# Appendix E A new measure of the diversification of the sources of income of the elderly, for Chapter 21 - The diversification and the privatization of the sources of retirement income in Canada

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by Long Mo

The concept of "the diversification of the sources of income of the elderly" concerns the tendency of the sources of income of the elderly to diversify. This phenomenon has essentially two features: with the diversification, the elderly possess more sources of income on the one hand, and count in a more balanced manner on those that are available on the other hand. This concept has been proposed by Rein and Rainwater in 1986, in the framework of the approach called "income package". From its beginnings, this innovative concept has been applied widely in studies on related problems about the income of the elderly. Nevertheless, we have not found until now, in the literature, an indicator that provides for measurement of the said tendency. For the purpose of measuring statistically this phenomenon in a single indicator, we propose "the index of diversification of the sources of income" (IDSI) as follows.

Suppose that the income available to the elderly is classified into n types of sources, and that the income available (R) of an aged person of is composed, in order importance, of income R1, R2, ., Rn, which arise from n types from source respectively. For convenience, if the person has no income from source number i, it is understood that Ri = 0. Then, for this person, we define

where,

G = Gini coefficient, which reflects the degree of concentration of the distribution of disposable income (R) among the n types of sources

Ri= Income arising from the source number i, i = 1, 2, ., n

By using the same principle, this index can be calculated at the level of the totality the population to measure the intensity of the diversification of the sources of income of the totality of the aged persons studied.

Given the nature of Gini coefficients, one knows that

Broadly speaking, for a given classification of sources of income, the more the sources of income are diversified, the greater is the value of the IDSI. As an example, in the case where the income is classified into five sources, IDSI = 1/5 for the person whose income concentrates in just one source. On the other hand, it is equal to one, if the income of the person comes in equal parts from all the five sources. In the end, reality is situated between these two extreme cases, the IDSI varying thus between 0.2 and 1.

Indeed, the IDSI is based on the Gini coefficient, a well-documented index and commonly used to measure the concentration of the distribution of a variable. That allows the IDSI to be an indicator of the regularity of the distribution of the income relative to the various sources, and thus of the above mentioned essential characteristic of the diversification of the sources of income.