德布拉吉瑞发展经济学chapter6

Professor Debraj RayCLA EC 320Sketches of Answers to Problems,Chapter 6.The answers below are brief and try to give you the basic idea of how to approach these problems.You will gain a lot more from studying these answers if you spend some time independently trying to work on the problems.(1)Read the discussion at the beginning of the chapter.(2)(a)In any one year,half the people have 100,and the other half have 200.The formula for the Gini coefficient is G =12n 2µm j =1m k =1n j n k |y j −y k |,where n is the total number of people,n i the number of people in income class i ,y i is the income of class i ,and µis the mean income.In our case,we can take n =1,and two groups 1and 2,with (y 1,y 2;n 1,n 2)equal to (100,200;1/2,1/2).Mean income µis therefore given by 150.Consequently,the Gini in this case isG =1300{14(200−100)+14(200−100)}=1/6.(b)Now calculate lifetime income.The expected income in the second period is 150for everybody (this is because there is a probability 1/2of getting a high job and a probability 1/2of getting the low job in period 2).Thus average income for someone currently holding a low job is 100+1502=125,while for someone holding a high job it is 200+1502=175.Note that there is a narrowing of average incomes relative to part (a),because of the mobility in the economy.You can calculate the Gini just as we did in part (a),and you will see that it is lower.(c)and (d)I will go ahead and do part (d)because it is a generalization of part (c)(but you should try part (c)separately).If you hold your current job with probability p ,then for a low income person (today),the expected income tomorrow is 100p +200(1−p ).F or a high income person (today)it is 100(1−p )+200p .Thus expected average incomes for the low income person and high income person are 50+50p +100(1−p )and 100+100p +50(1−p )respectively.The mean income in this society is still 150,as you can easily see by taking the 50-50average of these two incomes.So the Gini,calculated just as in part (a),isG =1300{14[50+50p −50(1−p )]+14[50+50p −50(1−p )]}=p 6.1Note that if p =1,this gives you exactly the same answer as in part (a),while if p =1/2,we get exactly the same answer as in part (b).This is as it should be.If p =1,there is no mobility at all (why?),so that the answer to overall inequality is the same as the answer to inequality within a single time period.In contrast,if p =1/2,there is perfect mobility,which is the case studied in part (b).As p goes up from 1/2to 1,mobility becomes progressively lower and lower,and in response the Gini goes up,signaling greater inequality in the presence of lower mobility.(3)Just apply the formulae and draw the Lorenz curves.This is good practice!Let me comment on the statement at the end of the question:understanding the implicit transfers that are moving us from one distribution to the other.(i)[a →b ]:there are no transfers,the inequality should be the same by the relative income principle.(ii)[b →c ]:there should be no change in inequality by the population principle.You should thus get the same Lorenz curves for distributions (a),(b),and (c).(4)This question essentially summarizes discussion in the text.What you need to know is that both the coefficient of variation and the Gini coefficient satisfy the Transfers principle.This means that when the Lorenz curves of two distributions do not cross,both the Gini and the coefficient of variation are in agreement with the Lorenz comparison —and therefore with each other.(5)If there are minimum survival needs,then the relative income principle may need to be suitably modified.Let us say that minimum needs cost 1000per year to satisfy (in some cur-rency).Then we could apply the relative income principle to the excess of incomes over 1000.The excess is what people have left to spend on other goods (apart from minimum needs),and changes in these that do not affect relative excesses should leave inequality unchanged.Here is the modified relative income principle that we might use:Let (y 1,y 2,...,y n ;n 1,n 2,...,n m )be an income distribution.Change every y i by multiplying the excess of y i over 1000(that is,y i −1000)by the same constant (independent of i ).Then inequality should remain unchanged.Note:This is only a suggestion for the purpose of this question and should not lead you to abandon the old relative income principle altogether.(6)(a)Take the first income distribution,x and let total income (which is just the sum over all the x i ’s)be denoted by X .The poorest k people in the population earn k i =1x i ,sotheir share of total income is k i =1x i X .Likewise,the share of the poorest k people in the y distribution is k i =1y i Y (again,Y is total income).If the Lorenz curve for x lies inside the one for y ,it must be the case that the former share is at least as large as the latter share for all k .This means that k i =1x i X ≥ k i =1y i Yfor all k ,but since X =Y (both distributions have the total income by assumption),it must2be thatk i=1x i≥ki=1y i(with strict ineuqlaity for at least one index k).This is what we needed to prove.(b)Omitted.(7)Omitted,not needed for course.(8)(a)The beginning income distribution is given by(1000,1000,1000,1000,1000,1000,1000,2000,2000,2000)and thefinal distribution is(1000,1000,1000,1000,1000,2000,2000,2000,2000,2000)Total income in thefirst case is13,000,and in the second case is15,000.Let’s scale incomes in the second distribution by using the relative income principle so that the sum is13,000. This means we multiply all incomes in the second distribution by13/15.This gives us the following distribution(which has the same inequality as the second(by the relative income principle):(867,867,867,867,867,1333,1333,1333,1333,1333)Now focus on individuals1through5.They have lost money relative to thefirst distribution. Look at individuals6and7.They have gained.Individuals8through10have lost(in relative terms).This is as if there has been a disequalizing transfer from1–5to6–7,and an equalizing transfer from8–10to6–7.These transfers run in opposite directions(and do not cancel each other out),so that the Lorenz curves must cross.You can also verify this directly by drawing the Lorenz curves.(b)Just apply the formulae.Check to see if the Gini and the coefficient of variation predict different directiuons of change in inequality.If so,why?(9)(a)F alse.An example of a Kuznets ratio is the ratio of the income share of the richest 20%of the population to that of the poorest60%of the population.Now suppose that there is a transfer of income from relatively poor to relatively rich within the poorest60%.Then it is clear that the Kuznets ratio will remain unaltered.(b)True.Both these measures satisfy the Dalton transfers principle.This means that when there are disequalizing or equalizing transfers,the measures go up or down respectively.But when two Lorenz curves do not cross,there mnust have been either a sequence of equalizing transfers,or a sequence of disequalizing transfers,so that the two measures must have reacted in the same direction.(c)alse.Here you need to give the example that we studied in class,of transfers on the same side of the mean.3(d)True.The Lorenz curve cannot go above the 450line at any point,say x .F or if it did,it would mean that the poorest x %of the population is earning more than x %of total income,which is a contradiction.(e)alse.These principles do allow us to make judgements when Lorenz curves cross,or equivalently,when there is a sequence of Dalton transfers all going in the same direction.When opposing Dalton transfers occur,we need a way to compare the strengths of these opposing transfers.This is not possible with the ethical principles stated here.Thus,for instance,the Gini and the coefficient of variation,which are both consistent with all these principles,nevertheless disagree with each other in their comparison of inequality across pairs of income distributions.(f)True.Imagine that the original income distribution is (y 1,y 2,...,y n ),written in ascending order.Now it is (y 1+100,y 2+100,...,y n +100).Using the relative income principle,rescale the new sitaution so that it has the same mean income:to do this,you simply multiply all new incomes by the fraction µµ+100,where µwas the mean income of the original income distribution (why?).Therefore the rescaled income for person i is justy i ≡y i +100µ+100.Note that y i goes up relative to y i ,if y i was less than mean income,goes down relative to y i if y i had exceeeded mean income,and stays unchanged if y i had been precisely at mean income.This means that we have a set of Dalton transfers from relatively rich (those above the mean in this example)to relatively poor (those below the mean in this example),which means that inequality must have fallen (by the Dalton principle).4。