Download Equilibrium interest rate and liquidity premium under

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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
EQUILIBRIUM INTEREST RATE And
LIQUIDITY PREMIUM LENDER
PROPORTIONAL TRANSACTIONS COSTS
Dimitri Vayanos
and
Jean-Luc Vila
WP #3508-92
Revised April 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
EQUILIBRIUM INTEREST RATE And
LIQUIDITY
PREMIUM LENDER
PROPORTIONAL TRANSACTIONS COSTS
Dimitri Vayanos
and
Jean-Luc Vila
WP
Revised April 1993
#3508-92
Massachusetts Institute of Technology
Revised:
December 1992
This version:
April 1993
'^'^^RIES
I
f
M.I.T.
AUG
LIBRARIES
3 1 1993
RECEIVED
Equilibrium Interest Rate and Liquidity
Premium Under
Proportional
Transactions Costs
Dimitri Vayanos
and
Jean-Luc Vila*
Massachusetts Institute of Technology
First version: April 1992
Tills version: April
•We would
like to
thank participants at the
participants at seminars at
Conference on Asset Pricing
in Philadelphia:
MIT, New York University and Wharton; Drew Fudenberg. Mark
John Heaton and Jean Tirole
financial support
NBER
1993
for helpful
comments and
suggestions.
We
also wish to
Gertier.
acknowledge
from the International Financial Services Research Center at the Sloan School of
Management. Errors
are ours.
Abstract
In this paper
we analyze
the impact of transactions costs on the rates of
return on liquid and illiquid assets.
We
consider an infinite horizon
with finitely lived agents along the lines of Blanchard (1985). In
agents face a constant probability of death, and the population
by an inflow of new
arrivals.
is
tliis
economy
economy
kept constant
Agents st&it with no financiad wealth and receive
a decreasing stream of lal)or income over their lifetimes. In addition they can
invest in long-term assets
two such
are
is
which pay a constant stream of dividends.
assets, the liquid asset
and the
The
illiquid asset.
There
liquid asset
traded without costs, while trading the illiquid asset entails proportional
costs. Neither asset can be sold short.
Agents buy and
sell
assets for lifecycle
motives. In fact, they accumulate the higher yielding illiquid asset for long-terni
investment purposes and the liquid asset
We
find that
when
for
short-term investment needs.
transactions costs increase, the rate of return on the
liquid asset decreases, while the rate of return
or decrease.
The
We
on the
illiquid asset
also find, quite naturzilly, that the liquidity
effects of treinsactions costs
may increase
premium
on the rate of return on the liquid
increases.
asset
on the liquidity premium, are stronger the higher the fraction of the
asset in the
returns
We
economy.
and on the
Finally, transactions costs
liquidity
have
first
and
illiquid
order effects on asset
premium.
evaluate these effects for reasonable parameter values.
^
Introduction
1
Although most of asset pricing theory assumes
frictionless
markets, transactions costs
are ubiquitous in financial markets. Transactions costs can be
rect transactions costs such as brokerage commissions,
taxes,
(ii)
bid-ask spread,
exchange
market impact costs and
(iii)
decomposed
(iv)
fees
into
and transactions
delay and search costs.
Aiyagari and Gertler (1991), report that typical (retail) brokerage costs for
2%
stocks average
amount
of the dollar
(i) di-
common
of the trade while the bid-ask spread for
actively traded stocks averages around .5%. Moreover, transactions costs vary across
assets
and over time. Money market accounts are
clearly
more
liquid than slocks.
In addition, deregulation as well as changes in information technology have reduced
(but not eliminated) transactions costs.
Empirical work on transactions costs documents not only their magnitude but
their
important
effect
on rates of return. Amihud and Mendelson
(198'6)
show that
the risk-adjusted average return on stocks
is
Even more
by comparing two assets with exactly the
direct evidence can be found
same cash flows but
different liquidity:
be publicly traded for 2 years
sell at
a
(i)
35%
positively related to their bid-ask spread.
restricted (''letter") stocks which cannot
discount below regular stocks^ and
(ii)
the
average yield differential between Treasury Notes close to maturity and more liquid
Treasury
Bills is
about .43%.^
The evidence above shows
returns
"
that liquidity
and should be incorporated into
impact of transactions costs on
issues as well.
capital gains
asset pricing theory.
assets' returns will
both reduce liquidity and therefore
In this paper
'See
an important determinant of
shed some
Understanding the
light
on some policy
we
will
affect assets' returns.
Amihud and Mendelson
Amihud and Mendelson
is
result,
analyze the impact of transactions costs on the rates of return
illiquid assets, in a general equilibrium
*More evidence
As a
change with additional welfare implications.
framework.
(1991a).
^See SUber (1991).
^See
assets'
Transactions taxes and differential taxation of long and short-term
investment decisions
on Uquid and
is
(1991a).
also presented in
Boudoukh and Whitelaw
(1991).
We
are interested in
questions such as:
On what
characteristics of the
economy does the Uquidity premium
(the difference between the rates of return on illiquid and liquid assets) depend? IIow
do transactions costs
effects first or
affect the rates of return
on liquid and
illiquid assets?
Are Ihese
second order effects?
Despite their importance for asset pricing, these questions have so far not been
satisfactorily addressed in the theoretical Uterature.
A
major reason
that to an-
is
swer them, one has to move away from the basic model of asset pricing, namely the
representative agent model. (One cannot understand the impact of trade frictions in
model where there
a
is
no trade.) Unfortunately, models with heterogeneous agents
(and trade), tend to be quite intractable analytically.
Since our objective
we take
risk
to
is
understand the
The
analysis of the joint effects of risk
an interesting question that we leave
There are many ways
may
agents. Agents
is,
they
may have
may have
on asset pricing,
out of the picture: All the assets that we consider (liquid or illiquid) pay
a constant stream of dividends.
is
effects of asset liquidity
liquidity
for future research.
to construct a deterministic
economy with heterogenous
trade because of differences in preferences or endowments, that
different preferences for current versus future
different labor
precisely, our
and
economy
is
consumption, or they
income paths. ^ In our economy both motives
exist.
More
a tractable version of a multiperiod overlapping generations
economy, the perpetual youth economy,
first
studied by Blanchard (1985).
We believe,
however, that our results on the 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
tion that
new
makes
arrivals.
of labor
things tractable), and the population
Agents
income over
start with
is
no financial wealth and receive a decreasing stream
their Ufetimes.
illiquid asset.
The
the key assump-
kept constant by an inflow of
In addition they can invest in long-tcnn
which pay a constant stream of dividends. There are two such
and the
is
liquid asset
is
assets, the
;issrts
hquid asspt
traded without transactions costs, while
trading the illiquid asset entails proportional transactions costs. Neither asset can be
*In a stochastic economy, differentiaJ uiformation together with liquidity shocks
trade (see
Wang
(1992)).
may also
generate
sold short. In this
economy agents buy and
they accumulate the higher yielding
and the liquid asset
We
find that
for
when
illiquid asset for
long-term investment purposes
short-term investment needs.
transactions costs increase, the rate of return on the liquid
asset decreases while the rate of return
We
assets for life-cycle motives. In fact,
sell
on the
may
illiquid asset
also find, quite naturally, that the liquidity
premium
increase or decrease.
The
increases.
effects of
transactions costs on both the rate of return on the liquid asset and the liquidity
of the illiquid asset in the
premium, are stronger the higher the fraction
Finally, transactions costs
have
first
order
effect.s
economy.
on asset returns and on the liquidity
premium.
The reason why the
rate of return on the liquid asset
increase in transactions costs, can be briefly
transactions costs increase from
same
stays the
asset
in equilibrium.
must increase (by the
cheaper. Agents
to
Then,
liquidity
now consume more
e
summarized
in
falls
response to an
as follows:
Suppose that
and that the rate of return on the Hquid
asset
the rate of return on the illiquid
in equilibrium,
premium) which implies that
this asset
becomes
since they face better investment opportunities
(they have the liquid asset at the same rate as before, and an additional investment
Moreover, they substitute consumption over time so that they buy
opportunity).
more
of the
cheaper
demand more
illiquid asset
securities for
and hold
two reasons. The
it
for a
first
longer period.
reason
is
Thus, they
will
that they have to finance
higher future consumption, selling the cheaper illiquid asset and paying transactions
The second reason
costs.
cheaper
illiquid asset
goes up.
The
In addition,
if
increase more,
We
is
and hold
that,
it
by substitution, they want
for a longer period.
As a
rate of return on the liquid asset has to
there are
more
and the rate
illiquid assets in the
of return
on the
to
buy more
result, total asset
fall
of the
demand
to restore equilibrium.
economy, total asset demand
liquid asset will
cannot infer whether the rate of return on the
have
to fall
by
will
ni'>r<-.
illiquid asset will incrcasr or
decrease, by a similar reasoning. Indeed, suppose that transactions costs increase trom
to
The
e
and that the rate of return on the
rate of return
liquidity
premium.
illiquid asset in
on the hquid asset then has to
fall
unaffected in equilibrium.
by an amount equal
to the
This time agents face worse investment opportunities since
(i)
the price of the liquid asset increases and
trading in the illiquid asset
(ii)
Agents' consumption shifts
to transactions costs.
down
also
securities
the
accumulate
less of
ambiguous.
is
more expensive
illiquid asset.
First, agents
effect
it
for a longer period.
on the
total
demand®
have to finance lower future consumption
liquid asset but they
Second, agents buy
less of
pay transactions
costs
when
for
selling
selling the
the liquid and illiquid assets, but hold the
a longer period.
illiquid asset for
Finally, the liquidity
illiquid asset.
The
the liquid asset.
subject
uniformly. Furthermore, by
substitution they accumulate less of the illiquid asset, but hold
They
is
It
premium depends on
minimum
the
holding period of the
increases with the fraction of illiquid assets in the economy, since
this period gets shorter.
There
a growing literature studying asset market frictions such as transactions
is
costs, short sale constraints or
basic questions.
The
first
policy given price processes
question
(ii)
is
to find the optimal
(i)
to derive the asset
to evaluate the cost that
The answer
"equihbrium implications" of market
frictions."
markets with
in particular)
frictions.
to
(ii)
is
The equilibrium determination
it
in this
i.e.
this question to
we take the
is
Finally, the third
to endogenize the financial
be a fundamental one we do not address
financial structure as given.
Most of the work on the equilibrium imphcations
*In
(and the imperfections,
also the question addressed in the present paper.
While we consider
paper
of
the second question raised in the Uterature on market
question addressed by the literature on market frictions
structure*.
for a particular price
sheds some light upon the
frictions, taking the financial structure
as given,
It is
demand
of Uterature
market imperfections impose upon market
participants (given the price process).
prices in
consumption/investment
and imperfections. The objectives of the body
addressing this question are:
process and
borrowing constraints. This literature addresses three
of
market
frictions, considrrs
number of shares.
'See for instance, Constantinides (1986), Davis and
Dumas and Luciano
(1992), Vila
(1991), Fleming et
al.
(1992),
(1989) and Ohashi (1992).
(1990). Duffie and Sun W^S^).
Grossman and Vila
and Zariphopoulou (1990), among many
*See for instance Allen and Gale (1988),
Norman
(1992),
Tuckman and
Vila
others.
Boudoukh and Whitelaw
(1992), Duffie
and Jackson
either a static
framework along the
among
Brennan
others,
and (1981)
and Vila (1992)
equihbrium analysis, and Fremault (1991) and Michaely
for a general
economy where agents hve
models give us usefid
questions
we
Moreover,
for all assets.
It
only two periods (Pagano (1989)).
for
in.
model
to
buy or
thus clear that
is
adequate
for
Although these
answering several of the
In static models, assets are not sold but only liquidated.
(as well as in a
when
agents cannot choose
equilibrium treatment) or an overlapping generations
insights, they are not
are interested
in a static
Goldsmith (1976), Levy (1978) and Mayshar (1979)
(1975),
for a partial
Asset Pricing Model (see
lines of the Capital
two period overlapping generations model),
sell assets,
many
and the holding period
of our results
is
the same
would not appear
in that
simplified framework.
In a context directly related to the present paper,
.\mihud and Mendelson (1986)
consider a dynamic model where investors have different horizons.
investors with, say, an investment horizon of 4 years
actions cost
when buying and
selling assets, will lose
year because of the transactions cost. Hence, they
who
face a
2%
They argue
roundtrip trans-
approximately 2/4% (.5%) per
will require a rate of
return of .5%
higher on illiquid assets than on Hquid assets. Consequently, the liquidity
on assets wliich appeal
to investors
The above reasoning
.5%.
by transactions
costs
that
premium
with a 4 year horizon must be approximately
implies that investors with longer horizons are less affected
and would
select higher yielding illiquid assets.
By
contrast,
investors with shorter horizons select low yield liquid assets. This clientele effect explains the empirical fact that the cross-sectional relation
and
asset returns
is
The
concave.^
analysis above, while insightful, takes investors'
horizons bs given and does not explain
transactions costs.
the liquid asset
costs
on the
Two
is
Moreover
assumed
between transactions costs
how they change
in
response to an increase in
since, as in the previous papers, the rate of return
for simplicity to
differentials of rates of return
be
on
fixed, only the effect of trnnsn< lions
and not on
their levels can be examinf-'l.
recent papers, one by Aiyagari and Gertlcr (11)91), and one
l)y
Ileaton and
Lucas (1992) consider dynamic models where investors" horizons are endogenous. In
their models, agents are infinitely lived, face labor
'By contrast,
if all
investors
had
tlie
same horizon
income uncertainty, and trade
this relation
would be
linear.
consumption-smoothing purposes.
assets for
premium puzzle
of return
(see
Mehra and
These papers seek
Prescott (1985)),
i.e.
to solve the equity
to explain the differential rates
between the stock and the bond market. Aiyagari and Gertler (1991) argue
that differential transactions costs between these
the equity
premium. In
the equity
premium
is
model
their
two markets account
(as in ours), the "stock'
due to transactions costs and not to
explains the fraction of the equity
They do not however analyze
premium which
is
is
riskless
risk.
for part of
and therefore
Hence
their
in fact a liquidity
model
premium.
the effect of transactions costs on the level of rates of
return as they take the rate of return on the liquid asset as given.
By
contrast with
Aiyagari and Gertler (1991), Heaton and Lucas (1992) allow for a truly risky asset
as well as for
aggregate labor income uncertainty. They argue that transactions costs
prevent investors from reducing the variability of their consumption by intertemporal
smoothing thereby raising the equity premium. In addition, they endogenize the rate
of return
on the liquid
asset
and fmd that
it falls
in
response to increased transactions
costs.
model agents save
Wliile in our
income uncerttiinty, our
the above two papers.
premium,
is
fall,
We
fmd
in particular that transactions costs create a liquidity
as in
Heaton and Lucas (1992). The contribution of our work
twofold: First, our closed form anedysis allows us to precisely identify the different
able to easily perform
The remainder
model.
5,
numerical simulation results of
and Gertler (1991), and that they cause the rate of return on
effects of transactions costs
3.
purposes rather than because of labor
results are consistent with the
as in Aiyagari
the Uquid asset to
for life-cycle
We
and interpret various comparative
of the paper
is
statics.
structured as follows: In section
2,
we
describe the
determine asset returns when there are no transactions costs
In section 4,
we
on asset demands and on rates of return. Second, we are
illustrate
we
consider the case where there are transactions costs.
our general results with some numerical examples. Section
concluding remarks and
all
proofs appear in the appendix.
in section
In sf-rtion
fi
contains
The Model
2
To analyze the impact
of transactions costs
on the return on assets and on the
A
premium, we have adapted Blanchard's (1985) model of perpetual youth.
liquidity
simpHfied
exposition of the original model can be found in Blanchard and Fisher (1989).
We
uum
consider a continuous time overlapping generations
of agents with total
mass equal
to
An
1.
probability of death per unit time, A.
In addition,
has a density function equal to \€xp(
lives in this
economy,
their
life
we assume
at a rate
— \i). Although
expectancy
is
that death
equal to
agents can
bounded and equal
inde-
is
Therefore the
A.
and the distribution
to one
mass equal
stationary, witii total
is
a contin-
agent in this economy faces a constant
pendent across agents and that agents are born
population
economy with
of age,
live arbitrary
t,
long
to 1/A.
Agents are born with zero financial wealth and receive an exogenous labor income
yt
over their Lifetimes.
We
assume that
The aggregate labor income
yt
declines exponentially with age
S>0
yt
=
ye-''\
Y
is
constant and equal to
y=
rXe-''y,dt =
Jo
The
financial structure in this
economy
time.
economy
is
-^y
A +
-
A;
(0
<
liquid asset
r
=
(2.2)
given as follows.
total supply of perpetuities
is
All assets in this
D
fc
is
D/p. The
<
1),
The
also the
can be exchanged without transactions costs. The price of the
illiquid asset, in total
R = D/ P^°.
transactions costs:
is
per unit
liquid asset, in total supply
denoted by p and the rate of return on liquid assets
return equal to
D
normalized to one so that
aggregate dividend. There are two such perpetuities.
1
(2.1)
are real perpetuities which pay a constant flow of dividends
The
(
Trading in the
when buying
must pay exP transactions
^°Note that we have defined
costs.
R
net of transactions coa<s depends
supply of
k.
has
a price
illiquid asset
is
equal to
is
F
denoted by
nnd
a
ml'-
'>l
subject to proportional
(or selling) x shares of the illiquid asset the agent
Because of transactions
costs, the rate of return
as the rate of return before transactions costs.
upon the holding period and
is
The
on
rate of return
therefore investor specific.
the illiquid asset and on the liquid asset will be different.
The
liquidity
premium
^i is
defined as
= R-r.
fi
none of these assets can be sold short. ^^
Finally,
Over the course of
stochastic
imposes a financial
it
way:
in the following
We
of
TTf/i
assume that
exchange
costlessly^"'
is
and
fully
and
(Hi)
is
perfectly competitive,
death
is
an idiosyncratic
We
full
a
of assets
For example an
premium
living participant.
We
assume that
(i)
insurance companies transfer assets
risk.
As a
result, the
both the liquid and
do not derive any
since, as previously indicated, agents
purchase
for
(ii)
costlessly insurable
the liquid asset will pay a
of, say,
is
of losing their
pay shares
to collect the share in the event of death.
be equal to the probability of death \dt,
utility
is
on their estate.
for a claim
that insures one share
the insurance market
will
this risk
additional shares of the liquid asset per unit time dt to
compensation
Its
Since death
upon the agents namely that
risk
there exist insurance companies which
to the Uving participants in
company
accumulate both assets.
their lives, agents
accumulated holdings. ^^
insurance
(2.3)
premium
must
illiquid asset. Finally
from
utility
Trdt
their estate they
insurance.
assume that agents maximize
time
at
function of their consumption
the expected value of a time separable
i.e.
u{ct)te-^Ut
/
(2.4)
.Jo
Since the only uncertainty comes from the possibility of death
^^If short sales
asset instead.
6,
which
is
were costless agents would not
Our
results
do not change
a reasonable assumption (see
and Vila (1992)
for evidence
if
sell
write equa-
the illiquid asset but would short the liquid
we assume
for instance
we can
that the
of short spUing
co<:t
Boudouiih and Whitelaw
(
\.^^l)
is
liielcr \\\n\\
anH Tn. kmmi
on short sale costs).
^^Agents do not leave any heir behind and care onlv about themselves.
^^The introduction of insurance companies
is
in the introduction suggests that our results
generations model with deterministic death.
a convenient way to close the model. Our discussion
would carry through
The
latter
is
in
much more
a multi-period overlapping
difficult to solve analytically.
as"
tion 2.4
Jo
We
also
assume that the
tution equal to 1/.4
function exhibits a constant elasticity of substi-
utility
i.e.
"(c)
In this
paper we focus on the stationary equilibria of
equilibrium, the rates of return
determination of
J, A.
Y
= ^-^c^-^i^
r,
R
and
//
r
and
R
are constant.
as functions of the
and D.
^<See Blanchatd
'^The case
A=l
and Fisher (1989)
for details.
corresponds to u{c)
—
logc.
10
(2.6)
this
economy. In a stationary
We
seek to understand the
parameters of the model:
e.
k. A, S,
The No Transactions Costs Case
3
In this section,
when
case
we analyze
the determination of the interest rate in the benchmark
transactions costs are equal to zero.
between the liquid asset and the
illiquid asset: r
= R and
/i
—
is
no difference
0.
The consumer's problem
3.1
The
financial wealth Wt of the consumer at date
consumer's assets. That
date
is
if j-,
is
the
number
defined as the value of the
is
t
consumer owns
of shares that the
at
t
Wt
At date
t,
Since he consumes
dwt
=
Dit
entails
(.3.1)
Dxtdt + Xpxtdt
From equations
in
2.5
dividend income plus
and
>
3.2,
+
{yt
0;
-
Wo
=
Ct)dt
=
(r
+
\)wtdt
>
Wt
0;
-\.rt
His financial
shares worth XpXf.
/5
+
(t/(
-
Ct)dt
0.
the consumer's problem
consumer with discount factor
of an infinitely lived
per unit time.
t/t
per unit time the dynamics of his wealth are
Ct
Ct
rate.
= pxf
the consumer receives a labor income
income (per unit time)
the
In this case there
(.3.2)
is
the optimization problem
+ A who
faces a constant interest
This constant interest rate equals the rate of return on the perpetuity,
premium paid by
r, pltts
the insurance company, A. Hence the consumer's problem can
be written as
r
max
»(c,)e-<'^+^'Vt
= max
Jo
s.t.
Jo
r Qe-''+''Vf = max H
Jo
The problem
^*To calculate
6
>w =
(/3
-
3.3
Hr*
1
—L-r'-'',-'''^'^'dt
I
- A
y,c''''^^"df:
«>
>
0.
Jo
above admits the following solutiou^^
this solution
we have assumed that the borrowing constraint
r)/A, and that the maiimuin in 3.3
hold in equilibrium. (See appendix
A
is finite, i.e.
for details.)
11
V'
=
''
+
'^
is
+ '«'>0.
not binding,
Both
i.e.
restrictions
.
= y^e-'
ct
(3.4)
with
(i)
=
r
+
A
4- 6
=
r
+
\
+
and
lb
Filially,
ijj.
the consumer's financial wealth at date
-
y
equals
—
e— _
p-^«
<
U-,
t
o
(3.5)
Equilibrium
3.2
In equilibrium the aggregate financial wealth
Xe-^'wtdt
(3.6)
Jo
/o
equals the market value of the perpetuities
p
i.e.
=
D/r.
Using equation 3.5 we can show that the equilibrium interest rate solves the equation
Illtl^
0*(\
where
r*, w*, 0*
and
i/'*
+
up with the discount factor
of aggregate financial
goes
down with
£
(3.7)
Y
denote the equilibrium values or
Equation 3.7 determines the interest rate
rate goes
=
u}')
r*
r. uj
uniquely.
.
6 and
d',
As expected the
3, the prol)ahility of d^aHi
A
greater incentives to save. Finally
goes up the interest rate goes
if
<*'.
interest
and thr
income over aggregate labor income D/Y. The
the rate of decline of labor income.
respectively.
interest
since an increase in
(*'
r;iii>>
ratp
leads to
the coefficient of elasticity of substitution, l/.l,
down provided
the interest rate goes up.
12
that
r*
be greater than 3. Otherwise
In equilibrium, agents use the financial markets to
their lifetime: they
buy
assets
when they
are
smooth
their
young and begin
consumption over
selling assets at
age r*
where t* solves
r'xvr'
From
3.5, r*
is
+yr- -Cr' =
=
u!'
T=
r
volume
dollar
\e-"\rtvt
Jo
^
(3.8)
given by
r'
The aggregate
0.^^
+ yt-
-^ogi—--].
-
in this
c,\dt
=
\ S
6
+
(3.9)
X
economy equals
=
2Auv.e-'^'
27—^e-<'+'-*'^'.
Note that we do not consider the payment of shares by insurance companies to be a
although the agent's portfolio
may
still
portfolio
grows at a rate lower than A
amounts
to defining
are different.
assets as
(3.10)
(j)*
Since
who
is
.
be growing (dW, >
opposed to cash and that
the agent
is
considered a spIIt
Ilein-f
if liis
In the absence of transactions costs, this eissumption simplv
called a seller
we have assumed
0).
traile.
and who
is
not.
With transactions
costs,
however, matters
that insurance companies pay living participants shares of
this transfer
is
13
costless, our definition of
a
seller is
the correct
Transactions Costs and Assets' Returns
4
In this section,
we determine
on the
illiquid asset,
costs.
We
the rate of return on the liquid asset,
R, and the liquidity premium
/(
+ (6-
r*
R{()
=
;/(t)
where
b
and m' are the
first
we
m*)f.
+b€ +
r'
= m'e +
o(e)
4-
o{e)
o(()
consider the case where the supply of the illiquid asset.
The
somewhat
different
and
is
first
write
order equilibrium effects that
assets are available to consumers.
presence of transactions
in the
order effect on equilibrium variables. For this purpose
=
the rate of return
small transactions costs and focus on their
will consider the case of
r(f)
r,
case where
we
is
/.-,
seek to calculate.
less
than one so both
assets are illiquid,
all
We
i.e.
k
=
1, is
studied in appendix E.
Before proceeding with the formal derivations,
show that
useful to
it is
in equilib-
rium the liquidity premium per unit of transactions costs n/e must be greater than
the rate of return on the liquid asset,
since agents are born without
illiquid asset at
dollar
some point
worth of asset
later. If
r,
or equivalently
he buys the liquid asset
rds for
his
s
shares.
Hence
illiquid asset,
+
consider an agent
(
and
date
at
t
date
and
t
+
t
is
because,
must buy the
who buys
for
one
At periods
sells it
+ At and
At.
given the transactions costs he
-1 at
(1
This
t
his cash flows are
R/{1 + €)d3
e).
cash flows are
between
+1
he buys the
Now
inclusive of transactions costs at date
-1 at
If
r(i
assets, in equilibrium they
any financial
in their lives.
R>
for s
date
between
-e)/(l +
e) at
14
t
t
and
date
t
+
t
4-
At and
At.
will get
I
[F(
I
I-
'
l|
UR <
r(l
buying the
-I-
e), i.e.
<
if /i
illiquid asset.
then buying the liquid asset always dominates
re,
Hence
/«
means that the
In particular, this
mium
is
prices,
IX
>
at least
a
first
liquidity pre-
a priori information about equilibrium
this
demand
investor's
on the
and
for liquid
illiquid assets
when
The consumer's problem
With transactions
consumer's financial wealth Wt
costs, the
Uquid portfolio, denoted
by At. Denoting by
and of the value of
l)y 0(,
(respectively
it
It)
asset (respectively illiquid asset), the
=
dat
dAt
Ct
From
4.2 above,
=
=
yt
=
itdt;
oq
\Atdt
+
Itdt;
Ao =
+ RAt -
it
-It-
see that the agent's
income
purchases of illiquid assets
With transactions
dynamics of
+
rat
It,
rat
is
the
sum
of the value of
denoted
his illiquid portfolio,
the incremental dollar investment in the liquid
Xotdt
-I-
we can
plus the dividend
i/t,
(4.1)
re.
4.1
his
re.i»
effect of transactions costs
With
order effect.
we next characterize the
>
Oj
and At are given by
0;
at
^
>
Aj
0;
e\Ii\-
Ct
(4.2)
>
0.
consumption equals the labor income
+ RAt, minus
purchases of liquid assets
minus transactions costs
it,
minus
e\It\-
the consumer's problem becomes far more complex.
costs,
Proposition 4.1 (proven in appendix B) describes the optimal policy of the consumer
for small transactions costs
i.e.
and
for a subset of values of r
and
R
that are of
intf-r'^st,
such that their equilibrium values belong to this subset.
Proposition 4.1
possible values,
infinity,
that
is
small and for
e
r
and
R
the optimal policy has the following
illiquid asset until
^'Tliis lower
For
bound
an age
is
when the
t^
.
He
belonging to a svbsft nf llnir
form:
then buys the liquid asset.
The consumer buys
He
next sells the liquid
reached asymptotically when the holding period of
fraction of illiquid assets, k, goes to zero.
15
the
illiquid assets
goes to
an age
asset until
He then
asset.
We find
Ti
+ A. At
age
start selling the illiquid asset until he dies.
that in equilibrium, agents will buy high yield illiquid assets for long-term
investment and low yield liquid assets
result
for the illiquid asset are the
asset are the agents of age
marginal investor
at
date
rj
-*-
is
A. As
premium
liquidity
Amihud and Mendelson
fairly intuitive
The
(1986).
clientele
agents of age less that
tj
while the clientele for the liquid
and the age
at
which they begin
between
Tj
who buys
the investor
in
short-term investment. This
for
consistent with the analysis of
is
he does not oxen any share of the liquid
+ A,
rj
the illiquid asset at date
Amihud and Mendelson
to sell
r^
The
it.
and
sells it
(1986), this investor determines the
(see below).
The Hquid and
illi([uid
portfolios as function of age
t
are plotted in figure
1.
Proposition 4.1 presents the qualitative properties of the optimal consumption/investment
sumption
how
what
In
policy.
at
date
follows, these qualitative properties will allow us to calculate con(, C(,
as a function of the initial
cq-
We
will also
show
the intertemporal budget equation, properly modified to account for transactions
consumption
costs, leads to the determination of the initial
how
the parameter
A
premium,
the sake of the presentation,
Over the course
of his
all
life
//,
and the
which
Indeed, consider a consumer
wants to have the same wealth
at
t
between
+
dt,
ie e^"".
and
show
will
level of transactions costs,
e.
For
relevant for the consumption-savings
is
is
^
=
+
1
5
we
the agent faces three interest rates.
Rl +
At
Finally,
technical details have been sent to appendix B.
First until age ri, the interest rate
decision
cq.
can be easily calculated as function of the rate of return on the
liquid asset, r, the liquidity
t
consumption,
and
t
t-rdf. he
consume
after
t
at
f
"^
Therefore, for
earlier rather
decide? to ronsum*" ^1
[O.r,]
+ dt. He then buys
he consumes the proceeds from avoiding
dt.
A.
e
l,'F(l4-e) illiquid
consumes the extra dividend flows
Hence by foregoing $1
t -\-
who
-I-
than
R
at
<
he gets $1
+
to
\dt
(£)
buy (I/P(l
+ (R/([ ^
Tf
+
e))dt
l'"--'^.
'"'l
s;rriiriii'-s.
l-i-''))^^'*
"" and
())e^'^' securities,
+
o(dt)
between
given, higher transactions costs increase the desire to
later.
The reason
16
is
that the consumer has to buy an
which
asset
is
more expensive, but pays the same dividend.
Second, between ages
Tj
and
therefore faces the interest rate r
Third and
illiquid assets,
finally,
+ A,
tj
+
the consumer invests in the liquid assets and
A.
+ A, when
after age rj
the consumer
\
-^
+
—
=
1
Indeed, suppose that at
between
5
(it
t
-\-
dt
t
same wealth
and
t
4- dt,
+
dt.
f
G
after
+ A,oo)
[ti
t
+
He
dt.
A.
e
he decides to consume $1
sells
1/^(1 —
f.)
For
R
$1 at
t
he gets $1
+
\dt
+
(/?/(!
-
{D/P{1 —
ejje"^*^'
The reason
())dt
that the consumer has to
is
but wants
+
sell
e))e'*'''~''
securities
o(dt)
given, higher transactions costs increase the desire to
rather than earlier.
less
less illiquid securities.
he consumes the extra dividend flow
he consumes the proceeds from selUng (1/P(I —
Hence by foregoing
t
divesting out of the
he faces a higher rate
Rb +
to have the
is
i.e.
between
consume
At
and
c^"^'.
t
and
later
a cheaper asset that
pays the same dividend.
We
denote by p{t) the interest rate relevant
p(t)
p{t)
=
pit)
and by
r
= Ri +
\ (oT
+
Ti
A for
t
<
date
p(t)
we indeed show
=
I
Jo
t i.e.
< n
<
Ti
+A
= Rb + Mot n + A <
and date
p(t) the discount rate betrveen date
In appendix B,
<
for
t
t i.e.
p{s)d3.
that the optimal consumption must satisfy
Q = coe-""
(I.:'.)
with
IM{t)
where the consumption
at birth cq
is
=
{i3
+ \)t-p{t)
A
derived from the intertemporal budget constraint
presented below.
17
Given proposition
path
tion
r
must
C{
4.1
and equation
4.2,
can easily
it
1
-I-
shown
tiiat
the consump-
satisfy the iniertemporal budget equation
r
yL-Sie-»(^^dt+ r^^(y, + {R-r)At-c,)e-''^'Ut+
Jo
Ije
Jri
e
^L-iie-^'^'cit
1
Jri+Ci.
'n
—
=
(4.4)
€
with
^0
1
+
e
Equation 4.4 says that the Net Present Value of lifetime savings net of transactions
costs
must equal
where we define savings
zero,
minus what must be reinvested
Between periods
lO, Ty\
and
and therefore savings equal
[rj
yt
in
+ A.
as total
income minus consumption
order for financial wealth to grow at the rate
oc
[,
this latter
~ Cf Between
Tj
f>{t).
quantity equals the dividend income
and
t^
+ A,
only a fraction rAt of the
dividends from the illiquid portfolio must be reinvested and thus savings equal labor
income,
minus consumption
yt,
plus excess dividends
c,
{R —
r)At.
Finally savings
are adjusted for transactions costs.
We now
show how the minimum holding period
calculated as a function of
A
are optimally chosen, this
illiquid asset
his
r,
change
and not doing
in utility
if
/.i
and
e.
Consider a consumer
consumer must be
so.
of the illiquid asset,
Given that he
indifferent
at age tj.
Since
between investing
u'(c^)(l
+
e)P
he buys one unit of the
+
u'(c..+^)(l
-
ti
e)Pe-''^
and
in the
starts selling the illiquid asset at r^
+A
illiquid asset at t^ is
rn + A
-
A, can be
+ r^""
u'{ct)De-^^'-^^dt
=
0.
(4.5)
Jri
'n
Use of equation
4.3
and simple algebra show that the above equation can be
rewritten as
!L^r'^'"\
e
Equation 4.6 shows that the
is
decreasing in
its
1
(4.B.
- e-^^
minimum
holding period of the illiquid asset, A.
excess rate of return over the liquid asset.
/(,
and increasing
in
transactions costs, e}^
^®We can
also derive equation 4.6 by noting that
18
between
n and n + A
the consumer invests
From equation
optimal choice of
4.6
Tj
it
follows that the intertemporal
and A,
budget constraint,
Net Present Value of consumption equals
states that the
the Net Present Value of income where the discount factor
Hiyt The
initial
consumption,
ct)€-''^'Ut
=
p(t), i.e.
is
0.
(4.7)
can be derived from equations 4.3 and 4.7.
Cq
Having characterized the solution to the consumer's problem we turn
librium determination of
r
an
for
to the equi-
and R.
Equilibrium
4.2
In equilibrium, the dollar
demands
and
for liquid
illiquid assets
Jo
and:
"
\e-^'Atdt
I
/o
equal the assets' market value, (1
As we stated
in the beginning of this section,
costs (small values of
we have defined
— k)D/r and kD/R
b
t),
and
= r*+(6-m*)e +
=
;((f)
tha.t r*, u;*. 4>*
and
i'*
We cdso define by T\{t)
e,
on
R
r,
and
/(.
Recall that
r'
=
+
be
7Tj*e
+
+
o(f)
and A(e)
(4.8)
(4.9)
oif)
(I.
o(€)
are the equilibrium values of
r.
^. o and v for
as the equilibrium values of t^
and by r* and A* the respective
in the liquid asset.
consider small transactions
and m* by
R{e)
of
will
find their first order effects
r(e)
and
we
respectively.
limits of ri(e)
and A(f
)
as
e
and
<=
-
KM
0.
A as a function
goes to zero. (Note
Therefore the Net Present Value rule applies, and the Net Present Value of
investing in the illiquid asset between these dates
is
19
zero.
that
when
e
equals zero, the liquid asset and the illiquid asset are the same asset and
therefore the holdings at
and At are not well
defined.) In other terms, r*
and A* are
the zero-th order effects of transactions costs on holding periods.
The next proposition
Proposition 4.2
of equations ^.S,
costs
on
J,-9
the liquidity
There exists an equilibrium where
and
and
premium,
return on the illiquid asset,
b
b.
r,
R
and
/t
b.
have the form
In equilibrium the first order effect of transactions
Jf.lO.
of return on the liquid asset,
The
m
characterizes the equilibrium values of
rn'
is
,
— m*,
positive while the first order effect on the rate
is
negative.
The
order effect on the rate of
first
has an ambiguous sign.
rigorous derivations o[ b
—
m\
and m*.
b
as well as explicit
formulae are
presented in appendix C.
Proposition 4.2, that we discuss in detail next, states that transactions costs
decrease the rate of return on the liquid asset but have an ambiguous effect on the
rate of return on the illiquid asset.
We
discuss the results of proposition 4.2 in subsections 4.2.1., 4.2.2. and
subsection 4.2.1,
we
of transactions costs
characterize the parameters r* and
on optimal consumption/savings
A*
we
in the
discuss the determination of the rates of return (or,
In
the zero-th order effect
policies. In subsection 4.2.2
go over the determination of the liquidity premium, and
4.2.3
i.e.
4. 2. .3.
we
rather long subsection
more
accurately, the
first
order effects of transactions costs on these variables.)
Optimal Switching Times
4.2.1
The age
at
which agents switch from the
illiquid asset to the liquid asset, r*
age at which they start selling the illiquid asset, r*
from the
with
6
=
limit case as
and
= R —
r
and
of the liquid
e
=
A*, can he easily interprftpd
goes to zero. Indeed consider the accumulation pfination^
r'
.
Given the investors
illiquid portfolios
At
At
-t-
and the
to,;e^<'-"'''
=
and
Wt
a,
total wealth «>
=
"f
+
<
t*
+ A*
-4,.
over time are given by
and
=
xvt
at
-
—
{or
t
< tau[
uv;f'''-"i'' for r^
<
t
l.J
the vahu-s
At
It
=
and
ivt
follows that the values of r^
financial wealth Wt
=
Oj
(ii)
+ A' <
t.
and A* can be calculated by noting
grows by fxp(AA*) between r* and t'
UV; + A-
and
for r*
=
+ A*
(i)
that total
i.e.
uv;e^^*
(4.11)
that aggregate liquid financicd wealth must equal the supply of hquid assets
i.e.
\e-''atdt=
/
Using the expression
Ae-''(u-, -ti;..e'"-^''Hi
above we obtain the values of
4.2.2
Liquidity
In appendix
premium,
from equation
for w,
r,*
C we show
different
that the
first
by
first
''
^^-^^^
-
from
its
value in equation 4.13.
demand
in
(i.e. if /t
were different
in the first
A. Therefore, there would a zero-th
for liquid versus illiquid assets.
(Although there would
order change in total asset demand.)
The reasoning
(and
1 + e-'"^*
= \.e-'^'
Rates of Return
4.2.3
b
order effect of transactions costs on the liquidity
would be a zero-th order change
order change in the
only be a
and 4.12
easy to understand the determination of m*. Equation 4.6 imphes that
It is fairly
order), there
3.5 as well as equations 4.11
Premium
in, is given
m* were
(4.12)
and A*.
'"*
if
= (l-A.-)--
b
— m*) we
r transactions
by m*€,
for the rates of return
for
m*
will
make
costs increase.
is
more
involved.
the following exercise:
To determine the parninf*fr
We
will
assume that
In order to preserve equilibrium,
given by equation 4.13.
We
21
will
then find by
R
for fixed
has to increase
how much
total asset
demand and supply change
demand
states that total asset
/~
+
Xe-^'iat
in the first
At)Jt
the equation that
r
=
\e-^'wtdt
=
- k)D/r + kD/R.
(1
(4.14)
Jo
Since this exercise
asset decreases
useful for understanding
is
when
To determine
why
transactions costs increase,
total asset
path of the agent
we
— m' by
b
equals total asset supply:
Jo
that
order and infer
demand, we must
the rate of return on the liquid
we go over
first
it
in
some
detail.
understand how the consumption
modified by the change in transactions costs and asset returns
is
are considering.
Lemma
4.3
(proven in appendix C) gives us the consumption
of the agent at age 0.
Lemma
Th^ cou sumption at date zero,
4.3
co
where
Cw
is
Cq is given by
— + eCw + tC, +
w'
=
y
o{e)
(4.1-5)
0'
given by
+ (77 ^
1
/"
)
^j(^
-^
^^^'"'^
-
(^
-^
u;-)e-'"']dij
(4.16)
or alternatively
ytl l{m' and Cs
is
(e--^*'
-
e-**')rft
+ (m* +
r')
H
{e""''
-
e-''')dt\
Jr-j.^'
Jo
\
4>*
/''
r*)
(4.17)
J
given by
Cs =
-\y—\(m'
The terms
A
<p*
Cw
-
\
r') f^' e-'''dt
Jo
+ {m'
4.r')
r
and Cs have a very intuitive interpretation.
interpreted as a wealth effect. For this
we need
to note that
!(A-^6)e-*'' -(A+u;*W-"^-"'|
'
0^
22
\i.\X)
e-'',It].
/r^+A-
/
First,
Civ
'"a.n
l)e
is
the present value of the dollar
case
=
e
amount
The consumer when buying
0.
of transactions
the illiquid asset between
When
transactions costs but pays a lower price.
is
we can
positive as
is
more attractive from
to rj.
A
e)
>
r,
Thus
(Agents buy the
term
this
is
Having interpreted the expression
investment opportunities,
illiquid asset
(i.e.
we can
for cq,
is
also
e
=
plus transactions costs), the
buys more of the
illiquid asset
4.4 (proven in appendix C)
4.4
Total asset
6
where W\v and
Ws
W^y
=
demand
-u;*
is
attractive from
how the
and holds
life
it
is
the wealth effect.)
more than compensates
it
are lower (lowpr price
for a longer period.
(he saves more) and he
the substitution
is
we determine
eir,v
+
eH',
e-»*+^'^.-)
o'r*
=
con-
total asset
In other
sells it at
a
effect.
demand.
given hy
r+
Ay
2-^(e-(--+^)^i -
e
saving
also clear that
It is
+
(4.l!l)
o{€)
are given by
^"Compared to the case
r.
paying transaction
more
up uniformly. (This
lower rate (he defers consumption for later). This
Lemma
this
consumer changes the slope of the consumption path
words, he buys more in the beginning of his
lemma
and
Because the consumer has better
0.
available at a lower price (which
is
+ €) >
briefly describe
transactions costs), and because the proceeds from selling
In
0,
kept
is
he has the liquid asset at the same price as before, and
the illiquid asset), his consumption path goes
so that he
=
€
t/'*,
negative.
sumption path changes compared to the case
illiquid asset
to the case
as a substitution effect. Since i?/(l
therefore deferring consumption for later
until death.
Because the
compared
which more than compensates them.)
costs but at a lower price
R/(l —
off
A
see in expression 4.17.
The term Cs can be interpreted
Tj 4-
+
ri
The term
shows. Clearly, since the rate of return in the liquid asset
consumer can only be better
constant, the
is
pays the
ti
seUiug the illiquid asset (from
price.
in the
equal to the present discounted value of these "extra''^° cash flows, times
as expression 4.16
term
and
he pays the transactions costs and receives a lower
until he dies),
Cw
between r and r + dr
_
!!!_
r'
0.
23
/"
Jo
\e-''Atdt +
(
\+u>j')(p'r* \
Jr^ + A'
Jo
/
(i.20)
and
A (A + a:'}(f}'r'
Jr^+A-
Jo
\
I
(4.21)
The term
\V\v represents the additional
the consumer
path
(i.e.
if
if
additional
wealth
for
(in the first order) of
the latter changes only the level but not the slope of his consumption
the wealth effect
change
to the
demand
is
in transaction
demand
present, but not the substitution effect) in response
costs
for (dollar)
and asset returns that we are studying.
This
wealth has an ambiguous sign because, on the one
hand, higher future consumption to be financed and transactions costs to be paid
when
more wealth, but on the other hand
selling assets require
illiquid assets are
cheaper.
The term
Ws
corresponds to the substitution
effect:
Indeed as
it
was said before
the consumer changes the slope of his consumption path so that he buys
illiquid asset
This term
is
and holds
it
positive
and
for a longer period.
its
more
of the
This implies more wefdth accumulation.
magnitude depends on the
elasticity of intertemporal
substitution.
Finally total asset supply (in the
€
The
difference
between
always positive.
is
discussion.
asset
/
It
total asset
is
order)
Xe-^'Atdt
decreases since the illiquid asset
It
and
—
first
is
is:
=
tW,^„iy
cheaper.
demand and supply
easy to understand
why
this
Higher future consumption to be financed by
is
is
may
\V\v
+ IK +
so. liased
selling the
and paying transactions costs requires a larger niiwber oi
(Although the dollar amount
(4.22)
\\ .„,.,,;„
on our
cheaper
securities to
'^arlif-r
illiqnirl
l)e
held.
be lower.) In addition, the agents change the slope
24
consumption paths
of their
longer period,
The
e
is
then easily deduced, and
discussion which explained
and that the rate
Then,
why
is
higher.
the rate of return on the liquid asset
on the hquid asset stays the same
of return
(in equilibrium) the rate of
they buy more of the
Agents
will
illiquid asset
demand more
thus
to finance higher future
it
it
securities for
up because there
their slope will
change so that
for a longer period (substitution effect).
two reasons.
First,
cheaper
selling the
because they have
illiquid asset
and paying
illiquid asset
and
for a longer period.
Although the
liquid asset
first
order effect of transactions costs on the rate of return on the
unambiguous
is
(6
of return on the illiquid asset
replicate
(more
costs increase,
above
and hold
consumption by
and
effect),
Second, because they want to buy more of the
transaction costs.
in equilibrium.
return on the illiquid asset must increase (in the
more investment opportunities (wealth
hold
for a
negative.
order) by m*t. Agents' consumption paths will shift uniformly
first
are
— m*
demand and supply even
asset
it
can be summarized as follows: Suppose that transactions costs increase from
falls,
to
order to buy more of the illiquid asset and hold
making the imbalance between
value of 6
The above
in
briefly) the
R
stays the
(r decreases
— m*
(i.e.
is
negative), the
the sign of 6)
is
first
ambiguous. In what follows, we
above exercise, assuming that
same and
order effect on the rate
this time, as transactions
r decreases in the first order, as
determined
by m't).
This time, agents face worse investment opportunities.
The
price of the liquid
asset increcises while trading the illiquid asset entails transactions costs. This (wealth)
effect implies
then that their consumption paths
hand, by substitution, agents accumulate
asset for a longer period.
The
effect
on the
shift
Indeed, the future consumption to be financed
demand
is
uniformly.
for securities
less of
the liquid and illiquid assets.
period.
25
the other
is
Ijiit
illiquid
ambiguous.
lower and the liquid assri
expensive, but on the other hand transactions costs have to be paid.
agents buy
On
both assets but hold the
less of
total
down
is
m<>rf
In a/ldilion.
hold the iUiquid asset lor
a
Iookt
Comparative
4.3
In this subsection
Statics
we study how the
depend on the parameters
6
— m' depend on
k,
effects of transactions costs
(More
of the model.
these parameters.)
we
precisely,
The parameter
that
is
the fraction of illiquid assets to the total stock of assets. In
appendix D) we examine how
Lemma
in k is
We
4.5 m* increases
,
b
and
in k, b
—
b
depend on
»n*
— m*
find
assets' returns
how m'.
b
and
of greatest interest
lemma
is
4.5 (proven in
k.
decreases in k while (he dependence of b
ambignous.
briefly discuss the results of this
The dependence
shorter.
The
m* on
of
economy imply
in the
ni*
on
A-
that the
liquidity
Lemma.
relatively simple to understand.
is
minimum
premium must
More
iUiquid assets
holding period of an illiquid asset becomes
increase so that consumers are wilUng to hold
illiquid assets for shorter periods.
The dependence
of 6
— m* on
k can be explained in the light of the analysis of the
was argued that to understand why the rate of return
previous subsection. There
it
on the liquid asset
response to increased transactions costs,
falls in
the following experiment:
to
e
could suppose that transactions costs increased from
and that the rate of return on the
order) by ni*e.
total
We
We
we could make
illiquid asset
had
could then study the difference between
to increase (in the first
demand and supply
of
wealth and infer the direction of change of the rate of return on the Uquid asset.
In fact,
asset,
we can
also infer the magnitude of
change of the rate of return on the liquid
studying the magnitude of the difference between total asset demand and total
asset supply.
As k increases, m* increases, therefore both the wealth
effect are stronger.
is
greater,
The
and the
effect
and the substitution
This implies that the difference between asset demand and
first
order effect on r
effects of the other
h
—
iv')
is
m\
h
and
6
(i.e.
parameters on
are not reported here.
26
bigger (in absoltitf'
— m'
!='tpply
valnr-i.
are of less intef^st
;iimI
Numerical Examples
5
In this section
we
present
previous sections.
In
all
some numerical examples
these examples
that the level of transactions costs
e
we assume
equals
3%
to illustrate the results of the
.4=1
that
which
figures are consistent
(i)
the Uquidity
down,
(iii)
is
is
positive,
(ii)
when k
When
k
is
lemma
4.5,
is
close to
is
significant
and
plot
These
namely that
the rate of return on the Hquid asset goes
(iv)
the
on the liquidity premium and the rate of return on the
eff"ect
liquid
1.
The main quantitative observations
premium
we
the supply of the illiquid asset.
the rate of return on the illiquid asset can go up or down,
large
liquidity
figures 2 to 4,
all
with the results in proposition 4.2 and
premium
of transactions costs
asset
fc,
logo)
consistent with empirical
is
evidence (see Aiyagari and Gertler (1991) for instance). In
various rates of return as a function of
=
(v{c)
When
1,
the
(about 10% ui the level of the rates of return),
(ii)
are as follows:
(i)
close to 1, transactions costs cause a non-trivial
fall
k
is
close to
in the rate of return
on
the liquid asset wliile the rate of return on the illiquid asset remains almost constant.
These quantitative
results
have important practical appUcations. To understand
the impact of a change in transactions costs in the economy,
how
derstand
assets are differently affected
by
this change.
A
it
is
important to un-
technological change,
such as a reduction in computer cost, can be assumed to reduce transaction costs for
all
assets
and
in
our model corresponds to the case k=l. Our results suggest that
rates of return will not
on one
change much. By contrast, a reduction of transactions costs
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 assets. Finally, a
transaction tax on a significant subset of existing assets (stocks, real estate
lower their value by an amount
analysis
which takes the
less
than suggested by a simple pnrtial
rates of return
on the other assets
27
as given.
..)
will
eq'iilibrinni
Conclusion
6
In this
work we have constructed
imperfect capital market.
push the rate of return on
a fairly simple general equilibrium
Our main
result
illiquid assets
assets goes
down
that while transactions costs tend to
is
upward, there
which tends to lower rates of return. The net
result
is
a general equilibrium effect
is
that the rate of return on liquid
while the rate of return on illiquid assets
may
believe that these results are robust to the specification of
income shocks'^ or taste shocks and
cycle, labor
life
Our model endogenously generates
This clientele effect
ity.
In fact,
(1986).
if
is
model of an
We
go up or down.
the trading motives:
(i)
the preferences.^^
(ii)
clienteles for assets with differential Hquid-
consistent with previous work
we generalized our model
1o allow for
l)y
Amihud and Mendelson
many
assets with different
transactions costs, we would obtain the concave relationship between rates of return
and transactions costs derived by these authors.
we assumed that transactions
In this paper,
costs were a pure destruction of
due to a transaction tax whose proceeds are distributed
resources. If instead, they are
to the agents, the results are similar.
Amihud and Mendelson
(1991b) 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 we do not dispute the
transaction tax will increase the liquidity
that the risk free rate will
premium
significantly, our results suggest
so that the stock price
fall
fact that a small
fall is
likely to
be somewhat
smaller.
This line of research can be pursued in (at least) two directions.
terection between risk and Hquidity
is
not fully understood.
It
First, the in-
would be interesting
to
construct tractable models to analyze the interaction between transactions costs and
risk
and examine
in particular
^^S^e for instance Ainihud et
al.
whether, as
spirit to the
period
is
the
companion
note,
model
herein. This
same
for all assets
has been areued. illiquid markft'^
nr'-
(1992).
^^Tlie treatment of the perpetual vouth
intractable. In a
it
model
for a general ulilitv function
seems to ns analvtirallv
we consider a two-period overlapping generations model
model
and
is
simpler but also
much
as a result the liquidity
le^s rich.
premium
the results are independent of the functioned form of preferences.
28
is
similar in
In particular, the holding
fixed. In this simple
model
more
volatile since investors find
more importantly, very
little is
it
more
costly to absorb liquidity shocks.
known about
Second and
the determination of the level of trans-
actions costs as well as the financial structure created to deal with these transactions
costs.
29
Appendix
A
The No Transactions Costs Case
when
This appendix considers the case
prove that in equilibrium 6 >
equilibrium
We
is
must
uj
and
=
!/'
We
transactions costs are zero.
+ A+u;>0. We
r
will
will first
then prove that the
unique.
calculate the optimal policy which entails solving
first
roo
m ax
r ^(o).-"^+^^/i = max T -J—c'-'^e''"^'''
Jo
Jo
s.t.
1
r
H
= max
cte-'^^'"dt
-
1
dt
.4
y,e-'^^'"Jt;
>
Wt
0.
(
A.l)
Jo
yo
From He and Pages
we know that
(1991),
bounded value
a
for
.3. .3
aliove exists
provided that
=
V'
In that case,
He and Pages show
C(
Ct
If V'
<
0,
=
=
yt for
every
r
+
then the value of 3.3
is
+ u;>0.
(A.2)
that
t
\[ uj
every
{yxl'/(p)e~'^'- for
A
—
t if u;
We
oo.
(3
—
<
^,
A >
S
and
with o
=
r
r) /
show below that
+
X
this
+
6.
cannot be the case
in equilibrium.
In equilibrium, the resource constraint implies that
Jo
Hence the equilibrium utihty
to the
of the representative agent
is
liounded by
thf^
soluiion
program below
/•oo
max
/
Jo
s.t.
r
1
I- A
c
Xe-^'ctdt
Jo
30
e
nt
= D+
Y.
(A.4)
.
easy to show that A. 4 has a finite value.
It is
U
this
>
(jj
is
8
then the agent does not buy any asset and therefore
=
0.
Obviously,
not consistent with the equilibrium condition
\e-^'wtdt
/;
/o
Hence
?(',
8
must be larger than
From the expression
u;
for Ct
and
3.2
follows easily that
it
=
_
^-st
(A.6)
.
y
yields the equilibrium condition
+
u;-)
_D
r'(8-^)
r'{8-^')
which must be solved
(A..5)
equilibrium.
in
e-u,t
<A*(A
—
r
^f^t
Combining A. 5 and A.6
=
(r'
+
X
+
8){\
+
'^)
(A.7)
Y
for r*.
Simple algebraic manipulations show that A.7 above has a unique positive solution
r*, that this solution is increasing in /?,
is
increasing in
A
if
r* is greater
than
/?
A and
D/Y
and that
and decreasing with
31
it is
,4
decreasing in
otherwise.
6. It
B
Proof of Proposition
The method
of proof
as follows:
is
We
4.1
define the control variables.
first
Step
The
1
Recall that
and that
is
indeed optimal.
control problem
the per unit time value of the liquid assets purchased at date
is
it
the per unit time value of the illiquid assets purchased at date
/( is
then
Next we construct a candidate
derive heuristic conditions for an optimal control.
optimal control and show that
We
t.
t
Recall
also equations 4.2
dot
dAt
ct
The
it
and
(ii)
it
constraint
/(
i{t)
= \A,dt +
yt ^
rot
>
Ct
and
It
We
0.
=
=
oq
ItdU
+ RAt -
.4o
it
-
that a control («(),/())
satisfies the
the payoff function,
=
-L i(dt',
It
0:
=
-
is
>
at
0;
i.e.
I(t).
is
maximize
adwissihle
no short sale constraints
denote by C the
i*
>
(B.l)
Q >
e\It\\
to
>
-4,
subject to the above dynamics of at and
we say
Formally,
it
\atdt
control problem faced by the consumer
controls
uous
-
—
0.
2.5 with respect to the
Af
piecewise contin-
if it is (i)
and At >
set of admissible controls
the utility that the consumer enjoys
if
as well as the
and by J( /()./())
he follows the controls
Using the fact that
at=
f i,e'^'-'U3
(B.2)
At= f I.e^^'-'\h
Jo
(B.31
Jo
and
the payoff function can be written as
J(i[)J[))
=
r
u{ijt
+
r /' ;,e'"-"(/.s
Jo
+ R
Jo
f I,c'"-'^ds
-
it
-I,-
c\It\U
"'^'^'df
Jo
(B.4
Hence the control problem
is
to
maximize J( /(),/()) with respect
ing to C.
32
to
(j(
),/()) belong
step
To
2
Heuristic optimality conditions
derive heuristic conditions for an optimal control,
control
(i( ),/()),
denote by
c() the resulting
we take
a candidate optimal
consumption and consider the possible
perturbations:
Suppose
(i)
first
sumption by Q
At
he
s
between
sells ae'^'".
that
>
a,
Suppose that
0.
and
+
t
He does
dt,
t,
the consumer changes his con-
and invests a more
dollars (per unit lime)
t
at
dollars in the liquid asset.
he consumes the extra dividend ore^''"'* and at
(
+
di
not change his investment decision thereafter. His payoff will
change by
a[-»'(c-)+e-"^+^"^'(e^^'4 rdt)u'{r,^,,)].
Since (i(),/())
and
dt
is
optimal, the change in payoff must be non-positive for every a
going to zero which yields the standard Euler equation
^
(ii)
Suppose now that
in the liquid asset
at
=
= (/i-rK(c.).
Then
0.
(B.5)
the consumer can only increase his investment
and therefore condition B.5 must be replaced by
^^<[l3-r)u'{ct).
(iii)
Consider a policy where At
>
for every
t,
(B.6)
which
will
be the case
in
our
candidate policy.
(iiia)
tion
Suppose
by a(l
the illiquid
will
+
/*
>
0. If
e) dollars
Jisset
between
t
and
+ dt,
t
the consumer changes his consump-
and does not change
his
a+
more
dollars in
investment decision thereafter,
his payoff
(per unit time), invests a (with
I*
>
0)
change by
a[-a'(o)(l
which must be non-positive
for
u'{ct){l
+
+
r
Rv'{c,]r-'''-''ds\,lt
any a and so
+ ^)=
r
Ru{c,)e->^^'-'^d3.
33
(B.7)
(iiib)
Suppose now that
consumption by
—
(1
/*
0.
between
If
t
and
t
+
dt the
e)a dollars (per unit time), invests
and does not change
dollars in the illiquid asset
his
<
consumer changes
a (with a +
investment decision
his
<
I'
0)
his
more
thereafter*'^
payoff will change by
a[-ti'(Q)(l
which must be non-positive
-
+ /~
e)
any a and so
for
r
u'{c,)(l-e)=
(iiic) If /,
=
0,
then the
u'{ct)(l
(B.8)
Ru'(c,)e-^'^'-'\ls.
order condition reads as
first
-
Rn{c,)t-'^^'-'^ds\dt
<
I"
Ru'(c,)e-'^^'-"d3
<
u'(ct)(l
+
(B.9)
f).
Step 3 Construction of the candidate policy
Given
Co, t^
A
and
we
define the candidate optimal consumption plan as in equa-
tion 4.3
—
CqC
(0
+ \)t-p{t)
Cf
with
u;{t)
This consumption plan
We now
be completely defined once we specify
will
buy shares
of the illiquid asset
shares of the liquid asset between
assets
from
Tj
+A
to oo.
^' ^'
Jo
-'Note that this perturbation
indeed be positive
for
tj
and
From equations
A
will
.
motivate our definition of these parameters.
plan, the agent will
sell
=
every
is
1
~
+
rj
4.2
optimalitv condition with respect to
/(
is
it
To finance
this
consumption
He
will
buy and then
to ri.
A. Finally he
will sell
the illiquid
follows tliat
it
t
However
the optimality condition with respect to
and A.
'
feasible only if A,
(.
-t-
from
cq, r^
is
tit
=
>
for
for every
<
< n and
/.
<
In the optimal policv,
> n + A. For
.4,
this reason,
written as an Euler equation (B.5 and B.6) while the
an integral condition.
34
for
t
<
Ti
=
At
for Ti
<
i
<
Ti
A..e^('-^'';
=
at
f {y, - c, + RA,)e''^'^-'>^'^ds
+ A and
At
=
/l.,+^e'""-'"^>+'^'
/'
+
h^L5le''(^^->'^')ds;
Jn+ti
for Ti
+ A <
From
1
at
=
e
t.
the above equations together with the transversality condition
=
lim ylfe-"*"
we
—
get
r yiZSi,-Mt)^t = e^^ r ^i^i^e-'^'c/f.
—
+
Vri+A
^0
1
e
I
(B.IO)
€
Finally
an+A =
We
define
first
A
by
r
6"'"'+^^
=
4.6 that
{yt
-Q+
(B.U)
RAt)e-''^'Ut.
is
R-T
^l
e
=
e
-rA
1
+
e
1
-
e-'''^
r
or equivalently
Adding equations B.IO and B.ll
define Co (as a function of ri,
which together with
e,
r
gives the intertemporal budget equation 4.4.
We
and R) from the intertemporal budget equation 4.4
4.6 yields the simpler equation 4.7 below
Jo
/o
Finally
we
We now
T
and
varies
R
define
Ti
show that
by equation B.ll.
this policy
is
well defined and admissible for
belonging to a subset of their equilibrium values.
smoothly with
e, r
and
m
where
m
is
35
defined by
We
e
small and for
will also
show that
it
R=
We
will
+ me.
r
show the following lemma
Lemma
B.l
Consider
r.
m
and m* such
m' / r' >
thai
1
(recall that r*
There exists
equilibrium interest rate when transactions costs are zero).
(q
is
the
such that
if
<
[i)
(ii)
the
—
\r'
r|
<
cq
and
r^
R=
the consumption plan for
proof
for the
collapses to the optimal one for
first
e
>
=
e
0.
is
uniquely defined,
is
C*
well-defined
is
If 6
is
in
=
and admissible.
(c.r.m).
0, this
consumption plan
The
admissible.
admissibility of
by continuity.
will follow
A
note that the equation defining
A
Cq
(C^)
simple.
is
<
-^mt
r
That plan
€-'^
Therefore
7T!|
are infinitely differentinhlc
Proof: The intuition
We
£o
\m' -
eo,
consumption plan defined above, where
Moreover, A.
<
e
-
=
can be written as
—
(B.13)
-.
m+r
(f,r,m) and
in
verifies
0<A<A<A<'X)
for eo sufficiently small.
The equation
'^y
1
giving Cq can be written as
Rl + X +
r
6
„-(Rr+\-*-uj,'lri
Rl +
\ +
+
+
ffs
6
,-(rX\4.^1^
1
r
iJJL
\
+
A -
+ A + ^Z
-(r-l-\J-..-lA
/("b
u.'
-
\
*
^0
(B.14)
with
u;^
=
^
and
36
u;^
=
i^.
From equation B.14,
It is fairly
uniquely defined and
is
we can
straightforward to show that
m
and
for all values of r
is
obvious that Cq
it is
in
>
restrict to
the set defined before, and for
is
C'°° in
(f,r,m,ri).
such that
6
all Ti
[0,oo[,
where
A',
a positive constant.
We now
turn to the equation defining
+ fV
1
Ri + \ +
This equation can be written as
Tj.
-
Co-
flg^A + cB
1-€V°
Ri +
8
Straightforward algebra shows that
\
+
Rb +
^
we can
L.jL
\
+ 8)
^
'
rewrite this equation as
'
y'
(B.16)
where
/(.,
., ., ., .)
is
a C°° function such that /
=
for
e=0 and
||f|</Oin[0,.o]
for all vedues of r,
m
and
Consider equation B.16
_
g-(>+<)n
The function
g{.)
:
t
(restricting €o
Tx
for
e
if
necessary). A'/
A>A>
a positive constant.
=
g-(A+u;)ri
-> e-(*+-^^'
_
g-(r + A+*)(n+A)
-
e-^'^+'^^*
0) that there exists C
>
_
g-(r + A+u;)(r,
+A
(B.17)
)
_
has the following graph (see figure 5)
Therefore this equation has a unique solution
(given
is
such that
Using the implicit function theorem, the fact that
in (O.r*).
rj
n
:^
r*
-
It
is
C aiul r^
easy to show
^
0<A<A<A<oc
\
_
t'
and thp
I
T.
l;u
I
that
||f|< A>in[0..ol
for all values of r, m.
for
e
<
Co
and
and
rj,
we can show
for all values of r
that
we can
and m. Moreover,
37
define a C"" function ri(€,r,m)
it is
easy to show that
2^ < AVin
1
m
uniformly in r and
n <t' -
(restricting cq
n +A>
C/2 and
r*
+
if
is
Indeed when
=
However,
particular meaning.
defined positive limits as
A(0,r.77j)
=
can be seen that
0, all assets are liquid
A(e,r,77?).
Given
wealth (when e=0) grows
+ A,oo[ and
We
between
at a rate A
it
Oj
is
is
m) and A(0.r.n?).
m
is
easy to calculate
the time such that accumulated
and
ri(0.r.777)
>
>
0, .4^
We
0.
to prove that
ri(0.r.777)
cq, t^
* A.
vary smoothly with
It is
in [0,ri], /,
<
in
above statements.
we have
easy to see that
uniformly. Since
n <
Co (restricting
+
r*
again
=
It
show by continuity
It\(=o
+
(/2 and
(q),
In [ri,ri -f A] again, at
easily
>
/(
recall that
RAt +
<
is
weU-defined and A,
we have
admissible,
briefly sketch proofs of the
In [0,ri],
e
=
ni.
To show that
[ri
e
A), one can calculate ri(0,r,77r) directly from
(or
Therefore, the consumption plan
;•,
close to its
seen above ri(e,r,m) and A(e,r,m) have well-
as
m
is
and the switching times do not have any
the no transactions costs case. This value ri(0,r,n!)
e.
m)
Co(e, r,
goes to zero. As seen from B.13, given
e
that.
small.
e
straightforward to interpret the values of ri(0,r,
less
e
it
This implies in particular
necessary).
C/2, for
Summarizing our discussion above,
value. It
[0,co]
I
h >
and
;,
0.
yt
-
ct
5(e,r,7n,f) where
/t|,=o
>
6»
>
g-
in [0. r*
which implies that
.4,
are very close to their
that
af
>
it
it
<
in [r*
>
-CI\.t' ^ CI ^]
in [ri,r*
in [t*
-
C/4]
4-C/4,ri
38
+
A].
f
for
j
=
€=0 and
C/21.
it
follows that for
0.
counterpart?. \\f
<:\\\
>
This implies that a,
In
[ti
+
+
in [ti,Ti
A].
A,oo[, simple calculations using
Jt
i
—
e
show that
^
''-
^^^
'
l-^Ua
+
A
A+u;b
.-u,r.r. .-w^,-u;„(t-(r.+A)l
.
'
'
Rb + X + ^.^'''
.-.„-6t
+ i^'
The continuity argument can be applied
that
It
<
while since 6
0,
—
u^b
\
+
>
tj
>
in a
compact
ioi co small.
It
will
set [r*
+
(12. T] to
be negative
if
T
is
show
large
enough.
Finally
1
-
\
€
Rb +
Rb +
^'B
and similar arguments show that At >
sible.
lemma
This ends the proof of
0.
\
+
S
Therefore the consumption plan
is
admis-
n
B.l.
Step 4 Optimality of the control
Having shown that our candidate optimal control
we
will
show that
it is
is
well defined and admissible,
indeed optimal. For this purpose, we
first
show
B.5-B.9.
Lemma
Proof:
B.2 Our candidate
It is
control satisfies conditions B.5-B.9.
obvious that
in [0,ri]
dlogit'ic)
dt
in [ri,ri
+
=
f3-Rc<1-
A]
dlogu'ict)
and that
in
\t\
+ A, oof
39
that
it
satisfies
dloca'ic,)
dl
Hence B.5-B.6 are
For
t
G
It
t
=
u'{ct)e-^^'-'^J3
follows that B.8
For
satisfied.
+ A,oo[
[ti
r
„
-,
=lS- RB<l3-r.
G [n,n
+
is
r
u'(ct)
e-^^^'-' )j^_
"!(^_ "lQ)(l-f]
R
Rb
satisfied.
A)
/ 1
_
^-'•(n+A-t)
r(ri-lA-n
4-(l-£)-
«'(q)
In addition
1-e <
R -
1
_
e-''^'^'^-"
:
r
From equation B.12 we
-r(r,+A-o
,
+
(1
e)'
<
-
R
^-""^
-rA
i_e-'-
e"
+
(1
'
r
- 0'
R
get that
+ (l-,f-— = !-Ll.
fl
R
(B.18)
^
r
The
-
'
inequalities B.9 follow.
Finally, for
t
€
[0,rij,
we have
1
_
e"— (^'-')
R
,,
,1 _e- rA
because of equation B.18.
This proves equation B.7 and ends our proof of
We now
Lemma
show that the candidate
B.3 The candidaie
Proof: To derive
this
policy
policy
lemma we
is
shall
is
lemma
B.2.
U
indeed optimal.
optimal.
show that no perturbation
policy can be utility enhancing.
40
of the candidate
Consider an alternative policy
We know
8ct
from equations
=
R
Jo
u(ct),
it
+
6ct)
next multiply equation B.19 by
u{c,
+
Ct
+ 6c,.
8it
-
Sit
-
e(|/(
+
8It\
-
\h\).
follows that
<
u{c,)
£ Siy^'-'^ds + rI^ Siy^'-'Us /
-
f 6I,e^^'-'^d3
u(ct
We
induces consumption
Jo
Using the concavity of
(r
+ 8it,It + SIt) which
B.2 and B.3 that
4.2,
j 8i,e^^'-'Us +
T
{it
Sc,)e-^^^^^'ds
<
Jo
6it
+
-
e"'''"'"-^",
u'{ct)
Sit
-
e(|/t
+
SIt\
integrate from
f u(c,)e-<^'+"'rf5
+
K,(t)
-
\It\)]
to
t,
+
(B.19)
.
and
get
Ki{t)
Jo
with
K,{t)
= j'
„'(c.)
(''
/' Sine'^'-'Uh -
Si,) e'^'^^'^'ds
and
Ki(t)
We
= fu'ic,)
(^R
I'
show that when
will
Shc'^'-^'Uh
t
-
SI,
-
f(\It
+
SIt\
-
|/,1)) e-^i^^^^'ds.
goes to infinity, Ki{L) and Ki{t) are asymptotically
non-positive.
Integrating the second term of Ki(t) by parts,
we
get
Jo
V
Jo
Jo
io
Jo
ds
Therefore, A',(t) equals
I^L\c,){r-^)+'^^^yi^Si,e-'\th)e-''ds-(f\
(B.20)
We
note that
Sa,e-^'
=
[' Sinc-^'^dh.
Jo
41
If a,
>
then condition B.5 holds and the integrand in the
0,
above must be zero.
—
If a,
0,
then the short sale constraint 6a,
B.6 ensure that this integrand
Sat
must
positive, for
large
be greater or equal than
also
t
large enough.
We
For
non-positive.
is
>
t
first
>
+ A,
Tj
term of B.20
and condition
tze
=
and thus
This implies that the second term
0.
have therefore proven that K,(t)
is
is
non-
non-positive for
t.
Consider
now
K[{t). Integrating by parts ihf
Jo
^-{J^
>
0, |/,
+
we
get
u\c,)€-^''dh)^'
-^
i,'{ci,)e-'''dh)6I,e-''ds.
J\f^
equal to
is
If /,
term,
Jo
6he-''dh){l^
Therefore K[{t)
first
—
RiT u'(c,)e->^'ds)(f SI,c-^'ds).
>
81,, the
+
«)
\I,\
>
—61,, the integrand
(-u'{c,){l
-
f)
6I,\
\I,\
('-u'(cj(l
+
integrand in B.21
is
(B.21)
less or
/"' Ru'{cH)e-'^^''-'^dh^ 61,
equal to
=
by condition B.7.
If /,
<
0, \I,
+
6I,\
—
+
is
less or
/~ i?u'(c/.)e-^"—
equal to
61,
=
Ru'(c, )c-'*''-'^dh^6I,
<
'.//i)
by condition B.8.
If /,
=
and 61, >
0,
the integrand
(-«'(c,)(l
+
€)
+ !"
is
by condition B.9.
If /,
=
and
61,
<
0,
the integrand
(^-u'(c,)(l
-
+
is
/~ i?u'(c/.)e"'-'"'"''^/')
42
^I,
<
by condition B.9.
Therefore the
first
term
in K[{t)
is
less or
equal to
0.
For the second term, note that
Jo
and that
1-e
Jt
for
^
>
Ti
it
R
B.
+ A.
Since 8 At
small,
_,/Ml-e
is
> -At,
{SAt
+
At
>
0),
At
--
+
>
t;
>
for e
follows that A';(0
is
asymp-
erpi-u^Bt) and A
u^-b
clear that the term:
-/2(^" u'{c,)e-^'ds){f
SI,e-''ds)
'0
is
smaller that an arbitrary ^
totically
non
>
for
i
large enough.
positive. This concludes the proof of
43
It
lemma
B.3.
D
C
Proofs of Proposition 4.2 and
Lemmas
and
4.3
4.4
Lemma
Proof of
C.l
Suppose that
r
=
r'
and that
4.3
R =
consumption
small, the optimal
at
+
r'
From appendix
Tn*e.
time
B,
we know
that for
given by equation 4.7, which as
is
e
we have
seen can be rewritten as
^
^
^-(fi,.4X4-6)r,
y
p-(R[,+ \ + WL
1
Ri r A f
The
first
)Ti
1
in
iZ£,
_(fij,+A4-#)rif
„-(r*-H+u-)A
r'
u;r,
term
g
^
+
+
A
+
]
_
^-(r*4-\4-u;)A
+
i?B
B
a-
A
+
u,'g
brackets (divided by y) can be written as
A
+
^
r'
+
A
+
6
iZi:,
+
A
+
6
Straightforward algebra shows that this term can be written as
1
1
_ ^nLzJl, ^ "'•--\e-^--r _ ^'+-\e-^-.n-^A-) +
(C.2)
o(.)
or equivalently, as
J_(l_c(Tn.*-r*)
Similarly, the second
P
e -*•'</«
-e(m*
4-
r-)
/*
c"**'-/*
+
o(6)
.
)
(CI)
term can be written as
l-U-t{i-^)({m'
-r')
p
e-'^'*'(/^
+
(n7*
+ r-)/" ,^~^''*A + ''^A
(C.4)
Combining C.3 and C.4 we
get equation 4.15
44
i.e.
4''
=
Co
y— + eCw
+
eC,
+
o{e)
<t>
with C\v and
Cs given by
4.17 and 4.18. Integrating 4.17 hy parts
('H'-r')
-(X+w*)f
(e
_
we
get
g-(A+*)(
JO
JQ
r*
j
-rV
+ (m* +
(e
r':
-(A + u-*)(
_
-(A+«)f^
Tj'+A-
+ /~
—
((A
+ 6)e-''+*''-(A +
Using equations 4.11 (substituting
in
from
it'f
brackets cancel out which yields equation
and
3.5)
4. 16.
u;-)e-''^'^*^'Wf)
4.1.3,
we
•
find that the
This concludes the proof of
terms
lemma
4.3.
Proof of Proposition
C.2
Consider r* and
4.2
m* where
7V
=
1
r
1
The
results of
Appendix B imply that
+
-
e
-r'\'
e-^-^*
for
e
sufficienty small
and
for r
and
m
be-
longing to a neighborhood of r* and m*, the consumption plan defined in Proposition
4.1
is
indeed well-defined, admissible and optimal. These results also imply that
h
is
'0
C°° in these variables.
We know
that
F(0,r*,m*)
=
((1
-k) — .k —
r'
It is
also easy to see that
'15
r'
).
^(0,r.m)=^(0,r.nO7^0
am
am
(if e
=
0. r is
kept constant and
ni
and the proportion of the wealth
changes,
as
it is
and
in liquid
the total wealth
if
is
kept constant
changes) and that
illiquid assets
—Ur^(0,r.m) + —or
^(0.7-.r77)^0
(a
change
in the
is
e
=
case). Therefore the Jacobian
invertible
It is
e,
We now
e
dFi
df:^
Or
Or
dFi
dF2
dm
din
how
and prove lemma
at the point
r
and
7tj
4.2.
wealth changes when
total
(O.r'.m*).
Moreover,
small there exists an equilibrium.
which establishes proposition
calculate
at least
matrix
and we can apply the impUcit function tlieorem
thus clear that for
are C°° in
C.3
non-zero effect on the total stock of wealth,
in interest rate has a
changes for fixed
e
r*
and m*
4.4.
Lemma
Proof of
4.4
Adding the equations describing the evolution of
diet
+
A,)
=
\{at
+
at
At)
and At we
+h+
get
It
dt
=
(A
+
r){at
+
At)
+ {R-
Multiplying by e"^', integrating from
appendix B) that
r
\{at
(at
+
Jo
Equation C.5
+
r)At
to oc
At)€~^' goes to zero as
At]e-'\{t
r=
- f"
e-''(c,
^
f
-^
]h
and
-
c,
-
nsiiis;
(I,.
tlie Inrt
goes to infinity,
fl/J
-
ijt
we
(.'staMi'
get
-{R-r)A,)dt
(C.5)
r Jo
will allow us to calculate total
assume that we are
at (e,r*,m*).
46
wealth as a function of
e.
We
will
/;e->.,.,../;-e-'.(^-,M,)-/;^..-«.(^ -...) =
(C.6)
=
Co
Co
^
m* -
/
+ ^-(x+w.K^
+ e-'^+-^K.L_i
)••
1
A +u;'
(C.7|
Replacing
for Co,
we
get
r
e-^'ctdt
=
Jo
g
^
0*(A+u;*)
(C.8)
Using C.5, C.6 and C.8,
Fi
+
it
is
F2 to the transactions costs
We
easy to find the sensitivity of total asset
e.
The equations
next derive the expression for b
—
in'.
in Ipuinia 4.4 follow.
For this purpose
we
differentiate the
equilibritim condition
Fi
+
F2
=
r
\e-^'{at
+
At)dt
Jo
= il- k)- +
r
47
(iriiuiiid
k—
+ me
r
I
Note that when
with respect to
e.
change. Hence
we
d{Fr
+
changes, the cquilibrimn mines of
F2)
,
+
1-°
Fe
a(Fi4-F,),
—IJT-''-*^
-{1 - k)-U=o{b -
-
/n')
+
"^
is
and
;7?*will
le=o
or
m
-
U=o
- m*) + £^|,.o +
'"*)•
dc
m
independent of
T
-I
1.^0^1,^0
J,n
\.=o{{b
,
dm
F,),
^
+
^
^
d(F--F,)
^2)
+
d{F,
.^
-
(r 4- 7ne)2
Using the fact that Fi + F2
^
at
^,
k
r
^(^1
r'
get
,
=
e
)
=
for
e
=
0,
6
-
D
r'
we have
m
)
m'D
k—
r*
.
r*
Using the notation of subsection 4.2.3 we get
We
can calculate
OiF +
F,)
,
5
U^o
from
F^
+
F2
S
=
(
and derive
(6
—
+
(jj
a-
)(f>
m').
The right-hand
is
A
—
side
is
positive. Therefore (6
negative and
— m')
is
it is
easy to see that the coefficient of (6- m*)
negative.
D
48
D
If
Proof of
Lemma
k increases, obviously
4.5
A* decreases. Moreover
t* increases
and
t'
+ A*
decreases.
These can be seen from equations 4.11 and 4.12 with very simple algebra. Equation
4.13 implies then that
m'
increases.
Simple algebra then shows that
decreases,
exp(—(\
+
i.e.
u!)t)
the effect of
— exp(-(A
e
4-
on
r*
(6
is
— m*) which
stronger.
is
to:
(Note that the function
6)t) increases in [0,r*]
49
proportional
and that
Ti
<
r*).)
g{.)
:
t
-^
D
The Case k=l
E
The
a
case
first
=
/c
1 is
We
slightly different.
find, as before, that transactions costs
The
order effect on the rate of return on the illiquid asset.
case (0
<
A;
<
that
1) is
if
that cannot be sold short,
by zero-th order term.
its
(i.e.
for this result is that the
Since
we introduce a
we have
of
economy
in zero supply,
minimum
The reason
a zero-th order liquidity premium.)
holding period has a
is
first
order length.
held in the form of the illiquid asset, we have:
.'It
The dynamics
difference with the
return will be lower than the return on the ilUquid asset
of the consumer's wealth
all
liquid asset in this
have
=
Wt
are described by:
iCt
(iw,
ct
=
yt
=
+
+
Xwfdt
Rtu't
-
Ifdt
-
It
(E-1)
t\It\
Proposition E.l describes the optimal policy of the consumer for small transactions
costs
and
for a subset of values of
R
that are of interest,
i.e.
such that
its
equilibrium
value belong to this subset.
Proposition E.l
asset until
T\
until
e
small and for
R
bclovging to a subset of
its
optimal policy has the following form.: The consumer buys the
the
values,
For
an age
an age
The proof
Tj
rj.
+
He does nothing
A
when he
(i.e.
he consumes his income
yt
possible
{illiquid)
+ Rwt) from
starts selling the asset until he dies.
of proposition E.l
analogous to the proof of proposition 4.1 and
is
is
therefore omitted.
In
will
what
follows,
we
will (briefly) discuss tlip implications of proposition
show how the consumption
Cj
as well as the
width of the inaction
f
perio<l
J. W--
A
'an
he derived.
In the case k
equation
4. .3
=
I,
the expression for consumption
to
ct
=
coe-'*'^"
50
t
<
r,
c,
must be changed from
Q = Rwt +
Again the
initial
consumption
co
n <
yt
A
< n +
i
(E.2)
can be obtained from the intertemporal budget
equation which in this case can be written as
r\y, r 'i^e-^^^Ut
+
-f
i
•'0
'o
e
c.)e-''<')c/.'
f
+
^l^.-"',, ^
y^+A
Jri
1
—
(E.3)
q
€
where
p{t)
p(t)
= Rl +
=
+
/?
p(0 -
X ioT
A for
/?s
+
<
ri
A for
<
t
<
<
Ti
Ti
ri
+ A
+ A <
i
and
^(0=
This intertemporal budget equation
evolution of
analysis
is
ivt,
Ti.
doing
utility if
is
<
very similar to the case
A
condition
This consumer
so.
is
<
1
and
is
omitted.
between investing
in the illiquid asset
starts selling the illiquid asset at
he buys one unit of the
illiquid asset at Tj
From equations
between consumption
fc
age
at
rj
E.l'*
«'n+A
is
and E.2 we
and consumption
=
^
=
n-.^r
at
-^
n +
also given
24
Note that
/(
=
between
t^
and
is
n+
of order
A.
51
e:
and not
A, the change
in his
by equation 4.5 and
get that the following relation
age
=
r^
+ A
.
.
Equations 4.5 and E.4 yield the following relation which shows that the
period of holding the illiquid asset
The
derived by the portfolio decision of a consumer of
indifferent
is
Given that he
equal to zero.
derived from the equation describing the
is
using our results on the investment policy of the consumer.
The parameter
age
tp{s)ds
Jo
'0
(L.ll
minimum
(\
Having characterized
tlie
2e
i/'*
A =
+
uj'){S
-u.-)
+
A
(E.5)
n(f)
solution to the consumer"? problem
we turn
to the equi-
hbriuni determination of R.
In equilibrium, asset
demand
r
Xe-^'ivtdt
/o
equals asset market value.
As before, we
D
I
R.
consider small transactions costs (small values of
will
their first order effects on R.
We
Proposition
b.
Proposition E.2
equation E.6, with
The proof
b
E.'2
and find
thus write:
will
=
R{c)
and calculate
e),
r'
+
gives us
6.
R
is
In equilibrium.
^
be
(E.6)
o(f)
uniquely determined.
has the form of
It
having an ambiguous sign.
of proposition E.2 as well as the analytic expression for 6 are again
omitted.
The
discussion on the determination of 6
we do not
present
it
here. Instead,
focus on the determination of the rate of return on a liquid asset that
in this
an
similar to the discussion offered at the
4.2.2, (the eflects are similar) so
end of subsection
we
is
economy
in zero supply.
Proposition
asset, as well as the implicit liquidity
Proposition E.3 In
the rate of return
Proposition
for this result
on
gives us the rate of return on such
premium.
and on
the liquid asset
that the
we have
minimum
the liquiditij
a zeroth order litiuidity premium.
holding period has a
=
r-
-
— 4-0(1)
A
=
r*
first
i^'*
52
ffrrl
The
order length.
^o(l)
-.4^
<
mi
prrwium.
show that
r
introduced
equilibrium, transactions costs hnrr a zrrolh order
E..3 states that
is
E..3
is
reason
We
can
a
= A-
+
53
0(1).
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Figure
1:
IIoldinEs of the liquid
;<
and
illiquid assets
o
0.0
0.:
0.1
0.3
Figure
0.4
2:
0.7
0.6
0.5
Rates of return as functions of k
This figure plots the rates of return as a function of the fraction,
assets.
costs.
dashed
The
solid line represents the
The dotted
0.9
0.8
bencmark case where there
line represents the rate of return
line represents the rate of
on the
/c,
are no transactions
illiquid asset
= 2%;
6
= 4%;
l3
while the
return on the liquid asset. For this figure we have
used the following parameter values:
A
of illiquid
=
0.2%;
.4
58
=
1;
D/Y =
50%:
r
= 3%.
1.0
o
0.0
0.1
Figure
3:
Rates of return as functions of k
This figure plots the rates of return as a function of the fraction,
The
assets.
costs.
solid line represents the
The dotted
line represents
bencmaik
k, of illiquid
case where there are no transactions
the rate of return on the
dashed line represents the rate of return on the liquid
illiquid asset
while the
asset. For this figure
we have
used the following parameter values:
A
Compared
=
in
6
= 40%;
(3
=
2%;
A=
to the previous case, the agent
shorter horizon.
than
20%;
As a
result, the interest rate
1:
is
D/Y =
50%:
f
= 3%.
more impatient and has therefore a
and the
liquidity
premium are higher
the previous figure. Qualitative results are however unchaiiKcd.
59
o
0.0
Figure
4:
Rates of return as functions of k
Tliis figure plots the rates of
assets.
costs.
dashed
The
return as a function of the fraction, k, of illiquid
solid line represents the
The dotted
line represents
bencmark case where
there are no transactions
the rate of return on the illiquid asset while the
line represents the rate of return
on the liquid
asset.
For this figure
we have
used the following parameter values:
A
=
2%;
6
= 4%;
/3
In this case (where financial
the following paradoxical
=
A =
0.2%;
income
phenomenon
is
1:
D/Y =
?.m%:
f
= 3%.
much morr import nnl than
labor
occurs: transactions rost? lower
return on both assets.
GO
tlic
iii<<tiiic|
rairs of
Oi
Figure
2935
5:
The graph
5
61
o( g(t)
MIT
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