### 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.

Original language | English |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Journal of Mathematical Sociology |

Volume | 28 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 Jan 1 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory
- Social Sciences (miscellaneous)
- Sociology and Political Science