Inequality of Realization of a Stochastic Dynamics Based on the Erdös Discrepancy Problem

This paper proposes a stochastic dynamics model in which people who are endowed with different discount factors chose to buy the capital stock periodically with different periodicities and are exposed to randomness at arithmetic progression times. We prove that the realization of a stochastic equilibrium may render to the people quite unequal benefits. Its proof is based on Erdös Discrepancy Problem that an arithmetic progression sum of any sign sequence goes to infinity, which is recently solved by Terence Tao [1]. The result in this paper implies that in some cases, the sources of inequality come from pure luck.

beyond the individual capacity (e.g. inheritance or pure luck), governmental or nongovernmental policies are considered to be required in many respects (tax, wage control, nationalization of institutions and so on) and the inequality is an important problem we must grapple with. What this paper concerns is the sources of inequality and especially we focus on the possibility that the inequality arises from pure luck. We provide a simple stochastic model in which the ex-post realization of the equilibrium stochastic process is quite biased among people.
To complete this purpose, we have to investigate the existence of some regularity within randomness. Intuitively, the realization of randomness from uniform distribution offers quite equal benefit among people in the long run, for example, in throwing dices or flipping coins, the same numbers realize in almost the same times as experiments continue infinitely. However, from a different mathematical viewpoint, it is possibly said that the same number arises in a regular manner so that the same numbers fall upon almost the same people. To support this aspect, we employ a monumental mathematical theorem which is recently solved. That theorem is the so called Erdös Discrepancy Problem, long time being conjecture from around 1932, which is proved by Terence Tao in [1]. This theorem roughly states that for any random sequence, the realization of which contains almost the same number periodically.
In this paper, we construct a stochastic equilibrium model in which consumers who have different discount factors buy periodically the capital stock so that they are exposed to randomness at arithmetic progression times. Therefore according to the Erdös Discrepancy Problem, there are some people who obtain high wages arbitrary larger times than low wages or who get low wages arbitrary larger times than high wages corresponding to their distinct discount factors. 1 The main feature in this paper is its approach to elucidating the inequality.
The existent models (such as [1]- [9]) basically attribute the inequality to intrinsic character such as productivity, ability and income resource. Since we aim to investigate the other resources that give rise to inequality, the model developed in this paper is in a class of its own though based on standard economics notions such as utility, production and equilibrium, and we draw the distinctive conclusion that the pure randomness possibly causes inequality. The underlying mathematics is the Erdös Discrepancy Problem which is deep and new theorem in the number theory. After Tao's proof [1], some papers clarify the substance of this problem (such as Soundararajan [10]).
The next section describes the stochastic model in which people who have different discount factors select capital stock with different periodicity. The third 1 The claim that the possession of capital becomes biased among people according to heterogeneous discount factors is apparently related to the Ramsey's conjecture, which says that the people who have the lowest discount factors own all the capital and is solved by many authors in various settings (e.g. Becker [11], Mitra and Sorger [12]). However, in our paper, the discount factor endowed by people who have much capital depends on the realization of stochastic processes and it is not necessarily the lowest discount factor's people who have the large capital.   : Consumers' objective function can be described by with 0 2 1 t + = where u and v stand for the utility function and disutility one respectively. In what follows, we assume that the utility and disutility functions are linear.
where t S is the price of stock, which is used for financing the capital or saving, and t θ ∆ means the increment of quantity of the stock at t. Notice that We assume that the price of stock has no trend.
Thus consumers prefer buying at most capital to saving something at the periods other than  due to the linearity of utility, presence of discounting ρ and no trend of stock prices. They save only when being in  and buy the capital using all the savings and current wages while being in other than  . Hence we can express as and  . Set the price process t S and t w by for some Since it needs to hold t t a w = in equilibrium due to the linearity of produc-  ( ) Thus the assumptions are consistent. From the latter part of Assumption 3 and due to Note from (1) and (2) that for We calculate as follows; for The second and fourth equalities come from (4) and the last inequality is obtained by (5). Hence consumers select 0 t l = for  , the same arguments apply. Hence we conclude that 0 for . Consider We see from (1) and (2) Hence from (2) and (3) it holds that for 1, 2, n =  , It suffices to know the sign of the numerator in (9) to determine the sign of the fraction (9). Note that Now we further put the following assumption on the parameters.   Thus we process the following arguments.
Although ( ) (since k rises and (10)), we see from (11) and (12) that ( ) In the same way, we have ( ) it follows for some ρ that    Roughly speaking, even under random environment, there may be a fixed member in a society who is almost always lucky or unlucky for large period of time. Note that in the case of 0 d = that attains the given C, we take periods, say, 2, 4, 6, , 2 , t n =   that 2 ϕ -people encounter and reinterpret it as original sequence, then we can take subsequence that attains the given C.

Conclusions
This paper proposes a stochastic dynamics in which people who are endowed with different discount factors buy the capital stock periodically and are exposed to randomness at arithmetic progression times. We prove that the realization of the stochastic equilibrium may render to the people quite unequal benefits. Its proof is based on Erdös Discrepancy Problem that an arithmetic progression sum of any sign sequence goes to infinity, which is recently solved by Terence Tao (2016). There are some people who obtain high wages arbitrary larger times than low wages or who get low wages arbitrary larger times than high wages corresponding to their distinct discount factors. The result in this paper implies that in a certain society, the sources of inequality come from pure luck.
Finally we note the topics that remain in future research. Inequality arising from realization of stochastic processes only identifies the most lucky or the least one and does not explain the distribution of various income realization. In addition, whether people face the fortunate case or not reflects observation of the finite time and we cannot say anything about what occurs beyond the periods.
The type of phenomena that is in this paper out of scope may be explained by other approach or more generalized mathematical theorem on the number theory or stochastic analysis.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.