Applied and Theoretical Issues in Income Redistribution

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Applied and Theoretical Issues in Income
Redistribution

• Mallorca summer course (June 2001)
• F. Bourguignon

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II . Optimal income redistribution in practice

• A reminder of the basic optimal (labor) income
  redistribution model (OIRM)
• Trying to implement empirically the OIRM
• Adequacy of the OIRM for redistribution analysis
  in developed and developing countries

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A) A reminder of the basic optimal (labor) income
redistribution model (OIRM)

• Original model due to Mirrlees (1971). More
  accessible statements may be found in various
  texts (Atkinson-Stiglitz, Stiglitz, Tuomala,
  Myles, )
• Purpose here is to recall the structure of the
  model, the first order conditions and their
  interpretation, and then to reflect on the ways
  to apply them to actual data, something which was
  not done until recently, most of the applied OIRM
  literature bearing on hypothetical distributions.
  

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Reminder of the OIRM (cont'd)

• Statement of the model
• Individuals in population are heterogeneous with
  respect to their productivity, w.
• Redistribution authority does not observe w but
  only labor income w.L.
• But it knows what are the (identical) individual
  preferences, U(c, L), over consumption,c and
  labor, L.
• It also knows the statistical distribution of w
  in the population. f() is the corresponding
  density function.

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Reminder of the OIRM (cont'd)
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Reminder of the OIRM (cont'd)

• 2) Indications on the solution of the general
  model
• Original variables and functions
• T(wL), Lw, T(), y wL - T(wL), V(w, T( )
  )
• New variables z wL, c z T(z), u U(c,
  w/w)
• Question what is the optimal correspondence
  between w and (z, c, u), or the optimal functions
  z(w), c(w), u(w) ?
• Redistribution function T( ) may then be
  recovered from the implied relationship between c
  and z.

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Reminder of the OIRM (cont'd)

• This leads to the optimal control problem (state
  variable u, control variables c, z)

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Reminder of the OIRM (cont'd)

• Preceding problem has a solution as long as the
  'agent monotonicity' property holds.
• This property requires that optimal consumption
  is an increasing function of productivity, w, in
  the absence of taxation. In other words,
  consumption gives a truthful information on
  (unobserved) productivity.
• Sufficient that consumption is not an inferior
  good for the preceding property to be satisfied.

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Reminder of the OIRM (cont'd)

• Some general properties of the optimal
  redistribution schedule and the need for
  simulation
• Optimal T' lies between 0 and 1
• T' 0 at the lowest and highest level of
  productivity but this may of little practical
  relevance
• It may be optimal to have the agents with the
  lowest productivity not working
• Pre-tax income and consumption are increasing
  functions of productivity

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Reminder of the OIRM (cont'd)

• Properties obtained by numerical simulation
• G(u) u1- ?/(1-??) with the limit Rawlsian case
  of ????
• U(y , L) CES with varying substitution
  elasticities
• f( ) Pareto, Log-normal
• marginal tax rates my be declining, increasing or
  constant w.r.t. productivity
• High marginal tax rates do not necessarily
  require a Rawlsian social welfare function
• Intermediate values for ? and for the CES
  elasticity of substitution lead to approximately
  constant T' () linear redistribution system.

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Reminder of the OIRM (cont'd)

• 3) The case of quasi-linear preferences (Diamond,
  Atkinson, ..)
• leading to optimal tax rates solution of
• Where t(w) marginal tax rate at w , S(w)
  average marginal social welfare G'() of agents
  with productivity greater than w and F( )
  cumulative of f( ) .

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Reminder of the OIRM (cont'd)

• Intuitive proof of optimal tax formula
• Increase the marginal tax rate at the
  productivity level w by ?t.
• All people with productivity greater than w pay
  the tax. Total receipt 1 F(w) ?t
• People concerned by the marginal tax rate change
  modify their labor supply L by ?L ?.L.?t
  /w(1-t). The corresponding change in tax
  payment then is t.w ?L t.w.?.L.?t /w(1-t)
• Density of people concerned by marginal tax rate
  change density of z wL at productivity level
  w. From z B.w1??(1-t)?, it may be shown that
  this density is given by f(w)/(1 ?).L
• People in a) loose on average a social welfare
  S(w), people in b) have no loss of welfare.
  Additional tax receipt distributed as an equal
  lump sum to all individuals, with an average
  welfare gain equal to S(w0).

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Reminder of the OIRM (cont'd)

• Intuitive proof of optimal tax formula
• Collecting all preceding terms together and
  noticing that the change in social welfare cannot
  differ from zero at the optimum leads to
• And then

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Reminder of the OIRM (cont'd)

• Analysis of this optimal redistribution system
  easy if social welfare function reasonably simple
• Full tax system obtained by integrating t(w)
  along z.
• Government budget constraint satisfied by
  modifying the constant of integration, T(0).
• - T(0) guaranteed disposable income at 0 income
  from labor

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II) Trying to implement empirically the OIRM

•   1) Motivation
• Various recent attempts at identifying features
  of optimal redistribution schedules based on
  'true' data i.e. true f(w). See Diamond
  (1998), Salanié (1998), Saez (2000), Bourguignon
  and Spadaro (2000)
• Should the optimal marginal tax rate curve be
  U-shaped, increasing, decreasing, flat ?
• What follows summarizes problems and results
  obtained by Bourguignon and Spadaro (2000)

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Trying to implement empirically the OIRM

•   2) Methodological issues
• a) Choosing a labor supply elasticity
•   - hours of work and labor supply
•   - heterogeneity of ? across various types of
  individuals
•   - 'household' productivity and labor-supply
  elasticity
•  
• b) Estimating the distribution of productivities
  f(w)
• - use observed wage rates 
• - Choose elasticity parameter, ?, and 'invert'
  observed EMTR and gross labor income, z, to get
  w for the household z B.w1??(1-t)? .  

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Trying to implement empirically the OIRM

•   c) Choosing and calibrating a social welfare
  function
• Probably the simplest social welfare function
  G(u) to use is piecewise linear. This facilitates
  the computation of the mean marginal social
  welfare S(w). 
• The function being used here is of the following
  charitable conservative type
• G(u) u if F(w) ? q
• G(u) bu if F(w) gt q with blt1
• This function has the usual properties of being
  increasing and concave

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Trying to implement empirically the OIRM

•  
• How to calibrate the social welfare function ?
• G(u) u if F(w) ? q
• G(u) bu if F(w) gt q with blt1
• b 1 ? Utilitarian
• b 0 ? Rawlsian
• Preferred choice b such that relative poverty
  would optimally be eliminated if the labor-supply
  elasticity were 'small'. This is equivalent to
  imposing T(0) - .5mean income.

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Trying to implement empirically the OIRM

•   d) Some complications
•  (i) Continuity and differentiability
• (ii) Household size
• (iii) Households with zero income and households
  with apparently 'irrational' behavior
• - 100 per cent marginal tax rate at minimum
  income - Less than 100 per cent marginal tax
  rate at minimum income
•  Drawing measurements errors to eliminate
  anomalies extrapolating productivity
  distribution for non-working individuals
• (iv) Lack of observations at the top of the
  distribution (approximation by a Pareto)

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Trying to implement empirically the OIRM

•   3) Results for 4 EU countries
• Data base Household surveys used in Euromod
  project
• Euromod tax-benefit software to compute marginal
  tax rates
•  

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Trying to implement empirically the OIRM

• Main Findings
• Optimal systems based on 'actual' data and
  reasonable assumptions are far from actual
  redistribution systems.
• Systematically declining marginal tax rates
  only way to have U-shaped curve is to have
  decreasing social marginal welfare at the top of
  the distribution.
• Related conclusion relatively low marginal tax
  rate at the top of the distribution
• Rapidly declining marginal tax rates at the
  bottom of the distribution (a shape that differs
  from what is implied by strict means-tested
  transfers).

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Trying to implement empirically the OIRM

• Preceding exercise is probably the limit of what
  can be done empirically to implement in a
  reasonable way the OIRM.
• Is it really conclusive ?
• After all, two main conclusions may be linked to
  assumptions made on unobserved or badly observed
  part of the distribution of productivities.

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Trying to implement empirically the OIRM

• 4) Revealing social preferences
• Instead of comparing the optimal system linked to
  'observed' f(w), arbitrary ?? and arbitrary
  social welfare function, G( ), one may go the
  other way.
• 'Inverse optimal problem' (Kurz)
• Observing the actual redistribution system
  through the marginal tax rates associated to
  productivity levels, w, is it possible to infer
  the social welfare function that would make that
  system 'optimal' ?

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Trying to implement empirically the OIRM

• Inverse optimal problem work in progress (BS)
• Results encouraging in the sense that actual
  redistribution systems observed f(w)
  assumptions for unobserved parts of f( )
  reasonable values for ? lead to
  'non-contradictory' shapes of G'(u).
• Revealed values of G'( ) decreasing with w, at an
  accelerated rate for highest values of w
• However, provisos with assumptions about
  unobserved partd of f( ) still apply.

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C) Adequacy of the OIRM for redistribution
analysis in developed and developing countries

• Overall, results are somewhat disappointing
  because they rely on rather stringent
  assumptions, which are more or less all open to
  debate.
• Lets' consider four kinds of issues
• Representation of the whole economy and
  functioning of the labor market.
• In particular, are all households on their
  labor-supply curve? Things would clearly be
  different if non-workers were constrained on the
  demand side of the market.

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Adequacy of the OIRM for redistribution analysis
in developed and developing countries

• b) Other sources of heterogeneity among
  households capital income, household size,
  gender issues within the household, etc
• To what extent actual systems reflect
  redistribution along these other dimensions?
• Unfortunately, it is hopelessly difficult to
  generalize OIRM to more than one dimension

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Adequacy of the OIRM for redistribution analysis
in developed and developing countries

• c) Concept of a social welfare function. What
  would happen if G( ) were to reflect the views of
  some decisive voter, or the outcome of some kind
  of political economy equilibrium?

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Adequacy of the OIRM for redistribution analysis
in developed and developing countries

• d) Static model, whereas actual systems may take
  into account dynamic features of individual
  welfare as well as uncertainty aspects.
• The insurance part of redistribution schemes.

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Conclusion

• OIRM was an extremely promising framework.
  Probably the best representation of the old
  efficiency-equity trade-off
• Yet it may be too simple for being of much
  practical use, beyond theoretical economic
  reasoning.
• Workfield still open. Most likely that modern
  computation techniques should permit to solve
  some of the technical problems listed previously.
• Remains true that good understanding of the
  actual functioning of economies remain a priority.