Subject Index

A

Abundances, 10, 14, 223, 229 Aggregate dynamics, 7, 100 binary models, 41-43 business cycles, 116 combinatorics and, 49-51 critical points, 47-49 demands, 103-104 Diamond model, 130-131 feedback in, 41, 69 fluctuation and, 35-36, 41-51, 131-132 Fokker-Planck equations, 25, 97 hazard function, 47-49 logistic process models, 35-40 master equation.

See also Clusters; specific models, parameters

Alarm clock model, 205

Alternatives, evaluation of, 52-65. See also specific models

Anonymous interaction, 41 Approximate evaluation, 60 Ascending factorial, 207, 217 Assemblies, 141, 143, 150-153 Asymmetric interactions, 80 Asymmetrical cycles, 128, 136-138 Asymptotic relations, 210-212 Asymptotic stability, 45

B

Backward Chapman-Kolmogorov equations. See Master equation

Bandwagon effects, 1

Bankruptcy, 78-79

Bare-bones models

business cycles, 86, 96-99 demand in, 90-94, 122-126 growth rate and, 99-117

Langevin equation, 118-122

nonlinear, 94-95

Poisson models, 87-88

Ramsey model, 93

sectoral, 95-96, 99-126 stochastic,96-99, 117-120 urn models, 88-89, 197 See also specific models, parameters

Beta distribution, 187, 197, 230-231

Binary models, 27, 63, 145

aggregate dynamics and, 41-43

closed, 30-32

dynamics of, 41-43

fluctuations and, 41-43

open, 32-35

See also specific types

Binomial coefficients, 46

Binomial distribution, 35, 81, 148 Binomialrandomvariable, 143 Birth-and-death process models, 78

bankruptcy and, 78

clustering and, 154

generating functions, 70-75 immigration and, 16, 32-34, 69-75, 148 nonstationary master equations, 69-75 stationary distribution, 70

transitionrates,73-74, 144-150

See also specific models

Black-Scholesequation, 121

Block structures, 151, 227

Bose-Einstein statistics, 149

Boundary-condition equations, 60 Brownian motion, 48

Business-cycle models, 32, 86, 96-99, 111

C

Capacity-limited processes, 150

Categories, 9

See also types

Cauchy formula, 151, 216, 218, 226 Centering constant, 214

Chapman-Kolmogorov equation.

Characteristic curves, 195-197

Characteristics, method of, 70

Chartists, 188

Choice, 142

set, 2

Closed models, 16, 18, 27, 30-32

Clusters

assemblies and, 150-153

capacity-limited processes and, 150 diffusion equation and, 122-126 discrete frequency spectrum, 177-178 dynamics of, 157-165

Ewens formula, and, 160-162, 184 expected value of, 166-169 frequency spectrum, 156-157, 172-178 Herfindahl index, 173-174

heuristic derivation, 17^-176

interaction patterns and, 141-179

joint probability density, 169-171

large, 165-171

market share models, 184-185

moment calculations, 171-172

multisets, 146-150

parameter estimation, 178-179

partition vectors and, 153

size distributions, 141-144

social, 154

transition rates, 144-153

Coalescents model, 124

Combinatorics, 8, 49-51, 141, 209-210

Community preference, 57

Component sizes, 141n1

Conditional limit theorem, 58

Configuration, 3

Congestion effects, 76, 91 Consistency, of partition, 229, 230 Consumption models, 32, 93-94 Continuous-time dynamics, 19-23

Critical points, 47- 49

Cumulant generating functions, 68-69,

73-75

Cycles. See Business cycles; Permutation cycles

D

Day-Huangmodel, 181

de Finetti theorem, 12-13, 221

Decision rules, 2

Decomposable structures, 8, 142

Defining events, 124

Demand, 86, 90, 104-105 Demography, 2, 5, 11, 143, 230 Detailed-balance conditions defined, 18 equilibrium distribution, 40, 157, 159, 161, 165, 202

Ewens formula and, 158 stationary solutions and, 26, 34, 66-67, 81, 91

Deterministic dynamical equation, 117-118 Deterministic process, 36, 40, 76-79 Deterministic share dynamics, 77-78, 95-96, 181

Deterministic state variables, 60 Developingeconomies, 142 Developmentalhistory, 124 Deviationalequation, 136-138

Diamond model, 3, 7 aggregate dynamics, 25, 130-131 asymmetrical cycles, 136-138 equilibrium selection, 138-139 expected value functions, 133-134 extensions of, 139-140 fluctuations, dynamics for, 131-132 mean of fraction, 130-131 mulitiple equilibria and cycles, 134-138 transition rates, 129-130 value functions and, 60, 132-134

Diffusion process, 23n3, 44, 55, 93, 122-123, 244

Dirichlet distributions, 122, 225, 230-236 exchangeability and, 166 generalized, 238

Dirichlet distributions (cont.)

marginal distribution of, 172, 233

Poisson-Dirichlet distribution, 233-234 size-biased sampling, 234-235

Disappearance, of goods, 90-93

Discount rate, 95, 132

Discrete choice theory, 53, 143

Discrete frequency spectrum, 172, 177 Discrete-time models, 25-26, 88 Disequilibrium analysis, 100

Dixit model, 32

Double exponential distribution, 214 Double-well potential, 99

Dynamic models, 9-19.

Dynamic programming, 132

Dynamic search models, 52

Dynamics, aggregate. See Aggregate dynamics

E

Ecologicalproblem, 10, 13-14, 173, 222

Economy-wide effects, 34

Ehrenfest model, 31, 145

Emergence, of sectors, 110-111

Empirical distributions, 11-12, 31, 225 Entropy, 31, 46, 50, 51

Entry/exit models, 60-61, 215

Markovprocessesand, 141-152, 181 probabilities for, 102, 200-205 sectors and, 110-112

Equilibrium cycles, 112

Equilibrium distribution, 67

Equilibrium models, 105-107, 127-128, 157

Equilibrium selection, 138-139

Ergodic assumption, 191

Error functions, 48, 53, 62-65

Euler's constant, 169, 185, 212

Ewens distribution

cluster and, 184-185

expected value functions, 185-186

Hoppe model, 185 market shares and, 181-183 parameter estimation, 178 partition vectors, 174

See also Ewens sampling formula

Ewens sampling formula, 141, 153, 199, 216,

225

ascending factorial, 216 clustering and, 162, 184 detailed-balance conditions, 158 developmental history and, 125 entry rate and, 102n4 examples, 162-164 partition vectors, 181

Poisson variables and, 215-219 sectors and, 110n6 urn model and, 198

See also Ewens distribution

Excess demands, 27-30, 104-105, 188-193

Exchangeability, 200, 220

Dirichletdistributionsand, 166 partitions and, 15, 221-225 sequences, 12-16

Exit rate, 37-38, 102, 147

Expected

fractions, 169-171, 185

values, 166-169

value functions, 133-134

Exponential generating functions, 151, 215,

227

Externality, 41

Extreme value distributions, 53, 57-59

F

Factor productivity models, 88

Factorial, ascending, 207, 217

Fashion, 1

Feedback effects, 69

Fermi-Dirac statistics, 150

Field effects. See Aggregate dynamics

Final goods, 93-95

Finitaryapproach, 143

Firms, 141

growthratesof, 119-120 stochastic model, 83-89

Fish-market example, 230

Fluctuations

aggregate dynamics and, 35-51 Diamond model, 131-132 dynamics for, 131-132 growth and, 85-126

logistic process models, 35-40 two-sector model, 27-30 underutilized production factors, 99-117 Fokker-Planck equations

aggregate dynamics and, 25, 80, 97 defined, 19, 24-25, 119

equilibria and, 38-39

Ito representation and, 120

Langevin equations and, 142 linear, 25, 80 master equation and, 19, 24-25, 97, 128 nonstationary, 142n3

random deviations and, 128, 131 See also Master equations ForwardKolmogorovequation, 122 Frequencies of frequencies, 10, 14, 223, 229 Frequency spectrum, 172-178

cluster size and, 156-157, 172-178 definition of, 173

derivation of, 174-176 discrete, 172, 177-178

Herfindahl index, 173-174 interaction patterns, 156-157, 172-178 logarithmic distribution, 156-157

Frequency vector, 221

Fundamentalists, 188

G

Gamma distribution, 232

Gamma function, 32

Gaussian distribution, 99, 138

GDP.

Griffiths-Engen-McClosky distribution

Generalized extreme value (GEV) functions, 65

Generating functions, 123 birth-death-with-immigration and, 70-75 characteristic curves and, 195-197 cumulant, 68-69, 74-75 master equation and, 70-73 methods of, 66-67, 75, 151 nonstationary distributions, 7 probabilities and, 67-68

Stirling numbers, 208

Generator matrix, 205

Genetics literature, 124

Geometric distribution, 35

GEV. See Generalized extreme value functions

Gibbs distribution, 47, 51, 57-59, 62-64 Gibbs inequality, 23

Goods, in markets, 87-93, 122-126 Griffiths-Engen-McClosky (GEM) distribution, 231, 235, 236-240

Gronwall inequality, 93

Gross domestic product (GDP), 99, 100, 107 Growth, 85-126

aggregate effects, 102-103 demand conditions and, 104-105 deterministic share dynamics, 95-96 diffusion approximation, 121-123, 244 emergence of new sectors, 110-117 equilibrium and, 104-107 exponential distribution of, 119-120 of firms, 119-120 fluctuations and, 85-126 holding times, 102-103 innovation and, 90 invention and, 123-126

Langevin equation and, 117-121 ofmarkets, 87-93, 122-126 partition vectors, 157

Poisson model, 87-88 production factors and, 99-117 random processes, 243 rates of, 3, 32, 119-120 sector size equilibrium, 104-105 stability analysis, 92-93 stationary distributions, 107-110, 117-119 stochastic business-cycle model, 96-99 time-dependent density, 121-122 transition rates, 101-102 urn model for, 88-90

See also specific models, parameters

H

Hamming distance, 72, 73 Harzard functions, 47-49

Heat equation, 121-123 Herding effects, 128n3

Herfindahl index, 142, 173-174 Heterorelevance coefficient, 201 Hicks-neutral progress factor, 90

Holding times, 28, 29

growthand, 102-103

Markov chains and, 27-29, 106, 202-205 production factors and, 102-103 sojourn time models and, 205

Homozygosity, 173

Hoppe urn model, 185, 197-199 Households model, 93-95, 141 Hypergeometric

distribution, 35 function, 123

I

Imitation process, 75-84

stochastic model for, 80-84

Immigration, 16, 32-34, 69-75, 148, 154

Industrial organization, 3, 116, 142, 173, 214

Infinite-dimensional simplex, 239 Information measure, 51, 53

Innovation, 75-80

deterministic process, 76-79

growth and, 90

imitation and, 77-78 invention and, 123-126 market shares by, 75-80

nonstationary equations, 75-80

stochastic dynamic model, 79-84

Interaction patterns

assemblies and, 141, 150-153 asymmetric, 80

capacity-limited processes, 150

cluster distributions and, 141-179 discrete, 177-178

frequency spectrum, 156-157, 172-178 Herfindahlindex, 173-174 heuristic derivation, 174-176 joint probability density, 169-171 large clusters and, 165-171 logarithmic series distributions and, 153-157

moment calculations, 171-172 multisets, 141, 146-150 parameter estimation, 178-179 partition vectors and, 153 r fractions, 169-171 selections, 141

stochastic, 41

symmetric, 84

time profiles, 1 transition rates, 144-153

See also Aggregate dynamics; Clusters

Interest rate, 61

Invention, 28, 123-126. See also Innovation

Ising model, 41

Ito representation, 118, 120

J

Jacobian matrix, 167, 240

Jensen’s inequality, 168

Johnson’s sufficientnesspostulate, 13, 102, 222

Joint probability density, 169-171

Jordan formula, 212

Jump Markov processes, 6-7, 9, 55

entry and exit rates, 141, 181 skeletal Markov chain, 204 time-homogeneous, 202

transition rates, 9

K

K-sector model, 27 Kelly’s theorems, 161 Kendall-Kellyidentity, 158, 165

Kimura solution, 122

Kingman representation theorem, 222, 225 Kingmantheory, 124, 146, 199

Kirman model, 47n4

Kiyotaki-Wright model, 3 Kolmogorov

criterion, 46 cycle conditions, 18

Kolmogorov forward equation, 21, 119, 122 Kullback-Leiblermeasure, 30, 51, 53

L

Labeling, 10, 143, 143n5

Lagrange multipler, 54

Langevin equations, 117-121, 142

Laplace

method of, 61

transform, 61, 167, 186 Large-deviation theory, 218 Lawler model, 205 Lead and lag operators, 43, 98 Lebesgue measure, 239, 241 Levy-Khinchin formula, 167, 232 Logarithmic

distribution, 156-157 series, 153-157

Logistic process models, 35-40, 63, 214 Lyapunov functions, 22

M

Macrosignals, 54

Market share models, 85, 153-154, 214

Ewens distribution, 181-183 excessdemand, 104-105, 188-193 expected value of fractions, 185-186 fish-market example, 230 groups of traders, 180-194 growth rates, 142

Herfindahl index, 3 imitation process, 77-78 innovation process, 76-77 joint deterministic process, 78-79 nonstationary equations, 75-80 number of clusters, 184-185 stochastic dynamic model, 79-80 transition rates, 181-183 volatility, 187-188

Markov processes

conditional probabilities, 17

entry and exit rates, 141, 181

finite states, 20-21

graph underlying, 22

holding times, 27, 102-103, 202-205 jump.

transition rates, 9

urn models, 197-200

Marshall quantity adjustment model, 99n3 Martingale convergence theorem, 197

Master equation, 6-7, 19-27

aggregate dynamics. See Aggregate dynamics

approximate solutions, 36-37 continuous-time dynamics, 19-23 discrete-time dynamics, 25-27 generating functions, 70-73 nonstationary, 66-75. See Nonstationary master equations

power-series expansion, 19, 23-25, 66, 130 probability distribution, 19

size effects, 2

stationary solutions, 7

stochastic model, 36

time evolution of, 7

time-varying transition rates, 73

See also Fokker-Planck equations; specific models

Maximum-likelihood, 47

McFadden model, 58

Mean field effects. See Aggregate dynamics Microshocks, 128n3

Moment calculations, 171-172

Money traders, 139, 140

Monte Carlo experiments, 111-112 Multi-sector model, 27, 111

Multiagent models, 13-14, 222

Multiple equilibria, 86, 127-128, 134 138 Multisets, 141, 143, 146-150

Mutator, 225

N

Negativebinomialdistribution, 35, 148 Neighborhood interactions, 41

Networks, electrical, 55

Newgoods, 87-90, 123

Newton series expansion, 207, 208

Niche effects, 34, 148

Non-Walrasian analysis, 100

Noninterference, 229-230

Nonlinearity, 37-41, 127-128 Nonstationary distributions, 7, 84 Nonstationary master equations birth-death processes, 69-75 cumulant generating functions, 68-69, 74-75

deterministic process, 76-79 equilibrium distribution, 67 finite number of firms, 83-84 generating functions, 67-68, 70-73 imitation process, 77-78 innovation process, 76-77 market share models, 75-80 open models, 66-69

Polya model and, 38-40

stationary probability distribution, 70 stochastic dynamic model, 78-84 time-inhomogeneous transition rates, 73-74

See also Master equations Normalizingconstant, 45, 108, 160

O

Occupancy problems, 5, 12, 143

Old-age effect, 91

Open models, 16, 27, 32-35, 66-69 Operator notation, 98

Optimalitycondition, 134 Option-pricing equation, 121

Order statistics, 15, 143,213-214

P

Pairwise interaction, 41

Parabolic partial differential equations, 119, 121

Parameter estimation, 178-179

Partition vectors, 14, 143, 151

cluster distributions and, 153 detailed-balance conditions and, 165 equilibrium distribution, 165

Ewensformula, 162, 174, 181 growth processes, 157 interaction patterns and, 153 partition process, 199 state vectors, 5-6, 10 transition rates and, 153 urn models, 198-199

See also Partitions

Partitions, 210

consistency and, 229-230

Partitions (cont.)

exchangeable, 14, 15, 220-224

finite sets, 210

noninterference of, 229-230 patterns of, 10, 164-165 permutations and, 225-229 random, 12-14, 166, 226-229 vectors.

Percolation models, 142n2 Permutations

consistency and, 230 cyclic products, 14, 16, 209-210, 223 noninterference and, 229-230 random partitions and, 226-229

Poisson approximation, 35, 214 Poisson-Dirichlet distribution, 8, 225,

233-234,237-238

Poisson distributions, 35, 81, 165, 218

Poisson growth model, 87-88

Poisson process, 129

Poisson random variables, 215-220 Polyadistribution, 26, 31-32, 38 40, 148 Polyamodel, 13, 88-89, 164, 197-198, 222-225

Population genetics, 10, 14, 28, 172, 223 Potential, definition of, 45 49

Power laws, 190, 241, 244

Power-series expansions, 19, 23-25 Pricing theory, 190-194 Probability-generating function method, 67-68

Product developments, 123-126 Production factors

aggregate outputs, 102-103 emergence of new sectors, 110-111 employment and, 129 equilibrium and, 105-107

excess demand conditions and, 104-105 growth and fluctuations, 99-117 holding times, 102-103

sector size equilibrium, 104-105 stationary probability distribution, 107-110

transition rates, 101-102 underutilization of, 99-117

Productivity across sectors, 86

Q

Quantity adjustment model, 99n4, 100 Quasilinearpartial differential equations, 195

R

Ramsey model, 93

Randomcombinatorial structures, 49-51, 141, 144, 146, 150

Random graphs, 146n6

Random growth processes, 243

Random partitions, 12-14, 166, 219, 226, 229

Random walks, 112, 145

Ranking function, 240n10

Recursion, 176-177, 207-209

Reversibility, 18

Reservation cost, 61, 129, 134

Residual allocation models, 235, 238

Riccati equation, 73

Risk, 128n3

S

S-shaped profile, 94

Sampling-of-species problem, 6, 13, 14, 15,

180, 222-223,229

Savings rate, 95

Scaling, 20,43-44

Schmookler study, 28

Search model. See Diamond model

Sector sizes, 28, 104-105, 110-111

Selections, 141, 143, 144-146, 150

Separation of variables, 131

Series expansion, 23-25

Shannon entropy, 30

Simple models, 27-40, 87

Simplexanalysis, 143, 231, 239

Singleton, 158

Size-biased distributions, 122-126, 141-144, 153, 231-240

Size distributions, 122-126, 141-144, 153

Skeletal Markov chains, 29, 103n5, 202-205

Social influence. See Aggregate dynamics

Sojourn time, 28, 102, 203, 205

Species sampling, 180

Stability analysis, 92-93

State-dependent models, 42

State vectors, 3, 5-6, 9-11

Stationary distributions, 70-73, 84, 107-110,

117-119

Step function, 134

Stirling numbers, 151, 206

approximation, 164 asymptotics and, 210-214 combinatorics and, 209-210 cycle structures, 209 explicit expressions, 210-212 finite-difference expressions, 207 first kind, 156, 176, 185 formula for, 30, 50, 156 generating function, 208 inverse relation between, 213 recursion and, 89, 207-209, 226 second kind, 207, 210, 226, 228 signless, 208

Taylor series and, 206

unsigned, 89

Stochastic difference equation, 241-243 Stochastic models

business cycles and, 96-99, 128n3 difference equations, 241-243 finite number of firms, 83-84 imitation or innovation, 79-84 market share models, 79-80 master equation, 36, 79-84 nonstationary equations, 79-80 power laws, 241

Stock markets, 13-15, 66, 75-80, 181-183

Strong Markov property, 203

Structure, 151

Sufficient statistics, 12

Sufficientness postulate, 13, 102, 222 Sums, approximate, 61-62

Symmetric interactions, 84

T

Tail behavior, 241

Taylor series, 7, 66

Technical progress, 28, 85, 90 Tilted probability distributions, 144 Time-dependentdensity, 121-122 Time dynamics, continuous, 19-23

Time homogeneity, 73-74, 203 Transition rates, 129

assemblies and, 150-153 capacity-limited processes, 144-153 changes oftype, 200-202

cluster size distributions and, 144-153 Diamond search model, 129-130 dynamic models, 16-17

entries/exits, 200-203

growth and fluctuations, 101-102 interaction patterns and, 144-153 jump Markovprocesses, 9 market shares and, 181-183

multisets, 146-150

partition vectors and, 153

production factors, 101-102

selections, 144-146, 150

simple, 33, 146

specifications of, 101-102 time-inhomogeneous, 73-74 types of, 144-153

Tree structure, 70, 145

Two-sector model, 105-111

Types see categories

U

Unanticipated knowledge, 222

Uncertainty, 32, 45, 47

Unemployment, 60, 129-135

Urn models, 31, 87, 88, 163

emergence of goods, 88-90

Ewens formula and, 198

growth and, 88-90

Hoppe model, 197-199

interpretation of, 125

Markov chains and, 197-199

partition vectors, 198-199

Polya model. See Polya model

Possionapproximation, 215

U.S.Patent Office, 28

Utilities, maximization of, 58

V

Value functions, 54-57, 132-134

approximate, 60

Diamondmodel, 132-134 evaluations of, 60 number of alternatives, 60

VanLint-Wilsontheorem, 151, 153,227-228

Volatility, 189-190

W

Watterson equation, 124125, 241

Wiener process, 118, 120

Wilks theorem, 235

Y

Yoshikawa model, 111

Z

Z-transforms, 98

Zero excess demand, 104, 188-189

Zipfdistribution, 153,244