ON THE OPTIMAL INCOME TAXATION Some further numerical

Journal of Public Economics 23 (1984) 351-366. North-Holland Publishing Company

ON THE OPTIMAL INCOME TAXATION

Some further numerical results

Matti TUOMALA*

University of Helsinki, SF-00100, Helsinki. Finland

Received June 1982, revised version received March 1983

Using numerical calculations we show in the optimal income tax model that a realistic value for the elasticity of substitution between consumption and leisure (a=0.5) leads to conclusions different from some of those drawn by Mirrlees (1971) and Atkinson (1973). The marginal tax rates are not so low and the shape of the tax schedule is for a great majority of the population substantially nonlinear. A new feature in our calculations is that both results come out without going to maximin, as done by Atkinson (1973). We study in special cases the locality of the zero limit of the marginal tax rate at the upper end of the ability distribution and we achieve a very definite conclusion: this result is really very local. We also present an interesting approximation result concerning the relationship between the level of the government revenue and the marginal tax rate for individuals with high ability. An important methodological remark analogous to Sen’s critique on the ethical measures of inequality is made on the interpretation of optimal income tax calculations.

1. Introduction

Since the seminal article on optimal income taxation by James Mirrlees (1971) there has been a series of analyses in this field.’ A striking feature of the optimal income tax literature is that there are only very few purely analytical results concerning the marginal tax structure. Roughly speaking there are, in the Mirrlees model, two analytical results; the marginal tax rates lies between 0 percent and 100 percent and it is zero at both ends of the distribution.’

It can be said that the very basic nature of income tax problems requires quantitative results. Mere general principles are not of much value. Thus, our

*This paper is part of my work on optimal income taxation for which James Mirrlees has provided most useful supervision. I am also grateful to Amartya Sen, Vidar Christiansen, the editor of this Journal and the referee for their kind comments. An earlier version of this paper was presented at the 1981 Winter Symposium of the Econometric Society (Copenhagen, Denmark). Financial support from Yrjo Jahnsson Foundation is gratefully acknowledged.

‘See Atkinson and Stightz (1980) for a survey. ‘This is true if all individuals are perfect substitutes.

0047-2727/84/$3.00 0 1984, El sevier Science Publishers B.V. (North-Holland)

352 M. Tuomala, Optimal income taxation

natural aim is to find in the optimal income tax models how to connect empirical measurements to specific numerical proposals. On the basis of numerical work done by Mirrlees (1971) and Atkinson (1973) in the nonlinear income tax model and by Stern (1976) in the linear case a number of conclusions have been drawn on the levels of tax rates and the shape of the tax schedule. The main conclusions in Mirrlees’ work can be seen as (a) the optimal tax schedule is approximately linear, (b) marginal tax rates are rather low, and (c) perhaps the most surprising feature of his numerical results is that the tax schedule is not progressive throughout in the sense of rising marginal tax rates. These results are based on utilitarian objectives.

Atkinson (1973), using the utility-based version of the Rawlsian difference principle, maximin, noticed that marginal tax rates are no longer so low and the tax schedule departs from linearity in the income range that covers the majority of the population. Both Mirrlees and Atkinson used in their calculations a log-linear utility function which implies the unitary elasticity of substitution between consumption and leisure. The effect on linear tax rates of relaxing the assumption of a log-linear utility function is examined by Stern (1976). He assumes that the utility function has a constant elasticity of substitution form. On the basis of empirical results Stern estimates that values of the elasticity of substitution between leisure and consumption, denoted E, around 0.5 are of interest. In fact, this corresponds to the portion of backward-bending labour supply curves on which labour supply elasticities are negative. Using this and other parameter values for E in solving the linear income tax model numerically, Stern finds that even with moderately egalitarian objectives, optimal tax rates are considerably higher than those found by Mirrlees.

One of our purposes is to find out how the above-mentioned conclusions achieved by Mirrlees and Atkinson change when 0.5 is used as an appropriate value for E. At the same time we can throw some new light on the question of progressivity in the marginal tax sense, which was not possible in Stern’s work.

Furthermore, we study by numerical methods how local the result is on the zero limit of the marginal tax rate at the top of the income range. We also present an interesting approximation result concerning the relationship between the level of the government revenue and the marginal tax rate for individuals with high ability. Finally, an important methodological remark analogous to Sen’s critique on the ethical measures of inequality is made on the interpretation of optimal income tax calculations.

The structure of the paper is as follows. In section 2 the optimal income tax model i la Mirrlees is recalled and the outline of the computational procedure is given. In section 3 numerical results are reported and discussed. Finally, there is a brief concluding section.

M. Tuomala, Optimal income taxation 353

2. The model

The original model is due to Mirrlees (1971).3 There are two commodities in this model, a consumption good and labour. Consumers are regarded as identical except that they vary in their ability to supply labour. This ability is assumed to be measured by the scalar n. Ability is distributed in the population with density function f(n). We assume f(n) > 0 for 121 n 5 ?i, where n> 0, tis Z. Utility is u(x, y), where x is consumption and y is labour, e.g. the time spent at work. It is assumed that u, > 0, u,~ <O and u is strictly concave. (Subscripts denote partial derivatives with respect to the appropriate arguments.)

Both producer prices, for goods and labour, will be set equal to unity. The wage rate by an n-individual is assumed to be n.4 Hence, his gross income z is hours worked times the marginal productivity of time in the job being done, i.e. z=ny. The tax schedule is represented by a function t(z). The budget constraint is then given by x =z-t(z).

Each individual chooses x, z and y so as to solve:

max 4x, Y)

s.t. x=z-L(Z), z=ny. (1)

When Y>O,~ the first-order characterization of the solution to (1) is:

u,x’ + u,/n = 0. G-7

Eq. (2) can be written x’=s, where s= -z+/nu, (= -u,/u,) is the marginal rate of substitution of consumption for labour.

The individual’s utility maximization condition (2) gives the trade-off between the two commodities faced by the individual at the margin. This is

‘The original paper was by Mirrlees (1971); a major extension is found in Mirrlees (1976) and (1977).

4A criticized feature of the standard optimal income tax model is that an individual’s hourly wage (=ability) is assumed to be independent of the tax policy. Feldstein (1973) relaxed this assumption but his numerical calculations with a linear income tax model changed only very little. Later, Allen (1982) showed that Feldstein’s results depend critically on the use of a Cobb- Douglas production function. Moreover, Allen (1982). Stern (1982), and Stightz (1982) using the same approach as Feldstein, have produced new qualitative results especially concerning the endpoint results.

51t can be shown that there is a critical n, such that:

y(n)=O, for nsn,,

y(n) > 0, for n > nW


an envelope condition for this economy, i.e. tangency of indifference curves with the budget constraint for each individual.

To simplify the government’s optimization problem x is expressed as a function of u and y. This inversion is permitted because u(n) is monotonically increasing in x(n). Denote x=h(u,y) and let du/dn=g(u,y). Differentiation with respect to y and u shows:

h, = - uy/u,, h, = l/u,:

g, = (n,,s + u,Jn = - nxsnr g, = %X/~X.6 (3)

After these preliminaries the explicit form of the envelope condition can be

calculated as follows:

duldn = - yu,/a7 (4)

The aggregate production constraint of this economy is:

X=/x(n)f(n)dn=M n

(5)

where M is assumed to be linear in our numerical examples. It is assumed that the government’s aims can be represented by a social

welfare function of the form:

S=jG(u)f(n)dn, ”

(6)

where S is an indicator of social welfare. In this model, the optimum is characterized by a pair of functions, u(n)

and y(n), which maximize social welfare (6) subject to the conditions of individual utility maximization, (4), and an overall resource constraint (5). Introducing Lagrange multipliers 2 and m(n) for the constraints (5) and (4) and integrating by parts the following first-order conditions are obtained:

(G’ - i/u,)f(n) - tl’ - au,&, = 0, (7)

‘Because u,<O, then du/dn>O. This means that.a person with higher n enjoys at least as high utility as a person with lower n. In other words, a person with higher n can always earn the same income as someone with lower n and at the same time enjoy more leisure.


(1 + ~J~%) = - ~u,s,l~f(n), (8)

along with the transversality conditions cc@ = a(E) = 0. Now (1 +u,/nu,) is just the marginal tax rate faced by an n-individual, so

(8) combined with the transversality conditions shows that the marginal tax rates will be zero at each end of the income distribution [see, Seade (1977)].8 An important requirement for these results is boundedness of income, boundedness away from zero, and infinity.

Furthermore, it can be shown that the marginal tax rate lies between zero and one [see, Mirrlees (1971)].’ By considering the first-order condition for individual choice, (2), we can see that at an interior solution x’>O, so that the marginal tax rate is less than 100 percent. Moreover, the sign of the marginal tax rate can be shown to be non-negative. To prove this it must be shown that a(n) 2 0. First, solving (7) for TV we obtain:

where

B,,=expj(u,,/u,)dn’zO. ”

CY(~) = 0 is used in deriving (9). ~$3) in turn implies that:

B,,f dm’=O.

(9)

(10)

We must have l/u,< G’/n in (9), since otherwise (9) would imply that a(n) falls continuously and hence violates the endpoint condition a(C) =O. Thus, IX(~) starts at zero, increases until l/u,= G’/1, and after that falls again to

‘First Phelps (1973) and then Sadka (1976) showed this result for the top person. Later, Seade (1977) showed it holds at both ends of distribution.

‘See also a recent paper by Seade (1982), where he clarifies and extends the result of the sign of the marginal tax rate.

“In the maximin case:

a(n)= -ib/drn (UXY = 0) _ x

and differentiation with respect to u(n,) yields:

[ 1 _ Md

--a(no) Mb) 1


zero. Hence, since a(n) 20, this implies with assumption of s, ~0’ 1 a positive marginal tax rate.

Now we can interpret formula (8). Roughly speaking it suggests that the marginal tax rate should be greater (a) the greater is the number of n’- individuals to n-individuals (n’>n), and (b) the greater is the difference between s,, and s,.

The result that there will usually be a group of people in the lower end of

the ability distribution ‘who ought to work only if they enjoy it’ [Mirrlees, (1971)] has also been regarded as qualitative. This result needs, however, some comments. Seade (1977) has noticed that the result depends on the value of -uy/nu,, the marginal rate of substitution of consumption for labour, when y=O. Two cases can be distinguished here. In the ‘bunching’ case, if the marginal rate of substitution remains strictly positive as y> 0, then there will be a group of people (O-n,,) who do not work provided that they face a non-negative marginal tax rate. In the ‘no-bunching’ case -u,./nu, =0 for any n>O so that if the marginal tax rates are less than one (100 percent), then all individuals with n>O prefer to work.

For numerical calculations it is assumed that the utility function is additive in consumption and hours. Also, it is useful to write (8) as a

differential equation as follows:

dr/dn = - r(2 + nf”/f’) +((G’/i.) -( l/u,))/n’, (11)

where

r= 1 + uJnu,

%a + Y%v

The formulae (11) and (4) form a pair of nonlinear differential equations in u and r. These two differential equations provide the solution to the optimal income tax problem together with the conditions (5) and a($=0.r2 This differential equation system can be solved only by numerical integration methods. In this study a fourth-order Runge-Kutta method is used. In eqs. (11) and (4) the unknowns are U, r and A. Instead of solving for i a value is assumed for it. The condition, cr(ti) = 0, makes the numerical solution difficult. A trial-and-error method is used to obtain a value n,, which enables us to satisfy the condition Y(n,) =O. Extra care was taken to obtain an accurate value for no and to satisfy cc(n) =O. The value for x at n, is solved from eq. (8) by using the Newton method. When i is given and x0 is solved, the integration of (11) and (4) can be started. When ui +’ and ri +I (i refers to an

“The assumption is equivalent to (B) in Mirrlees (1971, p. 182). ‘*In our calculations this transversality condition takes the form x+0. when n-+m because

we use the density function with infinite range.


iteration cycle) are obtained the new values for y and x are calculated from eq. (8) and from the utility function by using the Newton method. The range

of n was from n, to 1.5 at which point the integrated value off(n) was more than 0.9999.

3. Numerical results

The calculations were carried out for the case

u(x,Y)=(-l/x)-(1/(1-y)), all n, (12)

where a=OS and, for comparison with Mirrlees (1971) and Atkinson (1973), for the case

u(x, y) = log x + log (1 - y), all n, (13)

where a= 1.13

The social welfare function is the following concave transformation of each individual’s utility reflecting the government’s distributional preferences:14

(14)

where /? expresses the degree of inequality aversion. The results are given for three different forms of the social welfare function:

p =0 (in this case we define G= U) corresponding to a classical utilitarian case, /I= 1, and /I?= cc corresponding to the utility-based version of the Rawlsian difference principle (maximin), which attaches no weight at all to anybody’s well-being apart from the worst-off individual’s; and for three different tax revenue requirements R = 1 -X/Z1’ II is log-normally

i31t is known, proved by Stern (1976), that optimum taxation involves a marginal tax rate of 100 percent when the elasticity of substitution between consumption and leisure is zero.

14To use Sen’s terminology we can call this Mirrleesian approach ‘welfaristic’. Welfarism is the view that the goodness of a state of affairs can be judged entirely by the goodness of the utilities in that state. So utilitarianism and maximin are in this sense special cases of welfarism. Of course (14) does not cover all welfaristic cases. We could also take some other function of the utilities - other than the sum-total or the minimal element. For example, we could judge the welfare of society in terms of utility of the ‘median’ person in the utility ranking.

“We can interpret (1 -X/Z) as the proportion of GNP collected as government tax revenue. Government expenditure on public goods is approximately 10 percent of GNP in Finland [see Tuomala (1979)]. Thus, X/Z=O.9 might be viewed as an appropriate figure. If X/Z= 1.0, then taxation is purely redistributive. If X/Z= 1.1, then for instance the nationalized industries make positive profits.


distributed, where the mean ,u and the standard deviation of the distribution 6 are taken to be (- 1, 0.39) and (- 1, 1) in one case.

The optimum tax schedules computed for the model came out as shown in tables l-3 and in figs. 1 and 2. In tables l-3 we give n, and x0, the values of y, x, and z, the average tax rate, the marginal tax rate, and utility level at the 10, 50, 90, and 99 percent points of the ability distribution.

There are several features of these results that we would like to emphasize. First, we take up the question concerning the relationship between the marginal tax rate and the elasticity of substitution between consumption and leisure (E) in the nonlinear income tax model. It might be expected that the regressivity of Mirrlees’ results would disappear if lower labour responsiveness (lower E) were assumed. Yet, it will be seen that in all cases reported in tables l-3 the marginal tax rate falls as income increases, except at income levels within the bottom decile of the distribution. (The situation of the bottom decile can be seen only in figs. 1 and 2.) In other words, one of the basic features of earlier calculations [see, Mirrlees (1971) and Atkinson (1973)] is carried over in our calculations; the marginal tax rate is not monotonically increasing for most of the taxpayers; it is falling. In fact the reason why the numerical calculations yield a regressive marginal tax rate structure has come out already in the context of formula (8). Namely, if the income distribution thins out rapidly, marginal tax rates on high incomes

should be low because we have to take into account incentive effects. Although the marginal tax rate remains falling with income in the case of

s=OS, there are, however, substantial differences both in the levels of the marginal tax rates and in the general shape of the tax schedule between the (E = 0.5) case and Mirrlees’ Cob&Douglas examples. For example, we can see in the right-hand part of table 2, in the case where E =0.5, p= 1, and R = 0.1, that the marginal tax rates are very high in the middle and lower income ranges, median rates being over 60 percent. Thus, lower labour responsiveness would increase marginal tax rates and it would increase them by more in the middle and lower income range than at the upper end. For contrast, it is interesting to compare our results in the right-hand part of table 2 with Stern’s calculations. He found an optimum marginal tax rate of 43 percent for the linear case with e=0.5, v = - 1 (corresponding nearly to b= 1 in our calculations), and R =O. In our corresponding case the median marginal tax rate is 59 percent and at other income levels reported in table 2, except at the 99 percent point, the marginal tax rates are higher than 43 percent. What is surprising is that the differences are quite substantial. On the basis of our calculations it seems to be more plausible to conclude, as Stern did, that ‘an argument for very high tax rates must be based on the claim that E is very low rather than on extreme views of values (fi)‘. In fact we come to this conclusion in the case where E is not very low (s=OS) and the social objectives reveal quite mild egalitarian views (/?= 1). There is also

Tab

le

1

Uti

lita

rian

, w

ith

p =

0.”

F(n

)

Tax

rat

es (

%)

Tax

rat

es (

%)

Y

z x

Ave

rage

M

argi

nal

U

tili

ty

Y

z x

Ave

rage

M

argi

nal

U

tili

ty

&=

l B

=O

n

,=0.

04

x,=

0.03

F(

n,)=

O.O

X

/Z=

O.9

&

=O

S*

/I=

0 n

, =

0.0

07

x,, =

0.0

8 F(

n,)

=O

.O

XfZ

=

0.9

0.10

0.

42

0.09

0.

10

-2

25

- 2.

88

0.57

0.

13

0.15

-1

6 48

-9

.10

0.50

0.

46

0.17

0.

16

9 22

-

2.48

0.

57

0.21

0.

19

9 44

-

7.52

0.

90

0.48

0.

29

0.25

14

19

-

2.03

0.

56

0.34

0.

27

21

37

- 5.

98

0.99

0.

50

0.45

0.

39

15

16

-1.6

4 0.

55

0.50

0.

38

24

23

-4.8

4 .

Pop

ula

tion

av

erag

e 0.

18

0.17

0.

22

0.20

&=

l /?

=o

n,

= 0

.05

x0 =

0.0

4 F(

n,)=

O.O

X

/Z=

1.

0 &

=0.

5 /I

=0

n,

= 0

.009

x0

= 0

.09

F(n,

) =

O.O

x/

z=

1.0

0.10

0.

39

0.09

0.

11

- 19

23

-

2.74

0.

54

0.12

0.

16

-32

44

- 8.

42

0.50

0.

44

0.16

0.

16

-1

21

- 2.

39

0.55

0.

20

0.21

-2

41

-

6.98

0.

90

0.47

0.

29

0.26

8

18

-1.9

7 0.

54

0.33

0.

29

13

35

~ 5

.62

0.99

0.

50

0.45

0.

41

11

12

-1.5

9 0.

52

0.48

0.

39

19

29

-4.6

4

Pop

ula

tion

av

erag

e 0.

18

0.18

0.

22

0.22

&=

l B

=O

n

o =

0.0

6 x0

= 0

.05

F(n,

)=O

.O

X/Z

=

1.1

&=

0.5

/I=

0 n

,=0.

012

x,=

0.10

F(

n,)=

O.O

x/

z=

1.1

0.10

0.

36

0.08

0.

11

-42

21

-2.6

1 0.

52

0.12

0.

17

-46

41

- 7.

87

0.50

0.

42

0.15

0.

17

-12

19

- 2.

29

0.54

0.

20

0.22

-1

1 39

-

6.67

0.

90

0.46

0.

28

0.27

1

17

-1.9

0 0.

53

0.32

0.

30

7 33

-

5.45

0.

99

0.48

0.

44

0.41

7

15

- 1.

55

0.52

0.

47

0.41

13

23

-4

.51

Pop

ula

tion

av

erag

e 0.

17

0.19

0.

21

0.23

“6 =

0.39

.

Tab

le

2

Util

itari

an,

with

/?

= 1 .

a

F(n)

Tax

ra

tes

(7;)

T

ax

rate

s (%

)

Y

z x

Ave

rage

M

argi

nal

Util

ity

Y

: x

Ave

rage

M

argi

nal

Util

ity

e=l

B=

l no

= 0

.06

x0 =

0.0

4 F(

n,)

= 0.

0 x/

z =

0.9

0.10

0.

36

0.08

0.

09

-15

36

-2.8

2 0.

50

0.43

0.

16

0.14

9

32

- 2.

50

0.90

0.

47

0.29

0.

23

18

26

- 2.

08

0.99

0.

50

0.45

0.

36

20

21

-1.7

0

Popu

latio

n av

erag

e 0.

17

0.16

&=o

.s

B=l

n,

=0.0

7 x,

=0.0

5 F(

n,)=

O.O

X

/Z=

I.

0

0.10

0.

33

0.07

0.

10

-37

33

-2.6

8 0.

50

0.41

0.

15

0.15

-3

30

-2

.39

0.90

0.

46

0.28

0.

25

11

25

-2.0

0 0.

99

0.49

0.

45

0.38

15

18

-1

.64

Popu

latio

n av

erag

e 0.

17

0.17

&=0

.5

p=1

n, =

0.0

8 x0

= 0

.06

F(n,

)=O

.O

x/z=

1.

1

0.10

0.

30

0.07

0.

11

-62

30

-2.5

7 0.

50

0.39

0.

14

0.16

-1

4 28

-

2.30

0.

90

0.44

0.

27

0.26

5

24

-1.9

4 0.

99

0.48

0.

43

0.39

II

20

~

1.60

Popu

latio

n av

erag

e 0.

16

0.18

“ri =

0.3

9.

e=0.

5 /I

=1

“0 =

0.0

3 x,

=0.1

1 F(

n,)=

O.O

x/

z =

0.9

0.45

0.

10

0.15

-4

4 68

~

8.64

0.

52

0.19

0.

18

8 62

-

7.70

0.

56

0.34

0.

24

29

50

-6.3

9 0.

56

0.51

0.

34

33

35

- 5.

20

0.21

0.

19

c=o.

s /?

=I

n,=0

.03

X”

= 0.

12

F(Q

) =

0.0

X/Z

= 1.

0

0.44

0.

10

0.16

-6

0 65

-8

.10

0.51

0.

19

0.19

-2

59

-

7.24

0.

54

0.33

0.

26

22

47

- 6.

06

0.55

0.

50

0.36

28

29

--

4.9

9

0.20

0.

20

9=0.

5 /I

=1

n, =

0.0

3 X

,=0.

13

F(n,

)=O

.O

x/z=

I.

1

0.42

0.

09

0.17

-7

9 62

~

7.63

0.

49

0.18

0.

20

-13

56

-6.8

5 0.

53

0.32

0.

27

15

45

- 5.

78

0.54

0.

49

0.38

23

28

-4

.79

0.19

0.

21

Tab

le

3

Max

imin

.”

F(n)

Tax

ra

tes

(%)

Tax

ra

tes

(%)

Y

z x

Ave

rage

M

argi

nal

Util

ity

Y

z x

Ave

rage

M

argi

nal

Util

ity

&=

0.5

p=O

3 n,

=0.1

8 x,

=0.1

3 F(

n,)

= 0

.09

X/Z

=

0.9

&=0

.5

/?=c

c n,

=0.1

7 x,

=0.1

6 F(

n,)=

0.23

X

JZ =

0.9

0.10

0.

24

0.05

0.

14

- 15

5 86

-8

.67

0.50

0.

50

0.19

0.

17

11

70

- 8.

09

0.28

0.

10

0.18

-7

2 83

-

7.00

0.

90

0.57

0.

35

0.23

34

52

-6

.72

0.48

0.

63

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another lesson to be learnt from the comparison between our results and those of Stern. On the basis of calculations based on the linear income tax

model we cannot make reliable conclusions about how things are in the nonlinear model.

Next we turn to the question of the approximate linearity of the optimal

tax schedule which Mirrlees regarded as ‘perhaps the most surprising result in his numerical calculations’. Atkinson (1973) found in his numerical

calculations that the maximin criterion in the Mirrlees model yields substantially higher marginal tax rates for most of the income range and the tax schedule departs significantly from linearity in the range that covers the, majority of the population. Again, we can say that there is no need to go to the maximin objective in order to get a tax schedule which departs significantly from linearity for the great majority of people. Fig. 1 illustrates our results showing the income tax schedule in the case E =0.5, /I = 1, and

R = 0.1 to be considerably nonlinear. There is one further comment to be made on the maximin solution. It has

sometimes been thought that the maximin solution yields high marginal tax

rates at all income levels. For example, Mirrlees (1977) in criticizing the maximin principle writes that in the maximin case ‘the argument for high marginal rates could apply at all income levels’. The calculations presented in table 3 show that this is not the case [see, also Atkinson (1973)]. The marginal tax rates at the upper income levels are not higher than in the utilitarian solutions.

In addition to the maximin results Atkinson (1973) found that ‘comparing the Mirrlees results for fi=O with those for p= 1, it can be seen that the marginal tax rates are higher in the latter case, the difference is, however, much less marked than the p= 1 case and the maximin solution’. Therefore Atkinson concludes that increasing b above 1 in the utilitarian cases is likely to have a significant effect on the optimal solution. On the basis of our calculations this conclusion does not emerge when s=O.5. In fact, it can be seen that the difference in the marginal tax rates between the p= 1 case and the maximin solution is quite small, whereas it is much more marked between the /I=0 case and the p= 1 case.

What about if a larger tax revenue were to be raised, would it mean significant changes in the shape of the tax schedule? Our numerical results seem to suggest that the marginal tax rates tend to increase for most taxpayers with increasing net government expenditure. The shape of the tax schedule seems to remain quite similar in spite of changes in government expenditure.

In this context it may be useful to notice an approximation result concerning the relationship between the marginal tax rates at higher income levels and the net government expenditure. Assume an additive utility function in consumption and hours, then formula (10) can be written in the


form (utilitarian case):

l/A=j(l/u,)f(m)dm. L!

(15)

Suppose that n is so large that l/u, is much greater than l/i,, then the following approximation for the marginal tax rate is obtained:

marginal tax rate ‘v f(n) n,1(, _ -%Ux jJ’Wdm

. (16)

This means that the marginal tax rate for an individual with high n is independent of ;1 or government expenditure. However, strictly speaking we do not know how good an approximation this is likely to be.

In fact the marginal tax rate continues to fall beyond the range shown in tables l-3, and ultimately it tends to zero as we should expect on the basis of the earlier mentioned analytical result about the zero marginal tax rate at the top of the income scale. However, we have to remember that the zero

limit of the marginal tax rates does not hold for all ability distributions. We know from Mirrlees (1971) that the log-normal distribution has this property. Therefore we can examine, with the help of numerical calculations, how rapidly the marginal tax falls to zero. In this way it is possible to answer the important question concerning how local the zero marginal tax rate is at the top of the income scale. These results have been calculated using very small steps at the upper tail of the n-distribution. Our answer has been illustrated in fig. 2. In words, it can be interpreted to mean that the marginal tax rate is much above zero even for an individual whose income is

such that 99.9 percent of the population have a lower income.

Of course, it is possible to argue against the structure of income taxation shown in tables l-3 and figs. 1 and 2. For example, we could claim that high-income people are not so responsive to marginal tax rates or take up some other objections. Our purpose is not to go to this kind of critique which can be called ‘empiristic’. Instead of that we take up the methodological question. Our calculations illustrate very clearly that neither the maximin criterion nor utilitarianism turn out to be very egalitarian principles. Sometimes the Rawlsian maximin principle has been wrongly identified with egalitariansim. We think that the reason for the rather nonegalitarian outcomes in optimal income tax examples is not to be found directly in these principles. We see that the problem is similar to the methodological question of measurement of inequality raised by Sen (1978a).16 Namely, it is important to distinguish between two statements: (a)

‘60bviously, Atkinson (1980) has a similar remark to make


how upset we are about inequality of welfare, and (b) how much inequality

there is in society. Or, to put it another way, greater inequality may lead to

higher social welfare, since ‘equality’ and ‘social welfare’ are two quite distinct concepts. We think that this point - the need to distinguish between these concepts in the optimal income tax model ~ comes through very clearly and convincingly in our calculations.

4. Concluding remarks

Using numerical calculations we have obtained some new results in the optimal income tax model. First, we have shown that a quite realistic value for the elasticity of substitution between consumption and leisure (c=0.5) leads to conclusions that are quite different from some of those drawn by Mirrlees (1971) and Atkinson (1973). The marginal tax rates are not so low and the shape of the tax schedule is for a great majority of the population substantially nonlinear. A new feature in our calculations is that both results come out without going to maximin, as done by Atkinson (1973). Hence, the argument for very high marginal tax rates in the lower and middle income ranges and the nonlinearity of the income tax schedule can be based on a quite reasonable estimate of the elasticity of substitution between consumption and leisure and on utilitarian social objectives.

Secondly, we studied in special cases the locality of the zero limit of the marginal tax rate at the upper end of the ability distribution. In this question we achieved a very definite conclusion: this result is really very local.

By and large our calculations with different specifications yield outcomes which are far from egalitarian. We think that a partial explanation for this nonegalitarian outcome is that the optimal income tax model is not able to distinguish between the two following statements: (a) how upset we are about inequality in the distribution of welfare, and (b) how much inequality there is in society.

References

Allen, F., 1982, Optimal linear income taxation with general equilibrium effects on wages, Journal of Public Economics 17, 135- 143.

Atkinson, A.B., 1973, Maximin and optimal income taxation, Discussion paper no. 47, University of Essex, Department of Economics.

Atkinson, A.B., 1980, Book review of Personal income distribution (eds. by W. Krelle and A.F. Shorrocks), Journal of Political Economy, 815-819.

Atkinson, A.B. and J.E. Stiglitz, 1980, Lectures on public economics (McGraw-Hill, New York and London).

Feldstein, MS., 1973, On the progressivity of the income tax, Journal of Public Economics 2, 357-376.

Mirrlees, J.A., 1971, An exploration in the theory of optimal income taxation, Review of Economic Studies 38, 135-208.

Mirrlees, J.A., 1976, Optimal tax theory: A synthesis, Journal of Public Economics 6, 327-358.


Mirrlees, J.A., 1977, Labour supply behaviour and optimal taxes, in: Fiscal policy and labour supply (Institute for Fiscal Studies, London).

Phelps, ES., 1973, Taxation wage income for economic justice, Quarterly Journal of Economics 87, 331-354.

Sadka, E., 1976, Income distribution, incentive effects and optimal income taxation, Review of Economic Studies 43, 261-268.

Seade, J., 1977, On the shape of optimal tax schedules, Journal of Public Economics 7, 2033236. Seade, J., 1982, On the sign of the optimum marginal income tax, Review of Economic Studies,

637-643. Sen, A.K., 1978a, Ethical measurement of inequality: Some difficulties, in: W. Krelle and A.F.

Shorrocks, eds., Personal income distribution (North-Holland, Amsterdam). Sen, A.K., 1978b, Utilitarianism and welfarism, Journal of Philosophy 9, 4633489. Stern, N.H., 1976, On the specification of models of optimum income taxation, Journal of Public

Economics 6, 123-162. Stern, N.H., 1982, Optimum taxation with errors in administration, Journal of Public Economics

17, 181-21 I. Stiglitz, J.E., 1982, Self selection and Pareto efficient taxation, Journal of Public Economics 17,

2 13-240. Tuomala, M., 1979, Public expenditure and optimal linear income taxation, Yearbook of the

Finnish Society for Economic Research (Helsinki) [in Finnish].

Documents

ON THE OPTIMAL INCOME TAXATION Some further numerical