How Poverty is . . . Fuzzy Approach to . . . Towards Precise . . . Independence Semi-Heuristic Poverty Two Main Ways of . . . Measures Used by Main Definition Discussion and Main . . . Economists: Justification Conclusions Proof Motivated by Fuzzy Home Page Techniques Title Page ◭◭ ◮◮ Karen Villaverde 1 , Nagwa Albehery 1 , ◭ ◮ Tonghui Wang 1 , and Vladik Kreinovich 2 Page 1 of 14 1 New Mexico State University, Las Cruces, NM 88003, USA Go Back kvillave@cs.nmsu.edu, albehery@nmsu.edu, twang@nmsu.edu 2 University of Texas at El Paso, El Paso, TX 79968, USA Full Screen vladik@utep.edu Close Quit
How Poverty is . . . Fuzzy Approach to . . . 1. How Poverty is Measured Now Towards Precise . . . • Usually, there is a poverty threshold threshold z : a per- Independence son i with an income x i is poor ⇔ x i < z . Two Main Ways of . . . • Media: measures property by the proportion F 0 = H Main Definition N Discussion and Main . . . of poor people. Conclusions • Limitation: F 0 does not distinguish between very poor Proof ( x i ≪ z ) and simply poor. Home Page • To capture this difference, economists use special Foster- Title Page Greer-Thorbecke (FGT) property measures: ◭◭ ◮◮ H H F 1 = 1 and F 2 = 1 1 − x i 1 − x i � 2 � � � ◭ ◮ � � N · N · . z z Page 2 of 14 i =1 i =1 • Success: these measures are used to gauge the success Go Back of different measures aimed at reducing poverty. Full Screen • Problem: these measures are semi-heuristic, other mea- Close sures may be more adequate. Quit
How Poverty is . . . Fuzzy Approach to . . . 2. Fuzzy Approach to Poverty Towards Precise . . . • Poverty is a matter of degree: µ (0) = 1, µ ( z ) = 0. Independence • Simplest membership function – linear: µ ( x ) = 1 − x Two Main Ways of . . . z . Main Definition • The cardinality of a fuzzy set is defined as � µ ( x ) . Discussion and Main . . . x Conclusions • Thus, the cardinality of the set of all poor people is Proof H Home Page 1 − x i � � � . Title Page z i =1 ◭◭ ◮◮ • This sum is proportional to F 1 . ◭ ◮ • For a fuzzy property P , “very P ” is usually interpreted Page 3 of 14 as µ 2 ( x ). Go Back • Thus, the cardinality of the set of all very poor people H 1 − x i � 2 Full Screen � � is ; this sum is proportional to F 2 . z i =1 Close Quit
How Poverty is . . . Fuzzy Approach to . . . 3. Fuzzy Approach: Conclusion and Limitations Towards Precise . . . • Good news: all three FGT measures F 0 , F 1 , and F 2 Independence naturally appear in the fuzzy interpretation. Two Main Ways of . . . Main Definition • Each of F i is the ratio of the number of poor people to Discussion and Main . . . the population as a whole: Conclusions • F 0 appears when we consider poverty to be a crisp Proof property; Home Page • F 1 appears when we take that poverty is a fuzzy Title Page property; ◭◭ ◮◮ • F 2 appears when we count the number of very poor ◭ ◮ people. Page 4 of 14 • Limitation: the justification depends on a specific choice of linear membership function (and µ 2 ( x ) for “very”). Go Back • What we do: we go from an informal to a precise jus- Full Screen tification. Close Quit
How Poverty is . . . Fuzzy Approach to . . . 4. Towards Precise Definitions Towards Precise . . . • Natural requirement: Independence Two Main Ways of . . . – if we know poverty measures, populations, and num- Main Definition ber of poor in two subareas, Discussion and Main . . . – then we should be able to compute the property Conclusions measure for the whole area. Proof • Known result: all such measures have the form v f = Home Page H � f ( x i ) for some f ( x ). Title Page i =1 ◭◭ ◮◮ • Fact: different measures describe different aspects of ◭ ◮ poverty. Page 5 of 14 H � • So: we want to select k measures f j ( x i ), j = 1 , . . . , k . Go Back i =1 Full Screen Close Quit
How Poverty is . . . Fuzzy Approach to . . . 5. Independence Towards Precise . . . • In principle: based on two measures f 1 ( x ) and f 2 ( x ), Independence we can form a new measure Two Main Ways of . . . f ( x ) = f 1 ( x ) + f 2 ( x ) Main Definition . 2 Discussion and Main . . . Conclusions • In this case: Proof H H Home Page � � – once we know v f 1 = f 1 ( x i ) and v f 2 = f 2 ( x i ), i =1 i =1 Title Page H � – we can reconstruct v f = f ( x i ) as ◭◭ ◮◮ i =1 ◭ ◮ v f = v f 1 + v f 2 . Page 6 of 14 2 • Natural: assume that f 1 ( x ) , . . . , f k ( x ) are independent : Go Back none of the v f i ’s can reconstructed from the others. Full Screen Close Quit
How Poverty is . . . Fuzzy Approach to . . . 6. Two Main Ways of Helping the Poor Towards Precise . . . • One possibility is to allocate a certain fixed amount of Independence money (or goods) a to each poor person. Two Main Ways of . . . Main Definition • Example: US food stamps. Discussion and Main . . . • In this case, the original incomes change from x i to Conclusions x ′ i = x i + a . Proof Home Page • Another possibility is to provide tax deductions to all the poor people. Title Page • Example: tax deductions in the US. ◭◭ ◮◮ • Since taxes are usually proportional to the income x i , ◭ ◮ income increases to x ′ i = λ · x i , for some λ > 1. Page 7 of 14 • An efficient set of poverty measures should enable us Go Back to predict how these measures change when we help. Full Screen Close Quit
How Poverty is . . . Fuzzy Approach to . . . 7. Main Definition Towards Precise . . . An independent set of poverty measures f 1 ( x ) , . . . , f k ( x ) is Independence called efficient if the following two properties hold: Two Main Ways of . . . Main Definition H � • once we know all k poverty values v j = f j ( x i ) and Discussion and Main . . . i =1 a > 0, we can uniquely predict the new poverty values Conclusions Proof H � Home Page v ′ j = f j ( x i + a ); i =1 Title Page ◭◭ ◮◮ H � • once we know all k poverty values v j = f j ( x i ) and ◭ ◮ i =1 λ > 1, we can uniquely predict new poverty values Page 8 of 14 H Go Back � v ′ j = f j ( λ · x i ) . Full Screen i =1 Close Quit
How Poverty is . . . Fuzzy Approach to . . . 8. Discussion and Main Result Towards Precise . . . • Lemma: The set of FGT measures f 0 ( x ) = 1, f 1 ( x ) = Independence 1 − x 1 − x � 2 � Two Main Ways of . . . z , and f 2 ( x ) = is efficient. z Main Definition • We say that two independent sets of poverty measures Discussion and Main . . . f 1 ( x ) , . . . , f k ( x ) and g 1 ( x ) , . . . , g l ( x ) are equivalent if: Conclusions – each f j ( x ) depends on g 1 ( x ) , . . . , g l ( x ); and Proof Home Page – each g j ( x ) depends on f 1 ( x ) , . . . , f k ( x ). Title Page • Proposition: The set of FGT measures is equivalent to { 1 , x, x 2 } . ◭◭ ◮◮ ◭ ◮ • Theorem: Every efficient independent set of poverty measures f 1 ( x ) , . . . , f k ( x ) is equiv. to { 1 , x, x 2 , . . . , x k − 1 } . Page 9 of 14 • Corollary: Every efficient independent set of poverty Go Back measures f 1 ( x ) , f 2 ( x ) , f 3 ( x ) is equiv. to the FGT set. Full Screen • Conclusion: We have thus justified FGT measures. Close Quit
How Poverty is . . . Fuzzy Approach to . . . 9. Conclusions Towards Precise . . . • Several semi-heuristic poverty measures have been pro- Independence posed, e.g., Foster-Greer-Thorbecke (FGT) measures. Two Main Ways of . . . Main Definition • FGT measures have worked well on many situations to Discussion and Main . . . which they have been applied; however: Conclusions – to be sure that these poverty measures will work in Proof other situations as well, Home Page – it is desirable to supplement the empirical confir- Title Page mation with a theoretical justification. ◭◭ ◮◮ • In this talk: ◭ ◮ – we first use fuzzy logic to provide a commonsense Page 10 of 14 interpretation of the FGT measures, and – then we transform this commonsense explanation Go Back into a theoretical justification for these measures. Full Screen • This makes us more confident in using FGT measures. Close Quit
How Poverty is . . . Fuzzy Approach to . . . 10. Acknowledgment Towards Precise . . . • This work was supported in part: Independence Two Main Ways of . . . – by the National Science Foundation grant HRD- Main Definition 0734825 (Cyber-ShARE Center of Excellence), Discussion and Main . . . – by the National Science Foundation grant DUE- Conclusions 0926721, and Proof – by Grant 1 T36 GM078000-01 from the National Home Page Institutes of Health. Title Page • The authors are thankful to the anonymous referees for ◭◭ ◮◮ valuable suggestions. ◭ ◮ Page 11 of 14 Go Back Full Screen Close Quit
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