Hobson's Choice

free script provided by

Income Inequality-2: Empirical Evidence

January 25, 2005

[ 1 | 2 | 3 | 4 | 5 | 6 ]

The Evolution of U.S. Earnings Inequality: 1961–2002 (PDF; Zvi Eckstein & Éva Nagypál, Minneapolis Fed, Dec '04)

In the previous post I discussed the method of Modeling income distribution. My research has been confined to households, because income data is available for households (US Census, 1, 2), so if it doesn't match up with your recollection of personal data, please don't be surprised.

In general, income distribution is modeled with a normal-log distribution because this is what economic theory predicts: your household income is probably a function of several variables y(x₀,x₁,...,x_n), where

y(x₀,x₁,...,x_n) = (x₀^α)(x₁^β)....(x_n^ν)where α, β,..., ν are exponents; we could expect at least some of these to be normally distributed, leading to the pattern of normal-log distribution. Greater equality means the standard deviation σ is smaller (a σ of 0.795 corresponds to a Gini Coefficient of 0.404, which is what we have in the USA). Now, the question is: how does this correspond to reality?

In the image above, Census data was used to create a graph of [log] income by percentile ranking for European American (blue), Hispanic American (dark red), and African American (red) households. Notice the "fuzzy" blue and red lines; these represent idealized plots for European and African American households, with the closest possible match.

First, notice that it doesn't help much to represent the national income distribution as the sum of several normal-log functions for different cohorts. If it did, the separate cohorts would be a closer match to the idealized curves. In the graph below I plotted the predicted income (using the best fit of a normal-log distribution function) minus the observed income distribution for European Americans and African Americans. When the error term is positive, it means the corresponding percentile gets less income than would be predicted.

It looks like there's really massive segmentation within the cohorts of African Americans and Europeans, does it not? And if so, it's a bit surprising who is shortchanged: the very poorest, of course, whom we would expect to fall off the edge, but also the 71th-97th percentiles. For African American households, the error term is almost linear. As a casual observer, I would suppose that reflects the "tournament theory" of CEO pay in action (HC). Calmo mentions in comments that the tournament theory actually applies to a far smaller share of households than the top 2%; however, I meant the term a little more broadly, to include the highest ranks of managers in large, multi-establishment firms. In essence, the tournament theory holds that the executive gets more than his marginal product, in order to motivate subordinates to try to get his job. I used it to suggest that a big scoop of workers in the 71st-98th percentiles are "underpaid" so that superiors can enjoy the prerequisites office.

I've been inclined to suspect that marginal product (or marginal revenue product—see HC) of labor has relatively little to do with earnings, to say nothing of household income; on the contrary, I suspect a more accurate model is a labor market consisting of smallish pools (determined by job openings at any given time, region, seniority, tenure, race, sex, specialty, networks, and associations). While one might experience a normal distribution of income within a pool, the pools themselves have such sharply differing median incomes (relative to σ) that the curve of income distribution becomes rather choppy as one reaches the highest levels. (To be continued)

CAVEAT: Unfortunately, error analysis was difficult with Hispanics since there were too many lapses in the data; the Asian American population is not sufficiently large (at >4%) to influence the income distribution.

UPDATE: Calmo, in comments below, pointed out that I had been less than clear about what the second graph is showing. The first graph is supposed to compare the distribution of income by percentile, in comparison with a best-fit curve in which the natural log of income is normally distributed. This is for the year 2004 only. The second graph is supposed to show the difference between predicted and actual income distribution. The blue fuzzy line, with the mysterious upward bulge on the right, reflects the displacement of income from the 71-97th percentiles to the 98th-100th percentiles. The Evolution of U.S. Earnings Inequality: 1961–2002 (PDF; Zvi Eckstein & Éva Nagypál, Minneapolis Fed, Dec '04)