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Working Paper 8719

INTEREST RATE RULES ARE INFEASIBLE
AND FAIL TO COMPLETE MACROECONOMIC MODELS

by James G. Hoehn

James G. Hoehn i s an economist a t t h e Federal
Reserve Bank o f C l e v e l a n d .
Working papers o f t h e Federal Reserve Bank
o f Cleveland a r e p r e l i m i n a r y m a t e r i a l s
c i r c u l a t e d t o s t i m u l a t e d i s c u s s i o n and
c r i t i c a l comment. The views s t a t e d h e r e i n
a r e those o f t h e a u t h o r and n o t n e c e s s a r i l y
those o f t h e F e d e r a l Reserve Bank o f
C l e v e l a n d o r o f t h e Board o f Governors o f
t h e Federal Reserve System.

December 1987

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ABSTRACT

T h i s a r t i c l e r e c o n s i d e r s t h e r e c e n t l y c o n t r o v e r s i a l i s s u e s of w h e t h e r an
i n t e r e s t r a t e r u l e i s f e a s i b l e and whether i t l e a v e s nominal magnitudes
indeterminate.

I t i s shown t h a t i n t e r e s t r a t e r u l e s a r e i n f e a s i b l e u n l e s s t h e

p o l i c y a u t h o r i t i e s possess complete c u r r e n t i n f o r m a t i o n .

Furthermore,

i n t e r e s t r a t e r u l e s do n o t complete p r o t o t y p e macroeconomic models u n d e r
r a t i o n a l e x p e c t a t i o n s w i t h n o money i l l u s i o n .

Some i n f l u e n t i a l a n a l y s e s due

t o McCallum (1981, 1986) a r e r e i n t e r p r e t e d i n a manner c o n s i s t e n t w i t h t h e s e
propositions.

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INTEREST RATE RULES ARE INFEASIBLE AND FAIL TO COMPLETE MACROECONOMIC MODELS

I. I n t r o d u c t i o n

Sargent and Wallace (1975) p r e s e n t e d t h e f i r s t dynamic model showing t h a t

if an i n t e r e s t r a t e r u l e i s s u b s t i t u t e d i n p l a c e of a money s u p p l y r u l e , t h e n
nominal v a r i a b l e s a r e i n d e t e r m i n a t e .
Wallace i n d e t e r m i n a c y r e s u l t .

This a r t i c l e r e v a l i d a t e s t h e Sargent and

McCallum (1981) sought t o modify t h i s r e s u l t ,

c o n t e n d i n g t h a t i n t e r e s t r a t e r u l e s do n o t r e s u l t i n nominal i n d e t e r m i n a c y i f
t h e y a r e chosen w i t h any degree o f concern about t h e consequences f o r t h e
money s t o c k .

Canzoneri, Henderson, and Rogoff (1983) then argued t h a t t h e

a u t h o r i t i e s can peg t h e i n t e r e s t r a t e , even a t a c o n s t a n t l e v e l , and t h e
nominal determinacy problem w i l l n o t a r i s e , so l o n g as t h e a u t h o r i t i e s
announce a t r e n d f o r t h e money s u p p l y .

McCallum (19861, c o n s i d e r i n g t h i s

r e s u l t , c l a r i f i e d o r i n t e r p r e t e d i t t o mean t h a t p r i v a t e e x p e c t a t i o n s o f t h e
money s u p p l y a r e n o t anchored by an i n t e r e s t r a t e r u l e unless i t i s a 1 i m i t i n g
case o f an u n d e r l y i n g money s u p p l y f u n c t i o n , as t h e e l a s t i c i t y o f t h e money
s u p p l y w i t h r e s p e c t t o t h e i n t e r e s t r a t e increases w i t h o u t l i m i t .
A v i e w has developed t h a t i n t e r e s t r a t e r u l e s a r e f e a s i b l e and d e t e r m i n e

nominal magnitudes, i f t h e y a r e l i m i t i n g cases of a w e l l - s p e c i f i e d money
s u p p l y f u n c t i o n o r i f t h e y a r e d i r e c t e d toward t h e achievement of money
targets.

A number o f papers r e l y c r i t i c a l l y on t h e n o t i o n of t h e f e a s i b i l i t y

o f interest rate rules.

For example, Dotsey and K i n g (1983, 1986) f i n d t h a t

i n t e r e s t r a t e r u l e s have t h e disadvantage o f removing i n f o r m a t i o n s i g n a l s
o t h e r w i s e a v a i l a b l e t o p r i v a t e agents, by e l i m i n a t i n g any o b s e r v a b l e r e l a t i o n

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between t h e i n t e r e s t r a t e , which i s observable, and money, which i s n o t .

In

t h e i r model, t h e e l a s t i c i t y o f t h e money supply w i t h r e s p e c t t o t h e i n t e r e s t
r a t e i s e n t i r e l y i r r e l e v a n t , unless t h a t e l a s t i c i t y can be i n f i n i t e , i n which
case t h e s i g n a l s a v a i l a b l e t o agents a r e reduced.
The widespread popular i d e a t h a t p o l i c i e s designed t o c o n t r o l money,
p r i c e s , and/or o t h e r nominal v a r i a b l e s can be f o r m u l a t e d , d e s c r i b e d , and
executed i n terms o f predetermined i n t e r e s t r a t e s i s v u l n e r a b l e t o b o t h t h e
i n f e a s i b i l i t y and indeterminacy problems.

Unless and u n t i l these problems can

be r e s o l v e d , t h e r e can be no o b j e c t i v e b a s i s f o r such a p o l i c y designed i n
terms o f i n t e r e s t r a t e s .

These problems can be avoided by p l a c i n g f e a s i b l e ,

meaningful c o n s t r a i n t s d i r e c t l y on t h e money s t o c k v i a a money s u p p l y
function.

11.

Concepts and D e f i n i t i o n s

I n t e r e s t r a t e r u l e s a r e o f t e n seen as d e s c r i p t i o n s o f p o l i c y .

This

s e c t i o n p r e s e n t s t h e view t h a t i t i s more p r e c i s e t o d i s t i n g u i s h i n t e r e s t r a t e
r u l e s f r o m d e c i s i o n r u l e s , o r t r u l y s t r u c t u r a l equations f o r monetary p o l i c y .
The l a t t e r a r e e a s i e r t o m o t i v a t e on a f o r m a l l e v e l than a r e i n t e r e s t r a t e
r u l e s , so d i s c u s s i o n begins w i t h them.
A money s u p p l y f u n c t i o n i s a s t r u c t u r a l equation.

It specifies a decision

by t h e a u t h o r i t i e s concerning t h e q u a n t i t y o f money t o supply, c o n t i n g e n t on
t h e observed s t a t e o f the economy.

The observed s t a t e o f t h e economy i s t h e

s e t o f i n f o r m a t i o n a v a i l a b l e a t t h e t i m e o f t h e d e c i s i o n about t h e q u a n t i t y o f
money.

R e a l i s t i c a l l y , i t i n c l u d e s t h e c u r r e n t nominal i n t e r e s t r a t e and

lagged s t a t e v e c t o r .

An example o f a money supply f u n c t i o n a p p r o p r i a t e t o

s t a n d a r d l o g - l i n e a r macroeconomic models i s :
mt=qRt+pSt-,

(1)

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where m, i s t h e l o g o f t h e money s t o c k ,
R,

i s t h e l o g o f one p l u s t h e i n t e r e s t r a t e ,
i s t h e lagged s t a t e v e c t o r ,

and q and p a r e f i n i t e parameters r e p r e s e n t i n g p o l i c y choice.
q and p must be f i n i t e o r ( 1 ) i s n o t a v a l i d expression.

Incidentally,

t h e money supply f u n c t i o n can be renormal i z e d w i t h t h e nominal i n t e r e s t r a t e
on t h e l e f t - h a n d side, as i n :
~,=q-~m,-(q-'~)s,-,
so l o n g as q i s nonzero.

(2)

Regardless o f how t h e money supply f u n c t i o n i s

w r i t t e n , t h e c u r r e n t money s t o c k , m,,

must appear.

An i n t e r e s t r a t e r u l e i s a s t r i c t r e l a t i o n between t h e c u r r e n t nominal
i n t e r e s t r a t e and t h e lagged s t a t e v e c t o r .

I n o t h e r words, an i n t e r e s t r a t e

r u l e i s a requirement o r r e s t r i c t i o n t h a t t h e i n t e r e s t r a t e behave i n a
p a r t i c u l a r predetermined manner.

Under t h e r u l e , t h e c u r r e n t i n t e r e s t r a t e i s

i n v a r i a n t w i t h respect t o c u r r e n t innovations i n the s t a t e vector.

The

f e a t u r e t h a t d i s t i n g u i s h e s an i n t e r e s t r a t e r u l e from a money s u p p l y f u n c t i o n
i s t h a t t h e former excludes t h e c u r r e n t money s t o c k as an argument.

For

example, t h e equation

(3)

Rt=po+plRt-I+~2mt+p3mt-l
r e p r e s e n t s a money supply f u n c t i o n i f pi i s nonzero, b u t r e p r e s e n t s an
i n t e r e s t r a t e r u l e i f p2 i s zero.

Money supply f u n c t i o n s and i n t e r e s t r a t e r u l e s a r e f u n d a m e n t a l l y d i f f e r e n t
kinds o f relations.

The money supply f u n c t i o n i s a s t r u c t u r a l e q u a t i o n :

i s l o g i c a l l y p r i o r t o t h e s o l u t i o n t o t h e model.

it

The a n a l y s t can w r i t e down

t h e money supply f u n c t i o n and t h e n o n p o l i c y s t r u c t u r a l e q u a t i o n s and from them
d e r i v e s o l u t i o n s f o r endogenous v a r i a b l e s , as w i l l be i l l u s t r a t e d below.

On

t h e o t h e r hand, an i n t e r e s t r a t e r u l e i s a r e s t r i c t i o n on t h e b e h a v i o r o f t h e

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interest rate.

Unless t h e i n t e r e s t r a t e i s a d i r e c t i n s t r u m e n t o f p o l i c y ,

then an i n t e r e s t r a t e r u l e cannot adequately r e p r e s e n t an o p e r a t i o n a l p o l i c y .
What i s needed i s a d e c i s i o n r u l e f o r an a c t u a l choice v a r i a b l e .

Among

c a n d i d a t e v a r i a b l e s i n t h e s t r u c t u r e , o n l y money i s a p p r o p r i a t e l y conceived as
a f e a s i b l e choice v a r i a b l e .

Then f e a s i b l e p o l i c y can be d e s c r i b e d a d e q u a t e l y

o n l y i n terms o f a r u l e f o r t h e behavior o f t h e money s t o c k as a f u n c t i o n o f
t h e observed s t a t e .

I n o t h e r words, p o l i c y can be adequately and

o p e r a t i o n a l l y described o n l y i n terms o f a money supply f u n c t i o n .
An i n t e r e s t r a t e r u l e can be i n t e r p r e t e d o n l y as a r e s t r i c t i o n on t h e
s o l u t i o n o r outcome o f t h e model.

Indeed, t h e i n t e r e s t r a t e r u l e , i f i t i n

f a c t holds t r u e i n an economy, i s t h e reduced- form e q u a t i o n f o r the i n t e r e s t
rate.

111.

Nominal Determinacy i n a P r o t o t y p e Model

T h i s s e c t i o n presents a s i m p l e model and shows how money and p r i c e s a r e
determined by s o l v i n g t h e money supply f u n c t i o n and t h e n o n p o l i c y s t r u c t u r a l
e q u a t i o n s simultaneously.
Consider t h e f o l l o w i n g i l l u s t r a t i v e macroeconomic model.

L e t aggregate

demand be a f u n c t i o n o f t h e r e a l r a t e o f i n t e r e s t and a d i s t u r b a n c e , w h i l e
aggregate supply i s a c o n s t a n t .
y:

Formally:

= do-dCRt-(Et-,pt+l-pt)I+ut,

d>O

(4)

y: = y F

(5)

where y: and y: a r e aggregate demand and supply, r e s p e c t i v e l y ; p,
f

i s t h e log o f the p r i c e l e v e l ; y

i s a f i x e d o u t p u t supply; and u, i s a

n o n a u t o c o r r e l a t e d d i s t u r b a n c e t o aggregate demand.
operator, E,-,,

The lagged e x p e c t a t i o n

when a p p l i e d t o p t + , , r e t u r n s t h e o b j e c t i v e mathematical

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e x p e c t a t i o n o f t h e f u t u r e p r i c e l e v e l c o n d i t i o n e d on a l l lagged r e a l i z a t i o n s
o f the state vector.

'

Then t h e commodity market equi 1 ib r i u m c o n d i t i o n , o r

I S function, i s
Rt = r t + ( E t - l ~ t + ~ - p ~ )

(6)

where t h e (ex a n t e ) r e a l r a t e ,
r,

=

i s exogenous.

(l/d)C(do-yf)+utl,

(7)

( d o - y f ) w i l l be t a k e n as zero.

T h i s assumption i s

h e u r i s t i c o n l y ; i t has t h e e f f e c t o f making t h e mean o f t h e r e a l r a t e equal t o
zero, thereby s i m p l i f y i n g mathematical expressions t h a t f o l l o w .
L e t t h e money demand f u n c t i o n be
mt-pt=ao-aRt+et,

(8)

a>O,

where m, i s t h e q u a n t i t y o f money and e, i s a white- noise d i s t u r b a n c e .
For t h e purpose a t hand, i t w i l l s u f f i c e t o c o n s i d e r a s i m p l i f i e d v e r s i o n
o f t h e money supply f u n c t i o n analyzed by McCallum (1986):
mt=Xo+X(Rt-R*),

-(l+a)<X<m,

where R* i s an i n t e r e s t r a t e t a r g e t .

(9)
R* i s t r e a t e d as a c o n s t a n t t o s i m p l i f y

analysis.
The economic s t r u c t u r e i s composed o f t h e two n o n p o l i c y equations, t h e I S
f u n c t i o n , ( 6 ) , and t h e money demand f u n c t i o n , ( 8 ) , p l u s t h e p o l i c y e q u a t i o n ,
o r money supply f u n c t i o n , ( 9 ) .

I t i s easy t o show t h a t these t h r e e e q u a t i o n s

a r e s u f f i c i e n t t o p r o v i d e s o l u t i o n s f o r t h e t h r e e endogenous v a r i a b l e s , p,,
m t , R,,

as w e l l as t h e endogenous e x p e c t a t i o n E t - l p t + l .

I t i s useful

f o r what f o l l o w s t o show how t h e method o f undetermined c o e f f i c i e n t s can be
used t o s o l v e t h e model i f an a p p r o p r i a t e s t a t e v e c t o r can be i d e n t i f i e d .
Formally, a s t a t e v e c t o r i s a s e t o f dated v a r i a b l e s , b o t h predetermined
and exogenous, t h a t c o m p l e t e l y d e s c r i b e t h e p o s i t i o n , o r s t a t e , of a dynamic
system.

C u r r e n t r e a l i z a t i o n s o f endogenous v a r i a b l e s a r e s t r i c t f u n c t i o n s o f

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the state vector.

The concept o f s t a t e v e c t o r i s invoked d u r i n g a n a l y s i s o f

g i v e n systems i n o r d e r t o hypothesize a t r i a l s o l u t i o n , whose arguments a r e
t h e s t a t e v e c t o r and whose c o e f f i c i e n t s a r e f u n c t i o n s o f s t r u c t u r a l
Given a ( l i n e a r ) t r i a l solution- - based on an adequate s t a t e

parameters.

vector- - and g i v e n a c o n s i s t e n t ( l i n e a r ) economic s t r u c t u r e , t h e c o e f f i c i e n t s
of t h e t r i a l s o l u t i o n can be determined.

Operationally, a state vector i s a

s e t o f d a t e d predetermined and exogenous v a r i a b l e s t h a t , when i n c l u d e d as
arguments i n a t r i a l s o l u t i o n , p r o v i d e a c o n s i s t e n t s e t o f i d e n t i t i e s r e l a t i n g
s t r u c t u r a l parameters of t h e system t o t h e c o e f f i c i e n t s i n t h e t r i a l
solution.

If a t r i a l s o l u t i o n i s found t o be i n c o n s i s t e n t w i t h t h e s t r u c t u r e ,

then t h e s t a t e v e c t o r on which i t i s based i s inadequate and must be expanded
(assuming t h e s t r u c t u r e i t s e l f i s c o n s i s t e n t ) .
U n f o r t u n a t e l y , t h e s t a t e v e c t o r i s nonunique; t h e r e a r e an i n d e f i n i t e l y
l a r g e s e t of v e c t o r s adequate t o d e s c r i b e t h e c u r r e n t s t a t e .

From among

t h i s adequate s e t , t h e a n a l y s t may choose a p a r t i c u l a r s t a t e v e c t o r f o r
a n a l y t i c a l convenience.

McCallum (1983) has suggested t h e employment o f a

minimal s t a t e v e c t o r , which i s t h e smallest- dimensioned v e c t o r adequate t o
d e s c r i b e t h e c u r r e n t s t a t e o f t h e system.

A s t a t e v e c t o r i s a minimal s t a t e

v e c t o r i f none o f i t s elements can be l e f t o u t of t h e t r i a l s o l u t i o n w i t h o u t
making t h e t r i a l s o l u t i o n i n c o n s i s t e n t w i t h t h e s t r u c t u r e .
Exogenous and predetermined v a r i a b l e s e x p l i c i t l y appearing i n t h e
s t r u c t u r a l e q u a t i o n s a r e obvious candidates f o r i n c l u s i o n i n t h e minimal
state vector.
vector.

Sometimes, b u t n o t always, t h e y comprise a minimal s t a t e

I n g e n e r a l , however. t r i a l and e r r o r must be employed t o ensure

t h a t t h e s t a t e v e c t o r i s adequate.

I n t h e i l l u s t r a t i v e model, t h e s e t of

s t a t e v a r i a b l e s e x p l i c i t l y appearing i n i t w i l l c o n s t i t u t e an adequate s t a t e
v e c t o r t h a t r u l e s o u t bubbles, o r e x p l o s i v e p a t h s f o r r e a l money balances,

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b u t t h a t o t h e r w i s e does n o t r e s t r i c t t h e outcome o f t h e model beyond t h e
r e s t r i c t i o n s already present i n the s t r u c t u r e .

'

I n t h e i l l u s t r a t i v e model, t h e exogenous and predetermined v a r i a b l e s
appearing i n t h e s t r u c t u r a l r e p r e s e n t a t i o n a r e :
St={l ,rt,et),

(10)

including the u n i t "variable" t h a t i s i m p l i c i t .
of t h e undetermined c o e f f i c i e n t s , denoted as n , , s ,

A t r i a l s o l u t i o n i n terms

i s given i n equations

(ll),
(12), and ( 1 3 ) .

pt=fllo+nllrt+n12et

(11)

mt=n20+n21rt+I122et

(12)

Rt=n30+n31 r t + n 3 2 e t

(13)

Under t h e i n f o r m a t i o n assumptions and u s i n g (ll), h e e x p e c t a t i o n o f t h e
t
f u t u r e p r i c e must be
E t - ~ ~ t + l =o n l

( 1 4)

where use has been made o f E t ( r t ) = [ d o - y f l d > l = O and Et(et)=O.
The s o l u t i o n i s o b t a i n e d by s u b s t i t u t i n g ( l l ) ,
(121, (13), and (14) i n t o
t h e s t r u c t u r a l e q u a t i o n s (61, (8>, and (9) and s o l v i n g t h e r e s u l t i n g
i d e n t i t i e s f o r a l l ni, c o e f f i c i e n t s .

These i d e n t i t i e s a r e shown below.

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These i d e n t i t i e s a r e c o n s i s t e n t , so t h e s t a t e v e c t o r i s adequate.

The

c o e f f i c i e n t s a r e then determined t o be as shown below.

When these expressions f o r t h e n i j s a r e s u b s t i t u t e d i n t o ( l l ) ,( 1 2 ) ,
(13),

and

i t i s c l e a r t h a t t h e s o l u t i o n s f o r a l l endogenous v a r i a b l e s a r e

we1 1- defined.

IV.

I n t e r e s t Rate Rules Are I n f e a s i b l e

As argued above,

an i n t e r e s t r a t e r u l e cannot be i n t e r p r e t e d as a

s t r u c t u r a l e q u a t i o n ; i t i s a r e s t r i c t i o n on, t h e b e h a v i o r o f t h e i n t e r e s t

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rate.

I f t h e i n t e r e s t r a t e r u l e i s t o h o l d t r u e f o r an economy, t h e n t h e

reduced f o r m must be i d e n t i c a l t o t h e i n t e r e s t r a t e r u l e .

This s e c t i o n

shows t h a t t h e r e i s no f e a s i b l e p o l i c y choice- - no s e t of money s u p p l y
f u n c t i o n parameters- - that s a t i s f i e s t h e requirement t h a t i t r e n d e r s t h e
i n t . e r e s t r a t e reduced f o r m i d e n t i c a l t o t h e i n t e r e s t r a t e r u l e .
Consider the t r i v i a l i n t e r e s t r a t e r u l e
Rt=R*.

(17)

I f an economy obeys such an i n t e r e s t r a t e r u l e , t h e n t h e reduced form
c o e f f i c i e n t s i n t h e i n t e r e s t r a t e e q u a t i o n o f t h e t r i a l s o l u t i o n , ( 1 3 ) , must
be
1 3 0=R*
1
n
3

1=0

llsr=O.

The second and t h i r d o f these r e s t r i c t i o n s render t h e i n t e r e s t r a t e
predetermined; t h e f i r s t r e s t r i c t i o n ensures t h a t t h e i n t e r e s t r a t e t a r g e t
R* i s achieved.

I n view o f t h e s o l u t i o n values f o r t h e I13,s shown i n

(16), t h e i n t e r e s t r a t e r u l e r e q u i r e s
R*=O
l/(a+X+l)=O.

(19)

Then an i n t e r e s t r a t e r u l e i s f e a s i b l e i f and o n l y i f t h e r e e x i s t s a
p o l i c y e q u a t i o n such t h a t (19) h o l d s .
v e c t o r {Ao ,X,R*)

But t h e r e i s no a l l o w a b l e p o l i c y

s a t i s f y i n g these e q u a t i o n s .

i s no f i n i t e X t o s a t i s f y n3l=fl32=(a+~+l)-'=O.

Specifical l y , there
Nhile i t i s true

t h a t 113 and 113* approach z e r o as X approaches i n f i n i t y ,

I,
I

and

f13, a r e undefined a t X=a because i f X i s n o t f i n i t e ,

t h e n (9) i s n o t a

v a l i d e x p r e s s i o n of t h e s t r u c t u r a l p o l i c y e q u a t i o n .

And i f (9) i s n o t t h e

p o l i c y e q u a t i o n , something e l s e d e s c r i b i n g monetary p o l i c y must be s u p p l i e d t o

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complete t h e model.

The f e a s i b i l i t y o f an i n t e r e s t r a t e r u l e cannot be

assessed w i t h o u t a complete model.
Of course, t h e argument a g a i n s t t h e f e a s i b i l i t y o f an i n t e r e s t r a t e r u l e
depends, so f a r , on t h e assumption t h a t t h e money s u p p l y f u n c t i o n takes t h e
f o r m (9).

F o r t u n a t e l y f o r t h e argument, i m p o r t a n t g e n e r a l i z a t i o n s on ( 9 )

can be made w i t h o u t a f f e c t i n g t h e r e s u l t t h a t i n t e r e s t r a t e r u l e s a r e
infeasible.

Simple i n d u c t i v e evidence, i n t h e f o r m o f t r y i n g v a r i o u s

candidates and o b s e r v i n g t h e i m p l i c a t i o n s , suggests t h a t lagged r e a l i z a t i o n s
of any v a r i a b l e s appearing i n t h e model can be added as arguments t o t h e
p o l i c y r u l e ( t h e s e , o f course, may augment t h e minimal s t a t e v e c t o r ) w i t h o u t
affecting the i n f e a s i b i l i t y r e s u l t .
Apparently, t h e r e i s o n l y one money s u p p l y f u n c t i o n t h a t would make an
i n t e r e s t r a t e r u l e a feasible solution f o r the interest rate.

T h i s money

s u p p l y f u n c t i o n i s o f t h e form:
mt=Ao+X,rt+A2et

(20)

( w h i c h i s a l s o t h e reduced f o r m f o r money), where t h e Xs a r e p o l i c y c h o i c e
parameters.

Then t h e reduced f o r m f o r t h e i n t e r e s t r a t e i s :

Rt = [ ( I - X I )/(l+a)lrt+C(l-X2)/(l+a)let.

(21 >

I n view of t h e l a t t e r , i t i s c l e a r t h a t i f X1=A2=1, then Rt=R*=O.
Therefore,

i f p o l i c y can c o n d i t i o n t h e money s t o c k a r b i t r a r i l y on t h e s t a t e

v e c t o r , as i n ( 2 0 ) , then an i n t e r e s t r a t e r u l e i s f e a s i b l e - - t h e r e e x i s t s
{A, ,A,, A 2 ) such t h a t R, i s predetermined.

T h i s a l t e r n a t i v e money s u p p l y f u n c t i o n , (20), c o u l d be made o p e r a t i o n a l
o n l y i f t h e policymaker had complete c u r r e n t i n f o r m a t i o n , t h a t i s , i f t h e
a u t h o r i t i e s c o u l d observe t h e complete, c u r r e n t s t a t e o f t h e economy,
1 r

,

e

But i n t h e r e a l i s t i c case i n which t h e a u t h o r i t i e s do

n o t have f u l l knowledge o f t h e c u r r e n t u n d e r l y i n g shocks, because of

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i n f o r m a t i o n l a g s , no o p e r a t i o n a l money supply f u n c t i o n can be found t h a t
generates predetermined b e h a v i o r f o r t h e i n t e r e s t r a t e .

An i n t e r e s t r a t e

r u l e i s i n f e a s i b l e because i t i s i n c o n s i s t e n t w i t h t h e s t r u c t u r a l
assumptions o f t h e model (no money i l l u s i o n ) and t h e i n f o r m a t i o n a l
assumption ( p o l i c y a u t h o r i t i e s have incomplete c u r r e n t i n f o r m a t i o n ) .

V.

I n t e r e s t Rate Rules Are Not L i m i t i n g Cases o f Money Supply F u n c t i o n s

This s e c t i o n considers t h e i d e a t h a t an i n t e r e s t r a t e r u l e i s a l i m i t i n g
case o f a money supply r u l e , an i d e a t h a t has been a t t r i b u t e d t o
McCallum.

T h i s s e c t i o n d i s p r o v e s t h i s i d e a and ends w i t h an account o f

how t h i s i d e a a r i s e s f r o m a f a i l u r e t o d i s t i n g u i s h between reduced f o r m and
s t r u c t u r a l equations.
McCallum proves t h a t as X approaches i n f i n i t y , t h e reduced f o r m
c o e f f i c i e n t s , t h e II,,s,

remain w e l l - d e f i n e d .

The p o i n t here i s t h a t any

f i n i t e X, no m a t t e r how l a r g e , i s c o n s i s t e n t w i t h determinacy.

This p o i n t

i s unexceptionable, b u t t h e n e x t s t e p i n h i s argument i s t h a t , because t h e
c o e f f i c i e n t s on c u r r e n t i n n o v a t i o n s o f t h e reduced form f o r t h e i n t e r e s t
r a t e , 113, and

II32,

become a r b i t r a r i l y c l o s e t o z e r o as X g e t s

l a r g e r , t h a t one can come a r b i t r a r i l y c l o s e t o a c h i e v i n g an i n t e r e s t r a t e
rule.

Furthermore, i t then seems reasonable t o say t h a t , i n t h e l i m i t , t h e

money supply f u n c t i o n becomes t h e i n t e r e s t r a t e r u l e .
This i d e a i s i m p o r t a n t t o r e f u t e because i t has a p p a r e n t l y become
influential.

For example, Dotsey and K i n g (1986) analyze t h e e f f e c t s o f

i n t e r e s t r a t e r u l e s by s e t t i n g 1 equal t o i n f i n i t y :

"An i n t e r e s t r a t e peg

i s t h e l i m i t i n g case o f a contemporaneous response t o i n t e r e s t r a t e s ( i . e . ,

CXI=~)." (p. 37).

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To p r o v e t h a t i n t e r e s t r a t e r u l e s or pegs a r e

not l i m i t i n g

cases o f

money s u p p l y r u l e s , i t i s h e l p f u l t o r e c a l l t h e d e f i n i t i o n of a l i m i t .
l i m i t i s a f i x e d p o i n t t o which an i n f i n i t e sequence converges.

A

Not a l l

i n f i n i t e sequences have l i m i t s n o r , i f t h e y e x i s t , a r e l i m i t s n e c e s s a r i l y
elements o f t h e sequence.
A={ X I , XZ,

w i t h Xl<Xz<

1 3 ,

Consider an i n f i n i t e sequence o f X v a l u e s .

. ..,

Xj,

....

)

... < A j <....

The money s u p p l y e q u a t i o n ( 9 ) d e f i n e s a t r a n s f o r m a t i o n T:A->M where M i s
t h e i n f i n i t e sequence o f money s u p p l y f u n c t i o n s :
M={ m;=Xo+Xl (Rt-R*) , m:=Xo+X2 (R,-R*) ,

m;=Xo+X3(Rt-R*),

...

, mT=Xo+Xj<R,-R*),

....)

I f a l i m i t i n g money s u p p l y f u n c t i o n e x i s t e d as X j i n c r e a s e d w i t h o u t
l i m i t , i t would be t h e e x p r e s s i o n f o r m:=Xo+Xj(R,-R*)
sequence M converged as X j i n c r e a s e s w i t h o u t l i m i t .

t o which t h e
O b v i o u s l y , such a

l i m i t does n o t e x i s t .
How, then, can a r u l e such as ( 1 7 ) have been m i s t a k e n f o r a l i m i t i n g case
o f ( 9 > ? The e r r o r a r o s e f r o m a f a i l u r e t o d i s t i n g u i s h between t h e p o l i c y r u l e
( 9 > , w h i c h i s a s t r u c t u r a l e q u a t i o n , and t h e s o l u t i o n for t h e i n t e r e s t r a t e .
A n a t u r a l source o f t h i s c o n f u s i o n i s t h a t t h e l a t t e r

does

i n f a c t possess a

f i x e d p o i n t , or l i m i t i n g f u n c t i o n , as X approaches i n f i n i t y .
f o r m s o l u t i o n for R,,

The r e d u c e d

g i v e n by (13) and (161, has t h e l i m i t i n g f u n c t i o n or

f i x e d p o i n t i d e n t i c a l t o t h e i n t e r e s t r a t e r u l e , R,=O.

VI.

I n t e r e s t Rate R u l e s F a i l t o Complete t h e Model

An i n t e r e s t r a t e r u l e does n o t c o m p l e t e t h e model.

I n particular, the

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-1 3-

n o n p o l i c y s t r u c t u r a l equations ( 6 ) and ( 8 > , t o g e t h e r w i t h an i n t e r e s t r a t e
r u l e such as (17), do n o t p r o v i d e unique bubble- free s o l u t i o n s f o r t h e p r i c e
l e v e l and money s t o c k .
Consider t h e system comprised o f ( 6 ) ,

(8>,and ( 1 7 > , shown below f o r

convenience.
Rt=rt+(Et-lpt+l-pt)

(6)

mt-pt=ao-aRt+et

(8)
(17)

Rt=R*

The money demand e q u a t i o n , (81, i s o f no use i n d e t e r m i n i n g t h e p r i c e
l e v e l i n t h i s system.

( 6 ) and (17) t o g e t h e r a r e s u f f i c i e n t t o determine t h e

expected r a t e o f i n f l a t i o n ,
(Et-lpt+l-pt)=R*-rt,

(22)

b u t t h e p r i c e l e v e l , p t , i s undetermined, b e i n g dependent on an u n s u p p l i e d
terminal condition, E , - l p t + l :
pt=Et-lpt+l+R*-rt.

(23)

By s u b s t i t u t i n g (23) and (17) i n t o (81, i t i s seen t h a t m, i s a l s o
dependent one- for- one on t h e u n s u p p l i e d t e r m i n a l c o n d i t i o n and, hence, i s
a l s o indeterminate.
r e s p e c t t o p , and m,

The system ( ( 6 ) ,(8) , ( I 7 1 1 i s i n c o m p l e t e w i t h

.

McCal lum e v a l u a t e d t h e completeness o f systems such as ( ( 6 ) ,(8) ,(17))
u s i n g a t r i a l s o l u t i o n d e r i v e d f r o m t h e undetermined c o e f f i c i e n t s method
i l l u s t r a t e d above.

Systems f o r which t h e t r i a l s o l u t i o n s f o r m t and p,

a r e nonunique were considered n o m i n a l l y i n d e t e r m i n a t e .

McCallum focuses on

t h e c o n d i t i o n s under which t h e i n t e r c e p t c o e f f i c i e n t s o f p t and m,,
are

which

n l o and n r o , r e s p e c t i v e l y , f a i l t o be u n i q u e l y determined, because

t h i s i s t h e manner i n which i n d e t e r m i n a c y r e v e a l s i t s e l f i n t r i a l s o l u t i o n s .
Such a n a l y s i s i s i n v a l i d , however, because i t depends upon assumptions a b o u t

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t h e s t a t e v e c t o r t h a t cannot be j u s t i f i e d .

The s t a t e v e c t o r can be

determined o n l y f r o m a n a l y s i s o f t h e complete s t r u c t u r a l model.

If the

s t r u c t u r a l model o r any o f i t s components i s unknown, then t h e r e i s no
assurance e i t h e r t h a t any p a r t i c u l a r s t a t e v e c t o r i s adequate o r t h a t i t i s
consi s t e n t w i t h t h e s t r u c t u r e .

Demonstrations t h a t t h e reduced forms f o r

money and p r i c e s a r e i n d e t e r m i n a t e i n t r i a l s o l u t i o n s a r e i r r e l e v a n t i n an
a n a l y s i s o f t h e completeness o f a model, because t h e y assume
knowledge-- about t h e s t a t e v e c t o r - - t h a t cannot be assumed i f t h e
completeness o f t h e model i s i n doubt.
I t may seem t h a t t h e s t a t e v e c t o r assumption i s innocuous.

McCallum

i n t e n d s i t n o t as a r e s t r i c t i o n on t h e s t r u c t u r e , b u t r a t h e r as a
r e s t r i c t i o n on t h e reduced form--one t h a t l e t s t h e s t r u c t u r e speak f o r
itself.

Yet, i t a l r e a d y assumes t h a t an i n f i n i t u d e o f p o t e n t i a l s t a t e

v a r i a b l e s have been excluded as arguments i n t h e money supply f u n c t i o n
necessary t o complete t h e model.

Consider t h e general f o r m u l a t i o n f o r t h e

money supply:

m, = Xo+Xlrt+X2et

+gym,- +gym

-

+ . . . +8ym,- + . . .

+ e ~ p t - 1 + e ; p t - 2 + . . .+e;~,-~+..
.

+87rt-1+8;rt-2+...+e;tt-J+...
+8:et-l+8:et-2+.
+TI t + T 2 t L + .

. .+8;et-J+. . .

. .+TJtJ+. ...

N o t i c e t h a t r, and e t - - t h e c u r r e n t s t a t e variables- - must be arguments i n
t h e money supply r u l e i n o r d e r f o r t h e i n t e r e s t r a t e r u l e t o be f e a s i b l e .
There i s a f e a s i b l e i n t e r e s t r a t e r u l e a s s o c i a t e d w i t h any s e t o f z e r o
r e s t r i c t i o n s on t h e 8s and

TS.

I f a l l l a g s o f s t a t e v a r i a b l e s and a l l

t i m e t r e n d s a r e excluded a r b i t r a r i l y , then t h e s t r u c t u r e i s b e i n g r e s t r i c t e d

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such t h a t a l l 8s and

TS

i n t h e money s u p p l y f u n c t i o n a r e zero.

But i f

t h i s r e s t r i c t i o n can be made, i t i s because t h e money supply f u n c t i o n i s
known t o be o f t h e f o r m shown and because i t i s f u r t h e r known t h a t t h e
r e l e v a n t parameters a r e zero.

mt-,,

r t - , , e,-l,

The f a c t t h a t v a r i a b l e s such as p t - l ,

o r t do n o t appear i n e i t h e r t h e n o n p o l i c y equations

(6) and ( 8 ) o r i n t h e i n t e r e s t r a t e r u l e (17) does n o t mean t h a t such
v a r i a b l e s m i g h t n o t be s t a t e v a r i a b l e s ; t h a t i s s u e can be answered o n l y
a f t e r t h e money supply f u n c t i o n has been made e x p l i c i t t o complete t h e model.

R e i n t e r p r e t a t i o n o f a R e s u l t Due t o McCallum

VII.

McCallum's 1981 a r t i c l e sought t o show how an i n t e r e s t r a t e r u l e c o u l d
be c o n s i s t e n t w i t h determined values o f nominal v a r i a b l e s , so l o n g as t h e
i n t e r e s t r a t e was s e t i n o r d e r t o have some d e s i r e d e f f e c t on t h e money
stock.

Because h i s a n a l y s i s i s i n f l u e n t i a l , some r e c o n c i l i a t i o n between h i s

a n a l y s i s and t h a t o f t h i s a r t i c l e i s c a l l e d f o r .
McCallum assumed t h a t " . . . t h e

monetary a u t h o r i t i e s adopt a feedback r u l e

f o r t h e i n t e r e s t r a t e ...." (1981, p. 319)

I t i s n o t c l e a r whether t h i s

means t h a t t h e a u t h o r i t i e s make t h e i n t e r e s t r a t e predetermined t h r o u g h
c h o i c e o f an a p p r o p r i a t e money supply f u n c t i o n , o r whether i t means t h a t t h e
i n t e r e s t r a t e i t s e l f i s a p o l i c y i n s t r u m e n t o r a d i r e c t choice v a r i a b l e .

In

e i t h e r case, t h e assumption r u n s i n t o problems discussed above c o n c e r n i n g
t h e f e a s i b i l i t y o f an i n t e r e s t r a t e r u l e .

B u t even though such a r u l e

should n o t be considered a p o l i c y r u l e , and even though an i n t e r e s t r a t e
r u l e i s i n f e a s i b l e , McCallum l s a n a l y s i s can s t i l l be considered f o r what i t
says about whether i n t e r e s t r a t e r u l e s (however t h e y may come i n t o f o r c e )
complete t h e model i n t h e sense o f r e n d e r i n g nominal magnitudes

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determinate.

I t was t h i s i s s u e t h a t Sargent and Wallace e x p l i c i t l y

addressed.
The f o l l o w i n g d i s c u s s i o n presents McCallum's assumptions about monetary
p o l i c y i n a s i m p l i f i e d form, b u t one e s s e n t i a l l y adequate t o t h e i s s u e s a t
hand.

The o r i g i n a l a n a l y s i s i n v o l v e d a d d i t i o n a l , n o n e s s e n t i a l dynamic

elements, w i t h money t a r g e t s depending on t h e lagged s t a t e and w i t h t h e
i n t e r e s t r a t e r u l e c a l l i n g f o r smoothing o f i n t e r e s t r a t e movements.
McCallum m o t i v a t e d t h e i n t e r e s t r a t e r u l e i n t h e f o l l o w i n g way.

The

a u t h o r i t i e s , w i s h i n g t o e x e r t e f f e c t i v e c o n t r o l o v e r t h e money s t o c k , choose
t h e i n t e r e s t r a t e i n a manner c o n s i s t e n t w i t h a predetermined money t a r g e t .
L e t t i n g t h e money t a r g e t be a f i x e d v a l u e , P O , f o r s i m p l i c i t y , t h e
a u t h o r i t i e s then s e t R,

i n such a way t h a t

Et-lmt=po
v i a some a p p r o p r i a t e l y chosen i n t e r e s t r a t e r u l e .

(24)
Such an i n t e r e s t r a t e

r u l e can be found by s e t t i n g t h e e x p e c t a t i o n o f money demand equal t o t h e
,
money t a r g e t , p:

Et-lmt=Et-l{pt+ao-aRt}=po,

which i m p l i e s
Et-,Rt=(l/a){Et-lpt+a~-l-Io}.

The i n t e r e s t r a t e r u l e t h a t achieves t h i s i s then
R,=(lla){Et-,pt+ao-po}.

(27)

McCallum regarded t h i s l a s t e q u a t i o n as t h e s p e c i f i c a t i o n o f p o l i c y b e h a v i o r .
Given these assumptions, t h e r e i s a w e l l - d e f i n e d e q u i l i b r i u m f o r a l l
endogenous v a r i a b l e s i n terms o f a t r i a l s o l u t i o n i n t h e minimal s t a t e
vector.

From t h i s , McCallum concludes t h a t an i n t e r e s t r a t e r u l e i s

c o n s i s t e n t w i t h nominal determinacy.
However, t h i s l i n e o f r e a s o n i n g does n o t p r o v i d e an argument t h a t

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i n t e r e s t r a t e r u l e s a r e c o n s i s t e n t w i t h nominal determinacy, because t h e
c o n c l u s i o n i s e s s e n t i a l l y among t h e premises.

McCallum's d e m o n s t r a t i o n

hypothesizes t h a t a nominal v a r i a b l e - - t h e money stock - - is r e n d e r e d
d e t e r m i n a t e by a p p r o p r i a t e choice o f t h e i n t e r e s t r a t e r u l e w i t h o u t a c t u a l l y
e x p l a i n i n g how t h a t c o u l d come about.

Then i t i s shown t h a t t h i s assumption

i s adequate t o c o m p l e t e l y determine t h e s o l u t i o n t o the model, i n c l u d i n g a
unique bubble- free p r i c e l e v e l .
Formally, McCallum's argument has t h e f o l l o w i n g e s s e n t i a l s t r u c t u r e .
Hypothe.si s :
(HI)

There e x i s t s a s o l u t i o n i n terms of t h e s t a t e v e c t o r S t :
pt=nr O + ~~I r ~ +* e tl ~
f
mt=n20+n21rt+n2ret
Rt=n30+n31rt+f132et
Et-I~t+l=fllo

(HZ)

The i n t e r e s t r a t e obeys an i n t e r e s t r a t e r u l e ( a d i r e c t imp1 i c a t i o n o f
(27) > :
n
3

(H3)

1=n32=0

The p r i o r e x p e c t a t i o n of money i s predetermined ( i n t h i s example, an
exogenous c o n s t a n t ) :
Et-lmt=po

(H4)

The n o n p o l i c y s t r u c t u r e :
Rt=rt+(Et-lpt+ l-pt)
mt-pt=ao-aRt+et

(24)

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Conclusion:
(C1)

A l l t h e H i j S a r e u n i q u e l y determined, i n c l u d i n g lllo
and
0

.

Since t h e l l i j s a r e w e l l - d e f i n e d , no i n d e t e r m i n a c i e s

arise.
I f i t i s intended t o prove t h a t an i n t e r e s t r a t e r u l e i s f e a s i b l e , and

some m i g h t i n t e r p r e t McCallum's argument i n t h a t way, then i t o b v i o u s l y
should n o t make assumption HZ o f t h e hypothesis.

As s t a t e d above, however,

McCallum's argument can a l s o be examined f o r what i t says a b o u t completeness
and nominal determinacy, which i s a c o n c e p t u a l l y d i s t i n c t i s s u e .

But i f t h e

argument i s intended t o prove t h a t an i n t e r e s t r a t e r u l e completes t h e model
i n t h e sense o f r e n d e r i n g a l l nominal magnitudes determinate, and o f course
t h e argument i s used f o r t h a t purpose, then i t should n o t make assumptions
H1 o r H3.

Assumptions H1 and H3 d i r e c t l y i m p l y ll,o=po. T h i s , i n

c o n j u n c t i o n w i t h t h e money demand equation, ( 8 ) , i m p l i e s t h a t O l o = p o - ~ o .
Hence, i f I , i s determined -- which i t i s by t h e assumption H3--then
I,
IIlo a l s o determinate.
is

But i f nominal indeterminacy i s t o o c c u r , i t

w i l l a p p l y t o b o t h money and p r i c e s .

Hence, a c o n v i n c i n g argument would n o t

assume t h e determinacy o f e i t h e r mt o r p t , b u t r a t h e r would show how t h e i r
determinacy can come about under an i n t e r e s t r a t e r u l e .
The i d e a t h a t t h e i n t e r e s t r a t e can be manipulated f o r t h e c o n t r o l o f
money--the idea o f t h e i n t e r e s t r a t e as an i n s t r u m e n t f o r t h e c o n t r o l o f
money--is based on an i m p l i c i t c o n f u s i o n between r e a l and nominal q u a n t i t i e s .
I n v i e w o f t h e money demand function- - which i s a r e l a t i o n between r e a l money
balances, exp(mt-p,),

t h e i n t e r e s t r a t e , and a demand shock- - variations i n

t h e i n t e r e s t r a t e have t h e i r e f f e c t on t h e r e a l q u a n t i t y o f money, n o t on t h e
nominal q u a n t i t y .

The money demand f u n c t i o n i m p l i e s no necessary r e l a t i o n

between t h e nominal i n t e r e s t r a t e and t h e nominal q u a n t i t y o f money.

I f such

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a necessary r e l a t i o n d i d e x i s t , then i t would be p o s s i b l e t o d e s c r i b e a method
by which t h e assumption H3 c o u l d be implemented.
As i t i s , t h e r e i s no e x p l a n a t i o n o f f e r e d o f how t h i s c o n t r o l o v e r t h e
e x p e c t a t i o n o f money i s t o be achieved.

What, then, i s t h e s p e c i f i c a t i o n o f

p o l i c y u n d e r l y i n g t h e d e t e r m i n a t e s o l u t i o n and t h e i n t e r e s t r a t e r u l e ?

If it

i s assumed t h a t some money s u p p l y f u n c t i o n does complete t h e model, t h e n i t i s
p o s s i b l e t o i n f e r t h a t t h i s money supply f u n c t i o n i s
mt=Xo+rt+et.

(28)

Consider t h e " p o l i c y r u l e , " (27).

When t h e s o l u t i o n f o r Et-,pt=IIl,

is

substituted i n t o it, the r e s u l t i s
R t = 0.

5

(27'

I n o t h e r words, the " p o l i c y r u l e " b o i l s down t o a statement t h a t t h e i n t e r e s t
r a t e i s predetermined.

But i t has a l r e a d y been argued i n s e c t i o n I V t h a t

t h i s r e s u l t i s f e a s i b l e o n l y i f t h e a u t h o r i t i e s can observe t h e f u l l c u r r e n t
s t a t e and e f f e c t t h e money s u p p l y f u n c t i o n (28).
Formally, i f H5 i s added t o t h e hypothesis, then C2 can be added t o t h e
c o n c l u s i o n , where H5 and C2 a r e as f o l l o w s :
(H5)

The model i s complete ( a money supply f u n c t i o n e x i s t s ) .

(C2)

The money s u p p l y f u n c t i o n i s (28).

B u t t r e s s i n g t h i s c o n c l u s i o n i s t h a t t h e n,,s i n the s o l u t i o n a r e
p r e c i s e l y those t h a t o c c u r under t h e money supply f u n c t i o n (28).
I t seems, then, t h a t t h e i d e a t h a t t h e i n t e r e s t r a t e i s used as an
i n s t r u m e n t o r l e v e r f o r i n d i r e c t c o n t r o l o v e r money p l a y s no e f f e c t i v e r o l e
i n r e n d e r i n g p r i c e s o r money d e t e r m i n a t e .

I n s t e a d , determinacy a r i s e s

because s u f f i c i e n t s i d e r e s t r i c t i o n s , f o r example, H1 and H3, have been made
t h a t s u b s t i t u t e p e r f e c t l y f o r t h e assumption t h a t t h e money supply f u n c t i o n

i s (28), thus completing t h e model and p r o v i d i n g unique s o l u t i o n s f o r a l l

)

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variables.
Canzoneri, Henderson, and Rogoff (1983) a l s o make s u f f i c i e n t s i d e
assumptions- - that money has a f i x e d , predetermined t r e n d - - t o make t h e model
complete.

McCallum (1986) p r o v i d e d a demonstration t h a t t h e f i x e d - t r e n d

assumption i s n o t i m p l i e d by t h e i n t e r e s t r a t e r u l e .

This interesting p o i n t

i s a s p e c i a l case o f t h e more general p r o p o s i t i o n t h a t few r e s t r i c t i o n s on
t h e money supply f u n c t i o n o r on t h e behavior o f nominal v a r i a b l e s a r e i m p l i e d
by t h e incomplete system {(6>,(8),(17)).

VIII.

Summary o f Conclusions

I n t e r e s t r a t e r u l e s are i n f e a s i b l e .
An i n t e r e s t r a t e r u l e predetermines t h e i n t e r e s t r a t e ; t h a t i s , an
i n t e r e s t r a t e r u l e makes t h e i n t e r e s t r a t e a s t r i c t f u n c t i o n o f t h e lagged
state.

I f t h e i n t e r e s t r a t e i s t o be rendered predetermined, then t h e money

s u p p l y must somehow be a b l e t o o f f s e t t h e e f f e c t s o f c u r r e n t shocks on t h e
interest rate.

The i n t e r e s t r a t e r u l e i s i n f e a s i b l e because economic

d i s t u r b a n c e s a r e n o t contemporaneously observable by t h e a u t h o r i t i e s , so t h a t
t h e i n t e r e s t - r a t e - s t a b i l i z i n g money supply response t o t h o s e d i s t u r b a n c e s
cannot be f o r t h c o m i n g on t h e t i m e l y b a s i s r e q u i r e d .

Instead, the a u t h o r i t i e s

must depend on movements i n t h e i n t e r e s t r a t e i t s e l f t o convey the u n d e r l y i n g
disturbances a f f e c t i n g the i n t e r e s t r a t e .

Changes i n t h e money supply can be

c o n d i t i o n e d on v a r i a t i o n s i n t h e c u r r e n t i n t e r e s t r a t e t o any a r b i t r a r y ,
f i n i t e degree, b u t these money changes cannot a b s o l u t e l y e l i m i n a t e i n t e r e s t
r a t e movements, f o r then t h e r e would be no movements i n t h e i n t e r e s t r a t e
upon which t o c o n d i t i o n money s u p p l y movements i n t h e f i r s t place.

Hence,

i n t e r e s t r a t e r u l e s a r e i n f e a s i b l e unless t h e a u t h o r i t i e s can c o n d i t i o n money

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supply movements d i r e c t l y on t h e c u r r e n t economic d i s t u r b a n c e s .

But t h e y

cannot, because t h e y do n o t possess complete c u r r e n t i n f o r m a t i o n .

I n t e r e s t r a t e r u l e s f a i l t o complete p r o t o t y p e macroeconomic models w i t h
r a t i o n a l e x p e c t a t i o n s and a r e a l money demand f u n c t i o n .

I f t h e p o l i c y d e s i g n i s t o r e f l e c t any degree of concern about nominal
v a r i a b l e s , then- - with t h e p r e s e n t s t a t e of macroeconomic science- - the p o l i c y
must p l a c e d i r e c t c o n s t r a i n t s on t h e nominal money supply.

More p r e c i s e l y ,

t h e r e must be d i r e c t c o n s t r a i n t s p l a c e d on t h e r e l a t i o n between t h e nominal
money s t o c k and t h e observed s t a t e of t h e economy.

But any c o m p l e t e l y

s p e c i f i e d p o l i c y can be f o r m u l a t e d i n terms o f such a money s u p p l y f u n c t i o n .
Suppose, f o r argument, t h a t an i n t e r e s t r a t e r u l e i s f e a s i b l e .
i n t e r e s t r a t e r u l e predetermines t h e i n t e r e s t r a t e .

An

T h i s means t h a t t h e

r e l a t i v e p r i c e term i n t h e r e a l money demand f u n c t i o n i s predetermined.
There i s no way t o m a n i p u l a t e t h i s r e l a t i v e p r i c e so as t o c o n t r o l t h e
nominal money stock; i t can be used o n l y t o i n f l u e n c e t h e e x p e c t a t i o n o f t h e
r e a l money stock.

I f o n l y t h e r e a l money stock, o r r a t i o between nominal

money and t h e p r i c e l e v e l , i s determined, then n e i t h e r nominal money n o r t h e
p r i c e l e v e l i s determined.

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Footnotes
1. The issues o f t h i s a r t i c l e a r e n o t m a t e r i a l l y a f f e c t e d i f p r i v a t e agents
a r e g i v e n some o r a l l c u r r e n t i n f o r m a t i o n i n f o r m i n g f u t u r e p r i c e
e x p e c t a t i o n s , o r even i f t h e y a r e g i v e n p e r f e c t f o r e s i g h t .
2. T h i s i s n o t g e n e r a l l y t h e case. I n dynamic models w i t h r a t i o n a l
e x p e c t a t i o n s , a s t a t e v e c t o r adequate t o d e s c r i b e a l l i n t e r e s t i n g (non-bubble)
s o l u t i o n s f o r a g i v e n , w e l l - s p e c i f i e d model w i l l g e n e r a l l y i n c l u d e s t a t e
v a r i a b l e s n o t e x p l i c i t l y p r e s e n t i n t h e s t r u c t u r a l form. This p o i n t i s
discussed i n Hoehn (1986).
3. McCallum (1986) i s ambiguous as t o whether he means t h a t an i n t e r e s t r a t e
r u l e i s a money s u p p l y f u n c t i o n w i t h X=m, o r t h a t an i n t e r e s t r a t e r u l e i s
t h e 1i m i t i n g case o f a money supply f u n c t i o n as X approaches m. He
c l e a r l y means a t l e a s t one o f these statements, and p o s s i b l y b o t h . The
p r e v i o u s s e c t i o n argued t h a t i f X=a, t h e money s u p p l y f u n c t i o n i s
undefined so t h a t any expressions dependent on i t , such as t h e reduced f o r m
f o r t h e i n t e r e s t r a t e , a r e a l s o undefined.

4 . There seems t o be no assurance o f the e x i s t e n c e o f any f i n i t e l y
dimensioned s t a t e v e c t o r t h a t i s adequate t o d e s c r i b e t h e p o s i t i o n of t h e
system, unless some s o r t o f r e s t r i c t i o n on t h e money supply r u l e i s made.

5. Equations (27) and ( 2 7 ' ) a r e p a r t i c u l a r l y d i f f i c u l t t o t h i n k o f as means
f o r c o n t r o l l i n g money, because t h e y c a l l f o r a f i x e d i n t e r e s t r a t e . McCallum
i n t r o d u c e d n o n e s s e n t i a l dynamic elements i n t o t h e d e s c r i p t i o n o f p o l i c y and
d e r i v e d a d e t e r m i n a t e s o l u t i o n f o r nominal v a r i a b l e s t h a t was c o n s i s t e n t w i t h
i t . Dynamic elements were thereby i n t r o d u c e d i n t o t h e d e t e r m i n a t i o n o f money
and p r i c e s as w e l l as t h e i n t e r e s t r a t e . But whether o r n o t dynamic elements
a r e i n t r o d u c e d , t h e r e i s no sense i n which t h e i n t e r e s t r a t e r u l e e x e r t s
c o n t r o l o v e r money o r p r i c e s o r renders e i t h e r d e t e r m i n a t e .

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References
Canzoneri, Matthew B., D a l e W. Henderson, and Kenneth S. R o g o f f . "The
I n f o r m a t i o n C o n t e n t o f t h e I n t e r e s t Rate and Optimal Monetary P o l i c y , "
Q u a r t e r l y J o u r n a l o f Economics, v o l . 48 (1983). 545-66.
Dotsey, M i c h a e l , and R o b e r t G. K i n g . " Monetary I n s t r u m e n t s and P o l i c y
Rules i n a R a t i o n a l E x p e c t a t i o n s Environment," J o u r n a l o f M o n e t a r y
Economics, v o l . 12 (1983), 357-82.
Dotsey, M i c h a e l , and R o b e r t G. K i n g . " I n f o r m a t i o n I m p l i c a t i o n s o f I n t e r e s t
Rate Rules," American Economic Review, v o l . 76 (19861, 33-42.
Hoehn, James G. " D e s i g n i n g Monetary P o l i c y Under R a t i o n a l E x p e c t a t i o n s :
A n a l y s i s and P r a c t i c a l I m p l i c a t i o n s , " F e d e r a l Reserve Bank o f C l e v e l a n d
Workinq Paper 8612 (December 1986).
McCallum, B e n n e t t T. " P r i c e Level D e t e r m i n a c y w i t h an I n t e r e s t Rate P o l i c y
Rule and R a t i o n a l E x p e c t a t i o n s , " J o u r n a l of Monetary Economics, v o l . 8
(1981), 319-29.
McCallum, B e n n e t t T. "On Non-Uniqueness i n R a t i o n a l E x p e c t a t i o n s Models:
An A t t e m p t a t P e r s p e c t i v e , " J o u r n a l o f Monetary Economics, v o l . 11 ( 1 9 8 3 ) ,
139-68.
McCallum, B e n n e t t T . "Some I s s u e s C o n c e r n i n g I n t e r e s t Rate Pegging, P r i c e
Level D e t e r m i n a c y , and t h e Real B i l l s D o c t r i n e , "
Economics, v o l . 17 (1986), 135-60.
S a r g e n t , Thomas J . , and N e i l Wallace. " ' R a t i o n a l ' E x p e c t a t i o n s , t h e O p t i m a l
Monetary I n s t r u m e n t , and t h e O p t i m a l Money Supply Rule," J o u r n a l o f
P o l i t i c a l Economy, v o l . 83 (1975), 241- 54.