INTRODUCTION TO MATHEMATICAL MODELLING LECTURES 6-7: MODELLING WEALTH DISTRIBUTION

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1 January 004 INTRODUCTION TO MATHEMATICAL MODELLING LECTURES 6-7: MODELLING WEALTH DISTRIBUTION Project in Geometry and Physics, Department of Mathematics University of California/San Diego, La Jolla, CA ~ dmeyer/; dmeyer@math.ucsd.edu Rank from probability density In the previous lecture I introduced the Pareto distribution and shoed that it has a heavy tail, i.e., that its probability density function, f(x), is asymptotically x α as x []. We can see that this leads to a linear log-log plot of random variable value as a function of rank as follos: Since f(x) = αk α x α, for x k, (fraction of values at least ) = so for a random sample of N independent values, hich implies that (number of values at least ) rank() αk α x α dx, Nαk α x α dx, Nαk α x α dx = Nk α α, here e have used the fact that α > 0 so the integral converges. Taking the logarithm of both sides gives: log r log Nk α α log, hich implies that log α log Nkα log r. α Comparing this ith eq. 5., e see that the Forbes 400 data is approximated by a Pareto distribution ith /α , hich implies that α.785. We can also use the constant term to set (log Nk α )/α 4.054; using N.9 0 8, this gives k c 004

2 A simple model for the distribution of ealth In Lectures and 4 e modelled the amount of money in the pocket of a gambler on the ay out of a casino as a function of the initial amount and the outcomes of a sequence of random events (the coin flips) []. That model produced a binomial distribution of pocket money. To explain the Forbes 400 data, e must create a different model, that produces a different distribution. At first sight this might seem hopeless, since the economy in hich individuals acquire ealth is so complicated. But just as ith the flipping coin, perhaps e can avoid modelling many of the details, replacing them instead ith some reasonable probabilistic description. The folloing simple model attempts this, although its reasonableness is debatable. Let us suppose first that a model can be constructed that does not depend on population groth, i.e., that e can postulate a constant number of people, N. Let us suppose second that the total ealth of the society is constant, i.e., N W i (t) = W, i= here W is constant, and W i (t) is the ealth of individual i at time t. Each of these assumptions is only an approximation that describes the real orld over very short time scales. If the observed ealth distribution is a longer term effect, our model may not succeed; nevertheless, let us proceed. Real economic interactions beteen people can transfer ealth beteen them. We make to very simple (and unrealistic) assumptions about this for our model: that pairs of people interact at random, and one (chosen at random) transfers a fixed amount of ealth, δ, to the other. We implement these events at a sequence of times, so for each time t Z >0, A(t) and B(t) are random variables ith probability functions prob ( A(t) = i ) = N = prob( B(t) = i ), (6.) for i {,..., N}, and W i (t ) + δ if A(t) = i and W B(t) (t ) > δ; W i (t) = W i (t ) δ if B(t) = i and W B(t) (t ) > δ; W i (t ) otherise. (6.) The condition that W B(t) (t ) be greater than δ prevents any individual s ealth from ever becoming zero (or negative). This model is simple enough that e could compute prob(w i (t) = ) as a function of, t, and the model parameters N, W and δ. Rather than doing so, Figure 6. shos a movie of 0 5 timesteps for N = 400, W =.4N, δ = 0.6, starting from the initial condition W i (0) =.4. The vertical axis is log W i (t) and the horizontal axis is log(rank), just as in

3 Figure 6.. QuickTime movie of 0 5 timesteps of the simple model for ealth distribution. Each frame is 00 timesteps. Figure 6.. The distribution of the W i after t = 0 5 timesteps. log W i is plotted as a function of log(rank). Figure 5.8 []. Figure 6. shos the same plot at t = 0 5. Since the plot is not linear, this model has not produced a Pareto distribution for the W i. We can imagine modifying this simple model in many ays: the amount of ealth transfered need not be constant, the interactions need not be random, the direction of the transfer need not be random, ealth might gro over time, there could be taxes, etc. Homeork: Modify the model for ealth distribution described by equations in some ays that seem realistic to you. Do any of them lead to a Pareto distribution? A particularly simple version of this model, in hich an interaction beteen to individuals ends ith them sharing their combined ealth evenly, as interpreted by Melzak as describing a socialist economy []. Other versions have been investigated by Mainieri [4], and Ispolatov, Krapivsky and Redner [5]. To natural modifications are to transfer a constant fraction of the ealth of the individual ho is losing ealth during a transaction, and to include some rate of return on ealth. That is, the model is described by eq. 6., and ith W i (t) = ( + R i (t) ) W i (t ) + fw B(t) (t ) if A(t) = i; W i (t ) fw B(t) (t ) if B(t) = i; (6.) W i (t ) otherise, replacing eq. 6.. Here 0 f and R i (t) is a ne random variable that represents the rate of return on ealth during the time period from t to t. This model is more complicated than the previous one, but it is still easy to simulate numerically. Figure 6. shos a movie of the ealths of 400 individuals for 0 5 timesteps, using W i (0) = 0.6 (the minimum ealth on the Forbes 400 list), f = 0., and prob ( R i (t) = ±0.5 ) =

4 Figure 6.. QuickTime movie of 0 5 timesteps of the more complicated model for ealth distribution. Each frame is 00 timesteps. Figure 6.4. The distribution of the largest 400 W i, after t = 0 5 timesteps. log W i is plotted as a function of log(rank). hen t 0 mod 00 and R i (t) = 0 otherise. (That is, each individual only collects interest on his/her ealth every 00 timesteps.) As in Figures 6. and 6., the vertical axis is log W i (t) and the horizontal axis is log(rank). Figure 6.4 shos the same plot at t = 0 5. This plot is very similar to the plot of the Forbes 400 data shon in Figure 5.8, so it appears that e have constructed a reasonably good model. Figure 6.5, hoever, shos the results for the hole model. The simulation shon in Figure 6. actually has N = 000, ith only the 400 largest ealths plotted at each timestep. When all 000 ealths are plotted, the distribution looks - quite different; there is a sharp bend in the log-log plot, and there is a substantial -4 number of very poor individuals. Since e have not collected data on the true distribution of ealth in the U.S. beyond the ealthiest 400 people, e cannot kno if this model is consistent ith reality. But e do kno that it does not produce a Pareto distribution, since the log-log plot is not linear throughout its domain Figure 6.5. The distribution of all 000 W i, after t = 0 5 timesteps. log W i is plotted as a function of log(rank). 4

5 References [] D. A. Meyer, Introduction to Mathematical Modelling: Other distributions, ~ dmeyer/teaching/inter04/imm0404.pdf. [] D. A. Meyer, Introduction to Mathematical Modelling: Basic probability theory, ~ dmeyer/teaching/inter04/imm04009.pdf. [] Z. A. Melzak, Mathematical Ideas, Modeling and Applications, Vol. II of Companion to Concrete Mathematics (Ne York: Wiley 976) [4] R. Mainieri, lectures at Santa Fe Institute Complex Systems Summer School. [5] S. Ispolatov, P. L. Krapivsky and S. Redner Wealth distributions in asset exchange models, European Phys. J. B (998)