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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
EQUILIBRIUM INTEREST RATE And
LIQUIDITY PREMIUM
UNDER
PROPORTIONAL TRANSACTIONS COSTS
Dimitri Vayanos
and
Jean-Luc Vila
WP #3508-92
December 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
EQUILIBRIUM INTEREST RATE And
LIQUIDITY PREMIUM UNDER
PROPORTIONAL TRANSACTIONS COSTS
Dimitri Vayanos
and
Jean-Luc Vila
WP
December 1992
#3508-92
Massachusetts Institute of Technology
First version:
This version:
April 1992
December 1992
• t^Avtititr'.
JAM
1
9 1993
2
Abstract
In this paper
assets.
We
we
analyze the impact of transactions costs on the rates of return on liquid and
consider an infinite horizon
Blanchard (1985). In
is
this
economy agents
kept constant by an inflow of
new
economy with
finitely lived
illiquid
agents along the lines of
face a constant probability of death, and the population
arrivals.
Agents
start
with no financial wealth and receive a
decreasing stream of labor income over their lifetimes. In addition they can invest in long term assets
which pay a constant stream of dividends. There are two such
asset.
The
liquid asset
is
assets, the liquid asset
traded without transaction costs, while trading the
illiquid
illiquid asset entails
proportional transactions costs. Neither asset can be sold short. Agents buy and
cycle motives.
and the
sell assets for life-
In fact, they accumulate the higher yielding illiquid asset for long term investment
purposes and the liquid asset for short term investment needs.
We
find that
when
the rate of return
the liquidity
asset
on the
may
We
on the
premium
and on the
economy.
transactions costs increase, the rate of return
illiquid asset
increases.
liquidity
The
increase or decrease.
effects of transactions costs
liquid asset decreases, while
also find, quite naturally, that
on the
rate of return
premium, are stronger the higher the fraction of the
Finally, transactions costs
have
first
on the
liquid
illiquid assets in
the
order effects on asset returns and on the liquidity
premium.
We
evaluate these effects for reasonable parameter values.
We
thank participants in the NBER Conference on Asset Pricing
in Philadelphia; participants at seminars at MTT, New York University and Wharton; Drew
Fudenberg, Mark Gertler, John Heaton and Jean Tirole for helpful comments and suggestions. We
also wish to acknowledge financial support from the International Financial Services Research Center
at the Sloan School of Management. Errors are ours.
Acknowledgments
:
would
like to
I.
INTRODUCTION
Although most of asset piicing theory assumes
in financial
frictionless markets, transactions costs are ubiquitous
markets. Transactions costs can be decomposed into
brokerage commissions, exchange fees and transactions taxes
and
costs
(i)
(ii)
direct transactions costs such as
bid-ask spread,
(iii)
market impact
delay and search costs.' Aiyagari and Gertler (1991) report that typical (retail)
(iv)
brokerage costs for
common
stocks average
2%
of the dollar amount of the trade while the bid-ask
spread for actively traded stocks averages around .5%. Moreover, transactions costs vary across assets
and over time.
as well as
Money market
changes
Empirical work
on
in information technology
on
rates of return.
stocks
is
accounts are clearly more liquid than stocks. In addition, deregulation
transactions costs
have reduced (but not eliminated) transactions
documents not only
Amihud and Mendelson
their
magnitude but
their important effect
(1986) show that the risk-adjusted average return on
positively related to their bid-ask spread.
Even more
direct evidence can be
comparing two assets with exactly the same cash flows but different
stocks which cannot be publicly traded for 2 years sell at a
(ii)
35%
is
found by
liquidity: (i) restricted ("letter")
discount below regular stocks^ and
the average yield differential between Treasury Notes close to maturity and
Bills
costs.
more
liquid
Treasury
about .43%.'
The evidence above shows
that liquidity
is
an important determinant of
assets' returns
and should be
incorporated into asset pricing theory. Understanding the impact of transactions costs on assets'
'See
Amihud and Mendelson
^^See Silber
'See
(1991b).
(1991).
Amihud and Mendelson
"More evidence
is
(1991a).
also presented in
Boudoukh and Whitelaw
(1991).
4
returns will shed
some
light
on some
policy issues as well. Transactions taxes
and
differential taxation
of long and short term capital gains both reduce liquidity and therefore affect assets' returns. As a
result,
investment decisions
In this paper
assets, in a
we
illiquid
and
change with additional welfare implications.
analyze the impact of transactions costs on the rates of return on liquid and
We
general equilibrium framework-
characteristics of the
on
will
and
economy does
liquid assets)
How do
depend?
Are these
illiquid assets?
the liquidity
effects first or
are interested in questions such
as:
premium (the difference between the
illiquid
On
what
rates of return
transactions costs affect the rates of return
on
liquid
second order effects?
Despite their importance for asset pricing, these questions have so far not been satisfactorily
addressed in the theoretical literature.
from the basic model of asset
A major reason
pricing,
understand the impact of trade frictions
is
that to
answer them, one has to move away
(One cannot
namely the representative agent model.
in a
model where there
is
no
trade.)
Unfortunately, models
with heterogeneous agents (and trade), tend to be quite intractable analytically.
Since our objective
is
to understand the effects of asset liquidity
of the picture: All the assets that
The
we
on
asset pricing,
we
take risk out
consider (liquid or illiquid) pay a constant stream of dividends.
analysis of the joint effects of risk
and
liquidity
is
an interesting question that
we
leave for future
research.
There are many ways
to construct a deterministic
economy with heterogenous
trade because of differences in preferences or endowments, that
preferences for current versus future consumption, or they
'In a stochastic
(see
Wang
(1992)).
economy,
may have
is,
they
agents. Agents
may have
different labor
differential information together with liquidity shocks
may
may
different
income
paths.'
also generate trade
5
In our
economy both motives
multif)eriod overlapping
We
Blanchard (1985).
More
exist.
our economy
precisely,
a tractable version of a
is
generations economy, the perpetual youth economy,
believe,
however that our
on the
results
first
studied by
effects of transactions costs
on
asset
returns could appear in other contexts as well.
In our model, agents face a constant probability of death (this
things tractable),
no
and the population
financial wealth
the key assumption that makes
kept constant by an inflow of
is
a decreasing stream of labor
and receive
is
new
income over
arrivals.
their lifetimes.
they can invest in long term assets which pay a constant stream of dividends.
assets, the liquid asset
and the
illiquid asset.
The
liquid asset
Agents
start
with
In addition
There are two such
traded without transactions costs,
is
while trading the illiquid asset entails proportional transactions costs. Neither asset can be sold short.
In this
economy agents buy and
motives. In fact, they accumulate the higher
sell assets for life -cycle
yielding illiquid asset for long term investment purposes
and the
liquid asset for short
term investment
needs.
We find that when
the rate of return
transactions costs increase, the rate of return
on the
may
We
on the
illiquid asset
the liquidity premium increases.
The
effects of transactions costs
also find, quite naturally, that
on both the
rate of return
premium, are stronger the higher the fraction of the
liquid asset
and the
economy.
Finally, transactions costs
liquidity
increase or decrease.
liquid asset decreases, while
have
first
on the
illiquid asset in
the
order effects on asset returns and on the liquidity
premium.
The reason why
costs,
the rate of return
on the
liquid asset falls in response to
an increase
in transactions
can be briefly summarized as follows: Suppose that transactions costs increase from
that the rate of return
of return on the
on the
liquid asset stays the
illiquid asset
same
must increase (by the
in equilibrium.
liquidity
Then,
to e
and
in equilibrium, the rate
premium) which implies
that this asset
6
becomes cheaper. Agents now
same
rate as before,
consumption.
(cheaper)
face better investment opportunities (they have the liquid asset at the
and an additional investment opportunity), which leads
illiquid asset
first
and hold
reason
for a longer period.
it
is
that they
transactions costs
when
cheaper, they buy
more of it and hold
rate of return
more
Thus, they
The second reason
for a longer period.
it
the economy, total asset
fall
As
demand
is
will increase
the
of the liquid
same
asset.
transactions costs have to be paid
but hold the
Finally, the liquidity
if
there are
The
rate of return
and that the rate of return
on the
liquid asset falls
by an
as before, but, in addition there are transactions costs, while the
Agents' consumption
The
effect
the future consumption to be financed
illiquid assets,
In addition,
up.
more, and the rate of return on
to €
shifts
down
substitution they accumulate less of the illiquid asset, but hold
less
demand goes
is
premium. This time, agents face worse investment opportunities. The
price of the liquid asset increases.
accumulate
that, since the illiquid asset
the illiquid asset will increase or decrease, by a similar
unaffected in equilibrium.
price of the illiquid asset
is
a result, total asset
reasoning. Indeed, suppose that transactions costs increase from
to the liquidity
securities for
by even more.
We cannot infer whether the rate of return on
illiquid asset in
demand more
will
liquid asset has to fall to restore equilibrium.
the liquid asset will have to
amount equal
rise in their
have to finance higher future consumption and pay
selling the illiquid asset.
on the
illiquid assets in
on the
uniform
Moreover, they substitute consumption over time so that they buy more of the
two reasons. The
The
to a
on the
is
on the
total
demand
it
Furthermore, by
uniformly.
for a longer period.
for securities
is
They
ambiguous.
also
First,
lower and the liquid securities are more expensive, but
illiquid securities.
illiquid asset for a
Second, agents buy
less
of the liquid and
longer period.
premium depends on the minimum holding period of the
illiquid asset.
increases with the fraction of illiquid assets in the economy, since this period gets shorter.
It
7
There
is
a growing literature studying asset market frictions such as transactions costs, short sale
constraints or borrowing constraints. This literature addresses three basic questions.
is
body of
literature addressing this question are:
a particular price process and
(ii)
to evaluate the cost that
The answer
participants (given the price process.)
implications' of
market
frictions.*
The
to
(i)
to derive the asset
Finally, the third
(ii)
sheds some
paper
i.e.
upon the
light
we
demand
for
'equilibrium
equilibrium determination of prices in markets with frictions,
frictions. It
is
is
While we consider
this
the second question
also the question addressed in the present paper.
question addressed by the literature on market frictions
financial structured
in this
on market
question
market imperfections impose upon market
taking the financial structure (and the imperfections, in particular) as given,
raised in the literature
first
and imperfections. The
to find the optimal consumption/investment policy given prices processes
objectives of the
The
question to be a fundamental one
is
to endogenize the
we do
not address
it
take the financial structure as given.
Most of the work about the equilibrium implications of market
framework along the
lines
frictions, considers either a static
of the Capital Asset Pricing Model (see
Goldsmith (1976), Levy (1978) and Mayshar (1979
&
among
others,
Brennan (1975),
1981) for a partial equilibrium analysis, and
Fremault (1991) and Michaely and Vila (1992) for a general equilibrium treatment) or an overlapping
generations
economy where agents
live for
only two periods (Pagano (1989)). Although these models
give us useful insights, they are not adequate for answering several of the questions
in.
we
are interested
In static models, assets are not sold but only liquidated. Moreover, in a static model (as well as
two period overlapping generations model), agents cannot choose when to buy or
in a
sell assets,
and
See for instance, Constantinides (1986), Davis and Norman (1990), Duffie and Sun (1989), Dumas and
Luciano (1991), Fleming et. al. (1992), Grossman and Vila (1992), Tuckman and Vila (1992), Vila and
'
Zariphopoulou (1990), among many others.
'See for instance Allen and Gale (1988),
Ohashi (1992).
Boudoukh and Whitelaw
(1992), Duffie and Jackson (1989) and
8
the holding period
appear
is
same
the
in that simplified
for
all assets.
It is
model where investors have
approximately
-f
who
%
face a
assets
different horizons.
2%
Amihud and Mendelson
They argue
roundtrip transactions cost
illiquid assets
when buying and
cost.
selling assets will lose
Hence, they
will require a rate
By contrast,
less affected
by transactions costs and would
investors with shorter horizons select low yield liquid
This clientele effect explains the empirical fact that the cross-sectional relation between
transactions costs
and asset returns
is
concave.'
horizons as given and does not explain
Moreover
and not on their
how
The
analysis above, while insightful, takes investors'
they change in response to an increase in transactions
since, as in the previous papers, the rate of return
be
for simplicity to
Two
(1986) consider a dynamic
which appeal to investors with a 4 year horizon must be approximately .5%. The above
select higher yielding illiquid assets.
costs.
would not
than on liquid assets. Consequently, the liquidity premium
reasoning implies that investors with longer horizons are
assets.
results
that investors with, say, an investment
(-5%) per year because of the transactions
of return of .5% higher on
on
many of our
framework.
In a context directly related to the present paper,
horizon of 4 years
thus clear that
fixed,
levels
recent papers,
on the
liquid asset
is
assumed
only the effect of transactions costs on the differentials of rates of return
can be examined.
one by Aiyagari and Gertler
(1991),
and one by Heaton and Lucas (1992)
consider dynamic models where investors' horizons are endogenous. In their models, agents are
infinitely lived, face labor
income uncertainty, and trade
assets for
consumption-smoothing purposes.
These papers seek to solve the equity premium puzzle (see Mehra and Prescott (1985)),
explain the differential rates of return between the stock and the
i.e.
to
bond market. Aiyagari and Gertler
(1991) argue that differential transactions costs between these two markets account for part of the
'By contrast,
if all
investors had the
same horizon
this relation
would be
linear.
9
equity premium. In their model (as in ours), the 'stock'
is
due
to transactions costs
premium which
costs
on the
is
and not to
in fact a liquidity
risk.
Hence
is
their
riskless
and therefore the equity premium
model explains the
premium. They do not however analyze the effect of transactions
level of rates of return as they take the rate of return
contrast with Aiyagari and Gertler (1991),
variability of their
on the
Heaton and Lucas (1992) allow
well as for aggregate labor income uncertainty.
from reducing the
fraction of the equity
They argue
liquid asset as given.
for a truly risky asset as
that transactions costs prevent investors
consumption by intertemporal smoothing thereby
equity premium. In addition, they endogenize the rate of return
falls in
response to increased transactions
While
in
our model agents save for
By
on the
liquid asset
and
raising the
find that
it
costs.
life-cycle
purposes rather than because of labor income
uncertainty, our results are consistent with the numerical simulation results of the above two papers.
We
find in particular that transactions costs create a liquidity
(1991), and that they cause the rate of return
(1992).
The
contribution of our
work
is
on the
liquid asset to
fall,
in Aiyagari
as in
and Gertler
Heaton and Lucas
twofold: First, our closed form analysis allows us to precisely
identify the different effects of transactions costs
we
premium, as
on
asset
demands and on
rates of return. Second,
are able to easily perform and interpret various comparative statics.
The remainder of the paper
asset returns
when
is
structured as follows: In part
there are no transactions costs in part
there are transactions costs. In part
5,
we
Part 6 contains concluding remarks and
illustrate
all
2,
3.
our general
we
describe the model.
In part 4,
we
results with
We determine
consider the case where
some numerical examples.
proofs appear in the appendix.
10
n.
THE MODEL
To
analyze the impact of transactions costs
we have adapted
in
mass equal to
1.
An
agent
we assume
X. In addition,
in this
that death
t,
economy,
their life expectancy
bom
We
y.
y,
=
is
Y
=
normalized to one so that
f
D
liquid asset in total supply 1-k
is
continuum of agents with
independent across agents and that agents are
stationary, with total
bounded and equal
to
mass equal
to
bom
at a rate
one and the
distribution
long
lives in this
live arbitrary
1/A,
declines exponentially with age
financial structure in this
liquid asset
a
faces a constant probability of death per unit time,
y,
over their
t
is
(1)
constant and equal to
Xe-Ndt
economy
=
is
—
y
is
(2)
.
given as follows. All assets in this
perpetuities which pay a constant flow of dividends
is
premium,
ye*', fi^O.
labor income
Y
The
is
economy with
with zero financial wealth and receive an exogenous labor income
assume that
The aggregate
liquidity
A simplified exposition of the original
has a density function equal to Xe^\ Although agents can
Agents are
lifetimes.
economy
is
equal to k. Therefore the population
of age,
and on the
Blanchard and Fisher (1989).
consider a continuous time overlapping generations
total
assets
Blanchard's (1985) model of perpetual youth.
model can be found
We
on the return on
economy
are real
D per unit time. The total supply of perpetuities
also the aggregate dividend.
There are two such
(Osk^l) can be exchanged without transactions
denoted by p and the rate of return on
liquid assets
is
perpetuities.
costs.
The
denoted by r= D/p
.
The
price of the
The
illiquid
11
supply of
asset, in total
the illiquid asset
illiquid asset
return
is
a rate of return equal to
subject to proportional transactions costs:
the agent must pay e x
on the
P and
has a price equal to
k.
illiquid asset
P
and on the
when buying
R = D/P
'.
Trading
in
(or selling) x shares of the
transactions costs. Because of transactions costs, the rate of
liquid asset will
be
different.
The
liquidity premium
\i is
defined
as
ji
Finally,
none of these
Over the course of
financial risk
this risk
r.
(3)
be sold
assets can
and
company
short.'"
their lives, agents accumulate both assets. Since death
that of losing their
is
stochastic
living participants in
that insures
exchange for a claim on their
one share of say the
We
accumulated holdings."
costlessly insurable in the following way: there exist insurance
pay shares of assets to the
insurance
-
upon the agents namely
fully
is
= R
liquid asset will
We
assume that
(i)
premium «dt must be equal
the
'Note that
we have
defined
R
death
to the probability of death
Finally since, as previously indicated, agents
transactions costs
(iii)
A.dt,
is
estate.
is
an idiosyncratic
utility
from
as the rate of return before transactions costs.
depends upon the holding period and
is
For example an
is
to collect the
perfectly competitive,
for both the liquid
do not derive any
assume that
pay a premium of ndt additional
the insurance market
insurance companies transfer assets costlessly" and
imposes a
companies which
shares of the liquid asset per unit time dt to a living participant. Its compensation
share in the event of death.
it
risk.
and
As
(ii)
a result,
illiquid asset.
their estate they will
The
rate of return net
of
therefore investor specific.
were costless no agent would sell the illiquid asset and would short the liquid asset instead.
do not change if we assume that the cost of short selling is higher than t, which is a reasonable
assumption (see for instance Boudoukh and Whitelaw (1991) and Tuckman and Vila (1992) for evidence on
'If short sales
Our
results
short sale costs).
"Agents do not leave any heir behind and care only about themselves.
'^e introduction of insurance companies is a convenient way to close the model. As can easily be seen
from our discussion in the introduction our results would carry through in a multi-period overlapping
generations model with deterministic death. The latter would be much more complicated to solve analytically.
12
purchase
full
We assume
their
insurance.
that agents
consumption
maximize
at
time
the expected value of a time separable
e-" dt
]
utility
function of
i.e.
m
E
[
u(c
J
)
(4)
.
Since the only uncertainty comes from the possibility of death
J
We
also
assume that the
u(c)e^»-^>'dt
utility
u(c) =
^
In this paper
we
rates of return r
focus
and
^
on the
R
(5)
function exhibits a constant elasticity of substitution equal to
J- c'-
x
i-C-
''
(6)
.
1-A
stationary equilibria of this
are constant.
"See Blanchard and Fisher (1989)
A=l
write (4) as"
.
We
functions of the parameters of the model:
'"The case
we can
seek to understand the determination of
€, k, X, S, p,
for details.
corresponds to u(c)
economy. In a stationary equilibrium, the
= Log
c.
A,
Y
and D.
r,
R
and
\i
as
13
THE NO TRANSACTIONS COSTS CASE
in.
In this section,
we
analyze the determination of the interest rate in the benchmark case
transactions costs are equal to zero. In this case there
the illiquid asset:
3.1.
r=R
is
no difference between the
when
liquid asset
and
and ji=0.
The consumer's problem
The financial wealth
the
is if X, is
w, of the
number of
t,
at
date
=
dynamics of
his
owns
at
date
t
px,.
the investor receives a labor income
income plus
entails Dx, in dividend
defined as the value of the consumer's asset. That
t is
shares that the consumer
w,
At date
consumer
A.X,
y,
per unit time. His financial income (per unit time)
shares worth
Since he consumes
A.px,.
c,
per unit time the
wealth are
dw,
=
Dx,dt
+
+
A.px,dt
(y,-cOdt
=
(r+A.)w, dt
+
(y,-Ct)dt; c.jjO.
(7)
W(,
From
equations (5) and
=
(7),
0;
w, ^ 0.
the consumer's problem
is
the optimization problem of an infinitely lived
consumer with discount factor p + X and who faces a constant
equals the rate of return on the perpetuity,
Hence the consumer's problem can be
max
J
interest rate. This constant interest rate
plus the premium paid by the insurance company, X.
r,
written as
u(c )€-<»*«' dt
max
=
J
_L c'-^e-^-^'dt
(8)
s.t.
The problem
(8)
J
c e-<'*«'dt =
>
=
and that the maximum
equilibrium. (See appendix A for details.)
u)
(^-r)/A,
y^e-'^*"'dt
,
w_ ^
above admits the following solution^
^0 calculate this solution we have assumed
6
J
that the borrowing constraint
in (8)
is
finite,
i.e.
is
not binding,
V = r+A+w>0. Both
i.e.
restrictions hold in
14
15
buy assets when they are young and begin
+
r'w,.
From
(11), T*
The aggregate
is
y..
-
c..
=
selling assets at
age
t*
where
t'
solves
0.'*
given by
dollar
T- =
_!_
Log ^i:^
volume
in this
economy equals
T
=
(13)
.
fXe-^'Irw +y -c
I
dt =
2A.w
e"^'' =
2
vLlif e-'^-'>'"
.
(!'*)
<P
'*Note that we do not consider the payment of shares by insurance companies to be a trade. Hence
although the agent's portfolio may still be growing (dW,>0), the agent is considered a seller if his portfolio
grows at a rate lower than A. In the absence of transactions costs, this assumption simply amounts to defining
who
is
and who is not. With transactions costs, however, matters are different. Since we have
companies pay living participants shares of asset as opposed to cash and that this
our definition of a seller is the correct one.
called a seller
assumed
transfer
that insurance
is
costless,
16
IV.
TRANSACTIONS COSTS AND ASSETS' RETURNS
In this section,
we
derive the main results of this paper namely the determination of the rate of
return on the liquid asset,
in the
on
the rate of return on the
r,
presence of transactions costs.
on equilibrium
their first order effect
r(€)
=
+
r'
We will consider
+
(b-m')€
variables.
\i(e)
=
m*€
+
first
order equilibrium effects that
+
shall see, the case
purpose
o(€)
where
k, is less
we seek
unit of transactions costs
or equivalently R>r(l+e). This
i.e.
k=l,
is
\i/€
it is
useful to
is
somewhat
show
We
first
consider the
different. This case will
he buys the
because, since agents are
some
bom
point in their
at
liquid assets,
without any financial
Now
lives.
date
t
and
assets, in
consider an agent
sells
it
At periods
who
later.
liquid asset his cash flows are
- 1
at
date
rds for s
+
he buys the
that in equilibrium the liquidity
must be greater than the rate of return on
buys for one dollar worth of asset inclusive of transactions costs
flows are
to calculate.
than one so both assets are available to investors.
assets are illiquid,
all
equilibrium they must buy the illiquid asset at
If
write
in section 4.4.
premium per
If
we
o(€)
Before proceeding with the formal derivations,
r,
this
r*
b€
\i
o(€)
case where the supply of the illiquid asset,
be studied
For
=
+
R, and the liquidity premium
the case of small transactions costs and focus
R(e)
where b and m* are the
As we
illiquid asset,
1 at
illiquid asset,
t
between
t
and t+At and
date t+At.
given the transactions costs he will get 1/[P(1 +€)] shares.
Hence
his
cash
17
-
at date
1
t
Ry(H-€)ds
for
s
between
(l-€)/(l+€) at date
If
R
^ r(l+€),
t
t
and
t
+ At and
+ At.
^ re, then buying the liquid asset always dominates buying the illiquid asset.
i.e. if |i
Hence
>
ii
In particular, this
order
effect.
investor's
4.1.
means
With
demand
r€."
that the effect of transactions costs
a priori information
this
and
for liquid
liquidity
about equilibrium prices,
illiquid assets
when
premium
we
is
at least a first
next characterize the
\i>Te.
The consumer's problem
With transactions
portfolio,
dynamics of
I,)
a,
the
his illiquid portfolio,
sum of
the value of his liquid
denoted by
A,.
Denoting by
i,
and A, are given by
=
dA,
c,
(15c) above,
Xa,dt
=
ra,
a,
i,dt,
+
XA,dt
=y, +
we can
+
transactions costs
€|I,|.
With transactions
costs, the
"This lower bound
is
=
Ao =
I,dt,
+ RA,-i,
-I, -e|I,|,
c,
0,
0,
a,
(15a)
^
A, i
(15b)
(15c)
^ 0.
see that the agent's consumption equals the labor income y„ plus the
dividend income ra,+RA„ minus purchases of liquid assets
when
is
the incremental dollar investment in the liquid asset (respectively illiquid asset), the
da,
From
consumer's financial wealth w,
costs, the
denoted by a„ and of the value of
(respectively
is
on the
consumer problem becomes
reached asymptotically
the fraction of illiquid assets,
k,
when
goes to zero.
i„
far
minus purchases of illiquid
assets,
more complex. Proposition
4.1
minus
(proven
the holding period of illiquid assets goes to infinity, that
18
appendix B) describes the optimal policy of the consumer for small transactions costs and for a
in
subset of values of
r
and
R
that are of interest,
i.e.
such that their equilibrium values belong to
this
subset.
Proposition 4.1: For e small and for r and
R
belonging to a subset of their possible values, the
optimal policy has the following form: The consumer buys the illiquid asset until an age
buys the liquid asset
He next sells
Ti.
He then
the liquid asset until an age T|+A. At age T|+ A, he does not
any share of the liquid asset He then start selling the
illiquid asset until
he
own
dies.
We find that in equilibrium, agents will buy high yield illiquid assets for long term investment and low
yield liquid assets for short
term investment. This
fairly intuitive result is consistent
of
Amihud and Mendelson
c,
while the clientele for the liquid asset are the agents of age between
begin to
it
at
date
sell
Ti
it.
(1986).
The marginal
+ A. As
in
The
investor
with the analysis
clientele for the illiquid asset are the agents of age less that
is
the investor
Amihud and Mendelson
who
buys the
t,
and the age
illiquid asset at
at
date
which they
Ti
(1986), this investor determines the liquidity
and
sells
premium
(see below).
The
liquid
and
illiquid portfolios as
function of age
t
are plotted in figure
1.
Proposition 4.1 presents the qualitative properties of the optimal consumption/investment policy. In
what
follows, these qualitative properties will allow us to calculate the
a function of the initial consumption, c^
We will
also
consumption
at
date
c,.
rate of return
Finally,
on the
we will show how
liquid asset,
or the sake of the presentation,
r,
all
c„ as
show how the intertemporal budget equation,
properly modified to account for transactions costs, leads to the determination of the
consumption
t,
the parameter
A can be easily calculated
the liquidity premium,
technical details have
\i,
and the
been sent
initial
as function of the
level of transactions costs, £.
to appendix B.
19
Over the course of
First until
age
t,,
agent faces three interest rates.
his hfe the
the interest rate which
"-
relevant for the consumption-savings decision
is
is
1+e
who
Indeed, consider a consumer
He
wealth after t+dt.
at t6[0,T,] decides to
then buys \fP(\+e)
consume $1
At
illiquid securities.
s
but wants to have the same
less,
between
t
and t+dt, he consumes
the extra dividend flows (D/P(l+e))e^'"' and at t+dt, he consumes the proceeds /rom avoiding to buy
(l/P(l+€))e"' securities,
between
t
and t+dt. Therefore,
earlier rather than later.
t,
for
R given,
The reason
expensive, but that pays the
Second, between ages
Hence by foregoing
ie $e"'.
same
and
Tj
is
$1 at
he gets $
t
l
+ Xdt + (R/(l+e))dt+o(dt)
higher transactions costs increase the desire to
that the
consumer has
buy an
to
asset
which
consume
is
more
dividend.
+ A,
the consumer invests in the liquid assets and therefore faces the
interest rate r+A..
Third and
finally, after
age tj+A, when the investor
is
divesting out of the illiquid assets, he faces a
higher rate
R„
^
+ X =
i
+ X
.
1-e
Indeed, consider a consumer
wealth after t+dt.
He
who
at te[T„<») decides to
sells l/P(l-€) less illiquid securities.
extra dividend flow (D/P(l-€))e*<"' and at t+dt he
securities
For
R
reason
We
i.e.
e"'.
consume $1
Hence by
foregoing $1 at
t
At
that the
consumer has
between
t
but wants to have the same
and t+dt, he consumes the
consumes the proceeds from
he gets $
1
consume
to sell a cheaper asset that pays the
denote p(t) be the interest rate relevant for date
selling (l/P(l-e))e^'"
+ Xdt+(R/(l-e))dt+o(dt) between t and t+dL
given, higher transactions costs increase the desire to
is
s
less
t i.e.
later rather than earlier.
same
dividend.
The
20
and by
p(t)
= Rl+X
for
t
p(t)
=
r+A.
for
T,
^
p(t)
= Rb+X
for
T,
+A
<
T„
p(t) the discount rate between date
p(t) =
In appendix B,
s
T,
+ A,
t
and date
that the optimal
with
= c„ e-<"
where the consumption
<
t i.e.
Jp(s)ds
we indeed show
c
t
at birth Co
o>(t) =
is
consumption must
satisfy
^P^^^^'P^^^
(16)
derived from the intertemporal budget constraint presented
below.
Given proposition
1
and equation
satisfy the intertemporal
f
"L-^
(15),
it
can easily be shown that the consumption path
must
c,
budget equation
e-''"Mt +
(y
+(R-r)A -c)
e'-'-'dt +
f
II
.'
(l'^^)
e-^<"dt =
I*
with
Equation (17a) says that the Net Present Value of lifetime savings net of transactions costs must
equal zero where
we
define savings as total income minus consumption minus what must be
reinvested in order for financial wealth to grow at the rate p(t).
[t,+ A,+<»[,
T
this latter quantity
equals the dividend income and therefore savings equal
and T + A, only a fraction rA, of the dividends from the
savings equal labor income, y„ minus consumption
adjusted for transactions costs.
Between periods
c,
illiquid portfolio
[0,t,[
y,-c,.
and
Between
must be reinvested and thus
plus excess dividends (R-r)A,. Finally savings are
21
Some
simple algebra shows that the intertemporal budget equation can be written as
TJ •
-c)
(y,
[J
e-'-'-'dt +
I
A
(y^-c) e-^<"dt
+
J
-c
(y_
)
e'-'-'dt
]
+
A^e"''<'"[if(l-e-^')-€(l+e-''^)] =
The expression above
consists of
two terms. The
first is
the Net Present Value of labor income minus
the Net Present Value of consumption taken with discount rate p(t).
Present Value of investing
.
The second term
is
the present value of the excess return
in the illiquid asset, i.e.
the Net
p,
in the
period [ti,t,+A[ minus the present value of the transactions costs. Since the investor maximizes the
present value of his investment with respect to A,
choice of
r,
^
and A,
,
this
term must be equal to zero for an optimal
i.e.
^^^"^'
=
(18)
r
1-e-'*
€
Equation (18) can also be
easily derived as follows.
Consider a consumer at age
t,.
Since
t,
and A
are optimally chosen, this consumer must be indifferent between investing in the illiquid asset and
not doing
so.
Given
that
he
starts selling
unit of the illiquid assets at t,
the illiquid assets at
t,+A
J
u'(c)De"""''Mt
The equation above together with the equation determining
As
it
change
in utility
if
he buys one
is
-u'(c^^)(l+€)P+u'(c^^Je-»*(l-€)P+
a simple interpretation.
his
was
said, the
c,
consumer buys the
=
.
(16) gives us (18). Equation (18) has
illiquid asset for
long term investment
and the liquid asset for short term investment. The minimum time between buying and
illiquid asset,
A,
will,
obviously, be decreasing in
Equation (18) states precisely how
From equation
and A,
A depends on
its
selling the
excess rate of return over the liquid asset,
(i.
ji.
it
follows that the intertemporal budget constraint, for an optimal choice of t
states that the
Net Present Value of consumption equals the Net Present Value of income
(18)
22
where the discount
J
(y,
factor
-c)
For any value of
e-'C'dt +
A, the
t,
p(t),
is
i.e.
(y,-c) e-'^'Mt +
J
initial
4.2.
r
)
(^9)
(16), (19).
consumer's problem
the
to
=0.
e-'^'Mt
we
turn
to
the
equilibrium
E^quilibrium
demands
Xe-^'a
J
and
for liquid
As we
€),
illiquid assets (in dollar),
and
dt
equal their market value, (l-k)D/r and
J
kD/R
stated in the beginning of this section,
and find their
and that
+
(b-m*)€
=
^(€)
= m*€ +
and
r*
i|;'
+
be
+
and the
+
r,
X
e"^'
respectively.
we
R
will
and
|i.
consider small transactions costs (small values
Recall that
we have
defined b and m* as
(20a)
0(c)
(20b)
o(€)
(20c)
are the equilibrium values of
limits
Adt
o(€)
by t,(6) and A(€)
and A* the respective
liquid asset
r*
order effects on
R(€)
r', o)*, <p*
also define
first
=
r(€)
We
-c
and R.
In equilibrium, the
of
(y,
consumption can be derived from
Having characterized the solution
determination of
J
r,
oj, <p
and
as the equilibrium values of tj
i|f
for
€=0.
and A as a function of
of Ti(€) and A(€) as e goes to zero (Note that
illiquid asset are
the
same
asset
when
and therefore the holdings
a,
and by
t[
e equals zero, the
and A, are not well
defined.) In other terms, xj and A* are the zero-th order effects of transactions costs
periods.
€,
on holding
23
The
next proposition characterizes the equilibrium values of
Proposition
There
4.2.:
exists
an equilibrium where
R
r,
m
and
and
b.
have the form of equations (20a),
\i
(20b), (20c). In equilibrium the flrst order effect of transactions costs
m',
is
positive while the first order effect
on the
on the rate of return on the liquid
liquidity
premium,
asset, b-m', is negative.
has an ambiguous sign.
The
first
The
rigorous derivations of b-m", b and m*, as well as explicit formulas are presented in appendix C.
order effect on the rate of return on the illiquid asset,
we
Proposition 4.2, that
on the
liquid asset but
We discuss
discuss in detail next, states that transactions costs decrease the rate of return
have an ambiguous effect on the rate of return on the
the results of proposition 4.2 in subsections
characterize the parameters t, and A*
(or,
more
4.2.1.
the rather long subsection 4.2.3
in
accurately, the
first
4.2.1., 4.2.2.
and
4.2.3. In
illiquid asset.
subsection 4.2.1,
we
the zero-th order effect of transactions costs on optimal
i.e.
consumption/savings policies. In subsection 4.2.2
premium, and,
b,
we go over
we
the determination of the liquidity
discuss the determination of the rates of return
order effects of transactions costs on these variables.)
Optimal switching times
The age
at
which agents switch from the
start selling
illiquid asset to
the liquid asset,
t*,
and the age
at
which they
the illiquid asset, xi+A', can be easily interpreted from the limit case as e goes to zero.
Indeed consider the accumulation equations (15) with
wealth w,=a,+A„ the values of the liquid and
For
t
For
t! i
For
s x[
A
11
t
t
s T, +
^ t" + A*
1
and
= w^
a^
= w
A It.
A
.
A
Itw
=
e
e=0 and r=R=r'. Given
illiquid portfolios
=
over time are given by:
.
and
'
a,
1
and
=
a
t
.
the investor's total
= w, t
w
.
T^
e
24
It
follows that the value of x[ and A* can be calculated by noting
grows by e*" between
w
and
(ii)
.
w
=
.
.
e
tj
and xj+A'
that total financial wealth w,
i.e.
.lA-
(21a)
that aggregate liquid Gnancial wealth
*
*
Xe-^'a,dt=
J
(i)
must equal the supply of
i.e.
«
J
Xe-^'(w,-w^.e*'^'"'')dt =
(l-k)5
(21b)
.
*
•
•
liquid assets
Using the expression for w, from equation (11) as well as equations (21a) and (21b) above
we
obtain
the values of t\ and A*.
4.2.2.
In appendix B,
is
Premium
Liquidity
we show
that the
order effect of transactions costs on the liquidity premium, m,
first
given by:
m- =
r-
ilill'
(22)
.
1 -e-'**'
It is fairly
from
its
easy to understand the determination of m*. Equation (18) implies that
value in equation (22),
(i.e. if
\i
were
different in the
first
order), there
order change in A. Therefore, there would a zero-th order change in the
illiquid assets.
(Although there would only be a
first
if
m were different
would be zero-th
demand
for liquid versus
order change in total asset demand.)
4.23. Rates of Return
The reasoning
we
for the rates of return
study total asset
demand equals
demand and
supply,
more
i.e.
involved.
we
To determine
the parameter b (and b-m*),
use the equation that states that total asset
total asset supply:
m
J
The reason
is
m
Xe-^'(a+A)dt
for doing that
is
=
J
Xe'^'w^dt =
(l-k)H
r
+
kP
.
(23)
K
that to calculate the parameter b using the equation describing
25
equilibrium in one of the two asset markets,
previously, a
first
order change in
\i
only a
first
order effect
know
in total asset
order effect.)
first
demand
total asset
to
ji
to the
has a zero-th order effect in the
therefore a second order change has a
equation that states that
we need
This
demand
for
(A
first
if
we
order change
second order change
a
(As stated
each type of
not a problem
is
equals total asset supply.
demand, therefore
second order.
in
\i
asset,
use the
in
\i
has
has a second
order effect in total asset demand.)
To determine
b-m',
costs increase.
R increases
We
how much
is
total asset
total asset
Lemma
in
in
(it
We
we go over
demand, we must
why
it
first
in transactions costs
in
The consumption at date
<P'
C„ and
Q are given by
transactions
in the first
order and infer b-m'.
on the
liquid asset decreases
the rate of return
some
detail.
how
the consumption path of the agent
asset returns that
zero, Cg is given by:
r,
increases by m'e for m' given by equation (22)).
understand
and
for fixed
the zero-th order in each of the two markets,
appendix C) gives us the consumption of the agent
4.3.:
assume that
demand and supply change
useful for understanding
modified by the change
(proven
the following exercise:
determined above
transactions costs increase,
To determine
where
make
in the first order, as
Since this exercise
is
thus
Moreover, to preserve equilibrium
then find by
when
we
at
we
age
are considering.
0.
Lemma 4.3
26
r:r
,
^rn'
P"
'"'
{24b)
•*
(^ + 1)
+
[{X*6)e-*'--{X*w-)e-*'']dt
J
}
or, alternatively:
m
I
:^
=y^'
{
(m'-r*)
f [e*''-e-»'']d/ +
(m"+r")
T
[e-*''-e-''']df
}
and
i v^
lyl-
^5
The terms Cw and
For
effect.
this
r»i--rn f e-*''A
(m'+r-^
(m'-r-)
e-*''<ir + {m'+r')
I
{
Q have a very
we need
^-''A
e-*''df
f
f
V
}
intuitive interpretation. First
(24<i)
.
Cw
can be interpreted as a wealth
to note that
- |(A.-a)e -*''-(X^u)e-*''|
is
the present value of the dollar
consumer when buying the
When
price.
\|f
illiquid asset
The term Cw
between
is
be better off compared to the case
t, until
e
=
0,
and
this
Q can be interpreted as a substitution
from
T*,
(With dividends, agents buy the
which more than compensates them.)
is
also
more
attractive
t
and t+dx
in the case
6=0. The
pays the transactions costs but pays a lower
t,
he
dies),
he pays the transactions costs and
equal to the present discounted value of these cash flows
The term
for later
and
Clearly, since the rate of return in the liquid asset
*.
to
between
transactions
(from
selling the illiquid asset
receives a lower price.
times
amount of
It is
term
is
effect.
illiquid asset
is
kept constant, the consumer can only
positive as
Since R/(l +6)
until death.
>
r,
see in expression (24c).
saving
is
more
attractive
paying transaction costs but at a lower price
also clear that R/(l-€)
from tj+A*
we can
Thus
> r,
therefore deferring consumption
this
term
is
negative.
27
Having inteqireted the expression
compared
same
the liquid asset at the
(This
more than compensates
beginning of his
This
In
illiquid asset
Because the
the consumption path changes
illiquid asset
is
and holds
it
consumption path goes up
available at a lower price (that
it
are lower (lower
it
at a
that
he
more
in the
lower rate (he defers consumption for
later.)
In other words, he buys
for a longer period.
sells
he has
(i.e.
consumer changes the slope of the consumption path so
(he saves more) and he
life
how
transactions costs), and because the proceeds from selling
price plus transactions costs), the
buys more of the
briefly describe
price as before, and the illiquid asset), his
the wealth effect.)
is
we can
c^,
e=0. Because the consumer has better investment opportunities,
to the case
uniformly.
tor
the substitution effect.
is
lemma
Lemma
4.4 (proven in appendix
4.4.
Total asset
6-(j"
Y
demand
+
(W^
*
is
C) we determine
total asset
demand.
given by:
(2Sa)
iW^ +o(£)
(A+w*)(p'
where
'
w
-7JL{
(m'-r-) [[«-'' -e-»'']df+(m'+r')
f
[e'*''
-e-^'-dt]}
A+w'
(25b)
cpV
r'
\
and
W^
The term
latter
= 1.
A
-Jl-yJlL
A+w'
[
v'r-
(m'-r)
f
(
e-(^-> -e-*'')dt
demand
(m' ^r)
(e-(^-'>' -e-*'')dt
[
.J
4
W,,^ represents the additional
+
.(^^'=>
]
,
t,*A-
for wealth (in the first order) of the
changes only the level but not the slope of
his
consumption path
(i.e. if
consumer
if
the
the wealth effect
is
present, but not the substitution effect) in response to the change in transaction costs and asset
returns that
we
are studying.
This additional
demand
for (dollar) wealth has an
ambiguous
sign,
28
because,
when
on the one hand, higher future consumption
selling assets, require
The term Ws corresponds
be financed and transactions costs to be paid
more wealth, but on the other hand,
more of the
more wealth accumulation. This term
is
(illiquid) assets are
cheaper.
was said before the consumer changes
to the substitution effect: Indeed as
the slope of his consumption path so that he buys
period. This implies
to
illiquid asset
positive
and
and holds
its
it
for a longer
magnitude depends on
the elasticity of intertemporal substitution.
Finally total asset supply (in the
R
order)
first
-€l!i:fxe-Adt
=
'
rIt
decreases since the
The
is
r-
illiquid asset is
why
this
is
demand and supply
so,
addition, the
asset
number of
(26)
supply
cheaper.
is
Ww+Ws+W,upp,y and
securities to
in selling assets
it
always positive.
be held.
(and a lower price of the
It
for a longer period,
illiquid asset),
(Although the dollar amount may be lower.)
consumer changes the slope of his consumption path
and hold
is
based on our earlier discussion. Higher future consumption to
be financed, transactions costs to be paid
require a larger
-eW^
""^
2
r-
-1
difference between total asset
easy to understand
is:
in
order to buy more of the
making the imbalance between
asset
demand and
In
illiquid
supply even
higher.
The
value of (b-m*)
The above
summarized
return
is
then easily deduced, and
discussion which explained
as follows:
on the
mmt
negative.
the rate of return on the liquid asset
Suppose that transactions
liquid asset stays the
the illiquid asset
why
is
same
increase (in the
costs increase
in equilibrium.
first
Then,
order) by m'e.
from
falls,
can be
to e and that the rate of
(in equilibrium) the rate
of return on
Agents' consumption paths
will shift
29
uniformly up, because they face more investment opportunities (wealth effect), and their slope
change so that the agents buy more of the
effect).
The agents
will
demand more
illiquid asset
securities for
higher future consumption, pay transaction costs
and hold
two reasons.
when
and they hold more of
asset
Although the
First,
selling assets
is
is
negative), the
first
that they
(r
assets are cheaper.
buy more of the
illiquid
the liquid asset
order effect on the rate of return on the
we
replicate
(more
R
stays the
same and
that this time, as transactions costs increase,
briefly) the
r
is
illiquid asset (i.e.
above exercise, assuming
decreases in the
first
order, as
decreases by m*e).
This time, the consumer faces worse investment opportunities.
same
and because
it.
ambiguous. In what follows,
determined above
because they have to finance
order effect of transactions costs on the rate of return on
first
unambiguous (b-m*
the sign of b)
for a longer period (substitution
it
Second because they change the slope of their consumption path so
will
The
price of the illiquid asset
is
the
as before, but, in addition there are transactions costs, while the price of the liquid asset
increases.
The wealth
effect implies then that his
consumption path
shifts
down
uniformly.
On
the
other hand, the substitution effect implies that he changes the slope of his consumption path so that
he accumulates
accumulates
First,
less
less
of the
illiquid
of the liquid asset.
asset but
The
effect
the future consumption to be financed
but,
on the other hand
and
illiquid assets,
ambiguous.
he holds
is
on the
for a longer period,
total
demand
and so that he
for securities
is
ambiguous.
lower and the liquid securities are more expensive,
transactions costs have to be paid.
but they hold the
it
illiquid asset for a
Second, consumers buy
longer period.
less
of the liquid
Therefore, both effects are
30
43. Comparative Statics
we
In this section
study
how
the effects of transactions costs
parameters of the model. (More precisely,
The parameter
In
assets.
Lemma
We
that
lemma
4.5.
m'
is
of greatest interest
4.5 (proven in appendix
increases in
k,
imply that the
minimum
is
was argued that
first
D) we examine how
in
k while
we
order) by m*e.
and b-m* depend on
m', b
the dependence
ofb
in
k
is
illiquid asset
More
becomes
why the
could
to e
make
shorter.
rate of return
on the
The
b-m*)
the following experiment:
total asset
demand and
m' increases, therefore both the wealth
is
liquidity
premium
liquid asset falls in response
and that the rate of return on
This implies that the difference between asset
r (i.e.
economy
We
could suppose that
the illiquid asset had to
We could then study the difference between demand and supply
magnitude of the difference between
on
the
explained in light of the analysis of the previous subsection.
can also infer the magnitude of change of the rate of return on
increases,
ambiguous.
illiquid assets in
of total wealth and infer the direction of change of the rate of return on the liquid
As k
k.
willing to hold illiquid assets for shorter periods.
to understand
transactions costs increased from
increase (in the
and b-m* depend on these parameters.)
the fraction of illiquid assets to the total stock of
k,
holding period of an
to increased transactions costs,
m", b
relatively simple to understand.
The dependence of b-m* on k can be
it
how
depend on the
assets' returns
Lemma.
must increase so that consumers are
There
find
is
b-m' decreases
briefly discuss the results of this
The dependence of m' on k
we
on
bigger (in absolute value).
effect
asset.
In fact,
we
the liquid asset, studying the
total asset supply.
and the substitution effect are stronger.
demand and supply
is
greater,
and the
first
order effect
31
The
effects of the other
parameters on m', b and b-m* are of
less interest
and are not reported here.
32
V.
NUMERICAL EXAMPLES
In this section
In
all
present
these examples
all
equals
In
we
that
A=l
=
(u(c)
log c) and that the level of transactions costs e
we
(see Aiyagari and Gertler (1991) for instance).
plot various rates of return as a function of k, the supply of the illiquid asset.
and lemma
figures are consistent with the results in proposition 4.2.
liquidity
return
we assume
to illustrate the results of the previous sections.
3% which is consistent with empirical evidence
figures 2 to 4,
These
some numerical examples
premium
on the
positive,
illiquid asset
premium and the
The main
is
(ii)
the rate of return on the liquid asset goes down,
can go up or down,
rate of return
on the
(iv)
liquid asset
quantitative observations are as follows:
significant (about
cause a non-trivial
10%
fall
namely that
4.5,
the effect of transactions costs
is
(i)
large
When
of the level of the rates of return),
in the rate of return
on the
when k
(ii)
k
is
is
close to
close to
When
k
ts
1,
(iii)
(i)
the
the rate of
on the
liquidity
1.
the liquidity
close to
1,
premium
is
transactions costs
liquid asset while the rate of return
on the
illiquid
asset remains almost constant.
These quantitative
change
results
have important practical applications.
in transactions costs in the
affected by this change.
economy,
it is
important to understand
how
the impact of a
assets are differently
A technological change such as a reduction in computer cost can be assumed
to reduce transaction costs for
all
assets
which
in
our model corresponds to the case k=l. Our
suggest that rates of return will not change much.
one
To understand
By
results
contrast, a reduction of transactions costs
on
single asset (e.g. by the introduction of a derivative security) will increase the price of this asset
without any significant impact on the other
of existing assets (stocks, real estate
..)
will
assets. Finally, a transaction tax
lower their value by an amount
on
less
a significant subset
than suggested by a
simple partial equilibrium analysis which takes the rates of return on the other assets as given.
INSERT HGURES
2
TO
4
33
VI.
CONCLUSION
In this
work we have constructed
Our main
market.
result
assets
upward, there
result
is
is
is
that while transactions costs tend to
that the rate of return
We
the trading motives:
life cycle,
on
consistent with previous
many
to allow for
relationship
between
In this paper,
push the rate of return on
illiquid
liquid assets
goes down while the rate of return on
The
net
illiquid assets
believe that these results are robust with respect to the specification of
labor income shocks'* or taste shocks and
Our model endogenously generates
model
capital
a general equilibrium effect which tends to lower rates of return.
may go up or down.
is
model of an imperfect
a fairly simple general equilibrium
the preferences."
(ii)
clienteles for assets with differential liquidity. This clientele effect
work by Amihud and Mendelson
(1986). In fact,
assets with different transactions costs,
rates of return
we assumed
(i)
if
we would
we
generalized our
obtain the concave
and transactions costs derived by these authors.
that transactions costs
were a pure destruction of ressources.
If instead,
they are due to a transaction tax whose proceeds are distributed to the agents, the results are similar.
Amihud and Mendelson
(1990) argue that, holding the risk free rate constant, a .5% transaction tax
would lower the market value of the
NYSE
stocks by 13.8%. While
a small transaction tax will increase the liquidity
premium
risk free rate will fall so that the stock price fall
is
significantly,
likely to
Amihud
et. al.
not dispute the fact that
our
results suggest that the
be somewhat smaller.
This line of research can be pursued in (at least) two directions.
'*See for instance
we do
First,
the interection between risk
(1992).
''The treatment of the perpetual youth model for a general utility function seems to us analytically
companion note, we consider a two-period overlapping generations model similar in spirit
model is simpler but also much less rich. In particular, the holding period is the
assets and as a result the liquidity premium is fixed. In this simple model the results are
untractable. In a
to the
same
model
for all
herein. This
independent of the functional form of the preferences.
34
and
liquidity
is
not fully understood.
It
would be interesting
to construct tractable models to analyze
the interaction between transactions costs and risk and examine in particular whether, as
argued, illiquid markets are
more volatile since
Second and more importantly, very
little is
investors find
known about
it
more
it
has been
costly to absorb liquidity shocks.
the determination of the level of transactions
costs as well as the financial structure created to deal with these transactions costs.
35
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Rao and Mark
Individual Risk:
A Stage
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III
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New York University, mimeo.
Yu
Policy" Financial
(1992) "Income Uncertainty and Transaction Oasts",
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223-247.
Blanchard, Olivier and Stanley Fisher, Lectures on Macroeconomics,
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Boudoukh, Jacob and Robert Whitelaw (1991) "The Benchmark Effect
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in the
2,
1989.
Japanese Government
52-59.
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A Lesson
from the
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Quantitative Analysis, September, 483-496.
Constantinides, Georges (1986) "Capital Market Equilibrium v«th Transaction Costs", Journal of
Political
Davis,
Economy, 94(4), december, 842-862.
Mark and Andrew Norman
(1990) "Portfolio Selection with Transaction Costs" Mathematics
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DufGe, Darell and T. Sun (1989) 'Transactions Costs and Portfolio Choice
in a
Discrete-Continuous
36
Time
Setting"
mimeo, Stanford University.
Duffie, Darrel and
Financial Studies,
Matthew Jackson
2,
(1989), "Optimal Innovation of Futures Contracts" Review of
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Dumas, Bernard and
Elisa Luciano (1991) "An Exact Solution to the Portfolio Choice Problem Under
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Fleming, Wendell, Sanford Grossman, Jean-Luc Vila and Thaleia Zariphopoulou (1992) "Optimal
Portfolio Rebalancing with Transaction Costs" mimeo, Massachusetts Institute of Technology.
Fremault,
Anne
(1991) "Stock and Stock Index Futures Trading in an Equilibrium
Model with Entry
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Goldsmith, David (1976) "Transactions Costs and the Theory of Portfolio Selection", Journal of
Finance, 31, 1127-1139.
Grossman, Sanford and Jean-Luc Vila (1992) "Optimal Investment Strategies with Leverage
Constraints", Journal of Financial
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Heaton, John and D. Lucas (1992) "Evaluating the Effects of Incomplete Markets on Risk Sharing
and Asset Pricing" mimeo, Massachusetts Institute of Technology.
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Approach", mimeo.
A
Haim
Constraint on the
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in the Portfolio" American Economic Review, September, 643-658.
Levy,
Mayshar, Joram (1979), "Transactions Costs
Political
Economy,
in a
Model of
Capital
Number
A
Duality
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Market Equilibrium" Journal of
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Mayshar, Joram (1981) "Transactions Costs and the Pricing of Assets" Journal of Finance, 36, 583-597.
Mehra, R. and Edward Prescott (1985) "The Equity Premium:
Economics, 15, 145-162.
A
Puzzle" Journal of Monetary
Michaely, Roni and Jean-Luc Vila (1992) "Tax Heterogeneity, Transactions Costs and Trading
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Massachusetts Institute of Technology.
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Silber,
William (1991) "Discounts on Restricted Stock: The Impact of
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on Stock
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37
Financial Analysts Journal, July- August, 60-64.
Tuclcman, Bruce and Jean-Luc Vila (1992) "Arbitrage with Holding Costs:
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A Utility-Based Approach"
Jean-Luc and Thaleia Zariphopoulou (1990) "Optimal Consumption and Investment with
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raimeo, Massachusetts Institute
38
APPENDIX A
This appendix considers the case
when
equilibrium 6><i) and i|»=r+X + (ij>0.
We
must
first
We will
prove that
will first
then prove that the equilibrium
in
unique.
is
calculate the optimal policy which entails solving
max
_i_c'-^e-<»*«'dt
f
1-A
J
s.
,.t.
From He and Pages
=
He
+
r
A.
and Pages show
Cj
=
c
=
<p
=
'
(8)
c^e-<'*^"dt =
J
we know
(1991),
\|i
In that case,
We
transactions costs are zero.
that a
J
bounded value
w^ ^
,
for (8)
above
exists
.
provided that
+ (o>0.
that
for every
y,
t
—
=
-
w<
5
if to
^ 6
and
(Al)
_*
y-
y^e-'^-^"dt
for every
e'**
if
t
with
<P
If
t|f
r
sO, then the value of (8)
+ X + 8
is
.
We
+«.
show below
that this cannot
be the case
in equilibrium.
In equilibrium, the resource constraint implies that
J
Hence the equilibrium
A.c,e-" s
utility
(^)
D+Y.
of the representative agent
is
bounded by the solution
to the
program
(A3) below
max
f
i
s. t.
It is
If
easy to
show
-J_c'-^e-'»*«'dt
1-A
J
'
c,e-^'dt =
(A3)
D+Y.
that (A3) has a finite value.
0)^6 then the agent does not buy any asset and therefore w,=0. Obviously,
this
is
not consistent
39
with the equilibrium condition
Xe-^'wdt
f
Hence
ft
must be larger than
From (Al) and
(7)
it
o) in
=P
(A4)
.
equilibrium.
follows easily that
w
Combining (A4) and (A5)
(A5)
= y.
yields the equilibrium condition
r-(5-ili:)
A
r-(5-co-)
^
D
(A6)
A
which must be solved for
r*.
Simple algebraic manipulations show that (A6) above has a unique positive solution
solution
is
increasing in p,
A.
than p and decreasing with
and D/Y and that
A otherwise.
it is
decreasing in 6.
It is
increasing in
A
r',
that this
if r' is
greater
40
APPENDIX B
Proof of proposition
4.1
The method of proof
is
as follows:
We
conditions for an optimal control. Next
first
we
define the control variables.
We
then derive heuristic
construct a candidate optimal control and
show
that
is
indeed optimal.
Step
at
1:
date
The
t
control problem Recall that
i,
:
and that
I, is
the per unit time value of the liquid assets purchased
is
the per unit time value of the illiquid assets purchased at date
t.
Recall also
equations (15a), (15b) and (15c):
=
da,
dA,
c,
The
=
+
Xa, dt
=
+
A.A, dt
+
y,
ra,
i,dt,
-
control problem faced by the consumer
subject to the dynamics of
Formally,
we
and A,
a,
say that a control
(i.e.
and
\=
I,dt,
+ RA,
=
a,,
i,
is
€|I,1,
(15a)
0.
and A, s
- 1, -
to
^
a,
(15b)
0.
(15c)
c,iO.
maximize
(5) with respect to the controls
(i( •),!( •)) is
admissible
piecewise continuous
if it is (i)
if
he follows the controls
1(0) ^he payoff function,
J(i(-).
and
i,=i(t)
a,
=
J
and
I,
(Bl) and (B2)).
the no short sale constraints a,^0 and A,^0 as well as the constraint c,^0.
admissible controls and by
i,
I,=I(t).
i^e'"-"ds
Using the
A
and
i.e.
the
(ii) it
satisOes
We denote by C the set of
utility that
the consumer enjoys
fact that
=
J
I,e^"-'ds
the payoff function can be written as
•
J(i(-);I(-)) =
Hence the
J
t
t
u(y, +rJi,e^<'-'ds+Rjl,e^<'-'ds-i,-I,-€
control problem
is
to
maximize
J(i( •),!(•))
with respect to
|
IJ
(i( •),!(•))
)
e'^**^" dt
(^1)
.
belonging to
C
41
Step
2:
Heuristic optimality conditions
a candidate optimal control
:
(i( •),!(•)),
To
derive heuristic conditions for an optimal control,
denote by
c(-)
we
take
the resulting consumption and consider the
possible perturbations:
(i)
Supf>ose
unit time)
first
and
that a,>0.
invests
Suppose
that at
a more dollars
t,
the consumer changes his consumption by a dollars (per
in the liquid asset.
extra dividend ore*'"' and at t+dt he sells oe**.
thereafter. His payoff will
(i( •),!(•)) is
He
s
between
t
and t+dt, he consumes the
does not change
his
investment decision
change by
«[-u'(c)
Since,
At
+ e-'»**""(e"' + rdt)u'(c^,)].
optimal, the change in payoff must be non-positive for every a and dt going to
zero which yields the standard Euler equation
-^
du'(c)
/rn\
(B2)
=(P-r)u'(c.).
dt
(ii)
Suppose now
that a,=0.
Then
the consumer can only increase his investment in the liquid asset
and therefore condition (B2) must be replaced by
du'(c)
/D-JN
(B3)
—l-ll ^(P-r)u'(c).
(iii)
Consider a policy where A,>0 for every
(iiia)
Suppose I^>0.
If
between
t
dt,
his
will
be the case
in
our candidate
policy,
the consumer changes his consumption by o(l+€) dollars
o+I*>0) more
(per unit time), invests a (with
investment decision thereafter,
and
which
t,
dollars in the illiquid asset
and does not change
his
payoff will change by
a[ -(l+€)u'(c)
+
jRu'(c)e-'''-'>ds]dt
t
which must be non-positive
for any
o and so
(l+e)u'(c)
=
jRu'(c.)e-»'-"ds
.
(^)
t
(iiib)
Suppose now
that I*<0. If
between
t
and
dt,
the consumer changes his consumption by (l-€)o
42
dollars (per unit time), invests
his
o (with o+I'<0) more dollars
investment decision thereafter^ his payoff
o
[
-
-€
( 1
)
u' (c
)
will
change by
+
R u' (cj e-»'-'> ds
J
and does not change
in the illiquid asset
]
dt
t
which must be non-positive for any a and so
(l-€)u'(c)
=
jRu'(cJe-»(-"ds
(B5)
.
t
(iiic) If I,
=
0,
then the
first
order condition reads as
(l-€)u'(c)
Step
3:
i
jRu'(cJe-»'-"ds
(l+€)u'(c)
(B6)
.
Construction of the candidate policy:
Given c^
x,
and
A we
c,
define the candidate optimal consumption plan as in equation (16)
= c„ e-<''
o>(t) =
with
(^^^)^-P(^)
To
definition of these parameters.
illiquid asset
he
from
will sell
to
the
t,.
He
will
(16)
.
This consumption plan will be completely defined once
Finally
dt ^
we
specify
c^, t,
and A.
We now motivate our
finance this consumption plan, the agent will buy shares of the
buy and then
illiquid assets
from
sell
t,-I-A
shares of the liquid asset between t, and t,+A.
and +«. From equation (15a), (15b) and (15c)
it
follows that
^ote
that this perturbation
positive for every
to
i,
is
t.
is
However a,=0
feasible only
for t<r,
if
A,>0
for every
and t>ri+A. For
t.
In the optimal policy. A, will indeed be
this reason, the optimality condition
written as an Euler equation (B2)-(B3) while the optimality condition with respect to
condition.
I,
with respect
is
an integral
43
From (B7)
A
=
f
A
=
A
A, =
A
^J—J
e'"*"-""" ds
e"'"'''
f
for
+RA
(y -c
e'^'"-""'*"' +
t
^ t,
it^t+A,
fort,
)e"""-''""ds
a ,=
^llf! e''("-'"»ds;
f
,
for
t
^ t,
(B7)
+A
together with the transversality condition
Lim A^e"*'^
we
=
a
;
=
a
;
=
get
';
f
•i
^J—Je-'-c'dt
i:
:e-'""dt
= e^*
l+€
^'
^'
f
J
J
'
(B8)
^-"'"Hf
l-€
Finally
t;
We
first
define
A by
^l
=
=
,
e""'*''*
(18) that
^
R-r
€
€
e-- =
i
^
^A
f
(y -c
+RA)e-'"'dt
(B9)
=
.
is
^
l^e-^
(18)
1-e-^*
or equivalently
—
(BIO)
-
1
+e
Adding equations (B8) and (B9)
function of t„
e, r
we
define Co (as a
(y,
-c)
below
e-o'^'dt +
J
T,
Finally
We
and R) from the intertemporal budget equation (17a) which together with (18)
yields the simpler equation (19)
J
gives the intertemporal budget equation (17a).
define t, by equation (B8).
(y,
-c)
e'^'^dt +
(y,
J
ij
A
-c)
e-^^'dt
=
.
(19)
44
We
now show
that this policy
well defined and admissible for e small and for
is
to a subset of their equilibrium values.
where
m
We
show
will also
that
it
r
and
R
varies smoothly with
belonging
€, r
and
m
defined by
is
R
= r
+ me.
We will
show the
Lemma
Bl: Consider r,
following
(Bll)
lemma Bl below:
m and m' such that m'lr'>l
(recall that r is the equilibrium interest rate
when
transactions costs are zero). There exists eg such that if
0<e<ea
(i)
(ii)
Ir'-rj^e^ Im'-mj^Co
the consumption plan defined above, where
and
T,
Proof:
one
(C)
are infinitely differentiable
The
for
intuition for the proof
€=0. That plan
is
is
R=r+me is
well defined
and
admissible. Moreover, A, Cg
in (e, r, m).
simple. If €=0, this consumption plan collapses to the optimal
The
admissible.
admissibility of the candidate policy for
€>0
will follow
by continuity.
We
first
note that the equation defining
»-rA
Therefore
A
is
The equation
y
m
m
A
can be written as
-r
(B12)
+r
uniquely defined,
is
C
in (e.r.m)
and verifies 0< A ^ A s A < +«
giving Co can be written as:
+ e
f
R,
'
+
'
+X + 5
r
1
=
R„ + X + 6
+X+6
'-
°
1
^-C^L*^*"!)';
„
+ e
RL + A. + (i3L
with
for €o sufficiently small.
^
^
(B13)
^-C^L-i'-L)'/ ~-(r.JL.»)4
„-(r.l.»)A
1
+
'
r
+ X + o)
Rg + X + oj^
J
45
From equation
It is fairly
(B13),
it is
obvious that
straightforward to
show
that
\p\ .yK
for
all
values of
and
r
m
Cq
is
uniquely defined and
we can
restrict
€o>0 such
C"
in (€,r,m,T,).
that
[0,€j
in
defined before and for
in the set
is
all
Ti6[0,+«{ where K^
is
a positive
constant.
We now
l+€
turn to the equation defining
—
o
[y—Rj^
^-T+X +6
- ^0
p
"
g-(r.l-»)T, _g-(r.l.»),,
f(
•,•,•,•,•)
a
is
|!
I
—
,
.
This equation can be written as
1
=
Rl + >- + (JL
Straightforward algebra shows that
where
Tj.
we can
1-—
1-e
[
Co
—5—
. K,
Rg + X +
fi
rewrite this equation as
^ g-(r»l—),,-(l.»)A_^-(fi.«)x,-(l.«)4 ^
C" function such
-y
TT
Rb*^*<^
"
that
f=0
for
^
.
S
f
^
€ T
)
(B14)
e=0 and
[0,€j
in
I
O€
for
all
values of
r,
ra
and
t, (restricting €o if necessary).
Consider the equation for
C
The
function
g(-):
t
-
exists
Q>0
this
is
a positive constant.
€=0
-(!.»)»,
e"****'"
-
C
-(1.4.)x/
e"'"**"
-(l.»).,-(l»»)A
-(1.4.)t,-(l.«)4
_
~C
— C
has the following graph (see figure
INSERT
Therefore
K,
equation has a unique solution
(B15)
•
A)
HGURE A
Tj in ]0,t'[. It is
easy to show (given
A ^ A >0)
that there
such that x^^x'-C and Ti+A2t*+C. Using the implicit function theorem, the fact that
+oo>AiA^A>0
and the
fact that
||I| ^K,
in
[0,ej
46
for
all
all
values of
values of
m
r,
and m. Moreover,
r
^
I
unifomily in
T,+AiT*+
and
r
-f-
and t„ we can show that we define a C" function Ti(€,r,m) for e^Co and for
K
^
I
easy to
it is
if
that
[0,€j
in
ra (restricting Co
show
necessary). This implies in particular that t,st'-
and
for € small.
,
Summarizing our discussion above,
it
can be seen that Co(€,r,m)
is
close to
straightforward to interpret the values of Ti(0,r,m) and A(0,r,m). Indeed
liquid
-§-
its
6=0
value.
when €=0,
all
assets are
and the switching times do not have any particular meaning. However, as seen above T,(€,r,m)
and A(€,r,m) have well-deflned positive
limits as c
goes to zero. As seen from (B12), given
to calculate A(0,r,m) = A(e,r,m).
m
one can
Given
(or A),
transactions costs case. This value Ti(0,r,m)
grows
at a rate
between
A.
T,(0,r,m)
Therefore, the consumption plan
To show
We
that
it is
is
the time such that accumulated wealth (when
dA
_J
=
-
XA
well-defined and A,
Co,
tj
vary smoothly with
In
[0,Xi],
.
It is
of the above statements.
we have
T
RA
_
easy to see that
I,
0€
=
t
+y't -c
I,|,.o
*
+
e, r,
to prove that 1,^0 in [O.ti], 1,^0 in [t,-I-A,+oo{
dt
briefly sketch proofs
no
e=0)
and T,(0,r,m) + A.
we have
admissible,
is
m is easy
calculate T,(0,r,m) directly from the
recall that
I
We
It is less
t
g(€,r,m,t)
where g=0
for
e=0 and
m.
and
a,iO, A,^0.
47
\ and
uniformly. Since t,<;t'-
I,|,.oi0>O in [0,t'-
4
],
it
follows that for e^e^ (restricting again
e),
I,>0 which implies that A,iO.
In [t,,t, + A] again,
We
a,
can easily show
and
i,
€=0
are very close to their
counterparts.
that:
in [t'- -f ,T*+ -f
in [ti,t'- -f
],
a,>0 by continuity,
],
i,>0 by continuity and
in [t*+ -f ,T,+A], i,<0
by continuity.
This implies that a,^0 in [t„Ti + A]
In [t, + A, + <»{, simple calculations using
A
=
'
show
o-(p(')-''C»Hc
l-€
that
_J_r_JLL?_7e-"-^L!-!^c
8-^
=
I
l-e^R3
'
The
^'~^'
f
J
+
A.
Rg
+
+ X + co^"
e-^-'e-^e-**"-^''**"!^
continuity argument can be applied in a compact set [t'+
8-(0b ^ T\/l
>
for te[T,+«{,
I,
will
be negative
if
T
is
-§-
,T] to
show
that I,<0, while since
large enough.
Finally
A^
=
^
^ -t'j^-^^ -"«(-('/ ^))_
'^o
r
l-cRg + X +
o)^
+
A.
and similar arguments show that A,iO. Therefore the consumption plan
proof of lemma Bl.
Step
4:
Optimalitv of the control
:
^-Mi
y
Rb
+ 6
is
admissible. This ends the
48
Having shown that our candidate optimal control
it is
indeed optimal. For
Lemma
Proof:
purpose,
this
B2: Our candidate control
It is
we
show
first
satisfies
is
well defined and admissible,
that
it
satisfies
we
will
(B2)-(B6).
conditions (B2)-(B6).
obvious that
in [0,T,]
d log
u'(c,)
P-Rl
dt
that in
^
P-r
,
[t„Ti+A]
dlogu'(c)
dt
and that
in
[t,+A,+«»[
dlogu'(c)
=P-R3^P-r
'
^^
Hence (B2)-(B3)
.
are satisfied.
For te[T,+A,+<»{
"•*«""
ds
Ju'(c)e-»<-"ds = u'(c) Je
It
follows that (B5)
is
u'(c)
u'(c)(l-e)
R„
R
satisfied.
For te[T,,Ti+A]
Ju'(c)e-»<-"ds=u'(c,)[
e-^<-'ds +
J
=
J
e
"'''**"
e'''''*-''-**ds
.(i-€)i_^
u'(c)[-L:£
In addition
-r(t,.d-.,
l-€
l_e-^<''-'-"
R
From equation
r
we
-€):___
'
r
(BIO),
,,
(1
'
get that
^
^
R
R
R
j_^.,4
<:
,^
^_J_ *(l-€)r_.
^
r
R
show
that
49
Equation (B6) follows.
Finally, for te[0,T,],
Ju'(c.)e-»<-"ds
we
have:
*e^
=u'(c)[(l.€)i-L^^
=
'
!_£_ *(!-€)£
r-^
]
iJll
because of (B16).
This proves equation (B4) and ends our proof of
We now show
Lemma
Proof:
that the candidate policy
B3: The candidate policy
To
derive this
lemma we
is
lemma B2.
indeed optimal.
is
optimal.
shall
show
that
no perturbation of the candidate policy can be
utility
enhancing.
Consider an alternative policy (i,+5i„I,+8I,) which induces consumption Ct+5c,.
We know from (B4)
that
8c
=
Using the concavity of
r f Si
u(-),
e^"-'ds +
it
R
fsi e'"->ds -
u(c)
-51
- e
(
1
1
+
M
I
-
1
1
I
)
.
follows that
t
u(c, + 5cj i
5i
t
+ u'(cj[rj5i,e^"-"ds +
Rj6I,e^"-'ds
-
5i^
- 81^ - €
(
|I,
+ «I,
|
1
1
J
) ]
(B17)
We
next multiply equation (B17) by e<***", integrate from
Ju(c. + 8c)e-"'-^"ds
^
to
t,
and get
Ju(c,)e-<»*^"ds + K(t) + K,(t)
with
K(t)
= Ju'(c,)
[
rJsij^e^'-^Mh -
5i,
]e-<»*^"ds
,
50
K,(t) = Ju'(c.)[Rj5I,e^<->dh-5I.-€(|I..8I.|-iI.I)le-'»-^"ds.
We will
show
when
that
t
is
goes to
infinity, K,(t)
Integrating the second term of Kj(t) by parts,
we
and
K,(t) are asymptotically non-positive.
get
'
Ju'(c,)6i,e-'»-^"ds =
•
,^
dCu'CcJe-*")
ids.
-J(j5i,e-"'dh)_L_lj^
(J5i,e-"dh)u'(c.)e-
ds
Therefore, the K,(t) equals
J[u'(cJ(r-p)
We
8i^e-"dh
= e""' 8a,
J
5i,e-''ds)u'(c,)e-".
(^18)
.
a,>0, then condition (B2) holds and the integrand in the
If a,=0,
first
term of (B18) above must be zero.
then from the short sale constraint 6a,i0 and condition (B3) ensures that
non-positive. For
tiTj+A, a,=0 and thus
the second term
is
positive for large
t.
Consider
J
(
note that
J
If
_!:](j6i^e-"dh)e-»'ds -
+
now
non
K,(t). Integrating
J 8I^e^(-'"dh
u'(cj
positive, for
)
t
fia,
large enough.
by parts the
e-(»-*"ds
must also be greater or equal than
first
We
term,
we
0.
this
integrand
is
This implies that
have therefore proven that
Ki(t)
is
non
get
-(Ju'(cJe-»^dh)(J5I,e-^dh)
J(Ju'(cJe-"'dh)8I,e-^ds
(
I
Therefore K,(t)
J
[
(
is
-u'(0 *R J
equal to
u'(c,)e-»"'->dh)8I, - u'(c,)€
( |I,
+ 6IJ -
1
1.
)]e-'»*^"ds 1
R( Ju'(c.)e-»'ds) ( J
Sl.e'^'ds
(B19)
If I.>0, |I,+8I,|
-
|I,I
^ 8I„ the integrand in (B19)
[-u'(cJ(l+€)
by condition (B4).
+
is
less
or equal to
Rju'(cJe-»"'-'dh]8I,
=0
.
51
I,<0. |I,+fiI,|
If
-
|I,|
i -5L,, the integrand
[-u'(cJ(l-€)
+
less
is
or equal to
Rju'CcJe-^c-'dhlM,
=
by condition (B5).
I,=0 and fil,>0, the integrand
If
is
m
Rju'(cJe-»"'-'dh]M,
^
[-u'(c,)(l-€) + Rju'(cJe-»«'-'dh]M,
^
[-u'(cJ(l+€)
+
by condition (B6).
If
I,=0 and 5I,<0, the integrand
is
by condition (B6).
Therefore the
first
For the second
terra in K,(t)
terra,
is
less
or equal to
0.
note that
J51,e-^'ds = e-^'5A
and that
1
-€
..^n
l-€
fu'(c)e-»'ds = e-»'u'(c)i-Il = e-'^"i—i
Since fiA,i-A, (ftA,+A,iO), A,
•
-
R J u'(c,) e-^' ds
—
exp
(-(Oet)
for
t^T
+A
.
and X + coB>'no/2 for € small,
it is
clear that the term
t
J
81,
e"^ ds
t
is
smaller that an arbitrary
S>0 for
t
large enough.
This concludes the proof of lemraa B3.
It
follows that K,(t)
is
asymptotically
non
positive.
52
APPE^a)IX c
Lemma
Proof of
4.3:
Suppose that r=r' and
consumption
yf
at time
R=r*+m*€. From appendix
that
is
+ e
given by equation (19), which as
'-
e
1
first
Rj^ +
term
+ e
-(R, .1.-.).;
'^
^
r'
by
+X+6
y)
1
1
r
1
can be written
or equivalently,
+
ee
e-^"^"'"^'-' e-'''-^*->*
Rg + X + 6
-•%,'
'+
m'+r'
<p'
1
Rg + X + cc^
+ X + (o
as:
^-€6
r-
+X+5
as:
/pi>.
v*-^^
-•'(t/.A*)
*o(€)
J
<P'
as:
-L[ l-€(ra--r-) re-'''dt-€(m'+r-)
Similarly, the
e-<'"*^*->^
Rl + X + 6
ra'-r"
<p'
<P*
-
r-
Straightforward algebra shows that this term can be written
_[1- m'-r' €+
rewritten as
=
'
Rj^ + X + o:^^
in brackets (divided
X+6
1
-<'^'--^-*"'
f
*
The
-
we have seen be
Rb + ^ + »
+X+5
r'
,
that for e small, the optimal
+
'
R^ + X + fi
c„
we know
B,
second can be written
J
e-'''dt
+o(€)
]
(^)
.
as:
»/
—
[
1
-€ (l-i)(m--r-) Je-*''dt-€
(
l-l)(m-+r-)
T
Combining (C2) and (C3) we get equation (24a)
Co
with
Cw and
J
•
=
y-.
*€C^
i.e
-^€€3 *0(€)
Cj given by (24c) and (24d). Integrating (24c) by
parts,
we
get
e-*''dt+o(€)
.<
]
.
(^)
53
(m'-r-)
J
[e*''-e-''']dt + (ra'+f)
-(«••!)(
= (m--r-)
[e-»''-e-''']dt
J
_p-(&*l)l>
(m'-r-) r£l!((5+A.)e-<*-^"-(o)i
+
(m'+f)
dt
Using equations (21a) (substituting w, from (11)) and
(22),
we And
out which yields equation (24b). This concludes the proof of
Proof of proposition 4.2
Consider
r*
+ X)e-<-''^"
r'
that the terms in brackets cancel
lemma
4.3.
:
and m' where
m
.
=
.1-e-'**'
r
1-e-'^'
The
results
of Appendix
neighborhood of
r*
and
B
imply that for € sufficienty small and for
m', the
consumption plan defined
in
Proposition 4.1
admissible and optimal.
These
results also imply that
F:(€,r,m)
-
Jxe'^'a^dt,
(
is
C"
We
in
these variables.
know
that
F(0,r-,m-)
=((l-k)P,kP)
r*
It is
Xe'^'A^dt)
J
.
r"
also easy to see that
__!(0,r,m)
dm
=
-^(0,r,m) #0
dm
r
and
is
m
belonging to a
indeed well defined,
54
(If
€=0,
r
is
m
kept constant and
changes,
proportion of the wealth in liquid and
It is
is
it
as
illiquid assets
if
the total wealth
is
kept constant and the
changes.)
also clear that
^(0,r,ra)
+
^(0,r,m)
dr
(A change
*
0.
dr
in interest rate has a
non-zero effect on the total stock of wealth, at least
in the
e=0
case).
Therefore the Jacobian matrix
is
invertible
It is
Hi
!Ii
dT
dr
dF.
aF,
dr
8m
and we can apply the
theorem
implicit function
thus clear that for € small there exists an equilibrium.
at the point (0,r',m').
Moreover
r
and
m
are C" in € which
establishes proposition 4.2.
We now calculate how
Proof of lemma 4.4
total
when
wealth changes
and m'
€ changes for fixed r
i.e
prove lemma
4.4.
:
Adding the equations describing the evolution of
a,
and A, we get
d(a.-A)
=
X(a,+AJ+i,+I^
=
^(a.+AJ+ra^+RA,+y,-c-e|I,|
=
(X-r)(a,*A.)*(R-r)A^*y.-c,-€|IJ
-
X(a
dt
d(a.^AJ
+A)=
r(a
.A)*(R-r)A
,
.
+y,-c -e
|I.
dt
Multiplying by e
",
integrating from
(a,+A,)e*' goes to zero as
t
goes to
to
» and
infinity,
we
using the fact (established in appendix B) that
get
55
A.
Jx(a +A)e-^'dt
Equation (C4)
will
_
=
J*e-''[c
+€
|IJ -y,
-(R-r)A
allow us to calculate total wealth as a function of
e.
(C4)
dt
]
.
We will
assume that we are
at (€,r*,m*)
J
dA
(
r
€ |IJe-^'dt =
e[
J
e-'" *
2€
A
„
—y [e
2€
=
.'-XAJe-^'dt
J
€A
=
A
d
'c
:-A.A]e-^'dt -
€[
€A
e-^'"-''
(C5)
e'"''
,
-(1-4.').,'
'
-e
A
since
(
-(1.*).,'
'
,
]
+
A e'M
=
.
o(€)
.
TK*
|e-^'c^dt =cjJe-<^-^"dt.e-''-^"'
J
e-'^-'"-'"dt ^e-'^**^' e"'^*-'^
e-<'*-«'"-''-*'dt
J
.,.4
= Co[
+e*
r
_!_[(!+€
X + o)'
X + CD*
A
—
"^^^
)+e
^
]
'(l-(l+€
X + oj'
A
)) +
e
(l+€
'
X + o)'
A
-l)]+o(€)
c
JX
{
+
1
-co-
l[(m'-r-) fe-<^-'"dt+(m- + r-)
A
f
J
e-<^-'"dt
]
}
+o(€).
J
T.
•A
(C6)
Replacing for
fe-^'cdt
c,,
we
=y
get that
*!
{
1
+
€[(ra--r-)
l[(m--r-)
J
[(e"*'' -e-''')dt +
(m'+r*)
(e-(^-'"-e-*'')dt + (m'+r-)
J
f
(e'*'' -e-''')dt] +
(e-<^-'"-e-*'')dt
]
}
,;.A-
(C7)
Using (C4), (C5) and (C7),
transactions costs
e.
it
is
easy to find the sensitivity of total asset
The equations
in
lemma
4.4 follow.
demand F1+F2
to the
+o(€)
56
We
next derive the expression for b-m*. For this purpose
+R
F
=
with respect to
we
Note
e.
+A)e-^'dt
A.(a
r
i
when
that
differentiate the equilibrium condition
= (l-k:)--*-k:
r
'
'
we
^
(C8)
r+me
e changes, the equilibrium values of r'
and m'
change. Hence
will
get
a(F,*F,)
a(F,-F,),
d€
=
Using the
a(F,*FJ,
..o(b-m-)
l<.0
-
3m
ar
F,+F2
a(F,*F,)
d€
%
independent of
is
dm
j_
-k—
L^(b-m-)
(r+mc)'
-(l-k)5L.o(b-m-)
r
fact that
,.0
m
for
'Mi:IA\
ac
dm
De
-k
2
(r+m€)^
•"*
••"
- k
«•'
H
d€<•
Dm
(r+rac)
2
li"0
€=0, we have
„(b-m-)' =
'""^
ar
It.O
-^(b-m-)---kP.
{t'f^
'
r-
r-
Using equations (25a)-(25c), we thus get
^1
^
(b-m-)
[
ar
X +w
(r-)^J
<p'
AX +
<pT-
(p'r-
oj-
f
{
{
(e-'''-e-''')dt]
i
l_i_yX[(m-r-)f(e-'^-'>'-e-'')dt*(m*r-)
lAy(e''-'*'''''-e"***'"') *
We
T.V
-i_yj!-[(m--r-)r(e-*''-e-''')dt+(m-+r-)
(e-'^-*^' -e"*'' )dt]
f
f
.
can calculate
a(F,^F,)
ar
from
F *F
'
^~"
=
'
(X
+ <o)<p
and derive (b-m*).
The
right
hand side
Therefore (b-m')
is
is
negative and
negative.
it
is
easy to see that the coefficient of (b-m*)
is
positive.
•
57
APPENDIX D
Proof of lemma 4.5
If
:
k increases, obviously A* decreases. Moreover
tj
increases and Ti + A' decreases. These can be seen
from equations (21a) and (21b) with very simple algebra. Equation (22) implies then that m*
increases.
It is
thus clear that (b-m*) which
is
proportional to
m
'1
-
[
A.
+
0)-
<p'
i
1
A
f
(e-'''-e-''')dt
]
^
f
(e"''-'*'' -e-''**"') +
<p*r'
i
^ y-!!!L[(m--r-)
X + o)- <p-r-
^
_
-
_^y^[(m--r-)f(e-»''-e-''')dt*(m'+r-)
(e"'^-'" -e-*'')dt) +(m-+r-)
•;
Tt
decreases,
i.e.
the effect of e
on
increases in [0,t'] and Ti<t').
r'
is
stronger (note that the function
g(*):t
(e-'^-'>' -e-*'')dt]
f
f
- e'"*^"
A
-
.
e"****"
58
APPENDIX E
The Case k=l
The
case of
effect
we
k=l
on the
is
rate of return
on the
be lower than the return on the
find, as before, that transactions costs
illiquid asset.
economy
introduce a liquid asset in this
liquidity
We
slightly different.
this result
difference with the case
by zero-th order term.
is
that the
minimum
(i.e.
first
(0<k<l)
zero supply, that cannot be sold short,
in
illiquid asset
premium.) The reason for
The
have a
we have
its
is
order
that
if
return will
a zero-th order
holding period has a
first
order
length.
Since
all
of the consumer's wealth
A,
=
The dynamics of w,
dw.
c.
is
held in the form of the illiquid asset,
we
have:
Wj.
are described by:
=
=
y,
A.w,dt
+
+
(El a)
I,dt
R.-I.-e|I,|.
(Elb)
Proposition E.1 describes the optimal policy of the consumer for small transactions costs and for a
subset of values of
R
that are of interest,
Proposition E.1: For e small and for
policy has the following form:
nothing
(i.e.
R
i.e.
such that
its
equilibrium value belong to this subset.
belonging to a subset of their possible values, the optimal
The consumer buys
he consumes his income y,+Rw,) from
the (illiquid) asset until an age
t^ until
an age
Ti
+ A when
t^.
He
does
he starts selling the
asset until he dies.
The proof of proposition E.1
In what follows,
we
is
analogous to the proof of proposition 4.1 and
will (briefly) discuss the implications
of proposition E.1.
is
therefore omitted.
We will
show how the
59
consumption q
In the case
as well as the width of the inaction period
k=l, the expression
= c„e--<'>
c
t
I
this
initial
c,
must be changed from equation (16) to
(E2a)
T,itiT,
=c TJ*A e-<"**'-"'
c
consumption
derived.
ST,
Cj=Rw_+y^
Again the
for
A can be
consumption
+
(E2b)
A
^ T
t
1
c,
(E2c)
*A.
can be obtained from the intertemporal budget equation which
in
case can be written as
''
''*'
_
^1—^
f
e-o^'dt +
f
-c)
(y
e'-'-'dt +
"
_
f
!l_J
(E3)
e'-^'dt =
where
p(t)
=
p(t)
p(t)
Rl+A.
<
for
t
= R+X
for
T s
= Rb+A.
for
T+A
T,
t
<
s
T
+ A,
t
and
t
p(t) =
Jp(s)ds
.
This intertemporal budget equation
is
derived from the equation describing the evolution of w„ using
our results on the investment policy of the consumer. The analysis
and
is
is
very similar to the case
0<k<l
omitted.
The parameter A
consumer
is
condition
indifferent
is
derived by the portfolio decision of a consumer of age
between investing
selling the illiquid asset at t, + A, the
in the illiquid asset
change
in his utility
if
and not doing
so.
t,.
Given that he
he buys one unit of the
This
starts
illiquid asset at
60
-(l+€)Pu'(c^^)
and
is
From
equal to zero.
consumption
c
age
at
t,
- y
.
+
(l-€)Pu'(c^^^Je-»* +
J
equations (Ela)^' and (E2)
and consumption
at
age t,+A
=
_:i
c
»
_r!__Z_'=w ''•* =we-"
R
we
get that the following relation
between
- y
R
•'
(E4)
Du'(c)e"'<'""dt
/Trc\
(^)
llie-".
Equations (E4) and (E5) yield the following relation which shows that the minimum period of holding
the
of order €
illiquid asset is
A
t
=
(X + o>-)(5-o)-)
^A
* o(€)
Having characterized the solution
to
(E6)
.
the
consumer's problem
we
turn
to
the
equilibrium
determination of R.
In equilibrium, asset
demand
A.e-*'w^dt
J
equals asset market value,
As
before,
effects
we
on R.
will
and calculate
b.
=
.
consider small transactions costs (small values of
We will
R(€)
D/R
r-
+
and find their
first
order
thus write:
(^'^)
b€ +o(€)
Proposition E.2 gives us b.
Proposition EL2: In equilibrium,
b having an ambiguous
"Note
e),
R
is
uniquely determined.
sign.
that I,=0 between r, and r,+A.
It
has the form of equation (33), with
61
The proof of proposition E.2
The
discussion
as well as the analytic expression for b are again omitted.
on the determination of b
4.2.2, (the effects
are similar) so
we do
is
similar to the discussion offered at the
not present
of the rate of return on a liquid asset that
is
it
here. Instead,
we
focus
end of subsection
on the determination
introduced in this economy in zero supply. Proposition
E.3 gives us the rate of return on such an asset, as well as the implicit liquidity premium.
PropositioB EJ:
on the
! eqnilibriaiii, transactions costs have a zero-th order effect on the rate of retnm
liquid asset
and on the
Proposition E.3 states that
that the
minimum
r = r
-
we have
a zero-th order liquidity
holding period has a
2€
premium.
liquidity
,1.
+ o(l) =
.
r
-
first
.(X
Ai
A
11
=
order length.
+ a)*)(8-ci)')
—
t|i*
Li
Ai(X + o>")(6-o)*)
.
:
¥
+
,1,
o(l)
.
:
+
premium. The reason
We
can show that
.,.
o(l)
for this result
is
m
<
D
CO
CD
<
GAUSS
o
Tu* Oct :o 15:1J.:9 1992
0.0
Figure 2
This figure plots the rates of return as a function of the fraction
represents the benchmark case where transactions costs.
on the
illiquid asset
ic
The dotted
= 2%;
6
illiquid assets.
= 4%;
p
=
.2%;
A
=
1;
DA' = 50%;
63
e
= 3%.
The
line represents the rate
while the dashed line represents the rate of return
example considers the following numerical values:
A.
of
on the
solid line
of return
liquid asset. This
o
0.0
0.1
Figure 3
This figure plots the rates of return as a function of the fraction k of
represents the
benchmark case where transactions
costs are zero.
illiquid assets.
The dotted
The
solid line
line represents the rate
of return on the illiquid asset while the dashed line represents the rate of return on the liquid asset.
This example considers the following numerical values:
A.
= 20%;
Compared
As
6
to figure
= 40%;
1,
p
= 2%;
in this case the
A=
1;
agent
a result, the interest rates and the liquidity
is
DA' = 50%;
e
= 3%.
more impatient and has
premium
are however unchanged.
64
therefore a shorter horizon.
are higher than in figure
1.
Qualitative results
GAUSS
o
Tue Ocl :0 15;15.00 1992
0.0
Figure 4
This figure plots the rates of return as a function of the fraction k of
represents the
benchmark case where
transactions costs are zero.
illiquid assets.
The dotted
The
solid line
line represents the rate
of return on the illiquid asset while the dashed line represents the rate of return on the liquid asset.
This example considers the following numerical values:
k
= 2%;
In this case, the
1,
6
= 4%;
p
=
.2%;
A=
1;
D/Y = 300%;
€
= 3%.
importance of financial income compared to labor
the following paradoxical
phenomenom occurs:
such
when k
is
close to
transactions costs lower the rates of return
on both
assets.
65
is
that,
<
Q::
o
O)
768
-B7
Date Due
^f*-
26
mm
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