DEPARTMENT OF ECONOMICS ISSN DISCUSSION PAPER 20/07 TWO NEW EXPONENTIAL FAMILIES OF LORENZ CURVES

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1 DEPARTMENT OF ECONOMICS ISSN DISCUSSION PAPER /7 TWO NEW EXPONENTIAL FAMILIES OF LORENZ CURVES ZuXiang Wang * & Russell Smyth ABSTRACT We resent two new Lorenz curve families by using the basic model roosed by Sarabia, Castillo and Slottje (1999) We resent estimations which show that the models in our new families are very efficient when alied to data on income distribution for a range of countries from Shorrocks (198) JEL classification: D; C5 Keywords: Lorenz curve * Deartment of Economics, Wuhan University, Wuhan 47, China Deartment of Economcs, Monash University, Caulfield East, Vic 145, Australia 7 ZuXiang Wang and Russell Smyth All rights reserved No art of this aer may be reroduced in any form, or stored in a retrieval system, without the rior written ermission of the author 1

2 1 Introduction Research into finding satisfactory arametric Lorenz models has rogressed in recent years Several good erforming Lorenz models have been develoed Sarabia, Castillo and Slottje (1999) (hereafter, SCS, 1999) roose a basic Lorenz model along with a generalized Pareto family of Lorenz curves In a later contribution, Sarabia, Castillo and Slottje (1) (hereafter SCS, 1) develo an exonential family of Lorenz curves Wang, Ng and Smyth (7) (hereafter WNS, 7) rovide the condition for the SCS (1999) basic model to be a Lorenz curve and resent a few families of Lorenz curves The urose of this letter is to rovide two new Lorenz curve families by using the SCS (1999) basic model We resent estimations which show that the models in our new families are very efficient when alied to data on income distribution for a range of countries from Shorrocks (198) The ordered family of Lorenz curves A function L ( ) defined on [,1] is a Lorenz curve if: + L ( ) =, L ( 1) = 1, L ( ) and L ( ) for all [,1] The basic SCS (1999) Lorenz model is ~ α η L ( ) = L( ) where ( ) L is a Lorenz curve The main model in the SCS (1999) generalized Pareto family is: S where [ 1 (1 ) ] η α ( ) S( ) 1 (1 ) = (1) = is called the generating curve of the family The following result from WNS (7), which is a generalization of the results of SCS (1999), can be used to create more efficient Lorenz models: Theorem 1 Assume L ( ) is a Lorenz curve ~ α η L ( ) = L( ) is a Lorenz curve for any α and η 1 Furthermore, if L ( ) for all [,1], ~ then L ( ) is a Lorenz curve if α, η 1 and α +η 1

3 The main model in the SCS (1) exonential family of Lorenz curves is: α η S ( ) = L ( ) () where the generating model is e L ( ) = e, > which is a Lorenz curve roosed by Chotikaanick (199) Two new families of Lorenz curves We first describe some roerties of L ( ) Lemma 1 L ( ) ossesses the following roerties: (a) For any, L ( ) and L ( ) on [,1] (b) For any [,1], L ( ) if > and L ( ) if < L ( ) is convex if >, but is concave if < (c) L ( ) = L ( ) and, consequently, L ( ) = L ( ) = L ( ) Therefore, (d) lim L ( ) = Proof L ( ) (or L ( ) ) on [,1] because its numerator and denominator will always have the same sign when > and < The definition of L ( ) imlies that (c) is true and (b) can be imlied by (c) and (a) Meanwhile, (d) can be verified using the L Hosital s rule QED To obtain another family, change the generating model of S ( ) from ( ) to L T ( ) = 1 L (1 ) e = 1 e (1 ) () We consequently obtain our first family of Lorenz curves which contains T ( ) and

4 T [ 1 L (1 ) ] α 1( ) T = (4) [ 1 L (1 ] η ( ) ) T = (5) [ 1 L (1 ) ] η α ( ) = (6) T ( ) is the main model in the family with the others being secial cases Note that T ( ) = and T ( 1) = 1 for i =,1,, if α, >, > and η > i i While T ( ) differs from S ( ), it is closely related to S ( ) The generating curve S ( ) 1 (1 ) = of S ( ) can only be a convex curve on [,1] for any (,1] Since limt ( ) = 1 (1 ) from Lemma 1, T ( ) includes S ( ) as a secial case However, T ( ) can be concave as well as convex on [,1] from Lemma 1 and thus can be more flexible than S ( ) Thus, T = is a generalization of α η ( ) T ( ) S = and is therefore more α η ( ) S( ) flexible We exect that the erformance of T ( ) will be at least as good as S ( ), which itself is a very good model as ointed out by SCS (1999) For the same reason, T ( ) must be more flexible than α η S ( ) = L ( ), since L ( ) must reresent a convex curve Lemma Assume < T ( ) is a Lorenz curve with T ( ) if (,1] Proof Note that T ( ) satisfies T ( ) = L (1 ) L (1 ), [(1 ) L (1 ) L (1 ) L (1 )] T ( ) = L (1 ) (7) Lemma 1 imlies L ( 1 ), L ( 1 ) and L ( 1 ) if < 4

5 Therefore, the two equations in (7) imly that T ( ) is a Lorenz curve if (,1] Moreover, note that T ( ) = (1 )( ) L (1 ) + L (1 ) L (1 ) L (1 ) (8) (1 ) L (1 ) L (1 ) Therefore, T ( ) if (,1] QED Lemma Assume (, ln ] T ( ) is a Lorenz curve if (,1) T ( ) is a Lorenz curve with T ( ) if (,1 ] ln is the natural log of Proof Because L ( 1 ) L = (1 ) from Lemma 1, the term between the braces on the right-hand side of (7) is L (1 ) L (1 ) (1 ) [(1 ) L (1 ) L (1 ) ] = ( 1 e ) e But 1 (1 ) e is non-negative on,1] [ if (,ln ] and (,1) Therefore, we have T ( ) Thus the first statement of the lemma is true Furthermore, denoting the right-hand side of (8) as f ( ), we have ( e ) L (1 ) f ( ) = (1 )( ) e (1 ) = (1 )( ) ( e + (1 )( e (1 ) ) (1 ) ) The right-hand side of this equation is non-ositive if > and (,1 ] This imlies that f ( ) is a decreasing function on [,1] Therefore f ( ) and, consequently, T ( ) on [,1] because f (1) (1 )( ) L () > = Therefore the second statement of the lemma is also true QED 5

6 Theorem Assume α, η 1 and α +η 1 T ( ) is a Lorenz curve if: or (,1], < (9) (,1 ], (,ln ] (1) Proof Lemma, the second statement of Lemma and Theorem 1 together imly that T ( ) is a Lorenz curve if (9) or (1) holds QED Note that T ( ) is also a Lorenz curve if α, (, ) U (, ln ], (,1] and η 1 But the model with this arameterization aears inferior to the model with the arameter ranges given by Theorem Using the concet of the hybrid model roosed by Ogwang and Rao (), we suggest another Lorenz family, which is a generalization of both T ( ) and S ( ) : V ( ) { δ ( 1 L (1 ) ) + (1 δ ) L ( ) } η α = Drawing on the result of Lemma, Lemma and Theorem 1, we have: 1 Corollary Assume α, η 1, α +η 1, δ [,1] and V ( ) is a Lorenz curve if and satisfy either (9) or (1) 1 > Then There is one major difference between the cases where < and > in Theorem can be any negative number in (9) However, if is ositive, then its admissible range must deend on as described in (1) The arameter ranges for T ( ), T ( ) and T ( ) to be Lorenz curves, though, 1 are much simler since Lemma, the first art of Lemma and Theorem 1 imly: 6

7 Theorem : Assume (,1] and (, ) U (, ln ] Then (1) T ( ) is a Lorenz curve () T ( ) is a Lorenz curve for any α 1 () T ( ) is a Lorenz curve for any η 1 One drawback of V ( ) is that it does not have a Gini index formula with closed form While the Gini index formula for T ( ) can be obtained in a similar manner to that described in SCS (1) if <, the formula becomes tedious to calculate and, furthermore, a convergence roblem emerges if > Thus we use numerical integral to obtain Gini indices in our estimation tests in the next section 4 Some emirical results We give results for models T ( ) and V ( ), imosing η 1 Following SCS (1999, 1), we use MSE, MAE and MAXABS as error measures The Levenberg-Marquardt algorithm (Dennis & Schnabel, 198) is used to solve the nonlinear least square roblem of minimizing the sum of residual squares to obtain arameter estimates for the models Similar arameter transformations to those described in WNS (7) are adoted to enforce the arameter constraints Following SCS (1999, 1) we use the Shorrocks (198) income distribution data for a range of countries in the tests The results are reorted in Tables 1- Tables - contain the arameters of the two models, from which it can be seen that all the fitted curves satisfy the condition of the Lorenz curve Table 1 7

8 shows that the fitted results for T ( ) and V ( ) are very good The largest MSE values for both models are for the Norwegian data, being for T ( ) and for ( ) V resectively Since the MAXABS values are all quite small for V ( ), we can conclude that each fitted curve is a good global aroximation to the original data We also tested the models by using US income distribution data from Basmann et al (199), where the samle size is larger, and also obtained satisfactory results We do not reort these results to conserve sace, but they are available on request 5 Conclusion We have introduced two new families of Lorenz curves building on the models roosed in SCS (1999, 1) The estimation tests show that the new families erform well in ractice The results resented here rovide further evidence that the basic form suggested by SCS (1999) is very imortant and that the basic result of theorem 1 is useful in the creation of arametric Lorenz models References Basmann, R L, K J Hayes, D J Slottje, 199 Some new methods for measuring and describing economic inequality JAI Press, Greenwich, Connecticut Chotikaanich, D, 199 A comarison of alternative functional forms for the Lorenz curve Economics Letters 41, 19-8 Dennis Jr, J E, R B Schnabel, 198 Numerical Methods for Unconstrained Otimization and Nonlinear Equations, Prentice-Hall, London Ogwang, T and U L G Rao, Hybrid models of the Lorenz curve 8

9 Economics Letters 69, 9-44 Sarabia, J, E Castillo, D J Slottje, 1999 An ordered family of Lorenz curves Journal of Econometrics 91, 4-6 Sarabia, J, E Castillo, D J Slottje, 1 An exonential family of Lorenz curves Southern Economic Journal 67, Shorrocks, A F, 198 Ranking income distributions Economica 5, -17 Wang, Z X, Y-K Ng, R Smyth, 7 Revisiting the ordered family of Lorenz curves Deartment of Economics Monash University Discussion Paer No 19/7 9

10 Table 1 Error measures for T ( ) and V ( ) using Shorrocks (198) income data T Model V ( ) Model ( ) MSE( 1 6 ) MAE MAXABS Gini MSE( 1 6 ) MAE MAXABS Gini Brazil Columbia 681 Denmark Finland 5589 India Indonesia 658 Jaan Kenya 4711 Malaysia Netherland 464 s N Zealand 449 Norway Panama 15 Sri Lanka Sweden 644 Tanzania Tunisia 514 UK Uruguay Note: η 1 is imosed for both T ( ) and V ( ) 1

11 Table Parameter estimates for T ( ) using Shorrocks (198) income data Brazil Columbia Denmark Finland India Indonesia Jaan Kenya Malaysia Netherland s N Zealand Norway Panama Sri Lanka Sweden Tanzania Tunisia UK Uruguay α η η is imosed so that and satisfy (9) or (1) Note: 1 11

12 Table Parameter estimates for V ( ) using Shorrocks (198) income data Brazil Columbia Denmark Finland India Indonesia Jaan Kenya Malaysia Netherland s N Zealand Norway Panama Sri Lanka Sweden Tanzania Tunisia UK Uruguay α 1 δ η Note: η 1 is imosed so that and satisfy (9) or (1) 1