n economics the Lorenz curve is a function L(x) defined on the interval [0, 1] as follows. The x-axis is the proportion of people in a country’s population ordered from poorest to richest, e.g. x = 0.4 is the poorest 40% of the population, x = 0.9 is the poorest 90% – so all except the richest 10%. The y-axis measures the proportion of total income earned in the country. A point y = L(x) is on the Lorenz curve if the poorest 100x% of the population earn 100y% of the income. For example if L(0.6) = 0.35 then the poorest 60% of the population earn only 35% of the income in the country. It is always true that L(0) = 0 and L(1) = 1. If income is spread equally across the country, then L(x) is the straight line y = x. In such a country everybody earns the same. In reality the Lorenz curve is concave up and lies below the line y = x for all countries. The shaded area between y = L(x) and y = x measures how far income distribution is from being equal. The Gini coefficient for a country is defined as the ratio of the shaded area above divided by the (triangular) area between the line y = x and the x-axis on the interval [0,1]. If income is spread perfectly evenly in a country then the Gini coefficient will be 0. If all the income is earned by only one rich person while everybody else earns nothing, then the Gini coefficient would be 1. (a) Explain why L(0) = 0 and L(1) = 1 for any Lorenz curve. (b) Explain why the Gini coefficient is 0 if income is spread evenly across a country. (c) If in a given country L(x) = x2 then: • What percentage of total income is earned by the poorest 10% of the population? • What percentage of total income is earned by the richest 10% of the population? (d) Calculate the Gini coefficient for each of the three curves. (For L3 you will need to research an antiderivative for y = tan x.) Which of the curves represents a country with the most unequal income distri- bution? Explain. country’s Lorenz curve is L(x) = xn and the Gini coefficient is 78, calculate n. [ (d) Three possible Lorenz curves are given as follows: L1(x)=x^3 L2(x)=2^x−1 L3(x)=tan(π/4x) (e) i) Show that these three curves all pass through the points (0, 0) and (1, 1). ii) Show that these three curves are all concave up on the interval [0, 1].
In economics the Lorenz curve is a function L(x) defined on the interval [0, 1] as follows. The x-axis is the proportion of people in a country’s population ordered from poorest to richest, e.g. x = 0.4 is the poorest 40% of the population, x = 0.9 is the poorest 90% – so all except the richest 10%. The y-axis measures the proportion of total income earned in the country.
A point y = L(x) is on the Lorenz curve if the poorest 100x% of the population earn 100y% of the income. For example if L(0.6) = 0.35 then the poorest 60% of the population earn only 35% of the income in the country. It is always true that L(0) = 0 and L(1) = 1.
If income is spread equally across the country, then L(x) is the straight line y = x. In such a country everybody earns the same. In reality the Lorenz curve is concave up and lies below the line y = x for all countries. The shaded area between y = L(x) and y = x measures how far income distribution is from being equal.
The Gini coefficient for a country is defined as the ratio of the shaded area above divided by the (triangular) area between the line y = x and the x-axis on the interval [0,1]. If income is spread perfectly evenly in a country then the Gini coefficient will be 0. If all
the income is earned by only one rich person while everybody else earns nothing, then the Gini coefficient would be 1.
(a) Explain why L(0) = 0 and L(1) = 1 for any Lorenz curve.
(b) Explain why the Gini coefficient is 0 if income is spread evenly across a country.
(c) If in a given country L(x) = x2 then:
• What percentage of total income is earned by the poorest 10% of the population?
• What percentage of total income is earned by the richest 10% of the population?
(d) Calculate the Gini coefficient for each of the three curves. (For L3 you will need
to research an antiderivative for y = tan x.)
Which of the curves represents a country with the most unequal income distri-
bution? Explain.
country’s Lorenz curve is L(x) = xn and the Gini coefficient is 78, calculate n.
[
(d) Three possible Lorenz curves are given as follows:
L1(x)=x^3 L2(x)=2^x−1 L3(x)=tan(π/4x)
(e) i) Show that these three curves all pass through the points (0, 0) and (1, 1). ii) Show that these three curves are all concave up on the interval [0, 1].
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