A Generalized Solow-Swan Model

and Applied Analysis 3 where k∗ i , i = 1, 2, are the nonzero solutions of a i f(k) − b i k = 0, i = 1, 2. Since lim t→∞ a(t) = a, lim t→∞ b(t) = b, there exists T 1 such that a 1 < a(t) < a 2 , b 2 < b(t) < b 1 , t ≥ T 1 . Let ?̃? i (t), i = 1, 2, and ?̃?(t) be the solutions of (6) and (8) with the initial ?̃? 1 (0) = ?̃? 2 (0) = ?̃?(0) = k(T 1 ); then, from lim t→∞ ?̃? 1 (t) = k ∗ 1 , lim t→∞ ?̃? 2 (t) = k ∗ 2 , and ?̃? 2 (t) < ?̃?(t) < ?̃? 1 (t), t > 0, there exists T 2 such that


Introduction
In the classical Solow-Swan model, saving rate, technological level, capital depreciation, and population growth rate are assumed to be fixed positive constants [1][2][3].However, they vary in the process of the economic growth and appear in different forms in the different periods.The growth rate of population presents an inverted-U form in the demographic transition [4][5][6][7][8][9], the growth of technology appears in the Sshape in some periods [10], and the saving rate varies with the age structure [11,12].
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 [0, ∞) and a comparison theorem is provided.The solution of the differential equation is Lyapunov asymptotically stable when the generalized exogenous variables converge, which implies that the short-term exogenous shocks have no substantial impact on the long-term economic growth.
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 [2].The saving rate is assumed to be a variable of time in this paper and it is proved that the economy with higher saving rate will grow faster than that with lower saving rate and the economy presents fluctuation when the saving rate is persistent oscillation.

The Generalized Solow-Swan Model
where , , and  are saving rate, population growth rate, and the technological change rate, respectively, and () is the intensive production function satisfying In this paper, we consider the following nonautomatic differential equation: where (), () are continuous functions on [−, ∞) and satisfy and  are positive constants.
(2) It is directly derived by the differential inequality [13].

The Right Maximal Interval of the Solution.
For the classical Solow Model, we have the following lemma.
From the above theorem, we have following lemma.

Oscillation
Definition 7. The generalized exogenous variables () (or ()) are called persistent oscillation if, for any given  > 0, there exist , and one of the following inequalities holds: min Lemma 8.If the generalized variables () are persistent oscillation, then, for any given constants , ,  > 0, there exist  1 > 0 and   >  such that one of the following inequalities holds: Proof.Assume that the first inequality of ( 14) holds; then one of the following inequalities holds: min Without loss of the generality, we assume that the first inequality above holds and we have Taking Similarly, we can prove the case that the second inequality holds.This completes the proof of the lemma.

Theorem 9. If one of the generalized variables is persistent oscillation and the other converges, then the solution of the differential equation (8) does not converge.
Proof.Assume that () is persistent oscillation and lim  → ∞ () = .
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 [11].The population growth rate in demographic transition appears in an inverted-U form and can be described by a function of time , satisfying Corollary 15.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 1 and 9, we have the following.
Corollary 16 implies that an economy with higher population growth rate has lower per capita capital and consumption than that with lower population growth rate.
Corollary 18.If the population growth is persistent oscillation of an economy, then its economic growth is not stable.

Changeable Labor Force Participation.
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 [16].Here, we suppose that the aggregate population is stable; that is, () is a constant and (, ) is the Cobb-Douglas production function    1− ; then () = 0 and (26) turns into where () =   .From Theorems 1, 3, 4, 6, and 9, we have the following.
Corollary 21.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 [16] and we have the following.

Corollary 22. The economic growth of an economy is slowed down by population aging.
Theorem 23.If () is persistent oscillation, then the solution of (31) does not converge.

Corollary 24. If the labor participation of an economy is persistent oscillation, then its economic growth is not stable.
Remark 25.For the variable population growth rate, (31) becomes From Theorems 1, 3, 4, 6, and 9, we derive that the economic growth speeds up when the labor force participation rate increases and the population growth rate declines.Therefore, there exists "demographic dividend" in the late stage of demographic transition, in which the population growth rate declines and labor force participation rate increases.

Variable Saving Rate
Assume that the saving rate varies with time, the population growth rate is a constant  0 , and 0 <  1 ≤ () ≤  2 < 1; then the classical Solow-Swan model changes into From Theorems 1, 3, 4, 6, and 9, we have the following.
Corollary 28.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.
Theorem 29.If () is persistent oscillation, then the solution of (35) does not converge.
Corollary 30.If the saving rate of an economy is persistent oscillation, then its economic growth is not stable.

The Hicks Neutral Technology.
The Hicks neutral technological production function [1] is given by where () is an index of the state of the technology, and Ṫ ≥ 0.
Under this production function, the Solow-Swan model turns into which is a special case of the generalized Solow-Swan model (3).Therefore, from Theorems 1, 3, 4, 6, and 9, we have the following theorems and corollaries.
Theorem 31.If () is bounded and (0) > 0, then the extension interval of the solution of the differential equation then the solution of the differential equation (39) is Lyapunov asymptotically stable and converges to the nonzero equilibrium of the differential equation where  = lim  → ∞ ().
Corollary 33.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.
Theorem 34.If () is persistent oscillation, then the solution of (39) does not converge.
Corollary 35.If the technological level of an economy is persistent oscillation, then its economic growth is not stable.

The Harrod Neutral Technology.
For the Harrod Neutral technology [1], the production function is given by  = (, ()), which is also called the labor-augmenting technological progress, and the Solow-Swan model becomes where  = /() is the capital stock per unit of effective labor and () = Ȧ / is the technological growth rate.
Remark 41.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.
Corollary 42. (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.
Theorem 43.If () is persistent oscillation, then the solution of (43) does not converge.
Corollary 44.If the technological growth rate of an economy with the labor-augmenting technological progress is persistent oscillation, then its economic growth is not stable.

Summary
From the analysis in Section 2, we see that the stability of the nonautomatic differential equation depends on the generalized exogenous variables.If they converge, then the solution of the equation is Lyapunov asymptotically stable and does not converge if one of the generalized exogenous variables is persistent oscillation.Therefore, the economy described by the model stably grows when the generalized exogenous variables tend to a stable level and presents fluctuation when one of the generalized exogenous variables is persistent oscillation.
In Section 3, we analyze the effect of the demographic factors on the economic growth.One demographic factor is the change of population growth rate in the period of demographic transition.There does not exist substantial effect on the economic growth in long term for the population growth rate tends to zero after the demographic transition.However, the economic growth speeds up in the later period of the demographic transition in which the labor force participation rate rises and the population growth rate decreases.This implies that the "demographic dividend" appears in this period and theoretically confirms the evidence provided by Bloom et al. [11] through empirical analysis.
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 3.2, we see that population aging slows down the economic growth.
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 4. The per capita capital of the economy with higher final saving rate will excess that with lower final saving rate.Under the same initial per capita capital, the per capita capital of the economy with higher saving rate is bigger than that with lower saving rate in whole period of economic growth.If the saving rate is persistent oscillation, then the economy presents long-term fluctuation.
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.

Theorem 11 .
If one of the generalized variables is periodical oscillation and the other converges, then the solution of the differential equation (8) does not converge.Example 12.If the intensive productive function is the Cobb-Douglas productive function, then the differential equation (3) becomes k =  ()   −  () .