Question

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- 4 marks) In economics, a population's Lorenz curve L(2) gives the percentage (as a decimal) of the country's distribution of wealth among the lowest earning percentage 2 (as a decimal between 0 and 1; e.g., 35% means x = 0.35) of the population. As an example, if L(0.4) = 0.1, then this means that the lowest earning 40% of households have only 10% of the entire economy's wealth. The line y = x is called the line of equality since this would mean that the lowest earning 100.0% of households would have 100x% of the economy's wealth. Consider a country with a Lorenz curve given by L(x) = cek-1. (a) To 3 decimal places, evaluate each of the following and interpret the results. i) (0.25) = ii) L(0.5) = iii) L(0.75) = iv) L(1) = (b) Another measure of the distribution of wealth in an economy is given by the Gini Index. The lower the index, the more the wealth is spread throughout the population. As examples, Canada has an index of 32.1 and South Africa has an index of 62.5 (i.e. Canada's wealth is spread throughout a greater percentage of the population than South Africa's wealth is). Given that a country's Gini Index is defined to be two hundred times the area enclosed by the population's Lorenz curve L(x) and the line of equality, determine (to 1 decimal) the Gini Index for the country with Lorenz curve L(x) = xet-1. Note: You may find it helpful to plot the two curves using desmos.com/calculator .] (c) How does this country's distribution of wealth compare to Canada and South Africa?

Answer 1

Answeron (a) in 2 (0.25) = 0.250 0.25) 1 - 0.25 0.062-1 = 0.25 & -0.937 -0.25X0.392 lowest coa 0.098 have only 9.8% of the entire That is, 25% of households economy's wealth. (1) L(Ois) - Oise (0.5) - 1 - 0.5 & 0.25-1 -0.50 -0.75 - 0SX 0.472 a lowest 0.236 That is 50% of households have only 23.6% of the entire economy's wealth. (1) L (0.75) Octs (ons- -0.75 -0.437 lowest earning = 0.75 X 0.646 = 0.485 That is 75% of households have economy's wealth. liv Lv - le (1²-1 6 of the entire - RII = 1 That is 100% of households have economy's wealth 100% of the entire

Ladrox²-1 Line of equality (2x) Area under line of equality = 0.5 Area under lorenz cowe - det?- doc - Let t=x²-1 dta 2x dx at xdx -Seldt at Seldt let [ex2-1] = √ [etleb = ₃ [ 1 - 1 - 1 (1-0.37) = 1 106) = 0.3 Area between Line of equality & Lorenz wowe 0.5-0.3 Crinni Index = 0.2X200 = 40 = 0.2 - .
c) Country's wealth is spread throughout a greater percentage of population than South Africa but lesser percentage of population than Canada.