© Sanjay K. Chugh
105
Spring 2008
Chapter 8
Infinite-Period Representative Consumer
and Asset Pricing
Macroeconomic models, at least those based on micro-foundations, used in applied
research and in practice assume that there is an infinite number of periods, rather than just
two as we have been for the most part
assuming
.
A
two-period
model
is
usually
sufficient
for
the
purpose
of
illustrating
intuition
about
how
c
o
n
s
um
e
r
s
m
a
k
e
intertemporal choices, but in order to achieve the higher quantitative precision needed for
many research and policy questions, moving to an infinite-period model is desirable.
Here we will sketch the problem faced by an infinitely-lived representative consumer,
describing preferences, budget constraints, and the general characterization of the
solution.
In sketching the basic model, we will see that in its natural formulation, it
easily lends itself to a study of asset-pricing.
Indeed, this model lies at the intersection of
macroeconomic theory and finance theory and forms the basis of consumption-based
asset-pricing theories.
We will touch on some of these macro-finance linkages, but we
really will only be able to whet our curiosity about more advanced finance theory.
For
the most part, we will index time by arbitrary indexes
1, ,
1
tt
t
±
²
, etc., rather than
“naming” periods as “period 1,” “period 2,” and so on.
That is, we will simply speak of
“period
t
,” “period
t
+1,” “period
t
+2,” etc.
Before we begin, we again point out that “the consumer” we are modeling is a stand-in
for the economy as a whole.
In that sense, we of course do not literally mean that a
particular individual considers his intertemporal planning horizon to be infinite when
making choices.
But to the extent that “the economy” outlives any given individual, an
infinitely-lived representative agent is, as usual, a simple representation.
Preferences
The utility function that is relevant in the infinite-period model in principle is a lifetime
utility function just as in our simple two-period model.
As before, suppose that time
begins in period one but now never ends.
The lifetime utility function can thus be written
as
12345
(, ,, ,,
.
.
.
)
vc c c c c
.
This function describes total utility as a function of consumption in every period 1, 2, 3,
… and is the analog of the utility function
12
(, )
uc c
in our two-period model.
The
function
v
above is quite intractable mathematically because it takes an infinite number
of arguments.
Largely for this reason, in practice an
instantaneous utility function
that