abel_catching_up_with_the_joneses ( leer para trabajo final).pdf
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American Economic Association
Asset Prices under Habit Formation and Catching up with the JonesesAuthor(s): Andrew B. AbelReviewed work(s):Source: The American Economic Review, Vol. 80, No. 2, Papers and Proceedings of theHundred and Second Annual Meeting of the American Economic Association (May, 1990), pp.38-42Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/2006539.
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VOL. 80 NO.
2
A
GGREGA
TE
ASSET
PRICING
39
to
some
endogenous
time-varying
bench-
mark
vP.2Under the isoelastic
utility
func-
tion
in
(4),
the
expression
for
dU8/dc,
in
(3)
becomes
(5)
d8L/dct
=[1-ftyD(
Ct+llct
) -O(
Pt/ t+
1)
]
X
(
ctl
Pt) (ilct).
II. Equilibrium
Let
y,
be the
amount
of
the
perishable
consumption good per
capita produced by
the
capital
stock. In
equilibrium,
all
output
is
consumed
in
the
period
in
which it is
produced, as
in
Lucas. Because all
con-
sumers are
identical,
ct
=
Ct=y
in
every
period.
Now let x
y41
Yt+l/Yt
be the
gross
growth
rate
of
output.
Because
ct
=
Ct
=
Yt,
it follows that
ct+1/Ct
=
Ct+11Ct
=
xt+l
Therefore, equation (2)
implies
that
it+lvt
=
xY which
allows
us to
rewrite
(5)
as
(6)
dUtldct
=
Ht+jPt`
-lt
where
Ht+1=-1-#/yDxt+xt-y(l-a)
Note that
Ht+1
-1
if
yD
=
0,
which is the
case
for
both
time-separable
and relative
consumption preferences.3
Ill.
AssetPricing
To
calculate asset prices,
let us examine a
consumer who
considers
purchasing
an asset
in
period
t and then
selling
it
in
period
t
+
1.
If
asset
prices
are in
equilibrium,
this
pair
of
transactions does not affect expected dis-
counted
utility.
Suppose
that
a
consumer
reduces
c,
by
1
unit, purchases
an asset with
a gross rate
of return
Rt+1,
sells
the
asset in
period t +1, and increases ct+1 by Rt+1
units.
The
equilibrium
rate
of
return
Rt+1
must
satisfy
(7)
Et
{
-(
dUtldc,)
+
Rt+l( dUt+l/dct+1)}
=0.
Equation
(7)
can be
rewritten
as
(8)
Et({
Rt+(
dUt+l/dct+1)
/E{t dU/act}}
=1.
Equation
(8)
is
the familiar result that
the
conditional
expectation
of
the
product
of
the
intertemporal
marginal
rate of
substitution
and the
gross
rate
of return
equals
one.4
We
can obtain
an
expression
for
(dUt+l/
a
ct
+
)/Et
{
d
Ut
d
ct
}
using equation
(6)
to
divide
d
Ut+
/dct+
1 by
Et
{
d
Ut/dct
}
to
ob-
tain
(9) (dbUt+l/dct+)/Etf{ dUc/at}
=[Ht+ 2/Et
{
Ht+ 1
}]Xty'a
)t1
IV.
The
Priceof
Risky
Capital
Let
pts
be
the
exdividend
price of
a share
of
stock
in
period
t,
which is
a claim
to a
unit
of
risky capital. The
rate
of
return on
stock
is
R
s
1
pts+
+
yt+
)pts-
Let
wt
p_/yt
be the
price-dividend
ratio.
Therefore,
Pt
=
wtYt
and
Pts+i=
Wt+iyt+i
so that
(10)
Rst
+l
=(
+
Wt
+
)Xt
+
IWt-
Substituting
(10)
into
(8) yields
(1
1)
wt
=
aEt
(1
+
wt+l) xt+
L
x
(
dUt+lldct+,)IEt
{
dut/dct
}
2George Constantinides (1988), Jerome
Detemple
(1989), John Heaton (1989), and Suresh Sundaresan
(1989)
also
examine asset
prices
in
the
presence
of
habit
formation. James Nason (1988) includes a
time-varying
benchmark level
of
consumption that differs from
habit
formation in that
it
is
independent
of
an individual
consumer's own consumption.
3A
sufficient condition for dU1/d
c
>
0
when y
=
D
=
1
(habit formation ) is
1 + ln
f3/ln(max{ x }/min{
x)