Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:C Alculus in Context #5
Math 113, Chapters 5 & 6
How does our understanding of integration and the Fundamental Theorem of
Calculus allow us to compare income inequality?
The Gini index is a statistical measure of income distribution. It ranges from 0 (or 0%) for perfect equality to 1
(or 100%) for perfect inequality. Source:
https://data.worldbank.org/indicator/SI.POV.GINI?view=map
A country where every
1. Perfect Equality
2. Unequal
3. More Unequal
4. Total Inequality
100%
resident has the same
income would have a Gini
index of 0.
G1=0
ZZ00
G2>0
G3>G2
G4=1
A country in which one
resident earned all the
0%
0%
100%
0%
100% 0%
100% I
Cumulative
%Population
Cumulative
%Population
Cumulative 100% 0%
%Population
Cumulative
%Population
income, while everyone
else earned nothing, would have a Gini index of 1.
Final shaded Area then Mult by 2
Lorenz Curve - Ukraine 2020
The Gini index is defined to be two times the area between
the line of perfect equality (y = x) and the Lorenz curve. (The
Lorenz curve shows the percentage of income held by each
percentile of the country's population. As an example, the
Lorenz curve would plot a point at (50, 20) if the bottom 50%
of the population held 20% of the country's income.)
y = 1.5625x5 -2.5781x + 1.3438x³
+0.2406x² + 0.4313x
1
0.8
<-Ukraine -Line of Equality
0.6
0.4
0.2
Talks were underway for Ukraine to join the European Union
an alliance which currently includes the countries of Austria,
Belgium, Bulgaria, Croatia, Cyprus, Czechia, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Ireland, Italy,
Latvia, Lithuania, Luxembourg, Malta, Netherlands, Poland,
Portugal, Romania, Slovakia, Slovenia, Spain and Sweden.
0
0
0.4
0.8
1
0.2
0.6
Cumulative Percentage of the Population
Area = 112
We are going to compare the income distribution of Ukraine with that of Poland, a country in the EU which is
geographically close to Ukraine.
1. Above is a plot of the line of equality and the Lorenz curve for Ukraine in 2020. Shade the area we need to
find first so that we can calculate the Gini index. Then, write an expression for THIS AREA that includes a
definite integral.
2. Use the integral expression you wrote in #1 and the Fundamental Theorem of Calculus to find the Gini
index for Ukraine in the year 2020. Show the substitutions necessary, but use fnint to do the calculation.
Cumulative Wealth
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