Benhabib et al., "Avoiding Liquidity Trap",JPE2002 を読む

8/9/2019 Benhabib et al., "Avoiding Liquidity Trap",JPE2002

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Benhabib et al. Avoiding Liquidity Trap, JPE2002

1 Introduction 2

2 4

2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 9

4 9

5 10

5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 13

6.1

. . . . . . 136.2 . . . . 15

7 18

8 Reference 19

1


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Benhabib, Jess, Stephanie Schmitt-Grohe, and Martn Uribe, Avoiding Liquidity Traps,

Journal of Political Economy, 2002

1 Introduction

20

Talyer

(1993) 1.5 0.5

1

Clarida, Gal, and Gertler 1998

s

Levin, Wieland, and Williams (1999)

2

Rotemberg and

Woodford (1999)

Leeper (1991), Bernanke and Woodford (1997), and Clarida et al. (2000)

1

R

R = R()

R = r + r

1 R() > 1

2


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L

R(L) < 1

Benhabib, Schmitt-Grohe , and Uribe (2001b) DGE

L

(, R())

(L, R(L))

3


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Woodford (1999)

Krugman (1998)

Woodford (1999)

5 II

III IV

VVI VII

Gesell

2

2.1

4


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0

ertu(c,M

P)dt. (1)

cMP u

ucm > 0R(L)

y > 0, limm

um(y, m)

uc(y, m) R(L).

B R

y

P c + P + M + B = RB + P y.

m = M/Pa = (M+ B)/P

*1

c + + a = (R )aRm + y. (2)

=

P /P

y

No-Ponzi-Game

*1 a = (M+ B)/P t

a =M+ B

P

M+ B

P2P ,

= P /P a = (M+ B)/P

a =M+ B

P a,

Pc + P+ M+ B = RB + Py P

c + +M+ B

P=

RB

P+ y,

a 0 = RM/P RM/P

c + +M+ B

P a =

RB

P+

RM

P a

RM

P+ y,

a = (M+ B)P,m = M/P a = (M+ B)/P a

c + + a = (R )aRm + y,

5


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limt

eRt

0

(R(s)(s)

)dsa(t) 0. (3)

Ponzi gameNPG

NPG

(2) (3) a(0) (1)

(2) (3)*2

uc = , (4)um = R, (5)

= (r + R). (6)

2.2

R = R(). (7)

*2

maxc,m

Z

0ertu(c,m)dt,

c + + a = (R )aRm + y,

limt

eRt

0

`R(s)(s)

dsa(t) 0.

H

H(c,m,a,) = ertu(c,m) + [(R )a Rm + y c ],

Hc = 0, Hm = 0,Ha = ,H = a

Hc = e

rtuc = 0,

Hm = ertum R = 0,

= Ha = ( R),

a = H = (R )a Rm + y c .

uc = ert, um = Rert = ert

uc = , (4)

um = R. (5)

= ert + rert = ert, = ( R)

= (r + R). (6)

6


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R() > 1R() < 1

3, R() 0, R() 0

> r

> r : R() = r + , R() > 1. (8)

B = RB M P

a = (R )aRm . (9)

a(0)

a(0) =A(0)

P(0). (10)

A(0) = M(0) + B(0) > 0

limt

eRt

0[R(s)(s)]dsa(t) = 0. (11)

> 0

+ Rm = a. (12)

R 0 = R

Benhabib et al. 2001a

2.3

c = y, (13)

u (4) (5)m = l(c.R), lc > 0, lR < 0

*3 y R

*3 (4) (5) um(c,m) = Ruc(c,m)R cm

lm = l(c,R)um(c, l(c.R)) = Ruc(c, l(c,R))

cumc + ummlc = R(ucc + ucmlc).

lc

lc = Ruccumcumm Rucm

> 0.

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(4)R*4

= L(R), L < 0. (14)

*5

(7)(6) (14)R(14)(6)

*6

R() =L(R())

L(R())[R() r]. (15)

(R() > r)

R

> 0

(9) (7) (12)

a = [R() ]a, (16)

(7) (11)R

lR RummlR = uc + RucmlR.

lR =uc

umm Rucm< 0.

*4 m = l(y, R)(4) = uc(y, l(y, R))L(R) = uc(y, l(y, R))L

L(R) = ucmlR < 0.

*5

cm ucm > 0 BSUMonetary Policy and Multiple Equilibria, AER 2001

*6 (7)R = R() (6) (14)

= (r + R()), (6)

= L(R()). (14)

(14) t

= L(R())R(). (14)

(14) (14) (6)LR = L(r + R()).

(15)

R() =L(R())

L(R())[R() r]. (15)

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limt

eRt

0[R((s))(s)]dsa(t) = 0. (17)

(10) (P(0) > 0, A(0) > 0)

(15)(17) a

3

(15) R() = r + (8)

R() > 1

R(L) = r + L L <

R() R(L) > 0 L

L

L/LR (15)

R() rR() > 1

(t) =

L

4

L

2

9


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5

5.1

(12)

+ Rm = ()a, > 0. (18)

() > 0, (19)

(L) < 0. (20)

L

(18) (7) (9) a

a = [R() ()]a. (21)

a(t) = eRt

0[R((s))(s)((s))]dsa(0),

*7

limt

eRt

0[R(s)(s)]dsa(t) = a(0) lim

te

Rt

0((s))ds, (22)

*7

da

dt= [R() ()]a, (21)

da dt

daa = [R() ()]dt,

10


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*8

= (19)(22)

=

L (20) (22)

a(0)

5.1.1

Woodford (1999)

A

A= k, (23)

k

R(L) < k < R(). (24)

L

0 t

[log a(s)]t0 =

Zt0

[R((s)) (s) ((s))]ds.

log a(t) =

Zt0

[R((s)) (s) ((s))]ds + log a(0).

elog a(t) = a(t) = eRt

0[R((s))(s)((s))]ds+loga(0),

= eRt

0[R((s))(s)((s))]dsa(0). (22)

*8 (17) a(t) (22)

limt

eRt

0[R((s))(s)]dsa(t) = lim

te

Rt

0[R((s))(s)]dse

Rt

0[R((s))(s)((s))]ds log a(0),

= limt

eRt

0((s))dsa(0). (22)

11


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A/A (a/a) +

(9)*9

+ Rm = [R() k]a.

() R() k (18)

R() k (24)

(19) (20)

5.1.2

P = RB

P RB *10 RB P

a = (B/P) + m

+ Rm = R()a, (25)

() = R() (18)

()R()

(19) R() > 0 (20) R(L) > 0

limtt

0R(s)ds <

(11)

limt

a(0)eRt

0R(s)ds = 0.

*9

a

a+ = k = k

a

a.

(9) a = (R )aRm a = 0

0 = (R k +a

a)aRm ,

= (R k)aRm .

+ Rm = (R k)a.

*10 RB

12


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6

6.1

(12)

> r

M

M= , (26)

m

m= . (27)

(4)(5)(6)(13)(27) c,m,, 2

(17)

*11

m =

r + um(y, m)

uc(y, m)

1

m+

ucm(y, m)

uc(y, m)

. (28)

m 2m

mm = m

*11 (4) uc(c,m) = t uccc + ucmm = (13)c = yy

c = 0ucmm = (6)ucmm = [r + R](4) uc =

ucmm = uc[r + R],

ucm

ucm = r + R,

=ucm

ucm r + R.

(27)

mm = ucmuc

m + r R.

13


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r + =um(y, m)

uc(y, m).

ucm > 0 umm < 0m > m m > 0m < m

m < 0m

m

m

m+

ucm

ucm = + r R,

1

m+

ucm

uc

m = + r R,

m = + r R1

m+

ucm

uc

.

(4) (5)R = um/uc

m =

+ r um

uc1m

+ucm

uc

.

14


15/20

6.2

(7)

L

0 B(t) Begt, B 0. (29)

B(t) = 0

GDP

(B = 0.6y, g = 0

(29)g

(11)

(29)

(7)

m(4)(7) (13)

m =uc(y, m)

ucm(y, m)[r + (m)R((m))], (30)

(m)um(y, m)

uc(y, m)= R(),

(17) (10)

limt

eRt

0[R((m(s)))(m(s))]dsm(t) + e

Rt

0R((m(s)))ds B(t)

P(0)= 0, (31)

P(0)m(0) + B(0) = A(0). (32)

(29) A(0) > 0B (30)(32)m

P(0) > 0

3 (30) (15) 1

m L mL > m

15


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(30)mL mL

m mL

(31)g < R((mL)) B = 0m

mL

> 0 (26) (7)

(28)m (17)

B (29)M(0) > 0

limt

e

t0

[um(y, m)

uc(y, m)

]ds

M(0) + e

t0

[um(y, m)

uc(y, m)

]ds

B(t) = 0. (33)

m (33)

(29)

6.2.1

m m

16


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m m (7)

(26)

m m (30)m > m (28)

m =

uc(y, m)

ucm(y, m)[r + (m)R((m))] for m m[

1

m+

ucm(y, m)

uc(y, m)

]1 [

r + um(y, m)

uc(y, m)

]for m > m.

(34)

m (34) m

4 2 3 m (34) m

m < m < m < mL m < m < mL

L < < . (35)

m, m, mL

um(y, m)/uc(y, m

) = r + ,

um(y, m)/uc(y, m) = r + ,

um(y, mL)/uc(y, m

L) = r + L.

(m, mL) m m m

(28) (30) 4 (28)(30)

(m, mL

)

17


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m

(31) m (34) m

m

(33)

7

Benhabib et al.

(2001b)

VVI

Schmitt-Groh? and Uribe

(2000)

Benhabib et al. (2000)

Buiter and Panigirtzoglou (1999) Gesell

Gesell

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Gesell

Gesell

8 Reference

Benhabib, Jess; Schmitt-Grohe , Stephanie; and Uribe, Martn. Chaotic Interest Rate Rules.

Manuscript. Philadelphia: Univ. Pennsylvania, Dept. Econ., 2000.

. Monetary Policy and Multiple Equilibria. A.E.R. 91 (March 2001): 167?86. (a) . The Perils of Taylor Rules. J. Econ. Theory 96 (January/February 2001): 40?69. (b)

Bernanke, Ben S., and Woodford, Michael. In?ation Forecasts and Monetary Policy.J. Money,

Credit and Banking 29, no. 4, pt. 2 (November 1997): 653?84.

Brock, William A. Money and Growth: The Case of Long Run Perfect Foresight. Internat.

Econ. Rev. 15 (October 1974): 750?77.

.A Simple Perfect Foresight Monetary Model.J. Monetary Econ. 1 (April 1975): 133?50.

Buiter, Willem, and Panigirtzoglou, Nikolaos. Liquidity Traps: How to Avoid Them and How

to Escape Them. Discussion Paper no. 2203. London: Centre Econ. Policy Res., August 1999.

Clarida, Richard H.; Gal?, Jordi; and Gertler, Mark. Monetary Policy Rules in Practice: Some

International Evidence. European Econ. Rev. 42 ( June 1998): 1033?67.

. Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory.

Q.J.E. 115 (February 2000): 147?80.

Fuhrer, Jeffrey C., and Madigan, Brian F. Monetary Policy When Interest Rates Are Bounded

at Zero. Rev. Econ. and Statis. 79 (November 1997): 573?85.

Krugman, Paul R.Its Baaack: Japans Slump and the Return of the Liquidity Trap.Brookings

Papers Econ. Activity, no. 2 (1998), pp. 137?87.

Leeper, Eric M. Equilibria underActive and Passive Monetary and Fiscal Policies. J.

Monetar y Econ. 27 (February 1991): 129?47.

Levin, Andrew; Wieland, Volker; and Williams, John C. Robustness of Simple Policy Rules

under Model Uncertainty. In Monetar y Policy Rules, edited by John B. Taylor. Chicago: Univ.

Chicago Press (for NBER), 1999.

Lucas, Robert E., Jr. Money Demand in the United States: A Quantitative Review.Carnegie-

Rochester Conf. Ser. Public Policy 29 (Autumn 1988): 137?67.

Obstfeld, Maurice, and Rogoff, Kenneth. Speculative Hyperinflations in Maximizing Models:

Can We Rule Them Out? J.P.E. 91 (August 1983): 675?87.

Orphanides, Athanasios, and Wieland, Volker. Price Stability and Monetary

Policy Effectiveness When Nominal Interest Rates Are Bounded at Zero.Finance and Economic

Discussion Series, no. 98-35. Washington: Board Governors, Fed. Reserve System, June 1998.

19


20/20

Reifschneider, David, and Williams, John C.Three Lessons for Monetary Policy in a Low In?ation

Era. Finance and Economic Discussion Series, no. 99-44. Washington: Board Governors, Fed.

Reserve System, August 1999.

Rotemberg, Julio, and Woodford, Michael. Interest Rate Rules in an Estimated Sticky Price

Model.In Monetar y Policy Rules, edited by John B. Taylor. Chicago: Univ. Chicago Press (for

NBER), 1999.

Schmitt-Grohe , Stephanie, and Uribe, Martn. Price Level Determinacy andMonetary Policy

under a Balanced-Budget Requirement. J. Monetar y Econ. 45 (February 2000): 211?46.

Shigoka, Tadashi. A Note on Woodfords Conjecture: Constructing Stationary Sunspot Equi-

libria in a Continuous Time Model. J. Econ. Theor y 64 (December 1994): 531?40.

Stock, James H., and Watson, Mark W.A Simple Estimator of Cointegrating Vectors in Higher

Order Integrated Systems. Econometrica 61 ( July 1993): 783?820. Taylor, John B. Discretion versus Rules in Practice. Carnegie-Rochester Conf. Ser. Public

Policy 39 (December 1993): 195?214.

Wolman, Alexander L.Real Implications of the Zero Bound on Nominal Interest Rates.Man-

uscript. Richmond, Va.: Fed. Reserve Bank Richmond, December 1999.

Woodford, Michael. Monetary Policy and Price Level Determinacy in a Cashin-Advance Econ-

omy. Econ. Theor y 4, no. 3 (1994): 345?80.

.Price-Level Determinacy without Control of a Monetary Aggregate.Carnegie-Rochester

Conf. Ser. Public Policy 43 (December 1995): 1?46.

.

Control of the Public Debt: A Requirement for Price Stability?

Working Paper no.5684. Cambridge, Mass.: NBER, July 1996.

. Price-Level Determination under Interest-Rate Rules. Manuscript. Princeton, N.J.:

Princeton Univ., April 1999.

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Benhabib et al., "Avoiding Liquidity Trap",JPE2002 を読む