Is There a Kuznets Curve?


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1 Is There a Kuznets Curve? John Luke Gallup Portland State University 1721 SW Broadway Cramer Hall, Suite 241 Portland OR tel: fax: September 25, 2012 Abstract No. There has never been good evidence for a pattern of rising inequality in lowincome countries and falling inequality in higher income countries. The only evidence that appears to support the Kuznets hypothesis is the crosssectional pattern of inequality levels across countries, although the Kuznets hypothesis is an assertion about the path of inequality within countries. Numerous crosssectional studies established the Kuznets curve as a stylized fact, dominating empirical and theoretical research on the effect of economic growth on income inequality since then. New international panel data with the first internally consistent time series for a large number of countries shows no evidence of a Kuznets curve. The data show an anti Kuznets curve: inequality decline in lowincome countries, and inequality increase in highincome countries. The Ushaped pattern shows up strongly in a nonparametric trend, in stochastic kernel estimation, but weakly in a quadratic fixed effect trend. JEL Codes: D31, O15, O47 Keywords: Income inequality, Kuznets curve, Inequality panel data
2 1 Introduction In late 1954, Simon Kuznets gave a bold Presidential Address to the American Economic Association (Kuznets, 1955). He chose a topic which had been largely unstudied due to lack of data, and used the scant data he had created himself or found elsewhere to propose a law of motion for the distribution of income. Kuznets had data for the change in income distribution in the United States, United Kingdom, and two states in Germany. The change he was confident of was a sharp decline in inequality in the U.S. and the U.K. after World War I. He also noted substantial income growth in the two countries during the same period. Kuznets combined his observations about the U.S. and the U.K. with the historical shift from agriculture to industry in the course of economic development to propose a typical pattern of change for income distribution. Kuznets assumed that rural agricultural incomes are lower and more equally distributed than urban industrial incomes. In that case, a shift into nascent industry will raise income inequality, as a rising fraction of workers earn higher industrial wages. Beyond a tipping point, the predominance of industrial employment will improve income distribution, as most workers earn similar industrial wages. This theory predicts an inverted Ushaped relationship between income levels and inequality. The international income distribution data necessary to evaluate Kuznets s hypothesis remained severely limited until the mid 1970s. By that time, at least one national income distribution observation was available for enough countries to look at the crosscountry relationship between income levels and inequality. The crosssectional data were consistent with the Kuznets hypothesis: middleincome countries, especially in Latin 1
3 America, tended to have higher levels of inequality than low or highincome countries. This lead to scores of empirical papers over the next forty years looking for an inverted Ushaped curve in crosscountry data with gradual improvements in country coverage and econometric technique. A partial list of studies includes Kravis (1960), Paukert (1973), Chenery and Syrquin (1975), Ahluwalia (1976a, 1976b), Saith (1983), Papanek and Kyn (1986), Campano and Salvatore (1988), Ram (1988), Tsakloglou (1988), Bourguignon and Morrison (1990), Anand and Kanbur (1993), Randolph and Lott (1993), Bourguignon (1994), Ogwang (1994), Fields and Jakubson (1995), Ram (1995), Jha (1996), Dawson (1997), Eusufzai (1997), Mbaku (1997), Chang and Ram (2000), Savvidesa and Stengos (2000), Lin and others (2006), and Huang and Lin (2007). The most careful of these studies, such as Anand and Kanbur (1993) and Fields and Jakubson (1995) did not find robust support for the Kuznets hypothesis, but most did. None of this research tested Kuznets hypothesis directly: that income inequality would increase and then decrease as income grew within countries. If other factors can influence the level of income distribution in each country, country characteristics rather than a Kuznets process might explain the crosssectional pattern. The shear repetition of crosssectional results gradually convinced people that the Kuznets curve was an empirical reality, to the point where theorists creating models with income distribution dynamics made sure they were able to replicate a Kuznets curve in their models. By the turn of the century, Kanbur (2000) felt that the dominance of the Kuznets hypothesis in income distribution research had become counterproductive: In fact, in a strange way the framework set out by the originators may have by now become a straightjacket which inhibits fresh thinking, as every new attempt to model development and distribution does so with at least half an 2
4 eye on whether or not the model can, in principle, generate an invertedu relationship between inequality and development, while most empirical work keeps returning to the question of whether or not there is an invertedu pattern to be discerned in the data. Crosscountry panel data did not become available until the 1990s. Deininger and Squire (1996, 1998) assembled the first largescale dataset with enough observations to study the typical path of inequality within countries. Deininger and Squire took the estimates of Gini coefficients from hundreds of separate studies of inequality in individual countries to construct a large number of time series. The two main conclusions of their research were that more countries have inequality paths inconsistent with the Kuznets hypothesis than consistent, and that most countries inequality changes slowly over time. Using the Deininger and Squire dataset, researchers have found no support for a Kuznets curve once they control for country fixed effects (Deininger and Squire, 1998, Higgins and Williamson, 1999, Savvidesa and Stengos, 2000, and Barro, 2000). Barro s influential paper (updated in Barro, 2008) did find support for the Kuznets curve, but that was due to regressions which do not include country fixed effects. That is, Barro was reproducing the usual crosssectional patterns. When Barro did include country fixed effects, the quadratic trend in income is insignificant, although the trend becomes statistically significant when he adds a number of additional explanatory variables 1. Even then, Barro acknowledges that income levels explain little of the trend in inequality. 1 One can see from a footnote (fn. 24 on p. 31) that the quadratic term in a simple fixed effects regression of the Gini coefficient on log GDP per capita is statistically insignificant. 3
5 Criticism of the accuracy of the inequality time series constructed by Deininger and Squire cast a shadow over research using the data set. The measured level of inequality depends sensitively on the definition of income or expenditure, unit of observation, survey coverage, etc. Combining inequality estimates to construct a time series from studies using different definitions and methods risks introducing spurious jumps due to changes of definition. Atkinson and Brandolini (2003) were skeptical of the accuracy of time series constructed by Deininger and Squire. They showed that for many Western European countries, Deininger and Squire's time series often departed significantly from a series Atkinson and Brandolini constructed using consistent definitions, causing the series to have serious inaccuracies both in the level of inequality and the trend over time. Western European data should presumably be among the most accurate. Atkinson and Brandolini conclude that we are not convinced that at present it is possible to use secondary datasets [like Deininger and Squire] safely without evaluating the accuracy of each series within it. Atkinson and Brandolini s critique effectively ended use of Deininger and Squire s panel data in published research due to doubt about the internal consistency of its time series. This paper uses a new panel of inequality data constructed with consistent definitions and data sources within each country over time. The data are used to estimate two kinds of nonparametric models of the relationship of inequality and income: a nonparametric fixed effects trend and a stochastic kernel model. The estimations show a clear Ushaped relationship rather than Kuznets invertedu relationship. Several earlier papers have used nonparametric methods to look for a Kuznets curve. Deininger and Squire (1998) in effect does this by counting countries with invertedu shaped inequality paths. However it not clear that this is a test for a general Kuznets curve if the invertedu trajectory occurs over different ranges of income in different 4
6 countries. Does this imply that countries would undergo multiple invertedu shaped trajectories as their income rises? Frazer (2006) uses nonparametric regression to test for a Kuznets curve using panel data from an update of the DeiningerSquire dataset. The nonparametric specification does not control for different levels of inequality in each country, however, so his results are still dominated by the crosssectional levels of inequality. Two other papers (Lin and others, 2006, and Huang and Lin, 2007) use semiparametric methods to test the Kuznets hypothesis, but by using crosssectional data, they are also not able to evaluate the typical trend of inequality within countries. The next section discusses the data in Kuznets original paper. Section 3 describes the new inequality data and quadratic trend lines. Section 4 presents a nonparametric fixed effects estimate of the Kuznets curve. Section 5 evaluates change in the distribution of inequality of a panel of countries using stochastic kernel estimation. Section 6 concludes. 2 Kuznets data Kuznets hypothesis about the relationship between inequality and economic growth in his 1955 address was based on timeseries data for just three countries and his intuition about the mechanisms of economic development. His conjecture was audacious given the data he had to work with. He acknowledged this, saying that his paper is perhaps 5 per cent empirical information and 95 per cent speculation, some of it possibly tainted by wishful thinking (Kuznets, 1955, p. 26). Kuznets used data he helped collect for the United States combined with data for the United Kingdom and two states in Germany. He also found point estimates of inequality in India, Puerto Rico, and Ceylon. Figure 1 presents the inequality data from Kuznets's original article combined with historical estimates of GDP per capita from 5
7 Maddison (2010) to show the patterns he was visualizing. It is striking that his data do not provide much support for his own hypothesis, except that United Kingdom and United States had substantial declines in their inequality before World War II. Kuznets discusses the likelihood that inequality worsened in both of these countries in the nineteenth century before his time series start, but he had no empirical evidence for it. Besides the U.S. and the U.K, the other countries for which he has inequality estimates do not indicate an invertedu curve. Kuznets presumption was that inequality is very low in agrarian societies before the advent of industrialization. Premodern inequality data are hard to come by, but Milanovic and others (2007) were able to reconstruct inequality data for eleven preindustrial societies ranging from the Roman Empire in AD14 to China in the 1880s. As shown in Figure 2, inequality in these agrarian societies was not particularly low. All but three of the estimates of inequality are higher than the median inequality in countries today (calculated from the data described in the next section). Half the preindustrial Gini coefficients are above the 75 th percentile of modern inequality estimates, including the estimates of inequality in England and Wales in the 17 th and 19 th centuries. The data do not suggest that one can assume low inequality in preindustrial societies. Feudalism didn t promote particularly equal distribution. Kuznets showed that inequality fell in two highincome countries as they grew richer after World War I, but he had no evidence of rising inequality at low income levels. Ironically, Kuznets prediction that inequality will rise during the early stages of development, for which he had no evidence, is better remembered than his prediction that inequality will fall at higher incomes. One reason that Kuznets's paper had a big impact was his model explaining the path of 6
8 inequality. In its simplest form, if agricultural workers all earn a low wage and industrial workers earn an identical higher wage, then the transition from agriculture to industry will create an invertedu curve in inequality. The movement of the first workers out of agriculture into higher wage industry will increase inequality, but beyond a certain point, inequality will fall as the majority of workers receive the constant industrial wage. A vulnerability of Kuznets s theory is that minor changes in the story change the prediction. For instance, if agricultural incomes are more unequal than industrial wages, perhaps due to unequal land ownership, movement out of agriculture into industry could reduce inequality right away. Furthermore, all kinds of other dynamics of economic development are likely to have implications for inequality besides the movement of labor out of agriculture into industry. International trade, the spread of education, and transportation linking previously isolated regions, to name just a few dynamics, are all likely to have major impacts on income distribution in ways that do not naturally suggest an invertedu shaped curve. Besides Kuznets scanty data and model of inequality change, what caused his hypothesis to so thoroughly capture the imagination of economists? The country crosssectional data. Figure 3 shows recent estimates of inequality for 156 countries. The Gini coefficients are the most recent observation for each country from the panel data set used in this paper for 87 countries augmented with the most recent Gini coefficients for 69 other countries from the World Development Indicators (World Bank, 2011). 2 GDP per capita 2 The region categories in Figure 3 and other figures are OECD90, Latin America, Eastern Europe and the Former Soviet Union, Asia, and Africa. OECD90 indicates members of the Organization of Economic 7
9 estimates are from the Penn World Tables (Heston et al., 2010). This is the evidence, such as it is, that the Kuznets curve exists. A quadratic fit is significantly concave, although most of the curvature comes from the low inequality levels at high incomes rather than low inequality at low incomes. The curvature is also entirely dependent on using a logarithmic scale for income. Figure 4 shows that the Kuznets curve disappears when the quadratic fit is made using a unit GDP per capita scale rather than a logarithmic scale. The level of inequality in different countries relative to income does not necessarily tell us anything about the typical path of inequality within countries, which was the object of Kuznets hypothesis. Figure 5 shows that if countries differ in their level of inequality, inequality could follow a Ushaped trend within each country, but the quadratic fit across countries could be an invertedu curve. The level of inequality could differ across countries in a simple Kuznets model of inequality change if some countries have regional variation in incomes and others don t. If each region industrializes separately, the countries with regional income differences will have higher inequality (due to the interregional income differential) even if every region follows the trajectory of rising then falling inequality in the course of industrialization. Looking at withincountry inequality trends and allowing for different levels of inequality across countries requires panel data. The panel data assembled for this study is Cooperation and Development as of 1990, which includes the highest income countries of the world except for oil exporting countries. More recent OECD members Mexico, South Korea, Chile, Israel, Czech Republic, Slovakia, Estonia, Hungary, Poland, and Slovenia are excluded from the OECD90 group, since most of these countries still have income levels substantially lower than OECD90 members. The excluded OECD members countries are included in the other regional groups. 8
10 described next. 3 Data and quadratic trends The lack of comparable data across countries has always plagued research on income inequality. Unlike most other basic national statistics, there is no international organization that collects standardized measures of income distribution worldwide. Good time series of inequality require data from a uniform household survey design over time because the scope of the questions about household income or household expenditure can have a big effect on the calculated dispersion of income or expenditure. Surveys should cover the whole countries' population and all sources of income and/or expenditure. Inconsistencies over time in the method used to estimate inequality levels also cause inaccuracies in the time series. Only very recently have consistently constructed timeseries of income distribution become available for a large number of countries. Four organizations have created series of inequality statistics from raw survey data spanning more than a decade for a large number of countries in certain regions of the world. The four organizations are Eurostat (2011) for European Union members, the TransMONEE database created by UNICEF for Eastern Europe and former Soviet Union countries (TransMONEE, 2011), SEDLAC (2011) at the Universidad Nacional de la Plata for Latin America and the Caribbean, and the Luxembourg Income Study database (LIS, 2011) for selected highincome countries. These four organizations provide statistics for all of Europe (East and West), Central Asia, Latin American, and several additional highincome countries. 9
11 Major parts of the world still do not have organizations which collect standardized income distribution statistics: East and South Asia, the Middle East, and Africa. Of these, only the Asian regions have a large number of countries with the raw material for the statistics: household income and/or expenditure surveys spanning a substantial number of years. This study supplements the data from the four regional organizations above with statistics come from the UNUWIDER World Income Inequality Database (WIID 2011) and a few other sources. WIID, unlike the other organizations, compiles secondary data rather than generating statistics directly from household survey microdata. WIID is a continuation of the work of Deininger and Squire (1996, 1998) to collect inequality series from national statistical agencies or academic studies. Unlike the original Deininger and Squire data, however, this study does not patch together data from disparate studies into a time series. Crucially, the data in this study only includes country series which are internally consistent over time in terms of the source household survey and the method used to calculate the inequality statistics. The country times series used in this study hew to the following criteria: 1) They are calculated from surveys of household income (covering all income sources) or household consumption expenditure drawn from a national sample of all households. 2) The time series are calculated from surveys with the same survey design each year. 3 3) The time series of inequality statistics are calculated using the same method and definitions within each country. 3 Minor changes of survey questions and survey design still occur over time in many of the standard national surveys, but statisticians usually address these inconsistencies when they calculate a time series of inequality estimates. 10
12 Criteria 2 and 3 are particularly important for ensuring an accurate measurement of the change of inequality over time. Failure to meet these criteria is what caused inaccuracies in Deininger and Squire's inequality series. These data cover a later period than the data in Deininger and Squire. Most of the data from Eurostat, TransMONEE, and SEDLAC are derived from national surveys established in the 1990s which were not available when Deininger and Squire compiled their data. Even most Western European countries did not have annual household income surveys before the establishment of a Europeanwide survey in For the most part, lowincome countries measure inequality using household consumption, and highincome (and Latin American) countries measure inequality using household income. Different countries use different weighting methods for household members. Some use income per adult equivalent, some use income per capita, and a few use total household income. These differences can affect the level of inequality across countries, but that won t be an issue in this study due to controls for countryspecific inequality levels. 4 The data include time series from 87 countries, split regionally so that about a quarter of the countries come each from the OECD90 2, Latin America, Eastern Europe and the Former Soviet Union, and Asia and Africa combined. Asia and especially Africa are underrepresented. The data for Eastern Europe and the former Soviet Union before 1994 are excluded to avoid picking up the sudden inequality changes after the collapse of central planning. 4 See more details about the definitions of inequality in each country in Gallup (2012). It was this kind of inconsistent definitions within countries which caused the Deininger and Squire data not to have reliable time series. 11
13 Gini coefficients are paired with income levels for each country and year. Income levels are measured by gross domestic product (GDP) per capita from the Penn World Tables version 7.0 (Heston and others, 2011). The GDP per capita figures are adjusted for purchasing power parity and reported in 2005 constant international dollars. The inequality time series for 87 countries are graphed in Figure 6. One can see the invertedu shape of inequality levels similar to the cross section graph in Figure 3. However, if we control for inequality level differences in each country with a quadratic fixed effects estimator, there is a slight U shape to the fit, although not statistically significant, as shown in Table 1. There is no sign of the inverted U of the Kuznets hypothesis in the typical withincountry trend. Table 1: Fixed Effect Inequality Trend Gini coefficient ln(gdp per capita) (0.62) ln(gdp per capita) (0.68) Constant (2.93)** R N 852 * p<0.05; ** p<0.01 In the crosssection, the appearance of an apparent Kuznets curve depends on whether or not GDP per capita is transformed in logarithms. This suggests that testing for a 12
14 Kuznets curve may depend sensitively on the functional form used. The next section looks at the relationship of inequality to income level without parametric assumptions about the trend. 4 Nonparametric trend Kuznets hypothesis is about the change in inequality as income grows, not about the level of inequality. Inequality increases at a decreasing rate up to a middle income level and then decreases at an increasing rate. This relationship can be expressed as dg dy = f (y) where f (y) 0 for y y m and f (y) < 0 for y > y m. The slope of the trend of inequality, dg dy, increases up to a middle income, y m, and then decreases. The commonly used quadratic trend line takes f (y) = β 1 + β 2 y which implies that g = β 0 + β 1 y + β 2 y 2 for some β 0, which could be country specific. The Kuznets hypothesis is that β 1 > 0 and β 2 < 0. With nonparametric methods, it is not necessary to specify the functional form of f (y) ; the shape of f can be inferred from the data. Using m dg dy for the slope of the inequality trend, we can use nonparametric smoothing to fit the equation m it = f (y it ) + ε it (1) to data where ε it is a random error, i is the country indicator, and t is the time indicator. m it = g it g i,t 1 y it y i,t 1. Equation 1 is independent of the initial level of inequality in each country, so it can be used to estimate the typical trend in inequality across countries. 13
15 f ( ) is estimated by kernelweighted local polynomial smoothing (Stata, 2011, p ). This method takes a weighted polynomial regression of neighboring values to predict the level ˆf (y it ) for each level of income, where y it is the natural log of GDP per capita. This method fails due to the large influence of a small number of outliers. In some countries, GDP is virtually unchanged over the period of two inequality surveys. Due to sampling variation, the inequality estimates are different in the two periods, causing the estimate of the slope to explode: for g it g i,t 1 > 0 as y it y i,t 1 0, m it. A solution is to smooth each country s inequality trend separately beforehand, and calculate m it from the smoothed values. Since the ultimate purpose is to find the average smoothed trend in inequality, smoothing each country s trend first doesn t introduce any bias, but it does eliminate the large outliers due to sampling error. Figure 7 shows the smoothed trend in the inequality slope for different income levels. The dotted lines are the 95% confidence bounds. The results are the opposite of the Kuznets hypothesis. At low income levels, inequality falls with income, and at higher income levels, inequality rises with income. The preferred specification shown in the figure is a linear smooth (polynomial of degree 1) using an Epanechnikov kernel with a bandwidth determined by the ROT algorithm of Under certain assumptions, the ROT bandwidth is optimal (Fan and Gijbels, 1996). The smooth in Figure 7 turned out not to be sensitive to variations in bandwidth, kernel choice, or degree of polynomial. The bandwidth was varied from
16 to 10, with virtually no effect on the smooth. Gaussian, triangle, and rectangle kernels all produced qualitatively similar trends, although the rectangular kernel causes the point of inflection where the slope crosses the axis to occur at a lower income level. A polynomial of degree 0 (local mean smoothing) flattens the trend somewhat, and polynomials of degree 2 and 3 increase the confidence interval somewhat, but the upward trend remains the same. In each case, the parameters are used for presmoothing each of the individual country series are varied along with the final smooth. To help visualize the results, we can construct a curve in inequality levels from the estimated slope function. Given an estimated slope ˆm it = ˆf (y it ), the inequality level can be calculated recursively: ĝ it = ĝ i,t 1 + ˆm it (y it y i,t 1 ). The level of g i0 is fixed so that the average level of the curve is equal to the average inequality in the sample. Figure 8 shows the trend line of inequality superimposed on the smoothed country trends. The confidence bounds were calculated by bootstrapping. Like the estimated slope curve, the shape and precision of the level curve is not very sensitive to the choice of bandwidth, kernel, and degree of polynomial. The nonparametric trend in inequality in Figure 8 shows a strong downward trend in inequality up to relatively high income levels, where the trend becomes upward sloping. The nonparametric trend of inequality is clearly Ushaped, not invertedu shaped. The trend in Figure 8 is consistent with Kuznets data, such as it was, but it is also consistent with very long historical series that have recently been compiled for top income shares for almost two dozen countries over the course of the twentieth century. Almost all of 22 countries discussed in the survey by Atkinson, Picketty, and Saez (2011) have graphs of the share of income going to the richest 1% over the last hundred years which have remarkably similar shape to the graph in Figure 8, although some 15
17 middle European countries and Japan show no sign of the upturn at the end. 5 Stochastic Kernel Estimation The previous sections used quadratic and nonparametric trends in inequality (controlling for country fixed effects) to evaluate the change in average crosscountry inequality as income grows. A fuller assessment would consider the evolution of the whole distribution of country inequalities as income rises, rather than just the trend. A single trend, for instance, could mask a dynamic where some kinds of countries tend towards one level of inequality and other countries tend towards a different level of inequality. The data show a lot of diversity within countries in the path of inequality, with some countries distribution becoming more equal and others less equal whether they start out with high or low inequality. Stochastic kernel estimation is flexible enough to model the net outcome of complex dynamics of inequality change. It can incorporate a dynamic path where all countries tend towards a single level of inequality or where they tend towards two or more different levels of inequality. It can also incorporate a dynamic path where countries switch places between different levels of inequality, while the overall distribution of crossnational inequalities remains the same. Stochastic process models are typically applied to processes that evolve over time. However, most hypotheses about income distribution concern the path of inequality as income grows, not as time passes. For this reason, we model the crosscountry distribution of inequalities in the income domain rather than in the time domain. 16
18 The stochastic kernel model represents the evolution of continuous distributions from period to period. It is the continuous analogue of a Markov chain, which represents the evolution of discrete distributions. Quah (1997, 2007) explains stochastic kernel models and applies them to the distribution of income levels across countries. Since stochastic kernel models use continuous distribution functions, Quah defines them using measure theory instead of the more accessible matrix algebra of Markov chains. Continuous income distributions are attractive conceptually, but in practice, stochastic kernel models are estimated by discrete approximation. Digital computers must approximate continuous transition surfaces with discrete grids, so a stochastic kernel model is actually estimated as a finegrained Markov chain. Since the model is ultimately estimated using a discretized distribution, we will present stochastic kernel estimation using the simpler Markov chain notation. There is no loss of generality since a continuous distribution can be arbitrarily well approximated by a high dimension discrete distribution. Refer to Quah (1997, 2007) for the continuous formulation. The usual difference in practice between the stochastic kernel and Markov chain estimation is the method of calculating the transition matrix. Markov transition matrices are typically estimated from crude frequency counts of transitions. Stochastic kernel estimation, in contrast, typically uses bivariate kernel density estimation. 5 5 Although stochastic kernel estimation and kernel density estimation both include the term kernel, meaning distribution function, they are referring to different uses of a distribution function. Stochastic kernel estimation refers to the estimation of the transition from one period's distribution to the next period's distribution of the variable of interest (here, inequality). Kernel density estimation refers to the weighting scheme for averaging neighboring frequency observations. The weights decline moving away from the cell of interest according to the value of the kernel, or distribution function, chosen (e.g. Gaussian, Epanechnikov, triangle, etc.) 17
19 The stochastic kernel estimation of the transition matrix can be finergrained with more rows and columns because the kernel density estimation uses information from neighboring cell frequencies to smooth the density estimates. Whereas a typical Markov transition matrix of frequencies from a sample of several hundred observations might be a 5x5 matrix to avoid small sample sizes in each transition cell, a bivariate kernel density estimator would commonly produce a 50x50 matrix smoothly approximating the surface. So in practice a stochastic kernel estimation is a Markov chain estimation with a smoothed transition matrix. The Markov chain in this application models the evolution of the distribution of country inequalities with transitions from one income level to another, rather than the more conventional transition from one time period to another. First divide inequality into N possible levels, with values g 1,,g N. The set G = {g 1,,g N } is the state space, and countries move from one inequality level, g i, to another, g j, at each step of income. A fundamental assumption of the Markov model is the Markov property. Let X s G be the inequality level of a country at income level s. The Markov property is that assumption that E( X s+1 X 0,, X s ) = E( X s+1 X s ). That is, the inequality level at the next higher income level depends only on the inequality level at the current income level, and not on the earlier history of inequality levels at lower income levels. In addition, we assume homogeneity of transitions across income levels: E( X s+1 X s ) = E( X s X s 1 ) for all s. Then we can define the transition probability as p ij = E( X s+1 = g i X s = g j ). The N by N matrix of all the transition probabilities is P = p ij. Let u s be an Ndimensional probability row vector which represents the state of the Markov chain at income level s. The i th component of u s represents the probability 18
20 that the chain is at inequality level g i. Then u s+1 = u s P. If we assume that for every inequality level g i except for g 1 there is a positive probability of inequality falling to g i 1 at the next income level, and at every inequality level g i except for g N there is a positive probability of inequality rising to g i+1 at the next income level, these are sufficient conditions for the Markov chain to be ergodic, which means there is a possibility of going from every inequality state to every other inequality state, although not necessarily in one step. By Doeblin's Theorem (Stroock, 2000, p. 28), ergodic Markov chains tend towards a unique stationary probability vector as income levels increase without bound. The stationary probability vector w is defined by w = lim u 0 P s. w shows us where the distribution of inequalities will end up if the s current distributional dynamic continues indefinitely. The stationary distribution w is equal to the first left eigenvector of the transition matrix P, and is independent of the initial distribution u 0 (Theorem 8.6, p. 106, Billingsley, 1979). This Markov model allows for a broad range of inequality dynamics including churning between inequality levels. However, it does rule out a Kuznets curve, the tendency for inequality to rise at low income levels and then switches to moving towards lower inequality beyond a certain income threshold income. The assumption of homogeneity, E( X s+1 X s ) = E( X s X s 1 ), means that if inequality tends to rise with income, it does so at all income levels. To allow for a Kuznets curve dynamic where inequality increases at low income levels but falls at high income levels, separate Markov processes are estimated for observations below middle income and for those above middle income. The Kuznets hypothesis predicts convergence towards higher inequality levels in the lowincome sample and convergence towards lower inequality in the highincome sample. 19
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