!!!!!!! !!! !!!!!!! !!! 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, and Monetary Policy Masanori AMANO 千葉大学 経済研究 第25巻第2号(2010年9月) (239) 31
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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
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.
Oligopoly, Reaction Functions, and Monetary Policy
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.
千葉大学 経済研究 第25巻第2号(2010年9月)
(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,
Oligopoly, Reaction Functions, and Monetary Policy
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-
千葉大学 経済研究 第25巻第2号(2010年9月)
(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
Oligopoly, Reaction Functions, and Monetary Policy
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,
千葉大学 経済研究 第25巻第2号(2010年9月)
(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
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).
Oligopoly, Reaction Functions, and Monetary Policy
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-
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.
千葉大学 経済研究 第25巻第2号(2010年9月)
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�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
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.
Oligopoly, Reaction Functions, and Monetary Policy
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