Question

# a4. (5 marks) In the previous assignment, you learned about the Gini index of a country (a number measuring the country's income inequality) is closely related to the area under the country's Lorenz curve over the interval [0,1]. Let's recall that if the point (100, 100) is on the Lorentz curve, then this means that the bottom a% of households receive 6% of the country's total income. In reality, we do not have a formula for the Lorenz curve but a collection of data.

So, for example, the point (0.2, 0.068) is on the Lorenz curve L Canada. This means that the bottom 20% of Canadian households received 6.8% of Canada's total (household disposable)income in 2020. (a) (1 mark) Use the data points above to approximate the area under the Lorenz curve LCanada over the interval [0, 1] by L5, the left endpoint approximation. (b) (1 mark) Use the data points above to approximate the area under the Lorenz curveL Canada over the interval [0, 1] by R5, the right endpoint approximation. (c) (2 marks) Use the data points above to approximate the area under the Lorenz curveLCanada over the interval [0, 1] by T5, the Trapezoidal Rule. (d) (1 mark) It is known that Lorenz curves are always concave upward. Using this fact,determine if T5 is an underestimate or an overestimates.

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