Consider two individuals who take a very different view of life — and consider how this shapes their tastes over intertemporal tradeoffs.
A: Jim is a 25 year-old athlete who derives most of his pleasure in life from expensive and physically intense activities — mountain climbing in the Himalayas, kayaking in the Amazon, bungee jumping in New Zealand, Lion safaris in Africa and skiing in the Alps. He does not look forward to old age when he can no longer do this and plans on getting as much fun in early on as he can. Ken is quite different — he shuns physical activity but enjoys reading in comfortable surroundings. The more he reads, the more he wants to read and the more he wants to retreat to luxurious libraries in the comfort of his home. He looks forward to quiet years of retirement when he can do what he loves most.
(a) Suppose both Jim and Ken are willing to perfectly substitute current for future consumption — but at different rates. Given the descriptions of them, draw two different indifference maps and indicate which is more likely to be Jim’s and which is more likely to be Ken’s.
(b) Now suppose neither Jim nor Ken are willing to substitute at all across time periods. How would their indifference maps differ now given the descriptions of them above?
(c) Finally, suppose they both allowed for some substitutability across time periods but not as extreme as what you considered in part (a). Again, draw two indifference maps and indicate which refers to Jim and which to Ken.
(d) Which of the indifference maps you have drawn could be homothetic?
(e) Can you say for sure if the indifference maps of Jim and Ken in part (c) satisfy the single crossing property (as defined in end-of-chapter exercise 4.9)?
B: Continue with the descriptions of Jim and Ken as given in part A and let c1 represent consumption now and let c2 represent consumption in retirement.
(a) Suppose that Jim’s and Ken’s tastes can be represented by uJ (c1,c2) = α c1+c2 and uK (c1,c2) = β c1 + c2 respectively. How does α compare to β — i.e. which is larger?
(b) How would you similarly differentiate, using a constant α for Jim and β for Ken, two utility functions that give rise to tastes as described in A(b)?
(c) Now consider the case described in A(c), with their tastes now described by the Cobb-Douglas utility functions uJ (c1,c2) = cα1 c2(1−α) and uJ (c1,c2) = cβ1c2(1−β). How would α and β in those functions be related to one another?
(d) Are all the tastes described by the above utility functions homothetic? Are any of them quasi- linear?
(e) Can you show that the tastes in B(c) satisfy the single crossing property (as defined in end-of- chapter exercise 4.9)?
(f) Are all the functions in B(a)-(c) members of the family of CES utility functions?