Consider two brothers, Eddy and Larry, who, despite growing up in the same household, have grown quite different personalities.
A: Eddy is known to his friends as “steady Eddy” — he likes predictability and wants to know that he’ll have what he has now again in the future. Larry, known to his friends as “crazy Larry”, adapts easily to changing circumstances. One year he consumes everything around him like a drunken sailor, the next he retreats to a Buddhist monestary and finds contentment in experiencing poverty.
(a) Take the characterization of Eddy and Larry to its extreme (within the assumptions about tastes that we introduced in Chapter 4) and draw two indifference maps with “current con- sumption” on the horizontal axis and “future consumption” on the vertical — one for steady Eddy and one for crazy Larry.
(b) Eddy and Larry have another brother named Daryl who everyone thinks is a weighted average between his brothers’ extremes. Suppose he is a lot more like steady Eddy than he is like crazy Larry — i.e. he is a weighted average between the two but with more weight placed on the Eddy part of his personality. Pick a bundle A on the 45 degree line and draw a plausible indifference curve for Daryl through A. (If you take the above literally in a certain way, you would get a kink in Daryl’s indifference curve.) Could his tastes be homothetic?
(c) One day Daryl suffers a blow to his head — and suddenly it appears that he is more like crazy Larry than like steady Eddy; i.e. the weights in his weighted average personality have flipped. Can his tastes still be homothetic?
(d) In end-of-chapter exercise 4.9, we defined what it means for two indifference maps to satisfy a “single crossing property”. Would you expect that Daryl’s pre-accident and post-accident indifference maps satisfy that property?
(e) If you were told that either Eddy or Larry saves every month for retirement and the other smokes a lot, which brother is doing what?
B: Suppose that one of the brothers’ tastes can be captured by the function u(x1, x2) = min{x1, x2} where x1 represents dollars of current consumption and x2 represents dollars of future consumption.
(a) Which brother is it?
(b) Suppose that when people say that Daryl is the weighted average of his brothers, what they mean is that his elasticity of substitution of current for future consumption lies in between those of his borthers. If Larry and Daryl have tastes that could be characterized by one (or more) of the utility functions from end-of-chapter exercise 4.5, which functions would apply to whom?
(c) Which of the functions in end-of-chapter exercise 4.5 are homothetic? Which are quasilinear (and in which good)?
(d) Despite being so different, is it possible that both steady Eddy and crazy Larry have tastes that can be represented by Cobb Douglas utility functions?
(e) Is it possible that all their tastes could be represented by CES utility functions? Explain.