Problem Set 7
EC720.01 - Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2009
Due Thursday, October 29
This problem asks you to derive some of the implications of a model of stock prices and con-
sumption due originally to Robert Lucas, “Asset Prices in an Exchange Economy,”
Econo-
metrica
, November 1978, pp.1429-1445.
Consider an economy populated by a large number of identical consumers, each of whom
has preferences described by the additively-time separable utility function
∞
X
t
=0
β
t
u
(
c
t
)
,
(1)
where the discount factor satisﬁes 0
< β <
1 and the single-period utility function
u
is
strictly increasing, strictly concave, and satisﬁes lim
c
→
0
u
0
(
c
) =
∞
, this last assumption
allowing us to ignore nonnegativity constraints on consumption in all of the analysis that
follows.
Each consumer ﬁnances his or her consumption by trading equity shares in the economy’s
productive assets: let’s call them “fruit trees.” Each share in each tree provides a dividend in
the form of
d
t
pieces of “fruit” during each period
t
= 0
,
1
,
2
,...
, where fruit is the economy’s
only consumption good. Let
s
t
denote the number of shares carried by a representative
consumer into each period
t
= 0
,
1
,
2
,...
, and let
p
t
denote the price of each share in each
tree during each period
t
= 0
,
1
,
2
,...
.
Then, as sources of funds during each period
t
= 0
,
1
,
2
,...
, the representative consumer has
his or her dividend payments
d
t
s
t
and the total value
p
t
s
t
of the shares carried into the
period. And as uses of funds during each period
t
= 0
,
1
,
2
,...
, the consumer has his or her
consumption
c
t
and the value
p
t
s
t
+1
of the shares that he or she will carry into period
t
+ 1.