EC513 PhD Public Economics 2006/7

http://darp.lse.ac.uk/EC513.htm

Inequality and “Convergence”Inequality and “Convergence”

8 March 2007

Overview...

Introduction

Growth

World income inequality

Inequality and welfare

Inequality and convergence

The basis for the question

Focus of the lecture

Issues of distribution of economic wellbeingIssues of distribution of economic wellbeing focus of a large number of recent empirical studiesfocus of a large number of recent empirical studies also of new (?) theoretical workalso of new (?) theoretical work

Has always been a big question in public economicsHas always been a big question in public economics back to Pareto and analysis of income distributionback to Pareto and analysis of income distribution Gini, Lorenz and inequalityGini, Lorenz and inequality Pigou and welfare economicsPigou and welfare economics

Analysing it has always involved other fieldsAnalysing it has always involved other fields MacroeconomicsMacroeconomics FinanceFinance Development economicsDevelopment economics

Brief look at what is involved...Brief look at what is involved...

Basic accounting Begin with the basics of individual wellbeingBegin with the basics of individual wellbeing

Utility: depends on consumptionUtility: depends on consumption Resources: income and wealthResources: income and wealth

The agent’s incomeThe agent’s income composed of earnings + interest income + transferscomposed of earnings + interest income + transfers ignore transfers hereignore transfers here yyii = = wlwlii + + rkrkii

A simple decomposition of inequality?A simple decomposition of inequality? Components statistically uncorrelated?Components statistically uncorrelated? Income more widely dispersed than earnings?Income more widely dispersed than earnings? Evolution of distribution of Evolution of distribution of kk important for distribution of important for distribution of yy..

Basis of an income-determining modelBasis of an income-determining model Role of factor pricesRole of factor prices Role of accumulationRole of accumulation

Old models, modern themes

Why concern with these issues now?Why concern with these issues now? Trends in within-country distributionTrends in within-country distribution

big increase in inequality in US…big increase in inequality in US… ……and several European countriesand several European countries

Debate on globalisationDebate on globalisation growing inequality?growing inequality? ...or convergence?...or convergence?

Improved data availabilityImproved data availability within-country: mainly based on individual tax recordswithin-country: mainly based on individual tax records across country: improved comparability and repeated across country: improved comparability and repeated

observationsobservations

Growth theory became fashionable againGrowth theory became fashionable again

Background

Build on a connection with standard growth modelsBuild on a connection with standard growth models Role of capital and labourRole of capital and labour

Can in fact be made more generallyCan in fact be made more generally One accumulated factorOne accumulated factor One or more non-accumulated factorsOne or more non-accumulated factors

Role of factor pricesRole of factor prices focus on focus on ww and and rr assume competitive markets?assume competitive markets? will there be an efficient equilibrium?will there be an efficient equilibrium?

Overview...

Introduction

Growth

World income inequality

Inequality and welfare

Inequality and convergence

New insights from simple neoclassical models

An approach

Subject has become huge in recent yearsSubject has become huge in recent years see see Bénabou, R. (1996)

Here consider an early, simplified modelHere consider an early, simplified model incorporates growth and inequalityincorporates growth and inequality build on build on StiglitzStiglitz (1969) (1969)

Based on Solow-Swan type of modelBased on Solow-Swan type of model affine (linear) savings functionaffine (linear) savings function but with a twistbut with a twist

Focus on both macro and distributional issues:Focus on both macro and distributional issues: Accumulation of capitalAccumulation of capital Distribution of wealthDistribution of wealth

What does equilibrium look like?What does equilibrium look like? Outline of modelOutline of model Use modified Stiglitz notationUse modified Stiglitz notation

Basic model: overall production

Start with neoclassical growth modelStart with neoclassical growth model two factorstwo factors CRTSCRTS

Aggregate production in percapita termsAggregate production in percapita terms yy: output per person : output per person kk: capital per person: capital per person

Standard assumptions on Standard assumptions on ff increasingincreasing concaveconcave Inada conditionsInada conditions

Competitive factor rewardsCompetitive factor rewards interest rateinterest rate wage ratewage rate

Basic model: wealth classes

ii: indexes individual wealth classes: indexes individual wealth classes ii: capital in per person in group : capital in per person in group ii

yyii = = wlwlii + + rri i agent’s income in group agent’s income in group ii

Same labour endowment in every group Same labour endowment in every group ii

aaii: Proportion of population in group : Proportion of population in group ii

kkii: capital in group : capital in group ii as proportion of as proportion of

populationpopulation

Simple aggregationSimple aggregation

Savings and growth Affine savings functionAffine savings function

Wealth accumulation in class Wealth accumulation in class ii

Substitute savings in class Substitute savings in class ii

Substitute income in class Substitute income in class ii

Aggregate growth in capital/labour ratioAggregate growth in capital/labour ratio

Equilibrium: basic Solow model

k

y f(k)

k*

y = nk /

m

Behaviour of the extended model

Change in capital-labour ratioChange in capital-labour ratio

Substitute in for Substitute in for ww and and rr

Simple phase-diagram behaviourSimple phase-diagram behaviour

Equilibrium growth again

k

y f(k)

k* k**

y = [n

k− b]

/m

Basic results (1)

One or two balanced growth pathsOne or two balanced growth paths Depends on the shape of the savings functionDepends on the shape of the savings function Lower equilibrium is unstableLower equilibrium is unstable

On each path :On each path : stabilitystability capital labour ratio constantcapital labour ratio constant factor prices constantfactor prices constant Schlicht (1975)Schlicht (1975)

Basic results (2)

Aggregate accumulation does not depend on Aggregate accumulation does not depend on distributiondistribution

Follows directly from the savings assumptionFollows directly from the savings assumption Would also hold if savings had a linear component Would also hold if savings had a linear component

dependent on wealthdependent on wealth Linearity: a reasonable empirical assumption? Linearity: a reasonable empirical assumption?

Schmidt-Hebbel and Schmidt-Hebbel and ServenServen (2000) (2000) on cross-section and on cross-section and panel datapanel data

Use variety alternative inequality measuresUse variety alternative inequality measures Alternative savings definitions and various econometric Alternative savings definitions and various econometric

specificationsspecifications Income inequality does not have systematic effect on Income inequality does not have systematic effect on

aggregate savingaggregate saving

Wealth classes and distribution

Now examine what is happening with the individual wealth Now examine what is happening with the individual wealth classes classes ii..

The group in equilibrium at any overall The group in equilibrium at any overall kk, given by, given by

Gives critical personal wealth level as function of Gives critical personal wealth level as function of kk

The critical wealth levels

Implicitly define Implicitly define kk~~

Implicitly define Implicitly define kk^̂

Key relationshipsKey relationships

Critical wealth levels

k

y f(k)

k*

k̂ k~

Changes in wealth distribution

Rate of change of wealth Rate of change of wealth in group in group ii

Relative change for two Relative change for two groupsgroups

If If 11 < < 22 then … then …

……get convergence if get convergence if b+mw b+mw > 0> 0 You have to be to the right of You have to be to the right of kk^̂ for this for this

Basic results (3)

On the balanced growth path On the balanced growth path kk*:*: instability in aggregateinstability in aggregate instability in distributioninstability in distribution

On the balanced growth path On the balanced growth path kk**:**: stability in aggregatestability in aggregate stability in distributionstability in distribution

In range In range kk* to * to kk^̂ : : Overall capital-labour ratio is increasingOverall capital-labour ratio is increasing Converges on equilibriumConverges on equilibrium But wealth inequality becomes more unequal along the wayBut wealth inequality becomes more unequal along the way

In range In range kk^̂ to to kk**:**: Overall capital-labour ratio is increasingOverall capital-labour ratio is increasing Converges on equilibriumConverges on equilibrium Wealth inequality becomes more equal along the wayWealth inequality becomes more equal along the way

Role of taxation Assume a purely redistributive income tax Assume a purely redistributive income tax Disposable income isDisposable income is

Relative performance of two wealth classes is nowRelative performance of two wealth classes is now

Critical Critical k k value for convergence is now wherevalue for convergence is now where

The income tax makes a differenceThe income tax makes a difference

Tweaking the model (1)

Consider nonlinear savings functionConsider nonlinear savings function Reconsider convergence in the Stiglitz modelReconsider convergence in the Stiglitz model

is equality inevitable? is equality inevitable? is it desirable?is it desirable?

May get multiple equilibriaMay get multiple equilibria Equality may be Pareto dominatedEquality may be Pareto dominated

Bourguignon (1981)Bourguignon (1981)

Tweaking the model (2) Optimised savings: Optimised savings:

a many-agent modela many-agent model based on based on Ramsey (1928)Ramsey (1928)

Many-agent versionMany-agent version In steady states: a paradox?In steady states: a paradox? Agents discount future utility at different constant ratesAgents discount future utility at different constant rates All the capital owned by agents with the lowest discount rate. All the capital owned by agents with the lowest discount rate.

Outcome depends upon the borrowing constraints Outcome depends upon the borrowing constraints If high discount consumer can borrow against future wage If high discount consumer can borrow against future wage

income converge to zero consumption. income converge to zero consumption. No steady state need exist No steady state need exist Becker (1980)Becker (1980)

Bliss model (1) Focus on simplified multi-person modelFocus on simplified multi-person model

Bliss (2004)Bliss (2004) AgentsAgents

all types have the same tastesall types have the same tastes all supply the same quantity of labour in all periodsall supply the same quantity of labour in all periods all earn the same wageall earn the same wage

Capital marketsCapital markets uniform rate of returnuniform rate of return all have same access to capital marketall have same access to capital market

InformationInformation all have perfect foresightall have perfect foresight no stochastic effects in the model to upset convergenceno stochastic effects in the model to upset convergence

Bliss model (2) Bliss (2004) focuses on Koopmans- separable preferencesBliss (2004) focuses on Koopmans- separable preferences

Let Let cc := ( := (cc11, , cc22, ..., , ..., cctt, ...), ...) UU((cc) = ) = WW11( ( uu((cc11), (), (cc22, , cc33, ..., , ..., cc+1+1, ...)), ...))

A generalisation of usual definition of separabilityA generalisation of usual definition of separability Can be used recursively:Can be used recursively: UU((cc) = ) = WWtt( ( uu((cc11), ), uu((cc22), ..., ), ..., uu((cctt), (), (cctt+1+1, , cctt+2+2, ..., , ..., cctt++, ...)), ...))

Bliss takes a multi-agent version of RamseyBliss takes a multi-agent version of Ramsey Optimising agentsOptimising agents Infinite livesInfinite lives

Again get a version of the Ramsey paradoxAgain get a version of the Ramsey paradox many-agent Ramsey model without capital markets gives income many-agent Ramsey model without capital markets gives income

convergenceconvergence with capital markets , convergence of incomes cannot occurwith capital markets , convergence of incomes cannot occur Ramsey assumed separable additive preferencesRamsey assumed separable additive preferences Bliss: even without this assumption chaotic dynamics may result.Bliss: even without this assumption chaotic dynamics may result.

Alternative approaches

Role of technologyRole of technology Increasing returnsIncreasing returns Imperfect capital marketsImperfect capital markets But rational savings behaviour may be the keyBut rational savings behaviour may be the key

Alternative approaches Human capital in the production function. Human capital in the production function.

By itself this does not make a great differenceBy itself this does not make a great difference human capital accumulated optimally to combine with physical capital.human capital accumulated optimally to combine with physical capital.

Suppose accumulation of human capital cannot be financed by Suppose accumulation of human capital cannot be financed by borrowing. borrowing.

Imperfect capital mobility will assist income convergence. Imperfect capital mobility will assist income convergence. BarroBarro et al (1995) et al (1995) ))

Consider convergence in a special case Consider convergence in a special case One small low-wealth country converges to a steady state One small low-wealth country converges to a steady state ……where rest of the world occupies from the start. where rest of the world occupies from the start. Country is borrowing constrained all the way to steady stateCountry is borrowing constrained all the way to steady state. .

Consider general many-agent equilibrium, Consider general many-agent equilibrium, same model with borrowing constraints, same model with borrowing constraints, low wealth country having significant weight in the world equilibriumlow wealth country having significant weight in the world equilibrium convergence is not assuredconvergence is not assured

Overview...

Introduction

Growth

World income inequality

Inequality and welfare

Inequality and convergence

con / di-vergence?

Questions about growth and inequality Individual incomes and wealthIndividual incomes and wealth

Role of savings behaviourRole of savings behaviour What will happen to individual capitalist countries?What will happen to individual capitalist countries?

Wealth of nationsWealth of nations Convergence?Convergence? Is the standard growth model the right one?Is the standard growth model the right one?

Inequality between countriesInequality between countries Role of globalisationRole of globalisation Role of savings behaviourRole of savings behaviour Or is inequality increasing?Or is inequality increasing?

Inequality within countriesInequality within countries Convergence?Convergence? Related to countries economic policies?Related to countries economic policies?

Kuznets reborn?

Kuznets (1963) focused on a statistical curiosityKuznets (1963) focused on a statistical curiosity Suggested a speculative conclusionSuggested a speculative conclusion Inequality first rises, with industrialisation…?Inequality first rises, with industrialisation…? ……then falls, as workers become more productive?then falls, as workers become more productive?

He was working just with cross-section dataHe was working just with cross-section data Now have repeated data for individual countriesNow have repeated data for individual countries Micro-data for many countriesMicro-data for many countries Methods include:Methods include:

analysis of income decompositionsanalysis of income decompositions regression modelsregression models

A Pattern of inequality

Bourguignon and Bourguignon and MorrissonMorrisson (2002) (2002) investigate the distribution investigate the distribution of well being among world citizens in 19of well being among world citizens in 19thth, 20, 20thth centuries. centuries.

Inequality worsened from the beginning of the 19th century to Inequality worsened from the beginning of the 19th century to World War IIWorld War II

Then stabilized or to have grown more slowly. Then stabilized or to have grown more slowly. Composition of inequality changedComposition of inequality changed

In the early 19In the early 19thth century inequality mainly due to differences within century inequality mainly due to differences within countriescountries

Later differences between countries. Later differences between countries. Inequality in longevity also increased during the19Inequality in longevity also increased during the19 thth century century Trend reversed in the second half of the 20Trend reversed in the second half of the 20 thth century century

How is inequality related to growth? A regression approachA regression approach

Dollar and Dollar and KraayKraay (2002) (2002) Traditional literature does something like thisTraditional literature does something like this

where where yyctct is income in country is income in country cc at time at time tt Q1/0.2 is “share of first quintile group” Q1/0.2 is “share of first quintile group” ZZctct is vector of control variables in country is vector of control variables in country cc at time at time tt

Dollar and Dollar and KraayKraay (2002) (2002) argue that this produces ambiguous answers argue that this produces ambiguous answers Depends on econometric methodDepends on econometric method Depends on sampleDepends on sample

Use an explicit model of poor incomesUse an explicit model of poor incomes

Model incomes of the poor Model incomes of the poor this wayModel incomes of the poor this way

Regression is equivalent to modelling “first quintile share”:Regression is equivalent to modelling “first quintile share”:

Interested in two parameters:Interested in two parameters: 11 effect of overall income effect of overall income

22effect of other factorseffect of other factors

Is growth good for the poor?

Dollar-Kraay data set covers period from the 1960sDollar-Kraay data set covers period from the 1960s To take account of data on levels and changes use first differencesTo take account of data on levels and changes use first differences

Income share of poorest fifth does not change with average Income share of poorest fifth does not change with average incomeincome

Does not change with institutions or policies designed to help Does not change with institutions or policies designed to help the poorthe poor

Growth is “good for the poor”?Growth is “good for the poor”?

Inequality convergence? Regression method applied to a different problem

not relationship between income levels and inequality but the cross-country pattern of inequality Ravallion (2003)

Time-series model of within-country income inequalities for example the Gini Git Gi0 = a + bGi0 +et i = 1, . . . , N

A different type of convergence? slow convergence since the 1980s inequality is tending to fall (rise) in countries with initially high (low)

inequality Result seems robust

correcting for measurement error in initial inequality Gi0 … … has little effect

International trends

It is obviously debatable that huge countries like China get the same weight as very small countries ((Sala-i-MartinSala-i-Martin 2006) 2006)

Within-country disparities have increased Within-country disparities have increased not enough to offset reduction in cross-country not enough to offset reduction in cross-country

disparities. disparities. But the particular case effect is importantBut the particular case effect is important

What drives cross-country reductions in inequality?What drives cross-country reductions in inequality? Large growth rate of the incomes of the 1.2 billion Large growth rate of the incomes of the 1.2 billion

ChineseChinese

Inequality measures and US experience

Source: Source: DeNavas-Walt et al. (2005)

0

0.1

0.2

0.3

0.4

0.5

0.6

GiniGE0GE1A.25A.50A.75

Inequality measures and World experience

00.10.20.30.40.50.60.70.80.9

1

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

GiniGE0GE1A.50A1.0

Source: Sala-i-Martin (2006)Source: Sala-i-Martin (2006)

Inequality measures and World experience: breakdown

00.10.20.30.40.50.60.70.80.9

1

GE0GE0 betwGE0 within GE1GE1 betwGE1 within

Source: Sala-i-Martin (2006)Source: Sala-i-Martin (2006)

Overview...

Introduction

Growth

World income inequality

Inequality and welfare

Inequality and convergence

Convergence of general well being?

Problem: setting

"For most of the past 40 years human capabilities have been gradually converging. From a low base, developing countries as a group have been catching up with rich countries in such areas as life expectancy, child mortality and literacy. A worrying aspect of human development today is that overall state of converging is slowing and for a large group of countries divergence is becoming the order of the day."

Human Development Report (2005)

Another look at inequality trends

Income inequality across countries has increasedIncome inequality across countries has increased unweighted by population unweighted by population

World inequality take other welfare dimensions into account?World inequality take other welfare dimensions into account? Would these give the same pattern?Would these give the same pattern? Neumayer, E. (2003) examines individual dimensionsNeumayer, E. (2003) examines individual dimensions

inequality decreasing?inequality decreasing?

Becker et al (2005)Becker et al (2005) composite measure of well-being composite measure of well-being combine income and life expectancy.combine income and life expectancy. inequality decreasing?inequality decreasing?

Perhaps look at a standard empirical welfare conceptPerhaps look at a standard empirical welfare concept human development index? human development index? see see Anand and Sen (2000)

Unweighted inequality

Decancq et al (2007) use unweighted data: three main reasons

A benchmark consensus that unweighted income inequality has increased check whether this is true for well-being

Political economy argument countries seen as sets of national policies compare these in terms of wellbeing

Measurement error: population-weighted inequality sensitive to the performance

of China , India.

Human Development Index (1)

Four “raw” components Four “raw” components xxjj : : GDP per capitaGDP per capita life expectancy at birthlife expectancy at birth adult literacy rateadult literacy rate school enrolment rateschool enrolment rate

Each of “raw” is standardised to a number Each of “raw” is standardised to a number yyjj using using (possibly) a transformation (possibly) a transformation jj((••)) a lower bound a lower bound ℓℓjj

an upper bound an upper bound uujj

Then aggregated using a weight Then aggregated using a weight wwjj

Human Development Index (2)

Specification for the four components: Specification for the four components:

component component jj((xx)) uujj ℓℓjj wwjj GDP per capitaGDP per capita log(log(xx)) log(100)log(100) log(40000)log(40000) 0.333 0.333 life expectancy life expectancy xx 25 25 85 85 0.333 0.333 adult literacy rateadult literacy rate xx 0 0 100 100 0.222 0.222 enrolment rateenrolment rate xx 0 0 100 100 0.1110.111

Note the special transformation for incomeNote the special transformation for income Standardisation formulaStandardisation formula

jj((xxjj) ) ℓℓjj

yyj j == ──────────

uujj ℓℓjj

Overall index:Overall index:

j j wwj j yyjj

Inequality trends: alternative indicators

0

20

40

60

80

100

120

140

1975 1980 1985 1990 1995 2000

GDP/capita log(GDP/capita) Longevity LiteracyRatio Enrolment ratio

Inequality trend: Income/hd

Inequality trend: Log income/hd

Inequality trend: Longevity

Inequality trend: Literacy

Inequality trend: Enrolment rate

Welfare convergence? Lessons

Raw wellbeing concept is crucialRaw wellbeing concept is crucial contrast behaviour of income and longevitycontrast behaviour of income and longevity

Normalisation mattersNormalisation matters how should one modify income?how should one modify income? log-transformation responsible for trendlog-transformation responsible for trend

For standard parameter values inequality declinesFor standard parameter values inequality declines But other extreme assumptions can lead to other But other extreme assumptions can lead to other

patternspatterns

References (1) Anand, S. and Sen, A. (2000) “The income component of the Human

Development Index”, Journal of Human Development, 1, 83-106 BarroBarro, R.J., , R.J., MankiwMankiw, G. and , G. and Sala-i-MartinSala-i-Martin, X. (1995), X. (1995) “Capital mobility and “Capital mobility and

economic growth”, economic growth”, American Economic ReviewAmerican Economic Review, , 8585, 103-115, 103-115 Becker, G., Becker, G., PhilipsonPhilipson, T. and , T. and SoaresSoares, R. (2005), R. (2005) “The quantity and quality of “The quantity and quality of

life and the evolution of world inequality”, life and the evolution of world inequality”, American Economic Review American Economic Review 9595, , 277-291277-291

Becker, R. A. (1980)Becker, R. A. (1980) “On the long-run steady state in a simple model of “On the long-run steady state in a simple model of equilibrium with heterogeneous households”, equilibrium with heterogeneous households”, Quarterly Journal of Quarterly Journal of EconomicsEconomics, , 9595, 375-382, 375-382

BBéénabounabou, R. (1996), R. (1996) “Inequality and growth”, “Inequality and growth”, inin: Bernanke, B., and : Bernanke, B., and Rotemberg, J. (eds.), Rotemberg, J. (eds.), National Bureau of Economic Research National Bureau of Economic Research Macroeconomics AnnualMacroeconomics Annual MIT Press, Cambridge, Mass, 11–74. MIT Press, Cambridge, Mass, 11–74.

Bliss, C. (2004)Bliss, C. (2004) “Koopmans recursive preferences and income convergence”, “Koopmans recursive preferences and income convergence”, Journal of Economic TheoryJournal of Economic Theory, , 117117, 124-139, 124-139

References (2) Bourguignon, F. (1981) “Pareto superiority of unegalitarian equilibria in Bourguignon, F. (1981) “Pareto superiority of unegalitarian equilibria in

Stiglitz's model of wealth distribution with convex saving function”, Stiglitz's model of wealth distribution with convex saving function”, EconometricaEconometrica, , 4949, 1469-1475, 1469-1475

Bourguignon, F. and Morrisson, C. (2002) “Inequality Among World Citizens: Bourguignon, F. and Morrisson, C. (2002) “Inequality Among World Citizens: 1820-1992”, 1820-1992”, American Economic ReviewAmerican Economic Review, , 9292, 727-744, 727-744

DecancqDecancq, K., , K., DecosterDecoster, A. and , A. and SchokkaertSchokkaert, E. (2007), E. (2007) “The Evolution of World “The Evolution of World Inequality in well-being” Discussion Paper 0704, Department of Economics, Inequality in well-being” Discussion Paper 0704, Department of Economics, Katholieke Universiteit LeuvenKatholieke Universiteit Leuven

DeNavas-Walt, C., Proctor, B. D. and Lee, C. H. (2005) “Income, poverty, and health insurance coverage in the United States: 2004.” Current Population Reports P60-229, U.S. Census Bureau, U.S. Government Printing Office, Washington, DC.

Dollar, D. and Dollar, D. and KraayKraay, A. (2002), A. (2002) “Growth is good for the poor”, “Growth is good for the poor”, Journal of Journal of Economic GrowthEconomic Growth, , 77, 195-225, 195-225

Kuznets, S. (1963) “Quantitative Aspects of the Economic Growth of Nations: Kuznets, S. (1963) “Quantitative Aspects of the Economic Growth of Nations: Part VIII, Distribution of Income by Size”, Part VIII, Distribution of Income by Size”, Economic Development and Economic Development and Cultural ChangeCultural Change,11,11

NeumayerNeumayer, E. (2003), E. (2003) “Beyond income: convergence in living standards, big “Beyond income: convergence in living standards, big time”, time”, Structural Change and Economic DynamicsStructural Change and Economic Dynamics, , 1414, 275-296., 275-296.

References (3) Ramsey, F. P. (1928)Ramsey, F. P. (1928) “A mathematical theory of saving”, “A mathematical theory of saving”, The The

Economic JournalEconomic Journal, , 3838, 543–559, 543–559 Ravallion, M. (2003) “Inequality convergence”, Economics

Letters, 80, 351–356 Sala-i-MartinSala-i-Martin, X. (2006) , X. (2006) “The world distribution of income: “The world distribution of income:

Falling poverty and ... convergence, period”, Falling poverty and ... convergence, period”, Quarterly Journal of Quarterly Journal of EconomicsEconomics, 121 , 121

Schlicht, E. (1975) “A neoclassical theory of wealth distribution”, Schlicht, E. (1975) “A neoclassical theory of wealth distribution”, JahrbJahrbüücher fcher füür Nationalr Nationalöökonomie und Statistikkonomie und Statistik, , 189189, 8-96, 8-96

Schmidt-Hebbel, K. and Schmidt-Hebbel, K. and ServenServen, L. (2000), L. (2000) “Does income “Does income inequality raise aggregate saving?”, inequality raise aggregate saving?”, Journal of Development Journal of Development EconomicsEconomics, , 6161, 417-446, 417-446

StiglitzStiglitz, J. E. (1969), J. E. (1969) “Distribution of income and wealth among “Distribution of income and wealth among individuals”, individuals”, EconometricaEconometrica, , 3737, 382-397, 382-397