t = 0
4. [40 MARKS] Let t be the 7th digit of your Student ID. A consumer has a preference relation defined by the utility function u(x, y) = −(t + 1 − x)2 − (t + 1 − y)2. He has an income of w> 0 and faces prices pa 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 = {(x, y) = R2 : PxX+Py¥ ≤ w}.
(a) [4 MARKS] Draw the indifference curve that achieves utility level of -1. Is this utility function quasi-concave?
(b) [5 MARKS] Suppose Px, Py >0. Prove that B is a compact set.
(c) [3 MARKS] If p = 0, draw the new budget set and explain whether it is compact. Suppose you are told that utility on the budget set.
Px = 1, Py
=
1 and w =
15. The consumer maximises his
(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 w? You can provide an informal argument.