Problem Set 2 Solutions
Due January 27, 2010
Professor Michael Noel
University of California San Diego
True or False. Give a heuristic proof if true, counterexamples if not.
Non-satiation implies that a consumer always prefers one more slice of apple pie.
True – but only assuming that everything else in the bundle is being held constant. If extra apple pie
comes at the cost of losing something else, the new bundle may or may not be preferred.
Pam loves pizza, but it must have mushrooms. Between any two pizzas with (or without) mushrooms,
she cares of course about the amount of cheese, the more the better. But she prefers a pizza with more
mushrooms over any pizza with fewer mushrooms
regardless of the amount of cheese
preferences satisfy the four (critical) axioms of preference theory.
Preferences are complete. She is very clear about her ranking of bundles and can rank any two pizzas.
If both have the same number of mushrooms, Pam prefers the pizza with more cheese. Otherwise, she
prefers the pizza with more mushrooms. If they are identical, she is indifferent.
Preferences are reflexive. Pizza A is equally preferred to pizza A (itself) so clearly it is weakly
Preferences are transitive. Consider pizzas A, B, and C, where A is preferred to B and B is preferred
to C. If A is preferred to B, it must be that 1. A has more mushrooms than B or 2. A and B have the
same number of mushrooms, but A has more cheese. Assume first that A has more mushrooms than
B. Now, as B is preferred to C we know that B has (weakly) more mushrooms than C. Thus A has
(strictly) more mushrooms that C and is accordingly preferred to C. Now consider the case where A
and B have the same number of mushrooms but A has more cheese.
Because B is preferred to C, we
know C must have either 1. Fewer mushrooms than B, in which case A is preferred to C or 2. the
same quantity of mushrooms as both A and B, but less cheese than B (and hence less cheese than A),
in which case A is preferred to C. This is the famous mushroom proof.
continuous. Consider pizza A and B with the same number of mushrooms.
Assume A has lots and lots of cheese and B has very little cheese, so A is preferred to B. Now
consider a pizza C that is very very close to pizza A, with a tiny bit fewer mushrooms (tiny, tiny,
tiny). Continuous implies that we should be able to find a pizza C close enough to A in terms of
amount of cheese and mushrooms that it is necessarily preferred to B. But we cannot. Even an
infinitesimally small decrease in the number of mushrooms drops us down to something worse than