Asset Prices in an Exchange Economy
Author(s): Robert E. Lucas, Jr.
Source:
Econometrica,
Vol. 46, No. 6 (Nov., 1978), pp. 1429-1445
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1913837
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Econometrica, Vol.
46, No. 6 (November, 1978)
ASSET PRICES IN
AN
EXCHANGE ECONOMY
BY ROBERT E. LUCAS,
JR.'
This paper is
a theoretical examination
of the stochastic behavior
of equilibrium asset
prices in
a
one-good,
pure exchange
economy with identical
consumers. A general
method
of constructing equilibrium prices
is developed and
applied to a series of
examples.
1. INTRODUCTION
THIS
PAPER
IS
A
THEORETICAL examination
of the stochastic behavior
of
equi-
librium asset
prices
in a
one-good,
pure exchange
economy
with
identical
consumers.
The single good in this economy
is (costlessly)
produced in a number
of different
productive
units; an asset
is
a
claim
to all or
part
of the output of
one
of
these units. Productivity
in each unit
fluctuates stochastically
through time,
so
that equilibrium
asset prices
will fluctuate as well. Our
objective
will
be
to
understand
the
relationship between
these
exogenously
determined productivity
changes
and market determined
movements
in
asset
prices.
Most of our attention
will be focused
on
the derivation
and
application
of
a
functional equation
in
the vector
of
equilibrium asset prices,
which is
solved
for
price
as
a
function
of
the
physical
state of
the
economy.
This
equation
is a
generalization
of
the
Martingale property
of
stochastic
price sequences,
which
serves
in
practice
as
the
defining
characteristic
of market
"efficiency,"
as that
term is used
by
Fama
[7]
and others.
The
model thus serves as
a
simple
context
for
examining
the conditions under which
a
price series'
failure to
possess
the
Martingale property
can be viewed
as evidence
of
non-competitive
or "irra-
tional" behavior.
The
analysis
is conducted
under the
assumption that,
in
Fama's terms, prices
"fully
reflect
all
available information,"
an
hypothesis
which Muth
[13]
had
earlier
termed
"rationality
of
expectations."
As Muth
made
clear,
this
hypoth-
esis
(like utility
maximization)
is
not "behavioral":
it does
not
describe
the
way
agents
think about
their
environment,
how
they learn, process
information,
and
so
forth. It
is
rather a
property
likely
to
be
(approximately)
possessed by
the
outcome
of this
unspecified process
of
learning
and
adapting.
One would feel
more comfortable, then,
with
rational
expectations equilibria
if
these
equilibria
were accompanied
by some
form of
"stability theory"
which illuminated
the
forces
which move an
economy
toward
equilibrium.
The
present paper
also
offers
a convenient context
for
discussing
this issue.
The
conclusions
of this
paper
with
respect
to the
Martingale property precisely
replicate
those
reached
earlier
by
LeRoy (in [10]
and
[11]),
and not
surprisingly,
since
the economic
reasoning
in
[10]
and the
present
paper
is the same. The
lThis paper originated
in a conversation
with Pentti
Kouri,
who
posed
to me the
problem
studied
below.
I
would
also like
to thank Yehuda
Freidenberg,
Jose
Scheinkman,
and
Joseph
Williams
for
many helpful
comments.
1429
1430
ROBERT
E. LUCAS,
JR.
context used here
differs somewhat from
LeRoy's, however, and
the analytical
methods used differ
considerably.
The economy is
informally described in
the next section, and
equilibrium is
formally defined in
Section 3. In Section
4, the basic functional
equation for
prices is derived and
studied. Section
5
develops
a certain "duality"
property, on
which is based the
discussion of stability
in Section 6. Section
7 deals with
examples which are
simple enough to permit
either explicit solution
or some
"comparative static"
exercises. The role of
the Martingale property
is discussed
in
Section 8. Section
9 concludes the paper.
2. DESCRIPTION OF
THE ECONOMY
Consider an economy
with a
single
consumer, interpreted
as a
representative
"stand
in"
for a large
number of identical
consumers.
He wishes to
maximize
the
quantity
(1)
E1 E
3tU(ct)}
where
c,
is
a
stochastic
process representing
consumption
of
a
single good, ,B
is a
discount
factor, U()
is
a
current
period
utility function,
and
E{ } is
an
expec-
tations
operator.
The consumption
good
is
produced on
n distinct productive
units. Let
Yit
be
the
output
of unit
i in
period t,
i
=
1,
...,
n,
and let
yt
=
(yIt,
.... Ynt)
be
the
output
vector in t.
Output
is
perishable,
so that feasible
consumption
levels are
those which
satisfy
Production is
entirely "exogenous":
no
resources
are
utilized,
and
there is no
possibility
of
affecting
the
output
of
any
unit at
any
time.
The motion of
yt
will
be
taken to follow a
Markov
process,
defined
by
its transition
function
F(y', y)=
pr
IYt+?
:
Y'lYt
=
Y}.
Ownership
in these
productive
units is
determined
each
period
in a
competi-
tive
stock
market. Each unit has
outstanding
one
perfectly
divisible
equity
share.
A
share entitles its owner as of the beginning
of t to all of the
unit's
output
in
period
t. Shares are
traded,
after
payment
of
real
dividends,
at a
competitively
determined
price
vector
Pt
=
(PIt
.
. .
,
Pnt).
Let
zt
=
(Zlt.
. .
Znt)
denote a
consurner's
beginning-of-period
share holdings.
In this
economy,
it
is
easy
to determine
equilibrium quantities
of
consumption
and asset
holdings.
All
output
will be consumed
(ct
=
:iyit)
and
all
shares
will
be
held
(z,
(1,
1,
.
. .
1)=
1
for
all
t).
The
main
analytical
issue,
then,
will be the
determination
of
equilibrium price
behavior.
Our
attack on this
problem begins
from
the
observation that all
relevant
information
on the current and
future
physical
state of the
economy
is
sum-
ASSET
PRICES 1431
marized in the current output vector
y. Since, given recursive preferences, the
asset market "solves" a problem of
the same form each period, equilibrium
prices should (if they behave in a systematic
way at all) be expressible as some
fixed function p( ) of the state of the
economy, or
p,= p(y,)
where the ith
coordinate
pi(y,)
is the price of a share
of unit i when the economy is in the state
y,.
If
so, knowledge
of the
transition
function
F(y', y)
and
this
function
p(y)
will
suffice to determine the stochastic character
of the price process
{p,}.
Similarly,
one would
expect
a
consumer's
current
consumption
and
portfolio
decisions,
c,
and
z,,1,
to depend on
his beginning of period portfolio, z, the
prices he faces,
p,,
and the relevant
information he possesses on current and
future
states of the
economy,
y,
If
so,
his behavior can be described by fixed
decision rules
c(-)
and
g(-):
c,
=
c(z,,
yt, pt)
and
zt+1
=
g(zt, Yt,
Pt).
Now given perceived, future price
behavior F(y', y) and p(y), consumers will
be able
to determine
these decision rules
c( )
and
g( ) optimally.
In this
sense,
a
price function p determines consumer
behavior. On the other hand, given
decision
rules c( )
and
g( ), the
current period market clearing conditions
determine a
price function p( ).
In this
sense,
consumer
behavior determines
the
equilibrium price function. We close
the system with the assumption of rational
expectations:
the market
clearing price
function
p implied by
consumer behavior
is
assumed to be the same as the price
function p on which consumer decisions
are
based.
3.
DEFINITION OF
EQUILIBRIUM
The economy described
in the
preceding
section
is
specified by
the functions U
and
F
and the number
,B.
Assume 0
< 3 <
1. U:
R
+
--
R
+
is
continuously
differentiable, bounded, increasing,
and
strictly concave,
with
U(0)=
0.'
F:
E`'+
x
En
R
is
continuous; F(-,
y)
is a distribution function for each fixed
y,
with
F(O, y)=
0. Assume
that the
process
defined
by
F has a
stationary
dis-
tribution
f(-),
the
unique
solution
to
(y')
F
E(y',y)dq
(y),
and
that for
any
continuous
function
g(y),
J
g(y')
dF
(y',y)
is a continuous function
of
y.
An
equilibrium
will
be
a
pair
of
functions:
a
price
function
p(y),
as discussed
above,
and
an
optimum
value function
v(z, y).
The value
v(z, y)
will
be
inter-
preted
as the value
of the
objective (1)
for
a consumer
who
begins
in state
y
with
holdings z,
and follows
an
optimum
consumption-portfolio policy
thereafter.
2Rt
is the
set of
nonnegative
real numbers.
En
is n-dimensional
space.
E"
is the subset of
En
with all
components nonnegative (xE EE
and
x
-
0).
LU
is the set of
continuous,
bounded
functions
with domain
En,
and so
on.
1432
ROBERT E.
LUCAS,
JR.
DEFINITION:
An equilibrium
is a continuous
function
p(y): E+- E'+
and
a
continuous,
bounded
function
v (z, y):
E'+
x
E'
+
-
R
+
such
that
(i)
ttv(z,Y)=max
U(c)+,8{v(x,y')dF((y',y)
c,x
subject
to
c
+p(y)
x
S
y z
+p(y)
z,
c
BO,
O!x
!z,
where
z is a vector
with
components
exceeding
one;
(ii)
for each
y, v(i,
y)
is attained by c
=
Eiyi
and x
=
1.
Condition
(i)
says
that, given
the behavior of prices,
a consumer
allocates
his
resources y z+p(y)
z
optimally
among current
consumption
c
and end-of-
period
share
holdings
X.3
Condition
(ii)
requires
that these
consumption
and
portfolio
decisions be
market clearing.
Since
the
market is
always
cleared,
the
consumer will
never be
observed
except
in the state
z
=
1.
On the other
hand,
the
consumer
has
(though
he
always
rejects
it) the option
to choose
security
holdings
x #
1. To
evaluate these
options,
he
needs
to
know v(z, y)
for
all Z.4
4.
CONSTRUCTION
OF THE
EQUILIBRIUM
We begin
by studying
the consumer's
maximum
problem
(i)
for
given
price
behavior
p(y).
We have
the
following proposition.
PROPOSITION
1: For each
continuous
price function
p( )
there is a unique,
bounded,
continuous,
nonnegative
function
v(z,
y; p)
satisfying (i).
For each
y,
v(z,
y; p)
is an
increasing,
concave
function
of
z.
PROOF: Define
the
operator
Tp
on
functions v
(z, y)
such that
(i)
is
equivalent
to
Tpv
=
v.
The domain
of
Tp
is
the
nonnegative
orthant
L
2n+
of the
space
L2n
of
continuous,
bounded
functions
u:
E'+
x
E+-->
R,
normed
by
Ilull
=
sup lu(z,
Y)i.
z,y
Since
applying
Tp
involves
maximizing
a
continuous
function
over
a
compact
set,
Tpu
is well defined
for
any
u
e
L
Since
U(c)
is
bounded, Tpu
is
bounded,
and
by [2,
p. 116]
Tpu
is continuous.
Hence
T,:
L2n? L2n+.
T,
is monotone
(u
,v
implies
Tpu >
T,pv)
and for
any
constant
A,
T,p(u
+
A)
=
Tpu
+
PA.
Then from
[3,
Theorem
5]
Tp,
is
a contraction
mapping.
It follows
that
Tpv
=
v has
a
unique
solution v
in
L2n+
as
was to be
shown.
Further,
limn
Tun
=
v
for
any
u
E
L2n+.
3
The
bound
z on x is to assure
that the maximization
in (i) is
always over a compact
set, even
if
some components
of p(y)
are zero.
4This
is
not a "new"
concept
of
equilibrium.
It is
(though
no
proof
is offered) a
standard,
Arrow-Debreu
equilibrium
where the commodity
space
is
the
space
of all
possible
realizations
of
the
process
XYiyi.
See [12]
for
a full
development
of this
relationship
in
a
closely
related context.