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A note on the marginal tax rate in a finite economy
,Journal
of Public
Economics
A NOTE
28 (1985) 2677272. North-Holland
ON THE MARGINAL TAX RATE IN A FINITE ECONOMY Ailsa A. RoELL* London School of Economics, Massachusetts
Received
London WC2A
Institute
2AE, UK
of Technology
April 1984, revised version
received June 1985
The marginal tax rate is shown to be non-negative in Guesnerie and Seade’s model of nonlinear pricing in a finite economy under assumptions that are as mild as those customarily adopted in nonlinear tax models with a one-dimensional continuum population.
1. Introduction Guesnerie and Seade (1982) (henceforth G/S) have investigated the properties of the optimal tax schedule in a model with finitely many consumers. This note extends their analysis of the direction of the distortion induced by redistributive taxation (proposition 6, p. 172) to a more general setting. The argument is extremely simple and does not suffer from the messiness that led G/S to confine their attention to a more restrictive setting. Section 3 shows that our result, when applied to the standard optimal income tax problem, implies the non-negativity of the marginal tax rate in a setting that is fully as general as that customarily investigated in models with a continuum of types, such as that of Mirrlees (1971). In contrast, an additional assumption, such as separability of preferences in consumption and leisure, seems to be needed to meet the conditions of G/S’s original proposition 6. 2. Extension of Guesnerie and Seade’s proposition G/S consider an economy with two goods; a consumption bundle is denoted x = (a, b). Households are indexed by h = 0,...,H and preferences are represented by a continuously differentiable and strictly concave utility function uh(a,b), increasing in a, which for our purposes may be regarded as the consumption good ‘after-tax income’, and decreasing in b, to be thought *The author
would like to thank
0047-2727/85/$3.30
0
Jesus Seade and Nicholas
1985, Elsevier Science Publishers
Stern for their valuable
B.V. (North-Holland)
comments.
268
A.A. Riiell, Marginal tax rate
of as a measure of effort or ‘before-tax income’. The marginal rate of substitution sh(x) is given by -Z&X)/L&X). A feasible nonlinear tax schedule is defined by a set of distinct ‘corners’ {Ci}izO,,.,, T in (a, b) space and an assignment of consumers h to nonempty corner membership sets Hi such that every consumer h prefers the consumption bundle at the corner Ci for which h E Hi to any other bundle, and a fixed net revenue K is raised (& ah xhbhZ K). Due to the monotonicity of preferences in a and b, corners can be ranked unambiguously and indexed in order of ascending size of (ai, bi). Two principal assumptions are adopted by G/S to investigate the marginal tax rate (G/S, pp. 168 ff.): Assumption
B.
Consumers
h>h’osh(x)
can be indexed
in such a way that:
Vx,h,h’.
Assumption R. ([strong] redistributive assumption). For each pair of corners Ci, with i 0 of commodity a from (any) h’ E Hj, to (any) h E Hi, provided incentive effects are ignored (that is, provided each type could be forced to remain at his assigned corner). We will replace assumption R by the following suggested by G/S in footnote 6 (p. 170). Assumption corners Ci, there exists to agent h are ignored.
much
weaker
assumption,
VWR.’ ([ ver y weak] redistributive assumption). For each pair of Cj with i < j, given K > 0 small enough, for any h E Hi and h’ E Hj zh, zh’ E [0, K] such that it is desirable to distribute (K - zh, -z”) and (K -zh’, -z”‘) away from agent h’, provided incentive effects
One may regard K as the quantity of money that is indirectly redistributed from h’ to h, as depicted in fig. 1. For convenience we will adopt the usage that two corners Ci < Cj are leftlinked if uh(Ci) = uh(Cj) for some h E Hi and right-linked if this holds for some h E Hj.2 It is easily verified, as shown by G/S in lemma 2, that assumption B implies: (i) two distinct corners cannot be both left- and right-linked; (ii) only adjacent corners can be left- or right-linked at all. ‘As a point of detail, this assumption seems stronger than that of G/S in one respect: it is required to hold for all small enough redistributions rather than for some fixed redistribution. Observe, however, that G/S do allow themselves to set the assumed redistribution arbitrarily small in their calculations. *These terms refer to pairs of corners and should not be confused with G/S’s concepts of winner-linked and loser-linked corners, which refer to single corners. Under assumption B, for example, a corner j is winner-linked if it is right-linked to some corner ij, or both. For our purposes the concepts of W- and L-links are unnecessary, as are chains and their various properties.
A.A. Riiell, Marginal
269
tax rate
Fig. 1
We now derive our main result, that of proposition 6 of G/S, under assumptions B and VWR. The proof is rendered considerably more succinct by a suitable choice of the corners that are to be subjected to a redistributive experiment. Proposition 6’. Under corners are right-linked.
assumptions
B and
VWR,
all pairs
of successive
Proof Suppose that successive corners which are not right-linked do exist. Let (Cj_r, Cj) be the highest such pair and let (Ci,Ci+r) be the lowest such pair; we do not rule out the possibility that i=j1. We will construct a redistribution from members of Hj to members of Hi that satisfies all constraints and imrpoves the social objective. From the definition of Ci, Ci and Ci+ 1 are not right-linked. Also, if i # 0, Ci and Ci_ 1 are not left-linked since they are, by assumption, right-linked. Thus all consumers who are not at corner Ci (in particular, those at Ci+r and Ci_ r, if such a corner exists) strongly prefer their own position. By continuity of the utility function, they would still prefer their own position to any point in a small enough neighbourhood around Ci. Similarly, if j # T, Cj and Cj+ 1 are not left-linked; and C,_r and Cj are not right-linked. Hence any consumer heHj will prefer positions in a small enough neighbourhood around Cj to any other corners. Distribute K >0 to all consumers ~EH~ by assigning to each of them a bundle (ui + K - xh, bi - x”) that maximizes
subject
to 0 5 x 5 K, where (bi, ai) are the coordinates
of corner
Ci.
270
A.A. Riiell, Marginal tax rate
Define K’ by: K’ = K. card Hi/card Hj. Now redistribute K’> 0 away from all individuals consumption to (aj - K’ + xh’, bj + xh’) that maximizes
h’ E Hj by setting
their
uh’(aj - K’ + x, bi + x) subject to 0 5 x 5 K’. Note that all members of Hj are being shifted to new corners that involve less consumption and leisure than Cj. Thus nonmembers of Hj will not be tempted to follow suit. As argued above, for K (and hence K’) small enough, the suggested redistribution will not violate any incentive constraints connecting Ci or Cj and other corners. Also, since all agents within Hi and Hj are assigned their optimal value of xh subject to a common constraint, there is no problem with incentive constraints amongst the members of Hi and Hj, respectively. The fact that corners Ci and Cj may ‘explode’ into many points need not cause any concern. Lastly, the suggested move (for K suitably small) improves the Paretian social objective, given assumption VWR. For we have chosen the vectors (K -xh, -x”) and (-K’ + xh’, x”‘) optimally from the point of view of the utilities of the agents involved, given the amount K to be redistributed. Q.E.D. The non-negativity of the marginal corollary of this proposition. 3. Non-negativity
tax rate is, as shown
by G/S, a direct
of optimal marginal income tax
Consider assumptions B and R. The first of these assumptions corresponds to a standard assumption generally used throughout the optimal income tax literature; for example in the model of Mirrlees (1971) it is equivalent to assumption (B), p. 182. The second is, however, too restrictive. For, as noted by G/S, the marginal utility of commodity a (consumption) is not independent of the quantity of commodity b (labor, in efficiency units) consumed. Thus in general the desirability of redistributing utility need not imply the desirability of redistributing good a. In addition to the usual assumptions adopted in the standard optimal tax setting, an extra assumption, such as separability of preferences in consumption and leisure, would be needed in order to ensure that assumption R holds.3 3If preferences
are separable,
u(a, 4 = .OA:(a) -h(l)), for f and g increasing
and concave.
Then:
u,= f’(.k’(a). Thus, any agent with both higher utility and higher after tax income must have a lower marginal utility of consumption, regardless of his effort level. This implies assumption R.
A.A. Riiell, Marginal
tax rate
271
In contrast, assumptions B and VWR are weak enough to include the usual identical-preference one-dimensional skill models of optimal income taxation.4 In particular, they are implied by the model used by Seade (1982). To see this, let all members of society ;have the same concave utility function ~(a, I) defined over consumption (a) and leisure (I) and increasing in both. Pre-tax income, that is, labor in efliciency units, is given by b(h,I)= w,(L, - I), where both L, and w,, are nondecreasing in the skill level h. The functions uh used in our analysis are then given by
uh(a,b(h, 1)) = u(a, 1). Social welfare is a concave symmetric function of the individual utility levels. Leisure is assumed to be a normal good,5 that is -+, is nondecreasing in a. Clearly, given any opportunity set in (a, b) space, uh’>uh for h’>h,6 for individual h’ can choose the same position as h, obtain an equal amount of the consumption good and yet retain more leisure. Hence the SWF will always weight changes in the utility of the less skilled type more heavily. Also, assumption B ensures that more skilled types will choose a higher pretax income: if h wh, this redistribution also generates some extra revenue: the arrow at A can be taken to be somewhat longer than that at B. In case II, assumption VWR is satisfied by a redistribution of the consumption good (a) alone, provided that leisure is a normal good. For by concavity of U, an equal redistribution from B to C would be desirable. And the marginal utility of the consumption good is lower at A than at B: along
4We assume that efficiency units of labor of all skill types are perfect substitutes in social production. See Stern (1982) for an example of a model in which relative marginal products of different skill types are endogenous, and the marginal tax rate is not everywhere-non-negative. sThough Mirrlees (1971) does not seem to require normality of leisure, Seade (1982) shows that his -arguments implicitly use the more restrictive assumption of additive separability of tastes. Seade’s derivation of the non-negativity of the marginal tax rate in a continuum economy assumes normality of leisure, as we do. 6This is not enough to assure that redistribution from h’ to h is desirable, even for concave u; see Dixit and Seade (1979). Thus, assumption VWR remains to be verified.
272
A.A. R&d. Marginul
tax rate
Fig. 2
the indifference
curve,
du, = u,, dl+ u,, da
provided that leisure is not inferior. Thus assumptions B and VWR are satisfied, and by proposition 6’, the marginal tax rate is non-negative. This is exactly Seade’s (1982) theorem, for a model that is identical except that the continuum population is replaced by a finite one.
References Dixit, A. and J. Seade, 1979, Utilitarian versus egalitarian redistributions, Economics Letters 4, 121-124. Guesnerie, R. and J. Seade, 1982, Nonlinear pricing in a finite economy, Journal of Public Economics 17, 1577179. Mirrlees, J.A., 1971, An exploration in the theory of optimum income taxation, Review of Economic Studies 38, 175-208. Seade, J., 1982, On the sign of the optimum marginal income tax, Review of Economic Studies 49, 6377643. Stern, N.H., 1982, Optimum taxation with errors in administration, Journal of Public Economics 17.181~211.
of Public
Economics
A NOTE
28 (1985) 2677272. North-Holland
ON THE MARGINAL TAX RATE IN A FINITE ECONOMY Ailsa A. RoELL* London School of Economics, Massachusetts
Received
London WC2A
Institute
2AE, UK
of Technology
April 1984, revised version
received June 1985
The marginal tax rate is shown to be non-negative in Guesnerie and Seade’s model of nonlinear pricing in a finite economy under assumptions that are as mild as those customarily adopted in nonlinear tax models with a one-dimensional continuum population.
1. Introduction Guesnerie and Seade (1982) (henceforth G/S) have investigated the properties of the optimal tax schedule in a model with finitely many consumers. This note extends their analysis of the direction of the distortion induced by redistributive taxation (proposition 6, p. 172) to a more general setting. The argument is extremely simple and does not suffer from the messiness that led G/S to confine their attention to a more restrictive setting. Section 3 shows that our result, when applied to the standard optimal income tax problem, implies the non-negativity of the marginal tax rate in a setting that is fully as general as that customarily investigated in models with a continuum of types, such as that of Mirrlees (1971). In contrast, an additional assumption, such as separability of preferences in consumption and leisure, seems to be needed to meet the conditions of G/S’s original proposition 6. 2. Extension of Guesnerie and Seade’s proposition G/S consider an economy with two goods; a consumption bundle is denoted x = (a, b). Households are indexed by h = 0,...,H and preferences are represented by a continuously differentiable and strictly concave utility function uh(a,b), increasing in a, which for our purposes may be regarded as the consumption good ‘after-tax income’, and decreasing in b, to be thought *The author
would like to thank
0047-2727/85/$3.30
0
Jesus Seade and Nicholas
1985, Elsevier Science Publishers
Stern for their valuable
B.V. (North-Holland)
comments.
268
A.A. Riiell, Marginal tax rate
of as a measure of effort or ‘before-tax income’. The marginal rate of substitution sh(x) is given by -Z&X)/L&X). A feasible nonlinear tax schedule is defined by a set of distinct ‘corners’ {Ci}izO,,.,, T in (a, b) space and an assignment of consumers h to nonempty corner membership sets Hi such that every consumer h prefers the consumption bundle at the corner Ci for which h E Hi to any other bundle, and a fixed net revenue K is raised (& ah xhbhZ K). Due to the monotonicity of preferences in a and b, corners can be ranked unambiguously and indexed in order of ascending size of (ai, bi). Two principal assumptions are adopted by G/S to investigate the marginal tax rate (G/S, pp. 168 ff.): Assumption
B.
Consumers
h>h’osh(x)
can be indexed
in such a way that:
Vx,h,h’.
Assumption R. ([strong] redistributive assumption). For each pair of corners Ci, with i
much
weaker
assumption,
VWR.’ ([ ver y weak] redistributive assumption). For each pair of Cj with i < j, given K > 0 small enough, for any h E Hi and h’ E Hj zh, zh’ E [0, K] such that it is desirable to distribute (K - zh, -z”) and (K -zh’, -z”‘) away from agent h’, provided incentive effects
One may regard K as the quantity of money that is indirectly redistributed from h’ to h, as depicted in fig. 1. For convenience we will adopt the usage that two corners Ci < Cj are leftlinked if uh(Ci) = uh(Cj) for some h E Hi and right-linked if this holds for some h E Hj.2 It is easily verified, as shown by G/S in lemma 2, that assumption B implies: (i) two distinct corners cannot be both left- and right-linked; (ii) only adjacent corners can be left- or right-linked at all. ‘As a point of detail, this assumption seems stronger than that of G/S in one respect: it is required to hold for all small enough redistributions rather than for some fixed redistribution. Observe, however, that G/S do allow themselves to set the assumed redistribution arbitrarily small in their calculations. *These terms refer to pairs of corners and should not be confused with G/S’s concepts of winner-linked and loser-linked corners, which refer to single corners. Under assumption B, for example, a corner j is winner-linked if it is right-linked to some corner i
A.A. Riiell, Marginal
269
tax rate
Fig. 1
We now derive our main result, that of proposition 6 of G/S, under assumptions B and VWR. The proof is rendered considerably more succinct by a suitable choice of the corners that are to be subjected to a redistributive experiment. Proposition 6’. Under corners are right-linked.
assumptions
B and
VWR,
all pairs
of successive
Proof Suppose that successive corners which are not right-linked do exist. Let (Cj_r, Cj) be the highest such pair and let (Ci,Ci+r) be the lowest such pair; we do not rule out the possibility that i=j1. We will construct a redistribution from members of Hj to members of Hi that satisfies all constraints and imrpoves the social objective. From the definition of Ci, Ci and Ci+ 1 are not right-linked. Also, if i # 0, Ci and Ci_ 1 are not left-linked since they are, by assumption, right-linked. Thus all consumers who are not at corner Ci (in particular, those at Ci+r and Ci_ r, if such a corner exists) strongly prefer their own position. By continuity of the utility function, they would still prefer their own position to any point in a small enough neighbourhood around Ci. Similarly, if j # T, Cj and Cj+ 1 are not left-linked; and C,_r and Cj are not right-linked. Hence any consumer heHj will prefer positions in a small enough neighbourhood around Cj to any other corners. Distribute K >0 to all consumers ~EH~ by assigning to each of them a bundle (ui + K - xh, bi - x”) that maximizes
subject
to 0 5 x 5 K, where (bi, ai) are the coordinates
of corner
Ci.
270
A.A. Riiell, Marginal tax rate
Define K’ by: K’ = K. card Hi/card Hj. Now redistribute K’> 0 away from all individuals consumption to (aj - K’ + xh’, bj + xh’) that maximizes
h’ E Hj by setting
their
uh’(aj - K’ + x, bi + x) subject to 0 5 x 5 K’. Note that all members of Hj are being shifted to new corners that involve less consumption and leisure than Cj. Thus nonmembers of Hj will not be tempted to follow suit. As argued above, for K (and hence K’) small enough, the suggested redistribution will not violate any incentive constraints connecting Ci or Cj and other corners. Also, since all agents within Hi and Hj are assigned their optimal value of xh subject to a common constraint, there is no problem with incentive constraints amongst the members of Hi and Hj, respectively. The fact that corners Ci and Cj may ‘explode’ into many points need not cause any concern. Lastly, the suggested move (for K suitably small) improves the Paretian social objective, given assumption VWR. For we have chosen the vectors (K -xh, -x”) and (-K’ + xh’, x”‘) optimally from the point of view of the utilities of the agents involved, given the amount K to be redistributed. Q.E.D. The non-negativity of the marginal corollary of this proposition. 3. Non-negativity
tax rate is, as shown
by G/S, a direct
of optimal marginal income tax
Consider assumptions B and R. The first of these assumptions corresponds to a standard assumption generally used throughout the optimal income tax literature; for example in the model of Mirrlees (1971) it is equivalent to assumption (B), p. 182. The second is, however, too restrictive. For, as noted by G/S, the marginal utility of commodity a (consumption) is not independent of the quantity of commodity b (labor, in efficiency units) consumed. Thus in general the desirability of redistributing utility need not imply the desirability of redistributing good a. In addition to the usual assumptions adopted in the standard optimal tax setting, an extra assumption, such as separability of preferences in consumption and leisure, would be needed in order to ensure that assumption R holds.3 3If preferences
are separable,
u(a, 4 = .OA:(a) -h(l)), for f and g increasing
and concave.
Then:
u,= f’(.k’(a). Thus, any agent with both higher utility and higher after tax income must have a lower marginal utility of consumption, regardless of his effort level. This implies assumption R.
A.A. Riiell, Marginal
tax rate
271
In contrast, assumptions B and VWR are weak enough to include the usual identical-preference one-dimensional skill models of optimal income taxation.4 In particular, they are implied by the model used by Seade (1982). To see this, let all members of society ;have the same concave utility function ~(a, I) defined over consumption (a) and leisure (I) and increasing in both. Pre-tax income, that is, labor in efliciency units, is given by b(h,I)= w,(L, - I), where both L, and w,, are nondecreasing in the skill level h. The functions uh used in our analysis are then given by
uh(a,b(h, 1)) = u(a, 1). Social welfare is a concave symmetric function of the individual utility levels. Leisure is assumed to be a normal good,5 that is -+, is nondecreasing in a. Clearly, given any opportunity set in (a, b) space, uh’>uh for h’>h,6 for individual h’ can choose the same position as h, obtain an equal amount of the consumption good and yet retain more leisure. Hence the SWF will always weight changes in the utility of the less skilled type more heavily. Also, assumption B ensures that more skilled types will choose a higher pretax income: if h
4We assume that efficiency units of labor of all skill types are perfect substitutes in social production. See Stern (1982) for an example of a model in which relative marginal products of different skill types are endogenous, and the marginal tax rate is not everywhere-non-negative. sThough Mirrlees (1971) does not seem to require normality of leisure, Seade (1982) shows that his -arguments implicitly use the more restrictive assumption of additive separability of tastes. Seade’s derivation of the non-negativity of the marginal tax rate in a continuum economy assumes normality of leisure, as we do. 6This is not enough to assure that redistribution from h’ to h is desirable, even for concave u; see Dixit and Seade (1979). Thus, assumption VWR remains to be verified.
272
A.A. R&d. Marginul
tax rate
Fig. 2
the indifference
curve,
du, = u,, dl+ u,, da
provided that leisure is not inferior. Thus assumptions B and VWR are satisfied, and by proposition 6’, the marginal tax rate is non-negative. This is exactly Seade’s (1982) theorem, for a model that is identical except that the continuum population is replaced by a finite one.
References Dixit, A. and J. Seade, 1979, Utilitarian versus egalitarian redistributions, Economics Letters 4, 121-124. Guesnerie, R. and J. Seade, 1982, Nonlinear pricing in a finite economy, Journal of Public Economics 17, 1577179. Mirrlees, J.A., 1971, An exploration in the theory of optimum income taxation, Review of Economic Studies 38, 175-208. Seade, J., 1982, On the sign of the optimum marginal income tax, Review of Economic Studies 49, 6377643. Stern, N.H., 1982, Optimum taxation with errors in administration, Journal of Public Economics 17.181~211.