This paper considers the resource constraint commonly used in stochastic one-sector growth models. Shocks are not required to be i.i.d. It is shown that any feasible path converges to zero exponentially fast almost surely under a certain condition. In the case of multiplicative shocks, the condition means that the shocks are sufficiently volatile. Convergence is faster the larger their volatility is, and the smaller the maximum average product of capital is.Stochastic growth; Inada condition; Convergence to zero; Extinction

Takashi Kamihigashi

English

Research Papers in Economics

Forthcoming in Economic Theory
Almost sure convergence to zero in stochastic growth models¤
Takashi Kamihigashi
RIEB, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan
(email: tkamihig@rieb.kobe-u.ac.jp)
First Version: September 24, 2003
This Version: May 31, 2005
Summary: This paper considers the resource constraint commonly used in
stochastic one-sector growth models. Shocks are not required to be i.i.d. It
is shown that any feasible path converges to zero exponentially fast almost
surely under a certain condition. In the case of multiplicative shocks, the
condition means that the shocks are su±ciently volatile. Convergence is
faster the larger their volatility is, and the smaller the maximum average
product of capital is.
Keywords and Phrases: Stochastic growth; Inada condition;
Convergence to zero; Extinction
JEL Classi¯cation Numbers: C61, C62, E30, O41
¤I would like to thank Santanu Roy, John Stachurski, Lars J. Olson, and an anonymous
referee for helpful comments and suggestions. The general result in Section 2 owes much
to the referee's comments on an earlier version of this paper. Financial support from the
21 Century COE Program at GSE and RIEB, Kobe University is gratefully acknowledged.1 Introduction
In a seminal paper, Brock and Mirman (1972) showed that the optimal paths
of a stochastic one-sector growth model converge to a unique nontrivial sta-
tionary distribution. While various cases are known in which their theorem
can be extended,1 it does not seem to be well understood when the theo-
rem fails. Most of the extensions of the Brock-Mirman theorem assume that
the production function satis¯es the Inada condition at zero, i.e., that the
marginal product of capital goes to in¯nity as capital goes to zero.2
Although the Inada condition at zero is widely used in economics, the
only justi¯cation for its use seems to be mathematical convenience.3 In fact
it is known to have the rather unrealistic implication that each unit of capital
must be capable of producing an arbitrarily large amount of output with a
su±cient amount labor (e.g., FÄ are and Primont, 2002).
In this paper we consider the resource constraint commonly used in
stochastic one-sector growth models, focusing on the case in which the In-
ada condition at zero is not satis¯ed. Our framework encompasses stochastic
endogenous growth models as well as stochastic overlapping generations mod-
els. To accommodate nonconcave production functions, we assume that the
maximum (stochastic) average product of capital is always ¯nite, which is
equivalent to the violation of the Inada condition at zero in the concave case.
Under this assumption we show that any feasible path converges to zero
exponentially fast almost surely if there is a negative upper bound on the
long-run sample average of the logarithm of the maximum average product
of capital. In the case of multiplicative shocks, this general condition means
that the shocks are su±ciently volatile. Convergence is faster the larger their
volatility is, and the smaller the maximum (deterministic) average product
of capital is.
To our knowledge, this relationship between almost sure convergence to
zero and the volatility of shocks has not been documented in the stochastic
1For example, see Stachurski (2002) and the references therein.
2Notable exceptions are Hopenhayn and Prescott (1992, Section 6.B(i)) and Nishimura
and Stachurski (2005, Theorem 3.1). Our results o®er partial converses to their and
Boylan's (1979, Theorem 2) results.
3According to Barro and Sala-i-Martin (1995, p. 16), the Inada conditions f0(0) = 1
and f0(1) = 0 are named after Inada (1963). But actually he used these conditions
following Uzawa (1963). Neither Inada nor Uzawa provided an economic justi¯cation for
the conditions.
1growth literature, though technically similar results have recently been ob-
tained independently by Mitra and Roy (2003) (MR henceforth) and Nishimura,
Rudnicki, and Stachurski (2004) (NRS henceforth). Both MR and NRS study
optimal stochastic growth models. The advantage of our results is twofold.
First, we do not assume i.i.d. shocks, while both MR and NRS require shocks
to be i.i.d. and to satisfy additional assumptions. Second, unlike MR and
NRS, we establish (or notice) almost sure exponential convergence to zero
and provide an approximate rate of convergence. We must admit however
that we obtain these advantages mainly because we focus on one particular
phenomenon, while MR and NRS consider various other phenomena as well.
Another closely related result was shown by Athreya (2004, Corollary 1)
for a certain class of Markov processes with i.i.d. shocks. Our general result
can be viewed as an extension of his result to a nonstationary setting with
non-i.i.d. shocks.
Two other results in the literature are particularly relevant to this pa-
per. First, for a parametric model with logarithmic utility, Danthine and
Donaldson (1981) showed that increased uncertainty negatively a®ects the
expected value of the long-run capital stock. Second, Rothschild and Stiglitz
(1971, Section 2.A) showed that increased uncertainty may increase the sav-
ings rate, depending on the curvature of utility.4 Our result in the case of
multiplicative shocks makes it clear that regardless of the objective function,
and regardless of the savings rate (even if it is 100%), convergence to zero
occurs almost surely if there is su±cient uncertainty, as long as the maximum
average product of capital is ¯nite.
The rest of the paper is organized as follows. Section 2 proves a gen-
eral result for nonstationary one-sector growth models with non-i.i.d. shocks.
Section 3 focuses on the stationary case with multiplicative shocks, showing
and discussing a consequence of the general result that is easier to interpret.
4See Jones et al. (2005) for a recent treatment of related problems in the context of
endogenous growth.
22 The General Result
Let (­;F;P) be a probability space. Consider an in¯nite horizon economy
in which the resource constraint in period t 2 Z+ is given by
8! 2 ­; ct(!) + kt+1(!) = gt(kt(!);!); 5 (2.1)
where ct(!) is consumption in period t, kt(!) is the capital stock at the
beginning of period t, and gt : R+ £ ­ ! R+ is the production function
in period t, which is random and may also vary over time in a deterministic
way. We say that a nonnegative stochastic process fktg1
t=0 is a feasible path if
it satis¯es (2.1) for all t 2 Z+ for some nonnegative stochastic process fctg.6
Special cases of (2.1) are used as resource constraints in various stochas-
tic growth models, including optimal growth models, endogenous growth
models with externalities, and overlapping generations models.7 No further
equation is required for our results, which concern only feasible paths. (Ad-
ditional assumptions on gt are introduced in Section 3.) Note that optimal or
equilibrium paths are required to be feasible no matter how they are de¯ned.
Thus our results apply to any model in which (2.1) is required as a resource
constraint.
The following result provides a su±cient set of conditions for almost sure
convergence to zero.
Theorem 2.1. Suppose
8t 2 Z+;8! 2 ­; gt(0;!) = 0; (2.2)
8t 2 Z+;8! 2 ­; at(!) ´ sup
k>0
gt(k;!)
k
< 1; (2.3)
¹ ´ esssup
!2­
limsup
T"1
1
T
T¡1 X
t=0
lnat(!) < 0:8 (2.4)
Then any feasible path fktg converges to zero exponentially fast a.s. More
speci¯cally, for almost all ! 2 ­;8¸ 2 (0;¡¹);
9T 2 Z+;8t ¸ T; kt(!) < e
¡¸t: (2.5)
5The quanti¯er \8! 2 ­" may be replaced by \for almost all ! 2 ­" throughout
this paper. The distinction is negligible since we are only concerned with almost sure
convergence to zero.
6By a stochastic process fxtg1
t=0, we mean a sequence of functions xt : ­ ! R.
7In overlapping generations models, ct in (2.1) represents aggregate consumption.
8For a random variable x : ­ ! R, esssup!2­ x(!) ´ inffs 2 Rjx · s a.s.g.
3Proof. Let fktg be any feasible path. By (2.1), (2.2), and (2.3),
8t 2 Z+;8! 2 ­; kt+1(!) · gt(kt(!);!) · at(!)kt(!): (2.6)
Let
­1 = f! 2 ­j8t 2 Z+;at(!) > 0;kt(!) > 0g; (2.7)
­2 =
(
! 2 ­
¯
¯
¯
¯
¯
limsup
T"1
1
T
T¡1 X
t=0
lnat(!) · ¹
)
: (2.8)
By (2.2) and (2.6), 8! 2 ­ n ­1;kt(!) = 0 for su±ciently large t. By (2.4),
P(­2) = 1. Thus we may restrict attention to ! 2 ­1 \­2. Fix ! 2 ­1 \­2
for the rest of the proof. We write kt instead of kt(!), etc., for notational
simplicity.
Since ! 2 ­1, it follows from (2.6) that
8t 2 Z+; lnkt+1 · lnat + lnkt: (2.9)
Hence
8T 2 N; lnkT ·
T¡1 X
t=0
lnat + lnk0: (2.10)
Dividing through by T, we get
8T 2 N;
lnkT
T
·
PT¡1
t=0 lnat
T
+
lnk0
T
: (2.11)
Let ¸ 2 (0;¡¹), i.e., ¹ < ¡¸ < 0. Since ! 2 ­2, the right-hand side of (2.11)
is strictly less than ¡¸ for su±ciently large T. Thus for su±ciently large T,
ln(kT)=T < ¡¸; i.e., kT < e¡¸T.
Condition (2.2) is a standard restriction. Condition (2.3) says that the
maximum (stochastic) average product of capital is always ¯nite, which im-
plies the violation of the Inada condition at zero.9 Condition (2.4) means
that there is a negative upper bound on almost every long-run sample av-
erage of the logarithm of the maximum average product of capital. Since
exp(
PT¡1
t=0 lnat) =
QT¡1
t=0 at; and since at is the maximum possible gross
9Given (2.4), (2.3) is essentially redundant as long as ¹ is well-de¯ned. But (2.3) is
needed for ¹ to be well-de¯ned unless at > 0 a.s.
4growth rate of capital in period t, (2.4) implies that for almost every sample
path, the gross growth rate over a long horizon is less than one, i.e., the net
growth rate over a long horizon is negative.
If flnatg satis¯es the law of large numbers, then (2.4) reduces to E lnat <
0. This condition becomes easy to interpret particularly in the case of mul-
tiplicative shocks, which we consider in the next section.
3 Multiplicative Shocks
In this section we focus on the case of multiplicative shocks. In particular
we assume the following in (2.1).
Assumption 3.1. There exist f : R+ ! R+ and a stochastic process fstg
such that
8t 2 Z+;8! 2 ­;8k ¸ 0; gt(k;!) = st(!)f(k): (3.1)
Hence (2.1) can now be written as ct + kt+1 = stf(kt). Let us state and
discuss our other assumptions.
Assumption 3.2. 8t 2 Z+; (i) 8! 2 ­;0 · st(!) < 1; and (ii) Est = 1.
Assumption 3.3. f(0) = 0, and
m ´ sup
k>0
f(k)
k
< 1: (3.2)
Assumption 3.2(ii) is merely a normalization. If f is concave and di®er-
entiable, then (3.2) is equivalent to f0(0) < 1, the violation of the Inada
condition at zero. More generally, (3.2) means that the maximum (determin-
istic) average product of capital is ¯nite. Though m is required to be ¯nite,
it is allowed to be arbitrarily large.
Assumption 3.4. 9º 2 (¡1;1];8t 2 Z+;E lnst = ¡º.
This assumption means only that E lnst does not depend on t. By
Jensen's inequality and Assumption 3.2,
¡º = E lnst · lnEst = 0: (3.3)
Thus º is in fact nonnegative. Since it is the di®erence between lnEst (= 0)
and E lnst due to the strict concavity of the log function, º can be interpreted
as a measure of volatility.
5Assumption 3.5. We have
lim
T"1
1
T
T¡1 X
t=0
lnst = ¡º a.s.; (3.4)
where º is given by Assumption 3.4.
Assumptions 3.4 and 3.5 mean that flnstg has a constant mean and satis-
¯es the law of large numbers. These assumptions hold if flnstg is stationary
and ergodic with Ejlnstj < 1 (e.g., White, 2000, Theorem 3.34). For ex-
ample, flnstg may be an i.i.d. process, as typically assumed in the stochastic
growth literature. More generally, it may be a stationary ARMA process.
The following result is a consequence of Theorem 2.1.
Theorem 3.1. Let Assumptions 3.1{3.5 hold. Suppose
lnm < º: (3.5)
Then any feasible path fktg converges to zero exponentially fast a.s. More
speci¯cally, for almost all ! 2 ­;8¸ 2 (0;º ¡ lnm), (2.5) holds.
Proof. Assumptions 3.1{3.3 imply (2.2) and (2.3). Condition (3.5) together
with Assumptions 3.4 and 3.5 implies (2.4) with ¹ = lnm ¡ º. Hence the
conclusion holds by Theorem 2.1.
If º = 0, i.e., if there is no uncertainty, then (3.5) reduces to
m < 1: (3.6)
This means that the graph of f lies entirely below the 45± line, implying that
all feasible paths converge to zero. Since º ¸ 0 by (3.3), (3.6) implies (3.5)
even in the stochastic case. However, (3.5) holds even if the graph of f lies
entirely above the 45± line, provided that º is su±ciently large. Thus almost
sure convergence to zero occurs if shocks are su±ciently volatile. A simple
example illustrates this point.
Suppose st is unconditionally log-normal.10 Then by Assumption 3.2 and
log-normality,
1 = Est = E exp(lnst) = exp
µ
E lnst +
V ar(lnst)
2
¶
: (3.7)
10This is true, for example, if flnstg is i.i.d. normal or a stationary AR process with
normal innovations.
6Since º = ¡E lnst by de¯nition (recall Assumption 3.4), from (3.7),
º =
V ar(lnst)
2
: (3.8)
Hence (3.5) holds if and only if V ar(lnst) > 2lnm. Thus if V ar(lnst) is
large enough, any feasible path converges to zero exponentially fast a.s. A
similar example can easily be constructed in which the support of shocks
is bounded and bounded away from zero, as in the original Brock-Mirman
(1972) model.
Theorem 3.1 shows not only that (3.5) implies almost sure convergence to
zero, but also that an approximate rate of convergence is given by º ¡ lnm.
Hence convergence is faster the large the volatility of shocks is, and the
smaller the maximum average product of capital is.
As mentioned above, Assumption 3.3 holds if f is concave and violates
the Inada condition at zero. A special case of this is when f is linear. Hence
in a stochastic \AK" model, under (3.5), any feasible path converges to zero
exponentially fast a.s. regardless of the objective function.11
Though Assumption 3.1 rules out the case in which the resource constraint
is given by ct + kt+1 = stf(kt) + (1 ¡ ±)kt for some ± 2 (0;1), this case can
be dealt with using Theorem 2.1. Other cases that can be dealt with using
the general result include nonconvex stochastic growth models of the type
studied by Majumdar et al. (1989) and Nishimura et al. (2004) as well as
stochastic overlapping generations models.
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Almost sure convergence to zero in stochastic growth models

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