A Lorenz curve is a curve defined by some equation y = L(x) on the interval [0, 1], where each x-value represents the least wealthy x portion (i.e. 100x%) of a society, and the corresponding y-value represents the portion of the total wealth that is owned by that part of the society. Note that if two or more people are 'equally wealthy', then order them in any order. For example, L(0.3) = 0.10 would mean that the 'poorest' 30% of the population owns 10% of the total wealth. The line y = x is called the line of perfect equality. Optional: Think about why this is a suitable name. The area of the region enclosed by y = x and y = L(x) is sometimes used as a measure of how (un)equitable a society is. %3= (a) For any Lorenz curve, L(0) = 0 and L(1) = 1. Explain briefly why this is. (b) Suppose that for the population of Narnia, the Lorenz curve can be modelled by LN(x) = x². Find the area of the region between the curves LN(x) = x² and y = x. 1 (c) Suppose that for the population of Westeros, the Lorenz curve can be modelled by Lw(x) : Find the area of the region between the curves Lw(x) = =x²e and y = x. Make sure to justify how you know which curve is above the other. Find the area of the region between them. Make sure to justify how you know which curve is above the other. If you wish, you can do this with a graph, and using DESMOS (or other graphing software/calculator) is acceptable to produce the graph.
A Lorenz curve is a curve defined by some equation y = L(x) on the interval [0, 1], where each x-value represents the least wealthy x portion (i.e. 100x%) of a society, and the corresponding y-value represents the portion of the total wealth that is owned by that part of the society. Note that if two or more people are 'equally wealthy', then order them in any order. For example, L(0.3) = 0.10 would mean that the 'poorest' 30% of the population owns 10% of the total wealth. The line y = x is called the line of perfect equality. Optional: Think about why this is a suitable name. The area of the region enclosed by y = x and y = L(x) is sometimes used as a measure of how (un)equitable a society is. %3= (a) For any Lorenz curve, L(0) = 0 and L(1) = 1. Explain briefly why this is. (b) Suppose that for the population of Narnia, the Lorenz curve can be modelled by LN(x) = x². Find the area of the region between the curves LN(x) = x² and y = x. 1 (c) Suppose that for the population of Westeros, the Lorenz curve can be modelled by Lw(x) : Find the area of the region between the curves Lw(x) = =x²e and y = x. Make sure to justify how you know which curve is above the other. Find the area of the region between them. Make sure to justify how you know which curve is above the other. If you wish, you can do this with a graph, and using DESMOS (or other graphing software/calculator) is acceptable to produce the graph.
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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![Problems:
1. A Lorenz curve is a curve defined by some equation y = L(x) on the interval [0, 1], where each x-value represents
the least wealthy x portion (i.e. 100x%) of a society, and the corresponding y-value represents the portion of the
total wealth that is owned by that part of the society. Note that if two or more people are 'equally wealthy', then
order them in any order.
For example, L(0.3) = 0.10 would mean that the 'poorest' 30% of the population owns 10% of the total wealth.
The line y
= x is called the line of perfect equality. Optional: Think about why this is a suitable name.
= x and y = L(x) is sometimes used as a measure of how (un)equitable a
The area of the region enclosed by y
society is.
(a) For any Lorenz curve, L(0) = 0 and L(1) = 1. Explain briefly why this is.
(b) Suppose that for the population of Narnia, the Lorenz curve can be modelled by LN (x) = x².
Find the area of the region between the curves LN(x) = x² and y = x.
1
(c) Suppose that for the population of Westeros, the Lorenz curve can be modelled by Lw(x) = -x²e®
e
1
Find the area of the region between the curves Lw (x) = =x²e*° and y = x. Make sure to justify how you
know which curve is above the other. Find the area of the region between them. Make sure to justify how
know which curve is above the other. If you wish, you can do this with a graph, and using DESMOS (or
other graphing software/calculator) is acceptable to produce the graph.
e
you
(d) According to these models, which society is more equitable - Narnia or Westeros? Explain briefly. (Does a
larger area represent a more or less equitable society? Why?)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67fe0dd7-d089-4006-94e6-a1dd08a07477%2Fa811b664-a993-4479-aca2-f1f18aeb7433%2Fwy28i8c_processed.png&w=3840&q=75)
Transcribed Image Text:Problems:
1. A Lorenz curve is a curve defined by some equation y = L(x) on the interval [0, 1], where each x-value represents
the least wealthy x portion (i.e. 100x%) of a society, and the corresponding y-value represents the portion of the
total wealth that is owned by that part of the society. Note that if two or more people are 'equally wealthy', then
order them in any order.
For example, L(0.3) = 0.10 would mean that the 'poorest' 30% of the population owns 10% of the total wealth.
The line y
= x is called the line of perfect equality. Optional: Think about why this is a suitable name.
= x and y = L(x) is sometimes used as a measure of how (un)equitable a
The area of the region enclosed by y
society is.
(a) For any Lorenz curve, L(0) = 0 and L(1) = 1. Explain briefly why this is.
(b) Suppose that for the population of Narnia, the Lorenz curve can be modelled by LN (x) = x².
Find the area of the region between the curves LN(x) = x² and y = x.
1
(c) Suppose that for the population of Westeros, the Lorenz curve can be modelled by Lw(x) = -x²e®
e
1
Find the area of the region between the curves Lw (x) = =x²e*° and y = x. Make sure to justify how you
know which curve is above the other. Find the area of the region between them. Make sure to justify how
know which curve is above the other. If you wish, you can do this with a graph, and using DESMOS (or
other graphing software/calculator) is acceptable to produce the graph.
e
you
(d) According to these models, which society is more equitable - Narnia or Westeros? Explain briefly. (Does a
larger area represent a more or less equitable society? Why?)
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