Solutions Problem Set 1 Social Welfare

Transcription

1 s Problem Set 1 Socal Welfare Professor: Marcelo Ner TA: Tago Bonomo March 9, 017 Exercse A populaton s dvded nto four groups, each one wth four ndvduals. The ndvdual ncomes are: x 1 = [1, 1,, ] ; x = [1, 1,, ] ; x 3 = [,,, 8] ; x = [,, 8, 16] Calculate the two Thel measures of nequalty, verfyng that the between and the wthn group components are the same for both measures. For the two Thel measures, we use the formulas T = y ln Ny and L = 1 N ln Ny, where N s the sze of the populaton and y s the share of ndvdual s ncome n total ncome, that s, y = x x = x Nµ. Usng ndvdual ncomes, we have that T = 1 Nµ x ln x µ e L = 1 x N ln µ. For the total populaton, we have that N = 16 and µ = /16 =, therefore Nµ = 6. The Thel-T ndex for the total populaton s 1

2 T = ln ln 1 + ln + ln ln ln 1 + ln + ln + 1 ln 6 + ln + ln + 8 ln ln 6 + ln + 8 ln ln = = [ 1 ln 1 + ln + 5 ln + 8 ln ln 16 ] [ 1 ln 1 ] ln + 16 ln ln 1 = + 1 [ ln + ln + 8 ln + 16 ln ] 6 = [ ln ] = ln = 1 ln T = The Thel-L for the total populaton s L = 1 ln ln 1 + ln + ln 1 ln ln 1 + ln + ln 1 ln 16 + ln + ln + ln 8 1 ln 16 + ln + ln ln = 1 16 ln 1 + ln + 5 ln + ln 8 + ln 16 = 1 ln ln ln + ln 16 = 1 16 [ ln ] = 1 ln 8 ln = 16 L = Now let s decompose each ndex and calculate ther wthn and between groups components. We denote the sze of group h by n h and the share of group h n total populaton by π h = n h N. The share of total ncome for ndvdual from group h s y h = x h Nµ and the fracton of total ncome for group h s Y h = y h.

3 We have that h {1,, 3, } and n h = for each h. Therefore, π h = 0.5 for each h. We also have that Y 1 = Y = Y 3 = y 1 = 1 6 y = 1 6 y 3 = 1 6 x 1 = = 1 8 x = = 1 8 x 3 = = 8 Y = 1 Y 1 + Y + Y 3 = 1 8 = 8 For the Thel-T ndex, we note that s possble to decompose t as follows: T = T b + h=1 Y ht h, where T b = h=1 Y h ln Y h π h e T h = y h y =1 Y h ln n h h Y h. Let s calculate ts between groups component, T b. T b = 1 8 ln 1 8 0, ln 1 8 0, ln 8 0, ln 8 0, 5 = 1 ln ln 1 + ln + ln = 1 8 ln ln ln = 1 ln ln = 8 T b = Note that f we know T and T b, t s possble to obtan the wthn groups component: h=1 Y h T h = T T b = ln ln = ln = Note that, n ths specfc case, nequalty between groups and wthn groups are the same. For the Thel-L ndex, the decomposton s as follows: L = L b + h=1 π hl h, where L b = h=1 π h ln π h Y h and L h = 1 To calculate the wthn groups component, let s fnd {L h } h=1 frst: n h =1 ln Y h n h y h. 3

4 L 1 = 1 ln L = ln L 3 = 1 ln L = 1 ln 1/8 + ln 1/8 1 /6 + ln 1/8 1 /6 + ln 1/8 = 1 ln ln + ln + ln 1 ln = /6 /6 /8 + ln /8 /6 + ln /8 /6 + ln /8 /6 8 /6 /8 /6 + ln /8 /6 + ln /8 8 /6 + ln /8 16 /6 Note that L 1 = L = L 3 = L, that s, nequaltes wthn the groups are the same. Ths happened because the Thel-L ndex as well as the Thel-T and the Gn attends the ncome homogenety condton, also known as scale ndependence. Observe that x = x 3 = x = x 1, that s, the ncome dstrbuton of any gven group can be wrtten as the ncome dstrbuton of any other group multpled by a constant. The Thel-L wthn groups component s h=1 π h L h = 0, 5 ln = ln = Therefore, the between groups component s L b = L h=1 π h L h = ln ln = ln = We also verfy for the Thel-L ndex that nequalty between and wthn groups are the same. 1. The ndvdual ncomes for three groups are gven. In group 1, there are sx ndvduals wth ncomes x 11 = x 1 = 0.5; x 13 = x 1 = x 15 = 1; x 16 = 8 In group, there are fve ndvduals wth ncomes x 1 = x = x 3 = x = 1; x 5 = 16 Group 3 has only three ndvduals and ther ncomes are x 31 = x 3 = x 33 = 16 The three groups together consttute a total populaton of 1 ndvduals. Calculate the mean, medan, mode, ampltude and varance of the ncome takng nto account the 1 ndvduals. Calculate the Thel-T ndex related to the nequalty n each group, the ndex related to global nequalty and ts wthn and between groups componentes. Do the same for the Thel-L ndex. Do the same for the Gn ndex. = 1 = 1 ln + ln + ln 1 ln = ln ln + ln + ln 1 ln = ln

5 Frst, let s calculate the measures for the total populaton. We have that N = 1 and total ncome s h x h = 0, = 80 The mean s µ = 80 /1 = 0 /7. To fnd the medan, let s rank the dstrbuton from the lowest to the hghest ncome and see what s the central term of the dstrbuton. We have that x = [ 1 /, 1 /, 1, 1, 1, 1, 1, 1, 1, 8, 16, 16, 16, 16]. The medan s 1 and s the mean between the 7 o and the 8 o terms n the dstrbuton. We have therefore that half of the populaton have ncomes less than or equal to 1, whle the other half have ncomes bgger than or equal to 1. The mode s the most repeated value n the dstrbuton and s also 1. The ampltude s the dfferente between the extreme ncomes of the dstrbuton and therefore s 16 1 = We denote the varance by σ. We have that σ = x µ N = x 0 /7 1 = The Thel-T ndex for the total dstrbuton s T = 1 Nµ = 1 80 T = The Thel-L s x ln x µ [ 1 ln 1/ 0/7 ] ln + 8 ln + 16 ln 0/7 0/7 0/7 L = 1 ln x N µ = 1 ln 0, ln 1 + ln ln 0/7 0/7 0/7 0/7 L = Populatons n each group are n 1 = 6; n = 5; n 3 = 3 Shares of populaton by each groups are π 1 = 6 1 ; π = 5 1 ; π 3 = 3 1 Mean ncomes n each group are µ 1 = 0, = ; µ = = ; µ 3 = = 16 Shares n total ncome by each group are Y 1 = 6 80 = 1 80 = 3 0 ; Y = 5 80 = 0 80 = 5 0 ; Y 3 = = 8 80 = 1 0 Let s calculate the Thel-T ndexes wthn each group. We have that 5

6 T 1 = T = T 3 = n 1 =1 n =1 n 3 =1 x 1 ln x 1 0, 5 = n 1 µ 1 µ x n µ ln x µ = x 3 n 3 µ 3 ln x 3 µ 3 = 3 0, 5 ln 0 ln ln ln ln 8 = 11 ln = = 0 16 ln = 6 ln = The wthn groups component s therefore h Y h T h = 3 0 T T T 3 = The between groups component s T b = T h Y h T h = 0, T b = 0, The Thel-L ndexes wthn each group are L 1 = 1 n 1 L = 1 n L 3 = 1 n 3 ln x 1 = 1 ln 0, 5 µ 1 6 ln x µ = 1 5 ln x 3 µ 3 = ln 1 + ln 8 = = ln ln 3 ln 16 = 0 16 The Thel-L wthn groups component s therefore 3 h=1 The between groups component s L b = L π h L h = 6 1 L L L 3 = π h L h = 0, , 561 = h=1 For the global dstrbuton, we have that the Gn s 6

7 G = n µ x n h = 1 [1 + 0, ] 0 /7 G = Now let s decompose t n the wthn and between groups components. Let s consder the groups ranked from the lowes to the hghest share n total ncome and, wthn each group, we rank the ndvduals from the lowest to the hghest ncome. We denote group h s share n total ncome as Φ h = 1 h Nµ j=1 µ jn j and the share of group h s ncome of ndvdual as Φ h = 1 n h µ h j=1 x hj. We can decompose the Gn ndex as follows: n h µ h G = G b + h π hy h G h + G s, where G b = 1 h Φ h + Φ h 1 π h and G h = x h n h We can make use of the followng table: h n h µ h π h NµΦ h Φ h 1 1 6/1 1 1/80 0 5/1 3 3/ / We have that G b = 1 Φ h + Φ h 1 π h h [ = ] 80 1 G b = The Gn for each group s G 1 = n 1 µ 1 x 1 1 1n1 = 6 [0, ] = G = n µ x 1 1n = 5 [ ] = G 3 = = n 3 µ 3 = 0 x 3 1 1n3 = 3 [ ] The wthn groups component s therefore 7

8 π h Y h G h = G G h = Consder two populatons dvded nto three stratum each. In populaton A, the 0% poorest have 10% of total ncome, the 0% of the mddle have 0% and the 0% rchest have 50%. In populaton B, the three stratum 0% poorest, 0% of the mddle and 0% rchest have 0%, 0% and 60% of total ncome, respectvely. We suppose there s no nequalty wthn each stratum. Calculate the Gn ndex for each one of the two populatons. Do the same for the Thel-T ndex and for the Thel-L ndexes. Based on these results, verfy n each one of the two populatons the ncome dstrbuton s more unequal. Comment the results takng nto account the Lorenz curve for each populaton. We can calculate the Gn usng the formula of the between and wthn groups decomposton, notng that the wthn groups component s zero n ths case. Therefore, the Gn for each populaton s G 1 = 1 h G = 1 h Φ h + Φ h 1 π h = 1 [0, 1 0 0, + 0, 5 + 0, 1 0, , 5 0, ] = Φ h + Φ h 1 π h = 1 [0, 0 0, + 0, + 0, 0, , 0, ] = The Thel-T for each populaton s T 1 = T = y h ln y h π h y h ln y h π h = 0, 1 ln 0, 1 0, 0, 5 + 0, ln + 0, 5 ln 0, 0, 0, = = 0, ln 0, 0, 0, 6 + 0, ln + 0, 6 ln 0, 0, 0, = The Thel-L s L 1 = h L = h π h ln π h = 0, ln 0, 0, 0, + 0, ln + 0, ln Y h 0, 1 0, 0, 5 = π h ln π h = 0, ln 0, 0, 0, + 0, ln + 0, ln Y h 0, 0, 0, 6 =

9 We can t say unequvocally whch of the two ncome dstrbutons s more unequal. If we use the Gn or the Thel-L, we have that populaton A s more unequal but f we use the Thel T we have the converse result. The results are dfferent because the Lorenz curves cross each other. The three measures of nequalty are consstent wth Lorenz, that s, when the Lorenz curves for two dstrbutons don t cross, we can tell whch dstrbuton s more unequal. The dstrbuton closer to the lne of perfect equalty wll be the less unequal usng the three ndexes. 1. Answer true or false and comment. The dual permts to compare dfferent measures of nequalty n the same scales and study the average sensblty of nequalty to ncome transfers. The sentence s true. The dual of the Thel convert the ndexes n a measure that take values from 0 to 1 and s dmensonless, so we can compare t to the Gn ndex. Besdes that, as the dual permt us to compare populatons wth dfferent numbers of null ncomes, we can therefore nfer the mpact of ncome transfers on the dstrbuton. 1.5 IBGE Natonal Bureau of Statstcs and Geography has recently lauched the dual of the Thel ndex takng nto account the workng populaton wth postve ncome usng data from PNADs 00 and 003. Calculate the evoluton percentual of nequalty usng the dual of the Thel-T ndex takng nto account the actve age populaton therefore PIA 15 to 65 years old Varaton Dual of the Thel-T % % of PIA wth zero ncome % Calculate the Thel-T ndex for PNAD 003 Fndng the dual for the PIA consst n fndng the dual of a new dstrbuton obtaned addng a share of the populaton wth null ncome. For 00, the proporton φ 0 = of the new dstrbtuon wll be added wth null ncome. The new dual wll be: U 0 = φ + U 0 1 φ 0 = = For 003 we have that φ 03 = 0, 519. Therefore U 03 = φ + U 03 1 φ 03 = = Inequalty n the PIA as measured by the dual of the Thel-T has evolved as follows U 03 U 0 U 0 = %. 9

10 Inequalty measured by the Thel-T can be obtaned from ts dual usng the formula U T = 1 exp T Therefore, for 003 we have that 1.6 U 03 = 1 exp T 003 0, = 1 exp T 003 exp T 003 = T 003 = ln T 003 = ln = We present the mean and the nequalty of per capta ncome usng the Gn ndex of a hpothetcal country before and after a socalst revoluton. Calculate the evoluton of the well beng n ths socety takng nto account the functon proposed by Sen. Before: Mean Income 300 and Gn 0.6 After: Mean Income 50 and Gn 0.5 Sen s socal welfare functon s W = µ 1 G, where µ s the mean and G s the Gn ndex of the dstrbuton. We have that, W o = = 10 W 1 = = 15 Socal welfare, as measured by the Sen s functon, ncreased from 10 to 15, that s,.%. 1.7 Answer true or false and comment. The extenson of temporal varablty of observed ncome always nfluence nequalty of annual ncomes keepng constant the presente value of the ncome earned durng the lfe cycle. False. If the temporal varablty of observed ncome doesn t change the permanent ncome, t wll only affect the dstrbuton of current ncome, once a negatve shock n one perod wll be compensated by a postve shock n a future one. 10

11 1.8 Wrte down the formula and dscuss possble problems of the followng ndcators: Sen s Socal Welfare Functon Varance of the logs as a measure of nequalty Sen s Socal Welfare Functon We have that W = µ 1 G, where µ s the mean and G s the Gn ndex of the dstrbuton. The nequalty measure used n Sen s Socal Welfare Functon s the Gn ndex, measure that has a complex decomposton n between and wthn groups components. Except n the cases where thera are no ntersectons between the ncome brackets n the dfferent groups, we observe a resdual that s very hard to nterpret n the decomposton. Besdes that, the funcon consders that effcent and equalty has the same weghts, beng a partcular case of the Graaff s Socal Welfare Functon, gven by W = µ 1 G σ. In the Sen s case, we have σ = 1. Varance of the logs We have that V = 1 N [E[logy] logy] The man advantages are t s scale-nvarant and decomposable. The decomposton follows because of the propertes of the logarthmc functon, specally addtvty. However, t s now defned for null ncomes and t s not very senstve to changes n the top of the dstrbuton, once the logarthmc functon "smoothes" the dstrbuton. 1.9 Wrte down the two alternatve formulas and explan the logc and the ntuton behnd the Gn ndex. G = 1 µnn 1 N >j N j x x j The Gn corresponds to the rato between the average of the absoluto devatons of the ncomes of all the people n the sample and twce the average. Remember that there are NN 1 dstnct pars. Note that n the case of perfect equalty x = µ for all, we have that the sum s equal to zero and the Gn s also zero. On the other hand, n the case of perfect nequalty x = Nµfor one ndvdual and x = 0 for the other ones, we have that the Gn s equal to one. 11

12 G = N+1 N 1 NN 1µ N =1 ρ x where ρ s the rankng n the dstrbuton of ncome from the hghest to the lowest. It s clear that poorer ndvduals receve bgger weghts: the rchest n the dstrbuton has weght 1, whle the poorest has weght N. The frst term of the expresson aprroaches 1 as N approaches nfnty. The advantages of ths formula s that t s smpler to use and t s nsenstve to scale. The dsadvantage s that t s not fully decomposable. G = 1 1 n n =1 Φ + Φ 1 The Gn s tmes the area between the Lorenz curve and the perfect equalty lne. Alternatvely, as expressed by the above formula, the Gn s equal to 1 mnos tmes the are between the Lorenz curve and the perfect unequalty lne. For a dscrete dstrbuton, the area between the Lorenz curve and the perfect unequalty lne consst n trapezums wth bases Φ bgger and Φ 1 smaller and heght 1 /n Accordng to the emprcal evdence seen n class, the nfluence of the attrbute n the decomposton of the Thel-T ndex s bgger when we measure: The race attrbute usng household per capta ncome or ndvdual labor ncome? The gender attrbute usng total ndvdual labor ncome or labor ncome normalzed per hour? The nfluence of the race attrbute probably s hgher when we use household per capta ncome because an assortatve matchng effect mpacts postvely the probablty that most dscrmnated ndvduals lke blacks, for example marry ndvduals that are smlar, the same happenng for the most prvleged. The consequence s that the contrbuton of the race attrbute should be hgh for the nequalty between households. In Brazl, we have that the women work more than men despte earnng less. That means that f we use labor ncome normalzed per hour, we wll have that the nfluence of the gender attrbute wll be hgher than f we use total ndvdual labor ncome. Exercse.1 Calculate the nequalty ndexes seen n the course Thel-T, Thel-L, Gn and the duals accordng to the followng sample of ncomes: 1

13 x = [1, 1,, 6, 30] If we add one person wth zero ncome to the sample, how do the ndexes change? In the above dstrbuton, we have n = 5, µ = /5 = 8 and therefore nµ = 0. Thel-T 5 T = y ln ny = 1 5 x ln x nµ µ =1 =1 = 1 1 ln ln ln ln 30 8 = The dual s U T = 1 exp T = 1 exp = Addng an ndvdual wth zero ncome represents a share of φ = 1 6 wth zero ncome n the new dstrbuton. The dual of the new dstrbuton s U T = φ + 1 φ U T = = The Thel-T of the new dstrbuton s therefore Thel-L T = ln 1 U T = ln 1 0, = L = 1 n 5 =1 = 0, 90 ln x µ = 1 ln ln 8 + ln ln 30 8 It s not possble to calculate the changes n the Thel-L when we add a person wth null ncome because the ndex s not defned for zero ncomes. Gn G = n µ = 5 8 = x n

14 The dual s the Gn tsfelf. If we add an ndvdual wth null ncome, we wll have that. G = φ + 1 φ G = = Suppose that per capta ncome of household A, composed of only one ndvdual, s 8. Suppose also that there s only another household n the economy, wth ncomes {1, 1,, 6, 30}. Calculate the level of nequalty accordng to the followng concepts: Household per capta ncome between households Household per capta ncome between ndvduals Calculate the nequalty component of ndvdual ncome between groups of households.e., A and B. Assume now that ncome of household A s 7. Recalculate t. v Suppose there s no socalzaton of ncomes nsde the households. How much of total ncome nequalty s gong to be underestmated takng nto account both scenaros? Let s use the Thel-T ndex to calculate the level of nequalty. House per capta ncome for household B s x = x1+x+x3+x+x5 n = = 8. The share n total populaton for each household s π 1 = 1 6 and π = 5 6. The share n total ncome for each household s Y 1 = 8 8 = 1 6 and Y = 0 8 = 5 6. The Thel-T between households s, therefore T b = h Y h ln Y h π h = 1 6 ln 1 /6 1/ ln 5 /6 5/6 = 0 Because all ndvduals have household per capta ncome of 8, the Thel-T ndex between ndvduals wll also be 0. Consderng now x 1 = 7, we have that Y 1 = 7 7 and Y = 0 7. The Thel-T between households s T b = 7 7 ln 7 / /7 ln 1/6 7 = /6 Note that, n ths case, T = T b, because we stll have that the wthn group components are zero snce we are consderng household per capta ncome. If we consder ndvdual ncome, we have that 1

15 T = 1 x ln x nµ µ = 1 8 T = 1 nµ x ln x µ = ln 8 8 ln ln ln ln ln 7 + ln 1 + ln ln ln 7/6 7/6 7/6 7/6 7/6 We verfy that, n the case wth no socalzaton of ncomes wthn the households, nequalty would be substantally hgher. In the frst case, for example, the Thel-T ndex would rse from T = 0 to T = Wrte down the formulas and compare advantages and dsadvantages of the Thel-T and Gn nequalty ndexes. Thel-T T = 1 Nµ x ln x µ Gn G = n µ x n h The Gn ndex has the advantage of beng easy to nterpret because t takes values from 0 to 1, whle the Thel-T can take values from 0 to ln N. However, we cannot decompose t nto between and wthn group components as we can do wth the Thel-T.. What s the meanng and the mportance of the Prncple of Transfers Pgou- Dalton n the specfcaton of a Socal Welfare Functon? The Pgou-Dalton Prncple of Transfers s one of the desrable propertes of a socal welfare functon and t reveals a preference for equalty. Ignorng the effects of ncentves and allocaton restrctons, we have that the socal welfare functon s hgher the less unequal s the ncome dstrbuton, condtonal on havng the same mean. Ths prncple says that any progressve ncome transfer that keeps constant the rankng n the dstrbuton must reduce nequalty and therefore ncreases the socal welfare functon..5 Defne and llustrate the concept of Lorenz domnance. = =

16 Lorenz domnance s a crteron that permts us to compare two dfferent dstrbuton and say unequvocally whch one s more unequal. The domnance s verfed when a Lorenz curve s always above another one, that s, when they don t cross. If they cross, there s no Lorenz domnance and t mght be the case that dfferent nequalty ndexes mply dfferent results n terms of whch dstrbuton s more unequal. Therefore, we cannot say unequvocally whch one s more unequal. If s the case of Lorenz domnance, all the most relevant nequalty ncludng the Thel-T, Thel-L and the Gn ndexes wll pont n the same drecton. Exercse Emprcal Estmates For the model lny = α + β X + u, we have the followng estmate lny = X R = 0.56 where Y = ncome from the man actvty X = years of schoolng The numbers n brackets are the standard errors of the estmates. Interpret the slope coeffcent gve the formula, ts sgnfcance and the R of the regresson. We have that the slope coeffcent s gven by Note that the t statstc s ˆβ = CovX, lny V arx t = ˆβ = ep ˆβ so that the estmatve s statstcally sgnfcant. The nterpretaton s that each addtonal year of schoolng s assocated on average wth an ncrease n the wages of approxmately 15.%. The R of the regresson ndcates that approxmately 5% of the varaton n the wages s explaned by the varaton n the years of schoolng. 16

17 3. Usng the regresson above and the Thel-T ndex, dscuss and explan the logc of the role of educaton n the determnaton of the labor ncome nequalty n Brazl. The estmatves of the regresson above, consderng that the model s a good one, show us that educaton has an mportant role n determng the wages of the ndvduals. Actually, 15.3% s a bery bg number for the returns to educaton, much bgger than n most of the developed countres. Even wth ths bg educatonal premum, we stll have n Brazl very low educatonal levels, so there s a bg room for mprovement. We wll also have that educaton must have an mportant role n determng the Thel-T ndex for ncome nequalty, as wages represent the most mportant component n ndvdual or even household ncome. Exercse.1 Takng nto account the followng socal welfare functon, dscuss how to ncorporate the Prncple of Transfers n the measures of nequalty. What would be the case for the Gn and Atknson s wth ε = 1 measures? W = ux = uxwxfxdx 0 The Prncple of Transfer can be ncorporated f we consder bgger weghts for the poorest ndvduals, that s, wth lower x. In the case of the Gn, we have that wx = [1 F x]. Note then that the poorest ndvdual, for whom F x = 0, has weght wx =, whle the rchest ndvdual, for whom F x = 1, has weght wx = 0. Another possblty s to consder ndvdual utlty functons wth decreasng margnal utlty. In ths case, an ncome transfer from a relatve rch ndvdual to a relatve poor onde ncrease welfare, onde the ncrease n the utlty of the poorest of the two ndvduals s bgger than the decrease n the utlty of the rchest. Ths s the case of the utlty consdered n Atknson s Welfare Functon, n whch ux = lnx.. Calculate the Thel-T ndex between groups by gender for per capta ndvdual ncome usng the followng data. Interpret. Male: Per Capta Income and Populaton 91,507,99 Female: Per Capta Income and Populaton 96,686,391 17

18 Total: Per Capta Income and Populaton 188,19,383 Note that we are consderng two representatve ndvduals one for the male and one for the female. Then, we only have to do the same procedure as we dd n exercse., notng that the wthn groups component wll be equal to zero only onde representatve ndvdual n each group. Therefore, we wll have that the Thel-T ndex wll be equal to the between groups component. Consderng men as group 1 and women as group, we have that π 1 = 0.86 and π = 96,686,391 91,707,99+96,686,391 = = There Also, Y 1 = = and Y = fore, we have that 91,707,99 91,707,99+96,686,391 = T = T b = Y h ln Y h = ln ln0.367 = = π h h=1 Note that for the male we have that ther share n total ncome s hgher than ther share n the populaton, the converse happenng for the female..3 Explan how the followng dmensons affect the measurment of nequalty and ts decomposton between and wthn groups: a use dsaggregated ncome between ndvduals from the same household and calculate the Thel and Gn ndexes between households versus between ndvduals. a If we calculate the Thel and Gn ndexes between households takng nto account dssaggregated ncome between ndvduals, we wll have that the dfferences n ncome between ndvduals wll tend to be reduced when we add them all n the household. That s, the wthn groups component wll tend to be smaller because we are mplctly consderng that ndvduals "socalze" ncome nsde the households. In terms of the between groups component, t wll depend n the heterogenety of the households. The more dfferent they are n terms of per capta ncome, the hgher nequalty between groups wll be. If otherwse we calculate the ndexes between ndvduals, we wll have that the wthn and between group components wll be the same, as we are consderng dsaggregated ncome. 18

19 . - Emprc Consder the labor decomposton of ndvdual ncome takng nto account dfferent sources: a. What s the rate of unemployment n the PIA Actve Age Populaton - 15 to 65 years old? b. What s the fracton of the growth of the mean labor ncome n the PIA that s explaned for the rse n occupaton? c. If we assume a 0.5% per year growth of the PIA as a result of the recent demographc transton, what should be the growth of ncome from all sources? d. Compare the mpacts n total ncome of the demographc bonus wth the mpacts of the rse n average years of schoolng of the occuped educatonal bonus. For ths exercse, see handout 11. a. The rate of unemployment n the Economcally Actve Populaton therefore PEA for 009 was 1 ocup P EA = = We also know that parctpaton rate n the labor market corresponds to P EA P IA = Therefore, we can calculate the rate of unemployment n the PIA as followng unemp P IA = unemp P EA P EA P IA = = 0.13 That s, the rate of unemployment n the PIA for 009 was 1.3%. b. The rse n occupaton n PIA was ocup P IA ocup P IA ocup 009 P IA 003 = ocup P IA ocup P IA 003 ocup P IA 003 ocup P EA 009 P EA P IA ocup P EA ocup P EA 009 P EA 003 P IA P EA P IA 003 = = 6.3% We have that totncome 009 = and totncome laborncome 009 = Therefore, laborncome 009 = = 689. Dong the same for 003, we have that laborncome 009 = = 51. Therefore, the rse n labor ncome was laborncome 009 laborncome 003 laborncome 003 = = 7.3% We have then that the fracton of the growth n labor ncome n the PIA explaned by the rse n occupaton s = 3.13%. c. Let s remember the followng relaton totncome = totncome ocup laborncome hourlywage educ worktme P EA P EA P IA P IA pop Takng logs and takng the dervatve wth respect to tme, we have that the growth rate of total ncome s the sum of the growth rate of each component above, as we have on the table. Consderng a growth rate of 0.5% per year n the PIA, we have that growth rate n total ncome wll be approxmately 0.5% hgher per year, that s, t wll be 3.86% + 0.5% =.36% 19

20 d. The mpact of the demographc bonus s 0.5%.36% = 11.7%. The mpact of the rse n average years of schoolng of the occuped s.1%.36% = 8.6%. Therefore, the mpact of the educatonal bonus s more than tmes the mpact of the demographc bonus. 0