Almost Sure Convergence to Zero in
Stochastic Growth Models

Takashi Kamihigashi
RIEB
Kobe University
Rokkodai, Nada, Kobe 657-8501
Japan
Phone/Fax: +81-78-803-7015
Email: tkamihig@rieb.kobe-u.ac.jp

September 24, 2003

Abstract
This paper shows that in stochastic one-sector growth models, if the production function does not satisfy the Inada condition at zero, any feasible
path converges to zero with probability one provided that the shocks are sufﬁciently volatile. This result seems signiﬁcant since, as we argue, the Inada
condition at zero is diﬃcult to justify on economic grounds. Our convergence
result is extended to the case of a nonconcave production function. The generalized result applies to a wide range of stochastic growth models, including
stochastic endogenous growth models, overlapping generations models, andmodels with nonconcave production functions.
Keywords: Stochastic growth; Inada condition; convergence to zero.

1

Introduction

In a seminal paper, Brock and Mirman (1972) showed that the optimal pathsof a stochastic one-sector growth model converges to a unique nondegeneratestationary distribution. While various cases are known in which their theorem can be extended,1 it is not well understood when the theorem fails. Mostof the extensions of the Brock-Mirman theorem assume that the production
function satisﬁes the Inada condition at zero, i.e., that the marginal productof capital goes to inﬁnity as capital goes to zero.2
Although the Inada condition at zero is almost ubiquitous in economics,
the only justiﬁcation for its use seems to be analytical simplicity. According
to Barro and Sala-i-Martin (1995, p. 16), the Inada conditions f  (0) = ∞ andf  (∞) = 0 are named after Inada (1963), who in fact used these conditions
following Uzawa (1963). Neither Inada nor Uzawa, however, provided an
economic justiﬁcation for the conditions. We argue in Section 2 that the
Inada condition at zero is diﬃcult to justify on economic grounds since it has
an unrealistic implication. More speciﬁcally, the condition is shown to imply
that each unit of capital must be capable of producing any large amount of
output with a suﬃcient amount of labor.
Given this unrealistic implication of the Inada condition at zero, it seems
worthwhile to study the case in which the condition is not satisﬁed. We
show in Section 3 that if the Inada condition at zero is not satisﬁed, i.e., if
the marginal product of capital is ﬁnite at zero, then under an additional
condition, any feasible path converges to zero with probability one. This
statement itself is rather trivial since it is well-known that all feasible paths
converge to zero in the deterministic case if the marginal product of capitalat zero is less than one.
What we show, however, is that no matter how large the expected marginal
product of capital at zero is, as long as it is ﬁnite, any feasible path converges
to zero with probability one provided that the shocks are suﬃciently volatile.
To our knowledge, this result has not been documented in the literature.3
The result is extended to the case of a nonconcave production function. The
1

For example, see Stachurski (2002) and the references therein.
An exception is a convergence result shown by Hopenhayn and Prescott (1992). We
discuss their result in Section 3.
3
Phelps (1962, p. 736) made the relevant observation that growth is not guaranteed by
the condition that the expected marginal product of capital exceeds the reciprocal of the
discount factor.
2

1

generalized result applies to a wide range of one-sector models, including
stochastic endogenous growth models, overlapping generations models, andmodels with a nonconcave production function.
The rest of the paper is organized as follows. Section 2 argues that the
Inada condition at zero has an unrealistic implication. Section 3 shows our
main results. Section 4 concludes the paper.

2

An Unrealistic Implication of the Inada Condition at Zero

This section shows that the Inada condition at zero has an unrealistic implication. This well-known condition is that the marginal product of capitalgoes to inﬁnity as capital goes to zero. The natural way to examine its implications would be to consider a situation in which capital is made arbitrarily
small. However, since most capital goods in reality are more or less indivisible, it is not clear what an inﬁnitestimal amount of capital represents.
Thus in this section we regard capital as the capital-labor ratio and consider
a situation in which labor is arbitrarily large.
As in most neoclassical models, suppose output is produced by a linearlyhomogenous production function F (K, L) using capital K and labor L. Deﬁne f (k) = F (k, 1). We assume that F and f are concave and continuously
diﬀerentiable. The Inada condition at zero isf  (0) ≡ lim f  (k) = ∞.

(2.1)

k↓0

To discuss an implication of this condition, it is useful to obtain thefollowing equivalent expression for f  (0):(2.2)

∀K ≥ 0,f  (0) = lim [F (K + 1, L) − F (K, L)].
L↑∞

To see this, note that by linear homogeneity,

 
 
K
K +1
(2.3)
−f
F (K + 1, L) − F (K, L) = L f
LL1= Lf  (k̃) = f  (k̃),
(2.4)
L

2

where the ﬁrst equality in (2.4) holds for some k̃ ∈ [K/L, (K + 1)/L] by themean value theorem. As L ↑ ∞, k̃ → 0. Hence (2.2) holds.
By (2.2), the Inada condition at zero implies that an additional unit ofcapital is capable of producing any large amount of output as long as a
suﬃcient amount of labor is available. Since this is true for each additional
unit, it follows that every unit of capital must be unboundedly productive.
This, however, seems unrealistic.
For example, consider a closed economy in which there is one machinethat must be operated by one worker. For simplicity we assume that thismachine is the only capital in the economy.4 For the moment, assume(2.5)∀L ≥ 0,F (0, L) = 0.

This means that workers with no access to the machine produce nothing. If
there are three workers in the economy, then the machine can run 24 hours
a day with each worker working eight hours a day. If there are six workers,
output may rise since each worker, working only four hours a day, may beless tired and more eﬃcient in handling the machine. If there are twelve
workers, output may rise for the same reason, but perhaps to a lesser degree.
Adding more workers probably will not make much diﬀerence though they
can cooperate in various ways to increase eﬃciency. In any case, since workers
produce nothing when they are not using the machine, it seems reasonable
to assume that output will reach its upper bound fairly soon as more andmore workers are added. But if this is the case, the Inada condition atzero is violated since, as shown above, it together with (2.2) implies that themachine must be capable of producing any large amount if a suﬃciently large
number of workers are available.
Now suppose (2.5) is not satisﬁed, i.e., capital is not an essential factor of
production. In this case, total output grows unboundedly if more and moreworkers become available. But the above example works with appropriate
modiﬁcations. Note that if we deﬁne G(K, L) = F (K, L) − F (0, L), thenF (K, L) = G(K, L) + F (0, L), i.e., G is the relevant part of the production
function. It is easy to see that G satisﬁes (2.2) and (2.5) with G replacing F .
Thus we may apply the above argument to G to draw the same conclusion.
4

By linear homogeneity, it does not matter how many machines there are in the economy. Though we consider here the case in which (2.2) holds with K = 0 for simplicity,
the following argument is valid for any K ≥ 0, where K is the number of machines in theeconomy less one.

3

Hence, even without (2.5), the Inada condition at zero is violated aslong as the machine’s capacity is limited. In view of this result, it seems
diﬃcult to conceive of a real-life situation in which the Inada condition atzero is satisﬁed. Note that producing more machines is a part of capital
accumulation, so is repairing the existing machine.

3

Main Results

Let (Ω, F, P ) be a probability space, and {Ft }∞t=0 ⊂ F be a ﬁltration. From∞
here on, any sequence of the form {xt }t=0 is assumed to be a stochastic
process adapted to this ﬁltration. Consider an inﬁnite horizon economy inwhich the resource constraint in period t ∈ Z+ is given by(3.1)

ct + kt+1 = st f (kt ),where ct is consumption in period t, kt is the capital stock at the beginningof period t, and st is the productivity shock in period t. We say that anonnegative stochastic process {kt }∞t=0 is a feasible path if it satisﬁes (3.1) for
all t ∈ Z+ for some nonnegative stochastic process {ct }.
Since our results apply to any feasible path, no further structure is required. No matter what objective function is speciﬁed, optimal paths are
required to be feasible. Therefore our results apply to optimal paths as well.
We state our results after discussing our assumptions.
Assumption 3.1. (i) f : R+ → R+ is C 1 on R++ , concave, and strictlyincreasing. (ii) f (0) = 0. (iii) m ≡ limk↓0 f  (k) < ∞.
Parts (i) and (ii) are standard. Part (iii) means that f does not satisfy
the Inada condition at zero. Though m is required to be ﬁnite, it is allowed
to be arbitrarily large. Part (iii) seems reasonable given our discussion in
Section 2.
Assumption 3.2. ∀t ∈ Z+ , (i) st > 0 a.s. and (ii) Est = 1.
Part (i) is standard. Part (ii) is only a normalization, implying that m asgiven in Assumption 3.1 is the expected marginal product of capital at zero.
Assumption 3.3. ∃ν ∈ (−∞, ∞], ∀t ∈ Z+ , E ln st = −ν.

4This assumption means only that E ln st does not depend on t. By
Jensen’s inequality and Assumption 3.2,
(3.2)

−ν = E ln st ≤ ln Est = 0.

Thus ν is in fact nonnegative.
Assumption 3.4. We have1
ln st = −νlim
T ↑∞ T
t=0
T −1

(3.3)

a.s.,

where ν is given by Assumption 3.3.
Assumptions 3.3 and 3.4 mean that {ln st } has constant mean and satisﬁes
the law of large numbers. These assumptions hold if {ln st } is stationary andergodic with E| ln st | < ∞ (e.g., White, 2000, Theorem 3.34). For example,{ln st } may be an i.i.d. process, as typically assumed in the stochastic growth
literature; more generally, it may be a stationary ARMA process.
We are now ready to state the following.
Theorem 3.1. Suppose Assumptions 3.1–3.4 hold. Supposeln m < ν.

(3.4)

Then any feasible path converges to zero a.s.
Proof. Let {kt } be any feasible path. Let
(3.5)
(3.6)

Ω1 = {ω ∈ Ω | ∀t ∈ Z+ , st (ω) > 0, kt (ω) > 0},



T −1

1

ln st (ω) = −ν .
Ω2 = ω ∈ Ω  lim T ↑∞ T t=0

Note that ∀ω ∈ Ω \ Ω1 , kt (ω) = 0 for t suﬃciently large by (3.1) and Assumption 3.1(ii). By Assumption 3.4, P (Ω2 ) = 1. Thus to show the conclusion,
it suﬃces to verify that ∀ω ∈ Ω1 ∩ Ω2 , limt↑∞ kt (ω) = 0. For the rest of
the proof, we ﬁx ω ∈ Ω1 ∩ Ω2 . But we write kt instead of kt (ω), etc., fornotational simplicity.
By Assumption 3.1, ∀k ≥ 0, f (k) ≤ mk. Thus
(3.7)

∀t ∈ Z+ ,kt+1 ≤ st f (kt ) ≤ st mkt .
5Since ω ∈ Ω1 , it follows that
(3.8)

∀t ∈ Z+ ,ln kt+1 ≤ ln st + ln m + ln kt .

Hence
(3.9)

∀T ∈ N,ln kT ≤T −1ln st + T ln m + ln k0 .

t=0

Dividing through by T , we get(3.10)∀T ∈ N,

ln kT
≤
T

T −1
t=0 ln stT+ ln m +

ln k0
.
T

The right-hand side converges to ln(m) − ν as T ↑ ∞ since ω ∈ Ω2 . By
(3.4), ln(m) − ν < 0. Let b ∈ (ln(m) − ν, 0). Then for T suﬃciently large,ln(kT )/T ≤ b, i.e., ln kT ≤ bT . Thus limT ↑∞ ln kT ≤ limT ↑∞ bT = −∞; i.e.,
limT ↑∞ kT = 0.
The basic idea of the above proof is that (3.4) and (3.8) imply that kt
converges to zero a.s. A similar argument was used by Kelly (1992, Proposition 1) to show the convergence of output paths with diﬀerent initial capitalstocks in an endogenous growth model. Another similar argument was usedby De Hek and Roy (2001, Lemma 1) to obtain a suﬃcient condition forsustained growth in a stochastic growth model. Our contribution here is topoint out that without the Inada condition at zero, it is possible that every
feasible path converges to zero a.s. even if the marginal product of capital atzero is arbitrarily large.
If ν = 0, i.e., if st is non-random, then (3.4) reduces to(3.11)

m < 1,

which is exactly the condition under which all feasible paths converge to zeroin the deterministic case (for an arbitrary production function satisfying
Assumption 3.1). Since ν ≥ 0 by (3.2), (3.11) implies (3.4) even in thestochastic case. In the stochastic case, however, (3.4) holds even if m isarbitrarily large, provided that ν is suﬃciently large. Roughly speaking, themore volatile st is, the larger ν (= ln Est − E ln st ) is. Thus without the
Inada condition at zero, almost sure convergence to zero occurs if the shocks
are suﬃciently volatile. A simple example illustrates this point.
6

Suppose st is unconditionally log-normal.5 Then by Assumption 3.2(i)and log-normality,
V ar(ln st )(3.12)
1 = Est = E exp(ln st ) = exp E ln st +.
2
Recalling that ν = −E ln st , we obtain
V ar(ln st ).
2
Hence (3.4) holds as long as V ar(ln st ) is suﬃciently large. In other words,
as long as the production function violates the Inada condition at zero, any
feasible path converges to zero a.s. provided that the shocks are log-normal
with suﬃciently large variance.
It is easy to see that such shocks can be constructed even if their support
is required to be bounded and bounded away from zero, as in the original
Brock-Mirman (1972) model. One way to do this is by approximating a
suﬃciently volatile log-normal shock by a random variable whose support isbounded and bounded away from zero. As a simpler example, suppose {st }is i.i.d., and st = 1 − r with probability 1/2, and 1 + r with probability 1/2,
where r ∈ (0, 1). Then Est = 1 and(3.13)

ν=1−ν = E ln st = [ln(1 − r) + ln(1 + r)].
2
Thus ν → ∞ as r ↑ 1, so that (3.4) holds for r suﬃciently close to one.
Hopenhayn and Prescott (1992, Sec. 6.B(i)) extended the Brock-Mirman
theorem to the case in which the Inada condition at zero need not hold. Let
us clarify the relationship between their result and ours. They assumed thatthe production function f˜ satisﬁes β f˜ (0) > 1 for some β ∈ (0, 1) and that theshocks s̃t are i.i.d. with s̃t ∈ [1, α] for some α > 1. Under our normalization,i.e., Assumption 3.2(ii), their assumptions are expressed as follows:(3.14)(3.15)(i) f  (0) > θ/β,(ii) st ∈ [1/θ, α/θ],

where α > 1, θ ∈ (1, α), and β ∈ (0, 1).6 Since m = f  (0) and ν = − ln Est ,by (3.15),(3.16)

ln(m) − ν > ln(θ/β) + ln(1/θ) ≥ − ln β > 0.

5This is true, for example, if {ln st } is i.i.d. normal or a stationary AR process withnormal innovations.
6
Hopenhayn and Prescott’s assumptions can be recovered by setting s̃t = θst and˜
f = f /θ. Note that θ = Es̃t .

7

Thus in this case, (3.4) does not hold.
The critical assumption here is that st ≥ 1/θ. Without this lower bound,
Hopenhayn and Prescott’s result may fail. For example, let α ∈ (0, 1) andsuppose st ∈ [α/θ, α/θ] instead of (3.15)(ii). Suppose st = α/θ with probability p(α), and α/θ with probability with 1−p(α), where p(α) = (α−θ)/(α−α).
Then Est = 1 and(3.17)−ν = E ln st = p(α) ln(α/θ) + (1 − p(α)) ln(α/θ).

Since p(0) > 0, we have ν → ∞ as α ↓ 0. Hence (3.4) holds for α suﬃciently
close to zero, in which case every feasible path converges to zero a.s. by
Theorem 3.1.
Theorem 3.1 can easily be extended to the case of a nonconcave production function. In the proof of Theorem 3.1, the production function f (x)is estimated above by mk. Hence the proof goes through as long as thisestimation is valid. The following result does not assume Assumption 3.1.
Theorem 3.2. Suppose Assumptions 3.2–3.4 hold. Suppose there exists m >0 satisfying (3.4) such that
(3.18)∀k ≥ 0,f (k) ≤ mk.

Then every feasible path converges to zero a.s.
Proof. Identical to the proof of Theorem 3.1 except that (3.18) replaces Assumption 3.1 here.
Theorem 3.2 covers various cases. For example, if f (k) = mk for somem > 0, then the hypotheses of Theorem 3.2 obviously hold except for (3.4).
Hence in a stochastic “AK” model, under (3.4), every feasible path converges
to zero a.s. regardless of the objective function.7 More generally, Theorem
3.2 applies to any one-sector stochastic growth model with bounded marginal
product. It applies to nonconvex stochastic growth models of the type studied
by Majumdar, Mitra, and Nyarko (1989) as well as stochastic overlapping
generations models of the type studied by Wang (1993).87

Phelps (1962) was the first to study a stochastic “AK” model. The analysis was
extended by Levhari and Srinivasan (1969). These articles studied mainly the propertiesof the consumption policy function rather than the asymptotic properties of optimal paths.
8
In overlapping generations models, ct in (3.1) represents aggregate consumption.

8

4

Conclusion

This paper has shown that in one-sector stochastic growth models, if the
production function does not satisfy the Inada condition at zero, then any
feasible path converges to zero with probability one provided that the shocks
are suﬃciently volatile. This result seems signiﬁcant since, as we have argued,
the Inada condition at zero is diﬃcult to justify on economic grounds. Ourconvergence result has been extended to the case of a nonconcave production
function. The generalized result applies to a wide range of one-sector models, including stochastic endogenous growth models, overlapping generations
models, and models with a nonconcave production function.
Dealing with feasible paths, our results of course apply to optimal models. Optimal paths, however, converge to zero with probability one under aweaker condition.9 For example, if there is an upper bound on the marginal
propensity to save, then it can be used to estimate optimal paths (rather thanfeasible paths) from above as in the proof of Theorem 3.1. If optimal paths
are characterized by a stationary policy function, then our argument can beapplied to the policy function instead of the production function. Likewise
it can be applied to equilibrium models in which there is an upper boundon the aggregate marginal propensity to save, or in which the equilibria arecharacterized by a stationary function. Such extensions are left for future
research.

References
Barro, R.J., and X. Sala-i-Martin, 1995, Economic Growth. New York: McGrawHill.
Brock, W.A., and L. Mirman, 1972, “Optimal Economic Growth and Uncertainty: The Discounted Case,” Journal of Economic Theory 4, 479–
513.
De Hek, P., and S. Roy, 2001, “On Sustained Growth under Uncertainty,”
International Economic Review 42, 801–813.
Hopenhayn, H.A., and E.C. Prescott, 1992, “Stochastic Monotonicity and Stationary Distributions for Dynamic Economies,” Econometrica 60, 1387–
1406.
9
The properties of optimal paths including a condition for convergence to zero arestudied by Nishimura, Rudnicki, and Stachurski (2003) in a more specialized setup.

9

Inada, K., 1963, “On a Two-Sector Model of Economic Growth: Comments
and a Generalization,” Review of Economic Studies 30, 119–127.
Kelly, M., 1992, “On Endogenous Growth with Productivity Shocks,” Journal of Monetary Economics 30, 47–56.
Levhari, D., and T.N. Srinivasan, 1969, “Optimal Savings under Uncertainty,”
Review of Economic Studies 153–164.
Majumdar, M., Mitra, T., and Y. Nyarko, 1989, “Dynamic Optimization under Uncertainty: Non-convex Feasible Set”, in Joan Robinson and Modern Economic Theory. Hampshire: Macmillan Press, pp. 545–590.
Nishimura, K., R. Rudnicki, and J. Stachurski, 2003, “Stochastic Optimal Growthwith Nonconvexities,” mimeo, IER, Kyoto University.
Phelps, E.S., 1962, “The Accumulation of Risky Capital: A Sequential Utility Analysis,” Econometrica 30, 729–743.
Stachurski, J., 2002, “Stochastic Optimal Growth with Unbounded Shock,”
Journal of Economic Theory 106, 40–65.
Uzawa, H., 1963, “On a Two-Sector Model of Economic Growth II,” Review
of Economic Studies 30, 105–118.
Wang, Y., 1993, “Stationary Equilibria in an Overlapping Generations Economy with Stochastic Production,” Journal of Economic Theory 61, 423–
435.
White, H., 2000, Asymptotic Theory for Econometricians. San Diego: Academic Press.

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