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经济动态与分布：微分方程、最优控制及其应用。这份资料以务实启发式方法，为经济动态与分布现象构建通用数学基础。资料围绕三大谜题、变分原理、写作定位与教学理念展开论述，面向经济学高年级硕士、入门级博士及相关领域研究者。资料分为四部分共 19 章，依次介绍常微分方程、泛函与最优控制、偏微分方程、随机过程与随机最优控制，结合经济模型案例与习题。作者简化数学严谨性，侧重定性洞见，优先讲解可求出闭式解的模型。
Economic Dynamics and Distributions: Differential Equations, Optimal Control and Applications
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, 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 own 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 output and build 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 textbooks, and can only be found in somewhat specialized applied mathematics textbooks. 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 the common elements in several separate concepts and theories in economics dealing with dynamics, distributions and uncertainty. Part III presents an introduction to first-order and parabolic partial differential equations, and their related optimal control problems. I try to show that the usefulness of partial differential equations goes beyond their usual application in finance, and should be in the toolbox of economists for dealing with heterogeneity and aggregation in a comprehensive and powerful way. Part IV is also dedicated to an introductory presentation of stochastic processes, stochastic differential equations and stochastic optimal control. It provides an introduction to the relationship between forward and backward equations, and between the dynamic programming and the Pontriyagin’s approaches to stochastic optimal control via the envelope theorem. Clearly, the last two parts try to pinpoint some small drops in a literature that is wide as an ocean. Overall, an effort has been made to highlight the connections between both the solution methods, the characterization of the solutions among different types of differential equations, and their associated optimal control problems.
Too many years were needed to conceive and complete this resource. I acknowledge and thank the several cohorts of students for enduring and contributing to the several rounds of revisions of my lecture notes that were the stepping stones leading to this resource. I also thankfully acknowledge the following colleagues that, by several different types of interaction, influenced my work: Bernardino Adão, Costas Azariadis, Emmanuele Augeraud-Veron, Robert Becker, Jess Benhabib, Gaetano Bloise, Stefano Bosi, Raouf Boucekkine, Alan Champneys, Luís F. Costa, Hippolyte d’Albis, Rui Dilão, Jean-Pierre Drugeon, Fátima Fabião, Giorgio Fabbri, Gustav Feichtinger, Ana Fernandes, Rodolphe Santos Ferreira, Pedro Gil, Fausto Gozzi, Huw Dixon, Mike Jeffrey, Cuong Le Van, Teresa Lloyd-Braga, Giancarlo Marini, Leonor Modesto, Alfredo M. Pereira, Alessandro Piergallini, Xavier Raurich, Çağri Sağlam, Carlos D. Santos, Manuel S. Santos, Thomas Seegmuller, Gerhard Sorger, Harutaka Takahashi, Mario Tirelli, Stephen Turnovsky, Vladimir Veliov, Alain Venditti, and Bertrand Wigniolle.
I thankfully acknowledge FCT, I.P., the Portuguese national funding agency for science, research and technology, for its financial support over the years.

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