1

m ^ m

m⋅

0

Models With Money:

y = y (m) , c = y′ (m) m

π =

(σ − m

m

)p = U ′ (c) ;p

p=U ′′

U ′y′ (m) m = r − (y′ (m)− π)

= r −(y′ (m)−

(σ − m

m

))becomes:

m

(y′ (m)

(1 +

U ′ (c)

U ′′ (c)m

))=U ′ (c)

U ′′ (c)(r + σ − y′ (m))

(1)

If we define the elasticities

εc = −U′′ (c) c

U ′ (c), εm =

y′ (m)m

y (m)

m =m (r + σ − y′ (m))

1− εcεm(2)

1

“To summarize, the research proposed

her shows the surprising efficiency and

robustness of simple policy rules in which

the reaction of the interest rate is above a

critical treshold. The analysis also shows

that the estimated gains reported in some

research from following alternative rules are

not robust to a variety of models considered in

this paper”

John Taylor,“ The robustness and effi-

ciency of monetary policy rules as guidelines

for interest rate setting by the European

Central Bank,” May 1998.

2

1 Multiple equilibria

Presence of multiple equilibria (which can

be real or nominal) depends on:

I: The way in which money is modelled

a) Money in utility

b) Money in prod. fn.

c) Cash-in-advance,

cash and credit goods

II: Flexible vs. sticky prices

III: Type of monetary feedback rule

a) passive or active rule

b) timing of the rule

IV) Type of fiscal policy

(Ricardian-Non-Ricardian)

3

2 Introduction

λ

λ= r + π −R (π)

Steady State : r + π −R (π) = 0

Active policy R′ (π∗) > 1

Passive policy R′ (π∗) > 1

R (π) ≥ 0

4

3 A flexible–price model

Max U =

∫ ∞0

e−rtu (c,mnp) dt (3)

Assumption 1 (1) u(·, ·) is strictly increasing

and strictly concave, and c and mnp are nor-

mal goods.

Assumption 2 y(mp) is positive, strictly in-

creasing, strictly concave,

limmp→0 y′(mp) =∞, and limmp→∞ y

′(mp) =0.

Assumption 2′ y(mp) is a positive constant.

5

a ≡ (Mnp + M p + B)/P (4)

a = (R−π)a−R(mnp+mp)+y(mp)−c−τ .(5)

limt→∞

e−∫ t0 [R(s)−π(s)]dsa(t) ≥ 0 (6)

taking as given a(0) and the time paths of

τ , R, and π. The optimality conditions

associated with the household’s problem are

uc(c,mnp) = λ (7)

mp [y′(mp)−R] = 0 (8)

um(c,mnp)

uc(c,mnp)= R (9)

λ (r + π −R) = λ (10)

limt→∞

e−∫ t0 [R(s)−π(s)]dsa(t) = 0 (11)

6

Assumption together with equation (8) and

R > 0 implies

mp = mp(R), (12)

with mp′ ≡ dmp/dR < 0. Alternatively,

equation (8), R > 0, and assumption 2′ imply

that mp = mp′ = 0. Using equation (9)

and assumption , mnp can be expressed as

a function of consumption and the nominal

interest rate,

mnp = mnp(c, R), (13)

that is increasing in c and decreasing in R.

7

3.0 The government

R = ρ(π), (14)

where ρ(·) is continuous, non–decreasing, and

strictly positive and there exists at least one

π∗ > −r such that ρ(π∗) = r+ π∗. Following

Leeper (1991), we will refer to the monetary

policy as active if ρ′ (π∗) > 1 and as passive if

ρ′ (π∗) < 1.

The sequential budget constraint of the

government is given by

B = RB − Mnp − M p − Pτ,which can be written as

a = (R− π)a−R(mnp + mp)− τ . (15)

The nominal value of initial government

liabilities, A(0), is predetermined:

a(0) =A(0)

P (0). (16)

8

We classify fiscal policies into two

categories: Ricardian fiscal policies and non-

Ricardian. Ricardian fiscal policies are those

that ensure that the present discounted value

of total government liabilities converges to

zero—that is, equation (68) is satisfied—under

all possible, equilibrium or off–equilibrium,

paths of endogenous variables such as the

price level, the money supply, inflation, or the

nominal interest rate

Throughout the paper we will restrict

attention to one particular Ricardian fiscal

policy that takes the form

τ = Ra−R(mnp + mp) (17)

We will also analyze a particular non–

Ricardian policy consisting of an exogenous

path for lump–sum taxes

τ = τ . (18)

9

Equilibrium

c = y(mp). (19)

Using equations (12)–(14) and (69) to replace

mp, mnp, R, and c in equation (7),

uc(c,mnp) = λ = λ(π) (20)

λ′(π) = ρ′ [uccy′mp′+ ucm(mnp

c y′mp′+mnp

R )](21)

where mnpc and mnp

R denote the partial

derivatives of mnp with respect to c and R,

respectively. (12)–(14), and (69), equations

(10), (68), (15), and (17) can be rewritten as

π =λ(π)[r + π − ρ(π)]

λ′(π)(22)

a = [ρ(π)− π]a (23)

−ρ(π)[mnp(y(mp(ρ(π)), ρ(π)) + mp(ρ(π))]− τ

limt→∞

e−∫ t0 [ρ(π)−π(s)]dsa(t) = 0 (24)

τ = ρ(π) [a− [mnp(y(mp(ρ(π))), ρ(π)) + mp(ρ(π))]]

OR τ = τ

10

Definition 1 (Perfect–foresight equilibrium in

the flexible–price economy) In the flexible–price

economy, a perfect–foresight equilibrium is a

set of sequences {π, a, τ} and an initial price

level P (0) > 0 satisfying (16), (22)–(24) and

either if fiscal policy is non-Ricardian τ = τor τ = Ra − R(mnp + mp) if fiscal policy is

Ricardian, given A(0) > 0.

Given a sequence for π, equations (12)–

(14), (69), and (20) uniquely determine the

equilibrium sequences of {c,mnp,mp, λ, R}independently of whether the equilibrium

price level is unique.

Definition 2 (Real and Nominal Indeterminacy)

The equilibrium displays real indeterminacy if

there exists an infinite number of equilibrium

sequences {π}. The equilibrium exhibits nom-

inal indeterminacy if for any equilibrium se-

quence {π}, there exists an infinite number of

initial price levels P (0) > 0 consistent with a

perfect–foresight equilibrium.

11

In what follows, we restrict the analysis to

equilibria in which the inflation rate remains

bounded in a neighborhood around a steady–

state value, π∗, which is defined as a constant

value of π that solves (22), that is, a solution

to r + π = ρ(π). By assumption π∗ exists

and is greater than −r. Note that π∗ may

not be unique. In particular, if there exists a

steady state π∗ with ρ′(π∗) > 1, then since

ρ(·) is assumed to be continuous and strictly

positive there must also exist a steady state

with ρ′ < 1.

It follows from equation (22) that if the

sign of λ′(π∗) is the opposite of the sign of

1 − ρ′(π∗), any initial inflation rate near the

steady state π∗ will give rise to an inflation

trajectory that converges to π∗. If, on the

other hand, λ′(π∗) and 1 − ρ′(π∗) are of the

same sign, the only sequence of inflation rates

consistent with equation (22) that remains in

the neighborhood of π∗ is the steady state π∗.

12

Note that the case ρ′(π) = 0 for all πcorresponds to a pure interest rate peg. In

this case it follows from equation (21) that λis constant, and therefore the inflation rate is

also constant, which implies that under a pure

interest rate peg the economy exhibits real

determinacy.

13

Under a Ricardian fiscal policy the set

of equilibrium conditions includes equation

(??). Given a sequence {π} and an initial

price level P (0) > 0, equations (23) and (??)

can be used to construct a pair of sequences

{a, τ}. Because the fiscal policy is Ricardian,

the transversality condition (24) is always

satisfied. If instead the fiscal authority follows

the non–Ricardian fiscal policy given in (18),

combining (16), (23), and (24) yields

A(0)

P (0)(25)

=

∫ ∞0

e−∫ t0 [ρ(π)−π]ds

{ρ(π) [mnp(y(mp(ρ(π))), ρ(π)) + mp(ρ(π))] + τ} dswhich given A(0) > 0 and a sequence for

π uniquely determines the initial price level

P (0).

14

The above analysis demonstrates that

for the class of monetary–fiscal regimes

considered nominal determinacy depends only

on fiscal policy and not on monetary policy

— a result that has been emphasized in the

recent literature on the fiscal determination of

the price level and that we summarize in the

following proposition:

Proposition 1 If fiscal policy is Ricardian, the

equilibrium exhibits nominal indeterminacy. Un-

der the non-Ricardian fiscal policy given by

(18), the equilibrium displays nominal deter-

minacy.

15

Proposition 2 Suppose preferences are sepa-

rable in consumption and money (ucm = 0)

and money is productive (assumption 2 holds),

then if monetary policy is active (ρ′(π∗) > 1),

the equilibrium displays real indeterminacy, whereas

if monetary policy is passive (ρ′(π∗) < 1),

there exists a unique perfect–foresight equilib-

rium in which the real allocation converges to

the steady state.

Proposition 3 Suppose that money is not pro-

ductive (assumption 2′ holds) and consump-

tion and money are substitutes (ucm < 0). Then,

if monetary policy is active (ρ′(π∗) > 1), the

real allocation is indeterminate, and if mone-

tary policy is passive (ρ′(π∗) < 1) there ex-

ists a unique perfect–foresight equilibrium in

which the real allocation converges to the steady

state.

16

1.Real Indeterminacy in the Flexible-PriceModel

Monetary Non-productive money Productive money

Policy (y′ = 0) (y′ > 0)

ucm > 0 ucm < 0 ucm = 0 ucm > 0 ucm < 0 ucm = 0Passive (ρ′(π∗) < 1) I D D A D D

Active (ρ′(π∗) > 1) D I D A I I

Note: The notation is: D, determinate; I, indeterminate; A, ambiguous. (Under A the real

allocation may be determinate or indeterminate depending on specific parameter values.)

Proposition 4 Suppose that money is not pro-

ductive (assumption 2′ holds) and consump-

tion and money are complements (ucm > 0).

Then, if monetary policy is passive (ρ′(π∗) <1), the real allocation is indeterminate, and if

monetary policy is active (ρ′(π∗) > 1) there

exists a unique perfect–foresight equilibrium

in which the real allocation converges to the

steady state.

17

Combining the case of non-productive

money (assumption 2′) with preferences

that are separable in consumption and real

balances (ucm = 0) results in the continuous

time version of the economy analyzed in

Leeper (1991). In this case equation (7)

implies that λ is constant. It then follows

that π, R and mnp are also constant, and the

only equilibrium real allocation is the steady

state. This result differs from that obtained by

Leeper who finds that under passive monetary

policy the inflation rate is indeterminate. The

difference stems from the fact that in Leeper’s

discrete–time model the nominal interest rate

in period t is assumed to be a function of

the change in the price level between periods

t − 1 and t, whereas in the continuous time

model analyzed here, the inflation rate is

the right hand side derivative of the price

level, so its discrete–time counterpart is better

approximated by the change in the price level

between periods t and t + 1.

18

In fact, it is straightforward to show that if

in Leeper’s discrete–time model, with ucm = 0and an endowment economy, the feedback

rule is assumed to be forward looking —

that is, Rt = ρ(Pt+1/Pt) — the equilibrium

displays real determinacy.

Leeper

U ′ (c) =β

πt+1Rt+1 (πt)U

′ (c)

Rt+1 (πt) = R∗+a (πt − π∗) , R∗ = r+π∗

πt+1 = aβπt + (R∗ − aπ∗)3.0.1 Discrete-time models

ρcm(mt − mt+1) = Rt − πt+1

mt = −εmRRt

Rt = ερπt+1

πt+2 =

[1 +

ερ − 1

ερρcmεmR

]πt+1.

19

In an economy with with ucm = 0 and

productive money,

ρcc(yt+1 − yt) = ερπt+1 − πt+1

yt = −εyRερπt+1

πt+2 =

[1 +

1− ερρccεyRερ

]πt+1.

20

4 A sticky–price model

Specifically, we assume that there exists a

continuum of household–firm units indexed

by j, each of which produces a differentiated

good Y j and faces a demand function

Y dd(P j

P

), where Y d denotes the level of

aggregate demand, P j the price firm j charges

for its output, and P the aggregate price level.

Such a demand function can be derived by

assuming that households have preferences

over a composite good that is produced from

differentiated intermediate goods via a Dixit-

Stiglitz production function. The function

d(·) is assumed to satisfy d(1) = 1 and

d′(1) < −1. The restriction imposed on d′(1)is necessary for the individual firm’s problem

to be well defined in a symmetric equilibrium.

The production of good j is assumed to take

real money balances, mpj, as the only input

Y j = y(mpj)

where y(·) satisfies assumption .

21

The household’s lifetime utility function

is assumed to be of the form

U j =

∫ ∞0

e−rt

u(cj,mnpj)− γ

2

(P j

P j− π∗

)2 dt

(26)

π∗ > −r denotes the steady–state inflation

rate. The household’s instant budget constraint

and no-Ponzi-game restriction are

aj = (R−π)aj−R(mnpj+mpj)+P j

Py(mpj)−cj−τ

(27)

limt→∞

e−∫ t0 [R(s)−π(s)]dsaj(t) ≥ 0 (28)

In addition, firms are subject to the constraint

that given the price they charge, their sales are

demand-determined

y(mpj) = Y dd

(P j

P

)(29)

The household chooses sequences for cj,mnpj, mpj, P j ≥ 0 and aj so as to maximize

(26) subject to (58)–(29) taking as given aj(0),P j(0), and the time paths of τ , R, Y d, and P .

22

The demand for real balances for non-

production purposes can be expressed as

mnpj = mnp(cj, R) (30)

which by assumption is increasing in cj and

decreasing in R.

23

Equilibrium

In a symmetric equilibrium all household–

firm units choose identical sequences for con-

sumption, asset holdings, and prices. Thus,

cj = c, mpj = mp, mnpj = mnp, aj = a,

P j = P , λj = λ, µj = µ, and πj = π. In

addition,

uc(y(mp),mnp(y(mp), ρ(π))) = λ. (31)

mp = mp(λ, π); mpλ < 0, mp

πucm < 0(32)

Let η ≡ d′(1)<−1 denote the equilibrium

price elasticity of the demand function faced

by the individual firm.

λ = λ [r + π − ρ(π)]

γπ = γr(π − π∗)− y(mp)λ

[1 + η

(1− ρ(π)

y′(mp)

)]a = [ρ(π)− π]a− ρ(π) [mnp(y(mp), ρ(π)) + mp]− τ0 = lim

t→∞e−

∫ t0 [ρ(π)−π]dsa(t)

τ = −ρ(π) [mnp(y(mp), ρ(π)) + mp] + Ra

τ = τ

24

Definition 3 (Perfect–foresight equilibrium in

the sticky–price economy) In the sticky–price

economy, a perfect–foresight equilibrium is a

set of sequences {λ, π, τ , a} satisfying (??)–

(??) and either τ = τ if the fiscal regime is

non-Ricardian or τ = Ra−R(mnp +mp). if

the fiscal regime is Ricardian, given a(0).

Given the equilibrium sequences

{λ, π, τ , a}, the corresponding equilib-

rium sequences {c,mnp,mp, R} are uniquely

determined by (14), (69), (30), and (32).

25

Ricardian fiscal policy(λπ

)= A

(λ− λ∗π − π∗

)(33)

A =

[0 uc(1− ρ′)A21 A22

]A21 = −ucc

∗ ηR∗y′′mpλ

γy′2> 0

A22 = r +ucc∗η

γ

[ρ′

y′− R∗

y′2y′′mp

π

]Proposition 5 If fiscal policy is Ricardian and

monetary policy is passive (ρ′ (π∗) < 1), then

there exists a continuum of perfect–foresight

equilibria in which π and λ converge asymp-

totically to the steady state (π∗, λ∗).

Proposition 6 If fiscal policy is Ricardian and

monetary policy is active (ρ′ (π∗) > 1), then, if

A22 > 0(< 0), there exists a unique (a contin-

uum of) perfect–foresight equilibria in which

π and λ converge to the steady state (π∗, λ∗).

26

Consider a utility function that is sep-

arable logarithmic in consumption, so that

ucc∗ = 1. In this case, the trace of A is given

by

trace (A) = r +(1 + η)ρ′

γR∗(34)

Let ρ′ ≡ − r R∗ γ1+η denote the value of ρ′ at

which the trace vanishes. Clearly, ρ′ may be

greater or less than one. If ρ′ ≤ 1, then the

equilibrium is indeterminate for any active

monetary policy. We highlight this result in

the following corollary.

Corollary 7 Suppose fiscal policy is Ricardian

and preferences are log–linear in consumption

and real balances. If ρ′ ≡ − r R∗ γ1+η is less than

or equal to one, then there exists a continuum

of perfect–foresight equilibria in which π and

λ converge to the steady state (π∗, λ∗) for any

active monetary policy.

27

Periodic perfect–foresight equilibria

Proposition 8 Consider an economy with pref-

erences given by u(c,mnp) = (1− s)−1c1−s +V (mnp), technology given by y(mp) = (mp)α,

and monetary policy given by a smooth interest–

rate feedback rule, ρ(π) > 0, which for π > πtakes the form ρ(π) = R∗ + a (π − R∗ + r)where s > 0, 0 < α < 1, R∗ − r > π >R∗− r−R∗/a, a > 0, and R∗ > 0. Let fiscal

policy be Ricardian and let the parameter con-

figuration satisfy a ≡ −rαγη

(η

1+ηR∗

α

)1−αsα−1

> 1

and 1 < s < 1/α. Then there exists an infinite

number of active monetary policies satisfying

a < a for each of which the perfect foresight

equilibrium is indeterminate and π and λ con-

verge asymptotically to a deterministic cycle.

28

4.0.1 Non-Ricardian Fiscal Policy

Suppose now that the government follows

the non-Ricardian fiscal policy described in

equation (18), that is, a fiscal policy whereby

the time path of real lump–sum taxes is

exogenous. Using

a = [ρ(π)− π]a (35)

−ρ(π) [mnp(y(mp(λ, π)), ρ(π)) + mp(λ, π)]− τ .(??), and (35), which can be written as λ

πa

=

[A 0ε r

] λ− λ∗π − π∗a− a∗

(36)

where A is defined in (33) and ε is a one by

two vector whose elements are the steady–

state derivatives of R(mnp +mp) with respect

to λ and π.

29

Proposition 9 If fiscal policy is non–Ricardian

and monetary policy is passive (ρ′ (π∗) < 1),

then there exists a unique perfect–foresight equi-

librium in which {λ, π} converge asymptoti-

cally to the steady state (π∗, λ∗).

Proposition 10 If fiscal policy is non–Ricardian

and monetary policy is active (ρ′ (π∗) > 1),

then ifA22 > 0(< 0), there exists no (a contin-

uum of) perfect–foresight equilibria in which

{λ, π} converge asymptotically to the steady

state (π∗, λ∗).

In the case that monetary policy is active

and both eigenvalues of A are positive, there

may exist bounded equilibria that converge to

a stable cycle around the steady state. Note

that for the system (??), (??), and (35) the

dynamics of {λ, π} are independent of a, and

thus the analysis of periodic equilibria of the

previous section still applies. For example,

in the special case introduced earlier if cycles

30

exist, any initial condition for (λ, π) in the

neighborhood of the steady state will converge

to a cycle. To assure that a does not explode,

however, we must restrict ourselves to a one

dimensional manifold in {λ, π} . This follows

because while cycles restricted to the {λ, π}plane are attracting, in the three dimensional

space the cycle in {λ, π, a} will have only

a two dimensional stable manifold: initial

values of λ and π will have to be chosen to

assure that the triple {λ, π, a} converges to

the cycle and a remains bounded.

31

5

2.Real indeterminacy in the Sticky-PriceModel

Fiscal Policy

Monetary Policy Ricardian Non-Ricardian

Passive (ρ′(π∗) < 1) I D

Active (ρ′(π∗) > 1)

A22 < 0 I I

A22 > 0 I or D I or NE

Note: The notation is D, determinate; I, indeterminate; NE, no perfect-foresight equilibrium

exists.

32

5.1 Backward- and forward-

looking feedback rules

5.1.0 Flexible-price model

We now analyze a generalization of the

interest-rate feedback rule in which the

nominal interest rate depends not only on

current but also on past or future rates

of inflation. Consider first the following

backward-looking feedback rule

R = ρ(qπ + (1− q)πp); ρ′ > 0; q ∈ [0, 1](37)

where πp is a weighted average of past rates

of inflation and is defined as

πp = b

∫ t

−∞πeb(s−t)ds; b > 0 (38)

Differentiating this expression with respect to

time yields

πp = b(π − πp) (39)

The rest of the equilibrium conditions are

identical to those obtained earlier for a flexible

33

price system. In particular, we have that

λ′(R)R = λ(R)[r + π −R] (40)

where

λ′(R) = [uccy′mp′ + ucm(mnp

c y′mp′ + mnp

R )](41)

Using equation (42) to eliminate π from (44)

and (40) and linearizing around the steady

state results in the following system of linear

differential equations[Rπp

]=

[λλ′

(1ρ′q1− 1)− λλ′

(1−q1)q1

b 1ρ′q1

− bq1

] [R−R∗πp − π∗

]Let J denote the Jacobian matrix of this

system. Because R is a jump variable and πp

is predetermined, the real allocation is locally

unique if the real parts of the eigenvalues

of J have opposite signs, or, equivalently,

if the determinant of J is negative. On the

other hand, the real allocation is locally

indeterminate if both eigenvalues have

negative real parts, that is, if the determinant

of J is positive and its trace is negative. The

34

determinant and trace of J are given by

det(J) =λ

λ′b

ρ′q1(ρ′ − 1)

trace(J) =λ

λ′

(1

ρ′q1− 1

)− b

q1

35

. Now consider the two polar cases of

money entering only through preferences

(y′ = 0) and money entering only through

production (ucm = mnpR = 0). If money

enters only through preferences and money

and consumption are Edgeworth complements

(ucm > 0), then equation (41) implies that

λ′ is negative. It follows directly from the

above two expressions that the conditions

governing the local determinacy of R are

identical to those obtained under a purely

contemporaneous feedback rule. Namely, the

equilibrium is unique under active monetary

policy (ρ′ > 1) and is indeterminate under

passive monetary policy (ρ′ < 1).

When money enters only through pro-

duction or only through preferences with

consumption and money being Edgeworth

substitutes, λ′ is positive. Thus, the equi-

librium is always locally determinate under

passive monetary policy, as was the case un-

der purely contemporaneous feedback rules.

36

However, contrary to the case of purely con-

temporaneous feedback rules, if monetary

policy is active, then equilibria in which Rconverges to its steady state may not exist. To

see this, note that in this case the determinant

of J is positive, so that the real parts of the

roots of J have the same sign as the trace of

J . However, the trace of J can have either

sign. If the trace is positive, then no equilib-

rium exists. If it is negative, the equilibrium

is indeterminate. For large enough values of

ρ′ the trace of J becomes negative. Thus,

highly active monetary policy induces indeter-

minacy. Furthermore, the larger the emphasis

the feedback rule places on contemporane-

ous inflation (q close to one) or the lower the

weight it assigns to inflation rates observed

in the distant past (b large), the smaller is the

minimum value of ρ′ beyond which the equi-

librium becomes indeterminate. In the limit,

as q approaches unity or b approaches infinity,

the equilibrium becomes indeterminate un-

37

der every active monetary policy, which is the

result obtained under purely contemporane-

ous feedback rules. On the other hand, as the

monetary policy becomes purely backward

looking (q → 0), no equilibrium in which

R converges to its steady state exists under

active monetary policy.

38

5.2 How backward looking

should Taylor Rules be?

We now analyze a generalization of the

interest-rate feedback rule in which the

nominal interest rate depends not only on

current but also on past or future rates

of inflation. Consider first the following

backward-looking feedback rule

R = ρ(χπ + (1− χ)πp); ρ′ > 0; χ ∈ [0, 1](42)

where πp is a weighted average of past rates

of inflation and is defined as

πp = b

∫ t

−∞πeb(s−t)ds; b > 0 (43)

Differentiating this expression with respect to

time yields

πp = b(π − πp) (44)

5.2.0

39

Sticky-price model

The pattern that arises under sticky prices is

that if monetary policy is active, the intro-

duction of a backward-looking component

in monetary policy makes determinacy more

likely, whereas a forward-looking compo-

nent makes indeterminacy more likely. To

facilitate the analysis, we reproduce here the

equilibrium conditions for the sticky-price

model.

λ = λ [r + π −R]

π = r(π − π∗)− γ−1y(mp(λ,R))λ

[1 + η

(1− R

y′

)]where mp(λ,R) results from replacing ρ(π)by R in equation. Also, from the first

equations above:1

R = qρ′π + bρ′ (π − π∗)− b(R−R∗).1 Take the case:

R = a

[qπ + (1− q)b

(∫ t

−∞π (s) eb(s−t)ds

)− π∗

]+R∗

R = −b (R−R∗ − aqπ + aπ∗) + ab(1− q)π + aqπ

= −b (R−R∗) + ab (π − π∗) + aqπ

40

Using this expression and linearizing equa-

tions (??) and (??), the evolution of λ, π, and

R is described by the following system of

differential equations: λπ

R

= A

λ− λ∗π − π∗R−R∗

where

A =

0 uc −ucA21 r A23

ρ′qA21 ρ′(b + qr) −b + ρ′qA23

41

Note the sequence

−1 Trace(A) −B+Det(A)

Trace(A)Det(A),

For local determinacy we need change in sign

pattern from (−,−,−,−) to (−,−,+,−)as we vary b from∞ to 0. This is because,

DET remains negative, so no real roots change

sign but we go from no variations in the sign

pattern (no roots with positive real parts) to

two variations (two roots with real parts). (See

Gantmacher, vol 2, p.180: Routh’ scheme,

and also on page 196, Orlando’s formula)

Det(A) = bucA21 (1− ρ′)Trace(A) = r − b + ρ′qA23.

B = Sum of the principal minors of A

= −ucA21(1− qρ′)− b (r + ρ′A23) .

42

−B +Det(A)

Trace(A)

= (ucA21(1− qρ′) + b (r + ρ′A23)) +

(bucA21 (1− ρ′)r − b + ρ′qA23

)A21 = −ucc

∗ηR∗y′′mpλ

γy′2> 0

A23 =

(ucc∗η

γy′

)(1− R∗

y′y′′mp

R

).

We’ll assume

R = a

[χπ + (1− χ)b

(∫ t

−∞π (s) eb(s−t)ds

)− π∗

]+R∗

so ρ′ = a.Also assume χ = 0.

43

Transform variables:

q = `nλ, x = π − π∗ z = R−R∗Then

q = [r + x + π∗ − z −R∗] = x− zx. = rx− γ−1y(mp(eq, z + R∗))eq

·[

1 + η

(1− z + R∗

y′(mp(eq, z + R∗))

)]= rx + f (q, z)

z = bax− bz = b(ax− z)

Note that f (q, z) depends on the model:

money in production, in utility etc. The rest

is the same for different models. Also note

steady state is independent of b.

44

Steady State:

(q, x, z) = (q, 0, 0) where q solves1 + η

η=

R∗

y′(mp(eq, R∗))So

A =

0 1 −1A21 r A23

0 ab −b

and

Det(A) = bA21 (1− a)Trace(A) = r − b.

B = Sum of the principal minors of A = −A21−b (r + aA23) .

−B+Det(A)

Trace(A)= A21+b (r + aA23)+

(bA21 (1− a)

r − b

)Choose critical b to make the above quan-

tity equal to zero. Below b do we have

determinacy?

6

45

Conclusion

In this paper we have shown that the implica-

tions of particular interest rate feedback rules

for the determinacy of equilibrium depend not

only on the fiscal policy regime but also on

the structure of preferences and technologies.

An important consequence of this finding is

that the design of monetary policy should be

guided not just by the stance of fiscal policy

but also by the knowledge of the deep struc-

tural parameters describing preferences and

technologies, which significantly complicates

the task of the monetary policy maker.

46

6.1 Derivation of the π equation

The household’s lifetime utility function is

assumed to be of the form

U j =

∫ ∞0

e−rt

u(cj,mnpj)− γ

2

(P j

P j− π∗

)2 dt

(45)

where cj denotes consumption of the compos-

ite good by household j, mnpj ≡ Mnpj/Pdenotes real money balances held by house-

hold j for non-productive purposes, Mnpj de-

notes nominal money balances, and π∗ > −rdenotes the steady–state inflation rate. The

household’s instant budget constraint and

no-Ponzi-game restriction are

aj = (R−π)aj−R(mnpj+mpj)+P j

Py(mpj)−cj−τ

(46)

limt→∞

e−∫ t0 [R(s)−π(s)]dsaj(t) ≥ 0 (47)

In addition, given the price they charge, firm’s

47

sales are demand-determined

y(mpj) = Y dd

(P j

P

)(48)

The household chooses sequences for cj,mnpj, mpj, P j ≥ 0 and aj so as to maximize

utility subject to (58)–(29) taking as given

aj(0), P j(0), and the time paths of τ , R, Y d,

and P . The Hamiltonian is

e−rt

{u(cj,mnpj)− γ

2

(P j

P j − π∗)2

+λj[(R− π)aj −R(mnpj + mpj)

+P j

P y(mpj)− cj − τ − aj]

+µj[Y dd(P j

P

)− y(mpj)

] The first–order conditions associated with cj,mnpj, mpj and aj, are, respectively,

uc(cj,mnpj) = λj (49)

um(cj,mnpj) = λjR (50)

λj[P j

Py′(mpj)−R

]= µjy′(mpj) (51)

48

λj = λj (r + π −R) (52)

limt→∞

e−∫ t0 [R(s)−π(s)]dsaj(t) = 0 (53)

Combining equations (60) and (50), the

demand for real balances for non-production

purposes can be expressed as

mnpj = mnp(cj, R) (54)

which by assumption is increasing in cj and

decreasing in R. Consider now the first–order

condition with respect to P j. Define

K = e−rt

−γ2

(P j

P j− π∗

)2

+ λjP j

Py(mpj) + µjY dd

(P j

P

).Then the first–order condition with respect to

P j can be expressed as

KP j =dK .

Pj

dt(55)

where KP j and K .

Pj denote the partial

derivatives of K with respect to P j and P j,

49

respectively. Letting πj ≡ P j/P j, yields

KP j =e−rt

P j

[γ(πj − π∗

)πj + λjP j

P y(mpj)

+µj PjY d

P d′(P j

P

) ]K .

Pj = −γe

−rt

P j(πj − π∗)

dK .

Pj

dt=γe−rt

P j

[(r + πj)(πj − π∗)− πj

]Equation (55) can then be written as

λjP j

Py(mpj)+µj

P j

PY dd′

(P j

P

)= γr(πj−π∗)−γπj

(56)

50

Substituting λj = uc,λj[Pj

P y′(mpj)−R

]y′(mpj)

= µj

from above, and using P j = P in equilibrium,

γr(πj − π∗)− γπj

= ucy(mpj) +

(uc[y′(mpj)−R

]y′(mpj)

)Y dd′

(P j

P

)= ucy(mpj)

(1 + η

(y′ −Ry′

))= ucy(mpj))η

(1 + η

η− R

y′

)LHS has interpretation of marginal revenue

per unit (η = d′ in equilibrium are units, 1+η is MR, Ry′ = um

y′ucis MC per unit of foregone

utility from money).

51

7 Optimal Monetary

Policies

y is output, z is the natural rate, x is the output

gap in logs: xt = yt − ztIS-Euler eq.:2

xt = −φ [it − Eπt+1] + Etxt+1 + gtPhillips curve:

πt = λxt + βEπt+1 + utgt = µgt−1 + gtut = ρut−1 + ut

Iterating forward: (future beliefs affect current

output)

xt = Et

∞∑i=0

−φ [it+i − πt+1+i] + gt+i

πt = Et

∞∑i=0

βi [λxt+i + ut+i]

2 If Yt = Ct + Et where E is government consumption,

yt − et = −φ [it − Etπt+1] + Et [yt+1 − et+1]where et = − log

(1− Et

Yt

).Then

52

where ut+i is cost push, whereas xt+i is

marginal costs associated with excess demand.

ut+i allows variations in inflation not due to

excess demand.

Policy

max−1

2Et

∞∑i=0

βi[αx2

t+i + π2t+i

]Discretion:

Maxxt,πt −1

2

[αx2

t + π2t

]+ Ft

ST πt = λxt + ft

Ft = −1

2Et

∞∑i=1

βi[αx2

t+i + π2t+i

]ft = βEtπt+1 + ut

because under discretion, govt. reoptimizes

each period. Solution:

xt = −λαπt

53

Substituting into phillips curve:

πt = −λ2

απt + βEπt+1 + ut

= (1 +λ2

α)−1Etβ

[(1 + λ2

α )−1βEtπt+2

+(1 + λ2

α )−1ut+1

]

+(1 +λ2

α)−1ut

=

∞∑s=t

(1 +λ2

α)−(s−t+1)usβ

s−t

= (1 +λ2

α)−1

∞∑s=t

(1 +λ2

α)−(s−t)ut (ρβ)s−t

=ut(1 + λ2

α )−1

1− (1 + λ2

α )−1ρβ=

ut

(1 + λ2

α )− ρβ=

αut

λ2 + α (1− ρβ)≡ αqut

Similarly:

xt = −λqutNow find interest rule by substituting into IS

54

curve:

−λqut = −φ [it − Eπt+1]− λq (ρut) + gtit = Eπt+1 + φ−1gt − φ−1λq (ρut)

But πt = αqut, Etπt+1 = αqρut

φ−1λq (1− ρ)ut = φ−1(1− ρ)λ

αρEtπt+1

it =

(1 +

(1− ρ)λ

αρ

)Eπt+1 + φ−1gt

Note

(1 + (1−ρ)λ

αρ

)> 1, so active Taylor rule.

Under Commitment:

max−1

2Et

∞∑i=0

βi[

αx2t+i + π2

t+i+φt+i (πt+i − λxt+i − βπt+i+1 − ut+i)

]FOC

−φt+iβ + 2βπt+i + βφt+i+1 = 0

φt+i+1 = φt+1 − 2πt+i+1

2αxt+i+1

λ=

2αxt+iλ− 2πt+i+1

xt+i+1 − xt+i = −λαπt+i+1

xt = −λαπt

55

Now plug into IS

xt = −φ [it − Etπt+1] + Etxt+1 + gtit = Etπt+1 + φ−1 (Etxt+1 − xt + gt)

it = Etπt+1 + φ−1gt − φ−1λ

απt+1

= Etπt+1

(1− φ−1λ

α

)+ φ−1gt

Note the passive Taylor Rule. But note also

it = Etπt+1 + φ−1gt − φ−1λ

απt+1

= Etπt+1

(1− φ−1λ

α

)+ φ−1gt

= aqρut

(1− φ−1λ

α

)+ φ−1gt

so it is a function of the two shocks and

where rt = aqρut(1− φ−1λ

α

)+ φ−1gt can be

interpreted as a variable natural rate. To avoid

indeterminacy add any active Taylor Rule-(off

equilibrium threat) path to implement the

precise optimal equilibrium :

it = aqρut

(1− φ−1λ

α

)+φ−1gt+BEπt+1, B > 1

56

57

The Perils of Taylor Rules

Note: Figures can be found in the paper

58

“To summarize, the research proposed

her shows the surprising efficiency and

robustness of simple policy rules in which

the reaction of the interest rate is above a

critical treshold. The analysis also shows

that the estimated gains reported in some

research from following alternative rules are

not robust to a variety of models considered in

this paper”

John Taylor,“ The robustness and effi-

ciency of monetary policy rules as guidelines

for interest rate setting by the European

Central Bank,” May 1988.

59

8 Introduction

λ

λ= r + π −R (π)

Steady State : r + π −R (π) = 0

Active policy R′ (π∗) > 1

Passive policy R′ (π∗) > 1

R (π) ≥ 0

0 π

r+π

R(π)

0 π

r+ π

R(π)

0 π

r+π

R(π)

0 π

r+ π

R(π)

Taylor Rules, zero-bound on nominal rates,

and multiple steady states

60

9 A Sticky-Price Model

Firm j :

Y j = y(hj) = Y dd(P j/P )

d(1) = 1 d′(1) < −1.

U j =

∫ ∞0

e−rt

u(cj,mj)− z(hj)− γ

2

(P j

P j− π∗

)2 dt

π∗ > −r. (57)

aj = (R− π)aj −Rmj +P j

Py(hj)− cj − τ

(58)

and

limt→∞

e−∫ t0 [R(s)−π(s)]dsaj(t) ≥ 0, (59)

The household chooses sequences for cj, mj,

P j ≥ 0, and aj so as to maximize U j taking

as given aj(0), P j(0), and the time paths of τ ,

R, Y d, and P .

61

The first-order conditions associated with

the household’s optimization problem are

uc(cj,mj) = λj (60)

um(cj,mj) = λjR (61)

z′(hj)

= λjP j

Py′(hj)− µjy′(hj) (62)

λj

= λj (r + π −R) (63)

λjP j

Py(hj)+µj

P j

PY dd′

(P j

P

)= γr(πj−π∗)−γπj

(64)

limt→∞

e−∫ t0 [R(s)−π(s)]dsaj(t) = 0 (65)

where πj ≡ P j/P j.

62

9.0 Monetary and Fiscal Policy

R = R(π) ≡ R∗eAR∗ (π−π

∗) (66)

The instant budget constraint of the

government is given by

a = (R− π)a−Rm− τ , (67)

That is, the monetary-fiscal regime ensures

that total government liabilities converge

to zero in present discounted value for all

(equilibrium or off-equilibrium) paths of the

price level or other endogenous variables:3

limt→∞

e−∫ t0 [R(s)−π(s)]dsa(t) = 0. (68)

3 As discussed in Benhabib, Schmitt-Grohé and Uribe (1998), an

example of a Ricardian monetary-fiscal regime is an interest-rate

feedback rule like (73) in combination with the fiscal rule

τ + Rm = αa; α > 0. In the case in which α = R, this fiscal

rule corresponds to a balanced-budget requirement.

63

9.0 Equilibrium

c = y(h). (69)

m = m(c, R). (70)

Let η ≡ d′(1)<−1 denote the equilibrium

price elasticity of the demand function faced

by the individual firm

λ = λ [r + π −R(π)]

π = r(π − π∗)− y(h(λ, π))λ

γ

[1 + η − ηz′(h(λ, π))

λ y′(h(λ, π))

]We define a perfect-foresight equilibrium

as a pair of sequences {λ, π} satisfying (??)

and (??). Given the equilibrium sequences

{λ, π}, the corresponding equilibrium se-

quences {h, c, R,m} are uniquely determined

.

64

10 Steady-state equilibria

0 = r + π −R∗e AR∗ (π−π

∗)

0 = r (π − π∗)− λy(h(λ, π))

γ

(1 + η − η z

′ (h (λ, π))

λy′ (h (λ, π))

)

65

11 Local equilibria(λπ

)= J

(λ− λ∗π − π∗

)J =

[0 uc(1− A)J21 J22

]J21 =

yη

y′γ

[(z′′ − z′y′′

y′

)hλ −

z′

λ

]> 0

J22 = r +yη

y′γ

(z′′ − z′y′′

y′

)hπ

The sign of J22 depends on the sign of

hπ, which in turn depends on whether

consumption and real balances are Edgeworth

substitutes or complements. Specifically, J22

is positive if ucm ≥ 0, and may be negative if

ucm < 0.

If monetary policy is active at π∗ (A > 1),

the determinant of J is positive, and the real

part of the roots of J have the same sign.

Since both λ and π are jump variables, the

equilibrium is locally determinate if and

66

only if the trace of J is positive. It follows

that if ucm ≥ 0, the equilibrium is locally

determinate. If ucm < 0, the equilibrium

may be determinate or indeterminate. If

monetary policy is passive at π∗, (A < 1),

the determinant of J is negative, so that the

real part of the roots of J are of opposite

sign. In this case, the equilibrium is locally

indeterminate.

67

12 Global equilibria

u(c,m)− z(h) =

[(xcq + (1− x)mq)

1q

]ww

− h1+v

1 + vq, w ≤ 1, v > 0

The restrictions imposed on q and w ensure

that u(·, ·) is concave, c and m are normal

goods, and the interest elasticity of money

demand is strictly negative.Note that the sign

of ucm equals the sign ofw−q.The production

function takes the form

y(h) = hα; 0 < α < 1

In the recent related literature on de-

terminacy of equilibrium under alternative

specifications of Taylor rules, it is assumed

that preferences are separable in consumption

and real balances (e.g., Woodford, 1996; Clar-

ida, Galí, and Gertler, 1998). We therefore

characterize the equilibrium under this prefer-

ence specification first, before turning to the

more general case.

68

12.1 Separable preferences

q = w :

Throughout this subsection we will assume

that

R∗ = r + π∗.Let p ≡ π − π∗ and n ≡ ln(λ/λ∗).

n = R∗ + p−R∗e(AR∗)p (71)

p = rp− γ−1 (1 + η) (λ∗)ω eωnxαθ

+α−1γ−1η (λ∗)β eβnxθ(1+v) (72)

The main result of this subsection is: if

r, A − 1 > 0 and sufficiently small, then

there exists an infinite number of trajectories

originating in the neighborhood of the steady

state (λ∗, π∗) (where monetary policy is

active) that are consistent with a perfect

foresight equilibrium.

69

Proposition 11 (Global indeterminacy under

active monetary policy and separable prefer-

ences) Suppose preferences are separable in

consumption and real balances (q = w). Then,

for r and A− 1 positive and sufficiently small,

the equilibrium exhibits indeterminacy as fol-

lows: trajectories originating in the neighbor-

hood of the steady state (λ, π) = (λ∗, π∗),where monetary policy is active, converge ei-

ther to a limit cycle or to the other steady state,

(λ, π), where monetary policy is passive. In

the first case, the dimension of indeterminacy

is two, while in the latter it is one.

70

12.1.1 Simulations

u (c,m, h) = w−1(

(xcq + (1− x)mq)1q

)w− (1 + v)−1 h(1+v)

w ≤ 1, q ≤ 1; y (h) = hα; R (π) = R∗e(AR∗)(π−π∗)

where R∗ = r + π∗ so that we have a steady

state (y, n) = (0, 0) at the specied parameter

values. We start with:

R∗ = 0.07; r = 0.03; γ = 5;α = .7;A = 1.45;

η = −50; v = 1;w = .15; q = .15;x = .975

Note in particular, that money receives a very

minor in the utility function, with (1− x) =0.025. The coefficient of adjustment costs

γ = 5, and η = −50 reflecting a fairly

competitive economy. A = 1.45 is roughly

the value specified by Taylor. Setting ω = qgives a utility function separable in money

and consumption. The other parameters are

standard. In general we should note that little

71

changes in the simulations if parameters are

changed, except of course where changes of

stability of the steady state (0, 0) occurs,

corresponding to the curves H and P in Figure

5, but indeterminacy of some form always

persists

72

1.205 1.21 1.215 1.22 1.225 1.23 1.235 1.24­0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

λ

π

active steady state: λ=1.22, π=0.042passive steady state: λ=1.22, π=0.007

Separable preferences: Saddle connection

from the active to the passive steady state

73

1.18 1.2 1.22 1.24­0.02

00.020.040.06

λ

πA = 2

1.2 1.21 1.22 1.23­0.02

0

0.02

0.04

λ

π

π* = .03/4

1.214 1.216 1.218 1.22 1.222

0

0.02

0.04

0.06

λ

π

γ = 35

1.22 1.23 1.24 1.25

0

0.02

0.04

0.06

λ

π

c/m =  20/4

1.2 1.21 1.22 1.230

0.020.040.060.08

λ

π

r = .04/4

1.28 1.3 1.32 1.34

0

0.02

0.04

0.06

λ

π

η /(1+ η  ) =  1.2

Separable Preferences: Sensitivity analysis

74

75

12.2 Non-separable pref.(q 6= w)

n = r + π∗ + p−R∗e(AR∗)p

p = rp− 1 + η

γxαθ(λ∗)ωeωn[

(R∗)χ(

1− xx

)1−χeAχpR∗ + 1

]αξ+

η

αγx(1+v)θ(λ∗)βeβn[

(R∗)χ(

1− xx

)1−χe

AR∗χp + 1

](1+v)ξ

,

where n, p, β, ω are defined the previous

section and χ ≡ q/(q − 1), ξ ≡ (w −q)/[αq(1− w)] 6= 0, θ ≡ w/[αq(1− w)].

76

Proposition 12 Suppose w < 0. Then, the

steady states of the system satisfy: (i) for each

steady-state value of p there exists a unique

steady-state value of n; and (ii) the steady state

at which monetary policy is active is either a

sink or a source and the steady state at which

monetary policy is passive is always a saddle.

77

Proposition 13 For parameter specifications

(r, A) sufficiently close to (rc, 1), the economy

with non-separable preferences exhibits inde-

terminacy as follows: There always exist an

infinite number of equilibrium trajectories orig-

inating arbitrarily close to the steady state at

which monetary policy is active that converge

either to: (i) that steady state; (ii) a limit cy-

cle; or (iii) the other steady state at which mon-

etary policy is passive. In cases (i) and (ii) the

dimension of indeterminacy is two, whereas in

case (iii) it is one.

Corollary 14 (Periodic equilibria) If− 1−BB(1+v+α) <

ξ < 0, then there exists a region in the neigh-

borhood of (r, A) = (rc, 1) for which the ac-

tive steady state is a source surrounded by a

stable limit cycle. On the other hand, if ξ > 0or ξ < − 1−B

B(1+v+α), then stable limit cycles do

not exist.

78

1.205 1.21 1.215 1.22 1.225 1.23 1.235 1.24­0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

λ

π

active steady state: λ=1.22, π=0.042passive steady state: λ=1.22, π=0.007

1.Non-separable preferences, w<q: saddleconnection from the active to the passivesteady state

1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23 1.24­0.04

­0.02

0

0.02

0.04

0.06

0.08

λ

π

active steady state: λ=1.21, π=0.042passive steady state: λ=1.21, π=0.007

Non-separable preferences, w>q: saddle

connection from the active to the passive

steady state

79

12.3 A staggered contract model

of Calvo(1983)

R (π) =UmUc

c =c

εc(R (π)− r − π)

π = b (q − c)Let n = `n c :

n =1

εc(R (π)− r − π)

π = bq − cenIf we define

F (π)

dπ= (r + π −R (π))

dG (n)

dz= bq − cen

and the Hamiltonian Function:

H = F (π)−G (n)solutions are level sets ofH, and

dHdt

=0.

80

Theorem (Kopell and Howard, 1975,

Theorem 7.1): Let X = Fµ,ν(X) be a

two-parameter family of ordinary differential

equations on R2, F smooth in all of its four

arguments, such that Fµ,ν(0) = 0. Also

assume:

1. dF0,0(0) ≡ A has a double zero eigenvalue

and a single eigenvector e.

2. The mapping (µ, ν)→ (det dFµ,ν(0), tr dFµ,ν(0))has a nonzero Jacobian at (µ, ν) = (0, 0).

3. Let Q(X,X) be the 2 × 1 vector con-

taining the terms quadratic in the xi and

independent of (µ, ν) in a Taylor series

expansion of Fµ,ν(X) around 0. Then

[dF(0,0)(0), Q(e, e)] has rank 2.

Then: There is a curve f (µ, ν) = 0 such

that if f (µ0, ν0) = 0, then X = Fµ0,ν0(X)has a homoclinic orbit. This one-parameter

family of homoclinic orbits (in (X,µ, ν)space) is on the boundary of a two-parameter

family of periodic solutions. For all |µ|, |ν|sufficiently small, if X = Fµ,ν(X) has neither

81

a homoclinic orbit nor a periodic solution,

there is a unique trajectory joining the critical

points.

The following proposition shows that the

equilibrium conditions of the economy with

separable preferences satisfy the hypotheses

of this theorem.

82

Y. Kuznetsov, Elements of Applied

Bifurcation Theory, Springer 1998.

83

Discrete time sticky price:

maxa,m,h,pj

∞∑t=0

βt

(1− σ)−1 (ct)1−σ − (1 + φ)−1 (ht)

1+φ

−θ2

(pjtpjt−1− π∗

)2

ST

πtat + ct−1 ≤ Rt−1at−1 − (1−Rt−1)mt−1

+pjt−1

pt−1Y dt d

(pjt−1

pt−1

)− τ t−1

f (mt, ht) = Y dt d

(pjtpt

)

y = f (mt, ht) = Y dt d

(pjtpt

)= ct

y = (a (m− m)ρ + (1− a)hρ)1ρ

84

L = (1− σ)−1

(Rt−1at−1 − (1−Rt−1)mt−1

+pjt−1pt−1Y dt d(pjt−1pt−1

)− τ t−1 − πtat

)1−σ

− (1 + φ)−1 (ht)1+φ − θ

2

(pjt

pjt−1

− π∗)2

+λt

(f (mt−1, ht−1)− Y d

t d

(pjt−1

pt−1

))FOC:

(ct−1)−σ = β (ct)−σ Rt

πt

−λtfm(mt−1, ht−1) = β (ct)−σ Rt

πt+1= (ct)

−σ

λt = (ct)−σ Rt − 1

Rtfm(mt, ht)

(ht)−φ

fh(mt, ht)= λt = (ct)

−σ Rt − 1

Rtfm(mt, ht)

(ht)−φ

(ct)−σ = fh(mt, ht)

Rt − 1

Rtfm(mt, ht)= wt

Rt − 1

Rtwt=fm(mt, ht)

fh(mt, ht)

85

0 = (ct)−σ

(Y dt d

(pjtpt

)+pjtptY dt d′

(pjtpt

))(1

pt

)−θ(pjt

pjt−1

− π∗)(

1

pjt−1

)

−λtY dt d′

(pjtpt

)(1

pt

)+βθ

(pjt+1

pjt− π∗

)(pjt+1

pjt

)(1

pjt

)0 = (ct)

−σ (Y dt d + Y d

t d′)− θ (πt − π∗) πt

−λtY dt d′ + βθ (πt+1 − π∗) πt+1

Let η = d′

d .

0 = Y dt d (ct)

−σ (1 + η)− θ (πt − π∗) πt− (ct)

−σ Rt − 1

Rtfm(mt, ht)Y dt d′ + βθ (πt+1 − π∗) πt+1

0 = Y dt d (ct)

−σ(

1 + η − Rt − 1

Rtfm(mt, ht)η

)− θ (πt − π∗) πt

+βθ (πt+1 − π∗) πt+1

86

0 = Y dt d (ct)

−σ(

1 + η − wtfh(mt, ht

η

)− θ (πt − π∗) πt

+βθ (πt+1 − π∗) πt+1

(πt − π∗) πt = β (πt+1 − π∗) πt+1 +

θ−1Y dt d (ct)

−σ η

(1 + η

η− wtfh(mt, ht)

)Since Y d

t d(pjtpt

)= ct.

(πt − π∗) πt = β (πt+1 − π∗) πt+1 +

θ−1 (ct)1−σ η

(1 + η

η− wtfh(mt, ht)

)Eqs to solve for system in (c, π) :

(ht)−φ

(ct)−σ = fh(mt, ht)

Rt − 1

Rtfm(mt, ht)f (mt, ht) = ct

The two equations above give (mt, ht) as

functions of ct and Rt.The Taylor rule below

gives Rt as a function of πt.

87

Rt = g (πt)Then we can substitute into the two difference

equations (correspondences?) below to get a

two dimensional system.

(ct)−σ = β (ct+1)−σ

Rt

πt+1

(πt − π∗) πt = β (πt+1 − π∗) πt+1 +

θ−1 (ct)1−σ η

(1 + η

η− wtfh(mt, ht)

)

88

13

Avoiding Liquidity Traps

Jess Benhabib, New York University

Stephanie Schmitt-Grohe, Rutgers Uni-

versity

Martin Uribe, U. of Penn.

89

Policies to escape or avoid liquidity

traps

1.Expansionary Fical policy: Commit-

ment to deficit spending

2.Expansionary Monetary Policy: Com-

mitment to high monetary growth

90

Early Models of Indeterminacy in

Monetary Models

Brock (IER, 1975): Hyperinflation and

hyperdeflation

Calvo (IER, 1979) Local Indeterminacies

F. Black (JET, 1978) Indeterminacies

from monetary feedback rules

91

Model∫ ∞0

e−rtu(c,M/P )dt,

Pc + Pτ + M + B = RB + Py.Let m ≡ M/P denote real balances and

a ≡ (M + B)/P real financial wealth. Let

ucm > 0.

c + τ + a = (R− π)a−Rm + y,

where π ≡ P /P is the instant rate of inflation.

limt→∞

e−∫ t0 [R(s)−π(s)]dsa(t) ≥ 0

uc(c,m) = λ

um(c,m) = λR

λ = λ (r + π −R)

0 = limt→∞

e−∫ t0 [R(s)−π(s)]dsa(t)

92

Monetary Policy

R = R(π). (73)

We refer to monetary policy as active at an

inflation rate π if R′(π) > 1 and as passive if

R′(π) < 1.

Assumptions

R′(π) ≥ 0 ∀π.R(π) ≥ 0 ∀π.

∃ π∗ > −r : R(π∗) = r + π∗ and R′(π∗) > 1Examples

R = R(π) = R∗( ππ∗

) AR∗

R = R(π) ≡ R∗eAR∗ (π−π

∗)

93

FISCAL POLICY

a = (R− π)a−Rm− τ a(0) =A(0)

P (0)RICARDIAN

limt→∞

e−∫ t0 [R(s)−π(s)]dsa(t) = 0

τ + Rm = α aα is chosen arbitrarily by the government

subject to the constraint that α ≥ α > 0. This

policy states that consolidated government

revenues are always higher than a certain

fraction α of total government liabilities. A

special case is a balanced-budget rule: tax

revenues are equal to interest payments on the

debt: α = R, provided R is bounded away

from zero.

limt→∞

e−∫ t0 [R(s)−π(s)]dsa(t) = a(0) lim

t→∞e−

∫ t0 α(π(s))ds

NON-RICARDIAN

94

A(t)

P (t)=

∫ ∞t

e−∫ vt [R(π)−π]ds {R(π (v))m (π (v)) + τ} dv

LOCALLY RICARDIAN

τ + Rm = α(π)a; α′ > 0. (74)

Assume

α(π∗) > 0, α(πL) ≤ 0

95

FISHER EQ

R (π) = r + π

ππ*πL

π<0⋅ π>0⋅π>0⋅

 r+πR(π)

EQUILIBRIUM

c = y

Assuming that consumption and real balances

are Edgeworth complements (ucm > 0) and

that the instant utility function is concave in

real balances (umm < 0), equations

λ = L(R); L′ < 0.

π =−L(R(π))

L′(R(π))R′(π))[R(π)− π − r]

96

a = (R(π)− π − α)a

97

Example 1: Targeting the growth rate

of nominal government liabilities

A

A= k,

where k is assumed to satisfy

R(πL) ≤ k < R(π∗)

Expressing A/A as a/a + π and combining

the above fiscal policy rule with the instant

government budget constraint yields

τ + Rm = (R(π)− k)a

This fiscal policy rule is a special case of the

one given when α(π) takes the formR(π)−k..

Under this policy, the government

manages to fend off a low inflation equilibrium

by threatening to implement a fiscal stimulus

package consisting in a severe increase in

the consolidated deficit should the inflation

rate become sufficiently low. Interestingly,

this type of policy prescription is what the

U.S. Treasury as well as a large number of

98

academic and professional economists are

advocating as a way for Japan to lift itself out

of its current deflationary trap.

99

Example 2: A balance-budget require-

ment

Consider now a fiscal policy rule consist-

ing of a zero secondary deficit, that is:

Pτ = RB,

Recalling that a = B/P + m, we can rewrite

the balanced budget rule as

τ + Rm = R (π) a

It follows that a balanced budget requirement

is a special case of the fiscal policy rule

in which α(π) = R(π). Clearly, in this

case α(π) is increasing because so is R(π).Condition

α(π∗) > 0is satisfied because R(π∗) is by assumption

greater than zero. However, condition

α(πL) ≤ 0

is only satisfied ifR(πL) = 0. This means that

a balanced budget rule is effective in avoiding

a liquidity trap only in the case where the

Taylor rule leads the monetary authority to cut

nominal rates all the way to zero at sufficiently

100

low rates of inflation, so that the low inflation

steady state occurs at a zero nominal interest

rate.

101

Example 3. Monetary Policy Regime

Switch

M

M= µ > −r m

m= µ− π

m ^ m

m⋅

0

m =r + µ− um(y,m)

uc(y,m)

1m + ucm(y,m)

uc(y,m)

Transversality:

limt→∞

e−∫ t0 [

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

−µ]dsM(0)+e−∫ t0um(y,m)uc(y,m)

dsB(t) = 0,

102

mLm* m

m⋅

0

m =uc(y,m)

ucm(y,m)[r + π(m)−R(π(m))] ,

0 ≤ B(t) ≤ Begt; B ≥ 0.

103

mLm*  m~m ^ m

m⋅

0

m =

uc(y,m)ucm(y,m) [r + π(m)−R(π(m))] for m ≤ m[

1m + ucm(y,m)

uc(y,m)

]−1 [r + µ− um(y,m)

uc(y,m)

]for m > m

.

m∗ < m < m < mL

πL < µ < π∗.

um(y,m∗)/uc(y,m∗) = r + π∗

um(y, m)/uc(y, m) = r + µ

um(y,mL)/uc(y,mL) = r + πL.

104

14

Chaotic Interest Rate Rules

Jess Benhabib, New York University

Stephanie Schmitt-Grohe, Rutgers Uni-

versity

Martin Uribe, U. of Penn.

105

Policies to escape or avoid liquidity

traps

1.Expansionary Fical policy: Commit-

ment to deficit spending

2.Expansionary Monetary Policy: Com-

mitment to high monetary growth

106

Early Models of Indeterminacy in

Monetary Models

Brock (IER, 1975): Hyperinflation and

hyperdeflation

Calvo (IER, 1979) Local Indeterminacies

F. Black (JET, 1978) Indeterminacies

from monetary feedback rules

107

Model ∞∑t=0

βtct

1−σ

1− σ

Mt+1+Bt+1 = Mt+Bt(1+Rt)+ptF

(Mt+1

pt

)−ptct−ptτ t

ct =Mt

pt+Bt

pt(1+Rt)+F

(Mt+1

pt

)−τ t−

Mt+1

pt−Bt+1

pt(75)

Let at ≡ (Mt +Bt)/pt = mt + bt and let

the inflation rate be defined by πt = pt+1pt.To

prevent Ponzi games, households are subject

to a borrowing constraint, where

limt→∞

at∏t−1j=0((1 + Rj) /πj+1)

≥ 0 (76)

The marginal productivity of money at the

optimum is equal to the opportunity cost of

holding money:

F ′(Mt+1

pt

)=

Rt+1

1 + Rt+1

ct−σ =

β (1 + Rt+1)

πtct+1

−σ (77)

108

ct−σ(

1− F ′(Mt+1

pt

))=β

πtct+1

−σ (78)

109

yt+1 = F

(Mt+1

pt

)=

(a

(Mt+1

pt

)ρ+ (1− a) yρ

)1ρ

(79)

The implied money demand equation then is

given by(1− F ′

(Mt+1

pt

))=

1− a((

Mt+1

pt

)−1

yt+1

)1−ρ

= (1 + Rt+1)−1

for which real balances are a decreasing

function of the nominal rate of interest R.Note that the interest elasticity of real

balances in this formulation is 1− ρ, which is

the reason that we prefer it to a Cobb-Douglas

specification with unitary interest elasticity.

Finally the transversality condition for the

agent, the flip side of the no-Ponzi condition

is given by:

limt→∞

at∏t−1j=0((1 + Rj) /πj+1)

≤ 0 (80)

110

Monetary Policy

Rt = R∗(πt−1

π∗

) AR∗

; (81)

where 1 +R∗ ≡ π∗/β > 1, π∗ > 1, A > R∗.Under this rule the nominal rate Rt on bonds

Bt, set by the central bank in period t − 1,depends (in the terminology of Carlstrom

and Fuerst (2000)) on the forward-looking

inflation rate between t − 1 and t. Inverting

this equation, we obtain:

πt = π∗(Rt+1

R∗

)R∗A

(82)

where 1 + R ≡ π/β > 1.

111

FISCAL POLICYEach period the government faces

Mt + Bt = Mt−1 + (1 + Rt−1)Bt−1 − Ptτ tor

at =1 + Rt−1

πtat−1 −

Rt−1

πtmt−1 − τ t (83)

Total government liabilities in period t are

given by liabilities carried over from the

previous period, including interest, minus

total consolidated revenues. Consolidated

government revenues have two components,

regular taxes and seignorage revenue. We

assume that the fiscal regime consists of

setting consolidated government revenues as a

fraction of total government liabilities.Rt−1

πtmt−1 + τ t = ω (πt) at−1; (84)

If ω (·) > 0 for all π, this expression implies

limt→∞

at∏t−1j=0 (1 + Rj) /πj+1)

= 0. (85)

Therefore, government solvency is assured.

On the other hand, if limt→∞πt = πp on a

112

candidate equilibrium path and ω (πp) < 0,

transversality conditions will fail because

the limit of discounted government liabilities

held does not converge to zero. Such a fiscal

policy, with ω (πp) < 0, will be locally

non-Ricardian.

113

FISHER EQ

Let β = (1 + r)−1 . At the steady

state,1 + R ≡ π/β = π (1 + r) > 1

R (π) = (π − 1) + πr

114

EQUILIBRIUM

yt = ct =

(a

(Mt+1

pt

)ρ+ (1− a) yρ

)−σρ

F ′(Mt+1

pt

)=

Rt+1

1 + Rt+1(a

(Mt+1

pt

)ρ+ (1− a) yρ

)−σρ

=β (1 + Rt+1)

π∗(Rt+1

R∗

)R∗A

(a

(Mt+2

pt+1

)ρ+ (1− a) yρ

)−σρ

An equilibrium allocation is a sequence

{(Mt+1

pt

), πt, at}∞t=1, given a0, satisfying

Rt > 0, the Euler equation, the equation for

the dynamics of a,the Taylor Rule, and the

transversality and no Ponzi conditions. Can

write it as a difference equation in Rt+1.

115

Given a0 and a pair of sequences

{(Mt+1

pt

)t, πt}∞t=1 satisfying the Euler equa-

tion, if ω(·) > 0, the transversality condi-

tion is satisfied. Thus, in such a case we can

limit attention to the existence of sequences

{(Mt+1

pt

), πt}∞t=1 satisfying the Euler equa-

tion and the Taylor rule. On the other hand,

if ω(π) < 0 for some π, some of the candi-

date equilibrium trajectories will fail to satisfy

transversality, and may be ruled out.

116

Steady States

Consider constant solutions xt ≡(Mt

pt

)ρ= xss > 0 to the Euler equation.

Because xt is not predetermined in period

t (i.e., xt is a “jump” variable), such solu-

tions represent equilibrium real allocations.

The steady state solutions can be more easily

constructed first in terms of the nominal rate

Rt = Rss. The steady state solutions πss and

Rss = R∗(ππ∗

) AR∗must satisfy

1 + R∗( ππ∗

) AR∗

= β−1π

By construction one such pair is (π∗, R∗) .There

exists however another solution, (πp, Rp) ,where

πp solves the above equation with πp < π∗

and 0 < Rp < R∗ and πp ≡ β(1 + Rp). It

is straightforward to show that at the steady

state equilibrium R∗ monetary policy is ac-

tive (R′(π∗) > 1), whereas at the steady-state

equilibrium Rp monetary policy is passive

(ρ′(πp) < 1).

117

(a

(Mt+1

pt

)ρ+ (1− a) yρ

)−σρ

=β (1 + Rt+1)

π∗(Rt+1

R∗

)R∗A

(a

(Mt+2

pt+1

)ρ+ (1− a) yρ

)−σρ

Let xt+1 =(Mt+1

pt

)ρ, zt = axt + (1− a) yρ

, qt = `n zt, so that zt = eqt. Then the

Euler equation can be written as an explicit

difference equation in zt:

zt+1 =

βR∗

R∗A

π∗

(1− a

(azt

zt−(1−a)yρ

)1−ρρ

)−1

((1− a

(azt

zt−(1−a)yρ

)1−ρρ

)−1

− 1

)RA

∗

ρσ

zt

(86)

118

qt+1 = qt +ρ

σ`n

βR∗

R∗A

(1− a

(aeqt

eqt−(1−a)yρ

)1−ρρ

)−1

π∗

((1− a

(aeqt

eqt−(1−a)yρ

)1−ρρ

)−1

− 1

)R∗A

qt+1 = qt +

ρ

σh (eqt) ≡ qt +

(−ρσ

)f (qt) (87)

119

Let qt+1 = F∆ (qt) = qt + ∆f (qt) , ∆ =−ρσ , where f is continuous in R1.Assume the

differential equation y = f (y) has two sta-

tionary points, one of which is asymptotically

stable.

As pointed out in Yamaguti and Matano

(1979), this assumption implies that after a

linear transformation of variables, we have:

A) f (0) = f (u) = 0 for some u > 0.B) f (u) > 0 for 0 < u < u.C) f (u) < 0 for u < u < κ where the

constant κ is possibly +∞.Note that the two steady states x∗

and xp in the transformed variables qp and

q∗ will correspond to the zeros of f (qt) .Then defining ut = qt − qp provides the

appropriate linear transformation required

above. Furthermore, the assumption used

by Yamaguti and Matano (1979) is weaker

than the one above, requiring F∆ (qt) to have

at least two steady states, or f (u) to have

at least two zeros. Requiring exactly two

120

steady states, as above, assures that κ = +∞,since otherwise there would be some u > usuch that f (u) = 0, implying the existence

of a third steady state. We use the fact that

our stronger assumption of two steady states

implies κ = +∞ in Theorem 2 below.

THEOREM 1(Yamaguti and Matano(1979))

: There exits a positive constant c1 such that

for any−ρσ > c1 the difference equation is

chaotic in the sense of Li-Yorke (1975).

THEOREM 2(Yamaguti and Matano(1979)):

Suppose the assumptions above hold and

κ = +∞. Then there exists another constant

c2 , 0 < c1 < c2, such that for any 0 ≤ ρσ ≤ c2

(where ∆ ≡(−ρσ

)), the map F∆ : qt → qt+1

given by qt+1 = qt + ∆f (qt) , ∆ =(−ρσ

),

has a finite interval [0, α∆] such that F∆ maps

[0, α∆] into itself, with α∆ > u. Moreover,

if c1 <−ρσ ≤ c2,the chaotic phenomenon in

the sense of Li-Yorke (1975) occurs in this

confinement interval.

The theorems immediately apply to

121

our transformed equation since it has two

steady states. However for an interior

solution we must check that

(Mt+1

pt

)and

Rt+1 =(

1− F ′(Mt+1

pt

))−1

− 1, remain

non-negative. This implies that there is a

lower bound to

(Mt+1

pt

), and we must check

whether the corresponding restrictions on

the transforms of

(Mt+1

pt

), that is on xt, zt,

or qt hold along these trajectories. In the

simulations below real balances as well as

the nominal interest rate remain positive

and interior along the computed equilibrium

trajectories.

122

CALIBRATION: The target nominal

rate is 6%, which for a quarterly calibration

implies R∗ = 0.0015. The discount factor

β = (1.01/1.015) = 0.9951. Since the

target stationary inflation is given by π∗ =β (1 + R∗) , we have π∗ = 1.01, or an

annual inflation target of 4%.The target

nominal annual rate of 6% corresponds to

the average yield on 3-month T-bills over

the period 1960:Q1 to 1998:Q3. The annual

inflation rate of 4% matches the average

growth rate of the U.S. GDP deflator during

1960:Q1-1998:Q3, which is 4.2%.

123

The Taylor coefficient on the interest

rate at the active steady state is given by

A/π∗ = 1.60/1.01 = 1.5842. The interest

elasticity of money is given by (ρ− 1)−1 .We

set it to−0.1, implying a value of ρ = −9. The

coefficient a is a measure of the productive

effect of money, and can be calibrated

from money demand, eq. (??). We set

a = 0.000359, and the constant endowment

income y = 1.The equilibrium condition (??) implies

F ′(Mt+1

pt

)=

Rt+1

1 + Rt+1= a

((Mt+1

pt

)−1

yt+1

)1−ρ

where R1+R ≈ R. Taking logs it is easy to

show that

d `n(Mt+1

pt

)d `n

(R

1+R

) =1

ρ− 1

so that ρ = −9 seems reasonable.

Given an annual velocity of around 5.8,

or a quarterly velocity of 1.45, yields a value

for a = 0.0003597.For the intertemporal

124

consumption elasticity σ = 1.5, and we

explore other values as well.

125

In certain cases, for which conditions

are difficult to verify, the aperiodic chaotic

equilibrium trajectories may exhibit ergodic

chaos so that the limt→∞ n−1∑n

0 xt converges

to a limit independently of the initial condition

x(0). This raises the possibility of eliminating

chaotic trajectories as equilibria by designing

the fiscal policy underlying the function

ω (·) appropriately, in analogy to the case

where trajectories converging to the passive

steady could be ruled out since they do not not

respect government solvency or transversality

126

conditions. In general however cyclic and

aperiodic trajectories around the active steady

state take values both above and below the

steady state value, and it may not be possible

to rule out such cyclic or chaotic trajectories as

equilibria by the use of locally non-Ricardian

fiscal policies.

14.0.1

127

Alternative Taylor Rules and Timing

Conventions

14.0.2 A Linear Rule

It is important to realize however that the

local properties of the ”active” steady state do

not depend on the particular Taylor rule that

we adopted. Consider for example the linear

Taylor rule:

Rt = R∗ + A(πt−1 − π∗

π∗)

which, at the active steady state, has the

same slope(Aπ∗

)as our previous non-linear

Taylor rule that respected the zero lower

bound on the nominal rate. For σ = 2.75, for

intial conditions in the neighborhood of the

active steady state, this linear rule generates

the same time-series picture as in the upper

panel of Figure 1: the active steady state is

indeterminate. For σ = 2.65 however, the

active steady state is determinate, but now

trajectories starting in a small neighborhood

of it converge to a period 2 cycle, as in the

128

upper panel of Figure 2. The amplitude

of the cycle is zero at the bifurcation point

σ ≈ 2.709, and grows as σ is decreased.

Thus local indeterminacy, with convergence

either to the active steady state or the cycle

surrounding it, does not depend on the

non-linear Taylor rule.

14.0.3

129

Backward Looking Rules

We have also explored alternative formula-

tions with forward looking Taylor rules, and

with money entering the utility function in

various non-separable ways. In such cases an-

other potentially serious problem emerges: the

instantaneous or temporary equilibrium is of-

ten non-unique. Given the Taylor rule, money

balances at time t do not uniquely determine

money balances at time t+1, so that the dy-

namics of the model, calibrated by respecting

the standard parameter values, are ambigously

defined even before the usual indeterminacy

considerations come into play. To see thgis

consider a backward looking rule given by

Rt = R∗(πt−2

π∗

) AR∗

; aβ > 1, (88)

which implies that

πt = π∗(Rt+2

R∗

)R∗A

(89)

130

Substituting into the Euler equation we get(a

(Mt+1

pt

)ρ+ (1− a) yρ

)−σρ

(90)

=

β

(1− a

(a(Mt+1

pt

)ρ+ (1− a) yρ

)1−ρρ(Mt+1

pt

)ρ−1)−1

π∗

(

1−a(a(Mt+2pt+1

)ρ+(1−a)yρ

)1−ρρ(Mt++2pt+1

)ρ−1)−1−1

R∗

R∗A

·

(a

(Mt+2

pt+1

)ρ+ (1− a) yρ

)−σρ

(91)

which is a difference correspondence. There

often is not a unique

(Mt+2

pt+1

)defined for a

given

(Mt+1

pt

): instantaneous equilibrium is

not well-defined. The equilibrium trajectory

may jump from one branch of solutions to the

next, creating a more severe problem than the

standard difficulties associated with multiple

131

equilibria and indeterminacy.

14.0.4

132

Timing Conventions

A similar problem emerges if the production

function depends on the initial rather than end

of period balances: ct = yt = F(Mt

pt

)=

F (mt) .4 In such a case the money and bond

markets are in equilibrium if Fm (mt) =Rt.The Euler equation becomes:

(F (mt))−σ πt = β (F (mt+1))−σ (1 + Rt+1)

With a forward looking Taylor rule this yields

(F (mt))−σ π∗

(Fm(mt+1)

R∗

)R∗A

= β (F (mt+1))−σ (1 + Fm (mt+1))

Again, whether we have a Cobb-Douglas

or CES production function, mt may not

generate a unique mt+1. If the Taylor Rule is

backward looking,

(F (mt))−σ πt = β (F (mt+1))−σ (1 + Rt+1)

4 For a general discussion of various timing assumptions and

their impact on local determinacy properties, see Carlstrom and

Fuerst (2001)

133

π∗(Fm(mt+2)

R∗

)R∗A

= β (F (mt+1))−σ (1 + Fm (mt+1)) (F (mt))σ

Fm (mt+2) =

((R∗)

R∗A (π∗)−1 β (1 + Fm (mt))

(F (mt))σ (F (mt+1))−σ

) AR∗

Equilibrium dynamics are defined by a second

order difference equation and the solution can

be complicated.

134