Inequality Decompositions: A Reconciliation

1
Inequality Decompositions A Reconciliation

• Frank A. Cowell and Carlo V. Fiorio
• London School of Economics
• University of Milan and Econpubblica
• August 2009

2
Motivation

• Reasons for interest in decomposition
• understanding structure
• explaining inequality?
• policy design
• Reasons for interest in reconciliation
• separate strands that dont talk to each other?
• can we get a clear unified account of structure?
• Reasons for this approach
• show the links between apparent disparate
  methodologies
• use elementary approach based on recent
  contributions
• show practicality of unified approach

3
A priori approach

• Standard inequality / welfare axioms
• Requirement 1 specification of a collection of
  admissible partitions
• ways of dividing up the population into mutually
  exclusive and exhaustive subsets
• Requirement 2 a concept of representative income
• mean income
• ede income
• What inequality measure can be used?
• must satisfy a subgroup consistency or
  aggregability condition
• inequality in a component subgroup increases
  implies, ceteris paribus, that inequality overall
  goes up
• Applicable to the other principal type of
  decomposability
• the break-down by factor-source

4
Explanatory models

• Development of a structural model for inequality
  decomposition
• Bourguignon et al (RIW 2001), Dinardo et al
  (Ecmet 1996)
• Perhaps richest version
• Attractive for explaining inequality
• specifies a counterfactual
• examine the influence of each supposedly causal
  factor
• but data hungry? cumbersome modelling?
  sensitive to specification?
• Simple regression model
• Fields (2003), Fields and Yoo (2000), Morduch and
  Sicular (EJ 2002)
• A less ambitious version
• Again an advantage over a priori methods?
• may not need to model groups or income components
  separately
• such potential influences on inequality can be
  incorporated within an econometric model
• just need appropriate specification of the
  explanatory variables

5
Integrated approach?

• The a priori method can be developed
• from an exercise in logic
• to an economic tool that relevant to policy
• Use subgroup-decomposition to assign importance
  to factors affecting overall inequality
• personal characteristics
• social characteristics
• How to treat between-group inequality?
• focus on the types of partition that are relevant
  
• caution needed on the meaning of decompositions?
  (Kanbur JEI 2006)
• Should be some connection between the two
  approaches
• between-group/within-group breakdown in a priori
  approach
• the explained/unexplained variation in the
  regression approach

6
Basic model

• Data generating process
• If f is separable and linear
• In the sample
• OLS estimate

7
Factor source decomposition

• Let Ck bkXk, k 1,2,,K, CK1U
• Inequality of total income Y in terms of
  component incomes C1, C2,,CK1
• Shorrocks (Ecmet 1982)
• Using the explanatory model

8
Factor source decomposition estimation

• Replace population quantities with sample
  counterparts
• contribution of kth factor
• becomes
• Inequality factor-decomposition in the sample

9
Subgroup decomposition

• X1 is discrete takes values from X1,j j
  1,... , t1
• Equation for each subgroup
• Within-group inequality
• Hence between group inequality
• For the GE case

10
Implementation

• Use the estimated DGP
• The estimated between- and within group terms are

11
Application

• Applied to real data using the Luxembourg Income
  Study
• net disposable income
• United States and Finland
• mid-1980s and 2004
• Contrasting cases?

12
Subgroups by education
13
Subgroups by sex
14
Subgroups assessment and progress

• Cant disentangle changed contribution of one
  characteristic
• Maybe use a finer partition by interacting
  education and sex ?
• Cowell and Jenkins (EJ 1995)
• could become cumbersome if try to control for
  additional characteristics
• would need a discretisation of continuous
  variables
• would reduce the sample size in each subgroup
• What additional insights might a regression-based
  approach yield?
• estimate a model of equivalent disposable income
• Y is equivalised household income
• family variables ( earners, children under 18,
  tenure)
• head of household (age, sex, education dummies)

15
OLS equivalent income regression US Decomposition
16
OLS equivalent income regression Finland
Decomposition
17
Subgroup regressions

• Decomposition by education subgroups
• young children at home contribute similarly to
  inequality in education subgroups in the US and
  more highly-educated in Finland
• number of earners accounts for a large
  proportion of inequality in low educated
  households in both countries
• female penalty uniformly decreased across time in
  both countries
• female penalty low for higher levels of
  education.
• gender subgroups
• the highest level of educational attainment
  contributes to much more of the inequality among
  males than among females in the US
• college education accounts for roughly the same
  proportion of inequality in Finland
• Largest part of inequality accounting left in
  unobserved characteristics
• applies to all subgroup decompositions

18
Discussion

• An exact decomposition only if the residual is
  not ignored
• Computation of standard errors sometimes treated
  as a trivial problem
• this is far from being the case
• Should only be interpreted as a descriptive model
• single-equation model
• do better with a structural model?
• may be too demanding

19
Inequality decomposition income or predicted
income?

• What happened to the role of college in Finland?
• predicted income inequality not much
• observed income inequality big change role of
  unobserved component?

20
Conclusions

• Approach to reconciling decomposition analysis
  based on a single-equation regression
• builds on the Shorrocks (1982) methodology
• tool for understanding inequality when data do
  not allow structural specification
• Shares some features with Fields (2003) approach
• includes in the analysis the decomposition by
  subgroups
• shows how this is useful to identify differences
  in determinants of inequality
• fairly robust
• yields results consistent with other
  decomposition methods.
• Distinguish clearly between explanations of
  inequality
• those that focus on breakdown of the factors
  underlying predicted income
• breakdown of inequality of observed income.

21
(No Transcript)
22
Supplementary equations

• intermediate step from Shorrocks decomposition
• Within-group inequality, GE case

23
General case

• If corr (X1,j, X1,k) ? 0 or corr (X1,j, U) ? 0

24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)