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Working Paper 8505
DYNAMICS OF FIXPRICE MODELS
by E r i c A. Kades

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
expressed a r e those o f t h e a u t h o r and n o t n e c e s s a r i l y
those o f t h e Federal Reserve Bank o f C l e v e l a n d o r t h e
Board o f Governors o f t h e Federal Reserve System.
The a u t h o r wishes t o thank p r o f e s s o r s Truman
Bewley, James Tobin, and John Geanakoplos o f Yale
U n i v e r s i t y f o r a wide range o f i n s i g h t s , i n s p i r a t i o n s ,
and c o r r e c t i o n s .

September 1985
F e d e r a l Reserve Bank of C l e v e l a n d

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DYNAMICS OF FIXPRICE MODELS

Abstract

T h i s paper examines t h e dynamics o f a c l a s s o f d i s e q u i l i b r i u m models
developed i n an e a r l i e r paper (Working Paper 8504) and uses b o t h g r a p h i c s and
a n a l y s i s t o show t h a t non- Walrasian e q u i l i b r i a can be steady s t a t e s for
d i s e q u i l i b r i u m models.

I n p a r t i c u l a r , i t i s shown t h a t Keynesian ( g e n e r a l

excess s u p p l y ) steady s t a t e s a r e t h e most l i k e l y outcome i n t h e model.

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

This paper s t u d i e s t h e time- paths of b o t h p r i c e s and s t o c k commodities i n
g e n e r a l e q u i l i b r i u m n o n - s t o c h a s t i c macromodels.

Our o b j e c t i v e i s t o show how

parametric p r i c e constraints (short- run f i x e d prices) explain the s t y l i z e d
f a c t s o f a disequilibrium world.

Our main r e s u l t i s t h a t non- Walrasian

e q u i l i b r i a can be s t a t i o n a r y s t a t e s o f these models.
We have discussed o b j e c t i o n s t o t h e f i x p r i c e methodology elsewhere and
concluded t h a t t h i s approach i s no more c o n t r o v e r s i a l t h a n t h e assumption o f
i n s t a n t a n e o u s market c l e a r i n g i n a l l markets a t a l l t i m e s .

There a r e ,

however, some f u r t h e r g e n e r a l comments about modeling t h e dynamics o f a
d i s e q u i l i b r i u m economy t h a t should be mentioned a t t h e o u t s e t .

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Our focus on dynamics stems f r o m t h e s t r o n g case made by F i s h e r (1984)
t h a t b e f o r e comparative s t a t i c s can be used w i t h confidence, t h e s t a b i l i t y o f
e q u i l i b r i a must be shown.

F u r t h e r , t h e speed o f adjustment must be r a p i d

enough t o a l l o w c l o s e a p p r o x i m a t i o n by i n s t a n t a n e o u s a d j u s t m e n t .

This

o b s e r v a t i o n i s e s p e c i a l l y i m p o r t a n t t o f i x p r i c e dynamics, s i n c e b e i n g o u t o f
e q u i l i b r i u m f o r extended p e r i o d s of t i m e g r e a t l y complicates t h e p a t h o f t h e
economy between steady s t a t e s .

So a l t h o u g h a g r e a t amount has been w r i t t e n

about t h e comparative s t a t i c s o f f i x p r i c e models, a p r e r e q u i s i t e f o r t h i s work

i s dynamic s t u d i e s o f s t a b i l i t y and adjustment speeds.

T h i s paper examines

only s t a b i l i t y issues.
The dynamics o f f i x p r i c e models, f o r t h e most p a r t , have v e r y r e c e n t
roots.

But P a t i n k i n (1965) should be mentioned i n passing.

He was t h e f i r s t

t o mention and a t t e m p t t o s t u d y t h e e f f e c t s o f " s p i l l o v e r s " f r o m r a t i o n i n g on
one market t o demand on another market.

The c a n o n i c a l example i s t h e

Keynesian case, i n which t h e i n a b i l i t y o f t h e l a b o r e r s t o s e l l a d e s i r e d l e v e l
o f t h e i r s e r v i c e s ( t h u s l o w e r i n g t h e i r income under f i x e d wages) leads them t o
demand l e s s o f t h e goods manufactured i n t h e economy.

To m a i n t a i n o u r f o c u s ,

we i g n o r e a l a r g e l i t e r a t u r e s t u d y i n g these i s s u e s on a more fundamental l e v e l
(e.g.,

Veendorp l19751) and l i m i t study t o t h e dynamics o f o u r s p e c i f i c models.

Our general dynamic framework i s a sequence o f temporary e q u i l i b r i a
(Grandmont 1982).

We imagine a d i s c r e t e sequence of t r a d i n g dates where

goods and l a b o r a r e t r a d e d f o r money.

The d i s t i n g u i s h i n g f e a t u r e o f

f i x p r i c e models i s t h a t a t each d a t e p r i c e s a r e exogenously f i x e d and t r a d e s
must c l e a r by non- Walrasian methods.
periods.

P r i c e movements take p l a c e between

Although t h i s approach seems l i k e t h e o n l y s e n s i b l e framework f o r

most o f t h e f i x p r i c e l i t e r a t u r e , i t s use i s n o t made e x p l i c i t by a l l

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authors

[

e.g.,

van den Heuvel (19831, Muellbauer and P o r t e s C19781, and

Honkapohja C19791, among o t h e r s ) .
I n any dynamic model (even n o n s t o c h a s t i c ) , e x p e c t a t i o n s and i n f o r m a t i o n
a r e key f a c t o r s .

For s i m p l i c i t y , though, we s i d e s t e p these i s s u e s w i t h t h e

simple assumption t h a t a l l agents have complete 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 so t h a t e x p e c t a t i o n s do n o t need t o be d i s t i n g u i s h e d f r o m
outcomes i n o u r c e r t a i n t y model.

Note t h a t t h i s i n no way bars o u r model

f r o m y i e l d i n g Keynesian outcomes.
I n t h e dynamic s p e c i f i c a t i o n o f t h e model, we must c o n s i d e r t h e p a t h s o f
b o t h p r i c e s and s t o c k v a r i a b l e s f r o m temporary e q u i l i b r i u m t o temporary
P r i c e movements a r e by f a r t h e more c o n t r o v e r s i a l .

equilibrium.

L i k e most

work t o date, we do n o t s p e c i f y how p r i c e s a c t u a l l y change by t h e s p e c i f i c
a c t s o f agents i n markets.
and demand":

r

p r i c e s r i s e f o r goods i n excess demand and f a l l f o r goods i n

excess supply.
auctioneer.

We merely adopt t h e c o n v e n t i o n a l " l a w o f s u p p l y

I t must be emphasized t h a t t h i s does n o t r e i n t r o d u c e t h e

The model economy s t u d i e d does n o t m y s t e r i o u s l y f i n d an

e q u i l i b r i u m p r i c e v e c t o r ; we merely assume market forces work i n t h e usual
direction.
I n f i x p r i c e models, t h e r e a r e a number o f c o m p l i c a t i o n s beyond t h i s
common a r b i t r a r i n e s s .

F i r s t , the law of s u p p l y and demand does n o t c l e a r l y

a p p l y t o d i s e q u i l b r i u m economies where, under o u r d e f i n i t i o n s , a l l goods may
be i n excess s u p p l y ( o r demand).

I f we a r e i n t e r e s t e d i n r e l a t i v e p r i c e

movements, how do we s p e c i f y which excess i s g r e a t e r ?

And should t h i s a l o n e

i n f l u e n c e which r e l a t i v e p r i c e r i s e s ? Second, t h e v e r y d e f i n i t i o n o f excess
demand i n d i s e q u i l i b r i u m models i s n o t c l e a r ; t h e r e w i l l be a number o f
possibilities.

No consensus e x i s t s on t h e c o r r e c t measure

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Once these d e t a i l s have been c l e a r e d up, and we add i n t h e s t o c k
adjustment e q u a t i o n s , we w i l l f i n d t h a t we have n o t one, b u t many, s e t s o f
d i f f e r e n t i a l e q u a t i o n s t h a t may d i c t a t e t h e p a t h of t h e economy.
o f equilibria (i.e.,

Each t y p e

each d i s t i n c t c o n s t r a i n t s t r u c t u r e ) w i l l have i t s own

set o f d i f f e r e n t i a l equations.

(Each dynamic system i s c a l l e d a regime.)

T h i s w o u l d n ' t be a problem i f t h e economy c o u l d n o t move a l o n g a dynamic
p a t h f r o m one regime t o a n o t h e r , b u t i n p r a c t i c e t h e r e i s n o t h i n g t o p r e v e n t
t h i s except d i r e c t assumption t o t h e c o n t r a r y , which we f i n d t o o
restrictive.

Some models l a c k even c o n t i n u i t y as t h e economy moves f r o m one

regime t o a n o t h e r .
show.

Even assuming c o n t i n u i t y , convergence i s n o t easy t o

Standard methods do n o t a p p l y , and when we can r e v i s e them t o s u i t

o u r economy, we s t i l l need e x t r a o r d i n a r y assumptions t o e s t a b l i s h s t a b i l i t y .
Because m a t t e r s become so messy i n dynamic s t u d i e s , we w i l l f i r s t s t u d y
t h e dynamic b e h a v i o r o f o u r s i m p l e r s t a t i c models t o g a i n some i n s i g h t s
b e f o r e t r y i n g t o extend o u r r e s u l t s t o t h e most general model.
Of p a r t i c u l a r i n t e r e s t w i l l be what Hansen (1970) l a b e l e d
"quasi- equilibria."

These a r e dynamic paths where r e a l v a r i a b l e s a r e f i x e d

( i n e q u i l i b r i u m ) b u t nominal v a r i a b l e s move i n p r o p o r t i o n .

We w i l l f i n d

t h a t a l t h o u g h f u l l y s t a t i o n a r y p o i n t s a r e i m p o s s i b l e t o l o c a t e except a t t h e
Walrasian outcome, i n t e r e s t i n g non- Walrasian q u a s i - e q u i l i b r i a e x i s t .

11.

The S t a t i c Model

The b a s i c atemporal model c o n s i s t s of one aggregate household, one
aggregate f i r m , and a government s e c t o r .

The f i r m s e l l s t h e good t o t h e

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household and buys l a b o r s e r v i c e s f r o m t h e household.
profits;

households maximize u t i l i t y .

Firms maximize

The government f i n a n c e s i t s purchases

by t a x i n g a1 1 p r o f i t s o f t h e f i r m s and f i n a n c e s d e f i c i t s , i f necessary, by
p r i n t i n g money ( o r d e s t r o y i n g money i f i t r u n s a s u r p l u s ) .
N o t a ti o n
u n i t s o f labor transacted,
u n i t s o f good t r a n s a c t e d ,
nomi n a l wage,
nominal p r i c e o f good,
r e a l wage; w = W/p,
exogenous parameter v e c t o r ; i n t h i s model x=(p,W),
end o f p e r i o d money h o l d i n g s ,
b e g i n n i n g o f p e r i o d money h o l d i n g s ,
r e a l money h o l d i n g s ,
r e a l government spending,
end o f p e r i o d i n v e n t o r y h o l d i n g s ,
beginning o f period inventory holdings,
U:C
(1 )

(2)

R + : u t i l i t y f u n c t i o n o f household.
We assume t h a t t h i s u t i l i t y f u n c t i o n has a l l t h e usual p r o p e r t i e s :

- twice d i f f e r e n t i a b l e ,
- quasi- concave,
- p a r t i a l d e r i v a t i v e s have s i g n s U , < 0; U,

> 0; U, > 0 .

F(L> : p r o d u c t i o n f u n c t i o n o f f i r m ,
- twice d i f f e r e n t i a b l e ,
-Ft > 0,
-F" < 0.

I n t e r t e m p o r a l adjustments a r e d i c t a t e d by t h e f o l l o w i n g e q u a t i o n s :

Government e x p e n d i t u r e s a r e f i n a n c e d i n two ways.

First, a l l profits

o f t h e f i r m s a r e taxed so t h a t we need n o t worry about t h e f i r m s h o l d i n g
money.

Any r e s u l t i n g d e f i c i t o r s u r p l u s i s financed by t h e c r e a t i o n o r

d e s t r u c t i o n o f money i n t r a d e f o r t h e good.
b y t h e household as money s a v i n g s .

(4)

A M = pg - r

=

WL- PY.

T h i s d e f i c i t must be accepted

A n a l y t i c a l l y t h i s says:

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Government demand i s n e v e r r a t i o n e d .
To model t h e f i r m ' s d e s i r e f o r i n v e n t o r i e s , we add a " v a l u a t i o n o f
s t o c k s " f u n c t i o n ( v a n den Heuvel 1983) t o t h e i r o b j e c t i v e f u n c t i o n .
l a b e l t h i s f u n c t i o n v ( i > or e q u i v a l e n t l y v ( x ) .

v maps R+ i n t o R,.

We
We

assume:
(5)

- v l > 0,
-v" < 0,
-v i s t w i c e d i f f e r e n t i a b l e .
We t h e n d e f i n e t h e f i r m s o b j e c t i v e f u n c t i o n as t h e sum o f p r o f i t s and

valuation o f inventories:
(6)

R(x>

=

r(x) + v(x).

Then o u r m a x i m i z a t i o n problems a r e :

(7)

Households:

MAX U(L,Y,M)

s.t.

F i rms :

MAX R(L,Y,i)

s.t.

M =
-

i

=

M + W1 - pY
1 +F(1> - y

2 0,
2 0.

T h i s economy f i t s t h e Arrow- Debreu framework (Debreu 1959), and W a l r a s i a n
e q u i l i b r i a e x i s t i n t h i s economy.

To s i m p l i f y m a t t e r s i n t h e dynamic a n a l y s i s

below, we d e s i r e t h e uniqueness o f ( W a l r a s i a n ) e q u i l i b r i u m i n o u r model.
we assume g r o s s s u b s t i t u t a b l i t y f o r a l l goods.

So

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

f o r o u r model i s d i s c u s s e d i n Working Paper 8503; we f i n d t h a t i t i s n o t v e r y
restrictive.
We c a l l t h e W a l r a s i a n q u a n t i t y d e c i s i o n s o f t h e agents ( a t a g i v e n ,
u s u a l l y d i s e q u i l i b r i u m , parameter v e c t o r ) n o t i o n a l q u a n t i t i e s (Clower 1965).
N o t a t i o n a l l y , t h e s e a r e marked w i t h an a s t e r i s k s u p e r s c r i p t .
r e f e r e n c e d b y an h s u p e r s c r i p t ; f i r m s a r e denoted b y an f .

Households a r e
So, f o r example,

we denote n o t i o n a l l a b o r s u p p l y by L h * o r good demand b y Y h * .

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E q u i l i b r i u m t h e o r i s t s p o s i t t h a t t h e Walrasian p r i c e v e c t o r i s somehow
a t t a i n e d so t h a t n o t i o n a l d e s i r e s l e a d t o balanced t r a d e .

I n s t e a d o f assuming

t h a t t h i s v e r y s p e c i a l Walrasian p r i c e v e c t o r i s found, t h e f i x p r i c e approach
imagines t h a t t h e p r i c e v e c t o r i s t r u l y p a r a m e t r i c
w i l l almost never be Walrasian.

determine a c t u a l t r a n s a c t i o n s .
models i s v o l u n t a r y t r a d e :

at a g i v e n

t r a d i n g d a t e and

More s t r u c t u r e must then be imposed t o
The most b a s i c r e q u i r e m e n t imposed i n f i x p r i ce

no agent i s ever f o r c e d t o t r a d e ( s u p p l y o r

demand) more o f a good than he d e s i r e s .

But s i n c e markets w i l l n o t , i n

g e n e r a l , c l e a r i n d i s e q u i l i b r i u m , agents w i l l p e r c e i v e q u a n t i t y c o n s t r a i n t s i n
f o r m u l a t i n g demand.
Benassy demands.

(8)

Q u a n t i t y - c o n s t r a i n e d demands a r e c a l l e d e f f e c t i v e o r

They a r e d e f i n e d by:

Households:

Firms :

Lh+

=

MAX

U(L.T,X)s u b j e c t

to g

+ wL - Y 2 0,

-

Yh+ = MAX u(L,Y,;>

s u b j e c t t o g + wL

L + = MAX R ( L , ~ , X )

s u b j e c t t o l+F(L)-Y 2 0,

R(T,Y,X)

s u b j e c t t o l+F(L)-Y 2 0,

f

Y +
f

=

MAX

Y 2 0,

a r e p e r c e i v e d c o n s t r a i n t s on t h e o t h e r market when e f f e c t i v e
where C and
demands a r e formed on a g i v e n market.
These demands d e f i n e a v o l u n t a r y t r a d e s e t t h a t w i l l , i n g e n e r a l , have
a large intersection.
transactions.

So more r e s t r i c t i o n s a r e necessary t o determine

We assume t h a t o n l y one s i d e o f a market can be r a t i o n e d - -

t h e agent w i t h t h e s m a l l e r e f f e c t i v e demand w i l l always have t h i s demand
fulfilled.

T r a n s a c t i o n s a r e t h e n determined by t h e i n t e r s e c t i o n o f two

minimal e f f e c t i v e demand curves.

To i n s u r e uniqueness o f d i s e q u i l i b r i u m we

assume t h e m o n o t o n i c i t y o f demands and some r e s t r i c t i o n s on t h e f i r s t
derivatives.

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F i x g r i c e e q u i l i b r i a a r e c a l l e d b y c o n v e n t i o n non- Walrasian.

They a r e

c l a s s i f i e d i n aggregated macroeconomic models l i k e o u r s , a c c o r d i n g t o which
s e c t o r s a r e r a t i o n e d i n which markets.

I n t h e f o l l o w i n g t a b l e we summarize

t h e p o t e n t i a l outcomes o f t h e model and p r o v i d e names f o r each.

Table I
Goods Market
excess supply
excess s u p p l y
excess demand
excess demand
b a l anced

Labor Market
excess s u p p l y
excess demand
excess s u p p l y
excess demand
b a l anced

Equi 1i b r i um Type
Keynesian (KE)
C l a s s i c a l (CE)
Underemployment (UE)
I n f l a t i o n a r y (IE)
Wal r a s i a n (WE)

We w i l l be most i n t e r e s t e d i n KE and I E s i n c e i t i s n o t a t a l l c l e a r
what d i r e c t i o n r e a l p r i c e s ( t h e r e a l wage) should change t o a l l e v i a t e t h e
non- Walrasian s t r u c t u r e of e f f e c t i v e

demands.

The law o f s u p p l y and demand

f a i l s t o g i v e a ready answer, and we may f i n d s t a t i o n a r y r e a l p r i c e p a t h s
away f r o m t h e WE.
We d e r i v e (assume) t h e s i g n s o f t h e d e r i v a t i v e s o f t h e n o t i o n a l and
e f f e c t i v e demands o f t h e agents w i t h r e s p e c t t o t h e parameters.

(9)

aLh*/am, aLh+/am < 0,
aLh*/aw, aLh+/aw > 0,
aYh*/am, a Y h + / a m > 0 ,
ayh*/aw, aYh+/aw > 0 ,

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I n dynamic s t u d i e s , we a r e i n t e r e s t e d i n t h e convergence o f t h e
parameters ( o r s t a t e v a r i a b l e s ) t o steady- states- - dynamic e q u i l i b r i u m o f
money, i n v e n t o r y s t o c k s , and p r i c e s .

Thus we make use of graphs due t o

Malinvaud (1977>, which show t h e range of

parameter values f o r which each

t y p e o f e q u i l i b r i a o c c u r s (WE, KE, I E , CE, o r UE).

Under o u r assumptions we

know t h a t each s e t o f parameter v a l u e s i m p l i e s a unique e q u i l i b r i u m .
The v e c t o r o f s t a t e v a r i a b l e s i s x = ( w , m , i ) .

We show t h e p o s i t i o n s o f

t h e e q u i l i b r i a i n a l l t h r e e 2-member subsets o f t h e parameter v e c t o r C(m,w>,

(m,i>,

(i,w>l.

To f i n d these r e g i o n s , we examine which c o n s t r a i n t s a r e b i n d i n g a t t h e
boundaries between two s t a t e s , and use t h e i m p l i c i t f u n c t i o n theorem t o
s o l v e f o r t h e d e r i v a t i v e o f one of t h e s t a t e v a r i a b l e s i n terms o f t h e
other.

I n most cases t h e s i g n of t h e slope o f t h e border i s d e t e r m i n a t e

under t h i s procedure; we make c l e a r g r a p h i c a l l y t h e cases where t h i s i s n o t
true.

U s i n g t h e f a c t t h a t a l l f o u r such l i n e s must meet a t t h e Walrasian

e q u i l i b r i u m and t h a t we know which s t a t e s a r e a d j a c e n t t o which o t h e r s ( b y
comparing c o n s t r a i n t s t r u c t u r e s ) we a r e a b l e t o p l a c e t h e f o u r r e g i o n s i n
each parameter subspace.

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Doing t h i s f o r each boundary i n each parameter subspace we d e r i v e t h e
f o l l o w i ng diagrams:

Figure 1

Divisions o f parameter spaces by equilibrium type

A l t h o u g h t h i s complete model i s more s a t i s f y i n g t h a n e a r l i e r models
(Malinvaud C19771, Bohm 119781, Honkapohja C19791) t h a t l a c k a s t o c k
variable f o r the f i r m , the t h i r d state variable (inventories) g r e a t l y
c o m p l i c a t e s dynamic a n a l y s i s .

Thus, o u r p r e l i m i n a r y dynamic i n v e s t i g a t i o n s

w i l l be conducted on s i m p l e r models l a c k i n g one o f t h e s t o c k v a r i a b l e s .

We

g r a p h i c a l l y summarize t h e i n v e n t o r y l e s s economy t o c a p t u r e t h e e s s e n t i a l
d i f f e r e n c e s when one s t o c k v a r i a b l e i s o m i t t e d .
Without i n v e n t o r i e s , the sole c r i t e r i o n i n the f i r m ' s p r o f i t
m a x i m i z a t i o n problem i s e f f i c i e n t p r o d u c t i o n
curves (i.e.

I t s two e f f e c t i v e demand

t h e Senassy demands L ~ 'and Y f ' > c o l l a p s e i n t o t h e

p r o d u c t i o n f u n c t i o n i n t h e t r a d e space (L,Y>.

Then i t makes no sense t o say

t h a t the f i r m i s
c o n s t r a i n e d i n b o t h markets, and UE disappears.
AND CE.

We s t i l l have WE, KE, I E ,

I n t h e two- dimensional s t a t e space (w,m> we can i n f o r m a l l y d e r i v e

t h i s graph by c o l l a p s i n g t h e UE r e g i o n o u t o f t h e diagram i n (w,m> space
d e r i v e d above f o r t h e general model (see f i g u r e l a ) .

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Figure 2

E q u i l i b r i u m l o c a t i o n s i n t h e parameter space f o r t h e
i n v e n t o r y l e s s model

For t h e i n v e n t o r y l e s s model, t h i s s i n g l e graph summarizes t h e e n t i r e
system.

A s i m i l a r graph i n ( w , i )

space, l a c k i n g CE, d e s c r i b e s t h e model

w i t h o u t money.

111.

Dynamics

General D i s c u s s i o n
There a r e two d i s t i n c t dynamics i n t h e model.

Money and i n v e n t o r y

movements comprise s t o c k dynamics, w h i l e p r i c e movements a r e market f o r c e
dynamics.

We f i r s t examine s t o c k s .

The household r e t a i n s money.

I n t h e one- period model above, t h e

a c c o u n t i n g i d e n t i t y f o r r e a l money h o l d i n g s a t t h e end of a p e r i o d was
d e f i n e d i n terms o f i n i t i a l h o l d i n g s p l u s t h e n e t o f t r a n s a c t i o n s :

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We d e f i n e t h e savings f u n c t i o n as t h e increment t o t h e w e a l t h h o l d i n g s o f
t h e household:
(11)

S(x>

WL(x>-pY(x>.

=

Then o u r money s t o c k i d e n t i t y i n r e a l terms i s
(12)

H= y +

S(x>,

Then t h e d i s c r e t e v e r s i o n o f money dynamics i s :
(13)

A M = R - M = S(X>,

To a v o i d t h e messiness o f d i s c r e t e systems, we approximate a l l d i f f e r e n c e
e q u a t i o n s w i t h continuous analogs.

Here we have:

Adding government bonds and a l l o w i n g f o r a more r e a l i s t i c d i v i s i o n o f
f i s c a l and monetary p o l i c y would c o m p l i c a t e t h e model w i t h o u t changing t h e
essentials o f t h i s story.

For a s t e a d y - s t a t e , t h e b e h a v i o r o f t h e

government i n i s s u i n g o r r e t i r i n g debt i n a l l forms must c o i n c i d e w i t h
savings b e h a v i o r o f households.

On t h e o t h e r hand, i f households a r e

a l l o w e d t o h o l d o t h e r assets ( i n v e n t o r i e s o r newly i n t r o d u c e d forms o f
w e a l t h ) , then o u r simple a c c o u n t i n g i d e n t i t i e s break down, and t h e model
might y i e l d d i f f e r e n t r e s u l t s .
The f i r m c a r r i e s i n v e n t o r i e s across t r a d i n g d a t e s .

The one- period

model's inventory equation i s :
(15)

i =

I +f(L> -

Y.

For n o t a t i o n a l s i m p l i c i t y we d e f i n e t h e i n v e n t o r y accumulation f u n c t i o n :
(16)

I(x
= >
f(L)

-

Y.

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Then o u r i n v e n t o r y adjustment d i f f e r e n c e
rc.

(17)

Ai = i(x) -

I=

equation i s :

I(x),

and i t s c o n t i n u o u s c o u n t e r p a r t i s :
(18)

i

=

I(x).

Now we can c o m p l e t e l y d e s c r i b e t h e a c t i v i t y o f t h e government.

The

p r o f i t s o f t h e f i r m ( t o be taxed 100 p e r c e n t ) a r e g i v e n by:
r ( x ) = pY - WL + pF(O),

(19)

where F(O) i s government demand.
(20)

g

=

We can r e w r i t e (19) as:

pF(0) = r ( x ) + s ( x ) .

This i d e n t i t y says t h a t government e x p e n d i t u r e s a r e f i n a n c e d by p r o f i t s and
savings.

A steady s t a t e i n money and i n v e n t o r y stocks r e q u i r e s t h a t

government spending mesh w i t h t h e aggregate behavior o f t h e p r i v a t e s e c t o r .
As l o n g as t h e f i r m cannot c o n v e r t p r o f i t s i n t o o t h e r s t o r e s o f w e a l t h ,
t h e i n t r o d u c t i o n o f o t h e r a s s e t s w i l l n o t change t h e r e s u l t s o f t h e model.
However, i f t h e f i r m can h i d e p r o f i t s b y c o n v e r t i n g them i n t o a d i f f e r e n t
(non-taxed)

f o r m b e f o r e t h e t a x c o l l e c t o r a r r i v e s , then o u r a c c o u n t i n g

i d e n t i t i e s a g a i n would become i n v a l i d , and we would have t o model t h e
dispensation o f retained p r o f i t s .
P r i c e dynamics a r e much l e s s s t c a i g h t f o r w a r d than t h e almost a c c o u n t i n g
f o r m o f s t o c k dynamics.

There i s l i t t l e agreement on how p r i c e dynamics

should be d e r i v e d f r o m t h e p r i m i t i v e elements o f a general e q u i l i b r i u m
system.

Even i n t h e simple Arrow- Debreu model, p r i c e adjustment by t h e

tatonnement i s e n t i r e l y ad hoc.

A l t h o u g h Arrow (1959) c l e a r l y o u t l i n e d t h e

d i f f i c u l t i e s i n v o l v e d , progress i n t h i s area has been slow.
R e c e n t l y , some f r e s h e f f o r t s have been made t o f o r m u l a t e more r i g o r o u s
p r i c e dynamics based on t h e e x p l i c i t b e h a v i o r o f maximizing agents.

This

r e q u i r e s t h e abandonment o f a l l a r t i f i c i a l c o n s t r u c t s such as t h e
auctioneer.

Very d e t a i l e d d e s c r i p t i o n s o f i n d i v i d u a l a c t i o n s (beyond c h o i c e

c r i t e r i a ) must be g i v e n .

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F i s h e r (1984) has c o n s t r u c t e d models where agents r e a l i z e t h a t markets
do n o t f i n d Walrasian e q u i l i b r i a and, based on such r e a l i z a t i o n s , agents may
change p r i c e s themselves.

The framework reduces a n a l y t i c a l l y t o a Hahn

Process w i t h a Lyapunov f u n c t i o n i n t a r g e t u t i l i t i e s .

Agents i n i t i a l l y

b e l i e v e t h e y can t r a n s a c t a l l t h e y d e s i r e a t p r e v a i l i n g p r i c e s , and thus
have a t a r g e t ( n o t i o n a l ) u t i l i t y i n each t r ' a d i n g p e r i o d .

But d i s e q u i l i b r i u m

i s a l l o w e d , and these t a r g e t t r a n s a c t i o n s may n o t o b t a i n .

Then assuming (as

we do) t h a t o n l y one s i d e o f a market can be c o n s t r a i n e d , agents r e a l i z e
t h a t i f t h e y have excess t a r g e t demandlsupply on a market, so do o t h e r
agents; thus, market p r e s s u r e s a r e g o i n g t o move p r i c e s t o a l l a g e n t s '
detriment.

They may then change these p r i c e s themselves t o t r y t o unload

excess s u p p l i e s o r purchase unmet demand.

But none o f t h e " s u p r i s e s " i n

u n r e a l i z e d t a r g e t t r a n s a c t i o n s can be b e n e f i c i a l .

Target u t i l i t y i s always

f a l l i n g , and can be shown t o converge under weak c o n d i t i o n s .
A l t h o u g h F i s h e r ' s model i s a p p e a l i n g as a more s o l i d f o u n d a t i o n f o r
p r i c e adjustments than t h e usual law o f supply and demand, o u r model i s much
r i c h e r than F i s h e r ' s i n o t h e r ways (he does n o t model s t o c k s and d o e s n ' t
d i s t i n g u i s h among d i f f e r e n t t y p e s o f e q u i l i b r i u m ) .

Superimposing F i s h e r ' s

p r i c e dynamics on t h i s c l a s s o f d i s e q u i l i b r i u m models produces an
a n a l y t i c a l l y d i f f i c u l t s e t of e q u a t i o n s .
Shapley and Shubik (1977) have i n t r o d u c e d another a p p e a l i n g model o f
p r i c e f o r m a t i o n d e r i v e d f r o m e x p l i c i t asssumptions on t h e n a t u r e o f market
interactions.

The economy i s modeled as a noncooperative game w i t h a

commodity money.

Agents send q u a n t i t y s i g n a l s t o t h e market t h a t

subsequently determine p r i c e s i n terms o f t h e money commodity.

The model i s

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e l e g a n t , i n g e n i o u s , and much l e s s c o n t r i v e d than a u c t i o n e e r w o r l d s .
O b v i o u s l y a Walrasian outcome w i l l n o t n e c e s s a r i l y be reached;
disequilibrium states are allowed.

But t h i s model determines p r i c e s

endogenously w i t h i n each p e r i o d , and t h e r e i s no p r o d u c t i o n .

Further, i t i s

much more d e t a i l e d than o u r models i n s p e c i f y i n g market i n t e r a c t i o n s .

For

these reasons i t i s i n a p p l i c a b l e t o o u r p r i c e dynamics.
For l a c k o f a s u p e r i o r a1 t e r n a t i v e , we f o l l o w t h e r e s t o f t h e
l i t e r a t u r e , and use t h e s t a n d a r d law o f s u p p l y and demand t o model t h e
adjustment o f p r i c e s .

P r i c e s r i s e i n t h e f a c e o f excess demand and f a l l

when t h e r e i s excess s u p p l y .

Thus, i n o u r model we have f o r t h e r a t e s o f

change o f nominal p r i c e s :

where Z Y , Z ' a r e some measure o f excess demand and h , and h, a r e
sign- preserving f u n c t i o n s .

To s i m p l i f y t h e study of dynamics we r e s t r i c t

h , and h 2 t o l i n e a r f u n c t i o n s i n demands and s u p p l i e s .

We d e f i n e D and

S ( a s some measure o f ) demand and supply ( t h e agent i n each case i s

obvious).

Then we have:
*

(22)

p/p

=

h l l(DY)-h12(SY),

e

WIN = h2 ,(Dl)-h,z(S'

>.

These equations may be t h o u g h t o f as t h e l i n e a r a p p r o x i m a t i o n o f more
g e n e r a l p r i c e dynamics.

The weights h , , ,

...,

h z 2 can be i n t e r p r e t e d as

speeds o f adjustment f o r p r i c e s i n r e a c t i o n t o t h e d i f f e r e n t demands.

I n t h e c a n o n i c a l Arrow-Debreu model t h e r e i s o n l y one p o s s i b l e measure
o f excess demand (up t o t h e f u n c t i o n a l f o r m t h e unique demands and s u p p l i e s
take)

The s t r i c t u r e on d i s e q u i l i b r i u m t r a n s a c t i o n s e l i m i n a t e s f u r t h e r

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complications.

But i n t h i s model, excess demand c o u l d c o n c e i v a b l y i n v o l v e

n o t i o n a l demands, t h e l a r g e r o f e f f e c t i v e demand, and t r a n s a c t e d q u a n t i t i e s
( t h e l e s s o r o f e f f e c t i v e demands).
NJ
E
j
ZJ
J

:
:
:
:

Define t h e n o t a t i o n :

n o t i o n a l demandlsupply f o r good j ,
e f f e c t i v e demandlsupply f o r good j ,
t r a n s a c t e d q u a n t i t y of good j ,
excess demand.

Then t h e r e i s a number o f p o t e n t i a l d e f i n i t i o n s o f excess demand i n
disequilibrium:
ZJ

(23)

=

N~ - E J ,

ZJ = NJ

-

Zj =

- j.

Ej

j,

There i s no f o r m a l method f o r s e l e c t i n g any o f these.

We have n o t

modeled t h e market w i t h enough d e t a i l t o d e t e r m i n e p r e c i s e l y which demands
a r e communicated t o t h e market.
t h e mechanics o f p r i c e movements.

The LSD i s n o t a s p e c i f i c d e s c r i p t i o n o f
We i n t e r p r e t n o t i o n a l q u a n t i t i e s as

m e r e l y w i s h f u l t h i n k i n g t h a t i s never communicated t o t h e m a r k e t .

Effective

q u a n t i t i e s a r e t h e f o r c e s t h a t a r e f e l t by t h e economy, and t h u s d r i v e p r i c e
dynamics v i a t h e LSD.

F u r t h e r , s i n c e t h e l e s s e r of t h e two e f f e c t i v e

demands determines t r a n s a c t i o n s , o u r d e f i n i t i o n o f excess demands i n v o l v e s
t r a n s a c t i o n s as w e l l .

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

o f t h e d e f i n i t i o n s o f excess demand ( d i f f e r e n c e , r a t i o ,
arbitrary.

zy

)

remains

For s i m p l i c i t y , we define excess demand i n terms o f d i f f e r e n c e s

(linearly):
(24)

. . .

=

y h+ - y f t ,

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Then o u r p r i c e dynamics e q u a t i o n s a r e :

a

(25)

plp

=

hl(ZY> = hll(Yh+> - h12(Yf+),

Thus t h e d i r e c t i o n o f p r i c e movements depends on what t y p e o f
e q u i l i b r i u m p r e v a i l s ; the r e s u l t s are given by t a b l e 2:

Table 2:

P r i c e Movements Across E q u i l i b r i a

The l a c k o f s t r i c t s t a t i o n a r i t y a t any p o i n t except t h e WE has l e d many
t h e o r i s t s t o u n f a i r l y r e j e c t t h e LSD i n f i x p r i c e models.

Even as r e s p e c t e d

a t h e o r i s t as J. M. Grandmont disavows o u r approach because

". . .

the

s t a t i o n a r y s t a t e s of t h e r e s u l t i n g dynamic system cannot d i s p l a y
unemployment." (Grandmont 1982, p . 916)

Yet i t i s c l e a r t h a t t h e

wage

may be s t a t i o n a r y i n KE o r I E ( b o t h i n v o l v i n g "unemployment" r e l a t i v e t o t h e
WE).

Grandmont m i g h t mean t h a t i n such a case a f i x e d money s u p p l y ( o r a

s t o c k l e s s model) would n o t p e r m i t a s t a t i o n a r y s t a t e o u t s i d e o f WE.

But

w i t h money dynamics i n t h e model, we can have (as we show) a
q u a s i - e q u i l i b r i a where t h e r e a l wage and t h e r e a l money supply a r e b o t h
stationary.

Thus, a l t h o u g h t h i s o b j e c t i o n i s r i g o r o u s l y c o r r e c t when

" s t a t i o n a r y s t a t e s " i s i n t e r p r e t e d i n terms of nominal v a r i a b l e s , i t
e n t i r e l y misses t h e p o i n t t h a t r e a l parameter values may be steady i n t h i s

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model a t p o i n t s i n t h e KE o r I E r e g i o n s .
The i n d e t e r m i n a t e n e s s o f r e a l p r i c e movements i n t h e KE and I E r e g i o n s
makes o u r a n a l y s i s more q u a l i t a t i v e ; t h e s t a t i o n a r y l o c u s o f t h e r e a l wage
l i e s i n t h e KE and I E r e g i o n s , b u t we know l i t t l e about i t s shape. We can
d e t e c t general tendencies b u t cannot f i n d closed- form s o l u t i o n s .

The

s t a t i o n a r y wage l o c u s must go t h r o u g h t h e WE p o i n t , and t h i s p r o v i d e s some
structure.
We now d i s c u s s some o f t h e d i f f i c u l t i e s i n s o l v i n g these d i f f e r e n t i a l
equations.

The s a l i e n t d i f f i c u l t y i s t h a t t h e s p e c i f i c f u n c t i o n a l forms o f

these g e n e r a l i z e d r e p r e s e n t a t i o n s depend on which e q u i l i b r i a we a r e i n
( f o r example S(x> takes on a d i f f e r e n t f o r m i n t h e KE r e g i o n t h a n i n t h e I E
r e g i o n s i n c e L and Y have d i f f e r e n t f u n c t i o n a l f o r m s ) .

As t h e economy

e v o l v e s , t h e e q u i l i b r i u m t y p e may s w i t c h , and a new dynamic system w i l l t h e n
govern movements.

A l l c o n v e n t i o n a l techniques f o r s o l v i n g systems o f

d i f f e r e n t i a l e q u a t i o n s must be m o d i f i e d o r abandoned.

It i s difficult t o

p i n p o i n t t h e steady s t a t e s o f t h e model, y e t perhaps t h i s c o m p l e x i t y i s
u n a v o i d a b l e i n modeling d i s e q u i l i b r i u m .
Second, none o f o u r assumptions on t h e uniqueness o f f i x p r i c e
t r a n s a c t i o n s i n a g i v e n p e r i o d i n s u r e s t h a t t h e r e w i l l be a unique
s t a t i o n a r y p o i n t t o t h e dynamic system f o r o u r d i s e q u i l i b r i u m model.

We

have assumed t h a t t h e Walrasian dynamic (tatonnement) analog of o u r economy
has a unique e q u i l i b r i u m .

This f o l l o w s almost a u t o m a t i c a l l y f r o m t h e

assumption o f gross s u b s t i t u t e s and t h e equivalence o f a dynamic tatonnment
model w i t h an atemporal one.

However, s i n c e o u r dynamics cannot be t i e d t o

a t e m p o r t a l p r i c e movements (where p r i c e s a r e f i x e d ) , uniqueness does n o t
c a r r y over.

We may have a denumerable, uncountable, o r even g e n e r i c s e t o f

s t a t i o n a r y s t a t e s t o o u r dynamic model.

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Non- uniqueness means t h a t s i n c e t h e system w i l l move toward d i f f e r e n t
r e s t p o i n t s based on i t s i n i t i a l l o c a t i o n , and t h e s t r e n g t h ( n o t j u s t
d i r e c t i o n ) o f v a r i o u s f o r c e s becomes r e l e v a n t .

T h i s i s much more d i f f i c u l t

t h a n t h e unique case and r o b s us of t h e p u r e l y q u a l i t a t i v e Lyapunov
function.

We c o u l d assume uniqueness

in one o f

t h e r e g i o n s and a p p l y a

Lyapunov f u n c t i o n t o t h e s e t of d i f f e r e n t i a l e q u a t i o n s g u i d i n g b e h a v i o r a t
that point t o test for stability.

But even i f t h i s gave a p o s i t i v e r e s u l t ,

we c o u l d n o t be sure t h a t t h e system never e v o l v e d i n t o a n o t h e r r e g i o n
somewhere d u r i n g i t s e v o l u t i o n toward t h e unique r e s t p o i n t .

Thus, t h e

m u l t i p l e regimes p r e v e n t even s t r o n g assumptions f r o m a d m i t t i n g t h e use o f
t h e u s u a l Lyapunov Theorem.
Dynamics o f d i s e q u i l i b r i u m models l i k e o u r s a r e u s u a l l y analyzed
q u a l i t a t i v e l y because o f t h e s w i t c h i n g regimes problem.

Phase diagrams i n

t h e s t a t e v a r i b l e space w i l l be one t o o l i n s t a b i l i t y a n a l y s i s .

We w i l l

a l s o examine l i n e a r i z e d v e r s i o n s o f o u r system a t p o s i t e d e q u i l i b r i u m p o i n t s
and t e s t f o r s t a b i l i t y o f these a p p r o x i m a t i o n s t o t h e t r u e dynamic p a t h .
F o l l o w i n g o u r main body o f dynamic r e s u l t s , we w i l l show how Lyapunov's
Second Theorem can be m o d i f i e d ( E c k a l b a r 1980) t o analyze models l i k e o u r s ,
b u t o u r example w i l l show t h e s t r o n g assumptions necessary t o r e a c h
meaningful r e s u l t s v i a t h i s r o u t e .

F i n a l l y we w i l l examine t h e a p p l i c a t i o n

o f F i l l i p o v methods t o t h e model ( I t o and Honkapohja 1983>, b u t here t o o t h e
p o i n t i s t h a t more t e c h n i c a l methods f a i l t o improve on t h e c o n c l u s i o n s o f
simp1 e r qua1 i t a t i ve t e c h n i q u e s .
Almost a l l dynamic analyses o f d i s e q u i l i b r i u m models f o c u s on t h e
s p e c i a l case o f f i r m s t h a t c a r r y no s t o c k v a r i a b l e s .

The reason i s

h i s t o r i c a l ; t h i s v e r s i o n was f o r m u l a t e d and understood much e a r l i e r than t h e
more g e n e r a l model.

F u r t h e r , t h e c o m p l i c a t i o n s o f t h e general model

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encourage examination o f s i m p l e r cases.
s i m p l e r models.

For t h i s reason, we w i l l examine

We a n a l y z e t h e model f i r s t w i t h o u t i n v e n t o r i e s , t h e n we

w i l l add i n v e n t o r i e s b u t remove money.

T h i s w i l l g i v e us some b a s i c

i n s i g h t s i n t o t h e dynamics of each s t o c k v a r i a b l e a p a r t f r o m more general
complications.

I t m i g h t be hoped t h a t we c o u l d d i r e c t l y g a i n s o l u t i o n s t o

t h e genera1 system by combining these two subsystems t h a t t o g e t h e r comprise
t h e e n t i r e economy.

U n f o r t u n a t e l y savings and i n v e n t o r i e s a r e n o t

independent i n t h e d e t e r m i n a t i o n o f each p e r i o d ' s t r a n s a c t i o n .

Inventory

d e c i s i o n s a f f e c t l a b o r c o s t s , which a f f e c t b o t h consumer b e h a v i o r and
government f i n a n c e ( t h r o u g h p r o f i t s ) .

Savings d e c i s i o n s , s i m i l a r l y ,

i n f l u e n c e t h e f i r m and t h e government.

Thus, o u r system i s t o o i n t e r t w i n e d

t o admit s o l u t i o n by examining each s t o c k v a r i a b l e s e p a r a t e l y .

But these

subsystem i n v e s t i g a t i o n s can p o i n t t o where we s h o u l d and should n o t l o o k
f o r s o l u t i o n s t o t h e general system.
There i s an immediate i m p l i c a t i o n o f t h i s procedure t h a t r e i n f o r c e s a
p o i n t t h a t we have discussed above.

I n t h e model w i t h o u t i n v e n t o r i e s UE

disappeared s i n c e w i t h o n l y t h e p r o d u c t i o n f u n c t i o n d i c t a t i n g ( p r o f i t
maximizing) b e h a v i o r , t h e f i r m cannot be doubly c o n s t r a i n e d .

We w i l l see i n

t h e model w i t h o u t money t h a t CE disappears s i n c e now t h e household l a c k s a
s t o c k v a r i a b l e , and so maximizes u t i l i t y s u b j e c t o n l y t o e f f i c i e n t
consumption.

N o t i c e t h a t i n e i t h e r case KE and I E e x i s t ; t h e y a r e r o b u s t t o

d i f f e r e n t s t o c k s p e c i f i c a t i o n s o f d i s e q u i l i b r i u m models.

We have a l s o

observed t h a t KE and I E (and WE) a r e t h e o n l y r e g i o n s where t h e r e a l wage
may be s t a t i o n a r y .

Combining these two r e s u l t s , we w i l l f o c u s most o f o u r

a t t e n t i o n on t h e KE and I E r e g i o n s o f t h e s t a t e space i n o u r search f o r
steady s t a t e s .

We cannot c o m p l e t e l y i g n o r e t h e CE and UE r e g i o n s , s i n c e t h e

economy may move t h r o u g h these r e g i o n s , and t h i s may a f f e c t t h e u l t i m a t e

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s t a b i l i t y o f t h e economy.

Our reasons f o r i g n o r i n g t h e WE ( u n l i k e many

o t h e r a u t h o r s ) have been d i s c u s s e d p r e v i o u s l y .

IV.

Model w i t h o u t I n v e n t o r i e s

We have a l r e a d y sketched t h e d i v i s i o n o f t h e parameter space between
d i f f e r e n t e q u i l i b r i u m types f o r t h e i n v e n t o r y l e s s model i n f i g u r e 2.

In

o u r dynamic systems, we w i l l supress t h e v a r i a b l e p and c o n s i d e r o n l y t h e
movements of t h e r e a l wage w

=

Wlp and t h e r e a l money s u p p l y m =M/p.

So we must m o d i f y o u r p r i c e and money dynamics so t h a t t h e y a r e i n r e a l
terms.

Taking l o g s and d i f f e r e n t i a t i n g w=W/p we have:

Then o u r r e a l wage d i f f e r e n t i a l e q u a t i o n , u s i n g ( 2 1 > , i s :

b=

(27)

wCh2(Z1) - h l ( Z Y > l .

With t h e l i n e a r LSD ( 2 5 > , we have:
(28)

= wCh21(Lf+)

-

h22(Lh+)

-

hll(Yh+) + h12(Yf+)l.

A s i m i l a r d e r i v a t i o n on (14) and (21) y i e l d s o u r e q u a t i o n f o r t h e dynamics

o f t h e r e a l money s t o c k :

h=

(29)

g

- r(x> +

mh,CZY(x>l

With o u r l i n e a r LSD, t h i s reads:

(30)

h=

g - r ( x ) + mChl l ( Y h + )

-

h12(Yf+)I.

Since t h e LSD p r e v e n t s t h e CE r e g i o n f r o m e v e r c o n t a i n i n g a steady s t a t e ,
we w i s h t o s i m p l i f y o u r first s t u d i e s o f t h i s model by p r o h i b i t i n g t h e r e a l

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wage f r o m r i s i n g above t h e Walrasian l e v e l w*.

By t h e m o n o t o n i c i t y o f t h e

KEICE and t h e CEIIE boundaries, t h i s r e s t r i c t i o n on t h e wage p r e v e n t s t h e
system f r o m e v e r e n t e r i n g t h e CE r e g i o n .

We a r e thus l i m i t i n g o u r s e l v e s t o

KE, I E and WE outcomes.
We now d e r i v e t h e t h e s t a t i o n a r y l o c u s f o r t h e KE and I E r e g i o n s i n
(m,w>

space.

and Y=Y +.
f

We have i n KE t h a t L=L F + and Y=Yh+, w h i l e i n I E L=Lh+
Then u s i n g t h e savings e x p r e s s i o n f o r money dynamics f r o m (14)

we have t h e f o l l o w i n g d e r i v a t i v e s f o r money s t o c k s :

I n t h e KE r e g i o n , then:

and so t h e l o c u s slopes upwards e x c e p t f o r v e r y low wage l e v e l s .

I n the I E

r e g i o n we have :

and so t h e h=0 locus slopes downward h e r e except f o r low wage l e v e l s .

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F i n a l l y , we have n o t h i n g i n o u r assumptions t o show t h a t t h i s l o c u s w i l l
be c o n t i n u o u s across d i f f e r e n t regimes.
so we must assume i t .

But w i t h o u t c o n t i n u i t y we a r e l o s t ,

C o n t i n u i t y i s a n a l y t i c a l l y a minimal assumption.

w i l l n o t assume d i f f e r e n t i a b i l i t y a t t h e boundary s i n c e i t i s n o t a

s u p e r f l u o u s i s s u e and d o i n g so would e l i m i n a t e t h e s w i t c h i n g regimes
problem; t h e d i f f e r e n t systems would, under d i f f e r e n t i a b i l i t y , l i n k up t o
f o r m a c o n t i n u o u s l y d i f f e r e n t i a b l e model t h a t would be amenable t o normal
methods o f s o l v i n g d i f f e r e n t i a l e q u a t i o n s .
Roughly, then, we have t h e f o l l o w i n g p i c t u r e i n t h e parameter space:

Figure 3

S t a t i o n a r y money l o c u s i n i n v e n t o r y l e s s model

F i r s t , n o t e t h a t o u r r e s t r i c t i o n on t h e wage l e v e l leads t o p o s i t i v e
s a v i n g s a t t h e WE.

T h i s stems f r o m t h e h i g h e r wage a t WE (hence low

p r o f i t s ) t h a t f o r c e s d e f i c i t f i n a n c e and a l l o w s households t o accumulate

We

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wealth.

Moreover, any wage above w' ( t h e maximum o f t h e m=O l o c u s w i t h

r e s p e c t t o w > leads t o d i v e r g e n t i n f l a t i o n a r y outcomes.
Drawing i n t h e phase arrows as d i c t a t e d by o u r e q u a t i o n s , we a l s o see
t h a t s t a t i o n a r y savings s t a t e s i n t h e KE area w i l l be s t a b l e w h i l e those i n
t h e I E r e g i o n a r e always u n s t a b l e r e s t p o i n t s .

T h i s i s an i n d i c a t i o n t h a t

Keynesian s t a t e s may be p r e v a l e n t i n t h e model.
Now we e n r i c h t h i s model by a d d i n g p r i c e adjustments ( i n w, t h e r e a l
wage) t o t h e dynamics.

To summarize what l i t t l e we can be sure o f w i t h

r e s p e c t t o p r i c e dynamics under o u r i n d e f i n i t e assumptions about them!

We

know t h a t t h e w=O l o c u s must go t h r o u g h t h e unique WE p o i n t , and t h a t t h e
remainder o f t h i s s e t l i e s i n t h e u n i o n o f t h e KE and I E r e g i o n s .

Beyond,

t h i s nothing i s d e f i n i t e .
S u p r i s i n g l y no one has made a s t r o n g case f o r a v e r y p l a u s i b l e
possibility:

t h e e n t i r e KE/IE b o r d e r may be s t a b l e i n t h e r e a l wage.

This

would f o l l o w under t h e assumption t h a t excess demands a r e c o n t i n u o u s a c r o s s
regimes (though n o t n e c e s s a r i l y d i f f e r e n t i a b l e ) s i n c e b o t h goods a r e i n
excess supply i n KE b u t i n excess demand i n I E .

I n t h i s case, we can e a s i l y

see t h a t t h e i n t e r s e c t i o n o f t h e m=O and t h e w=O l o c i g i v e s a s a d d l e p o i n t
e q u i l i b r i u m on t h e

KEJIE b o r d e r :

Figure 4 Saddlepoint equilibrium o n KE/IE border
under stationary wage locus I

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T h i s r e s u l t c e r t a i n l y f i t s t h e s t y l i z e d f a c t s of r e c e n t economic performance
we1 1, w i t h movements f r o m Keynesian r e c e s s i o n s t o i n f l a t i o r n a r y booms.

And

s t a t e s o f I E generate unemployment as w e l l as p r i c e pressu.res and p o r t r a y
what has been dubbed s t a g f l a t i o n .
The o n l y r e f e r e n c e i n t h e l i t e r a t u r e t o t h i s i s s u e (Honkapohja 1979)
p o i n t s o u t t h a t a l o n g t h i s b o r d e r , t h e marginal p r o d u c t o f l a b o r (mpl)
exceeds t h e r e a l wage and t h a t t h i s p l a c e s upward p r e s s u r e on t h e wage.

Yet

t h i s m a r g i n a l c o n d i t i o n f o r e q u i l i b r i u m h o l d s o n l y i n WE, and i t i s n o t
c l e a r t h a t mpl > w induces t h e f i r m t o h i r e more l a b o r i n successive p e r i o d s
i n v o l v i n g non- Walrasian s t a t e s .

I t depends on t h e s t r u c t u r e and l e v e l o f

t h e c o n s t r a i n t s o f t h e economy a t t h e temporary e q u i l i b r i a .
C o n t i n u i t y i n parameters i s one o f t h e weakest c o n d i t i o n s imposed on
demands i n t h e 1i t e r a t u r e .

Since no one has adop'ted t h i s h y p o t h e s i s , t h e

i m p l i c i t consensus seems t o be t h a t demands a r e d i s c o n t i n u o u s a t t h e KEIIE
b o r d e r i n t h i s model.

Most a u t h o r s p o s i t a s t a t i o n a r y r e a l wage locus

something l i k e t h e f o l l o w i n g :

111

Figure 5

S t a t i o n a r y wage locus I 1

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Combining t h i s w i t h t h e e q u i l i b r i u m s a v i n g s l o c u s , i t i s n o t a t a l l c l e a r
t h a t a q u a s i - e q u i l i b r i u m p a t h e x i s t s a t a l l , and e x i s t e n c e i n t u r n does n o t
imply s t a b i l i t y .

Figure 6

Here a r e some o f t h e s i m p l e r p o s s i b i l i t i e s :

Money and p r i c e dynamics i n i n v e n t o r y l e s s model

The l a c k o f r e s t r i c t i o n s on t h e w=O l o c u s l e a d s t o an unmanageable
proliferation o f possibilities.

We have n o t even drawn any cases where t h e

*

l o c u s w=O i s n o t monotone i n t h e s t a t e space. The p r o b l e m i s t h a t t h e LSD,
w h i l e h a v i n g d e c i s i v e p r e d i c t i v e f o r c e i n t h e CE r e g i o n (and i n t h e g e n e r a l
model, t h e UE) possesses

no

power o f r e s o l u t i o n i n t h e KE and I E r e g i o n s .

However, we can r e v e a l what f a c t o r s c o n t r o l t h e shape of t h e s t a t i o n a r y wage
l o c u s i n t h i s space.
I n c r e a s e d money b a l a n c e s have two c o n f l i c t i n g e f f e c t s on t h e excess
demands t h a t d e t e r m i n e r e a l wage movements.

By i n c r e a s i n g w e a l t h t h e y

i n c r e a s e demand f o r t h e good, and t h u s h i g h e r money b a l a n c e s t e n d t o d e p r e s s
t h e r e a l wage.

B u t t h e y a l s o p r o v i d e a s u b s t i t u t e f o r l a b o r , and t h u s

i n c r e a s e t h e wage r e q u i r e d t o h i r e a g i v e n volume o f l a b o r .

I f we assume

t h a t t h i s second f a c t o r dominates t h e f i r s t t o a l a r g e enough e x t e n t , a

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unique

q u a s i - e q u i l i b r i u m i n t h e KE r e g i o n r e s u l t s as i n diagram 6(a>

above.

The phase diagram i n d i c a t e s s t a b i l i t y f o r a p p r o p r i a t e i n i t i a l s t a t e s .

We can develop a g r a p h i c a l t o o l t o i l l u s t r a t e t h e f a c t o r s d e t e r m i n i n g
t h e type o f s t e a d y s t a t e t h a t w i l l be reached.

Based on o u r f i n d i n g s above,

we f o c u s on s t e a d y s t a t e s o f excess s u p p l y i n a l l markets.
For t h e 1i near p r i c e dynamics we have

(34)

4

w = w ( h 2 , L F + - h Z 2 L h +- h l l Y h +
#

m

=

:

+ h12Yf+),

g - r ( x ) - m(hl l Y h + - h 1 2 Y f + > .

We s o l v e t h e money e q u i l i b r i u m e q u a t i o n f i r s t s i n c e i t c o n t a i n s o n l y goods,
s u p p l i e s , and demands.

I t can be r e w r i t t e n as:

h 1 2 Y f +- h l l Y h +

(35)

=

C r ( x > - gl/m,

or
Yf+

=

Cr(x)

-

gllmhl, + ( h l 1 / h 1 2 > Y h + .

T h i s d e f i n e s a l i n e i n (Yh+,Yf')

space; any p o i n t a l o n g t h i s l i n e

d e f i n e s demand and supply o f t h e good c o n s i s t e n t w i t h e q u i l i b r i u m i n r e a l
money s u p p l y .

We a r e i n t e r e s t e d i n h i g h l i g h t i n g what f a c t o r s m i g h t cause

p a r t ( o r a l l ) o f these p o i n t s t o g i v e excess s u p p l y i n e f f e c t i v e
demands--those p o i n t s l y i n g above t h e 45 degree l i n e .
To s t a r t o u t w i t h t h e s t r o n g e s t case, i f t h e slope i s g r e a t e r than one
and t h e i n t e r c e p t p o s i t i v e , then t h e e n t i r e l o c u s l i e s above t h e 45'
l i n e and o n l y excess supply i n t h e goods market may p r e v a i l i n steady
states.

The s l o p e i s g i v e n by:

(36)

hil/hlz.

For t h i s t o be g r e a t e r than 1 , means t h a t t h e p r i c e of t h e goods i s more
s e n s i t i v e t o demand f a c t o r s than t o supply f a c t o r s .

This complements t h e

nominal wage s t i c k i n e s s we encounter below i n a f f i r m i n g t h a t Keynesian
outcomes a r e a s s o c i a t e d w i t h " s u p p l y s t i c k i n e s s " i n i n t e r - p e r i o d dynamics.

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The i n t e r c e p t i s :
Cr(x> - g l / h l z m .

(37)

So p o s i t i v i t y means:
r ( x ) - g > 0.

(38)

T h i s r e f l e c t s two f a c t o r s a s s o c i a t e d w i t h KE.

Low l e v e l s o f autonomous

demand and l o w e r wages (hence h i g h e r p r o f i t s r ) t e n d t o p r o d u c e KE i n
a t e m p o r a l models, and t h i s i s t h e e x t e n s i o n t o t h e dynamic s e t t i n g o f t h e s e
ideas.

I f t h e s e t w o c o n d i t i o n s h o l d , o u r d i a g r a m appears as f o l l o w s :

Figure 7

Excess s u p p l y of l a b o r as t h e o n l y p o s s i b l e s t e a d y s t a t e

S o l v i n g t h e e q u i 1 ib r i um wage e q u a t i o n w i 11 g i v e us s i m i l a r c o n d i t i o n s

f o r excess s u p p l y i n t h e goods m a r k e t .

We s u b s t i t u t e i n ( 3 5 ) t o supress t h e

demands and s u p p l i e s f o r t h e goods m a r k e t i n t h e f o l l o w i n g d e r i v a t i o n :
(39)
=

*

w = w(hr ,lf'
- hz2Lh' - h i lYh' + h i Z Y f + )

w [ h 2 , L f + - h , , ~ ~ +- h , l Y h '

+ h l 2 ( r ( x ) - g l / m h l r + ( h i l / h 1 2 ) Y h + l = 0.

This leads t o :
(40)

hrlLf'

-

hzzLh'

-

h i lYh'

+ [ r ( x ) - gI/m + h, lYh'

=

0.

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So o u r l i n e i n ( L h + , L f + > space i s :
(41 1

L h + = ( h 2 1 / h 2 2 ) L f ++ [ r ( x ) - g1/mh2L.

Again a p o s i t i v e i n t e r c e p t and a slope g r e a t e r than u n i t y w i l l g i v e excess
s u p p l y o f l a b o r i n e f f e c t i v e demands as t h e o n l y steady s t a t e s f o r t h e r e a l
wage.

The i n t e r p r e t a t i o n o f t h e i n t e r c e p t c o n d i t i o n i s t h e same as i n t h e

goods market above, and t h e s l o p e c o n d i t i o n here i s t h e c l a s s i c a l Keynesian
case, i n which wages a r e i n e l a s t i c i n s u p p l y f a c t o r s .
These a r e s u f f i c i e n t c o n d i t i o n s f o r
s u p p l y i n b o t h markets (KE).
c o n d i t i o n s , KE c o u l d p r e v a i l .

steady s t a t e s t o e x h i b i t excess

I t i s easy t o see t h a t under more r e l a x e d
F u r t h e r t h e p r e c i s e p o i n t s e l e c t e d on b o t h

l i n e l o c i i s determined s i m u l t a n e o u s l y i n a general e q u i l i b r i u m t h a t cannot
be i l l u s t r a t e d here.

But t h e c o n d i t i o n s t h a t can g i v e r i s e t o Keynesian

s t e a d y s t a t e s a r e c l e a r ; we can s a f e l y study t h e i r s t a b i l i t y w i t h o u t
w o r r y i n g t h a t we a r e examining a vacuous case.
Having seen t h a t t h e e x i s t e n c e o f Keynesian steady s t a t e s i s n o t a
r a r i t y , we now s t a t e Theorem I. (For p r o o f , see appendix).

V.

Theorem I

Under t h e b a s i c assumptions made about t h e i n v e n t o r y l e s s economy, KE
steady s t a t e e q u i l i b r i a are s t a b l e .

T h i s p r o o f i s an improvement o v e r e a r l i e r attempts because no ad hoc
assumptions beyond t h e b a s i c s t r u c t u r e of t h e model a r e necessary f o r t h e
s t a b i l i t y proof.
techniques .

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

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The system will have imaginary roots and oscillate (stably, unstably, or
critically, depending on the sign of the real part of the characteristic
roots) if, and only if:
(42

>

( a , , + arz)' - 4(al larr - a l r a r

< 0.

It is difficult to pin down the sign of this expression without gratuitous
and economically meaningless asssumptions. Some authors (Malinvaud 119771,
Honkapohja [19791, and Bohm [19781) have posited the plausibility of cycling
in this model along these lines. Other authors have argued for cycling from
the existence of various saddle-point equilibria discussed above.

Blad and

Zeeman (1982) have constructed a stochastic model with expectations based on
past observations that produces cycling between the KE and IE regions.
Unfortunately they require extended assumptions that we are wary of making and
their modeling of expectations introduces undesirable controversies.
Sneessens, in estimating a variant of the model for the Belgian economy, found
that the model cycled between KE and IE states in the 1970s.
Returning to the stability of the inventoryless model, we briefly examine

KE in the case in which we remove our restriction on the wage level and admit
the possibility of periods of CE. If we retain all other assumptions, we have
the same outcome.

Figure 8 KE in the inventoryless model with unrestricted wage level

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The s l o p e o f t h e e q u i l i b r i u m savings l o c u s i n t h e CE r e g i o n cannot be signed
unambiguously, b u t s i n c e t h e e q u i l i b r i u m wage l o c u s does n o t e n t e r t h e CE
r e g i o n , t h i s i s n o t so d i s t u r b i n g .

The phase diagram suggests s t a b i l i t y as i t

d i d when we r e s t r i c t e d t h e l e v e l o f w.
I t i s bothersome t h a t o u r s t a b i l i t y r e s u l t s a r e e i t h e r i n d i c a t e d o n l y by

phase diagrams o r h o l d o n l y f o r a l i n e a r i z e d v e r s i o n o f t h e system, and so a t
best, establish local s t a b i l i t y .

The most common t o o l i n d e m o n s t r a t i n g

.

s t a b i l i t y f o r general ( n o n l i n e a r ) systems o f d i f f e r e n t i a l equations i s
Lyapunov's Second Theorem.

We d i s c u s s what must be done t o a p p l y t h i s

t e c h n i q u e t o o u r model w i t h regime s w i t c h i n g .
Assume a Lyapunov f u n c t i o n V proves t h e s t a b i l i t y o f a system x=g,(x)
a t x*.

Now l e t a new system x=g2(x) o f d i f f e r e n t i a l e q u a t i o n s be

d e f i n e d o v e r t h e same r e g i o n .

Assume:

- x * i s a1 so an equi 1i b r i u m o f t h e new system g, ;
- x * can be shown s t a b l e w i t h t h e same Lyapunov f u n c t i o n t h a t g i v e s
s t a b i l i t y f o r system g l .
Now d e f i n e a combination o f t h e two systems i n t h e same phase space:

where
Sl v

SZ

=

e n t i r e phase space.

Me must f u r t h e r assume t h a t :
(44)

gl(x)

=

~ z ( x > :x

E

[S1 n

S21.

Then we can t r i v i a l l y a p p l y Lyapunov's Theorem t o show t h e s t a b i 1 it y o f x *
i n t h e h y b r i d system.

F u r t h e r , t h e e x t e n s i o n t o many regimes w i t h t h e same

assumptions a p p l i e d t o each a d d i t i o n i s s t r a i g h t f o r w a r d .

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The p r i c e o f p u t t i n g t h e theorem t o such d i r e c t use i s e x o r b i t a n t i n terms
o f s u f f i c i e n t assumptions.

I t assumes a u n i q u e and common e q u i l i b r i u m p o i n t

t o t h e s e p a r a t e l y d e f i n e d systems; t h i s would be a f l u k e i n o u r model.
F u r t h e r , i t r e q u i r e s e q u a l i t y of t h e systems on t h e i r b o r d e r , w h i c h i s
s t r o n g e r t h a n t h e c o n t i n u i t y assumption t h a t we u t i l i z e .

E c k a l b a r (1980)

d e v e l o p e d and a p p l i e d t h e theorem t o a much s i m p l e r economy t h a n o u r s :
- no s t o c k s ,
- l a b o r s u p p l y e x o g e n o u s l y f i x e d , and
- t h e p r i c e a d j u s t m e n t e q u a t i o n t a k e s a s p e c i a l form g r a t u i t o u s t o
a p p l y i n g t h e theorem.

I t does n o t appear t h a t t h e theorem can be e x t e n d e d t o economies l i k e o u r s .
Honkapohja and I t o (1983) p r e s e n t F i l l i p o v ' s method as a more p o w e r f u l
t o o l f o r s o l v i n g problems w i t h regime s w i t c h i n g .

This generalization of

L y a p u n o v ' s method p e r m i t s t h e s o l u t i o n t o i g n o r e b e h a v i o r o f t h e system o n
any s e t o f measure z e r o , l i k e t h e b o u n d a r i e s o f o u r system.

Thus, t h e

method can be extended t o t h e more g e n e r a l case i n w h i c h even d i s c o n t i n u i t y
i s p e r m i t t e d on t h e b o r d e r s between r e g i m e s .

However, i t i s a p p l i e d t o an

economy s i m i l i a r t o t h e one E c k a l b a r s t u d i e d w i t h h i s s t r a i g h t f o r w a r d
Lyapunov f u n c t i o n and c a n n o t be used t o s o l v e o u r s e t s o f d i f f e r e n t i a l
equations.
Thus, a l t h o u g h a t t e m p t s have been made t o s t r e n g t h e n t h e c o n c l u s i o n s o f
dynamic a n a l y s i s o f d i s e q u i l i b r i u m models by a p p l y i n g more p o w e r f u l
m a t h e m a t i c a l methods, t h e s e s t u d i e s h a v e n ' t reached f r u i t i o n .
s t i l l n o e l e g a n t approach t o t h e r e g i m e - s w i t c h i n g problem.

There i s

I n l i g h t o f the

s h a r p l y d e c r e a s i n g m a r g i n a l r e t u r n s t o t h e use o f t h e more s o p h i s t i c a t e d
m a t h e m a t i c a l t o o l s , t h e r e s t o f o u r dynamic s t u d i e s s t i c k s t o b a s i c methods.
Perhaps s i m u l a t i o n s o f t h e s e economies o v e r a b r o a d range o f p a r a m e t e r
s e t s w i l l p r o v i d e more c o n v i n c i n g e v i d e n c e o f t h e i r dynamic t e n d e n c i e s .

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Malinvaud (1980) is the only author we have read who has pursued this avenue
of inquiry.

Keynesian outcomes abound in his simulations, although his

model is much different than ours, and he makes some very specific
assumptions that might not be necessary.

Model without Money
We now allow the firm a stock variable (inventories) and remove money from
the household (and government) sectors. Our procedures are parallel to the
case with only money.
This model requires some changes in our framework, since the government
deficits/surpluses cannot be financed without the debt instrument money. We
could allow any level of government expenditure and replace money with
inventories. A balanced budget would have g equa'l to the hypothetical profits
of the firm, wi th the government taxing a1 1 of these inventory profits.

When

the government ran a deficit, it would expropriate the required amount of the
good from the firm's normal inventories; a surplus would be managed by the
firm retaining 'excess' profits in the form of higher inventories. But there
would exist levels of g that could not be financed (depending on stocks and
production of the good), so we would have to restrict the size of the
government deficit/surplus. T o avoid these complications, we instead let the
level of profits define the size of (now always balanced) government
expenditures. We still tax profits 100 percent, but permit no deficits. We
no longer need money; all transactions are barters.
Since the household has no stock-variable decisions to make, it simply
maximizes utility by choice of the desired level of work (which immediately

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i m p l i e s consumption, s i n c e t h e r e i s no s t o r a g e ) .

Thus i n t h e moneyless model,

i t i s households t h a t c a n n o t c o n c e i v a b l y be c o n s t r a i n e d i n b o t h t h e good and
t h e l a b o r m a r k e t , and CE c a n n o t o c c u r i n t h i s model.

On t h e o t h e r hand,

i n v e n t o r y d e s i r e s m i g h t l e a d t o s i t u a t i o n s where t h e f i r m i s c o n s t r a i n e d i n
b o t h l a b o r purchases and good s a l e s , so UE r e a p p e a r .

Of c o u r s e KE and I E

remai n.
Our p a r a m e t e r space i s now x = ( i , w > .

The d i v i s i o n between t h e t h r e e

p o s s i b l e s t a t e s can be most e a s i l y seen by c o l l a p s i n g t h e CE r e g i o n o u t o f
t h e diagram i n ( i , w )

Figure 9

space f o r t h e g e n e r a l model i n f i g u r e l ( c ) .

D i v i s i o n o f p a r a m e t e r space i n moneyless model

S i n c e t h e r e a l wage i n c r e a s e s unambiguously i n t h e UE r e g i o n , we know
t h a t t h e r e c a n n o t be even a q u a s i - e q u i l i b r i u m t h e r e ; we a g a i n r e s t r i c t t h e
domain o f t h e wage, t h i s t i m e bounding i t below by w " so t h a t we do n o t have
t o c o n s i d e r t h e UE r e g i o n i n o u r f i r s t e x a m i n a t i o n o f t h i s model.

The

m o n o t o n i c i t y o f t h e KEIUE and IEIUE b o r d e r s assures us o f t h i s .
The s t a t i o n a r y i n v e n t o r y l o c u s i s d e r i v e d as i n t h e p r e v i o u s model w i t h
money.

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

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equilibrium condition, shows that the i=O locus slopes downward in the IE
region and upward in the Keynesian region for (i,w) space.

Figure 10 Stationary inventory locus in moneyless model

This partial model, like the previous one, immediately suggests that KE
are the most likely candidates for stable quasi-equilibria. Again this is
contingent on our assumption that the i=O locus is continuous on the KEIIE
border.
Our general comments on price dynamics will not be repeated. If we
accept the continuity of excess demands across regimes, we must have that
the w=O locus is the KEIIE border. The phase diagram indicates that an
equilibrium along the KEIIE boundary will be oscillatory, and stability is
not clear.

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Figure 1 1 Oscillatory equilibria in moneyless model
under stationary wage locus I
If we reject this version of the equilibrium real wage locus and posit a
more general form, we again only know that the w=O locus lies in the KE and

IE regions and goes through the WE point. This case may yield an
oscillatory KE. As in the previous model, there is an abundance of other
possi bi 1 i ties.

Figure 12 Dynamics of moneyless model with wage dynamics I 1

We can heuristically argue when the system has an oscilllatory
equilibria. Increased inventories (cetaris paribus) decrease the demand for

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l a b o r i n a g i v e n p e r i o d and t h u s depress t h e r e a l wage.

However t h e y a l s o

tend t o r a i s e t h e d e s i r e d s a l e s o f t h e f i r m , p l a c i n g downward p r e s s u r e on
p r i c e s and t e n d i n g t o push up t h e r e a l wage.

I f t h e second i n f l u e n c e

dominates t h e f i r s t o v e r an a p p r o p r i a t e range o f i n v e n t o r y l e v e l s , we have
an o s c i l l a t o r y Keynesian e q u i l i b r i u m as i n f i g u r e 12.
T r y i n g t o e s t a b l i s h t h e l o c a l s t a b i l i t y o f t h i s KE b y a n a l y z i n g t h e
l i n e a r i z e d system about t h e p o i n t proved u n e n l i g h t e n i n g .

Too many o f t h e

s i g n s a r e i n d e t e r m i n a t e , and s t a b i l i t y cannot be d i r e c t l y e s t a b l i s h e d as i n
t h e p r e v i o u s case.

We n o t e , however, t h a t i n s t a b i l i t y i s no more a p p a r e n t

than s t a b i l i t y when t h e l i n e a r i z e d system i s examined.
F i n a l l y , i f we remove t h e a r t i f i c i a l r e s t r i c t i o n on t h e r e a l wage l e v e l ,
so t h a t t h e dynamic p a t h may move through t h e UE r e g i o n we g a i n l i t t l e
information.

The slope o f t h e i = O l o c u s i s i n d e t e r m i n a t e i n t h e UE r e g i o n ,

b u t t h i s d o e s n ' t a f f e c t any o f o u r qua1 it a t i v e re'sul t s .

The General Model
We w i l l now examine t h e s t a b i l i t y o f KE i n t h e general model w i t h b o t h
money and i n v e n t o r i e s .

U n f o r t u n a t e l y o u r 'main t o o l - - t h e phase d i a g r a m - - w i l l

be u n a v a i l a b l e t o us.

With one s t a t e v a r i a b l e (one f i r s t o r d e r e q u a t i o n ) , a

phase diagram t r i v i a l l y g i v e s t h e s t a b i l i t y o f any e q u i l i b r i u m p o i n t .

In

two- dimensional systems, i t i s n o t always c l e a r , b u t i t does h e l p i l l u s t r a t e
general tendencies.

But as w i t h a l l g r a p h i c t o o l s i t i s almost c o m p l e t e l y

u s e l e s s i n t h r e e dimensions.
O f course t h e m o d i f i e d F i l l i p o v and Lyapunov techniques a r e even l e s s
h e l p f u l here t h a n t h e y were i n t h e s i m p l e r cases.

Thus, f o r t h e g e n e r a l

model, we f o l l o w t h e suggestions o f o u r a n a l y s i s above and p o s i t t h e

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existence of a KE quasi-equilibrium and examine its local stability by
studying the linearized version of the system about the point.
Our generic form for the differential system in the general model is:

Translating to the origin and taking the linear approximation of the system we
have :
*
m =

(46)

[aAl/amlm + [ a ~ ~ / a w i+w [aA,/aili,

1

We can denote this system by x

PI

(47)

x

la12al

=

Ax or more explicitly:

a z l a r r a rw3

=

L a 3 1 a , ~ a 3i3,

where the coefficients of the system are given by:
al

=

aAl/am = -ar/am + mh, , a ~ ~ + / mhl,aYh+/am,
am

a l z = aA1/aw = -ar/aw + m h l l a Y h + / a -mhlraYh+law,
w
a l , = aAl/ai = 0.

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a,,

a,,

=

aA,/am

=

a~,/aw =

( F ~ ) ~ L ~ +-/ aYf+/aw,
~W

=

aA,/ai

( ~ ' ) a ~ ~ + / -a iaY +/ai.

=

=

0,

h

We can then prove:
Theorem I1
Keynesian equilibria of the general model are stable. See appendix for
proof.

VI. Summary and Conclusions

The essence of Walrasian equilibrium theory is that prices clear
markets. The essence of non-Walrasian equilibrium theory is that they do
not; quantities adjust faster than prices, and some agents are rationed.
Both approaches have developed rigorous atemporal models proving the
existence of equilibrium. Although Walrasian static models are more
elegant, they agree less with the stylized facts of the world. We have seen
that unemployment is natural in non-Walrasian worlds. However, unemployment
must be forced into Walrasian models with ad hoc specifications on
information, utility functions, technology shocks, or other areas. At least
these extensive efforts to coax employment swings out of equilibrium models
show that Walrasian theorists realize the existence of unemployment. But we
believe non-Walrasian models capture a greater slice of the reality of
markets and the causes of unemployment.

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Simple Walrasian dynamics (which have not progressed beyond the
tatonnement) similarly cannot explain the persistent unemployment modern
economies experience, while we have seen that our model can exhibit
Keynesian unemployment as a steady state. Again equilibrium models can
exhibit prolonged unemployment with various modifications, but we find it at
least as plausible to postulate disequilibrium as to impose some derivative
restrictions on an equilibrium model.
However, equilibrium analysis and comparative statics (for Walrasian o r
fixprice worlds) are applicable only if the dynamics of a model are stable.
More emphasis should be placed on dynamics, whether equilibrium or
disequilibrium. The assumption of stabi 1 ity, like the assumption of
market-clearing prices, is justified as a necessary simplification in
developing tractable models. But both issues are crucial to the results of
stable flexprice models, and neither is theoretically or empirically clear.
This paper has explored discarding both assumptions.

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Appendi x

P r o o f o f Theorem I
From t h e r e a l money s t o c k and wage d i f f e r e n t i a l
(49

w = ;[h2,(~l+)

-

h Z 2 ( S 1 + )- hl,(DY')

e q u a t i o n ( 3 4 > , we have

+ h12(SY+)l,

He t r a n s l a t e t h e p o s i t e d KE t o t h e o r i g i n and t a k e a l i n e a r a p p r o x i m a t i o n o f
t h e dynamic system t o r e w r i t e o u r d i f f e r e n t i a l e q u a t i o n s as:

o r i n t h e shorthand:
(51 1

=

AX,

where o u r c o e f f i c i e n t s i n t h e m a t r i x A a r e g i v e n by:
(52)

al

=

- a r / a m + mh, ,ayh+/am -mhl ,ayh+lam,

a,

=

- a r / a w + mh,

a2 =

l a ~ h + / a-mh,
~

zaYh+/a~,

, a ~ ~ + / a-m h 2 2 a ~ h + / a -m h l l a ~ h + / a m -

W C ~ ,

a z 2 = W C ~ ~ ~ ~ L- ~h 2+ 2/ a ~~ hW + /-a h~l

hl , a ~ ~ + / a m l

l a ~ h + / -a ~h 1 2 a ~ f + / a ~ 1

Then t h e c h a r a c t e r i s t i c e q u a t i o n i s derive'd f r o m t h e d e t e r m i n a n t o f A-XI:

and t h e s t a b i l i t y o f t h e system depends on t h e n e g a t i v i t y o f t h e r o o t s o f
t h i s polynomial.

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I n s t e a d o f d i r e c t l y s o l v i n g t h e q u a d r a t i c e q u a t i o n f o r 1, we make use
o f t h e e q u i v a l e n t Routh- Hurwicz c o n d i t i o n s f o r t h e s t a b i l i t y o f l i n e a r
systems.
(54)

I n t h i s case we have s t a b i l i t y i f :
al

+ a z 2 < 0,

allazz - a12a21 > 0.
Under KE we have t h e f o l l o w i n g forms f o r t h e components o f t h e dynamic
equations :
=

Yh+ + g - wLft,

excess supply i n L:

2'

=

L ~ ' L ~ <+ 0,

excess s u p p l y i n Y:

ZY

=

y +
f

r(x>/p

(55) r e a l p r o f i t s :

-

y h + < 0.

We examine each t e r m o f t h e m a t r i x A and t r y t o p i n down a s i g n ; we do n o t
t r y t o compare q u a n t i t i e s i n e s t a b l i s h i n g e q u i l i b r i u m , s i n c e o u r model i s
e n t i r e l y qua1 i t a t i v e :
(56)

a l l = -ar/am

+ mhl l a ~ h t / a m - m h l z a ~ h + / a m
=

=

mh,

l a ~ h + / a m-mhlzaYh+lam

(-p +mhll - m h l

, ) a ~ ~ ' / a m-

(w)aLf+/am,

because ( 6 ) i m p l i e s

(57)

ar/am

=

(p)ayh+/am -

(~)a~~+lam.

By ( 9 ) we have
(58)

ayh+/am > 0;

(w)a~~+/am
= 0.

We w i l l show a l l > 0 by d e m o n s t r a t i n g t h a t :
(59)

(-p +mhll - m h I 2 )

< 0.

Define:
(60)

H = h l l + h12.

Then f r o m o u r l i n e a r p r i c e dynamics i n e q u a t i o n ( 3 2 ) we have:
(61 )

hl

=

(PIP

+

H Y ~ ~ ) / ( Y +~ y+ f + ) ,

h l L = ( H Y ~- p / p ) / ( y h + + y f ' ) .

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Then (59) can be expressed as:
(62)

-p + m[(p/p + H Y ~ + > / ( Y +~ +Y f + ) l
=

1-p(Yh+ + Y f + ) + mHYf+

-

-

mC(HYh

-

p/p)/(Yh+ + Yf')l

mHYh+l/(Yh+ + Y f + > ,

and s i n c e i n KE we have Y h + < Y f + we immediately have:
(63)

all < 0

(64)

ale

=

- a r / a w + mhl , a y h + / a w - ~ ~ ~ h ~ , a ~ ~ + / a w .

From ( 6 ) we have:
(65)

ar/aw <

trivially.
(66)

o

Assumption ( 9 ) g i v e s :

ayh+/aw > O;

a y f + / a w < 0,

and so we have:
(67)

a 1 2 > 0,

(68)

a,

= W C ~ ,

, a ~ ~ + / a-m h , , a ~ ~ + / a m -

h , ,ayh+/am - hlEaYf+/aml.

" Since:
(69)

a ~ ~ + / a=mO ;

ayf+/am < 0,

we w i l l show t h a t :
(70)

- h2,aLh+/8m - h , , a y h f / a m < 0,

t o demonstrate t h a t a z l i s p o s i t i v e .

(71 )

h l ,ayh+/am > - h Z 2 a ~ h + ~ a m .

I n t e g r a t i n g with respect t o m y i e l d s :

(72)

h, , y h + > - h Z 2 L h + .

T h i s i n e q u a l i t y can be r e w r i t t e n as:

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Since Y, L, and t h e h ' s a r e p o s i t i v e , t h i s c o n f i r m s t h e i n e q u a l i t y , and we
have :
(73)

a z l < 0,

(74)

a,,

= ~ [ h , , a ~ ~ +- l ha 2~2 a ~ h + /-a h~, , a y h + / a w

-

h12a~f+~a~].

Under assumption ( 9 ) we can s i g n each t e r m as f o l l o w s :
(75)

a ~ ~ + >/ O;a ~ ayh+/aw >

a ~ ~ + / a< wO;

0;

ayf+/aw < 0.

So we have:

< 0.

(76)

The b a s i c model, t h e n , q u a l i t a t i v e l y s a t i s f i e s t h e Routh- Hurwicz
c o n d i t i o n s for s t a b i l i t y :
(77 >

( i )a , , + a r 2 =
( i i )a l l a 2 ,

-

(->

aI2a2,

+

(->

< 0;

= (-)(-)

so we have s t a b i l i t y f o r a l l KE.

-

(-)(+) = (+)

From o u r l i n e a r i z e d i n v e n t o r y l e s s model we have:
a , , < 0;

a,,

> 0,

a z l < 0;

a z 2 < 0.

We a l s o have:
(79)

a 1 3 = a A , / a i = 0,
a,,

= aA2/ai = 0,

a,,

= aA3/am = 0.

(-)

> 0,

T h i s completes t h e p r o o f o f Theorem I .

P r o o f o f Theorem I1

(78)

-

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Then t h e o n l y unsigned terms a r e a s 2 and a s s ; t h e y a r e e a s i l y signed:
(80)

as2

=

( F ' ) ~ L ~ + I -~ W
ayf+/aw.

From assumptions ( 2 ) and (6) we have:

F I > 0; a ~ ~ + >/ O;a ~a y f + / a w < 0,

(81 )

and so we have:
a 3 >~ 0.

(92)

Under assumptions ( 2 ) and ( 6 ) we have:

and t h u s :

Then q u a l i t a t i v e l y o u r m a t r i x o f c o e f f i c i e n t s A f o r t h e l i n e a r i z e d
system i s :

I t i s t h e n easy t o show t h a t t h i s l i n e a r system i s s t a b l e .

We

demonstrate t h a t t h e r e a l p a r t of each e i g e n v e c t o r o f t h e m a t r i x must be
n e g a t i v e by showing t h a t A i s n e g a t i v e d e f i n i t e .

z = (zl,

Zr,

z?)

For any v e c t o r

we have q u a l i t a t i v e l y :

I t i s then s u f f i c i e n t t o show:

t o prove n e g a t i v e d e f i n i t e n e s s .

But t h i s i n e q u a l i t y i s t r i v i a l ; s q u a r i n g

b o t h sides y i e l d s t h e r e s u l t immediately.

So we have shown t h a t when i t

e x i s t s , the l i n e a r i z e d v e r s i o n o f o u r dynamic system a t a Keynesian
e q u i l i b r i u m w i l l be s t a b l e .

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