William J. Reed

The Pareto, Zipf and other power laws

Abstract Many empirical size distributions in economics and elsewhere exhibit power-law behaviour in the upper tail. This article contains a simple explanation for this. It also predicts lower-tail power-law behaviour, which is verified empirically for income and city-size data.

The Double Pareto-Lognormal Distribution—A New Parametric Model for Size Distributions

Abstract A family of probability densities, which has proved useful in modelling the size distributions of various phenomens, including incomes and earnings, human settlement sizes, oil-field volumes and particle sizes, is introduced. The distribution, named herein as the double Pareto-lognormal or dPlN distribution, arises as that of the state of a geometric Brownian motion (GBM), with lognormally distributed initial state, after an exponentially distributed length of time (or equivalently as the distribution of the killed state of such a GBM with constant killing rate). A number of phenomena can be viewed as resulting from such a process (e.g., incomes, settlement sizes), which explains the good fit. Properties of the distribution are derived and estimation methods discussed. The distribution exhibits Paretian power-law) behaviour in both tails, and when plotted on logarithmic axes, its density exhibits hyperbolic-type behaviour.

Optimal escapement levels in stochastic and deterministic harvesting models

Abstract A constant-escapement feedback policy is shown to be optimal in maximizing expected discounted net revenue from an animal resource whose dynamics are described by a stochastic stock-recruitment model, provided that unit harvesting costs satisfy certain conditions. The optimal escapement in this model is compared with that in the corresponding deterministic model and it is shown how the way in which unit harvesting costs vary with population abundance can be important in determining the relative sizes of the optimal escapements. In most cases, the optimal stochastic escapement is no less than the optimal deterministic escapement.

The effects of the risk of fire on the optimal rotation of a forest

Abstract The effects of the risk of fire or other unpredictable catastrophe on the optimal rotation period of a forest stand are investigated. It is demonstrated that when fires occur in a time-independent Poisson process, and cause total destruction, the policy effect of the fire risk is equivalent to adding a premium to the discount rate that would be operative in a risk-free environment. Other cases are also investigated and in each a modified form of the Faustmann formula is derived and a “marginal” economic interpretation given.

Consumption/pollution tradeoffs in an environment vulnerable to pollution-related catastrophic collapse

Abstract This paper considers the consequences of an avoidable risk of irreversible environmental catastrophe for societys optimal long-run consumption/pollution tradeoffs. The risk is assumed to be a nondecreasing function of pollution concentration which evolves as a dynamic environmental renewal process. The main objective of the paper is to derive qualitative insights regarding the effects of such risk on optimal management.

The tree-cutting problem in a stochastic environment : The case of age-dependent growth

Abstract Harvest policies are derived for growing biological assets (e.g., forests, livestock) subject to stochastic age-dependent growth and price uncertainty. The task posed is analyzed as a continuous-time optimal stopping problem for diffusion processes. Both ‘single’- and ‘ongoing’-rotation problems are considered. Qualitative comparative static and numerical results are provided.

The Pareto law of incomes—an explanation and an extension

A stochastic model for the generation of observed income distributions is used to provide an explanation for the Pareto law of incomes. Analysis of the model also yields a prediction of Paretian (power law) behaviour in the lower tail of the distribution and this is shown to occur in a number of empirical distributions. A tractable four-parameter distribution is derived, and shown to fit extremely well to a number of different empirical income distributions.

On the Rank-size Distribution for Human Settlements

An explanation for the rank-size distribution for human settlements based on simple stochastic models of settlement formation and growth is presented. Not only does the analysis of the model explain the rank-size phenomenon in the upper tail, it also predicts a reverse rank-size phenomenon in the lower tail. Furthermore it yields a parametric form (the double Pareto-lognormal distribution) for the complete distribution of settlement sizes. Settlement-size data for four regions (two in Spain and two in the U.S.) are used as examples. For these regions the lower tail rank-size property is seen to hold and the double Pareto-lognormal distribution shown to provide an excellent fit, lending support to the model and to the explanation for the rank-size law.

The decision to conserve or harvest old-growth forest

Abstract The decision to harvest or conserve old-growth forest is formulated as a stochastic decision problem in continuous time. Uncertainty in future amenity values for standing forest and in future timber revenues for harvested forest are included in the model, along with the risk of catastrophic destruction by fire, pest infestation, etc. It is shown how the decision problem can be expressed as an optimal stopping problem which can be solved analytically. The optimal decision rule is shown to depend on how the ratio of current timber value to the current expected present value of amenity benefits foregone through harvesting compares with some critical level. The effects of changes in uncertainty and other parameters on the optimal rule are discussed. Also it is shown how the cost-benefit analysis and certainty-equivalence procedures lead to premature harvesting, and the expected loss in survival time for these sub-optimal procedures is calculated.

Power-law behaviour and parametric models for the size-distribution of forest fires

Abstract This paper examines the distribution of areas burned in forest fires. Empirical size distributions, derived from extensive fire records, for six regions in North America are presented. While they show some commonalities, it appears that a simple power-law distribution of sizes, as has been suggested by some authors, is too simple to describe the distributions over their full range. A stochastic model for the spread and extinguishment of fires is used to examine conditions for power-law behaviour and deviations from it. The concept of the extinguishment growth rate ratio (EGRR) is developed. A null model with constant EGRR leads to a power-law distribution, but this does not appear to hold empirically for the data sets examined. Some alternative parametric forms for the size distribution are presented, with a four-parameter ‘competing hazards’ model providing the overall best fit.