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Research Division Federal Reserve Bank of St. Louis Working Paper Series
Interest on Reserves, Interbank Lending, and Monetary Policy
Stephen D. Williamson
Working Paper 2015-024A http://research.stlouisfed.org/wp/2015/2015-024.pdf
September 2015
FEDERAL RESERVE BANK OF ST. LOUIS Research Division
P.O. Box 442 St. Louis, MO 63166
______________________________________________________________________________________
The views expressed are those of the individual authors and do not necessarily reflect official positions of the Federal Reserve Bank of St. Louis, the Federal Reserve System, or the Board of Governors.
Federal Reserve Bank of St. Louis Working Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to Federal Reserve Bank of St. Louis Working Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.
increase
E0
1X
t = 0
�t [�Ht + u(xt )] ;
H t CM ; xt DM ; u(�)
u0(0) = 1 ; u0(1 ) = 0; �x u 00(x )u0(x ) < 1:
E0
1X
t = 0
�t [�X t + ht ; ]
X t CM ; ht
D M :
E0
1X
t = 0
�t [�X t + Ht :]
zbt CM
CM ; zmt CM
CM ;�t CM :
y CM CM t :
CM ;
DM ;
cb
a
� retail bank depositorsunconventional bank
depositors� c
DM1 � �
a DM :�
b 1 � �a CM ;
DM
;CM :
DM ;
DM :
DM
DM
D M ;
Ct M t Bt
�t
t CM
�0
�C0 + zm
0 M 0 + zb0B0
�� �0 = 0;
t = 1; 2; 3; :::;
�t
�Ct � Ct � 1 + zm
t M t � M t � 1 + zbt Bt � Bt �1
�� �t = 0:
Bt
Retail banks unconventional banks
(kr ; c; dr )kr
CM ;c CM
CM ; dr
CM :t; �
DM ;
Ur = �kr + �u
��c�
�
+ (1 � �)u (�dr ) :
kr + zf f r �zbbr �zm m� qr ��c��(1��)dr +�(m + br � f r )
�+ �qr ( + y) � 0
f r
CM ; zf
CM :br ; m; qr
CM ;CM
1 � � 0 < �< 1:
CM
�(1 � �)dr �f r (1 � I )
�+
(m + br � f r I )(1 � �)�
+ qr ( + y)(1 � �) � 0:
� = 0� > 0:
(kr ; c; dr );(f r ; br ; m; qr ) Ur f r > 0
; I = 0; f r < 0I = 1:
kr ; c; dr ; br ; m; qr � 0:
Uu = �ku + �u
��b0
�
�
+ (1 � �)u (�du ) ;
ku CMCM ; b0
CM CM ;du CM t + 1 CM
(ku ; b0; du ) ; (bu ; f u ; qu ); bu
f u ; qu
CMUu
ku + zf f u � zbbu � qu � �(1� �)du ��f u
�+
�(bu � �b0)�
+ �qu ( + y) � 0
�(1 � �)du �f u
�+
bu � �b0
�+ qu ( + y) � 0;
ku ; b0; du ; bu ; qu ; bu � �b0 � 0:
�u0(�dr ) � �� �r = 0
��
u0
��c�
�
� 1 = 0
zf ���
+1�
�r
zf ���
+(1 � �)�r
�
zb ���
+(1 � �)�r
�
zm =��
+(1 � �)�r
�
� + �( + y) + �r ( + y)(1 � �) � 0
�r
�u0(�du ) � �� �u = 0
��
u0
��b0
�
�
� zb = 0
u0
��b0
�
�
= u0(�du ) ; bu � �b0 > 0;
u0
��b0
�
�
� u0(�du ) ; bu � �b0 = 0;
zf =��
+1�
�u
� + �( + y) + �u ( + y) = 0
�c � �t Ct ; �m � �t M t ; �b � �t Bt :
�c + zm �m + zb�b = �0;�
1 �1�
�
�c +
�
zm �1�
�
�m +
�
zb �1�
��b = �;
�0 CM ; �CM
zm �m + zb�b+ �c = V;
V > 0
�0 = V �
V
�
1 �1�
�
+�m�
(zm � 1) +�b�
(zb � 1) = �
A stationary equilibrium with binding collateral constraints con-sists of quantities ( �m;�b; �c; dr ; du ; c; m; f r ; br ; qr ; bu ; b0; qu ; f u ); prices (zf ; ); mul-tipliers (�r ; �u ); and gross ináation rate �; satisfying (5) and (9) with equality,(11)-(23), (26), and market clearing,
�br + (1 � �)bu = �b; [government bond market clears]
��c = �c; [market in currency clears]
�qr + (1 � �)qu = 1; [market in Lucas trees clears]
�m = �m; [market in reserves clears]
�f r + (1 � �)f u = 0; [interbank market clears]
given Öscal policy V and monetary policy (zm ; V � zb�b):
�r �u
V
zm
V � zb�b;
�m; �b; �c
b D MCM
b CM ;
DM ;
DM :CM ;
b DMzi < 1
i = m; b; f :
DM : xr1
xr2 DM
ca xu
1 xu2
b
a
�i = �[u0(xi2) � 1]; i = r; u:
DM
u0(xi2) = 1; u0(xi
2) > 1:
bu > �b0;
zb =��
[(1 � �)u0(xr2) + �] =
��
u0(xu2 ) =
��
u0(xu1 ) :
� > 0; xr2 <
xu2 :
�= �u0(xr1);
DM :
=�u0(xu
2 ) y1 � �u0(xu
2 ):
�(1 � �)xr2 [(1 � �)u0(xr
2) + �]1 � �
+ (1��)xu2 u0(xu
2 )+ ��xr1u0(xr
1) = V +�u0(xu
2 ) y1 � �u0(xu
2 )
zb = zm = zf =u0(xu
2 )u0(xr
1)
xu1 = xu
2
(1 � �)u0(xr2) + �= u0(xu
2 ) :
xr1; xr
2; xu1 ; xu
2 ; Vzm : zm = zf = zb;
r =1
�u0(xu2 )
=1
�u0(xu1 )
=1
�[(1 � �)u0(xr2) + �]
xr2 [(1 � �)u0(xr
2) + �]
xr2; �x u00(x )
u0(x ) < 1:xr
2xu
2 ;
F (xr1; xu
2 ) = V;
F (�; �)x� u0(x�) = 1: x�
DM
V +�y
1 � �<
[1 � ��� �(1 � �)] x�
1 � �;
zm 2(0; 1): DM a
(xr1; xu
2 )zm V: F (�; �);
I C
M Pzm M P
CM
zf ���
�1�
�s =�
u0(xr1)
[1 � u0(xr2)] < 0:
�zf +��
+(1 � �)�s
�= 0:
DM
zm
k = V � zb�b; k
c DM
�c = ��xr1u0(xr
1):
zm �m = k � ��xr1u0(xr
1)
bDM
V � k � (1 � �)�xu2 u0(xu
2 ):
V � k � (1 � �)xu2 u0(xu
2 ) ��yu0(xu
2 )1 � �u0(xu
2 ):
k;
��xr1u0(xr
1) � k � V � (1 � �)�xu2 u0(xu
2 )
+ min
�
0; �(1 � �)(1 � �)xu2 u0(xu
2 ) +�yu0(xu
2 )1 � �u0(xu
2 )
�
:
(xr1; xr
2; xu1 ; xu
2 ) :
zm < 1
zm zm = zb =zf ;
(xr1; xu
2 ) F (�; �)zm = z1
zm z2 < z1;I C
M P1 M P2: xr1
xu2 xu
1 xr2
xr1
k;
b
�b0 = bu ;
zb =��
u0(xu1 ):
zm = zf =��
[(1 � �)u0(xr2) + �] =
��
u0(xu2 ) ;
zb � zf = zm :
r m =1
�u0(xu2 )
=1
�[(1 � �)u0(xr2) + �]
;
r b =1
�u0(xu1 )
:
xu1 � xu
2 r b � r m ;
�(1 � �)xr2 [(1 � �)u0(xr
2) + �]1 � �
+ (1 � �)(1 � �)xu2 u0(xu
2 )
+ ��xr1u0(xr
1) + (1 � �)�xu1 u0(xu
1 )
= V +�u0(xu
2 ) y1 � �u0(xu
2 )
zb =u0(xu
1 )u0(xr
1);
zm =u0(xu
2 )u0(xr
1):
(1 � �)u0(xr2) + �= u0(xu
2 )
kDM b
V � k = (1 � �)�xu1 u0(xu
1 )
�(1 � �)xr2 [(1 � �)u0(xr
2) + �]1 � �
+ (1 � �)(1 � �)xu2 u0(xu
2 ) + ��xr1u0(xr
1)
= k +�u0(xu
2 ) y1 � �u0(xu
2 )
zm k;
(xr1; xu
2 )G(xr
1; xu2 ) = k
G(�; �)I C
M P(xr
1; xu2 ); xr
2xu
1 V k:�c;
k
��xr1u0(xr
1) � k � ��xr1u0(xr
1) +�(1 � �)xr
2 [(1 � �)u0(xr2) + �]
1 � �;
a DM :
k; M PM P1 M P2; xr
1 xu2
xr2 xu
1 zb
1zm
�1zb
= u0(xr1)
�1
u0(xu2 )
�1
u0(xu1 )
�
:
xr1 xu
2 xu1 zm
zm
kI C I C1 I C2:
xr1 xu
2 xr2;
xu1 zb
DM ;
k
b DM :
zb = zf =u0(xu
2 )u0(xr
1):
zm =(1 � �)u0(xr
2) + �u0(xr
1):
zm � zb
(1 � �)u0(xr2) + �� u0(xu
2 ) :
u0(xr2) � u0(xu
2 ) :
r m =1
�[(1 � �)u0(xr2) + �]
;
r b =1
�u0(xu2 )
;
r m � r b:
k;
�(1 � �)xr2 [(1 � �)u0(xr
2) + �]1 � �
+ ��xr1u0(xr
1)
= k
k zm ; (xr1; xr
2):
DM ;
V � k = (1 � �)xu2 u0(xu
2 ) ��u0(xu
2 ) y1 � �u0(xu
2 );
xu2 = xu
1 k:
bD M ;
(1 � �)(1 � �)xu2 u0(xu
2 ) ��u0(xu
2 ) y1 � �u0(xu
2 )� 0;
I CM P zm
M P M P1
M P2: xr1 xr
2 kxu
2 xu1 = xu
2
DM ;
zb
k I C I C1 I C2; xr1
xr2 xu
2 xu1 = xu
2
zb
xr1
bDM
zf =u0(xu
2 )u0(xr
1):
zb =u0(xu
1 )u0(xr
1);
zm =(1 � �)u0(xr
2) + �u0(xr
1):
zb � zf � zm ;
r b =1
�u0(xu1 )
;
r l =1
�u0(xu2 )
;
r m =1
�[(1 � �)u0(xr2) + �]
:
b DM ;
(1 � �)(1 � �)xu2 u0(xu
2 ) ��u0(xu
2 ) y1 � �u0(xu
2 )= 0;
xu2 :
(xr1; xr
2)xu
2 xu1 zm k:
k xr1 xr
2xu
1 xu2
zm =(1 � �)u0(xr
2) + �u0(xr
1):
zf =u0(xr
2)u0(xr
1);
zf = zb =u0(xu
2 )u0(xr
1)=
u0(xu1 )
u0(xr1)
:
u0(xr2) > 1;
a zm < zf =zb:
r m =1
�[(1 � �)u0(xr2) + �]
;
r b =1
�u0(xu2 )
:
(1 � ��)xr2 [(1 � �)u0(xr
2) + �]
+ ��xr1u0(xr
1)(1 � �)
= V (1 � �) +�[(1 � �)u0(xr
2) + �] y1 � �u0(xr
2)+
�(V � k)u0(xr
2);
k;(xr
1; xr2) V zm ;
k:u0(xr
2) = u0(xu2 ) = u0(xu
1 )
xu2 = xu
1 = xr2
(xr1; xr
2; xu1 ; xu
2 )
��xr1u0(xr
1) +�(1 � �)xr
2 [(1 � �)u0(xr2) + �]
1 � �� k � V � (1� �)�xu
2 u0(xu2 );
k
k
b D M :
xr1 xr
2 xu2
xu1
1zm
�1zf
=1
zm
��[u0(xr
2) � 1]u0(xr
2)
�
:
�; �= 0: zm ;D M
u0(xr2); zm
D M
k;I C I C1 I C2; xr
1 xr2
xu1 xu
2zm
zf = zb
D M b
zb =u0(xu
1 )u0(xr
1);
xu1 � xu
2zb � zf > zm :
r b =1
�u0(xu1 )
;
r f =1
�u0(xr2)
;
r m =1
�[(1 � �)u0(xr2) + �]
:
[1 � ��� (1 � �)�] xr2 [(1 � �)u0(xr
2) + �]
+ ��xr1u0(xr
1)(1 � �)
= k(1 � �) +�[(1 � �)u0(xu
2 ) + �] y1 � �u0(xu
2 );
DM b
V � k = (1 � �)�xu1 u0(xu
1 )
u0(xu2 ) = u0(xr
2);
(xr1; xr
2)xu
2 ; xu1
��xr1u0(xr
1) +�(1 � �)xr
2 [(1 � �)u0(xr2) + �]
1 � �� k � V:
kxr
1 xr2 xu
2xu
1
xr1 xr
2xu
2 ; xu1 zb
zm xr1 zf
zo
b
b
zo � zm ;
zo�o
�o� k
k1
k2
k = k1 + k2:
(1 � ��) xr2 [(1 � �)u0(xr
2) + �] + ��xr1u0(xr
1)(1 � �)
= V(1 � �) +�[(1 � �)u0(xr
2) + �] y1 � �u0(xr
2)+
�(V � k1)u0(xr
2)
zo = zb = zf
V; k1; zm ;(xr
1; xr2) : xu
2 = xu1 :
k;k2; k1
�xr
1 xr2 xu
2 xu1
(xr1; xr
2)
[1 � ��� (1 � �)�] xr2 [(1 � �)u0(xr
2) + �]+ ��xr1u0(xr
1)(1��) = k(1��)+�k2
u0(xu2 )
:
xu1 xu
2 ; kzm ; k2; � xr
1; xr2; xu
2xu
1 zb zf
Liberty Street Economics,
Liberty Street Economics,
Econometrica
Journal of Monetary Economics
Journal of Monetary Economics
FederalReserve Bank of New York Policy Review.
Journal of Political Economy
American Economic Review
Journal of Economic Theory.
Figure 1
0
500
1000
1500
2000
2500
3000
3500
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
2008 2009 2010 2011 2012 2013 2014 2015
Lower Bound (left axis) Overnight Rate (left axis)
Upper Bound (left axis) Reserves (right axis)
Percent CA Dollars (Mil.)
Sources: Bank of Canada/Haver Analytics
Figure 2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Jan 2009 Jul 2009 Jan 2010 Jul 2010 Jan 2011 Jul 2011 Jan 2012 Jul 2012 Jan 2013 Jul 2013 Jan 2014 Jul 2014 Jan 2015 Jul 2015
Effective Federal Funds Rate
3-Month Treasury Bills, Secondary Market Rate
Interest Rate on Reserves
Percent
Sources: Federal Reserve Board/FRED
Figure 3
Figure 4
A B
IC
2ݔ௨
1ݔ ∗ݔ �
∗ݔ
MP
F
IC
2ݔ௨
1ݔ ∗ݔ �
ܯ 1
∗ݔF
ܯ 2
Figure 5
Figure 6
B
A 1ܥܫ
2ܥܫ
2ݔ௨
1ݔ ∗ݔ �
∗ݔ
MP
F
A B
IC
2ݔ�
1ݔ ∗ݔ �
ܯ 1
∗ݔF
ܯ 2
Figure 7
Figure 8
B
A 1ܥܫ
2ܥܫ
2ݔ�
1ݔ ∗ݔ �
∗ݔ
MP
F
B
A 1ܥܫ
2ܥܫ
2ݔ�
1ݔ ∗ݔ �
∗ݔ
MP
F