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to Federal Reserve Bank of St. Louis Working Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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