Optimal Income Taxation When Top
Income Inequality Increases
Matti Tuomala
22.9.2016
SIEP, LECCE
Taxes and Top income inequality
• At the same time with a large growth in top
income shares over the past few decades
there has been in many advanced
countries a shift in the burden of taxation
from the top to the further down in the
income distribution.
• Piketty et al. (2011,2014) find in OECD
countries a strong correlation between tax
cuts for the highest earners and increases
in the income share of the top 1 per cent
since 1975.
The World Wealth and Income Database
(http://www.wid.world, 1.4.2016)
Top tax rates 1967–2010). Data source: Piketty, Saez and Stantcheva (2011; 2014).
Top marginal rates in the Mirrlees framework
• How high should the top marginal tax rate be?
(Top marginal tax rate = marginal tax rate
applying to the highest incomes)
• MTR=0 at top; unfair and/or irrelevant!
• Only holds for the person with the very highest
income level and must be able to identify that
person
• Note: amount of revenue raised depends on
average tax not marginal tax rate although
MTR=0, average tax rates can be high
Revenue maximizing (or maximin) top income
tax rate
• t = 1/(1+ αe)
• e = the net-of-tax rate (1-t) increases by 1%,
income z increases by e%
• The Pareto upper tail: the mean zm above z* is
α /(α -1)z* so that α=zm/(zm-z*)
• Intuition: higher α (Pareto parameter) implies
lower tax rates, and conversely
• Example: if e=0,5 and α=2, t = 50%
What is missing from the analysis
above?
• There are good reasons to suspect that the labour
market of top income earners deviates from the standard
competitive model in a number of important respects.
They are not paid their marginal product.
• Persson and Sandmo (2005) propose a “tournament”
model.
• If the economic activity at high incomes is primarily
socially unproductive rent-seeking e.g, Piketty, Saez and
Stantcheva (2014), Rothschild and Scheuer 2012,
Lockwood, Nathanson, and Weyl (2012)
• Relativity or externality effect, e.g.Kanbur-Tuomala (2013
Sources of top incomes and
implications for tax design
• The joint taxation of labour and capital income: earning
capability n and inherited wealth ω, leading to observed
capital income k(n,ω) and labour income z(n,ω).
• How to design T(z,k)?
• Should we separate taxes on labour income and capital
income as in a Nordic dual income tax: Tz(z) + Tk(k)?
• Can the optimum be attained with an income tax: T(z +
k)?
• Note:until 1970s, top tax rates on capital income often
higher than top tax rate on labour income
Marginal tax rates in Finland
70
50
40
30
20
10
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
0
1990
Marginal tax rate, %
60
Marginal tax rate for income subject to taxation (earned income)
Marginal tax rate for capital income
Marginal tax rate for income subject to taxation (earned and capital income)
Capital income share and average tax rate for
top 1 per cent in Finland 1967-2012
Income share and average tax rate
70
60
50
40
30
20
10
0
1985
1990
1995
2000
Capital income share
2005
2010
Average tax rate
2015
Pareto parameters for labour income, capital
income and total income in Finland 1990-2014
4,5
4,0
Income subject to
taxation: earned
income
3,5
2,5
Income subject to
taxation: capital
income
2,0
1,5
Income subject to
taxation: total income
1,0
0,5
0,0
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
zm/(zm-z*)
3,0
Separable schedules in labour and
capital income: T(z) + T(k) Maximin case
• As in Christiansen-Tuomala (2008) we consider
a two-period model wherein an individual starts
out with the endowment ω.
• Individuals are free to divide their first period
(when young) endowment between
consumption, and savings.
• Each unit of savings yields a consumer
additional units of consumption in the second
period.
Separable schedules in labour and capital income: T(z) + T(k)
Maximin case with quasi-linear preferences
•
•
•
•
•
•
•
•
•
(I) The pattern of marginal labour income tax rates
t/(1-t)=(1/ez)[(mθ+zθ)/(θzθ)]
The last term; Champernowne distribution (m,θ)
ez = the elasticity of taxable labour income
(II) The pattern of marginal capital income tax rates
t/(1-t)=(1+1/ek)(1/αk)
Pareto parameter αk
ek=the elasticity of taxable capital income
i.e. linear tax
Champernowne distribution, Finland
f(z)=θ{mθ zθ-1/(mθ+ zθ)2}, where θ is a shape or ”Pareto”
parameter
3.0
Pareto coefficient θ
2.8
2.6
2.4
2.2
2.0
1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010
Maximin
marginal labour income tax rates (%), the
Champernowne distribution with different θ
є=1/3
є=1/3
є=1
є=1
F(n)
θ=2
θ=3
θ=2
θ=3
0.10
94.7
93.3
89.2
86.7
0.50
81.2
74.1
66.6
56.9
0.90
70.4
61.4
52.3
42.3
0.99
67.0
58.2
48.4
39.5
0.999
58.7
52.2
39.9
33.5
Maximin marginal capital income tax rates (%) with
Pareto distribution
є =1/3
є=1
є =1/3
є=1
α=1.5
α=1.5
α =2
α=2
72,7
57.1
66.6
50
Top income inequality and
average tax rates
• The top marginal rates are in itself
important.
• The more relevant tax rates for
redistributive purposes are average tax
rates.
• The computational techniques can be used
to say something about average rates.
The optimal labour income tax in Mirrlees
model with additive utility and income effects
• We should look at the whole schedule.
• For simplicity we assume that the utility
function is additive u=U(x)+V(1-z/n)
defined over consumption x and hours
worked y=z/n, with Ux>0 and Vy<0
(subscripts indicating partial derivatives).
• Individuals differ only in the pre-tax wage n
they can earn
Specifications
• (I) Individual preferences: the constant elasticity of substitution form:
the elasticity of substitution between consumption and leisure =0.5.
• U=-1/x -1/1-y
• (II) Social objectives
W(u)=-(1/β)e-βu
where β measures the degree of inequality aversion in the social
welfare function of the government (in the case of β=0 , we define
W=u).
• (III) Distribution of n
The Champernowne-Fisk distribution
f(n)=θ{mθ nθ-1/(mθ+ nθ)2}, where θ is a shape or ”Pareto” parameter,
The shape parameter θ in itself is an appropriate measure for
top income inequality.
• The lower θ means greater inequality.
Average tax rate curves
100
50
ATR%
0
0
10
20
30
40
50
60
70
80
90
100
θ=3.3
θ=2.5
θ=2.0
-50
-100
-150
F(n)
Average tax rate: Utilitarian, f(n)=Champernowne
distribution holding mean constant m=e-1
Average tax rate curves
100
50
0
ATR%
0
10
20
30
40
50
60
70
80
90
100
θ=3.3
θ=2.5
-50
θ=2.0
-100
-150
-200
F(n)
Average tax rate: Maximin,
f(n)=Champernowne
distribution holding mean constant m=e-1
100
50
0
ATR%
0
10
20
30
40
50
60
70
80
90
100
θ=3.3
θ=2.5
-50
θ=2.0
-100
-150
-200
F(n)
Average tax rate: Rank order preferences, f(n)=Champernowne
distribution holding mean constant m=e-1
Marginal and average tax rate curves
100
80
60
40
%
20
0
-20
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
1,2
-40
-60
-80
-100
z
Marginal and average tax rates: f(n)=Champernowne
distribution, u2,X/Z=0.9 and maximin β=∞
Marginal and average tax rate curves
100
50
0
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
1,2
%
-50
-100
-150
-200
-250
z
Marginal and average tax rates: f(n)=Champernowne distribution
holding mean constant m=e-1 , θ =2.5, ,X/Z=0.9 and utilitarian
Marginal and average tax rate curves
100
50
0
%
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
1,2
-50
-100
-150
-200
z
Marginal and average tax rates:f(n)=Champernowne distribution holding
mean constant m=e-1 , θ =2.5, ,X/Z=0.9 and maximin β=∞
Marginal and average tax rate curves
100
50
0
%
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
1,2
-50
-100
-150
z
Marginal and average tax rates: f(n)=Champernowne distribution
holding mean constant m=e-1 , θ =2.5, X/Z=0.9 and Rank order
preferences
Average tax rate curves
60
40
20
0
ATR%
1
10
20
30
40
50
60
70
80
90
99
-20
β=0
-40
β=∞
-60
-80
-100
-120
F(n)
Average tax rate: Utilitarian and maximin cases, f(n)=Champernowne
distribution θ=3,3 and holding mean constant m=e-1
Concluding remarks
• Numerical results suggest that the shift in tax
burden seen in many countries cannot be justified
in the Mirrlees model, which embodies
conventional assumptions about inequality
aversion and the trade of between equity and
efficiency.
• Figures show how average tax rates decline
among the bottom 70 percent when pre-tax
inequality increases.
• The main lesson for the tax policy is that we need
more redistributive taxation, to tax the highest
incomes more heavily.
Concluding remarks
• So large has been the increase in the share of
income going to the top 1 per cent that taxing
them more heavily would only return them to the
level of earnings they had a some decades ago.
• It is hardly so that slightly higher taxes will
drastically reduce labour supply among highearners.
• But we will need to deal with tax avoidance and
evasion, which is the more serious problem
when taxing the rich more heavily