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ECONOMIC GROWTH MODELING IN THE THEORY OF CYCLIC DEVELOPMENT
Abstract
Nowadays in the economy full of crises and general instability there appears the need to broaden and deepen the theory of cyclical development and economic growth. The development of the theory of economic growth shows that in the most general sense the economic cycle can be explained by different effects of feedback and multipliers acting in the economic system. Factorial structure of the growth is open enough, it is not limited by the interaction of labor and capital, or new technologies. The purpose of this article is the analysis of the main approaches to the modeling of economic growth. Under the conditions of the instability of the modern economy theories have greater diagnostic power, when they take into account a nonlinear nature of interrelation of development factors, the decision of which forms cyclic trajectories and also models that are directly based on the use of functions, reflecting the cyclic nature of economic development. To this end special attention is paid to dynamic models of economic growth.
Keywords
economic growth, modeling of economic growth, cyclical development
AUTHOR
Inna V. Babenko
Associate Professor Southwest State University. 94, 50 let Oktyabrya, Kursk, 305040, Russia.
E-mail: babenkoinny@gmail.com
Introduction
In modern economic literature the interest in the problems of the unevenness economic development and its cyclic recurrence has been increased. Economic development is formed under the influence of economic reasons, and because of the influence of external to the socio-economic system of factors. Economic development takes into account the institutional, legal and religious peculiarities of education and accumulation of human capital, labor market and capital. It includes the combination and interaction of economic growth with fluctuations, differing in size, amplitude and duration. The main component of economic development is economic growth. The
problem of economic growth has always been in the center of economists' attention. The economic growth is a result of the influence of long-term factors of the economic system development- savings level, labor force growth and technological shifts. In today's economy, saturated with economic crises, descending and ascending waves of different economic cycles and general instability, there appears a need to expand and deepen the theory of economic growth, its capacity in describing the descending power of feedbacks action in the economy. The definition of the uneven development of the cyclic economy means that the periods of growth are alternating with the periods of decline in production. The cyclicity is limited by these two extreme states. In economic literature fluctuations of economic activity are characterized as economic cycles and business cycles, which show a process of economic transition from one development stage to another. The sequence of changes like growth - recession development - stagnation has repetitive nature, but not periodic. The impulses, generating successive changes of the ups and downs naturally appear during the reproduction and are generated by the system itself. At the stage of nucleation of the cycle, the emerged fluctuation in structure needs initiates the appropriate changes in the distribution structure of productive forces. The category economic growth operates with the categories related to the productive forces of society and it is a narrower notion than economic development.
Research Methodology
Economic growth is reflected in the quantitative and qualitative improvement of the social product. Economic growth of social-economic system characterizes its activity and it is a derivative from the production potential of social-economic system. The production potential of social-economic system is a cumulative capacity of the system to carry out a productive, economic activity and the development of production, turn out products, provide with social needs etc.. I.e. the production potential of social-economic system is determined by its resources, producer's goods, labor potential, as well as accumulated wealth. The production potential can be considered as a potential volume of output that can be made with the optimal use of available resources. The resources that make up the production potential can be considered as factors of economic growth. In this case the main factors of economic growth are: natural resources, human resources, physical capital and production methods. At the same time operation factors of economic growth are not fixed theoretically output indicators, they depend on the achieved level of economic development in the country, the existing market structures and a number of other conditions. In the growth stage a dynamic correspondence between changes in the structure of needs and production resources, consumption and production is provided, while the growth rate of quantitative structural shift stabilizes at a peak.
Analysis of the current state of the economic growth theory and practice shows that there are two main directions of its modeling. The first is connected with the construction of production functions, linking economic growth with the dynamics of the factors of production. In using production functions the economy is considered as holistic unstructured measure, where at input resources are coming, and at the output there is a result of the economic functioning in the form of gross output. Here resources are treated as arguments, and gross output as a function.
The second direction involves the modeling of production and consumption based on multi-sector models. In this case, the economy is structured and consists of a finite number of sectors or clear industries producing one or more types of products. Here, the economic growth is modeled on the basis of the supply-demand balance factors in the economy.
In scientific literature there are many different models of economic growth based on production functions.
The studies of the developers of the first economic growth models were based on Leontief production function with two factors and fixed input-output coefficients costs.
Y = min{ aK, bL}
(1)
where a and b - the average capital and labor productivity.
This function assumes the capacity evaluation for growth in output in terms of the labor supply and capital resource availability separately.
Another simple production function is linear one
Y (t) = A(t)(akKK (t) + bLL(t) (2)
where Y(t) - coefficient of multifactor productivity;
K(t), L(t) - costs of capital and labor;
aK, bL - weight numbers calculated by comparing the costs of labor and capital in the production of the final product as an average measure during the study period, when aK+ bL=1.
In later studies, economic growth appears as a result of the combined effect of three main factors: labor resources, capital and technological progress. As a rule, modified Cobb-Douglas production function was used in such researches.
In conditions of instability theories, which take into account the non-linear nature of the interaction between development factors, have more diagnostic power. Their solution forms the cyclic trajectories as well as models, directly based on the use of functions, reflecting the cyclical nature of economic development, that in fact is transition considering static distribution of income to the dynamics of economic growth.
The main ideas of Dynamic Economics, well-founded by Harrod and Domar are based on Keynesian income distribution system according to which households received income Y, make demands C for consumer goods and services and determine the level of savings S. With the assumption that investments are not only a source of income, but also means of increasing the economic capacity, Harrod took investments equal to capital gains and moved from considering the static distribution of income to the dynamics of economic development, based on the following considerations:
I = dK / dt
(3)
The connection between aggregate demand Y with investment demand I in the simplest model of the economy is given by:
Y =1 / ^ (4)
Where I = S, (5)
1/s - Keynesian multiplier;
^ = S / Y
The saving rate is less than unity as
Y = S + C (6)
From this, due to (3, 4) the economic expansion rate (G) can be given by
Gy = s y / V (7)
where
v = dK/dY represents a marginal capital intensity of release On the other hand due to (3)
GK = (dK / dt)/ K = sy /ky
(8)
where £ = K/ Y - the average capital intensity
So, if we assume that there is the equality of marginal capital intensity v = ky , then the optimal economic growth due to Harrod is given by
Gwr = Gk = (sy /ky) (9)
where an investment amount is equal to the product of the average capital intensity of release on its gain: I = kydY / dt
To sum up, households consume goods and lay aside savings, which become investments and then provide a capital gain. In turn, when this capital gain moves along the optimal growth trajectory the national product increases due to the equation (9).
According to Harrod the economic growth, described by equality (9), i.e. where growth rate Gy is the same as capital growth rate Gk , is optimal, because it is defined by propensity for save and technological limits, which are expressed by the average capital intensity ky. With such increase all economic agents are satisfied with that fact they have produced exactly that amount of product that is necessary.
Further development of the economic dynamics was found in neoclassical economics in Solow's work. He supplemented the Harrod-Domar's hypothesis by including the production function, thereby inputting the description of the production process explicitly. It allowed to consider the possible steps of replacing labor and capital and to link natural growth rate with an optimal gain rate for Harrod. Solow's consideration of the replacing possibility between labor and capital has led to the appearance of a new class models in which it turned out to be possible to explore the growth path where the movement was stable.
According to Solow in the differential form the connection between output Y, capital K and labor L set with the neoclassical production function Y=F(K,L), is represented as:
dY/dt = FKdK/dt + FLdL / dt ^q^
Where Fk u Fl are partial derivatives of F with respect to K, L resp.
The equation (10) can be rewritten in this way:
gy =akgk +vlgl (11)
where aK,aL - partial logarithmic derivatives F with respect to K u L resp. and
aK+aL= 1
It is important to note that although Solow unlike Harrod considered only the situation of full employment, this restriction is not essential for this model, and L refers to a demand for labor, no more than its supply.
According to (11), the growth rate of release is equal to a convex combination of optimal Harrod gain rate of output and the growth rate of labor, with coefficients equal to the elasticity of output to capital and labor, resp. Supposing the equality of aggregate demand (4) and supply F (L, K), substituting (8) into (11), the last equation can be given by:
GY = aKsY /kY +aLGL (12)
So, according to (12), if the growth rate of labor coincides with the natural growth rate, then the growth rate of release is equal to a convex combination of optimal Harrod growth rate and natural growth rate.
Motion along a trajectory, optimal to Harrod, is stable (at a fixed propensity to save sY), as the possibility of replacement labor for capital make economic growth in the long term growth limited by labor. At a constant rate of growth of Gl a trajectory optimal for Harrod took place and at some moment of time as a result of abrupt growth propensity to save sy turned out Gy<Gk and according to (11)
GL < GY < GK (13)
At the same time, as it follows from (8) that
Gy = GK + (dGK /dt)/GK -Gs ^^
where Gs - is the growth rate sy then subtracting the last from (11) we have
(dGK / dt)/gk = a k (gk - Gl ) + Gs (15)
or considering the propensity to save sY fixed,
(dGK/dt)/GK = -aL(GK -Gl) (16)
This implies that the capital growth rate Gk, and also Gy, due to (13) tends to Gl. Similarly, we can show that if during some time the following inequality holds Gy>Gk, then the growth rate of release Gy also eventually tends to Gl.
Solow was the first to make a conclusion that in the neoclassical approach the economic growth in long-term period is limited by the labor growth rate .This conclusion destroyed the theory based on the equality (7) that the problem of growth is in increasing of the propensity to save.
It is necessary to note that for this sustainable trajectory the capital growth rate coincides with the labor growth rate, and Solow called this trajectory stable. At the same time stable trajectory in this model is optimal for Harrod (Gy=Gl) and natural growth trajectory (Gy=Gl). Thus, under neoclassical approach trajectory is stable only when it is at the same time optimal growth trajectory for Harrod and natural growth trajectory.
T. Haavelmo made a great contribution to the study of the economic dynamics growth under the neoclassical approach. In 1954 Haavelmo proposed a model of economic growth based on Cobb-Douglas production function, according to which he describes an
economy's production of output as a function of the stock of capital (K) and the level of employment (N). Haavelmo model is:
L * L
— = a-B—, a,B> 0
L Y
(17)
where L и K - functions of time;
а, В - constant known functions of time.
Y = c *La *Kl-a (18)
где Y -production output; L - the level of employment; K - the stock of capital.
Expression (18) Y is a real output produced with a constant elasticity (a) with diminishing return of labor power.
Expression (17) can be represented as an autonomous growth proportional to the level (а) less the level of per capita income.
So, increase in growth takes place with increasing per capita income and it is limited by the parameter (a). Connecting the expressions (17) and (18) we get an expression (19) that reflects the dynamics of employed workers:
L = aL--L2-a
(c *K'-a) (19) where a is the parameter of the differential equation.
The expression (18) has two stationary solutions: unstable L * = 0 and
a *c* K
symptomatically stable L *2 = (———)1-a.
Having the constant stable solution a general solution can be the following:
(K(L(0))a1 )ea(a-1)t
L(t)=(-a--)a-1
K
a * c * K
If the initial condition is L(0)> (———)1-a, then L and K will be monotonically
a*c*K
decreasing, if L(0)< (———)1-a, then L and K will be monotonically increasing i.e. they
a*c*K a*c*K
will be closer to their unique stable values (———)1-a and K(———)1-a.
In 1980 г. Statzer developed a nonlinear model of economic growth on the basis of the previous model presented by Haavelmo.
In his work Statzer replaced the function trajectories L (t) with the orbit trajectories {L, t}, and differential operator with finite difference N T +1 - N т and got nonlinear firstorder differential equation, describing the behavior of employees in time.
Lt+1 = (1 + a)Lt -ßL2r (20)
K
Carrying out an analysis on the basis of equation (19) will determine the optimal level of employment, at the same time there is a need for further conversions:
r K (1 + a)xT1-
ß (21)
The equation (19) is written the following way:
X=1 =(a + a) xt(1 - x'ia)
where
(22)
r K(1 + a).71-
Xt = L( (p ))1-a
The equation (21) is a nonlinear first-order differential equation and its form allows to carry out the further dynamic analysis. To find the equilibrium state and the intervals for which there is convergence to equilibrium, it is necessary to use the properties of local bifurcations.
f(x) = (1 + a)*(1 - x1^) (23)
The solution of equation (23) gives two equilibrium points (24):
f (x) = x, or (1 + a)*(1 - xla) = x (24)
1
* _ * /- ^^ N 1
co = 0, = (--)1-a
1 + a (25)
After conversions (25) with the use of (21) we have
L* = 0, L* = (aK)l-a
^ 2 ß
(26)
The next step is to study the convergence of (21) at the equilibrium point l* .
At the beginning of the XX-th century Western economists noted that the growth rate of labor productivity exceeds the growth rate of capital intensity. We take into account only two factors - labor and capital, from the position of the factors theory of production it is difficult to explain such economic growth. Therefore, it was concluded that there is another factor that influences the economic growth. In growth models based on the neoclassical production function the economic growth in the long term is limited by an extensive factor (the growth rate of the labor resources) and the increased propensity to save by increasing the pace of economic growth in the short term does not lead to increase of the economic growth rate in the long term.
After Solow had established this fact, economists began to pay more attention to intensive economic growth factors, the main of which is the scientific and technical progress.
Most part of the work of neoclassicism in the50-60s of the 20th century was devoted to exogenous factors of scientific and technological progress and didn't consider the problem of forming parameters of scientific and technological progress. In the late 50's early 60's there appeared a discarding of the exogenous technological progress concept in favor of endogenous parameters. In the late 80's and early 90-ies there was a direction of modern Neoclassicism, which was devoted to modeling the impact of innovative activity on the technological changes taking into account the accumulation of human capital.
In general, the model of economic growth within the parameters of technological progress can be represented the following way:
Yt - A^K"lLna (27)
where Kt - physical capital ;
Lyt - the total amount of human capital involved in the production of output;
At - the total stock of ideas available in the economic system.
At the same time 0 <a <1 and o > 0. Notice that there are constant returns to scale in K and Ly holding the stock of ideas A constant, and increasing returns to K, Ly, and A together. This assumption reflects the now common notion that ideas are nonrivalrous or "infinitely expansible."
It is necessary to pay attention to the fact that there is a constant output K, and LY, and K u Ly, and also
Physical capital is accumulated by forgoing consumption:
Kt - SkY -dK, K0 > 0 (28)
The variable sKt denotes the fraction of output that is invested (sKt is the fraction consumed), and d > 0 is the exogenous, constant rate of depreciation.
Next, aggregate human capital employed producing output is given by
LYt — ht^Yt (29)
where ht is human capital per person and Lyt is the total amount of raw labor employed producing output. An individual's human capital is produced by forgoing time in the labor force. Letting h represent the amount of time an individual spends accumulating human capital,
ht - ^, 0 (30)
The final factor in the production of output is the stock of ideas, A. In the model, ideas represent the only link between economies; there is no trade in goods, and capital and labor are not mobile. Ideas created anywhere in the world are immediately available to be used in any economy. Therefore, the A used to produce output in equation (27) corresponds to the cumulative stock of ideas created anywhere in the world and is common to all economies.
New ideas are produced by researchers, using a production function like that in Jones (1995a):
A - ^^At^t ,
A > 0 (31)
where La is effective world research effort, given by
M
LAt = ^h itLAit
i=1 (32)
In this equation, i indexes the economies of this world, LAi is the number of researchers in economy i, and 0. World research effort is the weighted sum of the number of researchers in each economy, where the weights adjust for human capital.
According to equation (29), the number of new ideas produced at any point in time depends on the number of researchers and the existing stock of ideas. We allow 0 < A < 1 to capture the possibility of duplication in research: if we double the number of researchers looking for ideas at a point in time, we may less than double the number of unique discoveries. We assume 9 < 1, which still allows past discoveries to either increase (9 > 0) or decrease (9 < 0) current research productivity.
Conclusion. The model given above studies the dynamics of economic growth based on innovations. Growth in any particular country is driven in the long run by the implementation of ideas that are discovered throughout the world. In the long run, the stock of ideas is proportional to worldwide research effort, which in turn is proportional to the total population of innovating countries.
To sum up, the theory of economic growth includes the methodology of describing the economic cycle, which is represented in the works of neoclassical and Keynesian economic branches, where the multiplicative effects and akselerativ mutual influence functions of investments, incomes and consumption, generating economic growth are described. However, in the theory of economic growth these cyclic iterations are short term in nature, and models which describe them are designed just to show how the market zigzag dynamics generates an upward wave of comprehensive income. Thus it is implicitly assumed that growth is a function of the economic cycle. The impulse to the convergence of growth theory and cycles theory and the conjuncture was set by the theory of growth during the reaction of its representatives from the side of alternative directions of economic science.
On the whole, the development of the economic growth theory shows that in the most general form the economic cycle can be explained by the different effect of feedback and multipliers acting in the economic system. At the same time growth factor structure is widely open and it is not limited by the interaction of labor and capital, or new technologies. Moreover, the economic growth is generated at different levels of the economy, therefore, it is necessary to take into account the behavior of economic entities of different levels, with particular attention to the mesoeconomic structures.
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