The eventual failure and price indeterminacy of inflation targeting
ABSTRACT In stark contrast to the previous literature, we find that IT leads to price indeterminacy even when the central bank uses a Taylor-like feedback rule to peg the nominal interest rate. We also find that there is no mechanism with IT to determine the current inflation rate or price level. We conclude that the previous literature has either committed mathematical errors involving infinity or misused the non-explosive criterion for ruling out speculative bubbles. To avoid making errors involving infinity, we analyze inflation targeting (IT) in a typical rational-expectations, pure-exchange, general-equilibrium model where the time horizon is arbitrarily large, but finite.
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ABSTRACT: The new-Keynesian, Taylor-rule theory of inflation determination relies on explosive dynamics. By raising interest rates in response to inflation, the Fed does not directly stabilize future inflation. Rather, the Fed threatens hyperinflation, unless inflation jumps to one particular value on each date. However, there is nothing in economics to rule out hyperinflationary or deflationary solutions. Therefore, inflation is just as indeterminate under "active" interest rate targets as it is under standard fixed interest rate targets. Inflation determination requires ingredients beyond an interest-rate policy that follows the Taylor principle.SSRN Electronic Journal 10/2007;
Page 1
MPRA
Munich Personal RePEc Archive
The Eventual Failure and Price
Indeterminacy of Inflation Targeting
Eagle, David
Eastern Washington University
22 November 2006
Online at http://mpra.ub.uni-muenchen.de/1240/
MPRA Paper No. 1240, posted 07. November 2007 / 01:37
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The Eventual Failure and
Price Indeterminacy of Inflation Targeting
David M. Eagle
Department of Management, RVPT#3
College of Business Administration
Eastern Washington University
668 N. Riverpoint Blvd., Suite A
Spokane, Washington 99202-1660
USA
Phone: (509) 358-2245
Fax: (509) 358-2267
Email: deagle@ewu.edu
Revised December 13, 2006
Abstract
In stark contrast to the previous literature, we find that IT leads to price
indeterminacy even when the central bank uses a Taylor-like feedback rule to peg
the nominal interest rate. We also find that there is no mechanism with IT to
determine the current inflation rate or price level. We conclude that the previous
literature has either committed mathematical errors involving infinity or misused
the non-explosive criterion for ruling out speculative bubbles. To avoid making
errors involving infinity, we analyze inflation targeting (IT) in a typical rational-
expectations, pure-exchange, general-equilibrium model where the time horizon is
arbitrarily large, but finite.
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2004, 2005 by David Eagle. All rights reserved. Copyright will be transferred to publishing
journal when accepted.
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The Eventual Failure and
Price Indeterminacy of Inflation Targeting
In the early 1960s, the Phillips curve was the fad in macroeconomics. However, in the
late 1960s and the 1970s, changing expectations led to a breakdown in the Phillips curve.
Today, the current fad in central banking is inflation targeting (IT). This paper argues that
changing expectations could lead to a failure of IT in a manner similar to the breakdown in the
Phillips curve.
Today many central banks follow some form of IT, including the Bank of England, the
European Central Bank, and central banks in New Zealand, Canada, Australia and several
developing countries. Also, Ben Bernanke, chair of the Federal Reserve Board, endorses
committing the Federal Reserve to an inflation target. The definition of IT is that (i) the central
bank target an inflation rate although it may have competing output-gap goals, (ii) the central
bank be very committed to its stated targets and goals, and (iii) the central bank be very
transparent in its goals and plans concerning monetary policy, including contingency plans. The
transparency of the central bank under IT is considered very important for the formation of
public expectations.
An alternative to IT is price-level targeting (PLT), where the central bank targets the
price level instead of the inflation rate. IT and PLT are very similar. While IT has explicit
inflation targets, we can derive implied price-level targets from those inflation targets. For
example, if the central bank targets a 2% inflation rate forever and the current price level is 1.0,
the implied path of price-level targets would be 1.02t where t is the number of periods from now.
Also, under PLT we can derive implied inflation targets from the explicit price-level targets. For
example, if the path of price-level targets is 1.02t, then the implied inflation targets would be 2%
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each year. Under PLT, the central bank probably would communicate its intentions by
announcing its implied inflation target rather than its path of price-level targets because the
public is more likely to understand the former than the latter. Therefore, just because the public
receives information on inflation targets does not mean the public understands that the central
bank is following IT rather than PLT.
There is a subtle but very important difference between IT and PLT. This difference
concerns the central bank’s contingency plans; it concerns how the central bank reacts to when
the actual inflation rate differs from its target. Under PLT, the price-level targets remain
unchanged regardless of the past inflation rate, whereas under IT they do change. For example,
assume that the current price level is 1.0 and the actual or implied inflation target is 2% per year
forever. Then the actual or implied path of price-level targets under both IT and PLT would be
1.02t. If the actual inflation rate over the first period turns out to be 3% instead of 2% , the
implied path of price-level targets under IT would change to 1.03(1.02t-1) whereas under PLT,
the price-level targets would remain the same. In other words, under PLT, if the price-level
differs from its target, the central bank will take action to bring the price level back to the central
bank’s original path of price-level targets. However, under IT, the central bank is more forgiving
and will respond to the higher price level by changing its implied price-level targets so to be
consistent with its targeted inflation rate and the actual price level that just occurred. (Note: This
paragraph does assume that output gap remains the same throughout these examples.)
In this paper, I argue that the contingency-plan difference between IT and PLT is very
important to the issue about whether monetary policy can determine prices. Contrary to the
previous literature, I argue that true IT cannot determine the price level of an economy when the
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central bank uses the nominal interest rate as its monetary instrument. 1 The stable price-level
targets under PLT provides a solid anchor to the public’s expectations of prices, at least if the
central bank shows a strong commitment to those price-level targets as it would if it followed a
strong McCallum-Woodford feedback policy rule for setting the nominal interest rate. However,
under IT, the public will be unable to form expectations about future price levels because the
central bank’s implied targeted price levels will change whenever the actual price level differs
from its target. This indeterminacy in expectations of future prices leads to an indeterminacy in
the current price level.
Consider this scenario: Central banks throughout the world are following IT, but the
general public is confused and thinks that the central banks are following PLT. Hence the public
forms its expectations based on the principles of PLT. Initially, price levels are stable.
However, over time the public will learn from experience about how the central banks handle
situations where the actual inflation rate differs from its target and come to realize that the
central banks are following IT instead of PLT. The public’s expectations will shift as a result of
this realization. If I am right that IT leads to price indeterminacy when the public understands
IT, then this shift of expectations would bring instability to the price levels of the world’s
economies. Thus, the issue of whether or not IT determines prices could have profound
implications to the world’s economies.
In practice, IT is complicated by the fact that central banks following IT also have goals
and objectives concerning output gap. However, economists often have to assume that “all other
things remain the same.” In particular, when this paper discussed the examples to illustrate the
difference between IT and PLT, we assumed that output gap remained the same. In this paper,
1 Almost all central banks today conduct monetary policy by pegging nominal interest rates rather than by setting
levels of monetary aggregates.
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we want to study IT without having to worry about the complex issues of handling output gap.
We can do so theoretically by using a flexible-price general equilibrium model. If the price level
is perfectly flexible to move the economy immediately to equilibrium, then there never is any
output gap. As a result, this paper studies IT in the context of a flexible-price general
equilibrium model, not because we believe the price level is in fact that flexible, but rather so we
can concentrate on the issue of price determinacy without being distracted by issues involving
output gap.
A necessary though not sufficient condition for a system of equations to determine the
unknowns of the system is that the number of equations be no less than the number of unknowns.
Therefore, if the number of equations in a system is less than the number of unknowns, we know
that we cannot determine all the values of the unknowns; in other words, some of the unknowns
will be indeterminate.
Most of the previous literature on the price-determinacy of IT has concluded that IT does
determine prices when the central bank follows a Taylor-like feedback rule for setting interest
rates (e.g., Woodford, 2003, and Dittmar and Gavin, 2005). However, the previous studies of the
price determinacy of IT have utilized infinite-horizon economic models. In his infinite-horizon
model, Woodford (2003, p. 73) states, “I then have a system of two equations at each date, (1.15)
and (1.21), to determine the two endogenous variables Pt and it ...” All that Woodford’s
statement means is there exists a one-to-one correspondence between the equations and the
unknowns in his infinite-horizon economy. Most mathematicians should react to Woodford’s
statement with grave suspect as that statement is meaningless when time is infinite.
Mathematicians will remember Galileo’s establishing a one-to-one correspondence between the
positive integers and their squares even though the set of squares is a proper subset of the
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positive integers.2 Mathematicians will also remember the errors that people can make when
working with infinity. For example, applying the associative law of addition with respect to the
infinite sequence 1-1+1-1+1-1… would erroneously lead to the conclusion that 0=1 ().
Given that Woodford’s statement is an indication of the possibility that the price-
determinacy literature has been making an error involving infinity, this paper takes an approach
that avoids any possibility of making logical errors involving infinity. This paper studies an
economy with an arbitrarily large finite horizon and then takes the limit as that finite horizon
goes to infinity. With a finite horizon, we can count the number of equations and unknowns to
determine whether all the unknown price levels can be determined.
By carefully counting equations and unknowns, this paper reaches a conclusion that is in
stark contrast to the existing literature. We find that the current price level cannot be determined
in a finite-horizon model under IT when the central bank pegs the nominal interest rate. In the
situations where we do find an equation containing the current inflation rate, we find only one
such equation. That equation, which is the same whether the economy has a finite or infinite-
horizon, reflects the central bank looking at the current inflation rate when it pegs the nominal
interest rate. This paper argues that it is absurd to think that a mechanism where the current
inflation rate affects the nominal interest rate is the mechanism that determines the price level.
The outline of the rest of this paper is as follows: Using a finite-horizon economic
model, Section II presents the Fisher-Euler equation that is the basis for the price-determinacy
literature. Then for a finite economy, section III analyzes PLT while Section IV analyzes “past
IT.” Section V extends the analysis to cover “current IT” and “expected IT”. Section VI tries to
anticipate some possible counterarguments and rebuts those counterarguments. Section VII
invalidates the non-exploding criterion, which the previous price-determinacy literature has used
2 See ???, p. ???.
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as the basis for their analysis and conclusions. Section VIII summarizes this paper’s conclusions
and explains why IT does not determine the current price level.
II. The Fisher-Euler Equation
The price-determinacy literature primarily depends on two equations – the Fisher-Euler
equation and a policy feedback rule. For a representative consumer without utility shocks, the
Fisher-Euler equation states
??
?
??
?
=
+
+
1
1)( '
P
)( '
P
t
t
tt
t
t
cU
ER
cU
β
(1)
where β is the time discount factor. As does Carlstrom and Freust (2001), we define Rt to be the
gross nominal interest rate from time t to time t+1. The gross nominal interest rate equals one
plus the nominal interest rate. The Fisher-Euler equation (1) states that the marginal utility per
“buck” today equals today’s gross nominal interest rate times the expected marginal utility per
“buck” tomorrow. 3
One rational-expectations model that leads to (1) is the following: Assume a
representative consumer4 who maximizes
[]
?
=
s
T
t
st
s
cUE)(
β
(2)
subject to
ttt
cPM ≥
(3)
3 Carlstrom and Fuerst (2001) argue that (4) only applies for what they call CWID timing. However, for a CIA
constraint where no money is held from one period to the next; (1) does apply.
4 Just because I am using a representative consumer in this paper’s model, does not mean that I condone its use. As
Eagle and Domian (2005) show, having diverse consumers even with “aggregatable” utility functions gives a much
more rich sense of the Pareto-efficiency involving those economies.
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()
11111
−−−−−
−++=+
ttttttttt
cPMRByPBM (4)
where (3) and (4) hold for t=0,1,2,…T; and Mt is the money held at the beginning of the period,
Bt is the amount of one-period bonds, and yt is the representative consumer’s endowment at
time t. Equation (3) is a cash-in-advance (CIA) constraint and
111
−−−−
ttt
cPM is the amount of
money held from period t-1 to period t. When the nominal interest rate is always positive, which
means Rt >1 for all t, the CIA constraint (3) holds with equality which implies that (4) can be
written as:
11
−−
+=+
ttttttt
RByPBcP (5)
It is relatively elementary to maximize (2) with respect to (5) to get (1) as the first order
condition.
Equation (1) can be justified in other ways as well such as including money in the utility
function.5 Also, Woodford (2003, p. 71) derives his equation (1.21), which is the same as (1)
except that he does include utility shocks.
As is often done in the price-determinacy literature, assume a pure exchange economy
without storage. Where Yt is the aggregate endowment, this implies that ct=Yt. Also, for the sake
of mathematical simplicity, assume that the representative consumer knows the future aggregate
endowments. Then (1) can be transformed into the following:
??
?
??
?
=
+
+
1
1
1
) ( '
) ( '
P
t
ttt
t
t
P
EYUR
YU
β
(6)
When the central bank pegs Rt, equation (6) represents one of the mechanisms by which
the current price level would be determined, if it is determined. If
??
?
??
?
+1
1
t
t
P
E
is determined, then
5 See Carlstrum and Fruest (2001).
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- 10 -
(6) will determine Pt. See Eagle (2005) for an explanation how (6) reflects
??
?
??
?
+1
1
t
t
P
E
and Rt
together affecting nominal aggregate demand which then affects the price level.
III. PLT in a Finite-Horizon Economy
With a system consisting of a finite number of equations and unknowns, we know that if
the number of equations is less than the number of unknowns, then it is impossible to determine
values for all the unknowns. In such a situation, we say that some of the unknowns are
indeterminate. However, with infinity, such simplicity is lost. For example, if we have a
sequence of systems of n equations with n+1 unknowns for n=1,2,3,…; then each finite system is
indeterminate. However, in the limit as n goes to infinity, both the number of equations and the
number of unknowns go to infinity. In fact, in the limit there is a one-to-one correspondence of
the number of equations to the number of unknowns. With an infinite number of equations and
unknowns, we no longer can compare the number of equations with the number of unknowns.
Contrary to the way most economists think about such a sum, mathematicians define
?
t
?
=
t
∞→
∞
≡
T
t
T
t
xx lim
0
. In this paper, we take a similar approach to address the issue of the price
determinacy. We study a rational expectations economy with a finite but arbitrarily large
horizon. Let T be this finite horizon. We can then study the limit of this economy as T goes to
infinity.
In the previous section, we discussed the Fisher-Euler equations upon which the previous
literature bases its price-determinacy analysis. However, the previous literature has assumed
infinite-horizon economies. We now deviate from the previous literature by assuming a finite
horizon, where the last period of the economy is period T. The Fisher-Euler equation (1) and
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- 11 -
hence (6) then apply for periods 0,1,2,…,T-1. These equations do not apply for period T because
there is no period T+1 for loans to be paid back. As a result, at time T there are no new loans or
bonds and hence no interest rate.
We now discuss price-level targeting (PLT) so that we can see a situation where the price
level is (technically) determined. Under PLT, the central bank has price-level targets for each
period, which it makes transparent to the general public. Following the symbolization of
Woodford (2003), define
*
tP to be the central bank’s price-level target for time t. Following
Bellman’s Principle, we start our analysis at time T and work backwards. At time T, there is no
interest rate, and therefore the central bank must set the money supply to achieve its price-level
target. Under the assumption of the CIA constraint, the central bank will set
TTT
YPM
*
=
. This
combined with cT=YT and the CIA constraint (3) holding with equality implies that
*
TT
PP =
.
Assume that for periods t=0,1,…,T-1; the central bank pegs the gross nominal interest
rate according to the following McCallum-Woodford feedback rule:6
τ−
β
)
??
?
?
??
?
?
?
?
?
?
?
?
=
+
+
+
+
1
*
1
*
1
*
1
P
( '
) ( '
Y
t
t
t
tt
tt
t
P
P
E
U
PYU
R (7)
where τ >0 is a parameter that reflects how sensitive this rule is to situations where the expected
value of next year’s price level differs from its target. Note that (7) is written in the spirt of
Carlstrom and Fruest except that I write it in terms of the price-level targets instead of some
steady state. Also, note that when
?
?
?
?
?
?
+
+
1
*
1
t
t
t
P
P
E
=1, we will say that the expected price level next
period is “on target”. Equation (7) is an “expected PLT” rule. After we discuss expected PLT,
we will then study “current PLT” and “past PLT.”
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- 12 -
Substituting (7) into (6) gives:
τ
−
+
+?
1
?
?
?
?
?
=
1
*
1
*
t
t
t
t
t
P
P
E
P
P
(8)
which applies for t=0,1,2,…,T-1. Remembering that the central bank sets the money supply at
time T to achieve
*
TT
PP =
, (8) implies that
*
11
−−=
TT
PP . By backwards recursion, we conclude
that
*
tt
PP =
for t=0,1,2,…,T. This means that the price level is technically determined in this
model under PLT regardless how large, but finite, T is. Hence, we conclude that in the limit as T
goes to infinity, the price-level is technically determined.
In this specific model, the actual price levels for all t are determined. In general, it should
be noted that for price determinacy, only P0 needs to be determined where 0 denotes the current
period. However, for P0 to be determined in a rational expectations model under PLT, all the
expected values of future prices must in some sense be determined as well.7
When the central bank pegs Rt, equation (8) reflects the two mechanisms by which the
current price is determined. Since
??
?
??
?
=
?
?
?
?
?
?
+
+
+
+
1
*
1
1
*
1
1
t
tt
t
t
t
P
EP
P
P
E
, (8) shows that if
??
?
??
?
+1
1
t
t
P
E
is
determined, then Pt will be determined. The “1” in the exponent of (8) reflects the mechanism of
the Fisher-Euler equation (6) by which the value of
??
?
??
?
+1
1
t
t
P
E
affects Pt. The “-τ” in the
exponent of (8) reflects the mechanism in the feedback rule (7) by which the
??
?
??
?
+1
1
t
t
P
E
affects Rt,
which in turn affects Pt in the Fisher-Euler equation (6). If we take expectations at time s of (8)
6 The current PLT version of this policy rule was initially proposed by McCallum (1981), but has more recently
been discussed by Woodford (2003).
7 For a relatively simple model, the “in some sense” may mean that E0[1/?t] be determined for all t; for other more
complex models, it may mean that more complex moments or expectations involving ?t will need to be determined.
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- 13 -
for s<t, we get
?
?
?
?
?
?
=
?
?
?
?
?
?
+
+
1
*
1
*
t
t
s
t
t
s
P
P
E
P
P
E
. This can also be written as
??
?
??
?
=
??
?
??
?
+
+
*
t
1
*
1
11
P
t
s
t
t
s
P
E
P
P
E
. This
equation shows how we can work backwards through time to determine
??
?
??
?
t
sP
E
1
from the value
of
??
?
??
?
+1
1
t
sP
E
.
If we solve (8) using backwards recursion, we get
tT
T
T
t
t
t
P
P
E
P
P
−
−
??
?
??
?
=
)1 (
*
*
τ
. Note that if
τ =0 then Pt depends very strongly on
??
?
??
?
t
t
P
E
1
in the sense discussed by Sargent and Wallace
(1975, p. ?). If τ >0, this dependence is diminished as T-t goes to infinity. The previous price-
determinacy literature such as Woodford (2003, p. 82) has argued that the price level in an
infinitely-lived economy is determined only if
0
>τ
.8 Eagle (2006a) shows that, if the public is
no longer certain the central bank will meet its price-level target at time T, then a similar
conclusion holds for PLT in a finite-horizon economy.
In addition to expected PLT, there are two other forms of PLT discussed in the literature
– “past PLT” and “current PLT.” Instead of (7), a central bank following past PLT will peg the
gross nominal interest rate according to the following feedback rule:
τ
??
?
β
)
?
??
?
?
=
−
−
*
+
+
1
1
*
1
*
1
P
( '
) ( '
Y
t
t
tt
tt
t
P
P
U
PYU
R
(9)
where τ >0 is a parameter that reflects how sensitive this rule is to situations where last period’s
price level differs from its target.
Substituting (9) into (6) gives:
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- 14 -
τ−
??
?
?
??
?
?
?
?
?
?
?
?
=
−
−
+
+
1
*
1
1
*
1
*
t
t
t
t
t
t
t
P
P
P
P
E
P
P
(10)
which applies for t=0,1,2,…,T-1.
Solving for the price levels under past PLT is considerably more complex than under
expected PLT. However, the price level is determined under past PLT. Since our discussion of
past PLT is really just a side note, I will now just give the solution without proof.9 Define
ττ=
) 1 ,( z and
) 1
−
,(
τ
1
),(
τ
+
=
sz
sz
τ
for s=2, 3, … . Then the solution to (10) is:
∏
=
s
??
?
?
??
?
?
=
−
+
−
−
−
tT
t
sz
t
t
P
P
P
P
1
1
),( ) 1(
1
*
1
*
τ
for t=0,l,2,…,T-1.
Also
*
TT
PP =
because at time T the central bank sets the money supply so to achieve this price
level. Note that if
*
1
−
1
−= PP, then the solution is that
*
tt
PP =
for t=0,1,…,T.
Still another form of PLT in the literature is “current PLT.” Under “current PLT” the
central bank uses the following “feedback rule” when it pegs the gross nominal interest rate:
τ
??
?
β
)
?
??
?
?
=
+
+
**
1
*
1
P
( '
)( '
Y
t
t
tt
tt
t
P
P
U
PYU
R
(11)
where τ >0 is a parameter that reflects how sensitive this rule is to situations this period’s price
level differs from its target.
Substituting (11) into (6) gives:
?
?
?
?
?
?
=
??
?
?
??
?
?
+
+
+
τ
1
*
1
1
*
t
t
t
t
t
P
P
E
P
P
(12)
8 See Woodford (2003, p. 82).
9 This can be proven using recursive methods.