Four siblings share a house. The utility function of each sibling is u(m;) = 2m;, where m; is income of sibling j, and four income levels are {1,1,9,9}. (a) Draw the Lorenz curve associated with this income distribution of the household and illustrate a measure of inequality using your graph. (b) Compute the household Gini coefficient using the formula: [5 marks] 2 G = Mm i=1 1 i(m - m), where M individuals from poorest to richest, m denotes mean and i is the "rank" in the income distribution (i = 1,2,3, ... . M), from poorest to richest. [5 marks] (c) The parents can intervene to redistribute income between siblings with the objective of maximising a social welfare function of the household. (i) What assumptions about utility are required for the parents to do this? [2 marks] (ii) Explain the rationale for redistribution if the parents are Utilitarian. How will they redistribute income? What is G after the redistribution? [8 marks] (iii) Explain the rationale for redistribution if the parents are Rawlsian. How will they redistribute income? What is G after the redistribution? [8 marks] (d) Suppose the utility function of the two siblings with income 9 is now u(m;) = 6m;. How does this alter the parents' willingness to redistribute income in the case of (i) Utilitarianism, and (ii) Rawlsianism? Comment on household inequality in either case. [12 marks] (e) Assume once more that all siblings have utility function u(m;) = 2.m;. The siblings express a preference for income distribution m = {4,4,6,6}. What might explain this? [10 marks]