We set up a generalized Solow-Swan model to study the exogenous impact of population, saving rate, technological change, and labor participation rate on economic growth. By introducing generalized exogenous variables into the classical Solow-Swan model, we obtain a nonautomatic differential equation. It is proved that the solution of the differential equation is asymptotically stable if the generalized exogenous variables converge and does not converge when one of the generalized exogenous variables persistently oscillates.

In the classical Solow-Swan model, saving rate, technological level, capital depreciation, and population growth rate are assumed to be fixed positive constants [

In this paper, a generalized Solow-Swan model is set up by introducing generalized exogenous variables into the classical Solow-Swan model, which is described by a nonautomatic differential equation. We use this model to inquire the effect of exogenous short-term shocks and long-term fluctuation on the economic growth.

Firstly, we analyze the dynamics of the generalized Solow-Swan model. It is proved that the right maximal interval of the nonautomatic differential equation is

The case that the differential equation has oscillational solution is also discussed in this paper. It is obtained that the solution of the equation does not converge when one of the generalized exogenous variables persistently oscillates. Therefore, the economy presents fluctuation when one of the generalized exogenous variables is persistent oscillation.

Secondly, we inquire the impact of the main three factors on the economic growth. The first one is the change of the population and we mainly study the effect of the variable population growth rate and labor participation on the economic growth. It is obtained that the economy with low population growth rate has higher per capita capital than that with high population growth rate and the economy tends stable if the population growth rate tends to a stable level. On the other hand, we obtain that the economy with high labor participation rate has higher per capita capital than that with low labor participation rate. This implies that there exists “demographic dividend” in the late stage of the demographic transition in which the population growth rate declines and the labor force participation rate increases and the population aging will slow down the economic growth. If the population growth or labor participation rate is persistent oscillation, the economic fluctuation will appear.

In the classical Solow-Swan model, the saving rate is a fixed parameter and the shock of the saving rate change on the economic growth is studied by Romer [

The effect of different types of technological change on the economic growth is inquired by using the generalized Solow-Swan model. It is proved that the economy with higher technological level has higher per capita capital than that with lower technological level under the Hicks or Solow neutral technology. For the Harrod neutral technology, we obtain that the per capita capital of the economy with higher final technological growth rate will exceed that with lower final technological growth whatever how high the initial per capita capital or technological level the latter has. This implies that a developing economy can catch up a developed economy if it keeps a higher technological growth rate. When the technological level (under Hicks or Solow neutral technology) or the technological growth rate (under Harrod neutral technology) is persistent oscillation, the economy presents long-term fluctuation.

Finally, a brief summery is given in Section

The classical Solow-Swan model is given by

In this paper, we consider the following nonautomatic differential equation:

If

(1) It is only to prove that there exits

(2) It is directly derived by the differential inequality [

For the classical Solow Model, we have the following lemma.

The right maximal intervals of the solutions,

The solution of the initial value problem

Let

If

If

For any given

Since

Let

From the above theorem, we have following lemma.

If

If

For any given

Denote that

Let

Since

The generalized exogenous variables

If the generalized variables

Assume that the first inequality of (

Without loss of the generality, we assume that the first inequality above holds and we have

Similarly, we can prove the case that the second inequality holds. This completes the proof of the lemma.

If one of the generalized variables is persistent oscillation and the other converges, then the solution of the differential equation (

Assume that

If the solution

Let

Since

If generalized variable

Since

For any given

From Lemma

If one of the generalized variables is periodical oscillation and the other converges, then the solution of the differential equation (

If the intensive productive function is the Cobb-Douglas productive function, then the differential equation (

Let

Therefore, the solution of (

When

Suppose that

Let

From

Assume that the labor participation is positive constant, that is,

Let

If

Theorem

One of the most distinct characteristics in population change is the demographic transition, which has occurred in almost all developed countries and most developing countries [

The growth of the economy which has undergone the demographic transition is stable and its per capita capital converges to the steady state of the economy with the zero population growth rate.

From Theorems

If

Corollary

If

If the population growth is persistent oscillation of an economy, then its economic growth is not stable.

There are two major reasons that result in the change of the labor force participation: one is the changes in the age structure and the other is the population aging [

From Theorems

The extension interval of the solution of the differential equation (

If

The economy with higher labor force participation has higher per capita capital than that with lower labor force participation under the same other conditions; furthermore, the per capita capital of an economy with higher labor force participation tends stably to a higher steady state when the labor force participation tends to a stable level.

One of the distinct characteristics in population aging is the decline of the labor force participation [

The economic growth of an economy is slowed down by population aging.

If

If the labor participation of an economy is persistent oscillation, then its economic growth is not stable.

For the variable population growth rate, (

From Theorems

Assume that the saving rate varies with time, the population growth rate is a constant

From Theorems

The extension interval of the solution of the differential equation (

If

The economy with higher saving rate has higher per capita capital than that with lower saving rate under the same other conditions; furthermore, the per capita capital of an economy tends stably to a higher steady state when the saving rate tends to a higher stable level.

If

If the saving rate of an economy is persistent oscillation, then its economic growth is not stable.

The Hicks neutral technological production function [

Under this production function, the Solow-Swan model turns into

If

If

The economy with higher technological level has higher per capita capital than that with lower technological level under the same other conditions; furthermore, the per capita capital of an economy tends stably to a higher steady state when its technological level tends to a higher stable level.

If

If the technological level of an economy is persistent oscillation, then its economic growth is not stable.

In the case of Solow neutral technology (capital augmenting technology) [

For the Harrod Neutral technology [

From Theorems

The extension interval of the solution of the differential equation (

The special case of this theorem has been proved by Zhou et al. [

If

Let

Let

If

If

(1) It is directly deduced by Theorem

(2) Since

Let

By Lemma

This theorem implies that a developing economy can catch up and surpass a developed economy provided it maintains a higher technological growth rate than the latter one.

(1) An economy with lower technological growth rate has higher per capita capital of effective labor than that with higher technological growth rate under the same other conditions.

(2) The per capita capital of an economy with higher final technological growth rate will excess that with lower final technological growth rate.

If

If the technological growth rate of an economy with the labor-augmenting technological progress is persistent oscillation, then its economic growth is not stable.

From the analysis in Section

In Section

The other demographic factor affecting economic growth that we inquired is population aging. The distinct characteristic of population aging is that the labor force participation rate declines when the total population is stable. From the analysis in Section

The third demographic factor is unstable population growth rate and unstable labor force participation rate. If one of them is persistent oscillation, then the economy presents long-term fluctuation.

The effect of the saving rate change on economic growth is discussed in Section

Three types of variable neutral technology with time are put into the model to analyze their effects on the economic growth. It is obtained that the economy with higher technological level (Hicks neutral technology or Solow neutral technology) grows faster than that with lower technological level. The per capita capital of an economy tends to a stable level if the technological level tends to a stable level and the economy presents long-term fluctuation if the technologic level is persistent oscillation.

For the Harrod neutral technology, we show that the per capita of an economy with higher technological growth rate will excess that with lower technological growth whatever how high initial per capita capital the latter had and how high technological growth rate in the early stage the latter had. This result implies that a developing economy can catch up a developed economy provided it maintains a higher technological growth rate than the latter in long term.

If the technological growth rate is persistent oscillation, the economic growth is not stable and the economy presents long-term fluctuation.

Let

Let

Let

Let

If the given system is autonomous, the reference to

Otherwise the solution

If a solution is stable for

Let

By Gronwall's inequality, we obtain

The authors declare there is no conflict of interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation of China (70871094, 71271158).