ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E19-11

Klein and Monti meet Weyl and Fabinger:
Imperfect competition in the banking sector
byTakanori Adachi

September 2019

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Klein and Monti meet Weyl and Fabinger: Imperfectcompetition in the banking sector∗
Takanori Adachi†
September 9, 2019

Abstract
The Monti-Klein model of monopolistic banking is extended by Weyl and Fabinger’s(2013) conduct parameter approach to include oligopolistic banking. By considering imperfect competition in loan and deposit markets, I show that whether an increase in theconduct parameter in one market also raises the equilibrium rate in the other market depends on whether the marginal cost of loan/deposit increases or decrease as the amountof the other “product” increases. Specifically, if the cross partial derivative of the costfunction is positive, then an increase in the conduct parameter in the loan market lowers
the deposit rate, whereas an increase in the conduct parameter in the deposit market
raises the lending rate.
Keywords: Banking Industry; Imperfect Competition; Conduct Parameter Approach.
JEL classification: D43; G21; L13.

1Introduction

In this note, I study the Monti-Klein model of monopolistic banking in the loan and thedeposit markets (Klein 1971; Monti 1972; Dermine 1986; Gunji and Miyazaki 2019) by using
Weyl and Fabinger’s (2013) conduct parameter approach to consider oligopolistic banking inboth loan and deposit markets. Banks “produce” loan and deposit services for firms andconsumers by incurring production cost. This is captured by the standard cost functions withtwo outputs. It is shown that the cross partial derivative of the cost function is key: if it is∗

I am grateful to a Grant-in-Aid for Scientific Research (C) (18K01567) from the Japan Society of thePromotion of Science. All remaining errors are my own.
†School of Economics, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan. E-mail:adachi.t@soec.nagoya-u.ac.jp

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positive (resp. negative), then an increase in the conduct parameter in the loan market lowers(resp. raises) the deposit rate, whereas an increase in the conduct parameter in the deposit
market raises (resp. lowers) the lending rate.

2ModelI follow Freixas and Rocht’s (2008, Ch. 3) exposition of the Monti-Klein model. The demand
for loans is described by L(rL ), which is downward sloping, whereas the supply of deposits
from households is given by an upward sloping function, D(rD ). The banking industry as
a whole provides deposit and loan services; its production technology is captured by a costfunction C(D, L), which satisfies the regularity conditions. Thus, the representative bank has
the following profit function:π(rL , rD ) = (rL − r)L(rL ) + [(1 − α)r − rD ]D(rD ) − C[D(rD ), L(rL )],where r ≥ 0 is the fixed rate on the interbank market, and α ∈ [0, 1) is the rate for compulsory
reserves. To study the equilibrium pricing, consider a small increase in the lending rate, rL .
Then, imperfectly competitive banks recognize that they capture only 100 × θL percent of theprofit gain by raising the lending rate, where θL ∈ [0, 1] is the conduct parameter for the lending
market, measuring the degree of imperfect competition in lending, with θL = 1 being monopolyand θL = 0 being perfect competition. Let ∆rL > 0 and ∆L < 0 be the increase in rL and theassociate change in the amount of lending, respectively. Then, given the equilibrium deposit∗
and the amount of deposits D∗ ,the bank equates the marginal profit gain and the loss
rate rDfrom raising rL (see Figure 1):∂C· (−∆L) = (rL − r)(−∆L) .
θL ∆rL L +{z}∂L{z} | Marginal|Loss
Marginal Gain

Similarly, given the equilibrium lending rate rL∗ and the amount of loans D∗ , the bank also
equates the marginal gain and the marginal loss from lower the savings rate in the deposit
market (see Figure 1):∂CθD (−∆rD )D +· (−∆D) = [(1 − α)r − rD ](−∆D) .
{z}|{z ∂D} |Marginal LossMarginal Gain

∗Then, formally, the equilibrium rates (rL∗ , rD) satisfy: ∗
∗(rL − r) − ∂C
[D(rD), L(rL∗ )]θL∂L=∗rLL (rL∗ )2

Figure 1: Bank’s Pricing Incentivesand∂C∗∗(1 − α)r − rD− ∂D[D(rD), L(rL∗ )]θD=,∗∗rDD (rD)

0L (rL )where L (rL ) ≡ − rLL(r> 0 is the price elasticity of demand in the loan market, and D (rD ) ≡L)
rD D0 (rD )> 0 is the price elasticity of supply in the deposit market.
D(rD )

3Analysis[D(rD ), L(rL )] L (rL ) − θL rL and G(rL , rD ; θD ,α)≡Now, define F (rL , rD ; θL ) ≡ (rL − r) − ∂C∂L∂C
(1 − α)r − rD − ∂D
[D(rD ), L(rL )] D (rD ) − θD rD . Then,∂F∂F∂rL∂F ∂rL ∂rD   ∂θL  ∂θL  = −, ∂G ∂G   ∂rD  ∂G ∂rL ∂rD∂θL∂θL|{z}≡K

which indicates


∂rL∂G
 ∂θL  − ∂rD rL , = −1  det(K)  ∂rD  | {z }  ∂G
rL<0∂θL∂rL

where3det(K) ≡∂F∂rL∂G∂rD

−=∂F∂rL∂G∂rD

−∂F∂rD 2∂ C∂L∂D∂G∂rL
2|

D0 L0 L D > 0{z}<0because it is assumed that∂F∂C 0∂ 2C 0= (rL − r) −
 + 1−L L − θL > 0∂rL
∂L L
∂L2and∂ 2C 0∂C 0∂G= (1 − α)r − rL −
 + −1 −D D − θD > 0.
∂rD
∂D D
∂D2Hence, an increase in θL raises the lending rate:∂GrL∂rL∂rD=>0∂θLdet(K)whereas whether it raises or lowers the deposit rate depends on the sign of the cross partial
derivative of the cost function, C(D, L):∂rD=∂θL∂ 2C· (L0 D rL )∂L∂D
| {z } | {z }<0≷0det(K)| {z }≶ 0.

>0

Specifically, if the marginal cost of loan or deposit increases for a larger amount of deposit or∂2Cloan, respectively ( ∂L∂D> 0), the deposit rate decreases if the conduct parameter in the loan
market θL increases.
Similarly, the effect of an increase in the conduct parameter in the deposit market θD on
the equilibrium loan and deposit rates is captured by∂ 2C0L rD )  ∂L∂D · (D∂rL{z } 
 | {z } | >0 ∂θD  ≷0
1=, ∂rD  |det(K)∂F{z } 
r
D>0
∂θD
∂rL
| {z }
>0which implies that if the marginal cost of loan or deposit increases for a larger amount of deposit
or loan, respectively, the lending rate increases if the conduct parameter in the deposit market4

θD increases.
Finally, it is shown that the an increase in the rate for compulsory reserves has a similareffect. Specifically,∂ 2C0L rD )  ∂L∂D · (D{z } ∂rL
 | {z } | >0≷0 ∂α 1,= det(K) ∂rD∂F| {z } 
r
D>0

∂α∂rL
| {z }
>0that is, an increase in α unambiguously raises the deposit rate rD , whereas whether it also
∂2C
raises the lending rate rL depends on the sign of ∂L∂D.

References
Dermine, J. 1986. “Deposit Rates, Credit Rates and Bank Capital: The Klein-Monti ModelRevisited.” Journal of Banking & Finance, 10(1), 99-114.
Freixas, Xavier, and Jean-Charles Rochet. 2008. Microeconomics of Banking, Second Edition,The MIT Press.
Gunji, Hiroshi, and Kenji Miyazaki. 2019. “An Industrial-Organization Approach to Conventional and Unconventional Monetary Policy.” Unpublished.
Klein, Michael A. 1971. “A Theory of the Banking Firm.” Journal of Money, Credit and Banking, 3(2), Part 1, 205-218.
Monti, Mario. 1972. “Deposit, Credit and Interest Rate Determination under Alternative Bank
Objective functions.” Giorgio P. Szegö and Karl Shell (Eds.), Mathematical Methods in Investment and Finance, North-Holland, 430-454.
Weyl, E. Glen, and Michal Fabinger. 2013. “Pass-Through as an Economic Tool: Principles ofIncidence under Imperfect Competition.” Journal of Political Economy, 121(3), 528-583.

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