On the Mean of Squared Deprivation Gaps

Transcription

1 On the Mean of Squared Deprivation Gaps Achin Chakraborty Institute of Development Studies Kolkata,1, Reformatory Street Calcutta University Alipore Campus, 5th Floor, Kolkata , India achin@idsk.org PRASANTA K. PATTANAIK Department of Economics, University of California, Riverside, CA 92521, U.S.A. Telephone: ; Fax: ; ppat@ucrac1.ucr.edu YONGSHENG XU Department of Economics, Andrew Young School of Policy Studies P.O. Box 3992, Georgia State University, Atlanta, GA , U.S.A. Telephone: ; Fax: ; yxu3@gsu.edu This version: May 4, 2005 Abstract. This paper provides an axiomatic characterization of the mean of squared deprivation gaps, which is a distinguished member of the important class of deprivation measures first introduced by Foster, Greer and Thorbecke (1984) and which has been extensively used in applied work on deprivation. JEL Classification Numbers: D3, I3, O1 We are grateful to Salvador Barberà and Buhong Zheng for helpful comments on an earlier version of the paper.

2 1. Introduction In recent years, there has been an increased interest in poverty and deprivation in general and their measurement in particular. The purpose of this paper is to explore the structure of a well-known measure of deprivation, namely, the mean of squared deprivation gaps. It is a distinguished member of the well-known class of measures of social deprivation that was first introduced by Foster, Greer, and Thorbecke (1984). The mean of squared deprivation gaps, which measures social deprivation by the mean of the squares of the individuals normalized deprivation levels, has been discussed extensively and used in many contexts in applied work (see, for example, Greer and Thorbecke (1986), Datt and Ravallion (1992, 1997), and Dutta and Pattanaik (2000)). The popularity of this measure is due to its many attractive features including its simplicity, its additive decomposability, and its sensitivity to poverty aversion (see Zheng (2000) for some discussions of the issue of sensitivity to poverty aversion). However, despite its importance and prominence in the literature on poverty and deprivation, there does not seem to be any comprehensive investigation of the properties of the mean of squared deprivation gaps. Our objective is to investigate the structure of the mean of squared deprivation gaps by presenting an axiomatic characterization for it. At the risk of emphasizing the obvious, we would like to clarify that we are not necessarily trying to provide a justification of the mean of squared deprivation gaps through our axiomatic characterization. The purpose of the characterization is to understand the structural properties of the measure so that one can assess which features of the measure are intuitively pluasible and which of them are of doubtful appeal. Indeed, our characterization shows that one crucial property, equivalent transfer, which the mean of squared deprivation gaps shares with only two other measures (the head count ratio and the average deprivation gap) belonging to the very large Foster-Greer-Thorbecke (1984) class of deprivation measures, constitutes the main intuitively arbitrary feature of the mean of squared deprivation gaps. The paper is organised as follows. In Section 2, we introduce some basic notation and definitions. In Section 3 we introduce certain axioms and provide a chraterization of the mean of squared deprivation gaps. The proofs are given in the appendix. 2. The Basic Notation and Definitions Consider a society consisting of n ( > n 4) fixed individuals. Let IR + = 1

3 [0, ), and IR n + be the n-fold Cartesian production of IR +. For all e IR n +, all i {1,..., n}, and all a IR +, e i denotes the vector (e 1,..., e i 1, e i+1,..., e n ), and (e i, a) denotes the vector (e 1,..., e i 1, a, e i+1,..., e n ). For all c [0, 1], let c [n] denote the n-dimensional vector with c for each of its components. Let N denote {1,, n}. For all i N, let z i IR + denote the achievement of individual i, and let z i IR + denote the deprivation benchmark for i : i is deprived if and only if z i > z i. It is often assumed that z i is the same for all i N. However, in some contexts, one may like to allow z i to take different values depending on the individual under consideration: for example, when fixing the income poverty line, one may like to have different values of z i depending on whether the individual lives in an urban area or a rural area. We may interpret the achievement of an individual, as well as the deprivation benchmark, differently. If one s concern is with the measurement of income poverty, z i and z i will be interpreted in terms of income. On the other hand, if one is interested in measuring deprivation in terms of some real indicator, such as housing, health or education, then the interpretation of z i and z i is to be interpreted in terms of that indicator. For all i N, define x i as follows: x i = 0 if z i z i and x i = (z i z i )/z i if z i < z i. Thus, we have an n-tuple vector x = (x 1,, x n ), which we will call the normalized deprivation vector and which shows each individual s normalized deprivation level. Clearly, x [0, 1] n. For all x, y [0, 1] n, and all N (φ N N), we say that x and y are N -variants iff [for all i N, x i y i, and, for all j N N, x j = y j ]. A deprivation measure for the society is defined as a function f : [0, 1] n IR +. For all x, y [0, 1] n, f(x) f(y) is interpreted as indicating that the degree of deprivation in situation x is at least as great as the degree of deprivation in situation y. Throughout this paper, we assume that the deprivation measure is cardinal in the sense of being unique up to an affine transformation. Definition 1. Let f be a deprivation measure. f is a member of the Foster-Greer- Thorbecke (FGT) class of deprivation measures iff, for some non-negative real number α, f(x) = 1 n i N xα i, for all x [0, 1] n. Clearly, the FGT class of deprivation measures includes the mean of squared 2

4 deprivation gaps (α = 2), the average deprivation gap (α = 1), and the head-count ratio (α = 0). They are well-known in the literature and hardly need explanation. 3. An Axiomatic Characterization of the Mean of Squared Deprivation Gaps The following definition introduces the properties that we shall use to characterise the mean of squared deprivation. Definition 2. Let f be a deprivation measure. We define the following properties of f. (i) Continuity (CON). f(x) is continuous at x 0 (0, 1) n. (ii) Normalization (NOR). f( 0 [n] ) = 0, f(1 [n] ) = 1. (iii) Anonymity (AN). Let σ be a one-to-one function from N to N. Then, for all x [0, 1] n, f(x) = f(x σ(1),..., x σ(n) ). (iv) Monotonicity (MON). For all x [0, 1] n, all i N, and all x i [0, 1], if x i > x i then f(x i, x i) > f(x). (v) Independence (IND). For all x, y [0, 1] n, all i N, and all t 0, if x i +t [0, 1], then f(x i, x i + t) f(x i, x i ) = f(y i, x i + t) f(y i, x i ). (vi) Uniform Scale-invariance (USI). Let x, y, x, y [0, 1] n. Suppose f(x) f(y) = f(x ) f(y ). Then f(kx) f(ky) = f(kx ) f(ky ) for every k > 0, such that kx, ky, kx, ky [0, 1] n. (vii) Transfer Axiom (TA). For all x, y [0, 1] n, all distinct i, j N, and all δ (0, 1], if [x and y are {i, j} variants], [x i > x j ], [ y i = x i + δ], and [ y j = x j δ], then f(y) > f(x). (viii) Equivalent Transfer (ET). Let x, y, w (0, 1] n, i, j, i, j N, and t [0, 1] be such that [x i x j = x i x j ], [x and y are {i, j} variants and x and w are {i, j } variants], [y i = x i t and y j = x j +t], and [w i = x i t and w j = x j +t]. Then f(y) = f(w). CON, NOR, AN, MON are fairly straightforward and well-known in the literature, and therefore need no further explanation. IND ensures that the deprivation measure 3

5 is additive, and is formally equivalent to the counterpart, in our context, of Foster, Greer, and Thorbecke s (1984) well-known property of additive decomposability with population share weights. The intuition of USI can be explained as follows. Consider two situations with respective normalized deprivation vectors x and y. Suppose there is an equiproportionate increase in the miseries of all the people in both situations. Then the difference between the levels of social misery in the two situations will change by an amount that will depend exclusively on the initial difference in social misery and the proportionality factor by which the individual miseries change. The transfer axiom requires that if, initially, individuals i and j are both deprived, but the normalized deprivation of i is higher than that of j, then, other things remaining the same, an increase of δ in the normalized deprivation of i, together with a decrease of δ in the normalized deprivation of j, will increase the deprivation of the society. Our transfer axiom is reminiscent of (though, formally, not identical to) Sen s (1976) transfer axiom which stipulates that a transfer of income from a poor person to anybody richer must increase the society s poverty. ET can be explained as follows. Consider four individuals i, j, i, and j, and a deprivation vector x, such that everybody is deprived in x and the normalized deprivation of i exceeds (resp. falls short of) the normalized deprivation of j by exactly the same amount by which the normalized deprivation of i exceeds (resp. falls short of) the normalized deprivation of j. Now, consider two alternative changes, starting with the initial situation of x. In the first case, other things remaining the same, the normalized deprivation of i decreases by t (t 0), the normalized deprivation of j increases by t, and i continues to be deprived after the change. In the second case, the normalized deprivation of i decreases by t, the normalized deprivation of j increases by t, and i continues to be deprived after the change. Then ET requires that the new level of social deprivation in the first case must be the same as the new level of social deprivation in the second case. Thus, given that in the initial situation the deprivation difference between i and j is the same as the deprivation difference between i and j, ET ensures that the change in social deprivation when normalized deprivation of t is transferred from i to j is the same as the change in social deprivation when normalized deprivation of t is transferred from i to j. It seems to us that, of all the properties introduced in definition 2, ET seems to have the least intuitive appeal: we cannot think of any deep intuitive reason why one would 4

6 like to postulate this property for a measure of deprivation. Theorem 1. A deprivation measure satisfies CON, NOR, IND, USI, AN, MON, ET and TA if and only if it is the mean of squared deprivation gaps. Proof: The proof is given in the Appendix. Note that the mean of squared deprivation gaps is one of a very few deprivation measures in the FGT class which satisfy ET. In fact, it can be shown that the mean of squared deprivation gaps, the average deprivation gap, and the head-count ratio are the only three deprivation measures in the FGT class, which satisfy ET. Among the deprivation measures in the FGT class, the mean of squared deprivation gaps ( α = 2), the average deprivation gap (α = 1), and the head-count ratio (α = 0) have a remarkably appealing simplicity of structure, and all satisfy CON, NOR, IND, USI, AN, and ET. However, it can be checked that neither the average deprivation gap nor the head-count ratio satisfies TA, and the head-count ratio does not satisfy MON. Thus, of all the deprivation measues in the FGT class, the mean of squared deprivation gaps seems to be the only one which has a very simple structure and which violates very few of the basic appealing properties that one can think of in this context. It is not, therefore, surprising that the mean of squared deprivation gaps is perhaps the single most popular deprivation measure in the literature. At the same time, as we have noted earlier, it is difficult to find intuitive justification for the property of ET, which the mean of squared deprivation gaps shares with the average deprivation gap and the head-count ratio. In fact, from an intuitive point of view, ET would seem to be the main arbitrary feature of the mean of squared deprivation gaps as a measure of deprivation. References Aczél, J. (1966). Lectures on Functional Equations and Their Applications, Academic Press, New York. Aczél, J. (1988). Cheaper by the Dozen: Twelve Functional Equations and Their Applications to the Laws of Science and to Measurement in Economics, in Eichhorn, W. (ed.), Measurement in Economics: Theory and Applications in Economic Indices, Heidelberg: Physica-Verlag. 5

7 Datt, G. and M. Ravallion (1992). Growth and Redistribution Components of Changes in Poverty Measures: A Decomposition with Applications to Brazil and India in the 1980s, Journal of Development Economics, Vol. 38, Datt, G. and M. Ravallion (1997). Macroeconomic Crises and Poverty Monitoring: A Case Study for India, Review of Development Economics, Vol. 1 (2), Dutta, I., and P. K. Pattanaik (2000). Housing Deprivation in a Village in Orissa, Journal of Quantitative Economics, Vol. 16 (1), Eichhorn, W. and W. Gleissner (1988). The Equation of Measurement, in Eichhorn, W. (ed.), Measurement in Economics: Theory and Applications in Economic Indices, Heidelberg: Physica-Verlag. Foster, J. E., J. Greer, and E. Thorbecke (1984). A Class of Decomposable Poverty Measures, Econometrica, Vol. 52 (3), Greer, J. and E. Thorbecke (1986). A Methodology for Measuring Food Poverty Applied to Kenya, Journal of Development Economics, Vol. 24, Sen, A. K. (1976). Poverty: An Ordinal Approach to Measurement, Econometrica, Vol. 44, Zheng, B. (2000). Mininum Distribution-Sensitivity, Poverty Aversion, and Poverty Orderings, Journal of Economic Theory, Vol. 95, Appendix Proof of Theorem 1. It can be checked that if f is the mean of squared deprivation gaps then it satisfies the axioms specified in Theorem1. Suppose now that f satisfies CON, NOR, IND, USI, AN, MON, ET and TA. It can be verified that, by IND and AN, there exists a function g : [0, 1] IR + such that, for all x [0, 1] n, f(x 1,, x n ) = g(x 1 )+ +g(x n ). NOR implies that g(0) = 0 and g(1) = 1/n. Since f satisfies CON and MON, g is then continuous at some t 0 (0, 1) and monotonic: for all a, b [0, 1], a > b g(a) > g(b). Let a, b, a, b [0, 1] and k > 0 be such that g(a) g(b) = g(a ) g(b ) and ka, kb, ka, kb [0, 1]. Then, we must have f(a [n] ) f(b [n] ) = ng(a) ng(b) = ng(a ) 6

8 ng(b ) = f(a [n] ) f(b [n] ), and hence, by USI, f(ka [n] ) f(kb [n] ) = f(ka [n] ) f(kb [n] ). It then follows that g(ka) g(kb) = g(ka ) g(kb ). satisfies the following: Thus, we have shown that g (1) For all a, b, a, b (0, 1] and k > 0 such that ka, kb, ka, kb [0, 1], [g(a) g(b) = g(a ) g(b ) g(ka) g(kb) = g(ka ) g(kb )]. Note that g satisfies (1) is equivalent to that g satisfies the following functional equation: (2) g(ta) g(tb) = h(g(a) g(b), t) for some function h, where t, a, b [0, 1]. Let u = g(a) g(b), v = g(b) g(c). Then, u + v = g(a) g(c). Note that u, v [ 1/n, 1/n]. From (2), we obtain h(u + v, t) = h(u, t) + h(v, t). This is the Cauchy equation and its solution is given by h(u, t) = ψ(t)u (see, for example, Aczél (1966) ). Therefore, (2) becomes (3) g(ta) g(tb) = ψ(t)(g(a) g(b)) for all t, a, b [0, 1]. For all c IR + with c 0, c can be written as c = p/q where p, q (0, 1]. Define G(c) = 1 g(p) n g(q), for c > 0, where c = p, and p, q [0, 1] q and G(0) = g(0) = 0, G(1) = g(1) = 1/n. The function G is well defined because: (i) for all c 1 = c 2 IR +, we can write c 1 = p 1 /q 1 and c 2 = (kp 1 )/(kq 1 ) for some k [0, 1] and some p 1, q 1 [0, 1]; and (ii) by (3), G(c 2 ) = 1 g(kp 1 ) g(0) n g(kq 1 ) g(0) = 1 ψ(k)(g(p 1 ) g(0)) n ψ(k)(g(q 1 ) g(0)) = G(c 1). Given that g satisfies (1), it is straightforward to check that G satisfies the following: (4) for all positive real numbers u, v, u, v, [G(u) G(v) = G(u) G(v)] implies G(ku) G(kv) = G(ku) G(kv) for all k > 0. Further, given that g is continuous at some t 0 (0, 1) and is monotonic, G is continuous at some point in IR + and is nonconstant. By the result of Eichhorn and Gleissner (1988) and Aczél (1988), we must have either (for some constants θ, α and 7

9 γ, g(c) = θc α +γ, for all c (0, 1]) or (for some constants θ and γ, g(c) = θ log c+γ, for all c (0, 1]). Note that G(0) = 0 and G(1) = 1/n, and that G is monotonic. Therefore, G(c) = cα, where α > 0. Note that, from the definition of G, G is the unique n extension of g. Therefore, g(a) = aα for all a [0, 1], where α > 0. Let δ (0, 1/3] n and x, y [0, 1] n be such that x and y are {i, j}-variants, x j = δ, x i = 2δ, y i = 3δ and y j = 0. By TA, we must have 3 α δ α > 2 α δ α + δ α. That is, 3 α > 2 α + 1. Therefore, α > 1. Consider x, y, z (0, 1] n and i, j, i, j N and t, t 1 (0, 1] such that x i = x j + t 1, x i = x j + t 1, x and y are {i, j}-variants, x and z are {i, j }-variants, y i = x i t = x j +t 1 t, y j = x j +t, z i = x i t = x j +t 1 t, and z j = x j +t. From ET, f(y) = f(z); that is, (x j +t 1 t) α +(x j +t) α +(x j +t 1 ) α +x α j = (x j+t 1 ) α +x α j +(x j +t 1 t) α +(x j +t) α. Therefore, (x j + t) α x α j t (x j + t 1 ) α (x j + t 1 t) α t = (x j + t)α x α j (x j + t 1) α (x j + t 1 t) α t t Noting that α > 1 and the function a α is differentiable, by allowing t to approach to 0, we have αx α 1 j α(x j + t 1 ) α 1 = αx α 1 j α(x j + t 1 ) α 1. Since α > 1, we then obtain (x j + t 1 ) α 1 x α 1 j = (x + t j 1 ) α 1 x α 1 j. Note that x j > 0 and x j > 0, t 1 t 1 and the function a α 1 is differentiable for positive numbers a. By allowing t 1 to approach to 0, we obtain (α 1)x α 2 j = (α 1)x α 2 j. Then, x α 2 j = x α 2 j from α 1 > 0. Note that the above equality holds for any arbitrary x j and x j (0, 1]. Therefore, α = 2. follows in 8