HD28
.M414
no.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
EVALUATING THE EFFECTS OF INCOMPLETE MARKETS
ON RISK SHARING AND ASSET PRICING
John Heaton
Massachusetts Institute of Technology
Deborah Lucas
Northwestern University
WP#3491-92-EFA
November 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
EVALUATING THE EFFECTS OF INCOMPLETE MARKETS
ON RISK SHARING AND ASSET PRICING
John Heaton
Massachusetts Institute of Technology
Deborah Lucas
Northwestern University
WP#3491-92-EFA
November 1992
M.I.T.
NOV
LIBRARIES
1
8 1992
EVALUATING THE EFFECTS OF INCOMPLETE MARKETS ON RISK SHARING
AND ASSET PRICING
First Draft:
This Draft:
September 1991
November 1992
John Heaton
Deborah Lucas
*M. I.T.
and UBER
Northwestern University, and NBER
We thank George Constantinides, Darrell Duffie, Mark Gertler, Narayana
Kocherlakota, Thomas Lemieux, Danny Quah and Jean-Luc Vila for helpful
conversations and comments.
We also thank participants at the Canadian
Macroeconomics Study Group,
the N.B.E.R.
Asset Pricing meeting,
the
Northwestern Summer Workshop in Economics, and seminar participants at
Cornell, Duke, the Federal Reserve, McGill, M.I.T., New York University,
Princeton, Queen's, Rutgers, Western Ontario and Wharton.
Part of this
research was conducted at the Institute for Empirical Macroeconomics in the
summer of 1991.
We would also like to thank Makoto Saito for research
assistance and the International Financial Services Research Center at
M.I.T. for financial assistance.
Introduction
1.
Incomplete markets In the form of an Inability to borrow against risky
future income has been proposed as an explanation for the poor predictive
With complete
power of the standard consumption-based asset pricing model.
individuals fully insure against idiosyncratic income shocks,
markets,
consumption
individual
proportional
is
exceed
that
of
aggregate,
the
aggregate
consumption.
2
With
individual consumption variability may
however,
limited insurance markets,
to
and
and
prices
asset
implied
the
may
differ
significantly from those predicted by a representative consumer model.
this
paper
contingent
we
study
on
future
economy
an
in
income
labor
which
agents
cannot
uncertainty in the form of dividend and systematic labor
also Idiosyncratic
buffered by
trading
in
Idiosyncratic
income risk.
labor
financial
limited by borrowing constraints,
securities,
but
write
contracts
face
aggregate
They
realizations.
In
income risk,
and
income shocks can be
the
extent
trade
of
is
short sales constraints and transactions
costs.
motivation
The
frictions
and
reviewing
the
Telmer
(1991)
asset
for
prices
findings of
insure
in
a
the
environment
this
number
interaction
of
recent
is
between
best
papers.
trading
understood
Lucas
(1991)
by
and
examine a similar model with transitory idiosyncratic shocks
and without trading costs.
cannot
considering
against
Surprisingly,
idiosyncratic
they find that even though agents
shocks,
predicted
asset
prices
are
For discussions of problems with the standard model, see for example,
Hansen and Singleton (1983), Mehra and Prescott (1985) and Hansen and
Jagannathan (1991).
2
See R. Lucas (1978), for example.
similar
to
with
those
markets.
complete
3
occurs
This
because
when
idiosyncratic shocks are transitory, consumption can be effectively smoothed
by accumulating financial assets after good shocks and selling assets after
bad shocks.
Aiyagari and Gertler (1991) consider a model with no aggregate
uncertainty and with transactions
costs,
relative returns on stocks and bonds,
Constantinides
shocks
no
trade occurs.
increases
trade
offset
to
and reduce the total volume of trade.
study
(1992)
the
case
permanent
of
Although agents have unrestricted access to financial
idiosyncratic shocks.
markets,
Duffie
and
which agents
Differential transactions costs affect the
transitory idiosyncratic shocks.
Finally,
in
in
When the conditional variance of
downturn
a
the
riskfree
rate
idiosyncratic
and
falls
equity
the
premium rises relative to the complete markets case.
These
results
suggest
that
the
quantitative
asset
predictions
price
from this class of models will depend critically on several factors:
the
extent of
trading frictions
in
securities markets,
the
(ii)
(i)
size and
the correlation structure of
persistence of idiosyncratic shocks,
and (iii)
idiosyncratic and aggregate shocks.
To address
(ii)
and
(iii),
we develop
an empirical model of individual income that captures both the size of the
idiosyncratic shocks and the persistence of
upon evidence from
the
Panel
Study of
these shocks over
Income
Dynamics
time,
(PSID).
based
The
time
series properties of aggregate income and dividends are estimated using the
National
Income
and
Product
Accounts.
This
process
is
then
used
to
calibrate the theoretical model.
Our theoretical model differs substantively from those discussed above
3
Using a more volatile aggregate income process, Marcet and Singleton
The equity premium rises in the presence
(1991) also calibrate this model.
of short sales constraints.
by considering transactions costs in an environment with both aggregate and
idiosyncratic
need
agents
Transactions
shocks.
frequently
trade
to
individual income.
play an
costs
order
in
important
buffer
to
role
shocks
because
their
to
transactions costs can have two effects on
As a result,
asset prices.
First,
rates
(gross)
return on securities may be altered because
of
agents require higher returns to compensate for
transactions costs.
effect of transactions cost was emphasized by Aiyagari and Gertler
Amihud
and
Mendelson
Constantinides
effect
small
idiosyncratic
on
transactions
and
that
transactions
returns.
agents
Vila
and
his
In
trade
only
(1992).
costs
model
in
should
have
only
which
there
is
rebalance
to
costs.
returns
are
However
individuals face in our model,
not
due
much
to
affected
their
is quite costly for
it
by
the
idiosyncratic
the
(1991),
contrast,
In
a
no
portfolios,
transactions costs by reducing the frequency of
asset
result,
a
asset
risk
agents avoid most
As
argued
(1986)
Vayanos
and
(1986)
This
trades.
presence
shocks
of
that
individuals to change
their asset trading patterns in response to trading costs.
A second,
indirect, effect of transactions costs is that they limit the
ability of agents
shocks,
so
aggregate.
that
to
use
asset
individual
markets
consumption
to
self-insure against
does
not
move
transitory
directly with
the
The increased volatility in individual consumption reduces each
individual's tolerance for the aggregate uncertainty reflected in dividends,
for the utility specifications that we consider.
could rise in response to
alone.
This paper appears
The implied equity premium
increases in transactions costs for this reason
to
be
the
first
to
evaluate the importance of
this mechanism.
We calibrate the model under a variety of assumptions about
and incidence of trading costs.
the
size
When trading costs differ across markets.
we find that agents readily substitute towards transacting in the lower cost
then agents
market
trade
primarily
transitory
effectively
smooth
have
little
costs
if transactions costs are only introduced in the stock
For example,
market.
affect
income
shocks.
required
on
market
bond
the
in
by
this
However
return.
of
means
transactions
case
this
In
rates
and
if
transactions costs are also introduced in the bond market in the form of a
wedge between the borrowing and lending rate,
then the bond return falls.
With a binding borrowing constraint or a large wedge between the borrowing
and lending rate, a small transactions costs in the stock market can produce
an equity premium that is close to the observed value, and a low
return
bor,
that is close to the observed return on U.S. Treasury securities.
The remainder of the paper
describe
the
model
borrowing
costs,
We
constraints,
and
model
trading
Simulation
constraints.
sales
of
Model
The Environment
The economy contains two
their
empirical
parameter izations for
the
short
the
In Section 2 we
Section 5 concludes.
4.
2.
1
presents
3
also discuss
results are reported in Section
2.
organized as follows:
Section
economy.
income and dividends.
is
labor
income
(classes of) agents who are distinguished by
realizations.
stochastic labor income Y
.
each
At
time
t,
agent
receives
i
agents are not allowed to write
By assumption,
contracts contingent upon future labor income.
Agents also receive income from investments
time
t,
a
share of
stock,
with price p
random dividends from time t+1 forward,
}
J
provides a risk-free claim
to
one
unit
stocks and bonds.
provides a claim
,
{d
in
of
to
flow of
a
The bond, with price p
"^t
"^
J=t+i
At
consumption at
time
t
+ 1.
,
The
.
agents
trade
these
securities
two
smooth
to
their
Trading is costly, with transactions cost function
and
(j(-)
in
the bond market.
consumption over
time.
in the stock market
»c(-)
Agents also face short
sales and borrowing
constraints.
At time
each agent's preferences over consumption are given by:
t,
,
(2.1)
(/'
(c'
—^
CO
p"^
= e\i
is
1
^
:
W=o
where ?(t)
^'^ -
)
?(t)L
I
time
the
information set,
t
3-
>
.
J
1-r
which is common across agents.
information is generated by a state variable,
This
In principle,
below.
r
Z
,
which is specified
could be allowed to differ across agents.
However,
since we want to interpret the two groups as similar except for realizations
idiosyncratic shocks,
of
groups.
,
t
to
equate y across
the
two
4
At each date
c
seems appropriate
it
agent
t,
stock share holdings
^
i
maximizes (2.1) via the choice of consumption
bond holdings
b
^
and
s
t+i
subject
^
^
t+1
to
the
flow
wealth constraint:
(2.2)
c'
+ p^s^
t
t
t+i
+ p^b^
'^t
t+i
t
)
+ u(b^
t
;Z
t+i
)
t
+d)+ti
+y t
t
t
s s {p
t
,s^;Z
+ k(s'
t+i
t
and short sales or borrowing constraints:
(2.3)
s'
£ K^
t
= 0,
1,
2,
.
(2.4)
b^
£
t
= 0,
1,
2.
..
t
4
/("
t
Dumas (1989) considers the implications
parameters in a complete markets setting.
of
different
risk
aversion
The
initial wealth d
of
components
'^
s
,
b
,
and market prices
and /
are
taken as given.
2.2 Trading Frictions
The
extent
which
to
shocks depends
income
idiosyncratic
individuals
will
use
the
size
on
asset
and
markets
to
incidence of
costs, and the presence of borrowing and short sales constraints.
buffer
trading
Since the
assumed form of these frictions qualitatively affect predicted asset prices,
and since there is little agreement about the exact form of these costs, we
consider several alternative cost structures.
Transactions
costs
in
stock
the
Both
market.
buyers
and
sellers
are assumed to face a quadratic transactions cost function:
,s^;p^) = k {(s'
k(s'
(2.5)
t+i
t
t
- s^)p^}^
t+i
t
t
t
the parameter k
In the simulations,
of the transactions cost.
1
function by (Is
of
to
-s
shares traded:
an offsetting
Is
k \s
t
the magnitude
an increase in the cost parameter
reduction
in
trade.
Dividing the
cost
gives the trading cost as a percent of the value
)p
|
to control
However, because the realized cost is endogenous,
the range of attainable costs is bounded;
eventually leads
is used
1
Is
It
-s \p
t+i
.
In
the
simulation results we report
the
average of this percentage cost.
We
use
simplicity.
sold,
a
quadratic
However,
agents must
individuals hold
it
sell
no
cost
also
function
captures
increasingly
stock
at
all
the
primarily
idea
that
illiquid assets.
suggests
that
there
for
as
computational
more
The
fact
may
be
assets
that
are
many
significant
fixed costs to entering this market.
To partially address this issue,
we
also estimate the income process conditioning on data from families who own
non-negligible
amounts
of
stock.
Another
possible
objection
quadratic form is that small changes in stock holdings may be at
costly (proportionally) as large changes.
consider,
will
however,
infinitesimal
to
the
least as
In the discretized state space we
shocks never occur so
limiting
this
case is not relevant.
Bond market transactions costs.
Bonds in this model represent private
While it seems sensible to treat transactions costs
borrowing and lending.
symmetrically for sales and purchases of stock, this is less true for loans.
Typically
consumers
pay
a
substantial
spread
over
the
lending
rate
to
is
a default
premium which does not
apply to the riskfree bonds of the model.
However,
a portion of the spread
borrow.
C2in
the observed spread
Part of
be attributed
to
costs of financial
intermediation or monitoring that
must be incurred even if the debt is ex post riskfree.
To capture the asymmetry between effective borrowing and lending rates,
the bond transactions cost function is assumed to have the form:
(2.6)
The
w(b^
parameter
;Z
= n min(0.b'
)
Q
borrowing at time
controls
t
the
p")^.
magnitude
of
the
cost.
By
is represented by a negative value for b
agent who borrows pays the transactions cost.
As with stocks,
reported as a percentage of the per capita amount transacted:
,
convention
so only the
the cost
n \b
is
\p /2.
We also will consider the implications of a symmetric quadratic cost
The effects of transactions costs of this form have been considered by
Saito (1992)
function in the bond market of the form:
w(b'
(2.7)
t+i
As we shall
;Z
= n (b'
)
t
t
see,
t+i
p'f.
t
the choice of
versus
(2.6)
will have a significant
(2.7)
affect on the predicted equity premium.
To
match
observed
the
income
dividend
and
process,
the
assumed to be stationary in aggregate income growth rates.
price
the
of
stock and
the
accommodate the growth in value,
be
linear
in
aggregate
bond
the
of
Y
income,
Because
.
to
the
As a result,
grow over
the borrowing constraint,
quadratic in the value of trade,
cost to income ratio,
value
face
economy
K
,
is
the
To
time.
is assumed to
transactions
costs
are
induce a constant average transactions
= k/Y
we assume that k
and
t
t
= £l/y^ where k and n
Zl
t
t
are constants.
Finally,
we
refer
frictionless model.
to
the
where
case
k(-)
=
and
u(-)
=
The frictionless model is similar to Lucas
as
the
(1992)
and
Telmer (1992).
Consumption smoothing may also
Borrowing and Short Sales Constraints.
be curtailed by institutional limits on the amount of borrowing.
of
credit
scenarios.
income.
rationing
represented
is
In the first,
In the second,
by
(2.4).
We
will
This type
consider
two
agents can borrow up to 10% of average per capita
agents are precluded from any borrowing; only stock
holdings can be used to buffer income shocks.
The choice of an appropriate upper bound on borrowing is not obvious.
The value of household collateral,
not easily measured.
which is a plausible limit on debt,
Since agents rarely hit
for the income shocks considered,
the assumed
lO'/i
upper bound
and since the agents always desire
8
is
to
do
.
the chosen limits appear to bracket the relevant range.
some borrowing,
No
(2.3).
sales are permitted
short
stock market
the
in
This is motivated in part by the observation that
individuals to take short positions.
K
that
so
it
=
in
costly for
is
allowing short sales would
Clearly,
increase the effective quantity of tradeable assets.
2.3
Equilibrium
At time
t,
aggregate output consists of the aggregate dividend, d
labor income V
the sum of individuals'
(2.8)
b^
+ b^ =
(2.9)
s^
+ s^ =
(2.10)
y
^1=1,2
^1=1,2
1
t
= 0,
1,
2,
t
= 0,
1,
2,
+ k(s'
{c'
t+1
t
,s';Z
t
)
+ (j(b'
t*l
t
Y
and
CM
Market clearing requires:
.
t
;Z )> = d
t
t
* y^
* y^
-^t
-^t
t=0,l,2.
.
.
Notice that in (2.8) we are assuming that bonds are in zero net supply.
When the short sales and borrowing constraints are not binding,
the
first order necessary conditions from the agent's optimization problem imply
that for all
(2.11)
and
1
t:
Ip" + K (s'
t
,s';Z )]u' (c')
t+i
1
t
^eL'
=
t
t
(.c^
t+l
1^
)[p^
*
t+l
d
*
t+l
K {s^
2
t+2
.s'
;Z )]
t+1
t
and
(2.12)
[p"
t
*
w
(b'
1
t+l
;Z )]u'{c^)
t
t
=
m
1^
u'(c'
t+l
)
I
?(t)l
J
I
?(t)l.
J
an agent
If
sale constraint
short
the
then
(2.3),
is replaced by:
(2.11)
(2.11'
constrained by
is
=
s'
)
K^
t
t
Similarly,
the
if
agent
constrained by
is
the
borrowing constraint
(2.4)
then (2.12) is replaced by:
(2.12'
)
At
b^
= k"
t
t
time
t
the unknowns are p
;
'^t
p
;
c
i=l,2;
,
and b
s ,i=l,2;
t
'^t
,
i=l,2.
t
t
The equations defining an equilibrium are the budget constraints (2.2),
the market
2;
clearing conditions
pricing equations
Walras' s
By
Law,
one
which
the
the
of
constraint is redundant.
in
(2.11'),
or
(2.11)
(2.9),
(2.8),
and
i=1.2,
market
and
and the asset
(2.10);
(2.12)
or
(2.12'),
conditions
clearing
i=l,
or
a
1=1,2.
budget
We restrict our attention to stationary equilibria
consumption
growth
prices are functions of the time
rate,
t
portfolio
state Z
,
equilibrium
and
rules,
which is described in detail in
the next section.
State Variables
3.
3.
1
Empirical Model of Labor and Dividend Income
Our empirical model
income process when
captures
growth,
the
and
labor
correlation
the
is
designed to capture
income
is
between
correlation
in
uninsurable.
aggregate
individual
10
and
labor
important
features of
For example,
individual
income
the
labor
shocks
over
the
model
income
time.
income process serves as an
Since this
solved
be
must
that
numerically,
input
into
an asset
a
relatively
choose
we
pricing model
parsimonious
specification.
The components of aggregate income include aggregate labor income,
and aggregate dividends,
denoted by Y^
•'
in
yVy^
,
and D
Y
,
aggregate
income,
,
is
The growth rate of Y^ and the logarithm of the share of D
.
t
t
assumed
5
tt-it
p
The sum of
.
t
is
Y^
D
7
=
follow
to
and X^ s
tt
D^'/Y^
bivariate
a
then X*
iog(5 )]',
[log(.r^)
is
y
=
assumed to be
t
t
t
Letting
autoregression.
generated by:
X^ =
(3.1)
jx^
+ A^ X^_j
*
8^ c^.
t = l,
3,
2,
where e* is a vector of white noise disturbances with covariance matrix
I,
t
the
matrix 9^
assumed
is
lower
be
to
and
triangular,
jx*
is
a
vector
of
We estimate these parameters using annual aggregate income and
constants.
dividend data from the National Income and Product Accounts (NIPA).
Individual i's labor income as a fraction of aggregate labor income is
given
by
^
-^
In other words,
ti'.
t
{v
t.
}
t
1
7)
=
t = l,CD
where Y
Y^/Y
t
t
income at time
1.
tj'
is assumed
implies
a
law
motion
of
for
where
since
t)
iog(7}^)
the
t
independent
T)
+ piog(7)^_J
are
{c^}
=1.
+7)
The
basic
t
that we consider is of the form:
tj
=
labor
the law of motion for
12
t
t
specification for
(3.2)
2
t)
i's
stationary process for each
to be a
In the two person economy that we are considering,
t
individual
is
t
t
+
individual
c|
shocks
that
have
mean
zero,
that
are
t=l,oo
over
time
and
independent
11
of
c
for
all
t.
The parameter p
captures persistence
individual's
each
in
income
Equation
share.
(3.2)
implies that there is no correlation between the aggregate state and shocks
to
each
evidence
individual's
this
that
income
labor
reasonable
a
is
assuming that the variance of
discuss an alternative
assumption
is
appendix
Also
assumption.
process
selection criterion for families,
present
we
are
by the aggregate state.
We
below
estimated
in
(3.2)
Section
in
using
3.3.
household
annual
and the construction of
describes the estimation of equations
7
For
(3.1)
and
the aggregate
(3.2),
income.
It
the
also
along with several
dynamics we use the point
estimates of the parameters of the vector autoregression,
in Table A.l.
we
The Appendix has a detailed discussion of
income data from the PSID.
alternative specifications.
the
In
is not affected
c
this
to
income
individual
The
share.
(3.1), as reported
For the individual labor income dynamics in the base case, we
use the average of the cross-sectional estimates of (3.2) which are reported
in Table A.
3.
We have so far assumed that the only tradeable assets in positive net
supply are claims
3.9% of
total
income,
tradeable assets.
etc.
may
,
also
to
be
the dividend
this
stream.
Since dividends
clearly understates
Assets such as government
sold
or
used
as
the
share
securities,
collateral
for
additional debt instruments could be incorporated,
average only
of
income
from
corporate bonds,
loans.
In
principle
but doing so complicates
represent labor income we are using a first-order autoregressive
that
the
MaCurdy
(1982)
and
Abowd
Card
(1989)
argue
and
autocorrelation in individual income can be captured by a first-order moving
the first-order
average model for income growth.
For our purposes,
autoregressive model is advantageous since it provides a small state-space
model that captures dependence in income growth.
To
model.
7
For instance, we estimate this law of motion conditional on income
data from households that actually own stocks.
The results are similar to
the
entire
those for
sample.
12
the analysis considerably.
to
a
more realistic
income so
level
tradeable
that
Instead,
we
tradeable asset holdings
increase
by grossing up the assumed fraction of dividend
income
is
157.
income
total
of
average.
on
For
comparison, capital's share of income in the NIPA averages about 30%.
Markov Chain Approximat ions
3.2.
To use this estimated income process as an input for simulations of the
Section 2
VAR
the
model,
approximated with a Markov
is
method of Tauchen and Hussey (1991).
income
process:
the
"Base
9
the
We focus on two specifications of the
and
Case"
using
chain
"Cyclical
the
Distribution
Case"
models.
The
Base
Case
model
equations 3.1 and 3.2.
corresponding
values
of
matrix
the
state
closely
takes on
times as
two
large as
estimated
the
process
We use a state vector with eight dimensions,
of
transition
variables
transition probability matrix.
income
approximates
values,
the value
in
probabilities.
the
different
Table
states
3.1
in
the bad
in
the
with
the
along
the good state
state.
and a
gives
Notice that the individual share of
with a value
in
that
is
The persistence of
labor
1.65
these
individual shocks is captured in the transition probability matrix.
In the Base Case model,
to
be
independent
of
the
idiosyncratic labor income shocks are assumed
aggregate
growth
rate
and
dividend
realization
g
Including
additional
debt
instruments
problematic
due
to
is
nonstationarities in the data.
For example, net interest payments from the
corporate sector to the household sector (as measured by "net interest" from
the NIPA) account for 1.4% of total income in 1946, and rise to 12.1% of
total income in 1990.
Computationally, adding assets has the disadvantage
that it greatly increases size of the state space.
9
The GAUSS code to implement these procedures was kindly provided to us
by George Tauchen.
13
s
.
discussed
as
because,
between
correlation
However,
below,
there
individual
Mankiw (1986),
aggregate
and
evidence
little
is
shocks
PSID
the
in
and Constantinides and Duffie
significant
of
data.
show that
(1992)
the
distribution of labor income over the business cycle can play an important
role when individuals cannot
trade claims
this
to
In particular,
income.
they show that if the distribution of labor income widens in a downturn then
individuals may demand a large equity premium to hold stocks.
this potential effect,
To examine
the Cyclical Distribution Case
we consider
(C.D.C.)
model
model, we set
In the C.D.C.
growth rate of aggregate income, r
low state for
in the
7)
of
result
The
7j.
transition matrix
as
We then choose two values of
is high.
.
of
states
3.1.
This
set
the
in
idiosyncratic shock) when the
(no
such that we maintain the unconditional variance
-^
is
=0.5
t)
Table
given
Table
in
concentrates
with
3.2,
the
the
idiosyncratic
shocks into low growth states for the aggregate economy.
'
There
some evidence
in fact
is
income widens during downturns.
standard deviation of
and
cr
log{Y
/Y
t +i
t
incorporated
iog(T)
is
)
)
the PSID
in
Letting
for each
distribution of
the estimated correlation between
when
However,
-0.06.
the
be the estimated cross-sectional
<r
t,
that
correlation
this
is
t
into
the
Markov
chain
model,
results
it
in
very
a
small
departure from the model of Table 3.1.
3.3
Summary of State Variables
The
exogenous
state
variables
include
y
according to the Markov process specified above.
the
state
is
summarized
holdings
by
can
portfolio composition.
agent
be
I's
derived
holdings
using
the
.
These
evolve
An endogenous component of
two
person economy,
stocks
and
bonds,
market
14
t)
our
In
of
and
5
,
clearing
this
since
agent
conditions
(2.8)
is
2'
and
We define the state vector of the economy by
(2.9).
=
Z
{r*.
t
b
Asset prices,
}.
a function of Z
5
,
ti
t
t
,
t
s
.
t
consumption policies, and trading policies are found as
.
t
Simulation Results
4.
In this section we report the results of Monte Carlo simulations of the
economy of Section
Section
We
3.
2,
using
solve
for
exogenous driving processes described
the
equilibrium
the
numerically
using
a
in
modified
version of the "auctioneer algorithm" described in Lucas (1991).
A number
of
summary statistics are reported.
First,
we consider
the
volatility of individual consumption implied by the model and compare this
to
the volatility of aggregate consumption.
because risk sharing is not complete and,
link
between
individual
and
aggregate
individual consumption behaves
like
This statistic is of interest
as a result,
consumption.
the aggregate
there is no direct
The
extent
indicates
which
the degree
which risk sharing is being accomplished by trading securities.
report the mean and standard deviation of implied
to
to
We also
equilibrium asset prices.
These results allow us to evaluate whether the model with incomplete markets
and transactions costs can explain the equity premium puzzle of Mehra and
Prescott
Finally,
(1985).
trading volume,
in order
we
report
to gauge
the
mean
and
the sensitivity of
standard deviation of
trade to the level of
transactions costs.
To compute these statistics,
we assume that initially each agent holds
half the shares of stock and no debt.
driven by
realizations of
the
The economy evolves for 1000 years,
exogenous
income
process.
trades and consumption growth between years 999 and
each history.
Asset
returns,
1000 are recorded for
The reported statistics are based on averages across 1500 of
15
these experiments
4.1.
Representative Agent Baselines
Before
insurance
turning
to
we
markets,
the
simulations
briefly
examine
markets version of the model.
implications
the
is very
This experiment
undertaken by Mehra and Prescott
(1985),
with
model
the
of
of
incomplete
complete
the
similar to the one
except that we assume a different
Mehra and Prescott equate dividends and
consumption and dividend process.
while we treat consumption as the sum of dividends and labor
consumption,
income.
reports the results of the complete markets version of our
Table 4.1
model for y = 1.5 and
labor
case
income.
which
and each of the specifications of individual
representative
the
The table also reports,
agent
in
the
assumed
is
the complete markets
to
the column labeled
returns from the S&P 500 from
returns using
for
is
consume
"Data",
aggregate
the
sample
The moments of stock returns were calculated using
moments from the data.
annual
= 0.95,
The column labeled "Aggregate"
income.
in
|3
CPI.
The moments
1947
of
1990 converted
through
the
bond
returns
were
to
real
calculated
using annual Treasury bill returns from 1947 through 1990 also converted to
real returns using the CPI.
Although the consumption process here differs
from those used in previous studies,
For example,
the
the implied asset prices are similar.
predicted average return on risk-free bonds
close to the stock return in each case whereas
return is quite low.
Further,
the
is
high and
observed average bond
the standard deviation of the stock return is
The long time horizon was chosen to ensure that the effect of initial
conditions was eliminated.
See,
for example, Mehra and Prescott (1985).
16
low relative to historical levels.
Notice that these result are independent
of the model of idiosyncratic income since this risk is completely shared.
To assess whether the presence of uninsurable
the
potential
to
representative
explain
agent
"Individual B.C."
for
poor
the
performance
consider
model,
the
the Base Case model
Cyclical Distribution (C.D.C.) model.
labor
columns
complete
the
of
income shocks has
Table
markets,
4.1,
labeled
and "Individual C.D.C."
for the
in
These statistics were calculated in a
representative agent model using one of the agent's labor income dynamics as
if they were
if
the aggregate
labor income dynamics.
These results show that
the representative consumer is forced to consume his
income process,
example,
idiosyncratic labor
then the predicted equity premium becomes much larger.
in the Base Case model
the premium is predicted to be 6.8%.
For
The
relatively high level of both the bond and stock returns could be corrected
by using a larger value of
p.
The standard deviation of the stock return is
also predicted to be 38.67, so that the stock return is quite volatile, which
is
consistent with the data.
Notice that in the C.D.C.
model,
the equity
premium is even higher because the idiosyncratic shocks are concentrated in
periods of low aggregate growth.
In
general
the
results
in
Table
4.1
indicate
that
the
model
with
uninsurable labor income has the potential to explain some of the observed
moments of asset returns.
Of
interest,
however,
is
whether these results
change once trading is allowed in the stock and bond markets.
4.2
Frictionless Trading
Table 4.2 summarizes the case in which individuals can trade costlessly
in both
the stock and bond market,
subject to the restriction that stocks
cannot be sold short, and that individuals can borrow only up to 10% of per
capita income.
In the Table,
the rows labeled "Avg Consump Growth" and "Std
17
give the average and standard deviation of consumption
Dev Consump Growth"
growth for agent
more
volatile
Table
much
is
consumption,
less
consumption is only slightly
individual
that
aggregate
than
and
4.1)
Notice
1.
volatile
than
"Aggregate"
the
(see
individual
column
income
in
the
(see
"Individual" columns in Table 4.1).
The result
costless
trading
is
(1991),
who
•
.
more
findings
the
independent over
correlated
persistent.
relative
it
strengthens
.fficult
over
12
time
hence
and
Telmer
assumption that
the
the
and
(1991)
our model,
In
trades when
labor
the
idiosyncratic
labor
income
shocks
are
This persistence should, via a wealth effect, make
self-insure
to
time.
Lucas
of
perform a similar experiment under
income shocks are
shares
idiosyncratic shocks are offset by asset
that
through
the
asset
markets.
A
striking
example of this is given by Constantinides and Duffie (1992), who construct
cases
in which
However,
occurs.
income shocks follow a random walk and no
labor
the
results
persiste ce that we consider
income)
agents
can
shocks.
As a
result,
the comj
•
Gi
a
still
Table
of
4.2
indicate
that
smoothing
with
(which matches the PSID observations of
effectively
smooth
their
idiosyncratic
the
labor
Income
equilibrium asset prices are virtually identical
to
e markets case reported in Table 4.1.
the
parameterization
used
here,
an
increase
in
predicted
consumption volatility appears to require the introduction of some form of
trading
friction.
This
observation
motivates
the
experiments
with
transactions costs and borrowing constraints that follow.
12
We
The estimated autocorrelation coefficient is approximately 0.5.
the
also experimented with income processes exhibiting higher persistence in
This
income shocks, allowing autocorrelation coefficients as high as 0.7.
affect on the results reported in Table 4.2.
had li*''
1
n
Transaction Costs in Both Markets, Asymmetric Bond Market Costs
4.3.
We now consider the effects of transactions costs on equilibrium prices
and
consumption.
transactions
We
costs
reported here,
focus
will
both
in
specifications
on
markets
simultaneously.
finding of
simulations
is
not
second
a
intertemporal consumption
returns
are
Constantinides
their
portfolio
largely unaffected.
(1986)
that
rebalancing and
This
transactions
is
costs
related
cause
therefore have only a
is
such
the
to
a ;Order
f
In the first specification that we consider,
market
to
,ents
sei
effect on asset returns.
bond
face
they tend to substitute towards trading in the lower cost market,
equilibrium
delay
In
agents
Because portfolio balance
order consideration for these agents relative to
and
which
imposing a friction in one market alone has a
we find that
negligible impact on asset prices.
smoothing,
in
only
that
the
borrower
the cost structure in the
pays
the
transactions
cost.
Recall that in this case the transactions cost functions are given by (2.5)
for the stock market and (2.6) for the bond market.
Q to vary from
set k = n/2.
to 2.0.
We allow the
To balance the average costs in the two
The results
13
of
these specifications are given
4.1a-d for the Base Case model and 4.2a-d for the C.D.C. model.
the figures the x-axis gives the value of Q.
-
irameter
in
"kets we
ii
Figures
.•*lv
,each of
<
15
First consider Figures 4.1a and 4.2a which give the average stock and
bond returns, along with the equity premium, as a function of Q for each of
the two models.
Due to the cost of borrowing there are two different rates
of return in the bond market:
a
lending rate and a borrowing rate.
13
In the
In constructing these results a grid of values for Q between
and 2.0
are used.
To smooth across these grid points we fit a third order
polynomial through the points.
This
This curve is reported in the figures.
smoothing of the results only serves to remove some very small vf lability
in the figures.
19
since the equity premium is
figures we plot the average return to lending,
typically measured
using
individuals cannot
borrow at
average
bond
return
the
falls
average
this
and
1
,
the stock market rises from
report
4.2c
the
of
Q
as
and
increases
the
widens.
as a percentage of
The
average
the value of trade,
increases the average costs
As Q
trading
drops
the value of
to 4.1% of
average
percentage of per capita
level
that
security,
For example in the Base Case model the average cost paid in
paid increase.
and
treasury
a
premium
equity
reported in figures 4.1b and 4.2b.
are
on
Notice
rate.
the
transactions costs paid by agent
return
level
As
income.
This
off.
trading
of
markets
two
transactions costs
the
reflected
is
the
in
Figures 4.1c
trade.
consumption volatility which can be seen
in
in
Figures
increase
higher
a
and
4. Id
give the standard deviation of consumption growth for agent
as
level
4. 2d,
a
the
of
which
Notice that
1.
when n = 2.0 individual consumption volatility is close to the volatility of
individual income.
In
in
this model an increase in transactions costs results
the average bond return but
same
at
all
levels
of
the
in a decrease
the average stock return remains
transactions
This
costs.
result
understood by considering the direct and indirect effects of
about
is
the
best
transactions
costs.
The Direct Effect of Transact ions Costs.
With moderate
transactions
costs
each market,
in
period to buffer their idiosyncratic income shocks.
the adverse idiosyncratic shock borrows
in
agents
The
trade
every
individual facing
the bond market and sells stock
in the stock market to partially offset the shock.
The agent receiving the
favorable shock buys both bonds and stocks to create a buffer against future
adverse shocks.
rate
The borrower is willing to pay a relatively high borrowing
including transactions costs.
For
20
example,
consider Figure 4.1a and
The average return on the bond is about
n=2.
0.37.,
but the borrower faces
Since the cost function is quadratic,
an average transactions cost of 3.3%.
the marginal transaction cost that the borrower faces is 6.6% which implies
The lender, however, only receives a
a net marginal borrowing rate of 6.9%.
0.3% rate of return on average.
return because
rate of
Figure
4. Id)
The lender is satisfied with this average
high consumption variability when n = 2
the
(see
creates a strong precautionary demand for bonds.
Notice that the stock return remains at a little over 8% for all levels
of
n
each
In
effect
direct
The
model.
equilibrium stock return is difficult
demands
price
lower
a
to
lower
than 8% due
transactions
for
transactions costs,
to
on
the
whereas
costs
the
Note that the net return from buying stock
seller demands a higher price.
is
costs
predict since the buyer of stock
to
compensate
transactions
of
although
in
equilibrium
the
The lower net return also reflects
observed return is not greatly affected.
a precautionary demand for assets due to higher consumption volatility.
In
for
sum,
transactions
costs
specification
cost
this
depresses
that
the
increases the observed equity premium.
return
net
due
to
a
precautionary
there
observed
a
is
bond
direct
effect
return
and
of
hence
Both stocks and bonds have a lower
demand
induced
by
consumption
higher
variability.
similar
A
effect
of
differential
stock and bond market
transactions
costs on relative observed returns is found by Alyagari and Gertler
However in those models there is no aggregate
and Vayanos and Vila (1992).
risk
so
that
differences
in
rates
of
return are generated
differences in transaction costs across asset markets.
is
a
further
indirect
(1991)
effect
of
transactions
costs
solely by
In our model
which
is
increase in the covariance between stock returns and consumption.
21
due
the
there
to
an
The Indirect Effect of Transactions Costs
To the extent that the increase in consumption volatility increases the
covariance between individual consumption and the net returns on stock,
premium
equity
net-of-transactions-costs
agents
that
widens.
demand
the
To
assess this effect we need to construct a net-of-transactions-costs equity
There are several ways to do this construction.
premium.
time
A transactions cost
liquidated at time t+1.
to be
t
consider
investor from investing in a unit of the stock
the payoff to an individual
at
First
is paid
in the
but the size of this cost depends on the
current period and at liquidation,
portfolio position today and the expected portfolio position tomorrow.
marginal transactions cost that individual
unit
stock
the
of
2k (s
is:
liquidated
-2k
(s
t+1
11
t+2
time
at
-s
t+1
)(p
)
1
-s )(p
As
.
t+1
a
2
s
(see
)
(2.5)).
If
t
in purchasing a
that
position
is
cost
is
t
t
transactions
marginal
the
t+1
2
s
1
t+i
t
pays at time
i
The
result,
net
the
(of
transactions costs)
one
period rate of return from a unit of investment in the stock on the part of
individual
i
is given by:
p^
,,
,,
r
(4.1)
s.net
*
-
d
2k (s'
t+1
t+1
s
t,t+l
s
^
+
p
t+2
t
„,
,
2k(s
t
1
1
t+1
)(p^
-s'
t+1
,
,
-s )[p
t
t+1
f
-
s,2
1
)
t
Notice that this rate of return satisfies
(
eIp
(4.2)
a' (c'
)
^^^
>.
+
(1
5(t)l =
r^'""''^)
1.
I
t
which
is just the
The
net
(of
Euler
equation
transactions
for
costs)
investment in the bond is given by:
individual
rate
of
i
return
(see (2.11)).
from
a
unit
of
l/p"
'
i
if b'
t+
i
-"•"'' -
(4.3)
t,t+l
I
l/lp" + 2n b'
V
t
'^t
t-t-i
(p")^]
^t
if b'
<
t+i
The indirect effect of transactions costs occurs to the extent that the
covariance between
r
'"*
This
4.2a.
measure
of
the
equity
which we calculate for individual
\
conditional
changing
transactions costs
covariance
increase.
In
between
the
the
"net premium"
premium
the
£-|r^'"'
consumption
and
'
growth
t,t+i
when n=2
Base Case model
However
in
the equity premium is 7% the net premium is 2%.
indirect effect does affect
4.1a
The "net premium" reflects the
1.
r
in Figures
given by:
is
premium is 7.8% but the net premium is only 1.3%.
model when Q=Z,
return on the stock
consumption and the net
this effect we plot
To assess
changes.
and
individual
the observed equity premium,
as
equity
the
the C.D.C.
As a result
although the
direct effect on the bond return dominates.
Another measure of the net-of-transactions-costs premium is to look at
the difference between average stock and bond returns,
Notice that a borrower
average realized transactions costs in each market.
must
pay
transactions
costs
on
the
entire
subtracting out the
borrowed
amount
each
period.
However in selling stock the transactions cost is small as a percentage of
the
total
return
on
stock
holdings,
since
only
portfolio typically is bought or sold each period.
a
portion
of
the
stock
The average transactions
cost in the stock market as a percentage of the value of the stock portfolio
are
small,
model.
rising
to
a
maximum of 0.46% when n = 2
is
in
the
C.D.C.
As a result the average net return on an individual's portfolio of
stocks is about 7.5% for n =
= 2
(k=l)
about
1%,
this
2.
implies a
Since the average return to a lender for n
sizeable net-of-transactions-costs equity
premium of 6%.
Which
these
of
two
measures
is
appropriate
more
a
measure
the
of
net-of-transactions-costs premium depends on the question being asked.
The
first measure directly reflects the relation between risk and return in the
while the second is of no obvious theoretical interest.
model,
comparing the predictions of the model
to
data,
However,
the second measure
in
can be
calculated using only information on returns and average transactions costs,
while the former would also require detailed information on consumption.
results
The
of
section
this
indicate
for
that
relatively
low
transactions costs the model does predict a substantial equity premium and a
low risk-free rate of return.
For example in the B.C. model when
average transactions costs are
1.77.
there
that
some
is
effect
=
1,
the
in the bond market and 2.3% in the stock
market and the equity premium is 4.1%.
so
12
of
The net premium is 0.5% in this case
consumption
increased
volatility
on
However the largest effect is
individuals attitude towards aggregate risk.
the drop in the required rate of return on the bond due to a precautionary
demand for savings.
4.4.
Transactions Costs in Both Market, Symmetric Bond Market Costs
To further investigate the effect of the form of transactions costs on
the predicted
costs
equity premium,
shared
are
equally by
we examine
the
lender
function (2.6) is replaced with (2.7).
the
and
case
the
in
which the borrowing
borrower;
the
bond
cost
In this case any difference between
the average bond and stock return should reflect differences in risk between
the two markets.
The
4.3a-d
results
for
the
of
this
specification of
Base
Case
model
and
4.4a-d
costs
for
are
the
in
Figures
model.
Since
reported
C.D.C.
transactions costs are paid by both the borrower and the lender in the bond
and set k equal to
market, we limit Q to the range [0,1],
This produces
Q.
average transactions costs that are similar to those in Figures 4.1 and 4.2.
so that the results of the two cost structures can be easily compared.
Notice
that
slightly as a result of
C.D.C.
to
be
4.3a
Figures
in
and
4.4a
increase
the
the
rises
premium
equity
consumption volatility.
in
only
the
In
model the premium widens more than in the Base Case model, which is
given
expected
the
fact
individual
the
that
are
Because transactions costs are paid
concentrated during economic downturns.
the drop in the bond return is not due
by both the lender and the borrower,
in the transactions costs structure
to differences
shocks
income
in
the
two markets,
but
instead to an increased precautionary demand for savings.
The
effect
demand
precautionary
of
can
seen
be
considering the net-of-transactions-costs bond return.
C.D.C. model with n =
market
transactions costs
return
because
For example
in
by
the
the average bond return is 6.3% and the average bond
1
As a result
2.7%.
is
investing in the bond market is 0.9%.
bond
clearly
most
the
of
form
the
return from
marginal
This is not reflected in the observed
the
of
Since
function.
cost
the
transactions costs are balanced in each market the equity premium in this
case
reflects
the
premium
'^
net
as
measured
by
£-|r^'""
t,t+i
-
I
r^'"*
r
'
^
V,
where
t,t+il
is given by (5.1) and
t,t+i
(4.4)
r
•"•'^
t,t+i
= £{l/[p°
t
+
2n
t
b'
t+i
{p°r]}
.
t
Notice that the net premium and the measured equity premium track each other
closely in this case.
In this specification of
transactions costs there is
some effect on the observed equity premium but it
is
less dramatic than in
the asymmetric bond market costs case examined in Section 4.4.
25
4.5.
Transactions Costs in Stocks, No Borrowing
we
Here
consider
a
structure
market
that
permits
costly
the borrowing constraint is at 0).
stocks but precludes borrowing (i.e.,
of substitution of the agent with a good idiosyncratic shock.
to close the bond market would be to
In
14
A second way
impose a very high cost on the borrower
This is equivalent to the case where borrowing is not
the bond market.
allowed.
in
the (shadow) price of bonds is calculated using the marginal rate
this case,
in
trading
Hence,
the
results
this
of
section
can
interpreted
be
as
approximating a situation where there is extreme credit rationing or where
there is a very large wedge between the borrowing and
Figures
borrowing,
of
4.5a-d
report
results
for
Base
the
lending
Case
rates.
in which the only cost parameter that varies is k.
the average costs
the effect
of
without
model
For any level
stock market transactions costs on the
predicted bond return and the equity premium is slightly larger than in the
case where costly borrowing
agents
because
cannot
is
allowed
substitute
(see Figures
towards
4.1a-d).
bond
market
borrowing
are
the
This occurs
to
avoid
transactions costs.
The
results
Figures 4.6a-d.
for
the
C.D.C.
model
without
reported
These results are similar to Figures 4.5a-d except that the
equity premium is larger for any level of average transactions costs.
occurs
before
this
during
economic
because
downturns.
shocks
individual
to
Notice
however
that
labor
the
income
only
additional
premium predicted by this effect is not large.
14
in
See Heaton and Lucas (1992) for a discussion of this issue.
26
As
occur
equity
Concluding Remarks
5.
In this paper we have examined asset prices and consumption patterns in
in which agents face both aggregate and idiosyncratic income shocks,
a model
Agents reduce consumption variability
and insurance markets are incomplete.
by trading in a stock and bond market to offset
idiosyncratic shocks,
transactions costs in both markets limit the extent of trade.
but
To calibrate
the theoretical model, we estimated an empirical model of labor and dividend
using data from the PSID and the NIPA.
income,
Although the agents in the model are not very risk averse,
predicts
equity
sizable
a
simultaneously
considering
premium
and
aggregate
and
riskfree
low
a
idiosyncratic
the model
By
rate.
shocks,
we
can
decompose this effect of transactions costs on the equity premium into two
The direct effect
components.
net-of-cost margins,
with
asset
an
so
is due
the fact
to
that
individuals equate
associated
lower
transactions
cost
will have a lower market rate of return.
The
size
of
direct
the
are assumed to be similar,
assume
that
the
structure
the
of
When the cost structure in the stock and bond market
transactions costs.
we
widely with
varies
effect
the direct effect
transactions
cost
the
in
market
bond
when
However,
is negligible.
creates
a
wedge
between the borrowing and lending rate, or that there is a binding borrowing
constraint,
total
premium.
(1991),
50/4
the direct
This
effect
is
can account
related
to
the
for
three quarters of
over
findings
of
Aiyagari
the
and Gertler
who report that realistic transactions costs may account for about
of the observed equity premium.
A second,
individual
aggregate
indirect effect occurs because transactions costs result in
consumption
consumption.
that
The
more
higher
closely
tracks
variability
27
of
individual
individual
income
than
consumption
increases the covariance between consumption and the dividend process,
increases the systematic risk of
hence
While the size of the
stock.
increases with the assumed level of transactions costs,
indirect effect
is
the
relatively insensitive to the structure of transactions costs.
case
base
and
analysis,
the
indirect
accounts
effect
about
for
20%
it
In
the
of
the
premium.
While the model can explain the observed first moments of stocks and
bonds,
does not explain observed second moment differentials.
it
the feature of many consumption-based models,
shares
It
that an increase in the equity
premium predicted by the model is not accompanied by a substantial change in
Typically in models that
the relative volatility of bond and stock returns.
fit
equity premium,
the
However
in
the model
implies
costs
a
the
volatility of
bond
return
return
bond
is
too
high.
considered in this paper an increase in transactions
widening
in
equity
the
premium,
observed returns does not greatly increase.
the
the
remains
fairly
low
as
but
the
volatility
of
In particular the volatility of
transactions
costs
increase.
The
volatility of the net-of-transactions-costs bond and stock returns do rise
in response to transactions costs due to the increase in the variability of
the
marginal
there
is
a
rate
of
substitution.
realistic
assumption
It
remains
about
open question whether
an
transactions
costs
that
can
simultaneously explain the low volatility of short bond rates and the high
volatility of stock returns.
Several extensions of
the analysis
are
left
future research.
to
Our
two-person economy is the simplest case of heterogeneity, and intuitively we
expect
most
of
the
results
to
carry
over
to
Conceptually the model could easily be extended to
Because
of
computational
the
practical
techniques,
an
difficulty
of
the
increasing
interesting question
28
more
the
is
general
case.
case of n agents.
n
using
whether
it
current
would be
possible to exploit a different modeling strategy to develop a tractable way
to
allow for a large number of
individuals.
This would allow for a more
detailed description of the income distribution and its evolution over time.
One potentially important factor that could be studied in this context would
be the impact of a small probability of a very bad idiosyncratic shock.
The
model also could be extended to include a production or storage technology,
and a labor/leisure decision.
29
n
Data Description and Estimation Results Using the PSID
Appendix:
Section 3.1 we briefly described the empirical model of aggregate
In
In this appendix we
labor income used in the calibrations.
and individual
describe the estimation in more detail, and describe a more general model of
labor income that was also investigated.
A.
Further Specification of the Model of Individual Labor Income
1
individual i's labor income as a fraction of aggregate
As in Section 3,
income
labor
given by
is
process for each
time
over
in
ii)
for
}
<t)
}
t = l,00
specification of
general
tj
account
must
shocks
correlation between
and
7)
assumed to be a stationary
is
_
t
most
The
state.
where
,
The model
i.
'^
tj
to
that
and
i)
we
correlation
for
aggregate
the
consider
of
the
that
are
of
the
is
form:
(A.l)
iOg(Tl^)
where
the
{e
,
{p
1
1
,
= 1,
>
The
,_
,
shock
time
a
^
for
c
p iOg{7}^_^) + C^,
-^
shocks
individual
,
individuals
across
and
,
,.
all
t.
have
that
r^f,
£{(c
1,2,1/2
_
=
>
)
zero,
independent
and
,_l,i = l,n
,-l,l = l,n
1
o-
mean
{t)
.
}
{^
,
>
and
.
are parameters.
parameters
{tj
~
}
'"
permanent
capture
The parameter vector
labor income.
to
C,'C^
are
t=l,oo
over
independent
+
T)
*"
>
t
aggregate
=
{C,
differences
relative
in
captures the degree to which shocks
}
individual i's relative income are correlated with the aggregate shocks.
Differences
in
differentially
<
across
affect
allow
individuals
The
individuals.
the
aggregate
parameter
p
shock
captures
to
the
persistence in shocks to individual i's labor income.
As
it
stands,
the
specification
30
in
(A.l)
does
not
use
all
of
the
cross-sectional information.
the parameter
For example,
across individuals in the model.
to vary
is free
t)
may be possible to link variation in
It
to observable family characteristics.
unless variation in
However,
linked to Just two distinct subgroups of the population,
ij
t)
were
is not clear how
it
this information could be used in our investigation of the
implications of
the two person economy of Section 2.
Data
A. 2.
The
PSID provides
panel
a
observations
annual
of
of
In using the PSID,
family income, consumption of food, and other variables.
we took family income per family member as of a measure of Y
Because
changing
the
of
character
of
many
the
of
and
individual
for our model.
we
families,
took
a
subsample from the PSID consisting of those families for which the head of
the household was male and for which neither the head nor his spouse changed
This selection was used so that we did not have to keep
over the sample.
track of new families,
complete
The
family.
family split-offs, and other dramatic changes in the
PSID
over-samples
members
poorer
the
of
U.S.
population due to a sample of poor individuals from the Survey of Economic
population,
make
To
Opportunity.
those
the
families
sample
who
closer
were
to
a
sample
random
originally
part
of
of
the
the
U.S.
Survey
of
Economic Opportunity were excluded from the sample.
Total labor income of the head of the household and his wife along with
total
transfers
to
the
family
was
used
for
total
family
income.
The
transfers included unemployment compensation, worker's compensation, pension
income, welfare,
child support and so on.
all of these sources in any year,
it
If a
family had zero income from
was excluded from the sample.
At this
point we had a sample of 860 families with income data spanning the years
1969
to
1984.
Because families differ
31
in
size,
family
income per family
member was created by dividing each family's income by the total number of
family members in each year
This measure of nominal family labor income
.
was weighted by the Consumer Price Index (CPI)
to obtain a measure of
real
labor income per family member.
The model of
sample,
income dynamics was estimated using this full
individual
and also using a subsample
restricted to households owning stock.
The sample was split because a Large segment of the population does not hold
In the asset pricing model
financial assets.
appropriate
more
clearly
individuals who
participate
Zeldes
the
(1990),
consider
to
in
income
split
based
stock holdings
family reported some holdings of
stocks
in
individual
upon
securities using questions about
dynamics
in
1984,
it
is
those
of
Following Mankiw and
securities markets.
was
sample
the
that we are examining,
the
it
1984
was
holdings
PSID.
Included
of
If
in
a
the
group called stockholders.
For both
the
constructed as y
complete sample and the stockholder sample,
/-Ij]
"
Y
t)
was
then
where n is 860 for the complete sample and 327
>
for the stockholders sample.
Along
with
observations
of
individual
labor
income
from
the
PSID,
measures of annual aggregate labor income and dividends were taken from the
NIPA for
the
years
1947
through
1989,
obtained from CITIBASE.
income we used "total compensation of employees".
weighted by the total U.S.
For
labor
The aggregate series were
population and the CPI
in
each year
to
obtain
real per capita labor income and dividends.
We also conducted the analysis where family income was weighted only
by the number of adults in the family.
The results were similar and hence
are not reported.
32
I
Empirical Results
II. 3.
As
described above,
measure of aggregate
our
income was
labor
taken
alternative would have been to use total
from the NIPA.
A natural
from the PSID.
However because of the limited time dimension in the PSID,
the aggregate dynamics could not be estimated with much precision.
income
Although
the NIPA measure of labor income does not exactly correspond to the measure
of labor income obtained from the PSID,
Figure
For example.
are similar.
A.
1
the two measures of aggregate income
gives a plot of the CITIBASE measure
of aggregate labor income growth [logiY^/Y^
measure
constructed from
the
The
PSID.
)]
along with the corresponding
PSID measure
aggregate
of
labor
income per capita was constructed by summing household labor income across
households in our sample and then weighting by the total number of people in
The correlation between the two series is 0.9.
our sample for each year.
As a result,
measuring aggregate labor income shocks from the NIPA series
should do a reasonable job in capturing aggregate labor income dynamics.
Table A.l reports ordinary least squares estimates of the law of motion
for
aggregate
labor
income
and
aggregate dividends
Using the fitted residuals from this estimation,
equation
(see
the model
(3.1)).
given by
(A. 2)
was estimated individual by individual using ordinary least squares.
All
Individuals.
The estimation was first performed for our
largest
A summary of these findings is given
subsample of households from the PSID.
in Table A. 2 which reports sample averages of the parameter estimates along
with
sample
consistent
(1989),
standard
with
errors.
results
for example.
In
estimates
The
reported
MaCurdy
in
particular,
the
reported
(1982)
in
and
Table
Abowd
A. 2
and
are
Card
estimated parameter values imply
that individual income growth is negatively correlated over time.
Abowd and
Card (1989) also argue that aggregate variation in income has little affect
on the autocorrelation structure of individual
33
income.
In our
sample from
on average only 16% of individual income variation is captured by
the PSID,
the aggregate shocks.
for the
's
T)
with
C,
which reports
A. 3
= 0.
the
we also estimated the law of motion
For this reason,
The results of this estimation are given in Table
cross
sectional standard errors.
section average
the
of
Tables
A. 4
for the stockholder subsample.
A. 3,
variance of
larger.
than
the
in
and
However,
(A. 2)
In comparing these results to tables A. 2 and
explained
by
the
regressions
the
for
o-
that
for
entire
the
sample.
stockholders
Also
is
the
slightly
time and in which there is a very important role for
On average
aggregate
12% of
as our base case
the variation in
over
t)
time
is
reported
in
shocks.
Based upon these results we use
A. 3
,
the
the results are generally consistent with a model in which
idiosyncratic shocks.
case
.
present results of estimating
A. 5
idiosyncratic shock,
is correlated over
Table
cross
notice that the average autoregressive parameters are smaller for the
stockholders
Tj
and
Notice that the average autoregressive parameter
is slightly larger as is the estimated variance of c
Stockholders.
parameters
the estimated parameters
in Section
3.
In Section 3 we
also discuss a
captures variation in the cross-sectional distribution of
income over the business cycle.
34
labor
References
J.M. and D. Card (1989). "On the Covariance Structure of Earnings and
Hours Changes", Econometrica 57, 411-445.
Abowd,
Amihud, Yakov and Haim Mendelson (1986).
"Asset Pricing and
Spread," Journal of Monetary Economics 17, 223-249.
Aiyagari, S. Rao and Mark Gertler (1991).
Costs and Uninsured Individual Risk:
of Monetary Economics 27, 309-331.
the
Bid-Ask
"Asset Returns with Transactions
A Stage III Exercise,"
Journal
Constantinides,
George
(1986).
"Capital
Market
Equilibirum
Transaction Costs," Journal of Political Economy, 94, 842-862.
with
Constantinides, George and Darrell Duffie (1992).
"Asset Pricing
Heterogeneous Consumers," manuscript, University of Chicago.
with
"Two-Person Dynamic Equilibrium in the Capital
Dumas, Bernard (1989).
Market,"
The Review of Financial Studies, Vol. 2, No. 2, 157-188.
Lars Peter and Ravi Jagannathan (1991).
"Implications of Security
Market Data for Models of Dynamic Economies," Journal of Political
Economy, 99, 225-262.
Hansen,
Lars Peter and Kenneth Singleton (1983).
"Stochastic Consumption,
Risk Aversion and the Temporal Behavior of Asset Returns," Journal of
Political Economy, 91.
Hansen,
Heaton,
John and Deborah J. Lucas (1992), "The Effects of Incomplete
Insurance Markets and Trading Costs in a Consumption-Based Asset
Pricing Model," Journal of Economic Dynamics and Control, 16, 601-620.
Lucas, Deborah J. (1991), "Asset Pricing with Undiversif iable Income Risk
and Short Sales Constraints:
Deepening the Equity Premium Puzzle,"
manuscript. Northwestern University.
Robert E. Jr.
Econometrica
Lucas,
(1978),
"Asset
Pricing
in
a
Pure Exchange Economy,"
MaCurdy, Thomas E. (1982).
"The Use of Time Series Processes to Model the
Error Structure of Earnings in a Longitudinal Data Analysis," Journal
of Econometrics, 18, 83-114.
Mankiw, N. Gregory (1986), "The Equity Premium and the Concentration
Aggregate Shocks," Journal of Financial Economics 17, pp 211-219.
and S.
Zeldes (1990),
"The Consumption
Non-Stockholders" NBER working paper.
of
Stockholders
of
and
Marcet, Albert and Kenneth J. Singleton (1991), "Equilibrium Asset Prices
and Savings of Heterogeneous Agents in the Presence of Incomplete
Markets and Portfolio Constraints," working paper, Stanford University.
Mehra, Rajnish and Edward C.
Prescott (1985), "The
Puzzle," JournaJ of Monetary Economics 15, 145-161.
35
Equity
Premium:
A
"Essays on the Risk Premium
Saito, Makoto (1992).
Market," Ph.D. dissertation, M.I.T.
G. and R. Hussey (1991).
Solutions
to
Approxmimate
Econometrica, 59, 371-396.
Tauchen,
in
the U.S.
Financial
"Quadrature-Based Methods for Obtaining
Pricing
Models,"
Nonlinear
Asset
Chris I. (1991), "Asset Pricing Puzzles and Incomplete
Queens University working paper.
Telmer,
Markets,"
"Equilibrium Interest Rate and
Dimitri and Jean-Luc Vila (1992).
Liquidity Premium Under Proportional Transactions Costs," manuscript,
Sloan School of Management.
Vayanos,
36
laDie
1
J.
Markov Chain Model for Exogenous State Variables
Base Case
States
State Number
a
1
0.991
1.047
0.991
1.047
0.991
1.047
0.991
1.047
1
2
3
4
5
6
7
8
V
S
r
0.3772
0.3772
0.3772
0.3772
0.6228
0.6228
0.6228
0.6228
0.1522
0.1517
0.1483
0.1478
0.1522
0.1517
0.1483
0.1478
Transition Probability Matrix
[p
r
1
where p
= Pr{state J at time t+1
1
state
i
at time t>
0.3562
0.2430 0.08506 0.05803
0.1237 0.08436 0.02953 0.02015
0.3392
0.3360 0.03370 0.03338
0.1178
0.1166 0.01170 0.01159
0.03338 0.03370
0.3360
0.3392 0.01159 0.01170
0.1166
0.1178
0.05803 0.08506
0.2430
0.3562 0.02015 0.02953 0.08436
0.1237
0.1237 0.08436 0.02953 0.02015
0.3562
0.2430 0.08506 0.05803
0.1166 0.01170 0.01159
0.3392
0.3360 0.03370 0.03338
0.1178
0.1166
0.1178 0.03338 0.03370
0.3360
0.3392
0.02015 0.02953 0.08436
0.1237 0.05803 0.08506
0.2430
0.3562
0.01159 0.01170
37
_
1
Table 3.2
Markov Chain Model for Exogenous State Variables
Cyclical Distribution Case
State Number
1
2
3
4
5
6
7
8
0.1478
1.047
Transition Probability Matrix
As in Table
3.
0.5
1
Table
4.
Moments Implied by the Complete Markets Cases
Indlvidiual
Moment
Aeeresate
Avg Consump Growth
0.021
Std Dev Consump Growth
0.029
Avg Bond Return
0.008
Std Dev Bond Return
0.026
Avg Stock Return
0.089
Std Dev Stock Return
0.173
Individual
Table 4.2
Momemts Implied by the Frictionless Model
Base Case
Moment
Avg Consump Growth
1.
Std Dev Consump Growth
Avg Bond Return
Std Dev Bond Return
Avg Stock Return
Std Dev Stock Return
Avg Bond Trades
(°/.
of Consumption)
0.058
Std Dev Bond Trades
Avg Stock Trades
('/.
Std Dev Bond Trades
0.070
of Consumption)
0.121
0.057
Cyclical Distribution
Table A.l
Aggregate Dynamics
y.
=
t
t-1
dVv^ and
X^ 2
[iog(/) logiS
X" = A^ X^
8'a
*
t-1
t
a
t
a
^
Ordinary Least Squares Estimates
A" =
0.0805
0.0626
(0. 1673]
(0.0344)
-0.5166
(0.2329)
16
0.0276
9° =
0.9480
0.0056
(0.0706)
0.2249
a
M
=
(0.1141)
-0. 1630
(0.2343)
16
)]'
t
StcUidard errors in parentheses.
41
0.0432
2
Table
A.
Individual Income Dynamics with Aggregates
All Individuals
Cross sectional means and standard deviations of coefficient estimates.
t
,
,
1
>
jOglT)^)
=
-1
1
+
7)
I
0-
Coefficient
-i
t
,
,
t
1
,
p 10g(T}^_^) +
,„,
= {£{C
,1/3
'C^
C,
1
* C^,
1,2,1/2
)
>
Cross-Sectional Mean
Standard Deviation
-3.5284
2.6094
0.0036
0.3600
-0.0071
0.0722
0.5032
0.1268
0.2292
0.1226
42
3
Table
A.
Individual Income Dynamics without Aggregates
All Individuals
Cross sectional means and standard deviations of coefficient estimates.
t
1 ,
iogCi)
)
=
T)
1
a-
Coefficient
t
-1
+
I
t
,
,
1
p iog(T)
,_,
= {£(c
)
+ e
,
1,2,1/2
)
>
Cross-Sectional Mean
Standard Deviation
Ti^
-3.3499
2.4134
p*
0.5290
0.3320
<r'
0.2508
0.1309
43
4
Table
A.
Individual Income Dynamics with Aggregates
Stockholders
Cross sectional means and standard deviations of coefficient estimates.
t
1
a-
t
t
,_, 1,2,1/2
- {£(c ) >
t
Coefficient
Cross-Sectional Mean
Standard Deviation
V
-4.6673
2.1603
C'
0.0113
0.0930
C'
^2
-0.0441
0.1676
p'
0.2308
0.3479
0-'
0.3587
0.1494
1
44
5
Table
A.
Individual Income Dynamics without Aggregates
Stockholders
Cross sectional means and standard deviations of coefficient estimates.
t
lOg(l)^)
=
TJ
1
0-
t
t
+ p Jog(T)^_^)
,_,
= {£(C
+
C^,
1,2.1/2
)
>
t
Coefficient
-1
11
Cross-Sectional Mean
-4.2409
Standard Deviation
1
.
8692
0.2987
0.3039
0.3826
0.1589
45
B.C., Returns
Figure 4.1a:
0.09
0.08
0.07
0.06
£
0.05
3
^
0.04
0.03
0.02
0.01
1
F
r
I
1
1
1
1
1
!
1
Figure 4.1c: B.C., Average Trading
0.14-
0.12
=0
V
0.06
<
0.04
0.02 ^
Figure 4.2a:
0.09
0.08
0.07
0.06
£
3
0.05
^
0.04
0.03
0.02
0.01
1
1
;
C.D.C., Returns
,
]
I
1
I
Figure 4.2c: C.D.C., Average Trading
Stock Market
=0
c
"P
0.08
u
2
Bond Market
0.06
<
0.04
0.02
0.
0.2
0.4
0.6
0.8
Omega
1
(k
Figure 4.2d: C.D.C,
o
0.06
Q
^
0.05
0.2
0.4
0.6
0.8
Omega
1.4
1.6
1.8
1.6
1.8
= Omega/2)
STD
1
(k
1.2
of Consumption
1.2
= Omega/2)
1.4
Figure 4.3c: B.C. Symmetric Costs, Average Trading
C
1
0.08
H
V
2
0.06
Bond Market
<
0.04
0.02
0.1
0.2
0.3
0.4
Omega
0.5
0.2
0.3
0.4
Omega
0.7
0.8
0.9
= Omega)
(k
Figure 4.3d: B.C. Symmetric Costs,
0.1
0.6
0.5
(k
STD
0.6
= Omega)
of Consumption
0.7
0.8
0.9
Figure 4.4a:
C.D.C. Symmetric Costs, Returns
0.09
0.08
Stock Return
0.07
Bond Return
0.06
E
0.05
^
0.04
0.03
0.02
Equity
Premium
.
0.01
.+
—
+...-"
-
Net Premium
^----f
0.1
0.2
0.3
0.4
Omega
0.5
(k
0.6
0.7
= Omega)
Figure 4.4b: C.D.C. Symmetric Costs, Costs
0.06
0.05
0.04
O
0.8
0.9
Figure 4.4c: C.D.C. Symmetric Costs, Average Trading
0.14
0.12
Stock Market
Bond Market
0.04
0.02
b
0.1
0.2
0.3
0.4
Omega
0.6
0.5
(k
0.7
0.8
= Omega)
Figure 4.4d: C.D.C. Symmetric Costs,
STD
of Consumption
0.02L
0.1
0.2
0.3
0.4
Omega
0.5
(k
0.6
= Omega)
0.7
0.8
0.9
1
Figure 4.5a:
B.C.
No
Bonds, Returns
Figure 4.5c: B.C.
No
Bonds, Average Trading
0.25
k
Figure 4.5d: B.C.
0.05
0.1
0.15
No
0.2
Bonds,
0.25
STD
of Consumption
0.3
0.35
0.4
0.45
0.5
Figure 4.6a:
0.09
0.08
0.07
0.06 h
E
3
0.05
4-*
^
0.04
0.03
0.02
0.01
C.D.C.
No
Bonds, Returns
Figure 4.6c: B.C.
No
Bonds, Average Trading
0.25
0.05
0.1
0.15
0.2
Figure 4.6d: B.C.
No
0.25
Bonds,
0.3
STD
0.35
0.4
of Consumption
0.45
0.5
Figure A,l
0.06
0.04
0.02
o
o
-0.02
-
1972
1974
1978
1976
Year
'>668
135
1980
1984
Date Due
NOV,
i-
'
'^^^
25
Lib-26-67
Mil
3
IIPRARIES
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