chapter 15 income taxation. reading essential reading –hindriks, j and g.d. myles intermediate...
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Chapter 15Income 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
• 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
Consumption
Before tax incomeb. Before tax income
*z
ztpx 1
*x
Figure 15.1: Labor supply decision
Consumption
Leisurea. Leisure
p
wLt1
*x
*L L
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
Consumption
Leisurea. Leisure
a
b
c
Consumption
Before tax incomeb. Before tax income
a
c
Figure 15.2: Effect of a wage increase
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
Consumption
Leisurea. Leisure
a
b
c
wzL *Consumption
Before tax incomeb. Before tax income
a
c
b
*z
Figure 15.3: A tax threshold
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
Consumption
Leisurea. Leisure
Consumption
Before tax incomeb. Before tax income
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
mT
Consumption
Leisurea. Leisure
T
mw
Consumption
Before tax incomeb. Before tax income
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
-0.52-0.18-0.36-0.98-0.22-0.45Income
1.280.650.130.950.650.90Compensated wage
0.760.53-0.230.030.430.45Uncompensated wage
UKUSUKUSUKUS
Lone MothersMarried MenMarried Women
-0.52-0.18-0.36-0.98-0.22-0.45Income
1.280.650.130.950.650.90Compensated wage
0.760.53-0.230.030.430.45Uncompensated wage
UKUSUKUSUKUS
Lone MothersMarried MenMarried Women
Table 15.1: Labor-supply elasticitiesSource: 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′
z
x
o45
Tzc
z
zT ˆx
Figure 15.6: Taxation and the Consumption function
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 z
x
z
x
Low-skill
High-skill
Figure 15.7: Agent monotonicity
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 z
x
Low-skillHigh-skill
a
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
z
x
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
z
x
1c
2c1h
2h
1l2l
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
z
x
A
B
C
D
b
o45
Figure 15.11: Zero marginal rateof 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– The change raises welfare
• The high-skill must be indifferent between al and ah
z
x
Low-skill
High-skill
laha
Figure 15.12: Allocations and theconsumption 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 lu(xl) + hu(xh) – [(sl+sh)/2slsh][xl + xh]
• Where l = (3sl – sh)/2sl and h = (sl+sh)/2sl
Two Specializations
• The welfare weights l and h 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) = (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 T′ at income z
• Tax payments increase from all those with income above z
• Holding labor supply constant the total increase is [1 – G(z)]zT′
• The tax increase reduces labor supply and leads to a revenue loss g(z)T′zsT′/(1 – T′) where s is the elasticity of labor supply
Two Specializations
• At the optimum the gain must equal the loss [1 – G(z)]zT′ = g(z)T′zsT′/(1 – T′) • Solving this equation
T′/(1 – T′) = [1 – G(z)]/sg(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)]/sg(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
• The social welfare function is utilitarian if = 0• Higher values of give more concern for equity
0,1
0
dsseW U
0,0 dssU
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 = 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 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 ( = 0)
16150.430.50
18140.340.40
19130.260.30
2190.180.20
24-50.100.10
26-340.070.055
23-0.030
Marginal tax (%)
Average tax (%)
ConsumptionIncome
16150.430.50
18140.340.40
19130.260.30
2190.180.20
24-50.100.10
26-340.070.055
23-0.030
Marginal tax (%)
Average tax (%)
ConsumptionIncome
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
20170.410.50
22160.340.40
25130.260.30
2870.190.20
32-340.120.10
34-660.080.05
30-0.050
Marginal tax (%)
Average tax (%)
ConsumptionIncome
20170.410.50
22160.340.40
25130.260.30
2870.190.20
32-340.120.10
34-660.080.05
30-0.050
Marginal tax (%)
Average tax (%)
ConsumptionIncome
Table 15.3: Some equityconsiderations ( = 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
2x
1hI
I
1x
p'p
2hI
Figure 15.13: Differetial taxationand 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 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
s
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) – sm
2)
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