Economic Dynamics and Distributions.pdf
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In recent years there has been a rapidly growing interest in the study of dynamic nonlinear phenomena in economic and financial theory, while at the same time econometricians and statisticians have been developing methods for modeling such phenomena. Despite the common focus of theorists and econometricians, both lines of research have had their own publication outlets. The new resource series is designed to further the understanding of dynamic phenomena in economics and finance by bridging the gap between dynamic theory and empirics and to provide cross-fertilization between the two strands. The series will place particular focus on monographs, surveys, edited volumes, conference proceedings and handresources on: • Nonlinear dynamic phenomena in economics and finance, including equilibrium, disequilibrium, optimizing and adaptive evolutionary points of view; nonlinear and complex dynamics in microeconomics, finance, macroeconomics and applied fields of economics. • Econometric and statistical methods for analysis of nonlinear processes in economics and finance, including computational methods, numerical tools and software to study nonlinear dependence, asymmetries, persistence of fluctuations, multiple equilibria, chaotic and bifurcation phenomena. • Applications linking theory and empirical analysis in areas such as macrodynamics, microdynamics, asset pricing, financial analysis and portfolio analysis, international economics, resource dynamics and environment, industrial organization and dynamics of technical change, labor economics, demographics, population dynamics, and game theory. This resource aims to provide the common mathematical foundations of several economic dynamic and distribution phenomena by adopting a pragmatic and heuristic approach. On perspective, several back-thoughts and concerns have influenced its writing. A first concern relates to the paradoxes arising when confronting economic time series and events, with the relevant economic theory. The first puzzle is that, although every economy embodies a complex interaction of an immense number of “moving parts” (agents, markets, institutions, countries), the shape of most aggregate economic time series—in particularly historical ones—is relatively monotonic and smooth. The second puzzle is related to the fact that, given a sufficiently extended time span, all variables seem to be endogenously related in the long run. However, what lies behind most applied economics is a description of the world in terms of the mechanics of cause and consequence, and potentially plagued by the problem of indeterminacy: several theories can be consistent with a particular “evidence”. The third puzzle concerns the never ending debate—with a bearing on economic policy—on the relative importance of the two main interacting drivers in an economy: the granular effect of incentives on the decisions of private agents and economic policy authorities, and the almost mechanical effect of the aggregate state of the economy constraining economic decision making. In an environment in which there is a proliferation of data, information, and debates, those puzzles justify the need to having a solid foundation in economic theory, providing low dimensional models incorporating a wide spectrum of qualitative features, and integrating consistently both dynamic and distributional components. A second concern relates to the fact that the variational principle provides a common mathematical foundation to several specialized areas in economics (as macroeconomics, microeconomics, growth economics, environmental economics, spatial economics, and statistics). In other words, several forms of calculus underlie most economic concepts and results. In order to bring together the common mathematical structures of different economic sub-fields, the double continuum of time and the domains of other types of heterogeneity distribution is the one that allows for a simpler joint consideration of both micro and macro dimensions, intertemporal and distributional components, and deterministic and stochastic environments. It combines both simplicity and low dimensionality of the models, and allows for invoking an immense pool of results from several fields in mathematics. A third concern relates to the fact that I was trained as an economist, and the potential readership of this resource are also economists with an interest in economic theory. Accepting the risk of displeasing readers with a solid mathematical competence, I follow a heuristic and somewhat informal approach, and restrain to presenting low dimension and/or linear models. I have sacrificed mathematical rigour and generality to the endeavour of providing useful qualitative insights that can be suggested by the different mathematical models. This choice is not only the one that makes sense for the goal of covering a wide range of mathematical structures, with existing or potential application in economic theory and statistics, but also the one that addresses my next concern. A last concern is pedagogical. As a lecturer on macroeconomics, growth theory, financial economics and mathematical economics—in the University of Lisbon (ISEG) and also episodically in the University of Porto—I come to the conclusion that it is better to fully solve and characterize some simple models, not evading the “hard” technical details, rather than having in the syllabus a huge number of recent research papers which, by their on goal to contribute to moving the research frontier, can only be lightly skimmed. Thus my preference goes to study models in which explicit (or closed form) solutions can be obtained, and providing the tools to enable a geometrical interpretation of their solutions. My hope is to develop a “clinical eye” that, for instance, allows for interpreting the simulation results of large quantitative macromodels (which sometimes have only a very small number of really driving mechanisms). I expect the main audience of this resource to include advanced master or introductory PhD students in economics. But I hope fellow researchers in economics, finance, and other areas using differential equations and optimal control theories may find it useful. In particular, readers interested in the dynamics of heterogeneity and distributions will find useful material. The resource is divided into four parts, assembling nineteen chapters. Each chapter is dedicated to a particular type of differential equation or to optimization problems constrained by different types of differential equations. Some applications related economic models are presented as examples or exercises. Part I presents several different types of low dimensional ordinary differential equations, in particular, linear, non-linear regular, piecewise smooth and constant, and singular differential equations. The last two types of ordinary differential equations are not absent in similar economic textresources, and can only be found in somewhat specialized applied mathematics textresources. I include them because they are useful in modeling switching and feedback effects (for instance generated by policy rules) in both intertemporal micro and macroeconomic dynamic models. Part II deals with functionals and functional calculus applied to “static” optimization problems, calculus of variations, and optimal control of ordinary differential equations. As I try to show, a background knowledge in functional calculus is required to both understanding