Lexique-­‐Grammaire

The concept of calculating a rate of decline is essentially a simple one. It involves taking the total decrease from the first initial peak of a distribution (usually on the median) until the flat line, or until the population density no longer increases³. Once total decrease in population density is established it is then divided into the sum of the elapsed or observed wealth spectrum. The Slope-shaped distrbuitons provide great test subjects due to their smooth curve and identifiable flatlinings (flatlinings here are taken to mean the negative Slope-shape from the first peak) however the U-shaped countries have often run into problems with

identifying an appropriate flatline. The rate of decline is then calculated by taking the combination of X changing over Y peroid. This is to say that as net wealth decreases, it does so over a given population density

3 With further work a threshold for the end point of capture would be established for the rate of decline. This may be on a percentile basis which would accommodate a distribution perfectly asymptotic to the X axis. It would enable one to definitely establish where the capture of the rate of decline would end by calculating whether or not the decrease was, for example, larger or not than one percent.

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decreasing which provides a rate of population density falling as the wealth ranges increase.

The rate of decline could arguably represent a proxy representation of a the middle class (this representation depends on ones definition of the

‘middle-class’). If one were to concieve of the middle-class as the wealth ranges after the most densely populated range to the least populated range one would have an emperically workable definition. This argument has been accepted as an assumption, however one may also use the rate of decline as a means for measuring how ‘quickly’ the population density falls past a given point. This notion of how sharply any given country’s decrease is has the ability to yield some very interseting information.

Assessing the different gradiants for different countries offers a

comparative example of the relative inequality within systems. The three main points of analysis from this may include:

1) The ‘length’ of wealth needed to observed a decline from the most densly populated range to the least populated range;

2) The steepness of the gradiant, as shown by the rate;

3) The rate of decline which indicates the change in population density over wealth.

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To emphasise this point further consider the following two examples which are abstractions of the population-based wealth distribution graph.

In the first example shown by figure 8 in the graph below there are two countries, let these be country X and country Y. The hard filled lines are the real distribution of wealth ascending across the X axis and the count of the population, or could be considered as the density, is shown in the Y axis. The dashed lines are each respective countries rates of decline.

Country X, in the turquoise, is very similar to many Slope-shape countries. They typically begin with a high density peak which tails off into

Figure 8 - Example of hypothetical rates of declines: high and low.

Figure 9 - Example of rates of declines: three hypothetical countries.

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the higher levels of wealth. Country X has a relatively high rate of decline of which the starting capture point is the top of the first peak and the point at which the tail end no longer increases. Compartively speaking, country Y has a more gradual rate of decline which is much lower than country X.

This rate of decline means that for every additional €xx,xxx observed, x.x% of the total population will not have net wealth above the new threshold.

The next example in figure 9 instead uses three unusual hypothetical countries, called country X, Y and Z and for all intended purposes the graph uses the same X and Y axis labels. In this example, country Y has 100% of its population within one confined wealth range.

For hypothetical purposes, propose that the range is €1,000 in which all the population resides with. It can then be seen by calculating the rate of decline that the country’s rate is at the local maximum steepness and thus is a population of complete inequality. Country X may look similar to the Lorenz curve shown in the literature review section however it is not. The reason for this is that the Lorenz curve is percentage based whereas this graph is absolutely based. For hypothetical reasons, if one wanted to make a similar comparison to the GINI coefficient one could compare the difference between country Y and country X as one is of perfect equality and the other not. Country Z is similar to country Y in that it has no rate of decline as the density of its population does not change as wealth

increases. Country Z has a perfectly equal relative distribution of its population across the wealth spectrum whereas country Y has a perfectly equal distribution of wealth across its population. These three hypothetical examples are interesting cases of some of the possibilities of wealth distributions and how a rate of decline could describe them.

A real example shown shown below will examines Portugal’s distribution of wealth and the calculations for creating a rate of decline over a certain range of wealth. As it is highlighted, the calculation is concerned with the range of data beginning with the median peak and ending when a flatline has been observed. A flatline in this context is observed when the population density of any single bracket no longer increases. The concerned brackets for Portugal are highlighed in pink.

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As described previously, calculating the rate of decline requires measuring the percentage decrease from the central peak to the flatline. In figure 10 below the columns highlighted in pink show the selected wealth range that observes the decline. This wealth range is from €0 -> €450,000 and

across this range the population declines from 696 to 19.The total change within these wealth ranges is 97.27% - table 2.

Table 2 - Portugal's rate of decline.

Range 1 Range 2 Total