Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation.


 Gwendolyn Carpenter
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1 Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation. 1
2 Without any doubts human capital is a key factor of economic growth because labour productivity has a strong impact on output. And the workers effectiveness is determined by their knowledge and skills. For the first time human capital was included in the neoclassical model of economic growth in the seminal work by R. Lucas in In the model human capital is a counterpart of the scientific and technological progress. The distinction in manner of Lucas specification from others, for example, the Uzawa s model) is the consideration of both internal (describing the direct increase in production efficiency, associated with an upgrade of workers professional skills) and external effects (which characterizes the human capital average value in the economic system in whole) of human capital. Human capital accumulation in Lucas model takes place within the peculiar educational sector of economics. This approach explains the difference between growth rates in different countries. Lucas model is the basis for further development and generalization of this field of economic growth theory. 2
3 This work presents the generalized mathematical model of endogenous growth with the human capital accumulation. There are two sectors in the model economy: production and educational sectors. The human capital h(t) of representative agent is formed in the second one. Let us suppose that the output sector externalities are determined not only by average level of human capital h a (t) (as in the original Lucas model), but also depend on the average share of nonleisure time which an agent spends in production u a (t). Furthermore, we assume that the externalities have an impact on the educational sector unlike the classical Lucas model. So this effect depends on both the average level of human capital h a (t) and the fraction of time devoted to human capital formation (education and trainings). Thus, it is assumed that the agent is interested in spending the same part of time for production activity as the average one at the market. Under these assumptions, the equation of dynamics of physical capital can be written in the following form (1) where A(t) is a function describing the exogenous technological progress (  the rate of technological progress is constant); K(t)  physical capital; N(t)  labor force; u(t)h(t)n(t)  the effective labor force (or internal effect of human capital); c(t)  consumption per capita;  the share of physical capital;  positive parameter; K  the depreciation rate of physical capital. The equation of dynamic of the human capital accumulation is given by (2) where is a positive technological parameter; p, s, q and r  nonnegative parameters (elasticities); h  depreciation rate of human capital. (It should be mentioned that equation (2) reflects the fact that at the microlevel the agent examines the functions u a (t) and h a (t) as given exogenously, so that production function of human capital in the righthand side of equation (2) demonstrates the presence of the diminishing scale effect at the microlevel and the lack of the scale effect at the macrolevel.) The initial conditions should be added to the system. It is assumed that the agent chooses his consumption level c(t) and the share of his active time spent in production u(t) in order to maximize a utility function of the form (3) where is the subjective rate of time preference, and 1/ is the constant intertemporal elasticity of substitution. The optimal control problem related to the economic growth model with the human capital accumulation (1)  (3) is the search of control variables c(t) and u(t) to maximize the objective functional (3) on admissible trajectories {K(t),h(t)} of the dynamic system (1), (2) subject to conditions (4) 3
4 There are at least two economic interpretations of conditions (4): 1. The social planner s problem. There is a hypothetical social planner who chooses an optimal path (trajectory) of the economic system from the standpoint of society as a whole, initially possesses the complete information about its development, and is able to influence on its allcomposite parts. Therefore we can assume that conditions (4) are fulfilled initially. 2. The competitive equilibrium. We consider the situation where firms and households do not have the complete information about the economic system development, but expect that the accumulation of human capital and the share of nonleisure time which an agent spends in production u(t) will follow known in every moment of time functions u a (t) and h a (t), which are exogenously given (from firm s point of view) and aren t affected by an agent. In these conditions, economic agents solve the optimization problem. The economic situation will be in equilibrium if the expected and the actual behaviors coincide, i.e. if equalities (4) are carried out. For each of these cases the appropriate optimization problems should be considered. 4
5 Furthermore we can write the generalized problem for them. After some transformations and transition to neoclassical variables per capita (k(t)=k(t)/n(t)) we get the following equations: (5)(8) and transversality conditions: (9) Substitution 1 =q, 2 =0, 3 =p gets the social planner s problem and 1 =1, 2 =, 3 =1  the problem of the competitive equilibrium. 5
6 It can be proved that system (5)(8) has first integral of such form (10) And it also has the second first integral for particular case when is equal to. The existence of the first integral allows us to reduce the system (5)(8) by one dimension. 6
7 Let us construct the system s balanced growth path (BGP) from (5)(8): the particular solution {k(t),h(t),c(t),u(t)} for which the growth rates of all variables are constant. Moreover u(t) is constant on BGP. We define these rates of growth as with appropriate index. They are solutions of the following simple algebraic system: (11) When the determinant of the system doesn t equal to zero, the system (11) has a unique solution and the balanced growth path is given by (12) If we consider our result in comparison with Lucas one, the difference between equilibrium growth rates of human capital is characterized by the equation: (13) Obviously the output sector externalities are quite small, and thus this difference is more than zero. In other words human capital in generalized model is smaller than in Lucas one. While too big human capital is one of the most important issues for criticism of the Lucas model. 7
8 The next step of our investigation is the reducing of the system by one dimension for simplification of analytical study. We can define the new variables, which are stationary on the BGP: (14) Some manipulations leads to: (15)(18) Here we can see that the system (15)(18) has the closed subsystem (15)(17) further which will be studied separately. 8
9 If we want to find the unique nonnegative steady state of the reduced model we should find the solution of the presented system. And you can see it at the slide as a function of the set of parameters : (19) The parameters {x e,q e } should are positive and u e should belong to in the range from 0 to 1. These conditions impose some additional restrictions to the set of parameters (lowercase  строчная) and allow us to select a subset BGP (capitalпрописная) in a space of admissible parameters. This subset is not empty in general case. 9
10 Let s talk about the local stability properties of the BGP and regions of the parameter space with different types of equilibrium, defined by eigenvalues of Jacobian of the reduced system (15)(17). The Jacobian evaluated at BGP is given by: (20) The type of the equilibrium point and, correspondingly, its stability, is defined by the amount of eigenvalues at the right half plane of the complex plane. Because of sufficient complexity of analytical results further study was realized numerically with MatLab software. 10
11 Our model can be used for the investigation of economic dynamics of developed counties, since human capital of such economies has decreasing return to scale. That is why it is expediently to research the dynamic of the system for the set of parameters which are equal or near to real values for developed countries (e.g., Germany or USA). It is worth to note that the problem of determination of real values of parameters for any economic growth model is one of the most urgent questions. In this study we use (taking into account some remarks) these assessments of parameters which is built on the basis of assessments from these works. The other parameters vector characterizes the output sector externalities and the scale effect in educational sector. Let s see how dynamic peculiarities of the system (15)(17) depend on vector choice for the competitive economic system. 11
12 Let us consider subsets (k), where k is amount of eigenvalues in the right half of complex plane. This figure illustrates the general view of the fragment of the set BGP and its subsets (k). It should be mentioned that all possible variants of steady states types are realized in this problem. 12
13 We can see that the model demonstrates the indeterminacy effect the situation when a trajectory converging to BGP is not unique. This situation takes place when there are one ( (1) ) or all three ( (3) ) eigenvalues lays at right half of complex plane or all three eigenvalues lays at left half ( (0) ). 13
14 For (2) set two eigenvalues lays at right half plane. Here the twodimensional manifold will be locally unstable and onedimensional manifold locally stable. That is why we could define initial conditions definitely, and so the local determinacy is observed. 14
15 In our model the arrangement of sets (2) and (0) has a very important feature they have a common boundary. Let s consider only one result which shows that the dynamics of the investigated system can be quite complex. This example takes into account the transition from (0) to (2) across curve H along the segment of a vertical line p=0.9 when =0.4 (decreasing scale effect on years of schooling, near to Lucas value). Let s look at the changes of the phase space structure near stationary point depending on q value, which characterize the scale effect of human capital level in educational sector. 15
16 Calculations show that there are subcritical AndronovHopf bifurcation and direct and inverted fold bifurcations of limit cycles in the system. Thus there are three limit cycles (two of which are unstable) near equilibrium point or two limit cycles (stable and unstable), and the one unstable limit cycle when q>q 0. When q<q 2 equilibrium point is unstable. (Stable objects are depicted by a firm line, unstable by a dotdash line). 16
17 Next two slides present as an example some trajectories near a steady state. They give a clear submission of the phase space structure of our system in the case of coexistence of three limit cycles. 17
18 Within the framework of the economic growth theory the balanced growth paths are peculiar turnpikes of economic development representing an equilibrium evolution of economic systems. In most cases they are unstable. Therefore, the fact that a stable periodic motion can exist in the neighborhood of the unstable balanced path ( a turnpike of development ) is very important and positive for the theory of economic growth. Essentially, it means that although the equilibrium evolution of the economic system is unrealized along the turnpike of development (because of its instability), some other oscillatory trajectories are quite feasible, sustainable and have the same trend as the turnpike of development. Thus, the evolution of the economic system can be realized almost along the turnpike of development within the bounds of a stable educational economic cycle. 18
19 The graphics for initial variables for trajectories relevant stable limit cycle are presented on the slide. 19
20 Thank you for your attention 20
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