4.} A consumer has a preference relation dened by the utility function u{x, y) = -{7+ 1 - xlz (7+ 1 - 3:12. He has an income ofw > 0 and faces prices 3),. and 133, 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 ={ (x, y) E R2+ : pxx +pyy s w}. {a} Draw the indifference curve that achieves utility level of -1. Is this utility function quasi-concave? {b} Suppose px, py > 0. Prove that Bis a compact set. (c) lfpx = 0, draw the new budget set and explain whether it is compact. Suppose you are told that pg: 1, py = 1 and w = 15. The consumer maximises his utility on the budget set. {d} Explain how you would obtain a solution to the consumer's optimisation problem using a diagram. {e} Write down the Lagrange function and solve the consumer's utility maximisation problem using the KKT formulation. {f} Intuitively explain how your solution would change ifthe consumer's income reduces to w = 5. {g} Is the optimal demand for good 1 everywhere differentiable with respect to w? You can provide an informal argument