A GENERATIVE MODEL OF INCOME DISTRIBUTION 2: INEQUALITY OF THE ITERATED INVESTMENT GAME

Abstract

This paper attempts to develop the model formalized by Hamada (1999b), which generates a distribution of incomes obeying the lognormal law by starting with an individual's rational choice at the micro level. As in the previous work, we employ an iterated investment game as a baseline model in which each player has a binary choice between investing and not investing. Hamada (1999b) showed only that if the game time n is sufficiently large and γ > 1/(R + 1), then the distribution of the profit of an iterated investment game is subject to approximately the normal distribution without cumulative effect on one hand and the lognormal distribution with cumulative effect on the other hand.

The present paper develops the previous work considerably and hence specifies the mathematical structure of the model to release special conditions. We succeed in showing that the proposition holds for every γ in (0, 1) by the change of variables theorem with respect to the probability density function of the multidimensional normal distribution. This generalization enables us to exhibit the relation between the Gini coefficient of the lognormal distribution derived from the model and the parameters determining a structure of the game, such as prize density γ and the rate of return R. As a result of analysis, we show some remarkable implications.

Keywords:

• Income distribution
• Iterated investment game
• Gini coefficient
• Lognormal distribution
• Relative deprivation
• Inequality

Notes

¹A random variable subject to a process of change is said to obey the law of proportionate effect if the change in the variable at any step of the process is a random proportion of the previous value of the variable.

²Although Kosaka (1986) did not assume B ₁ − C ₁ > B ₂ − C ₂ as an axiom, this is a natural assumption. For if B ₁ − C ₁ ≤ B ₂ − C ₂, then nobody chooses Move 1 and the consequence is trivial.

³ , where the sign ∼ means that the ratio of the two sides tends to 1 as n →  ∞ .

⁴The definition of the small o is that if f(n)/g(n) → 0 as n →  ∞ , then we write f(n) = o(g(n)).

⁵We omit long and elemental algebraic operations to specify an explicit form of mean ζ₁ and ξ₁.

⁶An alert reader may find that the variance of the lognormal distribution derived from our model will be divergent as a game time n increases, since we did not use the central limit theorem. In fact, the central limit theorem is available only if we standardize a random variable and the parameter γ appropriately. However, we found that if we apply the additional mathematical assumptions to use the theorem, most of the remarkable implications shown in the paper could not be derived. We gave priority to an abundance of implications for the purposes of sociological research rather than mathematical exactness and elegance.

⁷Let random variable X and Y be X ∼ N(μ, σ²) and Y = aX + b with constant a and b, then Y ∼ N(aμ + b, a ²σ²). This must be proved.

⁸The computational method of Gini coefficient of the model without cumulative effect was described in (Hamada 1999b). However, we do not have to use those cumbersome methods any longer, since the parameters of the distribution function can be expressed as a function of the parameters of the game by improvement.

⁹γ = 1/(R + 1) is the value that all group members begin to stake C ₁. As the prize density increases more over than γ = 1/(R + 1), then the rate of the relatively deprived decreases in a one-shot game (Kosaka 1986).

¹⁰Let S be a nonempty subset of the set of real numbers. If S is bounded below, then a number u is said to be a infimum of S (denoted inf S) if it satisfies the conditions, 1)u is an lower bound of S, and 2) if vis any lower bound of S, then v ≤ u. Similarly we use sup S in order to denote a supremum of S.

¹¹This statement can be summarized as lim _R→∞[alog(R + b) − cR] =  −  ∞ where a, b and c are constant.