Chapter 15
Income Taxation
Reading
• Essential reading
– Hindriks, J and G.D. Myles Intermediate Public Economics.
(Cambridge: MIT Press, 2005) Chapter 15.
• Further reading
– Blundell, R. (1992) ‘Labour supply and taxation: a survey’, Fiscal
Studies, 13, 15—40.
– Feldstein, M. (1995) ‘The effect of marginal tax rates on taxable
income: a panel study of the 1986 tax reform act’, Journal of
Political Economy, 103, 551—572.
– Hindriks, J. (2001) ‘Is there a demand for income tax
progressivity?’, Economics Letters, 73, 43—50.
– Kanbur, S.M.R. and M. Tuomala (1994) ‘Inherent inequality and
the optimal graduation of marginal tax rates’, Scandinavian
Journal of Economics, 96, 275—282.
Reading
– Myles, G.D. (2000) ‘On the optimal marginal rate of income tax’,
Economics Letters, 66, 113—119.
– Romer, T. (1975) Individual welfare, majority voting and the
properties of a linear income tax, Journal of Public Economics, 7,
163—168.
– Roberts, K. (1977) ‘Voting over income tax schedules’, Journal of
Public Economics, 8, 329—340.
– Tuomala, M. Optimal Income Tax and Redistribution. (Oxford:
Clarendon Press, 1990) [ISBN 0198286058 hbk].
• Challenging reading
– Diamond, P.A. (1998) ‘Optimal income taxation: an example with
a U-shaped pattern of optimal marginal tax rates’, American
Economic Review, 88, 83—95.
– Mirrlees, J.A. (1971) ‘An exploration in the theory of optimum
income tax’, Review of Economic Studies, 38, 175—208.
Reading
• Seade, J.K. (1977) ‘On the shape of optimal tax schedules’, Journal
of Public Economics, 7, 203—235.
• Saez, E. (2001) ‘Using elasticities to derive optimal tax rates’,
Review of Economic Studies, 68, 205—229.
• Weymark, J.A. (1986) ‘A reduced-form optimal income tax problem’,
Journal of Public Economics, 30, 199—217.
Income Taxation
• Income taxation is a major source of government
revenue
• It is also a major source of contention
– The income tax is a disincentive to effort and
enterprise so the rate of tax should be kept as low as
possible
– Income taxation is well-suited to the task of
redistribution which requires that high earners pay
proportionately more tax on their incomes than low
earners
• The determination of the optimal income tax
involves the resolution of these contrasting
views
Taxation and Labor Supply
• The effect of income taxation on labor supply
can be investigated using the standard model of
consumer choice
• This highlights the importance of competing
income and substitution effects
• Assume
– The consumer has a given set of preferences over
allocations of consumption and leisure
– The consumer has a fixed stock of time to divide
between labour supply and leisure
• The choice is made to maximize utility
Taxation and Labor Supply
• Preferences are represented by
U = U(x, L - ℓ) = U(x, ℓ)
• L is the stock of time, ℓ is labor supply, and x is
consumption
– Leisure time is L - ℓ
•
•
•
•
Labour is assumed unpleasant so ∂U/∂ℓ < 0
Each hour of labour earns wage w
Income before taxation is wℓ
With tax rate t the budget constraint is
px = (1 – t)wℓ
Taxation and Labor Supply
• Alternatively the preferences of the consumer
can be be written in terms of income
• Let z ≡ wℓ denote income before tax
• Utility in terms of income is
U = U(x, z/w)
• The budget constraint becomes
px = (1 - t)z
• The consumer’s indifference curves depend
upon the wage rate
Taxation and Labor Supply
Consumption
• Fig. 15.1a depicts the
choice between leisure
and consumption
• The budget constraint
depends on the wage
• Fig. 15.1b depicts the
choice between before
tax income and
consumption
• The indifference curves
depend on the wage
• In both cases the budget
constraint depends on the
tax rate
1  t wL
p
x*
Consumption
L  *
a. Leisure
L Leisure
px  1  t z
x*
Before tax income
z*
b. Before tax income
Figure 15.1: Labor supply decision
Taxation and Labor Supply
Consumption
• The initial choice is at a
• In Fig. 15.2a an increase
in w shifts the budget
constraint
• In Fig. 15.2b an increase
in w shifts the indifference
curve
• The choice moves to c
– a to b is the substitution
effect
– b to c is the income effect
• The total effect can raise
or lower labor supply but
increases income
c
b
a
Leisure
a. Leisure
Consumption
c
a
Before tax income
b. Before tax income
Figure 15.2: Effect of a wage increase
Taxation and Labor Supply
Consumption
• Income z* in Figs. 15.3a
and b is a threshold level
of income below which
income is untaxed
c
b
– The budget constraint is
kinked at b
• Points a and c are interior
solutions
• Point b is a corner
solution
• A consumer at a corner
may be unaffected by a
tax change
a
L z* w
a. Leisure
Consumption
Leisure
c
b
a
Before tax income
z*
b. Before tax income
Figure 15.3: A tax threshold
Taxation and Labor Supply
Consumption
• For many tax systems the
marginal rate of tax has
several discrete
increases
• Figs 15.4a and b display
the case of four marginal
rates
• The marginal rates
increase with income
• The budget constraint is
kinked at each point of
increase
Consumption
a. Leisure
b. Before tax income
Leisure
Before tax income
Figure 15.4: Several thresholds
Taxation and Labor Supply
Consumption
• It may not be possible to
continuously vary hours
of work
• A minimum working week
gives a choice between 0
hours and the minimum
ℓm
• This causes a
discontinuity in the
budget constraint
• Figs. 15.5a and b show a
discontinuity in labor
supply as the tax rate
changes
Consumption
T  m T
a. Leisure
Leisure
w m
Before tax income
b. Before tax income
Figure 15.5: Taxation and the
participation decision
Empirical Evidence
• The theoretical analysis of labor supply makes
three major points
– The effect of a wage or tax change depends on
income and substitution effects
– Kinks in the budget constraint can make behaviour
insensitive to taxes
– The participation decision can be sensitive to taxation
• The theory does not predict the size of these
effects
• Empirical evidence is required to provide
quantification
Empirical Evidence
• Evidence on the effect of income taxes can be
found in
– The results of taxpayer surveys
– Econometric estimates of labor supply functions
• Two points are important in choosing s survey
sample
– Labor supply is insensitive to taxation if working hours
are determined by the firm or by union/firm agreement
– The effect of taxation can only be judged when
workers who have the freedom to vary hours of labor
Empirical Evidence
• Surveys usually conclude that changes in the tax
rate have little effect on the labor supply decision
• If correct the labor supply function is
approximately vertical
– This results from the income effect almost entirely
offsetting the substitution effect
– This predicts taxation will have little labor supply
effect
• Different groups in the population may have
different reactions to changes in the tax system
• This is now considered by reviewing some
econometric analysis
Empirical Evidence
• Tab. 15.1 reports
estimates of labor supply
elasticities for three
groups
• The substitution effect
(compensated wage) is
positive but the income
effect is always negative
• The elasticity for married
men is the lowest
• The elasticity for
unmarried women is the
largest
– Participation effect
Married Women Married Men
Lone Mothers
US
UK
US
UK
US
UK
Uncompensated
wage
0.45
0.43
0.03
-0.23
0.53
0.76
Compensated
wage
0.90
0.65
0.95
0.13
0.65
1.28
Income
-0.45
-0.22
-0.98
-0.36
-0.18
-0.52
Table 15.1: Labor-supply elasticities
Source: Blundell (1992)
Optimal Income Taxation
• The optimal income tax balances efficiency and
equity to maximise welfare
• A interesting model must have the following
attributes:
– An unequal distribution of income so there is an
equity motivation for taxation
– The income tax must affect labor so that it has
efficiency effects
– There must be no restrictions placed on the optimal
tax function
• The Mirrlees model of income taxation is the
simplest that has these attributes
Optimal Income Taxation
• All consumers have identical preferences but
differ in their level of skill
• The level of skill determines the hourly wage
• Income is the product of skill and hours worked
• The level of skill is private information and
cannot be observed by the government
– This makes it impossible to tax directly.
– A tax levied on skill would be the first-best policy but
this not feasible
• The government employs an income tax as a
second-best policy
Optimal Income Taxation
• The government is subject to two constraints
when it chooses the tax function
– The income tax must meet the government’s revenue
requirement
– The tax function must be incentive compatible
• View the government as assigning to each
consumer an allocation of labor and
consumption
• Incentive compatibility requires that each
consumer must find it utility maximizing to
choose the allocation intended for them
– No alternative allocation should be preferred
Optimal Income Taxation
• If a consumer of skill level s supplies ℓ hours of
labour they earn income of sℓ before tax
• Denote the income of a consumer with skill s by
z(s)
• For a consumer with income z the income tax
paid is given by T(z)
– T(z) is the tax function the analysis aims to determine
• A consumer who earns income z(s) can consume
x(s) = c(z(s)) = z(s) – T(z(s))
Optimal Income Taxation
• Fig. 15.6 illustrates the
budget constraint
• Without taxation the
budget constraint is the
45o line
• T(z) < 0 when the
consumption function is
above the 45o line
• T(z) > 0 when the
consumption function is
below the line
• The gradient of the
consumption function is 1
– T′
x
T  zˆ 
xˆ
c  z T
45o
zˆ
Figure 15.6: Taxation and the
Consumption function
z
Optimal Income Taxation
• Preferences are assumed
to satisfy the agent
monotonicity condition
• At any point (z, x) the
indifference curve of a
consumer of skill s1 is
steeper than the curve of
a consumer of skill s2 if s2
> s1
• This is shown in Fig. 15.7
• Consumers of lower skill
are less willing to supply
labor
x
High-skill
Low-skill
xˆ
zˆ
Figure 15.7: Agent monotonicity
z
Optimal Income Taxation
• Fig. 15.8 shows the
consequence of agent
monotonicity
• The low-skill consumer
chooses a
• The indifference curve of
the high-skill is flatter and
cannot be at a tangency
• The choice for the highskill must be further to the
right
• Income is increasing with
skill
x
Low-skill
High-skill
a
z
Figure 15.8: Income and skill
Optimal Income Taxation
• Consider the
consumption function in
Fig. 15.9
• No consumer will locate
on the downward-sloping
section
• This part of the
consumption function can
be replaced by the flat
dashed section
• This shows c′(z) > 0 so 1 –
T′(z) > 0
– The marginal tax rate is
less than 100 percent
x
z
Figure 15.9: Upper limit on tax rate
Optimal Income Taxation
• Fig. 15.10 shows the
marginal tax rate must be
positive
• Start with c1 with c1′ > 1
and move to c2 with c2′ = 1
– c2 chosen so tax revenue is
unchanged
• High-skill moves from h1
to h2, low-skill from l1 to l2
– Consumption is transferred
from high skill to low skill
so welfare rises
• c1 could not be optimal
x
c1
h1
h2
c2
l2
l1
z
Figure 15.10: Lower limit on tax rate
Optimal Income Taxation
• The highest-skill
consumer should face a
zero marginal rate of tax
• In Fig. 15.11 ABC does
not have this property
• Replace with ABD where
BD has gradient of 1
– Highest-skill consumer
moves to b
– Utility rises but tax payment
is unchanged
– No-one is worse-off
• ABC cannot be optimal
x
D
b
C
B
A
45 o
z
Figure 15.11: Zero marginal rate
of tax
Optimal Income Taxation
• A tax system is progressive if the marginal rate
of tax increases with income
– A zero rate at the top shows progressivity cannot be
optimal
– Most tax systems are progressive
• This result is valid only for the highest-skill
consumer
– The implications for lower skills are limited
– Observed systems may only be ‘wrong’ at the very
top
• Result questions preconceptions about the
structure of taxes
Two Specializations
• There are two specializations of the general
model that provide additional insight
• The quasi-linear model restricts the form of the
individual utility function
• The individual utility function becomes
U = u(x) – z/s
• Rawlsian taxation adopts a specific social
welfare function
• Social welfare is evaluated by
W = min{U}
Two Specializations
• Assume there are just two consumers
– The high-skill is sh and the low-skill sl
• The optimal tax problem is equivalent to
choosing the allocations ah and al for these
consumers
• Incentive compatibility requires that the
consumer of skill i prefers allocation i
• The low-skill will never mimic the high-skill so
only one incentive compatibility constraint is
binding
u(xh) – zh/sh = u(xl) – zl/sh
Two Specializations
• Fig. 15.12 illustrates the
role of allocations
• The allocations al and ah
are incentive compatible
• These cannot be optimal
since xh can be reduced
and xl raised without
violating incentive
compatibility
x
High-skill
Low-skill
al
ah
– The change raises welfare
• The high-skill must be
indifferent between al and
ah
z
Figure 15.12: Allocations and the
consumption function
Two Specializations
• The resource constraint xl + xh = zl + zh and the
incentive compatibility condition can be solved to
give
zl = (1/2)[xl + xh – sh[u(xh) – u(xl)]]
zh = (1/2)[xl + xh + sh[u(xh) – u(xl)]]
• Using these the optimal allocation of
consumption for utilitarian social welfare solves
max blu(xl) + bhu(xh) – [(sl+sh)/2slsh][xl + xh]
• Where bl = (3sl – sh)/2sl and bh = (sl+sh)/2sl
Two Specializations
• The welfare weights bl and bh incorporate
incentive compatibility and resource implications
• For the high-skill the solution to the optimization
is u′(xh) = 1/sh so that MRSh = 1
– This captures the zero marginal rate for the highestskilled
• For the low skill u′(xl) = (sl+sh)/sh(3sl – sh) so MRSl
= sh (3sl – sh)/sl(sl+sh) < 1
– The low-skill faces a positive marginal rate of tax
Two Specializations
• Rawlsian taxation aims to maximize the utility of
the worst-off
• Assume all tax revenue is redistributed as a
lump-sum grant
• It can then be assumed that the optimal
Rawlsian tax maximizes the grant
• A consumer of skill s earns income z(s) so z-1(s) is
the skill level associated to each income
• If F(s) is the cumulative distribution of skill then
G(z) = F(z-1(s)) is the cumulative distribution for
income
Two Specializations
• Since revenue is maximized any small change in
the tax function must have no effect on revenue
• Consider a increase in the marginal rate of DT′ at
income z
• Tax payments increase from all those with
income above z
• Holding labor supply constant the total increase
is [1 – G(z)]zDT′
• The tax increase reduces labor supply and leads
to a revenue loss g(z)T′zesDT′/(1 – T′) where es is
the elasticity of labor supply
Two Specializations
• At the optimum the gain must equal the loss
[1 – G(z)]zDT′ = g(z)T′zesDT′/(1 – T′)
• Solving this equation
T′/(1 – T′) = [1 – G(z)]/esg(z)
• This implies the marginal tax rate (T′) will be high
at income z when
– The labor supply elasticity is low
– There are few taxpayers with income z
• Even for Rawlsian taxation there will not be
progressivity unless [1 – G(z)]/esg(z) increases in
z
Numerical Results
• The theory describes some characteristics of the
optimal income tax function
• A numerical analysis is required to generate
more precise results
• Numerical results employ the social welfare
function
 1 eU
W  0 e  s ds, e  0
e

 0 U s ds , e  0
• The social welfare function is utilitarian if e = 0
• Higher values of e give more concern for equity
Numerical Results
• The density function for the skill distribution is
given by f(s)
• This is assumed to be log-normal with a
standard deviation of s = 0.39
– This value is similar to that for observed income
distributions
– But skill and income may not have the same
distribution
• The individual utility function is Cobb-Douglas
U = log(x) + log(1 – ℓ)
Numerical Results
• Tab.15.2 presents the
optimal tax rates for a
utilitarian welfare function
• The average rate of tax is
negative for the lowskilled but increases with
skill
– The negative tax is an
income supplement
• The marginal tax rate first
rises with skill and then
falls.
– The maximum rate is
around the median of the
skill distribution
Income Consumption Average tax
(%)
Marginal tax
(%)
0
0.03
-
23
0.055
0.07
-34
26
0.10
0.10
-5
24
0.20
0.18
9
21
0.30
0.26
13
19
0.40
0.34
14
18
0.50
0.43
15
16
Table 15.2: Utilitarian case (e = 0)
Numerical Results
• The results in Tab. 15.3
involve a greater concern
for equity
• The average tax rate
starts lower but rises
higher
• The marginal tax rate is
higher for all income
levels
• The marginal rate is
highest at a low income
level
Income Consumption Average tax
(%)
Marginal tax
(%)
0
0.05
-
30
0.05
0.08
-66
34
0.10
0.12
-34
32
0.20
0.19
7
28
0.30
0.26
13
25
0.40
0.34
16
22
0.50
0.41
17
20
Table 15.3: Some equity
considerations (e = 1)
Numerical Results
• The outcome is a negative income tax with the
government supplementing income
• The maximum average rate of tax is low
• The marginal tax rate first rises with income and
then falls.
– The system is not marginal rate progressive
• The marginal rate of tax is close to constant
– The consumption function is almost a straight line
• The zero tax for the highest-skill consumer is
reflected in the fall of the marginal rate at high
incomes
Tax Mix: Separation Principle
• Governments use both income and consumption
taxes
• Chap. 14 showed that efficient commodity taxes
should be inversely related to the elasticity of
demand
– This implies a system of differential commodity
taxation
• The question to address now is the role of
differential commodity taxation when there is an
optimal nonlinear income tax
• The answer is dependent on the relation
between commodity demand and labor supply
Tax Mix: Separation Principle
• Recall that the success of the income tax is
limited by incentive compatibility
– The high-skill will mimic the low-skill
• Differential commodity taxes are justified if they
relax the incentive compatibility constraint
– This can be done by making the consumption bundle
of the low-skill less attractive to the high-skill
• If the utility function is separable between
consumption and labor incentive compatibility
cannot be relaxed
– Separable utility has the form U = U(u(x), ℓ)
Tax Mix: Separation Principle
• Fig. 15.13 displays
nonseparable
preferences
• Changing prices from p to
p′ makes the
consumption plan of the
low-skill less attractive to
the high-skill
• The utility of the low-skill
is not affected
• Incentive compatibility is
relaxed
x2
I h2 I h1
I
p
p'
x1
Figure 15.13: Differetial taxation
and nonseparability
Voting over a Flat Tax
• The political process determine the tax system
through voting
• Assume skills are distributed with cumulative
distribution F(s), mean s and median sm
• A vote is taken over a linear tax with lump-sum
benefit b and constant marginal tax rate t
• Consumer preferences are represented by the
quasi-linear utility function
U = x – (1/2)(z/s)2
Voting over a Flat Tax
• Given the budget constraint x = [1 – t]z + b the
chosen income of a consumer with skill s is
z(s) = [1 – t]s2
• The government budget constraint is
b = tE(z(s)) = t[1 – t]E(s2)
• Substituting for b and z in the utility function and
maximizing over t gives the optimal tax of the
median voter
tm = (E(s2) – sm2)/(2E(s2) – sm2)
Voting over a Flat Tax
• Using the choice of income the tax can be
written
tm = (E(z) – zm)/(2E(z) – zm)
• The model predicts the political tax rate is
determined by the position of the median voter
in the income distribution
• As income inequality rises (E(z) – zm increases)
the tax rate rises
• In practice median income is below mean
income so voters will vote for redistribution