A Stochastic Version of the Ramsey’s Growth Model

Gabriel Zacar

ıas-Espinoza

, Hugo Cruz-Su

arez

and Enrique Lemus-Rodr

ıguez

Departamento de Matem

aticas, Universidad Aut

onoma Metropolitana-Iztapalapa, Ave. San Rafael Atlixco 186,

Col. Vicentina,09340, M

exico D.F., M

exico

Facultad de Ciencias F

ısico Matem

aticas, Benem

erita Universidad Aut

onoma de Puebla, Ave. San Claudio y R

ıo Verde,

Col. San Manuel, Ciudad Universitaria, Puebla, Pue., 72570, M

exico

Escuela de Actuar

ıa, Universidad An

ahuac M

exico-Norte, Ave. Universidad An

ahuac 46,

Col. Lomas An

ahuac, 52786, Edo. de M

exico, M

exico

Keywords:

Ramsey’s Growth Model, Markov Decision Processes, Dynamic Programming, Euler Equation, Stability.

Abstract:

In this paper we study a version of Ramsey’s discrete time Growth Model where the evolution of Labor through

time is stochastic. Taking advantage of recent theoretical results in the ﬁeld of Markov Decision Processes, a

ﬁrst set of conditions on the model are established that guarantee a long-term stable behavior of the underlying

Markov chain.

1 INTRODUCTION

Ramsey’s Growth Model has a long and interesting

history. In order to give a context to the material of

the present paper, we brieﬂy outline it.

The original model presented by Ramsey in (Ram-

sey, 1928) (formulated in communication with the

famous economist Keynes) analyzes optimal global

saving in a deterministic continuous time setting, and

it is no surprise that it is solved using Calculus of

Variations. Since then, several variants have appeared

in the Advanced Macroeconomics literature, but, as

Prof. Ekeland points out (one of the leading experts in

Mathematical Economics): “To the best of my knowl-

edge and understanding, none of the solutions pro-

posed for solving the Ramsey problem is correct with

one exception, of course, Ramsey himself, whose own

statement was different than the one which is now in

current use (Ekeland, 2010)”. On the contrary, the

discrete time setting allows the straightforward use of

Dynamic Programming techniques, and therefore, al-

lows both researchers and practitioners to focus on the

analysis of the model itself and its properties. The

deterministic case is very clearly stated and analyzed

in (Brida et al., 2015), (Le Van and Dana, 2003) and

(Sladk

y, 2012).

Ramsey’s seminal work on economic growth has

been extended in many ways, but, to the best of our

knowledge, the study of a random discrete time ver-

sion is still in its initial phase. Such study will allow

a fruitful interaction between economists and mathe-

matician that will lead to better simulations and con-

sequently, to a better understanding of the effects of

the random deviations in the growth of an economy

and its impact on the population. And, in this paper

a ﬁrst random model is posed, where the population

grows in a stochastic manner.

In this paper a discrete-time stochastic Ram-

sey growth process is modeling as a discounted

Markov Decision Process (MDP) (Hern

andez-Lerma

and Lasserre, 1996) and (Ja

skiewicz and Nowak,

2011). The performance criterion of interest is the to-

tal discounted reward. The optimal control problem is

to determine a policy that optimizes the performance

criterion. The solution of the optimization problem is

analyzed through of the Euler Equation (EE) (Cruz-

arez and Montes-de Oca, 2008) and (Cruz-Su

arez

et al., 2012). Later, the EE is applied to study the er-

godic behavior of the stochastic Ramsey growth pro-

cess.

2 THE MODEL

Consider an economy in which at each discrete time

t, t = 0,1,..., there are L

consumers (population or

labor), with consumption c

per individual, whose

growth is governed by the following difference equa-

tion:

t+1

= L

, (1)

Zacarías-Espinoza, G., Cruz-Suárez, H. and Lemus-Rodríguez, E.

A Stochastic Version of the Ramsey’s Growth Model.

DOI: 10.5220/0005752503230329

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 323-329

ISBN: 978-989-758-171-7

 2016 by SCITEPRESS – Science and Technology Publications, Lda. All r ights reserved

323

it is assumed that initially the number of consumers,

, is known. In this case, {η

} is a sequence of inde-

pendent and identical distributed (i.i.d) random vari-

ables. The random variable η

, t ≥ 0, represents an

exogenous shock that affects the consumer popula-

tion, for example: epidemics, wars, natural disasters,

new technology, etc. Then, in this context, it will be

supposed that for each t ≥ 0: η

> 0, almost surely.

Remark 2.1. In the literature of economic growth

models is usual to assume that the number of con-

sumers grow very slowly in time, see, for instance,

(Le Van and Dana, 2003) and (Sladk

y, 2012). Ob-

serve that the model presented in this paper is a

ﬁrst step in an effort to weaken that constraint of the

model.

The production function for the economy is given

= F(K

is known,

i.e. the production Y

is a function of capital, K

, and

labor, L

, where the production function, F, is a ho-

mogeneous function of degree one. The output must

be split between consumptions C

= c

and the gross

investment I

, i.e.

+ I

= Y

. (2)

Let δ ∈ (0,1) be the depreciation rate of capital. Then

the evolution equation for capital is given by:

t+1

= (1 − δ)K

+ I

. (3)

Substituting (3) in (2), it is obtained that,

− (1 − δ)K

+ K

t+1

= Y

. (4)

In the usual way, all variables can be normalized

into per capital terms, namely, y

:= Y

and x

. Then (4) can be expressed in the following

way:

− (1 − δ)x

+ K

t+1

= y

= F(x

,1).

Now, using (1) in the previous relation, it yields that

t+1

= ξ

(F(x

,1) + (1 − δ)x

− c

t = 0,1,2,..., where ξ

:= (η

)

−1

Deﬁne h(x) := F(x,1)+(1 − δ)x, x ∈ X := [0, ∞),

h henceforth to be identiﬁed as the production func-

tion. Then, the transition law of the system is given

t+1

= ξ

(h(x

) − c

), (5)

= x known, (6)

where c

∈ [0,h(x

)] and {ξ

} is a sequence of i.i.d.

random variables with a density function ∆.

Observation 2.2. Observe that, if x

= 0, for some

t ∈ {0,1,2,..} then x

= 0 for each k ≥ t. This fact is

a consequence of relation (5), and in this case zero is

considered an absorbtion state.

A plan or consumption sequence is a sequence

π = {π

}

∞

n=0

of stochastic kernel π

on the control set

given the history

= (x

,··· , x

n−1

for each n = 0,1,·· ·. The set of all plans will be de-

noted by Π.

Given an initial capital x

= x ∈ X and a plan π ∈

Π, the performance index used to evaluate the quality

of the plan π is determined by

v(π,x) = E

∞

∑

n=0

U(c

)

, (7)

where U : [0,∞) → R is a measurable function known

as utility function and α ∈ (0,1) is a discount factor.

The goal of the controller is to maximize utility of

consumption on all plans π ∈ Π, that is:

V (x) := sup

π∈Π

v(π,x),

x ∈ X.

Throughout of this paper the model will be called

a Stochastic version of the Ramsey Growth (SRG)

model.

The following assumptions it will be considered

in the rest of the document.

Assumption 2.3. The production function h, satis-

ﬁes:

a) h ∈ C

((0,∞)),

b) h is a concave function on X,

c) h

> 0 and h(0) = 0.

d) Let h

(0) := lim

x↓0

(x). Suppose that h

(0) > 1 and

αh

(0) > E[ξ

−1

]. (8)

Assumption 2.4. The utility function U satisﬁes:

a) U ∈ C

((0,∞),R), with U

> 0 and U

< 0,

c) U

(0) = ∞ and U

(∞) = 0,

d) There exists a function ϑ on S such that E[ϑ(ξ)] <

∞, and



(h(s(h(x) − c)))h

(s(h(x) − c))s∆(s)



≤ ϑ(s),

(9)

s ∈ S, c ∈ (0,h(x)).

Observation 2.5. Observe that in Assumption 2.3 is

not considered the Inada condition in zero. In the lit-

erature, it is known to have the rather unrealistic im-

plication that each unit of capital must be capable of

producing an arbitrarily large amount of output with

a sufﬁcient amount of labor (Kamihigashi, 2006).

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

324

3 DYNAMIC PROGRAMMING

APPROACH

In this section it will be presented an analysis of

the optimization problem introduced in the previous

section. Dynamic Programming approach have been

used to study different type of problems and in various

context. In particular have been applied to Markov

Decision Processes (MDPs).

SRG can be identiﬁed as a MDP. In this case, the

space of states is X := [0,∞), the admissible action

space is A(x) := [0,h(x)],x ∈ X, in consequence, the

action space is A :=

x∈X

A(x) = [0,∞). The transition

law is given by the stochastic kernel, deﬁned as

Q(B

= x,a

= c) = Pr(x

t+1

∈ B

= x,a

= c)

w(x,y,c)dy,

with B ∈ B (X), (B (X) denotes the Borel sigma alge-

bra of X), where the function w : [0,∞)

→ [0,∞) is

deﬁned as:

w(x,y,c) = ∆



h(x) − c



h(x) − c

, (10)

for x,y ∈ X, c ∈ [0, h(x)) and ∆ is the density function

of the sequence {ξ

}. Deﬁne K := {(x,c)|x ∈ X, c ∈

A(x)}. Finally, the reward-per-stage function is iden-

tiﬁed as the utility function, U: X → [0,∞), deﬁned in

the previous section. Then the model is referred as the

quintuplet: M := (X,A,{A(x) : x ∈ X},Q,U).

As it was mentioned above, a plan is a sequence

π = {π

}

∞

n=0

of stochastic kernel deﬁned on A given

the history of the process. Furthermore, it is assumed

that π

(C(x

)|h

) = 1, n = 0,1,···, this assumption

guarantee that in each decision epoch, it is possible to

choose an admissible action. A particular class in Π

is the class of stationary plans,

F := { f : X → A| f (x) ∈ [0,h(x)], for all x ∈ X}.

In this case, a stationary plan π = ( f , f , ...) is denoted

by f .

Under Assumption 2.3 and Assumption 2.4, for

each x ∈ X, it follows that:

(a) The optimal value function V satisﬁes the follow-

ing equation (optimality equation)

V (x) = sup

c∈A(x)



U(c) + α

∞

V (y)w(x,y,c)dy



(11)

(b) There exists and optimal stationary policy f ∈ F

such that

V (x) = U( f (x)) + α

∞

V (y)w(x,y, f (x))dy.

(x) → V (x) when n → ∞,

where v

is deﬁned by

(x) = sup

c∈A(x)



U(c) + α

n−1

(y)w(x,y,c)dy



with v

(x) = 0.

Remark 3.1. The functions, v

, n ≥ 0, deﬁned on (c)

are known as value iteration functions, (Hern

andez-

Lerma and Lasserre, 1996).

4 MAIN RESULTS ABOUT SRG

4.1 Euler Equation

In this subsection it will be presented a functional

equation, which characterize the optimal value func-

tion. In the literature of MDP’s, this functional equa-

tion is known as Euler Equation (EE), (Cruz-Su

arez

et al., 2012). The validity of EE is guaranteed due

to properties of differentiability of the optimal value

function and the optimal policy, (Cruz-Su

arez and

Montes-de Oca, 2008). Then, it just is necessary to

veriﬁed that the optimal policy is interior, according

to Theorem 3.3 in (Cruz-Su

arez et al., 2011), this fact

is veriﬁed in the following result.

Lemma 4.1. The optimal plan f satisﬁes that f (x) ∈

(0,h(x)), for each x > 0.

Proof. Let x > 0 ﬁxed, if the optimal policy is f (·) ≡

0, then

V (x) = v(0, x) =

U(0)

1 − α

where v is deﬁned in (7).

Since U and h are strictly increasing (see Assump-

tion 2.2 and 2.3), it is obtained that

V (x) =

U(0)

1 − α

< U(h(x)) +

1 − α

U(0),

but this is a contradiction, given that

v(h,x) = U(h(x)) +

1 − α

U(0).

On the other hand, if h ∈F is the optimal policy,

then

V (x) = v(h,x)

= U(h(x)) +

1 − α

U(0).

Let g : [0,h(x)] → R be a function deﬁned as

g(c) := U (c) + αE[U(h(ξ(h(x)− c)))] +

1 − α

U(0).

A Stochastic Version of the Ramsey’s Growth Model

325

Observe that g is continuous and strictly concave

function. Then, there exists an unique c ∈ [0, h(x)],

which maximizes to g. If c 6= h(x), then

V (x) ≥ g(c) > g(h(x)) = V (x),

which is impossible. Therefore c = h(x).

Now, Assumptions 2.2 and 2.3 imply that if c ∈

(0,h(x)),

(h(ξ(h(x) − c)))h

(ξ(h(x) − c))ξ],

it follows that

lim

c→h(x)

Therefore, there exists

c ∈ (0, h(x)) such that

(

c) < 0. This implies that g is decreasing in [

c,h(x)]

which h(x) can not be the maximizer, i.e. it is a con-

tradiction.

Theorem 4.2. Under Assumption 2.2 and 2.3, it fol-

lows that:

a) V ∈ C

((0,∞),R) and the optimal plan f ∈

((0,∞)).

b) The value iteration functions satisﬁes

(x)

= αE



n−1





h(x) −U

−1



(x)





for each x > 0, where U

−1

(·) is the inverse of

(·) .

c) The optimal plan f satisﬁes the following Euler

equation:

( f (x)) = αE[U

∗

(x))h

(ξ(h(x) − f (x)))ξ],

for each x > 0, where c

∗

(x) := f (ξ(h(x) − f (x))).

Proof. The proof of this result is a consequence of

Lemma 5.2 and Lemma 5.6 in (Cruz-Su

arez et al.,

2011).

Remark 4.3. Observe that if f ∈ F satisﬁes (4.2) and

lim

n→∞



( f (x

))x



= 0,

then f is an optimal plan.

4.2 Stability of the SRG

It is known (Hern

andez-Lerma and Lasserre, 1996)

that if f ∈ F is the optimal plan then the optimal pro-

cess {x

} is a Markov process, where

n+1

= ξ

(h(x

) − f (x

)),

n = 0,1,2,..., x

= x ∈ X = [0, ∞). In addition,

Q(B|x, f (x)) =

w(x,y, f (x))dy

= E[I

(ξ(h(x) − f (x)))].

Furthermore, in the literature of economic growth

it was studied optimal process stability using Inada

conditions. However, (Nishimura and Stachurski,

2005) and (Stachurski, 2009) make use of the Euler

equation. In this spirit we present this subsection.

Deﬁne Γ for x ∈ (0,∞) as

Γ(x) := [U

( f (x))h

(x)]

1/2

+ x

+ 1, (12)

where p > 1. Let us consider to Γ as a weight func-

tion.

Let B

(X) be the space of measurable and Γ-

bounded function on X with norm

deﬁned as

:= sup

x∈X

g(x)

Γ(x)

for a measurable function g on X. Let ϕ be a signed

measure deﬁned on B(X). Then, for each g ∈ B

(X),

ϕg denotes

ϕg :=

g(y)ϕ(dy).

Observe that for the transition Kernel Q, Qg has the

form

Qg =

g(y)Q(dy

x, f (x)),

and the k-th transition kernel is

(B|x, f (x)) =

Q(B|y, f (y))Q

k−1

(dy|x, f (x)),

B ∈ B (X), for k ≥ 1 with Q

:= δ

, where δ

is Dirac’s

measure on x ∈ X.

Let µ be a probability measure on B(X). The mea-

sure µ is invariant with respect to the Markov chain

}, if µQ = µ, where

µQ(B) :=

Q(B|y)µ(dy), B ∈ B(X).

Deﬁne for each measure ϕ on B(X)

:= sup

≤1

ϕg

Deﬁnition 4.4. Given a set C ∈ B (X). C is a small set

with respect to the Markov chain {x

}, if there exist a

ﬁnite measure µ on B (X) and n ∈ N, such that for

each x ∈ C

x) ≥ µ(B),

for each B ∈ B(X).

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

326

Deﬁne for A ∈ B(X) the measure

Ξ(A) :=

∆(s)ds.

Lemma 4.5. The optimal process {x

} of SRG is Ξ-

irreducible and strongly aperiodic.

Proof. Let B ∈ B (X) such that Ξ(B) > 0 and x > 0.

Then, we know that h(x) − f (x) > 0 because f is an

interior point in the corresponding interval. More-

over,

Pr(x

∈ B) =

((h(x) − f (x))s)∆(s)ds,

where I

denotes the indicator function of the set B.

As ∆ is positive, it follows that Pr(x

∈ B) > 0. Con-

sequently, the optimal process Ξ is irreducible.

On the other hand, let a,b ∈ (0, ∞), a < b and con-

sider C := [a,b]. Once again, due to the fact that the

optimal policy takes values in the interior of the corre-

sponding interval, it turns out that h− f is an increas-

ing function. In fact, as the optimal value function

V is concave, we conclude that V

is decreasing and,

by the envelope formula (Cruz-Su

arez and Montes-de

Oca, 2008) it follows that:

(x) = U

(h(x) − f (x))h

(x).

If h − f is decreasing, for x,y ∈ X, x < y:

h(y) − f (y) ≤ h(x) − f (x)

and, as both U

and h

are decreasing and positive, we

get

(h(x) − f (x))h

(x) ≤ U

(h(y) − f (y))h

(y)

that is, V

(x) ≤ V

(y), contradiction. Then, h − f is

increasing. Consequently, for each x ∈ C we have

0 < h(a) − f (b) ≤ h(x) − f (x) ≤ h(b) − f (b).

Then, due to ∆ is positive function,

m := inf

(x,s)∈C×C

∆



h(x) − f (x)



h(x) − f (x)

> 0.

for µ a measure on B ∈ B (X) deﬁned by

µ(B) := m

(x)dx,

we have

Q(B

x) ≥ µ(B).

The set C is small, see Deﬁnition 4.4. Clearly µ(C) >

0: the process is strongly aperiodic.

Analogously to (Nishimura and Stachurski, 2005)

and (Stachurski, 2009), it is possible to show that the

optimal process is ergodically stable.

Lemma 4.6. Γ (see (12)) is a Lyapunov function.

Proof. Consider a ∈ R and

:= {x ∈ (0, +∞)|Γ(x) ≤ a}.

Suppose that a ≤ 1, as Γ(x) > 0, for each x > 0, then

0 and hence its closure is compact. On the other

hand, if a > 1 and {x

} is a sequence in N

such that

→ x, due to the continuity of Γ it follows that x ∈ N

and consequently, N

is a closed set. Due to Inada’s

condition on the model and the deﬁnition of Γ it is

immediate that N

is bounded and hence, compact,

and trivially, with compact closure.

As a is arbitrary we conclude that Γ is a Lyapunov

function.

Lemma 4.7. The optimal process {x

} converges Γ-

ergodically to a unique invariant probability measure

µ, that is, there exists non-negative constants R and ρ,

ρ < 1, such that for each k = 0,1,...



− µ



≤ Rρ

. (13)

The main idea to prove the previous lemma is ap-

plying the Euler equation to show that Γ is a Lya-

punov function. Then, there exist constants λ and b

such that λ ∈ (0,1) and for each x ∈ (0, ∞):

E[Γ(ξ(h(x) − f (x)))] ≤ λΓ(x) + b.

Furthermore, there exists a measure ϕ on B (X) such

that the optimal process is ϕ-irreducible and aperiodic

strongly. Finally, the result follows of Theorem 16.1.2

in (Meyn and Tweedie, 2009).

Theorem 4.8. The RSL optimal process converges in

to a random variable with probability measure µ

given in Lemma 4.7.

Proof. Let x

= x ∈ X, {x

} be the optimal process.

It is known that there exists constants λ and b with

λ ∈ (0,1) such that E[x

|x] ≤ λx

+ b. Hence, for

n = 0,1,...,

E[x

n+1

] ≤ λx

+ b. (14)

Iterating and applying standard conditional ex-

pectation properties in (14) for n = 0,1,..., and as

λ ∈ (0,1), it follows that for each n ∈ N:

E[x

n+1

] ≤ x

1 − λ

< ∞,

hence

sup

E[x

] < ∞.

A Stochastic Version of the Ramsey’s Growth Model

327

Moreover, as {x

} is almost surely positive it fol-

lows that the optimal process is uniformly integrable

(see (Peligrad and Gut, 1999), Theorem 4.2, p. 215).

Furthermore, by Lemma 4.7 it is known that the se-

quence {x

} converges in distribution to the invariant

probability measure µ.

Finally, by Theorem 5.9, p. 224 in (Peligrad and

Gut, 1999), the result follows.

5 EXAMPLES: COBB-DOUGLAS

UTILITY

Consider the following utility function:

U(c) =

for c > 0, where b > 0 and γ = 1/3. The transition

law is determined by

t+1

= ξ

− a

∈ [0,x

], t = 0,1,2,..., x

= x ∈ (0,∞). Observe

that in this case the production function h(x) = x,

x ∈ (0,∞). Suppose that {ξ

} is a sequence of i.i.d.

random variables independent of x

. Let ξ a generic

element of {ξ

} and consider that ξ with log-normal

distribution with mean 3/2 and variance 1. Then:

:= E[ξ

] = e

5/9

it is easy to see that 0 < αµ

< 1, where the discount

factor α < e

5/9

. Moreover



−1



= e

−1

and Assumption 2.2-d) holds.

Remark 5.1. Assumption 2.2-d) holds for a log-

normal distribution if and only if σ

< 2µ where µ

and σ

are mean and variance, respectively.

Deﬁne δ :=



αµ



1/(γ−1)

. It is shown in (Cruz-

arez et al., 2011) that

f (x) :=



δ − 1



x, (15)

x ∈ X, is the optimal plan.

Consider the process corresponding to the optimal

plan:

t+1

= ξ

− f (x

))

t = 0,1,2,..., x

= x ∈ X; easy calculations show that

t+1

iterating this last equation we get

t+1

t−1

∏

i=0

Taking the expectation and using the indepen-

dence of ξ

,..., ξ

t−1

, it yields that

E [x

t+1

] =





where µ := E [ξ] = e

Finally, if µ < δ then E[X

] increasing indeﬁnitely

whit respect the time; if µ > δ then E[X

] decreasing

to zero; if µ = δ then E[X

] = x/µ.

6 CONCLUSIONS

The study of the Ramsey Growth Model in a discrete

time and stochastic setting opens an interesting re-

search ﬁeld, where not only stochastic labor, stochas-

tic depreciation or other variants may be studied. For

instance, we believe that a multidimensional case (for

instance, where labor is disaggregated into two sub-

populations, regarding their different saving capabil-

ities) may be studied combining his techniques with

the Euler Equation approach.

REFERENCES

Brida, J. G., Cayssials, G., and Pereyra, J. S. (2015). The

discrete Ramsey model with decreasing population

growth rate. Dyn. Contin. Discrete Impuls. Syst. Ser.

B Appl. Algorithms, 22(2):97–115.

Cruz-Su

arez, H. and Montes-de Oca, R. (2008). An en-

velope theorem and some applications to discounted

Markov decision processes. Math. Methods Oper.

Res., 67(2):299–321.

Cruz-Su

arez, H., Montes-de Oca, R., and Zacar

ıas, G.

(2011). A consumption-investment problem modelled

as a discounted Markov decision process. Kybernetika

(Prague), 47(6):909–929.

Cruz-Su

arez, H., Zacar

ıas-Espinoza, G., and V

azquez-

Guevara, V. (2012). A version of the Euler equation in

discounted Markov decision processes. J. Appl. Math.,

pages Art. ID 103698, 16.

Ekeland, I. (2010). From Frank Ramsey to Ren

e Thom: a

classical problem in the calculus of variations leading

to an implicit differential equation. Discrete Contin.

Dyn. Syst., 28(3):1101–1119.

Hern

andez-Lerma, O. and Lasserre, J. B. (1996). Discrete-

time Markov control processes, volume 30 of Appli-

cations of Mathematics (New York). Springer-Verlag,

New York. Basic optimality criteria.

skiewicz, A. and Nowak, A. S. (2011). Discounted dy-

namic programming with unbounded returns: appli-

cation to economic models. J. Math. Anal. Appl.,

378(2):450–462.

Kamihigashi, T. (2006). Almost sure convergence to

zero in stochastic growth models. Economic Theory,

29(1):231–237.

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

328

Le Van, C. and Dana, R.-A. (2003). Dynamic program-

ming in economics, volume 5 of Dynamic Modeling

and Econometrics in Economics and Finance. Kluwer

Academic Publishers, Dordrecht.

Meyn, S. and Tweedie, R. L. (2009). Markov chains

and stochastic stability. Cambridge University Press,

Cambridge, second edition. With a prologue by Peter

W. Glynn.

Nishimura, K. and Stachurski, J. (2005). Stability of

stochastic optimal growth models: a new approach.

J. Econom. Theory, 122(1):100–118.

Peligrad, M. and Gut, A. (1999). Almost-sure results for

a class of dependent random variables. J. Theoret.

Probab., 12(1):87–104.

Ramsey, F. (1928). A matemathical theory of saving. Econ.

J., 38:543–559.

Sladk

y, K. (2012). Some remarks on stochastic version

of the ramsey growth model. Bulletin of the Czech

Econometric Society., 19:139–152.

Stachurski, J. (2009). Economic dynamics. MIT Press,

Cambridge, MA. Theory and computation.

A Stochastic Version of the Ramsey’s Growth Model

329