Universidad Michoacana de San Nicolas de Hidalgo From the SelectedWorks of Teresa M. G. Da Cunha Lopes Fall September 17, 2013 University population dynamics as a recontracting allocative proccess Teresa M. G. Da Cunha Lopes Jesús Martínez Linares Available at: https://works.bepress.com/teresa_dacunhalopes/4/
University population dynamics as a recontracting allocative proccess Jesús Martínez-Linares and Teresa María Cunha Lopez Escuela de Ciencias Físico-Matemáticas. Edificio B. Ciudad Universitaria. Apdo 2-71. Universidad Michoacana de San Nicolás de Hidalgo. 58060-Morelia, Michoacán, México. Facultad de Derecho y Ciencias Sociales Universidad Michoacana de San Nicolás de Hidalgo. 58060-Morelia. Michoacán, México. Keywords: Nonlinear Dynamics, Strategic action, Competing technologies Abstract. The recent structural change of economic growth caused by the development of new technologies have placed a leading role on positive-feedback hightech companies in modern economy. University institution can also be considered a high-tech economic agent, producing qualifyed specialist than can couple non-linearly to the market demand. Here, we explore the effect of positive-feedback on the population dynamics of University students. We find a bifurcation in the market share stationary probability distribution that can serve as a signature of complex behaviour. 1. Introduction The theory of Nonlinear Complex Systems has been a successful problem-solving approach in natural sciences, ranging from laser physics to biology. On the other hand, the Social Sciences are recognizing that social structures dynamics are global, non-lineal and complex too. Local changes in the economic or political system can cause a global crisis. Linear thinking, where large effects are ascribed to large causes is therefore obsolete. Recent years have witnessed a revival of interest in the application of Nonlinear Dynamics to Economics, which allows to understand economic fluctuations as a deterministic phenomenum, endogenously created by the non-linear interaction of market forces, technologies and preferences. A main reason for the current relevance of non-linear models in Economics is given by the recent structural change of economic growth caused by the development of new technologies. The traditional theory of Economics assumes negative feedback in the interaction of agents, so that Economics (jesusmartinezlinares@hotmail.com)
University population dynamics as a recontracting allocative proccess 2 is stabilized by counter-acting each reaction caused by an economic change. Economies with a negative feedback of decreasing income, such a traditional agriculture, leads to a single equilibrium point corresponding to an optimal result with respect to the particular surroundings. On the other hand, sectors depending on high technical knowledge earn increasing income (self-reinforcement). Availability of high-tech product generates new necessities and new schemes of relations. Thus modern high-tech industries must be described in dynamical models as generating positive feedback of increasing income. Along these lines, University must be considered as a high-tech industry itself. It transforms source materials (unqualified bachelors) into high-tech products (specialized profesionals) able to induce and modify the market by themselves. The mean innovative idea behind this paper is to treat University as a positive feed-back economic agent. Thus, non-linear dynamical models are needed to explain its interaction with the rest of surrounding agents, i.e., to treat University as a complex evolving system. Complex Theory is devoted to explain the emergence of certain macroscopic phenomena via the nonlinear interaction of the microscopic components. University institution can be viewed as an emergent organizational structure of society with two correlated main goals: the generation of knowledge, nurturing science and technology, and transmission of knowledge as a training system, i.e., the production of specialized human resources to supply the employment market demand. We will study University as a labor force consumer, therefore competing to attract people from the labor market by means of selling them expectations of job improvement. This expectations are amplified or deamplified depending on the success of the graduates, which, in turn, change the job market. The non-linear character of University population dynamics is based on this competitive behavior. Autocatalytical dynamical systems, i.e. systems with local positive feedback, tend to posses multiplicity of asymptotic states of emergent structures (multiple equilibria). Random fluctuations can push the initial state to select one of these atractors. A number of properties of nonlinear complex systems are to be explored also for University Dynamics. In this paper we look for a signature of such a complex behaviour in the population dynamics of university students. This paper is organized as follows. In section II, we describe the system model, treating the population dynamics as a recontractive allocative process between market shares. Section III is devoted to present the results assuming a given form of the nonlineal interaction. Finally, we end in Sec. IV with a summary and a discussion of the results. 2. System Model As a first step, we model the coupled population dynamics as a re-contracting allocative process. This class of models allow re-contracting within the market once it has formed, so we can take into account gains and losses from and into University. In order to set up concepts and notations, we consider in a first approximation the birth-death problem of population dynamics in University as a re-contracting process of
University population dynamics as a recontracting allocative proccess 3 economic markets. We follow a generation of fixed-size 2N bachelors divided into two categories: university students and others. These choices are not irreversible since a member of each category can re-contract into the other. Similar systems of fixed-size, Markov type transitions have been considered in Genetics [5], Epidemiology, Laser Physics [6], and voting process in Sociology [7]. Therefore, following [7], we consider a total market of 2N bachelor graduates, where N + m choose to be university students (A-category), and N m do not (Bcategory). Let P AB (m) be the probability per unit time that an university student gives up and leaves A-category into B-category. Conversely, P BA (m) will account for the reverse process. Note that the transition of units between categories depends on the market share m, and thus, self-reinforcement or self-inhibition are possible. Explicit dependence of transition probability of time, will be considered in the next section. Then the probability P (m, t) of finding the system at state m, at time t, evolves as P (m, t + 1) = P (m, t) (1 P AB (m) P BA (m)) + P (m + 1, t)p BA (m + 1) + P (m 1, t)p AB (m 1), (1) which yields the master equation dp (m, t) dt = P (m + 1, t)p BA (m + 1) P (m, t)p BA (m) P (m 1, t)p AB (m 1) P (m, t)p AB (m). (2) We can go to the continuous limit with the help of the variable x = m/n, defined in the continuous interval ( 1, 1) as N. Defining the quantities ɛ = 1, P (x, t) = NP (m, t), N R(x) = P AB(m) P BA (m), N Q(x) = P AB(m) + P BA (m), (3) N and taking the limit N, Eq. (2) can be rewritten as the one-dimensional Fokker- Planck equation P (x, t) t = x R(x)P (x, t) + ɛ 2 Q(x)P (x, t). (4) 2 x2 The equation above represents a Markovian diffusion process. So permanent lockin to one market position is not possible. In general the system displays punctuated equilibria in the neighborhood of local maxima and transition among them.
University population dynamics as a recontracting allocative proccess 4 3. Results Now, we assume an explicit form for the transition probabilities, namely P AB (m) = ν e δ+κm (N m) (5) P BA (m) = ν e δ κm (N + m), (6) where ν denotes the frequency of switches, δ allows for a preference bias and κ corresponds to a fashion effect produced by a nonlinear coupling between offer and demand. An analitycal solution for the stationary state of the birth-death master equation given on Eq. (2) can be obtained as [8] P S (m) = P S ( N) m i= N P BA (i 1). (7) P AB (i) where P S ( N) is fixed by normalization. The stationary probability distribution of market share given in (7) is plotted in Fig. 1 as a function of the nonlinear couplig κ.in the absence of κ, a larger population of one category, increases the chances of switches to the other. This reflects the effect of saturation of labor market for specialized (A-category) and non-specialized (B-category) units. Hence, the overall effect is a centralizing tendency in the probability distribution. This is offset by the conformity effect which reinforces a concentration of one type. When κ is small centralization dominates and the stationary distribution is unimodal. As κ increases, the distribution bifurcates and becomes bimodal. In this case, the market lingers at prevalence of one-type of category with intermitent transitions to prevalence of the other. 4. Conclusions In this paper we have proposed to model University as a positive-feedback economic agent. We have found that a signature of the nonlinear character of such a system can be probed in the stationary probability distribution of market share. After a critical value of the nonlinearity is reached the system bifurcates and the probability distribution becomes bimodal. We have found multiple equilibria for such a system, which shows the rich dynamical behaviour of ths systems. This result illustrates the difficulty to predict the evolution of university population when nonlinearities are introduced into the system. 5. References [1] Klaus Mainzer, Thinking in Complexity (Springer Verlag, Berlin, 1997). [2] R.M. Goodwin, Chaotic Economic Dynamics (Clarendon Press, Oxford, 1990). [3] P.W. Anderson, K.J. Arrow and D. Pines (eds), The Economy as a Evolving Complex System. Santa Fe Institute Studies in the Science of Complexity, vol. 3 (Addison-Wesley, Redwood City, 1988). [4] H.O. Peitgen, H. Juergens, D. Saupe, Chaos and Fractals, New Frontiers in Science. (Springer Verlag, NY, 1992).
University population dynamics as a recontracting allocative proccess 5 [5] W.J. Ewens, Mathematical Populations Genetics (Springer Verlag, NY, 1979). [6] H. Haken, Synergetics (Springer Verlag, NY, 1978). [7] W. Weidlich and G. Haag, Concepts and Models of Quantitative Sociology (Springer Verlag, NY, 1983). [8] C.W.Gardiner, Handbook of Stochastic Methods, pg. 237 (Springer Verlag, Berlin, 1985). Figure 1. Stationary market share probability distribution as a function of the nonlinear coupling κ. A bifurcation to a bimodal distribution can be observed as κ increases.