Chapter 2: Income Standards, Inequality, and Poverty
a ⎧ ⎡1 ⎤ N ⎛ xn ⎞ 1 ⎪ ⎢ ∑ n =1 ⎜ ⎟ − 1⎥ if a ≠ 0,1 ⎝ x⎠ ⎪ a(a − 1) ⎢⎣ N ⎦⎥ ⎪ N x ⎪1 ⎛x ⎞ IGE (x; a) = ⎨ ∑ n =1 n ln ⎜ n ⎟ if a = 1. x ⎝ x⎠ ⎪N ⎪1 ⎛ x⎞ N ⎪ ∑ n =1 ln ⎜ ⎟ if a = 0 ⎝ xn ⎠ ⎪⎩ N
(2.26)
At first glance, the formula above looks complicated. However, measures in this class are closely related to general means. Every index in this class, except one, can be expressed as a function of the arithmetic mean and the general mean of order a. For a ≠ 0, 1, the class of generalized entropy measures can be written as IGE (x; a ) =
⎛ ⎢⎣WGM (x; a ')⎥⎦a − ⎢⎣WA (x)⎥⎦a ⎞ 1 ⎜ ⎟, a a (a − 1) ⎝ [ W (x)] ⎠
(2.27)
A
where we replace the term x¯ by WA(x) (the arithmetic mean), and where WGM(x; a) denotes the general mean of order a. Thus, a generalized entropy measure for any a ≠ 0,1 may be easily calculated once we know the arithmetic mean and the general mean of order a. For a = 1, the generalized entropy is Theil’s first measure of inequality and can be written as IT1 (x) =
⎛ xn ⎞ 1 N xn ln ⎜ . ∑ N n =1 WA (x) ⎝ WA (x) ⎟⎠
(2.28)
This is the only measure in this class that cannot be expressed as a function of general means and does not have a natural twin-standards representation. For a = 0, the generalized entropy index is Theil’s second measure of inequality, which is also known as the mean logarithmic deviation and can be expressed as a function of the arithmetic mean, WA(x), and the geometric mean, WG(x), as follows: IT 2 (x) = ln WA (x) − ln WG (x) = ln
WA (x) . WG (x)
(2.29)
Besides the two Theil measures, the other commonly used measure in the entropy class is the index for a = 2, which is closely related to the coefficient of variation (CV). The CV is the ratio of the standard deviation and
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