Lecture 3 Understanding Inequality: Structure and Dynamics

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Lecture 3Understanding InequalityStructure and
Dynamics

• Course on Poverty and Inequality Analysis
• Module 5 Inequality and Pro-Poor Growth
• Francisco H. G. Ferreira
• DECRG

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Roadmap

1. Distributions
2. The Determinants of Inequality a conceptual
   overview
3. Inequality Decomposition Analysis
4. Income Distribution Dynamics statistical
   analysis
5. Income Distribution Dynamics towards economic
   decompositions
6. Summing up

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3.1. Distributions

• Social welfare, poverty and inequality summarize
  different features of a distribution.
• Distribution of welfare indicator per unit of
  analysis.
• Discrete y y1, y2, y3, ., yN
• Continuous The distribution function F(y) of a
  variable y,defined over a population, gives the
  measure of that population for whom the variable
  has a value less than or equal to y.

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The density function f(x)
The distribution function
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The quantile function yF-1(p)
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The Lorenz curve
or
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The Generalized Lorenz curve
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3.2. The Determinants of Inequality a
conceptual overview

• Inequality measures dispersion in a distribution.
  Its determinants are thus the determinants of
  that distribution. In a market economy, thats
  nothing short of the full general equilibrium of
  that economy.
• One could think schematically in terms of
• y a.r
• This suggests a scheme based on assets and
  returns
• Asset accumulation
• Asset allocation / Use
• Determination of returns
• Demographics
• Redistribution

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3.2. The Determinants of Inequality a
conceptual overview
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3.2. The Determinants of Inequality a
conceptual overview

• Modeling these processes in an empirically
  testable way is quite challenging.
• Though there are G.E. models of wealth and income
  distribution dynamics
• Historically, empirical researchers have used
  shortcuts, such as
• decomposing inequality measures by population
  subgroups, and attributing explanatory power to
  those variables which had large between
  components
• Decomposing inequality by income sources, to
  understand which contributed most to inequality,
  and why
• Decomposing changes in inequality into changes in
  group composition, group mean and group
  inequality.

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3.3 Inequality Decomposition Analysisa. By
Population Subgroups
Not all inequality measures are decomposable, in
the sense that I IW IB. The Generalized
Entropy class is.
Examples include Theil L Theil T 0.5 CV2
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3.3 Inequality Decomposition Analysisa. By
Population Subgroups
Let ? (k) be a partition of the population into k
subgroups, indexed by j. Similarly index means,
n, and subgroup inequality measures. Then if we
define
where
Then, E EB EW.
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An Example from Brazil
The Rise and Fall of Brazilian Inequality
1981-2004
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A cross-country example Race and ethnicity
decompositions.
Source WDR 2006
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3.3 Inequality Decomposition Analysisa. By
Population Subgroups

• The methodology was developed by
• Bourguignon, F. (1979) "Decomposable Income
  Inequality Measures", Econometrica, 47,
  pp.901-20.
• Cowell, F.A. (1980) "On the Structure of
  Additive Inequality Measures", Review of Economic
  Studies, 47, pp.521-31.
• Shorrocks, A.F. (1980) "The Class of Additively
  Decomposable Inequality Measures", Econometrica,
  48, pp.613-25.
• Reviewed in
• Cowell, F.A.and S.P. Jenkins (1995) "How much
  inequality can we explain? A methodology and an
  application to the USA", Economic Journal, 105,
  pp.421-430.
• Example from
• Ferreira, F.H.G., Phillippe Leite and J.A.
  Litchfield (2001) The Rise and Fall of
  Brazilian Inequality 1981-2004, World Bank
  Policy Research Working Paper 3867.

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3.3 Inequality Decomposition Analysisb. By
Income Sources

• Shorrocks A.F. (1982) Inequality Decomposition
  by Factor Components, Econometrica, 50,
  pp.193-211.
• Noted that
  could be written as

Correlation of income source with total income
Share of income source
Internal inequality of the source
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3.3 Inequality Decomposition Analysisb. By
Income Sources
Source Ferreira, Leite and Litchfield, 2006.
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4. Income Distribution Dynamicsa. Scalar
decompositions
Mookherjee, D. and A. Shorrocks (1982) "A
Decomposition Analysis of the Trend in UK Income
Inequality", Economic Journal, 92, pp.886-902.
Pure inequality
Group Size
Relative means
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The (obligatory) example from Brazil
The Rise and Fall of Brazilian Inequality
1981-2004
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4. Income Distribution Dynamicsb. A More
Disaggregated Look

• In practice, decompositions of changes in scalar
  measures suffer from serious shortcomings
• Informationally inefficient, as information on
  entire distribution is collapsed into single
  number.
• Decompositions do not control for one another.
• Can not separate asset redistribution from
  changes in returns.
• With increasing data availability and
  computational power, studies that decompose
  entire distributions have become more common.
• Juhn, Murphy and Pierce, JPE 1993
• DiNardo, Fortin and Lemieux, Econometrica, 1996

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3.4. Income Distribution Dynamicsb. The
Oaxaca-Blinder Decomposition

• These approaches draw on the standard
  Oaxaca-Blinder Decompositions (Oaxaca, 1973
  Blinder, 1973)
• Let there be two groups denoted by r w, b.
• Then and
• So that
• Or
• Caveats (i) means only (ii) path-dependence
  (iii) statistical decomposition not suitable for
  GE interpretation.

returns component
characteristics component
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3.4. Income Distribution Dynamicsb. Juhn,
Murphy and Pierce (1993)
Juhn, Murphy and Pierce (1993)
where
Define
Then
Returns component
Unobserved charac. component
Observed charac. Component.
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3.4. Income Distribution Dynamicsb. DFL and BFL
How and why does fA(y) differ from fB(y)?
One could decompose fB(y) - fA(y) into
where
A similar (but distinct) decomposition would be
obtained with
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3.4. Income Distribution Dynamicsb. DiNardo,
Fortin and Lemieux (1996)
Essentially, DFL propose estimating a
counterfactual income distribution such as
By appropriately reweighing the sample, as
follows.
where
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3.4. Income Distribution Dynamicsb.
Bourguignon, Ferreira and Lustig (2005).
Partition the set of covariates T into V and W,
where V can logically depend on W.
Replace the joint distribution of covariates by
the appropriate product of conditional
distributions, and the joint distribution of W.
Define a counterfactual distribution fsA?B(y
ks, ?A). I.e.
Note that the order of conditioning will
affect interpretation.
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3.4. Income Distribution Dynamicsb.
Bourguignon, Ferreira and Lustig (2005).
For each counterfactual distribution fs, the
difference between fA and fB can be decomposed as
follows     And it follows that
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3.4. Income Distribution Dynamicsb.
Bourguignon, Ferreira and Lustig (2005).

• Estimate - for each country or time period -
  simple econometric models of earnings,
  occupational structure, education and fertility
  choices.
• Simulate the effects of importing the parameter
  estimates of each model from county A into
  country B (individually or jointly).
• Decompose distributional differences into
• Price effects
• Occupational structure effects
• Endowment (or population characteristic) effects

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3.4. Income Distribution Dynamicsb. Brazil,
1976-1996. (Ferreira and Paes de Barros, 1999)
Level 1 y G (V, W, ? ?)   Aggregation
rule   Earnings   Occupational
Choice     Level 2 V H (W, ?
?)   Education MLE (E?A, R, r, g, nah
?)   Fertility MLC ( nch ?E, A, R, r, g, nah
?)    
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Comparing g(p) and gs(p) (i) The price effect.
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Comparing g(p) and gs(p) (ii) The price effect
and the occupational structure effect combined.
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Comparing g(p) and gs(p) (i) Price, Occupation,
Education and Fertility effects.
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3.5. Income Distribution Dynamics a. towards
economic decompositions?

• Generalized Oaxaca-Blinder decompositions such as
  those discussed above, whether parametric or
  semi-parametric, suffer from two shortcomings
• Path-dependence
• The counterfactuals do not correspond to an
  economic equilibrium. There is no guarantee that
  those counterfactual incomes would be sustained
  after agents were allowed to respond and the
  economy reached a new equilibrium.

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(a) Partial Equilibrium Approaches

• The first step towards economic decompositions,
  in which the counterfactual distributions may be
  interpreted as corresponding to a counterfactual
  economic equilibrium, is partial in nature.
• One example comes from attempts to simulate
  distributions after some transfer, in which
  household responses to the transfer (in terms of
  child schooling and labor supply) are
  incorporated.
• Bourguignon, Ferreira and Leite (2003)
• Todd and Wolpin (2005)
• (These two papers differ considerably in how they
  model behavior. Todd and Wolpin are much more
  structural.)

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(b) General Equilibrium Approaches

• However, a number of changes which are isolated
  in statistical counterfactuals such as changes
  in returns to education, or in the distribution
  of years of schooling are likely to have
  general equilibrium effects.
• Similarly, certain policies one might like to
  simulate may require a general equilibrium
  setting.
• There are two basic approaches to generate
  GE-compatible counterfactual income distributions
  (and thus counterfactual GICs)
• Fully disaggregated CGE models, where each
  household is individually linked to the
  production and consumption modules. E.g. Chen and
  Ravallion, 2003, for China.
• Leaner macroeconomic models linked to
  microsimulation modules on a household survey
  dataset. E.g. Bourguignon, Robilliard and
  Robinson, 2005, for Indonesia.

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Distributional Impact of Chinas accession to the
WTO. (Chen Ravallion, 2003)
GE-compatible counterfactual GICs corresponding
to a specific policy.
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(b) General Equilibrium Approaches (continued).

• In the Macro-Micro approach, some key
  counterfactual linkage variables are generated in
  a leaner macro model, whose parameters may have
  been calibrated or estimated from a time-series,
  and then fed into sector-specific equations
  estimated in the household survey, to generate a
  counterfactual GIC.

Macro model
Linkage AggregatedVariables (prices, wages,
employment levels)
Household income micro-simulation model
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The distribution of the impacts of the 1999
Brazilian devaluation (Ferreira, Leite, Pereira
and Pichetti, 2004)
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3.6. Summing up

• There has long been an interest in understanding
  changes in (or differences across) income
  distributions.
• Static and dynamic decompositions of certain
  measures of inequality (by population subgroup or
  income source) can shed some light on the
  structure of inequality, and on the importance
  of covariates.
• But decompositions of scalar indices are
  inherently informationally constrained.
  Disaggregated statistical decompositions based on
  entire counterfactual distributions
  (parametrically or semi-parametrically) help shed
  more light on changes (and to separate the
  effects of returns, participation and composition
  effects).
• A step beyond this sort of statistical analysis
  is to build counterfactual distributions that
  correspond to economic equilibria. If sensibly
  estimated, these would allow inference of
  causality and hence policy simulations.
• Some progress on simple partial equilibrium
  models.
• Harder with general equilibrium approaches, where
  CGEs or macro models are subject to many
  criticisms.