Statistical Equilibrium Wealth Distributions in an Exchange Economy with Stochastic Preferences

417

⁄0022-0531/02 $35.00

© 2002 Elsevier Science (USA)All rights reserved.

Journal of Economic Theory 106, 417–435 (2002)doi:10.1006/jeth.2001.2897

Statistical Equilibrium Wealth Distributions in anExchange Economy with Stochastic Preferences

The U.S. Government’s right to retain a nonexclusive royalty-free license in and to thecopyright covering this paper, for governmental purposes, is acknowledged.

Jonathan Silver1

1 To whom correspondence should be addressed. Fax: (301)402-0226.

Laboratory of Molecular Microbiology, National Institute of Allergy and Infectious Disease,National Institutes of Health, Bethesda, Maryland 20892

[email protected]

Eric Slud

Department of Mathematics, University of Maryland, College Park, Maryland 20742

and

Keiji Takamoto

Laboratory of Molecular Microbiology, National Institute of Allergy and Infectious Disease,National Institutes of Health, Bethesda, Maryland 20892

Received January 2, 2001; final version received June 25, 2001

We describe an exchange market consisting of many agents with stochastic pref-erences for two goods. When individuals are indifferent between goods, statisticalmechanics predicts that goods and wealth will have steady-state gamma distributions.Simulation studies show that gamma distributions arise for a broader class of pref-erence distributions. We demonstrate this mathematically in the limit of largenumbers of individual agents. These studies illustrate the potential power of a statis-tical mechanical approach to stochastic models in economics and suggest that gammadistributions will describe steady-state wealths for a class of stochastic models withperiodic redistribution of conserved quantities. Journal of Economic Literature Clas-sification Numbers: C15, C62, C73, D3, D5. © 2002 Elsevier Science (USA)

Key Words: equilibrium; wealth distribution; stochastic preferences; gamma dis-tribution; Markov model; exchange market.

1. INTRODUCTION

Academic models of financial markets since the beginning of the 20thcentury have noted the essentially random nature of successive changes in

asset prices. Yet classical economic theory describes deterministic transi-tions of the detailed states of an economic system which balance supplyand demand in response to changes in preference and in information (e.g.,in expectations of future returns). This dichotomy between (largely)deterministic detailed transitions and stochastic observed aggregates ineconomics [1, 2] is analogous to that in physics between motions of New-tonian particles and macro- level observations of Brownian motion andthermodynamic variables [3–5]. In both economics and physics, the limit-ing behavior for superposed effects of large numbers of individuals(particles) is due to probabilistic central limit theorems. The first suchtheoretical results asserted Gaussian-distributed displacements [1–4]. Morerealistic models in physics, going beyond Gaussian observations, haveincorporated interactions among particles [5]. In economics, statisticaldescription of aggregates based on probabilistic limit theorems has beenbroadened to include infinite-variance laws and stable-increments processes[6–8]. Economic models leading to non-Gaussian distributions ofteninvolve complex or heterogeneous interactions among market participants[9–11], but results with true statistical-mechanics flavor, deriving suchbehavior from detailed microeconomic-level mechanisms, are rare.

One early attempt to incorporate thermodynamic and statisticalmechanical reasoning in economics [12] was largely confined to analogywithout detailed mathematical content. Another approach [13], which welargely follow, formulates the problem of economic market equilibrium asthat of finding a time-invariant, nondegenerate probability distribution in ahigh-dimensional state space describing the fortunes of a large populationof firms and individuals. The authors of [13] propose that micro-economic production decisions be treated deterministically while marketfluctuations based on changing preferences and exogenous influences beconsidered a stochastic process. Reference [2] provides one of the fewcases of analysis with simplifying large-population approximations leadingto true invariant distributions of a large state-space Markov model in eco-nomics. There are several references, and a growing physics literature, withstatistical mechanics flavor in economic modeling, which describe equilibriavia maximum ‘‘entropy’’ configurations subject to constraints such as fixedtotal wealth [14–16].

In this paper, we adopt a combined theoretical and simulation approachto a simple model in which a fixed supply of goods is traded among indi-viduals whose preferences for goods are determined stochastically. Acentral feature of our model is that individuals periodically change theirpreferences for goods independently, according to a distribution of pref-erences in the population. The changes in aggregate demand translatein the next time-period into changes in the market-clearing price. Asindividuals adjust the mix of assets they hold to satisfy their changing

418 SILVER, SLUD, AND TAKAMOTO

preferences, the value of their goods waxes and wanes according to howwell, by chance, they anticipate future demand.

Using four separate lines of argument, of increasing generality, we showthat the model results in a probabilistic steady state, independent of initialconditions, with gamma distributions of goods and wealths. The first is asimulation study, which confirmed the steady-state behavior of the modelunder various assumed preference distributions and showed that wealthsand amounts of goods behaved like gamma random variables with param-eters depending only on the preference distribution. The second line ofargument follows the heuristic of [14] from statistical physics in calculat-ing maximum ‘‘entropy’’ configurations subject to the constraint of fixedtotal wealth. This argument is applicable only when individuals are indif-ferent between goods, in which case we, like [14], treat all microstates asequiprobable because all microstates accessible from each wealth-configu-ration are. The third line of argument is a simple consequence of a theoremof [17]: gamma-distributed wealths are implied by the assumption that insteady state, wealths are jointly independent with ratios independent oftheir sums. Our final argument, a mathematical development along thelines of the approach to statistical mechanics in [4], shows how the gammadistribution for wealths follows necessarily from the assumption that theindividual wealths have steady-state distributions characteristic of a set ofindependent identically distributed variables conditionally given their sum.The generality of our result that wealths have steady-state gamma distribu-tions, across our family of models, suggests that it may be applicable to avariety of stochastic models in which conserved quantities are periodicallyre-balanced.

2. THE MODEL

For simplicity, we begin by considering a market consisting of N indi-viduals and two goods, A and B, the total number of units of each beingconserved respectively at aN and bN. At the start, we assign to individuals(indexed by i) certain amounts ai0 of good A and bi0 of good B. At allinteger times t, individuals hold amounts of good A and B respectivelydenoted ait and bit, and measure their total wealths in terms of an externalnuméraire, such as gold or inflation-adjusted specie, the total quantity ofwhich is conserved, at wN. We use such a‘‘gold standard’’ in order tomaintain symmetry in the treatment of goods A and B, and with a view toallowing many more goods in further developments of the model. Asomewhat simpler and less artificial approach, which we also follow indefining notations, is to view units of A as numéraire, with Jt denoting thetime-dependent value of one unit of good B in units of good A at time t.

EQUILIBRIUM WEALTH DISTRIBUTIONS 419

Initially, the holdings ai0, bi0 are assigned; thereafter, at each time-periodt \ 1, individuals determine fractions of their wealths

wit=ai, t−1+Jtbi, t−1 (1)

to allocate to good A, with the remainder allocated to good B. This isequivalent to maximizing over ait ¥ [0, wi, t−1] an individual Cobb–Douglas[18] utility function of the form

Uit(ait)=(ait)fit (wit−ait)1−fit, (2)

where fit is a doubly indexed sequence of independent and identically dis-tributed random variables with values in the interval [0, 1]. The maxi-mization of (2) results in the assignment

fitwit=ait(1−fit) wit=Jtbit.

ˇ (3)

The so-far-undetermined price Jt is uniquely determined from variables attime t−1 together with {fit} via the market-clearing constraint

CN

i=1ait=aN, C

N

i=1bit=bN. (4)

This constraint immediately yields

CN

i=1fitwit=aN, C

N

i=1(1−fit) wit=JtbN (5)

from which there follows, upon substituting (1) for wit and solving for Jt,

Jt=CN

i=1(1−fit) ai, t−1;C

N

i−1fitbi, t−1. (6)

Substituting this expression for Jt back into (3) leads to the expression

ait=fitai, t−1+fitbi, t−1 ; j (1−fjt) aj, t−1/; j fjtbj, t−1bit=(1−fit)[bi, t−1+ai, t−1 ; j fjtbj, t−1/; j (1−fjt) aj, t−1].

4 (7)

Thus, individuals’ holdings of goods (7), wealth (1), and prices (6) at time tcan be expressed in terms of their previous holdings (ai, t−1, bi, t−1) and theN random variables {fit}, making this a Markov process discrete in timewith 2N continuous state-variables ait and bit (which satisfy two linearconstraints, making 2N−2 the effective state dimension).


The equations determining new state from old have the alternativeexpression (6) together with

RaitbitS=R fit

(1−fit)/JtS R 1Jt

S − Rai, t−1bi, t−1S (8)

or, after premultiplying the last equation by the row-vector (1, Jt+1),

wi, t+1=1fit+Jt+1

Jt(1−fit)2 wit. (9)

If values are expressed in units of a ‘‘money’’ numéraire of which there isa total quantity wN, the aggregate relations (5) imply that the total of(a+bJt) N units of numéraire A at time t has value wN, so that 1 unit ofA has monetary value w/(a+bJt) at time t, and all prices in units of A areconverted to money prices through multiplication by the factorw/(a+bJt). The wealth wit of agent i at time t in terms of units of A isthen equivalent to the ‘‘money wealth:’’

mit — witw/(a+bJt).

The simulation results in the next section are presented in terms ofmoney-wealths. As will be seen in Section 6 below, when N is large theratio w/(a+bJt) is distributionally indistinguishable from the constant 1,and thereore mit and wit have asymptotically the same distribution.

3. SIMULATION RESULTS

We began by examining, in computer simulations with two assets (A andB) and N=1000 individuals, whether our model with uniform preferencedistribution leads to a steady state and a recognizable limiting distributionfor individual wealths. In our simulations, the assets were initially eitherequally divided (ai0=bi0=1/N for all i) or alternatively 99.9 % of A and Bwere initially in the hands of one individual with the remaining assetsequally divided. We characterized the distribution of goods and wealthevery 50 cycles by evaluating their first five moments, i.e. the average overindividuals of aki , b

ki , and money-wealths mki , for k=1 to 5. After about

10 ·N (i.e., 10,000) cycles, the moments appeared to have reached steady-state values that were independent of the initial conditions. This steadystate is summarized in Table I, which gives average moments and standarddeviations, where averages were taken over 200 cycles (every 50th cyclefrom cycle 15,050 to 25,000). Since the moments were essentially identical for


TABLE I

Measured and Predicted Values of Weighted kth Moments and nk for Steady-State Goods andMoney Wealth Distributions Arising from Uniform Preferences

Weighted kth moment nk computed from moments

k Theory Simulation Theory Simulation

Mean St. dev. Mean St. dev.

Goods

1st 1 1.002nd 2 1.97 0.063rd 6 5.79 0.54 1 1.06 0.144rd 24 26.6 4.9 1 1.14 0.275th 120 111 48 1 1.31 0.41

Wealths

1st 1 1.002nd 1.5 1.48 0.033rd 3 2.90 0.17 2 2.09 0.244rd 7.5 7.14 0.9 2 2.14 0.415th 22.5 21.1 5.0 2 2.30 0.63

Note. Theoretical values in columns 2 and 5 are n−kC(n+k)/C(n) and n for the gamma dis-tribution (n=c=1) for goods and (n=c=2) for wealths. Simulation results are mean andstandard deviation for weighted moments in columns 3 and 4, and nk (see text for definition)in columns 6 and 7, calculated over every 50th cycle of the data from 15,050 to 25,000.

the ai and bi distributions, as expected by symmetry, only one set ofmoments is listed (goods distribution). The moments scaled as a−k forgoods and w−k for wealths; to eliminate these scaling factors, Table I givesweighted moments, equivalent to choosing a=w=1. The Gamma(n, c)probability density, with shape-parameter n, scale parameter c, and meann/c, is defined as p(x)=(cn/C(n)) xn−1 exp(−xc) for x > 0, and we foundthat the moments of the empirical goods distribution were in excellentagreement with those of Gamma(1, 1/a), while the empirical moments of(monetary) wealths agreed closely with those of Gamma(2, 2/w) (Table I,column 3).

We also performed simulations using nonuniform preference distribu-tions. In this case, the simulation results suggested that the steady-stategoods and wealth distributions were still closely approximated by gammadistributions, although the parameter n was no longer an integer. We per-formed simulations with two types of preference distributions: truncatedGaussians, and polynomial densities of the form p(f)3 (min(f, 1−f))c


for 0 < f < 1, where c=1, 2, 3, 4, 5, −1/2, −2/3, −3/4, and −4/5. TheGamma(n, n/w) distribution with mean w and shape parameter n has askth moment OmkP=(n/w)−k C(n+k)/C(n). It follows that the weightedratio of moments (1/w)OmkP/Omk−1P increases by 1/n when k increases by1. In the simulations, the values of nk calculated from the weighted ratios ofmoments were fairly constant for successive values of k (Table I, column 6,and Table II), which suggests that the distributions are close to Gamma.

TABLE II

Estimates of nk, Calculated from Differences in Successive Moment Ratios

Preference dist’n Goods dist’n Wealth dist’n

Mean St. dev. Mean St. dev.

Gaussian, s=0.125n1 8.03 0.69 17.4 2.34n2 8.13 0.85 16.7 1.77n3 8.05 1.08 16.8 1.79

p(f)=12f2, 0 [ f [ 0.5n1 4.56 0.41 9.03 0.76n2 4.49 0.66 8.90 1.00n3 4.45 0.95 8.53 1.13

p(f)=4f, 0 [ f [ 0.5n1 2.50 0.26 4.82 0.32n2 2.51 0.46 4.67 0.64n3 2.63 0.68 4.60 1.06

Gaussian, s=0.25n1 2.14 0.23 4.30 0.29n2 2.17 0.38 4.41 0.47n3 2.31 0.57 4.62 0.67

Gaussian, s=0.5n1 1.28 0.16 2.50 0.19n2 1.36 0.27 2.56 0.35n3 1.56 0.42 2.72 0.59

p(f)=1/`8f, 0 < f [ 0.5n1 0.45 0.09 0.88 0.12n2 0.53 0.16 0.98 0.22n3 0.69 0.25 1.17 0.35

p(f)=(108f2)−1/3, 0 < f [ 0.5n1 0.31 0.07 0.58 0.10n2 0.38 0.13 0.67 0.19n3 0.52 0.19 0.86 0.31

Note. ‘‘Gaussian’’ with indicated standard deviation s is truncated, i.e., conditioned to lie in[0, 1]. The power-law densities were all taken symmetric about 0.5.


FIG. 1. Comparison of theoretical and simulated wealth distributions for some of thepreference distributions shown in Tables I and II. Values of n were taken from the simulationresults as in Table II.

Figure 1 shows graphically, in the form of slightly smoothed histogramsplotted in Excel overlaid with gamma densities with mean w and shapefactors n taken from Tables I and II, that the steady-state wealth distribu-tions from the simulation data were very close to gamma. While the simu-lation results are suggestive, they do not prove that the wealths asymptoti-cally follow gamma distributions. Hence, we sought theoretical arguments.

4. HEURISTIC ARGUMENT COUNTING MICROSTATES

The nonlinearity of Eq. (7) makes the exact analysis of this systemintractable. However, for the particular case of a uniform preference dis-tribution for fit for all i and t, with density h(f)=1 for 0 < f < 1, theasymptotic behavior of the system for large N and t can be roughly under-stood by arguing as in [14]. In the following heuristic argument, there is aclose analogy between the distribution of wealth in the economic system


and the distribution of energy among molecules of an ideal gas in statisticalmechanics. In the latter case all ‘‘microstates’’ (specifications of the energyof each molecule) with the same total energy are deemed equally likely.One considers the number of ways W in which one can have Nj moleculeswith energy Ej, subject to the constraints that the total energy and numberof molecules are fixed. The constraints are handled via Lagrange mul-tipliers and the distribution of energy is determined by maximizing thefunction

G(Nj, Ej)=log W−l1 1CjNj−N2−l2 1C

jNjEj−E2

with respect to the Nj (see [19]). In the case of the economics model withuniform preferences, all ways of dividing an individual’s wealth betweengoods A and B are equally likely. If wealth is measured in discrete units(e.g., dollars), the number of ways of allocating money-wealth mi betweentwo goods is proportional to mi+1 (zero dollars to good A, or one, or two,... up to mi dollars). Hence, the number of ways one can have N1 individ-uals with wealth m1, N2 individuals with wealth m2, etc., is W(Nj, mj)=N!Pj((mj+1)Nj/Nj!). The factorials account for the fact that differentcombinations of individuals in a set of wealth bins represent differentmicrostates. There are N! permutations of individuals, but this overcountsthe number of microstates because permutations of individuals within awealth bin do not represent different microstates; the Nj! terms in thedenominator correct for this overcounting. The Nj can be made indepen-dent variables by the introduction of Lagrange multipliers l1 and l2 corre-sponding to the constraints that the number of individuals and the totalwealth are constant. Maximizing the function G(Nj, mj)=log W(Nj, mj)−l1(; j Nj−N)−l2(; j Njmj−Nw) with respect to the Nj, and then solvingfor l1 and l2 from the constraint equations, leads, in the limit as Nj Q.and sums are replaced by integrals, to the Gamma(2, 2/w) densityp(m)=(2/w)2 m· exp(−2m/w) for wealths. For the case of n goods insteadof two goods, this generalizes to Gamma(n, n/w). The derivation of thedistribution of goods (ait or bit) among individuals follows the same argu-ment except that the term analogous to (mj+1)Nj is omitted because thereis no degeneracy in the way the goods can be allocated; this leads toW(Nj, mj)=N!/PjNj! and to a Gamma(1, 1/a) density p(m).

5. A QUALITATIVE ARGUMENT

In this section, we apply a purely probabilistic result to our model, underan appealing qualitative assumption, to derive the gamma distribution for


steady-state wealths mit. The assumption is that arbitrary subsets of finitelymany wealth-variables at a time behave as though a steady-state distribu-tion exists (for large t), and that

(a) the wealth-variables in the subset are asymptotically independent(for large numbers N of agents and large time-index t), i.e., the total-wealth asymptotically for large N imposes no constraint upon them, and

(b) the random mechanism of wealth-partition asymptotically makesthe total wealth in a sub-population of individuals {i1, i2, ..., ik} approxi-mately independent of the way in which that wealth is subdivided amongthe individuals in that sub-population.

Then the conclusion is that no distribution for the wealth-variables exceptgamma is possible.

In many economic contexts, agents form coalitions or groupings. Here,groups of finitely many agents are artificial, since the underlying modelcontains no mechanism for cooperation in the setting of preferences.(However, such a mechanism would be interesting to include in laterrefinements of the model.) The assumption (a)–(b) above says that theaggregate wealth for a finite grouping of agents fluctuates statistically (insteady state) in a way which is statistically unrelated to the fluctuating par-tition of the grouping’s wealth among its individual members.

The underlying probabilistic result is the following theorem of [17]:

Theorem 1. If X, Y are independent positive random variables such thatX/(X+Y) and X+Y are independent, then X, Y are respectively distributedas Gamma(n1, c), Gamma(n2, c) for positive constants n1, n2, c.

This theorem of Lukacs [17] was cited for economic relevance in arather different context by Farjoun and Machover [13, pp. 63–72]. Theywere interested in the random variables profit-rate R and labor cost rate Zper unit of capital per unit time in an economy, obtained by aggregatingover all firms. They adduced Theorem 1 to conclude that these variablesare approximately Gamma distributed because they found it qualitativelyand empirically plausible (after some discussion) that the ratio R/Z of thegross ‘‘return to capital’’ R over ‘‘return to labor’’ Z should fluctuateapproximately independently of R+Z (the total return excluding rent).Farjoun and Machover recognized that to apply Lukacs’s Theorem in thissetting, R and Z would have to be approximately independent, but they didnot attempt to justify this independence.

The formal result which we derive from Theorem 1 is as follows.

Theorem 2. Suppose that the array of wealth variables wit (implicitlyalso indexed by the number of agents N in our model) is such that for large


N, t the joint distribution of each finite nonrandom set {i1, ..., ik} of therandom variables (wi1, t, wi2, t, ..., wik, t) converges (in the usual sense ofpointwise convergence of the joint distribution functions to a proper joint dis-tribution function at continuity points of the latter), and that under the limit-ing distribution of (W1, ..., Wk), the variables Wj are positive and jointlyindependent, and

Sk —W1+·· ·+Wk is independent of 1W1Sk,W2Sk, ...,Wk−1Sk2 .

Then all of the limiting Wj variables are gamma distributed with the samescale parameter c > 0.

Proof. For each fixed j=1, ..., k, X=Wj and Y=;ki=1 Wi−Wj are

independent random variables such that X/(X+Y) is independent ofX+Y. Thus Theorem 1 implies that Wj and ;k

i=1 Wi−Wj are gamma dis-tributed with the same scale parameter c > 0. Since it is well known that thesum of independent gamma variables with common scale parameter isgamma with the same scale, ;k

i=1 Wi is gamma with scale parameter c,which is therefore the same for all j, and our proof is complete. L

The limitation of this approach is that the appealing assumption ofindependence of sums and ratios of wealths of subsets is not readily dedu-cible from our model.

6. PROBABILISTIC THEORY

For discrete-time Markov processes with continuous state-spaces, suchas the 2N−2 dimensional process defined in (4) and (7) above, there is afully developed but not very well-known theory of long-run asymptotics interms of equilibrium or stationary distributions, for which see the book[20]. Under various sets of conditions, which are not easy to apply in thepresent context of the process defined by Eq. (7), such continuous-stateprocesses can be shown to possess a unique stationary distribution whichcan be found by solving a high-dimensional integral equation. Thechallenge for application of this structure to economic theory is to obtainsimple approximate conclusions for observables such as the distributions ormoments of the wealth-variables wit or mit.

The mathematical argument of this section proceeds in three main steps.Because of space limitations, we describe the main steps but omit somemathematical details. We assume throughout that for large N and t

1N

CN

i=1w2it(1−fit)

kQ m2, k, k=0, 1, 2 (A.1)


with probability 1, where the constant limits m2, k do not depend upon N ort (but definitely do depend on k and the distribution of fit). Denote

m2 — m2, 0= limN, tQ.

N−1 CN

i=1w2it, s2=Var(fit).

A further technical assumption required throughout is that the densityh(f) — pf(f) has finite third moment and has Fourier transform abso-lutely integrable on the real line.

First we apply a refinement of Central Limit Theory (the Edgeworthexpansion [21], pp. 533ff, as generalized for non-iid summands in [22]) toestablish the form of the conditional density of normalized price deviationsXt+1 —`N (Jt+1−J) up to terms of order N−1/2, where

J=a(1−m)/(bm), m=E(fit).

Second, we apply related limit theory along with an auxiliary assumption(A.2 below) to provide, again up to terms of order N−1/2, the joint densityof (Xt, w1t, f1t). Third, we show that gamma is the only possible form forasymptotic marginal distributions for w1t which is time-stationary withinthe model (7), that is, for which the marginal distribution of w1, t+1 within(Xt+1, w1, t+1, f1, t+1) is the same up to order 1/`N as that of wit.

For the precise formulation of the first step, consider N large, and cal-culate from equations (6), (5), and (3), that

Jt+1

Jt=

; j (1−m(fj, t+1−m)) fjtwjt; j (fj, t+1−m+m)(1−fjt) wjt

=J

Jt31− 1

aN(1−m)Cj(fj, t+1−m) fjtwjt 4

×31+ 1bNmJt

Cj(fj, t+1−m)(1−fjt) wjt 4

−1

.

Now, applying Theorem 19.3 of [22] to the nonidentically distributed butconditionally independent summands, given {fjt, wjt}

Nj=1, yields the

conclusion that the conditional density (at y) of Xt+1 given (Xt, {wjt, fjt})(at Xt=x) is, asymptotically for large N,

bm

s`m2f 1 ymbs`m2

2 11+c3(y)`N2 exp 1 bx

`Nq2(y)2+o(N−1/2), (10)


where f is the standard normal density, c3 is a cubic polynomial, q2 is aquadratic, and o(N−1/2) denotes terms of order less than N−1/2. In particu-lar

Xt —`N (Jt−J) %DN 10, s

2m2

b2m22 . (11)

Thus, the central limit theorem implies that for large N, prices will benormally distributed, and further shows how the variance of prices dependson the mean and variance of preferences and the variance of wealths. Inconformity with the assertion (11) of asymptotic normality, we remark thatwhen histograms were made from Jt variables simulated (with N \ 100,and fit ’ Uniform[0, 1]) from the model under study, these histogramslooked approximately normal, but the histograms of the variables log Jtlooked still more symmetric and therefore closer to normality. Further-more, as expected, in the limit as NQ., the prices become fixed. Sinceprice fluctuations drive the redistribution of wealth, it may seem puzzlingthat the initial wealth distribution is significantly altered for very large N.The explanation is that the number of time-steps required to reach the newsteady state increases with N, and is of the order of N according to thesimulation results.

The next step relies on an auxiliary assumption restricting the form ofthe conditional density of (w1t, f1t) given Xt (or equivalently, given Jt),namely:

(A.2) The variables wit have a ‘‘nonlattice distribution [22, condition(19.29)] with third moments bounded uniformly in N, t; and asymptoti-cally for large N, t, conditionally given Jt, the 2-vectors (wit, witfit)Œ behaveas N independent identically distributed random vectors subject to

N−1 CN

i=1

R witwitfitS=Ra+bJt

aS . (12)

This assumption can be compared with, but is quite different in spirit from,the independence hypothesis of Theorem 2. In our model, the conditions(12) impose a specific dependence on the wealths wit. However, as thenumber N of agents gets large, nonrandom finite subsets of them may havewealths which are hardly affected by the constraints (12) and may thereforebe approximately independent, as assumed in Theorem 2. The alternativeapproach of (A.2) is to argue that the variables wit may still behave inde-pendently conditionally given the sum-constraints. This latter approach ismotivated as in classical statistical mechanics (cf. [4]), where the exactequilibrium distribution of particle positions and momenta is intractablebut the individual phase-space coordinates behave like independent


random variables conditionally given a sum-condition fixing energy andpossibly some macroscopic integrals of the motion of a dynamical system.

By use of the same non-iid Edgeworth-expansion result of [22] citedabove, Assumptions (A.2) plus (A.1) lead to the following technicalconclusion: For a family of positive differentiable functions gN on [0,.)such that the integrals > w3gN(w) dw are uniformly bounded, and for somefunctions c1, c2, c3 which may depend upon x,

pw1t, f1t | Xt (w, f | x)=h(f) gN(w) 11+c1w+c2w(f−m)+c3

`N2+o(N−1/2),

(13)

where the final o term is understood as a function rN(f, w, x) such that foreach x, as N gets large, N1/2 >10 >.0 w3 |rN(f, w, x)| dw dfQ 0.

In fact, the derivation of (13) from (A.2) shows that

c1=bxs22, c2=

−mbxs2m2

, c3=−m1c1, (14)

where we define for convenience

m1=a/m, s22 — m2−m21.

Formula (13) turns out to be precisely what is needed for the third stepof our derivation of gamma-distributed wealths. It is much less stringentthan (A.2) in restricting only the joint behavior of (w1t, f1t, Xt). It can beunderstood as a first-order Taylor-series correction for large N to the con-ditional independence of w1t and f1t given Xt and (12). That the variablesw, f appear in this correction only through terms proportional to w, wf isplausible because (12) is the model’s restriction on independence of w1t, f1t.It can further be derived from (13) by use of (A.1), (5) and symmetricpermutability of the agents within the model (7), that c1, c2, c3 in (13) arenecessarily linear in x and have the form (14). Thus, (13) can replace A.2 asthe fundamental assumption describing the way in which wealths behaveindependently (to order N−1/2) despite having a fixed sum.

Our third step is to derive and solve the stationary equation expressingthe requirement that the joint density of (w1, t+1, Xt+1) be the same up toterms of order 1/`N as the joint density of (w1t, X). By (10), (11), and(13), letting pXt (x) and pXt+1 (y) respectively denote the approximatelynormal densities of Xt and Xt+1, we find that the joint density of(Xt, w1t, f1t, X1, t+1) is, up to terms of smaller order than N−1/2,

pXt (x) gN(w) h(f) pXt+1 (y) exp 3 bx`N5w−m1s22−wm(f−m)s2m2

+q2(y)64 ,(15)


where q2(y) is quadratic as in (10). However, as a function of(Xt, w1t, f1t, Xt+1), by definition of Xt and Xt+1 together with (9),

w1, t+1=11+1J+Xt+1/`N

J+Xt/`N−12 (1−f1t)2 w1t. (16)

Our task is to compare, up to terms of order 1/`N , the joint distribu-tion of (w1t, Xt) obtained from (15) with that for (w1, t+1, Xt+1) obtainedfrom (13) and (14), which at (w*, y) is

gN(w*) pXt+1 (y) 11+by(w*−m1)

s22 `N2+o(N−1/2).

We do this by calculating E(k(w1, t+1, Xt+1) in two ways, with respect tothe two density forms just given, and equating them, where k(w, y) is anarbitrary smooth and compactly supported function. With (16) substitutedfor w1, t+1, this gives

FFFF k(w 11−f) y−xJ`N

, y2 pXt+1 (y) pXt (x) h(f) gN(w)

31+ bx`N5q2(y)+

1s221w 11+(1−f) y−x

`N2−m1 2

−(f−m) mws2m211+(1−f) y−x

J`N264 dx df dy dw

=FF pXt+1 (y) gN(w) 31+by(w−m1)

s22 `N4 k(w, y) dy dw.

Now subtract from both sides of the last equation the term

FF pXt+1 (y) gN(w) k(w, y) dy dw,

multiply through by `N, and use the approximate identity > xpXt (x) dx=O(N−1/2), to find after using smoothness of k to translate a differencequotient into a derivative plus remainder,

FF pXt+1 (y) gN(w) 51−mJ(y−m1) w

“k

“w(w, y)+k(w, y)

b

s22(w−m1) m6 dy dw

%b

s22FF pXt+1 (y), gN(w) y(w−m1) k(w, y) dy dw,


where % means that the difference between the two sides converges to 0 asNQ.. This implies that

FF pXt+1 (y) gN(w) 5(y−m)(w−m1) k(w, y)

−(1−m) s22bJ

(y−m) w“k

“w(w, y)6 dy dw % 0.

Since k is an arbitrary smooth compactly supported function, and sinceintegration by parts says that

F wgN(w)“k

“w(w, y) dw=−F k(w, y)(wg −N(w)+gN(w)) dw,

it follows that, up to terms converging to 0 as NQ.,

FF pXt+1 (y) gN(w) 5(y−m)(w−m1) k(w, y)

+(1−m) s22bJ

(y−m) k(w, y)(wg −N(w)+gN(w))6 dy dw % 0.

Since k is an arbitrary smooth compactly supported function on[0,.)×[0, 1], we can equate 0 to the integrand of the last integrals toobtain the approximate equation

gN(w) 1m1(w−m1)s22

+12=−wg −N(w). (17)

All solutions to this equation have the form

log g(w)=−F 1m1s22+1−m21/s

22

w2 dw

=C−m1

s22w+1m

21

s22−12 log w

for some constant C, and the only such solution g which is a probabilitydensity on [0,.) is the Gamma(m21/s

22, m1/s

22) density, which has mean m1

and variance s22=m2−m21.

Thus, the equality up to order N−1/2 terms of (expectations with respectto) the distributions of (w1t, Xt) and of (w1, t+1, Xt+1) has been shown toimply that the wealth variables w1, t+1 must be approximately gamma. The


second moment of w1t is not predicted from this development, although itis connected to the limiting variance of the variables`N(Jt+1−J) through

Asymptotic variance of `N (Jt+1−J)=s2m2

b2m2.

7. DISCUSSION

We simulated the evolution of artificial markets in which individuals buyand sell goods in order to achieve a certain fraction f of their wealth ineach type of good. In our model, the prices of goods were determinedendogenously to equate fixed supply with variable demand. In contrast toan analysis in which individual preferences are fixed, we randomized pref-erences periodically, with preference probabilities determined by variousdistribution functions. The purpose of the repeated randomization was tosee how stochastic forces would alter the distribution of goods and wealth.While goods and wealth were continuously exchanged among individuals,their distributions came to an equilibrium, independent of starting condi-tions, determined by the form of the distribution of preferences for good Aversus B.

Remarkably, the equilibrium densities of goods and wealth had the samemathematical form p(x)=cnxn−1e−cx/C(n) for all preference distributionstested. While this form was suggested for the case of uniform preferences,it was not obvious that it should hold for non-uniform preferences.Nevertheless, we showed by an extension of the central limit theorem thatif individual wealths behave in steady state as identically distributed, inde-pendent variables conditioned on their fixed sum, then all preference func-tions satisfying reasonable continuity conditions lead to steady-stategamma distributions for wealth. The mathematical derivation (Eq. (11))shows how the shape factor n depends on the mean and variance of thepreference distribution and the variance of successive price ratios. Thelatter is clearly determined by the preference distribution, but we do not, atpresent, have a method other than simulation to calculate it. While theorypredicts the asymptotic behavior for large N, it is remarkable that simula-tions with as few as 1000 individuals led to equilibrium distributions veryclose to the asymptotic limit. Thus, characteristics of the ‘‘thermodynamiclimit’’ may already be apparent in systems with modest numbers of indi-viduals.

We do not propose the model outlined here as an accurate description ofhuman behavior in economic markets. Rather, we view the model as an aidto thinking about how statistical phenomena may influence economic out-comes such as the distribution of wealth. In the model, inequality in wealth


is seen to be a consequence of stochastic forces even when all individualsare equivalent. Wealth disparity increases, the greater the diversity in pref-erences, and decreases, the larger the number of goods traded. Since theshape parameter n of the gamma distribution corresponds to the number ofgoods traded in the case of uniform preferences, nonuniform preferencefunctions can be viewed as having different effective numbers of goodstraded, with more heterogeneous (U-shaped) preferences corresponding toa reduced number of goods. The similarities between the economics modeland the statistical treatment of an ideal gas in physics suggests ways inwhich the economics model might be modified to take into accountinteractions between individuals that allow correlation among their pref-erences. The method used to derive probabilistically the form of the steadystate depends on the particular feature of the transition mechanism thatindividual agents’ wealths evolve independently and symmetrically exceptfor rebalancing in each generation. Such a rebalancing mechanism seemscharacteristic of price-allocation within mathematical economics but canarise also in neural-network or machine-learning models.

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Economy & Finance

Statistical Equilibrium Wealth Distributions in an Exchange Economy with Stochastic Preferences