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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)

abel_catching_up_with_the_joneses ( leer para trabajo final).pdf

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