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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
FINANCIAL FRICTIONS INVESTMENT
AND TOBIN'S q
Guido Lorenzoni
Karl Walentin
Working Paper 07-1
April 25, 2007
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without char ge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=983421
Financial Frictions, Investment and Tobin's
Guido Lorenzonit
Karl Walentin*
MIT and NBER
Sveriges Riksbank
q*
April 25, 2007
Abstract
We
develop a model of investment with financial constraints and use
it
to investigate
q. A firm is financed partly by insiders, who
investors. When their wealth is scarce, insiders
the relation between investment and Tobin's
control
its assets,
and partly by outside
earn a rate of return higher than the market rate of return,
on invested
by two
capital. This rent
forces:
is
i.e.,
they receive a quasi- rent
priced into the value of the firm, so Tobin's q
is
driven
changes in the value of invested capital, and changes in the value of the
insiders' future rents per unit of capital.
This weakens the correlation between q and
investment, relative to the frictionlcss benchmark.
We
present a calibrated version of
the model, which, due to this effect, generates realistic correlations between investment,
q,
and cash
flow.
Keywords: Financial constraints, investment, Tobin's
JEL
codes: E22, E30, E44,
q,
limited enforcement.
G30.
*We thank
for useful comments Andrew Abel, Daron Acemoglu, Joao Ejarquc, Mark Gertler, Veronica
Hubert Kcmpf, Sydney Ludvigson, Martin Schneider, and seminar participants at New York
University, MIT, University of Oslo, EEA Meetings (Amsterdam), Minneapolis FED, Sveriges Riksbank,
Uppsala University, Norges Bank, SED Meetings (Vancouver), NBER Summer Institute 2006, and CEPRBank of Finland conference on Credit and the Macroeconomy. The views expressed in this paper are those
of the authors and not necessarily those of the Executive Board of Sveriges Riksbank.
Guerricri,
^E-mail: glorenzo@mit.edu.
E-mail: karl.walentin@riksbank.se.
Introduction
1
The standard model
the investment rate should be entirely explained by changes in Tobin's
generally been rejected in empirical studies, which
of current profitability have a strong predictive
Tobin's
many
This has been taken by
q.
movements
of investment with convex adjustment costs predicts that
This prediction has
q.
show that cash flow and other measures
power
for investment, after controlling for
authors as prima facie evidence of the presence
of financial constraints at the firm level.
and Ejarque (2003) have challenged
in
Gomes
In recent papers,
this interpretation.
(2001) and Cooper
They compute dynamic general
equilibrium models with financial frictions, calibrate them, and look at the relation between
Their results show that, even in the
Tobin's q and investment in the simulated
series.
presence of financial frictions, Tobin's q
explains most of the variability in investment,
still
and cash flow does not provide any additional explanatory power. This seems to echo a
concern raised by Chirinko (1993):
"Even though
market
financial
frictions
impinge on the firm, q
is
looking variable capturing the ramifications of these constraints on
decisions.
Not only does
In this paper
frictions
does
so,
amount
we analyze
this issue
which can be interpreted
The
shareholder.
he
is
some
or
all
financial
by building a model of investment with financial
as the entrepreneur, the
For each firm there
an
is
manager, or the controlling
insider has the ability to partially divert the assets of the firm and,
if
he
punished by losing control of the firm. This imposes an upper bound on the
of outside finance that the insider
is
able to raise. In this framework,
to fully characterize the optimal long-term financial contract,
of the state-contingent claims issued
by the
firm.
and to derive the
analytical result
between average
q,
is
we
are able
total value
This gives a measure of Tobin's q and
allows us to study the joint equilibrium dynamics of investment,
Our main
the firm's
1
caused by limited enforcement of financial contracts.
"insider,"
all
q reflect profitable opportunities in physical investment
but, depending on circumstances, q capitalizes the impact of
constraints as well."
a forward-
q,
and cash
flows.
that the financial constraint introduces a positive wedge
which corresponds to Tobin's q
in
our model, and marginal
q,
which
determines investment decisions. This wedge reflects the tension between the future profitability of
investment and the availability of internal funds in the short run.
quantitative side,
we use a
calibrated version of the
model to show that
this
On
wedge
the
varies
over time, breaking the one-to-one correspondence between investment and q which holds
in the frictionless
we can obtain
model.
When we
realistic coefficients
run standard investment regressions on simulated data
on
q
and cash
flow. Therefore, financial frictions
to reconcile models of firms' investment with the data.
'Chirinko (1993)
p.
1903.
do help
Aside from the enforcement
Hayashi (1982) model.
friction,
In particular,
it
our model
on the equilibrium behavior of investment and
costs, the coefficient of q in
The main
q.
way the
In the
difference
Ejarque (2003)
benchmark model with quadratic
The presence
by a factor of 6 and gives a large positive
between our approach and that
is
identical to the inverse of
of the financial friction reduces
coefficient
in
Gomes
the modeling of the financial constraint.
is
and constant
effect of the financial friction
investment regressions
the constant in front of the quadratic term.
this coefficient
virtually identical to the classic
features convex adjustment costs
returns to scale. This allows us to identify in a clean
adjustment
is
on cash
flow.
(2001) and Cooper and
They introduce a
constraint
on the flow of outside finance that can be issued each period. Here instead, we explicitly
model a contractual imperfection and
solve for the optimal long-term contract. This adds
a state variable to the problem, namely the stock of existing
liabilities of
the firm, thus
generating slow-moving dynamics in the gap between internal funds and the desired level of
investment.
As we
investment and
Our paper
shall see, these
(2004)).
2
for the empirical disconnect
between
q.
macroeconomic implica-
related to the large theoretical literature on the
is
tions of financial frictions (e.g.,
Holmstrom and
dynamics account
Bernanke and Gertler (1989), Carlstrom and Fuerst (1997),
Tirole (1997), Kiyotaki and
Moore
(1997), Cooley,
Marimon and Quadrini
In particular, our model provides a tractable framework that introduces long-
term, state-contingent financial contracts, into a standard general equilibrium model with
adjustment
costs.
The form
of limited enforcement
we adopt, and the
recursive characteri-
zation of the optimal contract, are related to the approach in Albuquerque and
(2004).
By
exploiting constant returns to scale,
we
Hopenhayn
are able to simplify the analysis of the
optimal contract, which takes a linear form, making aggregation straightforward. In this
sense, the
rich
model retains the simplicity
dynamics of net worth,
profits
of a representative agent model, while allowing for
and investment.
Following Fazzari, Hubbard and Petersen (1988) there has been a large empirical
lit-
erature exploring the relation between investment and asset prices using firm level data.
The
great majority of these papers have found small coefficients on Tobin's q and positive
and
significant coefficients
condition of a firm.
on cash
flow, or other variables describing the current financial
This result has been ascribed to measurement error in
caused by non-fundamental stock market movements.
the explanatory power of
is
q,
3
Measurement
and cash flow would then appear as
a good predictor of future
2
4
profits.
Gilchrist
error
q,
possibly
would reduce
significant, given that
and Himmelberg (1995) show that
this
it
is
See Bernanke, Gertler and Gilchrist (2000) for a survey.
E.g.
Gilchrist
and Himmelberg (1995),
Gilchrist
and Himmelberg (1998). See Hubbard (1998)
for
a
survey.
4
The debate
is
open whether non-fundamental movements
Chirinko and Shaller (2001), Gilchrist, Himmelberg and
in q
Huberman
should affect investment or not.
(2005) and Panageas (2005).
See
insufficient to explain the failure of q theory in
investment regressions. 5 They replace the
value of q obtained by financial market prices with a measure of "fundamental q" (which
employs current cash flow as a predictor of future
flow retains
its
profits),
and they show that current cash
independent explanatory power. The evidence
in this literature provides the
model (Section
starting motivation for our exercise. In an extension of the
4)
we introduce
firm-level heterogeneity and further explore the connection between our model and panel
data evidence.
The
idea of looking at the statistical implications of a simulated model to understand
the empirical correlation between investment and q goes back to Sargent (1980). Recently,
Gomes
(2001),
Cooper and Ejarque (2001, 2003) and Abel and Eberly
(2004, 2005) have
followed this route, introducing both financial frictions and decreasing returns and market
power to match the existing empirical evidence. This
literature concludes that decreasing
returns and market power help to generate realistic correlations, while financial frictions do
In this paper
not.
we show
way one models the
the
that the second conclusion
between our approach and these papers,
emphasized
reflect
in
On
financial constraint.
is
unwarranted, and depends on
the other hand, there are some parallels
in particular
with the "growth options" mechanism
Abel and Eberly (2005). Both approaches imply that movements
in q
can
changes in future rents that are unrelated to current investment. In our paper these
rents are not
due to market power, but to the scarcity of entrepreneurial wealth, which
evolves endogenously.
The paper
is
organized as follows. Section 2 presents the model, the derivation of the
optimal contract, and the equilibrium analysis.
Section 3 contains the calibration and
simulation results. In Section 4 we extend the model to allow for firm-level heterogeneity.
Section 5 concludes. All proofs not in the text are in the appendix.
The Model
2
The environment
2.1
Consider an economy populated by two groups of agents of equal mass, consumers and
entrepreneurs.
Consumers are
infinitely lived
and have a
fixed
they supply inelastically on the labor market at the wage
and have a discount
of death 7.
factor
Each period, a
j3
c Entrepreneurs have
.
Ie
m
the
'first
period of
also risk neutral,
life,
j3
-
is
of labor Iq,
Consumers are
which
risk neutral
with a constant probability
replaced by an equal mass of newly
with no capital and have a labor endowment
life
which gives them an
with a discount factor
t
finite lives,
fraction 7 of entrepreneurs
born entrepreneurs. Entrepreneurs begin
w
endowment
E <
j3
initial
c The
.
wealth wtlE-
last
Entrepreneurs are
assumption, together with the
See Erickson and Whited (2000) and Bond and Cummins (2001) for a contrarian view. See
evidence in favor of the financial frictions interpretation.
Rauh
(2006)
for recent
See Schiantarelli and Georgoutsos (1990) for an early study of q theory
market power.
in
a model where firms have
assumption of
finitely lived entrepreneurs, is
We
state with a binding financial constraint.
+ lh =
lc
to-scale technology
is
is,
1-
Starting in their second period of
lt
needed to ensure the existence of a steady
normalize total labor supply to one, that
A F (kt, h),
where
t
entrepreneurs have access to the constant-returns-
life,
the stock of capital installed in period
kt is
The productivity
labor hired on the labor market.
p.d.f.
n
We
(et).
Investment in new capital
dG (kt+i, k°) /dkt+i =
1
G
G (fci+i, k°)
is
and
equal across entrepreneurs and
where
an
et is
mean
normalize the unconditional
units of capital ready for production in period
adjustment cost function
is
— 1,
of
i.i.d.
A
t
to
shock drawn
1.
subject to convex adjustment costs. In order to have k t +\
is
used capital needs to employ
t
— T (At-i,et),
follows the stationary stochastic process At
from the discrete
A
t
=
if fct+i
where entrepreneurs can buy and
k°.
sell
+
an entrepreneur with k° units of
1,
units of the consumption
convex in
7
t
fct+i,
There
homogeneous
is
good
at date
of degree one,
and
t.
The
satisfies
a competitive market for used capital,
capital at the price q°, after production has taken
place. This allows individual entrepreneurs to choose k°
^
However, market clearing
kt-
in
the used capital market requires that the aggregate value of k° across entrepreneurs equals
the aggregate value of
An
kt,
denoted by
entrepreneur born at date
to
K
t
8
.
finances his current and future investment by issuing a
long-term financial contract, specifying a sequence of state-contingent transfers (which can
be positive or negative) from the entrepreneur to the outside investors, {dt}^Z tQ
t
=
to, his
budget constraint
form
it
+ G (k t+l ,k°) +
into capital ready for use in to
+
1-
+ G (fct+x, k°t +
)
< wt l E -
dt
.
to acquire used capital
and trans-
Furthermore, he can increase his consumption
and investment by borrowing from consumers,
remaining periods, the budget constraint
q°k°
consume and
uses his initial wealth to
cf
In period
is
cf
The entrepreneur
.
i.e.,
choosing a negative value for
dt
.
In the
is
q° (k°
t
-
kt )
< A F
t
(k u k)
-w
t lt
-
dt
.
uses current revenues, net of labor costs and financial payments, to finance consumption
He
and investment. At the beginning of each period
t,
the entrepreneur learns whether that
his last period of activity. Therefore, in the last period,
consumes the
all
the capital k t and
receipts, setting
= A F (kt
cf
7
he liquidates
is
t
To keep notation compact G(k +i,kf)
t
,
h)
-w
t
lt
+ q°h -
df
includes both the direct cost of investment and the adjustment
See (14) below, for the explicit functional form used in the quantitative part.
8
To simplify notation we do not introduce indexes for individual entrepreneurs, although the value of
will be different across entrepreneurs born at different dates r <t.
costs.
kt
From then
on, the
payments
are set to zero.
dt
The entrepreneur
Financial contracts are subject to limited enforcement.
and can,
firm's assets k t
he does
so,
run away, diverting a fraction
in each period,
he re-enters the financial market as
if
for a defaulting
entrepreneur
is
—
9) of
them.
he was a young entrepreneur, with
wealth given by the value of the diverted assets, and zero
punishment
{1
controls the
liabilities.
That
is,
If
initial
the only
the loss of a fraction 9 of the firm's assets. 9
Aside from limited enforcement no other imperfections are present, in particular, financial
contracts are allowed to be fully state-contingent.
Recursive competitive equilibrium
2.2
We
our attention on recursive equilibria where the economy's dynamics are fully
will focus
characterized by the vector of aggregate state variables
X =
the aggregate capital stock and Bt denotes the aggregate
to be defined in a
moment.
t
(At,Kt,B
liabilities of
t
),
where
K
is
t
the entrepreneurs,
In the equilibria considered, consumers always have positive
consumption. Therefore, the market discount factor
and the net present value of the
liabilities of
is
equal to their discount factor,
(3
C
,
an individual entrepreneur can be written as
IE*
s=0
The
variable
A
B
t
equal to the economy-wide aggregate of these
is
recursive competitive equilibrium
is
liabilities.
defined by law of motions for the endogenous
state variables:
and by two maps,
w (Xt)
of the entrepreneurs.
K
= K(Xt-i)\
B
t
= B (*,_!, et),
q° {Xt),
which give the market prices as a function of the
(ii)
The quadruple K.,B,w(.) and
q° () forms a recursive competitive
the entrepreneurs' optimal behavior
is
consistent with the law of motions
the labor and used capital market clear. In the next two subsections,
we
characterize entrepreneurs' decisions, and then aggregate and check market clearing.
first
We
(i)
if:
and B, and
t
Given these four objects, we can derive the optimal individual behavior
current state.
equilibrium
and
K
use
Xt
to denote in a
r,/C,
compact way the law
=
of
H (Xt-i,e
motion
t)
for
,
Xt derived from the laws
of
motion
and B.
Here,
we
just take this as an institutional assumption.
For a microfoundation, we could assume that
defaulting entrepreneurs are indistinguishable from young entrepreneurs.
dressing a
number
of informational issues, which
However, this would require adwould considerably complicate the analysis.
Optima] financial contracts
2.3
Let us consider
the optimization problem of the individual entrepreneur.
first
Exploiting
the assumption of constant returns to scale, we will show that the individual problem
This property
linear.
We
will greatly simplify aggregation.
describe the problem in recursive form, dropping time subscripts. Consider a contin-
uing entrepreneur, in state
Let
V (k, b, X)
denote
X, who
controls a firm with capital k
expected
his end-of-period
utility,
and outstanding
computed
after
liabilities b.
production takes
place and assuming that the entrepreneur chooses not to default in the current period.
entrepreneur takes as given the law of motion for the aggregate state
functions
w
c
Lemma
1
E+
G (k\ k°) + q° (X) k° < AF
allows us to rewrite
it
where q m (X)
is
unit of capital,
q
m (X)k'
new
the shadow price of the
on the
- w (X) + q°{X)k-
I)
l
w (X)
<R(X)k-d,
capital
and q° (X),
any
that satisfy the following conditions for
k',
(1)
R (X)
and
lemma
m (X)k' =
k'
and
mm{G(k',k )+q°(X)k
be discussed
A
or
q° (X).
The
variable q
m
(X)
is
m
R{X)
show that q m (X) and
R (X)
k,
},
exploits the assumption of constant returns to
and
the (gross) return per
(X) and
q°{X)k.
are independent of the current and future capital stocks, k and k'
w (X)
is
there are two functions q
R{X)k = max {AF{k, I) -w{X)l} +
This
d.
installed capital k.
Given the prices
q
(jfe,
as
E
1
The
and the pricing
constraint takes the form
c
Lemma
X
(X) and q° (X).
The budget
prices
is
and only depend on the
,
equal to marginal q in our model, and will
in detail below.
continuing entrepreneur can satisfy his existing
by promising future repayments. Let
the realization of the aggregate shock
b' (e')
e',
if
liabilities b either
denote next-period
tomorrow
is
by repaying now
liabilities,
contingent on
not a terminal date, and
let b'
L
(e')
denote the same in the event of termination. Then, the entrepreneur faces the constraint
b
where the expectation
is
The entrepreneur has
that,
if
=
d
+ Pc
((1
-7
)
E[6' (e')]
taken with respect to
+ iWl
(
>
2)
e'.
to ensure that his future promised
the entrepreneur defaults, his
(01)
liabilities
repayments are
are set to zero
credible. Recall
and he has access
to a
fraction {1
—
repayments
9) of
b' (e')
the capital. Therefore,
if
tomorrow
is
a continuation date, his promised
have to satisfy the no-default condition
,
V(k\b'{e'),X')>V((l-9)k ,0,X')
for all
e'
Throughout
.
X' stands
this section,
H (X,
for
R (X
the entrepreneur can either liquidate his firm, getting
or default
and get
—
(1
6)
R (X
1
)
tomorrow
If
e').
(3)
1
is
the final period,
k\ and repay his
)
liabilities,
Therefore, the no-default condition in the final period
k'.
takes the form
R (X')
k'
for all
e'.
which again needs to hold
We
are
now ready
to write the
V{k,b,.X)
=
-
b'L (e')
>
(1
-
9)
Bellman equation
max
c
E
R (X')
k',
(4)
for the entrepreneur:
+ /3 E (1 - 7 )E
\V
(k',b' U')
+P El E[R(X')k -b L
(1),
Notice that, except for constraint
k and
6,
and
in the choice variables c
the value function
is
(3), all
,
(3)
(2),
and
+
,
l
s.t.
,X')1
(P)
(e')}
(4).
constraints are linear in the individual states
k', b' (.)
and
b'
L
Let us
(.).
make
the conjecture that
and takes the form
linear
V{k,b,X) =<j>{X){R{X)k-b),
for
some
comes
positive, state-contingent function (f)(X).
linear as well,
and can be rewritten
This
is
Then, the no-default condition
(3) be-
as
< 9R
b' (e')
(5)
(A")
(3')
k'.
a form of "collateral constraint," which implies that an entrepreneur can only pledge
R(X')k. 10
a fraction 9 of the future gross returns
constraints in the literature
(e.g., in
The
crucial difference with similar
Kiyotaki and Moore (1997)),
is
the fact that
we allow
for fully state-contingent securities.
Before stating Proposition
and q°
{.)
problem
(3') is
is
and on the law
well defined
always binding.
and
of
3,
we impose some
restrictions
on the equilibrium prices
w (.)
motion H. These conditions ensure that the entrepreneur's
deliver a simple optimal contract
In subsection 2.4
we
where the
collateral constraint
will verify that these conditions are
met
in
equilibrium.
Suppose the law of motion
H admits
an ergodic distribution
'Constraint (4) immediately gives an analogous inequality for
b'L
.
for the aggregate state
X,
Assume that equilibrium
with support X.
hold for each
prices are such that the following inequalities
X eX:
/3
6f3
E E[R{X')]
c E[R(X')]
>
q
m
(X),
(a)
<
m
g
(X),
(b)
(1- 1 )(1-6)E[R(X')}
[C)
q™(X)-9(3 c E[R(X')} ^
Condition
(a) implies
that the expected rate of return on capital
E
[R (X')] /q m (X)
greater
is
than the entrepreneur's discount factor, so a continuing entrepreneur prefers investment to
consumption.
Condition (b) implies that "pledgeable" returns are insufficient to finance
the purchase of one unit of capital,
investment cannot be
i.e.,
funds. This condition ensures that investment
the entrepreneur's utility
is
is finite.
fully financed
with outside
Finally, condition (c) ensures that
bounded.
Before introducing one last condition,
we need
to define a function
(j>,
which summarizes
information about current and future prices.
Lemma
2
When
conditions (a)-(c) hold, there exists a unique function
<j>
:
X—
>
[l,oo)
that satisfies the following recursive definition
9{
'
This function satisfies
A
<p
p E (l-e)E[( 1 + (l- 1 )<l>(X'))R(X')]
m {X)-e/3 E[R{X')}
q
c
(X) >
1
for
all
[>
'
IeX,
further condition on equilibrium prices
is
then:
cj>(X)>^(X')
(d)
Pc
for all
IgX and
all
investment. Namely,
X'
it
—
H (X,e').
Condition
(d) ensures that
implies that they always prefer to invest in physical capital today
rather than buying a state-contingent security that pays in
The
function
</>
entrepreneurs never delay
defined in
Lemma
The next proposition shows that
some future
state.
2 will play a central role in the rest of the analysis.
substituting <f>{X) on the right-hand side of (5), gives us
the value function for the entrepreneur (justifying our slight abuse of notation). Define the
net worth of the entrepreneur
n(k,b,X)
= R(X)k-b,
which represents the difference between the liquidation value of the firm and the value of
its liabilities.
Equation
(5) implies
that expected utility
8
is
a linear function of net worth
and
its
<f>
(X) represents the marginal value of entrepreneurial net worth.
will
go back to
interpretation in subsection 2.5.
Proposition 3 Suppose
q°
We
(.)
the aggregate law of
are such that (a)-(d) hold, where
V (k, b,X)
takes the
form
and
(5)
c
E
<f>
H
motion
and
defined as in
is
the equilibrium prices
Lemma
2.
and
Then, the value function
the entrepreneur's optimal policy
=
w (.)
is
0,
R(X)k-b
[
q^(X)-ep c E[R(X')Y
=
b'(e')
The
L (e')=6R(X')k'.
b'
(8)
entrepreneur's problem can be analyzed under weaker versions of conditions. (a)-(d).
However, as we
shall see in a
moment, these conditions
are appropriate for studying small
stochastic fluctuations around a steady state where the financial constraint
2.4
'
is
binding.
Aggregation
Having characterized optimal individual behavior, we now aggregate and impose market
and on the used
clearing on the labor market
To help the reading
capital market.
of the
dynamics, we now revert to using time subscripts.
Each period, a
fraction
WtlE- Their net worth
is
7 of entrepreneurs begins
life
with zero capital and labor income
simply equal to their labor income. Moreover, a fraction
— Rth — bt- The
of continuing entrepreneurs has net worth equal to n%
(1
—
7)
aggregate net worth
of the entrepreneurial sector, excluding entrepreneurs in the last period of activity,
is
then
given by
N
t
= (l-
Using the optimal individual rules
gregate states Kt and
B
7)
(7)
(R
and
t
K
- Bt ) + jw t E
l
t
(8),
we
.
get the following
dynamics
for
the ag-
t
Kt+1 ~
Bt+
i
{1
-
{R
7)
t
K
-B
t
q?-ep c E
= p c 6Rt+ iKt+1
and
qt
=
~
l
t
m
l
l9j
}
(10)
dF{K
—
dl
u
A
+ 7w E
.
in the used capital
Wt
)
'
t
Finally, the following conditions ensure that the prices
clearing in the labor market
t
[Rt+1
u>t
and q° are consistent with market
market
l)
'
^
dG(Kt+1 ,Kt)
dK
^
>
'
{U
>
t
To
clarify the role of condition (12), notice that all continuing entrepreneurs
.
choose the same
ratio k°/kt+i,
and
this ratio
must
q°+dG (kt+i, k°) /dk° =
satisfy the first-order condition
0.
Market clearing on the used capital market requires that continuing entrepreneurs acquire
all
K /Kt+
the existing capital stock, so
Summing
up,
t
we have found a
equal to k°/kt+\. This gives us condition (12).
recursive equilibrium
the pricing rules for wt and q° satisfy
are satisfied.
is
i
and
(9) to (12),
The next proposition shows
if
the laws of motion
if
and
K,
B
and
they are such that conditions (a)-(d)
that an equilibrium with these properties exists
under some parametric assumptions. Let the production function and the adjustment cost
function be:
A F{ku
t
= A
lt )
G(kt+l ,k
=
t)
t
k?l\-
kt+1
a
(13)
,
-(l-6)kt +
^
kt+1
~
kt)
(14)
.
To construct a recursive equilibrium, we consider a deterministic version
omy
(i.e.,
an economy where
A
t
is
constant and equal to
state as a reference point. Let [A, A\ be the support of
1),
A
t
of the
same econ-
and use the deterministic steady
in the stochastic
economy.
Proposition 4 Consider an economy with Cobb-Douglas technology and quadratic
ment
costs.
pendix).
Suppose the economy's parameters satisfy conditions (A) and (B)
Then
there
is
a scalar
A >
such
-0
that, if
A—A < A
adjust-
(in the ap-
there exists a recursive
competitive equilibrium with aggregate dynamics described by (9)-(10).
The
conditions (A) and (B) presented in the Appendix ensure that the
cally stable deterministic steady state
economy has a
lo-
with binding financial constraints. These parametric
restrictions are satisfied in all the calibrations considered below.
Finally, as a useful
which
arises
when
9
—
benchmark,
1.
let
us briefly characterize the frictionless equilibrium
In the frictionless benchmark, equilibrium dynamics are fully
characterized by the condition
<?r
The
definitions of
q™ and Rt
are the
and so are the equilibrium conditions
entrepreneurs consume their wealth
in all future periods.
Investment
is
= p c ®t
same
(11)
w^e
m
(is)
[Rt+i]
as those given in the constrained economy,
and
(12) for wt
the
Average
q
and marginal
life
/?£
< 0q
and consume zero
by consumers, which explains why the
consumers' discount factor appears in the equilibrium condition
2.5
Given that
q°.
period of their
first
entirely financed
and
(15).
q
Having characterized equilibrium dynamics, we can now derive the appropriate expressions
for
Tobin's q and for marginal
q.
Marginal q
problem as the shadow value of new
is
immediately derived from the entrepreneur's
capital, q^
10
1
.
The
definition of
q™
in
Lemma
1
and the
equilibrium condition (12) can be used to obtain
q?
This
is
t
between the investment rate and the shadow price of new
To derive Tobin's
is,
t
dK,t+i
the standard result in economies with convex adjustment costs: there
relation
that
dG{K +i/K ,l)
=
the
sum
we
g,
is
a one-to-one
capital.
need to obtain the financial value of a representative firm,
first
of the value of
the claims on the firm's future revenue, held by insiders
all
(entrepreneurs) and outsiders (consumers). For firms in the last period of activity this value
is
zero. For continuing firms, this gives us the expression
= V(k
Pt
We
t
subtract the current payments to outsiders,
firm.
+ bt-d
,bt,Xt)
t
(16)
.
to obtain the end-of-period value of the
dt,
Recall that continuing entrepreneurs receive zero payments in the optimal contract
(except in the final date), so there
is
no need to subtract current payments to
insiders.
Dividing the financial value of the firm by the total capital invested we obtain our
definition of average q
_
Pt
Qt
In the recursive equilibrium described above,
liquidating firms both pt
and kt+i are
The next proposition shows
marginal q and average
q,
qt is
zero, so qt
is
the
same
for all continuing firms.
For
not defined for those firms.
that the financial constraint introduces a wedge between
and that the wedge
is
determined by
</>
,
t
the marginal value of
entrepreneurial wealth.
Proposition 5 In
same for
all
the recursive equilibrium described in Proposition 4,
continuing firms and
equal, the ratio
qt/q™
is
is
increasing in
greater than marginal
(f> t
q,
qt
>
q"'•
average q
is the
Everything
else
.
Proof. Substituting the value function
in the value of the firm (16),
and rearranging
gives
Pt
=
(<p t
-
1)
(Rth -
bt)
+ Rth -
a\.
(17)
Using the entrepreneur's budget constraint, constant returns to scale
librium properties of q°
and
g[™, gives
Rth-dt = G(k t+l ,k°) +
dG{k
dG\ t+u
dk t+l
—
for
It
fc
k°
t )
q° k°-t
k+1 +
t+l
11
dG(h + uk?)
dkt+1
K + qt kt
G, and the equi-
Substituting in (17) and rearranging gives
It
=
—
Notice that (7) implies that (Rtkt
>
4> t
1
and
bt
<
< Rth
9 Rtkt
/
,
(d>t
-
—
Rt kt
s
..
1)
b*
+
k
bt)/kt+\
€m
equal across continuing firms. Given that
is
the stated results follow from this expression.
Notice that in the frictionless benchmark investment
-
we have
to the
bt
— Rth,
.
(18)
which immediately implies
Hayashi (1982) model: average q
is
=
qt
is
fully financed
by consumers and
down
q™. In this case, the model boils
identical to marginal q
and
a sufficient statistic
is
for investment.
It is useful to
in
provide some explanation for the wedge between average q and marginal q
the constrained economy.
First, notice that this
wedge
the discount factors of entrepreneurs and consumers. In fact,
present value of the entrepreneur's payoffs {cf+
A
not due to the difference in
is
if
we evaluated the expected
using the discount factor
/3
C instead
of
P E we would get a quantity greater than V (kt,bt,Xt) and the measured wedge would be
11
larger.
The fundamental reason why the wedge is positive is that t > 1, the marginal
,
<f>
value of entrepreneurial wealth
on the right-hand side of
To
clarify the
of wealth.
(18)
Suppose he uses
consumes Rt+i/q™.
is
than one.
If
4> t
was equal to
1,
mechanism, consider an entrepreneur who begins
and consumes the receipts
is
larger
The
then the
first
term
would be zero and the wedge would disappear.
life
this wealth to start a firm financed only
at date
so the entrepreneur can install
which
is
t
+
1.
The (shadow)
1/q™ units of
capital.
(a).
with inside funds
price of a unit of capital
In period
value of the firm for the entrepreneur
greater than one, by condition
with one dollar
+
t
is
1
is
q™,
he receives and
then fi^Et [Rt+i] llT
In short, the value of a unit of installed capital
and
larger inside the firm than outside the firm,
this explains
why
q theory does not hold.
This discrepancy does not open an arbitrage opportunity, because the agents that can take
advantage of this opportunity (the entrepreneurs) are against a financial constraint. This
thought experiment captures the basic intuition behind Proposition
To go one
5.
step further, notice that the entrepreneur can do better than following the
strategy described above.
In particular, he can use borrowed funds on top of his
funds, and he can re-invest the revenues
made
at
£
+ 1,
rather than consume.
The
borrowing allows the entrepreneur to earn an expected leveraged return, between
11
For the quantitative results presented
in
Section
3,
of q (discounting the entrepreneur's claims at the rate
j3
12
we
c
also
t
own
ability of
and t+1,
experimented with this alternative definition
E ), with minimal effects on the results.
instead of
.
equal to 12
(1
-
6)
E
[Rt+1 J (q?
t
]
- ep c Bt
Iterating expression (6) forward shows that
returns discounted at the rate
fi
(p t
[Rt+i])
>
Et [Rt+ i] lq?-
a geometric cumulate of future leveraged
is
into account the fact that, as long as the en-
E taking
,
trepreneur remains active, he can reinvest the returns
made
in his firm.
Therefore,
when
borrowing and reinvestment are taken into account, one dollar of wealth allows the entrepreneur to obtain a value of
receiving q™kt+i
is
—
1
<j>
>
t
/3
EE
[Rt+i] Jq™
t
from outside investors
>
At the same time, the entrepreneur
1.
(recall that
he only has
funds). Therefore, the value of the claims issued to outsiders
which
must equal q™kt+i —
an entrepreneur with one dollar to invest can start a firm valued
clusion,
is
m
larger than the value of invested capital, g t
fc(
1 dollar of
+ i, given that
<f> t
at
>
(f> t
1.
internal
In con-
+ q™kt+\ — 1,
1.
Quantitative Implications
3
we examine the quantitative implications
In this section,
behavior of investment, Tobin's
q,
and cash flow
of the
model looking
a simulated economy.
in
at the joint
First,
we
give
a basic quantitative characterization of the economy's response to a productivity shock.
Second,
we ask whether the wedge between marginal
q
and average q
in
our model helps to
explain the empirical failure of q theory in investment regressions.
Baseline calibration
3.1
The production
in (13)
and
function
(14).
is
Cobb-Douglas and adjustment costs are quadratic,
The productivity process
is
given by
A —
t
e
at
,
as specified
where a t follows the
autoregressive process
at
with
tt
a Gaussian,
"Notice that, from
i.i.d.
(7),
—
9) Rt+ikt+i-
m -
1
To prove the
In
t
+
1
j3
c >
(E t [Rt+i])
f}
E
it
2
is
et
,
and simplify to obtain
> 6q?E
t
[Rt +1 ]
.
theoretical analysis can be extended to the case where
bounded, we set A t = A whenever e at < A and
bounds A and A are immaterial for the results.
At
+
the capital stock kt+i which can be invested by an
the entrepreneur has to repay 8R t+ ik t+ i and can keep
inequality, rearrange
inequality follows from (a) and
The
t
8j3 c ^i t \Rt+{\) is
l//(<ji
6/3 c
The
pa -i
shock. 13
entrepreneur with one dollar of wealth.
(1
=
A =A
13
t
tt
is
a continuous variable.
whenever
e°'
>
A.
As long
To ensure that
as a\
is
small the
Pc
0.97
He
0.96
a
0.33
5
0.05
t
8.5
P
0.75
9
0.3
7
0.06
Ie
0.2
Table
The
year, so
Baseline calibration.
1.
baseline parameters for our calibration are reported in Table
we
set
/3
C
The
close to that of the consumers.
match basic
values of £ and p are chosen to
The
the Compustat dataset.
(CFK)
0.51
rate, r(.)
a
values for
a and
<5
are
statistics,
obtained from
14
r
CFK
is
features of firm-level data on
cash flow and investment. In particular, we consider the following
where
The time period
to give an interest rate of 3%. For the discount factor of the entrepreneurs,
we choose a value smaller but
standard.
1.
[CFK)
a (IK)
a
0.061
0.128
denotes cash flow per unit of capital invested,
IK
denotes the (yearly) coefficient of serial correlation, and
denotes the investment
<r(.)
the standard devia-
We calibrate p so that our simulated series replicate the autocorrelation of cash flow
r(CFK) = 0.51. In our baseline calibration this gives us p — 0.75. We set £ to match
the ratio between cash flow volatility and investment volatility, a {IK) /a (CFK) = 0.48.
Given all the other parameters, this gives us £ — 8.5.
tion.
Finally, the
parameters
(1988) report that
we choose 9
=
0.3.
30%
and Ie are chosen
as follows. Fazzari,
of manufacturing investment
The parameters 7 and
2%, as
of
7 and Ie and found
Hubbard and Petersen
financed externally. Based on this,
is
Ie are chosen to obtain an outside finance
Bernanke, Gertler and Gilchrist (2000).
of
in
9, 7,
that, as long as the finance
We
premium
experimented with different values
premium remains
at
2%, the
specific
choice of these two parameters has minimal effects on our results.
Impulse responses
3.2
In the model, the net investment rate of the representative firm
IK = h+i
(1
- S)k
is
t
t
H
14
We
use the
same data from Compustat
as Gilchrist
U.S. stock market listed firms from 1978 to 1989.
We
each variable. The moments reported in
a (IK) /a (CFK)) is a ratio of such means.
statistics separately for
Any
ratio used (e.g.
and Himmclberg (1995). The sample consists of 428
use the code of Joao Ejarque to calculate firm-specific
14
this
paper are the means across
all
firms.
.
0.05
IK
0.5
\
-
\^\^
^^^^__
"
-
marginal q
—
-
1
1
1
10
20
15
CFK
Figure
and the ratio
1:
Responses of investment,
1
and cash flow to a technology shock.
of cash flow to the firm's capital stock
CFK
Figure
q,
plots the responses of IKt,
t
=
qt,
A F(k
t
t
,l t
-w h
t
and CFKt, following a positive technology shock.
All variables are expressed in terms of deviations
All three variables in Figure 1 increase
financial frictions.
)
is
from
on impact,
their steady-state values.
as in the standard
However, the dynamics of average q are now jointly determined by
marginal q and by the wedge qt/q™. Marginal q moves one
q initially rises with investment, but at
its
model without
some point
(3
for
one with investment. Average
periods after the shock)
it falls
below
steady-state value, while investment continues to be above the steady state for several
more periods (up
to period 6 periods after the shock).
As marginal
q
is
reverting towards
steady state the wedge remains large, thus pushing average q below the steady state.
slow- moving dynamics of the
wedge are responsible
average q and investment.
15
for
its
The
breaking the synchronicity between
4>
E[R]/q
-0.05
Figure
2:
In Proposition
to
cj)
t
,
5,
Responses of
<j>
and expected returns
we argued that the
cf>
t
to the
is
its
dynamics of
of
<j)
t
,
<j>
t
steady-state value.
closely related to the slow adjustment of
To understand the response
is
The top panel
same technology shock, showing that
the shock, and then slowly reverts to
wedge
ratio of average q to marginal q
the marginal value of entrepreneurial net worth.
response of
to a technology shock.
recall
(f> t
positively related
of Figure 2 plots the
decreases on impact following
The slow adjustment
in the
.
from the discussion
in subsection 2.5 that the
are closely related to those of the rate of return
<p t
is
E
t
[Rt+i] /q™, since
(f> t
a forward-looking measure which cumulates the discounted returns on entrepreneurial
investment in
all
The dynamics
future periods.
of
</>
t
reflect the fact that the rate of
return
on entrepreneurial investment drops following a positive technology shock, as shown
the bottom panel of Figure
2.
15
Two
opposite forces are at work here.
persistent nature of the shock, future productivity increases
and
this raises
First,
in
due to the
expected returns
per unit of capital, Rt+i- This tends to increase the marginal value of entrepreneurial wealth.
At the same time, entrepreneurs' net worth increases because
flow.
of the current increase in cash
This leads to an increase in Kt+i, which reduces Rt+i, due to decreasing returns to
capital,
and increases g m due to adjustment
(
,
costs.
These
effects
tend to reduce the marginal
value of entrepreneurial wealth. In the case considered, the second channel dominates and
the net effect
15
is
a reduction in
Since the market rate of return
premium" E
t
[/?t+i]
/q™ —
E
is
t
[i?t+i]
/q™ and
in
<p t
.
As we
will see in
subsection 3.5, this
constant and equal to 1//3 C this also implies that the "outside finance
,
l//3c decreases following a positive shock.
16
depends of the type of shock considered, and can be reversed
result
with greater persistence. For now, what matters
the one-to-one correspondence between
IK
and
t
if
we consider shocks
that the dynamic response of
is
<fi t
breaks
qt-
Investment regressions
3.3
We now
turn to investment regressions, and ask whether our model can replicate the
on
cients
q
and cash flow observed
To do
in the data.
so,
we generate simulated time
coeffi-
series
from our calibrated model and run the standard investment regression
IKt
The
2.
ao
+a
1
frictions (9
=
points,
we
model are presented
coefficients
coefficient
frictions,
in the first
row of Table
obtained by Gilchrist and Himmelberg (1995).
latter are representative of the orders of
Absent financial
(19)
t..
report the coefficients that arise in the model without financial
and the empirical
1)
+ a2 CFKt + e
qt
regression coefficients for the simulated
As reference
The
=
q
is
magnitude obtained
in empirical studies.
a sufficient statistic for investment, so the model gives a
on cash flow equal to
In this case, the coefficient on q
zero.
which, given the calibration above
is
is
equal to l/£,
equal to 0.118, a value substantially higher than those
obtained in empirical regressions. Adding financial frictions helps both to obtain a positive
coefficient
on cash flow and a smaller coefficient on
reported in Figure
between
it
and
qt,
1
help us to understand why.
q.
The impulse response
functions
Financial frictions weaken the relation
while investment and cash flow remain closely related, due to the effect
of cash flow on entrepreneurial net worth.
Model with
a\
ai
financial friction
0.018
0.444
model
0.118
0.000
0.033 (0.016)
0.242 (0.038)
Frictionless
Gilchrist
Table
Third
2.
and Himmelberg (1995)
Investment regressions.
line:
Standard errors
Notice that under the simple
relation between q
AR1
and investment
in parenthesis.
structure for productivity used here, a sizeable corpresent.
is still
Running a simple univariate regression
of investment on q gives a coefficient of 0.13, not too far
and an
R
of 0.5.
once cash flow
matically.
the
R
2
To
is
This
is
frictionless coefficient,
not surprising, given that only one shock
added to the independent
see this, notice that the
R
2
is less
than
1
is
present.
variables, the explanatory
power of q
of the bivariate regression
is
of a univariate regression of investment
explanatory power of q
from the
on cash flow alone
is
virtually
falls
1,
dra-
while
0.995. So the additional
percent of investment volatility.
17
However,
The
values of
R
just reported are clearly unrealistic
and are a product of the simple
one-shock structure used. Furthermore, idiosyncratic uncertainty and measurement error
we do not attempt
are absent from the exercise. For these reasons,
16
empirical coefficients for q and cash flow.
model can help generate
calibration of the
to exactly replicate the
Instead, our point here
3.4
To
for idiosyncratic uncertainty
is
that a reasonable
both q and cash
realistic coefficients for
introducing a time-varying wedge between marginal q and average
model that allows
is
q.
An
flow,
by
extension of the
discussed below.
Sensitivity
we experiment with
verify the robustness of our result,
tions, in a
different
parameter configura-
neighborhood of the parameters introduced above. Table 3 shows the
of the investment regression for a sample of these alternative specifications.
basic result holds under a large set of possible parametrizations.
coefficients
Note that our
Moreover, a number of
interesting comparative statics patterns emerge.
Baseline
a\
0-2
0.018
0.44
9
=
0.2
0.012
0.50
61
= 0.4
0.025
0.39
a=
0.2
0.022
0.45
a=
0.4
0.017
0.44
4
0.022
0.67
0.015
0.35
0.017
0.44
0.019
0.45
0.023
0.36
0.011
0.57
£
=
£ = 12
E = 0.1
l
l
= 0.3
= 0.6
= 0.9
E
,0
p
Table
3.
First, notice that increasing 9 brings the
Sensitivity analysis.
economy
reduces the wedge between marginal q and average
coefficient
closer to the frictionless
(which determines the
coefficient
16
coefficient
on cash flow
By changing
initial
A
This accounts
for the increase in the
on cash flow when we increase
all
9.
How-
parameter changes that bring the
wealth of the entrepreneurs) both the coefficient on q and the
increase.
17
This
is
consistent with the general point raised by
the model parameters, in particular increasing 6 and
similar result
benchmark and
benchmark. In particular, notice that when we increase Ie
and Himmelberg (1995).
emerges if we decrease 7.
coefficients in Gilchrist
17
q.
comparative static result does not apply to
ever, this
economy
on q and the decrease of the
closer to the frictionless
18
£, it is
possible to
Kaplan
match exactly the
and Zingales (1997), who note that the
not necessarily a good measure of
how
coefficient
on cash flow
in
investment regressions
tight the financial constraint
is
is.
Increasing £ reduces the response of investment to the productivity shock and decreases
the coefficients of both q and cash flow.
technology shock,
The
cash flow.
To further
clarify
compare the
in the persistence of the
tends to lower the coefficient on q and to increase the coefficient on
effect of
changing p
is
analyzed in detail in the following subsection.
Current and future changes
3.5
to
p,
an increase
Finally,
in productivity
what determines the wedge between marginal and average
shocks with different persistence.
effect of
q, it is
useful
Figure 3 plots the impulse-
response functions of average q and marginal q for two different values of the autocorrelation
coefficient, p.
They can be compared
we
In panel (a) of Figure 3
case, the effect of the
to the middle panel in Figure
1.
plot the effect of a very persistent shock (p
shock on future returns dominates the
effect
=
0.98). In this
on current cash
flow.
Entrepreneurial investment becomes very profitable while entrepreneurs' internal funds are
The wedge
only catching up gradually.
the financial constraint
—
shock {p
0).
increases in the short-run, reflecting the fact that
In panel (b)
initially tighter.
is
This shock has the opposite
effect
are higher, while future total factor productivity
equilibrium rate of return
adjustment
move
in
costs.
E
f
The wedge
[i?t+i]
falls,
is
we
temporary
plot the effect of a
on the wedge on impact: internal funds
unchanged. As investment increases, the
IqT ^& ^ s due to decreasing returns to capital and convex
and this effect is so strong that average q and marginal q
opposite directions. Marginal q increases, due to the increase in investment, while
average q
falls reflecting
The two
the lowered expected profitability of entrepreneurial investment.
plots in Figure 3
show that the wedge between marginal and average
q captures
the tension between the future profitability of investment and the current availability of
funds to the entrepreneur.
They
also suggest that the observed volatility of q
the types of shocks hitting the economy.
of q to the volatility of the investment rate,
comparison, the value of the ratio a
(q)
we report the
In Table 4
a{q)/a{IK),
ja {IK)
for
depends on
ratio of the volatility
For
for different values of p.
Compustat firms
is
equal to 27.
18
In the frictionless benchmark, the ratio between asset price volatility and investment
volatility
is
equal to
£,
which we are keeping constant
the presence of the financial friction tends to
higher values of
p,
asset price volatility
volatility of q doubles
data.
compared
is
at 8.5. For values of p lower
dampen
amplified.
asset price volatility.
For example,
to frictionless case, although
it is still
than 0.89
However,
when p
—
for
0.98 the
smaller than in the
Highly persistent shocks to productivity help to obtain more volatile asset prices,
by generating variations
in the long
run expected return on entrepreneurial
role of shocks to future productivity in
emphasized
in
Abel and Eberly (2005),
capital.
The
magnifying asset price volatility has recently been
in the context of
See footnote 14 for calculation method.
19
a model with no financial
frictions,
.
a\erage q
marginal q
-0.5
20
10
40
30
60
50
70
80
100
90
(b)
-
0.1
marginal q
\
\
n
-A_10
Figure
Panel
3:
(a):
20
15
Responses of q and qm to a technology shock
p = 0.98. Panel (b): p = 0.
for different degrees of persistence.
but with decreasing returns and market power. This exercise suggests that a model with
constant returns and financial frictions can lead to similar conclusions.
shock considered here
in future productivity.
productivity,
is left
is
The
explicit
treatment of pure "news shocks," only affecting future
work (Walentin
i
a(q)./a(IK)
a (IK) /a
In Table 4
we
{CFK)
(2007)).
0.25
.50
0.75
0.98
0.967
8.5
8.5
8.5
8.5
8.5
18
1.9
2.3
3.4
5.5
16.5
27
0.19
0.24
0.33
0.48
0.73
0.48
p
4.
Shock persistence and the
volatility of q.
also report the effects of different values of p
vestment a (IK) /a (CFK). High values of p tend to increase the
relative to the volatility of cash flow.
a (IK) I a
(CFK) —
When
on the
volatility of in-
volatility of
we increase p we can
investment
re-calibrate £ to keep
0.48 (as in the baseline calibration above) and this leads to a further
increase in the volatility of
q.
In particular, setting p
the empirical values of both a (IK) /a
ble 4).
highly persistent
a combination of a change in current productivity and a change
to future
Table
The
(CFK) and a
—
0.967 and £
(q)
ja (IK)
=
18, allows us to
match
column
of Ta-
(see the last
Although the model does well on these dimensions, the required adjustment costs
20
seems very high and
for
cash flow.
A
an excessive degree of
this parametrization delivers
relatively easy fix
serial correlation
would be to introduce a combination of both temporary
shocks and shocks to long-run productivity. This would allow the model to deliver
correlation, while at the
same time having
current investment. Again, this extension
richer set of shocks
is left
movements
with
in q that are uncorrelated
better developed in a model that allows for a
is
to future work.
Firm-level Heterogeneity
4
So
and
larger
less serial
far,
we have focused on an economy where
all
firms have the
same
productivity,
and only
aggregate productivity shocks are present. This, together with the assumption of constant
returns to scale, implies that the investment rate,
identical across firms.
The advantage
and cash flow (normalized by
q,
of this approach
is
that
makes
it
assets) are
easy to compare
it
our results to the classic Hayashi (1982) model. At the same time, this approach has
on the relation between q and investment
limitations, given that the evidence
based on panel data. Therefore,
it is
useful to consider variations of the
is
its
largely
model that allow
for cross-sectional heterogeneity.
An immediate
extension
is
to allow for multiple sectors.
capital are immobile across sectors, which
w
run, then
and
to the aggregate
may be
q° are sector-specific prices
dynamics studied above.
If
we assume that labor and
a reasonable approximation in the short
and each
dynamics are analogous
sector's
Therefore, under this interpretation,
all
the
In this section, we pursue an
results presented so far apply to the multiple sector case.
alternative extension, by introducing productivity differences across firms. Let Ajj denote
the productivity of firm
j.
from a given distribution
process Aj it
simple,
we
=T
(Ajj-i,€j
The
t
t)
From then
with
e^t
receive
an
is
drawn from the
is
discrete p.d.f.
1
it
(e^j).
To keep matters
always identical to the ex-ante distribution for each individual
details of this extension are presented in
the wage
and equal to
than
random draw Ajj
on, individual productivity follows the stationary
Appendix B.
Given the absence of aggregate uncertainty, aggregate capital
and so
initial
abstract from aggregate uncertainty and assume that the realized cross-sectional
distribution of the shocks
firm.
<J>.
Newborn entrepreneurs
and
is
1.
w
and the price of used capital
q°.
to simplify the problem, as
it
The assumption
constant in this economy
This also implies that q m
However, as long as the financial constraint
different across firms.
is
is
of constant returns to scale
implies that the investment rate, Tobin's
flow-to-assets ratio are independent of the individual firm's assets kj )t
variables are
now
functions of the firm's productivity Aj
t
t
_
[l-9)Rht -\
l-0/3 c E[J?jit+1 |A,-,
21
t]
-
q,
constant
is
greater
still
helps
and the cash-
However, these three
and are given by the following
three equations,
ht
is
binding, average q
'
qht
= pE {i-e)m
[(
7
+
-
(1
7)
</>
jlt+ i)
CFK = R
J)t
where the return per unit of
(j>j t
,
are
The
now
Rj
capital,
firm-specific variables.
three expressions above for
t
,
jit
-
+ Wc® [Rj,t+i\A
jt t]
,
q°,
and the marginal value of entrepreneurial wealth,
19
IKj t,
t
between current and future changes
Rj*+Mj,t]
Qj,t,
and CFKjj, emphasize once more the tension
On
in productivity discussed in subsection 3.5.
the one
hand, current returns, captured by Rj tt affect positively both the investment rate and cash
,
but have no
flow,
effect
on
q,
future returns, captured by
effects
on current cash
which
[.Rj^+ilA^t], affect positively
IE
On
a purely forward-looking variable.
is
q,
but have no
flows.
To study the implications
of the
model
for
investment regressions, we construct simulated
model described and run the investment regression
time-series from the
the other hand,
investment and
In Table 5
(19).
we report the
regression coefficients obtained from the simulated series, using the
parameters as
in Section 3.
Model with
..,
-
Table
Once more,
financial friction
ai
<Z2
0.116
1.023
Investment regression. Firm-specific shocks.
5.
financial frictions introduce a strong correlation
between cash flow and
and
a?,
are
now
larger than in the corresponding line of Table 2
than their empirical counterpart. This
is
not surprising, given that firms
in-
Notice that both
vestment, so that cash flow has a positive coefficient in the regression.
coefficients a\
same
now
and larger
face essentially
zero adjustment costs. In this model, adjustment costs are only due to aggregate changes
in the capital stock,
and with no aggregate uncertainty such changes are absent. 20 Another
implication of the absence of adjustment costs
a (IK) J a
data.
21
(CFK)
is
that investment
is
too volatile. The ratio
equal to 1.34 in the simulated series, more than twice as large as in the
In our model
firms to trade
is
we have
homogeneous
essentially
capital
with firm-specific shocks clearly
both to reduce investment
assumed "external adjustment
on the used capital market.
calls for
fully
by allowing
developed model
the introduction of "internal adjustment costs,"
volatility at the firm level
cients in investment regressions.
A
costs,"
and
to obtain
more
However, with internal adjustment costs we
realistic coeffi-
lose analytical
tractability, as optimal investment rules are, in general, non-linear.
I9
Both Rj jt and jt are functions only of A,,t, so the distributions of Rj,t+i and <t>j +i conditional on
can be obtained from the law of motion A~,t+i = T (Aj t,tj,t+\)20
The parameter £ is accordingly irrelevant for this version of the model.
21
Notice also that the frictionless model is not a very useful benchmark in this case, as it gives very
extreme and unrealistic results. Absent financial frictions all the capital stock in the economy would go,
each period, to the single firm with the highest expected return on capital, while q would be constant and
<j)
:t
Aj,t
equal to
:
1.
22
Conclusions
5
we have developed a
In this paper,
frictions
tractable framework to study the effect of financial
on the joint dynamics of investment and of the value of the
The model shows
firm.
that, in the presence of financial frictions, q reflects future quasi-rents that will go to the
This introduces a wedge between average and marginal
insider.
is
determined by the tension between current and future
q.
The
that the growth of
is
and
this raises the future
The paper
its
capital stock
is
A
profitability.
future productivity and low internal funds today will display a higher
this
size of this
wedge
firm with high
The reason
q.
for
constrained relative to expected productivity,
marginal product of capital.
focuses on the implications of the
model
between invest-
for the correlation
ment, q and cash flow. In particular, we show that a model with financial frictions can help
between q and investment, and the
fact that cash
flow appears with a positive coefficient in standard investment regressions.
However, the
to replicate the observed low correlation
model has a number of additional testable predictions on the response of investment and
asset prices to different types of shocks (shocks with different persistence, shocks affecting
current/future productivity), as discussed in Section
market power and decreasing returns
3.5.
As we
noticed, recent models with
at the firm level also display rich
ing shocks with different temporal patterns.
dynamics
follow-
Empirical work documenting the conditional
behavior of investment and q following these shocks, would provide an important testing
ground
for
both
classes of models.
Throughout the paper, we have maintained Hayashi's (1982) assumption of constant
returns to scale both in the production function and in adjustment costs.
advantages.
First,
it
greatly simplifies aggregation.
Second,
it
This has two
allows us to focus on the
"pure" effect of the financial friction on investment regressions.
Models with decreasing
returns at the firm level can produce deviations from q theory for independent reasons, so
it
is
useful, at this stage, to separate those effects
financial contracts.
At the same time,
from the
effects
this choice leaves aside a
due to imperfections
number
of interesting issues,
which seem especially relevant when one introduces firm-level heterogeneity,
Section
in
as
we did
in
4.
Finally, in the
steady state.
It is
paper we have focused on the case of small stochastic deviations from the
possible to extend the
to potentially interesting
model
phenomena. In
to allow for "large" shocks, opening the door
particular, with large shocks
a model where firms hold precautionary reserves,
in order to
buy
i.e.,
it is
possible to have
choose to reduce investment today
financial securities as insurance against future shocks. This
where equilibrium behavior
will
be very sensitive to the time
the firm.
23
is
another area
profile of the shocks hitting
'
.
Appendix
A. Proofs
Proof of
Lemma
1
Consider the problem
minG(/c',fc )+g°/c
=
Suppose k°
any
=
k°
k',
k
k
(q°)
we can
Therefore,
optimal
is
for a given q°
and
m
completing the proof of the
first
(X)
= G(n(q
=
r)(w,q°,A)
-
(u\ q°, A))
tj
is
equal to
(G
(k (q°)
,
1)
+
q°) k'
optimal
is
+ q°(X),
part of the lemma. In a similar way, consider the problem
any
k,
I
wt] (w, q°,
A)
+
to scale imply that, given
{AF (1,
optimum
{X)),l)
max AF(k,
I
Constant returns to scale imply that, given
1.
set
q
and suppose
=
k'
a solution to problem (20) and the
(q°) k' is
(20)
.
R (X) = AF (1, V (w (X)
q°k,
(21)
=
a given triple w, q°,A and k
for
=
-wl +
I)
rj{w,q°,A)k
is
1.
Constant returns
a solution to (21) and the optimum
is
q°) k. Setting
q°
,
(X)
,
- w (X)
.4))
r,
(w (X) q° (X)
,
,
.4)
+
q°
(X)
completes the proof.
Proof of
Let
B
Lemma
2
be the space of bounded functions
T ^ (x) = Pe (1
,
Let us
first
and P E
check that T<p 6
< Pc
g)
]
[
B
if
</>
</>
:
X—
[1,
»
oo). Define the
-
6)
for
any
(1
showing that T<p(X)
S B, so the
map
is
well defined. Notice that conditions (a)-(b)
4>
B
£
-
>
1.
(X)
<
M
for all
Next, we show that
ity of
T
is
X
1.
we have
Assumption
(c)
P E (l-9)E [R (H (X, e'))]
q™ (X) - 90 C E[R(H (X,e<))}
e X, then
l
}
T0 (X) < Mj
(1
-
7) for
1
1
all
-7
X
g X, completing the argument.
T satisfies Blackwell's sufficient conditions for a contraction.
easily established.
'
implies that
p E (l-8)E[R(H(X,e'))}
g- (X) - 80 C E [R (H (X, £'))]"
if cp
as follows
[(
(H (X, e'))) R (H (X, e'))}
7)
~
™(X)-60 c E[R(H(X,e'))]
E [(7 +
q
so
»
imply that
This implies that
(1
:
7 + (1 - 7) <P (H (X, e'))) R (H (X, e'))}
m
(X)-e/3 c E[R(H(X,e'))}
q
E
(l-8)0 E E[R(H(X,e'))}
>
q™(X)-ej3 c E[R(H(X,e>))}
E
map T B — B
To check that
it
satisfies
24
The monotonic-
the discounting property notice that
if
<j>
—
+
<(>
then
a,
-
T4> (X)
=
T4> (X)
where the inequality follows from assumption
exists
and
a
qm{x) _ g0cE[R{H{Xtel))]
(22) immediately shows that
<f){X)
for all
1
T
Since
(c).
>
< Pb°,
a contraction a unique fixed point
is
X.
Proof of Proposition 3
Let
is
4>
be defined as
V (k, b, X) —
Lemma
in
4>{X)
(R{X)
k
We
2.
—
proceed by guessing and verifying that the value function
no-default condition can be rewritten in the form
max
B +
c
c B ,k',b'(.),b'L (.)
E
-7
(1
)
c
+ qm
E
=
b
T
n
Therefore,
(3').
(H
(e') \4>
{X,
that,
under
(R (H (X,
e'))
this conjecture, the
we can rewrite problem (P)
-
e')) k'
b' (e'))l
as
+
'—f
+(3 E -f
s.t.
we have shown
In the text,
b).
{X)
+ (3 C
d
J2*(z')lK(H(X,e'))k'-b'L
< R (X) k-d,
k'
( (1
-7
Y; n
)
(e')}
(A)
(e') b' (e')
+7
£>
(e') b'L (e')
(/i)
,
J
(0 < 6R {H (X, e')) **
b'
b'
L
(e')
E >
c
fc'
where,
in parenthesis,
>
e',
(y
<eR(H(X,e'))k'foialle'\
0,
(e') tt (e'))
[y L (e')Tr(e'))
(r c )
0,
(r fc )
we report the Lagrange
multiplier associated to each constraint.
ers of the no-default constraints are normalized
for this
for all
by the probabilities
it
The
(e').
- A + tc =
0,
EM
-7
[(
7+
(1
)
0') R'}
- \q m (X) + 9E [(j/ +
A
(£') + A£ c (1- 7 7T (e') -P E (1 - 7)
-^ S 77r (£') + \0cyn (O - u L (e') w (e') =
^
)
(£')
u L )R'\
n
1
(e
)
+
=
first-order conditions
J?' and 0' are shorthand for R (H (X, e')) and <p (H (X, e')). We want
E
c ,k',b' and b'L in the statement of the proposition are optimal.
It
that they satisfy the problem's constraints.
= A—
1
>
0, Tfc
=
0,
and
v{t') ,vl
(e')
To show that they
>
for all
e'
.
m
q
which, by construction,
is
equal to
<fr
{X)-
tc
{X)
+
Setting t>
+ (1-7)0')^]
ep c E\R'}
Then we have
=
4>
{X)
25
-
1
>
o,
=
0,
0,
to
show that the values
is
immediate to check
are optimal
condition gives us
(l-fl)/? E E[( 7
r fc
0,
where
rc
multipli-
problem are
1
for
The
=
we need
to
show that
the second first-order
which follows from
Lemma
v
2,
(0 =
which follows from condition
(1
function
<t){X)
we obtain
>
1
- P E (H
7) (P c 4> (X)
<t>
(X,
>
e')))
0,
and
(d),
vL
which follows from
-
= (l-j)(P c <t>(X)-0 E )>O,
(e')
and
(3
—
b)
(X) (R (X) k
C >
j3
E
Substituting the optimal values in the objective
.
confirming our
initial guess.
Proof of Proposition 4
The proof
economy,
two
split in
is
in the second,
steps.
we
In the first step,
we construct an equilibrium
derive the steady state of the deterministic
of the stochastic economy. Conditions (A)
(B) will be introduced in the course of the argument. First,
Applying the envelope theorem to problems (20) and (21)
fact that, in equilibrium, the ratio k°/k'
and using condition
we obtain the
(12),
,?
-
Rt
=
equal to
1.
(see the proof of
following expressions for
^M
Lemma
g™ and
R
t
1 in
each period
(recall
We
model).
is
1),
using the
equal to
l/K
t
,
:
e0{
(23)
,
OK-t + 1
At
t
8G(Kt+1 ,K
8K
,l)
—dK
t
t)
(24)
•
t
(Deterministic steady state) Consider a deterministic model where
in the stochastic
and
derive a useful preliminary result.
equal to Kt/Kt+i, and the ratio l/k
is
0F(K
Step
we
that
1 is
will derive
the unconditional
t
is
constant and
of the stochastic process for
a steady state of this deterministic model and use
reference point for the stochastic case. Let the superscript
S
=
1
state the equilibrium conditions (12)
mean
A
and
,s
(23) give q°
it
A
t
as a
denotes steady-state values. In steady
m
S and q s = 1. The law of motion
—
'
for the capital stock (9) gives the steady-state condition
(1
and
- 90 C R S )
Ks
=
(1
-7
(1
)
-
6)
R S K S + ~tw s
l
E
(25)
,
(24) gives
,
Rs =
Substituting (26) in (25)
^^
dF (K s
,
1)
+ 1-5.
we obtain
K s =~\f a(0|gc + (l-7)(l-g)) + 7(l-tt)M
l-(90 c + (l--y)(l-e))(l-6) J
and substituting back
(26)
in (26)
we
get
Rs = a
(A'
5 )"" 1
26
+1-5.
^
(27)
]
[
'
We make
the following assumption on the model parameters
'•
The
l-(flg c + (l-7)(l-fl))(l-*)
+
" a(6(3 c + (1 - 7) (1 - 0)) + 7 (1 - a)l B
1-5 )>1.
(A)
following three inequalities follow from assumption (A):
s
The
(these correspond to assumptions (a)-(c) in Proposition 3).
To
diately.
(0 C 6
+
(1
first
inequality follows
imme-
show that the second inequality holds notice that assumption (A) implies that
-
-
7) (1
0)) (1
Rearranging equation
-
(25),
given that
5) is positive,
j3
E
(I
-
-
9)
Rs >
S)
<
Then,
1.
(27) gives us
K
s
1
—
>
0.
one can then show that
1
-
(3
C 6R
S
-
(1
-
7) (1
0,
which implies both the second and the third inequalities.
In steady state, the recursive definition of
(1-8)13^
s
1 - ep c R
k3
Rearranging
Step
2.
this equation
(X),
<fi
shows that
>
<p
(7
+(1-7)/).
Condition (d) holds immediately, given that
1.
(Stability) Substituting (11), (24)
and the lagged version of
following second-order stochastic difference equation for
/i
(1
t+1
~~
- 7)\n
(1 -
—m
dG(K
t
o\ ( A dF(K,,l)'
0) (A t
gK ;
-
K
+ 1 ,K7)
-
t
takes the form
(6),
K
Q'
v
jKt+'yAt
dG(K, + i,K )\
t
g^'
'
.
.
dF(K ",l),
1
t
t
'
(3
C
.
we obtain the
l
—
E
~j
l
+ 1 X)
1
made
>
9L
dC(K- + 2 ,K
a K^-
9F(K, + ,1)
00 h
\( A
op
c K t [[At+i gKt+1
ln7\f
E <
t
Linearizing this equation (under the functional assumptions
— \nK s
=
second order equation for
kt
(10) into (9),
(3
in the text)
JJ
we
get the following
,
/C(
+
Otikt+l
+
=
Q2^( + 2
0,
where
qo
ai
q2
=
^
+ a(l-a)(-/l E -(l- 1 )(l-e)){K s
= -e-l + /3ei? S -^(? +
= pec
a-1
)
Q'(l-Q)(A- s )
Q"1
+ (R s -0(l-l)(l'e),
)+(l-7)(l-^)e,
Pro\aded that
a\
it is
possible to
show that the steady
state
- a a2 >
Ks
is
saddle-path stable. Then, given sufficiently small
shocks we can construct a stochastic steady state where
gives us
an ergodic distribution
(B)
for the state vector
27
K
t
varies in
a neighborhood of
X, with bounded support.
We
Ks
.
This,
can then establish
the continuity of the function
in
[<p,
0].
with respect to the parameters
<j>
A—A
hold in the stochastic steady state. Finally,
4>
(X)
X
This guarantees that condition
B.
The model with
Let
w
(d)
(X)
is
bounded
can be set so as to ensure that the bounds for
C
> 0e$.
also satisfied.
is
firm-level heterogeneity
and q° denote the constant values
return per unit of capital
is
now
for the
wage and the price
where
77
t
t,
The
the labor to capital ratio.
is
=
max
c B ,k',b'(.),b' {.)
L
c
B
is
subject to
+ /3 E
.
c
=
(gross)
d
(1
- j)E[V
now
(fc',6' (e')
,T{A,e'))}
+
(e')}
<R {A) k -
d,
7 )E[6'( e ')]+7E[(/L
b'(e')
<
9R(T(A,e'))k',
(e')
<
6R(T(Ae'))k'.
L
,
„..
E +k'
+ /3 c ((l-
b'
}
characterized by the Bellman equation:
+P El E[R(r(A,e'))k'-b'L
6
The
state variables for an individual entrepreneur are
and Ajj. The entrepreneur's problem
V(k,b,A)
of used capital.
defined as:
R (Ajj) = max {Ajlt F (l,r?) - wr] + q
The
<fi
satisfy
/3
kj,t,bj
and show that
Since (a)-(c) hold in the deterministic steady state, a continuity argument shows that they
(e')]),
no-default constraints have been expressed as linear constraints, proceeding as
we did
in
Propo-
sition 3.
Now
the marginal value of entrepreneurial wealth,
A and we
j
have
0[
is
a function of the individual productivity
,
E
,
,
(i-P|(7 + (i-7)»(r(i,f')p(r(A, f '))]
l-ep c E[R(T(A, £ '))]
'
The analogues
<fi,
to conditions (a)-(d) are
now
P E E[R(r(A,e'))\
>
1,
ep c E[R(T(A,e'))}
<
1,
(i- 7 )(i-fl)E[fl(r(A,
e '))]
l-O0 c E{R(T(A,e')))
and
<P(A)>^4>(T(A,e')).
Pc
Under these conditions the optimal individual policy can be derived
28
as in Proposition 3,
and we
obtain the following law of motion for the individual capital stock
k
A
newborn entrepreneur has
$ and
the law of motion
T
,
e'),
l-ep c E[R(T(A,e'))]
wealth wis- Putting together these conditions, the distribution
initial
(A,
(i-e).R(A)
=
allows us to completely characterize the joint dynamics of k and
A. Then, under appropriate assumptions,
that the wage rate
w
is
we obtain an ergodic
=
[q{A)k}dJ{A,k)
where
rj
{A)
is
joint distribution
J
(A, k)
and check
consistent with the market clearing condition
1,
the optimal labor to capital ratio for a firm with productivity A.
Proceeding as
Pj,t
Substitute for dj
t
t,
we can
in subsection 2.5,
=
<t>j,t
(
define the financial value of a continuing firm
R3,tkJlt -
using the budget constraint dj
}
t
b ht)
=
+
bjtt
Rj,tkj
t
t
-
—
dj, t
j:
.
kj,t+i,
and the law of motion
for the
capital stock
kj t+1
=
- ep c E [2WMW
>
i
(Rj ' ht
" ht)
'
to obtain
P]
<
- {*" +
1
- 90 c E[Ru+1 \Au})
ht] " " Kt)
[
Dividing both sides by k Jtt +i and using the recursive property of
for
<f>^ t
gives the following expression
Tobin's q
q ht
= P E (l-9)E
[(
7
+
(1
- 7
) <t>jt
i+i) Rj,t+i \Ajtt ]
+
9j3 c
E
[R ht+l \A 3 A
For the investment rate notice that
which gives the expression
in the text. For cash flow notice that
29
Aj
t
F (kj
it
,
I
Jt )
—
wljj
=
Rj,t^j,t
~
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