Yuri S. Popkov. Mathematical. Demoeconomy. Integrating Demographic and Economic Approaches DE GRUYTER
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1 Yuri S. Popkov Mathematical Demoeconomy Integrating Demographic and Economic Approaches DE GRUYTER
2 Contents Preface Part I v General principles of demoeconomics 1 The population-economy system General characteristics of the population-economy system Mathematical modeling of the PE system: specific features Principles of mathematical modeling Nonlinear processes Temporal hierarchy Spatial hierarchy Forecastingof demoeconomic development 11 2 Probabilistic techniques in demoeconomic forecasting Uncertainty in the PE system Demoeconomic forecasting: the structure of probabilistic technique 19 Part II Foundations of spatial demography 3 The population system Key notions State indicators of population States evolution in a demographic process: general modeling principles Structuring based on sex and space Structuring based on sex, age and space 34 4 Demographic characteristics of fertility Phenomenology of newborns distribution by maternal ages Entropy model of age-specific fertility rate Iterative method olf age-specific fertility rate recovery Dynamics of fertility rates Dynamic model of total fertility rate Dynamic model of age-specific fertility rate 57 5 Demographic characteristics of mortality Phenomenology of mortality Entropy model of sex-age distribution of mortality rate 62
3 X Contents Model construction Model analysis Parameter Identification for the entropy model of mortality based on real data Entropy decomposition of age-specific distribution of mortality by classes of diseases Dynamic model of total mortality rate 82 6 Demographic characteristics of migration General phenomenology of migration Entropy-optimal distribution of migration flows Optimality conditions for entropy models of migration Parametric properties in entropy models of migration Parametric properties of the B-model with complete consumption of resources An example of analyzing the parametric properties of the B-model of migration flows Parametric properties of the F-model with complete consumption of resources Macrosystem models of population dynamics Dynamics of isolated population Deterministic functions of fertility and mortality Random functions of fertility and mortality Macrosystem dynamic model with linear reproduction of population and balanced emigration Stationary states Stability of stationary states Stable stationary states of spatial distribution of population: an example of scenario forecasting General macrosystem model of population size dynamics Stationary states Stability of stationary states 162 Part III Foundations of spatial economics 8 Modeling economics Political economy, micro- and macroeconomics, mathematical economics: objects and goals Behavioral models for economic agents Models of rational behavior 177
4 Contents xi Models of compromise behavior Models of stochastic behavior Evolutionary economics General principlesof evolutionary economics Market equilibrium and stability Innovation activity of economic agents Externa! Investments Internal Investments Economic growth Self-organization in economic systems General notions Phenomenology ofthe model of competitivefirms. Determination of transitions Construction of Utility functions. Evaluation oftransition rates Equations ofthe model. Stationary states Stability of stationary states Spatial interaction of economic systems Entropy model of spatial economic interaction Economic system with triangular spatial structure Selected models of spatial macroeconomics Entropy decomposition Spatial interaction of economic Clusters Static interaction Dynamic interaction Model of economic systems exchanging Investments Singular stationary states Stability of Singular stationary states Fluctuations in models of spatial economics Downturns and upturns in economic activity The Immersion method for periodic solutions Periodic solutions togenerating system: application ofthe Laplace transform 286 Part IV Macrosystem models of demoeconomics 14 Macrosystems concept in demoeconomics Phenomenology of demoeconomics 299
5 xii Contents The systems character of demoeconomic processes The individualand the collective Timescales Macrosystems concept of demoeconomics: model representation The Monte Carlo method in probabilistic macrosystem modeling of demoeconomic processes One-sector macrosystem demoeconomic model (MSDEM) Structure and basic variables of the model Equations of one-sector MSDEM The block IsEM The block MSDM An example of one-sector MSDEM Equations of the model Analytic treatment of the simplified one-sector MSDEM Computer experiments with the one-sector MSDEM Analytic treatment and Computer experiments with the one-sector PMSDEM Macrosystem demoeconomic model with regional localization of sectors (branches) Ns ~ MSDEM Structure and basic variables of the model Equations of Ns - MSDEM with resource exchange on regional markets The block NsEM The block MSDM The block TRM An example of analytic treatment of Ns - MSDEM Equations of the model Stationary states Computer analysis ofns - MSDEM Equations of the model Macrosystem model of labour market Quantitative State indicators of labour market Structure and equations of the model Competition amongcohorts Intrinsic competitive ability The comparative competitive ability Labour force requirement and supply of labour force 380
6 Contents 17.4 Identification algorithm for model Parameters Identification of model parameters based on real data Probabilistic macrosystem demoeconomic model Aggregated structure ofpmsdem and its spatiotemporal characteristics Realization of PMSDEM: the Monte Carlo methods Average Computing Random search Generation of random variables with given properties The POPULATION block Classification of population Biological reproduction of population (the R module) Migration (the M module) Dynamics of population (the DP module) Outputs ofthe POPULATION block The economy block Production economy (the PE module) Exchange of products (the Ex module) Prices (the Pr module) The Output variable of the ECONOMY block The interaction block Migration (the MPP module) Fertility (the TFR module and the ASFR module) Mortality (the TMR module and the ASMR module) 431 Part V Mathematical appendices A Some theorems of implicit functions 441 A.l Introduction 441 A.2 Local properties 441 A.2.1 Existence and continuity 441 A.2.2 Homogeneous forms and posinomials 443 A.2.3 Differentiability 447 A.3 Global properties 450 A.3.1 Existence. 450 A.3.2 Differentiability 453 B Estimating the local Lipschitz Constant of the entropy operator B v q 454 B.l Introduction 454 B.2 Definitions 454
7 xiv Contents B.2.1 The operator B vq The normal operator B vq 455 B.2.3 The relation between B vq and B q 455 B.3 Properties of the entropy operator q 457 B.3.1 Existence and uniqueness 457 B.3.2 Majorant construction 459 B.4 Estimating the norm of derivative of the entropy operator B q 461 B.5 Estimating the spectrat norm of the matrix [r ] C Estimating the local Lipschitz Constant of the entropy operator F vq 467 C.l Definitions 467 C.2 Properties of the normal entropy operator F q 468 C.3 Majorants of the operator _F q 470 C.4 EstimateZ F 473 D Zero-order multiplicative algorithms for positive solutions to nonlinear equations 475 D.l Introduction 475 D.2 Auxiliary estimates 476 D.3 Convergence analysis by continuous analogs of iterative algorithms 479 D.4 Convergence of zero-order multiplicative algorithms with m-active variables: nonlinear equations Convergence of zero-order multiplicative algorithms with m-active variables: convex programming 482 E Multiplicative algorithms for positive solutions to entropy-quadratic programming problems 489 E.l Problem Statement 489 E.2 Optimality conditions 490 E.3 Multiplicative algorithm 492 Bibliography 493 Index 498
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STOCKHOLM DOCTORAL PROGRAM IN ECONOMICS Helshögskolan i Stockholm Stockholms universitet Paul Klein Email: paul.klein@iies.su.se URL: http://paulklein.se/makro2.html Macroeconomics II Spring 2010 Lecture