Full text of Working Papers (Federal Reserve Bank of Richmond) : A General Model of Bank Decisions, Working Paper 74-6

View original document

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Working Paper 74-6

A GENERAL MODEL OF BANK DECISIONS

Alfred Broaddus

Federal-Reserve Bank of Richmond
April 1972

The views expressed here are solely those of
the author and do not necessarily reflect the
views of the Federal Reserve Bank of Richmond.

A GENERAL MODEL OF BANK DECISIONS
Alfred Broaddus*

Introduction

This paper presents a general theoretical model of individual
bank balance sheet management under conditions of uncertainty. The model
seeks to integrate and extend the existing body of microbanking theory,
most notably the work of Klein [17], Bell and Murphy [4], Karehen [16],
Morrison [20], Orr and Mellon [22], and Porter [23]. On the basis of
the assumption that banks seek to maximize the return from their activities,
solution of the model yields a bank's desired balance sheet position over
a given time period and specifies the determinants of this desired position.
By "desired balance sheet position" we refer to the bank's desired stocks
of particular types of assets (such as loans, securities, and reserves)
and particular types of liabilities (such as demand deposits and time
deposits). We express these stocks as dollar balances and identify them
as the bank's decision variables.

Hence, solution of the model yields a

desired balance sheet position of the following general form:

Liability Categories
Asset Categories
(in dollars)
Asset 1
Asset 2
.
.

Liability 1
Liability 2
.
.

Asiet N

'Liability N
Net Worth

*This paper is based on
"A Stochastic Model of Individual
The author is grateful to Michael
tensive comments and assistance.

the author's unpublished Ph.D. dissertation,
Bank Behavior," Indiana University, 1972.
A. Klein and Elmus R. Wicker for their exThey are not responsible for remaining errors.

This result, by its nature, specifies three distinguishable
decisions:

(a) the bank's desired operating scale, as measured by the

dollar volume of total assets or total liabilities; (b) the bank's desired
liability structure, as indicated by the proportion of total liabilities
accounted for by each liability category; and (c) the bank's desired asset
structure, as indicated by the proportion of total assets allocated to each
asset category. A principal,goal of the analysis is to demonstrate that if
the bank attempts to maximize its return, these decisions are not independent of one another but are mutually interdependent. For example, it is
shown that the asset and liability structures that maximize the bank's
return are not invariant with respect to bank size, but vary systematically
/
with bank size. As a second example, the bank's decisions regarding asset
structure are systematically related to its decisions regarding liability
structure, and conversely.

Such interdependencies have not been compre-

hensively analyzed in the existing literature.

In constructing and solving

the model, we indicate the character of these interdependencies and specify
why they exist.
The paper proceeds as follows.

The next section outlines the

general analytical framework to be employed.

In subsequent sections the

model is constructed and solved, and a set of conditions consistent with
optimization by a bank of its balance sheet position is derived.

The

analysis involves a number of restrictive and often unrealistic assumptions.

Such assumptions are necessary, however, in order to confine the

analysis within manageable bounds.

Framework of the Analysis and General Assumptions

The object of the analysis is the individual bank, a financial
institution which seeks and obtains funds from a variety of sources and

subsequently invests these funds in a variety of financial assets.

assume that the bank i'snot subject to legal restrictions of any sort.
The controlling operational assumption of the model is that the bank acts
to maximize expected additions to equity over a finite time span desig1
nated the "planning horizon."

The bank accomplishes this optimization

by managing its balance sheet position over the course of the planning
horizon.

In reaching decisions, the bank is not influenced by events

preceding the planning period in time or by expectations regarding events
following its close.2

In reaching its planning period decisions, the bank

has certain knowledge of all relevant economic variables and parameters
comprising its environment with the following three exceptions:

(a) the

level of deposit liabilities at any moment during the period, (b) the
market value at any moment during the period of securities held as
secondary reserves against deposit withdrawals, and (c) total repayment
by borrowers of outstanding loans maturing during the period:
level of loan defaults.

i.e., the

Although uncertain,with respect to these variables,

the bank is assumed to know the form and parameters of their probability
distributions precisely. Having listed these variables, we must note the
omission, at this point in the analysis, of a major element of uncertainty
facing banks in the real world:

unanticipated changes in loan demand.

This omission is designed to simplify the model, since the basic goals of

1As indicated below, the bank is assumed to operate under conditions of uncertainty. Therefore, a more general assumption would be that
the bank maximizes utility, where utility is a function of both expected
return and the variance of return. Such an assumption would greatly complicate the analysis while adding little to its ultimate conclusions. As
in the Morrison [20] and Porter 1231 models, the presence of uncertainty
will affect the bank's decisions through its effects on the bank's expected
cost and revenue flows during the planning period.
'This assumption, in one form or another, is characteristic of
the theoretical microbanking literature. A thorough summary of its implications is given by Porter [23, pp. 325-3261.

the analysis can be achieved without explicitly considering this aspect
In an appendix to the paper, we indicate how the

of bank operations.

incorporation of uncertain loan demand in the model affects the results
generated by the model.

Throughout the paper itself, however, the term

"bank liquidity" refers solely to the bank's ability to meet unexpected
deposit withdrawals.
We have assumed that the bank maximizes additions to net worth
over the planning horizon by managing its balance sheet.

Therefore, the

elements of the balance sheet are the bank's decision variables.

We now

describe the nature of these balance sheet elements and the manner in
which their manipulation influences the planning period change in net
worth.

Decision Variables:

The Balance Sheet Elements

In reality, banks gain the use of funds by accepting liabilities
of widely varying form.

They subsequently allocate these funds among an

equally wide variety of assets.

Any theoretical analysis of bank operations

must abstract from the complexity of real world financial instruments. We
assume that the bank balance sheet consists of several categories of assets
and liabilities and that the bank formulates decisions in terms of these
categories. As indicated below, we shall assume that the instruments comprising each category are internally homogeneous with the exception of
loans.

The broad characteristics and analytical roles of each category

are outlined here.

Additional assumptions will be introduced when the

model is constructed.
parentheses.
below.)

Symbols used to refer to each category are in

(A complete list of symbols used in the paper is provided

Loans (L).

Loans are assets that pay an explicit return but

present the risk of default.

We assume that no loan outstanding during

the planning period is marketable during the period and that no loan
matures before the end of the period.
2.

Bonds (B). We use the term bonds to represent long-term

investments for income.

Bonds pay the bank a constant explicit rate of

return regardless of the quantity held:

i.e., bonds are available to

the bank in perfectly elastic supply. Bonds are free of default risk.
We assume that the bank, in its decision process, does not contemplate
selling bonds for any purpose during the planning period.

Hence, the

bank holds bonds solely for income purposes. They provide an alternative
to loans in that they guarantee a constant and certain, although generally
lower, average return.
3.

Securities (S). Securities are assets that pay an explicit

return and are free of default risk but whose market value at any moment
during the planning period is a random variable.

An organized market for

these issues exists, and the bank can buy or sell in this market at any
moment during the period without influencing whatever market price exists
at that moment.

The bank holds securities as a secondary reserve against

unexpected deposit withdrawals. We shall define the security issue as a
cons01 for analytical simplicity, but it will play the role of a short-term
government instrument.
4.

Reserves (R).

pay no explicit return.

Reserves are perfectly riskless assets that

The bank holds reserves in order to meet unex-

pected deposit withdrawals.
5.

Demand Deposits (DD). From the standpoint of the bank, demand

deposit liabilities represent funds that may be withdrawn at any moment during

In keeping with' the generality of the model, we assume that
I
the bank must pay explicit interest as well as service and promotional
the period.

costs in order to attract demand deposits.
6.

Time Deposits (TD). Time deposits, like demand deposits,

are funds which may be withdrawn at any moment during the period and
which cause the bank to incur interest, service, and promotional costs.
We shall assume that the probability of a given time deposit outflow
differs, in general, from the probability of an identical demand deposit
outflow.

Further, we shall assume that the functional relationship be-

tween deposit costs and deposit volume differs, in general, between the
two deposit categories.
7.

Borrowed Funds, (BF).

from various nondeposit sources.
the planning period.

"Borrowed funds" are funds obtained

Such funds cannot be withdrawn during

The bank must pay explicit interest in order to

obtain.these funds, but it does not incur any other costs for their use.
This category denotes longer-term borrowing designed to support a sustained increase in the bank's lending and investing activity that is
planned in advance by the bank's management.

The category does not

4
include borrowing to meet unanticipated liquidity needs.
If the above balance sheet elements are to serve as decision
variables, the model must be constructed in a manner that permits the

3
For simplicity, the analysis omits liabilities of intermediate
withdrawal risk: i.e. funds that may be withdrawn only after a warning
period or, if called immediately, only by forfeiture of interest. The
effect of including such liabilities can be inferred by generalizing the
solution of the model as constructed.
4
Borrowing of the latter variety does enter the model, however,
as indicated below.

bank to control, quantitatively, the dollar stock each variable represents.
The major difficulty in this respect concerns deposits.
does not control its deposit flows on a daily basis.

Clearly a bank

It is equally clear,

however, that through advertising and other means, banks attempt to influence at least

the direction and approximate magnitude of net deposit

flows over longer time periods.

In the case of any particular bank, the

time required to actually exert such influence is an empirical question.
In order to cope with this conceptual difficulty, we define
the decision variables as the expected average values of the respective
balance sheet stocks over the course of the planning horizon.

More pre-

cisely, if we define a particular time path for any balance sheet stock
over the planning horizon as:

(1)

0 = OttI,

then, assuming continuity, the average value of the stock over the period
is:
t=b
(2) 0 =

O[t]dt.

The value of a given stock may follow a number of different time paths
due to fund inflow or outflow (e.g., deposits) or market price fluctuation (e.g., securities) during the period.

In reaching its decisions,

the bank is uncertain as to which path will appear as the planning period
unfolds, but we assume it is able to attach a definite probability to
every conceivable path.

The average value of the stock over the period,

F, is then a random variable, the distribution of which the bank knows.
We define the decision variables as the respective means of these distributions.

For those balance sheet elements such as borrowed funds and

bonds which present no possibility of withdrawal or market price change,
I
the definitions are limiting~cases of the general definition. We assume
that the planning period is bong enough to permit the bank to control all
I
balance sheet decision variables as just defined. In the case of demand
deposits, for example, we ar* assuming that the length of the period
provides enough time for bank actions to control the parameters of the
average demand deposit balance distribution over the course of the period.
It seems reasonable to consider the period relatively short, perhaps two
to three months in duration.1

An Operational Equation for ~theExpected Chance in Bank Net Worth

We have assumed thbt the bank's goal is to maximize the addition to net worth5 over the planning horizon.

Hence, it

is necessary to

specify the manner in which bank actions influence the change in net
worth.

It will be useful tombegin with accounting relationships and

develop from these an operational relation between the change in net
worth and the bank decision variables.
Let us indicate the last day of the previous period and the
last day of the planning period by the subscripts t-l and t, respectively.
The balance sheet identities~for each of these days are then:
(3)

Ltsl + J&l

+ Stal + R+

- DDt-l - TDt,l - BFt-1 - NW,,1 i 0;

(4) Lo + Bt + St + Rt - DD, - TDt - BFt - NWt = 0,

where NW is bank net worth.

By subtracting (3) and (4) and rearranging

5
For simplicity, it is assumed that net worth consists entirely
of capital stock (i.e., shareholder equity) and that the bank neither plans
nor makes dividend payments ~during the planning period.

terms, we can write the following expression for the change in net worth
over the planning period:
(5) ANW = [(It - Ltml) + (bt - Bt-1) + (St - St-l) + (Rt - Q-l)]

- [(DDt - DDtml>
+ UDt - TQl)
where AHW = NW, - NWt 1.

+ Wt - B&l I,

We proceed from accounting identity (5) to an

operational relation as follows.
Step 1.

Consider the final bracketed term on the right side

of (5). Calculation of the magnitude of this term eliminates inter-liability
substitution and gives the net inflow or outflow of total funds during the
period.
Step 2.

Consider now the first bracketed term on the right

side of (5). Calculation of its magnitude eliminates substitution among
assets and gives the net increase or decrease in total assets over the
period.
Step 3.

Subtract the result of step 1 from the result of

step 2, as indicated by the right side of (5). This operation eliminates
from the net change in total assets (step 2) that portion which results
directly from the net change in total funds (step 1).
In general, a residual (net) change in total assets remains
after performance of step 3.

This residual represents that portion of

the change in total assets which does not result from inflows or outflows
of funds.

Further, this residual equals ANW, the change in net worth

over the period.

For present purposes, we may assume that the residual

consists of the following three elements:

(a) loan defaults during the

period; (b) the change in security portfolio value that results from
market price fluctuation as opposed to sales or purchase transactions;

and (c) the net revenue flow!during the period arising from the bank's
operations.
I
In subsequent analysis, we shall treat loan defaults as a
reduction of the rate of return on loans.

Therefore, element (a) above

will be absorbed by element (c). Schematically, we can write:
(6) ANW = exogenous security portfolio change + net revenue flows.
Equation (6) is the operational relation facing the bank at the beginning
of the planning horizon.

The equation is operational because the flow

of net revenue over the period depends upon the decision variables, as
we shall indicate in constructing the model.

Because the decision vari-

ables have been defined as expectations, however, equation (6) must be
rewritten as:
(7)

EW'W

= expected exogenous security portfolio change + expected
net revenue flows.

As indicated below, the bank does not, in the probabilistic sense, expect
security prices to change during the planning period.

Therefore, equa-

tion (7) reduces to:
(8)

E@W

= expected net revenue flows.
The model construction in the next section consists, essentially,

of a detailed specification iof (8) with particular attention to the dependence of expected net revenue on the decision variables.

II.

Construction of the Model

In this section, we shall develop a detailed operational function that specifies the determinants of the bank's expected change in
net worth during the planning period.

This relation will serve as the

bank's objective function. 'The procedure will be to consider each decision

variable (i.e., each balance sheet element) in turn, noting its contribution to the function. We shall then summarize by writing the complete
function.

Solution of the model in the following section consists of

maximizing the complete function subject to a balance sheet identity
constraint.

Loans

In addition to the assumptions already outlined, we place the
following specific restrictions on the character of bank lending activities.

(a) All loans outstanding on the day preceding the planning period

mature on that date.

If the bank chooses to renew a portion of these

loans, it does so under contract terms prevailing at the beginning of
the planning period.

This assumption eliminates from the calculation

of expected planning period revenues the analytically unnecessary complication of loan carry-over from previous periods at previously contracted rates.

(b) The entire balance of each loan contracted during

the planning period falls due at the end of the period, and any loan
default occurs at that time.

This assumption insures that all prospec-

tive defaults on loans contracted during the planning period enter the
objective function of the model.

loan size, are identical across loans and are exogenous to the bank.

In standard microeconomic theory, the firm faces a demand curve
which specifies the manner in which average revenue from product sales
varies with total sales, We wish to introduce a similar relation for the

'The homogeneity assumption is for analytical simplicity. It
could presumably accomodate balancing trade-offs among the noninterest
terms of particular loans.

bank's lending activity.

The analysis is complicated, however, by the

fact that, from the standpoint of a bank, alternative borrowers differ
with respect to prospective default (i.e., credit rating) and the duration
and strength of their customer relationship with the bank.

Banks, unlike

many nonbank firms, are highly selective in choosing the particular customers to whom they "sell" their loan product, and they discriminate
7
among customers in establishing prices for this product.

We wish to

avoid treating the complex process by which banks select the particular
customers to whom they lend.
set of assumptions.
loan customers.

To do so, we introduce the following final

(d) The bank faces a particular set of potential

The prospect of default associated with each customer

is summarized by a probability distribution of total loan repayment.
The character of these distributions varies from one customer to another.
The parameters of each such distribution are exogenous to the bank but
known by the bank.

We assume that the bank lends to these customers -in

a fixed sequence determined by considerations outside the scope of the
analysis.

The bank extends loans in this sequence up to a point of its

choice where it ceases lendkng altogether.

The usefulness of these last

assumptions will become clear as the analysis proceeds.
Loan revenue conditions facing the bank are summarized by the
following expected average net rate of return function for loans:

(9)

rLILl = $[Ll - dLILl i cLbl,

7
Banks-discriminate further by differentiating the eh-aracter
of the loan product among customers, that is, by varying noninterest
lending terms.- We have eliminated this difficulty by assumption (c)
above, For general analyses-of-the determinants of bank lending behavior
see Hester [ll] and Jaffe and Modigliani [14]. For a detailed examination
of the "customer relationship" and its effect on bank lending see Hodgman
[13, pp. 97-1441.

where:

rLCL1= the expected average net rate of return on loans as a function
of total loans,
L=

8
average total loans held by the bank during the planning period,

r;.= the average gross contract rate on loans as a function of total loans,
dLLl

= the expected average default rate on loans as a function of total
loans, and

CLILI = the average cost of loans (expressed as a percentage rate on the
dollar) as a function of total loans.
Equation (9) plays an analytical role similar to that of a demand curve
in standard theory. That is, (9) specifies how revenue from the bank's
principal revenue-producing activity varies as the volume of lending
activity changes.9

We now examine each of the component functions on

the right side of (9).
1.

dLbl

The construction of this function can be described

with the aid of Figure 1, which plots total loan repayment on the vertical
axis against the decision variable L on the horizontal axis.

As indicated

above, the bank knows the probability distribution of loan repayment for
each customer. These individual distributions are marginal distributions
of a joint distribution that we assume the bank also knows.

This joint

distribution may or may not exhibit some degree of covariance among borrowers.

Given knowledge of this distribution and, by assumption (d), the

sequence in which loans are extended, the bank can construct a probability

8
L is the bank's lending decision variable and refers to the
dollar volume of loans the bank extends. Throughout the body of this
chapter, the bank controls this particular decision variable with certainty. That is, the bank can select the precise volume of loans it
wishes to extend, even though the level of repayment is uncertain. In
the appendix, we analyze the implications of an alternative assumption.
9
The function obviously differs from a conventional demand
curve in that revenue is here defined net of lending costs.

distribution of total loan repayment at any particular level of total
loans outstanding.
level Lo.

Such a dhstribution is depicted by Figure 1 for loan

This particular distribution is one member of a family of such

distributions for various total loan levels.

For simplicity, we assume

that each distribution within this family is continuous and symmetric
and that the respective means and limits of the distributions comprising
10

the family form the continuous lines radiating from the origin in Figure 1.
Total Loan
Repayment

L$~~(FuII

Repaymenr)

Expected
Repayment

FIGURE 1

In terms of Figure 1, the variable dL is (for loan level Lo) the

AB
ratio
OL,’

Clearly, the functional relation of dL to L depends on the shape

10
The upper limits (full repayment) of the distributions obviously
form the continuous 45' line. We are assuming that each distribution exhibits
a finite lower limit indicating the smallest total repayment to which the bank
attaches a positive probability.

of the figure's "expected repayment" line which, in turn, reflects the
manner in which each additional borrower in the fixed sequence of borrowers affects the distribution of total loan repayment facing the bank.
For example, if the expected repayment line is linear, dL is a constant.
In this case, later borrowers in the sequence do not increase the expected
default rate.

As Figure 1 is drawn, later customers present a greater

11
risk of default and cause dL to rise.
(10)

In general terms:

dL = dL[L; 51,

where zis

a vector of parameters summarizing the default risk charac-

teristics of the bank's loan customers. We place no specific restrictions on the mathematical form of (10). As a model parameter, however,
the vector Z will affect the solution of the model for the optimal values
of the decision variables.
2.

ri &I .

This function specifies how the average gross con-

tract rate varies with L in much the same way that a conventional demand
curve specifies the relation of average revenue to sales.

The mathematical

form of this function in any specific case reflects the influence of two
interrelated factors.

First, the function reflects the credit risk char-

acteristics of particular borrowers in our assumed fixed sequence, since
these characteristics determine the risk premiums the bank seeks to extract from each customer. Second, the function reflects competitive
conditions in the loan market within which the bank operates.

In a manner

similar to conventional demand analysis, competitive conditions affect the

1lThe term "risk," as used here, refers only to the relationship
of the first moment of the total loan repayment distribution to the levelof loans outstanding. We can reasonably assume, however, that the variance
of the repayment distribution increases if dL increases with L.

form of the function through their determination of the bank's market
power within the relevant loan market.

Here, competitive conditions

specifically affect the degree to which the bank
Therefore, we may write:

from its borrowers.
(11)

extract risk premiums

rt = ri[L; aL, Z],

where aL is a vector of parameters summarizing the competitive structure
of the bank's loan market.

Again, we place no specific restrictions on

the form of the function.

It is worth noting that, in contrast to tradri
> 0 if competitive conditions
ditional demand theory, we may have dL
are such that the bank can fully compensate for the increased default
risk associated with nonprime borrowers.
3.

CLbl

This function indicates how the average cost to

the bank of making and servicing loans, per dollar of loans, varies with
total loan volume.

We assume that the bank's real capital stock is fixed

over the planning period.

Hence, this function is analytically comparable

to a short-run average cost function in the standard theory of the firm.
There are, however, a number of conceptual difficulties in treating loan
costs in this fashion.

In the traditional theory of the firm under con-

tinuous production conditions, one derives the functional relationship
between short-run average costs per unit of output and total output by
minimizing costs at each level of output subject to (a) the technical
constraint imposed by a productiofifunction and (b) factor supply conditions facing the firm.

This procedure presupposes an unambiguous

definition of the relevant output upon which costs depend.
In the present analysis, as the above cost function indicates,
the "output" is the dollar amount of loans outstanding.

This choice is

necessary because we are treating balance sheet stocks as decision variables.

It is by no means clear, however, that this variable is the relevant

output on which lending costs depend.12

Broadly, a bank incurs lending

costs due to (a) credit risk investigations and (b) administrative services
surrounding the management of loan accounts. These activities constitute
the physical output flows upon which lending costs directly depend,l3

general, one would not expect the volume of these services to exhibit any
invariant relation to dollar loan volume.

If for a given bank, however,

all loan characteristics including loan size were identical across loans,
a fixed relation would exist between the flow of loan services and total
loan volume.

Under these conditions, costs as a function of dollar loan

volume would be a simple transformation of costs as a function of service
output.

In the present analysis, we have assumed that loans are identical

with the exception of borrower default risk characteristics. Therefore,
we may think of cL[L] as the sum of two independent components. The first
component is a simple transformation of a standard cost function with all
loan services other than those related to the credit risk of individual
borrowers defined as output.

The second comprises those costs, such as

credit investigation costs, that are related to borrower risk.

Because,

by assumption (d), the sequence and risk characteristics of borrowers are
predetermined, it follows that the relationship between the costs of
investigating borrower credit standing and the bank's total loan volume
is exogenous from the standpoint of the bank.

121n this regard see Broaddus [6, pp. 37-441.
13For a thorough discussion see Benston [5, pp. 522-5341.

On the basis of these considerations, we may write:
(12)

= cL[L; Ko, ;jb, ?I,'

where x0 is the bank's fixed,capital stock, To is the constant wage rate
facing the bank, and, again, Zis

a vector summarizing the credit risk

characteristics of the bank's borrowers.

Following our earlier procedure,

we impose no specific restrictions on these parameters.
Having discussed the components of the bank's loan revenue
function, equation (9) can be rewritten in general form as:
(13)

= rL[L; Ko, Co, aLs zl,

where, again, rL is the expected average rate of return on loans net of
default and loan costs.

Because we have not restricted the parameters

of (lo)-(12), it follows that we have not restricted the parameters of
In what follows, we shall drop the parameter notation and write

(13).

(13) as:
(14)

rL = rLILl

From the preceding discussion, however, we know that the form of this
function depends on the risk characteristics of the bank's loan customers,
the competitive structure of the market in which the bank operates, and
factor prices.
We can now write the bank's expected total net revenue from
loans as:
(15) ERL

= rLILl OJ.

Equation (15) is the first component of the objective function of the
model.

Bonds

The analytical role played by bonds was briefly outlined in an
earlier section.

The distinction we have introduced between "bonds" and

"securities' is designed to separate, in a gross fashion, nonloan investments made for income purposes from securities held as secondary reserves
to meet deposit withdrawals.
14
clear-cut.

In the real world, this distinction is not

Further, the composition of an actual bank's earning asset

portfolio reflects the term structure of interest rates and expectations
In our static,

with respect to the future course of interest rates.

single-period model, these dynamic considerations play no role.

stated above, the bank views bonds as an alternative to loans because
they are free of default risk and yield a fixed rate of return regardless
15
of the dollar volume held.
We introduce the following additional assumptions.
available to the bank are homogeneous consols.

(a) Bonds

(b) The price of an in-

dividual bond is constant over the planning period.16
the fixed coupon.rate FB over the planning period.

(d) Bond transactions

are costless.
On the basis of these assumptions, the total planning period
revenue from bonds is:
(16)

ERB = F$B),

14This is not to say that such distinctions are unrecognized within
the banking industry. See, for example, American Bankers Association [l, pp.
270-2711.
15The effect of asset supply conditions on bank decisions has been
relatively neglected in the theoretical banking literature. See, however,
Klein [18].
16
Since, as indicated earlier, the bank does not contemplate bond
liquidation during the planning period, it is analytically unnecessary to
introduce uncertainty with respect to bond prices.

where B, the decision variable, is the average value of the bank's bond
portfolio over the period.

Equation (16) forms the second element of

the objective function.

Securities

As stated earlier, the bank holds securities as a secondary
reserve to meet unexpected deposit withdrawals.

Securities are an alter-

native to reserves for this purpose, because we have assumed that securities, unlike reserves, pay an explicit return.
additional assumptions regarding securities.
to the bank are homogeneous consols.

We introduce the following
(a) Securities available

(b) The price of a security at the

beginning of the planning horizon is one dollar.

standpoint, the average price of an individual security over the planning
period is a random variable:
(17) Ps =I1 + w,
where w is a uniformly distributed random variable with mean zero, and
where -a<_w'a,O<a<L

The homogeneity assumption implies that

individual security prices are perfectly correlated. Hence the average
value of the bank's total security portfolio over the period is also a
uniformly distributed random variable.

The expected value of this latter

random variable, designated by the symbol S, is the bank decision variable
with respect to security holdings., (d) Each security pays the coupon rate
Fs over the planning period.

(e) Transactions in securities are costless.

On the basis of these assumptions, the bank does not expect any
capital gain or loss on security holdings, and the total expected explicit

17Therefore, the distribution of Ps is $(PS) = k
1 -aIPS51+a.
_.. .-

over the range

revenue from securities is:
(18) EBS = Ys(S).
Equation (18) is the third element of the objective function.
We turn now to the liabilities side of the balance sheet, returning to reserves at a later point.

Deposits

As stated earlier, the bank accepts two types of deposit
liabilities:

(a) demand deposits and (b) time deposits.

It is assumed

that two characteristics of the bank's deposit accounts influence the
bank's decisions:
costs.

(a) stochastic deposit variability and (b) deposit

Since these factors play a critical role in the analysis, it is

necessary to discuss each at some length.
Stochastic deposit variability
In reality, individual banks face continual inflows and outflows of funds due to depositor transactions. The pattern of these
flows determines a bank's total deposit stock at any moment in time and
the path followed by the balance through time. The time path of an
actual bank's deposit stock reflects, ultimately, the behavior of the
bank's depositors and the innumerable factors that condition this behavior.

Banks themselves can influence depositor behavior to some degree

through deposit interest payments (to the extent that such payments are
permitted by regulatory authorities), services provided depositors, advertising, and promotional campaigns. Further, real world banks can
predict, with considerable accuracy, deposit variation caused by cyclical
and seasonal movements in income and in other economic variables.

addition to these partially predictable deposit movements, however, all

22'

banks experience essentially random deposit fluctuations caused by a
I
myriad of unsystematic and unpredictable conditions influencing their
18
depositors.

For simplicity, and in keeping with the single-period

framework of the analysis, we assume in what follows that all factors
influencing depositor behavior other than the bank's own actions are,
in the bank's view, random.
We assume that the bank attracts demand and time deposits from
a diverse group of individual depositors.

The average balance of the

ith individual demand deposit account over the planning horizon can be
expressed as:

I
i=

Ui,

(19) DDI = BDD,i +

. ..,

NDD'

where NDD is the total number of demand deposit accounts, and u i is a
random variable having zero mean and following an otherwise unspecified
probability distribution.

Since ui has zero mean, it follows that

is, in the bank's view, the ith depositor's expected average

BDD,i

balance over the planning horizon.

We further define:

NDD
(20)

DDToTa

(BDD,i

Ui)

i=l

DDTOTm

is the bank's average total demand deposit stock over the plan-

ning horizon.

Since we have'not specified the joint distribution of the

Ui, we cannot fully specify the mathematical form of the distribution of
DDTOTAL*

We can, however, define the mean of this latter distribution as:

NDD
(21) DD = C BDD,i.
i=l

181n this connection see Dewald and Dreese [a].

DD is the bank's expected average demand deposit stock over the planning
horizon and is the bank's demand deposit decision variable.

We assume

that the bank controls DD through deposit interest payments,19 advertising, and other policies that influence the levels of the individual
BDD i and the total number of depositors maintaining demand deposit ac,
counts at the bank.
Similarly, we express the average balance of the iath time
deposit account as:
(22) TDi. = BTD i.
,

Vi’,

where NTD is the number of time deposit accounts held by the bank, and
v.0,
1

like ui, is a random variable having zero mean but following an

otherwise unspecified probability distribution. The bank's average
total time deposit balance is:
NTD
(23) TDTOTAL ;.zl (BTD,i' + Vi').
The mean of the unspecified distribution of TDTOTAL is then:
NTD
(24) TD = C B
i'x1 TD,i#'
TD is the bank's time deposit decision variable.
Using (20) and (23), the bank's average total deposit balance
(i.e., demand plus time deposits) is:

(25) DTOTAL = DDTOTU + TDTOTa

NTD
NDD
= C (BDD i + 'Ji)+ ' (BTD i' + Vi*)*
,
,
i'=l
i=l

19Since our analysis is general and abstract, the current
prohibition of explicit interest payments on demand deposits in the
United States is ignored. Time deposit rate ceilings are also ignored.

Our assumptions to this point do not permit us to indicate the form of
the distribution of DToTAL; however, these assumptions do permit specification of the distribution'smean as DD + TD.

We now define an additional

random variable that will play an important role in subsequent analysis:
NTD
NDD
(26) U = c ui + c Vi"
i='l
i*=1
U is the absolute deviation of the bank's average total deposit balance,
DTOTAL, from its mean value DD + TD.

The above assumptions imply that

U has zero mean; however, the form of its distribution cannot be specified
further.
The analysis that follows focuses considerable attention on the
risk of deposit variability faced by the bank during the planning period,
as measured by the dispersion of the random variable U around its mean
value.20

Of great importance in the analysis, we shall assume that the

degree of this risk depends systematically on (a) the bank's size and
(b) the structure of the bank's liabilities.

Further, we shall assume

that the bank knows the quantitative character of these relationships
and takes explicit account of them in reaching decisions.
Since we have postulated relationships between deposit variability, bank size, and liability structure, it is necessary to indicate
the rationale for believing that such relationships exist.

We now

develop this rationale.
In the context of the above discussion, a useful measure of
the deposit variability risk faced by the bank is the standard deviation

201n what follows, the terms "deposit instability" or "stability"
will always refer to the degree of this dispersion. It is deposit instability in this sense, rather than in the sense of high deposit turnover
rates or velocity, that generates the bank's need for liquidity. See
Morrison and Selden [21, p. 121.

of the random variable U,

For simplicity, all individual deposit accounts,

both demand and time, are assumed to have identical mean balances B.

is further assumed that individual demand deposit accounts have identical
variance var(u),,and that individual time deposit accounts have identical
variance var(v).

(27)

Under these conditions, the standard deviation of U is:

var(u) + WTD

NDDNDD
var(v) + I I
cOv(ui3
i 1
i#j

+cccov+,
v
N

N
TD TD

ND#TD

j-)

i#j

'
I

c
i'

COV("i,

Vi.>

'j)

4
1
,

where cov(ui, uj) is the covariance of the 1.th and j th demand deposit
accounts,

COV(Vi,,

vjA) is the covariance of the icth and

j*th

time

de-

posit accounts, and cov(ui, vi*) is the covariance of the ith demand
deposit account and the iOth time deposit account. We can now indicate
why a,,is likely to vary with liability structure and bank size.
Liability structure.

For present purposes, we define liability

structure as the relative allocation of the bank's total deposits between
demand and time deposits.

For the moment, we assume that bank size, as

measured by the expected average total deposit balance DD + TD, is constant.

We further assume that the bank's depositors distinguish between

demand and time deposits with respect to function. Specifically, it is
assumed that depositors use demand deposits primarily as a means of
payment, but that they use time deposits primarily as a store of wealth.2l
On these grounds, it is reasonable to assume that the variance of individual

21This strong distinction is made for analytical convenience.
In reality, the distinction is not absolute but a matter of degree. In
this regard see Hicks [12].

demand deposit accounts exceeds the variance of individual time deposit
accounts:

22
i.e., var(u) > var(v>.

Consider now the effects of a shift

in the bank's liability structure to a greater proportion of time deposits.
For simplicity, assume that this shift takes the form of individual demand
depositors closing their demand deposit accounts and using the funds to
open savings accounts.

Such transfers reduce the magnitude of the first

term in parenthesis on the right side of (27) and increase the magnitude
of the second term.
terms is negative.

With var(u) > var(v), the net change in these two
This net change tends to reduce overall deposit vari-

ability as measured by au.

We cannot specify the effect of the postulated

deposit transfers on the covariance terms in (27) without detailed knowledge
of the underlying joint distribution of individual deposit deviations.
In general, however, there is no a priori reason for supposing that resulting changes in these covariance terms will exactly offset the downward
effect of the transfers on au just specified.

Therefore, we have estab-

lished theoretical grounds for presuming that deposit variability, as
measured by au, varies with changes in the bank's liability structure.
The precise character of this relationship in any given case depends on
the form of the joint distribution of the bank's individual deposit accounts and on the manner in which changes in liability structure affect
this distribution.

The particular assumptions we have made suggest that,

with total deposit volume DD + TD constant, au is inversely related to
the ratio of time to total deposits.

22The validity of this assertion is, of course, an empirical
question. Limited evidence indicates that time deposits are, in fact,
more stable than demand deposits. See Morrison and Selden [21, pp. 12-191.

Bank size.

Equation (27) can also be used to analyze the rela-

tionship between deposit variability and bank size, where bank size is
measured by the bank's expected average total deposit volume DD + TD.
In general, changes in the bank's expected deposit volume can
result from (a) changes in the average balances held by existing individual demand and time depositors, (b) changes in the number of individual
demand and time deposit accounts held by the bank, or (c) some combination
of the above.23

Suppose first that DD + TD increases due to increases in

existing individual deposit balances.

The variance and covariance terms

that comprise au are measures of the dispersion of individual deposit
balances around their respective means and of the codispersion of pairs
of deposit balances around their respective means.

The magnitudes of

particular variance and covariance terms are likely to change following
increases in the corresponding means of individual deposit balances.
For example, the variance of a particular demand deposit account might
increase following an increase in the account's mean balance if the increased balance is accompanied by unsynchronized receipt and payment
transactions of greater absolute size.

As indicated by (27), any such

changes in individual variance and covariance terms directly affect the
value of au.

It follows that deposit variability as measured by au is

likely to change following an increase in DD + TD caused by increases in
the mean balances of existing accounts.

The exact functional relationship

between au and DD + TD in any given case depends upon the precise manner
in which the variance and covariance terms comprising au change following

23For the moment, we ignore changes in liability structure
that may accompany changes in total deposit volume. This possibility
will be considered below.

a given increase in deposit volume.
Alternatively, suppose that DD + TD increases due to the
opening of new demand or time deposit accounts at the bank.

Such an

occurrence adds a new variance term to au for each of the new deposit
accounts, and (if the total number of accounts held by the bank is at
all sizable) a large number of new covariance terms. As (27) again
indicates, these new variance and covariance terms directly affect the
value of au.

Therefore, au is likely to change following an increase

in the bank's deposit volume caused by an increase in the number of

deposit accounts the bank holds.
We have now indicated why it is reasonable to postulate that
deposit variability faced by the bank, as measured by uU, varies with
changes in the bank's expected average deposit volume.

The exact quan-

titative character of this relationship in any given case depends ultimately on the nature of changes in the joint distribution of individual
deposit account balances that accompany any given change in the bank's
deposit volume.24

The discussion above suggests, however, that uu is

more likely to vary directly than inversely with deposit volume.

240ur discussion has been concerned solely with the relationship
between deposit volume and absolute deposit variability, as measured by uu.
Equation (27) can also be used to show that relative deposit variability is
likely to vary with bank size. For this purpose, an appropriate measure of
relative variability is:
=lJ
Cl=-.
DD+TD
Suppose that DD + TD increases by some proportionate amount due to an increase in the number of accounts the bank holds. No &priori reason exists
for expecting the accompanying change in UU discussed in the text to be
proportionately equal to the change in deposit volume. Therefore, it is
reasonable to presume that, in general, 0 varies with deposit volume. Most
recent empirical studies of deposit variability focus on the relationship
between relative deposit variability and bank size. Several of these studies
suggest that relative deposit variability declines as bank size increases.
See Gramley [lo, pp. 41-531, Rangarajan [24], and Struble and Wilkerson 1251.

Joint effects of liability structure and bank size.

To this

point, we have analyzed the effects of liability structure and deposit
volume on aU separately. That is, we first analyzed the effect of a
change in liability structure on uu under the assumption that total
deposit volume was fixed. We then analyzed the effect of a change in
deposit volume on au without considering changes in liability structure
likely to accompany the change in deposit volume.

Unless a given change

in deposit volume is caused by proportionately equal changes in demand
and time deposits, some alteration of liability structure must accompany
the change in deposit volume.

We can reasonably presume that the quan-

titative effect on au of a given change in deposit volume depends on the
particular change in liability structure that occurs.
tion can be defended by a final example.

This last proposi-

Suppose that the bank's expected

total deposit volume increases by some given amount.

Further, assume that

this increase results entirely,from an increase in the number of demand
deposit accounts the bank holds:

that is, the increase in deposit volume

is accompanied by a shift of liability structure in favor of demand
deposits.

The new demand deposits add additional variance and covariance

terms to (27), causing, as indicated above, some change in au.

Alterna-

tively, assume that the increase in deposit volume results entirely from
an increase in the number of time deposit accounts:

that is, the increase

in deposit volume is accompanied by a shift of liability structure in
favor of time deposits.

The new time deposit accounts then add additional

variance and covariance terms to au in (27). As indicated above, it is
reasonable to assume that the variance and covariance properties of demand
deposit accounts differ systematically from the corresponding properties
of time deposit accounts.

It follows that these two alternative occurrences

will have systematically different quantitative effects on au.

For

example, if the variance and covariance of time deposit accounts is less
than the variance and covariance of demand deposit accounts, an increase
in time deposits contributes less to the bank's deposit variability as
measured by au than a quantitatively equal increase in demand deposits.
This argument can easily be extended to apply to any proportionate mixture of demand and time deposit change comprising a given change in total
deposit volume.
We have now completed our defense of the assumption that the
variability of the bank's average total deposit balance over the planning
period depends systematically on the structure of the bank's liabilities
and on the bank's size as measured by deposit volume.

It should be em-

phasized that the discussion has not led to specific conclusions regarding
either the directions or quantitative characteristics of these relationships.

Rather, the discussion has suggested that the nature of these

relationships depends on the exact form of the joint probability distribution of the bank's individual demand and time deposit account balances
in any given case, and on the manner in which changes in either DD or
TD affect this distribution.

Some of the examples given in developing

the discussion, however, were designed to suggest that under a wide
variety of specific conditions, deposit variability as measured by uu is
likely to increase with increases in deposit volume, but to increase at
a slower rate following a given increase in the bank's time deposit
stock than following an identical increase in the bank's demand deposit
stock.
We must now express the postulated relationships between deposit
variability, bank size, and liability structure in a form appropriate for

inclusion in the model.

In the preceding discussion, the absolute dis-

persion of the bank's average total deposit stock distribution was
measured by uu.

Due to the inventory theoretic character of subsequent

model construction, it is convenient to introduce an alternative measure
of the deposit variability risk that the bank faces.

This alternative

measure is the maximum range of possible variation in the bank's average
total deposit balance to which the bank attaches a nonzero probability.
We assume that the expected average total balance DD + TD is the midpoint
of this range, and we define the width of the range as 2K.

To illustrate,

if the expected average balance is $100 and K is $10, the probability
distribution of the bank's average total balance (i.e., the distribution
of the random variable DTCTAL defined by (25)) has limits $90 and $110
or, equivalently, the probability distribution of U has limits -$lO and
$10. As this illustration suggests, we assume that the range specified
by any given value of K is generally less than the widest conceivable
range of deposit variation.

In this respect it will be useful to think

of K as some multiple of uU.
Let us now specify K somewhat more formally. We define:
(28) K =IK[DD, TD].
As indicated above, K and -Kz5 are limits to the distribution of the
random variable U defined by (26). Equation (28) states that K is a
function of the bank's expected demand and time deposit balances.

There-

fore, in keeping with the discussion of (27), K, like au, is a function
of both bank size, as measured by the bank's total deposit volume DD + TD,
and liability structure, as measured by the relative magnitudes of DD and
TD.

We do not specify the explicit form of this function; however, we

25The limit -K is comparable to Richard C. Porter's "deposit low."
See Porter 123, pp. 37-441.

assume that the bank knows the form of the function precisely.26
This completes the ,discussionof deposit variability.

turn now to deposit costs.
Deposit costs
The same fundamental conceptual difficulty encountered earlier
with respect to lending costs arises in treating deposit costs.

That

is, because we have defined the dollar stocks of demand and time deposits
as bank decision variables, it is necessary to specify the functional
relationship between deposit costs and these stocks.

As in the case of'

loans, however, dollar volume is not, in general, the relevant "output"
on which all costs directly depend;27

We now outline a general procedure

for coping with this problem analytically.

Subsequently, we shall

specify the bank's cost functions for demand and time deposits.
The types of costs an actual bank incurs in attracting and
maintaining deposit accounts fall, roughly, into 4 categories:

(a) oper-

ating and service costs, (b) ,promotional and advertising costs, (c) explicit interest payments, and (d) service charges, a negative increment
to

costs.

In what follows, we shall ignore service charges.

Empirically,

service charges appear to be largely unrelated to actual bank costs or to
the factors underlying bank costs."

Further, for simplicity, we shall

26The discussion of equation (27) implied that the effects on
deposit variability of a given change in either DD or TD may vary depending
on whether the change in deposit volume results from changes in the average
balances of existing accounts, changes in the number of accounts the bank
holds, or some combination of the two. Therefore, the assumption that the
bank knows the explicit form of (28) implies that the manner in which a
given change in either DD or TD occurs is predetermined. We shall return
to this point in our discussion of deposit costs.
27See Broaddus [6, pp. 37-441.
28See Bell and Murphy [3].

group categories (b) and (c) above into a single category.

Operating

and service costs arise from the bank's deposit maintenance activities.
These costs depend on the technical characteristics of deposit service
production and on competitive conditions in markets for the basic factors
(labor and real capital) that the bank uses to produce these services.
Both promotional and explicit interest costs, on the other hand, arise
from the bank's efforts to attract deposits and reflect competitive conditions within deposit markets."

We now analyze each of these two

remaining cost categories in turn.
Operating and service costs.

In reality, the deposit services

banks provide individual depositors vary both qualitatively and quantitatively from one deposit account to the next.

Clearly, the qualitative

character of deposit services differs between deposit categories such as
demand and time deposits.

Further, service flows vary quantitatively

among individual deposit accounts within any deposit category due to
differences in account activity (i.e., variations in the number of credit
and debit transactions) and differences in account size,

In order to

abstract from these complexities, we continue to assume that all of the
bank's deposit accounts have identical expected average balances 5.

further assume that the levels of individual account activity are identical across all demand deposits and time deposits, respectively, and
that these activity levels are exogenous to the bank.3o
-

On these grounds,

29
It is of course true that banks can vary the level of account
services, and therefore service costs, as a competitive move to attract
deposits away from other institutions. As indicated below, however, we
shall assume the services supplied each depositor by the bank are exogenously determined.
3oFor an empirscal index of account activity see Benston [S,
pp. 515-5161.

the bank is assumed to produce one "unit" of identical demand deposit
services for each demand deposit account it holds over the planning
period, and, similarly, one unit of identical
each time deposit account.

time

deposit services for

The bank produces these service flows using

labor and capital inputs in accordance with the technical constraints of
production functions for both demand and time deposits.

For analytical

convenience we assume that these production functions are mutually independent:
produced.

that is, that demand and time deposit services are not jointly
Therefore, we can define the following short-run average

service cost function for deposit category I:

[NDi;
ito)co,,
AD.
, ii],
(“) ‘ii =ci*
1
i
where:
S' = the average service cost per "unit"
of deposit category 1 service
=Di
output,
ND

A
Di

= the total output of category i service units = the total number of
category i deposits,
= the bank's fixed stock of real capital,
= the (constant) wage rate, and
= an index of the identical account activity levels of category 1
deposits.

The preceding discussion of 'depositvariability indicated that the bank
controls its expected average demand and time deposit balances by influencing both the number and average size of individual accounts.

For

expository convenience, assume that the (identical) average balances of
individual accounts are exogenous to the bank.

The bank then controls

its expected average deposit stocks by acting to influence the number of
accounts it holds.

Therefore, ND

;

becomes the bank's deposit category i

control variable, and x becomes a model parameter as indicated by (29).
Assume also, again for convenience, that 3 equals unity.

If we now

define:
S
CDi

= the average service cost per dollar of the bank's deposit category
1 balance, and

= the expected average category i balance, our assumptions imply:

(30)

(31)

S'
'gi = =Di ;

NDi

Under these circumstances, we can substitute cs for cs'
Di
Di
ND in (29), obtaining:
i

and Di for

(32) c; = cs [Di; x, To, s , ii].
1
1
Di
Equation (32) expresses average service costs per dollar of category i
deposits as a function of the category 1 deposit stock held by the bank.
On the basis of these specifications, we may presume that a
service cost function of the general form (32) exists for both demand
and time deposits. We write these functions, respectively, as:

(33)

= ciD[DD; Zo, Go, XDD, 31;

(34)

ciD = c;,[TD; To, Yo, SD,

xl.

We do not specify the explicit form of either function. Total service
cost functions are then:

(35)

SCDD = ciD[DD](DD);

(36) SCTD = c;~[TD](TD).

Equations (35) and (36) will' enter the objective function of the model.
31
Promotional, advertising, and explicit interest expenses.
This category of costs arises not from deposit service production but
from the bank's attempt to attract deposit funds away from competing
financial institutions and money market instruments.

It is reasonable

to presume that these costs depend directly on the total dollar volume
of deposits the bank seeks to attract. We write the bank's average
promotional-interest cost function for deposit category i in general
form as:
(37) rD

= rD [D ; aD 1,
ii
1

where:
rDi

= average promotional-interest costs per dollar of category 1 deposits,
and

oDi = a vector of parameters'summarizing the competitive structure of the
category i deposit market within which the bank operates.
Equation (37) is analytically comparable to a factor supply function
facing a firm in standard theory.

The explicit form of (37) depends on

competitive conditions facing the bank in the category 1 deposit market.
These conditions are summarized by aD , which is exogenous to the bank.
i
The bank faces average promotional-interest cost functions of
the general form (37) for both demand and time deposits.

We write these

functions, respectively, as:
(38)

rDD = rDD[DD; aDDI;

3?The reader will recall that, in keeping with the generality
of the model, we are ignoring real world restrictions on explicit deposit
interest payments.

(39) rTD = rTD[TD; aTD].
Again, we do not specify the explicit form of either function. Total
promotional-interest costs for each category are then:
(40) RCDD = rDDIDDl(DD);

(41) RCTD = rTD[TD](TD).

Equations (40) and (41) will enter the objective function of the model.

The Expected Loss Function: Implicit Returns to Reserves and Secondary
Reserves and Implicit Deposit Costs

In reality, banks hold a variety of reserve assets (such as
vault casn and reserve balances at the central bank) that pay no explicit
return.

Banks also hold so-called "secondary reserve" assets (such as

short-term government securities) that commonly pay explicit returns well
below the yields available on other assets.

Banks typically hold both

reserve and secondary reserve assets in amounts that exceed legal requirements. Rational banks behave in this fashion because, in a world
of uncertainty, they attach positive economic value to the liquidity of
these assets.

Stated differently, reserve assets yield implicit flows

of income to the banks that hold them. Previous theoretical analyses of
individual bank behavior have employed several procedures to express this
implicit reserve asset return in analytically explicit form.

The most

powerful approach to this problem yet devised is the direct application
of formal inventory theory under conditions of uncertainty to a bank's
demand for reserves.
L

In what follows we shall expand on earlier work by
.-.
-.-+

applying the inventory approach to secondary as well as primary reserves,32
We shall also apply the approach to deposits and develop certain implicit
costs.

As indicated above, the bank of our model faces stochastic
variation in its demand and time deposit balances over the planning
horizon.

We assume that the bank is required by law to meet all deposit

withdrawals immediately with acceptable primary reserve assets.

Hence,

the bank is particularly concerned with the possibility of net deposit
outflows during the planning period.
possibility continuously.

Actual banks, of course, face this

For analytical convenience, the bank of the

model is assumed to face the threat of deposit withdrawal at only one
point in time, toward the end of the planning horizon.

We designate this

point in time the "moment of adjustment."
In an earlier section, the need to define the bank's decision
variables as average values over the course of the planning period was
indicated.

Consequently, the various random variables we have introduced

relating to security prices and the bank's demand and time deposit balances
were also defined as planning period averages.

For simplicity, we now

assume that, in the bank's view, the probability distribution of each
stochastic variable at the moment of adjustment is identical to the distribution of the average value of the variable over the entire planning
period.

That is, if the probability is .30 that the bank's average demand

deposit balance over the planning horizon will be $10 million, then the

32Throughout this section, the terms "reserves" and "primary
reserves" refer to assets (such as vault cash) that the bank can use to
meet deposit withdrawals directly. The term "secondary reserves" refers
to assets that the bank must first convert to primary reserves in meeting
deposit withdrawals. In the present model, the bank's secondary reserves
consist entirely of securities as defined earlier.

probability is also .30 that the instantaneous balance will be $10 million
at the moment of adjustment.
The reader will recall that the bank's expected average demand
and time deposit stocks are bank decision variables subject to bank control. With this specification in mind, we define a net deposit withdrawal
as deviation of the bank's total deposit balance below its expected level
(DD + TD) at the moment of adjustment.
net deposit withdrawal.

Suppose that the bank faces a

Stated differently but equivalently, suppose

that, at the moment of adjustment, the random variable U falls in the
negative portion of its range.

We assume that, faced with this situation,

the bank can fulfill its deposit obligations in one of four ways:

it can

(a) meet the withdrawal directly with primary reserve assets such as vault
cash, (b) sell securities in exchange for primary reserves, (c) borrow
primary reserves at the constant penalty rate n, or (d) employ some combination .of the above.

In general, the costs associated with each of

these alternatives differ.33
it incurs no loss or cost.
cost at the rate n.

If the bank uses primary reserves on hand,
If the bank borrows, it incurs a penalty

The cost of using security sales is unknown to the

bank due to the stochastic character of security prices.

We assume that

if, at the moment of adjustment, the price of a security is less than its
initial one dollar value (i.e., if the random variable w defined by (17)
falls in the negative portion of its range), the bank records a capital
34
loss at the rate of w percent per withdrawal dollar met by security sales.

33We refer here only to the direct and immediate costs of meeting
a deposit withdrawal.
34In contrast, it is assumed that the bank ignores the possibility
of capital gains from security sales when making its decisions. This asymmetry seems reasonable since most actual banks are probably more concerned
about the possibility of a capital loss than the possibility of a capital
gain
when they liquidate seCUritiesa

Therefore, security liquidation presents the bank with the possibility of
capital loss at an uncertain rate.
Let us assume that, in meeting a deposit withdrawal, the bank always
selects the least costly of the above alternatives first.

Therefore, the

bank always meets a deposit withdrawal initially with available primary
reserves. 35

If primary reserve stocks are insufficient to cover the entire

withdrawal, the bank meets the remainder by either selling securities or
borrowing, whichever is least expensive.

If the bank chooses to liquidate

securities before borrowing, but primary reserve and security holdings together are insufficient to meet the entire withdrawal, the bank must and
will resort to borrowing.
Since the bank is aware that a deposit withdrawal may occur at
the moment of adjustment, it must introduce an expression into its objective
function which captures, probabilistically, the possibility it may suffer a
penalty cost or capital loss flow.
expected loss function.

We designate this expression the bank"s

Although somewhat formidable at first glance, it is

a straightforward extension of similar functions employed in several of the
previous studies cited at the beginning of this paper.

We write the function

and then discuss its meaning in detail:
-P -R
(42)
EL[S,R,DD,TD] n(-U-R)+(w)B(U)dUdw

J’I
JJ
JJ

-a -K[DD, TD]
,d -El

-w(-U-R)$(w)B(U)dUdw

-n -(R+S(l-lw))
i -(R+S(l+w))

n(-U-(R+S(l+wj))3(w>a(u>ducw,

-n -K[DD, TD]

nrimzlrv

35If w 1 0, we assume the bank first meets a withdrawal with
~~CpII?IOP

where R is the bank's stock of primary reserves, 9(w) is the uniform
distribution of the random variable w, e(U) is the unspecified distribution of the random variable U, and all other symbols are as previously
36
defined.
Equation (42) can be explained most conveniently by considering
the three terms on the right side of the equation in turn.
terms involves integration with respect to both w and U.

Each of these

For each term,

the inner integration is with respect to U where U, as indicated earlier,
is the deviation of the bank's total deposit balance from the expected
value of the balance at the moment of adjustment. The outer integration
for each

term

is with respect to w, where w is the moment of adjustment

deviation of security prices from their expected value.

Recalling our

assumption that w varies over.the range -a I w L a, the three terms can
be most clearly interpreted as specifying the bank's expected loss due
to deposit withdrawal over particular portions of the range of w:

i.e.,

over particular portions of the range of possible security prices at the
moment of adjustment. Let us now indicate the meaning of these terms in
detail.
Suppose first that, at the moment of adjustment, (a) a net
deposit withdrawal occurs (i.e., U is in the negative portion of its
range), (b) security prices are below their expected value (i.e., w is
in the negative portion of its range), and (c) - w < -n.37

In this

case, the first term of (42) is relevant, as indicated by its range of
integration with respect to w.

With -w < -n, borrowing is less costly

36The distributions of w and U are assumed to be independent.
37Throughout this discussion we assume /al.> n.

per dollar than security liquidation.
will first give up primary reserves.

Under these conditions, the bank
If the withdrawal exhausts its

primary reserve stock, the bank will subsequently borrow.38

As the inte-

grand and inner range of integration for this term indicate, the bank
incurs penalty n for each dollar by which net withdrawals -U exceed the
39
bank's primary reserve holdings R.

The lower limit to the inner inte-

gration, U = -K, is the maximum withdrawal the bank faces under our
deposit variability assumptions. Actual integration over this term
yields the bank's expected loss due to deposit withdrawals for this
position of the range of w.
Consider now the last two terms of (42). These two terms
specify the bank's expected loss from net withdrawals for the portion
of the range of w where the algebraic value of w 1 -n.

Over this portion

of the range of w, security liquidation is less costly than borrowing to
meet deposit withdrawal obligations.
up primary reserves.

Here, the bank will initially give

If the withdrawal exhausts primary reserves, the

bank will then sell securities.

If the withdrawal exhausts both primary

and secondary reserves, the bank will borrow.

The second term of (42)

specifies the expected loss from security sales after primary reserves
are depleted, and the third term specifies the expected loss from supplemental borrowing should the withdrawal exhaust both primary and secondary
reserves.

381f w = -n, the bank is assumed to sell securities before borrowing. Therefore, because 'definiteintegration is defined over a closed
interval, specification of -,nas the upper limit of integration for the
first term of (42) is not strictly accurate. We ignore this minor difficulty.
39Throughout our discussion of (42) we shall be dealing with the
negative portion of the range of U, within which U < 0 and -U > 0.

Consider the second term.

The integrand and inner range of

integration for this term indicate that the bank incurs capital loss
penalty w for each dollar of deposit withdrawal exceeding the bank's
primary reserve stock, up to the point where the bank's security portfolio as well as its primary reserve stock is exhausted:
the point where U = -(R+S(liw)).40

i.e., up to

For this term, integration with

respect to w is restricted to the range -n FTJ L 0, because the bank
does not incur a capital loss from security liquidation if w > 0,
Consider now the third term.

The integrand and inner range

of integration for this term indicate that the bank incurs borrowing
penalty n for each withdrawal dollar exceeding (R+S(l+w)), where
(R+S(l+w)) specifies the value of the bank's reserve and secondary reserve balance at the moment of adjustment.

In contrast to the second

term, integration with respect to w here is over the range -n IW

L a,

because the bank incurs penalty n when.all primary and secondary reserves
are exhausted, regardless of the prevailing market price of securities.
To summarize, (a) actual integration of the three terms comprising (42) under any particular specifications of the probability
distribution a(U) and the function K = K[DD, TD], and (b) summation of
the results of these integrations yields the bank's total expected loss
due to the possibility of a net deposit withdrawal at the moment of
adjustment.

This total expected loss will enter the objective function

Of the model as a negative increment to the bank's expected change in
equity over the planning horizon.

40On the basis of our earlier assumptions, S(l+w) is the value
of the bank's total security portfolio at the moment of adjustment. Hence
(R+S(l+w)) is the value of the bank's primary and secondary reserve holdings
at the moment of adjustment.

We are now in a position to specify (a) the implicit returns
to primary reserves and secondary reserves and (b) the implicit costs
of demand and time deposits that arise from the possibility of a net
withdrawal at the moment of adjustment. As indicated by (42), the total
expected loss EL is a function of the four bank decision variables R, S,
DD, and TD.

Partial differentiation of (42) with respect to any one of

these variables indicates the marginal change in EL resulting from a
marginal adjustment of the decision variable in question.

We cannot,

in general, specify the signs of these partial derivatives.

That is,

we cannot generally indicate whether an increase in one of the decision
variables increases EL, decreases EL, or leaves EL unchanged.

The quali-

tative character of these effects depends in any given case on the explicit character of the unspecified functions 4(U)-and K[DD, TD] which
appear in (42). We can, however, make reasonable presumptions regarding
the respective directions of these effects that would be valid under a
wide variety of explicit specifications of e(U) and K[DD, TD].
Consider first the effects on EL of marginal changes in primary
aEL
It is
reserves and securities, as given by and aEL respectively.
'
aR
as’
reasonable to presume that, ceteris paribus, an increase in the stock
of either of these two assets would reduce the bank's expected loss due
to net deposit withdrawals at the moment of adjustment.

As an intuitive

justification for this presumption , consider the effect of a marginal
m
increase in the primary reserve stock R on each of the terms comprising
EL in (42). With K unchanged, an increase in R reduces the range of
integration with respect to U for both the first and the third terms.
That is, an increase in R (a) reduces the range of possible deposit
withdrawals over which such withdrawals force the bank to borrow and

(b) reduces the amount the bank would be forced to borrow to meet any

given net withdrawal. With respect to the second term of (42), an increase in R does not reduce the extent of the range of possible withdrawals over which the bank would be forced to borrow; however, an
increase in R shifts this range outward from the mean value of U 'to
encompass larger (and therefore, under a variety of reasonable specifications of e(U), less probable) net withdrawals.

To summarize, this

illustration suggests that a marginal increase in R would probably reduce the magnitudes of all three terms comprising EL and therefore reduce
EL.

A similar although more complicated argument can be given with

aEL
respect to as'

We now define:

3UT
(43) - F
= the implicit marginal return to primary reserves;
aEL

(44)- as = the implicit marginal return to securities.
On the basis of our presumption that F

< 0

and e

< 0, it follows

that the implicit marginal returns to reserves and securities are positive."
Consider now the effects on EL of marginal changes in the bank's
expected average demand and time deposit balances as given, respectively,
Our discussion in an earlier section indicated that we
by aEL and aEL
aDD
aTD*
can expect increases in either DD or TD to increase total deposit variability as measured by au and therefore by IKI, where K and -K are the

41The argument just given requires [R[cIKI and IR+S(~+~)I<IKI.
These conditions simply state that neither the bank's primary reserve
balance nor its total primary and secondary reserve balance exceeds the
maximum withdrawal that the bank considers possible. We shall indicate
below that where a local solution to the model exists, both of the marginal returns just defined must be positive.

limits to the distribution of U.

On these grounds, we can reasonably

presume that, ceteris paribus, an expansion of either DD or TD increases
the bank's expected loss due to net deposit withdrawals at the moment of
adjustment.

Let us now defend this last presumption. An increase in

either DD or TD has two distinguishable effects on the terms comprising
(42).

First, by increasing IKI with R and S constant, an increase in

either DD or TD extends the range of integration with respect to U for
the first and third terms.

That is, deposit expansion extends the range

of possible net withdrawals over which the bank must borrow.
fect tends to increase EL.
of e(u).42

This ef-

Second, increases in DD or TD alter the form

It is likely that the increased range of U resulting from

expanded deposit volume would reduce the probability of any given net
withdrawal.

Hence, this second effect of augmented deposit volume would

probably tend to reduce EL.

The total effect of an increase in DD or TD

on EL represents the net result of the two opposing effects just outlined.

We can reasonably presume that, under a variety of particular

specifications of K[DD, TD] and 6(U), the former effect would outweigh
the latter effect, with the result that increases in DD or TD would cause
EL to rise.
We define:
(45) gg

=I the implicit marginal cost of demand deposits;

(46) E

= the implicit marginal cost of time deposits.

42From our earlier discussion of deposit variability, the reader
will recall that the form of 8(U) depends on the joint distribution of the
individual account deviations ui and vi', and that the form of this joint
distribution changes with increases in deposit volume. Therefore, in
general, the form of O(U) varies with deposit volume. That is, e(U) is
itself a function of DD and TD.

Since we have presumed g
are positive.

> 0 and ?!& > 0, both of these marginal costs
aTD

One additional point should be made.

In our earlier dis-

cussion of deposit variability we concluded that the respective effects
of changes in (a) demand and (b) time deposit volume on deposit variability
faced by the bank differ.

In the present context, the implication of this

aK and -aK diverge.
conclusion is that the quantitative characteristics of aTD
aDD
Since differentiation of EL involves differentiation of K, it follows that,
aEL
aEL
in general, the quantitative characteristics of and diverge. That
aDD
aTD
is, the implicit costs of'demand and time deposits differ due to the nonidentical effects of changes in the respective balances of the two deposit
categories on deposit variability.

We shall return to this point below.

We have now specified (a) the bank's expected loss due to the
possibility of a net deposit withdrawal at the moment of adjustment and
(b) implicit marginal returns and costs that accrue to the bank as a
result of this expected loss.

This completes the development of the

objective function. The next section closes and solves the model.

III.

Solution of the Model

From the discussion in the preceding section, we can rewrite
objective function (8) in detailed form as:
(47) UAW)

= rL[L; aL, Z](L) + f@)

- 'DDIDD; aDDI

+ F,(s)

- rTD[TD; a,](TD)

- EL[S, R, DD, TD].
The bank seeks to maximize (47) subject to the balance sheet identity
constraint:

(48) L + B + S + R = DD + TD + NiJt-l.43

We perform the optimization using the standard Lagrangian technique.
Omitting model parameters for notational simplicity, the first-order
conditions for a local maximum are:
drL @I
dL (0 + r,[tl = X

a)
b)

FB = x

d)
(49)
4

- s
dCDDIDDl
dDD
(W

+ clDIDDl

BEL[S,R,DD,TD]. = x
3s

_ aEL[S,R,DD,TD]
aR

p x

+ aEL[S,R,DD,TD]
aDD

= )(

drDD
[EDI
+
dDD
+rDD[DD
3
1 [ (DD)

- s
dcTDITDI
dTD (TD) + czDITDl +

(TD) f rTD[TD]
+ aELfS,R,DD,TDj
aTD

= x

L+B+S+R-DD-TD-NiJtl=O.

Equations (49a)-(49g) form a system in the six decision variables and
the Lagrange multiplier X.

Solution of the system yields the bank's

desired average balance shee,tposition over the planning period, given
the values of the model's parameters and the explicit forms of the
various unspecified functions that appear in objective function (47).
The solution, in any given case, simultaneously establishes three

43A11 elements in the identity are expected planning period
averages. NWtel is the bank's equity at the beginning of the period.

fundamental characteristics of the bank's desired balance sheet:

(a) the

relative allocation of funds among alternative assets, (b) liability
structure, and (c) the scale of the bank's operations as measured by
total deposits.

The interdependence of these decisions is directly im-

plied by conditions (49). To illustrate, consider a model parameter
shift that alters the bank's desired demand deposit balance.

In general,

such a shift would directly affect the bank's optimal operating scale
and desired liability structure. Xoreover, because we have specified
the bank's expected loss due to deposit withdrawals as a function of
demand deposit volume, the marginal implicit returns to reserves and
securities which appear in conditions (49c) and (49d), and therefore
the bank's optimal security and reserve balances, are functions of
desired demand deposit volume.

Therefore, a change in the optimal

demand deposit stock would cause a corresponding change in the composition of the bank's desired asset portfolio.
System (49) possesses a clear economic interpretation. Considering each of the component equations in turn, the left side of (49a)
is the marginal expected net return on loans.

The left sides of (49b),

(494, and (49d) are the marginal returns to bonds, securities, and
reserves, respectively. The marginal return to reserves,

dEL
-Tr

consists

entirely of the implicit return derived and described in the preceding
section.

The marginal return to securities contains both an explicit

component, the coupon rate Fs, and the implicit component

aEL
-as

The

left sides of (49e) and (49f) are the marginal costs of demand and time
deposit balances, respectively.

In both cases, the first two terms are

the explicit marginal costs arising from service and promotional-interest
expenses, and the last term is the implicit marginal cost arising from

the expected loss function.

Equations (49a)-(49f) state that, at a

maximum, the total marginal 'returnof each asset and the total marginal
cost of each liability all equal X and hence are mutually equal.

That

is, the rational bank invests among alternative assets and acts to
attract funds of alternative liability form so as to equate, at the
margin, all return and cost flows that arise in connection with each
individual asset and liability during the planning period.

This is the

basic general result of the model and is analytically comparable to the
equilibrium conditions in the standard theory of the firm.
Given restrictions on the form of objective function (47),any
number of comparative statics experiments are conceivable.

Such experi-

ments would analyze the effects of specific parameter changes on the
solution of system (49).

In this connection, we might briefly indicate

the relationship of the present model to the money supply function literature by pointing out that the functional relationship between the solution
value for DD (the bank's desired demand deposit balance) and the parameters
of the model is a conceptually proper microsupply function for money.

Before proceeding to second-order conditions, we briefly note
certain additional characteristics of system (49). Since the bond return
TB is positive, (49b) implies A > 0.

With h > 0, (49d) implies that

where a local maximum exists, the implicit return to primary reserves,
aEL
must also be positive.
-Tic'

Further, if, as one would expect, the

coupon yield Fs on securities is less than 7 B, (49c) implies that the
dEL
implicit return to securities, - -,
as

is also positive.

44
See Kareken [16, pp. 1709-17101.

Second-order conditions insuring that the solution of (49) is
indeed a maximum can be expressed as restrictions on the algebraic signs
of a sequence of bordered Hessian determinants.

The determinants in

this sequence are of continuously increasing dimension and involve
second-order partial derivatives of the Lagrangian expression used to
solve the model.

We shall not attempt a full analysis of second-order

conditions; however, the first determinant of the sequence can be used
to derive several restrictions relevant to economic interpretation of
the model's solution.

This first determinant, of dimension 3, has the

following general form:
v

(50)

Iv1 =

ii v ij g i

Vji 'jj gi
gi

g.
J

where the V.. are second-order partial derivatives of the Lagrangian
=J
expression with respect to the ith and j th decision variables, i # j.
For a maximum, we must have IV1 > 0.

Since i and j may refer to any

decision variable, we can derive restrictions by selecting decision
variables in pairs and computing the resulting determinant IV]. Using
this procedure we have:

CL) + rLILl
dL

< 0.

Condition (51) states that, at a maximum, the marginal return on loans
must be a declining function of loan volume.

Similar conditions hold

for the marginal returns to reserves and securities. That is:

(52)

a [:f,-as

(53)

aEL
-- aR

---LJ

2s1
aEL

c 0.

< 0;
I
1

Further:
a
(54)

d&[DDl
dDD

(DD) + cDD[DDI
+ dr$DD1(~~)

f raD[DD] + $jj ' 0.
I

aDD
Condition (54) states that, !at a maximum, total marginal demand deposit
costs must be an increasing ~functionof demand deposit volume.

A similar

condition holds for time‘deiosits.
With these preliminary remarks concerning the solution, we
may develop the economic content of the solution in somewhat greater
detail.

Because of the model's generality, it possesses a variety of

implications regarding (a) the operational practices of actual banks and
(b) public regulatory policies toward banks.

Many of these implications

can only be derived by first restricting the model and introducing explicit assumptions concerning the form of its component functions.

Some

representative experiments of this nature are carried out in the broader
'45 The discussion below is confined to
study underlying this paper;
several results that follow,directly from first-order conditions (49).

Marginal Deposit Costs

In the preceding section, we postulated certain explicit and
implicit cost flows arising from the bank's deposit activities. We
divided the explicit cost flows for both demand and time deposits into

45See Broaddus [6, Chs. 4-61.

two components:
costs.

(a) operating-service costs and (b) promotional-interest

Further, we specified (3) implicit deposit costs arising from the

possibility of net deposit withdrawals during the planning period.

It is

well known that explicit interest rates paid by actual banks in the United
States on demand and time deposits, respectively, differ due to legal
restrictions on such payments.

Available data suggest that systematic

differences also exist between total service-promotional-interest outlays
by banks for the two types of deposits. That is, the sum of cost flows
(a) and (b) above generated by demand deposits differs systematically
46
from the corresponding flow generated by time deposits.

The present

model suggests several factors that might account for these differences,
47
including dissimilar competitive conditions in the two deposit markets
and characteristically different average account activity levels.

terms of the model, such divergences are captured by differences between
corresponding parainetersappearing in cost functions (33)-(34) and (38)-

(391, respectively.
Our model also suggests an additional factor that might account
for variations in deposit expenditures: namely, systematic differences
between demand and time deposit variability. To isolate this factor,
assume for the moment that (a) average service cost functions (33) and

46This difference has been established using data developed
through the Federal Reserve Functional Cost Analysis Program. Specifically, Klein [17, pp. 216-2171 cites 1967 data from this source which
indicate that, for 769 small banks, total cost rates net of service
charges but including all permitted interest payments averaged 1.6 percent for demand deposits and 4.3 percent for time deposits. Comparable
data compiled by the Federal Reserve Bank of Cleveland for small banks
in the Fourth Federal Reserve District during 1966 yielded rates of 2.2
percent and 4.0 percent, respectively.
47See Klein [17, p. 2171.

(34) are identical and (b) average promotional-interest cost functions
(38) and (39) are identical. That is:
(55)

ciD[DD] = @TD];

(56)

rDD [DoI 5 rTD[TD].

These assumptions eliminate the possibility of divergent costs due to
differences in deposit market structure or account activity.
Let us now focus our attention on the implicit marginal deposit
aEL
costs E
and aTD
aEL that appear in first-order condition equations (49e)
and (49f). It was suggested above that the variability of individual
demand deposit accounts held by the bank differs from the variability
of individual time deposit accounts.

To capture this distinction for-

mally, assume that each of the random variables ui defined by (19) is
uniformly distributed on the interval -k 1.ui I k, where k is a constant.
Assume further that each of the random variables via is uniformly distributed on the interval -pk 5 vi' I pk, where p is a constant, 0 < p < 1.
These assumptions imply that (a) all demand deposit accounts have identical
ranges of variation, (b) all time deposit accounts have identical ranges
of variation, and (c) the range of time deposit account variation is less
than the range of demand deposit account variation.

If, for convenience,

we continue to assume that all deposit accounts have mean balance % = 1,
it follows that the random variable U defined by (26) varies on the range:
(57) -k(DD + pTD) 2 IIL k(DD + pTD).
We have previously noted that K and -K are the limits to the distribution
of u.

Therefore, (57) specifies the form of the heretofore unspecified

function (28) as:
(58) K = k(DD + pTD).

Hence, we can now substitute the expression on the right side of (58)
for K in expected loss function (42).
aEL
aEL
In order to compare aDD and -,
aTD

it is necessary to specify

the form of the distribution e(U) that appears in the expected loss
function.

Let us pick two extreme alternatives.

individual deposit deviation variables u

First, if all of the

and v., are perfectly corre1

lated, then, using our assumptions in the preceding paragraph, U follows
the uniform distribution:
(59)

e(u)

= k

1
2k(DD + pTD)'

Alternatively, if the ui and v., are mutually independent, then U follows
1
the normal distribution:

(60)

em> -

where:

. %p

(61)

‘u -

[DD + P~TD]

-1.
VT
0

We can now derive the results we are seeking.

If we substi-

tute (a) the right side of (58) for K in expected loss function (42) and
(b) either the distribution (59) or the distribution (60) for Q(U) in
the same function, the composite-function rule for differentiation implies:
(62) E

> -aEL
aTD

48
for all values of DD and TD.

This result states that marginal increases

in the bank's demand deposit balance increase EL at a faster rate than
equivalent marginal increases in the bank's time deposit balance.

48This assertion is proved in Broaddus [6, appendix Cl.

This

result follows directly from the assumption in the present discussion
that demand deposits are less stable than time deposits.

Since (62)

holds under either of the extreme assumptions represented by (59) and
(601,

we can expect it to hold under a variety of other specifications

of e(u).
Where (62) holds, first-order conditions (49e)-(49f) imply:
dc;D[TD*]
(TD*) + cgD[TD*] +
c

dTD

drTD [TD*1
(TD*) + rTD[TD*]
dTD

+ cs [DD*] +
DD

drDD [DD*1
dDD

(DD*> + rDDIDD*l

I
where DD* and TD* are solution values for the bank's deposit decision
variables DD and TD, and the terms on the left and right sides of the
inequality are the marginal costs arising from service-promotionalinterest expenses for time and demand deposits, respectively.

In con-

junction with (55)-(56), result (63) states that, at an optimal balance
sheet position, a bank operating under the conditions outlined in this
section would be willing to incur higher marginal service-promotionalinterest expenses for time deposits than for demand deposits, even where
the underlying service and promotional-interest cost functions characterizing the two deposit categories are identical.

This result follows

directly from the assumption that the bank anticipates greater stability
in its time deposit accounts than in its demand deposit accounts during
the planning period.

As implied by (62), the greater stability of time

deposit accounts means that time deposits present the bank with a less
compelling inducement to hold primary and secondary reserve assets having
low yields.

In this sense, time deposits are more "productive" than

demand deposits from the standpoint of the bank.

Consequently, the bank

can afford to incur higher marginal costs to attract them.

Mondeposit Sources of Funds

To this point, the sources of bank funds have been restricted
to demand and time deposits.

This is obviously unrealistic. Banks in

the real world obtain funds from a variety of nondeposit sources.

recent years these sources have included commercial paper issued through
holding company affiliates, a variety of other domestic financial instruments, and Eurodollar borrowings.

For our purposes, the distinguishing

characteristic of these liabilities is that, in contrast to deposits,
they do not generally present the risk of unanticipated withdrawal.

For

simplicity, we group these nondeposit liabilities under the heading
"borrowed funds" and denote this liability category by the symbol.BF.
It is assumed that the bank is not required to repay funds in this
category until some point in time following the close of the planning
period.

BF can be treated as an additional bank decision variable.

For

convenience, we assume that the only expense the bank incurs in obtaining
borrowed funds is an explicit interest charge paid to the lender. We
further assume that the average interest charge is a function of the
amount borrowed:
(64) rBF = rBF[BF; aBFls

where aBF is a vector of parameters summarizing competitive conditions
facing the bank in the market or markets for borrowed funds. Total
costs of borrowed funds are then:
(65)

RCBF = rBF[BF](BF).

It is a simple matter to add borrowed funds to the model by adding (65)
as a negative increment to objective function (47) and constraint (48).
Solution of the augmented model yields first-order conditions consisting
of system (49) plus the additional equation:

(66)

drBF [BFI
(BF) + rBF[BF]
dBF
I

= h.

The left side of (66) is simply the marginal cost of borrowed funds.
Together with the original first-order conditions (49), (66) implies
that to maximize its planning period return the bank assumed nondeposit
liabilities up to the point where their marginal cost equals the marginal
cost of deposit liabilities and the marginal return to assets.
Adding borrowed funds to the model produces two interesting
results.

First, from (66), (49e), and (63), we have:

(BF*) + rBF[BF*]

drTD[TD*1

dctD[TD*]
(67)

dTD

1.
1

(TD*) + rTD[TD*]

(TD*) + ciD[TD*] +
dTD

S
dCDDIDD*l
drDDIDD*l
(DD) + rDD[DD*]
(DD*) + ciD[DD*] +
dDD
dDD

This result states that the bank is willing to pay more in interest
charges for borrowed funds at the margin than it is willing to pay in
service-promotional-interest outlays for either time or demand deposits.
The bank accepts higher marginal costs for nondeposit liabilities because
funds derived from these liabilities cannot be withdrawn during the planning
period and therefore do not contribute to the expected losses specified by

(42).

Inequality

(67) states that, at a maximum, the marginal costs

that the bank actually pays out for alternative liabilities stand in
inverse relation to the marginal contribution of each liability to expected losses.
Second, the introduction of borrowed funds changes the optimal
scale of the bank's operations.

This can be seen by studying the first-

order conditions before and after the introduction of borrowed funds.
Because the bond return ?, is constant and BF does not enter any firstorder equation of the augmented solution other than (66), the two solutions are identical except that in the augmented solution optimal bond
holdings increase by an amount equal to the volume of borrowed funds
added to the balance sheet.

In addition to the change in scale, this

result also implies that the optimal ratio of total loan and investment
assets (L* + B*) to total reserve and secondary reserve assets (S* + R*)
increases, a result consistent with the reduced withdrawal risk per
dollar of total liabilities. In this sense, the introduction of nondeposit liabilities is similar to a technical innovation in the standard
theory of the firm.

Lendinp Behavior of the Bank

Students of banking and bank regulatory agencies are particularly
concerned with bank lending activity because bank loans constitute a significant portion of total credit available to individual consumers and
small business firms.

The conditions that determine the volume of bank

lending are of obvious interest to policymakers, since it may be possible
to affect bank lending by influencing these conditions.

We can use first-order conditions (49) to derive the determinants of a bank's desired loan volume in the context of the present
model.

Conditions (49a) and (49b) indicate that the bank allocates

available resources to loans up to the point where marginal loan revenue
equals the constant bond return Yb.

This condition is depicted

graphically by Figure 2, where the downward sloping curve is the bank's
Expected
Return

Marginal Net
on Loans

Loans
FIGURE 2
marginal loan revenue, and the horizontal line represents the constant
bond yield.

The bank's des%red loan volume L* is established by the

intersection of these two lines.

Figure 2 implies that L* can be altered

(a) by policies that influence FB or (b) by policies that affect the

The reader will note that no decision variable other than L
appears in first-order conditions (49a)-(49b). Therefore, changes in
the optimal scale of the bank's operations occasioned by changes in desired liability stocks have no effect on the bank's desired loan volume.
This result follows from the assumption that bonds are in perfectly elastic
supply to the bank.

parameters of the marginal loan revenue curve and hence the position of
the curve.
As an example, consider a policy that might affect the bank's
lending activity by influencing the parameter Z in the marginal loan
revenue function. The reader will recall from the discussion of (10)
that S specifies the default risk characteristics of the bank's loan
customers.

Consider a bank facing loan applications for the purpose of

home improvements from several isolated potential borrowers, all of whom
reside in a given low income neighborhood. The bank is likely to consider
the default risk associated with these applications relatively high and
scale its lending accordingly.

Under these circumstances, several alter-

native government policies might alter the bank's assessment of the risk
it would incur by granting the loans.

Obviously, the bank's risk would

decline if a government agency agreed to insure the loans.

As an alterna-

tive to loan insurance, a policy might be designed to coordinate rehabilitation throughout the neighborhood.

Such a policy, by reducing externalities,

might increase the probability that individual home improvements would
produce increased property values.

Under these conditions, the bank might

consider the default risk associated with individual loan applications less
than in the absence of such a policy.

In terms of the model, the result

would be a change in the parameter -Z, an upward shift of the marginal loan
revenue function, and an increased volume of lending,50
-

50Broaddus [6, Ch. 61 analyzes in detail the effects of policies
designed to influence competitive conditions in loan markets as represented
by the parameter aL in (11).

IV.

Conclusion

In this paper we have constructed and solved a general, static
Under the assumption

model of individual bank balance sheet management.

that the bank acts to maximize the return to equity, solution of the
model indicated that the external'conditions specified by the model's
parameters simultaneously determine the bank's desired asset and liability structures and the optimal scale of bank operations.

The

interdependence of these decisions resulted largely, although not entirely, from two related aspects of the model's construction:

(a) the

fact that the risk of net deposit withdrawals during the planning
period, as measured by K, is functionally dependent on total deposit
volume and deposit structure, and (b) the fact that the bank's expected
loss due to the possibility of withdrawals is functionally dependent on
both deposit volume and the bank's reserve and secondary reserve balances.
The model obviously has limited operational value in its
present highly abstract form.

It would have to be modified extensively

to serve as the basis for detailed analysis of particular banking issues.
The model has the virture, however, of treating a number of diverse bank
decisions within a unified analytical framework.

Further, the model

demonstrates that these decisions are realted, at least in principle,
on the basis of generally accepted optimization criteria.

Only recently

has the individual bank as an economic unit begun to receive the microtheoretic attention it deserves in view of its pivotal role in modern
economies.

It is hoped that the model developed here may suggest a

useful approach to further research in this field.

APPENDIX

The development of the model in this paper excluded an important
element of uncertainty faced by actual banks:
garding the volume of future loan demand.

namely, uncertainty re-

In this paper we introduced

a net loan revenue function similar to the demand function of standard
theory.

In constructing this function, we assumed that the bank extends

loans to individual customers in a predetermined sequence up to some
point where it ceases lending.1

This approach was useful due to its

similarity to the treatment of demand in the standard theory of the nonfinancial firm.
In reality, however, banks usually attempt to meet as many

reasonable requests for loans as possible, particularly from established
customers.

In this connection, actual bankers use the term "liquidity"

to refer to a bank's ability to meet unanticipated loan demand as well
as unanticipated deposit losses.

Since our model is a stochastic theory

of individual bank behavior, it is necessary to consider how the model
might be altered to permit explicit treatment of uncertain loan demand
under the assumption that the bank seeks to meet all or nearly all loan
requests.

This appendix develops a procedure for incorporating uncertain

loan demand in the bank's objective function and indicates the effect of
this modification on the model's solution.
We assume that the model construction in the paper remains in
effect except for the portion pertaining to bank lending.2

For simplicity,

1Solution of the model indicated that the bank ceases lending at
the point where marginal loan revenue equals the constant bond rate.
2
See pp. 11-17.

we continue to assume that all loans outstanding on the day preceding
the beginning of the planning period mature on that date, and that all
noninterest loan terms including loan size are identical across loans
and exogenous to the bank.

The new assumptions are as follows.

the bank faces a finite set of borrowers.

First,

Second, the bank attempts to

satisfy all loan requests received during the planning period.

Third,

the loan demand of each borrower is, from the bank's standpoint, a
random variable.
We write the loan demand of the ith borrower as:
n

(Al) L; = BL i+
,

gi,

i =: 1,

.. ..

NL'

where NL is the number of borrowers, and gi is a random variable having
zero mean but following an otherwise unspecified probability distribution.
It follows that BL i is the'amount the bank expects the i
,
demand during the planning period.

borrower to

Aggregate.planning period loan demand

is then:
N
CA21

S
trDOTAL=ifrCBL,i + S-j.1

D
where LTOTAL is a random variable.

Because we have not specified the

form of the joint probability distribution of the gi, we cannot specify
D
/
the distribution of LTOTAL., We can, however, define the mean of the

L is the bank's loan decision variable under the new specifications.
The bank controls L by inducing changes in the individual BL i through
,
loan rate manipulation.

That is, L is a function of the loan rate, rL:

(A4) L = L[rL; a,],

where aL is a parameter summarizing the competitive structure of the loan
market.

In constructing the modified objective function, it will be

convenient to treat rL as a function of L.

We assume L[rL; aL] is monot-

onic decreasing and write its inverse as:

(A51 rL = rL[L; aLlo
Expected total loan revenue is then:
(A6)

ERL = rL[L; a,](L).

Equation (A6) will enter the modified objective function.
The reader has undoubtedly recognized the similarity of the
above specifications to the treatment of deposit variability in the body
of the paper.3

In a manner also similar to that treatment we define:

N
(A7) G = CL g..
i=l '
G is the random deviation of the aggregate demand for the bank's loans
from its mean value L.

If G is in the positive portion of its range,

the bank faces unanticipated loan demand; if G is in the negative portion
of its range, loan demand is less than expected.

Like the individual gi,

G has zero mean; however, we cannot specify the form of its distribution
further. G is comparable to the deposit deviation variable U.

It is

assumed that the distribution of G has limits H and -H and that these
limits are functionally related to expected loan volume.
(~8) H = H[L].

3See pp. 20-31.

That is:

The limit variable H is comparable to the limit variable K in the treatment of deposit variability,
We assume that all unanticipated loan requests are presented
to the bank at the same "moment of adjustment" at which unexpected deposit
withdrawals occur.

We further assume that, at this moment, the bank first

satisfies all deposit withdrawals in the manner described in the paper.4
Once this is accomplished, the bank follows an identical procedure to
satisfy unanticipated loan demand.5

That is, after all deposit withdrawals

are met, the bank first uses any remaining reserves to make loans.

reserves are exhausted, the bank meets whatever loan demand remains by
either selling securities or borrowing, whichever is least costly.

the basis of these assumptions, we can write an expected loss function
which captures the bank's expected loss due to unanticipated loan demand.
This function can then be added to the expected loss function for random
deposit deviation in the objective function of the model.

The expression

is algebraically complicated because, under our assumption that the bank
meets unanticipated deposit withdrawals before meeting unexpected loan
demand, the expression must take account of possible movements in three
random variables:

w (random security price deviation), U (random deposit

deviation), and G (random loan demand deviation).

Nonetheless, the ex-

pression merely extends the logic used to develop the expected loss
function for random deposit flows to random movements in loan demand.

4
See pp. 37-38,
5
We assume that if G is in the negative portion of its range,
so that loan demand is less than expected, the bank costlessly shifts
a portion of the funds it had planned to use for lending to securities
or other assets.

The expression is:
-n

(A9)

EL[L,S,R,DD,TD] =

H[Ll
J

n(G-(R+U))$(w)o(U)$(G)dwdUdG

WI
J 1
-a -R

x+u

nG$(w)B(IJ)J,(G)dwdUdG

/
+JJJ
JJJ
0

(R+U)+S(l*l)
-w(G-(K+U))$(w)B(U)$(G)dwdUdG

-n -R R+U
a

H&l

-n -R

nG( -((R+U)+S(l+w))) $(w)B(U)$(G)dwdUdG

(R-kU)+S(l+w)

-wGQ(w)8(U)$(G)dwdUdG

( )Ji:“‘)
s(l)( (
n G

R+S 1-I-w

R+U +

(R+U)+S(lhi))) Q(w)~(U)J,(G)~~CCX~G

-i-w

H[LI
nG+(w)3(U)$(G)dwdUdG,
0

where all variables are as previously defined, and IL(G) is the unspecified distribution of the loan demand deviation variable G.

The explanation of expected loss function (A9) is similar to
the explanation of expected loss function (42).6 Each term of (A9) gives
the adjustment cost the bank incurs in meeting unanticipated loan demand
when the three random variables w, U, and G fall in specified portions
of their respective ranges.
The first two terms cover the case where -w c -n.

Under these

circumstances, the bank prefers to meet unanticipated loan demand by
borrowing rather than by liquidating securities.

If deposit withdrawals

do not exhaust all primary reserves, so that some reserves are left over
to meet unexpected loan demand, the first term is relevant.

On the other

hand, if deposit withdrawals exhaust all reserves, the second term is
relevant.
The last five terms as a group cover the case where security
liquidation is less costly than borrowing.
bank sells securities prior to borrowing.

Under these conditions, the
The third and fourth terms

are relevant where deposit withdrawals do not exhaust primary reserves.
Under these circumstances, the bank first meets unanticipated loan demand
by exhausting primary reserves that remain.
securities.

Subsequently, the bank sells

If this security sale does not exhaust the bank's stock of

securities, the third term is relevant.

If securities are exhausted, the

bank must then borrow to meet remaining loan demand, and the fourth term
is relevant.

The fifth and sixth terms cover the case where deposit with-

drawals exhaust primary reserves and consume part but not all of the bank's
securities.

The fifth term is relevant where enough securities remain to

cover unanticipated loan demand.

6See pp. 38-45.

The sixth term is relevant where loan

demand exhausts remaining securities, forcing the bank to borrow.
Finally, the seventh term covers the case where the bank depletes its
entire stocks of primary reserves and securities in meeting deposit outflows, making it necessary to meet all unexpected loan demand through
borrowing.
We can now reformulate the bank's objective function on the
basis of the modified specifications introduced in this appendix.

Under

our new assumptions the function becomes:
(AlO)

E(ANW) = rL[L; aLI

+ yB(B) + Fs(s)

- ciD[DD;;iDD,El (DD)- c;~[TD;qD, %(TD)
- rDD[DD; aDDI

- rTD[TD; aTD](TD)

- ELD[S,R,DD,TD] -

ELL[L,S,R,DD,TD],

where ELD is the expected loss due to unanticipated deposit withdrawals,
and ELL 'is the expected loss due to unanticipated loan demand. 7

Maxi-

mization of (AlO) subject to the balance sheet identity constraint yields
the following modified first-order conditions:

drLLI
dL

(L) i- rL[L]

-’

aELDIS,R,DD,TD1
as

_ aELL[L,S,R,DD,TD] = x
as

_ aELD[S,R,DD.TDL
aR

_ 2ELL[L,S.R,DD,TDL = 1
aR

d)
S

- aELLrL~s;~*DD*TD~= 1

dcDzL:D](D~) + GD[~o]

drDD[DD]
dJJJ) (DD) + rDDIDDl

7
All variables appearing in (AJ.0)are as previously defined.

+ aELD[S,R,DD,TD]
aDD

dc;DiTDl
f>

dTD

(TD)

(Am

+ cGD[TD]

= x

ETDITDI
dTD (‘JW + r&T31

+ aELD[S,R,DD,TDL
aTD

+ %ELLIL,S,R,DD,TD1
aDD

+ 3ELL[L,S,R,DD,TD]
aTD

L+B+S+R-DD-TD-N!J

= x

=0
t-l

These conditions are identical to conditions (49) except for
the addition of partial derivatives of the new expected loss function
ELL with respect to the various decision variables.

The economic content

of these derivatives is similar to that of the corresponding derivatives
of ELD.8

L
That is, each derivative of EL indicates the marginal change

in the bank's expected loss due to unanticipated loan demand that results
from a marginal change in one of the bank's decision variables.

Hence,

these derivatives can be viewed as marginal revenues and costs just as
the derivatives of ELD were viewed as marginal revenues and costs.

There-

fore, the modified first-order conditions (All) yield the same broad
result as the original conditions (49):

that is, the profit-maximizing

bank selects the balance sheet position that equates the marginal revenues
and costs associated with the various assets and liabilities the bank holds.
The modified conditions merely incorporate within this result the marginal
revenues and costs arising from the bank's expected loss due to unanticipated loan demand.
This appendix has demonstrated how the model of this paper can
be altered to take account of uncertain loan demand.
accomplished at the cost

of increased complexity.

8See pp. 41-45.

The modification was

Nonetheless, it should

be clear that the model is fully capable of dealing with this important
aspect of actual bank operations.

LIST OF SYNBOLS*

Bank Decision Variables

average total bond balance

average "borrowed funds" balance

expected average total demand deposit balance

average total loan balance

average total reserve balance

expected average total securities balance

expected average total time deposit balance

Other Variables and Parameters

a,-a

limits to the distribution of w

'DD

an index of demand deposit account activity

'TD

an index of time deposit account activity

BDD,i

expected average balance of the ith individual demand
deposit account

BL, i Cappendix)

expected ith borrower loan demand

'TD,i'

th
expected average balance of the i'
individual time
deposit account

'DD

average total service-maintenance and promotionalinterest costs per demand deposit dollar

S
'DD

average service-maintenance costs per demand deposit
dollar
average lending cost

*This list is restricted to principal variables and parameters.
The word "average" is used in two senses in these definitions. Where the
symbol denotes a stock, the word means average quantity over the planning
period. (See pp. 6-8 ) Where the symbol denotes a flow, the word is used
in the usual sense of economic theory to refer to average flow per relevant
unit: for example, average loan return per loan dollar.

CrD

average total service-maintenance and promotionalinterest costs per time deposit dollar
average service-maintenance costs per time deposit
dollar

expected average default rate on loans

expected loss due to unanticipated deposit withdrawals

ELD (appendix)

expected loss due to unanticipated deposit withdrawals

ELL

expected loss due to unanticipated loan demand

G (appendix)

random "moment of adjustment" deviation of the bank's
total loan demand from its expected value

g, (appendix)

random "moment of adjustment" deviation of the i th
borrower's loan demand from its expected value

H,-H (appendix)

limits to the distribution of G

K,-K

limits to the distribution of U

k,-k

limits to the distributions of the random variables ui

penalty rate for reserve deficiencies

pS
iF
B
'BF

average price of an individual‘security
constant coupon rate paid on bonds
interest cost of "borrowed funds"

rDD

average promotional, advertising, and explicit interest
expenses per demand deposit dollar

expected average net rate of return on loans

average gross contract rate on loans

constant coupon rate paid on securities

rTD

average promotional, advertising, and explicit interest
expenses per time deposit dollar

random "moment of adjustment" deviation of the bank's
average total deposit balance from its expected average
value

random "moment of adjustment" deviation of the ith
demand deposit account balance from its expected
average value

74
Vi'

random "moment of adjustment" deviation of the 1..th
time deposit account balance from its expected average
value

random "moment of adjustment" deviation of Ps from
its expected average value

a vector of parameters specifying the default risk
characteristics of the bank's customers

aBF

a vector of parameters summarizing the competitive
structure of the market for borrowed funds

aDD

a vector of parameters summarizing the competitive
structure of the bank's demand deposit market

a vector of parameters summarizing the competitive
structure of the bank's loan market

OLTD

a vector of parameters summarizing the competitive
structure of the bank's time deposit market

standard deviation of U

REFERENCES

American Bankers Association.
Englewood Cliffs, N. J.:

The Commercial Banking Industry.
Prentice-Hall, Inc., 1962.

Arrow, Kenneth J.; Karlin, Samuel; and Scarf, Herbert, eds. Studies
in the Mathematical Theory of Inventory and Production. Stanford,
Calif.: Stanford University Press, 1958.

Bell, Frederick W., and Murphy, Neil B. "Bank Service Charges and
costs." The National Banking Review, IV (June 1967), 449-457.

Bell, Frederick W., and Murphy, Neil B. Costs in Commercial Banking:
A Quantitative Analysis of Bank Behavior and Its Relation to
Bank Repulation. Boston: Federal Reserve Bank of Boston, 1968.

Benston, George J. "Economies of Scale and Marginal Costs in Banking."
The National Banking Review, II (June 1965), 507-549.

Broaddus, Alfred. "A Stochastic Model of Individual Bank Behavior."
Unpublished Ph.D. dissertation, Indiana University, 1972.

Brunner, Karl. tA Schema for the Supply Theory of Money."
national Economic Review, II (January 1961), 79-109.

Dewald, William G., and Dreese, G. Richard. "Bank Behavior With
Respect to Deposit Variability." Journal of Finance, XXV
(September 1970), 869-879.

Edgeworth, F. Y. "The Mathematical Theory of Banking," Journal of
the Royal Statistical Society, LI (1888), 113-127.

Inter-

10.

Gramley, Lyle E. "Deposit Instability at Individual Banks." Essays
on Commercial Banking. Kansas City: Federal Reserve Bank of
Kansas City, 1962.

11.

Hester, Donald D. "An Empirical Examination of a Commercial Bank
Loan Offer Function." Yale Economic Essays, II (Spring 1962),
3-57.

12.

Hicks, John R. Critical Essays in Monetary Theory.
University Press, 1967.

13.

Hodgman, Donald R. Commercial Bank Loan and Investment Policy.
Champaign, Illinois: University of Illinois Bureau of Business
and Economic Research, 1963.

14.

Jaffe, Dwight M., and Modigliani, France. "A Theory and Test of
Credit Rationing." American Economic Review, LIX (December 1969),
851-859.

London:

Oxford

15.

Kane, Edward J., and Malkiel, Burton G. "Bank Portfolio Allocation,
Deposit Variability and the Availability Doctrine." Quarterly
Journal of Economics, LXXIX (February 1965), 113-134.

16.

Kareken, John H. "Commercial Banks and the Supply of Money: A
Market-Determined Demand Deposit Rate." Federal Reserve
Bulletin, LIII (October 1967), 1699-1712.

17.

Klein, Michael A. "A Theory of the Banking Firm."
Credit and Banking, III (May 1971), 205-218.

18.

Klein, Michael A. "Imperfect Asset Elasticity and Portfolio Theory."
American Economic Review, LX (June 1970), 491-494.

19.

Klein, Michael A., and Murphy, Neil B. "The Pricing of Bank Deposits:
A Theoretical and Empirical Analysis." Journal of Financial and
Quantitative Analysis, VI (March 1971), 747-761.

20.

Morrison, George R. Liquidity Preferences of Commercial Banks.
The University of Chicago Press, 1966.

21.

Morrison, George R., and Selden, Richard T. Time Deposit Growth and
the Employment of Bank Funds. Chicago: Association of Reserve
City Bankers, 1965.

22.

Orr, Daniel, and Mellon, W. G. "Stochastic Reserve Losses and Expansion of Bank Credit." American Economic Review, LI (September
1961), 614-623.

23.

Porter, Richard C. "A Model of Bank Portfolio Selection." Yale
Economic Essays, I (Fall 1961), 323-359.

24.

Rangarajan, C. 'Deposit Variability in Individual Banks."
Banking Review, IV (September 1966), 61-71.

25.

Struble, Frederick M., and Wilkerson, Carroll H. "Bank Size and Deposit
Variability." Monthly Review, Federal Reserve Bank of Kansas City,
November-December 1967, 3-9.

Journal of Money,

Chicago:

The National

Full text of Working Papers (Federal Reserve Bank of Richmond) : A General Model of Bank Decisions, Working Paper 74-6

FRASER