Bhattacharya R., Majumdar M. Random Dynamical Systems: Theory and Applications
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Author Index
Akerlof, G., 139
Allen, R.G.D., 104, 108
An, H.Z., 346
Anderson, T.W., 375
Arrow, K.J., 112, 113
Athreya, K.B., 217, 317, 343, 344,
348, 369
Azariadis, C., 19, 110, 114
Baccelli, F., 338
Bala, V., 106, 113
Bandyopadhyay, B., 119
Barnsley, M., 285
Baumol, W.J., 105, 108
Becker, R.A., 110, 111
Benhabib, J., 90, 105, 110, 112, 113,
118
Berge, C., 409
Berger, M., 305
Berry, D.A., 417
Bertsekas, D.P., 409
Bhaduri, A., 36
Bhattacharya, R.N., 217, 219, 238,
286, 293, 342, 343, 344, 346, 348,
373, 374, 375, 415, 433
Billingsley, P., 201, 203, 219, 223,
325, 347, 373, 374, 375, 419,
423, 427
Blackwell, D., 384
Block, H.D., 113
Blumenthal, R.M., 285
Boldrin, M., 112
Box, G.E.P., 375
Brandt, A., 305
Brauer, F., 106, 115
Bremaud, P., 338
Brillinger, D.R., 375
Brock, W.A., 111, 112, 263
Brockwell, P.J., 375
Brown, B.M., 373
Carlsson, N., 344
Cass, D., 104, 417
Castillo-Chavez, C., 106, 115
Chakraborty, S., 293
Chakravarty, S., 110
Chassaing, P., 348
Chesson, P.L., 348
Chipman, J.S., 417
Chung, K.L., 205
Clark, C.W., 28, 30, 32, 108
Collet, P., 106
Corson, H., 285
Dai, J.J., 317, 344, 369
Dana, R., 113
Danthene, J., 417
Dasgupta, P.S., 82, 108, 112
Davis, R.A., 375
Day, R.H., 34, 51, 105, 106, 110,
113, 118
Debreu, G., 96, 108, 111, 409,
419
Dechert, W.D., 108
Demetrius, L., 58
Deneckere, R., 112
Devaney, R.L., 14, 17, 34, 39, 106,
108, 110
Diaconis, P., 238, 281, 285, 294,
338
Diamond, P.A., 113
Dieudonn’e, J., 419, 423, 425
453
454 Author Index
Diks, C., 262
Doeblin, W., 205
Donaldson, J.B., 417
Doob, J.L., 205
Dubins, L.E., 282, 284
Dudley, R.M., 219
Dugundji, J., 108
Durrett, R., 205
Dutta, P., 111, 112, 113, 417
Dynkin, E.B., 203
Easley, D., 417
Eckmann, J.P., 106
Elaydi, S., 108
Ellner, S., 348
Elton, J.H., 285, 294
Etemadi, N., 433
Evans, G.W., 294
Feller, W., 204, 205
Fill, J.A., 238
Foias, C., 110
Foley, D., 265, 416
Frankel, J., 375
Freedman, D.A., 281, 282, 284,
294, 338
Friested, B., 417
Frisch, R., 104, 105
Furukawa, N., 416
Futia, C.A., 238
Gale, D., 111, 112, 113, 116
Galor, O., 113
Gantmacher, F.R., 111
Gittins, J.C., 417
Glass, L., 106
Goldberg, S., 108
Goodwin, R., 104
Gordin, M.I., 373, 375
Gordon, R.A., 104
Goswami, A., 348
Granas, A., 108
Grandmont, J.M., 105, 110
Granger, C.W.J., 262, 375
Green, J.R., 277
Grenander, U., 375
Guckenheimer, J., 34
Haavelmo, T., 348
Hakansson, H., 417
Hannan, E.J., 375
Hardin, D.P., 348
Harris, D.J., 36
Harris, T.E., 217
Hassell, M.P., 10
Heal, G., 108
Hellwig, M., 265, 416
Herstein, I.N., 108, 111
Hildenbrand, W., 387, 419
Holme, H., 105
Honkapohja, S., 294
Hopenhayn, H., 293
Huang, F.C., 346
Hurwicz, L., 113, 348
Hwang, E., 375
Ibragimov, A., 373, 375
Jacquette, D., 348
Jain, N., 217
Jakobson, M.V., 33
Jamison, B., 217
Jenkins, G.M., 375
Jones, L.E., 112
Kac, M., 205
Kakizawa, Y., 375
Kamihigashi, T., 108
Kaplan, D., 106
Karlin, S., 205
Katok, A., 275
Kelly, J.L., 425
Khas’minskii, R.Z., 238
Khinchin, A. Ya., 325
Kiefer, N.M., 417
Kifer, Y., 275, 293
Klein, L., 104
Kolmogorov, A.N., 201, 205
Koopmans, T.C., 67, 68, 112
Kot, M., 348
Kreiss, J.-P., 375
Kuller, R.G., 425
Kuratowski, K., 387
Kurz, M., 112
Lee, O., 286, 293, 346, 374
Lee, C., 346
Letac, G., 348
Levhari, D., 416
Li, T., 12
Author Index 455
Lifsic, B.A., 373, 375
Ljungqvist, L., 409
Loeve, M., 181, 301, 302
Lucas, R.E., 293
Main, B., 139
Maitra, A., 97, 410
Majumdar, M., 30, 32, 77, 80, 81,
91, 106, 108, 111, 112, 113, 238,
263, 277, 286, 342, 343, 344, 348,
369, 415, 416, 417
Mammen, E., 375
Mann, H.B., 348
Manuelli, R.E., 112
May, R.M., 1
McFadden, D., 111
McKenzie, L., 104
Metzler, L.A., 104, 105
Meyn, S., 217
Miller, B.L., 417
Mirman, L.J., 263, 416
Misiurewicz, M., 33, 109
Mitra, K., 112, 294
Mitra, T., 30, 32, 77, 80, 81, 91, 92,
106, 108, 111, 112, 263, 348, 367,
416
Montrucchio, L., 112, 113, 348, 367
Mora, M., 348
Morishima, M., 111
Nash, J.F., 409
Nelson, E., 201
Nermuth, M., 108, 111
Neveu, J., 202
Ney, P., 217
Nikaido, H., 59, 65, 111, 114, 419
Nishimura, K., 90, 91, 108, 110, 112
Norman, M.F., 294
Nowak, A.S., 417
Nummelin, E., 217
Nyarko, Y., 263, 416
Olson, L.J., 416
Orey, S., 217
Parthasarathy, K.R., 219, 427
Parthasarathy, T., 417
Pelikan, S., 112
Peres, Y., 367
Phelps, E., 417
Pianigiani, G., 34, 105, 110
Pollard, D., 219
Polya, G., 204
Prescott, E.C., 293
Pristley, M.B., 375
Privileggi, P., 348, 367
Radner, R., 108, 112, 113, 238,
342, 348
Ramsey, F., 60
Ranga Rao, R., 219
Rao, B.V., 293, 344
Ray, D., 108
Rebelo, S., 112
Reed, W., 348
Rhenius, D., 417
Ricker, W.E., 10
Rosenblatt, M., 375
Roskamp, K.W., 112
Ross, S.M., 90, 384, 409
Rothschild, M., 238, 417
Roy, S., 108, 408
Royden, H., 409, 419
Rudin, W., 15, 348
Ruelle, D., 2
Ryder, H.E., 113
Ryll-Nardzewski, C., 387
Saari, D.G., 113
Samuelson, P.A., 1, 4, 38, 58, 59,
101, 102, 104, 105, 113, 115,
277
Sandefur, J.T., 108
Sandmo, A., 417
Sargent, T.J., 409
Sawarigi, Y., 417
Scarf, H., 113
Schaffer, W.M., 348
Schal, M., 417
Shashahani, M., 285
Shlag, W., 367
Silverstrov, D.S., 294
Simon, H., 238
Singer, D., 33
Solomyak, B., 367
Solow, R.M., 51
Sorger, G., 91, 92, 110, 112
Spitzer, F., 338
Srinivasan, T.N., 93, 416
Stenflo, O., 294
456 Author Index
Stiglitz, J.E., 417
Stokey, N.L., 293
Strauch, R., 390
Strook, D., 238
Sundaram, R.K., 112, 113, 417
Tak ´ac, P., 348
Tamiguchi, M., 375
Taylor, A.E., 425
Taylor, M., 205
Terasvirta, T., 262, 375
Tong, H., 262
Topkis, D.M., 90
Torre, J., 110
Tweedie, R.L., 217
Uzawa, H., 93
Verhulst, P.F., 10
Wagner, D.H., 387
Wald, A., 348
Waymire, E.C., 205, 217, 219,
342, 344, 373, 374, 433
Webb, G.F., 348
Wendner, R., 113
Yahav, J.A., 255
Yano, M., 91, 112
Yorke, J., 11, 12, 14
Yoshikawa, T., 417
Young, A., 416
Zarnowitz, V., 105
Zilcha, I., 263, 416
Subject Index
accelerator coefficient, 105
after-n process, 143, 145
aggregative model
of optimal economic development,
60–66
of optimal growth, under uncertainty,
390–397
Alexandrov’s theorem, 219–221, 223, 258,
287
asymmetric random walk on Z
k
, 158
asymptotic productivity, 74
asymptotic stationarity, 121, 196–201
autoregressive moving-average model, 306
of order (k, q), 308
average productivity function, 24, 29
Baire category theorem, 16, 421
balanced growth and multiplicative
processes, 53–59
Bernoulli equation, 146, 211, 366
Bernoulli innovation, 330–336
Bernoulli state, 191
Bifurcation theory, 39–46
Billingsley–Ibragimov central limit theorem,
375
biological reproduction law, 26–28
Birkhoff’s ergodic theorem, 374
birth-and-death chains, 120, 164,
168–170
Blackwell, theorem of, 390
Boltzman’s kinetic theory of matter, 170
Bolzano–Weirstrass theorem, 422–423
Borel measurable function, 96, 125,
230–232, 408, 410, 415
Borel sets, 229, 237, 312, 426
Borel sigmafield, 122, 125, 176, 184, 191,
197, 224, 246, 260, 282, 284, 310, 380,
386, 426
Borel subset, 235, 260, 291
of a Polish space, 121–122, 179, 184, 214,
284, 380
Borel–Cantelli lemma, 280–281, 298,
300–302, 328, 430–432
canonical construction, 123
Cantor subset, 368
capital, 99–100
Cauchy sequence, 6, 180, 236–237, 254,
258, 281, 287, 420
central limit theorem, 121, 133, 360–365
Billingsley–Ibragimov, 375
classical, 192
for Markov chains, 187–191
martingale, 360, 375–378
Chebyshev’s inequality, 354, 361, 432
classical optimization models, 28
Cobb–Douglas economies, 101–104
compactness, 422–423
comparative statics and dynamics, 38–46,
273–275
competitive equilibrium, 115
competitive programs, 66–71
completeness, 420
complex valued function, oscillation of, 229
conditional Lindeberg condition, 373
conditional variance, of g(X
1
), 352
conservative thermostat behavior, 175
consistency condition, 122, 202
consumption program, 18, 61, 78
consumption–investment decisions, 93
continuous time Markov models, 414
contraction mapping theorem, 179
457
458 Subject Index
controlled semi-Markov model, 410–415
convergence, 431–435
in distribution in metric spaces, 121, 187
countable additivity property of probability
measures, 430
critical level, for survival of resource, 27
critical point, 33
decentralized economy, 66, 68
delay distribution, 206
delayed renewal process, 206
dense orbit, 16
diagonalization argument, 63, 423–425
Dirac probability measure, 179
discount factor, 60, 62, 77, 83, 92
discounted dynamic programming, under
uncertainty
applications of, 390–409
complementary theorems and details of,
409–415
dynamic programming, with compact
action space, 388–390
maximum theorem of, 385–388
model of, 380–385
disjoint communicating classes, 131
disjoint cyclic classes, 132
distributional identity, 330
Doeblin hypothesis, 361
Doeblin minorization, 359
condition, 120, 177
theorem, 260–262
duality theory, of dynamic competitive
prices, 60
Dubins and Freedman variants, of Markov
processes, 281–284
dynamic games, 95–98
dynamic multisector economy, 58–59
dynamic optimization model, 90
dynamical system
comparative statics and dynamics, 38–46
complexity theorem and approaches,
11–17
definition of, 3–11
equilibrium of, 4
evolution of, 3
fixed points in, 4
in Cobb–Douglas economies, 101–104
increasing laws of motion, 20–26
linear difference equations, 17–20
periodic points in, 4
quadratic equations of, 32–38
thresholds and critical stocks of, 26–32
dynamical system applications
in balanced growth and multiplicative
processes, 53–59
in dynamic games, 95–98
in dynamic programming, 83–95
in Harrod–Domar Model, 46–47
in multilicative processes, 53–59
in Solow model, 47–53
intertemporal equilibrium, 98–101
intertemporal optimization with a single
decision maker, 59–74
supplementary exercises, 113–119
Dynkin’s π –λ theorem, 203
Economic Dynamics, 23
economic exploitation, 32
Ehrenfest model, 187
of heat exchange, 120, 170, 204
eigenvalues, 307
eigenvector, 55–56
empirical distribution function, 355
ergodic chaos, 109
ergodic measure, 109
ergodic theorem, 349
Ergodicity of Harris recurrent processes,
214
Euclid’s algorithm, 324–325
Euclidean distance, 104
Euclidean norm, of a vector, 304
excess demand, for commodity, 102
exponential convergence, 120
extinction program, 30
as optimal program, 30
feasible production program, 29, 61, 65
Feldman–Mahalanobis model, 110–111
Feller continuity, 199, 276
Feller property, 191–192, 194, 246, 279
Feller’s renewal theorem, 208–210, 216
Finite dimensional weak convergence,
221–223
first passage time, 145
first-order nonlinear autoregressive
processes, 262–263
fixed points, 12, 33, 35–36
basic definitions, 3–11
hyperbolic, 8
non-hyperbolic, 45
Subject Index 459
repelling, 7
unique, 6, 44, 257, 264
Foster–Tweedie criterion, for irreducible
Markov processes, 346
Foundations, 104
Gale prices, 67
gamma innovation, 330
Gaussian hypothesis, 309
dynamic system of, 324–325
Gaussian processes, 123–124
general continued fractions, 325–330
global stability, 7
globally stable, 7
golden rule input, stock , and consumption,
71–72
Guckenheimer dependence, 34–35
Harris recurrent Markov processes, 184,
196, 210–214
Harrod–Domar model, 46–47
harvest program, 29
Hassell map, 11
heavy-tailed distributions, 303
Hilbert cube, 225
homeomorphic image S
h
, 227
immediate return function, 60
independent sequence of random variables,
123
infinite product space, 144
input program, 29, 61
interior optimal processes, 397–402
intertemporal equilibrium, 98–101
invariant distributions, 130
central limit theorems, 360–365
complement theorems and details of,
369–375
condition for
√
n-consistency, 351–360
estimation of, 350–351
nature of, 365–369
of Markov chain, 127, 141
supplementary exercises, 375–379
unique, 162, 167, 169
invariant probability, 130
inverse functions, 206
irreducible chains, 120
iterations
backward, 275, 323
of quadratic maps, 310–317
of quadratic maps, in random, 367–369
of random Lipschitz maps, 275–281
of real-valued offline maps, 297–304
Jakobson’s theorem, 34, 39, 109
Jensen’s inequality, 336, 340
Jordan–Hahn decomposition, of the signed
measure µ
n
, 181
Journal of Economic Theory, 111
Kolmogorov distance, 237–238, 251,
257–258, 271, 273–274, 286, 315, 317,
331
Kolmogorov measures, 275
Kolmogorov metric, 248, 403
Kolmogorov sigmafield, 123, 144, 145, 203
Kolmogorov’s consistency condition, 124
Kolmogorov’s existence theorem, 123–124,
201
Kolmogorov’s maximal inequality, 217–219
k th-order nonlinear autoregressive
conditional heteroscedastic model,
318–323
k th-order nonlinear autoregressive model,
317–323
Kuratowski theorem, 284
law of motion, 3, 380, 384
for cyclical forces, 268
increasing, 20–26
law of transition, 60
Lebesgue differentiation theorem, 348
Lebesgue measure, 34–35, 109, 182,
284–285, 312, 336, 348, 366–367, 369
Lebesgue’s dominated convergence theorem,
199, 278
Li–Yorke Chaos theorem, 2, 3, 11–14,
32–34, 91, 103, 106
robustness of complexicity of, 14–16
linear difference equations, 17–19
linear time series model, 304–310
Lipschitz coefficient, 275
Lipschitz constant, 284
Lipschitz maps, 275–281
Lipschitzian distance, 233
bounded, 276, 279
Lipschitzian function, bounded, 233–234
local attraction, 7
local stability, 7
long-run average returns, 415
460 Subject Index
m-step transition probability matrix, 129
Malinvaud prices, 67
Markov chain
aperiodic, 132, 136, 139
as irreducible, 131–132, 136, 139, 154
central limit theorem for, 187–191
having the transition probability matrix p
and initial distribution µ, 125
hitting time of, 144
Markov properties of, 145–150
on a finite state space, 158–159
positive recurrent, 185
recurrent state of, 154, 158–176
r th return time of, 145
stopping times of, 143–150
strong law of large numbers for, 185–187
transition probability of a, on finite space,
194
Markov models under heterogeneity,
estimation of, 137–139
Markov processes
φ-irreducible and φ-recurrent, 217–219
aperiodic, 184
asymptotic stationarity, 196–201
complement theorems and details, See
Markov processes and compliment
theorems
construction of stochastic processes,
122–126
convergence of steady states for, on finite
state spaces, 133–143
different states of Markov chain, 131–133
measurable state spaces, 176–185
metric spaces, 191–196
on polish space, 210–211
properties of Markov chains, 143–150
recurrence and steady state distributions
of Markov Chains, 130, 151–176
skeleton, 277
stable in distribution, 248
stationary, 130
strong law of large numbers and central
limit theorem, using, 185–191
supplementary exercises on, 239–245
transient and recurrent chains, 150–159
transition probability of, 119
unique invariant probability of, 252,
281–282, 344, 358–359
Markov processes and compliment theorems
Alexandrov’s theorem, 219–221
Ergodicity of Harris recurrent processes,
214–217
Feller’s renewal theorem, 209–210
Finite dimensional weak convergence,
221–227
Kolmogorov distance theorem, 237–238
Prokhorov’s theorem, 234–237
Scheffe’s theorem, 228
Tychonoff’s theorem, 227
Markov property, 126, 128, 130, 203
strong, 145, 173
Markov renewal model, 412
martingale difference sequence, 353
matrix in triangular form, 307
maximal solution of the system, 241–242
maximum sustainable consumption,
25–26
maximum sustainable harvest, 25
maximum sustainable stock for technology,
61
maximum theorem, 386–388
mean value theorem, 8, 23, 43
measurability, 323, 364
change of variable, 428–430
product spaces, 426–428
subspaces, 426
support of a measure, 428
measurable space, 176, 425
metric spaces
and diagonalization argument,
423–425
compactness, 422–423
completeness, 420
infinite products of, 423–425
nonempty compact, 7
separability, 420
mild discounting, 30
minimum safe standard of conservation,
32
monetary equilibrium, 116–117
monotone maps, 255, 257, 286, 322, 362,
389
Multiplicative shocks, 266–267
Newton’s law of cooling, 170
non-decreasing maps, 286, 288, 362
non-increasing maps, 286
nonautonomous systems, 19
nonclassical optimization model, 28
nonnegative definites, 123
Subject Index 461
nonnegative matrix, 55, 58
nonnegative random variables, 303, 323
nonnegative real roots, 30
nonnegativity constraints, 336–338
nonrandom sequence, 370
one-sector growth theory, 24
optimal consumption policy function, 64, 74,
76
optimal exploitation, of a fishery, 82–83
optimal for discounted reward, 412
optimal growth, in a productive economy,
74
optimal investment policy function, 71, 74,
76
optimal policy function, 380, 384
optimal programs
chaotic, 79–80
long-run behavior of, 31
periodic, 77–79
qualitative behavior of, 30
unique, 31, 62–64, 75
optimal stationary programs, 60, 71–72
optimal transition functions, 80
optimization with wealth effects, 77–83
orbit, from x , 4
output program, 29, 61
period-doubling bifurcation, 42
to chaos, in an overlapping generations
model, 115
period-two orbit, 35
periodic points, 12, 33
basic definitions, 3–11
hyperbolic, 8
periodicity, 139
permutation matrix, 53
Perron–Frobenius operator, 55
point process, 206
Poisson equation, 352, 359, 363–364
policy function, 96
positive recurrent state, 176
probability of survival, of economic agent,
339
product sigmafield, 123
product topology, 423
production function, 24, 28, 94, 98
Prokhorov metric, 236
Prokhorov’s theorem, 234–238, 276
pure accumulation program, 30–31
Q-uniformity class, 229
quadratic maps, 310–317
random iterations of, 367–369
stable periodic orbits, 33–37
Ramsey–Euler condition, 69, 71–72, 76–77,
401, 403–404, 406
Ramsey–Weizsacker criterion, 111
random continued fractions
Bernoulli Innovation, 330–336
compliment theorems and details,
342–349
dynamical systems of Gauss, 324
Euclid’s algorithm, 324
general continued fraction, 325–330
model with multiplicative shocks and
survival probability of an economic
agent, 338–342
nonnegativity constraints, 336–338
random dynamical systems, 121, 191–192
applications of, 262–275
compliment theorems and details,
284–294
contractions, 275–284
evolution, 246–247
of optimal inputs, 402–407
role of uncertainity, 248–250
splitting, 250–260
supplementary exercises, 294–296
random dynamical systems, special
structures of
compliment theorem and details,
342–399
iterates of quadratic maps, 310–317
iterates of real-valued affine maps,
297–304
linear autoregressive models, 304–310
random dynamical systems (cont.)
multiplicative shocks model, 338–342
NLAR (k) models, 317–323
NLARCH (k) models, 317–323
non negative constraints, 336–338
other linear time series models, 304–310
random contained fraction, See random
continued fractions
survival probability of an economic agent,
338–342
random Lipschitz coefficients, 275
recurrent chains, 120
regeneration program, 31–32
462 Subject Index
regeneration times, 210–211
regenerative structure, 210–211
relative stability, of a balanced-growth
solution, 58
renewal epochs, 206
renewal methods, 364
renewal process, 206
renewal times, 206
residual lifetime process, 206
Ricardian model, 36
Ricker map, 11
Riesz representation theorem, 202
risky capital, accumulation of, 407–409
Robinson Crusoe economy, 67
robust phenomenon, 80
Samuelson model, 105
Sarkovskii ordering, 14, 33
Sarkovskii theorem, 14, 32
Scheff´e’s theorem, 162, 228, 230
semi-Markov model, See Markov renewal
model
separability, 420
sequential decision-making process, 60
simple symmetric random walk on Z
k
,
154–155
Sinai–Ruelle–Bowen measures, 275
single decision maker, 59–74
Solow Model, 47–52
space the variation norm, 228
speed of adjustment parameter, 102
speed of convergence, 350
splitting class, 256
splitting conditions, 250–260, 266, 270,
313–315
splitting property, 316, 345
stability in distribution, 121
stable periodic orbits, 33–37
standard monotone class theorem, 203
standard state space, 121
state space, 3
state-dependent actions, 415–419
stationary distribution function, 130, 255
stationary dynamic programming problem,
59
stationary optimal stock, 78
stationary stochastic process, 130
stationary strategy, 96–97
stationary, definition of, 4
Stirling’s formula, 154, 157, 204
Stone–Weirstrass theorem, 202
stopping time, of events, 144, 173, 207
strict concavity, 395
strict contraction, 368
uniformly, 5
strict contractions, 7
strict inequality, 395
strictly positive program, 61–62, 65, 67
strong discounting, 30
strong law of large numbers, 121, 160, 163,
214, 276, 340
for Markov chains, 185–187
subsistence level of minimal consumption,
of a worker, 24
supermodularity, 90
symmetric definites, 123
symmetric Nash equilibrium, 97–98
tent map, 9
thermodynamical equilibrium, state of,
171
thermodynamics, 171
thermostat model of managerial behavior,
120, 174–176
three-period cycle, 91–92
tightness, 234
time invariant law of motion, 19
time-homogeneous process, 119
topological chaos, 80–83
topological conjugacy, 39
topological transitivity, of maps, 16–17
trajectories
and nondecreasing sequence, 21
and nonincreasing sequence, 21
from x , 4–5, 20, 27
global behavior of, 45
long-run behavior of, 17, 26
transition functions, 85, 88–90
transition probability density functions,
125
transition probability matrix, 126, 141–142
turnpike theorems, 61, 72–77
two-sector model, of optimal economic
growth, 93
Tychonoff’s theorem, 227
undiscounted continuous time model of
Ramsey, 60
unique probability distribution, 136
unique steady state, 121