ECONOMIC GROWTH AND INCOME CONVERGENCE: A ...

Economic Growth and Income Convergencecountries (or regions or economies) tend to converge one another and make aconvergence club at a low level of per capita wealth. On the other hand, richer nationslike United States, western European countries, OECD countries are grouped into a higherincomeper capita convergence level. Barriers such as educational limitations, lack ofresources or poor infrastructure prevent poor countries from moving to a higherconvergence club. These factors make it nearly impossible for a country in oneconvergence club to move to another convergence club. Convergence clubs are usefulfor examining economic development in a specific country, relative to other countries.These groups help to identify similarities and differences between countries, and assistresearchers in making generalized hypotheses.In bountiful studies of convergence test of economic growth and income, regressiongrowth models are used extensively. In the recent empirical growth literature on theconvergence of GDPs across regions and countries, the growth regression approach(Barro and Sala-i-Martin 1992a) is popularly used to establish empirics of convergence(and divergence).The neo-classical growth models for closed economies as presented byRamsey (1928), Solow (1956), Cass (1965), and Koopmans (1965), in which the per capitagrowth rate tends to be inversely related to the initial level of output or income per capita.This model, importantly, gives the directions by which economies reach at ‘steady-state’equilibrium situation, and by accepting the restrictive conditions two importantconclusions can be drawn – First, if the economics factors (technological growth rate,investment rate and growth rate of labor force) are identical or similar across economiesthen convergence will come into existence as towards a common ‘steady-state’.Second, if any economy is ‘below’ its ‘steady-state’, the faster this economy should grow,which leads the convergence literatures to obtain a more general idea that is poorereconomies will grow faster than richer ones. Barro and Sala-i-Martin (1992a) useneoclassical growth model as a framework for studying the determinants of economicgrowth across the 48 contiguous of U.S. states and provide clear evidence ofconvergence in the sense that poor economies tend to grow faster than rich ones in percapita. Young et al. (2008) follows the neoclassical model as the exposition of Sala-i-Martin (1996b) who considers this model as ‘a good candidate to account forconvergence results’, Barro and Sala-i-Martin (1995) present a similar exposition andrecently Fruceri (2005) presents a related demonstration based on an Ordinary LeastSquare (OLS) estimator of the coefficient on initial income, where it is assumed that β-convergence holds for economies i=1,….,N. Natural log-income of the ith economy canbe approximated byln(yit) = α + (1-β) ln(yi, t-1) + uit,…………………………………………………. (1)Where 0 0 implies a negative correlation between growth and initial log income.So, by the help of regression equation the neo-classical growth model is widely used forthe estimation of economic convergence. Within this framework, to test absolute (orunconditional) β-convergence, σ-convergence, and conditional β c -convergencefollowing regression models are regressed respectively.Let yitbe the natural logarithm of per capita GDP for economy i (i = 1,…, N) during periodt. absolute -convergence can be tested by running the following regression of growthof per capita GDP across economies:(yit - yi,t-T) = α+βyi,t-T + νt …………………………………………………………… (2)Where t indicates the end of the time interval and (t-T) is the beginning (initial) of the timeinterval and νt is the stochastic error term. In terms of equation (2) a significant negativevalue for βimplies unconditional beta-convergence, while β≥0 implies non-convergence.137http://www.bdresearchpublications.com/journal/

Uddin et al.If σtbe the standard deviation of yit across i at time t. Absolute σ-convergence can betested by estimating the following regression model:σt= α + βt + νt……………………………………………………………………… (3)Where, αandβ are parameters and νt is the stochastic error term. A significant negativevalue for β implies absolute convergence, while β≥0 implies non-convergence.The concept of conditional beta convergence (β c ) can be derived by augmentingequation (2) by including a set of control variables xi(e.g., investment, saving, population,openness etc.) that are expected to determine the steady-state growth of per capitaoutput. Thus, conditional beta convergence (β c ) can be tested by estimating thefollowing model:(yit - yi,t-T) = α + βyi,t-T + γxi + νt .........................................................................(4)In terms of equation (4) a significant negative β implies convergence holds conditionallywhenγ ≠ 0.Many researchers use the panel data regression models as robust estimators, namelyFixed Effects (FE) and Random Effects (RE) models. In these models OLS, Least SquareDummy Variable (LSDV) and Generalized Least Square (GLS) estimators are used underparametric approach for convergence test. The panel framework can provide dramaticimprovements in statistical power compared to performing a separate unit root test foreach individual time series. The panel unit-root test advanced by Quah (1992, 1994) andLevin and Lin (1992, 1993) was widely used in several applications on test for convergencehypothesis. Evans and Karras (1996) develop a different framework for testing that allowdifferences in trend growth rates across economies with heterogeneous intercepts validunder much less restrictive conditions. Using Monte Carlo methods, Goddard and Wilson(2001) suggested that a panel estimator outperforms both the unconditional andconditional cross-sectional and pooled OLS estimators in the presence of heterogeneousindividual effects.Levin and Lin (1992, 1993) and Im-Pesaran-Shin (IPS) (1997) consider general models(exogenous and endogenous growth models) that include individual specific interceptsand time specific common effects, and allow for possibility of correlated errors acrosseconomies and time series for panel data. The individual specific intercepts can be usedto predict different trend growth rates for a sample of countries. There models allow forheteroscedasticity and serial correlation across time series and they propose unit root testsfor dynamic heterogeneous panels based on the mean of individual unit root statistics.The Augmented Dicky-Fuller test (p) statistic for unit root tests is used to get rid of the serialcorrelation problem including some lagged-difference terms of the dependent variable inthe regression model.Presently, the econometricians widely use non-parametric approaches to test theconvergence hypothesis by assuming as more robust estimator where presumptions ofprobability distribution are avoided. The transition metric, Probability Density Function(PDF) and Cumulative Distribution Function (CDF) are using for estimation. Now-a-days,Kernel Stochastic density estimator is a most popular tool for non-parametric estimations.Feve and Pen (2000) use a switching regression approach with imperfect sampleseparation information to determine convergence clubs. They follow Lee and Porter(1984) and consider an intermediate situation where sample separation is available butimperfect. In this framework a dichotomous indicator is introduced for each economy.Following the cross-sectional approach and a regression tree procedure used by Durlaufand Johnson (1995), this indicator is obtained from extra information based on initial percapita GDP.For convergence club test they also follows transition probability matrix with anormal probability density function (p.d.f.).For testing convergence clubs among 15 OECD countries, Su (2003) uses a clusteringalgorithm to identify sets of countries for which convergence occurs.The convergence tests based on cross-section regressions, such as the use of growthregressions that express the growth rate as a function of initial income, have been138http://www.bdresearchpublications.com/journal/

Economic Growth and Income Convergencecriticized by Quah (1993a)on the grounds that modeling a conditional mean may beinadequate for analyzing the hypothesis of convergence. The first problem with thisregression is the assumption that the estimated coincident is the same for all economies.The second problem is known as ‘Galton’s Fallacy’, as pointed out by Friedman (1992)and Quah (1993a), who show that the negative coincident encountered in growthregressions may be a symptom of regression to the mean rather than implyingconvergence. Relaxing these assumptions of linearity and a given distribution, Laurini et al.(2005) test for convergence and model the dynamics of relative income for Brazilianmunicipalities using non-parametric methods. They have carried out sigma convergencetests using the traditional statistics of the ‘coefficient of variation’ and the Theil index,which measure the dispersion between incomes, obtaining the distributions of theseestimators using bootstrap methods. The beta convergence test, which uses the nonparametricsmoothing spline estimator, relaxes the linearity imposed by estimation usingordinary least squares, and they derive a convergence test based on the first derivative ofthis estimator. This test shows that the hypothesis of convergence, represented by anegative relationship between the growth rate and initial income, is not valid for all levelsof initial income, showing that there are signs of divergence for the relative incomes ofBrazilian municipalities. This result is consistent with the bimodality obtained in the nonparametricdensity estimation using a kernel function for income for the specific period.This bimodality, which may be interpreted as the formation of income convergence clubsas proposed by Quah (1996), is tested statistically through a test of multimodality that usesbootstrap methods.A recent study of Massoumi and Wang (2007) propose a new concept of convergencewhich is based on the metric entropy measure recently governed by Granger et al. (2004)to investigate economic convergence in China. This approach compares wholedistributions of growth rates across individual economies. Separately, based on this samemeasure, they implement cluster analysis to identify convergence clubs and concludethat there is no nation-wide convergence but exists a number of very small convergenceclubs.In another recent research, Liontakis and Papadas (2010) use stochastic convergenceapproach of panel data unit root tests but in order to capture sufficiently the evolvingdistributional dynamics, nonparametric econometric methods are implemented as well.An alternative conditional density estimator, proposed in the literature, is applied for thisreason. This estimator is chosen as superior, not only to the restrictive discrete Markovchain approaches but also to the usual estimators of conditional densities using stochastickernels.A very recent work of Bandyopadhyay (2012) observes the distributional dynamics ratherthan just beta or sigma convergence. This approach moves away from establishing simplyempirics of convergence and divergence, but describes the evolution of state-levelincome distribution over time. In doing so, one can identify the empirics of catch-up moreaccurately and the presence of long-run cohesive tendencies, polarization, stratificationor the emergence of convergence clubs. He then provides some explanations of theobserved dynamics: he finds several macroeconomic indicators to be significantlyassociated with the evolution of the income distribution and the formation ofconvergence clubs. The distribution dynamics approach (Quah, 1996) is based ontreating a single income distribution as a random element in a field of incomedistributions, called the random field. The density function of the income distribution isestimated at each point in time and is then observed how it evolves over time. A transitionprobability matrix records the probabilities of persistence and mobility across the incomedistribution. Stochastic kernels and transition matrices provide an estimate of intradistributionmobility taking place. An economy over a given time period (say, one year orfive years) either remains in the same position, or changes its position in the incomedistribution. The transition probabilities are then encoded in the transition probability matrixfor transitions over the income distribution. Low probabilities of transition indicatepersistence, while higher probabilities indicate mobility.For testing the convergence hypothesis, generally, annual time-series cross sectional dataor panel data or longitudinal data of per capita income or GDP or output are usedintensively. Presently annual panel data of inflation rate, interest rate etc. are using for139http://www.bdresearchpublications.com/journal/

Uddin et al.testing the convergence hypotheses. Basically, states’ or regional or national annualeconomic growth data for a period of time are collected from different national andinternational statistical (printed and online database) sources like IMF, World Bank,EUROSTAT, FAOSTAT, different version of PennWorld Tableetc.Baumol (1986) uses Madison’s (1982) data covering the period 1970-1979 to test theconvergence among 16 industrialized countries. Barro and Sala-i-Martin (1992a) use twomeasures of per capita income or product across the U.S. states. The first is per capitapersonal income since 1840. The U.S. Commerce Department has published annual dataon nominal personal income for the 48 continental states since 1929. The second type ofdata is per capita gross state product (GSP), which is available annually for each statesince1963.Sala-i-Martin (1996b) uses the data of the regions within various countriesextended to 1990 with the previously used in a number of papers of Barro and Sala-i-Martin for convergence analysis (see Barro and Sala-I Martin (1991, 1992a, 1992b, and1995)).Feve and Pen (2000) draw a sample of 92 countries from Penn World Table (PWT) ofMark 5.5 (1993) that covers the period 1960-1989. Kim (2001) obtains data from the PennWorld Table (PWT) 5.6 ofSummers and Heston (1991). The panel data consist of real GDPper capita for the selective 17 Asian countries from 1960 to 1992. Su (2003) uses two datasets. The first is the data set of Bernardand Durlauf (1995), consisting of annual real percapita GDP for 15 OECD countries ranging from 1900 to 1987. The second data set is fromMaddison (1995). But these data he also runs from 1885 to 1994.Chowdhury (2004) hasexamined the convergence issue by using per capita GDP annual data across 7 SouthAsian (SAARC) countries during 1960-2000. Laurini et al. (2005) use the database consists ofper capita incomes for 3,781 Brazilian municipalities for the years 1970 and 1996,constructed on the basis of income and population data obtained, respectively, from theIPEA and IBGE. Maasoumi and Wang (2007) use two data sources consists of 28 provincesof China: one is Hsueh and Li (1999) for the period 1952-1995, and the other is pooled fromChina Statistical Yearbook for the period 1996 – 2003. Mathur (2005) uses cross sectionaldata of GDP per capita levels and growth rates of 44 countries where 16 Europeancountries (EU15 +United Kingdom), 5 South Asian Countries, 8 East Asian and 15 CISCountries to test for 'absolute convergence' hypothesis for four different periods 1961-2001,1970-2001,1980-2001,1990-2001. Young et al. (2008) utilize per capita personalincome of the U.S. county-level data used by Higgings et al. (2006) and Young et al.(2007) to study income growth from 1970 to 1998. The data set includes 3,058 county-levelobservations and 50 individual state samples of various sizes. Rahman and Hossain (2009)use annual data on personal per capita income of six divisions at current factor cost fromthe different Statistical Yearbooks published by Bangladesh Bureau of Statistics (BBS) forthe period 1977 – 2000. They also take the data on factors affecting per capita incomeconvergence from World Development Indicators of World Bank. Rezitis (2010) uses thecross sectional and time-series dataset of Ball et al. (2001) for the United State and asample of nine European countries for the period of 1973 to 1993.Alexiadis (2010) exploitdata on Gross Value Added per worker in agriculture of EU-26 regions covering the period1995-2004.Liontakis and Papadas (2010) use monthly data of the Harmonized Indices ofConsumer Prices (HICPs) for the whole group of “Food and non-alcoholic beverage”, andfor eleven specific individual subgroups of food products. The data set cover a periodfrom January 1997 to May 2009.The work of Baumol (1986) draws a gigantic attention to the field of economic growthand income convergence. He analyzes Maddison's (1982) 1870-1979 data and empiricalresults show the remarkable convergence of productivities of industrialized marketeconomies, with convergence apparently shared by planned economies but not lessdeveloped countries. He argues that there are some forces accelerate the growth of theweak nations who take slowly place their industrialization process and economicdevelopment give rise to a tendency towards convergence of levels of precipitateproduct or, alternatively of per worker product. His works also initiates clearly the conceptof convergence clubs.Barro and Sala-i-Martin (1992a) show that the U.S. states provide clear evidence ofconvergence, but the findings can be reconciled quantitatively with the neoclassicalmodel only if diminishing returns to capital set in very slowly. The results for per capita grossdomestic product from a broad sample of countries are similar if a set of variables holdconstant that proxy for differences in steady-state characteristics.http://www.bdresearchpublications.com/journal/140

Economic Growth and Income ConvergenceAfter analyzing the Solow (1956) and Swan (1956) models for an estimation of conditionaland unconditional beta- and sigma-convergence, Maurer (1995) argues that given theimplications of this model a consistent estimation of conditional and unconditional betaandsigma-convergence should be possible. He says, depending on identical stochastic -these models imply unconditional beta- and sigma-convergence, if the cross sectionsample embodies only economies with identical steady state parameters. In case ofdifferent steady state parameters, both models imply conditional beta- and sigmaconvergence.Sala-i-Martin (1996b) argues that the concepts of β- and σ-convergence areindependently interesting and then extends the empirical evidence on regional growthand convergence across the United States, Japan, and five European nations, andconfirms the tendency of convergence surprisingly at a 2% annual rate. He also showsthat the interregional distribution of income in all countries has shrunk over time.Kim (2001) tests the null hypothesis of endogenous growth theories which predict crosscountrydifferences in trend growth rates against the alternative hypothesis of exogenousgrowth theories which predict the same trend growth rates. He uses the modified testprocedure with heterogeneous intercepts allowing different growth rates acrosseconomies. He applies the test to17 Asian countries and NIEs with panel data. The resultsare consistent with neoclassical growth theories which predict the convergence of the 17Asian countries and NIEs, but which imply that trend growth rates are different acrosseconomies. These results support the conditional convergence of the exogenous growthmodel against the endogenous growth model.Su (2003) investigate the question of convergence among 15 OECD countries, and getsanswer in several ways, as joint stationary of differences of log per capita real output.Since no convergence is found among the economies, a clustering algorithm is used andfound convergence.Chowdhury (2004) does not find the evidence of σ- convergence, β convergence andconditional β-convergence (β c ) in South Asia. He places the reasons for non-convergenceof per capita GDP are low and falling volume of intra-country trade, weak governanceand low level of growth achieved by the individual countries.Laurini et al. (2005) analyses the evolution of relative per capita income distribution ofBrazilian municipalities over the period 1970–1996.Analyses are based on non-parametricmethodologies and do not assume probability distributions or functional forms for thedata. Two convergence tests have been carried out – a test for sigma convergencebased on the bootstrap principle and a beta convergence test using smoothing splinesfor the growth regressions. The results obtained demonstrate the need to model thedynamics of income for Brazilian municipalities as a process of convergence clubs, usingthe methodology of transition matrices and stochastic kernels. The results show theformation of two convergence clubs, a low income club formed by the municipalities ofthe North and Northeast regions, and another high income club formed by themunicipalities of the Center-West, Southeast and South regions. The formation ofconvergence clubs is confirmed by a bootstrap test for multimodality.Mathur (2005) tests convergence hypotheses under four different periods and findsuniform evidence of absolute convergence in case of only EU and East Asian countriestogether in all period. While EU region shows significant evidence of absoluteconvergence in early two periods but does not show in the last two periods. The speed ofabsolute convergence in the four periods range between 0.99-2.56 % p.a. which favors tothe rate of 2% for the EU regions was worked out by Barro and Sala-i-Martin(1995), while itranges between 0.57-1.16 % p.a. for the countries in East Asia and EU regions together.However, there is no evidence of convergence among the South Asian countries in allperiods and some major CIS republics. There is however tendency for absoluteconvergence among countries of South Asia, East Asia and European Union togetherparticularly after the 1980s. He implies in his work that conditional convergence isprevalent among almost all pairs of regions in our sample except East Asian and SouthAsian nations together. Speed of conditional convergence ranges from 0.2 % in a year to22%.In the European nations, the speed of conditional convergence works out nearly 20 %unlike the speed of absolute convergence which hovered around 2 %. Moreover,141http://www.bdresearchpublications.com/journal/

Uddin et al.conditional beta convergence seems to be a better empirical exercise because it reflectsthe convergence of countries after controlling for differences in steady states.Additionally, conditional convergence is simply a confirmation of a result predicted bythe neoclassical growth model that countries with similar steady states exhibitconvergence.It does not mean that all countries in the world are converging to the samesteady state,only that they are converging to their own steady states.Furceri (2005) examines and compares in detail the concepts of β and σconvergence. Heprovides a mathematical relation of causality between these two concepts. His resultargues that a necessary condition for the existence of σ-convergence is the existence ofβ-convergence.Maasoumi and Wang (2007) propose a new concept of convergence which is based onthe metric entropy measure recently proposed by Granger et al. (2004) to investigateeconomic convergence in China. This entropy measure compares whole distributions ofgrowth rates across individual provinces. Separately, based on this same entropymeasure, they also implement cluster analysis to identify any convergence clubs. Theirfour main conclusions are: (1) while they certainly reject the null hypothesis that thereexists a nation-wide convergence, they do not find that there exist convergence clubs forboth the pre- and post-reform periods; (2) They find a number of very small convergenceclubs. In particular, there are eleven and six convergence clubs for the pre- and postreformperiods, respectively, (3) in comparing the number and size of convergence clubsfor both the pre- and post-reform periods, it could be argued that the extent ofconvergences more prevalent during the post-reform period than during the pre-reformperiod, (4)convergence groups cannot be simply characterized by such unique featuresas region or the extent of policy preference level that are commonly used in the literature.Young et al. (2008) outline the reasons for thinking the separate entities of the twoconvergence concept- β and σ convergence. They discuss the evidence of β-convergence in the U.S. but containing over 3000 observations at the county-levels crosssectionaldata cannot detect σ convergence, even considered separately the largemajority individual U.S. states, rather than find σ-divergence in many cases are statisticallysignificant.Alexiadis (2010) augment the research on regional convergence in Europe using theagricultural sector in terms of agricultural labor productivity convergence. They exhibit alow annual rate of absolute convergence over the period whilst a test for conditionalconvergence yields a higher rate of convergence. Thus, evidence of club convergence isapparent.Liontakis and Papadas (2010) recognize that concepts and developments in the literatureof economic growth and convergence have recently been adopted and used in thestudy of inflation rate convergence. They examine initially the existence of β-convergence, as mean reversion, of food price inflation rates in the European Union, usingthe stochastic convergence approach of panel data unit root tests. It examines also theexistence of σ-convergence but in order capture sufficiently the evolving distributionaldynamics, nonparametric econometric methods are implemented as well. An alternativeconditional density estimator, proposed in the literature, is applied for this reason. Thisestimator is chosen as superior, not only to the restrictive discrete Markov chainapproaches but also to the usual estimators of conditional densities using stochastickernels. Monthly data on the EU harmonized consumer price indices of food and elevenspecific food product subgroups are used, for the 15 older EU member states, coveringthe 1997-2009 period.Rezitis (2010) tests the agricultural total factor productivity (TFP) growth convergenceacross Europe and United States and shows the presence of β-convergence but not σ-convergence considering full period whilst both are present in case of latest sub-period.He also implies that a wide spectrum of unit root test results support the presence of longrunconvergence among the selected economies.Bandyopadhyay(2012) has examined the convergence of growth and incomes acrossthe Indian states using an empirical model of dynamically evolving distributions, andpresents some explanations of the observed dynamics. The model reveals ‘‘twin peaks’’dynamics, or polarization across the Indian states, over 1965–1997 – empirics who wouldhttp://www.bdresearchpublications.com/journal/142

Economic Growth and Income Convergencenot be revealed under standard empirical methods of cross-section, panel data and timeseries econometrics. He finds that the dominant cross-state income dynamics are that ofpersistence, immobility and polarization, with some cohesive tendencies in the 1960s, onlyto dissipate over the following three decades.This paper constructs the concepts of convergence mainly in terms of economic growthand income distribution. We have gone through some remarkable research article andsummarized the methodologies, data and findings. Our main aim is to explore theconcepts of economic growth and income convergence and divergence in the differentregions, nationals and economies across the world based on the evolving literature. Wefound neoclassical growth model is used extensively by the researcher. Baumol’s (1986)work has given the good location in this field of research, but a series of papers of RobertJ. Barro and Xavier X. Sala-Martin jointly and separately have given the broaden shape ofknowledge of convergence in the field of economic. At the same time Dunny Quah hasadded an important series of papers regarding this issue. Surprisingly, sometimes we seecontrary arguments among them. Moreover, a bountiful researcher has augmented thevolume of literature in this regard.We understand that convergence concerns poor economies catching up with rich ones.The focal point is that convergence exists among the economies ifpoorer economies grow at faster rates than richer economies. Consequently, alleconomies should eventually converge in terms of per capita income. Based on thisassumption, developing countries’ growth rates are more potential than developedcountries because of diminishing returns, in particular to capital, are not as strong as in richeconomies. In addition, poorer economies can use the same production methods,technologies, and institutions those are already used in developed economies.From the literature we draw two terms are widely used in the convergence literature as‘absolute (or unconditional) β convergence’ and ‘σ convergence’. One can separatethese two concepts from each other easily as “Beta-convergence” occurs when pooreconomies grow faster than rich ones. On the other hand, “Sigma-convergence” impliesa reduction in the dispersion (or minimize the gap of inequalities) of levels of incomeacross economies. More specifically, in case of β-convergence the partial correlationbetween growth in income over time and its initial level is negative, and σ-convergencesignifies when the dispersion of real per capita income across a group of economies fallsover time. Economists also focus on a third concept of convergence namely “Conditionalβ-convergence (β c )” is often talked about which takes into account when the growthrate of an economy declines as it approaches its steady state.We capture also the contrary arguments regarding the two concepts of convergencefrom the researchers. Some scholars argue that β-convergence and σ-convergenceoccur independently. On the other hand, some researchers show an inverse opinion thatthese two concepts are dependent each other for their existence. The important remarksare – (i) Beta-convergence is necessary but not sufficient condition for Sigmaconvergence;(ii) The existence of Sigma-convergence signifies the existence of Betaconvergence,i.e. the presence of sigma-convergence ensures the presence of betaconvergence;(iii) Convergence Club is made within the economies whose initialconditions are quite close. Some results show that though there is no evidence of absoluteconvergence among the economies together but presence conditional convergencewhen divided in to groups according to same economic conditions which indicates theformation of convergence clubs.ReferencesAlexiadis, S. (2010), Convergence in Agriculture: Evidence from the European Regions,Agricultural Economics Review, Vol. 11, No 2.Ball, V. E., Bureau, J. C., Butault, J. P. and Nehring, R. (2001) Levels of farm sectorproductivity: an international comparison, Journal of Productivity Analysis, 15, 5–29.Bandyopadhyay S. (2012), "Chapter 8 Convergence Club Empirics: Evidence from IndianStates", John A. Bishop, Rafael Salas, in (ed.) Inequality, Mobility and Segregation:Essays in Honor of Jacques Silber (Research on Economic Inequality, Volume 20),Emerald Group Publishing Limited, pp. 175 – 203.143http://www.bdresearchpublications.com/journal/

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