Optimal Taxation

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Optimal Taxation

• Professor Jane H. Leuthold
• Department of Economics
• University of Illinois at Urbana-Champaign

Econ 415 Fall 2001
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Topics for today

• The optimal tax problem
• Why not use lump-sum taxes?
• Optimal income taxation
• Optimal commodity taxation and the Ramsey rule

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The optimal tax problem

• How to raise a given amount of tax revenue in the
  best possible way.
• Best is defined in terms of a social welfare
  function.

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Why not use lump-sum taxes?

• If all individuals are identical, then a lump-sum
  tax would be the only efficient tax. Why not use
  it?
• A lump-sum tax burdens the poor.
• A lump-sum tax is impossible if certain
  individual characteristics (ability, leisure,
  risk aversion) are unobserved and therefore
  untaxable.

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Optimal income taxes

• How should the tax rate vary with income so as to
  raise a given revenue with least loss of
  efficiency?
• Assume
• No saving, all income from labor
• Linear tax structure

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Linear (flat-rate) income taxes
Tax

• Proportional taxT1 t1 YATR MTR t1
• Progressive taxT2 t2 (Y A) t2 Y t2
  AMTR t2 gt ATR t2 t2 (A/Y)

T2
T1
A
Income
Negative income tax
t2 A
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Average linear income tax rates
Tax rate
t2
ATR t2 t2 (A/Y)
t1
A
Income
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Deadweight loss with a proportional or
progressive tax
SC
w
w(1-t1)
Compensated labor supply curve
w(1-t2)
DWL/R ½ t2 ecS
Labor
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Deadweight loss with a progressive flat-rate
income tax
Income
24w
24w(1-t1)
Tax revenue from proportional tax
E
Leisure
24
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Optimal linear income tax

• More progressive tax systems will lead to greater
  deadweight loss.
• More egalitarian social welfare functions will
  choose more progressive tax systems.
• Policy design problem is to balance the
  objectives of efficiency and egalitarianism.

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Non-linear income tax
Tax rate
t3
Marginal bracket tax rates
t2
t1
b1
b2
Income
A
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Optimal nonlinear income tax

• Impose high average tax rates and low marginal
  tax rates
• Make as few people as possible face high marginal
  tax rates
• Impose high marginal tax rates on those for whom
  the tax is least distorting Stiglitz, p. 560

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Linear vs nonlinear tax structures
Tax

• Which taxpayers face the highest marginal tax
  rates?
• When will deadweight loss be less under the green
  tax than under the blue tax?

D
C
A
Income
tA
B
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Zero tax rate on the highest income individual
Income
With a zero tax rate, the same revenue can be
raised with a zero tax rate above E while making
the taxpayer better off.
Zero rate tax above E
A
A
E
After-tax budget constraint
E
Leisure
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Optimal commodity tax problem

• If everyone is the same (or equivalently if
  equity doesnt matter), how should commodities be
  taxed so as to best raise a given revenue?
• If equity matters, do the rules for optimal
  commodity taxation still hold?

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Optimal commodity taxes
Price

• Assume equity is not an issue.
• There is one untaxed commodity.
• How should taxes be arranged across commodities
  to minimize the sum of excess burdens?

DC
St
Pt
S
P
Q Q Rice
DWL ½ t2 PQ ?C
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Frank Ramsey

• Shown in 1925 (age 22) shortly before his death.
• Proposed what is now known as the Ramsey rule
  which governs the pattern of optimal commodity
  taxes.
• Taxes that follow this rule are known as Ramsey
  taxes.

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Optimal taxation the Ramsey Rule
If not all commodities can be taxed, then the
optimal taxes satisfy
If the cross-effects are zero, then the optimal
taxes satisfy
t1 constant / eC1 and t2 constant / eC2
which is known as the inverse elasticity rule.
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Ramsey Rule

• If commodities are independent, taxes should be
  inversely proportional to the compensated
  elasticities of demand.
• If commodities are not independent, the optimal
  tax structure is such that the percentage
  reduction in the compensated demand for each
  commodity is the same.

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Equity and the Ramsey Rule

• In general, the Ramsey Rule implies imposing
  taxes in an inverse relationship to the
  elasticity of demand.
• This would result in a regressive tax system
  since necessities would be taxed more heavily
  than luxuries.
• Optimal taxation requires balancing equity and
  efficiency objectives.

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A special case
If U a ln X1 (1- a) ln X2 P1 X1 a Y P2 X2
(1-a) Y where Y is income, Then eC1
eC2 and the optimal tax is uniform.
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Optimal taxation in developing economies

• Highly unequal income distributions suggest that
  equity is likely to be an important
  consideration.
• Information on the compensated price elasticities
  of demand may not be available.
• Differential taxes may cause tax administration
  problems.

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Discussion

• Developing countries often use taxes on luxuries
  as a major source of income. Is this consistent
  with the theory of optimal taxation? Why or why
  not?

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Lab 9

• Experiment with different combinations of the tax
  rates ?1 and ?2 to find out the combination that
  best raises 10 units of revenue.
• Convert the model into one where equity doesnt
  matter and where utility functions are
  Cobb-Douglas. Do your results change?

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Next Time
Lab 9 Optimal taxes in the CGE model Chat How
is the presence of negative externalities like
pollution likely to affect the optimal tax
rules? Next week Consumption taxes VAT