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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Incomplete Market Dynamics
Neoclassical Production
in
Economy
George-Marios Angeletos
Laurent- Emmanuel Calvet
Working Paper 04-41
November 2004
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E52-251
50 Memorial Drive
Cambridge,
a
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This paper can be downloaded without charge from the
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C^0 L^
MASSA C H U S
P
TE
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1
,TUTE
2005
LIBRARIES
Incomplete Market Dynamics
in a Neoclassical
Production Economy*
Laurent-Emmanuel Calvet
George-Marios Angeletos
MIT
and
NBER
Harvard,
angeletOmit .edu
HEC
Paris and
NBER
laurent_calvet@harvard. edu
This version: November 2004.
Abstract
We
investigate a neoclassical
idiosyncratic production risk.
rich effects that
do not
economy with heterogeneous
Combined with precautionary
arise in the presence of
agents, convex technologies
pure endowment
multiple growth paths and endogenous fluctuations can exist even
In infinite-horizon economies, multiple steady states
may
taking and interest rates instead of the usual wealth
and
savings, investment risk generates
arise
effects.
risk.
Under a
when
agents are very patient.
finite horizon,
from the endogeneity of
risk-
Depending on the economy's
parameters, the local dynamics around a stead}' state are locally unique, totally unstable or
locally
undetermined, and the equilibrium path can be attracted to a limit cycle. The model
generates closed-form expressions for the equilibrium dynamics and easily extends to a variety
of environments, including heterogeneous capital types
Keywords:
and multiple
sectors.
Entrepreneurial Risk, Precautionary Motive, Endogenous Fluctuations, Poverty
Traps.
JEL
*We
Classification:
received helpful
D51, D52, D92, E20, E32
comments from
P.
Aghion, A. Banerjee, R. Barro,
J.
Benhabib, R. Caballero, 0. Galor,
J.
Geanakoplos, J.M. Grandmont, O. Hart, A. Kajii, H. Polemarchakis, K. Shell, R. Townsend, and seminar participants
at
Brown, Harvard, Maryland, MIT, NYU, Yale, and the 2004
benefited from excellent research assistance by Ricardo Reis.
GETA
Conference at Keio University. The paper also
1
Introduction
How
does incomplete risk-sharing affect the level and volatility of macroeconomic activity?
investigate this question in a neoclassical
economy with missing markets and decentralized produc-
endowment
tion. Idiosyncratic technological risks, unlike
capital investment.
The
We
shocks, introduce private risk premia on
interaction of these premia with the precautionary motive can generate
endogenous fluctuations and multiple equilibria even when agents are very patient, technology
strictly convex,
We
(2003).
and the wealth distribution has no
conduct the analysis
Each agent
is
in the
effect
on endogenous aggregates.
GEI growth framework
introduced in Angeletos and Calvet
both a consumer and a producer, who can invest
in a private neoclassical
technology with diminishing returns to scale. In contrast to the traditional
uals are exposed to idiosyncratic shocks in productive investment
endowment. Agents
and partially hedge
securities are real
economy reduces
(1965),
also trade in financial markets.
their idiosyncratic risks
by exchanging a
Ramsey growth model with
finite
sales.
some exogenous
of risky assets.
All
markets are complete, the
identical agents, as in Cass (1965),
Koopmans
and Brock and Mirman (1972). With missing markets, on the other hand, the economy
CARA-normal
hood
in
individ-
or lend a risk-free bond,
number
When
cannot be described by a representative agent; explicit aggregation
We
Ramsey model,
and possibly
They can borrow
and there are no constraints on short
to a
is
specification for preferences
and
is
nonetheless possible under a
risks.
previously established two main results on macroeconomic performance in the neighbor-
of complete markets. First, idiosyncratic production shocks, unlike
discourage investment. Thus in contrast to Bewley models
(e.g.
endowment
risk,
tend to
Aiyagari, 1994; Huggett, 1997;
Krusell and Smith, 1998), incomplete risk-sharing can lead to lower steady state levels of capital
and output than under complete markets. 1 Second, financial incompleteness can increase the persistence of the business cycle. In the traditional
some persistence because agents seek
incomplete, productive investment
sumption. This can slow
is
Ramsey framework,
a negative wealth shock has
When
markets are
less attractive relative to
current con-
smooth consumption through time.
to
risky
and becomes even
down convergence
to the steady state,
and thus increase the persistence
of aggregate shocks.
The present paper extends our
of financial incompleteness
risk
and
earlier
work
in a
number
finite-horizon economies.
of useful directions, including high levels
We
show that idiosyncratic production
can generate rich dynamics that cannot be generated by endowment
We begin
risks.
by investigating finite-horizon economies. In contrast to the complete market Ramsey
model, agents accumulate large precautionary wealth in later periods, because shocks received
around the terminal date cannot be smoothed through time by borrowing and lending. In economies
1
See Ljungqvist and Sargent (2000) for an excellent discussion of Bewley models.
with large uninsurable
risks,
the anticipation of these movements lead to endogenous fluctuations
along the entire equilibrium path.
The
interaction between investment risk
monotonicities in the equilibrium recursion.
some economies with investment
We
and precautionary savings can
As a
also generate non-
result, there exist multiple
growth paths
in
risks.
next examine infinite horizon economies. Multiple steady states can then arise from idiosyn-
cratic production risks
risk-taking
is
and the endogeneity of
interest rates.
Under incomplete markets, individual
encouraged by the ability to self-insure against future consumption shocks, and thus
by the anticipation
of low borrowing rates in later periods.
This property can generate multiple
steady states in infinite horizon economies. In a rich equilibrium, a low real interest rate encourages
a high level of risk-taking and investment, which in turn implies a low marginal productivity of
capital
and therefore a low
interest rate.
real interest rates, high private risk
Conversely in a poor steady state or poverty trap, high
premia and low investment are
Poverty traps
self-sustained.
thus originate from the interaction between borrowing costs and risk-taking, a source of multiplicity
new
work has obtained
that, to the best of our knowledge,
is
poverty traps from wealth
building on the idea that poorer agents have lower ability to
effects,
to the literature. In contrast, earlier
borrow and invest and may thus be trapped at low wealth
1991; Galor and Zeira, 1993).
We
levels (e.g.,
eliminate wealth effects by assuming
highlight another channel through which financial incompleteness
an economy: the impact of
The
local
real interest rates
dynamics around the steady state have a
incompleteness
may
dogenous cycles
when
fi
is
under incomplete markets.
locally unique, totally unstable or lo-
agents are very patient.
We
observe that
This suggests that financial
l)/2]
2
~
0.38.
By
contrast, our
Cobb-Douglas technology
exists
if
the discount factor
GEI growth economy
j3
is
less
than
generates determinis-
for large values of the discount factor,
such as
0.95.
We
is
development of
Boldrin and Montrucchio, 1986). 2 For instance, Mitra (1996) and Nishimura
fluctuations with a
=
rich structure
state
and Yano (1996) prove that a period-three cycle only
—
and
help mitigate the role of heavy discounting in neoclassical growth models of en-
(e.g.
the constant [(\/5
preferences,
affect the
undetermined, and the equilibrium path can be attracted to a limit cycle.
the complicated dynamics arise even
tic
CARA
Newman,
on risk-taking.
Depending on the economy's parameters, a steady
cally
may
Banerjee and
observe that
like in finite
horizon case, complicated dynamics arise even though technology
convex and there are no credit constraints. Our results thus complements
how
(e.g.
equilibrium multiplicity and endogenous cycles
may
earlier
work examining
originate from production non-convexities
Benhabib and Farmer, 1994), overlapping generations (Benhabib and Day, 1982; Grandmont,
"Note that
it is
possible to observe fluctuations in a multi-sector growth
and Nishimura, 1979).
model
for
more patient agents (Benhabib
1985) or credit constraints (Galor and Zeira, 1993; Kiyotaki and Moore, 1997; Aghion, Banerjee and
Piketty, 1999; Aghion, Bacchetta
endowment economy
in the incomplete-markets
endowment
unlike
smoothing
risks,
tensions.
also induce novel
we show that the
dynamic
effects,
such a complementarity between future
framework
tractability of our
is
capital
and current consumption.
preserved under a
number
of ex-
For instance, we introduce a storage technology with fixed positive returns, as well as
randomness
human
Idiosyncratic production risks,
of Calvet (2001).
and current investment, and a negative feedback between future
Finally,
also extends the results obtained
generate cycles even though storage and productive capital can be used as
They
devices.
and Banerjee, 2000). The paper
in the depreciation of productive capital.
and multiple
capital,
The model
also generalizes to physical
and
Idiosyncratic production risks then also affect the alloca-
sectors.
tion of savings across different types of capital or different sectors, thus introducing additional
inefficiencies.
We
Section 2 presents the model.
solve the individual decision
equilibrium path under a finite horizon in Section
economy and examine the comparative
and
statics
In Section 4
3.
local
5.
We
state.
Extensions
conclude in Section
6.
Unless
proofs are given in the Appendix.
The Model
2
We
consider a neoclassical growth
by h £
t
all
investigate the infinite-horizon
dynamics of the steady
to multiple capital types and sectors are considered in section
stated otherwise,
we
problem and calculate the
€
and
2.1
all
Agents are born at date
{1, ...H}.
{0, 1,
..,
T}. The horizon
random
economy with a
is
either finite
variables are defined
(T
<
0,
and
number
and consume a
live
oo) or infinite (T
on a probability space
of heterogeneous agents, indexed
=
oo).
single
good
The economy
is
in dates
stochastic
(£l,J-, P).
Production and Idiosyncratic Risks
Each individual
is
an entrepreneur who owns
duction scheme. The technology
is
growth model,
if it
were not
idiosyncratic uncertainty.
for the following:
An
F
satisfies
F'
>
stock of capital and operates his
production
is
0,
own
pro-
subject to (partially undiversifiable)
investment of k\ units of capital at date
+
t
3
concave, and satisfies the Inada conditions.
function
own
These assumptions would lead to the standard neoclassical
units of the consumption good at date
The
his
standard neoclassical, convex, and requires neither adjustment
costs nor indivisibilities in investment.
3
=
t
finite
F" <
0,
F(0)
t
yields
1.
The production function
The
total factor productivity
=
0,
F'(0)
=
+co, F(+oo)
=
F
is
increasing, strictly
A^+1 and
the depreciation
+oo, and F'(+oo)
rate 6t+1 are
random shocks
which introduce investment
specific to individual h,
risk.
They
are
the key ingredients of the model.
The productivity shocks
of technological risks.
(-^t+iJ^t+i) a U°
The uncertainty
R&D. More
investment in capital or
w
of an entrepreneurial project obviously influences specific
generally, the riskiness of a worker's
wide range of decisions such as labor supply,
As
choices.
will
be seen
us to capture the impact on growth of a wide range
in Section 5, the
4
model can be conveniently extended
to explicitly include
physical capital.
For comparison purposes, we introduce two additional sources of income.
have access to a linear storage technology with gross rate of return p G
St
units of the good yields ps^ with certainty at date
stochastic
endowment stream
1
{ej
}^.
Income
t
+
1.
[0, 1].
2.2
unknown
is
at
An
investment of
e^+1 captures the effect of risks that are outside the
and revealed
t
First, individuals
Second, agents receive an exogenous
control of individuals and do not affect production capabilities. Like productivity
exogenous income e^+1
capital affects a
education, learning-by-doing, job search and career
human and
multiple sectors or the accumulation of
human
at
t
and depreciation,
+ 1.
Asset Structure and Preferences
Individual risks can be partially hedged by trading a limited set of real securities.
n G
{0, ..,N} is short-lived:
the good at date t+1. Security
=
The quantity Rt
l/7To,t
worth
it is
n
=
7r„ jt
is
good
units of the
riskless
and
at date
delivers do,t+i
denotes the gross interest rate between
—
t
t,
1
and
yields
Each
asset
dnt t+i units of
with certainty next period.
and t+1, and r t
=
Rt
—
1
is
the corresponding net rate. Assets are in zero net supply, there are no short-sale constraints, and
default
is
not allowed.
It is
convenient to stack asset prices and payoffs into the vectors
n
Without
loss of generality,
=
{^n,t)n=0
we assume that
and dt+l
=
(d n ,t+i)^Lo * s
{dn,t+l)n=0-
an orthonormal family of
L 2 (f2),
implying
that risky assets have zero expected payoffs and are mutually uncorrelated.
At the outset
of every period
t,
investors are informed of the realization of the asset payoffs df and
idiosyncratic shocks {{At,5 t ,e^)}^=1
on available information, agent h
We
Information
.
is
thus symmetric across agents. 5 Conditional
selects a portfolio 0^
assume by construction that
all
=
(9n t )n=o
m P er i°d
t.
the assets traded in one period are short-lived, in the sense
that they only deliver payoffs in the next period. In the next sections,
we
will focus
on equilibria with
zero risk premia and deterministic interest rate sequences {Rt}o<t<T- For this reason, equilibrium
See Marcet, Obiols-Homs and Weil (2000) for a recent discussion of the labor supply in a Bewley model.
°The
results of this paper
would not be modified under the weaker assumptions that income shocks are privately
observed and the structure of the economy
remains exogenous).
is
common knowledge
(provided of course that the financial structure
allocations
and
prices
do not change
unit of the good every period. 6
if
The
we introduce a
perpetuity
perpetuity,
n =
worth
is
t
namely a security delivering one
J^ s =t l/{Rt---Rs) at date
t
after
delivery of the period's coupon.
Each agent maximizes expected
lifetime utility
Uq
=
Eo Ylt=o P* 11 ^) subject
in all date-events
to the budget constraints
<£+.**
-
«£+i
The
+ a* + **.#
e*+1
w.
+A
h
h
t+1 F{k t )
(1)
+
(1
h
-
tf+1 )tf
+ ps +
t
d t+1
h
9t
(2)
variable w^, called wealth, represents the trader's net financial position at date
T
horizon
is finite,
we use the convention that
and 9 T
Sj.
instead impose the transversality condition limj^ooEo/? u(w^)
CARA-Normal
2.3
is
Under an
infinite horizon,
= 0.
= --exp(-rc),
the coefficient of absolute risk aversion.
The
asset payoffs dt+\
and idiosyncratic
shocks {(-At+i,^t+i,ef+i)}h are jointly normal and independent of past information. There
no distinction between the conditional and unconditional distribution of these
conditional) distribution of payoffs
may
and shocks can vary through time, which
variables.
is
thus
The
(un-
in future applications
prove useful to capture the dynamic effect of financial innovation or business cycle variations
in uninsurable risk (e.g.
We
Mankiw,
obtain a tractable model
1986; Constantinides
when investment
and
Duffie, 1996).
opportunities are symmetric across agents and
idiosyncratic shocks cancel out in the aggregate. Specifically,
cratic shocks
is
we assume that the mean
of idiosyn-
both constant and homogeneous across the population:
A = E^ h+1 >
t
We
we
utilities
u(c)
>
0.
the
Specification
Agents have identical exponential
where T
=
When
t.
rule out aggregate risk
5
0,
by imposing
= E5 h+1
t
in every
H
G
[0, 1]
,
e
=
Ee£+1 >
0.
period the cross-sectional restriction:
Ah
A
°t+l
6
1
H h=l
e
This assumption implies that the endogenous fluctuations observed in equilibrium will be unrelated
to aggregate shocks.
'More generally, traders can dynamically replicate a large
class of long-lived risky assets.
we must guarantee that
Finally,
idiosyncratic risk
^N
^+E:=i&+iM
+
h,t
=
°t+i
=
't+1
The
symmetrically distributed across the pop-
on the asset span available at date
ulation. For all h, project the idiosyncratic risks
Ah
is
^N
ft
h,t
5
+ En=l Ps',n dn,t+1 +
e
+ Eti»+i +
q
>ff
residuals (A^+1 ;S t+1 e?+1 ) represent the undiversifiable
;
dowment shocks
to individual h.
We
Ah
5t+l)
%
component of the investment and en-
~N
a 2A,t+i
(
a lt+l
o,
pi
<t
V
and that the standard deviations
i
assume that they are mutually uncorrelated:
*H+1
5 t+l
t:
—
crt+x
(0\A,i+i; °"<5,t+i;
)
a e,t+i) are identical
for all agents.
The com-
ponents of dt+i represent useful measures of financial incompleteness, which can deterministically
vary through time.
The CARA-normal
specification will ensure that, given a deterministic price sequence, aggregate
quantities are independent of the wealth distribution. This will overcome the curse of dimensionality
that arises
when
the distribution of wealth - an infinite-dimensional object - enters the state space
of the economy.
Equilibrium
2.4
Given a price sequence
A GEI
Definition.
adapted contingent plan {c£\
{Ttt}J= Q, each agent chooses an
equilibrium consists of a price sequence {^t}J= o
and a
fc^,
s
h
,
8t
,
w^}J= q.
collection of individual
h
plans {{ct,k£,s h ,9 ,w'l}'[=0 )i< h <H such that:
(i)
(ii)
Given
prices, each agent's plan
optimal.
Asset markets clear in every date-event:
We make two
rate
in
is
R
t
and
immediate observations.
invest
it
First,
^ h=
if
Rt
in the storage technology.
i
=
@t
<
p,
0-
agents want to borrow an infinite
The absence
any date-event. Second, the absence of aggregate
of arbitrage thus requires that
risk in the
CARA-normal framework
that deterministic price sequences {^t}J=o ca n be obtained in equilibrium.
price sequences satisfying these
two conditions.
amount
We
R >
t
at
p
implies
henceforth focus on
Equilibrium under a Finite Horizon
3
Decision Theory
3.1
Consider an exogenous, deterministic path of asset prices {itt}J=o satisfying the no-arbitrage condition
R >
utility of
in every period.
p
t
We
focus on a fixed agent h and denote by
wealth along the price path as of date
J h (w£,t)
=
value functions
The
J
(.,t)
Since
(2).
J h (w,T)
=
her indirect
Bellman equation:
l)
u(w) at the terminal date, the
can be solved by a backward recursion.
results are conveniently presented using
The function
contrast,
Lemma
<&(fc)
G(k,a,a)
investment
1
k,
$(fc)
=
AF(k)
+ (l-8)k +
e,
G{k,a,a)
=
2
${k)-ra[F{k) 2 a A
+
is
+
a 2e }/2.
the risk-adjusted level of non-financial wealth corresponding to productive
marginal propensity to consume next period
The value function belongs
coefficients at
2
k aj
denotes the expected non-financial wealth when investing k units of capital. In
J h {w,t)
The
Indirect utility satisfies the
u(c)+/3E t J h (w?+1 ,t +
max
subject to the budget constraints (1) and
h
t.
J h (w,t)
and
b^
CARA
to the
= -(ra
t
a,
and residual
risks
=
a
(pa, as,
°~e)
•
class in every period:
)- 1 exp[-r(a t u;
+
^)].
are defined recursively by the terminal conditions
ax
=
1, b
T
=
0,
and
the equations
at
= T77
dvT'
1 + {at+iRt)
(
i
3)
where
h
Mt =
~
*
G{k£,a t+ i,at+i)
"
The optimal
N
_
^_
kt
*
Rt
hi(/3i?
"
;
~fa^Rt
+
„
^
n
t
'
Pe,n
+ P A ,n b
k
\ t
)
~
P5,n L t
+ ^f^~
(5)
1
capital stock satisfies
k\ G
argmax
\
G{k, ot+i,
<J t+1 )
- Rtk + Rt^2 ^.'[^/(^ ~ $i% k
n=l
^
(
'
(
6)
and
strictly positive if
is
A+
Rt Y^n=l Kn
'
>
^An
^-
Individual consumption and portfolio choices
are determined by
c\
=
a t w?
t,t
=
-^-^tn F ^)+^ikl-R^n
+ b?,
XTie agent does not use the storage technology if
storage
if
=
Rt
>
p,
t
and
l{Ya t+l ),
is
Vn€{l
JV},
(7)
indifferent between the
bond and
p
Forward iteration of
(3) implies
the price of perpetuity:
of uninsurable risks
Sandmo,
Rt
,
=
at
<x t+1
that the marginal propensity to consume
+
1/(1
The
lit).
intercept b\ decreases with the standard deviations
an implication of the precautionary motive
,
The demand
1970; Caballero, 1990).
inversely related to
is
for
for savings (Leland, 1968;
investment exhibits more novel features. The
optimal capital stock decreases not only with productivity risks a^,t+i and (rgt+ii but also with
at+\
=
1/(1
+ IIt + i)
and therefore future
interest rates
Rs
>
(s
t
+
1).
This property reflects the
sensitivity of current risk-taking to the future cost of self-insurance.
Equilibrium Characterization
3.2
Let Ct, Wt,Kt, and St denote the population average of consumption, wealth, capital and storage
investment in a given date
t.
Initial
wealth
Wo =
w o/H
Ylh=i
ls
an exogenous parameter of the
economy.
Since idiosyncratic shocks cancel out in the aggregate,
risk premia.
prices:
7r n)t
7
we
focus on equilibrium paths with zero
Risky assets, whose expected payoffs are normalized to zero, therefore trade at zero
=
for all
n€
{1,
...,
kt
N}. Condition
(6)
then reduces to
G argmax[G(fc, a t +i,a t +i)
-
Rtk].
(8)
Individual investment maximizes risk-adjusted output net of capital costs.
the same marginal propensity to consume at
same
capital stock can
be chosen by
all
=
+ lit) and the same
k£ = K for all h. We
1/(1
agents:
t
Since
all
agents have
objective function (8), the
then aggregate across the
population and obtain the equilibrium dynamics in closed form.
Proposition
1
The aggregate equilibrium dynamics are deterministic and
assets are equal to zero.
As shown
in the
risk
All agents have identical marginal propensities to
Appendix, the
risk
premium
single type of idiosyncratic investment risk
(i.e.,
is
necessarily zero in any equilibrium
if
only productivity risk or only depreciation
premia on financial
consume and choose
the economy contains a
risk).
K
Macroeconomic aggregates
identical levels of investment.
K
Wt+
=
<H
-C
t
=
t)
HPR )/T + Ta t+1
t
While individuals are subjected
=
2
a +1 [F(
tive.
t )
2
o-
(11)
}
+K
=C +
aj>
=
1
A
t+1
The optimal
+K
2
=
and i^x
(13)
macroeconomic quantities are deterministic
economy cannot be represented by a representative agent with
and consumption growth (12) depend
(6)
d\ t+1
capital stock,
+ a\ t+1
which
],
a familiar consequence of the precautionary mo-
is
on the other hand,
t)
t
t
FOC:
the
satisfies
)a% t+x + Kta 2s>t+1 ].
=
o~s
t
equated with the interest
well-known
in
°~S,t+\
0),
difference
the interest rate
&(K —
t)
R =
rate:
t
=
&'(Kt)
—
1
t+i
5
+
=
0),
(14)
the marginal product of
AF'(Kt). This
which
relation,
complete market models, holds more generally in the presence of uninsurable
endowment shocks (a^t+i >
>
(12)
0.
In the absence of undiversifiable productivity risk (cr^^+i
is
/2
increases with the variance of future individual consumption, Vart(c^+1
t
is
\
t
R = l-5 + AF'(K - Ta t+l [F(K )F'(K
capital
a\ t+l + a\ t+x
+S
Kt
t
2
risk.
Mean consumption growth
2
(9)
(10)
[F(Kt ) a% t+1
consumption flow {Ct}, because equilibrium investment
on idiosyncratic
k].
1
2
to idiosyncratic shocks,
functions of time. Nevertheless, the
the recursion
t
+ (at^Rt)-
W
Wq = Wo,
t
t
= $(K + pS
2
boundary conditions
i/ie
i
1/[1
t
and
R
G argmax[G(A;,a t+ i,<T t+ i) -
t
Ct+l
satisfy in every period
Rt
on investment. Note that
it
cannot be
fully
hedged
+i[F(Kt)F'(Kt)o- it+l
K
+
2
t o-
t
,A represents a private risk
does not include the endowment coefficient
o~
&i t+\.
=
1/(1
+ Ht+\),
are anticipated in the near future, agents
productivity shocks by borrowing.
The
and thus to future
know
that
it
will
interest rates.
The
premium
Under idiosyncratic
production shocks, risk aversion thus induces the sensitivity of current investment
marginal propensity a t +\
(o~A t+l or
equal to the risk-adjusted return to capital Gk(Kt,<H+l,°~t+l)2
t
risks
t
is
= Fa
But when production
0).
When
K
t
to future
high real rates
be costly to smooth adverse future
anticipation of a high marginal propensity a t +\ then implies
low productive investment in the current period.
Idiosyncratic production risk, unlike
sensitive to future interest rates.
equilibrium.
The
This
endowment
risk,
effect introduces
thus implies that current investment
is
a dynamic complementarity in general
anticipation of low income, low savings, and high real interest rates in future
periods discourages current risk-taking and investment, which in turn imply low income and high
real interest rates in the future.
The expectation
of low
economic activity
is
thus partly or fully
)
In Angeletos
self-fulfilling.
and Calvet (2003), we highlighted how
this positive feedback
between
We
future and current investment can increase the persistence of the transitional dynamics.
below that
see
may
with the precautionary motive,
this complementarity, coupled
will
also generate
multiple growth paths and steady states.
We
finally
output G(k,
note that although the production function F(k)
a,
when a a >
a) need not be concave in k
0.
The
is
strictly concave, the risk-adjusted
function G, however,
is
single-peaked
under the following condition.
Assumption
There
(ii)
We
One
1
of the following conditions holds:
no idiosyncratic depreciation
is
risk
and p >
The function F(k) 2
(i)
1
—
is strictly
convex;
5.
then easily show
Lemma
2 The optimal capital stock k£
is
FOC
the unique solution to
condition (14).
Equilibrium Analysis
3.3
Equilibrium paths can be calculated by a backward recursion. To simplify the discussion, we
specialize to a stationary
°~t
=
cr S,,o~e)
{°~Ai
reduced vector zt
Lemma
(0,1) x
f° r all
=
economy,
t-
(at, Ct,
3 For any zt +\ G
R
2
x
Ri
which residual standard deviations are constant through time:
in
We
R
x
(0, 1]
easily
x
[e,
(at, Ct,
Wt, Kt,
St,
Rt) and the
show
+oo), there exists a unique macroeconomic vector x t £
satisfying the equilibrium recursion (9)
= H(z
This property defines the mappings xt
zt
The
=
Consider the macroeconomic vector xt
Wt).
t
—
(13).
+\) and
= H(z
w
(15)
).
equilibrium dynamics are thus fully characterized by the three-dimensional vector
The equilibrium
calculation
ar
conditions
Wq = Wq.
=
l
and Ct
[e,
instant
t
+oo).
>
0.
= H(z
Some
t
Wt
+i).
[e,
for the
Wt
complete-market Ramsey
wealth level provides the third boundary condition:
=
+oo), we can define zt
W
t
<
e
[e,
(1,
Wt, Wt) and
our model reduces
complete; a one-dimensional system in at
to:
is
recursively
+oo) does not guarantee that
and the algorithm stops at some
generate an entire path {zt)J= Q, leading to
dynamics
8
zt.
which implies the terminal
their wealth,
Note that the condition Wt+i G
of the equilibrium
specifically,
G
initial
all
terminal wealth levels imply that
Other values
The dimensionality
More
= Wt- The
For any choice of
calculate the path z t
capital.
method used
At terminal date T, individuals consume
model.
Wt G
similar to the
is
now
initial
wealth level
determined by the financial structure and the productivity of
a tuio-dimensional system in (Ct,Wt)
when agents do not produce (A
10
=
0,
5
=
1,
p
when A >
=
0,
aa
=
and markets are
ffs
=
0).
Wo- The recursion thus defines a function Wq(Wt), whose domain
wealth
initial
Wq
W {WT )= W
is
we extend the function Wq(-)
admits at least one solution
Proposition 2 There
We now
[e,
+00). Since the
Wt such that
.
behavior when
its
a subset of
exogenous, an equilibrium corresponds to a terminal wealth level
In the Appendix,
examine
is
Wt —
+00.
»
It is
mapping defined on
then easy to prove that
[e,
00),
and
Wo(Wr) = Wq
any Wo-
for
exists
Wt —
and
e
to a continuous
an equilibrium in any finite-horizon economy.
investigate the properties of equilibrium paths.
Growth Paths
3.4
Finite-Horizon
When
uninsurable risks are relatively small, we
growth path. In
close to the complete-market
may
fact,
GEI
expect that the
the
equilibrium remains
GEI growth path combines two
features:
a turnpike property and a strong precautionary effect around the terminal date, as illustrated in
1A 9
Figure
economy accumulates wealth and remains many
Starting from a low capital stock, the
GEI
periods in the neighborhood of the
equilibrium wealth normalized to
its
steady state. 10 This
is
evident in Figure IB, which plots
steady-state level. In the traditional
Ramsey model, aggregate
wealth progressively decreases around the terminal date T. Under incomplete markets, however,
individuals have a very strong precautionary motive in later periods, because shocks around
cannot be easily smoothed by borrowing and lending.
As a
result, aggregate
T
wealth overshoots
the steady state before being consumed in the very last periods. Note that the large accumulation
of wealth
is
financed by a reduction of consumption and an increase in productive investment
some periods before the terminal
market dynamics,
in
These
date.
results thus contrast with finite-horizon
which the capital stock never exceeds
its
complete
steady-state level in an initially poor
economy.
When markets
than
are very incomplete, the growth paths of the
Ramsey model. The
in the traditional
Consider
in the equilibrium recursion.
Wt+i
=
Kt
in the absence of storage.
3
>_1
(Wi_|_i)
and a lower
11
economy can be
startlingly different
differences originate in nonmonotonicities
for instance
embedded
the impact of a higher future wealth level
Higher wealth next period requires higher current investment
interest rate Rt
,
leading to an increase in the component
— ln(/3i?j)/r
of current consumption:
C =
t
9
The economy
r = 10, A =
I
We
II
The
1,
Ct+i
in Figure 1
a=
0.4,
=
is
-
HPR )/T - Ta t+1
2
t
specified by the
0.95, S
=
0.05,
K
2
o2
+
a 2]
/2.
= k a and
T = 1000.
Cobb-Douglas production function F(k)
p= 1 -
S,
discuss the steady state of the infinite-horizon
analysis corresponds to the case VKt+i
[F(Kt ) 2 a\ +
e =
0,
aA =
economy
< W"+ i,
0.5,
as
=
0,
ac =
as defined in the proof of
11
0,
in the next section.
Lemma
3.
(16)
the parameters
Under complete markets,
this
the only effect. Current consumption Ct and wealth
is
are increasing functions of future wealth
Wt
W +\.
The
t
Wt
= C + Kt
t
function Wq(-) monotonically increases in
an d there exists a unique equilibrium growth path. More generally, we show in the Appendix
that the monotonicity of Wq(-) and thus equilibrium uniqueness also hold in economies with only
endowment shocks
(a e
>
0,
o~
a
=
With uninsurable production
as
=
risks,
High output next period requires a high
0).
however, the equilibrium recursion need not be monotonic.
level of capital
investment Kt and therefore high individual
risk-taking in the current period. Agents then have a strong precautionary motive, which implies
equation
As a
(16).
result,
of future wealth Wt+i-
2
A + K^a\ and a possible reduction of current consumption in
consumption Ct and wealth Wt = Ct + Kt can be decreasing functions
a large risk adjustment F(Kt)
The
o~
anticipation of high wealth in the future
may
thus imply a precautionary
reduction in current consumption and wealth. In other words, the interaction of investment risk
with the precautionary motive can generate a negative feedback between future and current wealth.
The non-monotonicities generated by productivity shocks have two
and complicated dynamics, which we successively examine.
tiple equilibria
Proposition 3 There
robustly exist multiple equilibrium paths in
with idiosyncratic production risks. In contrast, equilibrium
complete markets or only endowment
Figure 2 illustrates the function
Given an
possible implications: mul-
initial
is
some
finite-horizon economies
unique in any economy with either
risks.
Wo(Wt)
economy's parameters. 12
for appropriate values of the
wealth level Wq, agents can coordinate on several paths. For instance, they can
choose a high level of risky investment, and low consumption for precautionary reasons. Conversely,
they can coordinate on low investment, which
is
matched by weak precautionary savings and
high consumption. Under uninsurable production risks, two economies with identical technologies,
preferences and financial structures can thus coordinate on different paths of capital accumulation,
savings and interest rates.
Complicated dynamics can also
a single equilibrium.
fluctuations.
markets
To
simplify the discussion,
The unique growth path
The graph
may
arise.
illustrates the
we
consider an
illustrated in Figure 3 displays large
necessarily require a high degree of impatience.
For instance
in
Figure
fluctuations along the unique equilibrium path with a high discount factor
'"The economy
r
=
13
in
10,
A=
10,
We
also
assume
a=
=
Figure 2
0.35,
in
j3
=
is
0.1).
13
Such behaviors do not
0.9, S
Large undiversifiable production
3,
(/3
we obtain endogenous
=
0.99)
and a low rate
risk thus generates
endogenous
= k a and the
= 0.5, p = 1 - 5, e = 0, a A = 100, as = 0, cr e = 0, T = 40.
T = 1.625, A = 10, a = 0.55, p = 0.3, e = 0, a A = 50, a s = 0, a e =
specified
Figure 3 that
endogenous
kind of complex dynamics that the introduction of incomplete
generate in an otherwise standard neoclassical economy.
of capital depreciation (5
economy with
by the Cobb-Douglas production function F(k)
12
parameters
0,
T=
50.
fluctuations with
more patient agents than
in earlier
models considered in the
observe in Figure 3 that the interest rate frequently reaches the lower
endowment and productivity
the equilibrium recursion
is
dampens
and macroeconomic aggregates.
economy
Overall, the analysis of the finite-horizon
idiosyncratic
also
bound imposed by the storage
technology. Thus, storage does not preclude the existence of endogenous fluctuations, but
variations in the interest rate
We
literature.
When
shocks.
monotonic and there
illustrates the strong distinction
exists a
between
investors receive only
endowment
unique growth path.
On
shocks,
the other hand,
the presence of undiversifiable productivity shocks generates non-monotonicities, multiple equilibria
and endogenous
cycles. Note,
however, that large idiosyncratic risks seem to be required for such
complicated dynamics. This suggests that when building macroeconomic models, investment risk
should probably be used
in
combination with traditional sources of endogenous fluctuations.
Equilibrium with Infinite Horizon
4
Decision Theory and Equilibrium
4.1
Given a deterministic sequence of asset prices {^t}t^o^ we calculate the optimal decision of an
individual investor h by taking the pointwise limit of the finite horizon problem.
in every period, let
at
=
+ n^)
(1
-1
11^
=
The convergence
Assumption
We
2.
of the series a t
defined by
and
b\
is
(5),
We
also consider the solution k^ to
and
guaranteed by the following sufficient conditions:
The sequences {^t}t^y {Rt}t^o an<^ {^t}t^o are bounded.
4 Under Assumption
J (W, t) = — (rai)
-1
[2],
exp[— T(atW
Since the optimal decision
is
+
the value function belongs to the
b^ )],
CARA
class every period:
and the consumption-portfolio decision
the limit of the
finite
satisfies (7).
horizon problem, Ponzi schemes are ruled out
the strongest conceivable way, along any possible path.
As
there
is
M/
1
can then show
Lemma
in
Y^s^t+i l/(Rt--Rs- l) denote the exogenous price of a perpetuity and
the implied marginal propensity to consume.
equation (14), the variables
Specifically
in the finite- horizon case,
is
we focus on GEI
no risk premium, capital investment
is
then straightforward to show that the vector
equilibria in
which asset prices are deterministic,
identical across agents,
zt
=
13
(a*, Ct,
Wt)
and Assumption
satisfies
[2]
the recursion z t
holds. It
= H{z
t
+\)
Assumption
in every period. Furthermore,
[2]
implies that a t
=
+
1/(1
11*) is
bounded away from
0.
Conversely, consider a sequence {-ij-^o satisfying equilibrium recursion (15) and the condition
inft at
>
0.
sequence
7r t
The perpetuity
price
=
are
(l/i? t 0,
,
..0)
=
lit
l/ a *
bounded
—
the corresponding interest rate
1,
across
By Lemma
t.
4,
R
t
and the price
each agent has a unique optimal
plan and markets clear in every date-event. This establishes
Proposition 4 Any sequence
GEI
H(z
t
We
henceforth focus on infinite horizon paths with these properties.
+i) defines a
a
is
=
equilibrium.
steady state
a fixed point of the recursion mapping H.
is
finite price IIoo
every period.
larger than unity:
R^,
—
1
m
The
net interest rate r
+ r^ >
1.
By Assumption
is
[2],
the perpetuity has
therefore positive,
and the gross rate
Since storage has a lower rate of return (p
agents never allocate their savings to this low-yield technology: Sx,
to
and the recursion zt
Steady State
4.2
A
>
{zt} satisfying the condition inft a t
consume
is
given by
a^ =
Proposition 5 In a steady
HRooP)
—
1
R^
It is
.
2
1
J
)
0.
1
<
i?oo),
The marginal propensity
then easy to show
state, the interest rate
= -r 2 (l- R-
=
<
and
2
[F(A'00 ) <T^
capital stock satisfy
+
A^ + ^]/2
(18)
R^ = l-5 + AF (K00 )-T(l-R^)[F'(K00 )F(K00 )a A + K00 aj]
2
,
The
equation corresponds to the stationary Euler condition for consumption, and the second
first
to the optimality of capital investment.
1//3 is
(19)
We
note that
reached when markets are complete. This result
R^ <
is
1/(3,
and that the upper bound
a possible solution to the low risk- free
rate puzzle, as has been extensively discussed in the literature (e.g.
Weil, 1992; Aiyagari, 1994;
Constantinides and Duffie, 1996; Heaton and Lucas, 1996).
The closed-form system
and comparative
functions
statics.
R\(K) and
and as
We
—
(19) allows us to conveniently analyze existence, multiplicity
observe that equations (18) and (19) define two weakly decreasing
R-2(K), which intersect at a steady state.
Proposition 6 There
We now
(18)
A
continuity argument implies
exists a steady state in every infinite-horizon
economy.
turn to comparative statics. As can be seen in equation (18), the idiosyncratic risks
all
increase the precautionary
and increase
capital investment.
The
demand
for savings,
capital stock
14
which tends to reduce the
<7 e
,
aa
interest rate
Aoo therefore increases with the endowment
risk
ae
,
as in Aiyagari (1994).
We
note, however, that uninsurable technological shocks
also reduce the risk-adjusted return to investment
While better insurance against endowment
effect,
risks.
may
aggregate activity
These conflicting
and can thus discourage
some
17,5
capital accumulation.
always reduces output through the precautionary
risk
actually be increased by
effects lead in
and
17,4
new instruments
for
hedging technological
cases to a non-monotonic relation between capital and
idiosyncratic technological shocks, as can be checked numerically.
The steady
state
unique when markets are complete, or more generally when agents are
is
>
exposed only to uninsurable endowment shocks (a e
by observing that Euler equation
R^.
real interest rate
Euler equation (18).
(18)
is
The
Proposition 7 There
functions
R\{K) and R2(K)
14
We
prove this property
are then both strictly decreasing in K, and
role in our
K^,
high, as seen in (18).
unique in economies with either complete
work
(e.g.
Banerjee and
Newman,
1991; Galor
and
model, Proposition 7 shows that the interaction between the bond
risk
agents have a
By
is
some economies with unin-
risk.
market and idiosyncratic production
is
0).
15
effects considered in earlier
a low capital stock
—
robustly exists a multiplicity of steady states in
markets or only endowment
no
°~5
then independent of the capital stock and implies a unique
surable production risks. In contrast, the steady state
Zeira, 1993) play
=
aa
In the presence of productivity shocks, however, productive investment affects
multiple steady states can arise.
While the wealth
0,
is
a possible source of poverty traps. In an
economy with
weak precautionary motive and the equilibrium
investment equation (19), the low capital stock
high marginal productivity and interest rate.
The economy
is
interest rate
also consistent with
thus stuck at a low wealth level as
is
agents coordinate on a large real interest rate on the bond market.
The
multiplicity of steady states can be viewed as a natural consequence of the
plementarity in investment discussed in Section
3.2.
When
low investment
rate i?oo are anticipated in future periods, investors are unwilling to
in
the present.
This implies
in
make a
i\~oo
dynamic com-
and high
interest
large risky investment
turn a high marginal productivity of capital, low precautionary
savings and thus a high interest rate in the current period.
urally generates multiple equilibria, as has
been emphasized
This type of positive feedback natin the literature
on macroeconomic
complementarities. Proposition 7 establishes that the complementarity between future and current
investment can impact the number of steady states.
It
can also
affect the local
dynamics, as
is
now
discussed.
14
Note that the uniqueness of the steady state
is
specific to the the
pure endowment version of our model.
In
general Bewley economies, multiple steady states can arise from the nonmonotonic impact of the interest rate on the
aggregate
15
It is
demand
for the
bond (e.g. Aiyagari, 1994).
number of steady states
easy to show that the
is
generically odd.
15
Local Dynamics
4.3
The
dynamics around the steady state can be examined by linearizing the recursion mapping.
local
Wealth Wt
the only predetermined variable in the model, while consumption and marginal
is
propensity are free to adjust.
the system
When
markets are complete, the stable manifold has dimension
determined, and the economy converges monotonically to a unique steady state.
is
1,
None
of these results need hold under incomplete markets.
We
begin by considering economies with a single steady state.
Proposition 8 When
mined depending on
We
show
A
the
is
continuum
of equilibria
indeterminacy, because there
We
result
either locally determined or locally undeter-
it is
economy's parameters.
is
and large idiosyncratic production
0.95)
unique,
property by computing numerically the eigenvalues of the linearized system in some
this
examples.
the steady state
is
obtained in an economy with patients agents
risk.
no obvious
Note that
it
focal equilibrium
provide in the Appendix an example of a supercritical
is
j3
=
not easy to rule out this form of
on which agents can coordinate.
bifurcation as
flip
obtained in a class of economies with fixed psychological discount factor
is
(e.g.
aa
/?
increases. This
=
0.95.
We
infer
that cycles of period 2 exist on an open set of economies.
Proposition 9 Some economies
These cycles
arise
robustly contain attracting cycles.
with reasonably patient investors, but their existence appears to require high
levels of idiosyncratic
production
risk.
Slightly different results are obtained in economies with only undiversifiable
(<r e
>
state
risk
0,
is
is
=
aa
locally
o"<5
=
0).
Let
ft*
=
1/e
«
0.368.
We
show
determined whenever the discount factor
in the
j3 is
endowment
risk
Appendix that the unique steady
larger
than
(3*
.
Uninsurable investment
thus indispensable to obtain indeterminacy and cycles with very patient agents. 16 Consistent
with the analysis of Section
3,
these findings also suggest that idiosyncratic production shocks
help to generate cycles under less heavy discounting than
is
may
required in nonconvex complete-market
growth economies (Boldrin and Montrucchio, 1986; Deneckere and Pelikan, 1986; Sorger, 1992).
We now
examine economies with multiple steady
states.
The Appendix provides an example
with three equilibria, in which the two extreme steady states (lowest and highest Woo) are locally
unique and the middle one
Proposition 10 When
When
/3
—
>
0,
is
totally unstable.
there are multiple steady states,
some can
be totally unstable.
indeterminacy and cycles are obtained in any economy with fixed production function
parameters T, A, F,
5, p,
aa
=
&5
=
0, <r e
>
0.
This result
is
and
consistent with the indeterminacy obtained in Calvet
(2001) for exchange economies with sufficient impatience and large uninsurable
16
F
endowment
shocks.
Several mechanisms help to explain the rich local properties uncovered in this subsection. First,
idiosyncratic production risks induce a
dynamic complementarity between future and current
in-
vestment. Second, we showed in Section 3.3 that endogenous fluctuations can arise from a negative
precautionary feedback between future and current wealth. While a precise decomposition
we
available for our nonlinear system,
infinite
is
un-
conjecture that both effects contribute to indeterminacy in
horizon economies. Intuition suggests that the negative feedback also helps to generate en-
dogenous
cycles, while complementarities in
that uninsurable production risk
is
in
investment induce multiple steady states.
We
observe
any case crucial to obtain the negative feedback and dynamic
complementarity, and thus the rich dynamics, along the infinite-horizon growth paths.
Extensions
5
We now
show that our framework remains tractable under multiple
Human
5.1
Physical and
We
consider a one-sector
economy with two types
only traded asset and there
the stocks of physical and
y£+1
where 5 k t+1 and 5 X
is
sectors.
of capital,
and investigate the impact of
efficiency.
To
simplify notation, the
specific
bond
is
the
no exogenous endowment or storage. Let k and x respectively denote
human
=
and
Capital
on investment and production
idiosyncratic risks
capital types
capital.
The output
4+ iHkt4) +
(i
of agent
- aiUi)*? +
t+1 are the stochastic depreciation rates
h
(!
is
-
given by
s *,t+i)4,
on each capital type. The agent
satisfies
the budget constraints
c
h
+
k
<!
h
+ xh +
h
Q /Rt
= w h^
= A ht+l F{klx h + {\-8l t+l )h t + {l-8l t+1 )x h +e h
>
t )
where 9 t denotes bond holdings
Agents have identical
CARA
in period
t
t
,
t.
preferences and are exposed to idiosyncratic shocks
(4,4,4) ~Aai,£),
where 1
=
(1, 1, 1)'
and
£=
al
The
coefficient
aa
specifies the idiosyncratic productivity risk
whereas a^ and a x quantify
specific depreciation shocks.
17
common
to both types of capital,
We
focus on equilibrium paths along which capital investment
macroeconomic aggregates
are deterministic. Risk-adjusted output
G (k, x,a)=F {k, x) We
—
risk- adjusted returns to physical
and
identical across agents
defined as:
is
F (k, xf o\ + k 2 a\ + x 2 a 2x
easily check that the optimal individual decision (Kf, Xt)
The
is
G (k, x, a
maximizes
t
+\)
and human investment are therefore equated: Gk
— Rt (k + x).
= Gx =
Rt,
implying
Undiversifiable depreciation risk can thus create a wedge between the marginal productivities of
physical and
human
^ Fx ). The
capital (Fk
allocation of savings
then
is
inefficient, in
the sense
that decentralized production plans do not maximize aggregate output for a given level of savings:
(Kt
,
X
$ arg max {F(k, x)
t)
s.t.
+x=
k
K
+ Xt )
t
.
(k,x)
To provide
where a
>
additional intuition, consider a Cobb-Douglas production function
>
0,n
+
and a
characterized by the relation
Proposition 11
<
r\
=
Xt/Kt
Ifo'k/o'x 7^ y/v/ a
The
1.
->
r\ja.
first-best
We
=
fc
a :r 1',
(complete-market) allocation of savings
is
can then show that under incomplete markets,
the equilibrium allocation of savings
F{K ,X < max {F{k,x)
s.t.
t )
t
F(k,x)
+x =
k
K
+
t
X
t
is inefficient:
}
.
(k,x)
In particular, Xt/Kt
The
< n/a
if
and only
if
<
crk/a x
y/n/a.
allocation of savings remains efficient under incomplete markets only
are "balanced", in the sense that
o^ja x
generally shifts savings from physical to
depends on the
relative
=
The
capital.
magnitude of depreciation
the type-specific risks
In an arbitrary economy, an increase in a^
yrj/a.
human
if
risks. If
o-\c
effect
on overall productivity then
<
y/n/a, an increase in a^ will
ju x
actually increase investment efficiency and output for any level of savings, whereas an increase in
ax
will decrease productivity.
This simple model of
human
capital thus confirms
(2003) in the context of endogenous growth.
human
capital depreciation, our
and extends the analysis conducted by Krebs
While Krebs considers only idiosyncratic shocks to
framework includes shocks to
to the specific depreciation rates of each type of capital.
the relative importance of undiversifiable shocks to
empirical question.
incompleteness
may
From a
This
human and
theoretical perspective, our
substantially influence the
overall individual productivity
model
may be
a useful extension, since
physical capital remains an open
also suggests that capital-specific
macroeconomic impact of
18
and
financial innovation.
Multiple Sectors
5.2
Our benchmark model
technologies j € {1,
easily extends to a neoclassical
As
J}.
...,
average depreciation rate
the riskless bond
t,
is
is
in the previous example,
the only traded asset.
When
The budget
A h+1 =
is
good and multiple
by assuming that the
zero, there
is
no storage, and
agent h invests k^ t units in technology j in period
=
Aj jt+ iFj(kj )t ),
a production function satisfying
is
constraints are then given by
h
c?
The shocks
simplify notation
an idiosyncratic technology-specific shock and Fj
the Inada conditions.
single
random output
Vj,t+i
is
we
equal to unity, the endowment shock
she obtains next period the
where Aj t+1
economy with a
EU
\pi
+
i
Uh
k
t
are jointly
(^t + i)j=i,...,j
_ „„/i
wt
oh In,
+
,
t/Rt
normal
ftf(l,
S)
J
where
,
£=
o
The
coefficient
<jj
>
4
...
j
measures idiosyncratic risk in sector
Consider the risk-adjusted output of technology
17
j.
j:
Gj (kj ,a) = Fj (kj )-TaFj (kj yaj/2.
We
show that the optimal
easily
condition
is
capital stock k^ t maximizes Gj(k,at+i)
—
Rtk.
The
first-order
then
—— (Kj t,at+
t
i)
=
Rt-
Agents equate the interest rate with the risk-adjusted return to capital in every
result, the cross-sectoral allocation of capital
sector.
As a
does not maximize aggregate output given the level
of savings.
Proposition 12 The
allocation of capital
is
generically inefficient: for almost every S, the indi-
i
J
vidual plan (Kj,t)i<j<J satisfies
J
Y, fj
(
Kit)
<
3=1
where
17
K
t
=
J2j=i
For instance
if
J
^
^
{
J
J2 F
i
(*y)
s t
-
-
Yl ki
=
K
*
*
3=1
lj=l
Kj.t-
= 2,
<7i
>
and oi
=
0,
we can
interpret sector
19
1
as private equity
and sector
2 as
public equity.
As an example,
capital
if
al
=
is
let
=
F\
F%
=
...
=
Fj. Aggregate output
allocated equally across sectors. This
a2
=
••
=
crj. If
is
is
then maximized
if
obtained under incomplete markets
and only
if
and only
instead risks are asymmetric across sectors, the allocation of capital
biased towards the least risky sectors.
if
is
In our benchmark one-sector model, incomplete markets
induce lower income and savings because production risk discourages capital accumulation. With
multiple sectors, the allocation of any given amount of capital
is
also inefficient
and can further
reduce aggregate output.
6
Conclusion
This paper investigates a missing market economy with decentralized production and idiosyncratic
technological risk. Agents are unconstrained in their borrowing and lending, and incomplete risk-
sharing
is
the only source of market imperfection. Macroeconomic aggregates and financial prices
are deterministic but can follow complicated dynamics and cycles even
tient.
The
results illustrate that
when
agents are very pa-
endowment and productivity shocks can have profoundly
implications for aggregate dynamics. Investment risk thus appears to be a promising
for business cycle research.
In multisector settings, cyclical variations in sectoral risks
lead to cyclical variations in the cross-sectoral allocation of resources
business cycle.
We
new
direction
may
also
and thus further amplify the
leave the further examination of these issues for future research.
20
different
Appendix
Proof of
Lemma
1
We establish the lemma by a backward recursion. The results trivially hold at terminal date T.
We now assume that the lemma has been proven for dates t + 1, t + 2,..., T, and seek to show that
it
also holds at date
An
t.
entrepreneur chooses consumption dp, capital expenditure k^, storage
s^ and asset holdings 6t that maximize
h
u{c t
)
+ p®t J
h
subject to the budget equality
kt
>
0, St
>
+
c^
The agent never
0.
storage and investing in the
The
+ A%+1 F($) +
\e t+l
bond
k\
+
s\
+
uses storage
if
R =
t
solution to the Bellman equation
-
(1
7r<
#+1 )*£ + ps h + ck +1
>
Rt
if
simplified
+
t
,
1
w^, and the non-negativity constraints:
and
p,
between investing
indifferent
is
then convenient to consider
p. It is
is
=
Q\
•
h
6t
t
0t
=
ps"
+
6q
t
in
.
by the following observations. Since u^+1
is
normal, we rewrite the objective function as
h
u{c t
Budget constraint
)+$J
(1) implies
*(*£)
that
ot
=
Ta t+1
+ \tRt{wt
—
1
c^
—
Vart{i
k£
—
),
't+i
X)n=i
7r
t
+
l
".*^nt)-
The
decision problem
thus reduces to maximizing
N
h
u{c t
)
+ (3J
Tat+i
*nA)
Vart (w?+1 );t +
l
71= 1
over the unconstrained variables (c^, tf, {9 n t)n=i)-
We
demand
begin by considering the
for risky assets.
Conditional on productive investment
and consumption, the optimal portfolio maximizes
N
Tat+l
h
-RtJ^ 71 n,t u n,t
N
Var
h
t
+ A*+1 F{kt) +
e t+l
(1
-
n=l
The
#+ i)tf + E < A,t+i
n=l
solution to this mean-variance problem
is
thus
<t = -/% - ^(tf) + #£** - i? 7rn
t
The agent
fully
,
t
/(rat+1 ).
(21)
hedges the endowment and productivity shocks, and takes a net position
assets to take advantage of possible risk premia.
We
plug asset holdings into
(2)
and
in risky
infer that
future wealth satisfies
N
"t+i
=
*(*£)
+ i*t$>».'
W) -#£# + #£ +
oh,t
'A,n
Rt^n,i
(22)
ra=l
h
h
h
+Rt(w? -4- tf) + F{k )A t+l - 5 t+l rt
t
Rt
e t+l
N
Ta t+: /
n=l
21
K-nfidn. t+l-
+
P
We now consider the
optimal
level of
.
productive investment. Substitute next-period wealth (22)
into the objective function:
»(<£)
G{kl
+ /jj
a t+1 ,a t+1 )
+
Rt
+Rt(i
The optimal
£ ti tt^'F^) - /3&fc* +
*?
h
Uh\
<H)^ 2ra£S+ V
2^rc=l ^n
ih,t
?
>JV
JV
__
ct
,
7T
t
ti\
;*
2
productive investment therefore maximizes the criterion
level of
+l
(23)
1
(6).
Since F'(0)
=
A + R En=i /^'J^n.t > °We finally turn to optimal consumption. We differentiate equation (23), take logs and infer that
consumption is a linear function of wealth: c\ = atw^ + b^, with slope at and intercept 6j given by
(3) and (4). The consumption-investment plan satisfies the Euler equation u'(c ) = /3 RtE v! (c t+ i)
= PR E u(ct+ Hence J h {W,t) = E Y%=t Su { cs) =
Since v! = -Tu, we infer that u(c
U )u(c ), or equivalently J h (W,t) = —(Ta )~ exp [— T(a W + bj )] We conclude that the lemma
+oo, we
>
infer that k£
if
t
t
t)
t
t
(.1
-i).
l
t
t
aj<
1
show the theorem by a backward
=
1
risk
premium
conditions for
Step
is
t
+
Step
l,..,T.
all
depreciation rate
individual vectors
<5
4+1
It is
.
=
'
f$
A
Assume now
0.
shows that when there
that the properties of the theorem
only one type of technological
is
(PXn)n=l- Since there
>
The corresponding
JV}, the
risk,
the
Step 2 derives the equilibrium
convenient to define the truncated price vector n t
1
9t
..,
T, agents have identical propensities
are exposed to shocks to total productivity A^ +1 but not to the
=
{1,
=
=
economies.
and therefore Ylh=i ™t Pa
n€
fcj.
t
necessarily zero in any finite-horizon equilibrium.
Assume that agents
1.
1
At date
recursion.
and make no risky investment:
hold at instants
.
t
t.
Proof of Proposition
We
1
t
holds in period
t
•
/
^-
...
Without
is
capital stocks provided
>
%
flf
by equation
h=l
This yields the vector equality
7r<
1 '*
(6)
>
...
then satisfy k\
i|2
_
T a t+1
Y2h=i F{kt)P a /(HRt)- Since
L
H
./i=l
h=L+\
'
P
HRt
22
^
> 9t f/.
ra t+ i
h=\
= — Tat+i
Pa =
we index households so that
market clearing of security n implies
H
{^n.t)n=i an d the
no aggregate shock, we know that ^/i=i
loss of generality,
L
> 9t p /
>
—
>
> 1$
.
For any
we
infer
2
||7r t
a
-
l|5?t »
We
conclude that
=
7r t
2.
-^r^
any equilibrium.
in
to depreciation shocks 5 t+l
Step
# + ELl+1
< -Tat+1 [eLi H^) %t
||
= — /?£'* — P^ n F(K + Pg'n K
t)
t
A
X>
t
tradable components of her production and
=
Kt >
/(ffifc), or
tf]
*' -
°"
'
argument holds
similar
kl}
%
)
but not to total productivity A%+1
,
Agents make the same investment
purchases 9^
i+1)]
F(fc*
-
+1
F(tf
if
agents are only exposed
.
Each agent
in the risky technology.
units of each risky asset; she thus sells at
endowment
risks.
Equations
and
(9)
no cost the
(11) are implied
by
individual decision, and equations (10) and (13) are obtained by aggregating the budget constraints.
The
first-order condition u'(c^)
=
/3i?fEtu'(c£+1 ) implies
^m
+
-Ta*t+l [F(Kt ?o\ t+l
=
+
<*
l
Aggregation of this relation then yields equation
Proof of
The
Lemma
— Rtk
objective function G(k, at + \, o~t+\)
condition F'(Q)
Condition
(i).
The optimal
Condition
The
+ (1-5= +oo
Since
strictly
t+1 ]
-
c?+1
.
(12).
+e-
t
is
conveniently rewritten as
Ta t+l [F{k) 2 a\ t+x
+
k
2
implies that the optimal capital stock
p
>
1
is
and
is finite,
R >
the optimal capital stock
F'(k)[A
Rt)k
objective function
capital stock
(ii).
<
2
AF(k)
The
+ K?cl+1 +
— 5,
FOC
concave in
k,
is
<
(24)
t+1 ]/2.
strictly positive.
and diverges to — oo
as k
—
>
+oo.
(14) has a unique solution.
the objective function (24) diverges to
Let ko
is finite.
strictly
+
a\ t+l
= F~
1
— oo
as k
—
>
+oo, and
[A/(Ta t +ia 2i )]. The function Gk{k,at+ i,o~ t+ i)
— Ta t +iF(k)a\ t+l + 1 — 5 is decreasing and larger than 1 —
smaller than 1 — 5 on [&o, +oo). We conclude that the equation
]
5
when
/c
£
(0, &o],
Gk{k, at+i,o~t+i)
=
and
—
is
Rt has
a unique solution.
Proof of
Let
Lemma
K% = min {k
:
3
Gk{k, at+i,crt+i)
following procedure. If
and
R =
t
=
W +\ < W" +1
*
t
G;~(.R"t,at + i,cr t+ i)
t
>
p.
,
On
*
p} an d W"t +1
=
3>(2£j*).
Any
solution x t can be calculated by
equations (9) and (10) imply that
the other hand
23
if
W +\
t
K
> W£+1 we
,
t
—
3>
infer
-1
(Wt + i), St
from
(9)
and
=
0,
(10)
=
that Rt
to a t
,
C
p,
and
t
=
Kt
W
t
[Wt+i
— &(K
t
Equations (11)
)]/p.
(a t ,Kt ,St,Rt)
we observe that
Finally,
.
=
an d St
A'i";
€ (0,1)
— (13)
then assign unique values
x!*+
xR +
x
\p,
+oo), while no
simple restrictions seem to be imposed on current consumption and wealth.
Proof of Proposition
Consider
oGl
2
and the mapping
Wb(Wr)
if
a
otherwise.
j
=
<p(WT )
Wo(Wr)
defined and larger than a,
is
<
I
When Wt — e,
productive investment Kt-\ and St-\ decline to zero, implying that Rt-i —* +co, a^—i — 1,
Ct-i — —oo, Wt-i — — co, and <p(e) = a. On the other hand when Wt — +co, we observe that
Rt = p for all t and ^>(Wt) — +co. This establishes that (^[e,+co) = [a, +co). Since the argument
Since
<p is
continuous on
domain
its
[e,
+oo),
its
image
is
an interval of the real
line.
*
>
>
*
>
applies to
any choice of
= Wo
we conclude that <p(Wt)
a,
has a solution for any
Wo £
K-
Proof of Proposition 3
Consider an economy with only undiversifiable endowment shocks [ae
>
and a a
=
&s
—
0)-
We
and thus equilibrium uniqueness by recursively showing
establish the strict monotonicity of Wq(-)
the following
Property.
The wealth function Wi(Wr)
productive investment
K^Kt)
is strictly
capital stock
is
true for
Kt-\
= $
t
_1
=
T.
Rt(Wx)
Assume
that
Wt, we
infer that a t -\
it is
true for period
=
$'(Kt-i)
is
1/[1
+
{a t
Ci-i
is
therefore increasing in
the terminal wealth level
t.
while marginal
In the absence of storage, the
decreasing in Kt-\ and thus in
St-i strictly increases with
=
Wt,
are weakly decreasing.
Wt-
We
R -i)~
1
t
=C
t
-
}
is
Wt-
In the presence of
Wt
while Rt-i and Kt-\ are locally constant.
Wt-
Since a t and Rt-\ are both decreasing in
storage, the safe investment St-\ locally increases in
+
in
(Wf) locally increases with future wealth Wt and thus terminal wealth
Wt', the interest rate Rt-\
In either case, Kt-\
Wt- Mean consumption Ct(WT),
and storage S^St) ore weakly increasing
propensity a t {WT) and the interest rate
The statement
increasing in
decreasing as well. Consumption
ln(fiRf-i)/T
conclude that
- Ta2t a 2e /2
W ~\ = Ct-i + K -\ + St-i
t
Wt-
24
t
strictly increases
with
Proof of
We
4
begin by checking the convergence of the series (17) defining
quences
a
Lemma
>
{at}^ and {R
and
istic
R>
sequence
0. It
t
are contained in
}^l
compact
is
form
intervals of the
and {Mj*}^
follows that the sequences {k^}^l
{^}^o
By Assumption
b^.
[a, 1]
and
are bounded.
[2],
the se-
[R, R],
where
The determin-
thus well-defined and bounded.
We easily check that the consumption-investment plan defined by (7) satisfies the Euler equation
= PRtE u'(ct+i). Since v! = —Tu, we infer that u(ct) = RtE u(ct+ i) The candidate plan
u'(c
h
or equivalently J (W,Q) =
thus generates utility J h (W,0) = E Et°^o^ u c*) = (1 + n )u(c
t)
t
ft
t
),
(
-(r ao )- 1
We
exp[-r(aoW +
must now
6^)].
verify that
J h (W,
0)
is
indeed the value function.
an admissible plan {c't ,k't s't ,9't W/}^ that
T E u(W^) =
We know that
,
limr-^oo
P
.
,
W
Wq =
satisfies
For a given
W,
consider
and the transversality condition
0.
J h (W,0) =
Max
u
{c,k,s,9,W}
h
/3EJ (Wi,l)
c
L
>u(c'Q )+PE J h (W{,l),
and by repetition
T-l
J h (W,0)>E
+ f3 T E J h (W^T).
Y.^<^
_t=o
(25)
bounded and exp(— TcltWj) < 1 + exp(— TW^), the transversality
-1 T
condition implies that /^EoJ^W^T) = -(ra r )
as
/3 E exp [-T(a T W^ + b^)] converges to
Since {exp(— r6^.)/ar}^_
T—
»
oo.
We
is
thus obtain that
+ CX)
J h (W,0)>E Q
X>V4)
(26)
t=0
for
any admissible strategy, and conclude that J (W,
0)
is
the value function.
Proof of Proposition 6
Letting tp(R)
= T~ 2
(1
- l/R)' 2 ln(i), we
p(i2)
F'(A)
The
a - rv,
function y(i?) decreases on
i
=
rewrite system (18)
[F(A')
2
*>^
(1, 1//3]
the decreasing function R\(K), which
and
^+K a
2
-r
2
i_
satisfies 93(1, 1//3]
25
(27)
AVI +
=
[0,
l
1
-5-R=
0.
(28)
+00). Equation (27) thus defines
2
(\,<p~ {a /2)\. Similarly, equation (28)
E
R2(K) that maps
(F')-\5/A).
(19) as:
a 2] /2
i
maps [0,+oo) onto
implicitly defines the decreasing function
+
-
(0,
K*\ onto [l,+oo), where
K* =
Consider the function
_1
(cr|/2),
(/5
0.
R2 {K)
The graphs
A(K) = R\{K) - R 2 (K). When
-> +00, and therefore
R2
Ri and
of the functions
A(K)
-> -co.
We
K
->
0,
we observe that Ri(K) ->
also note that A(A'*)
and there
therefore intersect
= Ri{K*) -
exists at least
1
>
one steady
state.
Finally,
We
ters.
we analyze the monotonicity
>
consider the case |i?^(i C00 )|
;
R\(K) and
An
|i?j(ivoo)|.
R 2 (K)
leaves the function
of the steady state with respect to the economy's
increase in
ae
or
unchanged. The steady state
/?
is
pushes down the function
therefore characterized
a lower interest rate and a higher capital stock. Similarly, an increase in
R2 (K),
in r,
and a higher capital stock.
also leading to a lower interest rate
R2 (K)
a a °r a & pushes down both R\{K) and
parame-
—
1
We
5
and
A
by
pushes up
note that an increase
and can have ambiguous
effects, as is verified
in simulations.
Proof of Proposition 7
It is
enough to provide an example of
equilibria
when Y =
A=
1.2,
a =
20,
We
multiplicity.
0.95,
/?
=
0.05, 5
=
numerically check that there exist three
0.5,
aA
=
9,
as
=
0,
ae
=
0, e
=
0.
Local Dynamics around the Steady State
There
is
no storage around the steady
without undiversifiable depreciation
K
The
iterated function
Rt
at
=l/[l +
C
= Ct+1 - H/3Rt)/T -
[A
t )
t
+
- Ta t+1 F(K
t
{a t+1
K
t
R
t
+1-5,
ra?+1 [F{Kt fa\
+ o*]
/2,
has Jacobian matrix
dCt
da-t+l
dCt
dat+l
dK /dWt+1 = l/&(K >
observe that
t
dR
t
= -TF(K )F'(K
t
dR
t
dW +1
t
we
0,
t )
t
dat+i
1,
)a A ]
.
dH =
>
then implicitly defined by:
},
dat
dWt+i
oa.t + 1
Since R,x,
is
focus on economies
)- 1
^±-
We
H
we
),
= F'(K
t
H
For computational simplicity,
1
W =C
function
risk.
=$- (Wt+1
t
t
The
state.
1
&{K
{F"(Kt
)a A
)
[A
<
1
1
dCt
dWt+i
djCt+Kt)
dWt+1
and 18
0,
- Ta t+1 F(Kt )a A ]
a t+1 [F'(Kt )}
2
Ta\} <
t)
infer that
A—
2
TatJr \F(Kt)<J A
>
on a neighborhood of the steady
26
state.
0.
'
Let
= l/il+v- ). We note that
= <p(at+\Rt), we infer that
1
<p(v)
Since at
=
<p(v)
2
da t
at
dR
a-t+i
Future propensity a t +i has an ambiguous
on current propensity a t There
effect
[F{Kt
t
2
cr\
)
dC
+ a2
we
/2,
]
dR
1
t
da t +i
Tat +1 [F(Kt ya A
dR
1
t
dw,t+i
increase in a t +i
TRt dWt+i
and Wt+j leads
on current consumption.
On
2
2
and time
t
+
t
The
roots of
P(+oo)
P
= — oo,
and only
dH
P(x)
if
1
da t+ i
+
has an eigenvalue in
^^(i^oo)!
•
Wt+
d(C +
t
may
\
implies that the
lead to a
generate endogenous fluctuations.
K
t )
x
dW t+i
dC
da
t
t
x
dWt+i + da +i dWt+
+
t
t
Thus when there
i
= +oo
and
>
if
Simple calculation shows that P(l)
is
a unique steady state, the Jacobian matrix
+oo), and the dimension of the stable manifold
(1,
which has a positive
xl) can be rewritten as
there always exists a real eigenvalue.
>
=
)
backward dynamical system. Since P{— oo)
are the eigenvalues of the
lit^.?^)!
)F'(Kt
The precautionary motive can then
1.
= det(dH —
da t
=
Since Ct
<7
the other hand, the increase in at+i and
decrease in current consumption and current wealth, which
P(x)
t
+
to a decline in the current interest rate,
agent bears more risk between time
characteristic polynomial
F(K
a t+1 a A -
i
The
both a positive
infer that
z
t
TRt dat+i
dCt
effect
is
.
complementarity of future and current consumption) and a negative indirect
C +i - \n{PRt)lT - Ta 2+1
An
.
t
precautionary motive causes a decline in the current interest rate Rt).
effect (the
2
<0
dWt+i
at+iRt
t
[<p(v)/v]
dat+i
at
8Wt+i
+ t;) 2 =
l/(l
at+i
at+iRt
da t
=
tp'(v)
dRt
-
Rt
da t+ i
direct effect (due to the
v/(l + v) and therefore
is
at least
1.
A. Complete Markets
We know
that
dat
=
da t+ i
and the characteristic polynomial
Q(x)
This implies that x
Q(0) >
=
and Q{\) <
(i is
0,
is
=
dC
/3,
of the
x
2
infer the
0,
1
form P(x)
+
t
=
(/3
—
x
>.0,
x) Q(x),
+
where
dK
t
dWtt+i
contained in the interval
is
polynomial
t
dWt+1
d(C + Kt)
dWtt+i
an eigenvalue, which
and
dC
t
da t+ i
Q
27
We
observe that
(0, 1)
and one root
(0, 1).
has one root in the interval
eigenvalue larger than
B.
Endowment Risk
The unique steady
The
1.
No
-
state
has two eigenvalues in the interval (0,1), and one
stable manifold has thus dimension
>
Productivity Shocks (a e
is
aa
0,
—
o's
1.
=
0)
defined by
MR°oP)
Poo
The
dH
Overall, the Jacobian matrix
in (l,+oo).
= -r J (i - /CT^/2,
(29)
=
(30)
tfCKoo).
characteristic polynomial simplifies to
^='5:
We note
1
and
infer that the
<
\x2\
TRt
1.
The economy
is
+
^|
to observe that P(0)
It is useful
the roots £2 an d £3 are complex, we
implies that
J_}
'
polynomial has at least one root xi
to characterize the other roots £2 and £3.
If
A\F"(K
Q
>
that P(l)
—
+
P
+
1
know
=
that £3
The
£2-
=
R^
=
and X1X2X3
X2 and £3 have the same sign. Furthermore since P(0)
<
1,
one of these roots
assume without
It
;
remains
(0, 1).
=
£i|x2|
2
<
1
then stable and locally determined.
1
We
^ )k
€
relation P(0)
>
interval (—1,1).
(
+oo).
in (1,
the roots X2 and £3 are real, the inequalities x\
If
F»
loss of generality that
X2 €
(
— 1,1). We
P(Q)
is
>
implies that
contained in the
can then consider two
cases:
• If X2
and £3 are both
(0, 1).
•
If
The steady
state
is
and P(l) >
>
positive, the inequalities P(0)
imposes that X2,X3 €
stable and locally determined.
on the other hand X2 and £3 are both negative, we know that £3 > —1
P(-l) >
0.
= -r
Since ln(Poo/3)
2
(l
- P~
2
1
)
<r
2
/2 at the steady state,
2
The condition Poo £
(l,/3
1
P(-l)
Let p*
When
=
1/e
w
]
>2
1
agents only face uninsurable
indeterminacy and
Conversely,
we can
flip
+
2A\F"{K\
/?
>
endowment
/3*
easily infer
shocks,
When a e
28
—
>
+
ln/3).
has a locally determined steady state.
show that indeterminacy can
patience and large uninsurable shocks.
;i
TRL
Re
bifurcations with patient agents
easily
if
implies that
Any economy with
0.368.
and only
MTU?)
1
^>->iH'+^i
we
if
it
is
(i.e.
thus impossible to obtain local
with
arise in
/3
>
/?*).
economies with
+00, the steady state
sufficient
satisfies
P^ —
im»
1,
A'oo
—
>
(F')
1
r 2 a^
{8/A) and
—
cr^
infer that
P(— 1) <
indeterminacy obtained
large uninsurable
for a sufficiently
in
characteristic polynomial
4q^
4+
We
The
21n(l//3).
»
(1
.b
low value of
then
i
;
This result
/?.
is
consistent with the
is
Calvet (2001) for exchange economies with sufficient impatience and
endowment
shocks.
C. Uninsurable Productivity Shocks
=
Consider the parameter values:
We
T =
0.95,
check numerically that the steady state
one eigenvalue x\
when a a =
>
1
and two eigenvalues
is
A =
2,
when a a <
in the unit circle
is
flip
bifurcation as
Finally, consider the
as
it
=
0, e
=
examined
aa
varies
economy T
in the
=
between 30.5 and
A =
20,
a
proof of Proposition
7.
1.2,
0.1,
=
as
ae
0,
=
0.
=
30.5.
On
the other hand
one eigenvalue in each of the interval
(— co,— 1), (—1,1) and (1,+co). This establishes Proposition
undergoes a
=
0.35, 5
unique and that the characteristic polynomial has
remains unique but there
31.5, the steady state
a =
10,
We
8.
31.5,
0.95,
also note that the
which proves Proposition
=
/3
=
0.05, 5
oA
0.5,
=
9,
system
9.
ae
=
0,
This economy has three steady states, and
can be checked numerically that the steady state with the intermediate interest rate
is totally
unstable, thus establishing Proposition 10.
Proof of Proposition 11
We know that G k (K ,X = Ru we infer that Fx {Kt
t
t
)
,
X
t)
[l-ra
ma
2
4
F {Ku X
t )}
= R +rat+1 a x2 X
t
t
,
which implies
l-Ta t+1 a A F(Ku X )>0.
t
Equation (20) can be rewritten as
We
consider parameters satisfying a\.ja x
side of (31)
side of (31)
is
is
On
weakly negative.
strictly positive
<
\Jrj/a,
X /K
the other hand since
a contradiction.
-
and assume that Xt/Kt > n/a. The left-hand
We
t
t
> n/a > a^/a x
conclude that
X /K
t
t
<
,
the right-hand
n/a.
Proof of Proposition 12
The proposition
is
a direct application of Sard's theorem.
there are two sectors (J
=
2).
For a given sequence {Rt}^Z
'
H{ki,h 2 ]ai,a 2 )
=
We
r)C
-^-(/ci,^)
- Rt
BC
;
29
,
assume
for notational simplicity that
consider the function
-^--(fo,^) -
R
t
;
F[(ki)
- F^fo
It is
easy to check that the Jacobian matrix of
conclude that the system
H=
H
has rank 3 whenever H{k\, k%\
=
<J\,<J2)
0.
We
has no solution for almost every choice of {(T\,a2).
References
Aghion,
Bacchetta,
P.,
P.,
Banerjee, A., 2000.
"Capital Markets and the Instability of
Open Economies,"
manuscript, Harvard University.
Aghion,
P.,
Dualism and Macroeconomic
Banerjee, A., Piketty, T., 1999.
Activity.
Quarterly Journal of
Economics 114, 1359-1397.
Aiyagari, S. R., 1994. Uninsured Idiosyncratic Risk and Aggregate Saving. Quarterly Journal of
Economics
109, 659-684.
"Idiosyncratic Production Risk,
Angeletos, G. M., Calvet, L. E., 2003.
NBER
Working Paper 9764. Forthcoming
Banerjee, A., and
Economic Studies
Benhabib,
J.,
Newman,
in Journal of
Cycle."
Monetary Economics.
Risk-Bearing and the Theory of Income Distribution.
A., 1991.
Review of
58, 211-235.
Day, R., 1982.
A
Characterization of Erratic Dynamics in the Overlapping Generations Model.
Journal of Economic Dynamics and Control
Benhabib,
Growth and the Business
4,
37-55.
J.,
Farmer, R., 1994. Indeterminacy and Increasing Returns. Journal of Economic Theory 63,
J.,
Nishimura, K., 1979.
19-41.
Benhabib,
in Multisector
The Hopf
Models of Optimal Economic Growth. Journal of Economic Theory
Boldrin, M., Montrucchio, L., 1986.
Economic Theory
On
21, 421-444.
the Indeterminacy of Capital Accumulation Paths.
Journal of
40, 26-39.
Brock, W., Mirman,
of
Bifurcation and the Existence and Stability of Closed Orbits
Economic Theory
Caballero, R., 1990.
L.,
4,
1972. Optimal
Economic Growth and Uncertainty: The Discounted Case. Journal
497-513.
Consumption Puzzles and Precautionary Saving. Journal of Monetary Economics
25,
113-136.
Calvet, L. E., 2001. Incomplete Markets
Cass, D., 1965.
Optimum Growth
in
and
Volatility.
Journal of Economic Theory 98, 295-338.
an Aggregative Model of Capital Accumulation. Review of Economic
Studies 32, 233-240.
30
Constantinides, G., Duffie, D., 1996.
Economy
Journal of Political
104, 219-240.
Deneckere, R., Pelikan,
Galor, 0., Zeira,
Grandmont,
Heaton,
Asset Pricing with Heterogeneous Consumers.
J.,
J.
J.,
1993.
J.,
Economic Theory
40, 13-25.
Income Distribution and Macroeconomics. Review of Economic Studies
On Endogenous
M., 1985.
Lucas, D.
1986. Competitive Chaos. Journal of
S.,
1996.
Pricing. Journal of Political
Competitive Business Cycles. Econometrica
60, 35-52.
53, 995-1045.
Evaluating the Effects of Incomplete Markets on Risk Sharing and Asset
Economy
104, 443-487.
Huggett, M., 1997. The One-Sector Growth Model with Idiosyncratic Shocks: Steady States and Dynamics.
Journal of Monetary Economics 39, 385-403.
Kiyotaki, N., Moore,
Koopmans, T. C,
1997. Credit Cycles. Journal of Political
J.,
1965.
On
Economy
the Concept of Optimal Economic Growth,
105, 211-248.
in:
The Economic Approach
to
Development Planning (North- Holland, Amsterdam).
Krebs, T., 2003.
Human
Capital Risk and Economic Growth. Quarterly Journal of Economics 118, 709-744.
Krusell, P., Smith, A., 1998.
Economy
Income and Wealth Heterogeneity
in the
Macroeconomy. Journal of
Political
106, 867-896.
Saving and Uncertainty: The Precautionary
Leland, H., 1968.
Economics
Economics
for Saving.
Quarterly Journal of
82, 465-473.
Ljungqvist, L., Sargent, T.
Mankiw, N.
Demand
G., 1986.
J.,
2000. Recursive
Macroeconomic Theory.
The Equity Premium and the Concentration
MIT
Press.
of Aggregate Shocks. Journal of Financial
17, 211-219.
Marcet, A., Obiols-Hom,
F., Weil, P., 2000.
Incomplete Markets, Labor Supply and Capital Accumulation.
Working Paper, Universitat Pompeu-Fabra.
Mitra, T., 1996.
An
Exact Discount Factor Restriction
for
Period-Three Cycles in Dynamic Optimization
Models. Journal of Economic Theory 69, 281-305.
Nishimura, K., Yano, M., 1996.
On
the Least Upper
Bound
Optimal Period-Three Cycles. Journal of Economic Theory
Sandmo,
A., 1970.
Sorger, G., 1992.
Economic Theory
The
On
Effect of Uncertainty
the
Minimum Rate
of Discount Factors
That Are Compatible with
69, 306-333.
on Saving Decisions. Review of Economic Studies
of Impatience for Complicated Optimal
37, 353-360.
Growth Paths. Journal of
56, 160-179.
Weil, P., 1992. Equilibrium Asset Prices in
Economic Dynamics and Control
Economies with Undiversifiable Labor Income Risk. Journal of
16, 769-790.
31
Figure 1A. Finite-Horizon Growth Path
Wealth Wt
Interest Rate R t
1.06
20
1.04
1.02
15
250
10
500
750
0.98
V
15)00
0.96
250
750
500
1000
Storage S t
Capital Kt
10
20
/!
/
15
8
s*~
6
10
5
4
/
/
2
250
500
750
250
10 00
1
J
/"*""
750
1000
M.P.C. at
Consumption Ct
2.25
500
0.024
0.022
2
000
1.75
(
0.018
250
We
500
750
1000
plot the time paths of wealth, interest rate, capital, storage,
consumption and
marginal propensity to consume for a finite-horizon economy (T = 1000) with
uninsurable idiosyncratic production risk (a A >
0).
Figure IB. Turnpike Property
In (Wt /Wa
0.3
0.2
0.1
400
800
600
L
10)0
-0.1
-0.2
-0.3
-0.4
The
figure plots
period
\n(W
t
/
/
Wx
),
which quantifies the distance between
the wealth
W in
t
of a finite-horizon economy (T = 1000) and the steady-state level of wealth
Wx under infinite horizon (T=
<x>).
The economy
is
the
same
as in Figure 1A.
Figure
2.
Nonmonotonicity of
W (WT
)
Wo
WT
10
20
patient agents
initial
and
((3
0.9).
final wealth.
to different
The
60
70
idiosyncratic production risk and
figure illustrates the
non-monotonic relation between
For any given
equilibrium paths.
50
economy with
Multiple equilibria in a finite-horizon
=
40
30
W
<e
[1,3], there are multiple
WT corresponding
Figure
3.
Endogenous Fluctuations
Wealth Wt
in Finite
Horizon
Interest Rate Rt
10
10
10
20
30
40
50
20
10
40
30
50
Storage S t
Capital Kt
0.8
0.6
0.4
0.2
^awwAMW/VwI
10
20
30
40
/VCV
10
t
50
AywAAA^AAAAl
Consumption Ct
20
30
40
50
M.P.C. a t
0.08
0.06
0.04
0.02
10
We
20
30
40
50
10
20
30
40
50
plot the time paths of aggregate wealth, interest rate, capital investment, storage,
consumption, and marginal propensity
to
consume
idiosyncratic production risk and patient agents
((3
in a finite-horizon
=
0.99).
economy with
ate
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