t=0 plz

4. [40 MARKS] Let t be the 7\" digit of your Student ID. A consumer has a preference relation defined by the utility function u(:c,y) = (t l 1 33)2 (t + 1 y)2. He has an income of w > 0 and faces prices 19$ and py of goods X and Y respectively. He does not need to exhaust his entire income. The budget set of this consumer is thus given by B = {(sc,y) 6R1 2p$93+pyy S w}. (a) [4 MARKS] Draw the indifference curve that achieves utility level ofl. Is this utility function quasiconcave? (b) [5 MARKS] Suppose pup], > 0. Prove that B is a compact set. (c) [3 MARKS] If 12$ = 0. draw the new budget set and explain whether it is compact. Suppose you are told that pa, = 1, pg = 1 and w = 15. The consumer maximises his utility on the budget set. (d) [6 MARKS] Explain how you would obtain a solution to the consumer's optimisation problem using a diagram. (e) [10 MARKS] Write down the Lagrange function and solve the consumer's utility maximisation problem using the KKT formulation. (f) [6 MARKS] Intuitively explain how your solution would change if the consumer's income reduces to w = 5. (g) [6 MARKS] Is the optimal demand for good 1 everywhere differentiable with respect to 10? You can provide an informal argument