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Working Paper 871 6
MONETARY POLICY UNDER RATIONAL EXPECTATIONS
WITH MULTIPERIOD WAGE STICKINESS AND
AN ECONOMY-WIDE CREDIT MARKET
by James G. Hoehn
James G. Hoehn i s an economist a t t h e Federal
Reserve Bank o f Cleveland. The a u t h o r g r a t e f u l l y
acknowledges t h e comments o f A l a n C. Stockman,
John L. Scadding, and W i l l i a m C. Gavin on d r a f t s .
Working papers o f t h e Federal Reserve Bank
o f Cleveland are p r e l i m i n a r y m a t e r i a l s
c i r c u l a t e d t o s t i m u l a t e d i s c u s s i o n and
c r i t i c a l comment. The views s t a t e d h e r e i n
a r e t h o s e o f t h e a u t h o r and n o t n e c e s s a r i l y
t h o s e o f t h e Federal Reserve Bank o f
C l e v e l a n d o r o f t h e Board o f Governors o f
t h e F e d e r a l Reserve System.
December 1987
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ABSTRACT
The e f f e c t s o f money s u p p l y responses t o b o t h t h e c u r r e n t i n t e r e s t r a t e and
t h e lagged s t a t e o f t h e economy a r e analyzed i n a dynamic IS-LM model w i t h
m u l t i p e r i o d wage s t i c k i n e s s .
A l l o w i n g p r i v a t e agents t o use t h e c u r r e n t
i n t e r e s t r a t e i n f o r m i n g e x p e c t a t i o n s c r e a t e s some p o t e n t i a l l y c o u n t e r i n t u i t i v e e f f e c t s o f shocks, b u t leaves p o l i c y i m p l i c a t i o n s l a r g e l y
unaffected.
A c o n f l i c t between f i r m - l e v e l and aggregate- level o u t p u t
s t a b i l i z a t i o n i s found t o be i n h e r e n t i n t h e m u l t i p e r i o d wage c o n t r a c t i n g
scheme.
P o l i c y r u l e s t h a t s t a b i l i z e o u t p u t o r serve o t h e r o b j e c t i v e s a r e
e x p l i c i t l y derived.
Because o f supply shocks, p r o c y c l i c a l money growth may be
r e q u i r e d t o s t a b i l i z e f i r m o u t p u t , p r i c e s , r e a l wages, o r o u t p u t r e l a t i v e t o
i t s full- information level.
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MONETARY POLICY UNDER RATIONAL EXPECTATIONS
WITH MULTIPERIOD WAGE STICKINESS AND AN ECONOMY-WIDE CREDIT MARKET
I. I n t r o d u c t i o n
T h i s a r t i c l e r e c o n s i d e r s t h e r o l e o f m o n e t a r y p o l i c y i n a Keynesian model
t h a t i n c o r p o r a t e s r a t i o n a l e x p e c t a t i o n s and d i s p l a y s t h e n a t u r a l r a t e
property.
I m p o r t a n t f e a t u r e s a r e m u l t i p e r i o d wage s t i c k i n e s s and t h e
a v a i l a b i l i t y , t o b o t h p r i v a t e a g e n t s and t h e p o l i c y m a k e r , o f c u r r e n t
i n f o r m a t i o n f r o m o b s e r v a t i o n s o f t h e nominal i n t e r e s t r a t e .
The model i s
e s s e n t i a l l y a l o g - l i n e a r i z e d v e r s i o n o f t h a t o f F i s c h e r (1977), augmented b y
an ( i m p l i c i t ) economy-wide c r e d i t m a r k e t .
As i n F i s c h e r ' s i n f l u e n t i a l p a p e r ,
f e e d b a c k f r o m t h e lagged s t a t e o f t h e economy i s c o n s e q u e n t i a l f o r o u t p u t
b e h a v i o r because o f t h e c o n j u n c t i o n o f m u l t i p e r i o d wage s t i c k i n e s s and
a u t o c o r r e l a t i o n o f shocks.
A t t h e same t i m e , because o f i n c o m p l e t e c u r r e n t
i n f o r m a t i o n , p o l i c y responses t o t h e c u r r e n t i n t e r e s t r a t e a r e a l s o o f
consequence f o r o u t p u t , as i n t h e famous a n a l y s i s o f P o o l e (1970).
The a n a l y s i s o f t h i s a r t i c l e i s o f i n t e r e s t f r o m t h r e e d i f f e r e n t
standpoints.
F i r s t , i t addresses t h e a n a l y t i c a l p r o b l e m o f c u r r e n t
i n f o r m a t i o n s e t s under r a t i o n a l e x p e c t a t i o n s .
The a n a l y s i s f i n d s t h a t p r i v a t e
use o f c u r r e n t i n f o r m a t i o n f r o m t h e i n t e r e s t r a t e t o u p d a t e e x p e c t a t i o n s does
n o t n e c e s s a r i l y have t h e d r a m a t i c i m p l i c a t i o n s f o r p o l i c y t h a t Dotsey and K i n g
(1983,
1986) o b t a i n i n c o m p l e t e l y f l e x i b l e - p r i c e models.
I n particular, while
p o l i c y responses t o t h e i n t e r e s t r a t e a r e i r r e l e v a n t f o r o u t p u t i n
f l e x i b l e - p r i c e models, such responses a r e p o t e n t i a l l y i m p o r t a n t i n f l u e n c e s on
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t h e b e h a v i o r o f o u t p u t i n s t i c k y - w a g e models.
Second, t h i s a r t i c l e e x t e n d s t h e c l a s s i c a n a l y s i s o f o p t i m a l p o l i c y u n d e r
u n c e r t a i n t y , i n a u g u r a t e d by Poole, t o an expanded IS-LM model i n w h i c h p r i c e s
a r e n o t f i x e d , t h e n a t u r a l r a t e p r o p o s i t i o n h o l d s , and s u p p l y shocks o c c u r .
S u r p r i s i n g l y , p o l i c y r u l e s t o s t a b i l i z e o u t p u t , o r t o serve o t h e r o b j e c t i v e s ,
i n models w i t h p l a u s i b l e s p e c i f i c a t i o n s of commodity s u p p l y b e h a v i o r
a p p a r e n t l y have n o t h e r e t o f o r e been d e r i v e d .
A t h i r d r e a s o n t h i s a r t i c l e i s of i n t e r e s t i s t h a t i t t r e a t s t h e p r o b l e m
o f i n c o n s i s t e n c y between t h r e e p o t e n t i a l p o l i c y o b j e c t i v e s :
(1) s t a b i l i z a t i o n
o f t o t a l o u t p u t , ( 2 ) s t a b i l i z a t i o n of t y p i c a l i n d i v i d u a l f i r m o u t p u t , a n d ( 3 )
s t a b i l i z a t i o n o f o u t p u t around i t s p e r f e c t f o r e s i g h t o r f u l l - i n f o r m a t i o n
level.
The i n c o n s i s t e n c y between t h e f i r s t and second o b j e c t i v e s has n o t been
noted before,
and i s i n h e r e n t i n t h e s t a g g e r e d wage c o n t r a c t i n g s e t u p .
The
c o n f l i c t between t h e f i r s t and t h i r d o b j e c t i v e s has been n o t e d b e f o r e , b u t i t s
i m p l i c a t i o n s f o r p o l i c y r u l e s i n s t i c k y - w a g e models have n o t been d e v e l o p e d as
f u l l y as i n t h i s a r t i c l e .
The a p p r o p r i a t e response o f p o l i c y t o a commodity
s u p p l y shock w i l l depend o n w h i c h o b j e c t i v e i s embraced.
If the t h i r d
o b j e c t i v e i s chosen, money s h o u l d be i n c r e a s e d i n response t o s u p p l y s h o c k s ,
and t h e same may a l s o be t r u e under t h e second o b j e c t i v e .
Numerical examples
show t h a t t h e b e h a v i o r o f money a p p r o p r i a t e t o e i t h e r o f t h e s e o b j e c t i v e s can
e a s i l y i n v o l v e a p o s i t i v e c o r r e l a t i o n between money and o u t p u t .
I n s t u d y i n g t h e e f f e c t s o f p r i v a t e a g e n t s ' use o f c u r r e n t i n f o r m a t i o n f r o m
t h e i n t e r e s t r a t e , t h i s a r t i c l e j o i n s B a r r o ( 1 9 8 0 ) , K i n g (1983), C a n z o n e r i ,
Henderson, and R o g o f f (1983), and Dotsey and K i n g (1983, 1986).
These p r i o r
s t u d i e s make s i m p l i f y i n g assumptions t o r e n d e r t h e a n a l y s i s t r a c t a b l e ,
i n c l u d i n g , most i m p o r t a n t l y , l i m i t s o n p o l i c y a l t e r n a t i v e s , a l l o w a n c e f o r o n l y
a f e w economic d i s t u r b a n c e s , or b o t h .
Even t h e s e s i m p l i f i c a t i o n s have n o t
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always p e r m i t t e d e x p l i c i t s o l u t i o n s t o t h e model.
The p r e s e n t a r t i c l e o f f e r s
closed - form s o l u t i o n s w i t h o u t imposing any a r b i t r a r y r e s t r i c t i o n s on monetary
p o l i c y , e n a b l i n g d e r i v a t i o n and q u a l i t a t i v e c h a r a c t e r i z a t i o n o f p o l i c i e s t h a t
s a t i s f y various objectives.
Also, t h e a n a l y s i s here i n c l u d e s t h r e e d i s t i n c t
aggregate d i s t u r b a n c e s , f o r commodity supply, commodity demand, and money
demand.
The e x p e c t a t i o n a l e f f e c t s o f c u r r e n t o b s e r v a t i o n o f t h e i n t e r e s t r a t e
c r e a t e t h e p o t e n t i a l f o r c o u n t e r i n t u i t i v e responses o f t h e economy t o some
disturbances.
For example, a p o s i t i v e shock t o t h e money s u p p l y may reduce
o u t p u t and p r i c e s .
However, i n c o n t r a s t t o t h e o t h e r s t u d i e s , t h i s a r t i c l e
f i n d s t h a t c u r r e n t i n f o r m a t i o n does n o t appear t o change t h e b e h a v i o r o f
o u t p u t i n t h e model i n a way t h a t i s l i k e l y t o be q u a n t i t a t i v e l y i m p o r t a n t ,
nor does i t seem t o have v e r y i m p o r t a n t o r 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
monetary p o l i c y .
The c o n c l u s i o n s and i n t u i t i o n s o f Poole and F i s c h e r
c o n c e r n i n g o u t p u t - s t a b i l i z i n g p o l i c y -n e e d n o t be i m p o r t a n t l y a f f e c t e d b y t h e
presence o f c u r r e n t i n f o r m a t i o n .
11.
Review o f Recent L i t e r a t u r e
T h i s s e c t i o n presents a s t y l i z e d p e r s p e c t i v e o f t h e l i t e r a t u r e on p o l i c y
effectiveness,
r a t i o n a l e x p e c t a t i o n s , c u r r e n t i n f o r m a t i o n , and sticky- wage
models as t h e y r e l a t e t o t h e i s s u e s o f t h i s a r t i c l e .
A major problem i n r a t i o n a l e x p e c t a t i o n s macroeconomic models i s t h a t , i f
p r i v a t e agents have p a r t i a l i n f o r m a t i o n about t h e c u r r e n t s t a t e o f t h e
economy, then unambiguous q u a l i t a t i v e c o n c l u s i o n s about t h e e f f e c t s o f
monetary p o l i c y a r e o f t e n d i f f i c u l t t o deduce.
Indeed, unless t h e s t r u c t u r e
o f t h e model i s s e v e r e l y r e s t r i c t e d , i t i s o f t e n d i f f i c u l t t o d e r i v e
a n a l y t i c a l s o l u t i o n s t o t h e model.
From t h e s t a n d p o i n t o f monetary p o l i c y
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a n a l y s i s , t h e problem of contemporaneous i n f o r m a t i o n appears e s p e c i a l l y
c r i t i c a l f o r whether o r how money s u p p l y responses t o t h e nominal i n t e r e s t
r a t e a f f e c t t h e b e h a v i o r o f o u t p u t and o t h e r v a r i a b l e s o f concern.
T r a d i t i o n a l l y , as i n Poole (1970), t h e monetary a u t h o r i t y has been assumed
t o use t h e c u r r e n t i n t e r e s t r a t e as w e l l as delayed i n f o r m a t i o n on t h e s t a t e
o f t h e economy i n s e t t i n g t h e money supply.
An o p t i m a l p o l i c y g e n e r a l l y t a k e s
the form
mt=qRt+pSt-,,
where m, i s t h e money stock, R, i s t h e c u r r e n t nominal i n t e r e s t r a t e , and
St-,
i s t h e lagged s t a t e o f t h e economy.
q and p a r e p o l i c y response
c o e f f i c i e n t s r e l a t i n g t h e money s u p p l y t o c u r r e n t and lagged i n f o r m a t i o n .
In
P o o l e ' s a n a l y s i s , t h e relevance o f q- - the response o f p o l i c y t o t h e i n t e r e s t
r a t e - - a r i s e s from t h e p o l i c y m a k e r ' s incomplete i n f o r m a t i o n c o n c e r n i n g t h e
s t a t e o f t h e economy, w h i l e t h e r e l e v a n c e o f p--the
response of p o l i c y t o
t h e lagged s t a t e o f t h e economy--arises from a u t o c o r r e l a t i o n i n economic
shocks, t o g e t h e r w i t h i m p l i c i t , p e r f e c t l y e l a s t i c supply b e h a v i o r ( f i x e d
prices).
P o o l e ' s p a r t i c u l a r c o n t r i b u t i o n was t o e x p l i c i t l y r e l a t e s t r u c t u r a l
parameter and s t o c h a s t i c assumptions t o t h e c h a r a c t e r i s t i c s o f a s t a b i l i z a t i o n
pol i c y .
Sargent and Wallace (1975), i n a model t h a t i n t r o d u c e d an e x p l i c i t s u p p l y
s e c t o r , brought t h e r e l e v a n c e of g i n t o q u e s t i o n , b u t preserved t h e
r e l e v a n c e o f q.'
used S t - , ,
They imposed r a t i o n a l e x p e c t a t i o n s on p r i v a t e agents, who
t h e lagged s t a t e , b u t n o t R,,
t h e c u r r e n t i n t e r e s t r a t e , as
i n f o r m a t i o n i n forming expectations o f f u t u r e p r i c e s .
R, b u t d i d n o t use i t t o update i n f l a t i o n e x p e c t a t i o n s .
P r i v a t e agents knew
The r e l e v a n c e o f q
i n t h e model o f Sargent and Wallace depended on t h i s v i o l a t i o n o f Muthian
r a t i o n a l expectations:
i n t e r e s t rate.
agents wasted a v a i l a b l e i n f o r m a t i o n c o n t a i n e d i n t h e
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Subsequently, Barro (1976) demonstrated that policy effectiveness in
then-extant macromodels with the natural rate property generally depended on
superior information of the policymaker relative to private agents.
It seemed
that, unless the policymaker knew more than private agents, both q and p
would be irrelevant for the behavior of output.
Because private agents do
actually have equal access to information on current interest rates and to
virtually the same information on the lagged state of the economy, the
relevance of monetary policy appeared dubious.
Subsequently, King (1983) and Dotsey and King (1983, 1986) found that, via
a mechanism involving private, heterogeneous information sets that included
local prices as well as the global interest rate, policy was an effective
determinant of output behavior after all .'
But only money-supply responses
to the past state of the economy, p, were effective; q was irrelevant for
real variables.
Dotsey and King argued that rational expectations
dramatically reverses the conclusions of Poole on the relevance of q , and also
reverses the conclusions of Sargent and Wallace on the irrelevance of p.
However, p is relevant via a very different mechanism than envisioned by
Poole. Also, the heterogeneous information mechanism creates analytical
intractabilities and ambiguities that prevent derivation of explicit solutions
for the model or qualitative knowledge of how various values of p would
influence economic outcomes, making it difficult to design appropriate
state-dependent money rules. Certainly, Dotsey and King's results provided no
basis for conventional countercyclical policies, and indeed were used to
suggest that rational expectations destroys the case for such policies.
However, these provocative arguments of Sargent and Wallace, Barro, King,
and Dotsey concerning policy were illustrated in flexible wage and price
models, in which output deviations from optimal levels were due strictly to
incomplete information.
Fischer (1977) showed that the relevance of p for
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o u t p u t behavior f o l l o w s f r o m t h e assumption t h a t wages a r e s t i c k y f o r l o n g e r
than t h e g l o b a l i n f o r m a t i o n l a g , o r f o r a t l e a s t two p e r i o d s .
However,
l a c k i n g a c r e d i t market, t h e i n t e r e s t r a t e d i d n o t e n t e r F i s c h e r ' s model, so
t h a t t h e relevance o f q c o u l d n o t be assessed.
Canzoneri, Henderson, and Rogoff (1983) added f u r t h e r a m b i g u i t y t o t h e
i s s u e o f whether q and g were r e l e v a n t .
Given t h a t wages a r e s t i c k y f o r one
p e r i o d and t h a t agents making demand d e c i s i o n s ( " i n v e s t o r s " ) use a l l t h e
i n f o r m a t i o n contained i n t h e c u r r e n t i n t e r e s t r a t e , Canzoneri, e t a l . d e r i v e
two s e t s o f values f o r (q,p)
t h a t r e s u l t i n the minimum o u t p u t v a r i a n c e :
one o f these has (q=O, g#O), w h i l e t h e o t h e r has (q#O, ~ = 0 > . ~
I n t h e n e x t s e c t i o n , a model i s presented w i t h two- period wage c o n t r a c t s
t h a t makes b o t h q and
r e l e v a n t , and i n much t h e same way as i n t h e
a n a l y s i s o f Poole, d e s p i t e t h e a d d i t i o n o f r a t i o n a l e x p e c t a t i o n s , i n f o r m a t i o n
sets t h a t include the c u r r e n t i n t e r e s t r a t e , a supply sector t h a t d i s p l a y s the
n a t u r a l r a t e p r o p e r t y , and supply shocks.
I n e f f e c t , t h e a n a l y s i s invokes
F i s c h e r ' s m u l t i p e r i o d wage s t i c k i n e s s t o rescue P o o l e ' s p o l i c y f r o m
v u l n e r a b i l i t y t o t h e i n e f f e c t i v e n e s s p r o p o s i t i o n s a r i s i n g f r o m r e c e n t work i n
expectations formation.
Although F i s c h e r h i m s e l f has a l r e a d y done so w i t h
r e g a r d t o p, t h e more r e c e n t arguments concerning contemporaneous
i n f o r m a t i o n r e q u i r e t h a t t h e relevance o f q be reexamined.
111.
A P r o t o t y p e Keynesian Model
The p a r t i c u l a r model was chosen t o implement t h r e e major c h a r a c t e r i s t i c s
as s t r a i g h t f o r w a r d l y as p o s s i b l e .
F i r s t , the model generates a s u b s t a n t i a l
r o l e f o r c o u n t e r c y c l i c a l monetary p o l i c y v i a a Keynesian m u l t i p e r i o d
sticky- wage mechanism.
Second, i t i n c o r p o r a t e s an e q u i l i b r i u m wage dynamic
and does n o t r e l y on money i l l u s i o n , p e r s i s t e n t e x p e c t a t i o n a l b i a s e s , o r
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ignored i n f o r m a t i o n among p r i v a t e agents.
natural r a t e property:
As a r e s u l t , i t d i s p l a y s t h e
mean values o f r e a l v a r i a b l e s a r e i n v a r i a n t w i t h
r e s p e c t t o monetary f a c t o r s , i n c l u d i ng monetary p o l i c y .
T h i r d , t h e model
possesses an i m p l i c i t economy-wide c r e d i t market t h a t c l e a r s s i m u l t a n e o u s l y
w i t h t h e commodity market, as i n t h e IS-LM model, i n t r o d u c i n g t h e i n t e r e s t
r a t e as a v a r i a b l e t h a t i s j o i n t l y endogenous w i t h o t h e r r e a l and nominal
variables.
T h i s f a c i l i t a t e s a n a l y s i s of monetary p o l i c i e s t h a t c o n d i t i o n t h e
money supply on t h e i n t e r e s t r a t e , as i n P o o l e ' s (1970) famous a n a l y s i s .
These t h r e e f e a t u r e s a r e implemented by combining t h e dynamic IS-LM model w i t h
t h e supply s e c t o r o f F i s c h e r (1977) and by imposing r a t i o n a l e x p e c t a t i o n s .
The o u t p u t s u p p l y s e c t o r i s a s t r a i g h t f o r w a r d l o g - l i n e a r a d a p t a t i o n o f
Fischer's.
Firms d e c i d e upon o u t p u t a f t e r o b s e r v i n g c u r r e n t p r i c e s and t a k i n g
i n t o account predetermined wages, s e t e i t h e r one o r two p e r i o d s ago.
To f i x
ideas, and w i t h o u t l o s s o f g e n e r a l i t y , t h e economy can be seen as composed o f
two groups o f f i r m s .
t h e end o f p e r i o d t-1,
The f i r s t group c o n s i s t s o f f i r m s s i g n i n g c o n t r a c t s a t
and t h e second group c o n s i s t s o f f i r m s t h a t signed
c o n t r a c t s a t t h e end o f p e r i o d t- 2.
Any p a r t i c u l a r f i r m w i l l be i n group one
i n t h e f i r s t y e a r o f i t s c o n t r a c t and i n group two i n t h e second year.
For
example, t h e c o n t r a c t signed by a f i r m a t t h e end of p e r i o d t - 2 p r o v i d e d f o r
wages i n t-1,
when i t was a group one f i r m , and f o r wages i n t, when i t i s a
group two f i r m .
L i k e w i s e , t h e c o n t r a c t signed a t t h e end o f p e r i o d t - 1
p r o v i d e s f o r group one wages i n p e r i o d t and f o r group two wages i n p e r i o d
t+l. Wages a r e always s e t t o match t h e expected p r i c e l e v e l , thus a t t e m p t i n g
t o s t a b i l i z e r e a l wages i n t h e face o f u n c e r t a i n i n f l a t i o n .
This scheme c a l l s
f o r wages t o change f o r e v e r y f i r m i n each p e r i o d , b u t o n l y t h e most r e c e n t l y
c o n t r a c t i n g f i r m s , group one, have wages t h a t r e f l e c t knowledge o f t h e s t a t e
o f the economy i n p e r i o d t-1.
Furthermore, no wages a r e a d j u s t e d i n immediate
response t o contemporaneous shocks, b u t o n l y a f t e r a new c o n t r a c t i s signed.
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-8-
Equations ( 2 ) s t a t e these assumptions f o r m a l l y .
w ~ , ~ - ~ = E and
~ - wZt=Et-,pt;
~ ~ ~ - ~
WI
t = E t - l p t and w , , t + l = E , - l p t + l ;
where w i t i s t h e l o g o f t h e nominal wage a t group i f i r m s
and p t i s t h e l o g of t h e p r i c e l e v e l .
Firms a r e assumed t o have a c o n s t a n t , p o s i t i v e e l a s t i c i t y o f s u p p l y w i t h
r e s p e c t t o r e a l wages.
Yi
t
Formally, f i r m s u p p l y f u n c t i o n s are:
= a+fi(pt-wit>+ut,
i=1,2,
(3)
R>O,
where y i i s o u t p u t o f group i, and u t i s an economy-wide supply shock.
The p r e s e r v a t i o n o f l i n e a r i t y o f t h e model i s achieved by a p p r o x i m a t i n g t h e
l o g o f t h e sum o f group o u t p u t s as:
yt=ko+(l/2>(ylt+yZt>.
(4)
F i n a l l y , a s u p p l y f u n c t i o n o r Phi 11i p s Curve t h a t r e l a t e s o u t p u t t o
p r i c e - l e v e l s u r p r i s e s i s found f r o m combining ( 2 > , (3), and ( 4 ) :
y t = (ko+a)+(l/2>fi~:,l(pt-E~-,p~)+~~.
(5)
The key f e a t u r e o f t h e supply f u n c t i o n ( 4 ) i s t h a t o u t p u t responds equal l y t o
d e v i a t i o n s o f t h e p r i c e l e v e l f r o m i t s e x p e c t a t i o n formed b o t h one and two
p e r i o d s ago, when c o n t r a c t s c u r r e n t l y i n e f f e c t were signed.
The commodity demand ( o r I S ) e q u a t i o n , ( 6 ) , r e f l e c t s t h e assumption t h a t
t h e commodity market and t h e c r e d i t market c l e a r s i m u l t a n e o u s l y i n t h e sense
t h a t agents know t h e economy-wide i n t e r e s t r a t e and t h e c u r r e n t p r i c e i n t h e i r
l o c a l market when making demand d e c i s i o n s .
Demand depends on t h e (ex a n t e )
real rate o f interest.
yt
=
bo-b~CRt-(E:-~pt+~-pt>l+~t
(6)
where
E : - ~ P ~ += ~ E C ~ Inti,
~ + ~
n,
and S
=
observable s t a t e of economy a t t i m e t
=
{Rt;St-I),
=
state vector (given a s p e c i f i c i d e n t i t y i n the next section).
(7)
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-9-
The nominal i n t e r e s t r a t e , R t , i s measured as t h e n a t u r a l l o g a r i t h m o f u n i t y
p l u s t h e coupon r a t e o f r e t u r n .
Ei-lpt,l,
The f u t u r e p r i c e e x p e c t a t i o n ,
i s c o n d i t i o n e d on t h e observed s t a t e o f t h e economy,
at,
an i n f o r m a t i o n s e t t h a t i n c l u d e s t h e c u r r e n t economy-wide i n t e r e s t r a t e , R,,
and t h e lagged s t a t e v e c t o r , S t - l .
6
Et-,p,+,
can d i f f e r f r o m
E , - ~ P ~because
+ ~
agents a r e p e r m i t t e d t o know t h e nominal i n t e r e s t r a t e t o
update t h e i r i n f l a t i o n e x p e c t a t i o n s .
x t i s a s t o c h a s t i c demand shock.
Money demand i s conventional .
mt-pt
=
a. - a l R t + a z y t + v t ,
(8)
where v, i s a random d i s t u r b a n c e .
Monetary p o l i c y i s c h a r a c t e r i z e d by a d e c i s i o n r u l e f o r t h e q u a n t i t y o f
money as a f u n c t i o n o f t h e observed s t a t e of t h e economy, o r a money s u p p l y
rule.
The s i m p l e s t adequate f o r m o f t h i s r u l e i s
mt=qRt+po+plut-l+prvt-1+p3~t-l.
(9)
The shocks u, v, and x a r e f i r s t - o r d e r a u t o r e g r e s s i v e processes w i t h
innovations
E,
q,
and X, r e s p e c t i v e l y .
A l l shocks a r e m u t u a l l y
uncorrelated.
/L
ut=plut-l+ct,
O<pl<l,
E,
N(O,u:)
vt=prvt-I+qt,
O<pr<l,
qt
N(O,oG)
Xt=p3Xt- l + X t ,
O<p3<1,
A~N(o,U:)
E(E,Q,)=E(~,X, )=E(E ,At )=O
E ( c t r t ~ l ) = E ( ~ t q , ~ l ) = E ( X t X t t , ) =i >OO,.
IV.
A n a l y s i s o f t h e Model
A l t h o u g h t h e model i s e x t r e m e l y simple, t h e c o n j u n c t i o n of contemporaneous
i n f o r m a t i o n and r a t i o n a l e x p e c t a t i o n s makes t h e s o l u t i o n r a t h e r complex.
Many
uncouth d e t a i l s o f t h e s o l u t i o n method a r e banished t o an appendix, b u t some
fundamental p o i n t s a r e discussed i n t h e t e x t o f t h i s s e c t i o n .
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-1 0-
The Endogenous V a r i a b l e s and t h e S t a t e Vector
A.
The endogenous v a r i a b l e s o f e s s e n t i a l concern a r e t o t a l o u t p u t , group 1
and group 2 o u t p u t , t h e p r i c e l e v e l , t h e money stock, and t h e i n t e r e s t r a t e
( y t , y l t , y r t , p,,
m,,
and R,).
U s i n g t h e method o f undetermined
c o e f f i c i e n t s , each endogenous v a r i a b l e w i l l be assigned i t s own t r i a l e q u a t i o n
i n terms o f t h e s t a t e v e c t o r .
From these t r i a l equations a r e d e r i v e d i m p l i e d
l i n e a r reduced- form equations f o r t h e endogenous e x p e c t a t i o n s v a r i a b l e s ,
E t - l p t , E t - r p t , and E L - l p t + l .
The s t a t e v e c t o r i s determined as f o l l o w s .
F i r s t , obvious c a n d i d a t e s f o r
i n c l u s i o n a r e a l l predetermined and exogenous v a r i a b l e s e x p l i c i t l y a p p e a r i n g
i n t h e s t r u c t u r a l equations, i n c l u d i n g t h e a u t o r e g r e s s i v e processes ( 9 ) .
These a r e t h e lagged l e v e l o f s t r u c t u r a l d i s t u r b a n c e s , u , - , ,
x t - , , and t h e c u r r e n t i n n o v a t i o n s , d , , q,, and A , .
v,-,,
and
A u n i t "variable"
i s a l s o i n c l u d e d among t h e s t a t e v e c t o r t o a l l o w f o r i n t e r c e p t s i n t h e
reduced- form e q u a t i o n s .
(The c o n s t a n t s a r e o f no i n t e r e s t f o r c y c l i c a l
a n a l y s i s and w i l l n o t appear beyond t h i s s e c t i o n . )
predetermined p r i o r e x p e c t a t i o n s , E t-,ut-,,
Et-,x
t-l,
Et-2vt-l,
The appearance of t h e
and
i s r e q u i r e d even though t h e y do n o t appear e x p l i c i t l y i n t h e
s t r u c t u r a l equations.
T h i s i s because, i f t h e s t a t e v e c t o r o f t h e t r i a l
s o l u t i o n does n o t i n c l u d e these terms, t h e n t h e r e q u i r e d i d e n t i t i e s cannot
hold.
'
E x p e c t a t i o n s formed two p e r i o d s ago g e n e r a l l y m a t t e r f o r t h e
endogenous v a r i a b l e s because o f t h e e x i s t e n c e o f c o n t r a c t s made two p e r i o d s
ago, based on t h e n - a v a i l a b l e i n f o r m a t i o n .
T h i s appearance o f predetermined p r i o r e x p e c t a t i o n s o f t h e exogenous shock
processes r a i s e s t h e q u e s t i o n o f why t h e p o l i c y e q u a t i o n of t h e p r e v i o u s
s e c t i o n was a b l e t o o m i t these a r t i f i c i a l s t a t e v a r i a b l e s .
The reason i s
t h a t , under any o f t h e p o l i c y c r i t e r i a examined i n t h i s a r t i c l e , money s u p p l y
c o e f f i c i e n t s on these s t a t e v a r i a b l e s a r e redundant and, hence, t h e i r v a l u e s
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-1 1 -
are indeterminate. They have been arbitrarily set to zero in this study in
order to determine the policy vector uniquely and to simplify analysis. If it
were desired to study policies directed toward a mixed or compound
objective--such as both output and price stability, or policies to stabilize
both interest rates and prices--then the pol icy rule should generally include
nonzero coefficients on E t - r u t - l ,Et-,vt-l,and E t - 2 ~ t - l .
Thus, the conclusion is that the state vector
St={ut-I, vt-1, xt-1;
~
t
~,
t
At;
,
Et-2ut-1, Et-2vt-1, Et-2xt-1)
is adequate to describe the position of the system, ruling out bootstraps.
Table 1 provides a glossary of endogenous and state variables that will
appear in the trial solution.
B.
The Trial Solution
Then the trial solution includes the following equations.
(11) through (16)
are implications of the assumed linearity and of the state vector St. (17)
and (18) are direct implications of (13).
y ~ t =fllo+fll
,ut-1+n12vt-1+n13xt-1+n14~t+n1~~t+~16lt
+ n 1 7 E t - Z U t - l + n 1 8 E t - f l t - l + k E t - 2 X t - l
Y2t=
~ 2 0 + ~ ~ 1 U ~ - l + ~ 2 2 V t - 1 + ~ 2 3 x t - l + ~ 2 4 ~ t + ~ 2 5 Q t + ~ 2 6 ~ t
+ n 2 7 E t - ~ ~ t - l + n 2 a E t - 2 ~ t - l + W t - ~ ~ t - 1
Pt=
+ n 3 7 E t - ~ ~ t - l + n 3 8 E t - ~ ~ t - 1 + n 3 q E t - ~ ~ t - 1
mt=
n40+fl4lut-l+fl42vt-l+n43Xt-l+n44Et+n4~~t+n46lt
+ n 4 7 E t - L ~ t - i + n 4 8 E t - 2 ~ t - 1 + n 4 q E t - z ~ t -1
Rt=
~ ~ 0 + ~ 5 l u t - l + ~ 5 2 v t - l + n 5 3 X t - l + n 5 4 ~ t + n 5 5 ~ t + ~ 5 6 ~ t
+ns7Et-2~t-I+nsaEt-~~t-l+bEt-fit-I
yt=
no+nlut-l+n2vt-l+n3Xt-l+n4Et+nsQt+n6Xt+
+ n7Et-2~t-l+n8Et-2~t-,+~9Et-fit-1
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-1 2TABLE 1
GLOSSARY OF VARIABLES I N THE TRIAL SOLUTION
ENDOGENOUS VARIABLES
y,.
y
I
Log o f t o t a l o u t p u t
,.
Log of o u t p u t o f group one
y,,.
Log o f o u t p u t o f group two
pt.
Log o f p r i c e l e v e l
m,.
Log of money s t o c k
R,.
Log of [ I + nominal i n t e r e s t r a t e 1
Endogenous E x p e c t a t i o n s
E t - ~ p , = E [ p t l S , - ~ I, i=1,2.
( P r i o r e x p e c t a t i o n s o f p,)
E;-lpt+,=ECpt+1 l ~ , l = E C p ~l R+t ~, S t t l I. (Updated e x p e c t a t i o n o f p t + l )
STATE VARIABLES
Predetermined
u
.
Observed l e v e l o f aggregate supply
v , .
Observed l e v e l of money demand
x
Observed l e v e l o f aggregate demand
.
-
E t - Z U ~ - 1
Et-zvt-I
Et-zxt-1.
Exogenous
E,.
I n n o v a t i o n t o aggregate supply
qt.
I n n o v a t i o n t o money demand
1,. I n n o v a t i o n t o aggregate demand
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E ~ - I P ~ = ~ I ~ O + ~ I ~ ~ U ~ - ~ + ~ I ~ ~ V ~ - ~ + ~ ~ ~ ( 1X7 )~ - ~ +
E ~ - , P ~ = I I ~ ~l + (I fI 3I r~) E t - ~ t -1 + ( n 3 z + f l 3 8 ) E t - 2 V t - 1 + ( f l 3 3 + n 3 q ) E t - z X t - 1
The d e r i v a t i o n o f t h e t r i a l s o l u t i o n f o r E:Llpt
comment.
(18)
deserves s p e c i a l
T h i s e x p e c t a t i o n i s composed o f t w o o r t h o g o n a l components:
E:-lpt+l=Et-lpt+I+CE:-Ipt+ I - E t - l p t + ~ l .
(19)
The f i r s t component o f ( 1 9 ) i s t h e p r i o r e x p e c t a t i o n o f n e x t p e r i o d ' s p r i c e
l e v e l , which i s c o n d i t i o n e d on l a s t p e r i o d ' s f u l l r e a l i z a t i o n o f t h e s t a t e of
t h e economy, S t - 1 .
processes,
(lo),
U s i n g ( 1 3 1 , and t h e assumptions a b o u t t h e e r r o r
t h i s i s seen t o be
E t - l p t + l = f 1 3 0 + ~(n3
l 1 + n 3 7 ) ~ t - I + ~ 2 ( ~ 3 2 + f 1 3 8 ) ~ t - l + ~ 3 ( ~ 3 3 + f 1 3 9( 2)0~) t - l ~
The second component o f ( 1 9 ) i s t h e r e v i s i o n t o t h e f u t u r e - p r i c e e x p e c t a t i o n
t h a t o c c u r s a f t e r a g e n t s o b s e r v e t h e c u r r e n t nominal i n t e r e s t r a t e , R t .
T h i s second term, t h e n , c o n s t i t u t e s an u p d a t i n g o f t h e f u t u r e p r i c e
e x p e c t a t i o n based o n t h e i n f o r m a t i o n c o n t e n t o f t h e nominal i n t e r e s t r a t e .
From t h e t r i a l s o l u t i o n s p e c i f i c a t i o n (131, i t i s c l e a r t h a t t h e i n f l u e n c e o f
current (period-t)
shocks o n p t + l must be
(pt+l-Et-Ipt+l)
=
(n31~t+rI32~t+n33Xt).
(21)
The e x p e c t a t i o n o f t h i s component o f t h e f u t u r e p r i c e l e v e l formed a t t i m e t - 1
i s c l e a r l y zero.
B u t a t t i m e t , agents can c o n d i t i o n t h i s component o n t h e
i n n o v a t i o n i n t h e nominal i n t e r e s t r a t e ,
(22)
(Rt-Et-,Rt)=(ns,~t+rI~s~t+rIsbXt).
The f u t u r e p r i c e e x p e c t a t i o n u p d a t i n g c o e f f i c i e n t ,
e=d[E:-lpt+~-Et-lpt+ll/d[Rt-Et-lRtl, i s e s s e n t i a l l y the slope o f the
least- squares r e g r e s s i o n l i n e r e l a t i n g ( p t + l - E t - l p t + l )
E{Cpt+l-Et-Ipt+l l[Rt-Et-IRtI}
8 = ..............................
E[Rt-Et-IRtlz
=
(n31 ~ 5 4 ~ : +zn 3~ 5 ~s G + f 13 3 ~ 60); 5
............................
(n:,o:+n: so:+~: ,u:>
t o (R,-Et-IRt).
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-1 4-
Then t h e updated f u t u r e p r i c e e x p e c t a t i o n i s
ELP,+
l = n 3 0 + p l ( n 31 + n 3 7 ) ~ t - 1 + p 2 ( n 3 2 + n 3 8 ) ~ t - , + ~ 3 ( n 3 3 + n 3 9 ) ~ t - l
+@(n54~t+nssqt+ns,lt).
(24)
The e f f e c t o f t h e assumption t h a t a g e n t s use t h e i n f o r m a t i o n c o n t e n t o f
t h e i n t e r e s t r a t e i s t o add t h e @-term, w i t h 8 d e f i n e d by ( 2 3 > , t o t h e
r i g h t - h a n d s i d e of ( 2 4 ) .
The e f f e c t o f t h e assumption can be seen i n t h e
s o l u t i o n f o r t h e model, i n t h e presence o f 8 .
C.
General P r o p e r t i e s
The appendix p r e s e n t s t h e i d e n t i t i e s among n - c o e f f i c i e n t s i m p l i e d b y t h e
s t r u c t u r e , t h e s o l u t i o n s f o r t h e s e c o e f f i c i e n t s , and t h e f i n a l f o r m s o l u t i o n
o f t h e model.
These s o l u t i o n s , (A.18) t h r o u g h (A.21),
r e f l e c t some i m p o r t a n t
g e n e r a l p r o p e r t i e s o f t h e model, due t o i t s t w o - p e r i o d F i s c h e r c o n t r a c t s .
The
e f f e c t o f money demand or commodity demand shocks two p e r i o d s o r more e a r l i e r
i s o n l y on t h e nominal v a r i a b l e s such as t h e p r i c e l e v e l , p,,
w,,
t h e i n t e r e s t r a t e , R,,
such as o u t p u t , y,,
wages, w , ,-p,,
i=1,2.
and t h e money s t o c k , m,,
nominal wages,
and n o t on r e a l v a r i a b l e s
t h e r e a l r a t e o f i n t e r e s t , CRt-(E:-IPt+l-P,)I,
Group one o u t p u t , y ,
and r e a l
,, i s d e t e r m i n e d by t h e e n t i r e
h i s t o r y o f commodity s u p p l y i n n o v a t i o n s ,
{E,):=-,,
demand and commodity demand i n n o v a t i o n s ,
q,
and A,.
and on p e r i o d - t money
But y l t i s unaffected by
a n t i c i p a t e d l e v e l s o f money demand and commodity demand shocks, and hence i s
independent o f lagged values o f
q
and A.
S i m i l a r l y , g r o u p two o u t p u t , y 2 ,, i s
d e t e r m i n e d by t h e e n t i r e h i s t o r y o f commodity s u p p l y i n n o v a t i o n s and o n
p e r i o d - t and ( t - 1 ) money demand and commodity demand i n n o v a t i o n s : q,,
A,,
and A t - l .
Q,-~,
The e f f e c t s o f each i n n o v a t i o n o n nominal v a r i a b l e s d i e o u t ,
beyond t h e second l a g , g e o m e t r i c a l l y w i t h a r a t e o f decay s t r i c t l y dependent
on t h e a p p r o p r i a t e a u t o c o r r e l a t i o n c o e f f i c i e n t .
P o l i c y p a r a m e t e r s {q
behavior of output.
,p2 ,p,) a r e a1 1 g e n e r a l l y c o n s e q u e n t i a1 for t h e
Monetary p o l i c y can i n f l u e n c e t h e o u t p u t e f f e c t s o f
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current (period-t)
shocks and can e l i m i n a t e o u t p u t e f f e c t s of shocks o c c u r r i n g
i n the previous period (t-1).
I t i s i n f e a s i b l e f o r monetary p o l i c y t o a f f e c t
t h e component o f the business c y c l e due t o supply shocks a f t e r t h e i r p r i c e
l e v e l i m p l i c a t i o n s become r e f l e c t e d i n a l l wage c o n t r a c t s ( t w o p e r i o d s ) .
Hence, supply shocks o c c u r r i n g i n t-j, j > l , have e f f e c t s on r e a l v a r i a b l e s ,
i n c l u d i n g o u t p u t , t h a t p o l i c y cannot m o d i f y .
Money supply shocks o r c o n t r o l
e r r o r s need n o t be f o r m a l l y i n t r o d u c e d i n t o t h e model; t h e y have e f f e c t s
i d e n t i c a l i n magnitude b u t o p p o s i t e i n s i g n t o money demand shocks.
Constant
terms can a l s o be excluded f r o m f u r t h e r c o n s i d e r a t i o n , s i n c e t h e y have no
i m p l i c a t i o n s f o r c y c l i c a l behavior.
V.
Constant Money Supply
Examining t h e h y p o t h e t i c a l constant- money case h e l p s make t h e m o d e l ' s
p r o p e r t i e s more t r a n s p a r e n t .
I t i s useful t o review the f a m i l i a r p r o p e r t i e s
o f c o n v e n t i o n a l IS-LM-type models under a c o n s t a n t money p o l i c y .
Usually,
commodity supply shocks have p o s i t i v e e f f e c t s on o u t p u t and n e g a t i v e e f f e c t s
on p r i c e s .
Commodity demand shocks have p o s i t i v e e f f e c t s on o u t p u t , p r i c e s ,
and t h e i n t e r e s t r a t e .
Money demand shocks have p o s i t i v e e f f e c t s on t h e
i n t e r e s t r a t e , b u t n e g a t i v e e f f e c t s on o u t p u t and p r i c e s .
The model can be
regarded as d i s p l a y i n g these usual p r o p e r t i e s o f IS-LM models i n a v e r y broad
and g e n e r a l way.
However, some o f t h e s i g n s o f c o e f f i c i e n t s o f t h e f i n a l f o r m
a r e f o r m a l l y ambiguous, and some anomalous cases a r e p l a u s i b l e .
I n t h e model a t hand, under t h e c o n s t a n t money p o l i c y ,
{q,pl ,p2,p2)=0, t h e s o l u t i o n f o r p r i c e s , o u t p u t , and t h e i n t e r e s t r a t e
reduces t o t h e equations below.
yt={l-BCal+a2bl(l-8)lJ}~t+p111-B(l+al ) ( a l + a 2 b l )GI
m
J
-blB(1-B)Jqt-blBp2(1+al)G2~t-l+BalJX,+Ral~l+al~p3G3At-l
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where
and
where
Z=2(l+al
and
c
=
(I-a2bl )(al+a2bl ) G l p l o ~
-bl (bl+B)G2p20G
+al ( l+azB)G3pso:
{
)
(l-a2bl~2u~+(bl+~~2u~+(l+a2~)20~>0.
Some p r e l i m i n a r y a n a l y s i s o f 8 and Z w i l l f a c i l i t a t e u n d e r s t a n d i n g t h e
b e h a v i o r o f t h i s system.
The f u t u r e p r i c e e x p e c t a t i o n u p d a t i n g c o e f f i c i e n t ,
8, w i l l t a k e t h e s i g n o f t h e t e r m i n t h e l a r g e b r a c k e t s i n t h e d e f i n i t i o n o f
Z, ( 3 1 ) .
Z can be p o s i t i v e o r n e g a t i v e , and g e n e r a l l y depends on a l l o f t h e
parameters o f t h e system, i n c l u d i n g e r r o r v a r i a n c e s .
For example, Z, hence
0, i s more l i k e l y t o be p o s i t i v e i f t h e v a r i a n c e and a u t o c o r r e l a t i o n i n
money demand shocks,
and p,,
a r e l a r g e r e l a t i v e t o t h e v a r i a n c e and
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- 1 7-
a u t o c o r r e l a t i o n o f commodity demand shocks, o < and p 3 .
8 i s more
l i k e l y t o be p o s i t i v e i f t h e e l a s t i c i t y o f commodity demand w i t h r e s p e c t t o
t h e r e a l i n t e r e s t r a t e , b , , i s l a r g e , and i f t h e p r o d u c t o f t h i s e l a s t i c i t y
and t h e income e l a s t i c i t y o f money demand, a,b,,
exceeds u n i t y .
Finally,
t h e e f f e c t on 8 o f t h e e l a s t i c i t y o f money demand w i t h r e s p e c t t o t h e
nominal i n t e r e s t r a t e , a , , depends on t h e r e l a t i v e s i z e s o f shocks and on
o t h e r parameters.
I n most reasonable cases, however, i n c r e a s e s i n a l a r e
a s s o c i a t e d w i t h increases i n 8.
I n extreme cases, 8 may n o t o n l y be p o s i t i v e , b u t g r e a t e r than u n i t y .
8>1 i f f a,(b,+B)Z>E.
(33)
Because t h e c o n d i t i o n (33) i n v o l v e s Z m u l t i p l i c a t i v e l y , t h e l i k e l i h o o d o f t h i s
case may be enhanced by t h e same s t r u c t u r a l c o n d i t i o n s j u s t mentioned as
d e t e r m i n i n g t h e s i g n o f 8 and Z.
The p o t e n t i a l importance o f h i g h v a l u e s o f
t h e e l a s t i c i t y o f money demand, a,, i s seen i n t h e f a c t t h a t f o r some a l
s u f f i c i e n t l y h i g h , 8 w i l l exceed u n i t y .
8.
But an upper bound can be p l a c e d on
The d e f i n i t i o n s o f J and 8 can be manipulated t o show t h a t , g i v e n
E>O ( s i n c e E i s a weighted sum o f t h e d i s t u r b a n c e v a r i a n c e s , i t i s
n e c e s s a r i l y p o s i t i v e ) , J>O and
8 < l+al(bl+B)/bl(l+a2B)
(using az
/
L
l+allarbl
I).
These i n e q u a l i t i e s h e l p determine signs of t o t a l d e r i v a t i v e s , o r
c o e f f i c i e n t s i n t h e f i n a l form.
I f 8>1, then money demand o r supply shocks
w i l l have a n t i - i n t u i t i v e e f f e c t s on the p r i c e l e v e l and o u t p u t .
For example,
an i n c r e a s e i n money demand under t h i s c o n s t a n t money p o l i c y w i l l a c t u a l l y
i n c r e a s e t h e p r i c e l e v e l and o u t p u t i n t h e contemporaneous p e r i o d .
I n the
f o l l o w i n g p e r i o d , and beyond, t h e e f f e c t on p r i c e s and o u t p u t i s n e g a t i v e .
The e f f e c t o f a g e n t s ' use o f t h e i n t e r e s t r a t e t o update f u t u r e p r i c e
e x p e c t a t i o n s can be i l l u s t r a t e d w i t h a h y p o t h e t i c a l temporary i n c r e a s e i n t h e
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-1 8-
money supply o f one u n i t a t time t .
( T h i s shock has t h e same e f f e c t s as a
o n e - u n i t decrease i n money demand, I-,,=-].>
unambiguously f a l l s by (bl+R>J.
E:-lpt+l,
The c u r r e n t p r i c e l e v e l , p t , changes by b l (1-8>J,
which can be n e g a t i v e i f 8 > 1 .
[E:-lpt+l-ptl,
-R(1-8>J,
The expected f u t u r e p r i c e l e v e l ,
i s changed by t h e same amount times 8, which can be
p o s i t i v e o r negative.
8<-(bl l R > .
The i n t e r e s t r a t e , R t ,
The expected r a t e o f i n f l a t i o n ,
changes by -(b1+88>J, which i s n e g a t i v e unless
The r e a l r a t e o f i n t e r e s t , r,=CR,-(Ef
- l p t + l - p t > I , changes b y
which i s n e g a t i v e unless 8>1.
I f t h e r e a l i n t e r e s t r a t e d e c l i n e s i n response t o a temporary money
i n c r e a s e , then aggregate demand i n c r e a s e s , which i s a s s o c i a t e d w i t h h i g h e r
c u r r e n t p r i c e s and o u t p u t .
The c o u n t e r i n t u i t i v e case, i n which t h e r e a l
i n t e r e s t r a t e increases w i t h a money supply shock, a r i s e s i f t h e necessary
d e c l i n e i n t h e nominal i n t e r e s t r a t e i s overshadowed by an even l a r g e r d e c l i n e
i n i n f l a t i o n e x p e c t a t i o n s based on o b s e r v a t i o n o f t h a t i n t e r e s t - r a t e d e c l i n e .
I n t h e b o r d e r l i n e case o f 8=1, observed r i s e s i n R, generate equal r i s e s
i n E:-lp,+l,
so t h a t i n f l a t i o n e x p e c t a t i o n s r i s e by an amount equal t o
t h e r i s e i n t h e nominal i n t e r e s t r a t e .
Therefore, t h e r e a l r a t e , o u t p u t , and
t h e c u r r e n t p r i c e l e v e l a r e unchanged.
I f 8>1, t h e n observed r i s e s i n Rt
a r e a s s o c i a t e d w i t h l a r g e r r i s e s i n E:-lpt+, , so t h a t expected
i n f l a t i o n r i s e s by more t h a n t h e r i s e i n t h e nominal i n t e r e s t r a t e .
Then,
r e a l r a t e s f a l l w i t h observed p o s i t i v e i n n o v a t i o n s i n t h e nominal r a t e ,
i n c r e a s i n g aggregate demand and r a i s i n g o u t p u t and t h e c u r r e n t p r i c e l e v e l .
I t i s i n t e r e s t i n g t o c o n s i d e r t h a t , under a simple e l a s t i c money- supply
p o l i c y , such as m,=qRt,
without l i m i t .
8 w i l l e v e n t u a l l y exceed u n i t y as q i s i n c r e a s e d
(Such a p o l i c y m i g h t be m o t i v a t e d by a n a i v e e f f o r t t o
s t a b i l i z e t h e i n t e r e s t r a t e , a l t h o u g h t h i s method o f d o i n g so would be
unnecessari l y c o s t l y i n terms o f o t h e r p o t e n t i a1 o b j e c t i v e s . )
Then money
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demand and s u p p l y shocks would have c o u n t e r i n t u i t i v e e f f e c t s on p r i c e s and
output.
The importance o f t h e cases o f c o u n t e r i n t u i t i v e contemporaneous e f f e c t s o f
shocks i s i n h e r e n t l y l i m i t e d by t h e s t r u c t u r e o f t h e model.
I f t h e sum o f t h e
e l a s t i c i t i e s o f money supply and money demand, al+q, i s h i g h enough, these
cases a r i s e .
But, then, money s u p p l y and demand shocks have l i t t l e
c'ontemporaneous e f f e c t on o u t p u t , as can be seen by n o t i n g t h a t t h e
c o e f f i c i e n t J i n t h e reduced forms approaches z e r o as al+q r i s e s w i t h o u t
limit.
So, i f a l + q i s h i g h enough t o c r e a t e a n t i - i n t u i t i v e responses o f
o u t p u t and t h e p r i c e l e v e l t o money shocks, then these responses w i l l t e n d t o
be s m a l l .
L i k e w i s e , these cases can a r i s e due t o a preponderance o f commodity
demand shocks r e l a t i v e t o money demand and money supply shocks.
But then, b y
hypothesis, t h e i n f l u e n c e s t h a t c r e a t e t h e a n t i - i n t u i t i v e e f f e c t s a r e
r e l a t i v e l y small.
Beyond t h e contemporaneous p e r i o d , t h e e f f e c t o f an i n n o v a t i o n on any
endogenous v a r i a b l e depends, n o t on 8, nor on r e l a t i v e s i z e s o f e r r o r
v a r i a n c e s , b u t s t r i c t l y on s t r u c t u r a l e l a s t i c i t i e s and on a u t o c o r r e l a t i o n
c o e f f i c i e n t s o f shocks.
The most s u r p r i s i n g p o s s i b i l i t y i s t h a t o u t p u t can
f a l l i n t h e wake o f a p o s i t i v e s u p p l y i n n o v a t i o n .
Formally,
d y , / d ~ , - ~ < Oi f f
Cl+a,(l-pl)lCB(al+a2bl~+2bl(l+al)l< B ( l + a l ) p l ( a l + a r b l ) .
8
(35)
The behavior o f t h e i n t e r e s t r a t e i n response t o shocks d i s p l a y s s e v e r a l
interesting characteristics.
The e f f e c t o f a l l i n n o v a t i o n s on t h e nominal
i n t e r e s t r a t e i s g e n e r a l l y nonzero a t a l l l a g s , a l t h o u g h i t d i e s o u t a t f i x e d
r a t e s , beyond t h e second l a g , t h a t depend s t r i c t l y on t h e p i .
o f s u p p l y i n n o v a t i o n s on t h e i n t e r e s t r a t e i s ambiguous.
The e f f e c t
This e f f e c t o f
s u p p l y i n n o v a t i o n s i n t h e contemporaneous p e r i o d i s n e g a t i v e i f t h e p r o d u c t o f
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-20-
t h e income e l a s t i c i t y o f money demand, a l , and t h e r e a l i n t e r e s t r a t e
e l a s t i c i t y o f commodity demand, b l , i s l e s s than u n i t y , b u t p o s i t i v e i f
albl>l.
rate.
A t one l a g , supply i n n o v a t i o n s may r a i s e o r lower t h e i n t e r e s t
I f p l i s near u n i t y , I3 i s l a r g e , o r a z i s s m a l l , then t h i s e f f e c t
i s more l i k e l y t o be n e g a t i v e .
Although money demand i n n o v a t i o n s unambiguously r a i s e t h e i n t e r e s t r a t e
contemporaneously, they may a c t u a l l y decrease t h e i n t e r e s t r a t e i n t h e
subsequent p e r i o d .
Such a case i s f o s t e r e d by h i g h a u t o c o r r e l a t i o n , p r ,
and h i g h values of a, and b , .
The i n f l u e n c e o f p r i v a t e a g e n t s ' use o f t h e c u r r e n t i n t e r e s t r a t e t o
update i n f l a t i o n e x p e c t a t i o n s i s seen i n t h e presence o f 8 i n t h e s o l u t i o n
equations.
I f agents d i d n o t use t h e i n t e r e s t r a t e , t h e economy would behave
as i f 8 were z e r o i n t h e s o l u t i o n equations.
Obviously, t h e i n f l u e n c e of
a g e n t s ' use o f t h e i n t e r e s t r a t e i s small i f 8 i s reasonably c l o s e t o z e r o ,
as w i l l be i l l u s t r a t e d w i t h a numerical example.
The d i f f e r e n c e s i n t h e
b e h a v i o r o f t h e system under t h e a l t e r n a t i v e assumptions about p r i v a t e
i n f o r m a t i o n p e r t a i n s t r i c t l y t o t h e contemporaneous e f f e c t s of i n n o v a t i o n s .
The e f f e c t o f c u r r e n t supply i n n o v a t i o n s on o u t p u t , dy,/ds,,
i s smaller
i f p r i v a t e agents use t h e i n f o r m a t i o n c o n t e n t o f t h e i n t e r e s t r a t e , if and
o n l y i f the sign o f [(I-a2bl>B1 i s negative.
money demand i n n o v a t i o n s on o u t p u t , dyt/dq,,
agents do n o t use R,,
The e f f e c t o f c u r r e n t
i s always n e g a t i v e i f
b u t can be p o s i t i v e i f agents use R t and 8>1.
If
8 ~ 0 ,then t h e a b s o l u t e magnitude o f t h e o u t p u t e f f e c t o f money demand shocks
i s magnified.
The e f f e c t o f c u r r e n t demand i n n o v a t i o n s on o u t p u t ,
d y t l d X t , i s always p o s i t i v e .
R , i n e x p e c t a t i o n formation.
I t i s i n c r e a s e d f o r 8 > 0 i f agents use
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V I.
Output S t a b i 1i z a t i o n Pol i cy
The economy-wide o u t p u t s t a b i l i z a t i o n p o l i c y i n v o l v e s feedback from t h e
c u r r e n t i n t e r e s t r a t e and p a s t economic shocks i n a manner t h a t i s b r o a d l y
c o n s i s t e n t w i t h t h e well- known analyses o f Poole and F i s c h e r .
I n d e r i v i n g t h i s p o l i c y , i t w i l l be e a s i e r t o begin by n o t i n g t h a t t h e
e f f e c t s on o u t p u t o f i n n o v a t i o n s o f more than one p e r i o d ago cannot be
m o d i f i e d by p o l i c y .
t - 2 o r e a r l i e r supply i n n o v a t i o n s w i l l have f u l l impact,
w h i l e d i s t a n t demand and money demand i n n o v a t i o n s have no e f f e c t .
n7=n8=n9=0 i n e q u a t i o n (16).
Then
S t a b i l i z a t i o n p o l i c y operates by
m i n i m i z i n g t h e e f f e c t s o f t and t-1 i n n o v a t i o n s .
I t i s c l e a r f r o m (16) t h a t
t - 1 shocks a r e prevented f r o m i n f l u e n c i n g y t i f 111=112=113=0,o r
n l = [ p 1 + ( 1 / 2 ) R ~ 31=0,
112=(1/2)01132=0,
and l13=(1/2)8n33=0,
(36)
u s i n g t h e r e l e v a n t equations among (A.1>, ( A . 2 > , and (A.6) i n t h e appendix.
( 3 6 ) r e q u i r e s t h e f o l l o w i n g r e l a t i o n s between t h e p i and q, i n view o f t h e
expressions f o r
n 3 , , n,,, and n s 3among e q u a t i o n s (A.10).
The c o n t r i b u t i o n o f p e r i o d - t shocks t o o u t p u t variance cannot be
e l i m i n a t e d , b u t i s minimized, g i v e n t h e Gaussian s t o c h a s t i c assumption, i f t h e
e x p e c t a t i o n o f the o u t p u t innovation i s zero.
Given a v a i l a b i l i t y of c u r r e n t
o b s e r v a t i o n s on t h e i n t e r e s t r a t e , t h i s i m p l i e s t h a t o u t p u t and i n t e r e s t r a t e
i n n o v a t i o n s w i l l be u n c o r r e l a t e d , o r
2
COVC(yt-Et-lyt) ,(Rt-Et=(fin34+l )fl54~;+0&
The s u b s t i t u t i o n o f
1
Rt)I = n4ns4~~+fl~ll5s~~+ll6fl5~
S n S
n 3 i terms
e q u a t i o n s among (A.1>, (A.2),
S U ~ + B I I ~ ~ ~ =~ S0 -~ U ~
for
ni
(38)
terms i n (38) uses t h e r e l e v a n t
and (A.6).
(38) can, w i t h t h e model, be used t o
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determine t h e v a l u e of q i n t h e f o l l o w i n g way.
(A.7) f o r
n4, n,,
and
There a r e t h r e e e q u a t i o n s i n
n6. There a r e a l s o t h r e e equations i n ( ~ . 1 2 ) f o r n s 4 ,
n5,, and ri56. Together w i t h t h e d e f i n i t i o n o f
8, (23), these c o n s t i t u t e seven
e q u a t i o n s i n e i g h t unknowns:
The l a s t e q u a l i t y o f (38) p r o v i d e s an e i g h t h and d e t e r m i n i n g e q u a t i o n .
The
s o l u t i o n s f o r q and 8 a r e as f o l l o w s .
where
8
=
2pl(l-azbl>C(al+q>(bl+R>+bl(l+a~)l~~
......................................................
(1-azbl )C(l-azbl >+2pl b l (l+a28)lu%+(bl+B)2u~+(l+a2B)zu<
8 w i l l take t h e s i g n of (1-a2bl).
I n t h e s i m p l i f y i n g case o f (1-a2bl)=0 ( o r , w i t h i d e n t i c a l
consequences, u:=O), 8 equal s z e r o and
I f , f u r t h e r , b,=a,=l,
then
2
2
2
2
q= Cuq-a
o x2 I /ux=uq
/ura
.
T h i s l a s t r e s u l t i s r e m i n i s c e n t o f P o o l e ' s a n a l y s i s o f a f i x e d - p r i c e IS- LM
model under comparable assumptions about t h e e l a s t i c i t y o f demand f o r goods
w i t h respect t o the i n t e r e s t r a t e .
The o u t p u t - s t a b i l i z i n g p o l i c y p r o v i d e s f o r
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a money supply w i t h i n t e r e s t r a t e e l a s t i c i t y t h a t has a lower bound o f -al
and no upper bound.
I t i s d i r e c t l y r e l a t e d t o t h e r a t i o o f money demand
v a r i a n c e t o commodity demand v a r i a n c e .
q decreases one- for- one w i t h i n c r e a s e s
i n al.
The o u t p u t - s t a b i 1 iz i ng response o f t h e money s u p p l y t o commodity s u p p l y
shocks, p l , i s o f i n d e t e r m i n a t e s i g n .
However, even i f p , i s
p o s i t i v e , money w i l l respond n e g a t i v e l y t o observed o u t p u t shocks when t h e
i n d i r e c t e f f e c t s v i a i n t e r e s t r a t e changes a r e considered.'
The f i n a l - f o r m e q u a t i o n s f o r t o t a l o u t p u t , t h e p r i c e l e v e l , and t h e
i n t e r e s t r a t e a r e shown below.
J, q, and 8 a r e as g i v e n i n (A.141,
(39),
and (40), r e s p e c t i v e l y .
i
(43)
yt={l-8Cal+q+a2bl (1-8) l J } ~ ~ + E " ? ~ p l s ~ - ~( -1-8)Jqt+13(al+q)Jlt
l3b~
pt=-[al+q+a2bl ( 1 - 8 ) 1 J ~ ~ - 2 ( /pR~) E ~ - ~
-E(2/B)+(al+a2bl+q)b;'
( l+al+q)-I
IC?=
i+bl ~
(1-8)Jqk+(al+q)Jlt
~;E~-
(44)
Rt=-(1-a,bl
+ { ( a l + q > - ' C a 2 P I - p l - ( 2 + a ~ ) ( p 1 / B ~ l - ( P l / b l ~ ~ l - a 2 b l ~ ~ l + a' -lI+) q~ ~t ~-1 i} ~ ~ ~ 2 p ~
+(bl+B)J~t+(l+a,O)J~+(l/bl~~?~lP~~t~i
VII.
(45)
F i r m Output S t a b i l i z a t i o n P o l i c y
Although o u t p u t d e t e r m i n a t i o n under t h e wage - contracting scheme i s
n e c e s s a r i l y Pareto - suboptimal, i t i s n o t c l e a r t h a t a p o l i c y t o minimize
economy-wide o u t p u t f l u c t u a t i o n s w i l l improve economic w e l f a r e .
Several o t h e r
c r i t e r i a would seem more adequate i n measuring t h e w e l f a r e l o s s e s f r o m wage
stickiness.
The f i r s t , d e a l t w i t h i n t h i s s e c t i o n , i s t h e v a r i a n c e o f f i r m ,
as opposed t o economy-wide, o u t p u t .
If workers and f i r m s d i s l i k e v a r i a t i o n s
i n t h e i r own o u t p u t and employment more t h a n t h e y d i s l i k e v a r i a t i o n s i n
aggregate o u t p u t and employment, then economic welfare w i l l be h i g h e r under a
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policy aimed at stability of output at the
firm
level.
The primary difference between policies designed to stabilize firm output
and those designed to stabilize aggregate output is in the policy response to
observed supply, p l . In the aggregate output policy, P I is chosen so
that the effect of last period's supply innovation on economy-wide output is
zero, or dytldst-,=0. Given that the effect on output of group one is
dy,
, l d ~ , - ~ = >O,
p , this necessarily requires dy2 ,Id&,- l=-pl<O, since
(in view of (5)) dy, is a simple average of dy,, and dy,,.
Thus, the
aggregate output policy uses its power to decrease group two output, in
response to positive supply shocks in t-1, by a magnitude equal to the necessary increase in group one output.
With regard to supply shocks, then, such a
policy succeeds in stabilizing aggregate output partly by destabilizing
firm-level output. Last period's supply shock, E,-,, which unavoidably
caused y , , to vary, has now caused y,, to vary as well, via an excessively
deflationary policy response.
A less countercyclical response to supply
shocks, or a numerically higher P I ,is appropriate for firm-level output
stabilization. The appropriate response will have the effect of allowing
enough deflation to offset the effects of the supply shock on group two firms'
output decisions.
This policy is derived in the following way. An appropriate measure of the
variance of output for a typical firm over a contract interval, during which a
firm will have one period as a group one firm, and another period as a group
two firm, is ECy, ,-Ey, ,I2+ECy2,-Ey2 ,IL. Using the relevant
equations among (25) and (26), and rearranging,
ECy, t-Ey, t12+ECy2t-Ey2 t12=p:(l+p:)/(l-p~)l~~
+[(fill3
,+PI)2~g+(6n32
)'0:+(8n3;)~&1
+2[(fin3,+1
>2a:+(~~3
5)2~:+(~n3b)'o~I.
This expression is the sum of three components.
First, p:(l+p:
(46)
)l(l-p~)o~
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i s a c o n s t a n t , o r deadweight term, and can be i g n o r e d i n p o l i c y c h o i c e .
A
second component i n v o l v e s t h e terms w i t h 1131, 113,, and 1133, and i s
reduced t o z e r o i f
RII3
I + ~ I = O , 8fl32=0, Rn33=0-
(47)
(Comparing ( 4 7 ) w i t h ( 3 6 ) , i t i s seen t h a t n 3 1d i f f e r s between f i r m and
economy-wide o u t p u t p o l i c i e s , b u t
II3r
and 1133 a r e i d e n t i c a l l y z e r o . )
These, i n c o n j u n c t i o n w i t h t h e r e l e v a n t e q u a t i o n s among ( A . 1 0 > , i m p l y
p l = p l C(ar/2)-(1/B)-(aI+q){[(l-pl
p 2 = p 2 >0,
p3=-p3(al+q)/bl
I t r e m a i n s t o d e t e r m i n e q.
)/Rl-(112)-[pl
(al+a2bl+q)/2bl ( l + a l + q ) l } l
<0.
(48)
The t h i r d component o f f i r m v a r i a n c e ,
[2(BI134+1) 2 0 ~ + 2 ( 8 1 1 3 5 ) 2 + 2 ( & ) 2 1 ,
i s equal t o 2ECyt-E,-
ly,lz.
Given t h e
Gaussian e r r o r s t r u c t u r e , t h i s component i s m i n i m i z e d i f t h e p o l i c y v e c t o r i s
such t h a t c o n d i t i o n ( 3 8 ) o f t h e p r e v i o u s s e c t i o n i s f u l f i l l e d .
Then, ( 3 8 ) ,
(471, ( 2 3 1 , and t h e r e l e v a n t s i x e q u a t i o n s from (A.10) and (A.12) can be
s o l v e d f o r q and 8.
T h i s c o n d i t i o n , i n c o n j u n c t i o n w i t h t h e above
expressions f o r
n 3 2 ,and 1133, and t h e model, i m p l i e s
n31,
q=(CICr-C3C4)I(C3C5-Cbc2)
and
where
C1=pl(l-a2bl~[al(bl+B)+bl(l+a2~~l~~
Cr=-bl (1-arb
>o~-bl~(bl+B)o~
C3=CB(l-a,bl > + p l b , ( l + a , B ) l ( 1 - a 2 b l
C4=-(1-a,bl
)o~+(bl+B)2~~+(l+a2B)2~i
~ b l ( l + a l ) o ,2 - b l B ( b l + B ) ~ ~ + a l R ( l + a , R ) ~ ~
Cs=-bl (1-a, b l ) o ~ + B ( l + a , B ) o i
Cb=pl(l-arbl > ( b l + B > o ~ .
I n t h e s i m p l i f y i n g case o f (1-a2bl)og=0,
(49)
t h e n q i s g i v e n b y ( 4 1 ) and 8
i s z e r o , p r e c i s e l y as f o r t h e economy-wide o u t p u t s t a b i l i z i n g p o l i c y .
I n comparison w i t h t h e economy-wide o u t p u t s t a b i l i z i n g p o l i c y , t h e f i r m
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o u t p u t s t a b i l i z a t i o n p o l i c y i s c h a r a c t e r i z e d by a ( n u m e r i c a l l y ) h i g h e r v a l u e
of P I . The d i f f e r e n c e between t h e s e p o l i c i e s w i t h r e g a r d t o q i s ambiguous.
pr i s i d e n t i c a l .
p3 d i f f e r s o n l y t o r e f l e c t any d i f f e r e n c e s i n q, l e a v i n g
t h e t o t a l responses of money t o a commodity demand shock, d m t / d X t - , ,
i>O,
t h e same.
VIII.
Price Stabilization Policy
Another i n t e r e s t i n g f o r m o f o u t p u t s t a b i l i z a t i o n i s m i n i m i z i n g t h e
variance of deviations i n output from i t s f u l l - i n f o r m a t i o n l e v e l .
This p o l i c y
c r i t e r i o n has been advocated by, among o t h e r s , B a r r o (1976) and McCallum and
Whitaker (1979).
The f u l l - i n f o r m a t i o n l e v e l o f o u t p u t i s t h a t w h i c h w o u l d
o b t a i n i f p r i c e e x p e c t a t i o n s were r e a l i z e d e x a c t l y .
I f , b a r r i n g unexpected
i n f l a t i o n , o u t p u t i s chosen by t h e f i r m s i n an o p t i m a l manner, t h e n an
a p p r o p r i a t e measure f o r t h e performance of an economy i s t h e degree t o w h i c h
o u t p u t " t r a c k s , " o r tends t o match, t h a t o p t i m a l l e v e l .
Supply- shock e f f e c t s
on o u t p u t may be a p p r o p r i a t e responses t o changing o p p o r t u n i t i e s , e s p e c i a l l y
t o t h e e x t e n t t h a t such shocks r e p r e s e n t p r o d u c t i v i t y s h i f t s .
The v a r i a n c e o f t h e d e v i a t i o n o f o u t p u t f r o m t h e f u l l - i n f o r m a t i o n l e v e l i s
m i n i m i z e d i f t h e v a r i a n c e s o f p r i c e e x p e c t a t i o n e r r o r s o v e r one- and
two- period horizons are minimized.
e r r o r s over
all h o r i z o n s
I f t h e variances o f p r i c e e x p e c t a t i o n
a r e m i n i m i z e d , t h e n t h e p o l i c y i s u n i q u e and i s
i d e n t i c a l to the price s t a b i l i z a t i o n p o l i c y .
T h i s same p o l i c y a l s o
n e c e s s a r i l y m i n i m i z e s t h e v a r i a n c e o f b o t h r e a l and nominal wages.
This
s e c t i o n d e r i v e s the unique p r i c e s t a b i l i z a t i o n p o l i c y .
The p r i c e l e v e l has minimum f e a s i b l e v a r i a n c e i f t h e p o l i c y r u l e i s such
t h a t , g i v e n t h e o b s e r v a b l e s t a t e o f t h e economy, t h e e x p e c t a t i o n o f t h e p r i c e
l e v e l i s i t s c o n s t a n t mean.
F o r m a l l y , s e l e c t t h e q and { p , } ? = l f o r w h i c h
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E:-
1
pt=ECpt
-27-
lilt l = O .
T h i s c o n d i t i o n a l e x p e c t a t i o n i s composed o f two n e c e s s a r i l y o r t h o g o n a l
components:
(51 >
E:-lpt=Et-lpt+CE:-lpt-Et-lptl.
The f i r s t o f these terms can be reduced t o a v a l u e o f z e r o o n l y if t h e e f f e c t
o f t-1 d i s t u r b a n c e s on p t i s removed.
I t i s c l e a r f r o m i n s p e c t i o n o f (13)
t h a t t h i s requires
n31=n32=n33=0.
(52)
(52) and t h e r e l e v a n t e q u a t i o n s among (A.10) i m p l y
pl=(pllbl)(al+arbl+q)>O,p 2 = p 2 > 0 , and ~ . t ~ = - p ~ ( a ~ + q ) I b , < O .
Given (531, i t i s r e a d i l y seen f r o m t h e l a s t t h r e e equations of (A.10)
n37=&8=n39=0.
eliminated.
(53)
that
Therefore, t h e e f f e c t s o f a1 1 p a s t shocks on p t a r e
The f i r s t t e r m on t h e r i g h t - h a n d s i d e o f ( 5 1 > , E t - , p t , i s made
equal t o z e r o and p r i c e s a r e rende red n o n a u t o c o r r e l a t e d .
C o n d i t i o n (52) and t h e d e f i n i t i o n o f 0, ( 2 3 ) , i m p l y
8=0.
The second term i n t h e c o n d i t i o n a l e x p e c t a t i o n (51) can be reduced t o
zero,
CE:-lpt-Et-
lptl=O,
(55)
by a p p r o p r i a t e choice of q.
(lo),
(55) i m p l i e s , g i v e n t h e s t o c h a s t i c assumptions,
that
COVC(pt-Et- l p t ) , (Rt-Et-
1
Rt)
Then, ( 5 4 ) , (56), and t h e s i x r e l e v a n t equations f r o m (A.10) and (A.12)
c o n s t i t u t e a system o f e i g h t e q u a t i o n s i n t h e e i g h t unknowns,
{n34,
n3.5, n3b,
n54,
n55,
n56,
q, 0 ) -
The s o l u t i o n f o r q i s
-(al+a2bl ) ( l - a 2 b l )o;+bl (bI+B)o-f,-al (l+a2B)o:
w h i c h completes t h e c h a r a c t e r i z a t i o n o f t h e p r i c e s t a b i l i z a t i o n p o l i c y .
(56)
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The s i g n o f q i s ambiguous and w i l l depend s t r i c t l y and d i r e c t l y on t h e
numerator o f t h e r i g h t - h a n d s i d e o f (57).
I t i s easy t o imagine s t r u c t u r e s
w i t h e i t h e r p o s i t i v e o r n e g a t i v e values f o r q a r i s i n g f r o m a p r i c e s t a b i l i z i n g
policy.
The p a r t i c u l a r l y s i m p l e case o f a,=bl=l
q= (1+B)(o~-ala:)/20:
has
(58)
>O i f f oG/o:>al.
I n t e r e s t i n g l y , t h i s s i g n c o n d i t i o n f o r q under t h e p r i c e s t a b i l i z a t i o n p o l i c y
i s i d e n t i c a l t o t h a t o f t h e o u t p u t s t a b i l i z a t i o n p o l i c y w i t h a,=bl=l.
However, t h e s c a l e o r magnitude o f q i s s m a l l e r o r l a r g e r i n t h e p r i c e p o l i c y
r e l a t i v e t o t h e o u t p u t p o l i c y , as (1+0)/2 i s more o r l e s s than u n i t y ,
respectively.
The minimum p r i c e - l e v e l v a r i a n c e a t t a i n a b l e i s
Min
ECpt-Ept 12=C(al+q+a2bl > 2 & + b ~ o ~ + ( a l + q ) 2 0 : l J 2 .
The f i n a l - f o r m equations f o r t o t a l o u t p u t , t h e p r i c e l e v e l , t h e i n t e r e s t
r a t e , and t h e money s t o c k a r e shown below.
(A.141,
J, q, and 8 a r e as g i v e n b y
(541, and (571, r e s p e c t i v e l y .
The f i n a l f o r m f o r t h e money s t o c k c a l l s f o r p o s i t i v e responses o f money
t o commodity supply and money demand shocks, and n e g a t i v e responses t o
commodity demand shocks.
A l l o f t h e signs o f t h e t o t a l d e r i v a t i v e s i n t h e
f i n a l f o r m a r e o f d e t e r m i n a t e signs, except d R t / d s t , whose s i g n depends
o n whether a,bl
exceeds u n i t y .
The i n t e r e s t r a t e always f a l l s i n response
t o observed s u p p l y shocks ( d R , / d ~ ~ - ~i >, O ) and always r i s e s i n response
t o demand shocks, even contemporaneously (dRt/dX,-i>O,
f o r nonnegative i ) .
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IX.
A Numerical Example
An example s t r u c t u r e i l l u s t r a t e s t h e workings o f t h e model economy under
various policies.
I t i s s p e c i f i e d by t h e f o l l o w i n g s e t o f values f o r t h e
s t r u c t u r a l parameters.
{ al=2, az=2/3, b l = l , D=1; p , = . 8 , i=1,2,3;
2
o,=l,
u:=5,
o<=2 )
The v a l u e o f a , i m p l i e s , f o r example, t h a t an i n c r e a s e i n t h e nominal
i n t e r e s t r a t e from 5 p e r c e n t t o 6 p e r c e n t would, f o r g i v e n l e v e l s o f income
and p r i c e s , lower r e a l money demand by a p p r o x i m a t e l y 1.9 p e r c e n t .
a,
A value o f
somewhat l e s s than one i s suggested by a p r i o r i t h e o r i z i n g on t h e
t r a n s a c t i o n s demand f o r money.
The commodity s u p p l y and demand e l a s t i c i t i e s
o f u n i t y were chosen m e r e l y because, of a l l ( e q u a l l y a r b i t r a r y ) values, t h e y
a r e t h e most s t r a i g h t f o r w a r d choices.
(Econometric evidence c u r r e n t l y
a v a i l a b l e does n o t p r o v i d e d i r e c t knowledge of s u p p l y and demand
elasticities.)
The r e l a t i v e s i z e s o f t h e d i s t u r b a n c e s attempted t o g i v e
c o n s i d e r a b l e scope t o demand-side i n f l u e n c e s on o u t p u t , and t o a l l o w f o r a
r e l a t i v e l y u n s t a b l e money demand f u n c t i o n .
Table 2 presents t h e p o l i c y parameters and some measures o f t h e s y s t e m ' s
r e s u l t i n g c y c l i c a l b e h a v i o r under t h e p o l i c i e s s a t i s f y i n g t h e f o u r p o l i c y
criteria.
3.
The signs of c o e f f i c i e n t s i n t h e f i n a l forms a r e d i s p l a y e d i n t a b l e
The example i s f r e e o f c o u n t e r i n t u i t i v e anomalies t h a t m i g h t occur w i t h
o t h e r numerical s t r u c t u r e s o r w i t h a r b i t r a r y p o l i c i e s .
I n t e r e s t i n g f e a t u r e s o f t h e example a r e t h a t s t a b i l i z a t i o n o f f i r m - l e v e l
o u t p u t c a l l s f o r a p o s i t i v e response of money t o lagged supply shocks and a
p o s i t i v e c o r r e l a t i o n between money and o u t p u t .
Under t h e p o l i c y t o s t a b i l i z e
p r i c e s , which a l s o minimizes o u t p u t d e v i a t i o n s f r o m t h e f u l l - i n f o r m a t i o n
l e v e l , money w i l l respond even more s t r o n g l y t o s u p p l y shocks, r e s u l t i n g
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- 30TABLE 2
OUTCOMES UNDER ALTERNATIVE POLICIES
I N THE EXAMPLE STRUCTURE
Pol icy
Parameters
C o r r e l a t i o n Between
Money and Output
P o l i c i e s Designed t o S t a b i l i z e :
Economy-Wi de
Money
Output
F i r m Output
P r i c e Level
-
-.33
+. 14
+.59
Economy-Wi de
Output
2.43
1.80
1.96
2.44
Firm Output1
3.22
2.44
2.12
2.40
15.47
12.86
3.69
0.62
2.22
1.84
0.88
0.62
0
26.91
24.42
36.00
3.26
4.33
4.81
5.86
Variance o f :
P r i ces
Real Wages'
Money
I n t e r e s t Rate
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TABLE 3: RESPONSES OF ENDOGENOUS V A R I A B L E S TO DISTURBANCE INNOVATIONS UNDER ALTERNATIVE P O L I C I E S
DISTURBANCE INNOVATION a n d P e r i o d o f O c c u r r e n c e
POLICY CRITERION
Endogenous V a r i a b l e
CONSTANT MONEY
Output
P r ices
I n t e r e s t Rate
ECONOMY-NIDE
OUTPUT S T A B I L I Z A T I O N
Output
Prices
I n t e r e s t Rate
Money S t o c k
F I R M OUTPUT S T A B I L I Z A T I O N
Output
Prices
I n t e r e s t Rate
Money S t o c k
P R I C E LEVEL S T A B I L I Z A T I O N
Output
Prices
I n t e r e s t Rate
M o n e y Stock
COMMODITY SUPPLY
t
t-1
t- 2
t
MONEY DEMAND
t-1
t- 2
COMMODITY DEMAND
t
t-1
t- 2
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in a correlation of +.59 between money and output. This policy actually
increases gross, economy-wide output variance relative to a constant-money
policy, reflecting the importance of supply shocks for output behavior,
despite the relative smallness of supply variance compared to demand and money
demand variance. However, the price-stabilization policy actually reduces
firm output variance relative to the policy that stabilizes economy-wide
output, illustrating the conflict between stability of aggregate output versus
firm output.
The effect of private agents' use of the information content of the
interest rate on output behavior is relatively slight in this example. Under
a constant money policy, the effect of current supply innovations,
dytlds,, is virtually unaffected, taking the value .529 if agents do not
use R, versus .534 if they do. The effect of money demand innovations,
dyt/dq,, is -.I76 if agents do not use R, and -.203 if they do. The
effect of demand innovations, dy,/dX,, is lowered from .353 to .331 if
agents use the interest rate. The behavior of prices and interest rates, like
that of output, is only slightly modified by the alternative assumptions about
use of current information. '
Output objectives, either local or global, require somewhat different
values for policy parameters depending on whether agents use the information
in the interest rate, but the behavior of the controlled system is largely
unaffected.
p,, p 2 , p3)
For example, the economy-wide output stabi 1 ization pol icy is (q,
= (1.44,
-.66, .8, -2.76) if agents do not use the
information in the interest rate, compared with (.92, -.82, .8, -2.34) if they
do. Similarly, the firm-level output stabilization policy is (q, P I ,
p2, p 3 ) = (1.44, 1.31, .8, -2.76) if private agents do not use the
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i n f o r m a t i o n i n t h e i n t e r e s t r a t e , compared w i t h (1 -16, 1.16,
t h e y do.
.8, - 2.53) i f
P r i c e , o u t p u t , and ex a n t e r e a l i n t e r e s t r a t e b e h a v i o r i s e n t i r e l y
u n a f f e c t e d by p r i v a t e use o f t h e i n f o r m a t i o n c o n t e n t o f t h e i n t e r e s t r a t e ,
g i v e n t h a t t h e p o l i c y r u l e i s chosen a p p r o p r i a t e l y .
Nominal i n t e r e s t r a t e
b e h a v i o r i s m o d i f i e d by p r i v a t e use o f t h e i n f o r m a t i o n c o n t e n t o f t h e i n t e r e s t
r a t e , b u t t h e m o d i f i c a t i o n i s s l i g h t i f 8 i s reasonably c l o s e t o z e r o .
The
response o f t h e nominal i n t e r e s t r a t e t o any contemporaneous i n n o v a t i o n i s
(1-8) times g r e a t e r i f agents do n o t use t h e i n t e r e s t r a t e t h a n i f t h e y do,
where 8 i s t h e u p d a t i n g c o e f f i c i e n t when agents do use t h e i n t e r e s t r a t e t o
update e x p e c t a t i o n s .
I n t h e example s t r u c t u r e , t h e r e l e v a n t v a l u e s o f 8 a r e
.15 and .08 i n t h e aggregate- level o u t p u t and f i r m - l e v e l o u t p u t s t a b i l i z a t i o n
p o l i c i e s , so t h a t t h e contemporaneous responses o f t h e i n t e r e s t r a t e t o shocks
a r e reduced by 15 and 8 p e r c e n t , r e s p e c t i v e l y , i f p r i v a t e agents i g n o r e t h e
information i n the i n t e r e s t r a t e .
As i s always t h e case under any p o l i c y
o b j e c t i v e , i n n o v a t i o n s i n p e r i o d t have e x a c t l y t h e same e f f e c t on a l l
endogenous v a r i a b l e s i n subsequent p e r i o d s t+j, j > O , r e g a r d l e s s of whether
agents use t h e i n t e r e s t r a t e .
I f t h e p r i c e s t a b i l i z a t i o n p o l i c y i s chosen, then t h e f u t u r e p r i c e
e x p e c t a t i o n i s i t s u n c o n d i t i o n a l mean and t h e r e f o r e cannot change w i t h c u r r e n t
shocks, even i f t h e l a t t e r a r e known.
Therefore, v a r i a t i o n s i n t h e i n t e r e s t
r a t e , w h i l e t h e y c o n t i n u e t o be i n f o r m a t i v e about c u r r e n t shocks, a r e
u n i n f o r m a t i v e about f u t u r e p r i c e s , so 8=0.
Consequently, if t h e p o l i c y
a u t h o r i t i e s choose t o s t a b i l i z e t h e p r i c e l e v e l , then a l l o w i n g p r i v a t e agents
t o observe t h e c u r r e n t i n t e r e s t r a t e has no e f f e c t on e x p e c t a t i o n s , and hence
no e f f e c t on t h e b e h a v i o r o f any endogenous v a r i a b l e s o r on t h e a p p r o p r i a t e
p o l i c y r u l e parameters.
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X.
Summary and Conclusion
While use o f contemporaneous i n f o r m a t i o n i s an i m p o r t a n t and p r o b l e m a t i c
t h e o r e t i c a l issue, t h e a n a l y s i s here suggests t h a t i t s p o l i c y i m p l i c a t i o n s may
n o t be q u a n t i t a t i v e l y i m p o r t a n t .
Indeed, i f t h e o b j e c t i v e s a r e l i m i t e d t o
p r i c e s t a b i l i z a t i o n , r e a l wage s t a b i l i z a t i o n , and s t a b i l i z a t i o n o f o u t p u t
around i t s f u l l - i n f o r m a t i o n l e v e l , then p o l i c y a u t h o r i t i e s need n o t concern
themselves w i t h the i s s u e (assuming t h e y know t h e economic parameters and
v a r i a n c e s ) , because t h e a p p r o p r i a t e p o l i c y r u l e i s i n v a r i a n t t o whether o r n o t
p r i v a t e agents use contemporaneous i n f o r m a t i o n .
The a n a l y s i s o f t h i s a r t i c l e r e a f f i r m s t h e c o n v e n t i o n a l e f f e c t i v e n e s s of
money s u p p l y responses t o t h e c u r r e n t i n t e r e s t r a t e , q, and responses t o t h e
lagged s t a t e o f t h e economy, p, i n a model w i t h two- period wage s t i c k i n e s s ,
even i f p r i v a t e agents use t h e i n f o r m a t i o n c o n t e n t o f t h e nominal i n t e r e s t
r a t e t o form rational expectations.
Thus, n e i t h e r s u p e r i o r i n f o r m a t i o n o f t h e
policymaker r e l a t i v e t o p r i v a t e agents nor i r r a t i o n a l e x p e c t a t i o n s a r e
necessary t o make b o t h q and p r e l e v a n t , unless p e r f e c t l y f l e x i b l e wages and
p r i c e s a r e assumed.
T h i s a r t i c l e a l s o analyzes a c o n f l i c t between s t a b i l i z a t i o n o f t o t a l
o u t p u t , aggregate o u t p u t , and o u t p u t r e l a t i v e t o i t s p e r f e c t - f o r e s i g h t
l e v e l - - a c o n f l i c t t h a t a r i s e s i n t h e presence o f supply shocks.
Stabilization
of o u t p u t around i t s f u l l - i n f o r m a t i o n l e v e l , r a t h e r t h a n i t s u n c o n d i t i o n a l
mean, may be more a p p r o p r i a t e and i s l i k e l y t o i m p l y a p o s i t i v e c o r r e l a t i o n
between money and o u t p u t .
The p o l i c y r u l e t h a t s t a b i l i z e s o u t p u t around i t s
f u l l - i n f o r m a t i o n l e v e l has t h e a d d i t i o n a l advantage o f b e i n g c o n s i s t e n t w i t h
m i n i m i z a t i o n of p r i c e and r e a l wage v a r i a n c e s .
Firm - output s t a b i l i z a t i o n may
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also require positive money-supply responses to supply shocks. Hence, while
multiperiod wage stickiness provides a conventional mechanism for the
effectiveness of pol icy (both q and p), it does not necessari ly, or even
probably, provide a persuasive rationale for countercyclical monetary policy,
if the latter is interpreted as a negative correlation between money and
output.
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Append i x
This appendix c o n t a i n s d e t a i l s o f the d e r i v a t i o n o f t h e s o l u t i o n t o t h e
model.
Some i n t e r m e d i a t e r e s u l t s a r e r e f e r r e d t o i n t h e t e x t i n p r o o f s of
p r o p o s i t i o n s and d e r i v a t i o n s o f p o l i c i e s s a t i s f y i n g v a r i o u s c r i t e r i a .
The s e t
o f t e n equations {(11),(12),(13),(14),(15),(16),(17),(18),(23),(24)}
constitutes the t r i a l solution.
Then c e r t a i n i d e n t i t i e s among t h e parameters
of t h e t r i a l s o l u t i o n a r e i m p l i e d by the s t r u c t u r a l equations (31, ( 6 ) , (81,
and ( 9 1 , and t h e a c c o u n t i n g i d e n t i t y ( 4 ) .
The f i r s t two s e t s o f i d e n t i t i e s ,
numbered (A.1) and (A.21, a r e i m p l i e d by t h e o u t p u t supply equations ( 3 ) .
(A.3) i s i m p l i e d by t h e money demand equation, (8); (A.4) i s i m p l i e d by t h e
aggregate demand equation, (6); (A.5) i s i m p l i e d by the money supply f u n c t i o n ,
(9); and (A.6) i s i m p l i e d by t h e accounting i d e n t i t y , ( 4 ) .
(A. 1 )
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These identities can be solved for the undetermined coefficients o f the trial
solution, the 17s.
Total output coefficients:
no=a+ko
n l = p l - 8 ( l + a l + q ~ C p l ( a l + a 2 b l + q ) -]GI
blpl
n2=-O(p2-p2>(l+al+q>GZ
n3=B(l+al+q)Cp3(al+q)+blp~lG3
n4=l-BCal+q+a2bl(l-8>1J
ns=-Rbl (1-B)J
n,=B(al+q)J
n,=-(B2/2b, ) ~ ( a , + a ~ b ~ + q ) ~]GI
-b~p~
n,=-(B2/2)(p2-p2 >(al+a2bl+q)G2
ns=(RL/2bl)(al+a2bl+q)Cp3(al+q)+blp31G3
Group one output:
Illo=a
n1 , = P I
n12=0
n13=0
nI4=l-OCal+q+a2bl
(1-WJ
II1s=-Rb1(1-8)J
nl,=O(al+q>J
fl17=0
n1 ,=o
n l,=o
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Group two output:
n 2
o=a
n2 l=PI-28(l+al+q)Cpl(al+a2bl+q)-blp~
]GI
1 1 *=~ 28p,bl (l+al+q)G2
r1~~=28(l+a~+q)Cp~(a~+q)+b~p~lG~
n,4=l-B[al+q+a2bl ( l - W J
I12s=-Bbl(l-8)J
n,,=R(a,+q)J
n,,=(8*/b1 ) ~ ~ ~ ( a ~ + a , b , + q ) * - ]GI
b~p~
n28=p282(al+a2bl+q)G2
n29=-(82/bl)(al+a2bl+q)[p3(a~+q)+b~p3]G3
(A. 9)
(A. 10)
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(A. 1 1
where
Gi={Cl+(al+q)(l-pi > l ~ ~ ( a l + a 2 b l + q ) + 2 b l ( l + a l + q l } 1i=1,2,3,
,
(A. 13)
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N W
b
7
u
n
-
rd
+
7
-
a
u
I 7
1 1
I I )
I +
I W
I
I
I
-
RIA
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Footnotes
1. Sargent and Wallace d i d n o t d e r i v e t h e o u t p u t s t a b i l i z i n g v a l u e o f q o r
analyze i t s dependence on t h e s t r u c t u r e . I n s t e a d , t h e y c o n t r a s t e d s t r i c t
money-supply r u l e s (e.g., money supply r u l e s w i t h q=O) w i t h i n t e r e s t r a t e
r u l e s ( e . g . , R t = f ( S t - , > > . One i m p o r t a n t r e s u l t was t h a t p r i c e s and t h e
money s t o c k were i n d e t e r m i n a t e under the l a t t e r r u l e . As argued i n Hoehn
(1987>, an i n t e r e s t r a t e r u l e i s n o t a f e a s i b l e p o l i c y o p t i o n i n models such
as t h a t o f Sargent and Wallace. I n l a t e r s e c t i o n s o f t h i s a r t i c l e , a complete
a n a l y s i s o f t h e o u t p u t - s t a b i l i z i n g and o t h e r o b j e c t i v e - s e e k i n g values o f q and
t h e i r dependence on t h e s t r u c t u r e i s p r o v i d e d .
2 . Other i m p o r t a n t papers c o n t r i b u t i n g t o a n a l y s i s o f contemporaneous,
heterogeneous i n f o r m a t i o n i n models w i t h a c r e d i t market i n c l u d e B a r r o (1980),
Weiss (1980), and K i n g (1982). The l a t t e r p r o v i d e s an e x c e l l e n t d i s c u s s i o n o f
t h e general i m p l i c a t i o n s o f heterogeneous c u r r e n t i n f o r m a t i o n f o r t h e p o l i c y
e f f e c t i v e n e s s p r o p o s i t i o n i n f l e x i b l e - p r i c e models, as w e l l as some f u l l y
worked- out numerical examples.
3. Dotsey and K i n g (1986) a l s o argued, i n e f f e c t , t h a t t h e b e h a v i o r o f o u t p u t
and o t h e r v a r i a b l e s depends on whether q i s f i n i t e o r i n f i n i t e , and i d e n t i f i e d
t h e l a t t e r case w i t h an i n t e r e s t r a t e r u l e (e.g., R t i s a s t r i c t f u n c t i o n of
S t - l , as i n R t = f ( S t - l ) ) .
But Hoehn (1987) argues t h a t i n t e r e s t r a t e
r u l e s a r e i n f e a s i b l e i n models such as t h e i r s and t h a t p o l i c i e s w i t h i n f i n i t e
q cannot be made o p e r a t i o n a l i n any model. I n any case, v a r i a t i o n s i n q a r e
i r r e l e v a n t i n t h e f l e x i b l e - p r i c e model o f Dotsey and King, so l o n g as q i s
f i n i t e . The d i s c u s s i o n i n t h e t e x t o f t h i s a r t i c l e r u l e s o u t i n f i n i t e q.
4. This nonuniqueness o f t h e o u t p u t - s t a b i l i z i n g p o l i c y c o u l d have been
e l i m i n a t e d i f , as i n t h e p r e s e n t a n a l y s i s , t h e p o l i c y r u l e ' s arguments e x c l u d e
elements t h a t augment t h e minimal s t a t e v e c t o r , S, beyond those necessary t o
achieve p o l i c y o b j e c t i v e s . Then, nonzero p would have been r u l e d o u t and
t h e c o n c l u s i o n would have been t h a t q i s r e l e v a n t f o r o u t p u t w h i l e p i s
i r r e l e v a n t f o r output.
5. McCallum (1987) argues c o n v i n c i n g l y t h a t Keynesian sticky- wage models t h a t
i n c o r p o r a t e e q u i l i b r i u m wage dynamics r e p r e s e n t a s i g n i f i c a n t advance o v e r
e a r l i e r models t h a t d i d n o t .
6. The f o r m u l a t i o n o f t h e r e a l ex a n t e i n t e r e s t r a t e employed here i s s i m i l a r
t o t h a t o f Canzoneri, Henderson, and Rogoff (1983). Indeed, t h e u p d a t i n g
s u p e r s c r i p t , "+", i s taken f r o m t h e i r n o t a t i o n . However, t h e y use
E L - , p t f o r t h e c u r r e n t p r i c e term, whereas I have assumed t h a t
demanders know t h e c u r r e n t p r i c e t h e y pay when making p u r c h a s i n g d e c i s i o n s .
The economy-wide p r i c e l e v e l , p t , i s t h e a p p r o p r i a t e c u r r e n t p r i c e t e r m f o r
aggregate demand i f agents know t h e p r i c e t h e y pay when making p u r c h a s i n g
d e c i s i o n s , even i f i n d i v i d u a l s ' purchasing p r i c e s d i f f e r . Although agents a r e
a l l o w e d t o know t h e i r purchasing p r i c e , i t i s assumed f o r s i m p l i c i t y t h a t t h e y
do n o t use t h a t i n f o r m a t i o n t o update t h e f u t u r e p r i c e e x p e c t a t i o n . T h i s
assumption i s a reasonable a p p r o x i m a t i o n i f i n d i v i d u a l p r i c e s a r e h i g h l y
v a r i a b l e so t h a t t h e y c o n t a i n l i t t l e g l o b a l i n f o r m a t i o n . The assumption c o u l d
be r e l a x e d o n l y i n t h e c o n t e x t o f a disaggregate model t h a t accounts f o r
d i f f e r e n c e s i n i n f o r m a t i o n a v a i l a b l e t o agents i n d i f f e r e n t markets. Such a
model r e q u i r e s a s e t of assumptions about l o c a l markets, i n t e r m a r k e t t r a d i n g ,
f a c t o r m o b i l i t y , market c l e a r i n g , and so f o r t h . A s p e c i a l problem i s t h a t
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these assumptions would have t o be c o n s i s t e n t w i t h t h e aggregate s u p p l y
behavior i n v o l v i n g wage s t i c k i n e s s . Such an a n a l y s i s , w h i l e d e s i r a b l e , i s
beyond t h e scope o f t h i s a r t i c l e , and does n o t seem p a r t i c u l a r l y necessary f o r
an a n a l y s i s o f t h e use o f g l o b a l i n f o r m a t i o n f r o m t h e i n t e r e s t r a t e , which i s
t h e most i m p o r t a n t source o f c u r r e n t i n f o r m a t i o n a v a i l a b l e t o monetary
authorities.
K i n g (1982, 1983) and Dotsey and King (1983) c o n s t r u c t e d d i s a g g r e g a t e
m a r k e t - c l e a r i n g models a l o n g l i n e s suggested by Lucas and B a r r o . T h e i r
a n a l y s i s focused on t h e e f f e c t s o f p r i v a t e a g e n t s ' use o f i d i o s y n c r a t i c
information from l o c a l p r i c e s i n generating a k i n d o f p o l i c y e f f e c t i v e n e s s
from feedback t o t h e lagged s t a t e of t h e economy, g . Dotsey and K i n g ( 1 986)
a l s o analyze a model i n which some agents know t h e f u l l s t a t e , o t h e r agents
know o n l y t h e lagged s t a t e , and a s i n g l e commodity market c l e a r s . Again, t h e y
f i n d g r e l e v a n t f o r o u t p u t v i a a heterogeneous i n f o r m a t i o n mechanism,
a l t h o u g h l i t t l e can be s a i d about t h e i n f l u e n c e o f p o t h e r than t h a t i t
exi sts.
A heterogeneous i n f o r m a t i o n mechanism i s n o t i n c o r p o r a t e d i n t o t h e p r e s e n t
a n a l y s i s f o r t e c h n i c a l reasons, b u t i t m i g h t c o e x i s t w i t h t h e mechanisms o f
p o l i c y e f f e c t i v e n e s s analyzed i n t h i s a r t i c l e , i n a f u l l y developed
disaggregate v e r s i o n o f t h e macroeconomic model employed.
7. I n p a r t i c u l a r , t h e t r i a l s o l u t i o n f o r t h e endogenous e x p e c t a t i o n o f t h e
p r i c e l e v e l a t t i m e t formed a t t i m e t- 2, Et-,pt, n e c e s s a r i l y i n v o l v e s terms
i n E ~ - Z U ~ -E~t -, Z ~ t - l , and E,-,X,-~,
even i f these a r e n o t i n c l u d e d i n t h e
s t a t e v e c t o r . T h i s f a c t i s apparent from e q u a t i o n (18) o f t h e t r i a l
s o l u t i o n , i n which l 1 3 1 E t - Z ~ t - l I, 1 3 2 E t - 2 ~ t - I ,and l 1 3 3 E t - 2 ~ t - w1 i l l
appear even i f ll,,,ll,,,and 1139 a r e s e t t o z e r o , as would be t h e case i f
E t - 2 ~ t - l , E t - Z ~ t - l , and E t - z x t - l were excluded f r o m S t .
8. For example, i f 8=2, aI=5, a2=213, and b l = 1 / 2 , then d y t l d s t - , approaches
- 2.84 as p l approaches u n i t y .
9. For an example i n which p l i s p o s i t i v e , c o n s i d e r a,=b,=l, al=5, 8=112,
{ p i = 1 / 2 ; i=1,2,3), o:=10,
and & = l .
Then o p t i m a l p o l i c y i s g i v e n by
{q, p l , p 2 , p 3 ) = { 5, 3, 112, -51, which i n v o l v e s p,>O. I n t h i s example,
d m t / d e t - l = p l + ( d R t / d s . - l > q = - 3 . 0 6 , showing t h a t money c o n t r a c t s i n response t o t h e
observed ( t - 1 ) s u p p l y shock.
10. I r o n i c a l l y , t h i s economy would, under a c o n s t a n t money p o l i c y , produce
o u t p u t t y p i c a l l y c l o s e r t o t h e n a t u r a l r a t e i f p r i v a t e agents d i d n o t use t h e
i n t e r e s t r a t e t o update i n f l a t i o n e x p e c t a t i o n s . The v a r i a n c e of d e v i a t i o n s o f
o u t p u t f r o m i t s n a t u r a l r a t e i s r a i s e d by .016 as agents use t h e i n t e r e s t
rate.
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References
B a r r o , R o b e r t J . " R a t i o n a l E x p e c t a t i o n s and t h e Role o f Monetary P o l i c y , "
J o u r n a l o f Monetary Economics, v o l . 2 (1976>, 1-32.
. "A C a p i t a l Market i n an E q u i l i b r i u m Business Cycle Model,"
Econometrica, v o l . 48 (19801, 1393-1417.
Canzoneri, Matthew B., Dale W. Henderson, and Kenneth S. R o g o f f . "The
I n f o r m a t i o n C o n t e n t o f t h e I n t e r e s t Rate and Optimal Monetary P o l i c y , "
Q u a r t e r l y J o u r n a l o f Economics, v o l . 98 (1983), 191-205.
Dotsey, M i c h a e l , and Robert King. " Monetary I n s t r u m e n t s and P o l i c y Rules
i n a R a t i o n a l E x p e c t a t i o n s Environment," J o u r n a l o f Monetary Economics,
V O ~ . 12 (1983), 357-82.
. " I n f o r m a t i o n a l I m p l i c a t i o n s o f I n t e r e s t Rate Rules," American
Economic Review, v o l . 76 (19861, 33-42.
F i s c h e r , S t a n l e y . "Long-Term C o n t r a c t s , R a t i o n a l E x p e c t a t i o n s , and t h e
O p t i m a l Money Supply Rule," J o u r n a l o f P o l i t i c a l Economy, v o l . 85 (1977),
191 - 205.
Hoehn, James G. " I n t e r e s t Rate Rules Are I n f e a s i b l e and F a i l t o C o m ~ l e t e
~ a c r o e c o n o mci Model s," Federal Reserve Bank o f Cleveland Worki ng' Paper 871 9
(December 1987).
King, R o b e r t G. "Monetary P o l i c y and t h e I n f o r m a t i o n Content o f P r i c e s , "
J o u r n a l o f P o l i t i c a l Economy, v o l . 90 (1982), 247-279.
. " I n t e r e s t Rates, Aggregate I n f o r m a t i o n , and Monetary P o l i c y , "
J o u r n a l of Monetary Economics, v o l 12 (1983>, 199-234.
McCallum, B e n n e t t T. "The Development o f Keynesian Macroeconomics,"
American Economic Review Papers and Proceedings, v o l . 77 (19871, 125-129.
, and J.K. Whitaker. "The E f f e c t i v e n e s s o f F i s c a l Feedback R u l e s
and Automatic S t a b i l i z e r s Under R a t i o n a l E x p e c t a t i o n s , " J o u r n a l o f
Monetary Economics, v o l . 5 (19791, 171-186.
Poole, W i l l i a m . " Optimal Choice o f Monetary P o l i c y I n s t r u m e n t s i n a Simple
S t o c h a s t i c ~ a c r oModel , " Q u a r t e r l y ~ o u r n a lo f ~ c o n o mcs,
i
v o l . 84 ( 1 9701,
197-21 6.
Sargent, Thomas J., and N e i l Wallace. " ' R a t i o n a l ' E x p e c t a t i o n s , t h e O p t i m a l
Monetary I n s t r u m e n t , and t h e Optimal Money Supply Rule," J o u r n a l o f
P o l i t i c a l Economy, v o l . 83, no. 2 (1975), 241-254.
Weiss, Laurence. "The Role f o r A c t i v e Monetary P o l i c y i n a R a t i o n a l
E x p e c t a t i o n s Model," J o u r n a l o f P o l i t i c a l Economy, v o l . 88 (19801, 221-233.
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