Tax Reform and Progressivity

The Economic Journal, 110 ( January), 50±68. # Royal Economic Society 2000. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA. TAX REFORM AND PROGRESSIVITY Michael Keen, Harry Papapanagos and Anthony Shorrocks The established theory of tax progressivity cannot handle basic tax reform questions, such as whether an increase in personal allowances makes the tax system more progressive, because the core results assume that tax liability is never zero. This paper generalises the core theory to allow for zero tax payments, and applies the new framework to the analysis of allowances, income-related deductions and tax credits. Log concavity of the tax scheduleÐa property quite distinct from any existing notion of progressivityÐemerges as the critical determinant of whether the distribution of the tax burden becomes more progressive as allowances are increased. Does an increase in the personal allowance1 make the tax system more, or less, progressive? On one hand, an increase in allowances has the attractionÐwhich Finance Ministers have never been slow to point outÐof removing poor people from the tax system. By eliminating the liabilities of the poorest taxpayers, it may seem that progressivity is increased. But there is another, less evident, effect that acts in the opposite direction: for a £100 increase in allowances is worth £20 of after-tax income to a taxpayer facing a marginal tax rate of 20%, but £40 to the better-off taxpayer facing a 40% marginal rate.2 It is not immediately obvious, therefore, whether progressivity rises or falls. For an answer to this apparently simple question, one might naturally look to the literature on tax progressivity in the hope of ®nding appropriate conditions on the tax schedule which enable unambiguous conclusions to be drawn. Yet one looks in vain.3 There have been some attempts to decompose global measures of progressivity in the presence of allowances and the like: see, in particular, Lambert (1993), Loizides (1988) and PfaÈhler (1990). But the more basic problem of establishing conditions under which increasing allowances has an unambiguous impact on tax progressivity seems not only to be unresolved, but to have remained unexplored.  This paper forms part of the work of the network `The distribution and redistribution of income', ®nanced by the European Community under contract #ERBCHRXCT940647. We thank network members, especially Patrick Moyes and Peter Lambert, and two referees for their helpful comments. The views expressed are not necessarily those of any of the institutions to which the authors are af®liated. 1 We use the term `allowance' in the sense of a ®xed amount to be subtracted from pre-tax income in arriving at taxable income: a deduction, that is, which is independent of income. This is often called an `exemption' in the United States, although Steuerle (1983) uses the term `family allowances' when referring to US exemptions such as the standard deductions for non-itemisers, whose evolution he documents from 1948 to 1984. By `threshold' we mean the level of income at which tax becomes payable. 2 Growing awareness of this in Britain has led to an increased tendency to restrict the bene®t of various allowancesÐsuch as mortgage interest tax relief and the married couple's allowanceÐto a common (low) rate. Such a restriction effectively turns allowances into non-refundable tax credits. 3 An exception is Lambert (1985), who considers the implications of increasing the threshold of a linear tax system (one, that is, with a constant marginal tax rate above the threshold). But this special case precludes the second consideration just mentioned: that the increased allowance is worth more to taxpayers facing higher marginal tax rates. [ 50 ] [ J A N U A R Y 2000] TAX REFORM AND PROGRESSIVITY 51 The same point applies more widely to the analysis of other structural tax reforms. For studies of tax progressivity typically do no more than take tax liability as a function t(u) of pre-tax income u,4 and proceed to relate the distributional impact to the shape of t. It is widely recognised, of course, that the schedule t in such analyses is akin to a reduced form, re¯ecting such structural features as: allowances and income-related deductions which determine taxable income ö(u); the rate schedule used to determine the tax assessment ô(ö); and any tax credits applied to yield the ®nal tax liability T(ô). Focusing on the convolution t(u)  T fô[ö(u)]g con¯ates these individual effects. For many purposes, this represents a gain in elegance and generality. But for othersÐnot least the analysis of the kinds of tax reforms one often observes in practiceÐit becomes important to distinguish between the different components. Indeed, one might argue that the failure to do so has left us with surprisingly little understanding of the progressivity implications of even the most basic structural reforms. The principal aim of this paper is to investigate the progressivity effects of key structural tax reforms. Speci®cally, we seek to establish conditions under which one can reach unambiguous conclusionsÐconclusions, that is, which hold for any conceivable distribution of pre-tax incomesÐas to the effects on progressivity of increasing allowances, deductions and/or credits whilst keeping the rate structure unchanged. The previous neglect of these obviously important issues re¯ects a striking feature of the core results in the theory of progressivity: they are simply not applicable to these problems. For attention has been restrictedÐboth invariably and implicitly, to the best of our knowledgeÐto situations in which tax liability is always strictly positive. This is manifestly not the case in practice. Moreover, as is clear from the example of allowances, zero tax payments are not merely a technical irritation but are a central concern in the design and reform of tax schedules. Thus the ®rst task in this paper is to extend the pivotal results of the theory of tax progressivity to circumstances in which tax payments at some income levels may be zero. Powerful and novel insights into the progressivity effects of allowances and the like then emerge as corollaries of these more general results. Section 1 of the paper provides background, while Section 2 develops the generalisations of previous core results. These general results are then applied to the analysis of personal allowances in Section 3, and, more brie¯y, of income-related deductions and credits in Section 4. Section 5 concludes. 1. Background Here we ®rst set out our formal framework and then develop both the need for and the context of our later extensions by recalling, brie¯y, the core results of the established literature on tax progressivity. 4 And perhaps also of such personal characteristics as age and number of dependents, a possibility that, for simplicity, we ignore in this paper: the consequences of such heterogeneity for progressivity comparisons are discussed by Lambert (1994) and Moyes and Shorrocks (1998a, b). # Royal Economic Society 2000 52 THE ECONOMIC JOURNAL [JANUARY 1.1. The Framework We con®ne our attention to a population of tax units distinguished only by their positive pre-tax incomes xi 2 R‡‡  (0, 1). The income distribution for n , an n-person population is represented by a vector x ˆ (x1 , x2 , . . ., xn ) in R‡‡ and the complete set of income distributions for populations of arbitrary size S n is denoted by X ˆ 1 nˆ1 R‡‡ . When incomes are subjected to the transforman tion f : R‡ ! R, we use f(x) ˆ [ f (x1 ), . . ., f (xn )], x 2 R‡‡ to indicate the distribution of transformed values. The tax structure is characterised by a tax schedule t(u), u 2 R‡  [0, 1), or its corresponding residual (or net) income function r (u)  u ÿ t(u), u 2 R‡ . We maintain throughout the assumptions: t is continuous on R‡ ; (1a) t and r are each non-decreasing on R‡ ; (1b) 0 , t(u) , u for some u 2 R‡ ; (1c) 0 < t(u) < u for all u 2 R‡ ; t has a right derivative t9(u) at all u 2 R‡ ; (1d) (1e) and denote by T the set of functions t: R‡ ! R which satisfy (1a)±(1e). Note that taxes and residual incomes are treated symmetrically, in the sense that t 2 T if and only if the corresponding residual income schedule r 2 T . The constraints imposed by (1a)±(1e) are relatively innocuous. Assumptions (1a) and (1b)Ðprecluding jumps and systematic re-rankingÐare familiar ones in the literature;5 Fei (1981), for instance, regards a non-decreasing r as a prerequisite for `incentive preservation'. Condition (1c) is included simply to rule out degenerate cases in which either t(u) or r (u) is always zero, while condition (1e) is a weaker (and much more realistic) variant of the standard assumptionÐreferred to by Thon (1987) as a `calculus hang-up'Ðthat tax schedules are differentiable everywhere.6 The special signi®cance of (1d) is that it admits tax schedules t for which the tax threshold z(t)  inffujt(u) . 0g (2) is greater than zero. It is in this simple but crucial respect that we depart from previous work on tax progressivity, which has considered only the subset T of T for which residual incomes r (u) andÐmore disturbinglyÐtaxes t(u) are strictly positive for all u . 0. By restricting attention to T the literature has 5 Which is not to say that they invariably hold in practice: in many countries, such as France and Britain, the structure of social security contributions includes discontinuities which have caused effective marginal tax rates (re¯ecting both social security contributions and income taxation) to exceed 100%. In practice, however, re-ranking usually arises not because marginal tax rates exceed 100%, but because tax units differ in non-income characteristics (such as household size), or receive income from a variety of sources (which may be taxed differently); neither circumstance applies here. 6 Condition (1e) will hold automatically if t satis®es (1a)±(1d) and is also convex: see Hardy et al. (1952, p. 91). We do not assume that t is convex, although it is a common feature of real-world tax schedules. For tax schedules in the set T , convexity is equivalent to the assumption that the marginal tax rate is everywhere non-decreasing, and is suf®cient to ensure that the schedule is progressive in the sense de®ned below. # Royal Economic Society 2000 2000] TAX REFORM AND PROGRESSIVITY 53 immediately removed from considerationÐunwittingly, it seemsÐall tax schemes likely to be encountered in practice. It is this obstacle which must be overcome before the progressivity effects of allowances, for example, can be examined; we resolve the problem in Section 2 below. As a ®nal preliminary, we de®ne a tax schedule t to be progressive if and only if the average tax rate t(u)=u is non-decreasing in u for all u . 0.7 1.2. Previous Results The established theory of tax progressivity rests on a series of results, due variously to Fellman (1976), Jakobsson (1976) and Kakwani (1977), which link the shape of the tax schedule with the distributions of pre-tax incomes, tax payments and residual incomes.8 These results occupy a central place in analyses of the distributional effects of taxation, and will be familiar to many readers. What is not widely recognised, however, is that they are crucially dependent on a tacit restriction of admissible tax schedules to the set T . To describe these results and to bring out their limitationsÐso setting the scene for our analysisÐwe ®rst recall that x 2 Rn (weakly) Lorenz dominates y 2 Rn (written x LD y) if and only if ì(x) . 0; ì(y) . 0; and k y k x P P (i) (i) > for k ˆ 1, . . ., n: iˆ1 ì(y) iˆ1 ì(x) (3) where ì(x) is the mean of x, and x(i) is its ith smallest element. If, for any given functions f : R‡ ! R and g : R‡ ! R, we have f(x) LD g(x), for all x 2 X , (4) we will say that f(x) uniformly Lorenz dominates g(x), and write (4) more n and succinctly as f(x) ULD g(x). With a pre-tax income distribution x 2 R‡‡ tax schedule t 2 T , the corresponding vectors of taxes and residual incomes are represented by t(x) ˆ [t(x1 ), . . ., t(xn )] and r(x) ˆ x ÿ t(x), respectively. One of the core results on tax progressivity may now be stated as: Proposition 1. Suppose t 2 T . Then t is progressive if and only if r(x) LD x LD t(x), for all x 2 X : (5) or, equivalently, r(x) ULD x ULD t(x). Thus if a tax schedule t 2 T is progressive according to the average tax rate criterion, then it is also `distributionally progressive', both in the sense that it always induces a distribution of post-tax income which Lorenz dominates the distribution of pre-tax income, and in the sense that the 7 For tax schedules in T , progressivity is suf®cient to ensure that residual incomes are strictly positive. For if r (v) ˆ 0 for some v . 0, then r (u) ˆ 0 for u > v by progressivity, and r (u) ˆ 0 for u < v by property (1b). Hence r (u) ˆ 0 for all u . 0, contradicting assumption (1c). 8 These core results have been extended and further re®ned by Hemming and Keen (1983), Eichhorn et al. (1984), Thon (1987), Latham (1988), and Thistle (1989) among others. # Royal Economic Society 2000 54 THE ECONOMIC JOURNAL [JANUARY corresponding tax payments are never distributed more equally than pretax incomes. Proposition 1 is proved by both Fellman (1976) and Kakwani (1977). Neither, however, notes that the reliance on the notion of Lorenz dominance limits its validity to t 2 T . The reason for this limitation is simple: if t(u) ˆ 0 for some u . 0, then there exists a pre-tax income distribution x such that ì(t(x)) ˆ 0, and x is not Lorenz comparable with t(x), as required in order for condition (5) to hold.9 The second set of key results concerns the relative progressivity of alternative tax schedules, making statements to the effect that one tax schedule t a is more progressive than another t b . These are precisely the kinds of statements one needs to make in order to gauge, for example, whether an increase in allowances leads to a rise in progressivity. The appropriate de®nition of relative progressivity is a more fundamental question than whether a given tax schedule is progressive. For once a criterion for relative progression has been established, it becomes natural to regard a proportional tax (i.e. one for which the average tax rate is constant over all incomes) as being neutral with respect to tax progression, and to say that t is progressive if it is no less progressive than any proportional tax. Relative progressivity is usually formulated in two alternative ways, one focusing on the distribution of tax payments (liability progression) and the other on the post-tax income distribution (residual progression). The former approach leads to the notion that the tax schedule t a 2 T is uniformly L- progressive relative to t b 2 T written t a LP t b , if tb (x) always weakly Lorenz dominates ta (x); in other words, tb (x) ULD ta (x). In contrast, the residual progression approach suggests that t a 2 T is uniformly R- progressive relative to t b 2 T , written t a RP t b , if ra (x) ULD rb (x). We take no position here on the merits of these two approaches, but apply both in our subsequent analysis.10 Using å( f , u)  uf 9(u)= f (u) to denote the elasticity of f at a point u such that f (u) 6ˆ 0, previous results establish: Proposition 2. Suppose t a , t b 2 T . Then t a LP t b if and only if å(t a , u) > å(t b , u), for all u . 0. Proposition 3. Suppose t a , t b 2 T . Then t a RP t b if and only if å(r b , u) > å(r a , u), for all u . 0. 9 This point is also overlooked by Lambert (1993) in his proof of the Jakobsson-Kakwani theorem, despite his explicit intention `to admit into the analysis the very relevant case of a tax which is zero below a threshold' (p. 147). 10 It may appear that the choice between the liability progression criterion LP and the residual progression criterion RP hinges on whether one is ultimately interested in the distribution of tax payments or the distribution of post-tax incomes. This is somewhat misleading. The real distinction between LP and RP lies in the tax schedules regarded as equally progressive. From the viewpoint of liability progression, a scale change in all tax liabilities changes the level of tax payments, but not the degree of progression. In contrast, residual progression sees t a and t b as equally progressive if post-tax incomes are proportional. Thus one might adopt the liability progression criterion whilst simultaneously agreeing the distribution of post-tax incomes has more welfare signi®cance than the distribution of tax payments. # Royal Economic Society 2000 2000] TAX REFORM AND PROGRESSIVITY 55 Propositions 2 and 3 are proved in the classic paper by Jakobsson (1976), but he does not note (or, it seems, notice) the restriction t a , t b 2 T . The resulting problems are most evident in Proposition 2. Even if liability progression LP can be suitably reformulated to avoid the earlier dif®culty with the Lorenz dominance condition, the elasticity test in Proposition 2 cannot remain as currently stated, for the simple reason that the elasticity å(t, u) is not de®ned when t(u) ˆ 0. Similar problems will also arise in the context of residual incomes and residual progression if post-tax income r (u) is zero for some u . 0. The restriction of Propositions 1±3 to t 2 T is extremely worrisome, since in practice almost all schedules exclude some incomes from tax altogether, and many reforms alter the set of incomes excluded from tax. The prevalence of allowances, and of reforms that change allowances, are obvious examples. To address such reforms, Propositions 1±3 must be extended to the wider set T , and that is the taskÐwhich proves more substantive than one might have expected, both technically and in the additional implications it yieldsÐto which we now turn. Of course the issue is more pressing in terms of liability progression than it is for residual progression, as it is exceptional for residual income to be zero. For completeness, however, and because our ultimate aim is to apply both approaches, we provide generalisations for each in turn. 2. Generalisations To extend Propositions 1±3 to tax schedules in the set T , it is ®rst necessary to ®nd a suitable alternative to uniform Lorenz dominance statements of the form f(x) ULD g(x), (6) so that comparisons can be made when the mean of f(x) or g(x) is allowed to be zero. One simple and obvious solution is to disregard the comparisons involving distributions with zero means, so that (6) becomes f(x) LD g(x), for all x 2 X such that ì[f(x)] . 0 and ì[g(x)] . 0: (7) We will refer to this as quali®ed Lorenz dominance, and write f(x) QLD g(x) when condition (7) is satis®ed. Another possibility is to follow Moyes and Shorrocks (1998a) and say that f(x) is never Lorenz dominated by g(x) if, for all x 2 X , f(x) is not strictly Lorenz dominated by g(x). Writing this as f(x) NLD g(x), (8) it is easy to see that (6) implies (7) which in turn implies (8). But the reverse implications are not valid in general, since (8) allows the Lorenz curves to intersect, and (7) allows the distributions to have zero means. For the purposes of this paper, the problems associated with zero tax ranges are resolved by replacing ULD with either QLD or NLD: conditions (7) and (8) are equivalent when f and g are members of the set T .11 Although the 11 See Lemma 1 below. Although condition (8) in principle allows the Lorenz curves for f(x) and g(x) to intersect, this possibility does not arise when f and g are non-decreasing. # Royal Economic Society 2000 56 THE ECONOMIC JOURNAL [JANUARY NLD criterion is less demanding and simpler to interpret, QLD provides a more transparent extension of ULD to situations involving zero mean distributions. For this reason, we frame our analysis below in terms of quali®ed Lorenz dominance, deriving the characteristics of tax schedules which ensure that the appropriate Lorenz dominance condition holds for all distributions with nonzero means. Our main results are obtained as corollaries of the following lemma: Lemma 1. For f , g 2 T , the following statements are equivalent: f(x) QLD g(x); (9a) f(x) NLD g(x); (9b) f (u) is non-increasing in u whenever g (u) . 0: g (u) (9c) Furthermore, the above conditions imply f (u) . 0 whenever g (u) . 0:12 (10) Proof. See the Appendix. One implication of Lemma 1 is that condition (9c) is both necessary and suf®cient to ensure that the distribution f(x) is never strictly Lorenz dominated by g(x), irrespective of the choice of x. The supplementary condition (10) follows from (9c) because f is a member of the set T , and hence has the properties listed in (1). Though (10) is technically redundant, it is worth drawing attention to this relation between f and g , as it plays an important role in the subsequent analysis. Lemma 1 enables the core results contained in Propositions 1±3 to be extended to the wider set T . By setting f (u) ˆ r (u) ˆ u ÿ t(u) and g (u) ˆ u in Lemma 1, and then choosing f (u) ˆ u and g (u) ˆ t(u), we immediately obtain: Proposition 19. Suppose t 2 T . Then t is progressive if and only if r(x) QLD x QLD t(x): (11) This is evidently a straightforward generalisation of Proposition 1, the only difference being that condition (11) disregards attempts to make Lorenz comparisons between pairs of distributions when the mean of either of the distributions is zero.13 12 Note that Lemma 1 has some similarities with Proposition B.1.b of Marshall and Olkin (1979, p. 129), although their result uses the Lorenz dominance criterion, and is framed in terms of ordered vectors rather than functions. 13 Lemma 1 enables statement (11) to be replaced by r(x) NLD x NLD t(x): (119) In principle, this condition is less demanding than the corresponding requirements (5) in Proposition 1 or (11) in Proposition 19, because it allows for the possibility that the Lorenz curves intersect. # Royal Economic Society 2000 2000] TAX REFORM AND PROGRESSIVITY 57 Turning now to progressivity comparisons between pairs of tax schedules, consider ®rst the approach based on liability progression. Here it is natural to extend the earlier ordering LP de®ned for tax schedules in T to the broader class T by saying that t a 2 T is uniformly L-progressive relative to t b 2 T (again written t a LP t b ) if tb (x) QLD ta (x). If we have t a LP t b but not t b LP t a , we will say that t a is uniformly more L- progressive than t b . Setting f ˆ t b and g ˆ t a in Lemma 1, we deduce that t a LP t b if and only if t b (u) is non-increasing in u whenever t a (u) . 0: t a (u) (12) In addition, using (10), we have t b (u) . 0 whenever t a (u) . 0 or, equivalently in terms of the associated tax thresholds, z(t a ) > z(t b ). Reinterpreting condition (12) in terms of elasticities then yields Proposition 29. Suppose t a , t b 2 T . Then t a LP t b if and only if z(t a ) > z(t b ); and (13a) å(t a , u) > å(t b , u), for all u . z(t a ): (13b) Proposition 29 offers a simple two-step procedure for establishing whether t a is uniformly L-progressive relative to t b : ®rst, check that t a does not have a lower threshold; then con®rm that its liability elasticity is at least as high at all incomes above that threshold. This result is clearly a generalisation of its counterpart in Section 1. For if t a , t b 2 T , we have z(t a ) ˆ z(t b ) ˆ 0, and condition (13) reduces to the elasticity test described in Proposition 2.14 Proceeding in a similar fashion for comparisons based on residual income progression, we extend the ordering RP to all tax schedules in T by saying that t a is uniformly R-progressive relative to t b (written t a RP t b ) if ra (x) QLD rb (x). The schedule t a is uniformly more R- progressive than t b if t a RP t b is true, but t b RP t a is false. Applying Lemma 1 as before, we deduce that t a RP t b if and only if r a (u) is non-increasing in u whenever r b (u) . 0, r b (u) (14) which when expressed in terms of elasticities yields: Proposition 39. Suppose t a , t b 2 T . Then t a RP t b if and only if å(r b , u) > å(r a , u) whenever r b (u) . 0:15 (15) This is again a generalisation of its earlier counterpart for t 2 T , though here 14 While it may seem that Proposition 2 can be resurrected by simply restricting the elasticity comparisons to income levels at which tax payments are positive under both tax schedules, this is true only if condition (13a) holds. For the same reason, it is impossible to combine conditions (13a) and (13b) into a single statement resembling Proposition 2 by assigning an arbitrary value (say, 0 or 1) to the elasticity at income levels where tax payments are zero. 15 The supplementary condition corresponding to (13a) would require that r a (u) . 0 whenever r b (u) . 0. As the statement and proof of Lemma 1 makes clear, this is an implication of condition (15) and may therefore be omitted. # Royal Economic Society 2000 58 THE ECONOMIC JOURNAL [JANUARY the economic signi®cance of the generalisation is rather less, since tax schedules typically leave all taxpayers with strictly positive net incomes. The result is stated here both for completeness and later reference. For, taken together, the results of this section provide the tools needed to address the issues raised at the outset. 3. Progressivity and Allowances Consider ®rst the progressivity comparison between an original tax structure t 2 T and the tax structure after reform given by t a (u) ˆ t(maxfu ÿ á, 0g), u . 0, (16) with á . 0. Here t a is characterised by the same rate schedule as t, but differs in that any pre-existing allowances (tacitly absorbed into the form of t)16 are increased by the ®xed amount á. Since t is a member of T , it follows that t a is also a member of T . Furthermore, it may be con®rmed that t a is progressive whenever t is progressive.17 However, our concern is not whether t a is progressive, but whether it is more or less progressive than t. 3.1. Allowances and Liability Progression We begin with the comparison from the viewpoint of liability progression. From (16) it is evident that z(t a ) ˆ z(t) ‡ á . z(t). So, by Proposition 29, t cannot be uniformly more L-progressive than t a : increasing allowances cannot lead to an unambiguous reduction in liability progression. This re¯ects the ®rst of the effects of raising allowances noted in the Introduction: taking the poor out of tax can only shift the tax burden away from them. But this does not mean that liability progression necessarily increases, for recall that there is a second effect operating against such an outcome: a given reduction in taxable income generates a larger reduction in tax payments when the marginal tax rate is higher. Using condition (13b) of Proposition 29, it is necessary and suf®cient for a uniform rise in L-progression that t9u å(t a , u) t9(u ÿ á) < ˆ whenever u . z(t a ) ˆ z(t) ‡ á: t(u) u t(u ÿ á) (17) For (17) to hold, it is suf®cient that the original tax schedule t be log concave in the sense that18 16 More generally still, of course, t may incorporate income-related deductions and/or credits of the kind discussed in Section 4 below. 17 The progressivity condition t a (u)=u < t a (v)=v for all u , v is trivially satis®ed when 0 , u < á; while for á , u , v, progressivity of t yields t a (u) t(u ÿ á) : u ÿ á t(v ÿ á) : v ÿ á t a (v) ˆ < ˆ : u uÿá u vÿá v v 18 In order to avoid unnecessary repetition below, we omit the quali®cation that log concavity applies only to income levels above the tax threshold. # Royal Economic Society 2000 2000] TAX REFORM AND PROGRESSIVITY t9u d ln t(u) ˆ is non-increasing in u, for all u . z(t); t(u) du 59 (18) in other words, the logarithm of t is concave above the tax threshold. Furthermore, if (17) is required to hold for any increase in the level of allowances (as well as any initial income above the tax threshold), then (18) is also seen to be necessary. Hence: Proposition 4. Increasing allowances within the schedule t 2 T : (a) Never leads to a uniform reduction in L-progressivity; (b) Always leads to a uniform rise in L-progressivity if and only if t is log concave. The potential signi®cance of log concavity for progressivity comparisons has not been recognised in the past. While initially unfamiliar, however, its importance in the present context is easily understood. As noted earlier, the reason why increasing allowances may not lead to a more progressive distribution of the tax burden is that their value to the taxpayer increases with her marginal tax rate. Under progressive taxationÐand for the moment we speak loosely, since progressivity does not imply an increasing marginal tax rateÐan increase in allowances tends to be worth more, in absolute terms, to the better off. Therefore, to guarantee a progressive effect from increasing allowances, the tax schedule must not be `too progressive'. Crudely speaking, the log concavity property meets this requirement by ensuring that the tax schedule itself is not too convex. Somewhat more precisely, if an increase in allowances is to lead to a more unequal distribution of tax payments, the critical requirement is that the proportionate reduction in tax payments be greater for the poor than for the rich. And since the proportionate reduction for a taxpayer with income u is simply t9(u)=t(u), the requirement is exactly log concavity. It should be emphasised that log concavity is an entirely distinct concept from that of progressivity: for t 2 T to be progressive it is neither necessary nor suf®cient that t is log concave.19 Nevertheless, it seems likely that tax schedules encountered in practice will be both progressive and log concave. Since d ln t(u) å(t, u) ˆ , for all u . z(t), du u a tax schedule will be log concave if and only if the elasticity of tax payments with respect to income does not increase faster than income itself. Progressivity, on the other hand, requires that å(t, u) is never strictly less than one. It is 19 The tax schedule t(u) ˆ 1 ÿ eÿu , u > 0, is strictly log concave but not progressive, while ( 2 1 ue u ÿ1 , u 2 [0, 1] t(u) ˆ 31 u.1 3 u, p provides an example of a progressive tax schedule which is strictly log convex on the interval [ 12, 1], and hence not log concave. It is a simple matter to check the properties of both these schedules, and to con®rm that both are members of T . # Royal Economic Society 2000 60 THE ECONOMIC JOURNAL [JANUARY relatively easy to construct realistic examples of tax functions which satisfy these two properties. For instance, the tax schedule which imposes a constant marginal tax rate ã above some threshold c, so that t(u) ˆ maxfã(u ÿ c), 0g, (19) is both progressive and log concave. 3.2. Allowances and Residual Progression The impact of increasing allowances on R-progressivity is more complex. In this case one cannot immediately conclude from the higher threshold that progressivity cannot fall. Indeed, as we now see, there can be either a uniform rise in R-progressivity or a uniform reduction. To establish the relevant circumstances, it will simplify matters to assume that both t and t a are progressive (and hence that post-tax income is never zero). Consider ®rst the possibilityÐcontrary, perhaps to one's initial expectationÐthat an increase in allowances always leads to a uniform reduction in Rprogressivity. From Proposition 39, via (14), we deduce that t RP t a if and only if r (u) u ÿ t(u) ˆ is non-increasing in u, for all u . 0: r a (u) u ÿ t a (u) (20) It is shown in the Appendix that for (20) to hold for any positive level of allowances it is both necessary and suf®cient that t9(u)=r (u) be non-decreasing in u, for all u . 0. Thus we may state: Proposition 5. Increasing allowances within a progressive schedule t 2 T always leads to a uniform reduction in R-progressivity if and only if t9(u) is non-decreasing in u, for all u . 0: u ÿ t(u) (21) The broad intuition is the converse of that developed above for Proposition 4: condition (21) ensures that the tax schedule is so progressive that the rich not only derive a greater absolute gain from increased allowances, but also bene®t more in relative terms. As a consequence, the distribution of post-tax income always becomes more unequal. It is surprising to discover that a rise in allowances can lead to an unambiguous reduction in tax progressivity, and thus natural to wonder whether condition (21) is ever likely to be satis®ed in practice. Since (21) demands that the marginal tax rate rises as fast as after-tax incomeÐwhich in turn implies that after-tax income is bounded aboveÐthe condition appears to be rather strong. However, there do exist progressive schedules which satisfy (21), for example those given by 1 t(u) ˆ u ÿ (1 ÿ e ÿcu ), c . 0: c # Royal Economic Society 2000 2000] TAX REFORM AND PROGRESSIVITY 61 Now consider the possibility that an increase in allowances always leads to a uniform rise in R-progressivity. This requires, again from Proposition 39, the converse of condition (20). Taking incomes below the new threshold, this implies in particular that u ÿ t(u) t(u) ˆ1ÿ is non-decreasing in u, for u 2 (0, z(t) ‡ á]: u ÿ t a (u) u (22) But progressivity of t ensures that 1 ÿ t(u)=u is everywhere non-increasing. It therefore follows that t(u)=u ˆ ã, say, for u 2 (0, z(t) ‡ á]. Furthermore, since t(u) . 0 for u . z(t), we must have ã . 0 and hence z(t) ˆ 0. Thus increasing allowances by á . 0 leads to a uniform rise in residual progressivity only if the original tax schedule t imposes a proportional tax in the income range u 2 [0, á]. Allowing for any level of á . 0 then implies that t is a proportional tax over the whole income range. Conversely, if t(u) ˆ ãu, then r (u)=r a (u) ˆ (1 ÿ ã)u=r a (u) is non-decreasing in u, and t a RP t by Proposition 39. We may therefore state: Proposition 6. Increasing allowances within a progressive schedule t 2 T always leads to a uniform rise in R-progressivity if and only if t is a proportional tax. The intuition behind the suf®ciency part of Proposition 6 is clear enough. When everyone initially pays taxes and faces the same marginal tax rate, the introduction of an allowance is akin to the payment of a poll-subsidy to those who continue to pay tax, while the percentage rise in net income is at least as great for those whose taxes are reduced to zero. So the proportionate increase in net incomes declines monotonically with income, and this clearly leads to a more equal distribution. The necessity part of the result, on the other hand, is far from obvious and surprisingly strong: given any variation in average tax rates, one cannot be sure that increasing allowances will always reduce post-tax income inequality. Once seen, however, the result is easily explained. For imagineÐquite plausiblyÐ that the average tax rate is constant (possibly zero) up to some income level, and strictly increasing thereafter. Increasing allowances to a level suf®cient to bring all taxpayers into the range over which the average rate is constant then eliminates any redistribution initially achieved, causing R-progression to fall. In the context of speci®c types of tax schedules, some of the above results may be stated more forcefully. For example, if t is characterised by a constant marginal tax rate above some tax threshold, as in (19), then Propositions 29 and 39 imply that an increase in allowances never leads (as opposed to `does not always lead') to a uniform reduction in L-progression, and never leads to a uniform rise in R-progression. # Royal Economic Society 2000 62 THE ECONOMIC JOURNAL [JANUARY 4. Deductions and Tax Credits The analysis in the preceding section enables these applications to be treated more brie¯y. 4.1. Income-Related Deductions Let t be any tax schedule in T , and suppose that a tax reform introduces additional income-related deductions d(u). These have the effect of reducing taxable income, so that the reformed tax schedule is given by t a (u) ˆ t[ö(u)], where ö(u) ˆ maxfu ÿ d(u), 0g, u > 0, (23) represents taxable income. For simplicity, we assume throughout that d: R‡ ! R is continuous, non-negative and right differentiable with d9(u) 2 [0, 1] for all u > 0. We also assume that taxable income ö(u) is not bounded above. These assumptions ensure that t a is a member of T . Note too that t a is progressive if t is progressive and if å(d, u) < 1 whenever d(u) . 0:20 intuitively, income-inelastic deductions are akin to a (taxable) income-inelastic subsidy to taxpayers, and to that extent have an equalising effect. 4.1.1. Deductions and liability progression Consider ®rst the comparison between t a and t from the perspective of liability progression. Since t a (u) ˆ t[ö(u)] < t(u), it follows that z(t a ) > z(t). Application of Proposition 29 therefore yields: Proposition 7. Introducing deductions within a progressive tax schedule t 2 T : (a) Leads to a uniform rise in L-progressivity if and only if å[t, ö(u)]å(ö, u) > å(t, u) whenever ö(u) . z(t): (b) (24) Leads to a uniform reduction in L-progressivity if and only if z(t a ) ˆ z(t); and å[t, ö(u)]å(ö, u) < å(t, u) whenever ö(u) . z(t): (25) (26) One reassuring implication of the elasticity condition in (24) is that incomeinelastic deductions (implying å(ö, u) > 1) lead to a more progressive distribution of the tax burden if the rate structure is proportional. This is what one expects, given the intuition following (23) above. More generally, by writing (24) as t9(ö) t9(u) [1 ÿ d9(u)] > whenever ö(u) . z(t), t(ö) t(u) (27) we obtain a generalisation of the condition which appears in Proposition 20 The latter condition implies d(u) . 0 for all u . 0, and hence å(d, u) < 1 for all u . 0. # Royal Economic Society 2000 2000] TAX REFORM AND PROGRESSIVITY 63 4(b)Ðone which suggests that a more income-sensitive pattern of deductions requires a `more log concave' tax schedule (in the sense that t9=t falls more rapidly with income) in order to ensure that progressivity rises when deductions are introduced. Intuitively, when deductions increase more rapidly with income it becomes more likely that rising marginal tax rates will confer the greatest gains from deductions on the rich, so tighter constraints must be placed on the upward slope of the tax schedule. One immediate implication of this result is that, compared to allowances, income-related deductions demand a lower degree of progressivity in the rate structure in order to guarantee a uniform rise in liability progression. Although part (b) of Proposition 7 admits the possibility of a uniform reduction in liability progression, this outcome is unlikely to occur in practice. The requirement (25) implies that d(t) ˆ 0 for u < z(t), and cannot therefore apply if there is a minimum level of deductions, as might typically be expected. In this case, the result obtained earlier in Proposition 4(a) continues to hold, and we can conclude that liability progression is never uniformly reduced. 4.1.2. Deductions and residual progression The implications of Proposition 39 for the impact of deductions on residual progression need not detain us long, as the arguments employed to derive Propositions 5 and 6 can also be applied to general types of deductions. Given that allowances are special cases of deductions, we deduce from Proposition 5 that deductions always lead to a uniform reduction in R-progression only if the marginal tax rate increases rapidly with income, in the sense that condition (21) holds. Furthermore, repeating the steps in the second half of the proof of Proposition 5 shows that condition (21) is also suf®cient.21 Thus the statement of Proposition 5Ðand the accompanying intuitionÐcontinues to apply if allowances are replaced by income-related deductions. By similar reasoning, Proposition 6 establishes that t must be a proportional tax if deductions always lead to a uniform rise in R-progression. The converse will also apply if t a is progressive, which, as noted above, will be true if deductions are income-inelastic. The intuition provided after Proposition 6 also applies here: the introduction of deductions can always eliminate the initial redistribution achieved by the tax system, causing post-tax income inequality to rise, unless there was never any (relative) redistribution in the ®rst place; in other words, unless t is a proportional tax. Summarising: Proposition 8. The introduction of income-inelastic deductions within a progressive tax schedule t 2 T : (a) Always leads to a uniform reduction in R-progressivity if and only if the 21 The only changes to the proof involve substituting the more general expression for ö(u) given by (23), and noting that ö9(u) < 1. # Royal Economic Society 2000 64 THE ECONOMIC JOURNAL [JANUARY marginal tax rate increases suf®ciently rapidly, in the sense that condition (21) above is satis®ed. (b) Always leads to a uniform rise in R-progressivity if and only if t is a proportional tax. 4.2. Tax Credits Consider ®nally the introduction into the tax schedule t 2 T of a nonrefundable tax credit: in other words, a credit which may eliminate any liability, but cannot result in a refund. We now have t a (u) ˆ maxft(u) ÿ c, 0g, (28) where it is assumed that 0 , c , supft(u), u 2 R‡ g in order to ensure that t a (u) . 0 for some u . 0. Clearly t a belongs to T , and is also progressive if t is progressive. Since z(t a ) . z(t), and since t(u)=[t(u) ÿ c] is non-increasing in u for u . z(t a ), it follows immediately from condition (12) that a tax credit always leads to a uniform rise in L-progressivity. Residual progression, on the other hand, is seen from Proposition 39 to rise uniformly if and only if 8 t(u) > > 0 , u < z(t a ) <1 ÿ u u ÿ t(u) (29) ˆ u ÿ t(u) c u ÿ t a (u) > > : ˆ1ÿ u > z(t a ) u ÿ t(u) ‡ c u ÿ t(u) ‡ c is non-decreasing in u, for all u . 0. Just as in Proposition 6, a progressive schedule t satis®es (29) only if it is a proportional tax in the range u 2 (0, z(t a )], and if c can take any positive value then t must be a proportional tax over the whole income range.22 Once this requirement is met, however, the tax credit will indeed lead to a uniform rise in residual progression. The intuition again follows closely that of Proposition 6. A suf®ciently large credit will reduce all tax liabilities to zero, and so actually increase inequality of post-tax incomes if the initial tax schedule was achieving any degree of equalization. Under a proportional tax, however, the credit has exactly the same effect as an allowance, and is equivalent to a uniform pollsubsidy to all those who continue to pay tax. Conversely, if a tax credit leads to a uniform fall in R-progressivity, then expression (29) must be non-increasing in u. This can happen only if r (u) is constant for u > z(t a ); in other words, only if the marginal tax rate is 100% above the tax threshold. The reason is simple. For those whose tax liability is not eliminatedÐwhich, for some distributions of pre-tax income, will be all taxpayersÐa credit is a poll subsidy, and will usually lead to an increase in post-tax income equality. The only case in which this does not happen is when they all have the same post-tax income. 22 Since t is progressive, and t9(u) is a positive constant for u , z(t a ), it follows that t is unbounded on R‡ . Hence all positive values of c are admissible. # Royal Economic Society 2000 2000] TAX REFORM AND PROGRESSIVITY 65 Combining these results therefore yields: Proposition 9. Introducing a non-refundable tax credit into a progressive tax schedule t 2 T : (a) Always leads to a uniform rise in L-progressivity. (b) Always leads to a uniform rise in R-progressivity if and only if t is a proportional tax. (c) Never leads to a uniform fall in R-progressivity if r(u) is strictly increasing for all u . 0. 5. Conclusions This paper has pursued two broad objectives. The ®rst was to reformulate the core results of the theory of tax progressivity in order to remove the debilitating requirement that tax liability is never zero. Relaxing this assumption is no mere technicality, but a prerequisite for any coherent discussion of the progressivity effects of, for example, reforms to the structure of personal income taxation. The generalisations we have obtained are more complicated to state and prove, but relatively easy to apply. They suggest a simple two-step procedure for the evaluation of reform: ®rst compare the tax thresholds before and after reform; then employ elasticity tests similar to those used in previous studies. The second objective was to establish conditions under which unambiguous conclusions can be drawn on the progressivity effects of some of the most common and fundamental kinds of tax reform, involving increases in allowances, deductions, and/or credits. Such conditions emerge as corollaries of the extensions referred to above. These new conditions provide novel and distinctive insights into the progressivity effects of structural tax reforms. Consider, for example, the very basicÐyet previously unansweredÐquestion raised at the outset: Does a rise in allowances increase or decrease progressivity? If one assesses progressivity in terms of the distribution of tax payments, the requirement for an unambiguous rise in progressivity has been shown to be that the tax schedule is log concave. The log concavity property restricts the progressivity of the tax schedule to a point which ensures that the proportionate reduction in tax payments from increased allowances is greater for the poor than for the rich, thus generating a more unequal distribution of tax payments. This property is quite distinct from any existing concept of progressivity, and its potential importance in tax analysis has not been noticed previously. In contrast, if progressivity is assessed in terms of post-tax income inequality, then there is effectively no condition which will ensure that progressivity always rises as allowances are increased. This can be seen by noting that allowances may rise by so much that all tax liabilities are reduced to zero, thereby restoring the inequality observed in pre-tax incomes and eliminating any equalisation produced by the original tax schedule. These conclusions, to# Royal Economic Society 2000 66 THE ECONOMIC JOURNAL [JANUARY gether with corresponding results for income-related deductions and tax credits, go some way towards providing the core framework for the progressivity analysis of structural tax reforms that has been so surprisingly absent from the literature. Throughout the paper, we have assumed that tax liability is always nonnegative. In practice, of course, the payment of bene®ts or refundable tax credits means that the net payments of some taxpayers are negative. Any complete assessment of the progressivity of the ®scal system must encompass the full tax-bene®t system, requiring a conceptual framework that can admit negative tax liabilities. Several obstacles remain in constructing such a framework: in particular, a suitable way must be found to extend the Lorenz-type criterion for distributional progressivity, and the corresponding elasticity conditions, to the negative income domain. While this paper has not attempted to deal with these issues, the discussion in Sections 1 and 2 offers guidance on how one might proceed towards the construction of a comprehensive framework for assessing the progressivity characteristics of all types of tax-bene®t reforms. International Monetary Fund, University of Essex and Institute for Fiscal Studies University of Kent University of Essex and Institute for Fiscal Studies Date of receipt of ®rst submission: November 1997 Date of receipt of ®nal typescript: July 1999 Appendix Proof of Lemma 1 Since (9b) follows directly from (9a), the ®rst part of Lemma 1 will be established by demonstrating that (9b) implies (9c), and that (9c) implies (9a). Suppose (9b) is true and consider any distribution x ˆ (u, v) such that v . u . 0 and g (u) . 0. Then f (v) > f (u) > 0 and g (v) > g (u) . 0, since f and g are nonnegative and non-decreasing. Now if f (v) ˆ 0 we have f (u)= g (u) ˆ f (v)= g (v) ˆ 0, which is consistent with (9c). Alternatively, we have f (v) . 0, in which case (9b) implies f (u) g (u) > : f (u) ‡ f (v) g (u) ‡ g (v) (A:1) It then follows that f (u) g (v) > g (u) f (v) . 0, and hence f (u)= g (u) > f (v)= g (v), as required in (9c). Hence (9b) implies (9c), To show that (9c) implies (9a), suppose (9a) is false. Then there is some n and some n such that ì[f(x)] . 0, ì[g(x)] . 0, and f(x) does not Lorenz dominate g(x). x 2 R‡‡ Without loss of generality, assume that the elements of x (and hence also the elements of f(x) and g(x)) are arranged in increasing order, and denote the partial sums of f(x) and g(x) by Fk ˆ k P iˆ1 # Royal Economic Society 2000 f (x i ) and G k ˆ k P iˆ1 , g (xi ), k ˆ 0, . . ., n: (A:2) 2000] TAX REFORM AND PROGRESSIVITY 67 Then F n . 0; G n . 0; and, since f(x) does not Lorenz dominate g(x), F k , èG k for some k , n, (A:3) where è :ˆ F n =G n . 0. Now de®ne l ˆ minfkjF k , èG k g . 0, and m ˆ minfkjk . l; F k > èG k g < n. Then we have F lÿ1 > èG lÿ1 and F l , èG l , so that f l ˆ F l ÿ F lÿ1 , è g l . In addition, F mÿ1 , èG mÿ1 and F m > èG m , so f m ˆ F m ÿ F mÿ1 . è g m > è g l . 0. Hence f (xl ) f (xm ) ,è, for some x l , xm , g (xl ) g (xm ) (A:4) which contradicts (9c). So (9c) is false when (9a) is false. This completes the proof of the equivalence between statements (9a), (9b) and (9c). To establish the link from (9) to (10), suppose (9c) is true and consider any u . 0 such that g (u) . 0. If f (u) ˆ 0, we have f (v) ˆ 0 for all v , u, since f is non-negative and non-decreasing; and also f (v) ˆ 0 for all v . u, since f = g is non-increasing. Hence f (v) ˆ 0 for all v. But that is impossible, because f 2 T satis®es (1c). So f (u) must be positive, as required in (10), and the proof of Lemma 1 is complete. Proof of Proposition 5 Rewrite condition (20) as r a (u) u ÿ t a (u) ˆ is non-decreasing in u, for all u . 0, r (u) u ÿ t(u) (A:5) and note that (A.5) is certainly satis®ed for u < á, since t is progressive. If (A.5) also holds for any income u . á and any level of allowances á . 0, we may set v ˆ u ÿ á to obtain the requirement that   1 r a (v ‡ á) t(v ‡ á) ÿ t(v) ÿ1 ˆ is non-decreasing in v, (A:6) á r (v ‡ á) ár (v ‡ á) for all v . 0 and all á . 0. Taking the limit in (A.6) as a tends to zero then yields t9(v) is non-decreasing in v, for all v . 0: r (v) (A:7) Conversely, for any á . 0 denote taxable income by ö(u) ˆ maxfu ÿ á, 0g, and de®ne h(u)  r a (u) u ÿ t[ö(u)] u ÿ ö(u) ‡ r [ö(u)] ˆ ˆ : r (u) r (u) r (u) (A:8) As t9(u)=r (u) is non-decreasing in u, it follows that t9(u) is non-decreasing in u, and for all u . 0 we have t9(u) t9[ö(u)] > and t9(u)[u ÿ ö(u)] > t(u) ÿ t[ö(u)]: r (u) r [ö(u)] (A:9) Hence r (u)2 h9(u) ˆ r (u)f1 ÿ t9[ö(u)]ö9(u)g ÿ fu ÿ t[ö(u)]g ‡ fu ÿ ö(u) ‡ r [ö(u)]gt9(u) > ÿ r (u)t9[ö(u)]ö9(u) ‡ t[ö(u)] ÿ t(u) ‡ ft(u) ÿ t[ö(u)] ‡ t9[ö(u)]r (u)g ˆr (u)t9[ö(u)][1 ÿ ö9(u)] > 0, and h(u) is non-decreasing for all u . 0, as required in (A.5). # Royal Economic Society 2000 (A:10) 68 THE ECONOMIC JOURNAL [ J A N U A R Y 2000] References Eichhorn, W., Funke, H. and Richter, W. F. (1984). `Tax progression and inequality of income distribution.' Journal of Mathematical Economics, vol. 13, pp. 127±31. Fei, J. C. H. 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