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Intermediate Macroeconomics class assignment

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Department of Economics
Spring

2

020

Homework 2

Economics UN32

1

3

Intermediate Macroeconomics

1. (Rising Income Inequality in the United States) Read the attached NYTimes article, “Why

you should not believe those GDP Numbers,” by David Leonhardt. This question is abou

t

the graph, entitled ‘The Trouble with GDP,’ which accompanied the article and is shown

below

.

Source: NYTimes, Dec. 16, 2019, Section A, Page 31 of the New York edition.

Assume the economy consists of I = 100 individuals in 1947 and population is constant over

time. Individual i has income

yi
t

in year t for i = 1, . . ., I and t = 1947, 1948, . . ., 2018.

Suppose, without loss of generality, that yi
t
≤ yi+1

t
. Let Yt denote total real GDP in year t,

which is given by

Yt

=

I∑

i=1

yi
t
.

Let

Y 90
t

the total income going the bottom 90 percent of the population,

Y 90
t

=
90∑

i=1
yi
t
1

and

Y 10
t

the total income going to the top 10 percent of the population,

Y 10
t

=
100∑

i=91

yi
t
.

We then have

Yt = Y
90
t

+ Y 10
t

.

To simplify the algebra, assume that both the grey and the yellow lines are equal to 100 in

1980 and that the grey line is equal to 300 and the yellow line to 200 in 2018. (This is roughly

in line with what the graph shows.) Then use this stylized version of the graph to answer the

questions below. Let αt denote the share of total income going to the bottom 90 percent in

year t, that is,

αt =
Y 90

t

Yt
.

(a) Show that by definition 0 < αt ≤ 0.9.

(b) Suppose in year t there is no income inequality, find the associated value of αt.

(c) Find Y 902018/Y
90
1980 and provide a verbal interpretation of your answer.

(d) Find Y2018/Y1980 and provide an interpretation.

(e) Find Y 10
2018

/Y 10
1980

in terms of α1980.

(f) Assume that α1980 = 2/3, which means that the total income of the bottom 90 percent

of the population amounted to 67 percent of total income. Then compute Y 102018/Y
10
1980

and interpret your finding.

(g) Find α2018 in terms of α1980.

(h) Let x90
t

= Y 90
t

/90 denote real per capita income of the bottom 90 percent in year t,

x10
t

=
Y 10
t

/10 denote real per capita income of the top 10 percent in year t and xt = Yt/100

denote real per capita income in year t. Continue to assume that α1980 = 2/3. Find

x10
t

/x90
t

for t = 1980 and t = 2018. Discuss how much more unequal the distribution of

income has become over this time period in the United States.

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