CUBO A Mathematical Journal
Vol.15, No¯ 01, (13–32). March 2013

Solow Models on Time Scales1

Martin Bohner and Julius Heim

Missouri University of Science and Technology,

Department of Mathematics and Statistics,

Department of Economics,

Rolla, MO 65409-0020, USA

bohner@mst.edu

julius.heim@mst.edu

Ailian Liu

Shandong University of Finance and

Economics,

School of Mathematics and Quantitative

Economics,

Jinan 250014, P. R. China

ailianliu2002@163.com

ABSTRACT

We introduce a general Solow model on time scales and derive a nonlinear first-order

dynamic equation that describes such a model. We first assume that there is neither

technological development nor a change in the population. We present the Cobb–

Douglas production function on time scales and use it to give the solution for the equa-

tion that describes the model. Next, we provide several applications of the generalized

Solow model. Finally, we generalize our work by allowing technological development

and population growth. The presented results not only unify the continuous and the

discrete Solow models but also extend them to other cases “in between”, e.g., a quan-

tum calculus version of the Solow model. Finally it is also noted that our results even

generalize the classical continuous and discrete Solow models since we allow the sav-

ings rate, the depreciation factor of goods, the growth rate of the population, and the

technological growth rates to be functions of time rather than taking constant values

as in the classical Solow models.

Keywords and Phrases: Time scales, Solow model, dynamic equation, Cobb–Douglas produc-

tion function, economics.

2010 AMS Mathematics Subject Classification: 91B64, 34C10, 39A10, 39A11, 39A12,

39A13.

1This paper is dedicated to Professor Gaston M. N’Guérékata on the occasion of his 60th birthday



14 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

RESUMEN

Introducimos un modelo general de Solow en escalas de tiempo y derivamos una ecuación

dinámica no lineal de primer orden que describe el mencionado modelo. Primero asum-

imos que no existe ni desarrollo tecnológico ni un cambio en la población. Presentamos

la función de producción de Cobb-Douglas en escalas de tiempo y la utilizamos para

entregar la solución de la ecuación que describe el modelo. Luego, mostramos varias

aplicaciones del modelo generalizado de Solow. Finalmente, generalizamos nuestro

trabajo permitiendo desarrollo tecnológico y crecimiento de la población. Los resul-

tados presentados no sólo unifican los modelos de Solow continuos y discretos, sino

que además se extienden a otros casos “entre medio”, es decir, una versión del cálculo

cuántico del modelo de Solow. Finalmente, también se menciona que nuestro resultado

también generaliza los modelos clásicos continuos y discretos, ya que permitimos tasas

de ahorro, el factor de depreciación de bienes, la razón de crecimiento de la población y

las razones de crecimiento tecnológico por ser funciones del tiempo más que asumiendo

valores constantes como es el caso de los modelos de Solow clásico.

1 The Classical Solow Model

Modern growth theory is mainly based on the works of Solow [12] and Swan [13]. In the Solow

model, it is assumed that the national income Y depends on consumption C and investment I, i.e.,

Y(t) = C(t) + I(t).

Moreover, it is assumed that the national income is a function of the capital stock K and the

product of the technological progress A and the population N, i.e.,

Y(t) = F (K(t), A(t)N(t)) , (1)

where the production function F satisfies the following conditions:

1. F(λK, λL) = λF(K, L) for all λ, K, L ∈ R+ (constant returns to scale);

2. F(K, 0) = F(0, L) = 0 for all K, L ∈ R+;

3. ∂F
∂K

> 0, ∂F
∂L

> 0, ∂
2
F

∂K2
< 0, ∂

2
F

∂L2
< 0;

4. lim
K→0+

∂F
∂K

= lim
L→0+

∂F
∂L

= +∞, lim
K→+∞

∂F
∂K

= lim
L→+∞

∂F
∂L

= 0.

Furthermore, the change of the capital stock in a particular period does not only depend on the

new investment but also on the depreciation of goods. In other words, we take for granted that

K′(t) = I(t) − δK(t) (2)



CUBO
15, 1 (2013)

Solow Models on Time Scales 15

with given initial capital stock K(0), where δ is the depreciation rate of the goods. As usual, we

presume that

S(t) = I(t) = sY(t), (3)

where s is the savings rate, and thus the savings S is the portion of the national income which

is not consumed. Plugging (1) and (3) into (2), we obtain the nonlinear first-order differential

equation

K′(t) = sF (K(t), A(t)N(t)) − δK(t). (4)

In this paper, we assume that the technological knowledge of the society grows exponentially with

rate r, that is,

A′(t) = rA(t), i.e., A(t) = ertA(0),

where A(0) is the initial technological standing of the nation. Similarly, we suppose that the

population of the nation grows with growth rate n and an initial population of N(0), hence

N′(t) = nN(t), i.e., N(t) = entN(0).

Both n and r can be positive or negative depending on which nation we are talking about. Of

course, it seems quite unrealistic that r is negative. Solow also took the capital stock per efficiency

of labor k into account, which he defined as the stationary variable

k(t) :=
K(t)

A(t)N(t)
.

We use constant returns in order to define the intensive version of the production function to be

y(t) :=
Y(t)

A(t)N(t)
=

F (K(t), A(t)N(t))

A(t)N(t)
= F(k(t), 1) =: f(k(t)).

This allows us now to rewrite (4) as

K′(t)

A(t)N(t)
= sf(k(t)) − δk(t). (5)

A simple calculation shows that
k′(t)

k(t)
=

K′(t)

K(t)
− r − n.

Therefore (5) turns into

k′(t) = sf(k(t)) − (δ + n + r)k(t). (6)

Table 1 summarizes all the variables with their meanings that appear in the Solow model. For a

more detailed discussion of stability and qualitative analysis of the Solow model, the reader might

consult [9]. The discrete analogue of Solow’s model is discussed in [11], featuring some results

similar to those in the continuous Solow model. So far this dynamic process was regarded either

as solely continuous or solely discrete. In this paper, we generalize these two theories in such a

way that the continuous and the discrete versions of the Solow model are only special cases of



16 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

Table 1: Explanation of Variables

Variable Explanation

Y national income

I induced investment

C consumption

S Savings

s savings rate

K capital stock

K0 initial capital stock

δ depreciation factor of goods

A level of technological knowledge

A0 initial technological progress level

r technological growth rate

N level of population

N0 initial population

n growth rate of population

k capital stock per efficiency unit of labor

our generalized Solow model. We do this using the time scales theory, an area of mathematics

which was originally introduced by Stefan Hilger in his PhD thesis [10]. The two books [7, 8] by

Bohner and Peterson offer an introduction with applications to time scales calculus along with

some advanced topics. Applications of time scales calculus can be found in many areas, also in

economics. Tisdell and Zaidi [14] in particular already generalized some economic topics, and also

Bohner et al. discussed multiplier-accelerator models and utility functions on time scales in [5, 6].

The set up of this paper is as follows. In Section 2, we give a brief introduction to the time

scales theory. In Section 3, we present the Solow model on time scales and derive the nonlinear

first-order dynamic equation that describes this model. In Section 4, we define the generalized

Cobb–Douglas production function on time scales and provide examples for various time scales.

Furthermore, we state a theorem that gives the solution of the nonlinear first-order dynamic

equation, and we also provide examples for several time scales. We finally state a result that

addresses asymptotic stability of the solution. Section 5 is used to show some important properties

of the production function, called the Inada conditions. Finally, in Section 6, we point out how our

model can be extended, assuming the presence of technological and population growth (or decay),

i.e., by assuming n 6= 0 as well as r 6= 0. We present the Cobb–Douglas production function for



CUBO
15, 1 (2013)

Solow Models on Time Scales 17

this more general case and also derive the equilibrium solution for this extended model.

It is also noted that our results even generalize the classical continuous and discrete Solow

models since we allow the savings rate, the depreciation factor of goods, the growth rate of the

population, and the technological growth rates to be functions of time rather than considering

constant values as in the classical Solow models.

2 Time Scales Preliminaries

In this section, we introduce some elements of time scales calculus. For a more rigorous time scales

introduction, we refer the reader to [7, 8].

Let T be a time scale, i.e., a nonempty closed subset of R. For t ∈ T, the forward jump

operator σ : T → T is defined by

σ(t) := inf {s ∈ T : s > t} ,

while the backward jump operator ρ : T → T is defined by

ρ(t) := sup {s ∈ T : s < t} .

In this definition, we set inf ∅ = sup T (i.e., σ(t) = t if T has maximum t) and sup ∅ = inf T (i.e.,

ρ(t) = t if T has minimum t). If σ(t) > t, σ(t) = t, ρ(t) < t, and ρ(t) = t, then t is called

right-scattered, right-dense, left-scattered and left-dense, respectively. The graininess function µ :

T → [0, ∞) is defined by

µ(t) := σ(t) − t.

We also need the set Tκ which is defined in the following way: If T has a left-scattered maximum

m, then Tκ = T − {m}. Else, Tκ = T.

Now let f : T → R be a function. If t ∈ Tκ, then f∆(t) is defined as the number (provided that

it exists) such that for every ε > 0, there exists a neighborhood U of t (i.e., U = (t − δ, t + δ) ∩ T

for some δ > 0) such that

∣∣[f(σ(t)) − f(s)] − f∆(t)[σ(t) − s]
∣∣ ≤ ε|σ(t) − s| for all s ∈ U.

We call this number f∆(t) the delta derivative of f at t. Moreover, f is called rd-continuous provided

it is continuous at right-dense points in T and its left sided limits exist (finite) at left-dense points

in T. The function fσ : T → R is defined by fσ = f ◦ σ. In our calculations, we use the so-called

“simple useful formula”

fσ = f + µf∆.

We denote the set of rd-continuous functions by Crd = Crd(T) = Crd(T, R). Next, f is said to be

regressive given that

1 + µ(t)f(t) 6= 0 for all t ∈ T



18 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

holds. The set of all regressive and rd-continuous functions is denoted by R = R(T) = R(T, R).

We also define the set R+ of all positively regressive elements by

R+ = R+(T, R) = {f ∈ R : 1 + µ(t)p(t) > 0 for all t ∈ T} .

Let now p, q ∈ R. We define the “circle plus” addition ⊕ on R by

(p ⊕ q)(t) := p(t) + q(t) + µ(t)p(t)q(t) for all t ∈ T

and the “circle minus” subtraction ⊖ on R by

(p ⊖ q)(t) :=
p(t) − q(t)

1 + µ(t)q(t)
for all t ∈ T.

We put

R(α) :=

{
R if α ∈ N,

R+ if α ∈ R \ N.

For α ∈ R and p ∈ R(α), we define

(α ⊙ p)(t) := αp(t)

∫1

0

(1 + µ(t)p(t)h)α−1dh. (7)

The time scales exponential function ep(·, t0) is defined for p ∈ R and t0 ∈ T as the unique solution

of the initial value problem

y∆ = p(t)y, y(t0) = 1 on T.

We have

ep(·, t0)eq(·, t0) = ep⊕q(·, t0) and
ep(·, t0)

eq(·, t0)
= ep⊖q(·, t0).

If α ∈ R and p ∈ R(α), then

eα⊙p = e
α
p

(see [8, Theorem 2.44]). Let α ∈ R \ {1}. We say that

x∆ =

[
q ⊖

(
1

α − 1
⊙ (gxα−1)

)]
x (8)

(see [8, Section 2.6]) is a Bernoulli equation on time scales.

3 Solow Model on Time Scales

Assume that F and f are production functions as defined in Section 1. We now introduce the

generalized Solow model on an arbitrary time scale:





Y(t) = F(K(t), A(t)N(t)),

K∆(t) = I(t) − δ(t)K(t),

I(t) = s(t)Y(t),

A∆(t) = r(t)A(t),

N∆(t) = n(t)N(t),

(9)



CUBO
15, 1 (2013)

Solow Models on Time Scales 19

where we require

δ(t) > 0 and s(t) > 0 for all t ∈ T (10)

and

n, r ∈ R. (11)

The economical meanings of δ, s, r, and n are the same as described in Table 1. If (K, Y, A, N, I)

solves (9), then

K∆(t) = s(t)Y(t) − δ(t)K(t) = s(t)F(K(t), A(t)N(t)) − δ(t)K(t). (12)

Define

k(t) :=
K(t)

A(t)N(t)
and y(t) :=

Y(t)

A(t)N(t)
, (13)

which are regarded as the capital stock per efficiency unit of labor and the production per efficiency

unit of labor, respectively. By (12) and (13), we have

K∆(t)

A(t)N(t)
= s(t)f(k(t)) − δ(t)k(t). (14)

Theorem 3.1. Assume (9), (10), and (11). If k is defined as in (13), then

k∆(t) =
s(t)

(1 + µ(t)r(t))(1 + µ(t)n(t))
f(k(t))

−

(
δ(t) + n(t)

(1 + µ(t)r(t))(1 + µ(t)n(t))
+

r(t)

1 + µ(t)r(t)

)
k(t).

(15)

Proof. The time scales quotient rule [7, Theorem 1.20 (v)] provides

k∆ =

(
K

AN

)∆
=

K∆AN − K
(
AN∆ + A∆Nσ

)

ANAσNσ

=
K∆

AσNσ
−

KN∆

NAσNσ
−

KA∆

ANAσ

(9)
=

K∆

AN(1 + µr)(1 + µn)
−

n

(1 + µr)(1 + µn)
k −

r

1 + µr
k

(14)
=

s

(1 + µr)(1 + µn)
f ◦ k −

(
δ + n

(1 + µr)(1 + µn)
+

r

1 + µr

)
k,

i.e., (15) holds.

Example 3.2. If T = R, then σ(t) = t and µ(t) = 0 for all t ∈ T. Thus (15) can be rewritten as

k′(t) = s(t)f(k(t)) − (δ(t) + n(t) + r(t))k(t),

which reduces to (6) provided δ, s, n, and r are constants.



20 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

Theorem 3.3. Assume (10) and (11). Then (15) holds if and only if

k∆(t) = s(t)f(k(t)) − δ(t)k(t) − (n ⊕ r)(t)k(σ(t)). (16)

Proof. Suppose k solves (16). Then we use the “simple useful formula” to obtain

k∆ = s(f ◦ k) − δk − (n ⊕ r)kσ

= s(f ◦ k) − δk − (n + r + µnr)
(
k + µk∆

)
.

Hence

(1 + µn)(1 + µr)k∆ = s(f ◦ k) − (δ + n + r + µnr) k.

Dividing by (1 + µn)(1 + µr) yields (15). If k solves (15), then (16) follows by reversing the above

steps.

Theorem 3.4. Assume (10) and (11). If (16) holds, then

(1 + µ(t)n(t))(1 + µ(t)r(t))kσ(t) = µ(t)s(t)f(k(t)) + (1 − µ(t)δ(t))k(t). (17)

If (17) holds and µ(t) 6= 0, then (16) holds.

Proof. Suppose k solves (16). Then we multiply (16) by µ(t) and use the simple useful formula to

obtain

k(σ(t)) − k(t) = µ(t)k∆(t)

= µ(t)s(t)f(k(t)) − µ(t)δ(t)k(t) − µ(t)(n ⊕ r)(t)k(σ(t)).

Hence

(1 + µ(t)(n ⊕ r)(t)) k(σ(t)) = µ(t)s(t)f(k(t)) + (1 − µ(t)δ(t))k(t),

which results in (17). If (17) holds at t ∈ T such that µ(t) 6= 0, then the above steps can be

reversed.

Example 3.5. If T = Z, then σ(t) = t + 1 and µ(t) = 1 for all t ∈ T. Thus (17) can be rewritten

as

(1 + n(t))(1 + r(t))k(t + 1) = s(t)f(k(t)) + (1 − δ(t))k(t).

This equation can be found in [11].

Example 3.6. If T = hZ with h > 0, then σ(t) = t + h and µ(t) = h for all t ∈ T. Thus (17)

can be rewritten as

(1 + hn(t))(1 + hr(t))k(t + h) = hs(t)f(k(t)) + (1 − hδ(t))k(t).

Example 3.7. If T = qN0 with q > 1, then σ(t) = qt and µ(t) = (q − 1)t for all t ∈ T. Thus

(17) can be rewritten as

(1 + (q − 1)tn(t))(1 + (q − 1)tr(t))k(qt) = (q − 1)ts(t)f(k(t)) + (1 − (q − 1)tδ(t))k(t)

.



CUBO
15, 1 (2013)

Solow Models on Time Scales 21

4 Analysis of the Basic Solow Model

In this section, we assume that (10) holds and that there is no technological development and no

population change, i.e., n = r = 0. Then (16) simplifies to

k∆(t) = s(t)f(k(t)) − δ(t)k(t). (18)

Let

0 < α < 1, w(t) =

(
1

α − 1
⊙

δg

s

)
(t), and g(t) = (1 − α)s(t). (19)

If

f̃(x) :=
δ(t) +

(
w ⊖

(
1

α−1
⊙ (gxα−1)

))
(t)

s(t)
is independent of t ∈ T, (20)

then we define the generalized Cobb–Douglas production function on time scales by

f(x) = xf̃(x). (21)

Theorem 4.1. Let t ∈ T. If µ(t) = 0, then

δ(t) +
(
w ⊖

(
1

α−1
⊙ (gxα−1)

))
(t)

s(t)
= xα−1. (22)

Proof. Assume µ(t) = 0. Then at t, we have

δ(t) +
(
w ⊖

(
1

α−1
⊙ (gxα−1)

))
(t)

s(t)
=

δ(t) + w(t) −
g(t)x

α−1

α−1

s(t)

=
δ(t) +

δ(t)g(t)
(α−1)s(t)

−
g(t)x

α−1

α−1

s(t)

=
δ(t) − δ(t) + s(t)xα−1

s(t)

= xα−1,

which shows (22).

Example 4.2. If T = R, then f̃(x) = xα−1, and thus (20) holds. Hence the Cobb–Douglas

production function is defined and equals

f(x)
(21)
= xf̃(x) = xxα−1 = xα.

Theorem 4.3. Let t ∈ T. If µ(t) > 0, then

δ(t) +

(
w ⊖

(
1

α − 1
⊙
(
gxα−1

)))
(t)

=
1

µ(t)

{

δ(t)µ(t) − 1 +

(
1 + (1 − α)µ(t)s(t)xα−1

1 + (1 − α)µ(t)δ(t)

) 1
1−α

}

. (23)



22 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

Proof. Assume µ(t) > 0. Then at t, we have

1

α − 1
⊙ (gxα−1)

(7)
=

1

α − 1
gxα−1

∫1

0

(
1 + µgxα−1h

) 1
α−1

−1
dh

=

(
1 + µgxα−1

) 1
α−1 − 1

µ
,

w =
1

α − 1
⊙

δg

s
=

1

α − 1
⊙ (δ(1 − α))

(7)
=

1

α − 1
δ(1 − α)

∫1

0

(1 + µδ(1 − α)h)
1

α−1
−1

dh

=
(1 + µδ(1 − α))

1
α−1 − 1

µ
,

and hence

w ⊖

(
1

α − 1
⊙ (gxα−1)

)
=

w −
(1+µgxα−1)

1
α−1 −1

µ

1 + µ
(1+µgxα−1)

1
α−1 −1

µ

=
w −

(1+µgxα−1)
1

α−1 −1

µ

(1 + µgxα−1)
1

α−1

=

(1+µδ(1−α))
1

α−1 −1
µ

−
(1+µgxα−1)

1
α−1 −1

µ

(1 + µgxα−1)
1

α−1

=
1

µ

{

−1 +

(
1 + µδ(1 − α)

1 + µ(1 − α)sxα−1

) 1
α−1

}

,

which shows (23).

Theorem 4.4. Let µ(t) > 0 for all t ∈ T. Assume (10) and suppose

s̃ := s(t)µ(t) and δ̃ := δ(t)µ(t) are independent of t ∈ T. (24)

Then (20) holds and the Cobb–Douglas production function is defined and equals

f(x) =
x

s̃





δ̃ − 1 +

(
1 + (1 − α)s̃xα−1

1 + (1 − α)δ̃

) 1
1−α





.



CUBO
15, 1 (2013)

Solow Models on Time Scales 23

Proof. Using Proposition 4.3, we see that

δ(t) +
(
w ⊖

(
1

α−1
⊙ (gxα−1)

))
(t)

s(t)

(23)
=

1

µ(t)s(t)

{

δ(t)µ(t) − 1 +

(
1 + (1 − α)µ(t)s(t)xα−1

1 + (1 − α)µ(t)δ(t)

) 1
1−α

}

=
1

s̃





δ̃ − 1 +

(
1 + (1 − α)s̃xα−1

1 + (1 − α)δ̃

) 1
1−α






is independent of t and therefore equals f̃(x). By (21), f(x) = xf̃(x).

Example 4.5. If T = hZ with h > 0 and δ, s are constants and satisfy (10), then Proposition 4.4

gives us that (20) is satisfied and that the Cobb–Douglas production function is defined and equals

(note that (24) is satisfied in this case with s̃ = sh and δ̃ = δh)

f(x) =
x

hs

{

δh − 1 +

(
1 + (1 − α)hsxα−1

1 + (1 − α)hδ

) 1
1−α

}

. (25)

Example 4.6. If T = Z and δ, s are constants and satisfy (10), then Example 4.5 gives us that

(20) is satisfied and that the Cobb–Douglas production function, i.e., the discrete version of the

classical Cobb–Douglas production function, is defined and equals

f(x) =
x

s

{

δ − 1 +

(
1 + (1 − α)sxα−1

1 + (1 − α)δ

) 1
1−α

}

.

Example 4.7. If T = qN0 with q > 1 and δ, s satisfy (10) and (24), i.e.,

s̃ := (q − 1)ts(t) and δ̃ := (q − 1)tδ(t) are independent of t ∈ T,

then Proposition 4.4 gives us that (20) is satisfied and that the Cobb–Douglas production function

is defined and equals

f(x) =
x

s̃





δ̃ − 1 +

(
1 + (1 − α)s̃xα−1

1 + (1 − α)δ̃

) 1
1−α





.

Using (21), we rewrite equation (18) in the form (8), i.e., as a Bernoulli equation on time

scales.

Theorem 4.8. Assume (10) and (20) and let f be defined by (21). Then (18) holds if and only if

k∆(t) =

{

w ⊖

(
1

α − 1
⊙ (gkα−1)

)}
(t)k(t). (26)



24 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

Example 4.9. If T = R, then w = −δ, and equation (26) is

k′(t) =
(
skα−1(t) − δ

)
k(t).

Hence equation (26) is indeed a generalized form of the continuous Solow model with the Cobb–

Douglas production function.

With the generalized Cobb–Douglas function, we can find the solution of the Solow model

(18) on time scales.

Theorem 4.10. Assume (10) and

κ :=
s(t)

δ(t)
is independent of t ∈ T (27)

and define p ∈ R by

p(t) := (1 − α)δ(t) for all t ∈ T. (28)

Then the solution of (26) with initial condition k(t0) = k0 > 0, where t0 ∈ T, is given by

k(t) =

{

κ +
k1−α0 − κ

ep(t, t0)

} 1
1−α

for all t ∈ T, (29)

provided the quantity in curly braces in (29) is always positive.

Proof. Suppose k solves (26) such that k(t0) = k0. Define x̃ := k
α−1. By [8, Theorem 2.37], we

have

x̃∆

x̃
= (α − 1) ⊙

k∆

k

= (α − 1) ⊙

{

w ⊖

[
1

α − 1
⊙
(
gkα−1

)]}

= [(α − 1) ⊙ w] ⊖
(
gkα−1

)

= (δ(1 − α)) ⊖
(
gkα−1

)
,

so

x̃∆ = (p ⊖ (gx̃)) x̃,

which shows that x̃ solves the logistic equation on time scales (see [1] and [8, Section 2.4]). Define

y := 1/x̃. Then

y∆ =

(
1

x̃

)∆
=

−x̃∆

x̃x̃σ
= − (p ⊖ (gx̃)) yσ =

gx̃ − p

1 + µgx̃
yσ

and hence

(1 + µgx̃) y∆ = gx̃yσ − pyσ,

i.e., using the “simple useful formula”,

y∆ + (yσ − y) gx̃ = gx̃yσ − pyσ,



CUBO
15, 1 (2013)

Solow Models on Time Scales 25

i.e.,

y∆ = −pyσ + g. (30)

Using

g = (1 − α)s = (1 − α)δκ = κp

and the variation of constants formula [7, Theorem 2.74], the solution of (30) is given by

y(t) = y0e⊖p(t, t0) +

∫t

t0

g(τ)e⊖p(t, τ)∆τ

= y0e⊖p(t, t0) +

∫t

t0

κp(τ)ep(τ, t)∆τ

= y0e⊖p(t, t0) + κ

∫t

t0

p(τ)ep(τ, t)∆τ

= y0e⊖p(t, t0) + κep(τ, t)|
t
t0

= y0e⊖p(t, t0) + κ(1 − e⊖p(t, t0)).

From the substitutions we performed, we have that y0 = k
1−α
0 as well as k(t) =

1

y(t)
1

α−1

, which

shows (29). Conversely, k given by (29) is easily seen to be a solution of (26).

Using Theorem 4.10, we obtain the asymptotic stability of the unique equilibrium point of

(26).

Theorem 4.11. Assume (10) and (27). If

∫∞

t0

δ(t)∆t = ∞, (31)

then any solution k of (26) in the form (29) satisfies

lim
t→∞

k(t) = κ
1

1−α =: κ,

and κ is the unique equilibrium point of (26).

Proof. We have p(t) > 0 for all t ∈ T, where p is defined in Theorem 4.10. Hence p ∈ R+. Thus,

by [4, Remark 2], we have

ep(t, t0) ≥ 1 +

∫t

t0

p(τ)∆τ = 1 + (1 − α)

∫t

t0

δ(τ)∆τ for all t ≥ t0.

Therefore, using (31),

lim
t→∞

ep(t, t0) = ∞,

and thus

lim
t→∞

k(t) =

{

κ +
k1−α0 − κ

limt→∞ ep(t, t0)

} 1
1−α

= κ
1

1−α = κ.



26 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

Now we show that κ is the unique nontrivial equilibrium point of (26): A point κ is a nontrivial

equilibrium point of (26) if and only if

w ⊖

(
1

α − 1
⊙
(
gκα−1

))
= 0,

which holds if and only if (use the definition of w, (27), and the properties of ⊖)

1

α − 1
⊙

g

κ
=

1

α − 1
⊙
(
gκα−1

)
,

which is true if and only if (use the properties of ⊙ and the definition of g)

1

κ
= κα−1,

i.e., κ = κ. This completes the proof.

Example 4.12. Let T = R and assume (10) and (27). Then (29) reads

k(t) =

{

κ +
k1−α0 − κ

e
(1−α)

∫
t
t0

δ(τ)∆τ

} 1
1−α

.

If, in addition, t0 = 0 and δ is constant (this implies that s is constant as well), then

k(t) =

{

κ +
k1−α0 − κ

e(1−α)δt

} 1
1−α

.

Example 4.13. Let T = hZ with h > 0 and assume (10) and (27). Then (29) reads

k(t) =






κ +
k1−α0 − κ

t/h−1∏

i=t0/h

(1 + (1 − α)hδ(ih))






1
1−α

.

If, in addition, t0 = 0 and δ is constant (this implies that s is constant as well), then

k(t) =

{

κ +
k1−α0 − κ

(1 + (1 − α)hδ)
t/h

} 1
1−α

.

Example 4.14. Let T = qN0 with q > 1 and assume (10) and (27). Then (29) reads

k(t) =






κ +
k1−α0 − κ

logq t−1∏

i=logq t0

(1 + (q − 1)qi(1 − α)δ(qi))






1
1−α

.



CUBO
15, 1 (2013)

Solow Models on Time Scales 27

If, in addition, t0 = 1 and δ̃ := (q − 1)tδ(t) is constant (this implies that (q − 1)ts(t) is constant

as well), then

k(t) =





κ +

k1−α0 − κ(
1 + (1 − α)δ̃

)logq t






1
1−α

.

5 Properties of the Production Function

In this section, we show that our Cobb–Douglas production function f given in (21) satisfies the

time scales Inada conditions (see [2, 3])





f(x) > 0, f̃′(x) < 0, f′′(x) < 0 for all x > 0,

f̃(x) > ζ(t) ≥ 0, f′(x) > ζ(t) ≥ 0 for all x > 0 and all t ∈ T,

lim
x→0+

f̃(x) = lim
x→0+

f′(x) = ∞,

lim
x→∞

f̃(x) = lim
x→∞

f′(x) = ζ(t) ≥ 0 for all t ∈ T,

(32)

where ζ : T → R is defined in the following lemma.

Lemma 5.1. Assume (10) and define

ζ(t) :=
1

s(t)

{

δ(t) +

(
1

α − 1
⊙ ((1 − α)δ)

)
(t)

}

.

Then ζ(t) ≥ 0.

Proof. If µ(t) = 0, then ζ(t) = 0. If µ(t) > 0, then, as in the proof of Proposition 4.3, we have

(
1

α − 1
⊙ ((1 − α)δ)

)
(t) =

(1 + (1 − α)µ(t)δ(t))
1

α−1 − 1

µ(t)
.

Now using the well-known Bernoulli inequality, we obtain

(1 + (1 − α)µ(t)δ(t))
1

α−1 ≥ 1 +
1

α − 1
(1 − α)µ(t)δ(t) = 1 − µ(t)δ(t).

This proves the claim.

Theorem 5.2. Assume (10) and (20) and define the Cobb–Douglas production function by (21).

If there exists t ∈ T such that µ(t) = 0, then the Cobb–Douglas production function satisfies the

Inada conditions (32).

Proof. By Theorem 4.1 and (20),

f(x) = xα and f̃(x) = xα−1. (33)

Clearly, f given by (33) satisfies the Inada conditions (32).



28 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

Theorem 5.3. Assume (10) and (20) and define the Cobb–Douglas production function by (21).

If there exists t ∈ T such that µ(t) > 0, then the Cobb–Douglas production function satisfies the

Inada conditions (32).

Proof. By Theorem 4.3 and (20), we have

f̃(x) =
1

µ(t)s(t)

{

µ(t)δ(t) − 1 +

(
1 + (1 − α)µ(t)s(t)xα−1

1 + (1 − α)µ(t)δ(t)

) 1
1−α

}

. (34)

In order to check that the Inada conditions (32) are satisfied, we use the following notation:






δ̃ := µ(t)δ(t), κ :=
s(t)
δ(t)

, α̃ := (1 − α)δ̃,

ζ :=
1

δ̃κ

{
δ̃ − 1 + (1 + α̃)

1
α−1

}
, z(x) :=

1 + α̃κxα−1

1 + α̃
.

(35)

Using (35), we rewrite (34) as

f̃(x) =
1

δ̃κ

{
δ̃ − 1 + (z(x))

1
1−α

}
. (36)

We obviously have

z(x) >
1

1 + α̃
for all x > 0. (37)

Using (37) in (36), we find

f̃(x) >
1

δ̃κ

{
δ̃ − 1 + (1 + α̃)

1
α−1

}
= ζ for all x > 0.

By Lemma 5.1, we also have ζ ≥ 0, and hence f̃(x) > 0 for all x > 0 so that f(x) = xf̃(x) > 0 for

all x > 0. Next, note that

z′(x) =
α̃(α − 1)κ

1 + α̃
xα−2. (38)

Using (38) in (36), we find

f̃′(x) =
1

δ̃κ(1 − α)
(z(x))

1
1−α

−1
z′(x)

=
1

δ̃κ(1 − α)
(z(x))

α
1−α

α̃(α − 1)κxα−2

1 + α̃

= −
α̃xα−2

(1 + α̃)δ̃
(z(x))

α
1−α < 0 for all x > 0.



CUBO
15, 1 (2013)

Solow Models on Time Scales 29

Using this, (36), (35), (37), and the product rule for f(x) = xf̃(x), we get

f′(x) = f̃(x) + xf̃′(x)

=
1

δ̃κ

{
δ̃ − 1 + (z(x))

1
1−α

}
−

α̃xα−1κ

(1 + α̃)δ̃κ
(z(x))

α
1−α

=
1

δ̃κ

{
δ̃ − 1 + (z(x))

1
1−α

}
−

(
z(x) −

1

1 + α̃

)
1

δ̃κ
(z(x))

α
1−α

=
1

δ̃κ

{

δ̃ − 1 +
1

1 + α̃
(z(x))

α
1−α

}

>
1

δ̃κ

{

δ̃ − 1 +
1

1 + α̃
(1 + α̃)

α
α−1

}

=
1

δ̃κ

{
δ̃ − 1 + (1 + α̃)

1
α−1

}
= ζ,

i.e.,

f′(x) =
1

δ̃κ

{

δ̃ − 1 +
1

1 + α̃
(z(x))

α
1−α

}

> ζ for all x > 0. (39)

Also, since

lim
x→0+

z(x) = ∞ and lim
x→∞

z(x) =
1

1 + α̃
,

we find from (36) and (39) that

lim
x→0+

f̃(x) = lim
x→0+

f′(x) = ∞

and

lim
x→∞

f̃(x) = lim
x→∞

f′(x) = ζ ≥ 0.

Finally, using (39) and (38), we obtain

f′′(x) =
1

δ̃κ(1 + α̃)

α

1 − α
(z(x))

α
1−α

−1z′(x)

=
1

δ̃κ(1 + α̃)

α

1 − α

α̃(α − 1)κ

1 + α̃
xα−2(z(x))

2α−1
1−α

= −
αα̃

δ̃(1 + α̃)2
xα−2(z(x))

2α−1
1−α < 0 for all x > 0.

This shows that all conditions in (32) are satisfied.

6 General Solow Model on Time Scales

Let us now assume (10) and (11) so that we allow now that the technology develops exponentially

and/or the population increases exponentially. All results from Section 4 and Section 5 still hold

true when we replace s and δ in (18) by

s

1 + µ(n ⊕ r)
and

δ + (n ⊕ r)

1 + µ(n ⊕ r)
,



30 Martin Bohner, Julius Heim and Ailian Liu CUBO
15, 1 (2013)

respectively, as (18) in this case results in (15). This means that w and g from (19) are replaced

by

w(t) =

(
1

α − 1
⊙

(δ + (n ⊕ r))g

s

)
(t) and g(t) =

(1 − α)s(t)

1 + µ(t)(n ⊕ r)(t)
,

respectively. Next, f̃(x) in (20) is replaced by

δ(t) + (n ⊕ r)(t) + (1 + µ(t)(n ⊕ r)(t))
(
w ⊖

(
1

α−1
⊙
(
gxα−1

)))

s(t)
.

Moreover, κ and p in (27) and (28) are replaced by

s(t)

δ(t) + (n ⊕ r)(t)
and (1 − α)

δ(t) + (n ⊕ r)(t)

1 + µ(t)(n ⊕ r)(t)
,

respectively. Finally, condition (31) turns into

∫∞

t0

δ(t) + (n ⊕ r)(t)

1 + µ(t)(n ⊕ r)(t)
∆t = ∞.

Received: January 2013. Revised: February 2013.

References

[1] E. Akin-Bohner and M. Bohner. Miscellaneous dynamic equations. Methods Appl. Anal.,

10(1):11–30, 2003.

[2] P. Barelli and S. de Abreu Pessôa. Inada conditions imply that production function must be

asymptotically Cobb–Douglas. Econom. Lett., 81(3):361–363, 2003.

[3] G. Beer. The Cobb–Douglas Production Function. Math. Mag., 53(1):44–48, 1980.

[4] M. Bohner. Some oscillation criteria for first order delay dynamic equations. Far East J.

Appl. Math., 18(3):289–304, 2005.

[5] M. Bohner, G. Gelles, and J. Heim. Multiplier-accelerator models on time scales. Int. J. Stat.

Econ., 4(S10):1–12, 2010.

[6] M. Bohner and G. M. Gelles. Risk aversion and risk vulnerability in the continuous and

discrete case. A unified treatment with extensions. Decis. Econ. Finance, 35:1–28, 2012.

[7] M. Bohner and A. Peterson. Dynamic equations on time scales. Birkhäuser Boston Inc.,

Boston, MA, 2001. An introduction with applications.

[8] M. Bohner and A. Peterson, editors. Advances in dynamic equations on time scales. Birkhäuser

Boston Inc., Boston, MA, 2003.



CUBO
15, 1 (2013)

Solow Models on Time Scales 31

[9] G. Gandolfo. Economic dynamics. North-Holland Publishing Co., 1997.

[10] S. Hilger. Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD thesis,

Universität Würzburg, 1988.

[11] L. Ljungqvist and T. J. Sargent. Recursive Macroeconomic Theory. MIT Press, 2000.

[12] R. M. Solow. A contribution to the theory of economic growth. The Quarterly Journal of

Economics, 70(1):65–94, February 1956.

[13] Swan. Economic growth and capital accumulation. Economic Record, 32:334–361, 1956.

[14] C. C. Tisdell and A. Zaidi. Basic qualitative and quantitative results for solutions to nonlinear,

dynamic equations on time scales with an application to economic modelling. Nonlinear Anal.,

68(11):3504–3524, 2008.