The following is intended to explore what kinds of own-price demand relationships are logically possible in a two-good model with exogenous income (unless otherwise specified).
A: For each of the following, indicate whether the relationship is possible or not and explain:
(a) Tastes are homothetic and the own-price demand relationship is positive.
(b) A good is inferior and its own-price relationship is negative.
(c) In a model with endogenous income, a good is normal and its own-price demand relationship is negative.
(d) In a model with endogenous income, a good is normal and its own-price demand relationship is positive.
B: Suppose that tastes can be represented by the Cobb-Douglas utility function u(x1,x2)= xα1 x2(1−α).
(a) Derive the demand functions when income is exogenous and illustrate that own-price demand curves slope down.
(b) Now suppose that all income is derived from an endowment (e1,e2). If e2 = 0, what is the shape of the own price demand curve for x1?
(c) Continuing with part (b), what is the shape of the own price demand curve for x1 when e2 > 0?
(d) Suppose tastes were instead represented by the more general CES utility function. Without doing any additional math, can you guess what would have to be true about ρ in order for the own-price demand for x1 to slope up when e1 > 0 and e2 = 0?