: the ratio of the reserves held
by the banks over total time and demand deposits.
The above items were published in:

The Federal Reserve Bulletin.

 

loan rate (P2): the prime loan rate.
security rate (P3): monthly rate on three months treasury
bill, rate on new issues.

CD rate (R monthly rate on three months CD's in the

CD) ‘
secondary market. The rate in the primary market was

not available for the whole period.

federal funds rate (RF): monthly effective rate on federal
funds.
rate on demand deposits (RD): RD was set to zero

The above items were published in:

Banking MonetarygStatistics, 1941-1970 [Tables 12.5,

 

12.6, 12.7].

Annual Statical Digest, 1971—1975 [Tables 24,26].

 

Annual Statical Digest, 1972-1976 [Tables 22,24].

 

Federal Reserve Bulletin [Tables A20-24, A29].

 

wage rate (PL): gross-earning of production or non-super-
visory workers in banking (average hourly earnings)
Published in:

Employment and Earning, U.S. Department of Labor,

 

Bureau of Labor Statistics.
price of capital (PK): the wholesale price for producers'
durable goods.

Published by:

99

Survey of Current Business, (p.S.8).

Data on some other variables do not exist and have to
be generated. Expected values, variances, and covariances
are of this category. Data on expected values are generated
by the time series models developed in the following chapter.
Each variance is measured as the sum of the squared
deviations of the variable from its mean over the preceding
four months. Each covariance is in the same way measured
as the sum of products of deviations of the two variables

from the corresponding means over the preceding four months.

CHAPTER SIX

TIME SERIES FORECASTING

6-1 Introduction

 

The purpose for this chapter is twofold. First to
develop time series models to forecast demand deposits, the
prime loan rate, the 90—day treasury bill rate, the federal
funds rate, and the CD rate. Second, to use these models
to generate data on expected values of these variables,
which can be used in the next chapter for empirical
estimation.

A time series is a set of observations generated
sequentially over time. Each time series is a realization
of a specific stochastic process. A stochastic process is
said to be stationary if its stochastic features are invar~
iant with respect to displacements in time. Most of the
time series encountered in economics are nonstationary;
however stationarity can usually be induced, by suitable
differencing of these series [ 32].

A time series model, in its most general form, can be
written as an integrated autoregressive moving average

process ARIMA (P, d, q):
¢(B)(l-B)dZ = 0(B)E
t t

where ¢(B) and 9(B) are autoregressive (AR) and moving

average (MA) Operators respectively, B is the backward shift

lOQ

lOl

operator, i.e. Bkzt = Zt-k’ d is the order of homogeneity,

and Et is a white noise residual. An AR Operator of order

P, and a MA Operator of order q, can be expressed as

follows:
_ _ _ 2 _ m
w(B) — l wlB wZB . . . . me
(P = ¢I e
m = PI q

The general ARIMA process subsumes the pure AR scheme when

d = 0 and 6(3) = 1, and the pure MA scheme when d = 0 and
¢(B) = l. The mixed autoregressive moving average process
ARMA is subsumed when the condition d = 0 is satisfied. If
this condition is violated, the pure schemes will be replaced

by integrated ones.

6-2 Time Series Model Building

 

Time series modelling is an iterative process. A
model is tentatively specified and estimated. The estimated
model is then subjected to diagnostic checks. If the model
is found to fit the data adequately, it can be used for
forecasting purposes, otherwise the process is repeated,

until a satisfactory model is found.

6-2€lIdentification

In the identification or specification stage, the

purpose is to determine the orders P, d, and q in the general

 

 

102
ARIMA (P, d, q) process. The main tools in this stage are
the autocorrelation and partial autocorrelation functions.
A feature of stationary series is that their autocorrelation
functions die out rapidly, while those of non-stationary
series remain high. It is also known that autocorrelation
and partial autocorrelation of each class of time series '
have unique patterns. For an AR(P) model, the autocorrela-
tion function tails Off, while the partial autocorrelation
function cuts at lag P. On the other hand, for a MA(q) model,
the autocorrelation function cuts at lag q, while the partial
autocorrelation function tails off. For mixed processes,
both functions tail Off and dampen geometrically [ 6 , p. 79].
These features can be used to determine the parameters P, d,
and q.

The time series models here, are constructed using
monthly data on the prime loan rate, the treasury bill rate
(rate on new issues of three months bills), the federal funds
rate (effective rate), the CD rate, and demand deposits, for
large weekly reporting banks in New York City. The sources
of data were described in the last chapter. Non-seasonally-
adjusted data are used for all the models to avoid the
effect Of such adjustments on the structural form of the
models.

The data on the prime loan rate are available since
1949, at which time the sample period for this variable
starts. For the treasury-bill rate, however, the starting

point is chosen to be 1955 to avoid the possible change in

103
behavioral structure of the rate due to the Fed-Treasury
accord. The sample chosen for the federal funds rate starts
from 1954. It should be remembered that federal fund
activity increased substantially in 19603 and the behavioral
structure of this rate might not have remained unchanged
during the sample period. The data on the CD rate are
available beginning in 1964 and the sample starts at that
time. The data for demand deposits start at April 1961.
The classification of banks as "large weekly reporting banks
of New York City" has been revised different times. The
last revisions occurred in April 1961 and May 1969.
Consequently, the sample period was chosen to begin with the
first revision and end with the second.

6-2-1-1.Seasonality

The autocorrelation function for the demand deposit
series exhibits a substantial amount Of seasonality. This
does not seem to be the case for other series. This
seasonality effect on demand deposits is eliminated by annual
differencing.

6-2-1-21The Order of Homogeneity

The fact that the autocorrelation functions for the
series do not quickly die suggests that series are non-
stationary. The autocorrelation function for first differ-
enced series, however, do typically drOp off fairly rapidly,
indicating that first differencing the data does indeed

generate a stationary time series.

104

6-2gbinutoregressive and Moving Average Orders

One common feature among the series under considera-
tion is that the autocorrelation Of these series lack
qualities that would lead to a definite clear-cut specifi-
cation. These autocorrelations have in fact few clear-cut
patterns. This makes the specification difficult and
imprecise. To determine the orders P and q for a series, it
is necessary to know whether the autocorrelation function
cuts off at a certain lag. This information can be Obtained
by testing for the significance of the estimated autocorre-
1ation. On the hypothesis that theoretical autocorrelation
is zero, the estimated autocorrelation coefficient, when
divided by its standard error, is approximately distributed
as a standard normal variate [ 6 , p. 178]. Therefore, if
this ratio has a value of two or larger, the theoretical
autocorrelation coefficient has a probability of 95% to be
non-zero. This approach provides a guide in finding the
significant autocorrelation and consequently the AR and MA
orders of the process. The orders adopted for our series,
by using this approach, are given in Table 9.

As shown in the table, the models corresponding to

the loan rate, P2, the federal funds rate, R and the CD

Fl

rate, R are represented by integrated AR processes. After

CD’
differencing, each model’has only one AR factor. The loan
rate model includes two terms of order 1 and 3. This

indicates that, the change in this rate can be expressed, by

 

105

a weighted average of its past values during last month and
three months ago. The AR processes expressing the changes
in the CD rate and the federal funds rate have single terms
of order 1 and 3, respectively. The treasury bill model
follows an ARIMA process. This process makes the first
difference in the bill rate a function Of its past value
during three months ago and the shock which occurred six
months ago. The demand deposit series, after annual differ-
encing for seasonal adjustment, follows a MA scheme which
makes.it a function of shocks occurring last month and last
year. Parsimonious as they may be, each model passes the Q
test for white noise. This test is discussed in detail

below.
6-2-2 Estimation

Time series models can be estimated by a conditional
maximum likelihood approach. If the disturbance terms are
independent normal, have a zero mean, and a constant
variance, the estimates will be the same as conditional least
squares estimates. The identification process provides a
set of preliminary estimates for the parameters of the models.
These estimates are used as initial values in the iteration
process. For our models, in all cases the estimates seemed
to converge rapidly following one or two iterations. With
the final estimates, the models can be written as in Table

9.

6
0
l

macawflmmu mo 00000 Gnoccoum

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M M aux
x mma mo GOHHOHOHHOO m.0ms©0mmu n 0 m 0 w c n O
x

u

mc000m>ummno mo Hogans H c

 

00.000 00 0.00 U.0..0000_00...000.u00 n on 00-00 0000-00 000.00 020 00 00
0000. 00 0.00 00 n uoom 00-000000.-00 00.00 000. 00 com
0000. 00 0.00 00 n 00000-0000000.a00 A0.00 000 000 00
0000. 00 0.00 b000000.00 n 000001000000.-00 00.0.00 0:000 000 00
0000. 00 0.00 00 u 00000-000000.-m00.n00 00.00 000 000 00

m. .0. 0.0.00 0 500m 000000.000 c mm00mm

 

mQMQOZ mZHMZMHIxom

m mqm<fi

107
6-2fi3Diagnostic Checking

Diagnostic checks are used to determine the adequacy
of the models; they are tests Of "goodness of fit." One way
to investigate the model adequacy is to study the residuals
of estimation. Since we have assumed that the disturbance
terms are distributed normally and independently, then, if
the model is specified correctly, we should expect the
residuals to be nearly uncorrelated with each other. If this
is true, the sample autocorrelations of these residuals are
expected to be zero or insignificant. This argument is the
basis of the portmanteau lack of fit test. In this approach
the significance of the first k autocorrelations Of‘ the
estimated residuals is tested. Under the null hypothesis of
independence between residuals, it can be shown [ 32, p. 491]
that the estimated autocorrelations of lag k of the residuals
(K = l,2....K) are independent normal, have a zero mean, and
a constant variance l/n. Following this, if the autocorrela-
tion of lag K is represented by r., the variate Q = k§l rk2
will be approximately distributed as XZk—p-q where n Is the
number of Observations on the stationary series and p and q
are autoregressive and moving average orders respectively.
On the other hand if the residuals are correlated, rk's are
XZk-p-q’ and the model is

inapprOpriate. For our models Q values are given in Table

large, Q exceeds the value of

9. These values indicate that the models do pass the

portmanteau test of fit.

108

6-2-4Forecasting

The models that have passed the diagnostic checks may
be used to make forecasts. The models developed in the last
section were used to make monthly forecasts over the period
June l969-December 1978. To make forecasts for each
successive period, a new actual Observation was fed into the
model, the forecasts, therefore, have an adaptive nature.
The forecasting performance of the models is difficult to
evaluate, except other models are available to which these
models can be contrasted. The sample periods for the fore-
casts is one of extraordinary fluctuations in interest
rates. The forecasting performances Of the corresponding
models should therefore be judged taking this point into
account.

One point should be emphasized: the forecastability
of the model can be properly judged only if it is used to
forecast beyond the estimation data period. If the same
set of data is used to estimate the model, and to make the
forecast, the performance of the forecasts does not denote
the adequacy of the model. Such an approach has been used

by Hester and Pierce [17 , Chapter 8].
6-3 Accuracy Analysis of Postsample Forecasting

A perfect forecast series is one that is identical
with the actual series. Performance of forecasting models

are therefore compared to such an ideal model. Means and

 

109

standard deviations Of forecast and actual (perfect forecast)
series are contrasted. The correlation coefficient between
the two is used as the basis of comparison, or some error
measures are adopted to indicate the degree of forecasting
imperfection.

For the models in Table 9, these measures are
displayed in Table 10. The means and standard deviations of
forecast series deviate from their actual counterparts by
very narrow margins. The correlation coefficient between
actual and forecast series, and the regression coefficient
of actual on forecast series are generally larger than .94.
The error measures are meaningful, when compared with a
particular realization, or the mean of the corresponding
series. The mean square error (MSE) may be decomposed into
three components as displayed in Table 10 [ 39].

The results reported in Table 10 indicate that the
smallest mean square error, mean absolute error, or mean
error do not necessarily correspond to series with least
variations. Some series may have larger variations, and
still be forecasted with more precision than others.
Variability should be distinguished from predictability.

The former should not be emphasized to the exclusion Of the
latter.

In addition to these measures, Theil [39 ] has
introduced an inequality coefficient (U) between actual and
forecast series which is the root mean square error, adjusted

to lie between zero and unity.

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where Pi and Ai represent forecast and actual values. Three

8, and U0 are developed, using this coefficient,

measures Um, U
to represent the fractions of error due to unequal central
tendency, to unequal variation, and to imperfect correlation

between actual and forecast series. These measures may be

expressed as:

Um = (F - X)2/MSE

s 2
U = (SF - SA) /MSE
C

U 2(l-r) SF SA/MSE

where F, K, SF, and SA are means and standard deviations of
forecast and actual series, and r is their correlation
coefficient. The size of the inequality coefficient and the
fraction of error due to bias, to different variation, and
to different covariation for our models are displayed in
Table 11. The inequality coefficient has its lowest value
for the demand deposits model, but it is small for all
models. With the minute size of the fractions of error due
to bias, it can be concluded that the models produce unbiased
forecasts. Since the fraction of error due to different
variations is also small, it can be said that the distribu-
tion of actual and forecast series are quite similar at

least up to the second moment. The fraction of error due to

112

 

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113
different covariations is large as desired. These values
should be compared with the ideal case of Um = US = 0, U0 = l,
which represents perfect forecasting.

In an alternative decomposition, the error is divided
to two fractions due to difference of regression coefficient
of actual on forecast from unity, and the other due to
residual variance. The size of these fractions for our
models indicate a satisfactory performance.

Plots of actual versus forecast series, may be used
for visual inspection of the forecasting performance. Plots
of actual and forecast series for demand deposits, the Prime
loan rate, the treasury bill rate, the CD rate, and the
federal funds rate, are given in Appendix 6. These plots
show that time series forecasts do capture the actual trends,
however, they do not always predict the sharp turns.

The plots for mid l973-mid 1975 are especially
interesting. All the interest rates show erratic behavior,
and wild month-to-month gyrations. An inquiry into the data
reveals, that within 1973-1975 period the treasury bill rate
rose more than 60 percent, while the loan rate, the federal
funds rate, and the CD rate, more than doubled. Month to
month variations in the interest rates go as high as two
full percentages (federal funds rate June to July 1973),
and a full one percent rise or fall is frequent. ’It should
be remembered that this wild gyration is just mirroring the
money supply during its most erratic course since the

creation of the Federal Reserve System. Money supply data

114
show that during 1973, money supply (Ml) sometimes grew at

an annual rate of 15 percent (May and June, 1973), and some-
times even declined in absolute value (August and September,
1973). Whether we consider these conditions an exception

to the course of conducting monetary policy or not, a time
series forecasting model should not, by its nature, be
expected to perform too well under these circumstances.

In general the performance of the models indicate
that demand deposits and interest rates are satisfactorily
predictable, although loan rate and treasury bill rate are
less volatile and more precisely predictable than the CD
rate and the federal funds rate. The forecast precision
of all the models could most likely be improved if time
series and structural models were combined, and/or multi-
variate time series techniques were utilized. In the next
chapter the forecast values generated by these models, are

used as data for the estimation of the bank behavior model.

115

CHAPTER SEVEN
THE EMPIRICAL RESULTS

In previous chapters a model of bank behavior was
developed and some comparative static results were derived.
In this chapter the magnitudes of these results will be
examined. In Chapter Five the model was linearized, the
adjustment mechanisms between actual and desired values were
hypothesized, and the properties of the disturbance terms
were specified. In the last chapter time series models
were developed to generate data on expected values of demand
deposits and interest rates to be used in empirical estima-
tion reported in this chapter.

After eliminating the equations describint A, p, K,
and L, on which data are not available, there remains a
system of five equations. The dependent variables are the
loans, the investment securities, the CD's, the net federal
funds sold, and the debits. The regressors for all the
equations are the same except for the lagged values of the
dependent variables which appear due to the partial adjust-
ment hypothesis. The remaining regressors include expected
demand deposits (55), the expected prime loan rate, the
expected treasury bill rate, the expected CD rate, the
variance of demand deposits, the variance of foretold
interest rates, the covariances between demand deposits in
one hand and loans, securities, net federal funds sold, and

CD's on the other, the reserve requirement ratio, the

116
initial net worth, and the prices of capital and labor inputs.
Besides, eleven monthly dummy variables are added to the set
of regressors to take account of seasonality.

To find the appr0priate estimation technique,
ordinary least squares was used to run a preliminary set of
regressions on the levels of the variables. The results
indicated that substantial serial correlation was present.
Therefore a first order autoregressive scheme was hypothe-
sized and the standard Cochrane-Orcut. technique was imple-
mented. The results of this set of regressions revealed
that the serial correlation paramater p was close to unity.
Hence, the estimation of the first difference version of
the system seemed to be appr0priate. The system was trans-
formed into first differences (except for seasonal dummies),
and each equation was estimated by ordinary least squares.
The estimates of the behavioral coefficients, and the
corresponding t ratios are displayed in Table 12. Besides
tests of the significance of regressors, these results shed
light on the actual, as opposed to predicted, behavior of
banks.

Clearly the empirical signs displayed in Table 12
are not all the same as those suggested by comparative
static results represented in Table 8. But it should be
remembered that the comparative static signs were determinate
only under the conditions specified for each result in the
latter table, so that they could have different signs under

other conditions. Therefore, although the empirical and

 

 

117

 

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121
the displayed comparative static signs are not always iden—
tical, this does not refute the model.

Some of the empirical results are worthy of inter-
pretation. The significance of the lagged values of the
endogeneous variables indicates the possibility of a lock-in
effect. The bank may find it difficult to alter the magni-
tude of the loans, CD's, federal funds, and clearance output
as much as it desires by one period. This does not seem to
be the case however (as might be expected) for government
securities. The assumption of partial adjustment behavior
seems therefore approPriate, at least for the former varia-
bles. Surprisingly, expected demand deposits are not a
"significant" determinant of loans. The comparative static
results of changes in demand deposits were indeterminate,
but the empirical result seems to be in accordance with the
Hester and Pierce result that demand deposits are allocated
to loans only after a lag [ 17, Chapter 9].

The variance of demand deposits, which may be taken
to represent deposit uncertainty, is shown to reduce the
loan output as expected, while the effect of variations in
interest rates are insignificant. The effects of covariances,
whenever significant, are in the directions predicted by the
model. The significance of variance of demand deposits and
significance, at least in some regressions, of covariances
between demand deposits in one hand and loans, securities,
CD's, and federal funds on the other lend support to the

risk-aversion hypothesis as opposed to risk-neutrality

122

implicit in the models which maximize expected profits. The
latter models may therefore be considered misleading.

The reserve requirement instrument of monetary policy
is shown to have the predicted restrictive impact on loans.
The initial net worth has a significant effect only on the
clearance output.

One interesting result is the significance of prices
of capital and labor on both the clearance and loan outputs.
Labor is found to be a normal input for both outputs since
its coefficients have a negative signs in both regressions.
Capital is an inferior input for both outputs. Besides, it
behaves as a substitute for CD's. The coefficients of the
capital input have positive signs in both loan and clearance
output regressions. Positive signs indicate that with an
increase in the price of capital, outputs will rise. This
result about loans makes sense if it can be argued that at
higher capital prices the banker transfers some of the cash
assets to loans in order to increase its earning. The
significance of the coefficients of capital and labor prices
supports the Pesek point of View that operating costs do
have an effect on a bank's decision making. The dummy
coefficients indicate substantial seasonality in loans and
clearance outputs. The seasonality in investment securities
is small, and CD's and federal funds hardly exhibit any
seasonality effects.

In general the model estimated for the large weekly

reporting banks in New York City does not show a good

123

empirical performance. Several explanations may be provided.
It should be recalled that estimation did not take place
under appropriate conditions. The model was developed in
terms of an individual bank, while the data refers to a group
of banks which are not even homogenous. By using aggregate
data, it is implicitly assumed that all the individual banks
involved have-the same objective functions, same constraints,
identical behavioral parameters, and identical expectations
about the future, while there is indeed some evidence to
suggest the contrary [ 2 , p. 219].

Besides, the model is represented by a system of
non-linear equations subject to some behavioral constraints.
The estimated unconstrained linear version may simply be
oversimplified and inaccurate. Multicollinearity between
regressors may also lead to insignificance of the coeffi-
cients. To investigate the effect of multicollenearity
between the first differences of interest rates included in
the model, two of the interest rates and the corresponding
variances were dropped from the set of regressors. The R2
delete was close to total R2 as in the presence of multi-
collinearity, but the regression results did not substan—
tially change. It was concluded that the adverse effect of
multicollinearity is slight. One final explanation may be
that the static single period theory can not satisfactorily

explain the multiperiod dynamic behavior of banks.

CHAPTER EIGHT

CONCLUSION

8-1 Results and Policy Implications

 

In this study a model of bank behavior under un-
certainty was developed. The bank was considered as a
multi product firm which produces its clearance output as a
direct-supplier, and loans and securities as a financial
intermediary. An expected utility maximization framework
was adopted, and portfolio selection, liquidity and liability
management, risk-aversion, jointness in outputs, and the
bank's Operating costs were incorporated into the model.

The comparative static results about regulation Q in
a two asset model indicate the presence of such regulation
reduces the percentage of the risky asset in the portfolio,
leading to an increase in liquidity. It is also shown that
this regulation, by allowing the cost of the clearance output
to be partially offset by interest free demand deposits,
leads to a reduction in the service charges. This arti-
ficially low price for the clearance output leads in turn to
over utilization of this product by demanders and a resource
allocation which is not socially Optimal. A trade-off may
therefore be said to exist between the share of the liquid
asset in portfolio, and an optimal allocation of resources.
It is desirable that the policy maker chooses the optimal

point on the trade-off possibility curve by choice of the

124

 

125
ceiling for regulation Q. The existing zero ceiling is not
necessarily appropriate over time. These results about
regulation Q hold true only so long as risk-aversion
prevails; models based on expected profit maximization cannot
therefore be expected to produce analogous results.

The comparative static results also indicate the
possibility of both complementarity and substitution between
outputs. More specifically, the model shows that two outputs
can be complements (substitutes) for a certain range of their
variations, while possibly having a different relationship
over another range or for another output mix. Uncertainty,
as measured by variances, was found to lead to both a
reduction in outputs, and a change in the output mix. This
feature should be considered in the making of the monetary
policy. Frequent and unusually large changes in monetary
policy instruments might increase uncertainty and consequently
strengthen or partially offset the policy's effect.

Reserve credit (reserve requirement) on assets, and
payment of interest on excess reserves were investigated as
potential monetary policy instruments. In a two asset model
it was shown that asset reserve credit on one asset will lead
the bank to increase that asset's share in its portfolio.
This instrument can therefore be used to favor or disfavor
a certain category of assets. With payment of interest
by the Fed on excess reserves the discount window operation
is complemented to work on excess reserves as well as reserve

deficiencies. The Fed's control over credit is consequently

126
broadened.

Lack of appropriate data does not allow a proper test
of the theoretical results. However, the empirical results
with aggregate data indicate that banks do display risk-
aversion, and do consider the Operating costs in their
decision makings. It may be concluded that depending on the
rate of inflation, a smaller or larger quantity of loans will
be made by the banking system and consequently a smaller
or larger quantity of demand deposits and money supply will
be created. This effect may distort the money supply from
its targeted level by the Fed. The accuracy of money supply
control may therefore improve if operating costs are consi-

dered in policy making.

802 Limitations

 

It should be recalled that the model's results are
based on the specific underlying framework. Exponential
utility, profit normality, and linear demand for the
clearance output are some of the restricting features of
the model. It is very desirable to determine the bank's
response directions under a general utility function, but
the problem is that determinacy is often lacking even for
specific utility functions. The relative determinacy of
the comparative static results in the present model is felt
worth the slight loss of generality.

The static, single period nature of the model is

another special feature. The empirical results support the

127

presence of a lock-in-effect in banks' assets and liabilities.
This suggests that bank behavior may embody adjustment lags
in response to stochastic variations in its portfolio. These
adjustments which reflect a multi period optimization behavior
are necessarily omitted in a single period context. In order
to capture the dynamic nature of bank behavior, multi‘period
models which consider the time path of variations in the
variables seem to be the most desirable models to be examined.
As another alternative a combination of time series and
structural models may be utilized to explain bank behavior.

In either case, the introduction of market imperfections and

interrelations in output demands would enrich the model.

APPENDICES

APPENDIX ONE
The Moments of an Incomplete Distribution (Borrowing)

For a group of banks the term free reserves (FR) may
be defined as the difference between excess reserves (ER) and
borrowing from the discount window (BR): FR = ER - BR.
Following this definition, for an individual bank which does
not hold excess reserves and borrow at the same time, free
reserves equal excess reserves when positive, and the
negative value of borrowing when negative. The portion of
the free reserves distribution to the right of point zero
refers to excess reserve holdings, and the portion to the

left, the negative value of borrowing.

      

L--'-o-o-

413'}? o Ffi FR
FIGURE 5
THE DISTRIBUTION OF FREE RESERVES
Since the variable free reserves is a linear function
of demand deposits, its distribution may be derived from
that of the latter. Utilizing the balance sheet identity,
the value and the moments of free reserves may be obtained

as follows:

FR = (l-rd)DD + W - (1-82)X2

O

128

129

Ffi' = (1—rd)55 + w - (1-52)x2

O

V(FR) = (1-rd)2 V(DD)

The borrowing variable does not have a well-defined
distribution. Mean and variance of this variable can be
considered as incomplete moments of the distribution of
free reserves, where the values to the right of point zero
are all treated as zero borrowing. To find the range of
demand deposits which correspond to negative values of
reserves (borrowing), free reserves may alternatively be

written as:
FR = Ffi + (l-rd) (DD - 65)

From this relation it can be seen that negative free
reserves correspond to the values of demand deposits between

zero and 55 - FR/l-rd, that is:

FR < 0 : DD < 56 - FR

l—rd
The moments of borrowing may consequently be defined as:
0' w

ER [FRdF(FR) - J0 dF(FR)

—CD

[FE + (l-rd) (DD — BB] g(DD)dDD

{6.5 - FR/l-rd
0

130

0 —— 2 0 —— 2
[(-FR-BR) dF(FR) + [(O-BR) dF(FR)
-0 o

V(BR)

o o
= FRZ dF(FR) + Efiz [dF(FR) + 25R IFRdF(FR)

_cn -oo
__2 m
+ BR {dF(FR)
0

= EBR2 + E§2(Pr) + 2§§(-§§) + §§2(1-Pr)

= 0002 - 032

Where Pr is the probability of borrowing, g(DD) is
the density function for demand deposits, and F(FR) is the

cumulative distribution of free reserves.

2. Partial Derivatives

The variations in BR and V(BR) come about because of
changes in the distribution of free reserves. To make
matters simple it may be assumed that the latter changes can
be prOperly represented by changes in mean and variance of
free reserves. This simplification can be justified by
assuming that free reserves have a two parameter distribu-
tion. The partial derivatives of the moments of borrowing

may consequently be obtained as:

66 - FR/l—rd

Q)
m
DU

_ ';_ [FE + (l-rd) (DD-BE) g(DD) dDD

l
'0)

Q)
'11
5d
O)
'11
$0

0

 

 

131

 

 

53-070100 _ _
= - Ig(DD)d(DD) = -G(DD-FR/l-rd) = -Pr
0
where G is the cumulative distribution of DD.
3V(BR) = aIEBRZ - Efiz]
affi aEE
BE - FR/l-rd ‘__2
a _._ _- BBR
= —_—_—__ [FR + (1-rd)(DD-DD)] g(DD) dDD '— _-
3 R BFR
0
BB - Efi/l-rd
= 2 [ES + (1-rd) (DD-55)] g(DD) dDD - 233 a—E—E
0 BFR

= -2§§ + ZER(Pr) = 2§R(Pr-l) < 0

To derive the second partial derivatives, the relation

Pr = 6(55 - FR/l—rd) may be utilized to conclude:

3P ' 8P
1'

 

 

r
__ < 0, .__— < O
aFR 3rd
and consequently:
affi afi
2 - 3P
3.!éggl = —§: [2§§(pr-1)1 == 2 1(P '1)(‘Pr) +‘§§ -:§ 1:
aFR aFR r BFR

has an indeterminate sign.
Since free reserves were assumed to have a two

parameter distribution, the total derivative of Pr can be

 

132

measured as follows:

ap ___ ap
dP = —:§ d(FR) + r dV(FR)
BFR av(FR)
where: V(FR) = (1-rd)2 V(DD)
dV(FR) = -2 (l-rd) V(DD)drd + (l-rd)2 dV(DD)
apr 39
The sign of -:: was shown to be negative. The sign of ——————
aFR _§V(FR)

is indeterminate. Notice that from dER = (l-rd)d55 - DD

drd - dX dW , the effects of changes in X2, rd, DD, W on

2 " O o
the probability of borrowing (Pr) are reflected through

changes in FR.

 

 

APPENDIX TWO

The Second Order Conditions

For second order conditions the Hessian matrix H
should be negative definite. This requires its bordered

principal minors to alternate in sign with the first being

 

 

positive.
’ l
O O FPl F2 0 FL PK
0 0 0 l-S2 l 0 0
PP 0 All 0 0 0 0
1
H = F2 l-S2 0 A22 0 O 0
0 l 0 0 A33 0 0
FL 0 0 0 O -AFLL 0
FK 0 O 0 0 0 -AFKK
L J
A = 2b - 2aV(X ) - lb 2 F
11 l 1 l 11
A22 = - 2 a V(PZ) - AFZZ
__. 2 __ BPr
A33 = RB BPr/BFR - 20 R BIPr(l-Pr) + BR —; 1
BFR
We need: H55,44,33 > 0 H55,44 < 0 H55>’0 H <
where Hii’jj indicates the minors of H obtained by deleting

rows and columns i and j.

133

 

 

 

 

 

 

V 0 0 F F ‘
P1 2
o o o 1 2
H I I = = F > 0
55 44 33 FPl o All 0 P1
F2 1 0 A22
L .
r 3
o o 3? F2 0
1
o o o 1 1
F o A o o 2
_ P 11 _ 2
H55'44 l ‘ A33F]?l +A11F2 ‘:0
F2 1 0 A22 0
o 1 o 0 A33
L 1
r W
o 0 PP F2 0 FL
1
o o o 1 1 o
FPl o All 0 o 0
H55 = FF 1 0 A22 0 o = FLL(H55'44)
2 2
+ FL A111333 > 0
o 1 o 0 A33 0
J

 

 

 

 

 

135

 

 

r W
0 0 PP Ed 0 FL FK
1
0 0 0 l l 0 0
FPl 0 All 0 0 0 0
F l 0 A 0 0 0
_ 2 22 _
O 1 O 0 A33 0 O - AF 2F A
K LL 11
F 0 O 0 0 -AF 0
L ‘ LL
”*2 + A33) < 0
PK, 0 0 0 0 0 -AFKK
L 1

For these relations to hold the following conditions are

sufficient.

Aii < 0 l = 1,2,3 FLL'FKK > 0

in turn for Aii < 0 i = 1,2,3 it is sufficient to have:

_39
> o P(1-P)+BR—£ < 0
r r _—

F ,F
11 22 BFR

the last condition will be satisfied if V(BR) declines at a
constant or an increasing rate when RR increases. (See
Appendix one).

Consider the marginal rate of product transformation
(MRPT) and marginal rate of substitution (MRS).

dX2 Fl
MRPT == 52’ = f_ (by implicit function theorem).
Xm 2

MRPT is increasing if:

 

 

 

a: F F 2 + F 2F
2 > 0 or 11 2 1 22 > O
a? 2 F 3
l 2
F
-dK L
MRS = .__—- = _
dL FK
MRS is decreasing if:
2 2
_g_ (F /F) __ FLLFK + FKKFL < 0
dL L K 3
PK

Since these conditions are satisfied by normalization
and second order conditions, it can be concluded that MRPT

is increasing and MRS decreasing.

APPENDIX THREE

Total Differentiation of The First Order Conditions

Condition 1:

a 066 + 2b dP

1 l l - {2al(alPl-RD)V(DD) + 2P1V(E1)

+ (4alP -2RD) COV(DD,E1) - 2alRB COV(DD,BR)}d0

1

- d{2ai V(DD)dPl - 2alV(DD) dRD + 2al(alPl - RD)dV(DD)

+ 2V(El) dPl + 2Pl dV(El) + 4 al COV(DD,El) dPl

-2 COV(DD'E1) dRD + (4alPl - 2RD) d COV (DD,El)

-2 al C0V (DD,BR) dRB - 2alRB d COV(DD,BR)} - FP d1

 

1
___ aal/at
-)LFllbl(al dDD + bldpl) - Abl a 55 + b P dt = 0
l 1 l
or:
2
-FPldA + dPl[2bl - 20V(xl) - Abl F11] —

(ZonaiPl - dalRD)dV(DD) + 2aPldV(El) + (4dalPl - ZdRD)
dCOV(DD,E1) - 2aCOV(DD,Xl) dRD - 2aa1 COV(DD,BR)dRB
- ZaalRB dCOV(DD,BR) + (Aalblsll - a1) dDD .

+ [2al(alP1 - RD) V(DD) + 2Pl V(El) + (4alPl - 2RD)

COV(DD,E1) - 2 alRB COV(DD,BR)] da

137

138

Condition 2:

01":

dC - dd - [2X2V(C2 - d ) - 2 RB COV (C2 - d BR)] dd

2 2 2 2'

d{2X2[dV(C2) + dV(dz) - 2 dcov(c2,d2)] + 2V(C2-d2)dxz

2RBdCOV(C2,BR)+-2RBdCOV(D2,BR)-2COV(C2-d2,BR)dRB}

dek - XFzde2 - (l-SZ) du + udS2 = 0

- dek - (1-82) du - [2dV(C2-d2) - AF22] dX2 =

~06 + dd + 20x dV(CZ) + 20x

2 2 dV (d2) - 40X dC

2 2 2 OV

(C2,d2) — ZdRB d CO (C2,BR) + ZaRBdCOV(d2,BR)

V

- 2aC (CZ-dleR) dRB - udS

ov 2 + [2X2 V(Cz'dz) ’ 2R3

COV(C2-d2,BR)] dd

Condition 3:

where:

2 ——- 2 ——
RBdPr + PrdRB + ZRB BR (l-Pr)dd + 201RB (l-Pr) dBR

2 —— —— ‘ _
- 2aRB BR DPr + 403R RB(l-Pr) dRB - dp — 0
SP __ 3P
dPr = —:§ dFR + ——£——— dV(FR)
BFR 3V(FR)
__ afiR __ ERR
dBR = air-i dFR + dV(FR) dV(FR)

dV(RF) = <1-rd)2 dV(DD) - 2(l—rd) V(DD) drd

 

 

 

139

CI”:
apr __ apr 2
- 00 + (RB - ZaRBBR){;;;> dFR + EVTFRT [(l—rd) dV(DD)
— 2(l-rd) V(DD) 0:01} + [Pr + 403R RB (l-Pr)] 0RB
2-—— 2 w 33R —— ERR
+ 2 RB BR(l-Pr) d0. " 20RB(1"Pr) {E dFR + W
[(1-r0)2 dV(DD) - 2 (l-rd)V(DD) 0:01} = 0
or:
apr
-00 + {[ - 20 BR] ——— - 20 (1- p rr)p } dFR =
RE RE aFR RB

ap
-dV(DD){[RB-2aR: RR](l-rd)2 EVIFR) + 20R§(l-Pr)(l-rd)

3BR .__ 2__
8V(FR) dRB [Pr + 403R RB<l-Pr)1 - ZRBBR (l‘Pr’ 00
+ dr {2[ -Za 2 RR](l-rd)V(DDyii£___ + 40 2(l-P )(l-rd)
d RE RE 3V(FR) RB r
aER
V(DD)BVTFRT

Condition 4:

- FLdA - AFLLdL = dP

Condition 5:

-FKdA - XFKKdK = dP

Condition 6:

FPl dPl + F2 dx2 + FLdL + FKdK = -(alFl-aD/DD)dDD
___ Bal
+ dy + log (alDD + blpl) ——— dt

at

140

Condition 7:

(1-82) dX2 + dFR = (l-rd)dDD - DDdrd + dWO + XZdSZ

APPENDIX FOUR

Comparative Static Results

21 = P1, x2, FF, L, K, -)L, -u

1 = 1, 2, 3, 4, 5, o, 00

321 = __H12

— H

ac2

821 = H12

- H

302

azi = x HiOO H12

582 2 H H

321 Hi1 H12
E = " 20LalCOV(DD,BR) T " ZGCOV(C2-d2,BR) T

-—— Hi3
"' [Pr + 4aBR RB (l‘Pr)] 71-—
32. H.
ERR = - 20COV(DD,xl) {fl-
D

321 = Hi4

aPL H

BZi = H15

39K H

- 20X 1
3V<C2) 2 "H—

141

142
32 H

 

 

 

 

 

 

1 _ 12
5V<02) ‘ 20LX2 "H
azi Hi1 apr
EVTEBT = 20al(alP1-RD) -{[RB-20RBBR](1— r0)2 SVTFRT
+ 201 2 (l-P )(l-rd)2 i§---E—-———}EI-3'-§-
RB r 3V(FR) H
32.
1 _ H.
my ' 20“’1 _._—$1
3Z1 H12
= - 40X ———
azi
= - 20
c V(C2,BR) RB
azi
c V(d2,BR) RB
aZi H11
c V(DD,BR) = ' 20‘a‘lRB H
32. H.
1 11
= 201(2al Pl ) -——
c V(DD'El) RD
321 —— HiOO 2—— apr
m = - DD —- + {[RB- ZGRBBR] [2(l_rd)v(DD)W7RR—)']
—— H.
2 BER 13
+ 40RB (1-Pr)(l-rd)V(DD) 3V(FR)} H
azi = HiOO
aw H

O

143

:11

az. ' H

 

1 _ _ _ —— i0 _ 11
3:: ‘ (alFl 0L133/DD) H + (AalblFll a1) H
DD
BZ1 2
M1 = [(2alPl - alRD)V(DD) + 2PlV(El) + (4alPl - 2RD) C0v
H11
(DD,E1) - 2 alRB COV(DD,BR)] T + [2X2V(C2 - (32)
H. H
_ _' 12__ 2 ——- _ i3
2% Cov‘Cz 02,0101 ‘0'“ 2% R ‘1 Pr) ‘0—
azi
7RF' = Hio/H
30 H. 3a /3
az./at=-1og(abfi+bp)_l_£9+xb (1t )
1 l 1 1 3t 11 l ——
a DD + b P
1 1 1
H11

H

APPENDIX FIVE
Moments of a Product of Random Variables

Consider two random variables P and Z which can be

written as:
P = R + U
z =§+V

where R and 2 represent mean values of P and Z respectively,

and g and y are independent random terms with zero mean and

constant variances.

r 1 r 1 where:
U o V(U) 0 V(U) = V(P)
v m o ' o V(V) V(V) = V(Z)
L 1 L 1

 

 

 

 

The moments of the product PZ can be measured as

follows:
P2 = (R+U)(§+V) = P2 + P6 + 26 + UV
(p2)2 = (RZ)2 + (RV)2 + (26)2 + (UV)2 + 2§2§v + zfiizu
_- _ 2 — 2
+ 4quv + 2PUV + zzu v.
E(PZ) = P2
E(Pz)2 = (RZ)2 + P2V(Z) + 22V(P) + v(P)V(z)
2 2 —2 —2
V(PZ) = E(PZ) - (E(PZ)) = P V(Z) + z V(P) + V(P)
V(Z)

144

145

C V(Pizi,Pij) = EPizinZj - E(PiZi)E(Pij)

= E{(Pi + Ui)(Zi + Vi)(Pj + Uj)(Zj + vj)} -

P.Z.P.Z.
1 1 J J

= E {(532, + Pflv. + E.U. + U.v.)(P{E. + Pflv.
1 1 1 1 1 1 1 1 j j j j

+ 2.U. + U.v.)} - P. 7.P '2
J J J J i 1P j 23

= E{PLELPZE. + P prgv. + P E §.U. + P E U.v.
1 1 j j 1 j 1 j 1 1 j j 1 1 1 j
“P“ . ‘
1 j 3 1 1 j 1 j 1 j 1 j 1 1 j j
+ ‘“’ " P _.U. . "— '
3311 311] 131] 11]]
+ —“ .U U v U U.V U v }
J J 1 1 J 1 1 J J 1 1
- P{— 51‘
1 1 J J

= PinZiE(Vj) + PiZiZjE(Uj) + PiZiCOV(Ui'Vj)

+ PinZjE(Vi) + PinCOV(Vi,vj) + Pisz0V

(Vi'Uj) + PiE(Uj)COV(Vi,Vj) + PijZiE(Ui)

Pj Zj MOV(U ,Vi ) + zizjcovmi ,Uj) + ZiCOV

(U. ’Uj )E(Vj ) + szjcovwi ’Vi) + PjE(Ui)COV

‘. . . . + . .
(Vi'vj) + ZjE(V1)COV(U1’U3) COV(Ul,UJ)COV

(ViIVj)

l4 '6’

Since: E(Ui) = E(Uj) = E(Vi)
c V(vi,vj). = COV(zi,zj)

C v(Ui,Uj) = C V(Pi’Pj)

c V(PiZi,Pij) = Pinc 0V(zi, zj)
+ 2.2 (Pi ,Pj ) + cO Vi(z

iZ j COV

,Zj )CO

E . =
. (V3)

V(Pi’Pj)

0

 

 

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

Q
on . .
O ./

 

o-c- ooooooooooo

 

Actual

DAJLForecasted
. . FIGURE 6:

GRAPH OF ACTUAL VERSUS FORECASTED

DEMAND DEPOSITS

 

 

?1 Actual
..Forecasted

FIGURE 7:

GRAPH OF ACTUAL VERSUS FORECASTED PRIME

LOAN RATE

‘1‘?

‘1

149

 

 

 

 

\
I
t! \
L

Actual
\ {.Forecasted FIGURE 3;

GRAPH OF ACTUAL VERSUS FORECASTED

TREASURY BILL RATE

 

150

 

\ O“......0.0o.oo.o......— ............................................................
\
\

Actual
., 1,.
\/ \/\ Forecasted FIGURE 9 .

GRAPH OF ACTUAL VERSUS FORECASTED FEDERAL

FUNDS RATE

1‘

‘2‘

151

 

‘5”) Actual
Vl\/\_‘Forecasted FIGURE 10;

GRAPH OF ACTUAL VERSUS FORECASTED CD

 

RATE

1‘

1'!

BIBLIOGRAPHY

 

BIBLIOGRAPHY

l

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