Public Economics

1
Chapter 15 Income Taxation
2
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, 1540.
• 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, 551572.
• Hindriks, J. (2001) Is there a demand for income
  tax progressivity?, Economics Letters, 73,
  4350.
• Kanbur, S.M.R. and M. Tuomala (1994) Inherent
  inequality and the optimal graduation of marginal
  tax rates, Scandinavian Journal of Economics,
  96, 275282.

3
Reading

• Myles, G.D. (2000) On the optimal marginal rate
  of income tax, Economics Letters, 66, 113119.
• Romer, T. (1975) Individual welfare, majority
  voting and the properties of a linear income tax,
  Journal of Public Economics, 7, 163168.
• Roberts, K. (1977) Voting over income tax
  schedules, Journal of Public Economics, 8,
  329340.
• 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, 8395.
• Mirrlees, J.A. (1971) An exploration in the
  theory of optimum income tax, Review of Economic
  Studies, 38, 175208.

4
Reading

• Seade, J.K. (1977) On the shape of optimal tax
  schedules, Journal of Public Economics, 7,
  203235.
• Saez, E. (2001) Using elasticities to derive
  optimal tax rates, Review of Economic Studies,
  68, 205229.
• Weymark, J.A. (1986) A reduced-form optimal
  income tax problem, Journal of Public Economics,
  30, 199217.

5
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

6
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

7
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 lt 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

8
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 consumers indifference curves depend upon
  the wage rate

9
Taxation and Labor Supply

• 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

Figure 15.1 Labor supply decision
10
Taxation and Labor Supply

• 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

Figure 15.2 Effect of a wage increase
11
Taxation and Labor Supply

• Income z in Figs. 15.3a and b is a threshold
  level of income below which income is untaxed
• 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

Figure 15.3 A tax threshold
12
Taxation and Labor Supply

• 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

Figure 15.4 Several thresholds
13
Taxation and Labor Supply

• 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

Figure 15.5 Taxation and the participation
decision
14
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

15
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

16
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

17
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

Table 15.1 Labor-supply elasticities Source
Blundell (1992)
18
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

19
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

20
Optimal Income Taxation

• The government is subject to two constraints when
  it chooses the tax function
• The income tax must meet the governments 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

21
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))

22
Optimal Income Taxation

• Fig. 15.6 illustrates the budget constraint
• Without taxation the budget constraint is the 45o
  line
• T(z) lt 0 when the consumption function is above
  the 45o line
• T(z) gt 0 when the consumption function is below
  the line
• The gradient of the consumption function is 1 T'

Figure 15.6 Taxation and the Consumption
function
23
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 gt s1
• This is shown in Fig. 15.7
• Consumers of lower skill are less willing to
  supply labor

Figure 15.7 Agent monotonicity
24
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 high-skill must be further to
  the right
• Income is increasing with skill

Figure 15.8 Income and skill
25
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) gt 0 so 1 T'(z) gt 0
• The marginal tax rate is less than 100 percent

Figure 15.9 Upper limit on tax rate
26
Optimal Income Taxation

• Fig. 15.10 shows the marginal tax rate must be
  positive
• Start with c1 with c1' gt 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

Figure 15.10 Lower limit on tax rate
27
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

Figure 15.11 Zero marginal rate of tax
28
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

29
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 minU

30
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

31
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
• The change raises welfare
• The high-skill must be indifferent between al and
  ah

Figure 15.12 Allocations and the consumption
function
32
Two Specializations

• The resource constraint xl xh zl zh and the
  incentive compatibility condition can be solved
  to give
• zl (1/2)xl xh shu(xh)
  u(xl)
• zh (1/2)xl xh shu(xh)
  u(xl)
• Using these the optimal allocation of consumption
  for utilitarian social welfare solves
• max blu(xl) bhu(xh) (slsh)/2slshxl
  xh
• Where bl (3sl sh)/2sl and bh (slsh)/2sl

33
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
  highest-skilled
• For the low skill u'(xl) (slsh)/sh(3sl sh)
  so MRSl sh (3sl sh)/sl(slsh) lt 1
• The low-skill faces a positive marginal rate of
  tax

34
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

35
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

36
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

37
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
• The social welfare function is utilitarian if e
  0
• Higher values of e give more concern for equity

38
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)

39
Numerical Results

• Tab.15.2 presents the optimal tax rates for a
  utilitarian welfare function
• The average rate of tax is negative for the
  low-skilled 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

Table 15.2 Utilitarian case (e 0)
40
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

Table 15.3 Some equity considerations (e 1)
41
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

42
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

43
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)

44
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

Figure 15.13 Differetial taxation and
nonseparability
45
Voting over a Flat Tax

• The political process determine the tax system
  through voting
• Assume skills are distributed with cumulative
  distribution F(s), mean 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

46
Voting over a Flat Tax

• Given the budget constraint x 1 tz b the
  chosen income of a consumer with skill s is
• z(s) 1 ts2
• The government budget constraint is
• b tE(z(s)) t1
  tE(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)

47
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