Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a a lognormal distribution. In trying

Zipf’s law is a very tight constraint on the class of admissible models of local growth. It says that for most countries the size distribution of cities strikingly fits a power law: the number of cities with populations greater than S is proportional to 1/S. Suppose that, at least in the upper tail, all cities follow some proportional growth process (this appears to be verified empirically). This automatically leads their distribution to converge to Zipf’s law. I.

A family of probability densities, which has proved useful in modelling the size distributions of various phenomena, including incomes and earnings, human settlement sizes, oil-field volumes and particle sizes, is introduced.

A transportation network is a complex system that exhibits the properties of selforganization and emergence. Previous research in dynamics related to transportation networks focuses on traffic assignment or traffic management. This research concentrates on the dynamics of the orientation of major roads in a network and abstractly models these dynamics to understand the basic properties of transportation networks. A model is developed to capture the dynamics that leads to a hierarchical arrangement of roads for a given network structure and land use distribution. Localized investment rules – revenue produced by traffic on a link is invested for that link’s own development – are employed. Under reasonable parameters, these investment rules, coupled with traveler behavior, and underlying network topology result in the emergence of a hierarchical pattern. Hypothetical networks subject to certain conditions are tested with this model to explore the network properties. Though hierarchies seem to be designed by planners and engineers, the results show that they are intrinsic properties of networks. Also, the results show that roads, specific routes with continuous attributes, are emergent properties of transportation networks.

When human capital accumulation generates pecuniary externalities across professions, and capital markets are imperfect, persistent inequality in utility and consumption is inevitable in any steady state. This is true irrespective of the degree of divisibility in investments. However, divisibility (or fineness of occupational structure) has implications for both the multiplicity and Pareto-efficiency of steady states. Indivisibilities generate a continuum of inefficient and efficient steady states with varying per capita income. On the other hand, perfect divisibility typically implies the existence of a unique steady state distribution which is Pareto-efficient.

This paper proposes a convenient measure of the degree of distributional overlap, both parametric and nonparametric, useful in measuring the degree of Polarization, Alienation and Convergence. We show the measure is asymptotically Normally distributed, making it amenable to inference in consequence. This Overlap measure can be used in the univariate and multivariate framework, and three examples are used to illustrate its use. The nonparametric Overlap Index has two sources of bias, the first being a positive bias induced by the unknown intersection point of the underlying distribution and the second being a negative bias induced by the expectation of cell probabilities being less than the conditional expected values. We show that the inconsistency problem generated by the first bias, prevalent within this class of Goodness of Fit measure, is limited by the number of intersection points of the underlying distributions. A Monte Carlo study was used to examine the biases, and it was found that the latter bias dominates the former. These biases can be diluted by increasing the number of partitions, but prevails asymptotically nonetheless.

The city size distribution of many countries is remarkably well approximated by a Pareto distribution. We study what constraints this regularity imposes on standard urban models. We find that under general conditions urban models must have (i) a balanced growth path and (ii) a Pareto distribution for the underlying source of randomness. In particular, one of the following combinations can induce a Pareto distribution of city sizes: (i) preferences for different goods follow reflected random walks, and the elasticity of substitution between goods is 1; or (ii) total factor productivities in the production of different goods follow reflected random walks, and increasing returns are equal across goods.

We study the dynamics of the distribution of wealth in an overlapping generation economy with finitely lived agents and inter-generational transmission of wealth. Financial markets are incomplete, exposing agents to both labor and capital income risk. We show that the stationary wealth distribution is a Pareto distribution in the right tail and that it is capital income risk, rather than labor income, that drives the properties of the right tail of the wealth distribution. We also study analytically the dependence of the distribution of wealth, of wealth inequality in particular, on various fiscal policy instruments like capital income taxes and estate taxes, and on different degrees of social mobility. We show that capital income and estate taxes can significantly reduce wealth inequality, as do institutions favoring social mobility. Finally, we calibrate the economy to match the Lorenz curve of the wealth distribution of the U.S. economy.

Abstract not found

A well established regularity is that the city size distribution in many countries is well approximated by a Pareto distribution. We study the class of Markov processes that can support a Pareto equilibrium, an equilibrium path along which the city size distribution is Pareto. We offer a sharp characterization, a generalized Gibrat’s law. This generalized "law" requires that (i) the expected growth rate of a city be independent of its size, x; and (ii) the variance of city growth be proportional to xδ−1,whereδis the exponent of the associated Pareto distribution. In contrast with Gibrats ’ law, our process can account for the diversity of Pareto exponents observed in the data. Moreover, our finding allow us to prove, under mild conditions, the equivalence between Zipf’s law and Gibrat’s law.