(2) [E] Draw Lorenz curves and calculate the Gini coefficient and the coefficien

Question

(2) [E] Draw Lorenz curves and calculate the Gini coefficient and the coefficient of variation for the income distributions (a)–(f). In each situation, the first set of numbers represents the various incomes, whereas the second set of numbers represents the number of people earning each of these incomes:

(a) (100, 200, 300, 400); (50, 25, 75, 25)

(b) (200, 400, 600, 800); (50, 25, 75, 25)

(c) (200, 400, 600, 800); (100, 50, 150, 50)

(d) (200, 400, 600, 800); (125, 25, 125, 50)

(e) (100, 200, 300, 400); (50, 15, 95, 15)

(f) (100, 200, 300, 400), (50, 35, 55, 35).

[Try to understand the implicit transfers that move you from one income distribution to the other (except for the first three, which should turn out to have the same inequality — why?).]

Answer only a and d only

Explanation / Answer

A)

(y1,y2,y3,y4) : (n1,n2,n3,n4) =  (100, 200, 300, 400); (50, 25, 75, 25)

n= 50+25+75+25=175

Mean income: 1/175 (100*50 + 200*25 + 300*75 + 400*25) = 42,500/175 = 242.86

The coefficient of variation:  1/242.86[50/175(100-242.86)2 + 25/175(200-242.86)2 +75/175(300-242.86)2 +25/175(400-242.86)2] = 11020/242.86=45.38

The Gini coefficient:

1/2*242.86*1752[ (100*100)|50-50| + (200*100)|25-50| + (300*100)|75-50| + (400*100)|25-50| + (100*200)|50-25| + (200*200)|25-25| + (300*200)|75-25| + (400*200)|25-25| + (100*300)|50-75| + (200*300)|25-75| + (300*300)|75-75| + (400*300)|25-75| + (100*400)|50-25| + (200*400)|25-25| + (300*400)|75-25| + (400*400)|25-25| ] = 22500000/14875175= 1.5

D)

(y1,y2,y3,y4) : (n1,n2,n3,n4) =  (200, 400, 600, 800); (125, 25, 125, 50)

n= 125+25+125+50=325

Mean income: 1/325 (200*125 + 400*25 + 600*125 + 800*50) = 150000/325= 461.5

The coefficient of variation:  1/461.5[125/325(200-461.5)2 + 25/325(400-461.5)2 +125/325(600-461.5)2 + 50/325(800-461.5)2] =51598/461.5= 111.8

The Gini coefficient:

1/2*461.5*3252[ (200*200)|125-125| + (400*200)|25-125| + (600*200)|125-125| + (800*200)|50-125| + (200*400)|125-25| + (400*400)|25-25| + (600*400)|125-25| + (800*400)|50-25| + (200*600)|125-125| + (400*600)|25-125| + (600*600)|125-125| + (800*600)|50-125| + (200*800)|125-50| + (400*800)|25-50| + (600*800)|125-50| + (800*800)|50-50| ] = 176000000/ 97428500= 1.8

therefore there should be the same Lorenz curves for distributions (a), (b), and (c).

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