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The optimal linear income tax
Journal
of Public
Economics
34 (1987) 379-390.
THE OPTIMAL
North-Holland
LINEAR
A Diagrammatic
Toshihiro
INCOME
TAX
Analysis
IHORI*
Osaka University, Osaka 560, Japan Received
March
1987, revised
version
received
September
1987
This paper clarifies the role of the tax possibility frontier and the social indifference curve in the comparative statics analysis of the optimal linear income tax. By a mostly diagrammatic derivation of the results we confirm the conventional conjecture that the optimal marginal tax rate increases with the government’s inequality aversion. On the other hand, we cannot always confirm analytically the conventional conjecture that the optimal marginal tax rate increases with the government’s budgetary needs.
1. Introduction
There are four main ingredients for a model of standard optimum linear income taxation: a social welfare function, a preference relation or labor supply function for individuals, an ability structure and distribution, and a revenue requirement for the government. There have been extensive analyses on the responie of the parameters of the optimal linear income tax to changes in the above four components. Mirrlees (1971), Atkinson (1973), and Stern (1976) have conducted numerical simulations of the effect on the optimal linear tax rate of varying the above components. Helpman and Sadka (1978), Ihori (1981), Balcer and Sadka (1982), and Hellwig (1986) have provided some comparative statics results. As was discussed by Atkinson and Stiglitz (1980), the standard conjectures may be summarized as follows:
(1) The optimal marginal tax rate increases with the government’s inequality aversion. (2) The optimal marginal tax rate decreases with the elasticity of labor supply. (3) The optimal marginal tax rate increases with the spread in abilities. (4) The optimal marginal tax rate increases with the government’s needs. In this paper we intend to reexamine conjectures (1) and (4) using a diagram of the tax possibility frontier and the social indifference curve. All of *I wish to thank
W7-2727/87/$3.50
two anonymous
0
referees for many helpful comments
1987, Elsevier Science Publishers
and suggestions.
B.V. (North-Holland)
380
7: Ihori, The optimal linear income tax
the conjectures (1)44) are not always analytically valid. It is now well known that the marginal tax rate is higher under Rawls’ criterion than under Bentham’s. Ihori (1982,1984) and Hellwig (1986) among others have shown that the optimal marginal tax rate is bounded above by the Rawlsian rate, which in turn is bounded by the revenue-maximizing rate. However, conjecture (1) has not been well established analytically. Furthermore, Helpman and Sadka (1978) have reported that the effect of a mean-preserving spread in abilities cannot be determined in general. We are still at the stage of attempting to understand the importance of the various components by conducting numerical calculations. The purpose of this paper is to contribute to the understanding of the structure of the optimal linear income tax model through diagrammatic examination of some comparative statics. The rationale for this approach is not that we believe that such numerical calculations are unimportant, Instead, the particular concern of this paper is to make the income tax problem more accessible for intuitive discussion and classroom presentation using a diagram of the tax possibility frontier and the social indifference curve. Most of the results of this paper have already been discussed elsewhere. Nevertheless, the mostly diagrammatic derivation of the results in this paper is quite helpful and intuitively appealing. This paper is organized as follows. Section 2 recapitulates Sheshinski’s (1972) formulation of the linear income tax problem and presents a diagram of the tax possibility frontier and the social indifference curve. Section 3 analyzes the response of the parameters of the optimal linear income tax to changes in the social objective function from Bentham’s sum-of-utilities to Rawls’ max-min. It is shown that conjecture (1) is analytically established. Section 4 analyzes the response of the parameters to changes in the government budgetary needs, using Sheshinski’s (1971) educational investment model. It is shown that conjecture (4) cannot be valid in general. Finally, section 5 concludes this paper. Section 3 corresponds to the comparative statics of the social welfare function (shift of the social indifference curve), and section 4 corresponds to the comparative statics of the tax requirement (shift of the tax possibility frontier). It is shown that once the tax possibility frontier shifts, the analytical results will become ambiguous in general. Note that conjectures (2) and (3) correspond to the situation where both the tax possibility frontier and the social welfare function shift. It seems that such comparative statics are so complicated that we may not obtain clear-cut analytical results in general. 2. The model This model is due to Mirrlees (1971) and Sheshinski (1972). For simplicity suppose there are two individuals who have the same preferences but
T Ihori, The optimal linear income tax
381
different skills. Let u(x, y) be the common utility function, where x>O is consumption and 0 5~5 1 is labor. It is assumed that u1 >O and u1 ~0, and u is strictly concave. We also assume normality of consumption and leisure. The skill of an individual is denoted by n. Namely, for the rich individual n = nH and for the poor individual n= n,_. From now on subscript H refers to the rich individual and L the poor individual. The wage rate earned by an n-man is assumed to be n. Hence, his gross income z is ny. Each consumer chooses x, z and y so as to solve:
(1)
max4~ Y) St. x = z - T(z) z=ny,
where T is the tax function. We consider a linear tax function:
(2)
T(z)= -z+(l-@z,
where c1 is the minimum guaranteed income and 1 ---/I is the marginal tax rate. We denote by x@n;a) and y(/?n: a) and n-man’s demand for consumption and his supply of labor, respectively. /?n is the after-tax real wage rate and CIis nonwage income. We also define an indirect utility function: u(/h; a) = u[@n; a), y(Bn; a)]. Let R be a pre-determined level of per-capita government spending, so that the government’s budget constraint is TH+ T,= R. Employing (2) this constraint reduces to:
R + 2~= (I- D)Cw(Lh; 4 + w(h;
4
(3)
Let us draw a diagram of the tax possibility frontier. In fig. 1 curve AB shows the government budget constraint for given R. When /I=O, y is zero and from (3) 01= -R/2 (OA= R/2). When j?= 1, a is also given by- R/2. For small values of /I, y increases with /I and c1 increases as well. However, as a rise of jI means a reduction of tax rate, the feasible guarantee eventually declines. M is the highest point of curve AB, and /7 is the associated j?. For B> fl, the negative effect on revenue of a decrease in the marginal tax rate (1 -fl) dominates the positive effect on revenue of an increase in work effort. We call curve AB the tax possibility frontier (TPF).’ ‘The tax possibility frontier may not be concave. Where more than one solution exists, a global comparison must be made of the levels of welfare. However, as far as the qualitative comparative statics results are concerned, the concavity of the tax possibility frontier is not crucial.
7: Ihori, The optimal linear income tax
382
Fig. 1
Mathematically,
we have
+ GY,~) -(~,YI_ + n,y,) -=da (1- B)(Q,Y~L~ @A,, Z-(1 -PhYLa+%Y,J ’ where yij = ayi/aj (i = I, H and j = LX, /I). Let us then draw an indifference curve of an individual, curve I. Curve I is downward sloping; an increase in c1will raise utility and this has to be offset by a decrease in p so as to maintain the same utility. Considering the first order condition of utility maximization, we have du/d/I, = - z < 0.
(5)
It is interesting to note that curve I is not necessarily strictly convex.2 The linear income tax problem may be written as
max W4 D)
(6)
s.t. (3) where W( .) is the social welfare function. In this paper the social welfare ‘In Atkinson and Stiglitz (1980) and Hellwig (1986) the indifference curve was drawn as a convex curve. This curve is strictly concave so long as the tax-induced disincentive effect on work effort exists.
‘I: Ihori, The optimal linear income tax
383
function is given by (7) where v 2.0. With v =O, we have the Benthamite utilitarian objective. With v= cc we have the Rawlsian case. For higher values of v the function is more concave. We now illustrate a social indifference curve in the (a,b) plane. The slope of the social indifference curve is given by da ---=
-
dBw
v, vvLB+
v~“v”p
v, vvLor+ v, VVHa’
where vii= &Jaj (i= L, H and j = a, /I). The social indifference curve is not necessarily convex. Fig. 1 illustrates the social optimal point E where curve W is tangent to curve AB. Once we know the tax possibility frontier and the social indifference curve, we can attain the optimal p0int.j
3. Shift of the social welfare function In this section we investigate the comparative statics of the weight of the social welfare function. When v changes, the social indifference curve will shift, but the tax possibility frontier will not shift. The optimal point moves on the initial tax possibility frontier. As shown by the movement from W to W’ in fig. 2, if the absolute slope of the social indifference curve increases at the same values of a and fi, the optimal point moves to the right: the optimal level of a decreases and the optimal level of fi increases. Therefore, it is useful to differentiate da/d/I with respect to v. d2a -__= dpdv
1 C(C %_a+ v, vQla) (v; “oLor + v; vvnJ2
x ( - v,fJv,v log VL- v,&qv,”log vu) + (u,VoL&q + v; “Vu&q) x (VL“ULor log UL+ u; vv”. log vn)]
31hori (1982,1984) and Hellwig (1986) use a diagram of the tax possibility social indifference curve in the optimal linear income tax problem.
frontier
and
the
7: Ihori, The optimal linear income tax
384
Fig. 2
We know
and vu>vL. We show that
n,
vLpvHa
-
vHj3”L,
VW VHB -<-. VLa
VHir
Using the envelope theorem, it is straightforward
to see that
Since z is an increasing function of n [see Sadka (1976)], it follows that the above inequality holds.4 Hence it is easy to see that the sign of (9) is negative. The absolute value of the slope of the social indifference curve decreases with v. Hence, the optimal value of fl decreases with v and the optimal value of a increases with v. When the social function approaches Rawls’ criterion as v--rco, the optimal tax parameters converge to the Rawls-optimal tax parameters. When the social welfare function approaches the Benthamite criterion as v+O, the 41 am grateful to a referee for suggesting proof in a previous version of the paper.
the above
proof,
which
is much
simpler
than
the
T Ihori, The optimal linear income tax
385
optimal tax parameters converge to the Bentham-optimal tax parameters. We confirm analytically the conjecture that the optimal marginal tax rate increases with the government’s inequality aversion5 4. Shift of the tax possibility frontier Let us examine how the optimal point changes when R is increased. In this case the tax possibility frontier will shift but the social indifference curve will not shift. From now on we concentrate on the case of the educational investment model which is due to Sheshinski (1971). Remember that the educational investment model is a special case of the labor incentive model; we have
4x7 Y)= ucx -dY)l and
(104
v, = li,
g(y) is the cost of education where g( .) is convex (i.e. there are increasing marginal costs). It is not necessary to assume that u is strictly concave here. In the educational investment model the income effect on the labor supply is assumed away: y, = 0. Substituting y, = 0 into (4), the slope of the tax possibility frontier is given by
dH_(I q?-
-B)h~,p+n,~,,d
-(YL+Y,)
2
(11)
As y is dependent only on /?, the slope of curve AB is determined solely by the level of p. Once we allow for the income effect in a more general framework of the labor incentive model, the slope of curve AB is dependent on c1as well. The comparative statics result will be related to the forms of the utility function, the distribution function of ability, and the cost function in such a complicated way that it is not easy to get clear-cut analytical results on this ‘The result obtained in this section was discussed in Helpman and Sadka (1978) and Hellwig (1986). Helpman and Sadka showed that an increase in the degree of inequality aversion raises the marginal tax rate of the Bentham-optimal tax. However, they cannot show that the optimal tax parameters converge to the Rawls-optimal tax parameters as v-0. Hellwig (1986) shows that the optimal marginal tax rate is arbitrary close to the Rawlsian rate if the government is sufficiently inequality averse. However, he does not show that the optimal tax parameters converge to the Bentham-optimal tax parameters as v+ a~.
7: Ihori, The optimal linear income tax
386
issue. A celebrated property of the educational investment model is that the slope of curve AB is independent of g, so that we can explore intuitive implications of an increase in R. An increase in R will shift curve AB downwards. It is easy to see that the combination of da/dR >O and dp/dR > 0 is not feasible. The marginal tax rate should be moderate in the sense that a decrease in 1 -b will not induce people to work in a way that it increases the tax revenue. An extra resource left to the government will not be available as the result of increases in c1and P. Therefore, we have the following three possibilities.
dfl
(4
&
and
dRLO,
(b)
$
and
da dR
(c)
SzO
and
dB E
We consider special cases of v= cc (maximin case) and v=O (utilitarian case) as for the social welfare function. Note that the analysis carried out for the case of v = 0 is valid also for all v 2 0. 4.1. Maximin case When the social welfare function is given by the maximin criterion, the relevant social indifference curve is given by the worst-off person’s indifference curve I,, which is strictly concave. Considering y,=O, its slope is dependent only on /?. Because the slope of curve AB is dependent only on fl as well, the new optimal point E’ in fig. 3 is just below the initial optimal point E. The optimal marginal tax rate in the maximin criterion 1 -pz is independent of R. We have case (a). According to the maximin criterion, an increase in the tax revenue requirement should be financed by a decrease in the income guarantee, while the marginal rate should be kept constant. Observe that this policy implication is valid as long as the social welfare is given by the representative individual of some given target ability A. As shown in the previous section, it is true that the optimal marginal tax rate is higher in the maximin case than in any other case. The optimal level of /? increases with target ability A. But, in any case, a less progressive tax structure is desirable when R is increased. 4.2. Utilitarian case Let us now consider the utilitarian case. Substituting
v =0 and (10) into
7: Ihori, The optimal linear income tax
387
M
Fig. 3
(8), the slope of the social indifference curve is reduced to
(12) The social indifference curve U is downward-sloping. However, curve U is not necessarily concave. Let us now examine the comparative statics of an increase in R. If da/d/?, is independent of c1as in the maximin case, the new optimum E’ is just below E in fig. 4, given by a point like F. If du/dfiu is an increasing function of tl, at point F the slope of curve U is steeper than that of curve AB; therefore E’ is to the southeast of E. Thus, if d2u/d/IdC(uz0, then we have case (a) and a less progressive tax structure is desirable. We have u;zr + r&Z”)(u; + u;1)- (ULZL+ u;izu) (u;l.+ u;;)]
388
7: Ihori, The optimal linear income tax
Fig. 4
(13) where r = - u”/u’ is absolute risk aversion. If absolute risk aversion is constant, (13) is zero and hence we have dbJdR =0 and du/dR ~0 as in the maximin case. If absolute risk aversion is non-decreasing, (13) is positive and we have case (a).” In order to illustrate the analysis, let us take the following example. At the initial optimum point E, zL = 1 and zn= 4, and the associated marginal utility is uL= 1 and t&=0.5, respectively. Thus, the slope of curve U is -( 1 x 1+ 0.5 x 4)/( 1 + 0.5) = - 2 at E. We examine the marginal utility associated with point F in fig. 4. At point F, which is just below E, /s is the same as at E but a is decreased compared with E. z remains the same as before, but x is decreased and hence u’ is increased. We now consider alternative utility functions (i) and (ii) in fig. 5. In case of the utility function (i) where u”’ is positive and rather high, u; is increased to a great extent (2) while u;( is increased to a small extent (0.55). In case of the utility function (ii) where u”’ is negative u; is increased 6Thisresult was derived Balcer and Sadka (1982).
in Ihori (1981) mathematically,
and corresponds
to Proposition
2 in
‘I: Ihori, The optimal linear income tax
389
Fig. 5
to a small extent (l.l), while u;l is increased to a great extent (0.8). In case (i) the absolute slope of curve U at F is 2 x 1 + 0.55 x 4 _ 4.2 < 2 2+0.55 2.55 In case (ii) the slope of curve U at F is 1.1 x 1+0.8x4 1.1+0.8
4.3 ‘E>2. .
Thus, the new optimum point E’ is between F and B’ in case (ii), while it is between M’ and F’ in case (i). This example suggests that if the utility function is such that the marginal utility of higher ability persons is relatively more weighted as the result of a decrease in u (an increase in R), the slope of the social indifference curve U becomes steeper, and hence it is more likely to have case (a); a less progressive tax structure is optimal. It should be stressed that if E’ is between M’ and F, it is still possible to have a less progressive tax structure; case (b). Even when r is increasing (but no so rapidly), a less progressive tax structure would be optimal.
5. Conclusion
This paper has considered the role of the tax possibility frontier and the social indifference curve in the comparative statics analysis. It is shown that
390
7: Ihori, The optimal linear income tax
when the social indifference curve shifts, the comparative statics result is analytically well investigated. We confirm the conventional conjecture that the optimal marginal tax rate increases with the government’s inequality aversion. On the other hand, if the tax possibility frontier shifts, the comparative statics result is rather ambiguous. Even if we employ an extreme case of the educational investment model, we cannot always confirm analytically the conventional conjecture that the optimal marginal tax rate increases with the government’s budgetary needs. This paper has explored how accessible the comparative statics results are and provided a useful diagram of the tax possibility frontier and the social indifference curve for intuitive presentation. Although most of the results of this paper were already discussed elsewhere, the mostly diagrammatic derivation of the results in this paper is quite helpful and intuitively appealing. It should be stressed that this paper considers a very special case of optimal income taxation. In doing so, however, I hope that we obtain more insight into the problem of income taxation.
References Atkinson, Anthony, B., 1973, How progressive should income tax be? in: Michael Parkin and A.R. Nobay, eds., Essays in modern economics (Longman, London) 9G109. Atkinson, Anthony, B. and Joseph E. Stiglitz, 1980, Lectures on public economics (McGraw-Hill, New York). Balcer, Yves and Efraim Sadka, 1982, Horizontal equity, income taxation and self-selection with an application to income tax credits, Journal of Public Economics 19, 291-310. Hellwig, Martin F., 1986, The optimal linear income tax revisited, Journal of Public Economics 31, 163-179. Helpman, Elhanan and Efraim Sadka, 1978, The optimal income tax: Some comparative statics results, Journal of Public Economics 9, 383-393. Ihori, Toshihiro, 1981, Sufficient conditions for the optimal degree of progression to decrease with the government revenue requirement, Economics Letters 9>100. Ihori, Toshihiro, 1982, Optimal degree of progression when the tax revenue requirement is increased, Keizai to Keizaigaku 25-34. Ihori, Toshihiro, 1984, Modern Japanese public economics (Toyokeizai, Tokyo) (in Japanese). Mirrlees, J.A., 1971, An exploration in the theory of optimal income taxation, Review of Economic Studies 38, 175-208. Sadka, Efraim, 1976, On income distribution, incentive effects and optimal income taxation, Review of Economic Studies 261-267. Sheshinski, Eytan, On the theory of income taxation, HIER Discussion Paper No. 172. Sheshinski, Eytan, 1972, The optimal linear income tax, Review of Economic Studies 49, 637643. Stem, Nicholas H., 1976, On the specification of models of optimum income taxation, Journal of Public Economics 6, 123-162.
of Public
Economics
34 (1987) 379-390.
THE OPTIMAL
North-Holland
LINEAR
A Diagrammatic
Toshihiro
INCOME
TAX
Analysis
IHORI*
Osaka University, Osaka 560, Japan Received
March
1987, revised
version
received
September
1987
This paper clarifies the role of the tax possibility frontier and the social indifference curve in the comparative statics analysis of the optimal linear income tax. By a mostly diagrammatic derivation of the results we confirm the conventional conjecture that the optimal marginal tax rate increases with the government’s inequality aversion. On the other hand, we cannot always confirm analytically the conventional conjecture that the optimal marginal tax rate increases with the government’s budgetary needs.
1. Introduction
There are four main ingredients for a model of standard optimum linear income taxation: a social welfare function, a preference relation or labor supply function for individuals, an ability structure and distribution, and a revenue requirement for the government. There have been extensive analyses on the responie of the parameters of the optimal linear income tax to changes in the above four components. Mirrlees (1971), Atkinson (1973), and Stern (1976) have conducted numerical simulations of the effect on the optimal linear tax rate of varying the above components. Helpman and Sadka (1978), Ihori (1981), Balcer and Sadka (1982), and Hellwig (1986) have provided some comparative statics results. As was discussed by Atkinson and Stiglitz (1980), the standard conjectures may be summarized as follows:
(1) The optimal marginal tax rate increases with the government’s inequality aversion. (2) The optimal marginal tax rate decreases with the elasticity of labor supply. (3) The optimal marginal tax rate increases with the spread in abilities. (4) The optimal marginal tax rate increases with the government’s needs. In this paper we intend to reexamine conjectures (1) and (4) using a diagram of the tax possibility frontier and the social indifference curve. All of *I wish to thank
W7-2727/87/$3.50
two anonymous
0
referees for many helpful comments
1987, Elsevier Science Publishers
and suggestions.
B.V. (North-Holland)
380
7: Ihori, The optimal linear income tax
the conjectures (1)44) are not always analytically valid. It is now well known that the marginal tax rate is higher under Rawls’ criterion than under Bentham’s. Ihori (1982,1984) and Hellwig (1986) among others have shown that the optimal marginal tax rate is bounded above by the Rawlsian rate, which in turn is bounded by the revenue-maximizing rate. However, conjecture (1) has not been well established analytically. Furthermore, Helpman and Sadka (1978) have reported that the effect of a mean-preserving spread in abilities cannot be determined in general. We are still at the stage of attempting to understand the importance of the various components by conducting numerical calculations. The purpose of this paper is to contribute to the understanding of the structure of the optimal linear income tax model through diagrammatic examination of some comparative statics. The rationale for this approach is not that we believe that such numerical calculations are unimportant, Instead, the particular concern of this paper is to make the income tax problem more accessible for intuitive discussion and classroom presentation using a diagram of the tax possibility frontier and the social indifference curve. Most of the results of this paper have already been discussed elsewhere. Nevertheless, the mostly diagrammatic derivation of the results in this paper is quite helpful and intuitively appealing. This paper is organized as follows. Section 2 recapitulates Sheshinski’s (1972) formulation of the linear income tax problem and presents a diagram of the tax possibility frontier and the social indifference curve. Section 3 analyzes the response of the parameters of the optimal linear income tax to changes in the social objective function from Bentham’s sum-of-utilities to Rawls’ max-min. It is shown that conjecture (1) is analytically established. Section 4 analyzes the response of the parameters to changes in the government budgetary needs, using Sheshinski’s (1971) educational investment model. It is shown that conjecture (4) cannot be valid in general. Finally, section 5 concludes this paper. Section 3 corresponds to the comparative statics of the social welfare function (shift of the social indifference curve), and section 4 corresponds to the comparative statics of the tax requirement (shift of the tax possibility frontier). It is shown that once the tax possibility frontier shifts, the analytical results will become ambiguous in general. Note that conjectures (2) and (3) correspond to the situation where both the tax possibility frontier and the social welfare function shift. It seems that such comparative statics are so complicated that we may not obtain clear-cut analytical results in general. 2. The model This model is due to Mirrlees (1971) and Sheshinski (1972). For simplicity suppose there are two individuals who have the same preferences but
T Ihori, The optimal linear income tax
381
different skills. Let u(x, y) be the common utility function, where x>O is consumption and 0 5~5 1 is labor. It is assumed that u1 >O and u1 ~0, and u is strictly concave. We also assume normality of consumption and leisure. The skill of an individual is denoted by n. Namely, for the rich individual n = nH and for the poor individual n= n,_. From now on subscript H refers to the rich individual and L the poor individual. The wage rate earned by an n-man is assumed to be n. Hence, his gross income z is ny. Each consumer chooses x, z and y so as to solve:
(1)
max4~ Y) St. x = z - T(z) z=ny,
where T is the tax function. We consider a linear tax function:
(2)
T(z)= -z+(l-@z,
where c1 is the minimum guaranteed income and 1 ---/I is the marginal tax rate. We denote by x@n;a) and y(/?n: a) and n-man’s demand for consumption and his supply of labor, respectively. /?n is the after-tax real wage rate and CIis nonwage income. We also define an indirect utility function: u(/h; a) = u[@n; a), y(Bn; a)]. Let R be a pre-determined level of per-capita government spending, so that the government’s budget constraint is TH+ T,= R. Employing (2) this constraint reduces to:
R + 2~= (I- D)Cw(Lh; 4 + w(h;
4
(3)
Let us draw a diagram of the tax possibility frontier. In fig. 1 curve AB shows the government budget constraint for given R. When /I=O, y is zero and from (3) 01= -R/2 (OA= R/2). When j?= 1, a is also given by- R/2. For small values of /I, y increases with /I and c1 increases as well. However, as a rise of jI means a reduction of tax rate, the feasible guarantee eventually declines. M is the highest point of curve AB, and /7 is the associated j?. For B> fl, the negative effect on revenue of a decrease in the marginal tax rate (1 -fl) dominates the positive effect on revenue of an increase in work effort. We call curve AB the tax possibility frontier (TPF).’ ‘The tax possibility frontier may not be concave. Where more than one solution exists, a global comparison must be made of the levels of welfare. However, as far as the qualitative comparative statics results are concerned, the concavity of the tax possibility frontier is not crucial.
7: Ihori, The optimal linear income tax
382
Fig. 1
Mathematically,
we have
+ GY,~) -(~,YI_ + n,y,) -=da (1- B)(Q,Y~L~ @A,, Z-(1 -PhYLa+%Y,J ’ where yij = ayi/aj (i = I, H and j = LX, /I). Let us then draw an indifference curve of an individual, curve I. Curve I is downward sloping; an increase in c1will raise utility and this has to be offset by a decrease in p so as to maintain the same utility. Considering the first order condition of utility maximization, we have du/d/I, = - z < 0.
(5)
It is interesting to note that curve I is not necessarily strictly convex.2 The linear income tax problem may be written as
max W4 D)
(6)
s.t. (3) where W( .) is the social welfare function. In this paper the social welfare ‘In Atkinson and Stiglitz (1980) and Hellwig (1986) the indifference curve was drawn as a convex curve. This curve is strictly concave so long as the tax-induced disincentive effect on work effort exists.
‘I: Ihori, The optimal linear income tax
383
function is given by (7) where v 2.0. With v =O, we have the Benthamite utilitarian objective. With v= cc we have the Rawlsian case. For higher values of v the function is more concave. We now illustrate a social indifference curve in the (a,b) plane. The slope of the social indifference curve is given by da ---=
-
dBw
v, vvLB+
v~“v”p
v, vvLor+ v, VVHa’
where vii= &Jaj (i= L, H and j = a, /I). The social indifference curve is not necessarily convex. Fig. 1 illustrates the social optimal point E where curve W is tangent to curve AB. Once we know the tax possibility frontier and the social indifference curve, we can attain the optimal p0int.j
3. Shift of the social welfare function In this section we investigate the comparative statics of the weight of the social welfare function. When v changes, the social indifference curve will shift, but the tax possibility frontier will not shift. The optimal point moves on the initial tax possibility frontier. As shown by the movement from W to W’ in fig. 2, if the absolute slope of the social indifference curve increases at the same values of a and fi, the optimal point moves to the right: the optimal level of a decreases and the optimal level of fi increases. Therefore, it is useful to differentiate da/d/I with respect to v. d2a -__= dpdv
1 C(C %_a+ v, vQla) (v; “oLor + v; vvnJ2
x ( - v,fJv,v log VL- v,&qv,”log vu) + (u,VoL&q + v; “Vu&q) x (VL“ULor log UL+ u; vv”. log vn)]
31hori (1982,1984) and Hellwig (1986) use a diagram of the tax possibility social indifference curve in the optimal linear income tax problem.
frontier
and
the
7: Ihori, The optimal linear income tax
384
Fig. 2
We know
and vu>vL. We show that
n,
vLpvHa
-
vHj3”L,
VW VHB -<-. VLa
VHir
Using the envelope theorem, it is straightforward
to see that
Since z is an increasing function of n [see Sadka (1976)], it follows that the above inequality holds.4 Hence it is easy to see that the sign of (9) is negative. The absolute value of the slope of the social indifference curve decreases with v. Hence, the optimal value of fl decreases with v and the optimal value of a increases with v. When the social function approaches Rawls’ criterion as v--rco, the optimal tax parameters converge to the Rawls-optimal tax parameters. When the social welfare function approaches the Benthamite criterion as v+O, the 41 am grateful to a referee for suggesting proof in a previous version of the paper.
the above
proof,
which
is much
simpler
than
the
T Ihori, The optimal linear income tax
385
optimal tax parameters converge to the Bentham-optimal tax parameters. We confirm analytically the conjecture that the optimal marginal tax rate increases with the government’s inequality aversion5 4. Shift of the tax possibility frontier Let us examine how the optimal point changes when R is increased. In this case the tax possibility frontier will shift but the social indifference curve will not shift. From now on we concentrate on the case of the educational investment model which is due to Sheshinski (1971). Remember that the educational investment model is a special case of the labor incentive model; we have
4x7 Y)= ucx -dY)l and
(104
v, = li,
g(y) is the cost of education where g( .) is convex (i.e. there are increasing marginal costs). It is not necessary to assume that u is strictly concave here. In the educational investment model the income effect on the labor supply is assumed away: y, = 0. Substituting y, = 0 into (4), the slope of the tax possibility frontier is given by
dH_(I q?-
-B)h~,p+n,~,,d
-(YL+Y,)
2
(11)
As y is dependent only on /?, the slope of curve AB is determined solely by the level of p. Once we allow for the income effect in a more general framework of the labor incentive model, the slope of curve AB is dependent on c1as well. The comparative statics result will be related to the forms of the utility function, the distribution function of ability, and the cost function in such a complicated way that it is not easy to get clear-cut analytical results on this ‘The result obtained in this section was discussed in Helpman and Sadka (1978) and Hellwig (1986). Helpman and Sadka showed that an increase in the degree of inequality aversion raises the marginal tax rate of the Bentham-optimal tax. However, they cannot show that the optimal tax parameters converge to the Rawls-optimal tax parameters as v-0. Hellwig (1986) shows that the optimal marginal tax rate is arbitrary close to the Rawlsian rate if the government is sufficiently inequality averse. However, he does not show that the optimal tax parameters converge to the Bentham-optimal tax parameters as v+ a~.
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issue. A celebrated property of the educational investment model is that the slope of curve AB is independent of g, so that we can explore intuitive implications of an increase in R. An increase in R will shift curve AB downwards. It is easy to see that the combination of da/dR >O and dp/dR > 0 is not feasible. The marginal tax rate should be moderate in the sense that a decrease in 1 -b will not induce people to work in a way that it increases the tax revenue. An extra resource left to the government will not be available as the result of increases in c1and P. Therefore, we have the following three possibilities.
dfl
(4
&
and
dRLO,
(b)
$
and
da dR
(c)
SzO
and
dB E
We consider special cases of v= cc (maximin case) and v=O (utilitarian case) as for the social welfare function. Note that the analysis carried out for the case of v = 0 is valid also for all v 2 0. 4.1. Maximin case When the social welfare function is given by the maximin criterion, the relevant social indifference curve is given by the worst-off person’s indifference curve I,, which is strictly concave. Considering y,=O, its slope is dependent only on /?. Because the slope of curve AB is dependent only on fl as well, the new optimal point E’ in fig. 3 is just below the initial optimal point E. The optimal marginal tax rate in the maximin criterion 1 -pz is independent of R. We have case (a). According to the maximin criterion, an increase in the tax revenue requirement should be financed by a decrease in the income guarantee, while the marginal rate should be kept constant. Observe that this policy implication is valid as long as the social welfare is given by the representative individual of some given target ability A. As shown in the previous section, it is true that the optimal marginal tax rate is higher in the maximin case than in any other case. The optimal level of /? increases with target ability A. But, in any case, a less progressive tax structure is desirable when R is increased. 4.2. Utilitarian case Let us now consider the utilitarian case. Substituting
v =0 and (10) into
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M
Fig. 3
(8), the slope of the social indifference curve is reduced to
(12) The social indifference curve U is downward-sloping. However, curve U is not necessarily concave. Let us now examine the comparative statics of an increase in R. If da/d/?, is independent of c1as in the maximin case, the new optimum E’ is just below E in fig. 4, given by a point like F. If du/dfiu is an increasing function of tl, at point F the slope of curve U is steeper than that of curve AB; therefore E’ is to the southeast of E. Thus, if d2u/d/IdC(uz0, then we have case (a) and a less progressive tax structure is desirable. We have u;zr + r&Z”)(u; + u;1)- (ULZL+ u;izu) (u;l.+ u;;)]
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7: Ihori, The optimal linear income tax
Fig. 4
(13) where r = - u”/u’ is absolute risk aversion. If absolute risk aversion is constant, (13) is zero and hence we have dbJdR =0 and du/dR ~0 as in the maximin case. If absolute risk aversion is non-decreasing, (13) is positive and we have case (a).” In order to illustrate the analysis, let us take the following example. At the initial optimum point E, zL = 1 and zn= 4, and the associated marginal utility is uL= 1 and t&=0.5, respectively. Thus, the slope of curve U is -( 1 x 1+ 0.5 x 4)/( 1 + 0.5) = - 2 at E. We examine the marginal utility associated with point F in fig. 4. At point F, which is just below E, /s is the same as at E but a is decreased compared with E. z remains the same as before, but x is decreased and hence u’ is increased. We now consider alternative utility functions (i) and (ii) in fig. 5. In case of the utility function (i) where u”’ is positive and rather high, u; is increased to a great extent (2) while u;( is increased to a small extent (0.55). In case of the utility function (ii) where u”’ is negative u; is increased 6Thisresult was derived Balcer and Sadka (1982).
in Ihori (1981) mathematically,
and corresponds
to Proposition
2 in
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Fig. 5
to a small extent (l.l), while u;l is increased to a great extent (0.8). In case (i) the absolute slope of curve U at F is 2 x 1 + 0.55 x 4 _ 4.2 < 2 2+0.55 2.55 In case (ii) the slope of curve U at F is 1.1 x 1+0.8x4 1.1+0.8
4.3 ‘E>2. .
Thus, the new optimum point E’ is between F and B’ in case (ii), while it is between M’ and F’ in case (i). This example suggests that if the utility function is such that the marginal utility of higher ability persons is relatively more weighted as the result of a decrease in u (an increase in R), the slope of the social indifference curve U becomes steeper, and hence it is more likely to have case (a); a less progressive tax structure is optimal. It should be stressed that if E’ is between M’ and F, it is still possible to have a less progressive tax structure; case (b). Even when r is increasing (but no so rapidly), a less progressive tax structure would be optimal.
5. Conclusion
This paper has considered the role of the tax possibility frontier and the social indifference curve in the comparative statics analysis. It is shown that
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when the social indifference curve shifts, the comparative statics result is analytically well investigated. We confirm the conventional conjecture that the optimal marginal tax rate increases with the government’s inequality aversion. On the other hand, if the tax possibility frontier shifts, the comparative statics result is rather ambiguous. Even if we employ an extreme case of the educational investment model, we cannot always confirm analytically the conventional conjecture that the optimal marginal tax rate increases with the government’s budgetary needs. This paper has explored how accessible the comparative statics results are and provided a useful diagram of the tax possibility frontier and the social indifference curve for intuitive presentation. Although most of the results of this paper were already discussed elsewhere, the mostly diagrammatic derivation of the results in this paper is quite helpful and intuitively appealing. It should be stressed that this paper considers a very special case of optimal income taxation. In doing so, however, I hope that we obtain more insight into the problem of income taxation.
References Atkinson, Anthony, B., 1973, How progressive should income tax be? in: Michael Parkin and A.R. Nobay, eds., Essays in modern economics (Longman, London) 9G109. Atkinson, Anthony, B. and Joseph E. Stiglitz, 1980, Lectures on public economics (McGraw-Hill, New York). Balcer, Yves and Efraim Sadka, 1982, Horizontal equity, income taxation and self-selection with an application to income tax credits, Journal of Public Economics 19, 291-310. Hellwig, Martin F., 1986, The optimal linear income tax revisited, Journal of Public Economics 31, 163-179. Helpman, Elhanan and Efraim Sadka, 1978, The optimal income tax: Some comparative statics results, Journal of Public Economics 9, 383-393. Ihori, Toshihiro, 1981, Sufficient conditions for the optimal degree of progression to decrease with the government revenue requirement, Economics Letters 9>100. Ihori, Toshihiro, 1982, Optimal degree of progression when the tax revenue requirement is increased, Keizai to Keizaigaku 25-34. Ihori, Toshihiro, 1984, Modern Japanese public economics (Toyokeizai, Tokyo) (in Japanese). Mirrlees, J.A., 1971, An exploration in the theory of optimal income taxation, Review of Economic Studies 38, 175-208. Sadka, Efraim, 1976, On income distribution, incentive effects and optimal income taxation, Review of Economic Studies 261-267. Sheshinski, Eytan, On the theory of income taxation, HIER Discussion Paper No. 172. Sheshinski, Eytan, 1972, The optimal linear income tax, Review of Economic Studies 49, 637643. Stem, Nicholas H., 1976, On the specification of models of optimum income taxation, Journal of Public Economics 6, 123-162.