## Abstract

This paper proposes a methodology for testing for whether tax reforms are pro-poor. This is done by extending stochastic dominance techniques to identify tax reforms that will be deemed absolutely or relatively pro-poor by a wide spectrum of poverty analysts. The statistical properties of the various estimators are also derived in order to make the method implementable using survey data. The methodology is used to assess the pro-poorness of possible reforms to Mexico’s indirect tax system. This leads to the identification of several possible pro-poor tax reforms in that country. It also shows how the pro-poorness of a tax reform depends on one’s conception of poverty as well as on the revenue and efficiency impact of the reform.

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## Notes

Quote taken from the 13 February 2006 edition of Sun Star (http://www.sunstar.com.ph/).

Different approaches have been proposed to separate the poor from the non-poor and to compute indices of “growth pro-poorness”. See, for instance, McCulloch and Baulch (1999), Ravallion and Datt (2002), Kakwani et al. (2003), Ravallion and Chen (2003), Klasen (2004), Son (2004), Essama-Nssah (2005), Araar et al. (2007) and Kakwani and Son (2008).

It is also a much revised version of a paper presented at a symposium held in Monterrey in October 2007; see Audet et al. (2007) for a preliminary version that appeared in the Symposium proceedings. The current paper expands on the theoretical framework, explores the links between efficiency and pro-poorness, and also presents statistical inference techniques that are then applied to Mexican data.

This is the common—though arbitrarily made—assumption in the literature; see Donaldson (1992) for a general discussion.

Consumption-dominance curves were introduced in Makdissi and Wodon (2002).

When

*γ*=1, we have*g*=*a*=0 since*X*_{ j }*dt*_{ j }+*X*_{ i }*dt*_{ i }=0.The analytical results can be extended to account for complex multi-stage sampling designs. Taking into account sampling design is indeed done in the Mexican illustration below, using analytical asymptotic methods along the lines of those described in Duclos and Araar (2006), Chap. 16. More details can be obtained from the authors upon request.

This is particularly true in the study of consumption data, where the second-order derivative of expected consumption at

*z*,*∂*^{2}*CD*^{1}(*x*_{ k };*z*)/(*∂z*)^{2}, may be expected to be small. For more on this, see for instance Härdle (1990), p. 101.For more details, see World Bank (2004).

In 2004, all foodstuffs were exempt of value-added taxes (VAT) in Mexico. A few of these goods were subsidized, however.

See also Navajas and Porto (1994) for a nice discussion of why it is the evolution of these shares across quantiles—and not the level differences of these shares across goods and services for a given quantile—that matter for optimal tax purposes.

Theoretically speaking, the dominance tests carried out in Sect. 4 must be applied over ranges varying between 0 and some

*z*^{+}. Statistically speaking, however, there is a general “information-less” problem in the tails of distributions that impedes such testing for values of*z*close to 0. Hence, statistically speaking, we must restrict the tests to a range that is lower-bounded somewhere above 0. See Davidson and Duclos (2013) for a discussion of this.Note that the poverty headcount at

*z*=0.145 is around 0.3 %. Very little statistical information is thus available below that value, an indication of the information-less problem mentioned in footnote 17. It would also require a pro-poor judgment that would be almost strictly Rawlsian to reverse the pro-poor judgments implied by the tests over 0.145–3 and 0.190–2.971.Regmi and Seale (2010) provide elasticity estimates for 114 countries using 1996 price data from the International Comparison Program.

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## Acknowledgements

This work was carried out with support from SSHRC, FQRSC and the Poverty and Economic Policy Research Network, which is financed by the Government of Canada through the International Development Research Centre and the Canadian International Development Agency, and by the Australian Agency for International Development. We are grateful to Sami Bibi, to two anonymous referees and to the Editor-in-Chief for useful comments. We are also grateful to Lourdes Treviño and Jorge Valero Gil for having facilitated access to the ENIGH data, for their invitation to present this paper at the Eight Symposium on “Capital Humano, Crecimiento, Pobreza: Problemática Mexicana” held in Monterrey, Mexico in October 2006, and for having made available a preliminary version of this paper in the Symposium proceedings—see Audet et al. (2007).

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## Appendix

### Appendix

### 1.1 6.1 Proof of Theorem 1

### Proof

Let

and

Using (5), (11), (12), (13), and (14), we obtain for *η*=*R*,*A*:

The sufficiency condition for *s*=1 is proved from (33) by noting that *p*
^{(1)}(*y*,*z*) is negative. We then need to integrate by parts \(\int_{0}^{\infty}p^{(1)} (y,z )\mathit{CD}_{i}^{\eta:1}(y)\,dy\):

We know that \(\mathit{CD}_{i}^{\eta:2} ( 0 ) =0\) and that *p*
^{1}(∞,*z*)=0. The first term on the r.h.s. of the above is thus nil. Consequently, equation (34) may be rewritten as

Now, assume that we have

Integrating by parts equation (36), we get

\(\mathit{CD}_{i}^{\eta:s} ( 0 ) =0\) and *p*
^{(s−1)}(∞,*z*)=0 is implied by the definition of ∞ and by (2). We can rewrite (37) as

Equation (35) obeys the relation depicted in (36). We have shown that if (36) is true then equation (38) is also true. This implies that equation (38) is true for *η*=*R*,*A* and for all integer *s*∈{2,3,…,*s*−1}. From equation (33) and (38), we get

This last equation together with equation (2) proves the sufficiency of the condition.

In order to establish necessity, consider the set of functions *p*(*y*,*z*) for which the (*s*−1)th derivative (with *p*
^{(0)}(*y*,*z*)=*p*(*y*,*z*)) is of the following form:

Poverty indices whose function *p*(*y*,*z*) has the particular above form for *p*
^{(s−1)}(*y*,*z*) belong to *Π*
^{s}. This yields

Imagine now that \(\mathit{CD}_{i}^{\eta:s} ( y )<0\) on an interval \([\overline{y},\overline{y}+\epsilon ]\) for \(\overline{y}<z^{+}\) and for *ϵ* that can be arbitrarily close to 0. For *p*(*y*,*z*) defined as in (40), expression (39) is then positive and the marginal tax reform induces a marginal increase of poverty. Hence, it cannot be that \(\mathit{CD}_{i}^{\eta:s} ( y )<0\) for \(y\in [ \overline{y},\overline{y}+\epsilon ] \) when \(\overline{y}<z^{+}\). This proves the necessity of the condition. □

### 1.2 6.2 Proof of Theorem 3

### Proof

\(\widehat{\mathit{CD}}^{s}(x_{k};z)\) is a consistent estimator of *CD*
^{s}(*x*
_{
k
};*z*) by the existence of the first population moment of \(x_{k}(y) ( z-y )_{+}^{s-2}\) and the law of large numbers. \(\widehat{\mathit{CD}}^{s}(x_{k};z)\) is *N*
^{0.5} consistent and asymptotically normal by the existence of the second population moment and the central limit theorem, with asymptotic variance given by (24) by simple calculation. □

### 1.3 6.3 Proof of Theorem 4

### Proof

Note first that \(E [ \widehat{\mathit{CD}}^{1}(x_{k};z) ] =\int \kappa_{h} ( z-y ) x_{k}(y)f(y)\,dy\). Denoting *t*=*h*
^{−1}(*z*−*y*) and expanding around *t*
_{0}=0, for small *h* this is approximately equal to

since ∫*κ*(*u*)*du*=1,∫*uκ*(*u*) *du*=0, and ∫*u*
^{2}
*κ*(*u*) *du*=*c*
_{
κ
}. Hence, the bias \(E [ \widehat{\mathit{CD}}^{1}(x_{k};z) ] -{\mathit{CD}}^{1}(x_{k};z)\) is given by \(0.5h^{2}\widehat{\mathit{CD}}^{1}{^{\prime \prime }}(x_{k};z) c_{\kappa }\).

By (25), note that \(\widehat{\mathit{CD}}^{1}(x_{k};z)\) is a sum of iid variables to which we may apply the central limit theorem and show asymptotic normality. We also have

where the last expression is obtained by substituting *u* for *h*
^{−1}(*z*−*y*). For small *h*, (43) is approximately equal to

Hence,

which concludes the proof. □

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Duclos, JY., Makdissi, P. & Araar, A. Pro-poor indirect tax reforms, with an application to Mexico.
*Int Tax Public Finance* **21**, 87–118 (2014). https://doi.org/10.1007/s10797-012-9260-x

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DOI: https://doi.org/10.1007/s10797-012-9260-x