The rest of the paper is organized as follows. In Section II we describe the test for lognormality and the dataset we employ. Section III reports the empirical results. Section IV derives the implications of lognormality for the poverty reduction roles of growth and inequality changes. Section V offers some concluding remarks.

# **II. Testing for lognormality of the size distribution of per capita income**

Attempts to model the size distribution of per capita income have a long tradition in the economics literature, dating back to Pareto (1897). The use of the lognormal function in this context was pioneered by Gibrat (1931), who found that if offered a good empirical fit to the observed distribution of income, and provided a theoretical justification based on a model in which individual incomes are subject to random proportionate changes.<sup>4</sup> Gibrat’s work was followed by a large literature extending his basic framework and offering additional empirical evidence. Kalecki (1945) modified Gibrat’s original setup making negative income changes less likely at low income levels than at high ones, to account for the fact that the variance of log income remained relatively constant over time. Rutherford (1955) expanded Gibrat’s model introducing birth and death considerations. He also presented empirical experiments based on the comparison of theoretical and observed quantiles of the distribution of income, searching for a functional form that would improve upon the lognormal.<sup>5</sup> On the theoretical front, other subsequent papers developed rigorous models that under fairly general conditions yield lognormal distributions of earnings and/or wealth (Sargan 1957, Pestieau and Posen 1979). In turn, on the empirical front, the fit of the lognormal function to the observed distribution of income was found to be somewhat less satisfactory at the upper end of the distribution (Hill 1959, Cowell 1977), specifically the top 3-4 percentiles (Airth 1985). This prompted attempts to fit more complex functional forms -- displaced and/or truncated versions of the lognormal density (Metcalf 1969, Salem and Mount 1974) or alternative functional specifications (Fisk 1961, Salem and Mount 1974, Singh and Maddala 1976, McDonald 1984). Both strategies pose a tradeoff between goodness-of-fit and analytical tractability (Metcalf 1969), as well as interpretability of the parameters (Lawrence 1988), which explains the continuing popularity of the lognormal specification.<sup>6</sup> More recently, Bourguignon (2003) has offered an indirect reassessment of the empirical validity of the lognormal approximation. He reports OLS regression estimates in a framework explaining the observed change in a selected poverty measure on the

> 4 Specifically, Gibrat (1931) argued that the good empirical performance of the lognormal density could be rationalized under the following three conditions: (i) in each period the distribution of income is derived from that of the previous period by assuming that the variable corresponding to each member of the distribution is affected by a small proportionate change; (ii) such proportions differ for different members of the distribution; and (iii) these differences are determined randomly according to a given frequency distribution.

> 5 Rutherford performed single-country estimations using data for the UK in 1949; USA in 1947 and 1948; Canada in 1947, Australia in 1951, Sri Lanka (Ceylon at the time the paper was written) in 1950 and Bohemia (modern Slovakia and the Czech Republic) in 1932.

> 6 For example, Dollar and Kraay (2002) resort to the assumption of lognormality in order to complete their data sample by generating quintile shares from Gini coefficients for those observations for which only the latter is available.

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