t = 1 i need it in 2-3 hours. please be quick. solve all the parts in short answer too no problem

4. [40 MARKS] Let t be the '7'\"1 digit of your Student ID. A consumer has a preference relation defined by the utility function utmy] = (t + 1 so): (t + 1 502. He has an income of w 2-:- and faces prices pm and pg, 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 = {lay} E R1 =p=m+pyy S w}. (a) [4 MARKS] Draw the indifference curve that achieves utility level of ~1. Is this utility function quasi-concave? (b) [5 MARKS] Suppose pmpy :> ID. Prove that B is a compact set. (c) [3 MARKS] If 311 a 0, draw the new budget set and explain whether it is compact. Suppose you are told that pr = 1, p3, = 1 and u; = 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 to? You can provide an informal argument