The following is intended to explore what kinds of cross-price demand relationships are logically possible in a two-good model with exogenous income.
A: For each of the following, indicate whether the relationship is possible or not and explain:
(a) A good is normal and its cross-price demand relationship is positive.
(b) A good is normal and its cross-price relationship is negative.
(c) A good is inferior and its cross-price relationship is negative.
(d) Tastes are homothetic and one of the good's cross-price relationship is negative.
(e) Tastes are homothetic and one of the good's cross-price relationship is positive.
B: Now consider specific tastes represented by particular utility functions.
(a) Suppose tastes are represented by the function u(x1,x2) = αlnx1 + x2. What is the shape of the cross-price demand curves for x1 and x2?
(b) Suppose instead tastes are Cobb-Douglas. What do cross-price demand curves look like?
(c) Now suppose tastes can be represented by a CES utility function. Without doing any math, can you determine for what values of ρ the cross-price demand relationship is upward sloping?
(d) Suppose tastes can be represented by the CES function u(x1,x2) = (0.5x1−ρ +0.5x2−ρ)−1/ρ. Verify your intuitive answer from part (c).