Chapter 15 Income Taxation. Reading Essential reading –Hindriks, J and G.D. Myles Intermediate...

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


• 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


• 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


• 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


• 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


a

c

Figure 15.2: Effect of a wage increase


• 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


a

c

b

*z

Figure 15.3: A tax threshold


• 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



• 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


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


• 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


• 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


• 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))


• 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


• 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


• 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


• 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


• 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


• 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


• 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)


• 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


• 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)


• 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

Chapter 15 Income Taxation. Reading Essential reading –Hindriks, J and G.D. Myles Intermediate...

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