DOI 10.1007/s00355-007-0237-0 O R I G I NA L PA P E R
Analyzing the impact of indirect tax reforms
on rank-dependent social welfare functions: a positional
dominance approach
Paul Makdissi · Stéphane Mussard
Received: 10 July 2006 / Accepted: 26 March 2007 / Published online: 23 May 2007 © Springer-Verlag 2007
Abstract A new approach is developed to identify marginal tax reforms for pairs of commodities and to test for the robustness of their impacts on Yaari’s dual social welfare functions. The rank-dependent social evaluation approach gives rise to a new device, the s-concentration curve, which is a generalization of the standard concen-tration curve. The s-concenconcen-tration curves are provided for every order of positional dominance and an illustration is performed using Canadian data.
We thank Jean-Yves Duclos, Peter Lambert, and an anonymous referee for insightful comments and suggestions. This paper was funded through the Social Science and Humanities Research Council of the Government of Canada and the Fonds Québécois de la Recherche sur la Société et la Culture of the Government of Quebec.
P. Makdissi (
B
)Department of Economics, University of Ottawa, P.O. Box 450, Station A, Ottawa, ON, Canada KIN 6N5 e-mail: [email protected]
S. Mussard
CEPS/INSTEAD Luxembourg, Differdange, Luxembourg e-mail: [email protected]
S. Mussard
GRÉDI, Université de Sherbrooke,
2500, boulevard de l’ Université, Sherbrooke, QC, Canada J1K 2R1 S. Mussard
GEREM, Département des Sciences Economiques, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan Cedex, France
1 Introduction
Due toHardy et al.(1929), the equivalence between Pigou–Dalton transfers and the concept of Lorenz dominance between pairs of income distributions F and G is well established in the sense that there is less inequality in distribution F if its corresponding Lorenz curve lies nowhere below that of G. Since then, the Lorenz curve has constituted the cornerstone of the theory of inequality measurement. If x represents the consump-tion expenditures, the Lorenz curve yields the total proporconsump-tion of x received by the first individuals ranked by ascending order of the x’s. On the other hand, inequalities may be computed with concentration curves, which yield the consumption proportion of x held by the first individuals ranked by ascending order of incomes. In our frame-work, this difference is crucial since the concentration curve confers the possibility to analyze taxation mechanisms and particular consumption habits, which may exhibit (notably for inferior goods) increasing and concave concentration curves on small intervals including the poorest individuals.
The non intersection between concentration curves is ordinarily presumed as a necessary condition to gauge the distributive effects of tax reforms (see e.g.Lambert 2001). In this perspective, the seminal investigations ofYitzhaki and Slemrod(1991) and afterwards those ofYitzhaki and Thirsk(1990) have highlighted the construction of welfare-improving indirect tax reforms for all social welfare functions respecting the Pigou–Dalton transfer principle. They identify pairs of commodities with non-crossing concentration curves, for which the analyst can select the differential in the efficiency cost of raising public funds throughout the two goods. Alternatively, Makdissi and Wodon (2002) have initiated the employment of the “Consumption Dominance Curve” concept (CD-Curve)1to capture the impact of marginal indirect tax reforms on poverty for any order of restricted stochastic dominance. This result has been subsequently generalized inDuclos et al.(2004), in which consumption domi-nance curves are implemented to test for the robustness of welfare-improving indirect tax reforms in a more general framework than the Pigou–Dalton transfer principle.
In this paper, following a different path, we aim at using rank-dependent social welfare functions (seeYaari 1987,1988), instead of the traditional utilitarian criterion, in order to evaluate the effects of marginal indirect tax reforms on these social welfare functions. These utility functions can be appealing since the property of differentiabil-ity, with alternative derivative signs, provides intuitive characterization of attractive transfer principles in both welfare, inequality and poverty frameworks (seeDuclos and Makdissi(2004)). However, the utilitarian ethical transfer principles are questionable (see e.g. Sect.2inChateauneuf et al. 2002, explained infra). First, the principle of diminishing transfer (Kolm 1976b) postulates that an income transfer, valued to be
δ, from a higher-income individual to a lower-income one yields a better impact on 1 The CD-Curve of order 1 for any given commodity i is the ratio between an individual consumption with income y and the aggregate consumption of good i : CD1i(y)=xi(y)/Xi. The CD-Curve of order 2,
CD2i(y)=0yC1(u)dF(u), is the share of total consumption of good i held by the persons whose income is less than y. CD-Curves of order s, CDsi(y)=0yCs−1du, are obtained by integrating over the cumulative distribution function of incomes, whereas concentration curves are obtained by integrating over population percentiles.
...
y1 y...2 y...3 y...4 y...5 y...6 y
δ ...
δ
Transfer 1 The principle of diminishing transfer
social welfare insofar as incomes(y)are the lowest possible. Hence, a decision maker respecting the diminishing transfer principle prefers the transfer from y2to y1rather than the one from y5to y3, given that the rank of the individuals remains unchanged after such transfers and y2−y1=y5−y3.
However, this principle entails some disproval arguments such as the insensibility towards individuals’ rank. Indeed, the transfer from y5to y3can be more appealing than the other since we take money from the second richest individual. Consequently, why not consider some rank-dependent criteria such as the principle of transfer sensitivity (seeMehran 1976;Kakwani 1980) also denominated principle of dual diminishing transfer (seeChateauneuf et al. 2002)? This principle supposes that an income transfer of amountδfrom a higher-income individual to a lower-income one yields a better impact on social welfare insofar as individuals’ ranks are the lowest possible. A deci-sion maker obeying the dual diminishing transfer principle adopts the transfer from y2to y1rather than the one from y4to y3, given that there is the same rank between individuals and that individuals’ ranking remains unchanged.
... y1 ... ... ... ... y2 ... y4 y6 δ y y3 δ ... y5
Transfer 2 The principle of dual diminishing transfer or transfer sensitivity
In spite of its appropriateness, an objection to this type of transfer is the insensi-bility towards the income gaps between individuals, particularly y2−y1= y4−y3. Accordingly, one can reasonably prefer the transfer from y4to y3since the income gap is higher.
In the light of this discussion,Chateauneuf et al.(2002) suggest a combination of the two above transfers in which both income gaps and rank gaps are relevant: the so-called principle of strong diminishing transfer. This ethical property is consistent with expected utility, rank-dependent expected utility and Yaari’s models, since it allows for computing the derivative signs of the underlying utility functions as well as those of the frequency distortion function—the rank-dependent function that assigns a weight to each individual in Yaari’s model. This yields the decision maker’s behavior under risk and uncertainty, showing that he or she can be risk loving or risk averse.2 2 Except inAaberge(2000), transfer principles are not clearly connected with the shape of the distribution. Indeed, he shows that the diminishing transfer principle is satisfied by Yaari’s social welfare functions (and their dual inequality measures) if, and only if, the absolute risk aversion in the sense of Arrow-Pratt is higher than a parameter reflecting the shape of the income distribution. It turns out that some income distributions (such that in Norway in 1996) imply that the social planner has to be highly risk averse towards what happens in the first percentiles for the principle of diminishing transfers to be fulfilled.
Up to the third-order derivative sign of the utility function and/or the frequency distortion function, it is possible to test how the decision maker behaves. But the determination of the s-order derivative sign in accordance with the strong diminish-ing transfer can be seen as an overcomplicated problem as far as s >3, whereas it can be easily realized with the principle of transfer sensitivity. Indeed,Aaberge(2004) proposed a generalization of such transfer (see Sect.2infra) in a so-called positionalist transfer sensitivity that provides the s-order derivative signs of the frequency distortion function. As observed inChateauneuf et al.(2002), the choice between the principle of diminishing transfers and positional transfer sensitivity is a matter of normative preferences. In spite of this, the latter remains intuitive since the resolution of the derivative signs enables s-order tests of inverse stochastic dominance to be character-ized by means of an analytical scheme, which encompasses the impact of indirect tax reforms on social welfare with respect to Yaari’s model (1987). This is precisely the aim of our article.
The rank-dependent social evaluation allows us to introduce a new approach, the s-concentration curve, which is a generalization of the usual concentration curve. In the same manner as the CD-Curves, the concentration curve of order 1 is the ratio between an individual consumption of good i and the aggregate consump-tion of good i , except that we consider individuals’ ranks instead of incomes. Con-trary to the CD-Curve of order 2, the concentration curve of order 2 is the share of total consumption of good i held by the persons whose income is less than a given percentile. Integrating successively the concentration curves over the percen-tile space yields s-concentration curves, which are good candidates to compare dis-tributions. Indeed, the purpose of rank-dependent tests of inverse stochastic domi-nance is to identify the minimum order of s-concentration, that permits non-inter-secting curves to be ranked, with its underlying ethical transfer principle. Indeed, considering pairs of commodities i and j and identifying the one for which the concentration curve of order s is below (let us say, i ) , enables decision makers to increase the tax on i and to use the subsidy to decrease the tax on j while enjoy-ing an increase of overall welfare.Yitzhaki and Slemrod(1991) provided necessary and sufficient conditions to perform indirect tax reforms with concentration curve dominance of order 2. Subsequently,Mayshar and Yitzhaki(1995) offered the pos-sibility to consider a multiple-good economy in which all social welfare functions respecting Dalton’s transfer are relevant with indirect tax reforms. Without consid-ering the multiple-good case, we restrict our attention to necessary and sufficient conditions to assess the impact of indirect tax reforms on rank-dependent social wel-fare functions for all pairs of commodities and for any order of inverse stochas-tic dominance based on: the generalization of the dual diminishing transfer (see Transfer2 supra) and the generalization of the well-known concentration curve of order 2.
The measurement of rank-dependent social welfare functions and the associated ethical criteria are discussed in the next section. Section 3 exposes the framework in which we analyze the impact of a marginal indirect tax reform via the concept of s-concentration curve. Section4presents an empirical illustration and a brief section follows to conclude and suggest further researches.
2 Measuring rank-dependent social welfare
Let1be the set of rank-dependent social welfare functions, which are expressed as W()= 1 0 (p) v (p)d p (1)
wherev (p)≥ 0 is the weight attached to an individual at the pth percentile of the distribution for all p∈[0,1] withv(1)=0, where (p)=infyE :FyE≥ pis the left inverse of FyE, FyEbeing the cumulative distribution function of
equiv-alent incomes yE defined over [0,a], which is a subset of nonnegative real incomes, and where a is the maximum conceivable equivalent income.
The concept of equivalent incomes yE has been introduced byKing(1983). To account for the effect of different prices across households/individuals,King(1983) used the utility function of a reference household as a basis for defining equivalent incomes, i.e., letψ(y,q,t)represent the indirect utility of household, endowed with exogenous income y, when facing prices qand tax system t. Next, consider a reference household R that faces prices qR. ThenKing(1983) defined the equivalent
income by the exogenous income y,tthat would allow the reference household facing
prices qRand tax system t to reach utilityψ(y,q,t):
ψR
y,t,qR,t
=ψ(y,q,t) . (2) Thus, if t1and t2denote the pre-reform and post-reform tax systems then y,t2−y,t1
can be considered as a money measure of the welfare change for household of changing the tax system from t1to t2.
In order to describe the normative implications underlying the functional form of
v (·), we define a set of positional ethical principles. The first principle stipulates that ordering two distributions of living standards is equivalent to make the living standards “parade” simultaneously alongside each other, and verifying if one parade weakly dominates the other—this exercise was first suggested byPen(1971). Since
v (p)≥0 for all p∈[0,1], all W()∈1satisfy this normative principle. The second principle is the Pigou–Dalton Principle of Transfers (Pigou 1912; Dalton 1920).
Definition 2.1 An income distributionY , whose left inverse cumulative distribution function is, is obtained from the distribution Y (of left inverse c.d.f.) by a pro-gressive Pigou–Dalton transfer if, a transfer of amountδ >0 occurs from yi to yj
such that yi >yj, letting their position unchanged: yi−1≤yi−δ, yj +δ ≤yj+1. A
social welfare index satisfies the Pigou–Dalton principle if
W()≤W(). (3)
A social welfare function W()∈1satisfies this normative principle if, and only if,v(p)≤0 for all p∈[0,1].
These two first dominance criteria are identical in both utilitarian and rank-depen-dent social welfare approaches. However, it would be attractive to impose more struc-ture on the social welfare function, permitting the decision maker to select between a wide range of transfer principles (see e.g.Gajdos 2002). As explained in the Introduc-tion, several transfer principles have been formulated from diverse perspectives such asKolm’s(1976a) Principle of Diminishing Transfers through an utilitarian design. Mehran(1976) andKakwani(1980) have adapted Kolm’s rule into a Positional Prin-ciple of Transfer Sensitivity (see Transfer1 supra). In order to expose the formal generalization of this principle, let us first definep,γW(δ, ).
Definition 2.2 The variation of social welfare induced by a progressive Pigou–Dalton transferδfrom the person at rank p+γ,γ >0 to the one at rank p is expressed as
p,γW(δ, ):=W()−W(). (4)
Remember that the Positional Principle of Transfer Sensitivity postulates that a small transfer from a higher-income person to a lower-income one, with a given pro-portion of the population between them, is more valuable if it occurs at lower in-come levels, formally:p,γW(δ, )≥p,γW(δ, )for all p>p.Fishburn and
Willig(1984) have suggested the generalization of this transfer principle in an utili-tarian scheme. Contrary to this,Aaberge(2004) has brought out a generalization in a so-called Principle of sth-degree (downside) Positional Transfer Sensitivity within a rank-dependent layout, based onsp,W(δ, ), which is recursively deduced:
2 p,2W(δ, ):=p,γ1W(δ, )−p+γ2,γ1W(δ, ) , (5) where2=(γ1, γ2),γi >0, ... s p,sW(δ, ):=s−1 p,s−1W(δ, )− s−1 p+γs,s−1W(δ, ) , (6) wheres =(γ1, γ2, . . . , γs),γi >0.
Definition 2.3 A social welfare function W() ∈ 1 satisfies the Principle of sth-degree Positional Transfer Sensitivity if,sp,sW(δ, ) ≥ sp,sW(δ, ) for all p> p.
The positional transfer principle of order s requires some assumptions on the weight function. Letv(i)(·)be the i th derivative of the functionv (·)=:v(0)(·), and consider the two following sets of rank-dependent social welfare functions:
s :=
W()∈1:vis continuous and s-time differentiable almost everywhere over [0,1], (−1)iv(i)(p) >0∀p∈ [0,1],
∀i =1,2, . . . ,s−1}, (7)
A social welfare function W()satisfies the Principle of sth-degree Positional Transfer Sensitivity if, and only if, W()∈s, ∀s∈ {3,4, . . .}.
It turns out that W()∈1satisfies the Pen Parade Principle, W()∈2⊂1 also satisfies the Pigou–Dalton Principle of Transfers, W()∈3⊂2⊂1also satisfies the Principle of 1st-degree Positional Transfer Sensitivity and W()∈s ⊂
s−1⊂ · · · ⊂ 3 ⊂2 ⊂1for all s ∈ {3,4, . . .}, also satisfies the Principle of sth-degree Positional Transfer Sensitivity (seeAaberge 2004, Theorem 3.2A). 3 Analyzing the impact of a marginal tax reform
Suppose the government is considering an increase in social welfare by marginally decreasing the indirect tax (or marginally increasing the subsidy) on good i and mar-ginally increasing the indirect tax on good j in order to keep the public budget constant. This marginal tax reform entails a variation in equivalent income (p)for an indi-vidual at rank p: d (p)= ∂ (p) ∂ti dti +∂ ( p) ∂tj dtj. (9)
As shown byBesley and Kanbur(1988), if the vector of reference prices used to com-pute equivalent incomes is the vector of prices before the reform, the change in the equivalent income induced by a marginal change in the tax rate of good i is
∂ (p)
∂ti = −x
i(p) , (10)
where xi(p)is the Marshallian demand of the individual at rank p for good i .
Assume that the average tax revenue is R=Kk=1tkXk where Xk is the average
consumption of the kth good Xk=
1
0 xk(p)d p. As shown inYitzhaki and Slemrod (1991), revenue neutrality dR=0 and constant producer prices imply
dtj = −α Xi Xj dti whereα= 1+X1 i K k=1tk∂∂Xtk i 1+ X1 j K k=1tk∂∂Xtjk . (11)
Wildasin(1984) interpretsαas the differential efficiency cost of raising one dollar of public funds by taxing the j th commodity and using the proceeds to subsidize the i th commodity. Substituting (11) and (10) in (9) yields
d (p)= − xi(p) Xi −α xj(p) Xj Xidti. (12)
Let us now introduce the concept of concentration curve of order s.
Definition 3.1 The first-order concentration curve, Ck1(p)=xk(p) /Xk, is the
con-sumption of good k for an individual at rank p divided by the average concon-sumption of the good. The s-concentration curve, Cks(p) = 0pCks−1(u)du, is given for all integers s∈ {2,3,4, . . .}.
It is worth mentioning that the second-order concentration curve, C2k(p), is the tra-ditional concentration curve, which represents the share of total consumption of good k consumed by the individuals whose rank in the income distribution is less than p.
Using our notation, Eq. (12) can be rewritten as
d (p)= −
Ci1(p)−αC1j(p)
Xidti. (13)
The total change in social welfare induced by the reform is then obtained by integrating (13): dW()= −Xidti 1 0 Ci1(p)−αC1j(p) v (p)d p. (14)
We can now prove a first positional necessary condition linking marginal tax reforms, s-concentration curve dominance and rank-dependent welfare variation.
Lemma 3.1 An average-revenue-neutral marginal tax reform dtj=−α
Xi
Xj
dti >0
implies dW()≥0 for all W()∈s, if
(i) Cis(p)−αCsj(p)≥0∀p ∈[0,1], s∈ {1,2,3, . . .}, (15)
(ii) Ciu(1)−αCuj(1)≥0∀u∈ {3, . . . ,s−1}.
Proof If we refer to Eq. (14), we easily realize that the condition for s=1 is proved by simply noting thatv (p)is nonnegative and that dtiis negative. To prove for s>1,
we first need to integrate by parts01Ck1(p) v (p)d p : 1 0 Ck1(p) v (p)d p= Ck2(p) v (p)1 0− 1 0 Ck2(p) v(p)d p. (16)
Now, assume that for some s>2, we have 1 0 Ck1(p) v (p)d p= s−1 u=2 (−1)u Cku(p) v(u−2)(p)1 0 +(−1)s−2 1 Cks−1(p) v(s−2)(p)d p. (17)
Integrating by parts Eq. (17), we get 1 0 C 1 k(p) v (p)d p= s u=2 (−1)u Cku(p) v(u−2)(p)1 0 +(−1)s−1 1 0 Cks(p) v(s−1)(p)d p. (18)
Equation (16) respects the relation depicted in Eq. (17). We have shown that if Eq. (17) is true then Eq. (18) is also true. This implies that Eq. (18) is true for all integers s∈ {2,3, . . .}. From Eqs. (14) and (18), we obtain for s∈ {2,3, . . .},
dW()= −Xidti s u=2 (−1)u Ciu(p)−αCuj (p) v(u−2)(p)1 0 +(−1)s−1 1 0 Cis(p)−αCsj(p) v(s−1)( p)d p ⎫ ⎬ ⎭. (19)
Ciu(0)=Cuj (0)=0 for all u ∈ {2,3, . . . ,s}and Ci2(1)=C2j(1)=1. Thus, we have for s =2: dW()=Xidti 1 0 Ci2(p)−αC2j(p) v(p)d p. (20)
Note thatv(p)is negative and that dti is negative. If Ci2(p)−αC2j(p)≥0 for all
p∈[0,1], then dW()≥0. This proves the proposition for s=2, which is the case studied byYitzhaki and Slemrod(1991). For s∈ {3,4,5, . . .}Eq. (19) becomes
dW()= −Xidti s u=2 (−1)u Ciu(1)−αCuj (1) v(u−2)( 1) +(−1)s−1 1 0 Cis(p)−αCsj(p) v(s−1)( p)d p ⎫ ⎬ ⎭. (21)
Note that(−1)s−1v(s−1)(p)is positive and that dtiis negative. If Ci3(p)−αC3j(p)≥0
for all p∈[0,1], then dW()≥0. The same reasoning applies for s∈ {4,5,6, . . .}, except that we must also check if Ciu(1)−αCuj (1)≥0 for all u∈ {3, . . . ,s−1}to ensure that dW()≥0.
Lemma3.1stipulates, whenα=1, that the marginal tax reform will increase social welfare if the s-concentration curve of good i dominates (lies above) the s-concentra-tion curve of good j , for any given percentile intervals. Ifα=1, we can still compare
the s-concentration curve of good i with the s-concentration curve of good j provided the latter is multiplied byα.
For s ≥ 4 an additional necessary condition is required at p = 1. This bound-ary condition can be seen as restrictive since the graphical approach that consists in controlling a dominance between two curves is not sufficient. While the traditional concentration curves can only be used to investigate some dominance at order s=2, this methodology enables the decision maker to test for any order of inverse stochastic dominance. However, in order to develop a more powerful result, which includes all degrees of positional dominance, we restrict our class of welfare indices tos.3 Theorem 3.1 An average-revenue-neutral marginal tax reform dtj = −α
Xi
Xj
dti >0 implies the following equivalence:
(i) d W()≥0 for all W()∈s,
(ii) Cis(p)−αCsj(p)≥0∀p∈[0,1], s∈ {1,2,3, . . .}.
Proof (ii)⇒(i): As noted in the proof of Lemma3.1, referring to Eq. (14) implies that the condition for s=1 is proved by simply noting thatv (p)is nonnegative and that dti is negative. For s =2, examine Eq. (20) and note thatv(p)is nonpositive
and that dtiis negative. If C2i (p)−αC2j(p)≥0 for all p∈[0,1], then dW()≥0.
For s= {3,4,5, . . .}, Eq. (8) implies dW()= −Xidti(−1)s−1 1 0 Cis(p)−αCsj(p) v(s−1)(p)d p. (22)
Note that (−1)s−1v(s−1)(p)is nonnegative and that dti is negative. If Cis(p)−
αCsj(p)≥0 for all p∈[0,1], then dW()≥0.
(i)⇒(ii): Consider the set of functions W()∈s for which the(s−1)th deriv-ative ofv (p)(v(0)(p)beingv (p)itself) is of the following form
v(s−1)( p)= ⎧ ⎨ ⎩ (−1)s−1 p≤ p, (−1)s−1(p+−p) p<p≤ p+, 0 p>p+. (23)
Sincev (p)is differentiable almost everywhere except at p and p+, it satisfies the conditions in (8). Thus, welfare indices whose frequency distortion functionsv (p) have the particular above form forv(s−1)(p)belong tos. This yields:
v(s)( p)= ⎧ ⎨ ⎩ 0 p≤ p, (−1)s p< p≤ p+, 0 p> p+. (24)
3 This condition is appropriate for the extended Gini family of social welfare functions introduced by Donaldson and Weymark(1980), for whichv(p):=ν(1−p)ν−1, whereν >1 is the aversion degree of inequality that provides a concave social welfare function if 1< ν <2 or a convex social welfare function ifν >2.
Imagine now that Cis(p)−αCsj(p) < 0 on an interval[p,p+]forthat can be arbitrarily close to 0. Forv (p)defined as in (23), expression (22) is then negative and the marginal tax reform induces a marginal decrease of welfare. Hence, it cannot be that Cis(p)−αCsj(p) <0 for p∈[ p,p+].
Note that if we denote by G Lσt1theσth-order generalized Lorenz curve (as defined byAaberge 2004) of pre-reform equivalent income and by G Lσt2theσth-order gener-alized Lorenz curve of post-reform equivalent income, then condition (ii) is equivalent to saying that G Lst2+1(p)≥G L
s+1
t1 (p)for all p∈[0,1].
4 Empirical illustration
We illustrate our technique on Canadian data using the Survey of Household Spending (2002), which involves 14,655 households. In order to compare different family sizes, the following equivalent scale has been adopted: single = 1 standard adult, couple = 1.4 standard adults. For couples with additional persons, 0.3 is added per child younger than 16 years old, and 0.4 for each additional member (older than 16 years old). The sole difference with the scale of Canada’s National Statistical Agency is that we use 0.4 for each additional member older than 16 years old instead of 18. The application is concerned with the following pairs of commodities: tobacco versus alcoholic drinks, tobacco versus education, and housing versus electricity.
The interpretation ofαis important. AsYitzhaki and Slemrod(1991) pointed out,
α <1 (α > 1) indicates, as a consequence of the tax reform, whether a diminution
(a rise) of the excess burden occurs. The case for whichα=1 is appealing since it yields neither efficiency gain nor efficiency loss for the government but can be welfare improving. Figure1depicts an example of dominance between concentration curves of order 2 of alcoholic drinks and tobacco assumingα=1.
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0, 0 3 0, 0 5 0, 0 8 0, 0 0 1, 0 3 1, 0 5 1, 0 8 1, 0 1 2, 0 3 2, 0 6 2, 0 8 2, 0 1 3, 0 4 3, 0 6 3, 0 9 3, 0 1 4, 0 4 4, 0 6 4, 0 9 4, 0 2 5, 0 4 5, 0 7 5, 0 9 5, 0 2 6, 0 4 6, 0 7 6, 0 0 7, 0 2 7, 0 5 7, 0 7 7, 0 0 8, 0 2 8, 0 5 8, 0 8 8, 0 0 9, 0 3 9, 0 5 9, 0 8 9, 0 Percentiles Cumul order 1 A. Drinks Tobacco
Fig. 1 Dominance with s-concentration curves of order 2. Source: Authors’ estimation using Canadian Survey of Household Spending (2002)
Subsequently, as the alcoholic drinks curve dominates that of tobacco, our results stipulate that increasing the tax on tobacco and using the subsidy to decrease the tax and alcoholic drinks improves, surprisingly, the overall welfare. Furthermore, as Yitzhaki and Slemrod (1991) have pointed out, the recourse to the “difference in concentration curve” approach (the DCC approach) is interesting. Indeed, making the difference between the highest curve and the lowest one allows to compute the house-hold percentage concerned with this welfare improvement, approximately 95% of the households that smoke and/or drink alcohol.
Instead of looking for one exogenous particular efficiency parameter that serves to derive the degree of inverse stochastic dominance, together with its ethical transfer principle, one may choose a given order of positional dominance and then compute an interval, for which a wide range of efficiency parameters are operational. Welfare indices belonging to1cannot be Pen improving ifα≥1, cannot be Dalton improv-ing for W() ∈ 2if α > 1, and cannot be welfare improving for W() ∈ s
∀s ∈ {3,4, . . .}if α > 1 (see Duclos et al. 2004). Therefore, one may examine
whether the order of inverse stochastic dominance test is robust with respect to the choice ofα∈ [0,1]. Let us remember thatαis the differential efficiency cost of raising one dollar of public funds by taxing the j th commodity and using the proceeds to sub-sidize the i th commodity. Precisely, the lowerαis, the higher the efficiency of taxing good j is. Then, for any given s ∈ {1,2,3, . . .}, the set of efficiency parameters for which the dominance and the welfare improvement are guaranteed can be determined as follows: arg min α {C s i (p)−αC s j(p)≥0:s∈ {1,2,3, . . .}} (25) s.t.∀p∈[0,1], s.t.α∈ [0,1].
The linear program has been conducted over education and tobacco consumption expenditures. Increasing the tax on tobacco and decreasing the one for education provides α ∈ [0,0.99 =: α∗]. Under the same ethical transfer principle (s = 2), increasing the tax on electricity and subsidizing housing yieldsα ∈ [0,0.59](see Fig.2). Then, the taxation scheme is more efficient since the maximum conceivable parameterα∗=0.59 is closer to 0 than the one concerned with the reform on education and tobacco.
Finally, our methodology provides, for any given interval of efficiency parameters
α∈ [0, α∗], the minimum order of stochastic dominance and its corresponding ethical
transfer principle: arg min s {C s i (p)−αCsj(p)≥0,∀α∈ [0, α∗]} (26) s.t.∀p∈[0,1], s.t. s∈ {1,2,3, . . .}.
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 3 0, 0 6 0, 0 9 0, 0 2 1, 0 6 1, 0 9 1, 0 2 2, 0 5 2, 0 8 2, 0 1 3, 0 4 3, 0 7 3, 0 4, 0 044, 74,0 5,0 35,0 05,6 5,09 26,0 56,0 86,0 027, 57,0 87,0 18,0 08,4 78,0 9,0 39,0 69,0 1 Percentiles Cumul order 1 Housing Electricity * 0.59
Fig. 2 Maximum effciency parameter and positional dominance of order 2. Source: Authors’ estimation using Canadian Survey of Household Spending (2002)
0 0,00000002 0,00000004 0,00000006 0,00000008 0,0000001 0,00000012 0,00000014 0,00000016 0,00000018 0,0000002 0 3 0, 0 6 0, 0 9 0, 0 2 1, 0 5 1, 0 8 1, 0 2 2, 0 5 2, 0 8 2, 0 1 3, 0 4 3, 0 7 3, 0 4, 0 034, 4,06 94,0 25,0 05,5 5,08 26,0 56,0 86,0 017, 47,0 77,0 8,0 38,0 08,6 98,0 29,0 59,0 89,0 Percentiles Cumul order 9 Housing Electricity
Fig. 3 Minimal order of positional dominance (s = 10). Source: Authors’ estimation using Canadian Survey of Household Spending (2002)
As can be seen in Fig.3, the 10-order inverse stochastic dominance is the minimal degree of positional dominance that guarantees a non crossing between housing and electricity curves, for allα∈ [0,1].
5 Conclusion
Our methodology—the s-concentration curve—allows decision makers to gauge whether the increase in social welfare induced by a marginal tax reform for pairs of commodities is robust over a large set of rank-dependent social welfare functions. This approach brings out two types of generalization. It generalizes the common con-centration curve of order 2 and its corresponding “difference in concon-centration curve” of order 2 to any order of inverse stochastic dominance, based on the generalization of the principle of positional transfer sensitivity proposed byAaberge(2004). Consequently, for any given efficiency parameter, instead of looking for non-intersecting concentra-tion curves at the order 2, the posiconcentra-tional dominance approach enables decision makers to choose either the order of positional dominance of interest or the minimal one. Con-versely, for any given order of positional dominance, the set of efficiency parameters [0, α∗]can be derived in order to compare the power of the taxation mechanism.
Accordingly, this can contribute to open the way on new issues. Indeed, the s-concentration curves can be used to adapt Duclos et al.(2005) tests for the tar-geting and allocative efficiency of public transfer programs in a positionalist frame-work. Moreover, as the positional stochastic dominance test, expressed in terms of quantiles, is compatible withDavidson and Duclos’s(2000) inference method based on “ p-approach to dominance”, it might be useful to determine wether the welfare improvement is significant or not. Finally, as we study overall welfare only , it would be reasonable to focus on population subgroups such as professional categories and their own consumption habits in order to obtain stronger welfare orderings based on a decomposed but generalized test of restricted inverse stochastic dominance.
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