The Cyclical Nature of Family Income Distribution in the United States: An Empirical Note
Naghshpour, Shahdad, Journal of Economics and Finance
This note utilizes a cubic polynomial to model the cyclical behavior of income inequality in the U.S. from 1947 to 2000. Linear and quadratic models have been used to explain the pattern, but they are not capable of detecting more than one extreme point. The simplest model capable of detecting a cycle is a cubic polynomial. By detecting the inflection points, the model can "predict" the turning points, from convergence to divergence. The model performs better with nominal data than with real data. (JEL 015, 040)
Researchers have examined the trends and the determinants of income inequality.1 The object of this note is to detect and forecast the changes in income inequality using the cyclical nature of the income distribution. Regression analysis of cubic and higher degree polynomials is used to detect the inflection points of the cycle of income inequality in order to predict the turning points. The inflection points signal the approach of changes in convergence or divergence. Indeed, if it is possible to detect the direction of the trend without having to wait until the "determinants" become available, such information would be useful for policymaking. This note extends the work by Amos (1986), Hsing and Smyth (1994), Dawson (1997), Matyas ct al. (1997), and Mushinski (2001) by using income to estimate the turning points associated with the cyclical nature of income inequality.
Patterns of Income Inequality
ICuznets (1955) asserted that income inequality worsens in the early stages of economic development but improves as economic development advances. The earlier research by Kuzncts (1955), Al-Samarrie and Miller (1967), Jonish and Kau (1973), and Nelson (1984) generally supports Kuzncts's assertion of an inverted-U distribution of income inequality. Their assertion, however, was not based on a quadratic model, as the hypothesis claims, but instead the downward sloping portion of the invcrted-U curve.
At least, at first glance, it seems that income inequality has a cyclical pattern. Most cycles can be approximated by simple functions with relatively few variables. In the presence of two or more peaks and troughs, a polynomial of third or higher degree is necessary. Cubic functions can predict the turning point, i.e., the maximum and the minimum points, as well as the inflection points that signal a change in the rate of change. These points signal the approach of a turning point, where inequality changes from convergence to divergence. Existing methods are not able to detect inflection points. In most instances and under "well-behaved" conditions, the inflection points are in the middle of the minimum and maximum points of the cycle.
While the invertcd-U hypothesis requires a quadratic function, the earlier studies chose the linear model with a negative slope, providing evidence to support both the inverted-U hypothesis and the use of the simplified model. Nelson (1984) was first to use a quadratic function, followed by Amos (1986), who augmented the inverted-U to account for the divergence of inequality. Since then, quadratic models have become more common. They are also used in studies of the determinants of inequality such as Dawson (1997), Hsing and Smyth (1994), Lcvernier et al. ( 1995), and Partridge et al. (1996).
In equation (1), "I" is the per capita family income, which is used as a proxy for economic development. Although the model is commonly used, it is not suitable when more than one extrerna is present, as in the case of the augmented inverted-U. Advocates of the augmented-U hypothesis argue that a negative coefficient for income (β^sub 1^ < 0) and a positive coefficient for squared income (β^sub 2^ > 0) indicate a (global) minimum (within the "relevant" range). However, I and I^sup 2^ are not independent; hence, the first and the second derivatives of the quadratic function with respect to income are not β^sub 1^ and β^sub 2^, respectively. …