Mathematical Demoeconomy (eBook)

Preface 5
Part I General principles of demoeconomics 15
1 The population-economy system 17
1.1 General characteristics of the population-economy system 17
1.2 Mathematical modeling of the PE system: specific features 21
1.2.1 Principles of mathematical modeling 22
1.2.2 Nonlinear processes 23
1.2.3 Temporal hierarchy 23
1.2.4 Spatial hierarchy 24
1.3 Forecasting of demoeconomic development 25
2 Probabilistic techniques in demoeconomic forecasting 30
2.1 Uncertainty in the PE system 30
2.2 Demoeconomic forecasting: the structure of probabilistic technique 33
Part II Foundations of spatial demography 37
3 The population system 39
3.1 Key notions 39
3.2 State indicators of population 43
3.3 States evolution in a demographic process: general modeling principles 46
3.3.1 Structuring based on sex and space 47
3.3.2 Structuring based on sex, age and space 48
4 Demographic characteristics of fertility 50
4.1 Phenomenology of newborns distribution by maternal ages 50
4.2 Entropy model of age-specific fertility rate 53
4.3 Iterative method of age-specific fertility rate recovery 58
4.4 Dynamics of fertility rates 63
4.4.1 Dynamic model of total fertility rate 63
4.4.2 Dynamic model of age-specific fertility rate 71
5 Demographic characteristics of mortality 74
5.1 Phenomenology of mortality 74
5.2 Entropy model of sex-age distribution of mortality rate 76
5.2.1 Model construction 76
5.2.2 Model analysis 79
5.3 Parameter identification for the entropy model of mortality based on real data 81
5.4 Entropy decomposition of age-specific distribution of mortality by classes of diseases 90
5.5 Dynamic model of total mortality rate 96
6 Demographic characteristics of migration 101
6.1 General phenomenology of migration 102
6.2 Entropy-optimal distribution of migration flows 106
6.3 Optimality conditions for entropy models of migration 118
6.4 Parametric properties in entropy models of migration 122
6.4.1 Parametric properties of the B-model with complete consumption of resources 126
6.4.2 An example of analyzing the parametric properties of the B-model of migration flows 130
6.4.3 Parametric properties of the F-model with complete consumption of resources 140
7 Macrosystem models of population dynamics 146
7.1 Dynamics of isolated population 146
7.1.1 Deterministic functions of fertility and mortality 146
7.1.2 Random functions of fertility and mortality 153
7.2 Macrosystem dynamic model with linear reproduction of population and balanced emigration 156
7.2.1 Stationary states 158
7.2.2 Stability of stationary states 161
7.3 Stable stationary states of spatial distribution of population: an example of scenario forecasting 167
7.4 General macrosystem model of population size dynamics 171
7.4.1 Stationary states 175
7.4.2 Stability of stationary states 176
Part III Foundations of spatial economics 181
8 Modeling economics 183
8.1 Political economy, micro- and macroeconomics, mathematical economics: objects and goals 184
8.2 Behavioral models for economic agents 190
8.2.1 Models of rational behavior 191
8.2.2 Models of compromise behavior 194
8.2.3 Models of stochastic behavior 201
9 Evolutionary economics 205
9.1 General principles of evolutionary economics 205
9.2 Market equilibriumand stability 206
9.3 Innovation activity of economic agents 210
9.3.1 External investments 213
9.3.2 Internal investments 216
9.4 Economic growth 220
10 Self-organization in economic systems 225
10.1 General notions 225
10.2 Phenomenology of the model of competitive firms. Determination of transitions 227
10.3 Construction of utility functions. Evaluation of transition rates 231
10.4 Equations of the model. Stationary states 234
10.5 Stability of stationary states 240
11 Spatial interaction of economic systems 246
11.1 Entropy model of spatial economic interaction 246
11.2 Economic system with triangular spatial structure 257
12 Selected models of spatial macroeconomics 262
12.1 Entropy decomposition 262
12.2 Spatial interaction of economic clusters 270
12.2.1 Static interaction 272
12.2.2 Dynamic interaction 275
12.3 Model of economic systems exchanging investments 278
12.3.1 Singular stationary states 281
12.3.2 Stability of singular stationary states 283
13 Fluctuations in models of spatial economics 295
13.1 Downturns and upturns in economic activity 295
13.2 The immersion method for periodic solutions 296
13.3 Periodic solutions to generating system: application of the Laplace transform 300
Part IV Macrosystem models of demoeconomics 311
14 Macrosystems concept in demoeconomics 313
14.1 Phenomenology of demoeconomics 313
14.1.1 The systems character of demoeconomic processes 314
14.1.2 The individual and the collective 315
14.1.3 Time scales 315
14.2 Macrosystems concept of demoeconomics: model representation 317
14.3 The Monte Carlo method in probabilisticmacrosystem modeling of demoeconomic processes 319
15 One-sector macrosystem demoeconomic model (MSDEM ) 324
15.1 Structure and basic variables of the model 324
15.2 Equations of one-sector MSDEM 327
15.2.1 The block 1sEM 327
15.2.2 The block MSDM 330
15.3 An example of one-sector MSDEM 334
15.3.1 Equations of the model 335
15.3.2 Analytic treatment of the simplified one-sector MSDEM 338
15.3.3 Computer experiments with the one-sector MSDEM 341
15.3.4 Analytic treatment and computer experiments with the one-sector PMSDEM 347
16 Macrosystem demoeconomic model with regional localization of sectors (branches) Ns-MSDEM 353
16.1 Structure and basic variables of the model 353
16.2 Equations of Ns - MSDEM with resource exchange on regional markets 358
16.2.1 The block NsEM 358
16.2.2 The block MSDM 363
16.2.3 The block TRM 365
16.3 An example of analytic treatment of Ns - MSDEM 366
16.3.1 Equations of the model 366
16.3.2 Stationary states 368
16.4 Computer analysis of Ns - MSDEM 372
16.4.1 Equations of the model 372
17 Macrosystem model of labour market 385
17.1 Quantitative state indicators of labour market 385
17.2 Structure and equations of the model 387
17.3 Competition among cohorts 389
17.3.1 Intrinsic competitive ability 390
17.3.2 The comparative competitive ability 393
17.3.3 Labour force requirement and supply of labour force 394
17.4 Identification algorithm for model parameters 395
17.5 Identification of model parameters based on real data 397
18 Probabilistic macrosystem demoeconomic model 406
18.1 Aggregated structure of PMSDEM and its spatiotemporal characteristics 406
18.2 Realization of PMSDEM: the Monte Carlo methods 411
18.2.1 Average computing 411
18.2.2 Random search 412
18.2.3 Generation of random variables with given properties 413
18.3 The POPULATION block 414
18.3.1 Classification of population 414
18.3.2 Biological reproduction of population (the R module) 416
18.3.3 Migration (theMmodule) 419
18.3.4 Dynamics of population (the DP module) 425
18.3.5 Outputs of the POPULATION block 425
18.4 The economy block 426
18.4.1 Production economy (the PE module) 426
18.4.2 Exchange of products (the Ex module) 430
18.4.3 Prices (the Pr module) 432
18.4.4 The output variable of the ECONOMY block 434
18.5 The interaction block 435
18.5.1 Migration (the MPP module) 435
18.5.2 Fertility (the TFR module and the AFRR module) 439
18.5.3 Mortality (the TMR module and the ASMR module) 445
Part V Mathematical appendices 453
A Some theorems of implicit functions 455
A.1 Introduction 455
A.2 Local properties 455
A.2.1 Existence and continuity 455
A.2.2 Homogeneous forms and posinomials 457
A.2.3 Differentiability 461
A.3 Global properties 464
A.3.1 Existence 464
A.3.2 Differentiability 467
B Estimating the local Lipschitz Constant of the entropy operator Bv,q 468
B.1 Introduction 468
B.2 Definitions 468
B.2.1 The operator Bv,q 468
B.2.2 The normal operator B0v,q 469
B.2.3 The relation between Bv,q and B0v,q 469
B.3 Properties of the entropy operator B0v,q 471
B.3.1 Existence and uniqueness 471
B.3.2 Majorant construction 473
B.4 Estimating the norm of derivative of the entropy operator B0v,q 475
B.5 Estimating the spectral norm of the matrix [I0?]-1 478
C Estimating the local Lipschitz Constant of the entropy operator Fv,q 481
C.1 Definitions 481
C.2 Properties of the normal entropy operator F0v,q 482
C.3 Majorants of the operator F0v,q 484
C.4 Estimate lF 487
D Zero-order multiplicative algorithms for positive solutions to nonlinear equations 489
D.1 Introduction 489
D.2 Auxiliary estimates 490
D.3 Convergence analysis by continuous analogs of iterative algorithms 493
D.4 Convergence of zero-order multiplicative algorithms withm-active variables: nonlinear equations 494
D.5 Convergence of zero-order multiplicative algorithms withm-active variables: convex programming 496
E Multiplicative algorithms for positive solutions to entropy-quadratic programming problems 503
E.1 Problem statement 503
E.2 Optimality conditions 504
E.3 Multiplicative algorithm 506
Bibliography 507
Index 512