Please sketch the indifference map for (i) and answer both parts!

4. Social Preferences. Consider a socialist planner who has preference over social welfare' vectors w e RW, with the interpretation that the i-th component w; is the welfare of the i-th person in the society. Suppose that the social planner's only concern is to maximize the welfare of the least well-off person in the society; she does not care about anything else. i. Write-down a welfare function W : R + R that represents the social planner's preference relationship over the welfare vectors. Ex- plain briefly whether or not this preference relationship is continuos, monotone, strongly monotone, convex or strictly convex. It may help to sketch the indifference map. ii. Now suppose that the planner decides that, if the least well off indi- vidual in w and w' are equally well off, then she prefers w to w' if the second-least worse off individual in w is better off than the second- least worse off individual in w', and if this is a tie then she shifts her concern to the third worse off person, etcetera. Explain whether or not this preference relation is continuous, monotone, strongly mono- tone, convex and strictly convex. 4. Social Preferences. Consider a socialist planner who has preference over social welfare' vectors w e RW, with the interpretation that the i-th component w; is the welfare of the i-th person in the society. Suppose that the social planner's only concern is to maximize the welfare of the least well-off person in the society; she does not care about anything else. i. Write-down a welfare function W : R + R that represents the social planner's preference relationship over the welfare vectors. Ex- plain briefly whether or not this preference relationship is continuos, monotone, strongly monotone, convex or strictly convex. It may help to sketch the indifference map. ii. Now suppose that the planner decides that, if the least well off indi- vidual in w and w' are equally well off, then she prefers w to w' if the second-least worse off individual in w is better off than the second- least worse off individual in w', and if this is a tie then she shifts her concern to the third worse off person, etcetera. Explain whether or not this preference relation is continuous, monotone, strongly mono- tone, convex and strictly convex