PARETO DISTRIBUTION OF THE PROBABILITY OF SURVIVAL
The Pareto probability distribution was originally intended to represent the allocation of wealth in an economy. Later on it was used to describe, among other things, the health structure of populations of individuals, the uneven distribution of human settlement, the frequency of occurrence of individual words in a text when decoding secret messages, and the size distribution of sources or deposits of raw materials. In physics it has been used to describe certain phenomena at temperatures close to absolute zero. In all these applications it has the advantage of being asymmetric.
FIRST-ORDER PARETO PROBABILITY DISTRIBUTION
If we assume that an agent's probability of survival is directly proportional to the ratio of his margin (relative to the extinction zone boundary b) to his income d, we arrive at a first-order Pareto probability distribution with the asymmetric distribution function:
The probability density function for this probability distribution has the following shape:
The plots of the probability distribution function F(d) and the probability density functioned) for the first-order Pareto probability distribution with a unit extinction zone boundary b are shown in Figure 1.
Figure 1: The first-order Pareto probability distribution with certain- extinction-zone boundary b = 1
The first-order Pareto probability distribution has a zero probability for income at or below the subsistence level b and a probability converging to one as income tends to infinity. Unlike higher-order Pareto distributions, the first-order Pareto distribution does not have a final mean or variance. Its median is m = 2b.
We use the first-order Pareto distribution to express the subjective probability of survival in most chapters of our book. Only in the final chapter, where preferences are the deciding factor for the survival of politicians and those preferences are linked to growth in (rather than the level of) the standard of living, do we work with the assumption that the probability of survival is directly proportional to the derivative of the relative margin with respect to income. This assumption is consistent with the second-order Pareto probability distribution.
SECOND-ORDER PARETO PROBABILITY DISTRIBUTION
According to the psychological Weber-Fechner law individuals in many cases decide not according to the intensity of a stimulus, but according to the change in the intensity of the stimulus. Individuals' assessment of their own satisfaction is often derived from the dynamics rather than the level of a utility indicator (wealth, threat): people in societies with low but rising living standards paradoxically tend to be more satisfied than those in societies with higher but flat or falling living standards. The incorporation of this law into the problem of economic threat (or the subjective feeling of threat) leads to the assumption that the subjective estimate of the probability of personal extinction is linked not directly with the relative margin, but with its derivative
So, if it is true that the determining factor for the strength of the subjective feeling of threat is the increase (decrease) in the margin relative to the subsistence level in response to a (small) unit change in income, the second-order Pareto probability distribution is the right one to use for the distribution of the subjective probability of extinction. For this distribution it holds that the risk of extinction decreases in proportion to the square of the distance from the extinction zone. In this case the distribution function representing the probability of survival is
and the probability density function for this distribution has the following shape:
Figure 2 shows the probability density function f(d) and the distribution function F(d) for the second-order Pareto distribution.
Figure 2: The probability density functioned) and distribution function F(d) for a second-order Pareto distribution with extinction-zone boundary b = 1
The second-order Pareto distribution has a zero probability for income not exceeding the boundary of the survival zone and a probability convening to one as income tends to infinity. It has mean and median . This distribution does not have a final variance.
GENERAL PARETO PROBABILITY DISTRIBUTION
The general Pareto distribution of order α with boundary b has the distribution function
The probability density function of this distribution has the shape:
The mean for second- and higher-order Pareto distributions is
The standard deviation of a Pareto distribution of order α > 3 is
We obtain the Dirac delta function S(d - b) from the α-th-order Pareto distribution function as the limiting case for α→∞.
The following figure compares Pareto distributions of various orders and the Dirac delta function:
Figure 3: Comparison of the characteristics of Pareto distributions of orders 1, 2 and 3 with extinction zone boundary b-1 (the dotted line shows the Dirac delta function δ(d - b))
-  Outside economics the first-order Pareto probability distribution is sometimes called the Bradford distribution.
-  See, for example, Frank, R. H,: Microeconomics and Behavior. New York: McGraw-Hill, 1994, chapter 8, p. 276.
-  Whereas for the first-order Pareto distribution the risk of extinction decreases in proportion to the distance from the extinction zone.
-  When used for the distribution of wealth this parameter is called the Pareto index.