PREFACE ....................................................... VII
1 Introduction ................................................. 1
1.1 Definition .............................................. 1
1.2 Functional equations .................................... 2
1.3 Economic dynamics: past and future ...................... 3
1.4 References .............................................. 5
I LINEAR DIFFERENCE EQUATIONS .................................. 7
2 Difference Equations: General Principles ..................... 9
2.1 Definitions ............................................. 9
2.2 Linear difference equations with constant
coefficients ........................................... 11
2.2.1 The homogeneous equation ........................ 12
2.2.2 The non-homogeneous equation .................... 14
2.3 Determination of the arbitrary constants ............... 15
2.4 References ............................................. 16
3 First-order Difference Equations ............................ 19
3.1 Solution of the homogeneous equation ................... 19
3.2 Particular solution of the non-homogeneous equation .... 23
3.2.1 g(t) is a constant .............................. 23
3.2.2 g(t) is an exponential function ................. 24
3.2.3 g(t) is a polynomial function of degree m ....... 25
3.2.4 g(t) is a trigonometric function of the sine-
cosine type ..................................... 25
3.2.5 g(t) is a combination of the previous
functions ....................................... 26
3.2.6 The case when g(t) is a generic function of
time. Backward and forward solutions ............ 26
3.3 General solution of the non-homogeneous equation ....... 29
3.4 A digression on distributed lags and partial
adjustment equations ................................... 30
3.5 Exercises .............................................. 33
3.5.1 Example ......................................... 33
3.5.2 Other exercises ................................. 34
3.6 References ............................................. 35
4 First-order Difference Equations in Economic Models ......... 37
4.1 The cobweb theorem ..................................... 37
4.1.1 The cobweb model and expectations ............... 40
4.1.1.1 The normal price ....................... 41
4.1.1.2 Adaptive expectations .................. 42
4.2 The dynamics of multipliers ............................ 45
4.2.1 The basic case .................................. 45
4.2.2 Other multipliers ............................... 47
4.2.2.1 A foreign trade multiplier ............. 48
4.2.2.2 Taxation ............................... 49
4.3 Exercises .............................................. 50
4.4 References ............................................. 53
5 Second-order Difference Equations ........................... 55
5.1 Solution of the homogeneous equation ................... 55
5.1.1 Positive discriminant (Δ > 0) ................... 56
5.1.2 Null discriminant (Δ = 0) ....................... 57
5.1.3 Negative Discriminant (Δ < 0) ................... 58
5.1.4 Stability conditions ............................ 60
5.2 Solution of the non-homogeneous equation ............... 62
5.2.1 The operational method .......................... 63
5.3 Determination of the arbitrary constants ............... 65
5.4 Exercises .............................................. 67
5.4.1 Example ......................................... 67
5.4.2 Other exercises ................................. 70
5.5 References ............................................. 71
6 Second-order Difference Equations in Economic Models ........ 73
6.1 Multiplier-accelerator interaction: the prototype
model (Hansen-Samuelson) ............................... 73
6.1.1 Graphical location of the roots ................. 75
6.2 Market adjustments and rational expectations ........... 77
6.3 Hicks' trade cycle model ............................... 78
6.3.1 The workings of the model ....................... 83
6.4 Exercises .............................................. 88
6.5 References ............................................. 90
7 Higher-order Difference Equations ........................... 93
7.1 Solution of the homogeneous equation ................... 93
7.2 Particular solution of the non-homogeneous equation .... 94
7.2.1 The operational method .......................... 95
7.3 Determination of the arbitrary constants ............... 97
7.4 Stability conditions ................................... 97
7.4.1 Necessary and sufficient stability conditions
(Samuelson's form) .............................. 98
7.4.2 Necessary and sufficient stability conditions
(Schur-Cohn form) ............................... 99
7.5 Exercises ............................................. 102
7.5.1 Example ........................................ 102
7.5.2 Other exercises ................................ 103
7.6 References ............................................ 103
8 Higher-order Difference Equations in Economic Models ....... 105
8.1 Inventory cycles (Metzler) ............................ 105
8.2 Distributed lags and interaction between the
multiplier and the accelerator (Hicks) ................ 108
8.3 Exercises ............................................. 110
8.4 References ............................................ 112
9 Simultaneous Systems of Difference Equations ............... 113
9.1 First-order 2 × 2 systems in normal form .............. 113
9.1.1 General solution of the homogeneous system:
first method ................................... 113
9.1.2 General solution of the homogeneous system:
second (or direct) method ...................... 116
9.1.2.1 Unequal real roots .................... 116
9.1.2.2 Equal real roots ...................... 118
9.1.2.3 Complex roots ......................... 119
9.1.3 Particular solution. Determination of the
arbitrary constants ............................ 120
9.2 First order nxn systems in normal form ................ 121
9.2.1 Direct matrix solution. The Jordan canonical
form ........................................... 124
9.2.2 Stability conditions ........................... 127
9.2.2.1 A digression on not-wholly-unstable
systems ............................... 130
9.2.2.2 Proof of the stability conditions ..... 132
9.2.3 Particular solution ............................ 134
9.2.3.1 The operational method ............... 134
9.2.4 Determination of the arbitrary constants ....... 137
9.3 General systems ....................................... 138
9.3.1 First-order systems not in normal form ......... 138
9.3.2 Higher-order systems ........................... 140
9.3.2.1 An example ............................ 140
9.3.2.2 The general case ...................... 141
9.3.2.3 Transformation of a higher-order
system into a first-order system in
normal form ........................... 142
9.3.2.4 Stability conditions for higher-
order systems ......................... 144
9.4 Exercises ............................................. 145
9.4.1 Example ........................................ 145
9.4.2 Other exercises ................................ 145
9.5 References ............................................ 146
10 Simultaneous Difference Systems in Economic Models ......... 149
10.1 Cournot oligopoly ..................................... 149
10.2 Multiplier effects in an open economy ................. 152
10.3 Oligopoly and international trade ..................... 155
10.3.1 The Equilibrium Solution ....................... 156
10.3.2 Stability ...................................... 158
10.4 Exercises ............................................. 159
10.5 References ............................................ 160
II LINEAR DIFFERENTIAL EQUATIONS .............................. 161
11 Differential Equations: General Principles ................. 163
11.1 Definitions ........................................... 163
11.2 Linear differential equations with constant
coefficients .......................................... 164
11.2.1 The homogeneous equation ....................... 165
11.2.2 The non-homogeneous equation ................... 166
11.3 Determination of the arbitrary constants .............. 168
11.4 References ............................................ 170
12 First-order Differential Equations ......................... 171
12.1 Solution of the homogeneous equation .................. 171
12.2 Particular solution of the non-homegeneous equation ... 174
12.2.1 g(t) is a constant ............................. 174
12.2.2 g(t) is an exponential function ................ 175
12.2.3 g(t) is a polynomial function of degree m ...... 175
12.2.4 g(t) is a trigonometric function of the sine-
cosine type .................................... 176
12.2.5 g(t) is a combination of the previous
functions ...................................... 176
12.2.6 g(t) is a generic function of time. The
method of variation of parameters .............. 177
12.3 General solution of the non-homogeneous equation ...... 178
12.4 Continuously distributed lags and partial adjustment
equations ............................................. 179
12.5 Exercises ............................................. 181
12.5.1 Example ........................................ 181
12.5.2 Other exercises ................................ 183
12.6 References ............................................ 183
13 First-order Differential Equations in Economic Models ...... 185
13.1 Stability of supply and demand equilibrium ............ 185
13.2 The neoclassical growth model ......................... 191
13.2.1 Existence of a growth equilibrium .............. 192
13.2.2 Stability of growth equilibrium ................ 194
13.2.3 Refinements .................................... 197
13.2.3.1 Depreciation and technical progress ... 197
13.2.3.2 Golden rule ........................... 199
13.2.4 Further developments ........................... 200
13.2.4.1 Adjustment time or, how long is the
long run? ............................. 200
13.2.4.2 β-convergence, σ-convergence,
and all that .......................... 203
13.2.4.3 Endogenous growth ..................... 205
13.3 Exercises ............................................. 205
13.4 References ............................................ 208
14 Second-order Differential Equations ........................ 209
14.1 Solution of the homogeneous equation .................. 209
14.1.1 Positive discriminant (Δ > 0) .................. 210
14.1.2 Null discriminant (Δ = 0) ...................... 211
14.1.3 Negative discriminant (Δ < 0) .................. 212
14.1.4 Stability conditions ........................... 214
14.2 Particular solution of the non-homogeneous equation ... 215
14.2.1 Variation of parameters ........................ 216
14.3 General solution of the non-homogeneous equation ...... 218
14.4 Determination of the arbitrary constants .............. 219
14.5 Exercises ............................................. 219
14.5.1 Examples ....................................... 219
14.5.2 Other exercises ................................ 221
14.6 References ............................................ 221
15 Second-order Differential Equations in Economic Models ..... 223
15.1 The second-order accelerator .......................... 223
15.2 Exercises ............................................. 226
15.3 References ............................................ 228
16 Higher-order Differential Equations ........................ 229
16.1 Solution of the homogeneous equation .................. 229
16.2 Solution of the non-homogeneous equation .............. 231
16.2.1 Variation of parameters ........................ 231
16.3 Determination of the arbitrary constants .............. 234
16.4 Stability conditions .................................. 235
16.4.1 Necessary and sufficient stability conditions
(Routh-Hurwitz) ................................ 239
16.4.2 Necessary and sufficient stability conditions
(Liénard-Chipart) .............................. 240
16.5 Exercises ............................................. 241
16.5.1 Example ........................................ 241
16.5.2 Other exercises ................................ 242
16.6 References ............................................ 242
17 Higher-order Differential Equations in Economic Models ..... 243
17.1 Feedback control and stabilisation policies ........... 243
17.1.1 Introduction ................................... 243
17.1.2 Three types of stabilisation policy ............ 244
17.1.2.1 Proportional stabilisation policy ..... 247
17.1.2.2 Mixed proportional-derivative
stabilisation policy .................. 248
17.1.2.3 Integral stabilisation policy ......... 249
17.2 Exercises ............................................. 250
17.3 References ............................................ 251
18 Simultaneous Systems of Differential Equations ............. 253
18.1 First-order 2 × 2 systems in normal form .............. 253
18.1.1 General solution of the homogeneous system:
first method ................................... 254
18.1.2 General solution of the homogeneous system:
second (or direct) method ...................... 256
18.1.2.1 Unequal real roots .................... 257
18.1.2.2 Equal real roots ...................... 259
18.1.2.3 Complex roots ......................... 260
18.1.3 Particular solution. Determination of the
arbitrary constants ............................ 261
18.2 First order nxn systems in normal form ................ 261
18.2.1 Solution of the homogeneous system ............. 263
18.2.1.1 The matrix exponential and the
Jordan canonical form ................. 265
18.2.2 Stability conditions ........................... 269
18.2.2.1 D-stability, and stabilisation of
matrices .............................. 272
18.2.2.2 Sensitivity analysis .................. 274
18.2.2.3 A digression on not-wholly-unstable
systems ............................... 277
18.2.2.4 Proof of the stability conditions ..... 281
18.2.3 Particular solution ............................ 282
18.2.3.1 Variation of parameters ............... 283
18.2.4 Determination of the arbitrary constants ....... 283
18.3 General systems ....................................... 285
18.3.1 First-order systems not in normal form ......... 285
18.3.2 Higher-order systems ........................... 287
18.3.2.1 An example ............................ 287
18.3.2.2 The general case ...................... 288
18.3.2.3 Transformation of a higher-order
system into a first-order system in
normal form ........................... 289
18.3.2.4 Stability conditions for higher-order
systems ............................... 292
18.4 Exercises ............................................. 292
18.4.1 Example ........................................ 292
18.4.2 Other exercises ................................ 294
18.5 References ............................................ 295
19 Differential Equation Systems in Economic Models ........... 297
19.1 Stability of Walrasian general equilibrium of
exchange .............................................. 297
19.1.1 Static stability ............................... 299
19.1.2 Dynamic stability .............................. 301
19.2 Human capital in a growth model ....................... 304
19.3 A digression on 'arrow diagrams ....................... 309
19.4 Balanced growth in a multi-sector economy ............. 311
19.5 Exercises ............................................. 316
19.6 References ............................................ 319
III ADVANCED TOPICS ........................................... 323
20 Comparative Statics and the Correspondence Principle ....... 325
20.1 Introduction .......................................... 325
20.2 The method of comparative statics ..................... 326
20.2.1 Purely qualitatively comparative statics ....... 330
20.2.2 The inverse comparative statics problem ........ 330
20.3 Comparative statics and optimizing behaviour .......... 331
20.4 Comparative statics and the dynamic stability of
equilibrium ........................................... 334
20.4.1 Criticism and qualifications ................... 336
20.5 Extrema and dynamic stability ......................... 338
20.5.1 An application to the theory of the firm ....... 343
20.6 Elements of comparative dynamics ...................... 344
20.7 An illustrative application of the correspondence
principle: the IS-LM model ............................ 345
20.8 Exercises ............................................. 349
20.9 References ............................................ 349
21 Stability of Equilibrium: A General Treatment .............. 351
21.1 Introduction .......................................... 351
21.2 Basic concepts and definitions ........................ 353
21.2.1 Stability ...................................... 353
21.2.2 Further definitions ............................ 357
21.2.3 Structural stability ........................... 359
21.3 Qualitative methods: phase diagrams ................... 363
21.3.1 Single equations ............................... 364
21.3.2 Two-equation simultaneous systems .............. 368
21.3.2.1 Introduction: phase plane and phase
path .................................. 368
21.3.2.2 Singular points ....................... 369
21.3.2.3 Graphical construction of the
trajectories .......................... 372
21.3.2.4 Linear systems ........................ 378
21.4 Quantitative methods .................................. 382
21.4.1 Linearisation .................................. 382
21.5 Elements of the qualitative theory of difference
equations ............................................. 387
21.5.1 Single difference equations .................... 388
21.5.2 Two simultaneous difference equations .......... 393
21.5.2.1 Linear systems ........................ 394
21.6 Economic applications ................................. 396
21.7 Exercises ............................................. 396
21.8 References ............................................ 398
22 Liapunov's Second Method ................................... 401
22.1 General concepts ...................................... 401
22.2 The fundamental theorems .............................. 402
22.3 Some economic applications ............................ 407
22.3.1 Global stability of Walrasian general
equilibrium .................................... 407
22.3.2 Rules of thumb in business management .......... 414
22.3.3 Price adjustment and oligopoly under product
differentiation ................................ 415
22.4 Exercises ............................................. 419
22.5 References ............................................ 420
23 Introduction to Nonlinear Dynamics ......................... 423
23.1 Preliminary remarks ................................... 423
23.1.1 A digression on existence and uniqueness
theorems ....................................... 425
23.2 Some integrable differential equations ................ 426
23.2.1 First-order and first-degree exact equations ... 426
23.2.2 Linear equations of the first order with
variable coefficients .......................... 429
23.2.3 The Bernoulli equation ......................... 431
23.2.4 The Riccati equation ........................... 432
23.3 Limit cycles and relaxation oscillations .............. 434
23.3.1 Limit cycles: the general theory ............... 434
23.3.2 Limit cycles: relaxation oscillations .......... 436
23.3.3 Kaldor's non-linear cyclical model ............. 439
23.3.3.1 The model ............................. 439
23.3.3.2 Kaldor via relaxation oscillations .... 443
23.3.3.3 Kaldor via Poincare's limit cycle ..... 446
23.4 The Lotka-Volterra equations .......................... 448
23.4.1 Construction of the integral curves ............ 453
23.4.2 Conservative and dissipative systems, and
irreversibility ................................ 455
23.4.3 Goodwin's growth cycle ......................... 457
23.4.3.1 The model ............................. 457
23.4.3.2 The phase diagram of the model ........ 460
23.4.4 Palomba's model ................................ 463
23.4.4.1 The model ............................. 463
23.4.4.2 Conclusion ............................ 466
23.5 Exercises ............................................. 466
23.6 References ............................................ 469
24 Bifurcation Theory ......................................... 473
24.1 Introduction .......................................... 473
24.2 Bifurcations in continuous time systems ............... 473
24.2.1 Codimension-one bifurcations ................... 475
24.2.2 The Hopf bifurcation ........................... 479
24.2.3 Sensitivity analysis and bifurcations:
a reminder ..................................... 484
24.2.4 Kaldor's non-linear cyclical model again ....... 485
24.2.5 Oscillations in optimal growth models .......... 486
24.2.5.1 The model ............................. 486
24.2.5.2 The optimality conditions ............. 488
24.2.5.3 Emergence of a Hopf bifurcation ....... 489
24.2.6 Cycles in an IS-LM model with pure money
financing ...................................... 491
24.3 Bifurcations in discrete time systems ................. 493
24.3.1 Codimension-one bifurcations ................... 494
24.3.2 The Hopf (or Neimark-Sacker) bifurcation in
discrete time .................................. 496
24.3.3 Kaldor's cyclical model in discrete time ....... 498
24.3.4 Liquidity costs in the firm .................... 499
24.3.4.1 The model ............................. 499
24.3.4.2 The dynamics .......................... 502
24.3.5 Expectations and multiplier-accelerator
interaction .................................... 504
24.4 Hysteresis and bifurcations ........................... 507
24.4.1 General ........................................ 507
24.4.2 Dynamical systems .............................. 509
24.4.3 Economics ...................................... 510
24.5 Singularity-induced bifurcations ...................... 511
24.6 Exercises ............................................. 513
24.7 References ............................................ 515
25 Complex Dynamics ........................................... 519
25.1 Introduction .......................................... 519
25.2 Discrete time systems and chaos ....................... 522
25.2.1 The logistic map ............................... 522
25.2.2 Intermittency .................................. 528
25.2.3 The basic theorems ............................. 529
25.2.4 Discrete time chaos in economics ............... 531
25.2.4.1 Chaos in growth theory ................ 531
25.2.4.2 Exchange rate dynamics and chaos ...... 533
25.3 Continuous time systems and chaos ..................... 535
25.3.1 The Lorenz equations, strange attractors, and
chaos .......................................... 535
25.3.2 Other routes to continuous time chaos .......... 537
25.3.2.1 The Rössler attractor ................. 537
25.3.2.2 The Shil'nikov scenario ............... 538
25.3.2.3 The forced oscillator ................. 538
25.3.2.4 The coupled oscillator ................ 539
25.3.3 International trade as the source of chaos ..... 542
25.3.4 A chaotic growth cycle ......................... 544
25.4 Significance and detection of chaos: Stochastic
dynamics or chaos? .................................... 545
25.5 Control of chaos ...................................... 549
25.6 Other approaches ...................................... 552
25.6.1 Introduction ................................... 552
25.6.2 Fast and slow, and synergetics ................. 552
25.6.3 Catastrophe theory ............................. 556
25.7 Exercises ............................................. 558
25.8 References ............................................ 560
26 Mixed Differential-Difference Equations .................... 567
26.1 General concepts ...................................... 567
26.2 Continuous vs discrete time in economic models ........ 568
26.3 Linear mixed equations ................................ 573
26.4 The method of solution ................................ 574
26.5 Stability conditions .................................. 579
26.6 Approximate methods ................................... 580
26.7 Delay differential equations and chaos ................ 581
26.8 Some economic applications ............................ 582
26.8.1 Kalecki's business cycle model ................. 583
26.8.1.1 The model ............................. 583
26.8.1.2 The dynamics .......................... 585
26.8.2 A formalization of the classical price-
specie-flow mechanism of balance of payments
adjustment ..................................... 589
26.8.2.1 The model ............................. 589
26.8.2.2 Stability ............................. 591
26.9 Exercises ............................................. 593
26.10 References ........................................... 594
27 Dynamic Optimization ....................................... 597
27.1 Introduction .......................................... 597
27.2 Calculus of variations ................................ 599
27.2.1 Particular cases ............................... 601
27.2.2 Generalizations ................................ 603
27.3 The maximum principle ................................. 604
27.3.1 Statement ...................................... 604
27.3.2 Proof .......................................... 606
27.3.3 Transversality conditions ...................... 610
27.3.3.1 The case with infinite terminal
time .................................. 611
27.3.4 Effects of parameter changes on the optimal
solution: the costate variables ................ 612
27.3.5 Discounting .................................... 614
27.3.6 Particular cases ............................... 615
27.3.6.1 The bang-bang control case ............ 615
27.3.6.2 Linear-quadratic problems ............. 616
27.3.7 The maximum principle in discrete time ......... 618
27.4 Dynamic programming ................................... 619
27.4.1 Dynamic programming in discrete time: multi-
stage optimization problems .................... 622
27.4.2 Dynamic programming and nonlinear programming .. 626
27.4.3 Infinite terminal time ......................... 627
27.4.3.1 Solution by conjecture ................ 628
27.4.3.2 Solution by iteration ................. 632
27.4.3.3 Solution by the envelope theorem ...... 633
27.5 Maximum principle vs. dynamic programming ............. 637
27.6 Exercises ............................................. 639
27.7 References ............................................ 641
28 Saddle Points and Economic Dynamics ........................ 643
28.1 Saddle points in optimal control problems ............. 644
28.2 Optimal economic growth ............................... 644
28.2.1 Optimal growth: traditional .................... 644
28.2.1.1 The setting of the problem ............ 644
28.2.1.2 The optimality conditions in the
basic neoclassical model .............. 647
28.2.1.3 Saddle-point transitional dynamics
in the basic neoclassical model ....... 651
28.2.1.4 Optimal and sub-optimal feedback
control ............................... 653
28.2.1.4.1 The sub-optimal feedback
control rule ............... 655
28.2.2 Optimal growth: endogenous ..................... 656
28.2.2.1 A model of optimal endogenous growth .. 656
28.2.2.2 The conditions for optimal
endogenous growth ..................... 658
28.2.2.3 Optimal endogenous growth: saddle-
point transitional dynamics ........... 660
28.3 Optimal endogenous growth in an open economy .......... 664
28.3.1 The Net Borrower Nation ........................ 669
28.3.1.1 Steady-State Stability and
Comparative Dynamics .................. 671
28.4 Rational expectations and saddle points ............... 674
28.4.1 Introduction ................................... 674
28.4.2 Rational expectations, saddle points, and
overshooting ................................... 677
28.4.2.1 A discrete-time equivalent ............ 682
28.4.3 Rational expectations and saddle points: the
general case ................................... 684
28.5 Indeterminacy and sunspots ............................ 686
28.5.1 Indeterminacy and fiscal policy ................ 688
28.5.1.1 Firms ................................. 688
28.5.1.2 Households ............................ 689
28.5.1.3 Government ............................ 689
28.5.1.4 The optimality conditions ............. 690
28.5.1.5 The singular point and its nature ..... 693
28.6 Exercises ............................................. 697
28.7 References ............................................ 699
Bibliography .................................................. 701
Index ......................................................... 731
Answers to Exercises .......................................... 751
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