Consider again the family of homothetic tastes.
A: Recall that essential goods are goods that have to be present in positive quantities in a consumption bundle in order for the individual to get utility above what he would get by not consuming anything at all.
(a) Aside from the case of perfect substitutes, is it possible for neither good to be essential but tastes nevertheless to be homothetic? If so, can you give an example?
(b) Can there be homothetic tastes where one of the two goods is essential and the other is not? If so, give an example.
(c) Is it possible for tastes to be non-monotonic (less is better than more) but still homothetic?
(d) Is it possible for tastes to be monotonic (more is better), homothetic but strictly non convex (i.e. averages are worse than extremes)?
B: Now relate the homotheticity property of indifference maps to utility functions.
(a) Aside from the case of perfect substitutes, are there any CES utility functions that represent tastes for goods that are not essential?
(b) All CES utility functions represent tastes that are homothetic. Is it also true that all homothetic indifference maps can be represented by a CES utility function? (Consider your answer to A(a) and ask yourselves, in light of your answer to B(a), if it can be represented by a CES function.)
(c) True or False: The elasticity of substitution can be the same at all bundles only if the underlying tastes are homothetic.
(d) True or False: If tastes are homothetic, then the elasticity of substitution is the same at all bundles.
(e) What is the simplest possible transformation of the CES utility function that can generate tastes which are homothetic but non-monotonic?
(f) Are the tastes represented by this transformed CES utility function convex?
(g) So far, we have always assumed that the parameter ρ in the CES utility function falls between −1 and ∞. Can you determine what indifference curves would look like when ρ is less than −1?
(h) Are such tastes convex? Are they monotonic?
(i) What is the simplest possible transformation of this utility function that would change both your answers to the previous question?