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0 0 0 9
Forward-Looking Versus
Backward-Looking Taylor Rules
by Charles T. Carlstrom and
Timothy S. Fuerst
FEDERAL RESERVE BANK
OF CLEVELAND
Working Paper 00-09
Forward-Looking Versus Backward-Looking Taylor Rules
by Charles T. Carlstrom and Timothy S. Fuerst
Charles T. Carlstrom is at the Federal Reserve Bank of Cleveland.
Timothy S. Fuerst is at Bowling Green State University, Bowling Green, Ohio and is
a visiting research associate at the Federal Reserve Bank of Cleveland.
Working papers of the Federal Reserve Bank of Cleveland are preliminary materials
circulated to stimulate discussion and critical comment on research in progress. They
may not have been subject to the formal editorial review accorded official Federal Reserve
Bank of Cleveland publications. The views stated herein are those of the authors and are
not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of
Governors of the Federal Reserve System
Working papers are now available electronically through the Cleveland Fed's home page
on the World Wide Web: http://www.clev.frb.org.
August 2000
Forward-Looking Versus Backward-Looking Taylor Rules
by Charles T. Carlstrom and Timothy S. Fuerst
This paper analyzes the restrictions necessary to ensure that the policy rule used by the central bank
does not introduce real indeterminacy into the economy. It conducts this analysis in a flexible price
economy and a sticky price model. A robust conclusion is that to ensure determinacy the monetary
authority should follow a backward-looking rule where the nominal interest rate responds
aggresssively to past inflation rates.
Forward-Looking Versus
Backward-Looking Taylor Rules*
Charles T. Carlstrom1 and Timothy S. Fuerst2
1Federal Reserve Bank of Cleveland, Cleveland, OH 44101, USA.
2Bowling Green State University, Bowling Green, OH 43403, USA;
and Federal Reserve Bank of Cleveland.
April 10, 2000
Abstract: This paper analyzes the restrictions necessary to ensure that the policy
rule used by the central bank does not introduce real indeterminacy into the
economy. It conducts this analysis in a flexible price economy and a sticky price
model. A robust conclusion is that to ensure determinacy the monetary authority
should follow a backward-looking rule where the nominal interest rate responds
aggressively to past inflation rates.
I.
Introduction.
The celebrated Taylor (1993) rule posits that the central bank uses a fairly simple
rule when conducting monetary policy. This rule is a reaction function linking
movements in the nominal interest rate to movements in endogenous variables (eg.,
inflation). Recently there has been a considerable amount of interest in ensuring that such
rules do no harm. The problem is that by following a rule in which the central bank
responds to endogenous variables, it increases the likelihood that the central bank may
introduce real indeterminacy and sunspot equilibria into an otherwise determinate
economy.1 These sunspot fluctuations are welfare-reducing and can potentially be quite
large.
This paper analyzes the conditions under which a simple Taylor rule, one that
responds only to inflation, will be determinate and hence avoid sunspot fluctuations. In
particular we ask whether the central bank should respond proactively to movements in
expected future inflation, or should they look backwards and base interest rate changes on
past movements in inflation. What about Taylor rules that respond to current inflation?
These questions are of more than academic interest. Several central banks around
the world currently use inflation forecasts as an important part of their decision-making
on policy issues.2 For example, the official position of the Central Bank of New Zealand
is that: “The Bank’s inflation projections relative to the inflation target range are the
1
The term “sunspot” is in one sense misleading since these shocks are accommodated by monetary policy.
But we use the term since the central bank introduces real indeterminacy by responding to public
expectations which can be driven by sunspots.
2
Countries with explicit inflation targets (Canada, New Zealand, and the UK, for example) all base their
policy actions on inflation forecasts.
1
critical input in the quarter by quarter formulation of monetary policy.”3 Alan Greenspan
has commented that “Implicit in any monetary policy action or inaction is an expectation
of how the future will unfold, that is, a forecast.”4
A standard argument in the literature is that to avoid real indeterminacy the central
bank must respond aggressively to either expected inflation (see Bernanke and Woodford
(1997) and Clarida, Gali, Gertler (1997)) or current inflation (see Kerr and King (1996)).
These analyses are all reduced-form sticky price models, where the underlying structural
model is a labor-only economy and money is introduced via a money-in-the-utility
function (MIUF) model with a zero cross-partial between consumption and real balances.
In sharp contrast to the standard result, this paper demonstrates that any forward-looking
or current-looking interest rate rule will always produce real indeterminacy.
The fundamental differences between our model and the above are two-fold. First
as demonstrated in Carlstrom and Fuerst (1999a), one important factor affecting the
determinacy issue is the assumption about which money balances enter the utility
function. The traditional MIUF assumption is that end-of-period balances enter the
functional. But a direct extension of a typical cash-in-advance (CIA) economy suggests
that the money the household has left after leaving the bond market and before entering
the goods market is more appropriate. Remarkably this timing distinction is critical for
questions of determinacy. As argued in Carlstrom and Fuerst (1999a), the CIA timing is
a more natural choice so we adopt it here. We utilize a rigid CIA constraint, but
3
See Huxford and Reddell (2000) “Implementing Monetary Policy in New Zealand.”
Greenspan, A. Discussion of Goodhart, C., Capie, F., and Schnadt, N., “The Development of Central
Banking,” in Capie, F., Goodhart, C., Fischer, S., and Schnadt, N. (eds), The Future of Central Banking,
Cambridge University Press, 1994, quoted in Sir Alan Budd, “Economic Policy, with and without
Forecasts,” published in the Bank of England’s Quarterly Bulletin, November 1998 (from a speech given at
4
2
following the arguments of our earlier paper, the results generalize to an arbitrary MIUF
environment with CIA timing.
The second major difference is in the type of sticky price model analyzed. The
pricing model in the earlier papers is adopted from Calvo (1983). The assumption in that
model is that a fixed fraction of firms adjust their prices every period. This implies that
after any arbitrarily long but finite number of periods along a deterministic path, not all
prices will not have adjusted to the levels implied by a flexible price model. This pricing
arrangement is problematic for issues of determinacy since an equilibrium is determinate
if perturbations from the equilibrium path lead to explosive inflation dynamics. But surely
this Calvo pricing arrangement would not continue to hold along these out-of-equilibrium
hyperinflationary paths, so that imposing this arrangement along the path seems quite
artificial.
Instead this paper makes the methodological point that for issues of determinacy
the more appropriate modeling strategy is to assume that along a hyperinflationary price
path the pricing mechanism becomes perfectly flexible in finite time. For example, in the
model analyzed below firms preset their prices in advance for some finite but definite
time period. This model has the property that for any set of initial conditions (pre-set
prices) the deterministic dynamics converge to the corresponding flexible-price model in
a finite time period. In particular, along a hyperinflationary price path the pricing
mechanism becomes perfectly flexible in finite time. Remarkably, the conditions for real
determinacy in a model with finite stickiness are quite different than in a model with
forever stickiness (eg., Calvo). This result is analogous to the so-called “folk theorem” in
the Sir Alec Cairncross Lecture for the Institute of Contemporary Bristish History and the St. Peter’s
3
game theory that a game that lasts for a finite but know period of time is fundamentally
different than a game that lasts forever.
A related contribution of this paper is the demonstration that in a model with
finite stickiness a necessary and sufficient condition for real determinacy is for the
corresponding flexible price economy to have both real and nominal determinacy. This
implies that with sticky prices either a forward or current-looking interest rate rule will
always be indeterminate. Finally this paper demonstrates that a necessary and sufficient
condition for real determinacy is for the monetary authority to react aggressively to past
movements in inflation.
The outline of the paper is as follows. The next section introduces a flexible price
model. This model lays the groundwork for understanding indeterminacy in section III
where firms preset their prices in advance. The principle conclusion is that the only way
to avoid both nominal and real indeterminacy in a flexible price model, and hence real
indeterminacy in a finite-lived sticky price model, is to respond aggressively to past
inflation. Section IV concludes. An appendix proves the main propositions.
II.
A Flexible Price Model
The economy consists of numerous households and firms each of which we will
discuss in turn. Since we are concerned with issues of determinacy without loss of
generality we limit the discussion to a deterministic model. As is well known, if the
deterministic dynamics are not unique, then it is possible to construct sunspot equilibria
in the model economy.
College Foundation, October 27, 1998).
4
Households are identical and infinitely-lived with preferences over consumption
and leisure given by
∞
∑
t =0
βtU(ct,1-Lt),
where β is the personal discount rate, ct is consumption, and 1-Lt is leisure. The utility
function is given by:
U(c,1-L) ≡ (c1-σ-1)/(1-σ) - L1+γ/(1+γ).
For the analytical results in this paper we will be working with the Hansen-Rogerson
indivisible labor formulation so that γ = 0. Although the exact quantitative results are
sensitive to this assumption, numerical calculations confirm that the qualitative results are
the same for alternative calibrations.
To purchase consumption goods, households are subject to the following cash-inadvance constraint:
Pt ct ≤ M t + Pt wt Lt − N t
where Pt is the price level, Mt denotes beginning-of-period cash balances, wt denotes the
real wage, and Nt denotes the household's choice of one-period bank deposits. These
deposits earn nominal rate Rt that is paid out at the end of the period. The household's
intertemporal budget constraint is given by:
M t +1 =M t + Pt {wt Lt +[rt + (1 − δ )]K t } − Pt ct − Pt K t +1 +Π t + N t ( Rt − 1) .
Kt denoes the households accumulated capital stock that earns rental rate rt. Note that we
are assuming that asset accumulation occurs at the household level and that cash in
advance is not needed to finance its purchase. The former assumption is without loss of
5
generality; the second assumption is quite important as it implies that the nominal interest
rate acts as a consumption (but not investment) tax.5 Π denotes the profit flow from
firms and financial intermediaries.
Firms in this economy utilize a production function employing labor and capital:
yt = f(K t , H t )
where Ht denotes hired labor, and f is a CRS Cobb-Douglas production function with a
capital share of α and a labor share of (1-α).6 To finance its wage bill the firm must
acquire cash and does so by borrowing cash short term from the financial intermediary at
(gross) rate Rt.
The intermediary in turn has two sources of cash, the cash deposited by
households and the new cash injected into the economy by the central bank. Hence, the
loan constraint is:
Pt wt H t ≤ N t + M ts (Gt − 1)
where Gt denotes the (gross) money supply growth rate, Gt ≡ Mst+1/ Mst. Note that
monetary injections are carried out as lump sum transfers to the financial intermediary.
We restrict our attention to equilibria with strictly positive nominal interest rates so
that the two cash constraints are binding. A recursive competitive equilibrium is given by
stationary decision rules that satisfy these two binding cash constraints and the following:
U c (t )
U c ( t + 1)
= β ( f k ( t + 1) + 1 − δ )
R t +1
Rt
5
(1)
This assumption is quite important for the results with flexible prices but not for the model with sticky
6
U L (t ) f L (t )
=
U c (t )
Rt
(2)
U c (t )
U c ( t + 1)
= β Rt
Pt +1
Pt
(3)
ct + K t +1 = f ( K t , Lt ) + (1 − δ ) K t .
(4)
Equations (1)-(2) illustrate the nominal rate distortion that is so important here.
Both the labor and capital margins are distorted by the nominal interest rate. The nominal
rate can thus be interpreted as a consumption tax of (1+tct) = Rt. This occurs because
households must acquire cash before buying the consumption good, which has an
opportunity cost of Rt. A high nominal rate is equivalent to a high consumption tax, and
a low nominal rate is equivalent to a low consumption tax.
To close the model we need to specify the central bank reaction function. In what
follows we assume a reaction function where the current nominal interest rate is a
function of inflation. We will consider three variations of this simple rule:
π
Rt = Rss t +i
π ss
τ
π
, where τ ≥ 0, Rss = ss ,
β
(5)
prices.
6
The constant returns to scale assumption while standard is also critical to the exact quantitative results.
7
where i = -1 is a backward-looking rule, i = 0 is a current-looking rule, and i = 1 is a
forward-looking rule.
Under any such interest rate policy the money supply responds endogenously to be
consistent with the interest rate rule. It is this endogeneity of the money supply that
increases the possibility of indeterminacy. If the private sector responds to sunspots, then
the central bank must passively vary the money supply to keep the nominal rate on target.
There are two types of indeterminacy that may arise. First, there is nominal
indeterminacy—are the initial values of the price level and all other nominal variables
pinned down? In our notation this corresponds to the question of whether πt ≡ Pt/Pt-1 is
determined (where t is the initial time period). This nominal or price level determinacy is
a standard occurrence under many interest rate operating procedures, the most celebrated
example being an interest rate peg. This nominal indeterminacy is of no consequence in
and of itself, but is important only if it leads to real indeterminacy.
By real indeterminacy, we mean a situation in which the behavior of one or more
real variables is not pinned down by the model. This possibility is of great importance as
it immediately implies the existence of sunspot equilibria which, in the present
environment, are necessarily welfare reducing. In the flexible price model of this section,
real indeterminacy manifests itself as an indeterminacy in expected inflation or the initial
nominal interest rate. Since this rate distorts real behavior, this is a form of real
indeterminacy. In the sticky price sections that follow, real indeterminacy can arise in
both the initial marginal cost and nominal interest rate. Since both of these distort real
behavior, this is also a form of real indeterminacy.
8
For simplicity, we begin the analysis with a labor-only economy, and then
demonstrate how the results extend to an economy with capital. Suppose that production
is a linear function of labor, f(L) = BL. Assuming preferences are linear in leisure (γ=0)
then inserting the labor margin (2) into the Fisher equation (3) yields:
~
Rt +1 − π~t +1 = 0
(6)
where the variables are expressed as log deviations from the steady-state. The generality
of this result is suggested by the capital accumulation equation (1). If output is instead
linear in capital then (1) also collapses to (6).
First suppose that monetary policy is forward-looking. Substituting the monetary
policy rule (5) into (6) we have
1
π~t + 2 = π~t +1 .
τ
(7)
Notice that (7) starts with πt+1, so that the initial price level is free, πt is free. Thus the
economy always suffers from nominal indeterminacy. At this point, however, nominal
indeterminacy is of no importance since it does not impact real behavior. For real
determinacy we need this mapping to be explosive. Hence, there is real determinacy if
and only if 0 < τ < 1. If this is the case then πt+j is determined for j ≥ 1. From the policy
rule this pins down Rt+j for j ≥ 0 so that real behavior is pinned down. At least in a
flexible price model the results of Clarida, Gali, Gertler (1997) have essentially been
turned on their head.
9
The intuition for this real indeterminacy is clear. Under this policy rule increases
in expected inflation increase the nominal rate but depending on the elasticity τ these
increases may or may not increase the real rate. Rewriting the above we have
(τ − 1) ~
~
Rt − π~t +1 = (τ − 1)π~t +1 =
Rt .
τ
For aggressive policies (τ > 1), nominal rate increases are associated with increases in the
real rate of interest. Thus, we have an implicit consumption tax (the nominal rate)
correlated tightly with expected consumption growth (the real rate). The self-fulfilling
circle goes like this. A sunspot-driven increase in current consumption lowers today’s
real interest rate. With τ > 1, the nominal rate (consumption tax) falls with this real rate
movement. This increases current consumption, which completes the circle. The initial
increase in consumption is therefore rational.7
The fact that πt is free for all values of τ is just a manifestation of nominal
indeterminacy, i.e., there is nothing to pin down the initial growth rate of money. This
innocent remark has some interesting implications for a current-looking policy rule.
Substituting the current-looking policy rule (5) into (6) yields
τπ~t +1 − π~t +1 = 0 .
(8)
Notice that πt is free, but future inflation rates are nailed down (when τ ≠ 1) since π~t + j =
0 for all j ≥ 1. From the policy rule this pins down Rt+j for j ≥ 1. But since πt is free, this
current-based policy implies that Rt is also free. Since Rt acts like a tax on consumption,
real behavior is not pinned down. The nominal indeterminacy from before is now real.
7
Schmitt-Grohe and Uribe (2000) and Carlstrom and Fuerst (1999c) report similar results for forwardlooking Taylor rules in a flexible price setting.
10
The reason for this indeterminacy is because the policy rule is responding to current
inflation, which is not pinned down for standard nominal indeterminacy reasons.
Because of the policy rule this nominal indeterminacy becomes real. More generally the
potential for indeterminacy in both the current- and forward-looking rules arises
whenever policy responds to endogenous variables. This is why an interest rate peg
(τ = 0) has real determinacy.
This discussion suggests that the central bank should look backwards so that it
responds only to predetermined variables. Remarkably, by looking backwards the
conditions for determinacy are (almost) entirely flipped on their head from when the
Taylor Rule is forward-looking. Substituting the backward-looking monetary policy rule
(5) into (6) we have
π~t +1 = τπ~t
(9)
Unlike earlier we may have nominal determinacy. If the monetary authority responds
aggressively to past inflation (τ > 1) initial inflation and hence Rt+1 is pinned down (the
policy rule implies that Rt is predetermined by last period’s inflation rate). Hence, there
is real and also nominal determinacy if and only if τ > 1. (As noted earlier, an interest
rate peg of τ = 0 yields real but not nominal determinacy.)
The intuition why an aggressive backward-looking policy can eliminate nominal
and real indeterminacy is as follows. Suppose there is an increase in the current price
level πt of 1%. This implies that next period’s nominal rate must rise by τ%. This
increase in the future nominal rate (consumption tax) leads to an increase in current
consumption. This implies that the real rate falls. The policy rule implies that the current
11
nominal rate does not respond to πt. Hence, the decline in the real rate must lead to an
increase in πt+1 that is greater than the initial increase in πt (see (9) with τ > 1). This
behavior is explosive, and thus eliminates this as a possible equilibrium path.
The idea that responding to a nominal variable can pin down prices is not new.
This result is a general equilibrium generalization of McCallum’s (1981) earlier result.
He argued that because an interest rate peg suffered from nominal indeterminacy the
monetary authority needed a nominal anchor, which could be accomplished by
responding to a nominal variable. This analysis confirms this but shows that merely
responding to a nominal variable, like past inflation, is not enough. The monetary
authority has to aggressively respond to past inflation to ensure both real and nominal
determinacy.
The preceding results were demonstrated in a model with linear labor. The
following propositions show that the linearity exploited above to demonstrate
indeterminacy is remarkably general. A CRS production function with linear leisure
provides this same linearity so that the conditions for determinacy are identical for all
three monetary reaction functions considered above. As noted earlier, for more general
calibrations of labor supply the quantitative results differ, but the qualitative results are
the same—only passive forward-looking rules and aggressive backward-looking rules are
consistent with real determinacy, and only the latter eliminate nominal indeterminacy.
Proposition 1: Suppose that prices are flexible, γ = 0 (linear labor,) and that monetary
policy is given by the following interest rate rule given by:
12
π t+ j
Rt = Rss
π ss
τ
π
, where τ ≥ 0, Rss = ss .
β
A. With a forward-looking interest rate rule (j=1) there is real determinacy if and only if
0 < τ < 1. In any event, there is always nominal indeterminacy as πt is free.
B. With a current-looking interest rate rule (j=0) there is real indeterminacy for all
values of τ ≠ 0.
C. With a backward-looking interest rate rule (j=-1) there is real determinacy if and only
if τ = 0 or τ > 1. In the case of τ > 1, there is also nominal determinacy.
Proof: See the appendix.
Although at this stage nominal indeterminacy is merely a nuisance, its presence
becomes critical in the next section when we consider how nominal rigidities affect the
above results. In particular we will show that if there is nominal indeterminacy in the
flexible price model, then in the corresponding sticky price model there is real
indeterminacy.
III.
A Sticky Price Model.
In this section we consider a popular model of monetary non-neutrality—a model
with sticky prices. We utilize the standard model of imperfect competition in the
intermediate goods market.8 We omit any discussion of household behavior as it is
symmetric with before.
Final goods production in this economy is carried out in a perfectly competitive
13
industry that utilizes intermediate goods in production. The CES production function is
given by
1
Yt = { ∫ [ y t (i ) (η −1)/η ]di}η /(η −1)
0
where Yt denotes the final good, and yt(i) denotes the continuum of intermediate
goods, each indexed by i ∈ [0,1]. The implied demand for the intermediate good is thus
given by
P (i )
yt (i ) = Yt t
Pt
−η
where Pt(i) is the dollar price of good i, and Pt is the final goods price.
Intermediate goods firm i is a monopolist producer of intermediate good i.
Fraction ν of these firms set their prices flexibly within each period, while without loss of
generality the remainder (1-ν) are assumed to set their price one period in advance. The
variable ν is thus a measure of price flexibility: (1) with ν = 1, this is a flexible price
model, (2) with ν between zero and one, this is a sticky price model, and (3) with ν = 0,
this is a model with rigid prices. Other than the difference in the timing of pricing, the
firms are all symmetric, so we will henceforth drop the firm-specific notation. Let
Pt f denote the flexible price, while Pt s will denote the pre-determined (or sticky) price.
The final goods price (or aggregate price level) is given by the appropriate average of
these two prices:
{
Pt = (1 − ν ) Pt
8
s (1− η )
+ ν Pt
f (1− η )
1
1− η
}
See, for example, Chapter 5 of Walsh (1998).
14
.
(10)
The flexible price is given by a constant mark-up over the marginal cost (zt) of
production:
η
Pt f =
Pt z t .
η − 1
(11)
The term in brackets will appear frequently below, so we define z ≡ (η-1)/η < 1. In a
model with flexible prices, equation (11) and the assumption of symmetry implies that zt
= z. Combining (10)-(11), we have
1− ν
s
Pt = h( z t ) Pt , where h( z t ) ≡
1−η
zt
−
1
ν
z
1/(1−η )
(12)
The function h is increasing so that innovations in marginal cost correspond with changes
in the price level.
The intermediate goods firm is owned by the household, and pays its profits out to
the household at the end of each period. Because of the cash-in-advance constraint on
household consumption, the firm discounts its profits using µt+1 ≡ βUc(t+1)/Pt+1, the
marginal utility of $1 in time t+1. Therefore the sticky price is given by the solution to
the following maximization problem:
−η
Pt s Pt s
Pt =arg max Et −1 µ t +1 Pt Yt − z t
Pt Pt
s
where Et-1 is the expectation conditional on time t-1 information. The firm's optimal
preset price is thus given by:
{
η Et −1 µ t +1 Ptη +1 z t Yt
Pt =
η
η − 1 Et −1 µ t +1 Pt Yt
s
{
}
} .
(13)
15
Using (12), this can be written as
Et −1 zµ t +1Yt [h( z t )]η = Et −1 z t µ t +1Yt [h( z t )]η +1 .
(14)
In a model without pre-set prices, this equation would hold at time-t, and thus imply that
zt = z.
As for production, the intermediate firm rents capital and hires labor from
households and utilizing the CRS production function from before. Imperfect competition
implies that factor payments are distorted. With zt as marginal cost, we then have rt =
ztfK(Kt,Lt) and wt = ztfL(Kt,Lt)/Rt.
A recursive competitive equilibrium is given by stationary decision rules that
satisfy (12), (13), (14), and the following:
U L (t ) z t f L (t )
=
U c (t )
Rt
(15)
U (t + 1)[z t +1 f K (K t +1 , Lt +1 ) + (1 − δ )]
U c (t )
=β c
Rt
Rt +1
(16)
U ( t + 1)
U c (t )
= β Rt c
Pt
Pt +1
(17)
ct + K t +1 = f ( K t , Lt ) + (1 − δ ) K t
(18)
16
The labor and capital accumulation margins are once again distorted by the
nominal rate of interest (the consumption tax). The novelty is that the marginal
production cost zt also distorts behavior. Continuing with the public finance analogy, this
marginal cost or imperfect competition distortion manifests itself as a tax on wage and
rental income. Thus we now have two implicit taxes: the nominal interest rate or
consumption tax, and marginal cost or income tax. The fact that both of these taxes are
endogenous suggests that it will be more difficult to ensure determinacy.
For stability we once again turn to the deterministic model. A key insight is that
since prices can adjust after one period zt+j = z for all j ≥ 1, but zt need not equal z
because some prices are predetermined. From t+1 onwards, therefore, the deterministic
version of the model is isomorphic to the flexible price model with a constant income tax
rate of 1-z. This implies that a necessary condition for real determinacy is that the
corresponding flexible price model be determinate for real variables. If the flexible price
model has real indeterminacy then, obviously, the sticky price model will also suffer from
real indeterminacy.
But this flexible price determinacy is only necessary. For sufficiency we also
need the flexible price economy not to have nominal indeterminacy. This is because we
need an extra condition to pin down the initial marginal cost zt, which from (12) is a
function of the initial price level Pt.
To help understand the intuition of this result, we return to the special case in
which production is a linear function of labor, f(L) = BL. Assuming preferences are
linear in leisure (γ=0) then inserting the labor margin (15) into the Fisher equation (17)
yields:
17
[
]
~
~
z t + Rt +1 − π~t +1 = 0 and
(19a)
~
Rt + j − π~t + j = 0 for all j ≥ 2.
(19b)
z t + j = 0 for all j ≥ 1 (because prices are fixed for only one period). Notice that
where ~
because prices are flexible from period 2 on we have split the conditions into their sticky
(19a) and flexible price (19b) periods. Equation (19a) represents one restriction on the
initial zt. The second restriction comes from (12) which implies that zt is a function of πt
= Pt/ Pt-1.
Is zt pinned down? Consider first the flexible price part (19b). If the flexible
price economy has both real and nominal determinacy, then (19b) pins down πt+1. If this
occurs then (12) and (19a) imply that zt and πt are both determined. But if (19b) suffers
from nominal indeterminacy so that πt+1 is not pinned down, then (19a) implies that zt and
πt will also be free. Remarkably the presence of nominal indeterminacy in the flexible
price economy necessarily implies that expected inflation (and thus real variables) are
indeterminate in a sticky price economy.
Why does the forward-looking policy rule always produce real indeterminacy?
Consider a sunspot increase in expected inflation. The monetary policy rule implies that
today’s funds rate must increase in response. To achieve this the monetary authority
lowers today’s money growth. This temporarily lowers output and hence consumption
thus increasing the real interest rate. Given this monetary contraction, when tomorrow
comes, firms’ pre-set prices will be too high. The monetary authority, therefore, must
increase tomorrow’s money growth to keep the nominal rate in a neutral position. This
increases today’s expected inflation thus completing the circle.
18
This analysis suggests that, perhaps, indeterminacy could be avoided if the
monetary policy rule completely stabilized expected inflation (τ=∞). Such a rule would
make the economy isomorphic to a flexible price economy (see Carlstrom and Fuerst
(1998)). From the flexible price model developed earlier we know that such a rule will
always suffer from indeterminacy since the real rate and the nominal rate would move in
tandem.
As in the previous section, the results of this simple example extend to a wider
environment with a CRS production function. In particular, we have the following:
Proposition 2: Suppose that monetary policy is given by either a forward-looking or a
current-looking Taylor rule. Then in the sticky price model there is real indeterminacy
for all values of τ.
Proof: see the appendix.
Proposition 3: Suppose that monetary policy is given by a backward-looking Taylor rule.
Then in the sticky price model there is real determinacy if and only if τ > 1.
Proof: see the appendix.
There are of course many convex combinations of possible interest rate rules that
we do not address above. We will conclude with two interesting cases. First, suppose
that policy is both forward- and backward-looking:
~
Rt = τ [επ~t −1 + (1 − ε )π~t +1 ] .
19
In this case, a necessary and sufficient condition for real determinacy in the sticky price
model is that τ > 1 and ε >1/2. That is, policy can look forward if and only if the weight
of the policy rule is on the past.
Second, consider policy rules that contain an inertial component:
~
~
Rt = ρ Rt −1 + τ π~t +i .
Under such a rule, forward-looking rules (j = 1) are always subject to sunspots. Currentlooking rules (j = 0) are determinate if and only if 1 < τ < (1+ρ)/(1-ρ). Backward-looking
rules (j = -1) are determinate if and only if τ > 1.9
The results in this paper also generalize to other pricing arrangements. The basic
argument applies to any longer-lived but finite stickiness. All that is necessary is that for
any set of initial conditions (Pt-1), the deterministic dynamics converge to the flexibleprice model in a finite time period. In particular, along a hyperinflationary price path, the
aggregate pricing mechanism becomes perfectly flexible in finite time. This is clearly not
the case for Calvo (1983) pricing, nor is it generally true for Taylor (1980) pricing.
However, it is the case for Fischer (1977) pricing.
While both Fischer and Taylor employ staggered contracting, the difference
between Fischer and Taylor pricing is that in the former firms set their prices some finite
number of periods in advance but they can choose different prices in each period. With
Taylor pricing they are constrained to choose the same nominal price in every period. We
argue that in much the same way that Calvo pricing is nonsensical along these
hyperinflationary paths so too is Taylor pricing. In both cases firms are selecting a price
9
See Carlstrom and Fuerst (1999b) for the proof of these results in a small open economy. These proofs
extend to a closed economy using the methods outlined in the appendix.
20
that is wildly out of line with the general price level even though these movements in the
general price level are deterministic. Why should we utilize such an assumption for
issues of real determinacy?
IV.
Conclusion.
The central issue of this paper is to identify the restrictions on the Taylor interest
rate rule needed to ensure real determinacy. A standard result in the literature is that an
aggressive response to either forward or current inflation is necessary and sufficient for
determinacy. This standard result derives from reduced form MIUF models. The
essential point of this paper is that this result does not stand up to more careful structural
modeling.
In our view, these models ignore two central issues: (1) the appropriate timing for
money demand modeling, and (2) the instability of the Phillips curve along a
hyperinflationary path. This paper considers both of these issues and overturns the
standard result.
We believe that both of these issues are of the first importance. Basing policy
advice on a model that does not include these issues seems quite premature. We have
argued elsewhere (Carlstrom and Fuerst (1999a)) for a reassessment of the timing
conventions used in monetary models. We will not review those arguments here.
As for the Phillips curve, we argue that for issues of determinacy the appropriate
modeling strategy is a model with finite stickiness. While the Calvo setup may be more
appropriate for positive analysis, clearly the assumption that prices have to be forever
21
sticky is especially troublesome for issues of determinacy. This can be understood by
remembering that an equilibrium is determinate if perturbations from the equilibrium path
lead to explosive inflation dynamics. But surely the Calvo pricing arrangement would not
continue to hold along such a path. If instead we assume that for any set of initial
conditions, the deterministic dynamics converge to the flexible-price model in a finite
time we are back to a world with finite stickiness. This argument is in the spirit of
McCallum (1994) who questions the robustness of any model that violates the natural rate
hypothesis.
Even if one remained wed to a Calvo pricing arrangement for these determinacy
issues, there are other problems. In a companion piece, Carlstrom and Fuerst (2000) add
investment spending to the Clarida, Gali, Gertler (1997) environment. The role of
investment across the business cycle has a long tradition in monetary economics so that
adding this to the environment seems like a good idea. The end result of Carlstrom and
Fuerst (2000) is that for anything but the most extreme parameter values, adding
investment spending to the Clarida, Gali, Gertler (1997) environment implies that
monetary policy must respond aggressively to past inflation to generate determinacy.
Yet whichever model is used the essential point of this paper remains. To avoid
doing harm, the central bank should place the most weight on past movements in the
inflation rate. As long as this link between current interest rates and past inflation is
aggressive enough, the central bank can eliminate the possibility of self-fulfilling
behavior. An immediate implication of this analysis is that inflation targeting over short
horizons, which necessarily implies the use of forecasts, is potentially a dangerous policy
as it is prone to sunspots.
22
Appendix
Proposition 1: Suppose that prices are flexible, γ = 0 (linear labor,) and that monetary
policy is given by the following interest rate rule given by:
π
Rt = Rss t +i
π ss
τ
π
, where τ ≥ 0, Rss = ss .
β
A. With a forward-looking interest rate rule (i=1) there is real determinacy if and only
if
0 < τ < 1. In any event, there is always nominal indeterminacy as πt is free.
B. With a current-looking interest rate rule (i=0) there is real indeterminacy for all
values of τ ≠ 0.
C. With a backward-looking interest rate rule (i=-1) there is real determinacy if and only
if τ = 0 or τ > 1. In the case of τ > 1, there is also nominal determinacy.
A. Proof: The relevant equilibrium conditions are
PtU c ( t + 1)
Pt +1U c ( t + 2 )
= β ( f k ( t + 1) + 1 − δ )
Pt +1
Pt + 2
(1)
U L (t ) θ t f L (t )
=
U c (t )
Rt
(2)
U c (t )
U c ( t + 1)
= β Rt
Pt +1
Pt
(3)
23
ct + K t +1 = f ( K t , Lt ) + (1 − δ ) K t
(4)
The outline of the proof is that we are going to collapse these four equations into three by
substituting out the labor equation (2).
Given the assumption that utility is linear in leisure (γ = 0) the first order conditions (1)
and (2) can be written as
U (t ) x α
L
β 1 = t ,where xt = t
Kt
Rt (1 − α )
U (t )
1 = β
Rt
(A1)
U 1 (t + 1) 1−α
{αxt +1 + (1 − δ )}
Rt +1
(A2)
where (A1) is the revised labor equation and (A2) is the revised capital accumulation
equation (1). Substituting (A1) and equation (A1) scrolled forward one period into (A2)
yields:
xtα = (αβ xt +1 + β (1 − δ ) xtα+1 )
(A3)
or the newly rewritten capital accumulation equation. The resource constraint provides
another equation:
K t +1 = K t xt1− α +(1 − δ) K t − ct .
(A4)
We wish to substitute (A1) into both (A4) and the Fisher equation (3). Note that (A1)
implies that ct depends only on Rt and xt. Inverting the monetary policy rule yields:
1
π t +1 Rt τ
=
π ss Rss
Using this and (A1), we can express the Fisher equation in terms of Rt+1, xt, xt+1, and Rt.
24
We are thus left with three equations:
xt+1 = F(xt)
Kt+1 = G(xt,Rt,Kt)
Rt+1 = H(xt,xt+1,Rt)
The characteristic matrix is
Fx − e
Gx
H x Fx + H x
t
t +1 t
GK − e
GR
0
H R − e
0
0
The three eigenvalues are
e1=
α
1
1 − β (1 − α )(1 − δ )
< 1 , e2 =
> 1 , e3 = .
1 − β (1 − α )(1 − δ )
αβ
τ
Since there is only one predetermined variable, Kt, for the economy to be determinate two
eigenvalues need to lie outside the unit circle. This is satisfied if and only if τ < 1. This,
however, only pins down πt+1, πt is free. QED
B. Proof:
Suppose τ > 0. The proof proceeds exactly as in Proposition 1. The only difference arises
because
1
τ
π t +1 Rt +1
.
=
π ss Rss
Therefore plugging (A1) into (3) and using the above relationship yields the following
revised Fisher equation
Rt+1 = H(xt,xt+1).
25
The absence of Rt from the above relationship coupled with the absence of Rt+1 from the
xt+1 and Kt+1 equations imply that e3 = 0 .
Since there is only one predetermined variable, Kt, for the economy to be determinate two
eigenvalues need to lie outside the unit circle. This is clearly never satisfied.
Suppose τ = 0. With the nominal interest rate given the real economy is given by (1), (2),
and (4). This is just the Canonical RBC model perturbed by a constant consumption tax
and is thus unique. QED
C. Proof:
Without loss of generality, assume τ > 0. Proceeding as in Proposition 1, we have
xt+1 = F(xt)
Kt+1 = G(xt,Rt,Kt)
Rt+2 = H(xt,xt+1,Rt+1)
where the last equation is of a different form because the inverted monetary policy rule is
πt
π ss
1
R τ
= t +1
Rss
which we have scrolled forward one period to solve for πt+1. The characteristic matrix is
Fx − e1
Gx
H xt +1 Fxt + H xt
0
0
G K t − e2
0
0
0
0
H Rt +1 − e3
1
0
G Rt
0
e4
The four eigenvalues are
e1=
α
1 − β (1 − α )(1 − δ )
< 1 , e2 =
> 1 , e3 = τ , e 4 = 0 .
1 − β (1 − α )(1 − δ )
αβ
26
Now there are two pre-determined variables, Kt and Rt, so that we only need two
explosive roots. Hence, we have real determinacy if and only if τ > 1. In this case, Rt+1 is
pinned down implying that πt is determined. QED
Proposition 2: Suppose that monetary policy is given by either a forward-looking or a
current-looking Taylor rule. Then in the sticky price model there is real indeterminacy
for all values of τ.
Proof: Consider the forward-looking rule (the other case is symmetric). After the initial
period, the deterministic dynamics of the sticky price model are identical to that of the
flexible price model. Hence, a necessary condition for the sticky-price model to be
determinate is for the flexible price model to be determinate. Assume that τ is chosen so
that this is the case. The flexible price determinacy implies that xt+1 and Rt+1 are unique
functions of Kt+1. For sufficiency, we must turn to the initial period. The initial period
Euler equations are:
xt+1 = F(xt,zt)
Kt+1 = G(xt,Rt,Kt,zt)
Rt+1 = H(xt,xt+1,Rt)
Since xt+1 and Rt+1 are unique functions of Kt+1, we have three equations in four
unknowns xt, Rt, Kt+1, and zt. The only other restriction is that zt is a function only of πt ≡
Pt /Pt-1. Hence, we have real indeterminacy. QED
Proposition 3: Suppose that monetary policy is given by a backward-looking Taylor rule.
Then in the sticky price model there is real determinacy if and only if τ > 1.
27
Proof: After the initial period’s price stickiness, the model’s dynamics are exactly that of
the flexible price economy (zt = z after the initial period). Hence, under the conditions in
Proposition 3 (τ > 1), there is real determinacy from time t+1 onwards. This implies that
xt+1 and Rt+2 are unique functions of Kt+1 and Rt+1 (Rt+1 is predetermined since the nominal
rate is backward-looking). We can now turn to the initial period. The initial Euler
equations are:
xt+1 = F(xt,zt)
Kt+1 = G(xt,Rt,Kt,zt)
Rt+2 = H(xt,xt+1,Rt+1)
Since xt+1 and Rt+2 are unique functions of Kt+1 and Rt+1, these represent three equations in
Kt+1, Rt+1, xt, and zt. But Rt+1 and zt are both functions only of πt ≡ Pt /Pt-1. Hence, we
have three independent linear equations in three unknowns Kt+1, xt, and πt. QED
28
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30
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