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OBWEY
5
5
Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
IDIOSYNCRATIC PRODUCTION
AND THE
RISK,
GROWTH,
BUSINESS CYCLE
George-Marios Angeletos
Laurent E. Calvet
Working Paper 02-10
February 2002
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn. com/paper.taf?abstract_id=xxxxx
loO
Massachusetts Institute of Technology
Department of Econonnics
Working Paper Senes
IDIOSYNCRATIC PRODUCTION
AND
RISK,
GROWTH,
THE BUSINESS CYCLE
George-Marios Angeletos
Laurent E. Calvet
Working Paper 02-10
February 2002
Room
E52-251
50 Mennorial Drive
Cambridge,
MA
02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn. com/paper.taf?abstract_id=xxxxx
MASSACHUSEHS
INSTITUTE
OF TECHNOLOGY
MAR 1
5 2002
LIBRARIES
Idiosyncratic Production Risk,
Growth, and the Business Cycle
George-Marios Angeletos and Laurent E. Calvet*
MIT
and Harvard University
This draft: February 2002
We
examine a neoclassical growth economy with
markets.
Under a CARA-nomial
general equilibrium
in
idiosj'ncratic
production risk and incomplete
specification for preferences
and
risk,
we characterize the
closed form. Uninsurable production shocks introduce a risk
premium on
private equity, and typically result in a lower steady-state level of capital than under complete
markets. In the presence of such
risks,
rates discourages current risk-taking
the anticipation of future low investment
and feeds
and high
interest
bcick into low investment in the present.
endogenous macroeconomic complementarity thus
arises,
An
which can amplif\' the magnitude and
the persistence of the busmess cycle. These results - contrasting sharply with those of Aiyagari
(1994) and Krusell
risks
and Smith (1998) - highlight that
can have significant adverse
effects
idiosyncratic production or entrepreneurial
on capital accumulation and aggregate
volatility.
Keywords: Entrepreneurial Risk, Investment, Growth, Fluctuations, Precautionary Savings.
JEL
Classification:
D5, D9, E2, E3, 01.
'Correspondence: 50 Memorial Drive, Cambridge,
MA
02138, laurent_calvet@harvard.edu.
MA
This manuscript
02142, angelet@mit.edu; Littauer Center, Cambridge,
is
a thoroughly revised version of an earlier working
paper, "Incomplete Markets, Growth, and the Business Cycle".
Robert Barro and John Campbell
from D. Acemoglu,
P.
for detailed
Aghion, A. Alesina,
J.
We
are especially grateful to Abhijit Banerjee,
feedback and extensive discussions.
Benhabib, A. Bisin, R. Caballero,
We
also received helpful
F. Caselli, C. Foote,
comments
O. Galor,
J.
Geanakoplos, M. Gertler, J.M. Grandmont, O. Hart, N. Kiyotaki, D. Laibson, N.G. Mankiw, M. Mobius, G. Moscarini,
H. Polemarchakis, M. Schwarz, K. Shell, R. ShiUer, C. Sims, R. Townsend, T. Vuolteenaho, and seminar participants
MIT, NYU, Yale,
Summer Macro Workshop.
at Brown, Harvard, Maryland,
2001 Minnesota
the 2000 Conference of the Society of Economic Dynamics, and the
Introduction
1
The standard
model
neoclassical growth
Ramsey-Cass-Koopmans assumes complete markets,
of
implying that private agents can fully diversify idiosyncratic risks in their production and invest-
ment
However, large undiversifiable entrepreneurial and production
activities.
paramount
risks are not only
developed countries, but also cjuantitatively significant even in the most ad-
in less
vanced economies. In a recent study of
document that entrepreneurs
extreme dispersion
US
private equity, Moskowitz
and Vissing-j0rgensen (2001)
face a "dramatic lack of diversification" in their investments
in their returns.^
Similarly, moral hazard
and an
and agency problems imply that the
executives of publicly traded firms are very exposed to the risk in the production and investment
decisions they
make on
accumulation of
A
human
natural question
behalf of the shareholders. Large idiosyncratic risks are also evident in the
or intangible capital.
is
then how the presence of large undiversifiable idiosyncratic risks in en-
trepreneurial, production, and investment choices, matters for
paper we propose that incomplete
macroeconomic outcomes.
risk sharing leads to substantial
underaccumulation of capital
and generates a powerful amplification and propagation mechanism over the business
We
The technology
is
standard neoclassical and
and each agent can invest
as
to idiosyncratic uncertainty.
much
If
both a consumer and a producer.
convex. There are no credit-market imperfections
as he likes in his production.
However, production
and the model
wo\ild reduce to a standard
is
subject
fully diversify
Ramsey economy.
markets are incomplete, however, individuals must bear at least part of the idiosyncratic
risk in their
production and investment choices.
Our model belongs
tlie
to the class of general-eqmlibrium economies with incomplete markets
United States, private companies accounted
Moskowitz and Vissing-Jorgensen
for
fully
is
markets were complete, indi\'iduals would be able to
their idiosyncratic production risks,
'111
cycle.
introduce idiosyncratic production risk and incomplete insurance in an otherwise standard
neoclassical growth economy. In our model, each private agent
When
In this
whom
also observe that
for
about half of production and corporate equity
"About 77 percent of
all
private equity
is
and
in 1998.
owned by households
private equity constitutes at least half of their total net worth. Furthermore, households with private equity
ownership invest on average more than 70 percent of their private holding in a single private company,
household has an active management interest.
[...]
in
which the
Survival rates of private firms are around 34 percent over the
first
10 years of the firm's hfe. Furthermore, even conditional on survival, entrepreneurial investment appears extremely
risk)',
generating a wide distribution of returns."
"Castenada
et al.
(1998) and Quadrini (2000) have argued that entrepreneurship
is
wealth distribution- Heaton and Lucas (2000) have shown that entrepreneurial income
to explain portfohos
and
risk premia.
important
is
in
explaining the
also important empirically
Models of
heterogeneous infinitely-lived agents.
from the "curse of
this class typically suffer
mensionality" because the wealth distribution - an infinite-dimensional object variable.
By
considering exponential preferences and Gaussian risks,
gates independent from the wealth distribution.
is
we render
di-
a relevant state
the macro aggre-
The equilibrium dynamics are then characterized
in closed form.
The
analytical tractability of the
model allows us to
clearly identify
how uninsurable
idiosyn-
both individual choices and aggregate outcomes. Because exposure to
cratic production risk affects
production risk can be controlled through the level of investment, production risk has an ambiguous
impact on precautionary savings. Most importantly, production risk introduces a
private equity and therefore unambiguously reduces the aggregate
sult,
demand for
risk
premium on
investment.''
As
a re-
the steady-state levels of capital and income typically decrease with the variance of uninsurable
production shocks.
Perhaps more
surprisingly, idiosyncratic production risks introduce a kind of
endogenous dy-
namic macTveconomic complementarity, which can slow down convergence to the steady
state,
amplify exogenous aggregate shocks, and increase the persistence of the business cycle.
This
macroeconomic complementarity has a simple
origin.
When
uncertainty, an individual's willingness to invest depends
on
is
subject to idiosyncratic
his ability to self-insure against un-
Risk premia and investment
diversifiable shocks to future production.
anticipated future credit conditions.
production
In our model, high interest rates
demand
thus depend on
and low investment
in
period feed back into high risk premia, low risk-taking and low investment in earlier periods.
one
Low
investment, low income, and low savings can thus be self-sustaining for long periods of time.
The
basis of this
mechanism
is
a particular pecuniary externality. If agents save too httle in
one period, they discourage risk-taking and investment
This externality arises only
complementarity
is
in the presence of idiosyncratic
periods via the interest rate.
production
risks.
The macroeconomic
thus a genuine general-equilibrium implication of a market imperfection.
The macroeconomic complementarity
ing constraints.
in earlier
Our mechanism
is
arises in
our model even though agents face no borrow-
thus independent from - and
is
in fact
complementary to -
the amplification and propagation mechanisms that are generated by credit-market imperfections.^
Important examples include Aiyagari (1994, 1995), Constantinides and Duffie (1996), Heaton and Lucas (1996),
Huggett (1993, 1997), Krusell and Smith (1998), and Rios-RuU (1996). See Rios-RuU (1995) and Ljungqvist and
Scirgent (2001) for a review of this Hterature.
'
Our model thus complements
earlier research,
which has stressed the impact of aggregate production shocks on
investment. See for instance Acemoglu and Zilibotti (1997) for the complete market case.
^See for instance Bernanke and Gertler (1989, 1990), Kiyotaki and
Piketty (1999).
Moore
(1997),
and Aghion, Banerjee and
We
share with this hterature the idea that market imperfections can have important imphcations
we
First,
modeh however,
outcomes. Our
for aggregate
differs
from these earher approaches
in three ways.
focus on incomplete insurance, which affects the willingness to invest, rather than borrow-
ing constraints, which also affect the ability to invest.
(OLG)
rather than overlapping generations
on finite-horizon
effects
Second,
we
consider infinitely-lived agents
economies; this ensures that our results do not depend
and makes our framework directly comparable to the standard
RBC frame-
work. Third, the pertinent literature has centered around the dependence of aggregate outcomes
on the wealth distribution as the only channel through which market incompleteness can generate
propagation and amplification; the pecuniary externality identified in this paper has not been considered before and
it is
thus an innovative finding on
its
own
that incomplete markets can generate
amplification and propagation even in the absence of the wealth-distribution channel.
Our
analysis concludes that incomplete insurance generates untfeT-investment in the steady state
and increases both the magnitude and the persistence of aggregate fluctuations. Aiyagari (1994) and
Krusell and Smith (1998) also examined the effect of incomplete markets in the neoclassical growth
model, but reached a dramatically different conclusion.
They argued that incomplete insurance
generates overinvestment in the steady state and has no important effect on the business cycle.
The
contrast between their results and ours has a simple origin.
production uncertainty as the main source of
consider instead the effect of idiosyncratic
individual investment. Idiosyncratic
do not introduce a
in risk taking.
This explains
and business-cycle
The
risk
premium on
rest of the
effects
paper
why
Aiyagari (1994) and Krusell and Smith (1998)
endowment shocks, which do not
endowment shocks
affect
the returns on
influence only precautionary savings.
They
private equity and thus do not generate a pecuniary externality
Aiyagari and Krusell and Smith did not identify the steady-state
documented
is
risk,
While we view idiosyncratic
in this paper.
organized as follows. Section 2 introduces the economy and Section 3
analyzes the individual decision problem. In Section 4 we characterize the general equilibrium in
closed form, analyze the steady state,
and describe the propagation and amplification mechanism
that arises in the presence of idiosyncratic production risk.
simulations.
We
and directions
2
Section 5 presents
some numerical
conclude in Section 6 with some remarks on the Ukely robustness of our findings
for future research. All proofs are in the
A Ramsey Economy
Appendix.
with Incomplete Risk Sharing
This section introduces a neoclassical growth economy with decentrahzed production,
erences, Gaussian idiosyncratic uncertainty, and an exogenous incomplete asset span.
CARA
pref-
Technology and Idiosyncratic Risks
2.1
Time is
discrete
and
infinite,
indexed by
t
N=
G
variables are defined on a probability space
They
are
all
born at date
Eacli individual
using his
own
0, live forever,
(
,
A:/
and
satisfies the
investment.
The
Tlie
economy is
The technology
and
single
who
is
endowment
f{k)
=
The output
F
is
deterministic and
The
F{k,l).'
individual controls
same labor supply,
through
his
J}.*^
in every date.
own production scheme
operates his
standard neoclassical.
con-
It exhibits
labor, decreasing returns with respect to the
of his production in period
l^
total factor productivity
fc^
random
G J ={1, •,
We
individual can invest in a single type of capital.
t.
individuals have the
all
j
all
Inada conditions. There are no adjustment costs and no
in period
The production function
by
consumption good
the stock of capital that individual j holds at the beginning of period
labor
and
stocliastic
P). Individuals are indexed
,
and consume a
stant returns to scale with respect to capital
divisibilities in
T
...},
also a producer, or entrepreneur,
is
labor and capital stock.
capital/labor ratio,
{0, 1,
common
—
Aj
across
all
t,
is
t
and by
l\
in-
denote by
his effective
given by AlF{kl,lf).
We
individuals.
assume that
and thus conveniently consider the function
I,
a random shock specific to individual
is
investment at date
t-1,
while A^
is
Production and returns are thus subject to idiosyncratic uncertainty.
The
j.
observed only at date
This risk
is
what we
t.
call
idiosyncratic production risk (or, technological, entrepreneurial, or investment risk, depending on
our preferable interpretation).
For comparison with production
e^
introduce endowment
risks, it is useful to also
We
risks.
let
denote the exogenous stochastic endowment of consumable good that individual j receives in
period
t.
These shocks model
and do not
risks that are outside the control of individuals
production or investment opportunities.
The
overall non-financial
income of individual j
in
affect
period
t is
yi
The random shock A^
is
=
Aif{ki)
+
endowment
multiplicative, while the
(1)
ei.
risk e^ is additive.
Asset Structure
2.2
Idiosyncratic risks can be partially hedged by trading a limited set of short-lived securities, which
are indexed
amount
The
of
by
m
e
{0, l,..,Af}.
Purchasing one unit of security
consumption dm,t+i at date
vectors
The model
ttj
=
iTTm,t)m=o ^^"^ '^
—
t
+ l. The
>
>
f"{k)
Vfe
G
at date
price of this security at date
i^,t)m=o
directly extends to economies with a
'/ satisfies f'{k)
m
^i'^'
continuum
(0,oo), Umfc„o/'(fc)
=
t
t is
yields a
denoted by
respectively, the price vector
of agents.
oo,
and lmik-.oof'{k)
=
0.
random
TTm.t-
and the payoff
vector at date
t.
=
1
delivers dot
gross
that
t
and Rt =
and
t
+
assets
m
and
1/ttoj
M}
e {1
?(
=
Default
At the outset
contemporaneous realization of
is
The bond
Rt-l denote respectively the
is
rule out
Finally, for simplicity,
of every period
asset payoffs, dt,
We
not allowed.
any other kind of credit-market imperfections.
{(yl(,ej)}jgj. Information
are risky.
1.
assets are in zero net supply.*
all
are informed of the
t,
individuals
and idiosyncratic shocks,
thus homogenous^ across agents and generates a filtration {!Ft}tZo-
denote by Et the expectation operator conditional on Tt.
Preferences
2.3
To
t,
and
exogenous and generally incomplete.
is
short-sales constraints or
We
a riskless bond,
between
interest rate
asset span
we assume
is
with certainty in every
and the net
The
m=
Security
distinguish between intertemporal substitution
and
risk aversion,
we adopt a preference
cation that belongs to the Epstein-Zin non-expected utility class. Consider
utilities
U
A
and T.
specifi-
two concave Bernoulli
stochastic consumption stream {c(}^o generates a stochastic utility stream
{ut}^o defined by the recursion
Uiut)
where CEt{u)
=
=
Uict)
pU[CEt{ut+i)] Vf >
+
T"-'[EtT(u)] denotes the certainty equii-alence of u.
governs risk aversion, while the cur\-ature of
U
measured
=
T thus
When T = [/,
The curvature
of
Eq ^J^q/3'{/(c(). Finally, note that
consumption units.
in
CARA-Normal
2.4
(2)
governs intertemporal substitution.
these preferences reduce to standard expected utility, U{uo)
Ut is directly
0,
Specification
Closed-form certainty-equivalent measures are not possible for general risks and general preferences,
but can be obtained
Assumption
1
in
CARA-normal
the
(Exponential Preferences) Agents have
U{c)
Ricardian equivalence holds
'
debt
is
financed by
case.
lump-sum
in
= -* expC-c/*).
T(c)
=.
identical recursive utility (2) with
-(1/r) exp(-rc).
(3)
our model because there are no credit-market imperfections. As long as public
ta.xation, there is thus
no
loss of generality in
assuming that the
riskless
bond
is
in zero
net supply.
The
results of this
paper would not be modifed under the alternative assumption that income shocks are privately
observed, provided that the structure of the
economy remains common knowledge.
Assumption
2 (Gaussian Risks)
The
idiosyncratic risks {(>!(, e^)}jgj and asset returns dt are
jointly normal.
Note that a high ^ corresponds to a strong willingness to substitute consumption through time,
while a high
T
implies a high risk aversion.
for analytical tractability,
paper.
They can
also
The above two
but do not appear to be
restrictions are
critical for
motivated by the desire
any of the main arguments
in this
be viewed as a kind of approximation to an economy with more general
preferences and risk distributions.
CARA-normal economy
to
Section 5 will propose a cahbration
match a constant
method that permits our
and a constant
elasticity of intertemporal substitution
degree of relative risk aversion at the steady state.
We make
two additional assumptions.
Assumption 3 (No Persistence) The
residuals of the projection of {(Aj, e^)}jeJi
on the span of
dt are serially uncorrelated
This will imply that an individual's saving and investment choices are independent of his con-
temporaneous idiosyncratic income and productivity shocks, a property that
aggregation but
analysis,
we
will
is
certainly not important for any of our qualitative results.
mimic persistence with a long time
Assumption 4 (No Aggregate Uncertainty)
and ^^ei/J=E(_ie^'
=
In
for our quantitative
interval (see Section 5).
all
dates and events,
^jAl/.J — E(_i^j = A
0.10
This restriction, too, only serves the tractability of the model;
all
As
will greatly simplify
macro variables are deterministic
in equilibrium.
We
it
will
will discuss,
imply that asset prices and
however, the implications of
our results for an economy with aggregate uncertainty.
3
Decision Theory
This section examines the decision problem of an entrepreneur.
We
show that the
portfolio choice
reduces to a mean-variance problem, and derive the optimal saving and investment rules.
This assumption follows naturally from the
agents facing independent shocks.
Law
of
Large Numbers when the economy contains an
infinity of
The
3.1
Individual's
Consider an individual
by
his non-financial
i\,
(^mf)m=o-
-H^^
j in
Problem
period
t.
income by
budget constraint
Denote
y^
and
,
Iris
consumption by
is
y[ is given
by
(1).
in
dfe{,
Capital accumulates according to
e{
then restate the budget constraint
A:j_,_j
we
satisfies
the budget constraint
where
i\,
6
G
[0,1]
conveniently rewrite the decision
^f6{=4-
K+i +
(5),
We
(5)
=
{c[, k^^^, ^t'
max
*^^* maximizes
Uic',)
.
Bellman equation,
+0U (cE,[V,^i«i)l)
and the transversality condition hmt_oo/?'Et[/[V'/(w^)]
sponding optimal consumption rule by
Wlf^o
(5).
indirect utility of wealth V/ (w) satisfies the
U[V/{u,i)]
=
(6)
,
0.
We
denote the corre-
c^(ty)-
Consumption-Investment Decision
3.2
CARA
sets
+
{l-6)k^
inclusive of the capital stock.
i.
Given a price sequence {ttjJ^q, agent j chooses an adapted plan
subject to
=
(4) as
4+
and
(4)
+ dfe\
represents the agent's total wealth (or cash-in-hand) at date
The
—
terms of stock variables. The quantity
y,l=AU{kl)+{l-8)l4 +
utility
risky assets by 0^
is
the fixed depreciation rate of capital. To simplify notation,
problem
pliysical-capital in\-estment
bond and the
his portfolio of the
i+il+TTfe[=yl +
where
c^. iris
preferences, coupled with no short-sales constraints, imply that the
and productive
distribution,
which
capital are
is
of course stochastic,
together with Assumption
we present
independent of wealth.
4, will
ensure that
in this section the decision
the interest rate
is
wiU not
all
We
affect the
macro
theory when the
demand
can thus anticipate that the wealth
aggregate dynamics. This property,
variables are deterministic. For this reason,
risk
premium on
financial assets
"See Angeletos and Calvet
premium on
is
zero
and
deterministic.^^
(2000) for the general analysis.
There, we
first
characterize individual choices for
arbitrary asset prices and then prove that there exists an equiUbrivun such that the interest rate
the risk
for risky as-
all
assets
me
{1, ...,M} is zero.
is
deterministic and
An
educated guess
consumption
is
equihbrium price path, the value
that, along the
where al,al >
and
M
and ^,6( €
4{w) = aiw + bl
non-random
are
where Tl
=
t+
1.
Without
payoffs.
be determined.
coefficients to
We
VtUwU,] =
K.
Ei<i-f^Vari«j)/2l
measures absolute risk aversion in period
raj_,_j
We
then infer from
(8)
,
with respect to wealth variation in
t
henceforth call Fj the effective degree of absolute risk aversion at date
we normalize
loss of generality,
Since there
hedging purposes.
is
project .Aj^j and
1) to
t.
have zero expected
no risk premium, the assets have zero prices and are thus only used
For any
ej_,_2
(m >
risky securities
all
(c^,A;^
the optimal portfolio (^mt)m=i
j,6'qj),
^^
for
therefore chosen to
This has a simple geometric interpretation.
minimize the conditional variance of wealth.
We
on the asset span:
M+i =
The
(7)
(7) that
CEt
period
and the optimal
rule are linear in wealth:
V,'{w)=aiw+V,,
(3)
iiuiction
dt+i
i^i+i
projections k^^j c^+i and ^^
j
+
4+1
4+1^
=
+ 4+v
dt+i
^l+i
(^t+i
represent the diversifiable components of the idiosyncratic
The
residuals
production and endowment shocks.
and
t7(_,_j
Only the
oi df+i, correspond to the undiversifiable components.
which are orthogonal to the span
e^+j,
latter
matter
for either individual
choices or aggregate outcomes. Their variances,
The
are thus useful measures of financial incompleteness.^'^
corresponds to a^
=
and a^
=
0.
The incomplete-market economies
and Krusell and Smitli (1998) correspond to
but no idiosyncratic production/investment
(production/investment
fXg
>
but
risks).
Vart(ui^_^j)
'"The implicit assumption that
relaxed. Besides,
A and
<Te
case
(additive idiosyncratic income risks
we
are
most interested
is
when a a >
—
(j^
+
w^_,_i
=
/(^+i)'^cr^.
(A-l-77^+i)/(A:^)-l-(l-(5)fcj -|-£^_,_j-f ^q
It
is
convenient to define $(fc)
j
=
{l-6)k as the expected production function; and
Gik,
(J
=
considered by Aiyagari (1994)
risks).
and has conditional variance
+
it_4
The
After optimal hedging, individual wealth reduces to
Af{k)
case of complete Arrow-Debreu markets
if
we wanted
tjJ^j
and
f = ^k)-f
)
t\j^\ Eire
[al
+ jikfa"^
stationary, so that
/2
g a and a e are independent
of
to captiire the fact that idiosyncratic risk worsens during recessions,
be functions of aggregate wealth.
t,
can be easily
we could
also let
as the nsk-adjusted output.
By
(8), (r^,fc^_^j. 0q
thus maximizes
+ pu{v]i, [G(ki^„fi) +
u{cl)
subject to the budget constraint,
j)
c^
+
fc^_|_j
+
0^
JRt =
eQ}
urj
Under complete markets, the optimal investment equates the marginal product of
the interest rate: Rt
=
^'{kt+\).
return on investment
is
discounted for
capital with
In the presence of uninsurable production risks, however, the
risk.
The FOCs with
respect to
and ^q
k^
imply the key
,
condition for investment demand:
Rt
The
= ^{kU,St) =
^'(fc^+i)-r^/(A-/+i)/'(^i+i)ai
agent equates the nsk-adjusted return with the risk-free rate.
pected marginal product of capital and the risk-free rate,
on private
equity.
This premium
is
The
$'(A-j_^j )-/?;,
(9)
difference
between the ex-
represents the risk
premium
proportional to the uninsurable production risk (t^ and the
effective risk aversion Tt-
The FOC with
respect to the riskless rate implies the Euler equation,
Etci^,-c>
=<i>HpRt) +
rVar,(c;^i)/2.
(10)
Expected consumption growth thus increases with the variance of consumption. This
reflects the
standard precautionary motive for savings (Leland, 1968; Sandmo, 1970; Caballero, 1990; Kimball,
1990).
The
sensitivity of sa\dngs to
sensitivity to the interest rate
The consumption
rule
is
to a{.
The
risk is
governed by
risk aversion (F), while their
governed by intertemporal substitution (^).
and the budget constraint imply that future consumption
in current wealth wj, with slope
must be equal
consumption
MFC
{dc[j^_-^l dw\j^-^){dwlj^-^l
dw^)
—
a\_^-^Rt{l-a\)
therefore satisfies the recursion al
=
1/[1
-I-
.
By
c^_,.j
is
linear
(10), this slope
{a^^^^RtY^]. Iterating
forward yields
The
MFC is thus the inverse of the price of a perpetuity, which delivers one unit of the consumption
good
in each period s
>
t.
Finally, the envelope condition -after simple
F(
=
Fa^^_j.
We
1
'a\
thus infer from (11) that the effective absolute risk aversion F(
function of future interest rates.
Proposition
manipulation- implies that
We
summarize our
results
=
is
a\
and thus
an increasing
below
(Individual Choice) For any path {/^jJ^q, the value function and the consump-
tion rule are linear in wealth, as in
(7),
and the
10
coefficients al
and
2^ are
equal and satisfy
(11).
The demand
investment
for
i?i
Consumption and savings
=
is
given by
+ il-6)-fif{ki^,)f{ki^,)aX
Af'iki^,)
(12)
are characterized by the Euler ecjuation,
'
Et4+,-4=^\ni0Rt) + ^VaTt{4+,),
where
Varf(c^_^i)
aversion Fj
=
=
3.3
is
{al_^_^)'^[al
+
/(A:^+i)V^]. Finally, the efFective risk
interest rates increase the effective risk aversion Fj
private investment.
current risk-taking
=
increases with future interest rates.
raj_,.j
Note that higher future
premium on
(aj_^j)^Var((u;j^j)
(13)
much more
We
and thus the
risk
expect than the feedback from future credit conditions to
general than our model, as will be discussed in Section 4.3.
Comparative Statics
Consider the impact of incomplete markets on investment. The optimality condition (12) defines the
optimal investment as a function of the contemporaneous interest rate, the effective risk aversion,
and the production
risk:
and the production
risk a^^,
optimal investment
k^_^_^
fc(_,_j
=
but
A;(i?f,cr^, Ff).
is
This function decreases with the interest rate Rt
independent from the endowment risk a^. Wlien a a
also decreases with the effective risk aversion F^.
aversion increases with future interest rates.
By
>
0,
the
(11), effective risk
Incomplete markets therefore introduce a negative
feedback from future interest rates to current investment demand.
Proposition 2 (Investment) Investment demand decreases with idiosyncratic production
9A;j_,_j/5fT4
dk^j^-y/dRs
<
<
0.
When a^ >
Vs >
i.
0.
higher future interest rates reduce investment
Moreover, production
with respect to their impact on investment:
Under incomplete markets,
in future periods.
means that
A
risk
9^fc^
and future
,
interest rates are
j/(5a"^5i?s)
<
Vs >
risk-taking in one period depends on the ability to
risk:
demand:
complementary
t.
smooth consumption
higher borrowing rate, or a more stringent credit market, in any given period
self-insurance
is
more
difficult in
the period. This in turn raises risk premia and discour-
ages investment in earlier periods. In Section 4.3,
we
will
demonstrate that
this
feedback generates
a kind of dynamic macroeconomic complementarity, which can be the source of amplification and
persistence over the business cycle.
11
Proposition 2 also states that a high level of production risk and a high level of future interest
rates reinforce each other's impact
ment
invest-
we document
in this
and bad credit conditions, the amplification and persistence
risks
paper are
on investment. Because recessions tend to predict high
likely to
effects
be particularly important during recessions.
Consider next the impact of incomplete markets on savings. Although the effect of an increase
in
endowment
risk
is
income
(13),
when
(7,4
positive, the effect of production
is
endogenous.
It is
risk /(^t+i)^cryi actually decreases despite
indeed
much
increases too much, the individual scales back investment kj^^ so
that
the increase in a a- From the Euler equation
we conclude:
Proposition 3 (Savings) An increase
and consumption growth,
aA
is
(^A
a—
in
endowment
(E(C^^j-C().
generally ambiguous.
capital share
1/2, there
On
erg,
raises
both wealth
risk, Vart(W(_,_j),
the other hand, the impact of the production risk
For example,
is
risk,
a threshold
in the case of
ct>i
such that
a Cobb-Douglas technology with
dV art{w^_^^) / da a <
if
and only
if
>'^A-
While the
literature has focused
observe that this channel
effect
unambiguously
is
small or even ambiguous, because exposure to production risk
possible that,
his
on precautionary savings
risk
is
on the
effect of
incomplete markets on precautionary savings, we
ambiguous in the presence of production
risks. In contrast,
the direct
on private investment unambiguously reduces investment, and seems hkely to dominate
in
equilibrium.
4
General Equilibrium and Steady State
In this Section
we characterize the general equilibrium and the steady
state of the
economy
in
closed form.
4.1
Definitions
Recall that there
CARA-normal
is
no exogenous aggregate uncertainty (Assumption
specification, the wealth distribution
which prices are deterministic.
these assets in equilibrium.
of the uninsurable
If
is
irrelevant.
We
and that, because of our
thus focus on equilibria in
asset prices are deterministic, there can be
FUsky assets thus play only one role in
component
4)
of idiosyncratic production
12
no
risk
premium on
the model - the definition
and endowment
risks.
Furthermore by
Assumptions
out in the aggregate and are normally distributed:
2-4, undiversifiable risks cancel
,40
cj^
and
are thus sufficient statistics for the structure of risks and the
(Tg
terized by
=
£
incomplete- markets equilibrium
lection of stoie-conim^ent plans ({c^,
maximizes the
tion,
what
4.2
We
follows,
we
let C(,
fc^_|_i,
each agent
utility of
Wt and Kt+i
wealth and capital. The
for the
economy can be parame-
{(3,r,'ii,F,A,6,aA,cre)-
An
Definition
In
Vj,i.
initial
is
a deterministic price sequence {ttjI^q
^j, u't }t^o)i6J
and
j;
asset
such that:
markets clear
and a
col-
t«^}^
The plan
{c^,
in every
date and event,
A;j_j.j,
O^,
respectively denote the population averages of consump-
mean
wealth,
Wq =
Y1,j=\^qI J^
'^ ^^^
exogenous parameter
economy.
Equilibrium Characterization
first
observe that (11) implies that
aversion a\
—
(12). First,
because preferences exhibit
investment
(A:j_,_j)
and T\
at
is
=
Tat for
all
all j, f.
agents share the same
We
and the same
effective risk
next observe two important properties of condition
CARA and there
are no borrowing constraints, the optimal
independent of contemporaneous wealth [wl). This ensures that the wealth
distribution does not matter for equilibrium allocations.
investment risk
MPC
(cr^),
and
Second, because technology
(/, 6,^4),
effective risk aversion {Tat+i) are identical across agents, all agents
choose the same level of investment
k^j^-^
=
Kt + \
for all j,t.
Aggregation
is
thus straightforward
and we conclude:
Theorem
1
(General Equilibrium) There
the macro path {Ct, ^t+i,
H^t, i?j}J^Q is
exists an incomplete-markets equilibrium in
deterministic and
productive investment. The equilibrium path
Rt
Ct+i-Ct
=
all
satisfies in all i
agents choose identical levels of
>
^'{Kt+iyrat+if{Kt+i)f'{Kt+i)(j\
= * Hl3Rt) +
Ta^+, [ol
+ f{Kt+ifa\]
««=[!+ ES(^*-R*+i--^t+s)"T'
Ct
+
Kt+i
Wt+i
=
which
.
(14)
/2,
(15)
(16)
Wt,
(17)
= HKt+i).
(18)
13
Conditions (14) and (16) follow directly from the individual decision problem. Equation (15)
tained by aggregating the individual Euler equations.
production
risks, (14)
markets Euler equation, U'{Ct)
constraint
and the production
=
endowment
ob-
there were no undiversifiable idiosyncratic
If
=
would reduce to the familiar complete-markets condition Rt
in addition there were no undiversifiable
is
risks,
^'{Kf+i).
If
then (15) would reduce to the complete-
PRtU'(Ct+i)- Finally, conditions (17) and (18) are the resource
economy, which of course remain the same under
frontier of the
either complete or incomplete markets.
Under incomplete markets, the aggregate consumption growth increases with the variance
down
individual consumption. This reflects the standard precautionary motive and will push
of
the
equilibrium risk-free rate. More interestingly, the presence of uninsurable idios5Ticratic production
risks introduces a risk
premium on
given risk-free rate. This risk
premium
and represents a gap between the
We
{pi) in
finally note
period
rates raises the risk
equal to
is
social
R^
premium on
In the next subsection,
=
p,
down aggregate investment
^'{Kt+i)~Rt
=
{Tat+i) f {Kt + i)
for all s
>
f.
risk aversion (Tat+i)
That
is,
and thus the
an anticipated increase
private equity and thereby decreases the
we show that
for
any
f {K t+i)cr'^^
and the private return to investment.
from (11) that the effective
increase with
t
private equity, which pushes
this feedback generates a
risk
premium
in future interest
demand
for investment.
kind of dynamic macro economic
complementarity, which induces persistence and amplification in the transitional dynamics.
Propagation and Amplification:
4.3
An Endogenous Dynamic Macroeconomic
Complementarity
Consider an economy which, starting from the steady state,
pated negative wealth shock.
is
quite familiar.
The impact
is
hit at a given date
of such a shock in a complete -markets
Consumption and investment
fall,
interest rates rise,
t
by an unantici-
Ramsey economy
and the economy converges
monotonically and asymptotically back to the steady state. The transition
is
slow under complete
markets only because agents seek to smooth consumption through time. But, when markets are
incomplete, the intertemporal substitution effect
is
complemented by a risk-taking
effect.
Anticipat-
ing high interest rates in the near future, private agents are less willing to invest in risky production.
This
as
effect amplifies the fall in initial
compared
investment and slows down convergence to the steady state
to complete markets.
Below, we illustrate this mechanism in a simplified version of the model.
Example.
To
Suppose that markets are incomplete
clarify the presentation,
we denote by
It
=
in period
but complete at
i
Kt+i the gross investment chosen by
14
G {l,..,oo}.
all
investors
in
period
Given wealth
t.
VK],
we can
solve for the
= C*{Wi)
Fq = Fai, is
Ramsey
>
equilibrium at dates
t
=
By
and write
1,
period-1 consumption as Ci
and period-1 investment as
risk aversion in
an increasing function of future interest rates, {Rt}'^i- In
any period
period
>
t
0,
the interest rate
1,
Since investment at
t
>
7]
I*{Wi).
(11), effective
—
equal to the marginal productivity of capital, Rt
is
2 increases with Ii, interest rates at all
i
>
1
decrease with h.
$'(/{).
We
can
therefore express the period-0 effective risk aversion as a function of period-1 investment alone,
To
=
We
r*(/i).^''
note that the functions C* and /* are increasing
while T* decreases
in wealth,
with investment.
We
suppose
for expositional simplicity that
zero in every period
t
^
Wealth
1.
period
in
the aggregate endowment
1 is
Wi =
$(/o)
+
^i-
et
=
is
Xlj ^t/-^
equal to
Using the notation introduced
above, we infer that the period-1 investment,
/i=r($(/o)+ei),
increases with
in period
1.
Iq.
This
generates more wealth and therefore more investment
More investment in period
is
the familiar wealth effect due to intertemporal consumption smoothing.
Since markets are incomplete at
i
=
0,
the initial investment /q satisfies
^'{Io)-p{IoJi)
where p{Io,h)
the aggregate
if
=
aA
0,
=
This
is
is
is
the risk
=
Ro,
premium on
(20)
private equity. This relation defines
investment Iq as a function of Rq and
for
but dio/dli
investment 7o
premium
a\f{Io)f'{Io)^*{h)
demand
premium on
risk
(19)
>
>
if cr_4
private equity
0.
If
production risks are
We
observe that dIo/dI-[
and the demand
p(7o, 7i) is a decreasing function of 7i,
because the anticipation of low (high) im-estment
(low) interest rates in later periods,
in
When
I\.
for 7o
is
=
instead
We
stress that
the
aA >
0,
the risk
therefore increasing in
7i.
the future leads investors to expect high
and current investment expressed by (20)
propagation and amplification mechanism.
0),
=
The optimal
Ro-
and thus discourages (encourages) risk-taking
future
=
fully insurable (174
zero and condition (20) reduces to $'(/o)
is
then independent of the future investment
The complementarity between
ii-^^
it
arises only
when
is
cr^
in the present.
the heart of our
>
and that
it
hinges simply on the property that high risk premia on private equity predict recessions in the near
future.
The functions C*(VV) and
I'
{W) =
W U C*(W)
standard Ramsey model. The interest rate
a'(I)=
1/ (l-|-{$'(/)a'(*(7))}'"-^^V
The second-order condition
It
follows that
dIo/dRo
=
is
$'(/).
are simply the optimcJ consumption and saving rule in the
MPC
The
and then ?'(/)
s
function a*
is
determined by the functional equation
r-a'(I).
of the individual decision problem implies 9[$'(/o)
l/(9^G/9/o)
<
0;
and dio/dh
=
15
{ff')/(d^G/dIo)
f"
•
D
p(/o, Ii)]/dIo
cx^
>
0, since
sd
?*'
<
G/dIo <
0.
0.
We now
To be
1.
When
Ro
IT
we assume that the slump originates
specific,
endowment
=
4
For simphcity, we also treat the
ei.^'^
0,
by (20) the optimal investment
we conclude that dio/dei
fixed,
of an anticipated recession or investment slump at date
consider the impact at date
=
0,
an exogenous decrease in date-1 aggregate
in
and investment
On
is
0;
=
dW\/de\
and dl\lde\
1,
the other hand,
when a a >
0,
h
the investment levels 7o and
scale
reduction in
Wi =
impact of the
and dli/dei >
We
The
effect (19).
feedbacks to even lower
overall
Thus, when markets
does not affect
in period 1
and the impact of the exogenous wealth shock on contemporaneous income
and induces private agents to
by the wealth
I*'.
/o.
initial
down
<&(/o)
are
complementary by
+
^i'
The reduction
which in turn implies a further reduction in
anticipation of the (endogenous) further reduction in
h
and another round of feedbacks between
exogenous shock can be quite
large. Indeed,
and
\\'\
and
/o is initiated.
dlo/dei
>
0,
/]
The
dWi/dei >
1,
ay\.
conclude that, in the presence of undiversifiable idiosyncratic production
is
(20).
signals high future interest
1
their risky investment in period 0.
These impacts are stronger the higher
I*'.
aggregate shock
With
in 7i.
not amplified.
in Iq implies a further
/i
=
slump
The anticipation of an exogenous negative wealth shock in period
rates
as exogenously fixed.-"'
independent of the expected decline
/q is
are complete, the anticipation of a recession or an investment
investment in period
Rq
initial interest rate
risks,
any given
propagated from one period to another via the dynamic complementarity in
investment; the contemporaneous impact of the shock
shock becomes more persistent, since
it
is
amplified;
and the dynamic impact of the
takes more time to revert to the steady state.
Several remarks are in order about the kind of amphfication and propagation in our model.
incomplete markets generate a particular form of pecuniary externality. In the presence of
First,
uninsurable production
decide
how much
risks, risk
taking depends on futiure interest rates. But,
when
private agents
to save and invest in any period, they do not internalize the impact that their
choices have on the equilibrium interest rate and therefore on aggregate risk taking
and investment
in earlier periods.
Second, the pecuniary externality generates a kind of dynamic macroeconomic complementarity.
'The case
effect
'
f
=
of aggregate productivity shocks is very similar but slightly harder to analyze because of the direct
on the marginal product of capital.
This
is
equivalent to assuming an infinitely elastic supply for savings.
is
derived from the contemporaneous Euler equation.
The
In our
model, the suppb' for savings at
anticipation of a recession in period
1
(lower ei)
increases the period-0 supply of savings under either complete or incomplete markets, but decreases the period-0
demand
for
but not to
investment under only incomplete markets. The endogeneity of the interest rate thus tends to dampen
offset the eimphfication
and propagation mechanism we are proposing. The cahbrations of Section
show that the mechanism can indeed be quite powerful
in general
16
equihbrium.
5 will
Because interest rates are endogenous and influence
low aggregate
risk taking, the anticipation of
investment in one period feedbacks into low aggregate investment
in earlier periods.
Low
levels of
aggregate investment can thus be self-sustaining for long periods of times. This dynamic macroeco-
nomic complementarity
the basis of amplification and persistence over the business cycle.
is
reader must be familiar with a standard example of macroeconomic complementarity - an
with production exfernalities, as
world, an individual's marginal productivity of capital
stock of capital. There
is
economy
Bryant (1983) and Benhabib and Farmer (1994).^^
in
is
assumed to be increasing
thus complementarity in investment,
in the
The
In this
aggregate
our model. Unlike our model,
like in
however, production externalities are exogenous and ad hoc. The complementarity in our model
is
instead endogenously generated by the pecuniary externality
we analyzed
above;
it is
a genuine
general-equilibrium implication of a market imperfection.
Third, this endogenous macroeconomic complementarity hinges on the fact that idiosyncratic
uncertainty affects production/investment.
(1994) and Krusell and Smith (1998),
who
thus not present in the economies of Aiyagari
It is
consider only
endowment
could not have found the propagation and amplification mechanism
We
also
risks.
we
want to emphasize the fundamental premises driving our
This explains
why they
identify in this paper.
result. First,
the willingness
to invest in risky production depends on the ability to insure or self-insure against future variations
in the return of
are obviously
such investment. Second, this ability
much more
is
lower during recessions. These two premises
general than our specific model. ^*
to invest in risky production
is
lower
when
a recession
is
They alone imply that the
anticipated to persist in the near future,
which can slow down recovery and make the recession partially
that our results are robust to
We
willingness
self-fulfiUing.
more general frameworks and empirically
We
are thus confident
relevant.
^^
conclude that the presence of undiversifiable idiosyncratic production risks generates a
powerful dynamic macroeconomic complementarity. In the presence of aggregate uncertainty, this
mechanism can amplify the impact
of an adverse aggregate wealth or productivity shock on con-
temporaneous output and investment, and can increase the persistence
in
investment and output
over the business cycle.
'^Cooper (1999) provides an overview of macroeconomic complementarities.
For example, in models with risk averse agents, incomplete insvirance and credit markets imperfections, the
anticipation of a higher interest rate for borrowing, a higher probabihty to need credit, or a higher probabihty to face
a binding borrowing constraint,
will
most
likely
induce the agent to take
less risk
and thus to
invest less, even
if
his
current funds for investment are not constrained.
'
The heart
in principle
of our
argument
is
simply that higher
and casual observation suggests
it is
risk
premia on private equity predict recessions. This
probably
true.
17
is
testable
Steady State
4.4
We now
demonstrate how the presence of
capital stock at the steady state.
system (14)-(17).
Theorem
We
A
undiversifial)le idiosyncratic
steady state
production risks reduces the
a fixed point (Cqo, ^'oa,
is
I'^oo,
^oo) of the dynamic
easily show:
The consumption
2 (Steady State) There exists a steady state. ^'^
level
Coo
is
—
^{Kao)-I^oo, while the interest rate and the aggregate capital stock satisfy
Roo
=
$'(/^oo)-Poo,
ln(/3i?oo)
The
first
=
—al
al
demand
p^
are complete {a a
>
~
and/or Ug
<^e
>
=
0);
but
R^ <
+ f{K^fa\]
(21)
(22)
.
second
for productive investment; the
We
premium on private
=
note that i?oo
1//?
when markets
that the risk-free rate
is
below the discount rate under
incomplete markets has been proposed as a possible solution to the low risk-free rate puzzle
Weil, 1992; Huggett, 1993; Constantinides
The comparative
and
equity;
1/ p in the presence of undiversifiable idiosyncratic risks
The property
0).
[ol
the risk
is
the standard deviation of indi\-idual consumption.
is
{a A
= (l-K^f
equation corresponds to the aggregate
corresponds to the aggregate supply of savings;
Oc
= m-TC)f{K^)f'{K^)a%
Poo
and
statics of the steady state
Duffie, 1996;
(e.g.,
Heaton and Lucas, 1996).
can easily be derived from (21)-(22):
Proposition 4 (Comparative Statics) The capital stock (Aoo) unambiguously increases with
the endowment
risk {(Je), the
discount factor
other hand, the effect of the production risk {(Ja)
We
will
demonstrate
in
and the mean productivity
(/3),
is
(^4).
On
the
generally ambiguous.
Section 5 that, for most reasonable parameter values, K^o
is
actually
decreasing in a a- Proposition 4 and our numerical findings in Section 5 establish that the distinction
between endowment and production
aA
=
0,
risks
is
Consider
critical.
first
the case where ^e
as in Aiyagari (1994) and Krusell and Smith (1998). In this case, there
on private investment; the marginal product of capital
is
is
no
risk
>
but
premium
always equal to the interest rate; and the
steady state reduces to
o
'^ =
A
higher
sa\dngs,
"
endowment
^'/r^
risk implies a higher
and thus reduces the interest
(l^l/i^^)2
consumption
rate.
stability.
18
="^^--2
risk, increases
And, since the
See the Appendix for a discussion on uniqueness and
r
ln(/?i?oo)
-dJ
N
^^''-^
.
the precautionary supply of
interest rate
is
equal to the marginal
product of capital, the capital stock necessarily increases. This
is
indeed the finding of Aiyagari
(1994).
Consider now the case where
cr^
>
Production
0.
premium on
risk introduces a risk
private
investment and breaks the one-to-one correspondence between the stock of capital and the interest
aa
rate. Indeed,
G^).
If
both the supply of savings
affects
down
agents scale
their production
even ambiguous. The investment
in aj\ raises the risk
effect,
premium on
interest rate. It should then
when a a
is
when the
of savings or the
is
demand
high ^.
We
implausibly low.
An
increase
effect
effect typically
dominates, implying
typically negative.
has to dominate.
One
is
when the precautionary
interest elasticity of the steady-state supply of savings
for investment.
is
in
consumption
not very small,
not very sensitive to shifts in either the supply
is
In terms of our model, the latter case corresponds to
verify these insights in the numerical simulations of the next Section,
which demonstrate that K^o decreases with o^ unless the
is
unambiguously negative.
be no surprise that the investment
implying that the steady-state interest rate
sufficiently
is
weak, implying a small response of the supply of savings to increases
itself is
Another
investment (unlike
private investment and decreases the capital stock for any given
There are two cases where the investment
risk.
for
increases, the savings effect can be small, or
on the other hand,
that the general-equilibrium impact of a a on K^o
motive
and the demand
(like Ue)
Therefore, while Aiyagari (1994) et
invests under incomplete insurance,
elasticity of intertemporal substitution
al.
suggested that the economy over-
we instead conclude that the most
likely case is that of
under-
investment.
Calibration and Numerical Results
5
In this section,
income
we
economy and simulate the impact of uninsurable
calibrate our infinite-horizon
on the steady state and the speed of convergence to the steady
risks
state.
Calibrated Economies
5.1
We first
K" a e
,
consider technology and risks.
(0, 1).
For instance,
of the
mean
We
when a a
=
production,
replace Oe by Cef{Koo)-
We
A=
next normalize
0.25, the
f{K^).
We
We
choose a Cobb-Douglas production function, f{K)
which permits a a to be interpreted as a percentage
rate.
standard deviation of gross output, crAf{Koo), represents
25%
\,
We accordingly
rescale Oe
by the steady-state output; that
can thus interpret both a a and Ce as percentage
next consider preferences.
=
An
important
difficulty
is,
we
rates.
with exponential preferences
is
that the
degree of relative risk aversion and the elasticity of intertemporal substitution are not invariant; at
19
consumption
level C,
the former
a meaningful calibration,
way
+
and the
adjust F and
Under complete markets,
for
to
7 and an
of
r and ^
<l!
=
'i>
/C. To remedy this problem and obtain
we perturb the
parameters
in
example, the steady-state level of consumption
C;,(q,/3,
5).2i
every {a,
If for
such a
and intertemporal
of risk aversion
intertemporal substitution equal to
;/;
we
(3,8)
at the steady state.
more complicated
similar under incomplete markets, but slightly
is
rest of the
is
C^ =
calibrate
T
=
C^{a,f3,8), we can then match a degree of relative risk aversion equal
tp
elasticity of
as
'I'
=
6-1)/q-(5][q/(/?"^ +6-1)]1/(1'°)
j/iC^{a,P,6) and
latter is
some meaningful measures
that the steady state matches
substitution.
[{P'^
we
C
T
is
The
calibration
for the reasons
described in Appendix B.
Overall, a calibrated
/3
is
economy
is
(/3,
7,
V',
i, 5,(7^,
the discount rate, 7 measures the aggregate degree of relative risk aversion,
aggregate elasticity of intertemporal substitution, q
ation rate,
and a a and
endowment
5.2
=
parametrized by a vector €"^
(Tg
is
where
measures the
ip
the income share of capital, 6
(Je),
is
the depreci-
are the standard dexiations of uninsurable idiosyncratic production and
risks, respectively, as
GDP.
percentages of
Appendix B
(See
for
further details.)
Calibrated Steady State
Consider a calibrated economy £^™'
=
a,
tp,
7,
(/?,
6,
a a,
(^e)-
We
can easily characterize the compar-
ative statics of the steady state near complete markets
Proposition 5 (Calibrated Steady State) As we move away from complete markets, Roo decreases with either a^ or a a; Koc increases with
ip>±, where
As we
oA
is
result,
it
+
(/3'^-l) [(/?"^-l)
discussed earlier, the
demand. As a
risk
±=
endowment
raises the risk
premium on
force dominates depends
5 provides a lower
private equity
for
R^
and increases
First, the
risk.
[ip),
A
=
if
effect of the productivity
impact of an increase
in
a a on
for
investment.
Which
technology (a), and markets {a a and 0^). Proposition
high
Therefore, as long as
-'This foUows from C'^
The
and therefore reduces the demand
i) is
*(/Ci,)-i^i, and ^'(K^^)
is
consistent with
implies a high interest elasticity of the supply of
xp
savings, which in turn implies that the steady-state interest rate
consumption
A'oo-
the elasticity of intertemporal substitution, which
our earlier discussion in Section 4.4.
in
and only
be moderated by a decrease in investment. Second, an increase in a a
on preferences
bound
if
risk a^ affects precautionary savings but not investment
unambiguously reduces
may
and K^o decreases with a^
il-a)6] l{2a^).
fundamentall}- different for two reasons.
precautionary savings
(Jg;
is
not very sensitive to variations
not too small, the investment effect dominates.
=
i?;,
20
=
1//3 =>
X^ =
[a/(/r^
+
6-l)]^/''"°\
We
^
note that the lower bound
example,
is
less
than 0.20
we consider annual frequency with a narrow
if
then
(.95, .35, .05),
^=
and thus
of capital,
0.02. Similarly,
set
suggest an EIS around
1,
{(5,
=
a, 5)
if
we consider
and certainly well above
model that Koo may increase with
definition for capital,
and thus
set
(/9,
=
a, 8)
a longer time interval and a broader definition
^ =
then
(.75, .70, .25),
For
for all plausible values of (/3,q,(5).
0.20,
Since most empirical evidence
0.14.
seems extremely unlikely in terms of our
it
ct^.
Numerical Simulations of the Steady State
5.3
We
have run various simulations and we have identified the following patterns as we vary a^.
When V
very small (typically
is
<
Kao
0.20),
a single-peaked function of a a-
is
completing the markets increases capital accumulation
but
may
for
moderate or high
decrease capital accumulation
V'
>
(typically
if
we
globally decreasing function of a a. In this case,
Figure
1 illustrates
is
we
start from highly incomplete markets,
are too close to complete markets.
0.20), the
accumulation. Besides, the impact of Ue
if
investment
more
risk
In that case,
effect
On
the other hand,
always dominates and
K^
is
a
sharing unambiguously increases capital
always to reduce
Rx
and increase Koo-
the monotonicities of the capital stock Koo and the net interest rate Too
=
RBC models in annual frequency. We set = 0.95, 7 = 4, ^ = 1,
A) or a — 0.35 (Panel B). Panel A thus corresponds to a broad
i?oo~l for a specification typical of
—
6
0.05,
and a
—
0.70 (Panel
definition of capital (physical
and human capital) and Panel B to a narrow
capital only). In each graph, the solid line corresponds to Ce
In
K^o monotonically decreases as a^
all cases,
higher the higher the income share of capital.
aA
= 100%
lower
is
when a
about 25% lower than
=
varies
We
=
and the dashed one to Ue
from
to
100% and the drop
can not be dramatic. The
effect
persistent, the abihty to self-insure
or adjustment costs.
is
should be
limited,
much
and
stronger
and investment
is
=
0.35 (Panel B),
and 40%
Hence, the
such
idiosyncratic shocks are highly
if
subject to long irreversibilities
is
if
we think
of
human
capital formation, large
projects, or other investments that involve specialization, indivisibilities,
we can not
irreversibilities or
effect of
With
Persistence in idiosyncratic production shocks and long irreversibilities are
certainly empirically valid assumptions, especially
Unfortunately,
Koo
that an idiosyncratic production shock lasts only one year.
short transitory shocks and infinite lives, agents can easily self-insure.
R&D
in
50%.
0.70 (Panel A).
The above simulations assume
risks
=
observe, in particular, that the capital stock at
complete-market value when a
its
definition (physical
explicitly
and long horizons.
introduce in the model persistent idiosyncratic production risks
adjustment costs,
for
then the model loses
its
tractabihty.^^ Nonetheless,
"'With persistent productivity shocks, an individual's optimal investment becomes a function
21
of his
we
contempo-
can implicitly capture these effects in our simulations, by simply increasing the length of a time
We
period.
then interpret the interval between
and the average
t
and
t
+
as the horizon of an investment project
1
of an idiosyncratic productivity shock.
life
Figure 2 considers the example of a 5-year investment horizon. As previously,
rate
over the 5-year period;
now
5%
and the depreciation rate to be approximately
really dramatic.
insurance
when a =
we
also set
=
At a a
'y
—
4,
i/j
=
and a €
i,
100%, the capital stock
0.35 (Panel B);
15% of
it is
per year, implying
is
{0.35, 0.70}.
30%
of
what
it
—
/3
The
we
let
the discount
=
0.75 and 6
K^
on
effect of aj^
0.25
would have been under
the complete-markets level
when a —
is
full
0.7 (Panel
A).
Figures
1
and 2 demonstrate another related
markets imply both a low
point. In contrast to Aiyagari (1994), incomplete
and a low capital stock.
risk free rate
And, the
risk free rate in
an
incomplete-markets economy can be a largely downward biased proxy of the marginal product of
capital. In Panel
aA
=
B
of Figure 2, for example, the marginal product of capital
100%, as compared to a risk-free rate of just
The simulations
duction
to
is
(7e
risks.
=
0.
We
also provide a useful insight
The dashed
when
year
per year.
on the interaction between endowment and pro-
and
lines in Figures 1
4%
18% per
is
2 correspond to
o-g
= 50%
and the
solid ones
observe that the steady state becomes less sensitive to a^ as a a increases.
because when aA
self-insure against
is
large, individuals are already holding a buffer stock that
both investment and endowment
The
risks.
This
can be used to
precautionary effect of a^ similarly
diminishes with a^, implying that the investment effect dominates more easily
when
there
is
already
a lot of precautionary saving in the economy.
We
that of
also observe that the
endowment
risk
the capital stock K^o at
impact of production
when
a^ —
on the
risk
the two types of risk take comparable value.
— 50%
(^e
is
dominate
capital stock tends to
In Figures
well below the complete-markets one.
That
1
and
2,
when
is,
production and endowment risks make equal contributions to idiosyncratic income variation, the
adverse investment effect of production risk tends to dominate the precautionary savings effect of
both types of income risk. This finding
is
suggestive of the fact that the dynamic macroeconomic
complementarity we have discussed works as a 'multipher' for the steady-state impact of a^-
We
conclude that the quantitative impact of a a. on
raneous idiosyncratic productivity. This
is
K^
is
^''
clearly quite large within our model.
obviously not important for our qualitative findings, but
it
complicates
aggregation and precludes a closed-form solution of the general equilibrium.
-''The numerical results
we
present in Figures
the effect of a a on Roo and strengthens
its
other hand, 7 tends to have a small ambiguous
the risk
premium on
1
and
impact on
2 are not very sensitive to either
Kcx>,
because
effect, since
in'vestment.
22
it
i/'
or 7.
A
higher
ip
raises the interest elasticity of savings.
weakens
On
the
a higher 7 increases both the precautionary motive and
Unfortunately, we do not
know
of
any ob\aous empirical analogues of a a or
We know,
(Tg.
however,
that idiosyncratic production, entrepreneurial, and investment risks appear to be quite substantial
the
first
34%
For example, survival rates of private firms in the United States are only
in reality.
10 years of a firm's
to entrepreneurial activity
is
And, even conditional on
life.
On
extremely wide.
over
survival, the distribution of returns
the other hand, private savings are overall very
low in the United States. These facts suggest that substantial underinvestment
is
indeed the most
likely scenario.
Numerical Results on Persistence
5.4
We
argued earlier that the presence of uninsurable production
dynamic macroeco-
risks generate a
nomic complementarity, which can be the source of amplification and propagation over the business
We now
cycle.
illustrate this effect
by examining the
effect of
ct
4
on the speed
of convergence to
the steady state.
In
Appendix C, we
economy
S"^''
=
(/?, ip,
and approximate the
linearize the
7, a, 5,
local
dynamic system (14)-(18) around the steady state of a calibrated
a a, de).
We
dynamics by \og{Kt+\l K^o)
1-A. Incomplete insurance slows
down convergence
Numerical simulations show that such
Consider our earlier example of a
with
(3
=
^ —
1,
and a 6
0.75 (discount rate
w 5%
•^'*
as
5- year
—
we then
effects
to the steady state
rate
is
1-A decreases with a^-
if
plot the convergence rate
and the
half-life of
an
to 100%. Clearly, the convergence rate decreases
to 100%.
=
We calibrate the model
« 5% per year), 7 = 4,
2).
0.25 (depreciation rate
definition of capital (a
both physical and human capital (a
These are substantial
^^og{Kt/ K^). The convergence
investment project (Figure
per year), 6
almost doubles as a a increases from
—
the case for a wide range of plausible parameter values.
we vary a^ from
With a narrow
rapidly with ca-
aflfects
is
{0.35, 0.70}. In Figure 3
aggregate wealth shock,
calculate the stable eigenvalue A of the linearized system
The
0.7,
=
0.35,
effect
is
Panel B), the
even stronger
half-life of
a shock
when incompleteness
Panel A).
on the convergence
rate,
which suggest that imdiversifiable pro-
ductivity risks can play a useful role in generating endogenous persistence over the business cycle
standard
in
RBC
models with aggregate uncertainty.
depend on idiosyncratic
risks affecting
'The
hajf-life
T
of a deviation from the steady state
is
we
discussed before. ^^
defined by
A''^
=
1/2; that
Figure 3 demonstrates the asymmetry between production and endowment
risk
again that these results
production and investment choices, thereby introducing the
kind of dynamic macroeconomic complementarity
'
We finally stress once
risk.
is,
why our
That
is
T=
logj A.
On
D
persis-
the one hand, endowment
does not introduce the kind of dynamic macroeconomic complementarity that production risk does.
On
the other
hand, the precautionary motive tends to boost up savings below the steady state and thus to speed up convergence.
23
tence findings could not have been identified by Krusell and Smith (1998).
Concluding Remarks
6
This paper examined a standard neoclassical growth economy with heterogeneous agents, decen-
and uninsurable production and endowment
tralized production,
specification for preferences
the aggregate dynamics.
and
We
risks,
we obtained
Under a CARA-normal
risks.
closed- form solutions for individual choices
found that uninsurable production shocks introduce a
on private equity and reduce the aggregate demand
As a
for investment.
risk
and
premium
result, the steady-state
capital stock tends to be lower under incomplete markets, despite the low risk-free rate induced
by
the precautionary motive. Undiversifiable idiosvTicratic production risks also generate a powerful
dynamic macroeconomic complementarity between future and current investment. Originating
a pecuniary externality, this
mechanism
output and investment, slows
of the business cycle.
amplifies the impact of
down convergence
The complementarity
in
an exogenous aggregate shock on
to the steady state,
and increases the persistence
rehes critically on the fact that uninsurable risks affect
private returns to production. This explains the contrast between this paper
and
earlier research
on economies with endowment shocks (Aiyagari, 1994; Krusell and Smith, 1998).
The model
tion.^^
rules out
We expect
that wealth effects should strengthen our results in
instance, investment
rates
and stringent
poor and
any wealth effects on risk-taking because of the chosen
demand would be weak during
CAFL\
specifica-
more general frameworks. For
a recession not only because high borrowing
credit conditions are expected in the near future, but also because agents are
less willing to
engage
More
in risky projects.
generally, the
CARA-normal
specification
appears to play no essential role in our argument that undiversifiable production risks introduce a
negative feedback firom future borrowing rates and credit conditions to current investment.
Credit-market imperfections and non-convexities
in
production have been viewed by
many
au-
thors as a souixe of persistence in the business cycle. Although these departures from the neoclassical
growth model are not considered here, we
to generate underinvestment
find that incomplete risk sharing alone
Therefore, while the convergence rate tends to decrease with a a,
dynamic macroeconomic complementarity appears
"
when
Krusell
not miss
when
a^i
to
and a^ are equal, the convergence rate
abstracting from them.
We
it
is
Introducing borrow-
tends to increase with
dominate the convergence
and Smith (1998) have argued that wealth
much by
sufficient
and introduce a powerful propagation mechanism. The presence of
uninsurable production risks reduces the individual's willingness to invest.
instance,
is
effect of
ffe.
Furthermore, the
precautionary savings. For
lower than under complete markets.
effects are quantitatively small.
This
may
suggest that
we do
expect, however, that wealth effects will be quantitatively significant
crecht constraints are accompanied by idiosyncratic technological shocks.
24
ing constraints would in addition restrict the ability to undertake risky projects, which should
make investment more
sensitive to future credit conditions.
We
thus expect that credit constraints
would only reinforce both the steady-state and the business-cycle
Moreover, our findings stress that
it is
effects
documented
in the paper.
not only contemporaneous credit conditions that matter for
investment, but also anticipated future credit conditions.
A
detailed quantitative assessment of our
RBC
dard
mechanism wo\ild require the calibration
economy with decentralized production, borrowing
and both aggregate and idiosyncratic productivity shocks.
framework of
and
this
of a stan-
constraints, isoelastic preferences,
This can be done by combining the
paper with the numerical analysis of Krusell and Smith (1998). Wealth
effects
financial constraints are then likely to reinforce our quantitative findings.
and Krusell and Smith (1998), we
Finally, like Aiyagari (1994)
as exogenous to the model.
There
in private information,^^ while
markets
a long tradition in models where incomplete markets originate
is
some more recent contributions study economies where incomplete
Although beyond the scope of this paper,
it is
work to production economies with idiosyncratic production
risk
risk sharing originates in the lack of
important to extend this
also treated incomplete
line of
commitment.
and endogenous incomplete markets.
"'See for example
Townsend
(1982),
Green (1987), Banerjee and
Newman
(1991), Atkeson
Cole and Kocherlakota (2001).
See
Kehoe and Levine
(1993), Kocherlakota (1996),
and Alvarez and Jermann (2000).
25
and Lucas (1992), and
Appendix A: Proofs
Proof of Proposition
We
(Individual Choice)
1
A more
have already presented the main steps the main text.
below, maintaining the assumption that there
is
no
premium on
risk
detailed derivation follows
treatment of the individual's decision problem, under arbitrary prices,
and Calvet
premium,
it is
w.l.o.g to
follows that Tr^j
it
max
assets
U{cl)
+
We
1.
>
0,\/m
1.
In the absence
thus rewrite the Bellman equation (6) as
Tflf+i
f3U
E;tw
V.i
We
min
t+\
Var((u'j_|_j)
(23)
successively soh'e for the optimal portfolio of
risk\'
and the optimal consumption-investment choices.
(c'^
,
wealth, Vart(w/+i)
k^ 6^
,
=
f)
,
the optimal portfolio (0^ j)^_j minimizes the conditional v^ariance of
Var^ [yl^+i/(fc^+i)
^L.t
=
-Coi-t
+ 4+i + E/=i 4t+i^'J The FOCs
dm,t+i
on the asset span. This yields A(^j
=
+ i^^i+i ^^^ deterministic constants and
orthogonal to the span of
;
+
>lt+i/(fc^+i)
kJ^j
t;^
The optimal
rff+i-
imply
•
This result has a natural geometric interpretation.
i<^t
=
assume that Efd^.j+i
— 0,Vm >
sugge.sts a two-step solution
Given any
ej_,_j
presented in Angeletos
is
the portfolio problem reduces to a simple mean-variance problem. Since the asset
(8),
structure includes a riskless bond,
The above
general
(2000).
Following
of a risk
A
financial assets.
For
d^+i
+
all
we can
t.
and
jy^^j
t+i^^t+i ^^^
Vm >
ef+i
project (regress)
=
ej_j.j
random
1.
^^^j
\-ariables,
di+i
+
/l(_,_j
^l+i'
and
where
^j+i-measurable and
portfolio hedges fully the diversifiable
component
of
idiosyncratic risks
i&inX=i =
The
and Ug
=
(A + vUi)f{K+i)
+
—
Vart(ej^.j)
(l-'5)fc/+i
Var,«,)^a2 + /(fc^^^)2a2,.
We now turn to the optimal
d
/(^-f+i)-f+i+e(+1
residuals represent the undiversifiable risks.
Var(77j^j),
Var(£^^.j).
+ 4+1 + ^if
We let
By
EstAj^-^
=
A.
E(ej_,_j
(24), wealth after
Thus, Etwi^,
=
(24)
t+i-
=
0.
a\ =
Var((/7j_^j)
hedging reduces to
Af{f4^,)
+
(l-<5)fc^+i
+
wj_^_-^
+
optimal
We define $(fc) =
E^w^+j = $(A;/+i) + 9{t
consumption, savings, and investment decision.
{l-6)k and G{k,rd)
j.
(c^, /cj_^j,
0Q
=
=
6^^ and
= $(fc)-^ [al + f{k)^a\] .It follows that
Combining with (23), we conclude that
Efty^_^i-raj_^jVar((wj^i)/2 = G{kf_^_^,Ta'^^y) +9^
Af{k)
and
-
the
J maximizes
[/(c^)
+ 0U
{y/+i [G(fc^+i, Fa^+j)
26
+
e^o
J}
,
(25)
subject to
c^
+
+
/cj_,_j
—
6q^/ Rt
w\.
The FOCs
with respect to k^ and ^qj give
dG
}s^+i^(fc^+i,ra^+i),
(Without
loss of generality,
we assume
= dG/dk,
above two yields Rt
that
Next, the envelope condition
reduces to
= a^Wf + h^-^ \na[. We infer
FOC with respect to ^qj as
=
Qt+i'
that a\
—
a^
and
bl
— U'{c{). Using (3) and (7), this
= 6(-* Ino^. Using (3) and (7), we
= pRtU'[v^^, [E,<i^ra^'+jVari«j)/2]
U'{4)
a(_|_j
dVi{vj{)ldvj
U'[Vf{wl)]
is
^+i~^
~
^^^t+i
^f
^^'-^
} al^,
=
^^^ consumption rule, the above reduces to
= mtU'
U'(4)
Combining the
condition (12).
is,
c^
rewrite the
Using
interior solutions throughout our analysis.)
[Et4+,-TYait{4+,)/2
.
This gives the Euler condition (13).
consumption rule and the budget constraint imply that d^+i/dwf
Finally, the
Plugging this relation
yields (11).
in the Euler condition,
we
infer that a^
=
l/[l
+ {al^-^Rt)~^].
—
aj^ji?j(l-aj).
Iterating forward
QED
Proof of Proposition 2 (Savings)
=
Since Var(«;)
0.
Let f{k)
miplies k
o\ >
=
a'i+a\f{kf,dVaT{w)/dal >
=~A' I {2R
2i?/f.
=
^/k and, w.l.o.g., 6
+ Pct^)".
but 5Var(«;)/af7^
Then, G{k,f)
1.
Hence, Var(u;)
=
al
=
=
f{k)^+[2a^^fik)f'{k)][dk/da\]
R=
{2Vky'^iA~falVk] and
dG{k,f)/dk
+ o\A^ l[2R + Va\)~ and dV&i{w)lda\ <
-^
QED
Proof of Proposition 3 (Investment)
By
/xF
and dk/dT —-^o\, where
d'^G/dk'^
<
and thus ^ >
0.
dependence of ^ on F and a\.
proportional to
(11),
R=
the implicit function theorem, the first-order condition,
~cr'^-
dft/dRs >
(The
for all s
ji
t.
f{k)f'{k)/{-d'^G/dk'^).
With some but not
We
first is
>
=
The second-order
serious loss of generality,
then conclude that dk/da'^
exact
when F ~
QED
27
dG/dk, implies dk/do\ =-
is
condition implies
we can ignore the
proportional to
and the second when
ct^
~
-F and dk/dT
0.) Finally,
is
from
Proof of Theorem
Existence
Here,
is
(General Equilibrium)
1
proved
Angeletos and Calvet (2000), by talcing the limit of finite-horizon economies.
in
we only characterize the general equilibrium.
at, ^t,j.
Given
this, (12) implies
A,-^
Finally, (18)
get (15).
Assumption
G
j
J,
—f^ l^e
and using the
and
and
=
—
at
a^
—
(13) reduces to
+ }{Kt + ,)o%
fact that
E E.c^+1 = eJ ^ E <:-) E kUr^t
^
we
all
(11) implies a^
(12) then reduces to (14)
A"(, Vi, j.
= * ln(/3i?0 +
¥.tc>t+i~<
Aggregating the above across
=
we note that
First,
E ^n
)
=
= Wtf+i--'^(+i
Kt+x —
Ct.
^t+i,
from aggregating the budget constraints and using
(17) follow
QED
4 (no aggregate uncertainty).
Proof of Theorem 2 (Steady State)
The steady
Since aoo
>
0,
or equivalently
state
we
^
K=
K^o <
and K.
[o-2,cr2
The second condition
defined by the system (21)-(22).
also have Roo
bounded between
m2[0, A')
is
>
Since Roo
>
1,
the
first
{F')^^{8/A). Therefore,
R^
is
We
1-
find
+ f{Kfo\),
it
equation implies AF'{Koo)
bounded between
useful to consider the functions mi(l, ^'^]
defined by mi(i?)
=
(0,
—
>
+
>
1,
A'oo
is
1-^
and
[0,
l//?-
+oo) and
m2{K) =
We observe that mi is decreasing in R and m2 is increasing in K For any K G [0, K),
mi{R) = Tn2{K) has unique solution, Ri{K) = mj^[m2(A')], which maps [0, +oo) onto
(l,mj^((Tg/2)]
maps
1//3
(2*/r)(l-i?-i)-2ln[l/(i?^)] and
al + f{K)^a\.
the equation
and
1
<
implies R^o
.
C
the
(1,/?"^]. Similarly,
A] onto
[l,-|-oo).
equation (21) imphcitly defines a function /?2(A) that
first
This function
is
decreasing, due to the second-order condition of the
individual's problem.
Consider the function
and
R2iK) —
>
-|-oc',
implying
the functions Rj and
The above proves
and
also multiplicity
A(A) =
R2
A(A)
existence.
stability.
cycles, or
-*-oo.
We
and
A ^ 0. we observe that Ri(K)
also note that
The system
We
(21)-(22)
earUer discussed
If this
A(A) =
indeterminacy
and the
/?i(
A)-l >
may
local
bounded
The graphs
of
dynamics allow us to analyze
is
strong enough, multiple steady states,
In Angeletos and Calvet (2000),
arise.
cr_4
and
ae-
When
instead
(jyi
stability of the incomplete-markets steady state follow,
uniqueness and stability of the complete-markets steady state. In this paper,
.
0.
is
how incomplete markets introduce an dynamic
complementarity
such complex dynamics arise only for very large
zero, uniqueness
When
therefore intersect at least once and there exists at least one steady state.
macroeconomic complementarity.
endogenous
/?i(A)-7?2(A').
28
we show
and Ug are
that
close to
by continuity, from
we consider only
the
where a 4 and a^ are not
plausible case
steady state
in
Section 5
is
This also ensures that our calibration of the
too large.
QED
meaningful.
Proof of Proposition 5 (CompEirative Statics)
We
used the functions
7?i
and R2 define
in
the proof of
Theorem
We
of the steady state with respect to the economy's parameters.
which
when the steady-state
2 to analyze the monotonicity
>
consider the case \R'2{Koo)\
An
increase in a^ or
/?
pushes down the function Ri{K) and leaves the function R2{K) unchanged. The steady state
is
|i?j(/Coo)|,
is
necessarily satisfied
is
unique.
therefore characterized by a lower interest rate and a higher capital stock. Similarly, an increeise in
1-6 and
A
pushes up R2{K), also leading to a lower interest rate and a higher capital stock.
increase in
F
or
a^ pushes down both R\{K) and R2{K),
Ta^
reflecting the fact that
An
enters in
both the demand for investment and the supply of savings. F and a a can therefore have ambiguous
QED
effects, as verified in simulations.
Proof of Proposition 6 (Calibrated Steady State)
—
Consider a calibrated economy 8'^
the output- capital ratio at date
The
state.
q*
=
{l3~^-\
calibration of
+
i5)/a.
A=
F and
implying f'{Koo)
^
=
ere)-
=
(l-^)
+
and u
[(i?oo-l)/i?oo][goo/(goo-i5)]
A=
{1-P)q* /{q*-6). u
standard Ramsey model.
in Rx)
and
(7A2)/(i/'i/)
dqao/d{cr^^)
goo are
rf(cr^)
>
When
^
2,
ct'^
=
R^ ^
and
(T^
A
obtained by keeping
and
dq^o
=
2{q* -6) / {qoo-6)
=
R*
1/P,
and
f{Kt)/Kt
=
Kf~^ denote
Cod K^o — ('?oo^<5) at the steady
FCoo — 7 and '^/Koo — i'i<l*-6),
and thus dKoo/daA <
if
F
and
1/
In Angeletos
8"^
q^o
Appendix we describe the calibration
_^.
=
(26)
and Calvet (2001) we
When ct^ = o-g =
= q* ^ (/3"^-l + 6)/a,
.
constant in (26).
and only Hip >
and
characteristics of our exponential-utilities
.
are positive but close to
7Agoo(l-:^/^)f^(cr^), where 3^
Appendix B: Calibrating
In this
=
\n{moo)^-^{o\ + ol).
agoo(l^Aa2^),
prove that a steady state exists for any calibrated economy
markets),
qt
Q^co and
Appendix C) implies
(see
Let
The steady-state system (21)-(22) thus reduces to
i?oc
where
f,
{P,'y,ip,a,8,aA,
0.
(complete
hke
in the
the first-order variations
We
thus get d(lni?oo)
{q*-6)(fr^-l)/{2a).
It
=-
foUows that
QED
*
of
F and $ that permit us
economy
utiUties economy.
29
to
map
the preference
to the preference characteristics of
an
isoelastic-
We
consider F.
first
consumption
of aggregate
economy S
so that FCoo
risk aversion at the
We
We
observe that the degree of relative risk aversion at the steady-state level
=
'I'.
We
pick
some constant 7 and
The parameter 7 thus corresponds
7-
aggregate
next consider
We
TCoo-
is
level,
to a constant measiu'e of relative
which permits a meaningful calibration of the model.
observe that the elasticity of intertemporal substitution (EIS) at the
^ /Coo-
steady-state level of aggregate consumption
is
we could
S such that ^/Coo =
problematic
C)
is
some constant V and
pick
restrict
the following reason.
for
= l-2{l +
The above
The convergence
/?+;^/?2(i-Q)(/ri+5-l) + A/
a function of
is
rate g depends as well on
'^/K^ and
and
17,4
a A,
there
is
1
This calibration, however,
+ P+^(3\l-a){(r^+d~l)
is
Appendix
direct effect of 174
we have discussed
on the convergence rate g captures
in Section 4.
^ I Koo
g.
to vary with a a-
Instead,
we want
CARA-normal
We
But, since
through the dependence of g on
also an indirect effect
-4/?
markets are incomplete, the convergence
caHbrate £ so that 'f/Coo remains invariant at some prespecified
our
^-
for risk aversion,
rate under complete markets (see
When
{a, 0,8).
The
Oe-
the macroeconomic complementarity
on
what we did
In analogy to
given by
g
of
incomplete-markets
restrict the
level,
'^
K^
jK^.
is
If
and
specification)
is
we were to
then we would be permitting
would thus be compounding the direct and the indirect
to control for the indirect effect (which
a function
effect of
an unfortunate implication of
partial out the direct effect (which captures the
macroeconomic complementarity introduced by uninsurable idiosyncratic production
control for the indirect effect,
we must keep
meaningful parametrization of 'i/Koo-
where
q*
to equal
suggests
=
'^ /
Koo constant
We observe
(/3"^-l-t-(5)/a. Therefore, restricting
at the
how
invariant at xp{q*-6).
We
pick
The parameter
tp
we vary a a- This
dynamic
risk).
To
in turn requires a
—
that under complete markets Coo/Koo
{q*-6),
the elasticity of intertemporal substitution, 'I'/Coo,
complete-markets steady state implies
to calibrate *.
as
aa
some constant
'^ I
ip
K^ = ijjCac/Kx =
and
restrict
4'{q*-8)-
8 so that
'^
The
latter
/K^c remains
thus measures the elasticity of intertemporal substitution
around the complete-markets steady state and our caUbration permits a meaningful assessment of
the impact of market incompleteness on the convergence rate.^^
We
think that the cahbration of
'I'
we propose above
the incomplete-markets steady state so that "P/Coo
rate
is
=
tp
is
the most reasonable one.
The
alternative of cahbrating
yields similar results in the following sense:
a decreasing function of cr^ and an increasing function of ^/Kao- Because
K^
the overall effect of a a on g turns out to be non-monotonic in some simulations. However,
if
we look
between the incomplete-markets convergence rate and the shadow complete-markets convergence
latter is defined as the convergence rate of a
complete-markets economy that has the same
30
The convergence
typically decreases with
'i
a a,
at the difference
rate,
where the
/Koo as the incomplete-
we chose a Cobb-Douglas
Finally, as discussed in Section 5,
rescale
17,4
and a^
as percentages of output.
Definition (Calibrated Economies)
A
We
^/Koo =
Appendix
We
first
vector
zt
Lemma
[0,
^P{q*-6), q*
=
(r'-l
calibrated
+
and
thus define:
sponds to an incomplete-markets economy 8
7,
specification for the technology
S)/a,
economy £™'
=
=
(/3,7,
^, a,
{j3,T,'^, F,6, A,a'j^, a'^)
F{K,L) - iC^L^-",
A =
(5,
ct^,
Ce) corre-
=
such that FCoo
1,
a\
=
oa, and
Dynamics Around the Steady State
C: Local
observe that equilibrium paths can be calculated by a backward recursion of the state
=
{at,Ct,Wt).
(Equilibrium Recursion) For any
=
state vector 2f+i
+00), there exists a generically unique
{at, Ct,
{ot+i, C't+i, W^t+i)
Wt, Kt+i,Rt) € {0,1) x R^ x
G
(0, 1]
R^
x
i?
x
satisfying
the equilibrium recursion (14)-(17).
Proof. Given zj+i
we
=
(a^+i, Ct+i, H^t+i) for
solve equation (14),
=
namely Rt
some
t,
and with Lj
dG{Kt+\,Tat+\) / dK,
=
for
IVi,
we proceed as
follows. First,
Kt+\ as a function of {Rt,at+\).
This equation might assume multiple solutions. However, generically, only one of them corresponds
maximum of the individuals problem. Second, we substitute Kt+i from the above into
namely Wj+i = ^{Kt+i, 1), to obtain an equation in {Rt, at+\, Wt+i). We solve this equation
to a global
(18),
for
Rt as a function of
Kf+i as a function
and Wt.
We
(aj+i,
Wt+i)-
We
=
then plug this back into Rt
dG{Kt+i,Tat+i)/dK, to get
of {ot+i, Wt+i). Finally, equations (16)-(17) then assign unique values to at, Ct,
QED
then
let
H denote the equilibrium recursion
markets economy, then
+
mapping;
zt
=
H(zt-)-i) is implicitly defined
at
=\l[l
Ct
=Q+i-*ln(^/?i)-Fa2^J/(X,+a)24+a2]/2,
Wt
=Ct+Kt+i,
this difference
is
by
{at+xRt)-\
always negative and increasing
indeed to slow down convergence.
31
in cr^.
That
is,
the
'true"
impact
of
oa
is
where Rt
= dG{Kt+-iSat + i)/dK and
=
A',+i
'(li'f+i).
<I>
_daj_
ac,
D.H
of
H
is
da,
Q
9a, + ]
The Jacobian
ac.
1
awt+i
.ML
We
observe that dKt+i/dWt+-i
dRt
=
l/^'{Kt+i)
>
0.
and
-Vf{Kt+i)f'{Kt+i)o\ <
dat+i
Let x{y)
dRt
1
dWt+i
^'(Kt+i)
^
{f"{Kt+i) [A-rat+if{Kt+i)<jl] -at+i[f'{Kt+ifral} <
1/(1
+
y~'^)
v/{l
+
v)
and note that
x'(^)
~
+
1/(1
=
^0^
0.
[x(''-')/^]^-
Since at
=
x{at+\Rt), we infer that
dRt
dat
at
dat+i
at+iRt
Rt-at^
dat+i
dRt
dot
at+i
dWt+i
\at+iRt
Future propensity Ut+i has an ambiguous
direct effect (due to the
on current propensity
effect
a^.
There
is
both a positive
complementarity of future and current consumption) and a negative indirect
effect (the precautionary
motive causes a decHne
^\n{PRt)-ra^+^ [f{Kt+i)^a^^ + aj]
/2,
we
the current interest rate Rt). Since Ct
in
Tat+iifiKt+iYa'^
Rt
dat+i
Rt
91V', +1
dCt
dRt
dWt+:
=
Ct+i-
infer that
dRt
dCt
dat+i
An
<0.
d\\' (+1
2
la* + 1".4
+
ai]
f{Kt+,)f'{Kt+i)
>i>'iK,+i)
increase in at+i and W't+i leads to a decline in the current interest rate, which has a positive
effect
on current consumption.
On
the other hand, the increase in Oj+j and Wt+i implies that the
agent bears more risk between time
t
and time
f
+
1.
Let I denote the identity matrix. The characteristic polynomial, P{x)
=
det(D2H-a;I), can be
rewritten as
P(x)
=
dat
d{Ct
da t+i
The
roots of
P
+
Kt+i]
OK,f+i
1
dWt +
dWt +
i
1
+
X
dCt
dat
dat+i dWt+i'
are the eigenvalues of the backward dynamical system. (The eigenvalue A considered
in Section 5 thus satisfies
P{l/X)
—
and 1/A >
32
1.)
Since P(-oo)
= +oo
and P{+oo) =-oo,
There always
<
\K2{Roo)\
When
Thus when
\K[{Roo)\
an eigenvalue
in (1,
there
=
markets are complete, a
=
~
{<yA,'^e)
dCt
0,
dat +1
(7
Obviously, x
=
=x2-x|l +
We
to an unstable manifold.
has one root in the interval
two eigenvalues
in
(0, 1)
and one root
(0, 1)
We
note that,
We
1.
has
1.
+
in (1,
>
0.
(T=0
dKt +1
dWt +1
(7=0
contained in the interval
and Q{1) <
(0, 1)
0.
We
and thus corresponds
infer
the quadratic
+00). Overall, the Jacobian matrix
The
1.
D^H
Q
has
stable manifold under
can explicitly calculate the root that corresponds to
when markets
dKt+i
D^H
if
(/3-x) (3(x), where
and one eigenvalue larger than
complete markets has thus dimension
>
dW,t+i
(7=0
next observe that (5(0)
the interval
the stable manifold.
is
=
Kt+,)
dW,f+i
the one eigenvalue. This
j3 \s
+
d{Ct
at least
is
dCt
and
=
characteristic polynomial thus reduces to P{x)\^_q
Q{x)
and only
we know that
0,
/?,
dat +1 (7=0
if
a unic}ue steady state, the Jacobian matrix
is
+00), and the dimension of the stable manifold
dat
The
Simple calculation shows that P(l) >
exists a real eigenvalue.
are complete,
R^ =
$'(Koo)
~
1//3
and
1_
(3,
dWt+i
(7
=
dRt
dWt+,
(7=0
dCt
dWt+,
The polynomial Q{x)
(7
production function
(1-q)(/?"^
+ 5-1)/ K^.
eigenvalue
is
Finally,
(7e
A
=
^
{ 1
^Al3'f"{K^)>0.
dWt+,
(7=0
therefore reduces to
Q{x)
If the
Rn
=
is
=
x2-
[1
+
f3-^i>P^Af"{K^.)] X
Cobb-Douglas, f{K)
=
K'^, then
+
p.
A/"(Xoo)
= (a-l)Af'{K^)/K^ =-
Therefore, with complete markets and a Cobb-Douglas technology, the stable
1/x and
+ p+-j^pHl-a){p-'+6-l)+ J
when markets
1
+ P+-^p^{l-a){p-'+6-l
are incomplete, the stable manifold
is
-4/3
one- dimensional as long as a a and
are not very large. (Locally, this follows from our finding that the cubic P{x) has only one root
outside
(0, 1)
when
(ja
—
<7e
=
and by continuity
33
of the
P{x)
in
a a and a a- More generally, we
check numerically that this
convergence rate
a A and
a,,
per
convergence
is
se.
the case for
a function of
Since P{x)
We omit
rate.
is
is
'i// I\r^,.
our simulations in Section
like
5.)
The incomplete-markets
the complete-markets one, but
a cubic, there
is
it is
it is
too long
and not interesting. For our
5,
we use Mathematica to solve analytically
it
for the particular
and then evaluate
also of function of
a closed-form solution for the incomplete-markets
the formula, however, because
numerical simulations in Section
rate in general
all
for the
convergence
"Was Prometheus Unbound by Chance?
Risk, Diver-
numerical
\-alues.
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Figure l.A (a = 0.70)
Interest Rate
Capital Slock
and
MPK
% par icar
Keo
MPK
7
6.5
6
5.5
5
4.5
Oa
0.2
0.4
0.6
0.2
O.i
0.4
0.6
0.8
Figure l.B (a = 0.35)
Interest Rate
Capital Stock
K>
6.75
%
pi
and MPK
yeer
—— —
"
6.5
/
---i^^
6.25
6
>«
^^-^^/
^^.^-^^
^
\X ^
5
\
5.25
0.6
_
— "T!"::!————..,..^
_=
—
"
0.2
0.8
of intertemporal substitution
is
1.
Panel B. The solid lines correspond to o^ =
= 50%
(of steady-state
GDP). The
100%
of steady-state
GDP.
~i]^^^^^^
The income share of
capital
0.6
0.4
is
of one year. The discount
4, and the
and 35% in
and the dashed ones to Oj
risk aversion
70%
(no idiosyncratic endowfment risk)
0.£
in
Panel
is
A
show the steady-state level of the capital stock, the interest rate,
(MPK), as idiosyncratic production risk Oa varies between zero and
plots
and the marginal product of capital
^"~-«i,i.,,,__^
4.5
FIGURE 1. We perform an RBC calibration of the model with a time period
rate is 5% per year, the depreciation rate is 5% per year, the degree of relative
elasticity
/
5.5
XX
5.5
0.4
^^y^
6
\\ \
5.75
0.2
yy
6.5
^^^*N^^
MPK
Figure 2.A (a =0.70)
Capital Stock
Interest
K^o
% per
Rate aiid
MPK
^yBer
17.5
2.5
MPK
15
2
12.5
1.5
10
7.5
1
5
0.5
2.5
Oa
0.2
0.4
0.6
0.2
0.8
0.4
0.6
0.8
Figure 2.B (a =0.35)
Capital Stock
Interest
K„
0.5
% per
^ ^ -.
'
0.4
Rate and
MPK
year:
MPK
17.5
^-<:-
15
12.5
^^"<;^
^^^^
0.3
0.2
10
7.5
5
0.1
2.5
Oa
0.2
FIGURE
2.
We
0.4
0.6
0.2
0.8
assume the same parameter values
as in Figure
1,
but
0.4
now
0.6
0.8
use a five year time period
and the duration of an idiosyncratic production shock). The
solid lines correspond to Oe =
and the dashed ones to Oe = 50%. The plots show the steady-state level of
the capital stock, the interest rate, and the marginal product of capital tMPK), as idiosyncratic production
risk Oa varies between zero and 100%.
(for both the length of an investment project
Figure 3.A (a =0.70)
Half Life of a Shock
Convergence Rate
years
40
35
30
y
/
25
20
Oa
0.2
0.8
0.6
0.4
0.2
0.4
0.8
0.6
Figure 3.B (a =0.35)
Half Life of a Shock
Convergence Rate
% per year
14
7
/
6
/
10
5
y
/
4
Ot.
0.2
FIGURE 3.
0.4
Assuming
the
0.6
same parameters
of the deviation from the steady
The
solid lines correspond to a^
0.2
0.
as in Figure 2,
we
0.6
0.8
plot the convergence rate and the half-life
state as idiosyncratic production risk
=
0.4
and the dashed ones to o^ = 50%.
Oa varies between zero and 100%.
Date Due
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MIT LIBRARIES
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