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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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(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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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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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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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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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.