MATHEMATICAL MODELS Vol. III - Mathematical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang

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1 MATHEMATICAL MODELS Vol. III - Matheatical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang MATHEMATICAL MODELING IN SOCIAL AND BEHAVIORAL SCIENCES Wei-Bin Zhang Ritsueikan Asia Pacific University, Juonjibaru, Beppu-Shi, Oita-ken, Japan Keywords: Social and behavioral sciences, Optiization, Operations research, Job aenity, Gae theory, Differential equations, Chaos, Altruis, Coparative statics analysis, Nash equilibriu, Neoclassical growth theory, Logistical for, Downs odel, Huan capital, Malthus population theory, Public good, Stochastic process, Utility function Contents 1. Introduction 2. Optiization Theory Job Aenity and Moonlighting 3. Operations Research The Job Assignent Proble 4. Gae Theory Political Copetition 5. Differential Equations Econoic Consequences of Altruis 6. Chaos Theory Population Dynaics Acknowlegeents Glossary Bibliography Biographical Sketch Suary This chapter presents soe applications of atheatics in social and behavioral sciences. It provides a general overview of atheatical approaches to different social and behavioral probles. Since the literature on the topic is vast, we can illustrate only a few areas of applications in this chapter. We introduce applications of a few branches of atheatics including optiization theory, operations research, gae theory, differential equations, and chaos theory. To depict how these theories can be applied to social and behavioral sciences, in each case we solve one siple exaple of a proble of social/behavioral sciences by applying one of these branches of atheatics. Firstly, we apply optiization theory and coparative statics analysis to decide the optial behavior of workers with ultiple interests and exaine the ipact of job aenity on oonlighting. Secondly, operations research is applied to solve the job assignent proble. Thirdly, we exaine political copetition, applying gae theory. Then, as an exaple of applying differential equations, we investigate long-run socioeconoic consequences of altruis and explain under what conditions the opposite opinions of Malthus and Keynes on the issue are valid. Finally, chaos theory is applied to identify socioeconoic chaos fro a siple one-diensional population odel built on Malthus population theory. 1. Introduction Matheatics is an integral part of conteporary social and behavioral sciences. Many of today s profound insights into huan behavior could hardly be obtained without the help of atheatics. It ay be said that the ain advance in odern social and behavioral sciences is Encyclopedia of Life Support Systes(EOLSS)

2 MATHEMATICAL MODELS Vol. III - Matheatical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang characterized by applying atheatics to various social and behavioral probles. The concepts of equilibriu versus non-equilibriu, stability versus instability, and steady states versus chaos in the conteporary literature are difficult to explain without atheatics. In soe sense, one can hardly properly appreciate achieveents of conteporary social and behavioral sciences without a thorough training in atheatics. The purpose of this chapter is to present soe atheatical odels that support the field of social and behavioral sciences. It provides a general overview of atheatical approaches to different social and behavioral probles. The variety of odels as well as the nuber of recent contributions is quite ipressive. Since the literature on the topic is vast, we will provide only a few areas of the applications in this chapter. We introduce exaples of applications of a few branches of atheatics including optiization theory, operations research, gae theory, differential equations theory, and chaos theory. Section 2 applies optiization theory and coparative statics analysis to decide the optial behavior of workers with ultiple interests and exaine the ipact of job aenity on oonlighting. Section 3 applies operations research to solve the job assignent proble. Section 4 exaines political copetition, applying gae theory. As an exaple of applying differential equations, Section 5 investigates long-run socioeconoic consequences of altruis and explains under what conditions the opposite opinions of Malthus and Keynes on the issue are valid. Section 6 shows how chaos theory can be applied to identify socioeconoic chaos fro a siple one-diensional population odel built on Malthus population theory. The bibliography at the end of the chapter provides the reader with guidelines for further investigations. 2. Optiization Theory Job Aenity and Moonlighting Optiization is a branch of atheatics, which encopasses any diverse areas of iniization and axiization. It enables us to deterine the ost profitable or least disadvantageous choice out of a set of alternatives. Typically the set of alternatives is restricted by several constraints on the values of a nuber of variables and an objective function locates an optiu in the reaining set. The ethod is largely used in operations research and systes analysis, that is for exaple, for optial scheduling of production processes, for deterining the best way for transporting a certain coodity. Optiization theory is closely related to the calculus of variations, control theory, convex optiization theory, decision theory, gae theory, linear prograing, Markov decision chains, network analysis, optiization theory, queuing systes, etc. This section uses an exaple of how people rationally deterine their occupation(s) to illustrate a generic process of applying optiization theory to solve a decision-aking proble. Different jobs, such as officer, police, university professor, factory worker, are associated with different social status and aenities. Differences in aenities give rise to copensating wage differentials. Assuing that different jobs bring about different aenities and disaenities, Lundborg (1995) builds soe odels to explain why soe people would have ultiple occupations. We now introduce Lundborg s general odel of aenities and oonlighting to show how traditional optiization theory can be applied to explain behavior of workers. Assue two sectors, one with jobs with soe aenity and one traditional anufacturing Encyclopedia of Life Support Systes(EOLSS)

3 MATHEMATICAL MODELS Vol. III - Matheatical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang sector with no aenity, denoted as the a-sector and the -sector, respectively. We express a typical individual s utility function as U = U( X, a, l), where X is his coodity consuption, a is the share of his total available tie spent for work in the aenity sector and l is the share of tie for leisure. Assue that the total tie T is divided between leisure tie L, work tie in the a-sector, A, and work tie in anufacturing, M so that T = L + A + M. We thus have l + a + = 1 (2.1) where l L/ T, a A/ T, and M / T. Assue that U rises in all three arguents. Consuption X depends on the individual s earnings such that X = w (1 a l) + wsa (2.2) where we use (2.1), w is the individual s wage in anufacturing work, w is his wage in the a-sector and S (1 + s) where s is a subsidy rate provided by the governent. We assue that w and w are fixed for the individual and w > Sw. Since the job with aenity provides ore pleasure than the job with no aenity, it is reasonable to require that the wage rate in the sector with no aenity is higher than the net wage rate, Sw, in the sector with aenity. Otherwise, all the tie available for work would be spent in the sector with aenity. The individual s rational behavior is described by the following axiization proble Maxiize U = U( X, a, l) (2.3a) X, a, l subject to w = X + ( w ws) a + w l, (2.3b) 0 a, l 1. (2.3c) There are then three goods, X, a and l, with prices 1, ( w Sw), and w respectively. Define the Lagrange function Γ [ ] Γ( X, al,, λ) U( X, al, ) + λ w X ( w wsa ) wl where λ is a Lagrangian ultiplier. The first-order conditions for the axiization proble are obtained as Encyclopedia of Life Support Systes(EOLSS)

4 MATHEMATICAL MODELS Vol. III - Matheatical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang Γ X Γ a Γ l U = λ = 0, (2.4) X U = λ( w ws) = 0, (2.5) a U = λw = l 0, (2.6) Γ = w X ( w ws) a wl = 0. (2.7) λ Calculating the total differentials of Eqs. (2.4)-(2.7) with w, w and S as paraeters and expressing the results in atrix for yields [This for is coonly used in social sciences. See Chiang (1984)], dx da D 0 λ λs λsds = dl λ dλ λ1 Sa wads where a1 l + a 1 and UXX UXa UXl 1 UXa Uaa Ual w 0 D UlX Ula Ull w 1 w0 w 0 (2.8) where the variables with double suffixes are second (partial) derivatives with the variable(s) and w0 w ws. It is known that a sufficient condition for a solution of the first-order conditions to be a axiu is that the deterinant D of the atrix is negative[see Chiang (1984)]. This is guaranteed if the utility function is strongly concave in X, a, and l. We require this condition to be satisfied in the reainder of this section. We first exaine the effects on the share of the work tie spent in the aenity sector as a result of an increase in the anufacturing wage. With = ds = 0, we rewrite (2.8) as Encyclopedia of Life Support Systes(EOLSS)

5 MATHEMATICAL MODELS Vol. III - Matheatical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang dx da 0 λ D = dl λ a1 d λ This is a linear equation. Applying Craer s rule, we solve for the ipact of a change w on a as follows da ( ) U W λ U w ws + U U xx XX ll al = (2.9) det D where ( λ ) W w au + w aw U U 0 1 ll 1 al XX. The effect of an increase in anufacturing wages on part-tie in the a-sector is abiguous. In line with the Slutsky equation, the effect can be broken down into an incoe effect and a substitution effect. An increase in anufacturing wages iplies that the incoe level has increased. As the aenity sector work is a noral good, there is a tendency to spend a larger share of total work in the a-sector. On the other hand, as the anufacturing wage w rises, consuption of aenity becoes ore costly since the price of aenity sector work, w ws rises. Since the two effects are opposite, the net effect is abiguous., Siilarly, the effects of an increase in dx ( ) 2 λ + U au + λw au λwu w on consuption of X are given by, 0 aa 1 ll 1 al 0 al = > 0, (2.10) det D in which λ ( wu U w ) 0 λ. 0 ll la As the anufacturing wage is increased, consuption of the coodity is increased. There is a positive incoe effect on X as w rises. As consuption of aenity work becoes ore costly as the price of a-sector work rises, the individual substitute consuption of the a-sector work with consuption of X. The effects on leisure of increases in w are Encyclopedia of Life Support Systes(EOLSS)

6 MATHEMATICAL MODELS Vol. III - Matheatical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang dl ( 1 ) λ ( ) 2 U XX aw 1 Uaa w0 + w awu 1 0 al + Ual Uaa λ =. (2.11) det D The effects are abiguous. The incoe effect is positive, but the substitution one is negative. We can siilarly carry out the coparative statics analysis with regard to the a-sector wage and the subsidy rate. This exaple shows the standard procedure for analyzing rational behavior by optiization theory. First we describe atheatically rational behavior of individuals as an optiization proble. Then, we find the first-order conditions and solve the associated equations. Finally we check the second-order conditions and exaine the ipact of changes in soe paraeters Bibliography TO ACCESS ALL THE 25 PAGES OF THIS CHAPTER, Visit: Becker, G.S. (1976). The Econoic Approach to Huan Behavior. Chicago: The University of Chicago Press. [A collection of iportant essays on the econoic approach to huan behavior written by the winner of the Nobel Prize in econoics in 1992, Gary S. Becker] Chiang, A.C. (1984) Fundaental Methods of Matheatical Econoics, 3 rd edition. London: McGraw- Hill. [A user-friendly introductory book on atheatical econoics. It includes eleentary concepts and techniques of atheatics for social and behavioral sciences as well.] Collins, L. (2001, edited) The Use of Models in the Social Sciences. London: Routledge. [A collection of papers showing applications of atheatics to varied fields in social sciences.] Diaand, M.A. and Diand, R.W. (1997) The Foundations of Gae Theory, in three volues. Cheltenha: Eard Elgar. [The collection gathers together priary sources on the developent of gae theory fro its beginnings until 1960.] Filar, J.A., Gaitsgory, V., and Mizukai, K. (2000, edited) Advances in Dynaic Gaes and Applications. Boston: Birkhäuser. [The book presents a coprehensive survey of recent developents and advances in dynaic gaes and their applications] Haavelo, T. (1954). A Study in the Theory of Econoic Evolution. Asterda: North-Holland. [This book presents an early contribution to forally odeling dynaic interactions between econoic developent and population.] Isaacs, R. (1965) Differential Gaes A Matheatical Theory with Applications to Warfare and Pursuit, Control and Optiization. New York: John Wiley and Sons. [This is the original book on differential gaes. The book introduces all the ain concepts and equations in differential gaes. A central feature is the use of any and varied exaples of differential gaes; the hooicidal chauffeur, wars of attrition, pursuit with a rocket, etc.] Kuhn, H. W. (1953), Extensive Gaes and the Proble of Inforation, pp in Contributions to the Theory of Gaes, Volue II, edited by Kuhn, H.W. and Tucker, A.W. Princeton: Princeton University Press. [The seinal paper includes the forulation of extensive for gaes which is currently used, and also soe basic theores pertaining to this class of gaes.] Encyclopedia of Life Support Systes(EOLSS)

7 MATHEMATICAL MODELS Vol. III - Matheatical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang Lundborg, P. (1995) Job aenity and the incidence of double work. Journal of Econoic Behavior and Organization 26, [This paper explains why oonlighting exists in labor arkets by taking account of job aenities in behavior odeling] Peigen, H.O., Jürgens, H., and Saupe, D. (1992) Chaos and Fractals New Frontiers of Science. Berlin: Springer. [The book docuents discoveries in chaos theory with plenty of atheatical detail, but without alienating the general reader. It uses hundreds of iages and graphics to illustrate key concepts. The book covers overview of fractals and chaos theory, feedback and ultiple reduction copy achines, the Cantor Set, the Sierpinski Gasket and Carpet, the Pascal Triangle, the Koch Curve, Julia Sets, siilarity, easuring fractal curves, fractal diensions, transforations and contraction apping, iage copression, chaos gaes, fractals and nature, L-systes, cellular autoata basics, attractors and strange attractors, Henon s Attractor, Rössler and Lorenz Attractors, randoness in fractals, the Brownian otion, fractal landscapes, sensitivity and periodic points, coplex arithetic basics, the Mandelbrot Set, and ultifractal easures.] Roeer, J. E. (2001) Political Copetition: Theory and Applications. Mass., Cabridge: Harvard University Press. [The book presents a theory of political copetition between parties under varied circustances, including whether these parties are policy oriented or oriented toward winning, whether they are certain or uncertain about voter preferences, and whether the policy space is uni- or ultidiensional.] Rosser, J.B.Jr. (1991). Fro Catastrophe to Chaos: A General Theory of Econoic Discontinuities, Boston: Kluwer Acadeic Publishers. [The book provides soe illustrative exaples of nonlinear econoic phenoena such as catastrophe, bifurcations, and chaos.] Saari, D.G. (1994) Geoetry of Voting. Berlin: Springer-Verlag. [This is an advanced book presenting a geoetric theory of voting] Shapley, L. S. and M. Shubik (1954), A Method for Evaluating the Distribution of Power in a Coittee Syste, Aerican Political Science Review 48, [The paper is one of the earliest applications of gae theory to political science. It uses the Shapley value to deterine the power of the ebers of the UN Security Council.] Stark, O. (1995). Altruis and Beyond An Econoic Analysis of Transfers and Exchanges within Failies and Groups. Cabridge: Cabridge University Press. [A coprehensive introduction to the literature on the econoic analysis of altruis] Stutzer, M. (1980). Chaotic dynaics and bifurcation in a acro odel. Journal of Econoic Dynaics and Control 2, [This presents an early foral odel of socioeconoic chaos. It is a atheatical description of Malthus theory of interdependence between the population and econoic growth.] Taha, H.A. (1997). Operations Research An Introduction. London: Prentice-Hall International. [A preliinary introduction to operations research] Varoufakis, Y. and Housego, A. (2001) Gae Theory: Critical Concepts in the Social Sciences. London: Routeledge. [A collection of papers showing the vital of gae theory in social sciences] Von Neuann, J. and Morgenstern, O. (1953) Theory of Gaes and Econoic Behavior. Princeton: Princeton University Press, 3 rd edition. [The book created gae theory, although since its publications the theory has been further developed along any new directions. The book requires substantial level of ath for a full understanding] Waldrop, M. (1992) Coplexity: The Eerging Science at the Edge of Order and Chaos. Glasgow: Sion & Schuster. [The book provides a coprehensive introduction to coplexity theory, the terinology and explains the basic concepts. It also depicts captivating portraits of soe people who created the science] Weidlich, W. (2000). Sociodynaics: A Systeatic Approach to Matheatical Modeling in the Social Sciences. The Harwood Acadeic Publishers. [A priary source for an advanced approach to socioeconoic dynaics] Zhang, W.B. (1999). Capital and Knowledge - Dynaics of Econoic Structures with Non-Constant Returns. Heidelberg: Springer. [A new analytical fraework for studying varied issues of socioeconoic Encyclopedia of Life Support Systes(EOLSS)

8 MATHEMATICAL MODELS Vol. III - Matheatical Modeling in Social and Behavioral Sciences - Wei-Bin Zhang dynaics.] Zhang, W.B. (2002). Econoic consequences of altruis. Altruistic Reveries Perspectives fro the Huanities and Social Sciences (ed. B.K. Kapur and K-C. Chong), pp Boston: Kluwer Acadeic Publishers. [This constructs a socioeconoic odel to investigate possible econoic consequences of altruis. It also presents a coprehensive discussion on what conditions the opposite opinions of Malthus and Keynes on consequences of altruis are valid.] Biographical Sketch Dr. Wei-Bin Zhang, born in 1961, teaches and does research at Ritsueikan Asia Pacific University in Japan. His ain research fields are nonlinear socioeconoic dynaics, social science, regional and international econoics, urban econoics, East Asian econoic developent, Chinese philosophy, and ethics. He has published ore than 100 acadeic articles. His books include Econoic Dynaics (1990), Synergetic Econoics (1991), Japan versus China in the Industrial Race (1998), Confucianis and Modernization (1999), and Capital and Knowledge (1999), Ada Sith and Confucius - The Theory of Moral Sentients and The Analects (2000), A Theory of International Trade (2000), An Econoic Theories of Cities (2002), The Aerican Civilization Portrayed in Ancient Confucianis (2003), Econoic Growth Theory (2005), and Discrete Dynaical Systes, Bifurcations and Chaos in Econoics (2006). He was brought up and received his university education in ainland China. He studied at Kyoto University in Japan for four years. He conducted research in Sweden fro 1987 to Encyclopedia of Life Support Systes(EOLSS)