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Work1ng Paper 861 2

DESIGNING MONETARY POLICY UNDER RATIONAL EXPECTATIONS:
ANALYSIS AND PRACTICAL IMPLICATIONS

By James G. Hoehn

James G. Hoehn i s an economist a t t h e Federal
Reserve Bank o f Cleveland. The author thanks
HI 11lam T. Gavin, Mark S. Sniderman, and Alan
C. Stockman f o r comments on d r a f t s .
Working papers o f the Federal Reserve Bank of
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 discussion 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 the author and n o t n e c e s s a r i l y
those o f t h e Federal Reserve Bank of
Cleveland o r o f the Board o f Governors o f t h e
Federal Reserve System.

December 1986

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DESIGNING MONETARY POLICY UNDER RATIONAL EXPECTATIONS:

ANALYSIS AND PRACTICAL IMPLICATIONS

Thi s paper describes and attempts t o general i z e t h e p r a c t i c a l imp1i cat i o n s f o r the design o f monetary p o l i c y o f some o f the most popular macroeconomic models i n c o r p o r a t i n g r a t i o n a l expectations.

Perhaps the most impor-

t a n t o f these i m p l i c a t i o n s i s a s h i f t i n the focus o f p o l i c y from o u t p u t o r
i n t e r e s t - r a t e s t a b i l i z a t i o n toward p r i c e - l e v e l s t a b i l i z a t i o n .
The r a t i o n a l expectations assumption r u l e s o u t systematic expectational
errors.

The assumption proved necessary t o ensure consistency o f models w i t h

the n a t u r a l r a t e property- - that t h e average l e v e l o f o u t p u t i s i n v a r i a n t w i t h
respect t o monetary p o l i c y and o t h e r monetary phenomena.

Introduction o f

r a t i o n a l expectations o v e r t u r n s the case f o r conventional c o u n t e r c y c l i c a l
policies.
New macroeconomic model s, which combine r a t i o n a l expectations w i t h
e l t h e r incomplete i n f o r m a t i o n o r nominal contracts, o f f e r a seemingly u n i n t e l l i g i b l e v a r i e t y o f r e s u l t s and i m p l i c a t i o n s f o r p o l i c y .

The purpose o f t h i s

paper i s t o describe the ways i n which r a t i o n a l expectations fundamentally
change monetary p o l i c y a n a l y s i s and t o attempt t o g e n e r a l i z e t h e i m p l i c a t i o n s
o f such analysis.

To do so e f f e c t i v e l y , some c a r e f u l development of mathe-

m a t i c a l concepts i s indispensable.

For example, i n r a t i o n a l expectations

model s, expectations are forward- looki ng r a t h e r than backward-looki ng.

There-

f o r e , p o l i c y must be s p e c i f i e d as a contingent r u l e o f behavior; t h a t i s , an
equation r e l a t i n g instruments t o observed outcomes.
I n p r a c t i c e , t h e a n a l y t i c a l problems o f f i n d i n g t h e optimal p o l i c y r u l e
are v i r t u a l l y insurmountable except i n s i m p l i f i e d cases.

The dimensions of

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the models or the range of pol icy rules considered must be severely reduced in
some arbitrary way t o obtain results.

T o place various results in perspec-

tive, it is essential that the significance o f particular simp1 ifying assumptions be understood.

Otherwise, interpretations are insecure, ambiguous, and

seemingly contradictory.

In principle, such problems exist in any class of

models, but the dynamics o f rational expectations models pose new and i 1 lunderstood analytical problems.

A willingness t o grapple with some tricky

analytical difficulties is essential to the practical application of rational
expectations models t o policy analysis.
The analytical problems may in part account for the continued popularity

of pre- rational- expectations IS-LM models. These are the source of persistently popular notions concerning policy and form the basis of much empirical
work, including the large-scale macroeconometric models.

In these models, a

reasonably well-defined policy can improve welfare by stabilizing aggregate
demand.

Under thi s conventional countercyclical pol icy, the money stock

depends upon the last observations o f the (currently unknown) state o f the
economy.

A1 so, under plausible assumptions about parameters and relative

disturbance variances, the money stock should respond positively to the
(currently known) interest rate.

These IS-LM models had serious problems,

symptomatic of which was inconsistency with the natural rate property, that
could be corrected only by introducing rational expectations.
Sargent and Wallace (1975) introduced rational expectations into an
otherwise conventional IS-LM model.

The main result was that, once policy

effects operating through systematic expectational errors were ruled out,
money supply responses to the state o f the economy were o f no consequence for
output behavior.

Nevertheless, positive money stock responses to the current

interest rate could still be helpful, as in the p r e - r a t i o n a l - e x p e c t a t i o n s

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IS-LM model.

But an " i n t e r e s t r a t e r u l e , " or money supply p o l i c y t h a t makes

t h e money s t o c k i n f i n i t e l y e l a s t i c 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 , would
leave t h e p r i c e l e v e l and t h e money s t o c k i n d e t e r m i n a t e .
Subsequent r a t i o n a l e x p e c t a t i o n s models considered d i f f e r e n t i n f o r m a t i o n
assumptions and d i f f e r e n t s t r u c t u r a l c h a r a c t e r i s t i c s , such as long- term cont r a c t s o r intertemporal substitution o f leisure.

P o l i c i e s t o improve w e l f a r e

i n these models depend too much on p a r t i c u l a r s t o p r o v i d e unambiguous d e s c r i p t i o n s of o p t i m a l p o l i c y r u l e s .

Indeed, d e r i v a t i o n o f t h e o p t i m a l p o l i c y

becomes an a n a l y t i c a l l y i n t r a c t a b l e t a s k , w i t h o u t somewhat a r b i t r a r y r e s t r i c t i o n s on e i t h e r t h e s t r u c t u r e o f t h e economy o r t h e range o f p o l i c y c h o i c e s
considered.
However, r a t i o n a l e x p e c t a t i o n s models ( e x c e p t c e r t a i n cases t h a t do n o t
possess t h e n a t u r a l r a t e p r o p e r t y ) have two c h a r a c t e r i s t i c s w i t h p r a c t i c a l
policy implications:

(1)

an o p t i m a l p o l i c y i s e q u i v a l e n t t o one t h a t m i n i -

mizes t h e p r i c e l e v e l u n c e r t a i n t y o f s u p p l i e r s o v e r v a r i o u s h o r i z o n s t h a t a r e
determined by i n f o r m a t i o n l a g s and/or c o n t r a c t l e n g t h s , and (2) " i n t e r e s t r a t e
r u l e s " make t h e p r i c e l e v e l and money s t o c k i n d e t e r m i n a t e .
The f i r s t s e c t i o n t o f o l l o w c o n s i d e r s t h e way p o l i c y o b j e c t i v e s and
choices a r e s p e c i f i e d i n r a t i o n a l e x p e c t a t i o n s models and notes some l i m i t a t i o n s and u n r e s o l v e d a n a l y t i c a l problems, which a r e i l l u s t r a t e d i n l a t e r
sections.

Next, t h e b a s i s o f p r e v a i l i n g concepts o f p o l i c y i s shown t o be

r o o t e d i n t h e pre- rational- expectations IS-LM models, and t h e shortcomings o f
those models a r e discussed.

I n t h e n e x t s e c t i o n , i t i s shown t h a t r a t i o n a l

e x p e c t a t i o n s d e s t r o y s t h e case f o r t h e c o n v e n t i o n a l c o u n t e r c y c l ic a l p o l i c i es ,
b u t does n o t l e a d t o v e r y s p e c i f i c c o n c l u s i o n s about t h e o p t i m a l p o l i c y r u l e
w i t h o u t p a r t i c u l a r assumptions about i nforrnation avai lab1 e t o p r i v a t e and
p u b l i c agents.

Non- market- clearing models, such as those i n which s e l l e r s o f

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goods o r l a b o r agree t o accommodate demand a t a predetermined nominal p r i c e ,
are found t o have unconventional, b u t few general , pol i c y impl i c a t i o n s .

Fin-

a l l y , s i m i l a r ambiguities are found i n intertemporal s u b s t i t u t i o n models.

A

f i n a l s e c t i o n i n t e r p r e t s the p r a c t i c a l p o l i c y impl i c a t i o n s o f r a t i o n a l expect a t i o n s models as arguing f o r p r i c e s t a b i l i t y and a g a i n s t p o l i c i e s t h a t seek
t o s t a b i l i z e output o r i n t e r e s t rates.

The Pol i c y Problem

What Should Monetary P o l i c y ' s O b j e c t i v e ( s 1 Be?
A t t h e most general l e v e l , t h e -p o l i c y o b j e c t i v e can be taken as t h e

enhancement o f the welfare o f the r e p r e s e n t a t i v e agent (consumer/factor supplier).

I m p l i c i t i n d i f f e r e n t macroeconomic models t h a t g i v e a r o l e f o r

monetary p o l i c y are d i f f e r e n t c o n s t r a i n t s t h a t impede agents' attainment o f
the f i r s t - b e s t economic outcomes.

Then p o l i c y can improve welfare by r e d u c i n g

the effectiveness o f these c o n s t r a i n t s .
I m p l i c i t i n a l l the major competing models i s a common set o f microeconomi c assumptions, which can be b r i e f l y described as f o l lows.

The welfare

of t h e r e p r e s e n t a t i v e i n d i v i d u a l i n the economy i s s p e c i f i e d by h i s u t i l i t y
f u n c t i o n , which he maximizes subject t o various c o n s t r a i n t s .

His u t i l i t y

depends p o s i t i v e l y upon the amounts o f consumption and l e i s u r e he enjoys i n
each p e r i o d , w i t h f u t u r e amounts discounted according t o how soon they w i l l
occur.

I n a d d i t i o n , consumption and l e i s u r e provide d e c l i n i n g marginal

u t i 1it y , so t h a t i n d i v i d u a l s d i s p l a y r i s k aversion- - they tend t o p r e f e r , f o r
example, more s t a b l e p a t t e r n s o f consumption and l e i s u r e , o v e r unstable ones,
f o r given present discounted values o f consumption and l e i s u r e streams.
I n d i v i d u a l s have access t o a production f u n c t i o n e i t h e r d i r e c t l y , o r

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i n d i r e c t l y through a l a b o r market i n which p r o f i t - m a x i m i z i n g f i r m s are
buyers.

The p r o d u c t i o n f u n c t i o n has l a b o r and a t l e a s t one o t h e r ( c a p i t a l o r

f i x e d ) f a c t o r i n p u t , and d i s p l a y s decreasing marginal p r o d u c t i v i t y and cons t a n t r e t u r n s t o scale.

I n the absence o f t h e o t h e r c o n s t r a i n t s t o be d i s -

cussed, i n d i v i d u a l s can t r a d e f r e e l y i n v a r i o u s markets, s u b j e c t t o budget
constraints.
Changes i n p r o d u c t i v i t y and technology alone are e n t i r e l y capable of
generating cycles o f the k i n d a c t u a l l y observed i n developed, market economies, even i f agents optimize, and even w i t h o u t the a d d i t i o n a l c o n s t r a i n t s
assumed i n monetary models o f the business cycle.

Recent work on

real

business c y c l e models suggests important l i m i t s on the scope o f f l u c t u a t i o n s
a t t r i b u t a b l e t o monetary phenomena i n general and monetary p o l i c y i n p a r t i c ular.

However, these models are n o t the s u b j e c t o f study here because t h e i r

i m p l i c a t i o n s f o r t h e conduct o f monetary p o l i c y i n a c y c l i c a l framework are
r e l a t i v e l y s p e c u l a t i v e and do n o t f i t w e l l w i t h i n the discussion.
Within the c l a s s o f models t h a t do g i v e an important r o l e t o monetary
p o l i c y i n generating cycles, the c r i t i c a l d i f f e r e n c e s between the a l t e r n a t i v e
models do n o t i n v o l v e the assumptions about u t i 1i t y and p r o d u c t i o n f u n c t i o n s .
I n s t e a d , the main d i f f e r e n c e s l i e i n c e r t a i n a d d i t i o n a l c o n s t r a i n t s faced by
agents.

Usually, i t i s assumed t h a t business cycles r e f l e c t some f a i l u r e o f

the market economy t o reach a Pareto-optimum.

To e x p l a i n t h i s f a i l u r e , models

have placed agents under c o n s t r a i n t s o f one o f two types:

( 1 ) incomplete

information, of which money i l l u s i o n can be considered an extreme special
case, o r ( 2 ) f a i l u r e o f markets t o c l e a r , o f which nonexistence o f markets can
be thought o f as an extreme s p e c i a l case.

These c o n s t r a i n t s determine the

mechanism by which p o l i c y e x e r t s i t s i n f l u e n c e , i n ways t o be e x p l i c a t e d
below.

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Given the u t i l i t y , p r o d u c t i o n f u n c t i o n , and market f a i l u r e assumptions,
each model imp1i e s a s e t o f d e c i s i o n r u l e s d e s c r i b i n g how agents respond t o
i n f o r m a t i o n a v a i l a b l e t o them.

These d e c i s i o n r u l e s i m p l y market demand and

supply f u n c t i o n s , u s u a l l y expressed as 1i n e a r approximations, t h a t can be used
t o r e l a t e the behavior o f t h e aggregate o f i n d i v i d u a l s t o t h e s t a t e o f t h e
economy.

I t i s these representations o f the models- - linear supply and demand

functions- - that have proven most a n a l y t i c a l l y t r a c t a b l e . '

This representa-

t i o n i s termed t h e " s t r u c t u r a l " form o f the model.
A serious problem i n h e r e n t i n these representations i s t h a t they a r e

r a r e l y , i f ever, i n v a r i a n t w i t h respect t o the class of p o l i c y i n t e r v e n t i o n s
considered.

This p o i n t was made f o r c e f u l l y by Lucas (19761, who demonstrated

t h a t the orthodox IS-LM models were n o t i n v a r i a n t t o changes i n p o l i c y i n t h e
presence o f r a t i o n a l expectations.

A s a p r i n c i p l e , t h i s p o i n t i s uncontrover-

s i a l , although i t s p r a c t i c a l i m p l i c a t i o n s are t r o u b l i n g :

e i t h e r p o l i c y analy-

s i s must be regarded as impossible, o r the s e n s i t i v i t y o f the r e p r e s e n t a t i o n
t o the range o f p o l i c i e s considered must be assessed.

This s e n s i t i v i t y

a n a l y s i s cannot be performed w i t h o u t making e x p l i c i t the microeconomic foundat i o n s of the model.

Recognition o f t h i s p r i n c i p l e gave added v i g o r t o t h e

ongoing search f o r microeconomic foundations.

What I s Monetary P o l i c y ?
The very existence o f monetary p o l i c y r e q u i r e s some s e t of r e g u l a t i o n s
and/or l e g a l tender r e s t r i c t i o n s a f f e c t i n g the f i n a n c i a l and payments
systems.

The nature o f these r e g u l a t i o n s and r e s t r i c t i o n s i s a c r i t i c a l p a r t

of t h e model i n which p o l i c y choices are made.

The r e g u l a t i o n s and r e s t r i c -

t i o n s are, however, n o t w e l l understood and are taken as f i x e d i n the most
common form o f p o l i c y a n a l y s i s .

S u f f i c i e n t assumptions f o r the existence o f

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monetary p o l i c y a r e t h a t t h e monetary policymaker (a) c o n t r o l s base money
t h r o u g h a r o l e as s o l e p r o v i d e r and (b) s i m u l t a n e o u s l y c o n t r o l s t h e r a t e of
r e t u r n o f base money r e l a t i v e t o o t h e r assets through r e s e r v e requirements and
o t h e r c o n t r o l s o v e r t h e payments system.

The f i r s t assumption g i v e s p o l i c y a

nominal q u a n t i t y i t can c o n t r o l ; t h e second one g i v e s r i s e t o a demand
f u n c t i o n f o r t h a t q u a n t i t y , which g i v e s m a n i p u l a t i o n o f t h a t q u a n t i t y
conceivable l e v e r a g e o v e r t h e macroeconomy.
Given t h i s l e g a l and i n s t i t u t i o n a l framework, some program of changes i n
t h e s t o c k o f money (or base money) c o n s t i t u t e s an i n s t r u m e n t f o r m i n i m i z i n g
t h e e f f e c t o f t h e market f a i l u r e s on t h e r e p r e s e n t a t i v e p r i v a t e agents' welfare.

This o p t i m i z a t i o n takes p l a c e w i t h r e f e r e n c e t o a1l conceivable

c o n t i n g e n t b e h a v i o r s f o r money.

These behaviors can be s p e c i f i e d most gen-

e r a l l y i n terms o f parameters o f a r u l e l i n k i n g t h e q u a n t i t y o f money t o 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 t o t h e policymaker.
To f o r m a l i z e , c o n s i d e r p o l i c y r u l e s t h a t a r e a l i n e a r f u n c t i o n of t h e
information set.
(1)

Then t h e r u l e may be w r i t t e n as:

mt = H I t ,

where m t i s t h e l o g o f t h e money s t o c k as o f t i m e p e r i o d t, H i s a v e c t o r of
c o e f f i c i e n t s c h a r a c t e r i z i n g t h e p o l i c y responses, and Iti s t h e i n f o r m a t i o n
set available
-

t o t h e policymaker a t time t. I, m i g h t i n c l u d e t h e " v a r i a b l e "

1 (one) and any powers o f t ( t i m e i n d e x ) .

itself.

By convention, Itexcludes mt

Although t h e l a t t e r i s observable, i t i s a l r e a d y i n c l u d e d on t h e

l e f t - h a n d side- - merely a k i n d o f n o r m a l i z a t i o n .
mt-I

However, Itmay i n c l u d e

o r o t h e r lagged money terms, and g e n e r a l l y w i l l .
Equation ( 1 ) s t a t e s t h e obvious t r u t h t h a t policymakers can o n l y respond

t o t h e i n f o r m a t i o n t h e y have a t a p o i n t i n time.
sented by choice o f v a l u e s o f H.

Choice o f p o l i c y i s r e p r e -

Furthermore, c h o i c e o f t h e elements of H

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t h a t are c o e f f i c i e n t s o f u n i t y and powers o f time w i l l be i r r e l e v a n t f o r
n e a r l y a1 1 purposes, a t l e a s t i n r a t i o n a l expectations models, o r any o t h e r s
d i s p l a y i n g the n e u t r a l i t y o f money.
A s w i l l be seen, i t i s a n a l y t i c a l l y u s e f u l i n the f o l l o w i n g d i s c u s s i o n

t o d i s t i n g u i s h c a r e f u l l y between p o l i c y responses t o contemporaneous informat i o n versus responses t o delayed i n f o r m a t i o n .

This d i s t i n c t i o n can be

effected by segmenting the i n f o r m a t i o n s e t , I,,i n t o c u r r e n t ( p e r i o d - t )
r e a l i z a t i o n s and previous ( p e r i o d t - 1 ,

t- 2, . . . I r e a l i z a t i o n s .

I n particular,

t h e c u r r e n t i n t e r e s t r a t e ought t o be considered contemporaneously observable,
w h i l e o u t p u t and p r i c e s a r e known o n l y w i t h a l a g .

Then the p o l i c y r u l e may

be w r i t t e n as:

(2)

m, + qRt = p + F(L)Yt-,,

where R i s the nominal i n t e r e s t r a t e , Y i s a v e c t o r o f s t a t e v a r i a b l e s , q i s a
s c a l a r , and F(L) i s an n- dimensional vector polynomial i n the l a g o p e r a t o r , L
(defined such t h a t L k Y t - I

=

Y,-,-,>.

Trends o r polynomials i n t have

been excluded i n (2) because they o n l y c l u t t e r the r e s u l t s w i t h u n i n t e r e s t i n g
terms.

p i s t r e a t e d as an exogenous constant; i t s value does n o t bear on

t h e issues addressed.

The pol i c y choice i s then represented by t h e j o i n t

choice o f q and F(L>.

I n each period, the p o l icymaker observes Y t - I

l i g h t o f Yt-,,

Yt-,,

...,

and, i n

chooses a l i n e a r sum o f money and the i n t e r e s t

r a t e t h a t w i 11 serve as t h e c r i t e r i o n f o r money p r o v i s i o n i n p e r i o d t.
Rules o f form (2) a r e commonly encountered i n the l i t e r a t u r e on monetary
p o l i c y , and are o f t e n termed money supply function^.^
b a s i s of the money supply curve i n f i g u r e 1.

This f u n c t i o n i s t h e

The p o s i t i v e slope t h e r e r e -

f l e c t s a negative q, i n d i c a t i n g t h a t the money stock i s increased contemporaneously w i t h r i s e s i n the nominal i n t e r e s t r a t e , f o r given r e a l i z a t i o n s o f
Y - , Y ,

. ...

The i n t e r c e p t , (p+F(L)Y

,-, > / q ,

v a r i e s w i t h the

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Figure 1
Money Supply Function

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observed (lagged) state o f the economy.

For example, the choices of

F-elements might tend t o reduce the intercept in response to output declines
and increase it after output increases.

These two features, especially the

cyclical intercept, characterize what can be termed the conventional
stabilization policy.

What Range o f Policy Choices Is Relevant for Analysis?
This representation o f the range o f pol icy choices must be simp1 ified
careful ly in the context of each model in order t o proceed with analysis.
Unfortunately, no general procedure exists for determining an adequate, yet
sufficiently parsimonious specification o f an optimal policy rule under
rational expectations.

The appropriate specification will depend on both the

structure and the assumptions about information sets avai lable t o the pol i cymaker and private agents.

This problem seems t o limit analysis t o cases in

which an adequate policy specification can be confidently determined.

In

practice, this has meant certain restrictions on the structure and information
sets that limit the generality o f results. Some considerations in the choice

of the appropriate specification are the subject of this section.
First, the specification of the policy rule might include as arguments
only variables in the set I t that are also in the set o f minimal state variables, denoted M,.

M t contains all variables, treating lags as distinct

variables, appearing explicitly in (nonpolicy) equations o f the model in its
structural form (see McCallum C19831).

In models in which period t-1 vari-

ables appear in the structural equations, but variables dated earlier do not,
the minimal-state-variable criterion will serve t o truncate F(L) t o a vector

of scalars.
(3)

m,

Then equation (2) can be rewritten as:

= p -qRt

+ FYq-,,

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where {Rt,

Ys-,)

= {Mt fl
It).
Except where o t h e r w i s e noted, t h i s

r e p r e s e n t a t i o n s h a l l be t r e a t e d as t h e a p p r o p r i a t e r e p r e s e n t a t i o n o f t h e
policy rule.
The m i n i m a l - s t a t e - v a r i a b l e approach has p r a c t i c a l advantages and may
serve as an a p p r o p r i a t e s t a r t i n g p o i n t f o r i d e n t i f y i n g r e l e v a n t v a r i a b l e s f o r
the p o l i c y r u l e .

C e r t a i n l y , v a r i a b l e s i n M t a r e prime candidates f o r i n c l u -

sion i n the p o l i c y r u l e .

And l i m i t i n g those i n c l u d e d t o t h e m i n i m a l - s t a t e -

v a r i a b l e s e t r u l e s o u t an i n d e f i n i t e l y l a r g e number o f t r i v i a l v a r i a b l e s ,
which a n a l y s i s would u l t i m a t e l y f i n d t o be i r r e l e v a n t anyway ( t h e i r o p t i m a l
c o e f f i c i e n t s i n t h e p o l i c y r u l e would be zero).

The l i m i t a t i o n t o minimal

s t a t e variables i s a l s o thought t o r u l e out inclusion o f i n t r i n s i c a l l y i r r e l e v a n t v a r i a b l e s , termed b o o t s t r a p v a r i a b l e s , t h a t a n a l y s i s would f i n d r e l e v a n t
o n l y i f t h e y were i n c l u d e d i n t h e setup o f t h e problem, e i t h e r i n t h e p o l i c y
r u l e or i n p r i v a t e r a t i o n a l e x p e c t a t i o n s f o r m a t i o n .
U n f o r t u n a t e l y , t h e m i n i m a l - s t a t e - v a r i a b l e approach w i l l n o t n e c c e s a r i l y
r e s u l t i n an adequate p o l i c y r u l e .

A m b i g u i t i e s about which v a r i a b l e s a r e

r e l e v a n t g e n e r a l l y a r i s e u n l e s s p a r t i c u l a r assumptlons a r e made concerning
p r i v a t e agents' i n f o r m a t i o n s e t s , denoted St. M t may n o t c o n t a i n a l l t h e
v a r i a b l e s p r o v i d i n g r e l e v a n t c o n d i t i o n i n g i n f o r m a t i o n t h a t agents use t o f o r m
r a t i o n a l expectations.

Hence, some v a r i a b l e s n o t i n

{Mtn It)
may be

r e l e v a n t s t a t e v a r i a b l e s a f t e r a l l and should appear i n t h e r e p r e s e n t a t i o n o f
the policy rule.

For example, p r i c e s i n t h e l a s t p e r i o d may n o t appear i n t h e

" s t r u c t u r a l " form o f t h e model ( s u p p l y and demand e q u a t i o n s ) ; y e t , if p r i v a t e
agents f o r m i n g e x p e c t a t i o n s have an i n f o r m a t i o n s e t c o n t a i n i n g o n l y lagged
p r i c e s , then those p r i c e s w i l l g e n e r a l l y i n f l u e n c e supply and demand d e c i s i o n s
v i a e x p e c t a t i o n s , and should appear i n t h e o p t i m a l p o l i c y r u l e .

B u t if, on

t h e o t h e r hand, p r i v a t e agents were endowed w i t h a d i f f e r e n t i n f o r m a t i o n s e t ,

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i n c l u d i n g t h e c u r r e n t p r i c e l e v e l , t h e y m i g h t f i n d lagged p r i c e s uninformative.

Then lagged p r i c e s m i g h t n o t be a r e l e v a n t s t a t e v a r i a b l e and need n o t

appear i n t h e p o l i c y r u l e .

The importance o f t h i s example i s t h a t , r e g a r d l e s s

o f t h e s p e c i f i c a t i o n o f St, lagged p r i c e s a r e n o t a minimal s t a t e v a r i a b l e ,
because t h e l a t t e r depends o n l y on t h e s t r u c t u r a l equations, n o t on St.
Yet, a change i n St a f f e c t s t h e r e l e v a n t s e t of v a r i a b l e s t h a t g e n e r a l l y
appear i n t h e o p t i m a l p o l i c y r u l e .

Hence, i n c l u s i o n o f a l l minimal s t a t e

v a r i a b l e s does n o t assure a s u f f i c i e n t r e p r e s e n t a t i o n f o r an o p t i m a l r u l e .
A l l t o l d , parsimony i n t h e setup o f t h e o p t i m i z a t i o n problem, and
p a r t i c u l a r l y t h e v a r i a b l e s i n c l u d e d i n t h e p o l i c y r u l e , i s b o t h e s s e n t i a l and
f r a u g h t w i t h dangers.

C h i e f among these dangers i s t h a t r e l e v a n t v a r i a b l e s

( i n c l u d i n g v a r i o u s l a g s ) may i n a d v e r t e n t l y be l e f t o u t o f t h e p o l i c y r u l e .
Yet, unless t h e a n a l y s t i s sure t h a t ( a t l e a s t ) a l l t h e r e l e v a n t v a r i a b l e s a r e
i n c l u d e d , t h e f o r m o f t h e r u l e p o s t u l a t e d may exclude t h e o p t i m a l p o l i c y o r
policies altogether.

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

t h e setup o f t h e a n a l y s i s .

Or, an unduly r e s t r i c t e d p o l i c y space may i n c l u d e

o n l y some 6 f t h e members o f t h e c l a s s o f o p t i m a l p o l i c i e s , b u t n o t a l l .

Then,

f e a s i b l e p o l i c i e s w i t h v e r y d i f f e r e n t c h a r a c t e r i s t i c s may be j u s t as d e s i r a b l e
as t h e b e s t w i t h i n t h e r e s t r i c t e d p o l i c y space, y e t t h e a n a l y s t m i g h t i n c o r r e c t l y argue a g a i n s t them on t h e b a s i s o f h i s l i m i t e d r e s u l t s .
H y p o t h e t i c a l i l l u s t r a t i o n s o f t h e problems a t t e n d i n g undue r e s t r i c t i o n
o f t h e p o l i c y space a r e conveyed by f i g u r e 2 f o r two d i f f e r e n t economic
models.

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

assumed t o be
(4)

m, = -qRt

+ fyt-, ,

where y i s o u t p u t .

The u n r e s t r i c t e d p o l i c y space i s q x f , o r R 2 .

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Figure 2
The Policy Space

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If the e x c l u s i o n r e s t r i c t i o n f=O o r q=O i s a r b i t r a r i l y imposed, the r e s t r i c t e d

optimum i s (q* lf=O) o r ( f * lq=O), r e s p e c t i v e l y .

W i t h o u t these undue

r e s t r i c t i o n s , the optimal p o l i c y s e t i s t h e l i n e MM i n t h e model corresponding
t o the upper panel, and the p o i n t M i n t h a t o f the lower panel.

I n the f i r s t

case, p a r t i a l analysis leads t o an optimum; b u t the a n a l y s t might, on t h e
basis of h i s r e s u l t s , argue f a 1 l a c i o u s l y against o t h e r optimal p o l i c i e s t h a t
had e i t h e r q o r f negative.

I n the second case, p a r t i a l a n a l y s i s does n o t

a r r i v e a t a g l o b a l optimum, o r even a c o r r e c t e v a l u a t i o n o f the signs o f opt i m a l p o l i c y parameters.

Problems o f both kinds can e a s i l y occur.

examples o f t h e f i r s t k i n d w i l l be given i n what f o l l o w s .

Formal

The second t y p e o f

problem was i n f o r m a l l y i l l u s t r a t e d above; i t i s l i k e l y t o a r i s e w i t h o u t part i c u l a r r e s t r i c t i o n s on i n f o r m a t i o n and t h e s t r u c t u r e .

These problems a r e a

major, i f inadequately acknowledged, p i t f a l l o f a n a l y s i s o f optimal p o l i c y
under r a t i o n a l expectations.

Often, a n a l y s i s has avoided t h i s problem o n l y by

somewhat a r b i t r a r y r e s t r i c t i o n s on information sets I, andlor St.
Another, v a l i d , r e s t r i c t i o n on the r e l e v a n t p o l i c y space i s t h a t which
r u l e s o u t indeterminacy o f important v a r i a b l e s .

The values o f q and F of t h e

p o l i c y r u l e s p e c i f i e d by t h e general form (3) cannot be specified a r b i t r a r i l y ,
f o r (3) must s u f f i c e t o complete the economic model i n the sense o f r e n d e r i n g
a1 1 the endogenous v a r i a b l e s determinate.

The f o r c e o f t h i s r e s t r i c t i o n

o b v i o u s l y depends on the o t h e r aspects o f the model.

An important example t o

be given r e l a t e s t o indeterminacy o f money and p r i c e s under a " pol i c y " o f
pegging the i n t e r e s t r a t e .
Throughout the formal a n a l y s i s t h a t f o l l o w s , i t w i l l be f u r t h e r assumed
t h a t q and F a r e n o t f u n c t i o n s o f time.'

This assumption i m p l i e s t h a t t h e

p o l icymaker i s a b l e t o make a commi tment t o a time- consi s t e n t r u l e of behavior.

Under r a t i o n a l expectations, the a b i l i t y t o so commit i s necessary t o

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the attainment o f the optimal outcome; otherwise, p o l i c y w i l l be unable t o
en1 i s t the support o f p r i v a t e expectations.
innocuous and i t s r e a l i s m i s d o u b t f u l .

This assumption i s f a r from

A new and growing l i t e r a t u r e attempts

t o deal w i t h the design o f second-best p o l i c i e s i n models i n which the p o l i c y maker i s constrained by an inabi 1it y t o precommi t.
C19861).

(See, f o r example, B a r r o

Nevertheless, even i f the f i r s t - b e s t p o l i c y i s i n f e a s i b l e , the

macroeconomic issues involved i n i t s design w i l l s t i l l be r e l e v a n t .
The a n a l y s i s t o f o l l o w w i l l r e s t r i c t a t t e n t i o n t o steady- state propert i e s o f a1t e r n a t i v e s t o c h a s t i c model s, because o n l y these p r o p e r t i e s are
determined by r a t i o n a l expectations models.

For a given model and a g i v e n

o b j e c t i v e f u n c t i o n , there i s a mapping from each element i n t h e p o l i c y space
t o t h e value o f the o b j e c t i v e f u n c t i o n .

The optimal p o l i c y i s c h a r a c t e r i z e d

by t h e element ( o r , i f nonunique, s e t o f elements) i n the p o l i c y space associa t e d w i t h the o p t i m i z a t i o n o f t h e o b j e c t i v e f u n c t i o n .

This p o l i c y w i l l serve

t o minimize the e f f e c t i v e n e s s o f , o r u t i l i t y l o s s pursuant t o , the c o n s t r a i n t s
on i n f o r m a t i o n o r market- cleari ng t h a t prevent the economy from a t t a i n i n g a
Pareto- optimal a l l o c a t i o n o f resources.

I n the f o l l o w i n g a n a l y s i s , the v a r i a -

t i o n s i n the c o n s t r a i n t s on p r i v a t e u t i l i t y maximization t h a t d i f f e r e n t i a t e
prominent macroeconomic models are shown t o imply v a r i a t i o n s i n the o p t i m a l
pol i c y rule.

Pre- Rational- Expectations IS-LM Models

For a number o f r e l a t e d reasons, i t i s u s e f u l toqtbegin a n a l y s i s w i t h

pre- rational- expectations IS-LM models.

F i r s t , they have pedagogical value i n

t h a t t h e i r a n a l y t i c a l simp1 i c i t y sets the stage f o r easier understanding o f
more complex models.

Second, these IS-LM models generate most conventional

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views on optimal p o l i c y .

They serve as e s s e n t i a l representations o f most

macroeconometric models.

An assessment o f the shortcomings of these pre-

r a t i o n a l IS-LM models helps motivate the assumption o f r a t i o n a l expectations.
F i n a l l y , understanding how optimal p o l i c y i s designed i n p r e - r a t i o n a l models

w i l l a l l o w insight- provoking contrasts w i t h the r a t i o n a l expectat.ions models.

The F i xed-Pri ce Model
I n the i n f l u e n t i a l treatment o f monetary p o l i c y o f Poole (19701, t h e
model was o f the simple textbook IS-LM form, w i t h f i x e d , o r a t l e a s t exogenous, p r i c e s . The aggregate commodity demand f u n c t i o n , o r I S curve, was

(5) y t

= do

+ dtRt + dzyt-1 +

~

l

t

,

dI<O<dz<l,

where y was output, and the money demand o r LM curve was

(6) m,

= a.

+

a,Rt + a,yt

+ et,

u t and e t were disturbances.
(7)

al<O<a,.

Output was determined s t r i c t l y by demand:

y, = y:.
Generally, i f , as i n t h i s case, there are no expectations i n a model,

t h e p o l i c y r u l e need i n c l u d e , a t most, the minimal s e t o f s t a t e v a r i a b l e s t h a t
are a l s o i n t h e policymakers' i n f o r m a t i o n set.
yt-,,

R,,

and m,,

policymaker.

(8) m,

The minimal s e t includes y t ,

but y t i s n o t contemporaneously observable t o the

Therefore, an optimal p o l i c y w i l l take the form:

= P O - qRt +

where q and f

are scalars.

Since output i s determined s t r i c t l y by demand,

r e f l e c t i n g the f i x e d - p r i c e assumption, and since the u t i l i t y f u n c t i o n embodies
r i s k aversion, the appropriate c r i t e r i o n i s minimization o f deviations o f o u t p u t around i t s optimal l e v e l , where the l a t t e r depends on i m p l i c i t and f i x e d
p r o d u c t i v i t y and t a s t e s .

An appropriate value o f p i s needed t o make

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average output equal t o the optimal o u t p u t l e v e l , because i n t h i s model, t h e
average l e v e l o f output depends on the average l e v e l o f money.

(Obviously,

t h i s i s a p a r t i c u l a r l y crude v i o l a t i o n o f the n a t u r a l r a t e property.)

Then,

using a quadratic l o c a l approximation f o r the u t i l i t y f u n c t i o n , the appropria t e o b j e c t i v e i s the minimization o f o u t p u t variance around t h i s optimal
output average.

The o p t i m i z a t i o n problem i s thus separable i n t o a l e v e l and a

variance problem, and the l a t t e r w i 11 occupy the f o l l o w i n g discussion.
Then the o p t i m i z a t i o n can be represented as t h a t o f minimizing t h e v a r i ance o f output w i t h respect t o ( q , f l ) ,

subject t o equations (51, (61, and

(7).
The reduced form s o l u t i o n f o r output i s

(9) y t

= dl(p-ao)Jt

+ ( a l d 2 + q + d l f l > J l y t - l + (al+q)Jlut

-

dlJlet,

where J l = (al+a2dl+q>-'
w i t h a steady- state variance o f
(10)

=

C(a,+q)'

0:)

where oG= E(yt-Ey

+ d:
) ',

2
UI=
E ( u ~t-Eu1

o:=

t)',

E(et-Eet>',

J 2 = (al

I J2,

0%

and

+ a 2 d 1 ) ' - (ald2 + d f 1 I 2 + 2q(al+a2dl-ald2-dlfl>2.

The p o l i c y space q x f l i s R 2 excluding q

=

-al-a2dl.

Assuming the

disturbances are uncorrelated, the f i r s t - o r d e r c o n d i t i o n s imply the optimal q
and f l are given by
(11)

q = ( d , / a 2 > ( o ~ / o : > - al

and
(12)

f l = -aldz/dI + q / d l .
These expressions show t h a t the s t a t i c IS- LM model supports conventional

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views on t h e a p p r o p r i a t e d e s i g n o f monetary p o l i c y .

The o p t i m a l v a l u e of q i s

n e g a t i v e i f , as commonly supposed, t h e v a r i a n c e o f money demand d i s t u r b a n c e s
i s " l a r g e " ( i n a loose sense t h a t depends on d l / a z and a l ) r e l a t i v e t o
commodity demand d i s t u r b a n c e s .

Then money supply should be p o s i t i v e l y r e l a t e d

t o the current i n t e r e s t r a t e (given y t - l ) .

The v a l u e o f f l w i l l be nega-

t i v e , i m p l y i n g t h a t c o u n t e r c y c l i c a l v a r i a t i o n s i n money h e l p s t a b i l i z e o u t p u t .

I t i s noteworthy, f o r comparisons w i t h l a t e r models, t h a t t h e o p t i m a l q and
f l a r e unique.

Also, t h e c h o i c e o f q (contemporaneous responses) i s separ-

a b l e f r o m t h e choice o f f , (lagged responses), i n t h e sense t h a t t h e o p t i m a l
c h o i c e o f q can be f o u n d w i t h o u t c o n s i d e r i n g t h e o p t i m a l v a l u e o f f ,

.

The

o p t i m a l choice o f q i s a l s o unaffected by t h e magnitude of d 2 , t h e c o e f f i c i e n t l i n k i n g commodity demand t o i t s p a s t .

I n o t h e r words, t h e dynamics o f

t h i s model a r e such t h a t t h e y do n o t become a c o n s i d e r a t i o n i n t h e c h o i c e of
t h e slope o f t h e money s u p p l y f u n c t i o n d e p i c t e d i n f i g u r e 1, b u t o n l y i n t h e
c h o i c e o f i t s state- dependent i n t e r c e p t .
One o f t h e most g l a r i n g shortcomings o f t h e s t a t i c IS-LM model i s t h a t

i t 1eaves p r i c e s undetermined, o r exogenous.

One s imp1e, and c o n v e n t i o n a l ,

means o f making p r i c e s endogenous i s t o i n t r o d u c e t h e " l a w o f supply and
demand,
(13)

"

p t = pt-1 + ~ ( y t - y c ) , v>O,

under which i n f l a t i o n v a r i e s d i r e c t l y w i t h "demand pressure," equal t o r e a l
demand, y,,

minus " f u l l employment" o r " c a p a c i t y " o u t p u t , y c .

This

a l t e r a t i o n i n v i t e s placement o f p, i n t h e money demand f u n c t i o n ,
(14)

m,

-

p t = a. + alR,

+ a z y t + e t , al<O<az.

However, t h i s method o f making t h e p r i c e l e v e l endogenous has severe
problems.

I f n e i t h e r buyers nor s e l l e r s can be f o r c e d t o t r a n s a c t , then t h e

demand- determination o f o u t p u t i m p l i e s t h a t p r i c e s must be t o o h i g h t o c l e a r

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the market a t f u l l employment.

To make rudimentary sense o f t h i s model, some

a d d i t i o n a l e x p l a n a t i o n o f supply behavior i s r e q u i r e d .

Suppliers must e i t h e r

face money i l l u s i o n or be under some type o f non- price r a t i o n i n g c o n s t r a i n t ,
o r p r i c e s and t h e i n t e r e s t r a t e would f a l l immediately t o t h e l e v e l t h a t would
c l e a r t h e commodity market and o u t p u t would be constrained by supply, r a t h e r
than by demand.

Even w i t h r a t i o n i n g o r money i l l u s i o n , p r i c e s w i l l f a l l per-

s i s t e n t l y over time, as l o n g as aggregate demand i s t h e c o n s t r a i n t on o u t p u t .
The optimal p o l i c y would seem t o be an increase i n the money stock adequate t o
force i n t e r e s t r a t e s down low enough and d r i v e o u t p u t up t o i t s supply cons t r a i n t , a t which time d e f l a t i o n would h a l t .

I t i s i n t e r e s t i n g t o note t h a t

t h e r e i s no t r a d e - o f f between m a i n t a i n i n g f u l l employment and s t a b i l i z i n g t h e
p r i c e l e v e l i n t h i s model.
capacity output levels.

F a l l i n g p r i c e s i n v a r i a b l y r e f l e c t less- than-

I t i s a1 so noteworthy t h a t aggregate demand i s m i s-

s p e c i f i e d i n t h i s model, and ought t o have t h e r e a l r a t e o f i n t e r e s t r a t h e r
than the nominal r a t e as i t s argument, unless expectations o f i n f l a t i o n
( a c t u a l l y , d e f l a t i o n 1 are f i x e d .

This problem can be r e s o l v e d o n l y by i n t r o -

ducing p r i c e - l e v e l expectations.

The Adaptive Expectations Model

An e x p l i c i t commodity supply f u n c t i o n , together w i t h some mechanism f o r
r e c o n c i l i n g demand and supply- - either by market- clearing p r i c e and i n t e r e s t r a t e adjustments, o r by some r a t i o n a l e f o r p r i c e s t i c k i n e s s o t h e r than r a t i o n i n g o f demand among suppliers--seemed necessary elements f o r macroeconomic
models w i t h d e s i r a b l e microeconomic (and e m p i r i c a l ) i m p l i c a t i o n s .

The f i r s t

major attempt, a t t r i b u t a b l e t o Phelps (19671, Friedman (19681, and Lucas and
Rapping (19691, r e l i e d on workers' confusions between r e a l and nominal wages
which were e x p l o i t e d by employers.

Wages were slow t o a d j u s t t o a c t u a l

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i n f l a t i o n caused by monetary shocks because workers were l e s s than f u l l y
informed about t h e p r i c e l e v e l .

Thus, t h e y were t r i c k e d i n t o w o r k i n g h a r d e r ,

a t lower a c t u a l r e a l wages, whenever t h e p r i c e l e v e l r o s e r e l a t i v e t o p r e v i o u s
expectations.

This n o t i o n i s i n c o r p o r a t e d i n t h e f o l l o w i n g dynamic v e r s i o n o f

t h e IS-LM model, i n which e x p e c t a t i o n s adapt s l o w l y and m e c h a n i s t i c a l l y .
I n t h e IS-LM model w i t h a u t o r e g r e s s i v e e x p e c t a t i o n s , p t appears i n t h e
money demand f u n c t i o n , as i n (14).
(15)

y t = d o + dirt

The aggregate demand f u n c t i o n becomes

+ d z y t - I + u 1t ,

dl<O<dz<l,

where
(16)

r t = Rt-Et(pt+l-pt)

and t h e aggregate supply f u n c t i o n becomes
(17)

yz = s o + Xyt-1 + ~ ( p -t E t p t ) + u z t ,

The lagged o u t p u t term, y t - , ,

s>O,

O<X<l.

i n (17) can r e p r e s e n t c a p a c i t y e f f e c t s o f

p r e v i o u s o u t p u t l e v e l s , o r c o s t s o f adjustment i n employment l e v e l s , as i n
Sargent (1979).
(18)

To (15) and (17) i s added a m a r k e t - c l e a r i n g e q u a t i o n

y: = y t

and e x p e c t a t i o n s equations o f t h e a d a p t i v e type, such as
(19)

Etpt

=

and E t p t + ,

+o++Pt-r
=

+o(l-+)++2pt-1,

0<+<1.

According t o (191, agents p r e d i c t p r i c e s a c c o r d i n g t o a f i r s t - o r d e r a u t o r e gression.

I t i s noteworthy, f o r l a t e r comparisons, t h a t p r i c e e x p e c t a t i o n s

a r e backward- looking:

t h e y a r e u n i q u e l y determined a t t i m e t by t h e i n i t i a l

condition pt-,.
I n g e n e r a l , an adequate y e t parsimonious r e p r e s e n t a t i o n o f an o p t i m a l
p o l i c y r u l e i s easy t o determine i n models w i t h ad hoc e x p e c t a t i o n s formations.

An adequate p o l i c y r u l e need i n c l u d e o n l y those v a r i a b l e s i n t h e

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pol 1 cymaker's information set that are either minimal state variables o r variables upon which private expectations are conditioned.

p t - I is the only

expectations-condi tioning variable, according to (18) and (19).

S o the rule

should include m,, R t , Y,-~, and P,-~:
(20) mt

=

p-qRt + f l ~ t -+ ~fzpt-l.

Many economists initially treated output stabilization as the appropriate criterion for policy in this model.

That criterion is inappropriate, as

will be argued later, when the microeconomics o f the supply function are
considered.

Nevertheless, to understand the implications for theory and

policy o f rational expectations--as distinct from the advances in microeconomic foundations o f supply behavior that occurred more or less concomitantly--it is useful to consider how policies might be designed to control
output in the adaptive expectations model.
The reduced-form equation for output is
(21) y,
for

=

no+n,~,-l+n2p,-l+n,~1t+n4~2t+n5et

no=d~Jp+(dk+do-++o>,
where k=Cs(-ao+l++o-a2do+a2++o)-do+~+olJ,
nl=dl(X+sfl-d2(l+a2s))J+d2,

n2=Cs(f2-+>+dl+(+-l)(l-a2s)1J+dl+(l-41,
n3=s(al+q)J,
n,=dJ,
and

n5=-sdJ,

where
J=[dl+s(q+al+a2dl)I-'.
In this model, the relevant policy space includes all combinations of
(q,f,,f,) except those for which q=-dl/s-a,-a2dl, because the latter would

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The p o l i c y t h a t minimizes output v a r i -

v i o l a t e the completeness r e s t r i c t i o n .
ance i s s p e c i f i e d by:
(22)

q = (dlo~+s2o~)/(l+a2s)sd-at,

(23) f ,=-s-'C~-d~(l+a~s)l-d~s-'d;'J-~,
and
(24)

f,=

~C1+(~-l)(dl/s)(l-azs+J-l)l.

Somewhat s u r p r i s i n g l y , the signs o f the optimal values o f f
a r e ambiguous, w i t h o u t extensive e m p i r i c a l i n f o r m a t i o n .

and f 2

However, a counter-

c y c l i c a l p o l i c y , by which i s meant nonzero F=Efl f,l, can o b v i o u s l y be
e f f e c t i v e i n t h i s model.

Indeed, the business cycle- - characterized by p e r s i s -

t e n t h i g h and low values o f o u t p u t r e l a t i v e t o trend--can be eliminated, g i v e n
complete knowledge o f the s t r u c t u r a l parameters.

The m i n i m i z a t i o n o f o u t p u t

v a r i a t i o n s (corresponding t o these choices o f q, f l , and f,) i s c o n s i s t e n t
w i t h t h i s e l i m i n a t i o n o f the business cycle.

w i 11

both make

So, t h e p r o p e r l y designed p o l i c y

o u t p u t innovations ( s t o c h a s t i c f l u c t u a t i o n s n o t a t t r i b u t a b l e t o

tendencies o f previous such f l u c t u a t i o n s t o p e r s i s t ) as small as p o s s i b l e
e l i m i n a t e any tendency toward persistence.

That these two p r o p e r t i e s a r e

found simultaneously i n the optimal r u l e f o r this- - and o t h e r p r e - r a t i o n a l expectations models--may seem t r i v i a l .

However, t h i s coincidence i s

not a

general f e a t u r e o f r a t i o n a l expectations model s.

A s i n the s t a t i c IS-LM model, the optimal choice o f q can be determined
w i t h o u t reference t o the choice o f

f r

and f,.

A l s o l i k e the s t a t i c IS-LM

model, the output- variance- minimizing value o f q i s independent of the
dynamics o f the model (d,,

4, f l , and f, do n o t appear i n equation (22)

),

w h i l e F depends on both the dynamic elements (d, and $1 as w e l l as q.

I t i s noteworthy t h a t conventional macroeconometric models are essent i a l l y an admixture o f the s t a t i c and dynamic IS-LM models described i n t h i s

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section. The aggregate supply behavior is much 1 ike that o f the dynamic IS-LM
model with autoregressive expectations (see McCallum C19801 and the references
cited there).

Instead o f suppliers' prices responding to adapting price

expectations, though, price behavior is described as responding t o labor and
product market conditions.

This difference in description is essentially

inconsequential for the qua1 i tative analysis of optimal pol icy.

However, the

aggregate demand specification o f conventional models is that of the static
model , with the nominal rather than the real interest rate as an argument.
This latter difference is consequential.
The IS-LM model with autoregressive expectations and an expectational
Phi 1 1 ips curve seemed, initially, to satisfy objections t o the earlier simple
IS-LM model.

Prices are no longer exogenous, but respond t o the same set of

shocks as does output.

If e t and

U,

have variances that are "large"

relative to that of u z t (in a loose sense that depends upon the structural
parameters d l, s, a,, a 2 , q, and $1, then output and prices wi 1 1 be
positively correlated.

Yet, an occasionally "large" supply shock

result in coincidence o f high inflation and low output.

Unt

could

These features gave

the model greater empirical credibility than the earlier IS-LM models in which
supply behavior was not made explicit.
Yet, despite these improvements, the IS-LM model with autoregressive
expectations retained one fatally implausible microeconomic implication:
was inconsistent with the natural rate hypothesis.

it

In the representation

shown above, in which price-level expectations are stationary, any regular
increase in the money stock would bring about a permanent increase in output.
In the more popular "accelerationist" representation, price-level expectations
are stationary in growth rates.

In these, a regular increase in the growth

rate o f money would bring about a permanent increase in output.

If auto-

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regressive expectations are s p e c i f i e d as s t a t i o n a r y i n the d t h d i f f e r e n c e ,
then s p e c i f i c a t i o n o f a p o l i c y r u l e o f ( d + l ) t h - o r d e r s t a t i o n a r i t y w i l l be suff i c i e n t t o make t h e model i n c o n s i s t e n t w i t h t h e n a t u r a l r a t e hypothesis.
This inconsistency a r i s e s because p o l i c y can render expectations biased.

To

e l i m i n a t e t h i s inconsistency, economists have found i t necessary t o adopt t h e
assumption o f r a t i o n a l expectations.

The IS-LM Model w i t h Rational Expectations

The fundamental p o l i c y i n s i g h t o f r a t i o n a l expectations i s t h a t , t o t h e
e x t e n t p o l i c y e f f e c t s depend on expectational e r r o r s , they cannot be systematic.

This p r o p o s i t i o n f o l l o w s , i n l a r g e p a r t , from the expectational

P h i l l i p s curve, o r supply f u n c t i o n (171, under which output i s e n t i r e l y
i n e l a s t i c w i t h r e s p e c t t o expected i n f l a t i o n .

Therefore, a supposedly

c o u n t e r c y c l i c a l p o l i c y o f , f o r example, i n c r e a s i n g money growth when a recess i o n i s observed, w i l l n o t be e f f e c t i v e i n s t a b i l i z i n g o u t p u t because s e l l e r s
w i l l f u l l y a n t i c i p a t e the i m p l i e d v a r i a t i o n s i n p r i c e s .

This p r o p o s i t i o n was

f r e q u e n t l y described as " p o l i c y i n e f f e c t i v e n e s s , " seeming t o suggest t h a t
choice o f any one p o l i c y r u l e i s as good as another ( a t l e a s t w i t h i n t h e c l a s s
o f r u l e s serving t o complete the model).

However, c a r e f u l a n a l y s i s below w i l l

show t h a t the relevance o r i r r e l e v a n c e of Q and F depends on p a r t i c u l a r
assumptions about i n f o r m a t i o n a v a i l a b i l i t y o r endowments.

The r e s u l t i n g

ambiguities are l a r g e l y a resu1.t o f problems r e l a t e d t o the aggregate demand
function.
Rational expectations, i n t h e i r strong Muthian form, are those generated
by u s i n g the i n f o r m a t i o n a v a i l a b l e and i n f u l l knowledge o f the model, i n c l u d i n g the money supply r u l e .

Formally, r a t i o n a l expectations o f c u r r e n t and

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future p r i c e s are
(25)

E t p t + l = E C P ~ +( S~t ] ,

i=1,2,

where the symbol E denotes the ( t r u e ) mathematical expectation d e r i v e d w i t h i n
the model and S t i s the i n f o r m a t i o n s e t c o n d i t i o n i n g expectations a t time
t.

The r a t i o n a l expectations assumption prevents any systematic e r r o r s i n

p r i c e expectations.

I n p a r t i c u l a r , i t replaces the autoregressive expecta-

t i o n s mechanism o f the previous s e c t i o n w i t h a mechanism t h a t i s e x p l i c i t l y
dependent on the s t r u c t u r e , i n c l w d i n g t h e p o l i c y r u l e .

This ensures t h a t t h e

e f f e c t s o f contemplated changes i n t h e p o l i c y r u l e do n o t r e l y on e x p l o i t a t i o n
o f systemati c expectational e r r o r s .
Sargent and Wallace (1975) demonstrated t h a t replacement o f autoregress i v e w i t h r a t i o n a l expectations i n t h e IS-LM model described above i m p l i e d two
r a d i c a l i m p l i c a t i o n s about the optimal p o l i c y r u l e :

(1) t h a t the values o f

f l and f 2 were i r r e l e v a n t f o r o u t p u t variance, and (2) t h a t p r i c e s and t h e
money stock were indeterminate under a pure " i n t e r e s t r a t e r u l e . "
To show the f i r s t p r o p o s i t i o n , consider t h a t reduced-form s o l u t i o n s f o r
p t (expressions f o r t h e l a t t e r are f u n c t i o n s , n e c e s s a r i l y l i n e a r , o f t h e
s t a t e v a r i a b l e s , o r predetermined and exogenous v a r i a b l e s e n t e r i n g model
equations) must take the t r i a l s o l u t i o n form:
(26)
and

Yt=n10+~llyt-1+r12~t-1+n~~~~~+n~~~~~+~~~e~
pt=nz0+n21 y t - l + n 2 2 p t - l + ~ lt+fi24u2t+n25et

f o r some nl,s,
model.

where the l a t t e r are f u n c t i o n s o f parameters o f the

The complete reduced-form s o l u t i o n s f o r a1 1 endogenous v a r i a b l e s

can be obtained by assigning them t r i a l s o l u t i o n forms, s u b s t i t u t i n g them i n t o
model equations, and s o l v i n g the i m p l i e d i d e n t i t i e s f o r the r l , s .
a l expectations are imposed by a p p l y i n g (25) t o (26).

Ration-

I n t h i s application,

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it was assumed by Sargent and Wallace that the information set contained only
lagged realizations of variables,
(27) St={pt-1 ,yt-1,.. .).
Agents were ignorant of u 1t , u Z t , and et, which for simplicity are
assumed here to be nonautocorrelated and independent.
(28) E t u lt=Etu2t=Etet=Eu t u 2t=Eu2tet=Eultet=O.
Then, (25) through (28) imply
(29) Etpt

=

r~~+w~lYt-l+w~~Pt-l

and Etpt+l=(rl
o+wll w 2 0 + w 1 o)+nl (w21+w1

2
I ~ Z Z + ~ I

2)pt-1.

It is by the derivation of the expectations expressions in (291, and their use
in the trial solution, that rationality of expectations is imposed on the
model .
The supply function (17) and the trial solutions (26) and (29) imply
(30) y t = s o + ~ y t - l + s w 1 3 u l t + ( l + s w 1 4 ~ ~ 2 t + s r l s e t .
y t - l appears in this expression with the fixed coefficient X, and p t - l does
not appear at a1 1 , hence, f and f, do not i nf 1 uence aytlayt- or
ayt/apt-,. The partial derivatives
(31

and

ayt/au1 = s w 1 3

=n3=s(al+q>J,

aytiauZt = 1 + ~ ~ 1 4

=nq=dJ,

ayt/aet = s w l s

=ns=-sdJ,

turn out to be identical to those for the version with autoregressive expectations, shown in equation (21).
Two implications for (q,fl,f,> of these results are immediate.
First, the optimal f l and f, are nonunique; indeed, those parameters are
irrelevant for output, as claimed by the first proposition of Sargent and
Wallace.

Second, the output- variance- minimizing value of q is the unique

value given by (22), and has the properties attributed to it there, including

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i t s independence o f dynamic elements.
(q,fl,f2)

I n f a c t , these two i m p l i c a t i o n s f o r

a r e e n t i r e l y unaffected by a d d i t i o n a l dynamic elements, such as

the i n t r o d u c t i o n o f a d d i t i o n a l lagged terms o r a u t o c o r r e l a t i o n s o f e r r o r terms
i n the model's equations.
I n t h i s model, even nonmonetary i n f l u e n c e s on aggregate demand t h a t can
be f o r e c a s t by agents i n advance o f t h e i r occurrence a r e impotent, being neut r a l i z e d v i a changes i n p r i c e s .

For example, f i s c a l p o l i c y t h a t operates

through the mechanism o f changing aggregate demand (as p a r t o f u l t > i s
i r r e l e v a n t f o r o u t p u t i f announced i n advance (regardless o f the monetary p o l i c y r u l e adopted).

I n IS-LM type model s , o n l y unexpected monetary o r f i s c a l

p o l i c i e s matter f o r o u t p u t , aside from the automatic s t a b i l i z e r s i n h e r e n t i n
graduated- rate t a x systems, i n the case o f f i s c a l p o l i c i e s , and the choice of
q, i n the case o f monetary p o l i c i e s . '

" I n t e r e s t Rate Rules"
The second major r e s u l t i n Sargent and Wallace's model i s t h a t an
" i n t e r e s t r a t e r u l e " does n o t serve t o complete the model i n t h a t i t leaves
p r i c e s and money indeterminate.

This r e s u l t w i l l r e q u i r e a modified a n a l y s i s

t o derive, because such " r u l e s " are n o t representable i n t h e p o l i c y space
( q , f l ,f,> i n R 3 .

Consider t h e " r u l e s " o f form:

(32) R t = g o + g i y t - ~ + g z p t - ~ .
(The indeterminacy r e s u l t would a l s o occur i n any g e n e r a l i z a t i o n o f (32) i n
which mt i s l a c k i n g ) .

(32) does n o t s p e c i f y corresponding elements i n t h e

( q , f l ,f,) space, because t h e r e i s no unique transformat'ion from
(gO,gl ,g2) i n t o (q,fl , f 2 ) . Therefore, the s o l u t i o n s f o r endogenous
v a r i a b l e s under the money supply r u l e (20) cannot be used t o determine o u t comes under (32).

'

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Before t u r n i n g d i r e c t l y t o t h e problem o f indeterminacy o f nominal v a r i ables, i t i s worth n o t i n g t h a t a k i n d o f p o l i c y i r r e l e v a n c e continues t o h o l d
true.
(33)

I n f a c t , o u t p u t under (32) would be
y t = so+Xyt-t+su1t

(which happens t o c o i n c i d e w i t h the l i m i t of expression (30) as q approaches
i n f i n i t y ) , an expression devoid o f the g i s .
To show t h a t p t and mt are indeterminate i n t h i s case, i t s i m p l i f i e s
matters t o n o t i c e f i r s t t h a t the money demand equation, (141, does n o t , i n
view o f (321, h e l p determine p t .

I t o n l y determines mt

mined by t h e o t h e r equations o f the model.

if

p t i s deter-

So (14) need n o t be p a r t o f t h e

a n a l y s i s o f determinateness o f p r i c e s i f the s p e c i f i c a t i o n of " p o l i c y " i s
(32).
Now t h e determinacy o f p r i c e s could be e s t a b l i s h e d by demonstrating t h a t
t h e t r i a l s o l u t i o n o f (26) f o r p t i s unique; t h a t i s , t h a t the w 2 , s are
f i n i t e and unique.

The r e s t r i c t i o n s on the r z l s i m p l i e d by the model can

be i n f e r r e d i n the f o l lowing way.

Equating y:,

(1 51, and y:,

(17), and

u s i n g (321,
(34) spt = ( d o + d g o - ~ o ) + d g l y t - 1 + ( d 2 - X > Y t - t + ( ~ r

t-~zt)-dEtpt+l+(d+~)Etpt.

Then, s u b s t i t u t i o n o f (26>, (291, and (33) i n t o (34) i m p l i e s the f o l l o w i n g
identities:
(35)

sw20=(do+dlgo-so)-dI ( n 2 0 ( 1 + n 2 2 ) + s 0 ~1)+(d1+s)w20
2
sn2 l=(d1gl+d2-X>-dl~21 ( X + ' ~ 1 2 2 ) + ( d 1 + ~ ) ~ 1 ,

S W 4=-1
~
s d 2

5=0

These i d e n t i t i e s o b v i o u s l y provide unique values f o r n z 3 ,w z 4 , and r 2 5 . 9

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However, the t h i r d i d e n t i t y , f o r n z 2 ,i s q u a d r a t i c , so n , , can take on two
d i s t i n c t values.

T h i s nonuniqueness i s contagious because t h e s o l u t i o n

values f o r r z O
and r 2 depend on n z 2
. Hence, i t i s already c l e a r t h a t t h e
model w i t h (32) does n o t place s u f f i c i e n t r e s t r i c t i o n s on the n,,s
determinacy.

t o provide

(However, the s o l u t i o n f o r o u t p u t remains uniquely determinate. 1

To s i m p l i f y f u r t h e r study o f the i d e n t i t i e s i n (351, consider t h e
r e s t r i c t i o n o f g2, which appears i n t h e t h i r d i d e n t i t y , t o zero.
r e s t r i c t i o n i s , i n view o f (331, i r r e l e v a n t f o r o u t p u t .

This

Under t h e minimal-

s t a t e - v a r i a b l e approach, t h i s r e s t r i c t i o n would be imposed by e l i m i n a t i n g
p t - l from the t r i a l s o l u t i o n , so t h a t nZ2=0, by assumption.

Then t h e

"solution" f o r prices i s
(36) ~ ~ = ( ~ ) + ( d o + d l g o - ~ o ) ~ 0 ~ d 7 ~ y t - l + ~ -~ '- su-l' u Z ~ ,
where the

(m)

symbol i n d i c a t e s t h a t the i n t e r c e p t r z O
i s undefined.

This i n d e f i n i t e i n t e r c e p t i n d i c a t e s p t i s indeterminate.
On the o t h e r hand, i f the minimal- state- variable s e t i s augmented by
inclusion o f pt-,,

n.,,

can take on the values of z e r o o r u n i t y .

former case, (36) i s again the s o l u t i o n .

I n the

I f IT^^=^, then the s o l u t i o n i s

(37) pt=E(do-so)+dl (go-sogl ) - ~ ( d 2 - x ) 3 ( 2 + 2 d l ) - '
+ C g l + ( d 2 - X > d ; ' l y t - i + ~ t - i + ~ - ~ ~ lt - s - l u z t .

This s o l u t i o n , having i3pt/i3pt-l=l,

i m p l i e s a time s e r i e s f o r p t t h a t i s

stationary only i n f i r s t differences.

Nevertheless, t h i s s o l u t i o n i s d e t e r -

m i nate.
McCallum (1986) shows t h a t a reduced form f o r Rt s p e c i f i e d by (32) can
r e s u l t from two d i f f e r e n t money supply r u l e s , one which i s s t a t i o n a r y i n mt,
as i n (201, and one which i s s t a t i o n a r y o n l y i n f i r s t differences,

as i f m t ,

the l eft- hand- side v a r i a b l e i n (20), were p r e f i x e d by the difference o p e r a t o r

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(1-L).

However, i t could a l s o be shown t h a t t h e r e are an i n d e f i n i t e l y l a r g e

number o f money supply r u l e s , i n higher o r d e r d i f f e r e n c e s o f money, t h a t would
make R t behave as s p e c i f i e d by (321, as t h e b o o t s t r a p v a r i a b l e s p t - I ,
p t - * , P ~ - ~ ., .. are allowed t o e n t e r the s o l u t i o n .

Consequently, as noted

a t the s t a r t o f t h i s section, " i n t e r e s t r a t e r u l e s " do n o t adequately specify
the p o l i c y r u l e and hence f a i l t o complete the model.
I n summary, e i t h e r t h e r e i s no determinate s o l u t i o n f o r p r i c e s , as when
bootstraps a r e r u l e d o u t , o r there are an i n d e f i n i t e l y l a r g e number o f m u l t i p l e s o l u t i o n s , i n c l u d i n g one i n which the i n t e r c e p t i s indeterminate.

It i s

hard, indeed, t o regard these two p o s s i b i l i t i e s as meaningfully d i s t i n c t .

Any

behavior o f p r i c e s can be c o n s i s t e n t w i t h t h e model under " p o l i c y r u l e " (32)
and r a t i o n a l expectations.

A n a l y t i c a l Problems Under A l t e r n a t i v e I n f o r m a t i o n Constraints
Money supply r u l e s , except i n p a t h o l o g i c a l cases, leave a1 1 v a r i a b l e s
determinate.

But, f o r a n a l y t i c a l reasons, c h a r a c t e r i z a t i o n o f an optimal r u l e

o r s e t of r u l e s i s d i f f i c u l t o r impossible except i n a very r e s t r i c t i v e c l a s s
of models, o f which the Sargent and Wallace model i s an example.

The assump-

t i o n s about information c o n s t r a i n t s on p r i v a t e agents made p o l i c y i m p l i c a t i o n s
easy t o d e r i v e i n t h a t model.

A l t e r n a t i v e i n f o r m a t i o n s t r u c t u r e s i n which

p r i v a t e agents observe c u r r e n t r e a l i z a t i o n s o f v a r i a b l e s , such as R t , i n
forming t h e expectations o f t h e c u r r e n t and f u t u r e p r i c e l e v e l , can l e a d t o
s u r p r i s i n g i m p l i c a t i o n s f o r t h e optimal p o l i c y r u l e .
This p o i n t i s n e a t l y i l l u s t r a t e d by an i n s i g h t f u l a n a l y s i s by Canzoneri,
Henderson, and Rogoff (1983).

F i r s t , they simp1 i f y the s t r u c t u r a l form o f t h e

Sargent and Wal l a c e model by dropping terms i n y t -, from both commodity
supply and demand f u n c t i o n s , and e l i m i n a t i n g supply shocks (uzt=O f o r a11 t ) .

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Then they consider a1 t e r n a t i ve assumptions about private- agent use o f t h e nomin a l i n t e r e s t r a t e .

I f t h e e x p e c t a t i o n o f t h e c u r r e n t p r i c e l e v e l , present i n

t h e supply f u n c t i o n , i s c o n d i t i o n e d on
(38)

St"

=

{ R t , a1 1 lagged r e a l i z a t i o n s o f s t a t e variables),

then t h e p o l i c y r u l e i s i r r e l e v a n t f o r o u t p u t .
q is indeterminate!

I n t h i s case, even t h e optimal

I n essence, suppl i e r s , whose behavior c o u l d be systemati-

c a l l y influenced by p o l i c y o n l y because o f contemporaneous p o l i c y responses t o
t h e i n t e r e s t r a t e , are able t o estimate such responses by l o o k i n g a t t h e
i n t e r e s t r a t e , and t h e market w i l l t h e r e f o r e a d j u s t p r i c e s t o n e u t r a l i z e them
after all.
An a l t e r n a t i v e i n f o r m a t i o n s p e c i f i c a t i o n assumes t h a t e x p e c t a t i o n s of
t h e r a t e o f i n f l a t i o n , present i n t h e demand f u n c t i o n , are made w i t h knowledge

of t h e c u r r e n t i n t e r e s t r a t e :
(39)

E t ( p t + l - p t ) = EC(pt+l-pt) lS:+'I.

I n t h i s case, t r a c t a b i l i t y r e q u i r e s f u r t h e r ad hoc and unmotivated r e s t r i c t i o n s on the pol i c y space.

I n t h e examples explored by Canzoneri , Henderson,

and Rogoff, e i t h e r q o r F i s r e s t r i c t e d t o z e r o o r a zero v e c t o r , respectively.
(40)

I f F-0 i s imposed, as i n t h e p o l i c y r u l e

mt= p

-

qRt,

then t h e expression f o r t h e o p t i m a l q i s t h e same as f o r the Poole model,
g i v e n i n equation (11).

The o t h e r example they explore considers p o l i c y r u l e s

o f t h e form
(41)

mt= p

+ f,mt-l

+ f4Rt-l.

I n t h e l a t t e r case, Canzoneri, Henderson, and Rogoff show t h a t , if

f3

is

r e s t r i c t e d t o u n i t y , t h e optimal f, i s
(42)

f4=l+Cal ( l + a 2 s ) / a l s l C ~ ~ / ~ ~ l ,

whose s i g n cannot be determined w i t h o u t f u r t h e r knowledge about parameters and

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disturbance variances.
I n t e r e s t i n g l y , the behavior o f o u t p u t i s the same i n these cases:
e i t h e r w i t h (40) and (ll),
o r w i t h (411, f,=l,

and (42).

The r e s u l t i n g

o u t p u t variance i s probably the lowest a t t a i n a b l e , but o t h e r p o l i c y r u l e s
w i t h o u t these values o f q and F are a l s o l i k e l y t o r e s u l t i n the same v a r i ance.

This case i s reminiscent o f the upper panel o f f i g u r e 2.
This example shows t h a t the optimal p o l i c y r u l e may n o t be unique, and

t h a t t h i s c h a r a c t e r i s t i c may n o t always be easy t o discover, unless the anal y s t i s w i l l i n g t o t r y o u t many d i f f e r e n t representations, some o f which may
n o t be obvious o r i n t u i t i v e .

The r e s t r i c t i o n f,=l

o r f 3 = 0 has commonly

been necessary t o impose even i n very simple models, w i t h few behavioral parameters and disturbances.

l o

And i f t h e supply shock, u Z t , i s r e i n t r o d u c e d

and a u t o c o r r e l a t i o n i n various shocks o r lagged output terms are allowed i n
s t r u c t u r a l equations, the optimal values o f q and F become hopelessly ambiguous.

Even t h e i r relevance o r i r r e l e v a n c e and optimal signs w i l l depend on t o o

many p a r t i c u l a r s .
Although ambiguities concerning p o l i c y e f f e c t i v n e s s a r i s e from g e n e r a l i z a t i o n s o f the Sargent and Wallace model, i t was important i n d r a m a t i c a l l y
demonstrating t h a t r a t i o n a l expectations, whose imposition was r e q u i r e d t o
make the conventional IS-LM models c o n s i s t e n t w i t h the n a t u r a l r a t e p r o p e r t y ,
had q u i t e r a d i c a l i m p l i c a t i o n s f o r those models, r a t h e r than r e p r e s e n t i n g a
t e c h n i c a l advance those models might o r might n o t u s e f u l l y i n c o r p o r a t e .

In

p a r t i c u l a r , previous demonstrations t h a t conventional p o l i c i e s could s t a b i l i z e
o u t p u t were found dependent on i m p l a u s i b l e e x p l o i t a t i o n o f biased expectations.

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Non-Market-Cl e a r i ng Model s

Market c l e a r i n g i s the equation o f commodity demand and supply v i a
adjustments i n p r i c e s and i n t e r e s t r a t e s .

This assumption has been a t t h e

h e a r t o f most microeconomic theory, and might be considered t h e n a t u r a l mechanism by which supply and demand are reconciled.

But p r i c e changes are o n l y

one conceivable means o f r e c o n c i l i n g q u a n t i t i e s demanded and supplied.

Many

economists see the economy as l a b o r i n g under c o n s t r a i n t s t h a t can be u s e f u l l y
described as c o n s t r a i n t s on adjustments o f p r i c e s .
imposed on the model , p o t e n t i a1 ( n o t i o n a l

As t h i s c o n s t r a i n t i s

demanders and suppl i e r s a r e f r u s -

t r a t e d , unable t o make t r a n s a c t i o n s they both d e s i r e a t m u t u a l l y agreeable
p r i c e s , because t r a n s a c t i o n s a t these p r i c e s are r u l e d o u t .

This f r u s t r a t i o n

can be q u i t e p e r s i s t e n t , i f p r i c e adjustments are slow, causing p e r s i s t e n t
output fluctuations.
This c o n s t r a i n t cannot be taken l i t e r a l l y .

I t i s intended o n l y as a

u s e f u l representation o f some hard- to- specify problems.

I t i s not necessarily

t h a t agents are somehow f o r c e d t o t r a n s a c t o n l y a t s t i c k y p r i c e s o r t o s i g n
c o n t r a c t s because o f c o n s t r a i n t s o t h e r than those a r i s i n g from technology o r
tastes.

I t i s , instead, t h a t aspects o f technology o r t a s t e s a r e n o t ade-

q u a t e l y captured by neoclassical production and u t i l it y f u n c t i o n s .

Then,

s t i c k y p r i c e s o r c o n t r a c t s may help, o r r e f l e c t attempts by, agents t o o p t i mize.

For exampl'e, nominal c o n t r a c t i n g may r e f l e c t " t e c h n o l o g i c a l " d i f f i c u l -

t i e s i n developing c r e d i b l e Pareto- optimal agreements a r i s i n g from some k i n d
o f information, monitoring, enforcement, o r c o o r d i n a t i o n problems.

Further

work i n t o the microfoundations o f " s t i c k y p r i c e s " may succeed i n r e p l a c i n g
p r i c e adjustment o r c o n t r a c t i n g " c o n s t r a i n t s " w i t h a more s a t i s f a c t o r y

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r e p r e s e n t a t i o n o f the u n d e r l y i n g problems.

Analysis a t t h i s deeper l e v e l may

show t h a t the t r u e c o n s t r a i n t s a r e p o o r l y represented by present s t i c k y - p r i c e
macroeconomic models.

Hence, any p r a c t i c a l p o l i c y conclusions o f present

models should be regarded as speculative.

Also, t h e i r o n y of t r e a t i n g such

phenomena as v o l u n t a r y c o n t r a c t s as a " c o n s t r a i n t " preventing, r a t h e r than
a i d i n g , o p t i m i z a t i o n should be noted.
Proceeding on t h e more s u p e r f i c i a l o r d e s c r i p t i v e l e v e l , recent work on
non- market- clearing models has incorporated r a t i o n a l expectations and imposed
a tendency o f p r i c e s t o a d j u s t t o market- clearing l e v e l s , a l b e i t slowly.
These two f e a t u r e s have made them c o n s i s t e n t w i t h the n a t u r a l r a t e p r o p e r t y .
Consequently, t h a t p o l i c y responses t o the s t a t e o f the economy can sometimes
be e f f e c t i v e

i n these models i s i n t e r e s t i n g , and they seem t o provide a poten-

t i a l l y pursuasi ve negation o f t h e f i r s t proposi t i o n o f Sargent and Wall ace.
Of course, the relevance o f t h i s negation depends on whether markets
actual l y clear.

Rational expectations models w i t h incomplete i n f o r m a t i o n have

a l t e r e d economists 1 views concerning the p l a u s i b i l i t y o f the market- clearing
assumption.

F a i l u r e o f markets t o c l e a r had o f t e n been considered apparent

from the slow movements and/or d i s c o n t i n u i t y o f i n d i v i d u a l p r i c e s .

However,

incomplete i n f o r m a t i o n among buyers and s e l l e r s can l e a d t o s l o w l y moving
p r i c e s under market- clearing, o r t o changes a t d i s c r e t e i n t e r v a l s when new
i n f o r m a t i o n i s received.

Skepticism regarding market- clearing had a l s o a r i s e n

from the persistence o f above- o r below-average measured unemployment r a t e s .
Then again, i t i s n o t obvious t h a t such phenomena r e f l e c t f a i l u r e of wages t o
a d j u s t t o c l e a r " the" l a b o r market.

Labor immobi 1i t y and i n t e r i n d u s t r y l a b o r

demand s h i f t s can combine t o c r e a t e such f l u c t u a t i o n s , even i f wages a r e n o t
sticky.

And, again, some c y c l i c a l behavior o f unemployment could be observed

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if information problems e x i s t e d , as i n the Sargent and Wallace model.

In

short, i t i s d i f f i c u l t o r impossible t o determine e m p i r i c a l l y whether observed
p r i c e behavior r e f l e c t s non- market- clearing.

The issue may be p u r e l y meta-

physical.
Yet, consider the view t h a t the markets do n o t c l e a r .

The i m p l i c a t i o n s

f o r pol i c y cannot be discovered w i t h o u t s p e c i f y i n g the nonprice elements o f
the mechanism by which the r e c o n c i l i a t i o n between demand and supply i s
effected.

I n a p e r f e c t l y competitive, many-good economy, i f the vector of

p r i c e s f o r a l l commodities i s somehow n o t equal t o the vector ( o r n o t an
element of the s e t of vectors, i f nonunique) t h a t achieves market- clearing,
outputs and l e i s u r e w i l l diverge from a Pareto-optimum.
o b t a i n such a r e s u l t would c e r t a i n l y be remarkable.

For the economy t o

Hypothetical methods of

doing so, such as the Walrasian auctioneer, are l i t e r a l l y implausible and
often r i d i c u l e d .

Also, i t i s reasonable t o suppose t h a t , i f exchange i s v o l -

untary, such discrepancies o f p r i c e s from the market- clearing p r i c e s w i 11
r e s u l t i n a s h o r t f a l l o f commodity o u t p u t from i t s Pareto-optimum ( f u l l
employment?).

This i s because e i t h e r some marginal s e l l e r s o r some marginal

buyers i n each good market w i l l b a l k a t the terms o f exchange, and transact i o n s are two-sided.

Monetary pol i c y can then a f f e c t the workings of t h e

economy depending on how the r e a l q u a n t i t y o f money appears i n the p r o d u c t i o n
and u t i 1 it y functions, the nature o f p r i c e adjustments, and how monetary
pol i c y i s specified.

But adequate models a t t h i s l e v e l o f g e n e r a l i t y have n o t

been constructed and w i 1 1 probably prove e l u s i v e .
A t a 1ess general 1 eve1, non-market-cl e a r i ng has been coup1ed w i t h t h e

assumption t h a t s e l l e r s s a t i s f y the demand a t c u r r e n t p r i c e s .

The simi 1a r i t y

o f t h i s type o f model w i t h the e a r l i e r IS- LM models l i e s i n the demanddetermi n a t i o n o f o u t p u t .

The newer types o f non- market- clearing model s a r e

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different,

however, i n t h a t t h e y a1 low p r i c e s t o be determined i n a manner

t h a t , i n c o n j u n c t i o n w i t h r a t i o n a l e x p e c t a t i o n s , a1 lows model c o n s i s t e n c y w i t h
the natural r a t e property.

I n p a r t i c u l a r , o u t p u t w i l l f l u c t u a t e around a

" n a t u r a l " o r average r a t e t h a t i s u n i n f l u e n c e d by t h e monetary p o l i c y r u l e and
t h a t i s f r e q u e n t l y supposed t o be t h e optimum o u t p u t l e v e l .

I n t h i s k i n d of

model, t h e n a t u r e o f p r i c e d e t e r m i n a t i o n i s c r u c i a l .
I t i s a n a l y t i c a l l y u s e f u l t o d i s t i n g u i s h between two types of p r i c e
adjustment mechanisms:

those i n which a l l p r i c e s change i n each p e r i o d , and

those i n which some p r i c e s do n o t change i n each p e r i o d .

Stabilization policy

appears t o have 1i t t l e o r no scope f o r e f f e c t i v e n e s s i n t h e f i r s t case, b u t i t
can g e n e r a l l y be e f f e c t i v e i n t h e l a t t e r .

One i n s t r u c t i v e example o f t h e

f i r s t type i s t h e case f o r which p r i c e s a r e completely predetermined one
p e r i o d i n advance.

To e n f o r c e consistency w i t h t h e n a t u r a l r a t e p r o p e r t y , i t

w i l l be s u f f i c i e n t t o assume t h a t s e l l e r s s e t p e r i o d - t p r i c e s a t t h e l e v e l
r a t i o n a l l y expected t o c l e a r t h e market, c o n d i t i o n e d on r e a l i z a t i o n s o f
variables i n p e r i o d t-1:
(43)

pt = Etpt,

where p r i c e e x p e c t a t i o n s a r e c o n d i t i o n e d on St and p t i s t h e s o l u t i o n t o
(44)

y:(pt,

EtM:-'1

= y:(pt,

I n t h e l a t t e r equation, M ' - '
omission o f p t .

EtM:-',

EtPt+l>.

denotes t h e minimal s t a t e v e c t o r t r u n c a t e d by

By s u b s t i t u t i o n o f (15), (161, and (17) i n t o (441, and

u s i n g (43) and i t s obvious i m p l i c a t i o n p t = E t p t , i t can be shown t h a t
p r i c e s are determined a c c o r d i n g t o
(45)

pt=Etpt+l-EtRt+(so-do)dY1+(1-d2)dY1yt-1.

Then, s o l v i n g t h e system o f equations (71, ( 1 4 ) , (15), (161, (20), and (451,
and imposing r a t i o n a l e x p e c t a t i o n s , i t can be shown t h a t o u t p u t has t h e
representation:

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(46)

yt=so+Xyt-l+(q+al )J1ul ,-dlJlet
(J1=(al+a,d,+q)-'

1.

I t i s immediately apparent t h a t f , and f, a r e i r r e l e v a n t f o r o u t p u t d e t e r m i n a t i o n , i m p l y i n g t h a t t h e f i r s t p r o p o s i t i o n o f Sargent and Wallace h o l d s i n
t h i s model.

I n t e r e s t i n g l y , t h e c o n d i t i o n a l mean E t y t i s determined

u n i q u e l y by s u p p l y behavior, whi l e t h e d e v i a t i o n ( y t - E t y t ) i s determined
u n i q u e l y by demand behavior, and i n p r e c i s e l y t h e same f a s h i o n as i n t h e s t a t i c IS-LM model (compare equations (46) and (9)).

The o p t i m a l v a l u e of q i s

g i v e n i n e q u a t i o n (11).
A s f o r t h e second p r o p o s i t i o n o f Sargent and Wallace, i n d e t e r m i n a c y o f

p r i c e s and money under an " i n t e r e s t r a t e r u l e " o f t h e f o r m ( 3 2 ) , i t m i g h t seem
t h a t p r e d e t e r m i n a t i o n o f p r i c e s would i m p l y determinateness.
however, t h a t such an i n t u i t i o n i s i n c o r r e c t .

I t turns out,

The s o l u t i o n f o r t h e p r i c e

l e v e l i s i n d e t e r m i n a t e , as i n t h e m a r k e t - c l e a r i n g case, w i t h one s o l u t i o n havi n g an undefined i n t e r c e p t , and an i n f i n i t e number o f a l t e r n a t i v e s o l u t i o n s if
b o o t s t r a p components a r e n o t r u l e d o u t .

The i r o n i c l a c k o f determinateness

despi t e p r e d e t e r m i n a t i o n a r i ses f r o m t h e f o r w a r d - l o o k ing n a t u r e o f expectat i o n s i n t h i s ( o r any) r a t i o n a l e x p e c t a t i o n s model.

P r i c e s expected t o c l e a r

t h e market a r e dependent one- for- one on expected p r i c e s o f t h e p e r i o d a f t e r
t h a t , and so on i n t o t h e i n d e f i n i t e f u t u r e .
anchor any of these e x p e c t a t i o n s .
non- market- clearing models.

An i n t e r e s t r a t e r u l e does n o t

This indeterminateness a l s o occurs i n o t h e r

Indeed, i t seems a necessary f e a t u r e of a1 1

models i n which r a t i o n a l e x p e c t a t i o n s p l a y a n o n t r i v i a l r o l e .

1 1

Another t y p e o f p r i c e - a d j u s t m e n t e q u a t i o n i n which a l l p r i c e s change
each p e r i o d i s t h e f a m i l i a r p a r t i a l - a d j u s t m e n t mechanism,
(47)

(pt-pt-,

>

=

p(pl-pt-l)

+

qt,

O<p<l,

where q i s a n o n a u t o c o r r e l a t e d random v a r i a b l e .

McCallum (1978) c o n s i d e r s

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this case at length, and finds that, given q-0, f l and f, are irrelevant
for output. l 2

Unfortunately, the optimal pol icy rule or, more likely, set

of policy rules, cannot be adequately characterized, because the ad hoc
restriction q=O is needed t o render analysis tractable.
In the second set of non-market-clearing models, some prices d o not
adjust each period.

Fixed-price models were an example treated above.

Models

of staggered, multiperiod contracts are particularly interesting because they

can simultaneously possess the natural rate property and imply policy effectiveness, even when the authorities have n o superior information.

In these

models, sellers, usually of labor services, agree t o accept wages o r prices
predetermined for more than one period in advance.
T o illustrate, suppose that the economy comprises t w o equally numbered
groups of perfectly competitive firms.

In each period, output consists of the

sum of the outputs of group 1 , Y1,, and of group 2, Y Z t :
(48) Yt=Ylt+YZt.

Without loss of generality, let group 1 consist of the firms that signed
contracts at the end of period t-1, t o remain in effect during period t and
t+l, while group 2 consists of those whose contracts were signed at the end of
period t-2, and expire at the end of period t.

Wages are set for group i

based o n available information at the end of period t-i.

Further assume the

production function:
(49) Y , ,=ZtN?,,

i=1,2,

y>O.

Z is a global productivity variable whose log is z,=k+c,, for some
constant k and disturbance
scedastic.

E,.

The latter is nonautocorrelated and homo-

N is the (unlogged) employment level.

In the spirit of the

contracting models, let the wages contracted for period t by group i equal the
expected marginal (physical) product times the price level:

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(50)

W , ,=E,-,

(ZtyN:7-l 'P,),

i=1,2,

where P i s the unlogged p r i c e l e v e l .

Firms make o u t p u t d e c i s i o n s based on

contemporaneous knowledge o f p, W , and Z.

Then each f i r m ' s p r o f i t s are maxi-

mized by equating the predetermined wage w i t h the a c t u a l marginal ( p h y s i c a l )
product times the p r i c e l e v e l :
(51

W , t=ZtyN:7-1 ) P t ,

i=1,2.

These two equations, the p r o d u c t i o n f u n c t i o n (49), and the l i n e a r approximation:
(52)

y t = ( y i t+y2 ,)I2

imply
(53)

yt=~o+~(pt-Etpt>+~(pt-Et-lpt)+~2t,

where s=y/2( 1-y) >0,
and u ~ ~ = ( ~ - ~ ) - ' E ~ .
Equation (53) suggests, i r o n i c a l l y , t h a t m u l t i p e r i o d c o n t r a c t models can be
represented as a p a r t i c u l a r k i n d o f " g e n e r a l i z a t i o n " o f t h e Sargent and
Wallace market- clearing model.
The design o f the optimal p o l i c y r u l e i s an i n t r a c t a b l e problem i n t h i s
model, unless some ad hoc r e s t r i c t i o n , such as q=O, i s imposed.
some c h a r a c t e r i s t i c s o f the optimal p o l i c y can be deduced.

Nevertheless,

I n t h i s model, as

i n t h e Sargent and Wallace model, the choice o f q t o reduce o u t p u t d e v i a t i o n s
i s dependent s t r i c t l y on the variances o f disturbances and the s t r u c t u r a l
parameters 1i n k i n g them t o c u r r e n t output.

The more i n t e r e s t i n g p r o p e r t y ,

however, i s t h a t f , and f 2 a r e r e l e v a n t f o r o u t p u t determination, u n l i k e
i n t h e Sargent and Wallace model.

This i s because f i r m l w o r k e r combinations

under the o l d e r c o n t r a c t s are n o t making f u l l use o f the i n f o r m a t i o n s e t St,
b u t , i n a p a r t i c u l a r sense, a r e a c t i n g a s i f they k n e w o n l y S t - l . ' 3

The

policymaker can use the i n f o r m a t i o n t h a t i s " ignored," i n e f f e c t , by t h e

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firmlworker combinations w i t h o l d e r contracts, thereby s t a b i l i z i n g output.

To

e x p l a i n simply, consider t h a t p o l i c y can i n f l u e n c e output, according t o (531,
v i a p r i c e l e v e l s u r p r i s e s over one- and two- period horizons.

Then the r e l e -

vance f o r output o f f , and f, and l i k e terms i n the p o l i c y r u l e can be
i n d i r e c t l y assessed by asking whether output i s influenced by expected p r i c e s .
I n f a c t , any tendency toward persistence o f o u t p u t f l u c t u a t i o n s could be
e l i m i n a t e d by a p o l i c y r u l e t h a t made expected p r i c e s vary i n the f o l l o w i n g
way:

(54)

Etpt = E t - l p t

-

CX(1-y)/ylyt-l.

As s t a t e d above, the money supply r u l e o r s e t o f r u l e s t h a t e f f e c t s t h i s predetermined v a r i a t i o n i n p r i c e s i s d i f f i c u l t t o d e r i v e .
Despite the e x i stence o f contracts, many economists suggest they a r e
l a r g e l y facades.

Perhaps they are merely means o f exchanging i n f o r m a t i o n

between f i r m s and workers, b u t do n o t create undesired f l u c t u a t i o n s i n o u t put.

C e r t a i n l y , i n c e n t i v e s e x i s t t o e l i m i n a t e , o r a t l e a s t minimize, these

undesired f l u c t u a t i o n s .

Nonprice mechanisms f o r r a i s i n g and lowering r e a l

wages may, a t l i t t l e c o s t i n e f f i c i e n c y , s u b s t i t u t e f o r changes i n e x p l i c i t
wages.

Contract p r o v i s i o n s regarding overtime pay, h i r i n g p r a c t i c e s , and

o t h e r aspects a f f e c t i n g l a b o r costs may tend t o face the f i r m w i t h a marginal
l a b o r c o s t schedule n e a r l y matching the r i s i n g d i s u t i l i t y o f work.

To t h e

e x t e n t these match, l a b o r c o n t r a c t s are c o n s i s t e n t w i t h optimal o u t p u t d e t e r mination.

I f the match i s imperfect, then the optimal p o l i c y w i l l depend on

t h e d e t a i l s of the mismatch, how the mismatch i s a f f e c t e d by the p o l i c y r u l e ,
and variances and parameters.

While optimal q, f , , and f z w i l l g e n e r a l l y

take nonzero values, even t h e i r signs w i l l be hard t o assess.

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Output Versus P r i c e S t a b i l i z a t i o n as t h e O b j e c t i v e

As e x p l a i n e d i n t h e p r e v i o u s s e c t i o n , t h e m u l t i p e r i o d c o n t r a c t i n g model
r e s t o r e s a t r a d e - o f f between o u t p u t and p r i c e s t a b i l i t y somewhat r e m i n i s c e n t
of pre- rational- expectations models.

Yet, i m p l i c i t i n b o t h t h e c o n t r a c t i n g

models and t h e Sargent and Hal l a c e model i s a source o f doubt concerni ng t h e
relevance o f t h i s t r a d e - o f f .

This doubt a r i s e s f r o m r e c o n s i d e r a t i o n o f o u t p u t

v a r i a n c e as an adequate r e p r e s e n t a t i o n o f t h e obj'ective.

As mentioned ear-

l i e r , r e a l business- cycle model s can d e s c r i b e f l u c t u a t i o n s i n o u t p u t as
o p t i m i z i n g responses t o changing p r o d u c t i o n o p p o r t u n i t i e s f a c i n g i n d i v i d u a l
agents.

T h i s p o s s i b i l i t y i s e i t h e r r u l e d o u t o r obscured i n models i n which

o u t p u t is demand-determi ned (determined by exogenous spending p r o p e n s i t i e s and
monetary and f i s c a l p o l i c y ) .
However, i n t h e models i n which supply behavior p l a y s a n o n t r i v i a l r o l e ,
i t becomes i m p o r t a n t t o ask what r o l e p r o d u c t i v i t y changes p l a y .

For example,

i n t h e supply e q u a t i o n (53) o f t h e c o n t r a c t i n g model, t h e terms s ( p t - E t p t )
s(pt-Et-lpt)

and

represent the deviations i n output r e s u l t i n g from the

i n a b i l i t y o f workers and f i r m s t o develop Pareto- optimal c o n t r a c t s ; ones i n
which o u t p u t i s determined by t h e a p p r o p r i a t e marginal c o n d i t i o n s .

Likewise,

under incomplete i n f o r m a t i o n among s u p p l i e r s t h a t i m p l i e d t h e aggregate s u p p l y
f u n c t i o n (171, t h e t e r m s ( p t - E t p t )
f r o m i t s optimum.

represents the d e v i a t i o n o f output

I n e i t h e r case, t h e e f f e c t i v e n e s s o f t h e a d d i t i o n a l con-

s t r a i n t on t h e economy t h a t p r e v e n t s f u l l o p t i m i z a t i o n i s r e l a t e d t o t h e
component o f p r i c e s t h a t c o u l d n o t be a n t i c i p a t e d i n advance.

I f mu1t i p e r i o d

c o n t r a c t s a r e considered i m p o r t a n t , then t h e i r l e n g t h w i l l h e l p determine t h e
h o r i z o n o v e r which p r i c e - l e v e l u n c e r t a i n t y should be minimized.

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The somewhat ironic conclusion i s that mu1 ti period nominal labor contracts appear to argue even more strongly for price stability over longer
horizons as a policy criterion, rather than suggesting a policy trade-off
between output and price stability.

In general, it is intuitive that the

degree o f efficiency o f (incompletely indexed) nominal contracts o f length n,
whether for labor, capital, or other factor services, will depend on pricelevel predictability over horizons from one to n periods.

At least, this

statement is obviously true if contracts couple predetermination (or incomplete indexation) o f nominal factor payments with Pareto-suboptimal demanddetermination o f factor quantities, as in the labor contracts supposed empirically relevant for unionized firms.
This conclusion does not immediately provide the optimal policy rule, of
course, because, at least in the models considered, price-level stabilization
cannot be perfect.

Also, there may be trade-offs between price stabilization

over various horizons.

For example, attempts t o return prices quickly t o a

long-established target might clash with the desire to avoid problems under
existing contracts, which may already reflect the existing deviation of
prices. The weight to be put on price uncertainty over various horizons will
depend on the c~nstraintsthat incomplete information and contracting models
describe the economy as being under.

If both kinds o f constraints are rele-

vant, both kinds of models can contribute to our understanding o f how monetary
policy should be designed.
One "small" change in the models could make the optimal policy virtually
unambiguous.

If the pol icymaker i s a1 lowed to observe the price level p t

contemporaneously, it can, t o any arbi trari ly exact degree o f accuracy, set it
o n any pre-announced course that eliminated forecast errors over the relevant
horizons.

Such a rule is:

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(55) m t = p - 8 p t ,
where 8 is a positive magnitude large enough t o prevent significant price
movements, but not so large as t o imply a contradiction:

some deviations in

prices must be observed in order t o practice this policy.
Even modest measurement and information-delay problems concerning prices
would seem t o call for something less than complete reliance o n current
prices.

Then, price stability would be best achieved by some more complicated

rule, about which there is inadequate knowledge, and about which economists
with different models could disagree.

Nevertheless, consensus among competing

models on the price-stability criterion is useful in the absence o f full
know1 edge or agreement.

Intertemporal Substitution Models

The models discussed in previous sections are variants o f IS-LM models.
Characteristic o f these is an asymmetry between agents' behavior as suppliers
and demanders in the commodity market.

This might be rationalized, as imp1 ic-

itly in contracting models, by inefficiencies on the supply side arising from
principal-agent problems.

But firms maximizing the welfare o f the representa-

tive owner-worker would make output respond t o (ex ante) real rates of return
t o labor (the variable input).

Models in which both supply and demand respond

t o real rates of return are termed intertemporal substitution models and have
an appealing basis in microeconomic theory.

Such behavior would make supply

behavior symmetric with respect t o demand behavior, as in:
(56) y:

= S;

+

S:

CRt-Et(pt+l-pt>l + Xyt-i + uzt.

Then the classical dichotomy would hold--that is, output would not be influenced by monetary factors--as shown in the solution for output:

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A p a r t i c u l a r l y i n t e r e s t i n g aspect o f t h i s s o l u t i o n i s t h a t i t i s i n v a r i a n t t o
t h e s p e c i f i c a t i o n o f e x p e c t a t i o n s f o r m a t i o n , so l o n g as t h e y a r e formed i n t h e
same manner by agents when making supply and demand d e c i s i o n s .
On t h e o t h e r hand, d i f f e r e n c e s i n e x p e c t a t i o n f o r m a t i o n a f f e c t i n g s u p p l y
versus demand d e c i s i o n s would generate a real- nominal i n t e r a c t i o n .

However,

some r a t i o n a l e f o r such a seemingly b i z a r r e double- consciousness among agents
would be needed t o p r o v i d e p l a u s i b i l i t y .
possess i n f o r m a t i o n o t h e r s do n o t have.

A l t e r n a t i v e l y , some agents may
Various forms o f h e t e r o g e n e i t y i n t h e

i n f o r m a t i o n s e t s a v a i l a b l e t o agents, St, have been used t o generate d i s p a r a t e i m p l i c a t i o n s f o r t h e o p t i m a l p o l i c y r u l e ; none appear t o have g e n e r a l ity.

I n most analyses, t h e p o l i c y space i s a r b i t r a r i l y r e s t r i c t e d i n ways

unmotivated, except by a n a l y t i c a l t r a c t a b i l i t y .

For example, p o l i c y r u l e s a r e

o f t e n c o n f i n e d t o those p r o v i d i n g t r e n d - s t a t i o n a r i t y t o nominal v a r i a b l e s , o r
r u l i n g o u t c y c l e s i n them.

Also, j u s t as i n t h e Sargent and Wallace model,

seemingly small changes i n t h e i n f o r m a t i o n assumptions can c r e a t e i n t r a c t a b i l i t i e s o r r e v e r s e t h e s i g n s o f o p t i m a l p o l i c y parameters.
Very r e c e n t l y , some deeper a n a l y s i s has been undertaken o f t h e microeconomic underpinnings o f p o t e n t i a l real- nominal i n t e r a c t i o n s i n i n t e r t e m p o r a l
s u b s t i t u t i o n models.

Models i n v o l v i n g a r e a l balance e f f e c t can d e s t r o y t h e

c l a s s i c a l dichotomy i n i n t e r t e m p o r a l s u b s t i t u t i o n models.

A real- nominal

i n t e r a c t i o n a r i s e s i f t h e t r a n s a c t i o n s e r v i c e s o f money a r e considered, and i f
demand and supply a r e d i f f e r e n t i a l l y s e n s i t i v e t o t h e r e a l balance and r e a l
r a t e arguments.

But t r a n s a c t i o n s s e r v i c e s have p l a y e d l i t t l e r o l e i n t h e

business c y c l e models and a r e t h o u g h t t o be e m p i r i c a l l y u n i m p o r t a n t as d e t e r minants o f c y c l i c a l v a r i a t i o n s i n output.

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I f o n l y the o r d i n a r y wealth (nontransactions service) e f f e c t s o f r e a l
money balances on i n d i v i d u a l s ' wealth are considered, t h e p o s i t i v e r e l a t i o n
between r e a l v a r i a b l e s and money surprises disappears.

False ( b u t r a t i o n a l )

i n d i v i d u a l perceptions o f h i g h e r r e a l wealth, due t o a s u r p r i s e increase i n
nominal balances, would have two e f f e c t s on employment and o u t p u t t h a t tend t o
be o f f s e t t i n g .
work.

F i r s t , a d i r e c t wealth e f f e c t increases l e i s u r e and reduces

But, second, t h i s e f f e c t would create an excess demand f o r c r e d i t a t

t h e i n i t i a l (ex ante) r e a l i n t e r e s t r a t e .
i s necessary t o c l e a r the commodity market.

Therefore, a h i g h e r i n t e r e s t r a t e
The e f f e c t o f t h i s h i g h e r r e a l

r a t e on i n d i v i d u a l decisions i s t o increase employment v i a an i n t e r t e m p o r a l
substitution effect.

(Whether t h i s r i s e w i l l mainly take t h e f o r m o f changes

i n nominal r a t e s o r o f expected i n f l a t i o n w i 11 depend, among o t h e r t h i n g s ,
upon the p o l i c y r u l e . )

Barro and King (1984) show t h a t i f t h e r e are no s t o r -

a b l e goods and u t i l i t y i s time- separable, then the wealth and s u b s t i t u t i o n
e f f e c t s must i d e n t i c a l l y cancel, so t h a t the c l a s s i c a l dichotomy i s confirmed.

Relaxing the assumptions t o a l l o w f o r c a p i t a l goods seems t o suggest

r e a l r a t e s w i 11 respond by l e s s i n the face o f i n i t i a l r e a l money balance
changes, implying t h a t o u t p u t would a c t u a l l y f a l l from money increases n o t
perceived by incompletely informed agents.

Hence, conventional s t a b i l i z a t i o n

p o l i c i e s are n o t i n d i c a t e d .

Practical Policy Implications

The r o l e o f monetary pol i c y i n r a t i o n a l expectations models a r i s e s f r o m
c o n s t r a i n t s on p r i v a t e o p t i m i z a t i o n i n the form o f incomplete i n f o r m a t i o n o r
l i m i t s on p r i c e adjustments.

P r o p e r t i e s o f p o l i c y r u l e s t h a t minimize the

w e l f a r e losses associated w i t h these c o n s t r a i n t s are very s e n s i t i v e t o

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particular aspects o f the constraints, about which adequate information is
unavailable.

Also, there are unresolved analytical problems. Thus, the opti-

mal policy rule is unknown, and probably unknowable.
A characteristic o f a pol icy that would be ideal, albeit infeasible, is
that it would prevent any price expectation errors over relevant horizons.
Information lags facing private agents and the length o f contracts would
largely determine the relevant horizons.

Long-term contracts, for example,

argue for policies that achieve longer-term price predictability.

But this

hypothetical ideal is not entirely feasible, because the pol i cymaker faces
information constraints regarding current prices.
The uncertainties and analytical problems in designing an optimal policy, however, seem to roughly correspond to the problem o f designing a means
of minimizing pri ce-level uncertainty.

That minimization o f price-level

uncertainty will be a property o f an optimal policy does not necessarily help
determine an optimal policy, if prices are not contemporaneously observable.
However, the price stabi 1 i ty criterion may help rank concrete policy a1 ternatives and evaluate actual policy performance.

And, to the extent that the

price level can be observed by the monetary authorities without significant
information delays, price stabilization may be considered not only a good
pol icy but a reasonably specific and practical one. Certainly, it is more
specific and practical than vague notions that policy should "lean against the
wind" of undesired output fluctuations. Adequate knowledge does not exist to
differentiate desirable from undesirable fluctuations, or to know how to offset them.

Hence, such a "policy" is too obscure to be a practical, discuss-

able alternative.
The major practical alternatives seem to be constant money growth rules
and predetermined price-level targets.

The former have often been chosen over

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t h e l a t t e r because o f the supposedly superior r e l a t i v e c o n t r o l l a b i l i t y o f
money.

This c o n s i d e r a t i o n cannot be represented f o r m a l l y i n models o f t h e

s o r t e x h i b i t e d i n t h i s paper and can have l i t t l e p r a c t i c a l f o r c e i n the curr e n t environment, i n which money stock measurement, o r even conceptual iz a t i o n ,
i s fraught with d i f f i c u l t i e s .
The problems o f a p o l i c y o f close p r i c e c o n t r o l a r e made l e s s formidable
by r a t i o n a l expectations.

To the e x t e n t t h a t such a p o l i c y i s p r a c t i c e d con-

s i s t e n t l y , p r i v a t e agents w i l l tend t o make d e c i s i o n s t h a t n e i t h e r r e f l e c t
expectations o f , nor serve t o encourage, f l u c t u a t i o n s i n p r i c e s .

And such a

p o l i c y can be p r a c t i c e d w i t h o u t concern f o r any supposed o u t p u t - i n f l a t i o n
trade- off.
Exi s t i n g macroeconomic model s w i t h r a t i o n a l expectations provide 1it t l e
support f o r conventional monetary p o l i c i es.

Output- stabi 1i z a t i o n p o l i c i es are

n o t o n l y extremely d i f f i c u l t t o design, but, t o t h e e x t e n t t h a t they are
" e f f e c t i v e , " tend t o i n t e r f e r e w i t h t h e economy's e f f i c i e n t responses t o
changing p r o d u c t i v e o p p o r t u n i t i e s .

F i n a l l y , p o l i c i e s n a i v e l y d i r e c t e d toward

i n t e r e s t - r a t e s t a b i l i z a t i o n o r s t a t e d i n terms o f " i n t e r e s t r a t e r u l e s " a r e
e i t h e r i n f e a s i b l e o r i n v i t e unknown, and probably undesired, consequences.

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FOOTNOTES
1. The l i n e a r i t y r e s t r i c t i o n , here and elsewhere i n t h i s a n a l y s i s , i m p l i e s a
general r e s t r i c t i o n on u t i l i t y and p r o d u c t i o n f u n c t i o n s and on d i s t u r b a n c e
d i s t r i b u t i o n s . For example, q u a d r a t i c u t i l i t y f u n c t i o n s , l i n e a r p r o d u c t i o n
f u n c t i o n s , and Gaussian d i s t u r b a n c e s ( i n l o g a r i t h m s o f v a r i a b l e s ) may suff i c e . Unless t h i s general r e s t r i c t i o n i s upheld, l i n e a r i t y must be regarded
as a l o c a l approximation.

2. I f t h e r e p r e s e n t a t i o n i s n o t so s e n s i t i v e , then i t i s considered, accordi n g t o a newer t e r m i n o l o g y , a s t r u c t u r a l model w i t h r e s p e c t t o t h a t range of
interventions.

3. Because, i n e x i s t i n g macroeconomic models, t h e d i s t i n c t i o n between base
money and money i s i r r e l e v a n t , t h e term money w i l l be used here w i t h o u t l o s s
o f g e n e r a l i t y . See Hoehn (1984) f o r a d i s c u s s i o n o f some ways i n which t h e
d i s t i n c t i o n between base money ( a c t u a l l y , bank r e s e r v e s ) and money becomes
i m p o r t a n t i n t h e o p t i m i z a t i o n problem under a number o f i n t e r e s t i n g o r h i s t o r i c a l l y r e l e v a n t r e g u l a t o r y and i n s t i t u t i o n a l frameworks.
4. Hoehn (1984) d e a l s a t l e n g t h w i t h t h e d e r i v a t i o n o f money supply f u n c t i o n s
i n e m p i r i c a l l y r e l e v a n t and r e l a t i v e l y d e t a i l e d i n s t i t u t i o n a l s e t t i n g s .
5. A c t u a l l y , t h i s assumption was i m p l i c i t i n t h e r e d u c t i o n o f e q u a t i o n (1) t o
e q u a t i o n ( 2 ) . The requirement t h a t p o l i c y i n v o l v e an a b s o l u t e commitment t o a
t i m e - c o n s i s t e n t r u l e i m p l i e s a r e s t r i c t i o n on t h e p o l i c y space considered.
The u n r e s t r i c t e d p o l i c y space i s (Q X F) X T, where T i s t h e i n f i n i t e dimensional t i m e v e c t o r . The p o i n t (Q,F) c o u l d be s p e c i f i e d f o r each p e r i o d
o f time. Indeed, t h e o p t i m a l p o l i c y would i n v o l v e a s p e c i f i c a t i o n o f (Q,F) as
a f u n c t i o n o f time, where t h a t f u n c t i o n depended on t h e i n i t i a l c o n d i t i o n s .
But, i n subsequent p e r i o d s , t h e i n i t i a l c o n d i t i o n s would change i n a way t h a t
can p a r t l y be p r e d i c t e d on t h e b a s i s o f t h e c u r r e n t s t a t e . Then i t w i 11 be
o p t i m a l t o make a new s p e c i f i c a t i o n o f (Q,F) as a f u n c t i o n o f time. But
agents, under r a t i o n a l e x p e c t a t i o n s , w i l l expect t h i s r e p l a n n i n g and t a k e i t
i n t o account, r e n d e r i n g u n a t t a i n a b l e t h e macroeconomic outcome e n v i s i o n e d when
t h e (Q,F> as a f u n c t i o n o f t i m e was o r i g i n a l l y s p e c i f i e d . Then, t h e u s u a l
methods o f a t t e m p t i n g t o determine t h e o p t i m a l p o l i c y - - o p t i m a l c o n t r o l
theory- - are i n a p p l i c a b l e . Here, a t t e n t i o n i s r e s t r i c t e d t o p o l i c i e s t h a t a r e
t i m e - c o n s i s t e n t and must be chosen p r i o r t o o b s e r v a t i o n o f t h e i n i t i a l c o n d i t i o n s . The a s t u t e r e a d e r w i l l n o t e t h e i r o n y t h a t t h e lack o f a c o n s t r a i n t o n
policymakers t o precommit reduces t h e p o l i c y space o v e r which a n a l y s i s need
search f o r an optimum. This p r o v i d e s another example o f t h e way i n which
r a t i o n a l e x p e c t a t i o n s poses new a n a l y t i c a l issues.
I n c i d e n t a l l y , t h e o p t i m a l values o f q and F c o u l d v a r y o v e r t i m e w i t h o u t
i m p l y i n g t i m e i n c o n s i s t e n c y , i f t h e model i s m o d i f i e d i n c e r t a i n ways. F o r
example, i f t h e s t r u c t u r a l parameters were known, n o n s t o c h a s t i c f u n c t i o n s o f
time, then a precommi tment t o f i x e d paths f o r q t and F t would g e n e r a l l y be
a p p r o p r i a t e . If parameters were s u b j e c t t o s t o c h a s t i c v a r i a t i o n , and agents
l a c k e d p e r f e c t f o r e s i g h t r e g a r d i n g such v a r i a t i o n , then t h e o p t i m a l p o l i c y
would i n v o l v e precommitment t o a r u l e f o r changing q t and F, i n response
t o a v a i l a b l e i n f o r m a t i o n . F i n a l l y , i f s t r u c t u r a l parameters a r e unknown, b u t

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agents, i n c l u d i n g t h e p o l i cymaker, a r e capable o f a d a p t i v e l e a r n i n g b e h a v i o r ,
then t h e o p t i m a l p o l i c y again i n v o l v e s a precommitment t o a r u l e f o r changing
q t and F,.
(Some p o t e n t i a l problems o f model nonconvergence a r e here
ignored.) Hence, t h e i d e n t i f i c a t i o n o f t i m e c o n s i s t e n c y o f a p o l i c y w i t h t i m e
constancy of q and F i s s p e c i f i c t o t h e c l a s s of models examined i n t h i s
paper, whi ch impose s t r u c t u r a l parameter constancy and parameter c e r t a i n t y for
convenience. I n g e n e r a l , time constancy o f q and F i s s u f f i c i e n t , b u t n o t
necessary, f o r t i m e consi stency o f p o l i c y .
6. B o o t s t r a p components i n pt-,, p t -,,... a r e o m i t t e d i n (26). This
omission i s i n c o n s e q u e n t i a l f o r study o f t h e b e h a v i o r o f y o r t h e i r r e l e v a n c e
of F. Such b o o t s t r a p s would m a t t e r f o r p r i c e b e h a v i o r , as discussed l a t e r i n
t h i s section.

are

7. I n c i d e n t a l l y , preannounced changes i n f i s c a l p o l i c y
e f f e c t i v e i n nonIS-LM models, such as t h e i n t e r t e m p o r a l s u b s t i t u t i o n and r e a l business c y c l e
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 incomplete i n f o r m a t i o n . Indeed, i n
these models, changes i n taxes o r spending programs may have even l a r g e r
e f f e c t s on p r i v a t e agents i f t h e l a t t e r have t i m e t o p l a n t h e i r responses t o
t h e changes i n i n c e n t i v e s i m p l i e d by f i s c a l p o l i c y changes.

8. Sometimes, s o l u t i o n s under (32) a r e taken t o be t h e l i m i t i n g cases of
expressions f o r t h e s o l u t i o n s under (201, as q approaches i n f i n i t y . The
mathematical concept o f a 1i m i t i s a p p r o p r i a t e t o use when t h e v a l u e o f a
f u n c t i o n i s u n d e f i n e d a t some p o i n t i n t h e range and values a r b i t r a r i l y c l o s e
t o t h a t p o i n t a r e o f i n t e r e s t . U n f o r t u n a t e l y , i t s use cannot be f o r m a l l y
j u s t i f i e d as r e p r e s e n t i n g t h e v a l u e o f t h e f u n c t i o n a t t h a t v e r y p o i n t i n t h e
range, i f t h e v a l u e i s undefined t h e r e . I n t h e case a t hand, i f q were
a c t u a l l y i n f i n i t e , r a t h e r than a r b i t r a r i l y l a r g e i n magnitude, then a money
supply f u n c t i o n such as (20) would n o t e x i s t . Hence, any s o l u t i o n s d e r i v e d
u s i n g (20) would be i r r e l e v a n t , and t h e l i m i t s o f such expressions as q
approaches i n f i n i t y cannot be regarded as outcomes under an " i n t e r e s t r a t e
r u l e . " As noted, o t h e r a n a l y s t s have n o t taken t h i s view, and indeed do
analyze an " i n t e r e s t r a t e r u l e " as a l i m i t i n g case o f a money supply r u l e , as
q approaches i n f i n i t y . This d i f f e r e n c e o f view leads t o some r a t h e r s u b t l e
d i f f e r e n c e s o f i n t e r p r e t a t i o n . I n p a r t i c u l a r , McCallum i n t e r p r e t s h i s r e s u l t s
(1981) as 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 , b u t o n l y as l i m i t i n g cases of
money supply r u l e s . F u r t h e r , these money supply r u l e s a r e n o t unique
(McCallum C19861), hence t h e a s s o c i a t e d i n t e r e s t r a t e r u l e i s n o t an adequate
s p e c i f i c a t i o n o f p o l i c y . My i n t e r p r e t a t i o n i s t h a t " i n t e r e s t r a t e r u l e s " a r e
n o t a c t u a l l y p o l i c i e s , b u t merely r e p r e s e n t outcomes f o r t h e i n t e r e s t r a t e
t h a t o c c u r under (nonunique) money supply r u l e s , where o n l y t h e l a t t e r a r e
a d m i s s i b l e p o l i c i e s . These d i f f e r e n c e s i n i n t e r p r e t a t i o n a r e p r o b a b l y w i t h o u t
p r a c t i c a l s i g n i f i c a n c e f o r two reasons. F i r s t , e i t h e r i n t e r p r e t a t i o n recommends p o l i c y r u l e s t h a t i n c l u d e m t as an argument, t h a t i s , money s u p p l y
r u l e s . Second, an o p t i m a l p o l i c y would g e n e r a l l y have a f i n i t e $ non- zero
v a l u e o f q even i f determinacy were n o t a t i s s u e . I n o t h e r words, p o l i c i e s
t h a t c o m p l e t e l y predetermine money o r i n t e r e s t rates- - often d e s c r i b e d as
p o l i c i e s u s i n g money o r i n t e r e s t r a t e " i n s t r u m e n t s u - - g e n e r a l l y a r e suboptimal
as l o n g as t h e p o l icymaker can observe b o t h contemporaneously, even a s i d e from
problems o f determinacy.

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9. To a v o i d inconsistency, i n the f o r m o f two d i f f e r e n t values f o r a,,,
go and g 1 must obey: go=sodo(dlgl+d2-X>+dl ( 1-X)(do-sold--1
I-'.
This r e s t r i c t i o n e s s e n t i a l l y ensures e q u a l i t y between t h e average r e a l r a t e o f
i n t e r e s t and t h e n a t u r a l r e a l r a t e o f i n t e r e s t , determined i n t h e commodity
market.
10. Goodfriend (1985, 1986) considers a p o l i c y space i n which f, i s n o t
constrained i n t h i s way.
11. " Rational expectations models" i n which p r i c e s are exogenous are conceivable. Expectations would then be e f f e c t i v e l y exogenous as we1 1, hence
anchored. Other examples o f r a t i o n a l expectations models w i t h o u t the i n d e t e r minacy problem may o r may n o t e x i s t , b u t a r e unl i k e l y t o have t h e n a t u r a l r a t e
p r o p e r t y o r o t h e r acceptable microeonomic i m p l i c a t i o n s .
12. Nominal indeterminacy under an " i n t e r e s t r a t e r u l e " could a l s o be shown
t o be a f e a t u r e o f t h i s model.
13. The m o d i f i e r " i n a p a r t i c u l a r sense" i s included because f i r m l w o r k e r
combinations which determined o u t p u t i n a P a r e t o - e f f i c i e n t manner, s u b j e c t t o
the c o n s t r a i n t t h a t they could n o t observe St, would n o t behave i n q u i t e t h e
same manner as i n the nominal c o n t r a c t i n g models, which i m p l i c i t l y assume
Pareto-inefficiency.

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