Giancarlo Gandolfo · 2010. 12. 13. · Giancarlo Gandolfo Economic Dynamics Fourth Edition 4y...
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Giancarlo Gandolfo Economic Dynamics Fourth Edition 4y Springer
Transcript of Giancarlo Gandolfo · 2010. 12. 13. · Giancarlo Gandolfo Economic Dynamics Fourth Edition 4y...
Giancarlo Gandolfo
Economic Dynamics
Fourth Edition
4y Springer
Contents
PREFACE L .. _._ VII
Introduction 11.1 Definition 11.2 Functional equations 21.3 Economic dynamics: past and future 31.4 References 5
LINEAR DIFFERENCE EQUATIONS 7
Difference Equations: General Principles 92.1 Definitions 92.2 Linear difference equations with constant coefficients 11
2.2.1 The homogeneous equation 122.2.2 The non-homogeneous equation 14
2.3 Determination of the arbitrary constants 152.4 References 16
First-order Difference Equations 193.1 Solution of the homogeneous equation 193.2 Particular solution of the non-homogeneous equation 23
3.2.1 g(t) is a constant 233.2.2 g(t) is an exponential function 243.2.3 g(t) is a polynomial function of degree m 253.2.4 g(t) is a trigonometric function of the sine-cosine type . 253.2.5 g(t) is a combination of the previous functions 263.2.6 The case when g(t) is a generic function of time. Back-
ward and forward solutions . . . .' 263.3 General solution of the non-homogeneous equation 293.4 A digression on distributed lags and partial adjustment equa-
tions 303.5 Exercises 33
3.5.1 Example 333.5.2 Other exercises 34
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3.6 References 35
4 First-order Difference Equations in Economic Models 374.1 The cobweb theorem 37
4.1.1 The cobweb model and expectations 404.1.1.1 The normal price 414.1.1.2 Adaptive expectations 42
4.2 The dynamics of multipliers 454.2.1 The basic case 454.2.2 Other multipliers 47
4.2.2.1 A foreign trade multiplier 484.2.2.2 Taxation 49
4.3 Exercises 504.4 References 53
5 Second-order Difference Equations 555.1 Solution of the homogeneous equation 55
5.1.1 Positive discriminant (A > 0) 565.1.2 Null discriminant (A = 0) 575.1.3 Negative Discriminant (A < 0) 585.1.4 Stability conditions 60
5.2 Solution of the non-homogeneous equation 625.2.1 The operational method 63
5.3 Determination of the arbitrary constants 655.4 Exercises 67
5.4.1 Example 675.4.2 Other exercises 70
5.5 References 71
6 Second-order Difference Equations in Economic Models 736.1 Multiplier-accelerator interaction: the prototype model (Hansen-
Samuelson) 736.1.1 Graphical location of the roots 75
6.2 Market adjustments and rational expectations 776.3 Hicks' trade cycle model 78
6.3.1 The workings of the model 836.4 Exercises 886.5 References 90
7 Higher-order Difference Equations 937.1 Solution of the homogeneous equation 937.2 Particular solution of the non-homogeneous equation 94
7.2.1 The operational method 957.3 Determination of the arbitrary constants 97
Contents XIII
7.4 Stability conditions 977.4.1 Necessary and sufficient stability conditions
(Samuelson's form) 987.4.2 Necessary and sufficient stability conditions
(Schur-Cohn form) 997.5 Exercises 102
7.5.1 Example 1027.5.2 Other exercises 103
7.6 References 103
8 Higher-order Difference Equations in Economic Models 1058.1 Inventory cycles (Metzler) 1058.2 Distributed lags and interaction between the multiplier and
the accelerator (Hicks) 1088.3 Exercises 1108.4 References 112
9 Simultaneous Systems of Difference Equations 1139.1 First-order 2x2 systems in normal form . . / 113
9.1.1 General solution of the homogeneous system: firstmethod 113
9.1.2 General solution of the homogeneous system:second (or direct) method 1169.1.2.1 Unequal real roots 1169.1.2.2 Equal real roots 1189.1.2.3 Complex roots 119
9.1.3 Particular solution. Determination of the arbitraryconstants 120
9.2 First order nxn systems in normal form 1219.2.1 Direct matrix solution. The Jordan canonical form . . 1249.2.2 Stability conditions 127
9.2.2.1 A digression on not-wholly-unstable systems . 1309.2.2.2 Proof of the stability conditions 132
9.2.3 Particular solution 1349.2.3.1 The operational method 134
9.2.4 Determination of the arbitrary constants 1379.3 General systems 138
9.3.1 First-order systems not in normal form 1389.3.2 Higher-order systems 140
9.3.2.1 An example 1409.3.2.2 The general case 1419.3.2.3 Transformation of a higher-order system into
a first-order system in normal form 1429.3.2.4 Stability conditions for higher-order systems . 144
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9.4 Exercises 1459.4.1 Example 1459.4.2 Other exercises 145
9.5 References 146
10 Simultaneous Difference Systems in Economic Models 14910.1 Cournot oligopoly 14910.2 Multiplier effects in an open economy 15210.3 Oligopoly and international trade , 155
10.3.1 The Equilibrium Solution S . .-. 15610.3.2 Stability 158
10.4 Exercises ; 15910.5 References 160
11 LINEAR DIFFERENTIAL EQUATIONS 161
11 Differential Equations: General Principles 16311.1 Definitions 16311.2 Linear differential equations with constant coefficients 164
11.2.1 The homogeneous equation 16511.2.2 The non-homogeneous equation f 166
11.3 Determination of the arbitrary constants 16811.4 References 170
12 First-order Differential Equations 17112.1 Solution of the homogeneous equation 17112.2 Particular solution of the non-homegeneous equation 174
12.2.1 g(t) is a constant 17412.2.2 g(t) is an exponential function 17512.2.3 g{t) is a polynomial function of degree m 17512.2.4 g(t) is a trigonometric function of the sine-cosine type . 17612.2.5 g(t) is a combination of the previous functions 17612.2.6 g(t) is a generic function of time. The method of vari-
ation of parameters 17712.3 General solution of the non-homogeneous equation 17812.4 Continuously distributed lags and partial adjustment equations 17912.5 Exercises 181
12.5.1 Example 18112.5.2 Other exercises 183
12.6 References 183
Contents XV
13 First-order Differential Equations in Economic Models 18513.1 Stability of supply and demand equilibrium 18513.2 The neoclassical growth model 191
13.2.1 Existence of a growth equilibrium 19213.2.2 Stability of growth equilibrium 19413.2.3 Refinements 197
13.2.3.1 Depreciation and technical progress 19713.2.3.2 Golden rule 199
13.2.4 Further developments 20013.2.4.1 Adjustment time or,, how Jong is the long run? 20013.2.4.2 /3-convergence, cr-convergence, and all that . . 20313.2.4.3 Endogenous growth ' 205
13.3 Exercises 20513.4 References 208
14 Second-order Differential Equations 20914.1 Solution of the homogeneous equation 209
14.1.1 Positive discriminant (A > 0) 21014.1.2 Null discriminant (A = 0) 21114.1.3 Negative discriminant (A < 0) 21214.1.4 Stability conditions 214
14.2 Particular solution of the non-homogeneous equation 21514.2.1 Variation of parameters 216
14.3 General solution of the non-homogeneous equation 21814.4 Determination of the arbitrary constants 21914.5 Exercises 219
14.5.1 Examples 21914.5.2 Other exercises 221
14.6 References 221
15 Second-order Differential Equations in Economic Models 22315.1 The second-order accelerator 22315.2 Exercises 22615.3 References 228
16 Higher-order Differential Equations 22916.1 Solution of the homogeneous equation . 22916.2 Solution of the non-homogeneous equation 231
16.2.1 Variation of parameters 23116.3 Determination of the arbitrary constants 23416.4 Stability conditions 235
16.4.1 Necessary and sufficient stability conditions(Routh-Hurwitz) 239
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16.4.2 Necessary and sufficient stability conditions(Lienard-Chipart) 240
16.5 Exercises 24116.5.1 Example 24116.5.2 Other exercises 242
16.6 References 242
17 Higher-order Differential Equations in Economic Models 24317.1 Feedback control and stabilisation policies 243
17.1.1 Introduction 24317.1.2 Three types of stabilisation policy 244
17.1.2.1 Proportional stabilisation policy 24717.1.2.2 Mixed proportional-derivative stabilisation pol-
icy 24817.1.2.3 Integral stabilisation policy 249
17.2 Exercises 25017.3 References 251
18 Simultaneous Systems of Differential Equations 25318.1 First-order 2x2 systems in normal form 253
18.1.1 General solution of the homogeneous system: firstmethod 254
18.1.2 General solution of the homogeneous system: second(or direct) method 25618.1.2.1 Unequal real roots 25718.1.2.2 Equal real roots 25918.1.2.3 Complex roots 260
18.1.3 Particular solution. Determination of the arbitraryconstants 261
18.2 First order nxn systems in normal form 26118.2.1 Solution of the homogeneous system 263
18.2.1.1 The matrix exponential and the Jordan canon-ical form 265
18.2.2 Stability conditions 26918.2.2.1 D-stability, and stabilisation of matrices . . .27218.2.2.2 Sensitivity analysis 27418.2.2.3 A digression on not-wholly-unstable systems . 27718.2.2.4 Proof of the stability conditions 281
18.2.3 Particular solution 28218.2.3.1 Variation of parameters 283
18.2.4 Determination of the arbitrary constants . 28318.3 General systems 285
18.3.1 First-order systems not in normal form 28518.3.2 Higher-order systems 287
Contents XVII
18.3.2.1 An example 28718.3.2.2 The general case 28818.3.2.3 Transformation of a higher-order system into
a first-order system in normal form 28918.3.2.4 Stability conditions for higher-order systems . 292
18.4 Exercises 29218.4.1 Example 29218.4.2 Other exercises 294
18.5 References 295I
19 Differential Equation Systems in Economic Models 29719.1 Stability of Walrasian general equilibrium of exchange 297
19.1.1 Static stability - 29919.1.2 Dynamic stability 301
19.2 Human capital in a growth model 30419.3 A digression on 'arrow diagrams' 30919.4 Balanced growth in a multi-sector economy 31119.5 Exercises 31619.6 References 319
III ADVANCED TOPICS 323
20 Comparative Statics and the Correspondence Principle 32520.1 Introduction 32520.2 The method of comparative statics 326
20.2.1 Purely qualitatively comparative statics 33020.2.2 The inverse comparative statics problem 330
20.3 Comparative statics and optimizing behaviour 33120.4 Comparative statics and the dynamic stability of equilibrium . 334
20.4.1 Criticism and qualifications 33620.5 Extrema and dynamic stability 338
20.5.1 An application to the theory of the firm 34320.6 Elements of comparative dynamics 34420.7 An illustrative application of the correspondence principle: the
IS-LM model 34520.8 Exercises . 34920.9 References 349
21 Stability of Equilibrium: A General Treatment 35121.1 Introduction 35121.2 Basic concepts and definitions 353
21.2.1 Stability 35321.2.2 Further definitions 357
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21.2.3 Structural stability 35921.3 Qualitative methods: phase diagrams 363
21.3.1 Single equations 36421.3.2 Two-equation simultaneous systems 368
21.3.2.1 Introduction: phase plane and phase path . . 36821.3.2.2 Singular points 36921.3.2.3 Graphical construction of the trajectories . . 37221.3.2.4 Linear systems 378
21.4 Quantitative methods 38221.4.1 Linearisation 382
21.5 Elements of the qualitative theory of difference equations . . .38721.5.1 Single difference equations 38821.5.2 Two simultaneous difference equations 393
21.5.2.1 Linear systems 39421.6 Economic applications . 39621.7 Exercises 39621.8 References 398
22 Liapunov's Second Method 40122.1 General concepts 40122.2 The fundamental theorems 40222.3 Some economic applications 407
22.3.1 Global stability of Walrasian general equilibrium . . . .40722.3.2 Rules of thumb in business management 41422.3.3 Price adjustment and oligopoly under product, differ-
entiation 41522.4 Exercises 41922.5 References 420
23 Introduction to Nonlinear Dynamics 42323.1 Preliminary remarks 423
23.1.1 A digression on existence and uniqueness theorems . . 42523.2 Some integrable differential equations 426
23.2.1 First-order and first-degree exact equations 42623.2.2 Linear equations of the first order with variable coeffi-
cients 42923.2.3 The Bernoulli equation 43123.2.4 The Riccati equation 432
23.3 Limit cycles and relaxation oscillations 43423.3.1 Limit cycles: the general theory 43423.3.2 Limit cycles: relaxation oscillations 43623.3.3 Kaldor's non-linear cyclical model 439
23.3.3.1 The model 43923.3.3.2 Kaldor via relaxation oscillations 443
Contents XIX
23.3.3.3 Kaldor via Poincare's limit cycle 44623.4 The Lotka-Volterra equations 448
23.4.1 Construction of the integral curves 45323.4.2 Conservative and dissipative systems, and
irreversibility 45523.4.3 Goodwin's growth cycle 457
23.4.3.1 The model 45723.4.3.2 The phase diagram of the model 460
23.4.4 Palomba's model , 46323.4.4.1 The model S 46323.4.4.2 Conclusion 466
23.5 Exercises 46623.6 References 469
24 Bifurcation Theory 47324.1 Introduction 47324.2 Bifurcations in continuous time systems 473
24.2.1 Codimension-one bifurcations 47524.2.2 The Hopf bifurcation 47924.2.3 Sensitivity analysis and bifurcations: a reminder . . . . 48424.2.4 Kaldor's non-linear cyclical model again 48524.2.5 Oscillations in optimal growth models 486
24.2.5.1 The model 48624.2.5.2 The optimally conditions 48824.2.5.3 Emergence of a Hopf bifurcation 489
24.2.6 Cycles in an IS-LM model with pure money financing . 49124.3 Bifurcations in discrete time systems 493
24.3.1 Codimension-one bifurcations 49424.3.2 The Hopf (or Neimark-Sacker) bifurcation in discrete
time 49624.3.3 Kaldor's cyclical model in discrete time 49824.3.4 Liquidity costs in the firm 499
24.3.4.1 The model 49924.3.4.2 The dynamics 502
24.3.5 Expectations and multiplier-accelerator interaction . . 50424.4 Hysteresis and bifurcations 507
24.4.1 General 50724.4.2 Dynamical systems 50924.4.3 Economics 510
24.5 Singularity-induced bifurcations 51124.6 Exercises 51324.7 References 515
XX Contents
25 Complex Dynamics 51925.1 Introduction 51925.2 Discrete time systems and chaos 522
25.2.1 The logistic map 52225.2.2 Intermittency 52825.2.3 The basic theorems 52925.2.4 Discrete time chaos in economics 531
25.2.4.1 Chaos in growth theory 53125.2.4.2 Exchange rate dynamics and chaos 533
25.3 Continuous time systems and chaos 53525.3.1 The Lorenz equations, strange attractors, and chaos . . 53525.3.2 Other routes to continuous time chaos 537
25.3.2.1 The Rossler attractor 53725.3.2.2 The Shil'nikov scenario 53825.3.2.3 The forced oscillator 53825.3.2.4 The coupled oscillator 539
25.3.3 International trade as the source of chaos 54225.3.4 A chaotic growth cycle 544
25.4 Significance and detection of chaos: Stochastic dynamics orchaos? 545
25.5 Control of chaos 54925.6 Other approaches 552
25.6.1 Introduction 55225.6.2 Fast and slow, and synergetics 55225.6.3 Catastrophe theory 556
25.7 Exercises 55825.8 References 560
26 Mixed Differential-Difference Equations 56726.1 General concepts 56726.2 Continuous vs discrete time in economic models 56826.3 Linear mixed equations 57326.4 The method of solution 57426.5 Stability conditions 57926.6 Approximate methods 58026.7 Delay differential equations and chaos 58126.8 Some economic applications 582
26.8.1 Kalecki's business cycle model 58326.8.1.1 The model 58326.8.1.2 The dynamics 585
26.8.2 A formalization of the classical price-specie-flow mech-anism of balance of payments adjustment 58926.8.2.1 The model 58926.8.2.2 Stability 591
Contents XXI
26.9 Exercises 59326.10References 594
27 Dynamic Optimization 59727.1 Introduction 59727.2 Calculus of variations 599
27.2.1 Particular cases 60127.2.2 Generalizations 603
27.3 The maximum principle 60427.3.1 Statement • .60427.3.2 Proof . \ : T. 60627.3.3 Transversality conditions 610
27.3.3.1 The case with infinite terminal time 61127.3.4 Effects of parameter changes on the optimal solution:
the costate variables 61227.3.5 Discounting 61427.3.6 Particular cases 615
27.3.6.1 The bang-bang control case 61527.3.6.2 Linear-quadratic problems 616
27.3.7 The maximum principle in discrete time 61827.4 Dynamic programming 619
27.4.1 Dynamic programming in discrete time: multi-stageoptimization problems 622
27.4.2 Dynamic programming and nonlinear programming . . 62627.4.3 Infinite terminal time 627
27.4.3.1 Solution by conjecture 62827.4.3.2 Solution by iteration 63227.4.3.3 Solution by the envelope theorem 633
27.5 Maximum principle vs. dynamic programming 63727.6 Exercises 63927.7 References 641
28 Saddle Points and Economic Dynamics 64328.1 Saddle points in optimal control problems 64428.2 Optimal economic growth 644
28.2.1 Optimal growth: traditional 64428.2.1.1 The setting of the problem 64428.2.1.2 The optimality conditions in the basic neo-
classical model 64728.2.1.3 Saddle-point transitional dynamics in the ba-
sic neoclassical model 65128.2.1.4 Optimal and sub-optimal feedback control . . 653
28.2.1.4.1 The sub-optimal feedback controlrule 655
XXII Contents
28.2.2 Optimal growth: endogenous 65628.2.2.1 A model of optimal endogenous growth . . . .65628.2.2.2 The conditions for optimal endogenous growth 65828.2.2.3 Optimal endogenous growth: saddle-point tran-
sitional dynamics 66028.3 Optimal endogenous growth in an open economy 664
28.3.1 The Net, Borrower Nation 66928.3.1.1 Steady-State Stability and Comparative Dy-
namics 67128.4 Rational expectations and saddle ̂ points 674
28.4.1 Introduction ".". 67428.4.2 Rational expectations, saddle points, and overshooting 677
28.4.2.1 A discrete-time equivalent 68228.4.3 Rational expectations and saddle points: the general
case 68428.5 Indeterminacy and sunspots 686
28.5.1 Indeterminacy and fiscal policy 68828.5.1.1 Firms 68828.5.1.2 Households 68928.5.1.3 Government 68928.5.1.4 The optimality conditions 69028.5.1.5 The singular point and its nature 693
28.6 Exercises 69728.7 References 699
Bibliography 701
Index 731
