Consider my tastes for paperclips and “all other goods” (denominated in dollar units).
A: Suppose that my willingness to trade paper clips for other goods does not depend on how many other goods I am also currently consuming.
(a) Does this imply that “other goods” are “essential” for me?
(b) Suppose that, in addition, my willingness to trade paperclips for other goods does not depend on how many paperclips I am currently consuming. On two graphs, each with paperclips on the horizontal axis and “dollars of other goods” on the vertical, give two examples of what my indifference curves might look like.
(c) How much can the MRS vary within an indifference map that satisfies the conditions in part (b)? How much can it vary between two indifference maps that both satisfy the conditions in part (b)?
(d) Now suppose that the statement in (a) holds for my tastes but the statement in part (b) does not. Illustrate an indifference map that is consistent with this.
(e)How much can the MRS vary within an indifference map that satisfies the conditions of part (d)?
(f) Which condition do you think is more likely to be satisfied in someone’s tastes — that the willingness to trade paperclips for other goods is independent of the level of paperclip consumption or that it is independent of the level of other goods consumption?
(g) Are any of the indifference maps above homothetic? Are any of them quasilinear?
B: Let paperclips be denoted by x1 and other goods by x2.
(a) Write down two utility functions, one for each of the indifference maps from which you graphed indifference curves in A(b).
(b) Are the utility functions you wrote down homogeneous? If the answer is no, could you find utility functions that represent those same tastes and are homogeneous? If the answer is yes, could you find utility functions that are not homogeneous but still represent the same tastes?
(c) Are the functions you wrote down homogeneous of degree 1? If the answer is no, could you find utility functions that are homogeneous of degree 1 and represent the same tastes? If the answer is yes, could you find utility functions that are not homogeneous of degree k and still represent the same tastes?
(d) Is there any indifference map you could have drawn when answering A(d) which can be represented by a utility function that is homogeneous? Why or why not?