Consider a consumer with preferences represented by a utility function u(x1, x2) = x1 + x2, where > 0 is a parameter. Fix prices p = (p1, p2) > 0 and income m > 0.

(a) [2] Illustrate the consumer's indifference curve through the bundle (0, 1).

(b) [2] Show that the consumer's preferences are convex. (

c) [2] Write down the consumer's utility maximisation problem.

(d) [2] Argue that the consumer's budget constraint holds as an equality at any optimal consumption bundle (x 1 , x 2 ). (e) [2] When is x 2 = 0 optimal for the consumer?