answer only (e) and (f) only if you can please, and with math only

1. Consider a consumer with utility function u(x1, x2) = min ( 4 x1 + x2, x1 + 2 x2)

(a) Draw indifference curves passing through points (2; 2), (1; 2) and (4; 2) (Note: these points may lie on different indifference curves). Make sure you correctly determine kink points. (b) Determine all properties of the preferences that you can deduce from the shape of indifference curves or utility function. For each claimed property, provide either a formal proof or a graphical visualization that will clearly indicate that the claimed property holds. (c) When X -> R2+, does UMP have a solution when Pk = 0? What property of the preference relation did you use to get your answer? (d) Assume that prices are positive. Derive the Walrasian demand of each good. Is the Walrasian demand always single valued? [Hint: graphically depicting the UMP can pin down the maximizing bundles. If p1=p2 > 4 what can you say about the location of the utility-maximizing consumption bundle? What is the location if 4 < p1=p2 < 1=2? What about prices such that p1=p2 < 1=2?]

(e) Let p1 = p2 = 1 and w = $60. Suppose that the consumer receives a $10 voucher from the government that he can spend only on good 1. Draw the new budget set of the consumer and calculate the quantity of each good demanded by the consumer. Does receiving the voucher make consumer better-off?

(f) Suppose instead that the government allows the consumer to choose between a cash payment of $10 that can be spent on both goods and a $10 voucher that can be spent on good 1 only. Which one would the consumer choose and why? Would your answer change if the government's assistance were $30? Explain your answer.