5.5*† Consider 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 they 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?