Gandolfo Economic Dynamics Pdf 1

Giancarlo Gandolfo (born November 17, 1937) is an Italian economist. He has been a Professor of International Economics at Sapienza University of Rome from 1974 to 2010. Gandolfo is notable for his popular graduate-level textbook on dynamic economic theory.[1]

"This fourth edition of Gandolfo' s masterful book on economic dynamics is the premier source on dynamic mathematical tools for economists, with illustrations from many areas of current economic research. Not only is the book valuable as an encyclopedic reference book for researchers but is an excellent choice for a textbook on economic dynamics. Gandolfo has managed to provide background in even the most advanced areas of nonlinear dynamics in a readable manner avoiding unnecessarily advanced notation. As a result, the book is broadly accessible while including coverage of many of the deepest areas of current research in economic dynamics. A glance through the table of contents is enough to indicate to almost any serious economist that this is a book to buy."

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Economic Dynamics: Methods and Models aims to give a simple but comprehensive treatment of mathematical methods used in economic dynamics and show how they are utilized to build and to analyze dynamic models. The text also focuses on methods, and every mathematical technique introduced is followed by its application to selected models. The book is divided into three different parts. Part I: Different Equations discusses general principles; first-order, second-order, higher-order equations; simultaneous systems; and their economic applications. Part II: Differential Equations also discusses the same areas as those in Part I, but instead features differential equations, as what the section name suggests. Part III: More Advanced Material covers comparative statistics and the comparative principle; stability of equilibrium and Liapunov's second method; and linear mixed differential and difference equations, as well as its other related topics. The text is recommended for mathematicians and economists who have an idea on advanced mathematics and would like to know more about its applications in economics.

2 Contents PREFACE \ VII Introduction Definition Functional equations Economic dynamics: past and future References 5 LINEAR DIFFERENCE EQUATIONS 7 Difference Equations: General Principles Definitions Linear difference equations with constant coefficients The homogeneous equation The non-homogeneous equation Determination of the arbitrary constants References 16 First-order Difference Equations Solution of the homogeneous equation Particular solution of the non-homogeneous equation g{t) is a constant g(t) is an exponential function g(t) is a polynomial function of degree m g(t) is a trigonometric function of the sine-cosine type g(t) is a combination of the previous functions The case when g(t) is a generic function of time. Backward and forward solutions General solution of the non-homogeneous equation A digression on distributed lags and partial adjustment equations Exercises Example Other exercises 34

3 XII Contents 3.6 References '. >' First-order Difference Equations in Economic Models The cobweb theorem The cobweb model and expectations The normal price Adaptive expectations The dynamics of multipliers The basic case Other multipliers * A foreign trade multiplier Taxation Exercises References 53 5 Second-order Difference Equations Solution of the homogeneous equation Positive discriminant (A > 0) Null discriminant (A = 0) Negative Discriminant (A < 0) Stability conditions Solution of the non-homogeneous equation The operational method Determination of the arbitrary constants Exercises Example Other exercises References ' Second-order Difference Equations in Economic Models Multiplier-accelerator interaction: the prototype model (Hansen- Samuelson) Graphical location of the roots Market adjustments and rational expectations Hicks' trade cycle model The workings of the model Exercises References 90 7 Higher-order Difference Equations Solution of the homogeneous equation Particular solution of the non-homogeneous equation The operational method Determination of the arbitrary constants 97

8 Contents XVII An example The general case Transformation of a higher-order system into a first-order system in normal form Stability conditions for higher-order systems Exercises Example Other exercises References ' Differential Equation Systems in Economic Models Stability of Walrasian general equilibrium of exchange Static stability Dynamic stability Human capital in a growth model A digression on 'arrow diagrams' Balanced growth in a multi-sector economy Exercises References 319 III ADVANCED TOPICS Comparative Statics and the Correspondence Principle Introduction The method of comparative statics Purely qualitatively comparative statics The inverse comparative statics problem Comparative statics and optimizing behaviour Comparative statics and the dynamic stability of equilibrium Criticism and qualifications Extrema and dynamic stability An application to the theory of the firm Elements of comparative dynamics An illustrative application of the correspondence principle: the IS-LM model Exercises References Stability of Equilibrium: A General Treatment Introduction Basic concepts and definitions Stability Further definitions 357

9 XVIII Contents Structural stability '".' Qualitative methods: phase diagrams Single equations Two-equation simultaneous systems Introduction: phase plane and phase path Singular points Graphical construction of the trajectories Linear systems Quantitative methods Linearisation Elements of the qualitative theory of difference equations Single difference equations Two simultaneous difference equations Linear systems Economic applications Exercises References Liapunov's Second Method General concepts The fundamental theorems Some economic applications Global stability of Walrasian general equilibrium Rules of thumb in business management Price adjustment and oligopoly under product differentiation Exercises References Introduction to Nonlinear Dynamics Preliminary remarks A digression on existence and uniqueness theorems Some integrable differential equations First-order and first-degree exact equations Linear equations of the first order with variable coefficients The Bernoulli equation The Riccati equation Limit cyctes and relaxation oscillations Limit cycles: the general theory Limit cycles: relaxation oscillations Kaldor's non-linear cyclical model The model Kaldor via relaxation oscillations 443

10 Contents XIX Kaldor via Poincare's limit cycle The Lotka-Volterra equations Construction of the integral curves Conservative and dissipative systems, and irreversibility Goodwin's growth cycle The model The phase diagram of the model Palomba's model The model Conclusion Exercises References Bifurcation Theory Introduction Bifurcations in continuous time systems Codimension-one bifurcations The Hopf bifurcation Sensitivity analysis and bifurcations: a reminder Kaldor's non-linear cyclical model again Oscillations in optimal growth models The model The optimality conditions Emergence of a Hopf bifurcation Cycles in an IS-LM model with pure money financing Bifurcations in discrete time systems Codimension-one bifurcations The Hopf (or Neimark-Sacker) bifurcation in discrete time Kaldor's cyclical model in discrete time Liquidity costs in the firm The model The dynamics : Expectations and multiplier-accelerator interaction Hysteresis and bifurcations General Dynamical systems Economics Singularity-induced bifurcations Exercises References 515

11 XX Contents 25 Complex Dynamics ** Introduction Discrete time systems and chaos The logistic map Intermittency The basic theorems Discrete time chaos in economics Chaos in growth theory Exchange rate dynamics and chaos Continuous time systems and chaos The Lorenz equations, strange attractors, and chaos Other routes to continuous time chaos The Rossler attractor The Shil'nikov scenario The forced oscillator The coupled oscillator International trade as the source of chaos A chaotic growth cycle Significance and detection of chaos: Stochastic dynamics or chaos? Control of chaos Other approaches Introduction Fast and slow, and synergetics Catastrophe theory Exercises References Mixed Differential-Difference Equations General concepts Continuous vs discrete time in economic models Linear mixed equations The method of solution Stability conditions Approximate methods Delay differential equations and chaos Some economic applications Kalecki's business cycle model &1.1 The model The dynamics A formalization of the classical price-specie-flow mechanism of balance of payments adjustment The model Stability 591

12 Contents XXI 26.9 Exercises References Dynamic Optimization Introduction Calculus of variations Particular cases Generalizations The maximum principle Statement Proof Transversality conditions The case with infinite terminal time Effects of parameter changes on the optimal solution: the costate variables Discounting Particular cases The bang-bang control case Linear-quadratic problems The maximum principle in discrete time Dynamic programming Dynamic programming in discrete time: multi-stage optimization problems Dynamic programming and nonlinear programming Infinite terminal time Solution by conjecture Solution by iteration Solution by the envelope theorem Maximum principle vs. dynamic programming Exercises References Saddle Points and Economic Dynamics Saddle points in optimal control problems Optimal economic growth Optimal growth: traditional The setting of the problem The optimality conditions in the basic neoclassical model Saddle-point transitional dynamics in the basic neoclassical model Optimal and sub-optimal feedback control The sub-optimal feedback control rule 655

13 XXII Contents Optimal growth: endogenous ^.-.' A model of optimal endogenous growth The conditions for optimal endogenous growth Optimal endogenous growth: saddle-point transitional dynamics Optimal endogenous growth in an open economy The Net Borrower Nation Steady-State Stability and Comparative Dynamics Rational expectations and saddle points Introduction Rational expectations, saddle points, and overshooting A discrete-time equivalent Rational expectations and saddle points: the general case Indeterminacy and sunspots Indeterminacy and fiscal policy Firms Households Government The optimality conditions The singular point and its nature Exercises References 699 Bibliography 701 Index 731 Answers to Exercises 751 be457b7860