By Masanao Aoki
This publication contributes substantively to the present cutting-edge of macroeconomic modeling via delivering a mode for modeling huge collections of heterogeneous brokers topic to nonpairwise externality referred to as box results, i.e. suggestions of combination results on person brokers or brokers utilizing state-dependent techniques. Adopting a degree of microeconomic description that retains tune of compositions of fractions of brokers by means of "types" or "strategies", time evolution of the microeconomic states is defined by way of (backward) Chapman-Kolmogorov equations.
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Additional info for New Approaches to Macroeconomic Modeling: Evolutionary Stochastic Dynamics, Multiple Equilibria, and Externalities as Field Effects
The space, £2, is a space of paths, with the usual Borel cr-field, and P is the fair-coin-tossing probability. The root of this tree stands for the unit interval. At level 1, we have a node labeled 0 and a node labeled 1 corresponding to the binary intervals [0, 2" 1 ] and [2" 1 ,1]. , 8j where 8k is 1 or 0, * = 1, 2 , . . , j . At this level, binary intervals [i2'j, (i + 1)2"^], / = 0, 1 , . . 2J: — 1, are associated with each node. See Fig. 2. , Xn(a>), where Xk(co) is the value of the node hit by co at level k.
This notion is explained in an introductory book by Schikhof (1984) and the notion of p-adic numbers behind it is explained in Mahler (1973). -adic numbers was developed by K. Hensel in 1897. 14 Rammal and Toulouse (1986) credit M. Krasner as the person who coined the word ultrametric in 1944. Hierarchical structures are self-similar in that their subsets have the same form as the original structure. Schikhof (1984) has an interesting illustration of figures within figures in the inside cover of his book.
By (ns + gs)\ because ns is expected to be large. The maximum is realized by nc = Y* - 1 ' where, for convenience, we set ys = a + PysOn substitution of this expression into the definition of the entropy Eq. 2 The partial derivative of the entropy with respect to N is equal to as and the partial derivative with respect to the other macroeconomic signal Y gives P 2 BY' See our discussion of Sanov's theorems. 16 Simple illustrative and motivating examples In carrying out the partial derivatives, dependence of these parameters on N and y, such as da/BY, cancel out.