Question

70.7 1. Judy's Marshallian demand for oranges is 10.4, where pa is the price of apples, po is the price of oranges, and I is Judy's income. Suppose I = 100, Pa = 2, and p. = 1. (a) Find and interpret the income elasticity for the demand for oranges. Are oranges an inferior or normal good? (b) Find the own price elasticity of demand for oranges. Discuss how the price elas- ticity varies with po. (c) Find the cross price elasticity for oranges. Are oranges and apples gross substitutes or gross complements?

Answer 1

Given,

$q=\frac{I^{0.7}}{2Po(Pa+3)^{0.4}}$

a)

$\frac{dq}{dI}=0.7\frac{I^{-0.3}}{2Po(Pa+3)^{0.4}}$

Income elasticity for demand is given by

$\varepsilon _{I}=\frac{dq}{dI}*\frac{I}{q}=0.7\frac{I^{-0.3}}{2Po(Pa+3)^{0.4}}*\frac{I}{\frac{I^{0.7}}{2Po(Pa+3)^{0.4}}}$

$\varepsilon_{I} =0.7\frac{I^{-0.3}*I*I^{-0.7}}{2Po(Pa+3)^{0.4}}*\frac{2Po(Pa+3)^{0.4}}{1}=0.70$

Income elasticity for orange demand is positive, we can say that orange is a normal good.

b)

$\frac{dq}{dPo}=-\frac{I^{0.7}}{2Po^{2}(Pa+3)^{0.4}}$

Own price elasticity for demand is given by

$\varepsilon _{po}=\frac{dq}{dPo}*\frac{Po}{q}=-\frac{I^{0.7}}{2Po^{2}(Pa+3)^{0.4}}*\frac{Po}{\frac{I^{0.7}}{2Po(Pa+3)^{0.4}}}=-1$

We can see that own price elasticity is independent of Po. It remains constant i.e. -1 for all values of Po

c)

$\frac{dq}{dPa}=-0.4*\frac{I^{0.7}}{2Po(Pa+3)^{1.4}}$

Cross price elasticity of orange demand is given by

$\varepsilon _{pa}=\frac{dq}{dPa}*\frac{Pa}{q}=-0.4*\frac{I^{0.7}}{2Po(Pa+3)^{1.4}}*\frac{Pa}{\frac{I^{0.7}}{2Po(Pa+3)^{0.4}}}=\frac{-0.4Pa}{1+Pa}=\frac{-0.4*2}{2+1}=-0.27$

Cross price elasticity is negative, it means that orange and apples are gross complements.