Oligopoly,ReactionFunctions,and MonetaryPolicy

��

��

１．Introduction

It has long been recognized that assuming the market structure

surrounding commercial banks to be perfectly competitive does not

properly represent reality（see Klein １９７１; Prisman, Slovin, and

Sushka １９８６; and Freixas and Rochet １９９７）. Berger and Hannan

（１９８９）, among others, have found statistical relationships between the

concentration ratios（as measured, e.g., by the Herfindahl index）in

deposit markets and deposit interest rates, and between the ratios in

loan markets and the corresponding interest rates.

Concerning the formal, theoretical models, on the other hand, the

analysis of relationships among bank market structure, the level of in-

terest rates, and the effectiveness of monetary control has also a fairly

long tradition, as is summarized in Alhadeff（１９６７）for the earlier

work and Freixas and Rochet（１９９７）for more recent work. Some

past decades have seen a resurgence of interest in this area,

prompted by such work as Tobin（１９６９）and Brunner and Meltzer

（１９７２）. See, for example, Startz（１９８３）, which deals with a situation

where the deposit market is in Chamberlinian monopolistic competi-

Article

Oligopoly, Reaction Functions, andMonetary Policy

Masanori AMANO

千葉大学経済研究第２５巻第２号（２０１０年９月）

（２３９）３１

tion, focusing on the effect of deposit rate ceilings on money demand

and so on, while leaving the loan（security）rate to be a parameter or

competitively determined.

This paper constructs a general equilibrium model of financial mar-

kets where the deposit and loan markets are both in oligopolistic com-

petition, and banks are assumed to entertain one of the three kinds of

conjectural variations. This model setting seems plausible and appro-

priate because the two market structures are closely related to the

number of banking firms in those markets. Also, since the previous lit-

erature in this area deals only with extreme market structures（pure

monopoly or perfect competition）or with partial equilibrium frame-

works, a further examination of a general equilibrium model with oli-

gopolistic interactions within the markets surrounding commercial

banks appears a necessary and interesting step to be taken next.

Our discussion proceeds in the following manner. In the next sec-

tion（Section２）I describe profit maximizing behavior of a representa-

tive bank, which faces the markets for loans and deposits character-

ized by Cournot, Bertrand, or market-share oligopoly, and which

therefore entertains corresponding conjectural variations. Throughout

the paper, the number of banks in deposit and loan markets is treated

as a parameter, corresponding to licencing and chartering require-

ments that are implemented, e.g., in the U.S. and Japan, and corre-

sponding to an obvious fact that adjustments of the bank number in

response to changing bank profits will take a long time.

Section３ examines the effects on the endogenous variables of

changes in policy and other parameters. The most interesting results

from the following discussion would be that, in the short- or medium-

Oligopoly, Reaction Functions, and Monetary Policy

３２（２４０）

run horizon, if the banks’marginal cost is constant（i.e. if their tech-

nology exhibits constant returns to scale）, then the change in the

number（or the size of the whole industry）of Bertrand or market−

share banks has no consequences on the size of total money supply,

the money multiplier, and rates of interest on securities and deposits.

Only when the technology exhibits decreasing returns so that banks’

marginal cost is increasing, does the larger bank number give rise to a

lower security rate, a higher deposit rate, as well as larger money sup-

ply and the money multiplier.

Section４concludes the paper with a summary and some remarks.

２．Bank behavior under three types of oligopoly

Before modeling profit maximizing behavior of an individual bank,

let us look at the accounting framework for financial markets which is

relevant to our banks, and which will help us visualize the general

equilibrium nature of the asset markets.

Financial assets are assumed to consist of high-powered money and

homogeneous deposits and securities, the last of which include bank

lendings to the nonbank public. The price level of physical capital and

Table１. The Accounting Framework

Sectors

Assets

Cetnral

bank

Commercial

banks

Nonbank

public

Exogenous

supply

High-powered money -R−Mp ＋R（＝＋nr）＋Mp ０Deposits ０－D（＝－nd）＋D ０Securities ＋Sc ＋Sb（＝＋ns）－Sp ０Physical capital ００＋K ＋K

Net worth ００＋A ＋A＝＋K


（２４１）３３

commodities is assumed constant and is set equal to unity. In the ta-

ble, entries with upper-case letters are aggregative amounts, n is the

number of banks（a parameter）, and r, d, and s are the amounts for

individual banks. As is described later on, all banks are assumed to be

of the same size. A plus（minus）sign before each quantity implies

that it is an asset（a liability）for the sector. The subscripts c, b, and p

stand for the central bank, commercial banks, and the nonbank public,

respectively. For R and D, the subscripts are omitted for later conven-

ience.

The vertical sum in each sector represents the net worth of that

sector. Hence, for example, the balance sheet of the central bank

reads

Sc＝R＋Mp.

The horizontal sum over the three sectors for each asset means the

excess demand for the asset if each entry is seen as an ex ante quan-

tity. Entry０ in the exogenous supply of securities reflects the current

assumption that government bonds are assumed away.１）

The profitsπof the representative bank are written as

π＝s・rs（Sp）－d・rd（D）－c（d）, （１）

where s is the demand for securities（supply of funds）and d is the

supply of deposits（demand for funds）both by the bank, rs（・）is the

（inverse）demand function for securities, with rs′＜０（recall that the

demand for securities includes the supply of bank loans）, rd（・）is the

（inverse）supply function of deposits, with rd′＞０. The function c（・）

１）On assuming that private and government bonds are homogeneous, assum-ing away the latter is not harmful.


３４（２４２）

is the cost of managing the assets and liabilities for the bank, whose

balance sheet has size d（see Table１）.２）

It is assumed that

c′＞０, c″≧０. （２）

c″＝０（resp.＞０）corresponds to the bank’s constant-returns-to-scale

（resp. decreasing-returns-to-scale）technology regarding its factors of

production.３）

In the short-run or medium-run horizon which our analysis is con-

cerned with, the banks will normally have some fixed factors of pro-

duction, a typical example being a capital stock. Hence, increasing re-

turns to scale with c″＜０can plausibly be ruled out from the following

discussion.

From the market equilibrium for securities which is shown in the

table,

２）The literature on this kind of cost functions is extensively reviewed in Bal-tensperger（１９８０）.

３）Here I briefly show that if the bank’s technology has constant returns（resp.decreasing returns）to scale, then its cost function has a constant（resp. in-creasing）marginal cost. Assume now that the factors of production are de-noted as u and v, and that their constant rates of remuneration are w and x,respectively. Output is represented by deposits d, which is a size of the bal-ance sheet.Then one hasαt d＝f（αu ,αv）,α＞０, where f is the production function.When t＝１, then f exhibits constant returns to scale（CR）, while if t＜１, fshows decreasing returns to scale（DR）. Write c０＝wu＋xv. Let us assumethat u and v are increased byβ times. Then, under CR, output and cost in-crease byβ times, implying that c is proportional to d and marginal cost isconstant. Under DR, when u and v increase byβ times, output increases byless thanβ times. Hence if the bank wants to increase its output byβ times,it needs to increase inputs byγ times, whereγ＞β. With this increase, thecost has to be increased toγc０which is larger thanβ c０, so that the cost hasincreased more than proportionally. This means that under DR, the marginalcost is increasing, i.e., c″＞０.


（２４３）３５

Sp＝Sb＋Sc. （３）

It is assumed that the sizes of individual banks are similar enough and

can be regarded as identical, so that

Sb＝ns.

This then gives

Sp＝ns＋Sc.

Assume also that cash holdings of the commercial bank consist only

of legally required reserves, i.e. that

r＝kd,

where k（０＜k＜１）is the required reserve ratio which is applied to all

the homogeneous deposits. Then, the bank’s balance sheet becomes

（１－k）d＝s. （４）

From（１）,（３）, and（４）, the Lagrangean L for the bank’s optimality

can be written as

L＝srs（Sb＋Sc）－drd（D）－c（d）＋λ［（１－k）d－s］,

whereλ is a Lagrangean multiplier. In the following, the first and sec-

ond order conditions for profit maximization of the representative

bank are derived for each of the three conjectural variation cases.

（i） Cournot conjectural variation

The Cournot assumption on conjectural variations implies that

dSb

ds＝１, （５）

namely, the bank conjectures that its variations of security demand

are not followed by other banks. For a similar reason, regarding the

deposit supply,


３６（２４４）

dDdd＝１. （６）

The marginal revenue of security purchase is, by using Cournot as-

sumption（５）,

d［srs（・）］ds

＝rs＋srs′dSb

ds＝rs＋srs′, （７）

while the marginal cost of deposit supply is, from（６）,

d［drd（・）］ds

＋c′＝rd＋drd′dDdd＋c′＝rd＋drd′＋c′ （８）

Hence, writing�L／�s＝Ls and so on, the first order conditions for

maximization ofπare

Ls＝rs（ns＋Sc）＋srs′（ns＋Sc）－λ＝０,

Ld＝－rd（nd）－drd′（nd）－c′（d）＋λ（１－k）＝０,

��

（９）

Lλ＝（１－k）d－s＝０,

where Sb＝ns and D＝nd are used for later comparative static analysis.

The multiplierλ denotes the shadow price of bank funds, which also

equals the marginal revenue from security purchase; hereafter,λ is

assumed to be positive.

The second order condition is that the following bordered Hessian

be positive:��

０

－１

１－k

－１

nσ＋rs′

０

１－k

０

－（nδ＋rd′＋c″）

��

＞０, （１０）

whereσ≡rs′＋srs″andδ≡rd′＋drd″. It will be assumed throughout that

σ＜０ andδ＞０. Assumptionσ＜０ implies that the bank’s marginal

revenue from security purchase should decrease when other banks in-

crease their security holdings, with securities s of the bank held con-


（２４５）３７

stant. A similar condition is assumed in Hahn（１９６２）in another con-

text, and is called a generalized Hahn condition by Dixit（１９８６）in dis-

cussing product oligopoly models. Regarding the marginal cost of de-

posit supply, assumptionδ＞０can be interpreted in a similar way.

Upon expansion, the second order condition（１０）becomes

Hc≡nδ＋rd′＋c″－（１－k）２（nσ＋rs′）＞０, （１１）

which is ensured under the current assumptions thatδ＞０,σ＜０; rd′＞

０, rs′＜０; and c″≧０.

（ii） Bertrand conjectural variation

Here, the representative bank conjectures that other banks’interest

rates will remain unchanged when it changes its security holdings s .

Hence it is the same as the bank’s conjecture under the（purely）com-

petitive environment. In notation, the Bertrand conjectures for the

two markets are written as

dSb

ds＝０, dD

dd＝０.

Hence, for the same profit function and the Lagrange expression, the

marginal revenue from security purchase and the marginal cost of de-

posit supply are rs and rd＋c′, respectively（see the left-hand equality

of（７）and（８））. The first order conditions are, therefore,

rs（ns＋Sc）－μ＝０,

－rd（nd）－c′（d）＋μ（１－k）＝０,

��

（９）

（１－k）d－s＝０,

whereμ is a Lagrangean multiplier attached to the balance sheet.

The second order condition is that


３８（２４６）

��

０

－１

１－k

－１

nrs′

０

１－k

０

－nrd′－c″

��

＞０,

or, upon expansion,

Hb≡nrd′＋c″－（１－k）２nrs′＞０,

which holds under our current assumptions so far made.

（iii） Market−share conjectural variation

In this case the representative bank conjectures that its change of s

or d will induce an equiproportionate changes in rivals’s or d. Hence

if the bank increases its security by s, the increase in rivals’security

is（n－１）s ; or the total increase is ns（including the bank’s）. An ex-

actly similar conjecture is applied to variations in deposits. In notation,

the market-share conjectures are

dSb

ds＝n ,

dDdd＝n

Hence, in view of（７）and（８）the above conjectures yield the mar-

ginal revenue rs＋srs′n and the marginal cost rd＋drd′n＋c′. Writing the

Lagrangean multiplier asν, the first order conditions for the bank are,

therefore,

rs（ns＋Sc）＋srs′（ns＋Sc）n－ν＝０,

－rd（nd）－drd′（nd）n－c′（d）＋ν（１－k）＝０,

（１－k）d－s＝０.

The second order condition for profit maximization is��

０

－１

１－k

－１

nσ′

０

１－k

０

－nδ′－c″

��

＞０,


（２４７）３９

where the generalized Hahn conditions in this case,σ′＜０, andδ′＞０４）,

are assumed, and the above inequality reduces to

Hm≡nδ′＋c″－（１－k）２nσ′＞０.

This relationship holds under our current assumptions.

３．Bank market structure and monetary policy

In this section I am concerned with the impacts on security rate rs,

deposit rate rd, their difference rs－rd, money supply M, money multi-

plier m（M and m are defined shortly）, and bank profitsπ, of changes

in policy and other parameters, for the three alternative conjectural

variations, and compare the response patterns of the endogenous vari-

ables among those three modes.

Total money supply M is defined as usual by

M≡Mp＋D＝Sp,

where the second equality follows from the balance sheet of the non-

blank public along with A（net worth of the nonblank public）＝K.

Then, by recalling the equilibrium for securities（３）,

M＝Sb＋Sc＝ns＋Sc. （１４）

Also, high-powered money R＋Mp equals Sc by the balance sheet of the

central bank. Therefore, money multiplier m, defined as the ratio of

money supply to high-powered money, can be written as

m＝ns＋Sc

Sc（１５）

From here onward, I describe and compare the comparative statics

results from the three conjectural variation cases. Total differentiation

４）The detailed expressions areσ′＝２γs′＋nsrs″andδ′＝２rd′＋ndrd″.


４０（２４８）

of the first order conditions（９）for the Cournot case yields��

０

－１

１－k

－１

nσ＋rs′

０

１－k

０

－（nδ＋rd′＋c″）

��

��

dλ

ds

dd

��

＝

��

ddk

－σdSc－sσdn

dδdn＋λdk＋c′vdv

��

,

（１６）

where dv＞０, with c′v＞０, in the last row denotes an exogenous in-

crease in the marginal cost of bank operation due, for instance, to a

rise in wage or material costs. Alternatively, c′v＜０ can be viewed as

technical progress in bank operation.

Our first target is to find the effect of open market purchase, dSc

（＞０）. It is noted that, in view of the central bank’s balance sheet, the

open market purchase is accompanied by extra supply of high-pow-

ered money of the same amount. Applying Cramer’s rule to（１６）one

finds

�s�Sc＝（１－k）２σ

Hc＜０,

where Hc＞０from（１１）. Therefore,

�rs

�Sc＝rs′（n

�s�Sc＋１）＝rs′

Hc［nδ＋nrd′－nrs（１－k）２＋c″］＜０, （１７C）

where C in the equation number implies it refers to the Cournot case,

while B and M in equation numbers to appear below mean they refer

to Bertrand and market-share oligopoly cases, respectively.

For the Bertrand case, using the matrix equations shown in Appen-

dix（A）, it can be seen that

�s�Sc＝（１－k）２rs′

Hb＜０. �rs

�Sc＝rs′

Hb（nrd′＋c″）＜０. （１７B）

All the appendices are omitted from the paper but they are available


（２４９）４１

from the author upon request. However, for the market-share case

one has

�s�Sc＝１

Hm（１－k）２（rs′＋nsrs″）,

which has an uncertain sign. But substituting the above into the mid-

dle term of（１７C）, one finds

�rs

�Sc＝rs′

Hm［nδ′－nrs′（１－k）２＋c″］＜０. （１７M）

implying again that open market purchase reduces the security rate

of interest. As for the effects on money supply and the money multi-

plier, referring to（１４）and（１７C）one finds for the Cournot case that

�M�Sc＝n�s�Sc＋１＞０. （１８C）

Also, from（１５）,

�m�Sc＝１

Sc２（nSc

�s�Sc－ns）＜０. （１９C）

Thus, the extra supply of high-powered money with open market op-

eration lowers security rate rs, increases total money supply M, and re-

duces money multiplier m. The last result occurs because commercial

banks’security holdings are reduced with an increase in Sc.

The sign patterns in（１８C）and（１９C）apply to the other conjec-

tural variations as well, i.e.,

�M�Sc＞０, �m

�Sc＜０. （１９B, M）

For the first inequality in the Bertrand case, see Appendix（A）, and

for the same inequality in the market-share case, see Appendix（B）.

To seek for the effect on the deposit rate for the Cournot case, I


４２（２５０）

first find from（１６）,

�nd

�Sc＝n�d�Sc＝n（１－k）σ

Hc＜０.

Hence,

�rd

�Sc＝rd′

�nd�Sc＜０. （２０C）

When the banks have other conjectural modes, the same pattern is

obtained:

�rd�Sc＜０; （２０B, M）

however, the market-share case again needs a linear security demand

function for the above definite sign.

The effect on the rate differential rs－rd, which is approximately the

net interest revenue per bank funds,５）cannot be signed definitely for

the Cournot case, but if rs（・）and rd（・）are linear（i.e. rs″＝rd″＝０）,

then one has

�（rs－rd）�Sc

＝１Hc［rs′｛rd′－（１－k）２rs′＋c″｝＋rd′nkrs′］＜０（２１C）

implying that the extra supply of high-powered money narrows the

rate differential. The other conjectural modes yield the same sign pat-

terns as above, for general rs and rd functions in the Bertrand case, but

only for linear rs and rd functions in the market-share case:

�（rs－rd）�Sc

＜０. （２１B, M）

５）If the net interest revenue per bank funds isρ, using（４）one hasρ≡（rss－rdd）／d＝（rss／d）－rd＝（１－k）rs－rd～～rs－rd, where ～～ implies‘approximatelyequals.’


（２５１）４３

See Appendix（A）for a detailed derivation of（２１B）, and Appendix

（B）for more description of（２１M）.

I next turn to how the endogenous variables will react to an in-

crease in bank number n . As is seen below, only the manners the

changes in n affect the endogenous variables depend on the cost

structure of banks. Starting as before with the Cournot case, one finds

�s�n＝１－k

Hc［（１－k）sσ－dδ］＜０.

Hence,６）

�ns�n＝s＋n

�s�n＝ s

Hc［rd′－（１－k）２rs′＋c″］＞０.

Recall that c″≧０ is assumed throughout. Therefore,

�rs

�n＝rs′

�ns�n＜０. （２２C）

The Bertrand and market-share conjectures, on the other hand,

yield

�rs

�n＝rs′�ns�n～（rs′sc″）≦０ according as c″≧０, （２２B, M）

where‘～’implies‘has the same sign as.’See Appendix（A）for

more detailed derivation. The above two results are counter-intuitive

because a larger bank number brings about the lower security rate

when the marginal cost is increasing. They occur because, under in-

creasing marginal cost, increasing n entails a smaller decrease in s for

the banks to get maximum profits in a new situation, so that ns in-

creases and rs declines. Note that, in Bertrand and market-share cases

６）In deriving the second equality, the bank’s balance sheet（４）is used.


４４（２５２）

（but not in the Cournot case）, if the banks have a constant-returns-to-

scale technology, so that c″＝０, then changes in bank number n leave

the security rate unchanged.

Also, from（１４）and（１５）it is immediate for the Cournot case that

�M�n＝�ns�n＞０, �m

�n＝１

Sc

�ns�n＞０（if c″≧０）. （２３C）

The other two cases can be seen in parallel manners:

�M�n＝�ns�n～sc″≧０according as c″≧０,

��

（２３B, M）�m�n＝１

Sc

�ns�n～sc″≧０according as c″≧０.

See Appendix（A）for detailed derivations of the above relations.

It is interesting to note that when the banks have Bertrand or mar-

ket-share conjectures, the larger bank number results in the larger

money supply and money multiplier, only when their marginal cost of

operation is increasing. The economics behind this result may be ex-

plained in the same way as for（２２B, M）. In other words, if the bank-

ing technology shows constant returns to scale regarding its inputs, so

that the cost function（the independent variable being the amount of

deposits as a proxy for production levels）is linear in its deposits, the

banking market size does not affect security and deposit interest rates

（see also（２４B, M））, money supply, and the money multiplier.

Concerning the effect on the deposit rate, one finds for the Cournot

case that

�d�n＝１

Hc［（１－k）sσ－dδ］＜０,

and


（２５３）４５

�nd�n＝d＋n

�d�n＝ d

Hc［rd′－（１－k）２rs′＋c″］＞０（if c″＞０）.

When the Bertrand or market-share conjecture is entertained, refer-

ring to the Appendix（B）one finds

�rd

�n＝rd′dc″≧０according as c″≧０. （２４B, M）

As for the rate differential rs－rd in the Cournot case, it follows di-

rectly from（２２C）and（２４C）that

�（rs－rd）�n

＜０（if c″≧０）, （２５C）

which implies that an increase in the bank number reduces the net in-

terest rate, if c″≧０.７）The other two conjectural variations yield, from

（２２B, M）and（２３B, M）,

�（rs－rd）�n

≦０according as c″≧０, （２５B, M）

i.e., increasing bank number n reduces（resp. leaves unchanged）the

rate differential if the marginal cost is increasing（resp. constant）.

The third parameter I am concerned with is the required reserve

ratio k .

To start with the Cournot case, an increase in k yields

�s�k＝－１

Hc［d（nδ＋rd′＋c″＋λ（１－k）］＜０,

and so,

�rs

�k＝nrs′

�s�k＞０. （２６C）

７）See, e.g., Heggestad and Mingo（１９７６）for corresponding empirical evidence.


４６（２５４）

Also,

�M�k＝n�s�k＜０and �s

�k＝n

Sc

�s�k＜０. （２７C）

A larger k results in the contraction not only of the total money sup-

ply but also of the money multiplier for Cournot banks.

If the banks entertain Bertrand or market-share conjecture, one ob-

tains

�rs

�k＞０. （２６B, M）

Hence it follows that

�M�k＜０and �m

�k＜０（２７B, M）

The effect on the deposit rate cannot be signed for Cournot banks

because

�d�k＝－１

Hc［λ＋（１－k）d（nσ＋rs′）］＞―＜０; i.e.

�rd

�k＞―＜０.

Further, for the Bertrand conjectural form, it has an uncertain sign,

but for the market-share case it becomes definite（see the matrix

equations in Appendices（A）and（B））, i.e.,

�rd

�k＞―＜０（２８B）; �rd

�k＞０. （２８M）

As for the effects on the rate differential, the three conjectural

forms yield８）

�（rs－rd）�k

＞０. （２９C, B, M）


（２５５）４７

implying that the tight monetary policy with larger reserve ratio k

tends to widen the rate differential, which is in accordance with simi-

lar policy via open market sale（dSc＜０）of securities; see（２１C）.（Note

that the Cournot and the market-share cases require the linear rs and

rd functions for the definite sign; see note８and Appendix B.）

The next objective is to search for the consequence of an autono-

mous increase in the marginal cost of bank operation, c′v＜０（or, of

technical progress occurring in bank operation, c′v＜０）. Starting with

the Cournot and market-share cases, one finds

�s�v＝－１

J（１－k）c′v＜０（J＝Hc, Hm）.

while, for the Bertrand case,

�s�v＝－１

Hb（nrd′＋c″）c′v＜０;

Hence, for all the three cases,

�rs

�v＝rs′n

�s�v＞０, �M

�v＝n�s�v＜０, and �m

�v＝n

Sc

�s�v＜０. （３０C, B, M）

On the other hand, irrespective of the conjectural forms,

�d�v＝－１

Jc′v＜０（J＝Hc, Hb, Hm）,

�rd

�v＝rd′n

�d�v＜０, （３１C, B, M）

and the effect on the net interest rate for a unit fund is obviously

８）The detailed expression for the Cournot case is, assuming that the rs and rd

functions are linear,�（rs－rd）�k

＝１Hc［（n＋１）rs′－rd′n（s－d）－rs′ndc″＋nλ｛rd′－（１－k）rs′｝］＞０.

See Appendix（A and B）for other conjectural patterns. In the market-sharecase also, this effect does not have a definite sign unless the rs and rd func-tions are linear（i.e. unless rs″＝rd″＝０）.


４８（２５６）

�（rs－rd）�v

＞０.

Irrespective of which conjecture the bank entertains, with an increase

in the marginal cost of operation, it will guard itself by raising the se-

curity rate and reducing the deposit rate（i.e. contracting the size of

its balance sheet）, thereby bringing about the falls in both total

money supply and the money multiplier.

How is the level of representative bank’s profits affected when each

parameter changes？ This is our next and final concern. The argu-

ment is very similar among the three conjectural forms, so it can be

dealt with in a unified way. Although the maximization ofπ is subject

to balance sheet（４）, substituting s of（４）into（１）leaves us an uncon-

strained maximization problem. Then, from（１）where s＝（１－k）d,

writing x for either Sc, k , n , or v,

dπdx＝（�π�s

�s�d＋�π�d）�d

�x＋�π�x

＝�π�x

by the envelope theorem（see, e.g., Varian１９９２）. This means that the

partial derivative ofπ with respect to each parameter shows the di-

rection of change inπwhen the parameter increases. Thus,

�π�Sc＝srs′＜０, and

�π�k＝�π�s

�s�k＝ω（－d）＜０

whereω＝λ,μ, orν, and where use is made of the first equations of

the three first order conditions（（９）etc.）,（４）, and the assumption

thatω＞０.９）Also,

９）In the first equation of the optimization regarding the Lagrangean form foreach conjectural pattern,ω（＝λ,μ, or v）is the marginal revenue of securityholding.


（２５７）４９

�π�n＝s２rs′－d２rd′＜０, and

�π�v＝－cv. （３３C, B, M）

It remains to find the sign of the second relationship. I have assumed

that

０＜c′v＝�２c／�d�v＝�２c／�v�d＝dcv／dd,

or that dcv／dd＞０. Then, one has

cv＝∫d ＊

d

dcv

dddd＝∫d ＊

ddcv, （３４C, B, M）

where d ＊ is an optimal level of d after an increase in v occurs, and d

is some minimum d（＞０）. Since the integrand in the above relation

（dcv／dd）is positive, it follows that cv＞０. Hence in（３３C, B, M）,

�π�v＜０, （３５C, B, M）

which means that with an increase in its marginal cost, the bank’s

profits will diminish in spite of its reaction in terms of price- and quan-

tity-adjustments.

４．Conclusions

This paper has set up and examined a general equilibrium model of

Table２. Effects of Policy and Parameter Changes

Var’s rs rd rs－rd M m π

Para’s（C）（B）（M）（C）（B）（M）（C）（B）（M）（C）（B）（M）（C）（B）（M）（C）（B）（M）Sc ― ― ― ― ― －l －ll ― －ll ＋＋＋l ― ― －l ― ― ―

n― －０－０＋＋０＋０ ― －０－０＋＋０＋０＋＋０＋０ ― ― ―

＠＠＠＠＠＠＠＠＠＠k ＋＋＋？？？＋ll ＋＋ll ― ― ― ― ― ― ― ― ―

v ＋＋＋ ― ― ― ＋＋＋ ― ― ― ― ― ― ― ― ―

Notes:１）In the second row,（C）,（B）, and（M）refer to Cournot, Bertrand, and market-share conjec-tural modes, respectively.２）l : in the case of a linear rs function.３）ll : in the case of linear rs and rd

functions.４）＠: according as c″≧０.


５０（２５８）

financial markets where commercial banks are close enough in size

and they entertain one of the Cournot-, Bertrand-, or market-share-

type conjectural variations for their change in security demand and

deposit supply. The profit maximizing bank’s reactions to external pol-

icy and other parameter changes are summarized in Table ２. The

sign ＋ or － indicates the direction of change in the relevant variable

following an increase in the parameter.

Among the signs listed in the table, the most interesting ones are

probably those that arise from increases in bank number n , which im-

ply that if the Bertrand- or market-share-bank’s production function

exhibits constant returns to scale, so that the cost function is linear in

its scale of operation（i.e., c″＝０）, then the change in the number of

banks（or the scale of the bank markets）has no effect on the security

and deposit rates of interest, total money supply, and the money mul-

tiplier. If the bank’s technology obeys decreasing returns（c″＞０）, in-

creasing bank number n（or larger size of banking markets）leads to

lower security rate rs, higher deposite rate rd, a lower rate margin,

larger money supply M as well as larger money multiplier m. Those

latter results also apply when the banks have a Cournot conjectural

variation irrespective of whether c″＞０or c″＝０. The above analytical

findings will be worth the subject of empirical inquiries.

Secondly, in the three conjectural forms, easy monetary policy（an

increase in Sc or a fall in k）lowers the interest rate margin, although

one needs linear security demand and deposit supply functions in the

Cournot and market-share cases to have this effect. Third, the money

multiplier becomes smaller with expansionary open market operations,

but goes larger with a smaller reserve ratio for all the congectural


（２５９）５１

forms. Fourth, as for the effects of the two expansionary policies on

bank’s profits, they are reduced by open market purchases, but are

made larger by a smaller reserve ratio. Fifth and finally, an exogenous

increase in the marginal cost of bank operation raises the security

rate, lowers the deposit rate, enlarging the interest rate margin, and

reduces money supply, the money multiplier, as well as bank profits.

Needless to say, technical progress in the form of smaller marginal

cost brings about the effects that are opposite in direction to the

above.

Our inquiry so far, however, did not include a similar examination of

a stationary state where the bank number becomes endogenous

through entry and exit activity and bank profits are reduced to zero.

In view of the recent trends still going on in many countries toward

further liberalization of financial environments, focusing on this situ-

ation may be an interesting item for future research.

References

Alhadeff, D.A.,“Monopolistic Competition and Banking Markets.”In Monopolistic Competition

Theory: Studies in Impact, edited by R.E. Kuenne. New York: John Wiley,１９６７.

Baltensperger, E.“Alternative Approaches to the Theory of the Banking Firm.”Journal of

Monetary Economics１（January１９８０）,１―３７.

Berger, A.N. and T.H. Hannan.“The Price Concentration Relationship in Banking.”Review of

Economics and Statistics７１（May１９８９）,２９１―２９９.

Brunner, K. and A. Meltzer.“Money, Debt, and Economic Activity.”Journal of Political Econ-

omy８０（September-October１９７２）,９５１―９７７.

Dixit, A.“Comparative Statics for Oligopoly.”International Economic Review ２７（February

１９８６）,１０７―１２２.

Freixas, X. and J.-C. Rochet. Microeconomics of Banking. Cambridge, Mass.: The MIT Press,


５２（２６０）

１９９７.

Hahn, F.H.“The Stability of Cournot Oligopoly Solution.”Review of Economic Studies２９（Oc-

tober１９６２）,３２９―３３１.

Heggestad, A.A., and J.J. Mingo.“Prices, nonprices, and concentration in commercial bank-

ing.”Journal of Money, Credit, and Banking８（February１９７６）,１０７―１１７.

Prisman E.Z., M.B. Slovin, and M.E. Sushka.“A general model of the banking firm under

conditions of monopoly, uncertainty, and recourse.”Journal of Monetary Economics １７

（March１９８６）,２９３―３０４.

Startz, R.“Competition and interest rate ceilings in commercial banking.”Quarterly Journal

of Economics９７（May１９８３）,２５５―２６５.

Tobin, J.“A general equilibrium approach to monetary theory.”Journal of Money, Credit,

and Banking１（February１９６９）１５―２９.

Varian, H.R. Microeconomic Analysis, Third Ed. New York: Norton,１９９２.

（Received: August２５,２０１０）


（２６１）５３

Summary


Masanori AMANO

This paper examines a general equilibrium model of financial mar-

kets where the commercial banks act as either Cournot, Bertrand, or

market-share oligopolists. Based on the banks’profit maximization, I

study the effects of changes in monetary policy tools, the bank num-

ber, and technology in bank operation, on the rates of interest, money

supply, the money multiplier, and bank profits. I found, among others,

that when the banks act as Bertrand or market-share oligopolists and

their technology shows constant returns to scale, so that their cost

functions have constant marginal cost, then money supply, the money

multiplier, and rates of interest become independent of the bank num-

ber or of the size of financial markets.

Summary

２９２（５００）

Documents

Oligopoly,ReactionFunctions,and MonetaryPolicy