m ^ m - New York UniversityS02).pdf · In what follows, we restrict the analysis to equilibria in...
Transcript of m ^ m - New York UniversityS02).pdf · In what follows, we restrict the analysis to equilibria in...
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