In this exercise, we explore the concept of elasticity in contexts other than own-price elasticity of (uncompensated) demand. (In cases where it matters, assume that there are only two goods).
A. For each of the following, indicate whether the statement is true or false and explain your answer:
(a) The income elasticity of demand for goods is negative only for Giffen goods.
(b) If tastes are homothetic, the income elasticity of demand must be positive.
(c) If tastes are quasilinear in x, the income elasticity of demand for x is zero.
(d) If tastes are quasilinear in x1, then the cross-price elasticity of demand for x1 is positive.
(e) If tastes are homothetic, cross price elasticities must be positive.
(f) The price elasticity of compensated demand is always negative.
(g) The more substitutable two goods are for one another, the greater the price elasticity of compensated demand is in absolute value.
B. Consider first the demand function x = αI/p that emerges from Cobb-Douglas tastes.
(a) Derive the income elasticity of demand and explain its sign.
(b) We know Cobb-Douglas tastes are homothetic. In what way is your answer to (a) simply a property of homothetic tastes.
(c) What is the cross-price elasticity of demand? Can you make sense of that?
(d) Without knowing the precise functional form that can describe tastes that are quasilinear in x, how can you show that the income elasticity of demand must be zero?
(e) Consider the demand function x1(p1, p2) = (αp2/p1) β. Derive the income and cross-price
elasticities of demand.
(f) Can you tell whether the tastes giving rise to this demand function are either quasilinear or homothetic?