Optimal Redistributive
Taxation
Louis Kaplow
Book Overview

Scope: Conceptual Framework


Focus on major structural features
Emphasizing relationships among the pieces

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contrast existing Public Economics Handbook (4+ vols.)
and surveys of specific subjects
Not:
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Empirical
Political economy
Macroeconomics (stabilization, ...)
2
Contents
I.
II.
III.
IV.
V.
Framework
Optimal Income Taxation
Optimal Commodity Taxation
Applications
Tax Equity
3
Contents
I.
II.
III.
IV.
V.
Framework
Optimal Income Taxation
Optimal Commodity Taxation
Applications
Tax Equity
4
Framework

Purposes of taxation


Redistribution
Revenue-raising

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identical individuals: trivial result that uniform
(lump-sum) levy optimal; no distortion
distortionary income tax (etc.) because
distribution matters
“equity-efficiency” tradeoff thus central
Corrective (Pigouvian)
5
Integrated view (ch. 2)


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Optimal use of instrument A depends on
whether instrument B available
Notably, income tax (and transfers) are core
instrument for revenue/redistribution
Other tools (commodity taxes, estate/gift tax,
...) viewed as special-purpose supplements


assessed on efficiency grounds – income tax
adjustments can hold distribution constant
Likewise for public goods, regulation, …
6
Approach



Distribution-neutral adjustment to the
income tax
Isolates “distinctive” effects of policies
(estate/gift taxation, public goods,
Pigouvian taxation)
Developed in detail, below, with
commodity taxation
7
Social objective (ch. 3)
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Explicit reference to social welfare function
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surprisingly high use of fuzzy proxies (ability to pay) and
often unspecified notions of fairness
inappropriate, and often indeterminate in any event
value of tracing all to welfare is large (e.g., family unit,
estate/gift tax, administration/error)
“Welfarism”: notion that all of social
relevance can be traced to effects on
individuals’ welfare


some defense in chapters 13-15
see generally Fairness versus Welfare (2002) (w/ Shavell)
8
SWF
SW ( x)   W ui ( x) f (i )di
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Ex.: Utilitarianism: sum of utilities
Decreasing marginal utility in u
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Ex: u = ln y
MU = 1/y
Utilitarian SWF: marginal $ worth 10 times
to those with 1/10 the income
9
Common formulation
ui ( x ) 1 e
SW ( x )  
f (i )di , for e  1
1 e
  ln ui ( x ) f (i )di , for e  1

e : inequality aversion parameter
Also decreasing MU, in the u function
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Weight on redistribution depends on both
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But arguably mainly the latter
10
Contents
I.
II.
III.
IV.
V.
Framework
Optimal Income Taxation
Optimal Commodity Taxation
Applications
Tax Equity
11
Optimal taxation setup
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Income taxation
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Really a labor income tax (one-period model)
Roughly equivalent to a “wage” tax or a cash-flow
consumption tax in this one-period setting
Commodity taxation


Think of as incorporating all manner of differential
taxation (including of savings)
Tax expenditures (and penalties) can be analyzed
as differential commodity taxation
12
Optimal income taxation (ch. 4)

Problem statement: Choose tax system to
max SWF, subject to:
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Individuals choose labor supply to max U
Exogenous revenue requirement
Source of equality/efficiency tradeoff:
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“Type” (earning ability w) assumed unobservable
Tax income, a signal of ability
Hence distortion because not taxing ability directly
13
Illustration of tax/transfer fn
Nonlinear Income Tax and Transfer Schedule
14
Linear income tax
Linear Income Tax Schedule, t = 40% and g = $12,000
15
Linear income tax: F.O.C.
cov ( w), y( w)
t

1 t
 y( w) ( w) f ( w)dw
 ( w) 

W uc ( w)

  l ( w) 
 tw

 g 
Stern (1976): t=54%, g=34% of ave. inc.
16
Two-bracket income tax
Slemrod et al. (1994)
Two-Bracket Income Tax Schedule
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Upper bracket w/ lower marginal rate
More so as weight on equality increases
17
Nonlinear income tax: concept
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Problem
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Perturbation thought experiment:
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Determine marginal tax rate at all levels of income
Determine intercept: i.e., grant to individual earning zero
Suppose we raise the proposed marginal tax rate on all income in the
range (50,000, 50,100) by 1% (one percentage point)
Two effects:

Inframarginal: all individuals with y (wl) above 50,100 pay $1 more in
tax
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Marginal: all individuals with y in the interval (50,000, 50,100)

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They all face same marginal tax rate as before: no substitution effect
(Income effect: all work a tiny bit more)
Sum: obtain significant revenue, and from higher-income individuals (where high is
relative to $50,100 in this case)
Pay $0 to $1 more if behavior unchanged
They face higher marginal tax rate, so substitution effect induces to work less (is also
an income effect)
Questions: how large is each effect, what does it depend on,
and how does it relate to social welfare?
18
Nonlinear income tax: F.O.C.

T ( w * l*)
1  F ( w*)

1  T ( w * l*)  * w * f ( w*)
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marginal folks (with income w*l*)
inframarginal folks (income > w*l*)
(1-F)/f:
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1  F ( w*)
Perturbation thought experiment
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 W (u( w))uc 
 f ( w)dw
w*1 


1-F is fraction of population above marginal rate in question – collect
more revenue inframarginally from them as raise marginal rate at a
given point
f is density at margin, those who are distorted
ξ*, w*
Second term: average welfare cost to the group that pays the
increase (for case with no income effects)
Many simulations have high marginal rates that trail downward at
top
19
Ability taxation (ch. 5.C)
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Ideal tax is ability tax
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Income taxed as observable signal of
ability
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Explanation of Pareto improvement
Could have full equalization
Source of incentive tradeoff
Ability tax surrogates possible?

Height? Age?
20
Transfer payments (ch. 7)
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Optimal transfers: optimal income tax
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Categorical assistance
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Model as type-specific income tax (ability taxation, above)
Results: phase-outs
Work inducements
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I.e., if have figured out optimal income tax, including
(plausibly negative) taxes on the poor, why aren’t we done?
Hard to justify
Informational assumptions
Cash versus in-kind (raises different types of
questions)
21
Contents
I.
II.
III.
IV.
V.
Framework
Optimal Income Taxation
Optimal Commodity Taxation
Applications
Tax Equity
22
Commodity taxation (ch. 6)

Problem: choose relative tax rates on different
commodities, given an income tax that can be
adjusted
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Result: no differentiation optimal in basic case
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Atkinson & Stiglitz (1976) showed true with optimal income
tax; Kaplow (2006) extends to when income tax nonoptimal
Intuition: adds additional distortion without helping
on the redistribution/distortion tradeoff
Example: inefficiency of luxury taxes, exemptions for
necessities (e.g., for food in a VAT)
Assumptions and qualifications
23
Proof sketch
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Step 1: begin with differential commodity tax
Step 2: abolish tax differentials and adjust income
tax to hold utility level constant for each taxpayer (at
each level of prior income)
Step 3: this combined reform has no effect on labor
supply

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intuition: utility return to every level of labor effort same (for
everyone); hence labor same (see diagram)
Step 4: a surplus is produced (due to reducing the
distortion of consumption choices)
Step 5: pro rata rebate of surplus: Pareto
improvement
24
Figure 6.1
Labor Supply with Offsetting Income Tax Adjustment
Summary of intuition

Distribution can be ignored

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Labor supply effects can be ignored
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Because by construction is distributionneutral
Because are none (qualifications coming)
What’s left?: the “distinctive” efficiency
consequences of the reform
26
Two-step decomposition
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What if a reform is not distributionneutral?
Decompose as follows:
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(1) Construct distribution-neutral reform
(2) Move from that to actual reform

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Note : latter is purely redistributive
Comments
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Clarifies analysis
Facilitates specialization
27
Qualifications
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Commodities that interact with utility of labor-leisure
choice:
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Other qualifications
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higher tax rate on books, which are leisure complements,
because discouraging leisure reduces labor supply distortion
subsidize central city amenities
Goods that signal ability (as distinct from income): opera?
Externalities – e.g., pollution, gifts, … (more below)
…
Still, it provides general way to assimilate many fiscal
issues into a common framework
28
Contents
I.
II.
III.
IV.
V.
Framework
Optimal Income Taxation
Optimal Commodity Taxation
Applications
Tax Equity
29
General implications

Ramsey taxation principles (inverse elasticity rule)
essentially wrong in this setting

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All analysis based thereon (e.g., of capital income taxation)
not right when is an income tax that may be adjusted
Examples

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Exemption of food from VAT (groceries vs. restaurant meals)
Taxation of gifts: viewed as one sort of consumption
expenditure
Public goods: can compare “price” to marginal WTP
Taxation of savings (income versus consumption tax, etc.):
standard income tax is equivalent to pure labor income tax
plus differential taxation on commodities consumed in
different periods . . .
30
“Income” vs. consumption Tax
and taxation of capital (ch. 9)

Earn in period 1, consume in periods 1 and 2; model
difference in terms of budget constraints (illustration
with flat-rate tax, no grant)
c2
wl (1  t )  c1 
1 r
c2
wl (1  t )  c1 
1  r (1  t )
31
Analysis

Interpret as differential commodity
taxation

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Consumption in different periods =
consumption of two different commodities
Atkinson-Stiglitz result applies

Note: not pro-rich because doing it
distribution-neutral (income tax adjusts)
32
Discussion
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Capital taxation generally
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Including wealth taxation
Corporate income tax
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Recent dynamic Mirrlees literature
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As a form of differential capital taxation
“Intertemporal wedge”
Intuition: multi-period, future labor income
uncertainty, induces precautionary savings, but
that dampens future labor supply (wealth effect)
Administrative arguments bearing on income
tax vs. consumption tax
33
Corrective taxation (ch. 8.G)
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Pigouvian taxes and subsidies
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Similar features for much of regulation
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Qualifications?
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Including liability through the legal system
Distributive weights
Labor supply distortion (“double dividend,” etc.)
Neither – again

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Analogy to differential commodity taxation
Distribution-neutral income tax adjustment
34
Transfer (estate/gift) taxation (ch. 10)
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Description/model: differential commodity taxation
based on own- versus other-consumption
Analysis
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Gift externalities
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again, presumptively inefficient
Positive, to donee
Negative, income effect on donee
Note: again, integrating w/ income tax (which can
tax rich however would like) transforms question and
produces qualitatively different analysis & results
Extension: charitable giving
35
Unit of taxation (ch. 12)

Marrieds vs. singles; number of children
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Controversial
Qualitative changes over time
Differences across countries
Huge common denominator problem – need
link to objective function (social welfare
function)
Approach
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Positive model of family (e.g., models of sharing,
as in Becker 1974)
SWF with each individual counting once
36
Analysis: distribution
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Does unequal sharing clearly favor
“individualized” treatment? (no)
Do scale economies clearly favor less
generous treatment of marrieds? (no)
Should child benefits be fixed amounts /
declining with income? (no)
Note: form of SWF and degree of riskaversion (rate of declining MU of
consumption) have qualitative effect here
37
Analysis: incentives

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Work
Marriage
Child-bearing
38
Contents
I.
II.
III.
IV.
V.
Framework
Optimal Income Taxation
Optimal Commodity Taxation
Applications
Tax Equity
39
Framework (ch. 13 & 14)

Welfarism
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Restate definition
Other norms can lead to Pareto conflicts
Proxy rationales
Choice of social welfare function
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“Original position”
Sufficiently egalitarian?
Whose welfare included?
40
Other tax equity norms (ch. 15)
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E.g.,

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Ability to pay
Horizontal equity
Benefit principle
Sacrifice principle
Definitions as norms (e.g., use of Haig-Simons definition of
“income”)
Criticism

Need root in social welfare

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Vague, incoherent
Conflict with Pareto principle
Possible value as proxy principles
41