In: Economics
Explain all answers and label any graphs clearly. Be sure to define any notation you introduce.
1. Suppose that there are two commodities which are available in nonnegative amounts. Draw
some pictures of indifference curves for preference relations having the following properties:
(a) Strictly (or strongly) convex but not monotone.
(b) Convex and strictly monotone but not strictly (or strongly) convex.
(c) Convex but not strictly (or strongly) convex AND monotone but not strictly (or strongly)
monotone.
(d) Not convex.
2. Show that any preference relation that can be represented by a utility function must be reflexive
and transitive.
3. Show that a preference relation that can be represented by a concave utility function must be at
least weakly convex.
4. Show that if a preference relation can be represented by a utility that is quasiconcave, then
every utility function that represents the preference relation must be quasiconcave.
5. Give an example of a preference relation that can be represented by a continuous utility
function and a utility function that fails to be continuous. Show explicitly two utility functions
(one of which is continuous and one of which is not continuous) that make your claim true.