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Book of Proceedings published by (c)
International Organization for Research and Development – IORD
ISSN: 2410-5465
Book of Proceedings ISBN: 978-969-7544-00-4
IRC-2015
Istanbul-Turkey.
The oligopolistic foundation for the prime rate: A price-setting
game in banking
Shigeru Nishiyama
Kyushu International University, [email protected]; [email protected], Kitakyushu, Fukuoka, Japan
Abstract
First, the optimization of interest rate charged by banks on their lending is formulated as a price-setting game. Second, the prime
lending rate and its pricing is analytically investigated under a set of reasonable assumptions, by means of applying the
equilibrium properties of Bertrand and Stackelberg games. The goal of this paper is to clarify how, by which bank and with
which property the prime lending rate is offered in the duopolistic banking market, as well as to grasp the strategy in the market,
consistent with the most-favored-customer pricing policy, under which the prime lending rate is offered.
© 2013 European Journal of Research on Education by IASSR.
Keywords: Prime rate, bank loans, commercial banks; Stackelberg game, Bertrand game, duopoly.
1. Introduction
The aim of this paper is to study the prime lending rate and its pricing from the perspective of an oligopolistic
price-setting game. It is well known that the prime lending rate has been received a considerable interest in financial
research as a benchmark on bank loans, as a close tracker with monetary policy instruments and as an indicator of
several other credit conditions. On the other hand, the prime lending rate has such a distinctive feature that it is
typically uniform across banks in the banking market of advanced economies. Although it is possible that individual
banks determine and change their prime lending rate at any time depending on credit conditions, the rate tends to
behave closely in line with market interest rates. This feature strongly suggests that the determination of the prime
rate is influenced by the relationship between banks (the interbank relationship), more specifically their strategic
interdependence with respect to lending rate, in the banking market. In spite that explanations for the prime lending
rate have pointed out a variety of its leading characteristics including its asymmetries in upward and downward
changes (Dueker, 1998, 2000; Dueker and Thornton, 1994, 1997; Thompson, 2006), the relationship the rate has
with the cost of bank liabilities (Goldberg, 1982; Hafer, 1983), and the frequency and effect of prime rate changes
(Mester and Saunders, 1995; Slovin, Sushka and Waller, 1994), they have paid little attention to the interbank
relationship and its significance for the prime rate. This paper provides analytical evidence on the prime lending
rate and its pricing in the light of strategic interdependence between duopolistic banks, with the intention to examine
the effect of interbank relationship on the determination of the rate. In detail, it focuses on the equilibrium
properties of Bertrand and Stackelberg games to clarify the essential characteristics of the prime lending rate as a
market price under strategic interdependence. Thereby, the paper investigates how, by which bank and with which
property the prime lending rate is offered in the duopolistic banking market. To apply the equilibrium properties of
Bertrand and Stackelberg games, the notion of the prime rate is referred to as the original indication of the lowest
interest rate charged by banks under their most-favored-customer pricing policy in the paper, without loss of
generality.
Shigeru Nishiyama
2. Duopolistic Banks and Their Price-Setting Games
2.1. Payoff Function
Let there be two banks with negligible cost of funds (the deposit rate). Bank i ’s payoff (profit) function is
 i   i  ri , rj   ri yi  ri , rj 
i, j  1,2 ; i  j ,
(1)
where yi : R2  R is the demand function of funds that bank i faces, and ri denotes the interest rate on bank i ’s
lending. yi is assumed to be twice continuously differentiable.
2.2. Bertrand Game
Each bank i ( i  1,2 ) determines its lending rate such that first the order condition
 i
y
 yi  ri i  0
ri
ri
i  1,2
(2)
is satisfied. In order to have (2) for any ri  0,  ,
yi
 0.
ri
(A3)
The second order condition
 2 i
y
2 y
 2 i  ri 2i  0
2
ri
ri
ri
i  1,2
(A4)
is assumed to be satisfied. Solving (2) with respect to ri , I obtain bank i ’s reaction function
ri  i  rj   arg max  ri yi  ri , rj 
ri  0
i, j  1,2 ; i  j ,
(5)
where
 yi
i rj    
 rj
 ri
 2 yi 

ri rj 
 yi
 2 y   0
 ri 2i  
2
ri   0
 ri
as
yi
 2 yi
 ri
rj
ri rj
 0
 .
 0
(6)
At a Bertrand equilibrium, bank i ’s interest rate ri B is specified as ri B  i  rjB  , i, j  1,2 ; i  j , then its payoff  iB is
 iB   i  ri B , rjB 
i, j  1,2 ; i  j .
(7)
As a common stability condition to assure the uniqueness of the Bertrand equilibrium, assume now that
2
If i  rj  , rj   R
, then 0  i rj   1 .
i, j  1,2 ; i  j .
2
(A8)
The oligopolistic foundation for the prime rate: A price-setting game in banking
2.3. Stackelberg Game
In the Stackelberg game, bank i determines its lending rate, taking account of bank j ’s reaction function.
Bank i thus maximizes its payoff (profit)  i   i  ri , j  ri   with respect to ri . The first order condition to be
satisfied is
 y y

d i
 yi  ri  i  i  j  ri    0


dri
 ri rj

i, j  1,2 ; i  j .
(9)
Assume that the second order condition
 y y

 2 y
2
d 2 i
 2 yi
2 y
 2  i  i  j  ri    ri  2i  2
 j  ri   2i  j  ri     0
2
 ri rj

dri

r

r

r

r
i
j
j


 i

(A10)
holds.
Hence the Stackelberg leader’s payoff (profit) is specified as
 iL   iL  ri    i  ri , j  ri    ri yi  ri , j  ri  
i, j  1,2 ; i  j ,
(11)
which is assumed to be a strictly concave function under the assumption (A10). At the Stackelberg equilibrium, the
lending rate of bank i as a Stackelberg leader, ri L , and the rate of bank j as a follower, ri F , are specified as
ri L  arg max  iL  ri 
(12)
rjF   j  ri L  .
(13)
ri  0
and as
3. The Prime Rate in the Duopolistic Banking Market
From (9), (A10) and (11), obviously
d iL
dri
0.
(14)
ri  riL
Taking account of the first order condition for the Bertrand equilibrium (2), I obtain
d iL
dri
 ri B
ri  ri
B
yi
 j  ri B  .
rj
(15)
3
Shigeru Nishiyama
Since (A10) holds,  iL  ri  is a strictly concave function,
ri L  ri B if
d iL
dri
0,
ri L  ri B if
ri  riB
d iL
dri
0.
(16)
ri  riB
Assuming identical demand functions between two banks to advance the analysis,
yi y j

ri rj
i, j  1,2 ; i  j .
Consider the two cases based on the sign of
(A17)
yi
( i, j  1,2 ; i  j ).
r j
yi
0
rj
(Case I)
i, j  1,2 ; i  j
(18)
From (15), (16) and (18)
ri L  ri B
ri L  ri B
 j  ri B   0
as
i, j  1,2 ; i  j .
 j  ri B   0
(19)
Under the assumption of identical demand functions,
ri B  rjB  r B
i, j  1,2 ; i  j .
(20)
from (2) and (A17). On the other hand, as
yi y j

rj ri
i, j  1,2 ; i  j
(A21)
holds under identical demand functions, I have, from (9),
ri L  rjL  r L
i, j  1,2 ; i  j .
(22)
Though the reaction function of bank i as a Stackelberg leader does not exist, the assumption of identical demand
functions leads to
rjF   j  ri L   ri F  r F
i, j  1,2 ; i  j .
(23)
Thus, in the light of (A8), (19), (20), (22) and (23),
rB  rF  rL
rL  rB  rF
as
 j  ri   0
 j  ri   0
i, j  1,2 ; i  j .
4
(24)
The oligopolistic foundation for the prime rate: A price-setting game in banking
Taking account that  iL  ri  is a strictly concave function, by definition,
 iL  r L    L   iL  r B    B regardless of the sign of  j  ri 
i, j  1,2 ; i  j .
(25)
i, j  1,2 ; i  j .
(26)
As identical demand functions are being assumed,


rjF y j  j  ri L  , ri L  rjF y j  rjF , ri L   r F yi  r F , r L    iL  r F    F
In view of (6), (A11), (12), (24), (25) and (26),
B F L
yi
 2 yi
 ri
0
rj
ri rj
as
F B L
i, j  1,2 ; i  j .
yi
 2 yi
 ri
0
rj
ri rj
(27)
To obtain the equilibrium lending volume, the reaction functions must be classified into two cases: upward
sloping and downward sloping.
First, the case of  j  ri   0 . Using the mean value theorem,
yi  r L , r F   yi  r F , r L    r L  r F 
yi
y
 r F  r L  i
ri
rj
i, j  1,2 ; i  j
 y y 
 yi  r , r    r  r   i  i  j 
 r r 
j
 i

F
L
L
(28)
F
(28) coupled with (A3), (18) and (24) yields
yi  r L , r F   yi  r F , r L 
i  1,2 .
(29)
In the same way as (28),
yi  r L , r F   yi  r B , r B    r L  r B 
yi
y
 r F  r B  i
ri
rj
(30)
 y y 
 yi  r B , r B    r L  r B   i  i  j 
 r r 
j
 i

and
yi  r F , r L   yi  r B , r B    r F  r B 
yi
y
 r L  r B  i
ri
rj
i, j  1,2 ; i  j .
 y
y 
 yi  r B , r B    r L  r B   i  j  i 
 r
rj 
 i
5
(31)
Shigeru Nishiyama
Now, from (30) and (31), in the light of (A3), (A8), (18), (24), (29) and  j  ri   0 ,
yi
ri
yi  r B , r B   yi  r L , r F   yi  r F , r L 
if 0  
yi  r L , r F   yi  r B , r B   yi  r F , r L 
if  j  
yi  r L , r F   yi  r F , r L   yi  r B , r B 
if 1  j  
yi
ri
yi
  j ,
rj
yi
 1  j ,
rj
yi
ri
yi
,
rj
i, j  1,2 ; i  j .
(32)
Second, the case of  j  ri   0 . Since (A3) and (18) holds,
dyi yi yi


 j  ri   0
dri ri rj
i, j  1,2 ; i  j
(33)
is satisfied. Thus, from (24),
yi  r F , r L   yi  r B , r B   yi  r L , r F 
(Case II)
yi
0
rj
i  1,2 .
(34)
i, j  1,2 ; i  j
(35)
In the same way as the case of (18), from (16),
rL  rB
rL  rB
as
 j  ri B   0
 j  ri B   0
i, j  1,2 ; i  j .
(36)
Thus,
rL  rF  rB
rF  rB  rL
as
 j  ri   0
i, j  1,2 ; i  j .
 j  ri   0
(37)
Hence,
B F L
F B L
as
yi
 2 yi
 ri
0
rj
ri rj
yi
 2 yi
 ri
0
rj
ri rj
i, j  1,2 ; i  j .
(38)
Moreover, in the case of  j  ri   0 ,
dyi yi yi


 j  ri   0
dri ri rj
i, j  1,2 ; i  j
6
(39)
The oligopolistic foundation for the prime rate: A price-setting game in banking
holds, which, coupled with (37), leads to
yi  r B , r B   yi  r F , r L   yi  r L , r F 
i  1,2 .
(40)
In the case of  j  ri   0 , taking account of the following classification,
yi
ri
yi  r L , r F   yi  r F , r L   yi  r B , r B 
if 0 
yi  r F , r L   yi  r B , r B   yi  r L , r F 
if  j 
yi  r B , r B   yi  r F , r L   yi  r L , r F 
if 1  j 
yi
  j ,
rj
yi
ri
yi
 1  j ,
rj
yi
ri
yi
,
rj
i, j  1,2 ; i  j .
Let us summarize the analysis up to here. In the market where demand for loans is substitutive (
(41)
yi
 0 ), if banks
rj
act coordinately with respect to their lending rates (  j  ri   0 ), the Bertrand rate ( r B ) is accepted as the prime
lending rate, in which case the relative volume of lending varies based on the slope of the reaction function and the
size of profit is smaller than at the other duopolistic rates. Under substitutive demand, if banks act reversely with
respect to their lending rate (  j  ri   0 ), the Stackelberg leader rate ( r L ) is accepted as the prime lending rate, at
which the largest volume of lending is provided and the largest profit is obtained among the duopolistic rates. On
the other hand, in the market with complementary demand (
yi
 0 ), if banks act coordinately, the Stackelberg
rj
leader rate is accepted as the prime rate, which makes the both volumes of lending and profit largest. If banks act
reversely, the Stackelberg follower rate ( r F ) is accepted as the prime rate, where the relative volume of lending
varies based on the reaction function’s slope with the smallest profit.
4. Conclusion
This paper has formulated and analyzed the prime lending rate and its pricing as a price-setting game, by means
of applying the equilibrium properties of Bertrand and Stackelberg games. Its main results are given in (24) and
(37), as well as in (27), (32), (34), (38), (40) and (41). These results provide one plausible explanation for that the
prime lending rate is typically observed as uniform across banks: the prime rate is offered through Bertrand strategy
by banks with positive reaction functions, facing substitutive demand for loans.
Acknowledgements
The research for this paper has been supported by JSPS Grant-in-Aid for Scientific Research (C), No. 24530381.
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