Chapter 2: Income Standards, Inequality, and Poverty
a parameter denoted by a, which can take any value between − ∞ and + ∞. Unlike the quantile means and the partial means, general means take into account the entire income distribution, but emphasize lower or higher incomes depending on the value of a. Parameter a is familiar in the literature as the order of general means. For income distribution x, we denote the general mean of order a by WGM(x; a). It is defined as ⎧⎛ x a + x a + …+ x a ⎞ 1a N 1 2 ⎪⎪⎜ ⎟⎠ if a ≠ 0. N WGM (x; a ) = ⎨⎝ ⎪ 1 ⎪⎩(x1 × x 2 × …× x N ) N if a = 0
(2.3)
Although a may take any value between − ∞ and + ∞, four means in this family are more well known than others: arithmetic mean, geometric mean, harmonic mean, and Euclidean mean. • For a = 1, WGM is known as the arithmetic mean (denoted by WA) or the average x¯ of all elements in x and can be written as5 WA (x) =
x1 + x 2 + L + x N . N
(2.4)
• For a = 0, WGM becomes the geometric mean (denoted by WG) of all elements in distribution x and can be expressed as WG(x) = (x1 × x2 × ... × xN)1/N.
(2.5)
If we take a natural logarithm on both sides of equation (2.5), we find WL (x) = ln WG (x) =
ln x1 + ln x 2 + L + ln x N . N
(2.6)
WL(x) is the average of the logarithm of all incomes in distribution x. The logarithm of incomes is frequently used for various analyses by labor economists. • For a = –1, WGM becomes the harmonic mean (WH) of distribution x and can be expressed as −1
⎛ x −1 + x 2−1 + L + x N−1 ⎞ WH (x) = ⎜ 1 ⎟⎠ . N ⎝
(2.7)
63