chapter 15 income taxation. reading essential reading –hindriks, j and g.d. myles intermediate...
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Chapter 15Income Taxation
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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))
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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
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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
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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
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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
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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
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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
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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
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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}
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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)
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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
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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
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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)
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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
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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
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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)
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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