Abstract
Masanao Aoki developed a new methodology for a basic problem of economics: deducing rigorously the macroeconomic dynamics as emerging from the interactions of many individual agents. This includes deduction of the fractal/intermittent fluctuations of macroeconomic quantities from the granularity of the mezoeconomic collective objects (large individual wealth, highly productive geographical locations, emergent technologies, emergent economic sectors) in which the microeconomic agents selforganize. In particular, we present some theoretical predictions, which also met extensive validation from empirical data in a wide range of systems:

The fractal Levy exponent of the stock market index fluctuations equals the Pareto exponent of the investors wealth distribution. The origin of the macroeconomic dynamics is therefore found in the granularity induced by the wealth/capital of the wealthiest investors.

Economic cycles consist of a Schumpeter ‘creative destruction’ pattern whereby the maxima are \(\varLambda \)shaped cusps. In between the \(\varLambda \)cusps, the cycle consists of the sum of 2 ‘crossing exponentials’: one increasing (corresponding to the creation of the ‘new’) and the other one decaying (corresponding to destruction of the ‘old’).
This unification within the same theoretical framework of short term market fluctuations and long term economic cycles offers the perspective of a genuine conceptual synthesis between micro and macroeconomics. Joining another giant of contemporary science—Phil Anderson 1972—Aoki emphasized the role of rare, large fluctuations in the emergence of macroeconomic phenomena out of microscopic interactions and in particular their non selfaveraging, in the language of statistical physics. In this light, we present a simple stochastic multisector growth model.
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Notes
In this paper we use for fractal fluctuations the definition of Mandelbrot for Lévy flights (Mandelbrot 1982): a random walk in which the steplengths have a power law (Pareto) probability distribution. For intermittency we use the definition of Zel’dovich (1987): “Some specific structures in which a growing quantity reaches record high values typically arise for instabilities in random media. Despite the rarity of these concentrations, they dominate the integral characteristics of the growing quantity (the mean value, the mean square value, etc.). The appearance of such structures is called ‘intermittency’.” Thus we reserve the term ‘intermittent’ for random processes totally dominated by their largest collective objects and rarest events while for processes that are merely outside the basin of attraction of the normal distribution we use the term fractal.
We shall call occasionally \(A(x,t)\) and \(K(x,t)\) ‘densities’ when discussing them in the macro approximation where \(A(x,t)\) and \(K(x,t)\) are considered as functions defined on a space where the locations \(x\) are real numbers. However one should not forget that the main point of this paper is that the macro/continuum approximation has severe limitations and the correct and binding formulation is the discrete one, where \(A(x,t)\) and \(K(x,t)\) are just the number of agents \(a\) and \(k\) at the discrete location \(x\) at time \(t\).
A few decades after Malthus, Verhulst (1838) included in Eq. 3 a nonlinear term \(K^2\) corresponding to \(k\)’s competition, confrontation, limited resources and called the new equation ‘the logistic equation’. In terms of the agents, the competition between \(k\)’s is expressed by the reaction: \( k+k \rightarrow k; \; c. \) The Verhulst nonlinear term has the effect of diminishing the \(K(t)\) growth, \({dK}/{dt}\), as \(K(t)\) increases and in fact eventually leads to the saturation of the \(K\) population growth. Thus, the solution of the logistic equation starts increasing in time with the same exponential growth rate \(g\) as the Malthus solution Eq. 5, but eventually it curbs down and saturates. In the current discussion we will not discuss the effects of the Verhulst term.
We will use occasionally the integral \(\int _x \) and differential \(\varDelta \) notations when discussing the continuum approximation (Eqs. 8, 10) . However, one should remember that it is one of the main claims of the present paper that this approximation fails in crucial ways and that the correct formulation is the discrete one. In particular using discrete sum instead of the integral.
In higher dimensions \(d\), the condition for the \(k\) population to survive and proliferate is \(s/D_A > 1P_d\). Where the Polya constant \(P_d\) is the probability that an \(a\) will eventually return to its site of origin (at least once) if one waits an indefinitely long time. In \(d=1\) and \(d=2\) dimenstions \(P_d=1\).
In fact one can compute it using the Poisson formula \(P(A(x,0)) = A^{A(x,0)} e^{A} / A(x,0)!\)
Of course, while the exponent \(\alpha \) in the distribution (14) is stationary, the actual order of the individual \(K_i (t)\)’s changes in time. In fact we will see that in the regime \(\alpha < 1\) the events by which the largest agent \(K_1 (t)\) changes place with another \(K_i(t)\) correspond to very important global changes in the system.
Actually due to the finite number of components \(i=1,...,N\), the Levy distribution is truncated for very large values of \(S\). For such extreme values the distribution decays much faster (Huang and Solomon 2001; Mantegna and Stanley 1999). In order to avoid these complications, we express here the fractal properties of the \(K(t)\) evolution in terms of the central peak corresponding to very small fluctuations \(\varDelta K(t) =0\).
This is not a necessary assumption. Smaller shocks like the ones in Finland and Britain discussed in the Sect. 5.3 do not lead to negative \(g_{11}\) but only to an exchange in the relation between the intrinsic growth rates of different parts of the economy: from \(g_{11} > g_{22}\) to \(g_{11} < g_{22}\). To detect with precision such shocks one needed finer GDP time resolution (Challet et al. 2009): quarterly rather than annually.
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Solomon, S., Golo, N. Microeconomic structure determines macroeconomic dynamics: Aoki defeats the representative agent. J Econ Interact Coord 10, 5–30 (2015). https://doi.org/10.1007/s1140301401353
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DOI: https://doi.org/10.1007/s1140301401353