Abstract
This chapter introduces two flexible functional form approaches to approximate Lorenz curves. The first approach expands the inverse function of an income distribution in an exponential polynomial series and derives the Lorenz curve from it. The required convexity condition can be imposed using a Bayesian method. The second approach approximates the Lorenz curve with a sequence of Bernstein polynomial functions. The required convexity condition is automatically established in this approach. We compare these approaches with another well known fixed functional form approaches. We evaluate the performance of these functional forms by comparing approximation errors, maximum error, and the estimates of the Gini coefficient produced by various approaches.
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Notes
This chapter follows Ryu and Slottje (1996a).
World Development Report (1990) shows the upper 10 per cent of Brazil income earners get 50.6 per cent of total income while the same 10 per cent of income earners in Japan get only 22.4 percent of income.
See Zellner and Highfield (1988) and Ryu (1993) for a justification of introducing the exponential polynomial functional form for a probability density function.
The inverse function of an income distribution function can also be considered as another distribution function if the range of income is rescaled to [0, 1]. Since a distribution function is much smoother than a density function, if a certain flexible functional form is found to be useful for describing a density function, then this same functional form can be useful in describing a distribution function, too.
This is essentially a curve fitting exercise and the least squares method is widely used. For efficient parameter estimation, the generalized least squares method can be used if the structure of the covariance matrix is known. However, the difference will be negligible when the observed sample size is large.
The first three polynomials explicityly written out, etc.
If the function L is bounded, say | L(z) | ≦ M in 0 ≦ z ≦ 1 and x a point of continutiy, for a given ε > 0 we can find a δ > 0 such that | z-z‱ | < δ implies | L(z)-L(z ′) | < ε. Lorenz(l986) shows | BI (z)-L(z) | ≦ ε+ M (2Iδ2)-1 < 2ε as I → ∞ If L(z) is continuous in [0, 1], then the above inequality hold with a a independent of x, so that B1 →L uniformly.
If k = 1, we obtain Here L((v + 1) / I)-L(v / I) is the first order difference ΔL(v / I) of the function L(z) evaluated at z = v / I.
A large number of observations, I, are placed into 100 grid cells and the Lorenz curve is derived using these averaged values.
If we approximate a convex function with a weighted averaged sequence, then the approximated value will always be larger than the true function value. In (4.22), we sum over point z′ = v / I with function values L(v / I) and weights.
The Gini coefficient for Korea has been reported by many researchers. These values ranged between 0.34 and 0.38 which is smaller than our value of 0.41. In the previous surveys, extremely large values of measured income were considered as outliers and removed from the data set. In comparison, our data set collected by KDI includes al the sample observations.
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© 1998 Springer-Verlag Berlin Heidelberg
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Ryu, H.K., Slottje, D.J. (1998). Some New Functional Forms For Approximating Lorenz Curves. In: Measuring Trends in U.S. Income Inequality. Lecture Notes in Economics and Mathematical Systems, vol 459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58896-9_4
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DOI: https://doi.org/10.1007/978-3-642-58896-9_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64229-9
Online ISBN: 978-3-642-58896-9
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