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John Heaton
Massachusetts Institute of Technology

Deborah Lucas
Northwestern University

WP#3491-92-EFA November 1992






John Heaton
Massachusetts Institute of Technology

Deborah Lucas
Northwestern University

WP#3491-92-EFA November 1992

NOV 1 8 1992



First Draft: September 1991
This Draft: 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.

1. Introduction

Incomplete markets In the form of an Inability to borrow against risky
future income has been proposed as an explanation for the poor predictive
power of the standard consumption-based asset pricing model. With complete

markets, individuals fully insure against idiosyncratic income shocks, and


individual consumption is proportional to aggregate consumption. With

limited insurance markets, however, individual consumption variability may
exceed that of the aggregate, and the implied asset prices may differ
significantly from those predicted by a representative consumer model. In
this paper we study an economy in which agents cannot write contracts
contingent on future labor income realizations. They face aggregate
uncertainty in the form of dividend and systematic labor income risk, and
also Idiosyncratic labor income risk. Idiosyncratic income shocks can be
buffered by trading in financial securities, but the extent of trade is
limited by borrowing constraints, short sales constraints and transactions

The motivation for considering the interaction between trading
frictions and asset prices in this environment is best understood by
reviewing the findings of a number of recent papers. Lucas (1991) and
Telmer (1991) examine a similar model with transitory idiosyncratic shocks
and without trading costs. Surprisingly, they find that even though agents
cannot insure against idiosyncratic 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).

See R. Lucas (1978), for example.

similar to those with complete markets. This occurs 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, in which agents trade to offset

transitory idiosyncratic shocks. Differential transactions costs affect the

relative returns on stocks and bonds, and reduce the total volume of trade.

Finally, Constantinides and Duffie (1992) study the case of permanent

idiosyncratic shocks. Although agents have unrestricted access to financial

markets, no trade occurs. When the conditional variance of idiosyncratic

shocks increases in a downturn the riskfree rate falls and the equity

premium rises relative to the complete markets case.

These results suggest that the quantitative asset price predictions
from this class of models will depend critically on several factors: (i)
the extent of trading frictions in securities markets, (ii) the size and
persistence of idiosyncratic shocks, and (iii) the correlation structure of
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 these shocks over time, based
upon evidence from the Panel Study of Income Dynamics (PSID). 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


Using a more volatile aggregate income process, Marcet and Singleton

(1991) also calibrate this model. The equity premium rises in the presence

of short sales constraints.

by considering transactions costs in an environment with both aggregate and
idiosyncratic shocks. Transactions costs play an important role because
agents need to trade frequently in order to buffer shocks to their
individual income. As a result, transactions costs can have two effects on
asset prices.

First, (gross) rates of return on securities may be altered because
agents require higher returns to compensate for transactions costs. This
effect of transactions cost was emphasized by Aiyagari and Gertler (1991),
Amihud and Mendelson (1986) and Vayanos and Vila (1992). In contrast,
Constantinides (1986) argued that transactions costs should have only a
small effect on asset returns. In his model in which there is no
idiosyncratic risk and agents trade only to rebalance their portfolios,
agents avoid most transactions costs by reducing the frequency of trades.
As a result, asset returns are not much affected by the presence of
transactions costs. However due to the idiosyncratic shocks that
individuals face in our model, it is quite costly for 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 to use asset markets to self-insure against transitory
shocks, so that individual consumption does not move directly with the
aggregate. 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. The implied equity premium
could rise in response to increases in transactions costs for this reason
alone. This paper appears to be the first to evaluate the importance of
this mechanism.

We calibrate the model under a variety of assumptions about the size
and incidence of trading costs. When trading costs differ across markets.

we find that agents readily substitute towards transacting in the lower cost
market. For example, if transactions costs are only introduced in the stock
market then agents trade primarily in the bond market and by this means
effectively smooth transitory income shocks. In this case transactions
costs have little affect on required rates of return. However 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 bor, return
that is close to the observed return on U.S. Treasury securities.

The remainder of the paper is organized as follows: In Section 2 we
describe the model economy. Section 3 presents the empirical model of
income and dividends. We also discuss the parameter izations for trading
costs, borrowing constraints, and short sales constraints. Simulation
results are reported in Section 4. Section 5 concludes.

2. Model

2. 1 The Environment

The economy contains two (classes of) agents who are distinguished by
their labor income realizations. At each time t, agent i receives
stochastic labor income Y . By assumption, agents are not allowed to write
contracts contingent upon future labor income.

Agents also receive income from investments in stocks and bonds. At

time t, a share of stock, with price p , provides a claim to a flow of

random dividends from time t+1 forward, {d } The bond, with price p ,

J J=t+i "^ "^t

provides a risk-free claim to one unit of consumption at time t + 1. The

agents trade these two securities to smooth their consumption over time.
Trading is costly, with transactions cost function »c(-) in the stock market
and (j(-) in the bond market. Agents also face short sales and borrowing

At time t, each agent's preferences over consumption are given by:

, CO (c' ) ^'^ - 1 ^

(2.1) (/' = e\i p"^ —^ : I ?(t)L 3- > .

W=o 1-r J

where ?(t) is the time t information set, which is common across agents.
This information is generated by a state variable, Z , which is specified
below. In principle, r could be allowed to differ across agents. However,
since we want to interpret the two groups as similar except for realizations

of idiosyncratic shocks, it seems appropriate to equate y across the two


At each date t, agent i maximizes (2.1) via the choice of consumption

c , stock share holdings s and bond holdings b ^ subject to the flow
t ^ t+i ^ t+1 ^

wealth constraint:

(2.2) c' + p^s^ + p^b^ + k(s' ,s^;Z ) + u(b^ ;Z )

t t t+i '^t t+i t+i t t t+i t

s s {p +d)+ti +y
t t t t t

and short sales or borrowing constraints:

(2.3) s' £ K^ t = 0, 1, 2, . .

(2.4) b^ £ /(" t = 0, 1, 2. ..

t t


Dumas (1989) considers the implications of different risk aversion

parameters in a complete markets setting.

The components of initial wealth d , s , b and / and market prices are


taken as given.

2.2 Trading Frictions

The extent to which individuals will use asset markets to buffer
idiosyncratic income shocks depends on the size and incidence of trading
costs, and the presence of borrowing and short sales constraints. 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 the stock market. Both buyers and sellers
are assumed to face a quadratic transactions cost function:

(2.5) k(s' ,s^;p^) = k {(s' - s^)p^}^
t+i t t t t+i t t

In the simulations, the parameter k is used to control the magnitude
of the transactions cost. However, because the realized cost is endogenous,
the range of attainable costs is bounded; an increase in the cost parameter

eventually leads to an offsetting reduction in trade. Dividing the cost

1 Is
function by (Is -s | )p gives the trading cost as a percent of the value

1 Is

of shares traded: k \s -s \p . In the simulation results we report the

t t+i It

average of this percentage cost.

We use a quadratic cost function primarily for computational
simplicity. However, it also captures the idea that as more assets are
sold, agents must sell increasingly illiquid assets. The fact that many
individuals hold no stock at all suggests that there may be 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 to the
quadratic form is that small changes in stock holdings may be at least as
costly (proportionally) as large changes. In the discretized state space we
will consider, however, infinitesimal shocks never occur so this limiting
case is not relevant.

Bond market transactions costs. Bonds in this model represent private
borrowing and lending. While it seems sensible to treat transactions costs
symmetrically for sales and purchases of stock, this is less true for loans.
Typically consumers pay a substantial spread over the lending rate to
borrow. Part of the observed spread is a default premium which does not
apply to the riskfree bonds of the model. However, a portion of the spread
C2in 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) w(b^ ;Z ) = n min(0.b' p")^.

The parameter Q controls the magnitude of the cost. By convention
borrowing at time t is represented by a negative value for b , so only the
agent who borrows pays the transactions cost. As with stocks, the cost is
reported as a percentage of the per capita amount transacted: n \b \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:

(2.7) w(b' ;Z ) = n (b' p'f.
t+i t t t+i t

As we shall see, the choice of (2.6) versus (2.7) will have a significant

affect on the predicted equity premium.

To match the observed income and dividend process, the economy is

assumed to be stationary in aggregate income growth rates. As a result, the

price of the stock and the face value of the bond grow over time. To

accommodate the growth in value, the borrowing constraint, K , is assumed to

be linear in aggregate income, Y . Because the transactions costs are

quadratic in the value of trade, to induce a constant average transactions

cost to income ratio, we assume that k = k/Y and Zl = £l/y^ where k and n

t t t t

are constants.

Finally, we refer to the case where k(-) = and u(-) = as the
frictionless model. The frictionless model is similar to Lucas (1992) and
Telmer (1992).

Borrowing and Short Sales Constraints. Consumption smoothing may also
be curtailed by institutional limits on the amount of borrowing. This type
of credit rationing is represented by (2.4). We will consider two
scenarios. In the first, agents can borrow up to 10% of average per capita
income. In the second, 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, which is a plausible limit on debt, is
not easily measured. Since agents rarely hit the assumed lO'/i upper bound
for the income shocks considered, and since the agents always desire to do


some borrowing, the chosen limits appear to bracket the relevant range.

No short sales are permitted in the stock market so that K = in
(2.3). This is motivated in part by the observation that it is costly for
individuals to take short positions. Clearly, allowing short sales would
increase the effective quantity of tradeable assets.

2.3 Equilibrium

At time t, aggregate output consists of the aggregate dividend, d and
the sum of individuals' labor income V Y . Market clearing requires:

^1=1,2 t CM

(2.8) b^ + b^ = t = 0, 1, 2,

(2.9) s^ + s^ = 1 t = 0, 1, 2,

(2.10) y {c' + k(s' ,s';Z ) + (j(b' ;Z )> = d * y^ * y^
^1=1,2 t t+1 t t t*l t t -^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 1 and t:

(2.11) Ip" + K (s' ,s';Z )]u' (c')
t 1 t+i t t t

= ^eL' (.c^ )[p^ * d * K {s^ .s' ;Z )] I ?(t)l.
1^ t+l t+l t+l 2 t+2 t+1 t J


(2.12) [p" * w (b' ;Z )]u'{c^) = m u'(c' ) I ?(t)l
t 1 t+l t t 1^ t+l J

If an agent is constrained by the short sale constraint (2.3), then
(2.11) is replaced by:

(2.11' ) s' = K^
t t

Similarly, if the agent is constrained by the borrowing constraint (2.4)
then (2.12) is replaced by:

(2.12' ) b^ = k"
t t

At time t the unknowns are p ; p ; c , i=l,2; s ,i=l,2; and b , i=l,2.

'^t '^t t t t

The equations defining an equilibrium are the budget constraints (2.2), i=l,
2; the market clearing conditions (2.8), (2.9), and (2.10); and the asset
pricing equations (2.11) or (2.11'), i=1.2, and (2.12) or (2.12'), 1=1,2.
By Walras' s Law, one of the market clearing conditions or a budget
constraint is redundant. We restrict our attention to stationary equilibria
in which the consumption growth rate, portfolio rules, and equilibrium
prices are functions of the time t state Z , which is described in detail in
the next section.

3. State Variables

3. 1 Empirical Model of Labor and Dividend Income

Our empirical model is designed to capture important features of the
income process when labor income is uninsurable. For example, the model
captures the correlation between aggregate and individual labor income
growth, and the correlation in individual labor income shocks over time.


Since this income process serves as an input into an asset pricing model
that must be solved numerically, we choose a relatively parsimonious

The components of aggregate income include aggregate labor income, 7 ,

and aggregate dividends, D . The sum of Y and D , aggregate income, is

denoted by Y^ . The growth rate of Y^ and the logarithm of the share of D
•' t t t

in Y^ is assumed to follow a bivariate autoregression. Letting y =

yVy^ , 5 = D^'/Y^ and X^ s [log(.r^) iog(5 )]', then X* is assumed to be
tt-it tt t t t

generated by:

(3.1) X^ = jx^ + A^ X^_j * 8^ c^. t = l, 2, 3,

where e* is a vector of white noise disturbances with covariance matrix I,

the matrix 9^ is assumed to be lower triangular, and jx* is a vector of

constants. We estimate these parameters using annual aggregate income and

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 ti'. In other words, tj' = Y^/Y where Y is individual i's labor
^ -^ t t t t t

income at time t. {v } is assumed to be a stationary process for each

t t = l,CD

1. In the two person economy that we are considering, the law of motion for

1 2 12

7) implies a law of motion for t) since t) +7) =1. The basic
t t t t

specification for tj that we consider is of the form:

(3.2) iog(7}^) = T) + piog(7)^_J + c|

where the {c^} are individual shocks that have mean zero, that are

t t=l,oo
independent over time and independent of c for all t. The parameter p


captures persistence in each individual's income share. Equation (3.2)
implies that there is no correlation between the aggregate state and shocks
to each individual's labor income share. In the appendix we present
evidence that this is a reasonable assumption. Also in (3.2) we are
assuming that the variance of c is not affected by the aggregate state. We
discuss an alternative to this assumption below in Section 3.3.

The individual income process is estimated using annual household
income data from the PSID. The Appendix has a detailed discussion of the
selection criterion for families, and the construction of income. It also

describes the estimation of equations (3.1) and (3.2), along with several

alternative specifications. For the aggregate dynamics we use the point

estimates of the parameters of the vector autoregression, (3.1), as reported

in Table A.l. 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 to the dividend stream. Since dividends average only

3.9% of total income, this clearly understates the share of income from

tradeable assets. Assets such as government securities, corporate bonds,

etc. , may also be sold or used as collateral for loans. In principle

additional debt instruments could be incorporated, but doing so complicates

To represent labor income we are using a first-order autoregressive
model. MaCurdy (1982) and Abowd and Card (1989) argue that the
autocorrelation in individual income can be captured by a first-order moving
average model for income growth. For our purposes, the first-order
autoregressive model is advantageous since it provides a small state-space
model that captures dependence in income growth.

For instance, we estimate this law of motion conditional on income

data from households that actually own stocks. The results are similar to

those for the entire sample.


the analysis considerably. Instead, we increase tradeable asset holdings
to a more realistic level by grossing up the assumed fraction of dividend
income so that tradeable income is 157. of total income on average. For
comparison, capital's share of income in the NIPA averages about 30%.

3.2. Markov Chain Approximat ions

To use this estimated income process as an input for simulations of the

Section 2 model, the VAR is approximated with a Markov chain using the

method of Tauchen and Hussey (1991). We focus on two specifications of the

income process: the "Base Case" and the "Cyclical Distribution Case"


The Base Case model closely approximates the estimated process in
equations 3.1 and 3.2. We use a state vector with eight dimensions, and a
corresponding matrix of transition probabilities. Table 3.1 gives the
values of the state variables in the different states along with the
transition probability matrix. Notice that the individual share of labor
income takes on two values, with a value in the good state that is 1.65
times as large as the value in the bad state. The persistence of these
individual shocks is captured in the transition probability matrix.

In the Base Case model, idiosyncratic labor income shocks are assumed
to be independent of the aggregate growth rate and dividend realization


Including additional debt instruments is problematic due to
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.


The GAUSS code to implement these procedures was kindly provided to us

by George Tauchen.


because, as discussed below, there is little evidence of significant

correlation between individual and aggregate shocks in the PSID data.

However, Mankiw (1986), and Constantinides and Duffie (1992) show that the

distribution of labor income over the business cycle can play an important

role when individuals cannot trade claims to this income. In particular,

they show that if the distribution of labor income widens in a downturn then

individuals may demand a large equity premium to hold stocks. To examine

this potential effect, we consider the Cyclical Distribution Case (C.D.C.)

model .

In the C.D.C. model, we set t) =0.5 (no idiosyncratic shock) when the

growth rate of aggregate income, r . is high. We then choose two values of

7) in the low state for -^ such that we maintain the unconditional variance

of 7j. The result is the set of states given in Table 3.2, with the

transition matrix as in Table 3.1. This concentrates the idiosyncratic

shocks into low growth states for the aggregate economy.

' There is in fact some evidence in the PSID that the distribution of

income widens during downturns. Letting

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