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1. Download LorenzData.xlsx. It contains data from 336 micro-borrowers in Thailand, on this
year’s “Income” and next year’s “Expected Income”. Graph the Lorenz curve for both
income and expected income, and turn in the graph. According to the Lorenz criterion, is
there more inequality in income or expected income? (Ignore the small regions where the
curves cross.) Why might this be?
[HINT: Consider the income data. First, make sure to sort this column of data. Next, next to it
create a column containing the cumulative sum of income – i.e. that has the sum of the first
income in the first row, sum of the first two incomes in the second row, the sum of the first
three incomes in the third row, etc. For cumulative sum, you can use something like a
“=sum($A$2:A2)” command and then copy and paste to the next 335 rows, e.g., where the $signs make sure the first cell in the summation does not change with the copy and paste. Next,
next to that create a cumulative fraction of income column, which is the cumulative sum
divided by the total sum of income. (You can use the same $X$n command to fix the total
sum of income and use cut and paste.) Put an extra 0 at the top of this column to make sure it
will start at 0. Finally, make a column that contains the fraction of borrowers included up to
that point. You can do this by starting at 0 (on the same row that has the 0 in the cumulative
fraction of income column) and adding 1/336 every row. Now, repeat the process for
expected income. Then, graph the Lorenz curves. You can do this (in Office 2007) by, for
example, opening a second sheet and inserting a basic line chart (Insert, Line Chart, First
Category). Then choose “Select Data”. You can then “Add” the columns on the previous
sheet that you want to graph in the “Series” category (namely, income, expected income, and
helpful also to graph fraction of the population, which gives a 45-degree line) and the column
you would like to graph against (which is the fraction of the population) in the “Horizontal
Axis” category. You can edit the graph by choosing the first option under “Chart
Layouts” and clicking on the various text items and adjusting the labeling.]
2. a. Country A has an income distribution of (3,5,6,10) and country B has an income
distribution of (1,2,4,5). Graph both Lorenz curves, labeling the key points, and say which is
more unequal using the Lorenz criterion.
b. Repeat part a. when B has a distribution of (2,4,5).
3. Graph and label the Lorenz curve for a country in which 1/4 of the population earns 3/4 of
the income, 1/2 of the population earns 1/10 of the income, and 1/4 of the population earns
3/20 of the income. Assume that every individual within each class has the same income.
4. In Mexico in 1950, the richest 5% earned 40% of the income and the poorest 40% earned
14.3% of the income. In 1957, the richest 5% earned 37% of the income and the poorest 40%
earned 11.3% of the income. In 1963, the richest 5% earned 28.8% of the income and the
poorest 40% earned 10.1% of the income. Has inequality gone up or down between 1950 and
1957, between 1957 and 1963, and between 1950 and 1963, according to the Lorenz criterion?
Justify answer graphically.
5. Calculate the gini coefficient and the coefficient of variation for each of the following
distributions. [Hint: It will save significant time if you make use of the inequality principles
(population, relative income, etc.) to simplify where possible.]
a. Two people earn 3, one earns 6, and three earn 15.
b. 2 million people earn 3, 1 million earn 6, and 3 million earn 15.
c. 2 million people earn 3,000, 1 million earn 6,000, and 3 million earn 15,000.
d. 250 million people earn 600, 1 billion people earn 300.
e. The richest 20% earn 20% of the income. [Hint: it may help to draw the Lorenz curve.]
6. Consider an economy of 5 people and 2 sectors, modern and traditional. The annual
income for a worker in the modern sector is $20k and in the traditional sector is $10k.
Suppose that each year, the economy grows by moving one person from the traditional
to the modern sector. That is, in year one the distribution is ($10k, $10k, $10k, $10k, $10k),
in year two it is ($10k, $10k, $10k, $10k, $20k), in year three it is ($10k, $10k, $10k, $20k,
$20k), and so on, until in year six it reaches ($20k, $20k, $20k, $20k, $20k).
a. Plot the gini coefficient on the vertical axis against average income on the horizontal axis
for this economy. You should calculate a gini coefficient for each average income level $10k,
$12k, $14k, ... $20k. Based on the gini, is inequality varying with growth as predicted by the
Kuznets hypothesis? How do you see this?
b. Graph the Lorenz curves corresponding to each income level. According to the Lorenz
criterion, what has happened to inequality as growth occurred? Say as much as possible that
can be said unambiguously.
7a. Assume that skilled wages are twice that of unskilled wages, and that half of the
population fits into each category, skilled and unskilled. Graph the Lorenz curve and
calculate the gini coefficient.
b. Assume that skilled wages become 3x that of unskilled wages, as demand for skilled
workers increases due to new technology. The population breakdown into skilled and
unskilled remains the same. Graph the new Lorenz curve and calculate the new gini
coefficient.
c. Assume now that there is a supply response to the higher skill premium, in that the fraction
of skilled workers goes up from one half to two thirds of the population. Also, given the
increased supply of skilled workers and decreased supply of unskilled workers, the ratio of
skilled wages to unskilled wages drops to 2.5. Graph the new Lorenz curve and calculate the
new gini coefficient.
d. What has happened to inequality according to the Lorenz criterion? According to the gini
coefficient? Are all comparisons unambiguous?
8. Imagine a country divided into four equally sized classes, poor, lower middle, upper
middle, and upper. Within each class, incomes are the same. Imagine incomes of the poor
class increase by 20%, of the lower middle class increase by 30%, of the upper middle class
increase by 27%, and of the upper class increase by 23%. Assume the average increase in
income is 24.5%. Further, assume the class orderings are unchanged, and that population size
has not change. Has inequality increased, decreased, or changed in an unrestricted way,
according to the Lorenz criterion? Justify your answer fully. [Hint: beware of solving this
question using an example distribution. For example, if you use (100,200,300,400) and then
apply the changes, you realize the mean went up by 25.3%, not 24.5% as stated in the
question. This exercise can be solved just by using the definition of the Lorenz curve; it
helps to focus on the bottom and top income segments.]
Income
11450
276840
88060
82600
62780
335070
124900
43450
28050
77000
172790
60661
32435
49900
33600
13000
63400
7200
34100
24700
124363
20785
49070
29990
23900
49400
18399.5
519430
26200
38000
22700
58800
46350
30660
45100
137700
10000
6000
35800
37000
40300
194170
122824.5
33990
76900
15660
9465
31500
12050
71442
60390
Expected_Income
10000
180000
70000
60000
50000
225000
100000
30000
23510
60000
100000
50000
30000
40000
30000
10000
50000
6000
30000
20000
100000
19000
40000
25050
20000
40000
15000
300000
20000
30000
20000
50000
40000
26000
37000
100000
8000
5000
30000
30000
30000
120000
90000
30000
60000
10000
8000
27520
10000
55800
50000
55628
49000
108900
87775
51550
108500
198500
49200
5300
13160
139610
130660
195450
48595
29427
313725
2500
15000
10800
27520
85200
23510
320100
16960
17960
19650
34000
23800
16000
43400
70812
18500
23800
21500
20250
420250
48790
91737
6680
72040
22608
60500
18800
71600
186607.5
99133
40980
28415
109750
103328
53800
21680
45000
40000
80000
70000
40000
80000
120000
40000
5000
10000
100000
100000
120000
40000
25000
200000
3000
10000
8000
23000
66133
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15000
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16000
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20000
12000
30000
55000
15000
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6000
59200
20000
50000
16000
56000
120000
71442
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25000
80000
80000
43000
20000
16900
21700
124215
7600
545800
14965
21990
77700
11450
79732
31600
136500
81480
123200
16590
660480
11050
42740
17950
33925
10800
71800
17800
27010
160900
81300
54630
163865
29650
54480
39720
3625
59650
186025
99726
8400
57300
27850
2424.5
127652
33350
15000
199400
63329.5
48120
43500
94467.5
287990
121000
536600
71320
16100
14965
20000
90000
6165
400000
10000
20000
60000
10000
60000
27600
100000
60000
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15000
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100000
60000
44600
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25000
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8000
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2000
100000
30000
10000
130000
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200000
90000
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10000
5500
18700
67348
31800
124300
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204200
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205150
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254060
56000
49000
179810
52715
68450
55822
17700
15200
50600
60855
31670
196310
77000
58752.5
53350
52400
37800
30440
22700
84500
15000
5000
16200
84480
83180
35830
59006
70439.5
178700
109922.5
9100
9000
30500
45990
21800
8000
5000
15840
52000
29900
95000
30000
150000
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661540
569702.5
28540
22115
27200
138600
45580
10462.5
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20210
1200
64200
87000
31000
49925
42700
58000
33360
77735
12820
15840
20600
156820
8000
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367630
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115617
20100
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282893
183849.5
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85807
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66180
11937.5
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20950
120275
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