# What consumer preferences exhaust the budget?

Suppose that a consumer has a complete and transitive preference relation over R+. Further suppose that the consumer faces prices p = (p1,p2,...,p) 0 and has income w >> 0. Which of the following statements are false?

(a) If ≿ is continuous, then the consumer exhausts her budget. (b) If ≿ is locally nonsatiated, then the consumer exhausts her budget. (c) If ≿ is strictly monotonic, then the consumer exhausts her budget. (d) If ≿ is convex, then the consumer exhausts her budget. (e) If ≿ is strictly convex, then the consumer exhausts her budget.

a.) Consider utility with a bliss point. It may represent continuous preferences, but clearly the budget will not bind if you can afford more than the bliss point.

b.) This statement is true, and there is a proof here.

c.) Strict monotonicity implies local non-satiation, so same as in b.)

d.) An example of convex preferences is indifference everywhere. So the budget would not bind in this case.

e.) Strict convexity rules out thick indifference curves like in d.), but does not rule out bliss points. Consider concentric indifference curves moving in towards a bliss point. Each of those upper countour sets are convex shapes.

• Why does strict convexity rule out bliss points? – Theoretical Economist Dec 3 '16 at 3:58
• @TheoreticalEconomist I've made a careless mistake. I'll correct my answer. – Kitsune Cavalry Dec 3 '16 at 4:16
• why would the budget not bind for d.)? thanks – bs2293 Dec 4 '16 at 22:19