5.9 Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good, x, the income share is defined as sx ¼ pxx/I. In this problem we show that most demand elasticities can be derived from corresponding share elasticities.

a. Show that the elasticity of a good’s budget share with respect to income ðesx , I ¼ @sx=@I & I=sxÞ is equal to ex, I – 1. Interpret this conclusion with a few numerical examples.

b. Show that the elasticity of a good’s budget share with respect to its own price ðesx , px ¼ @sx=@px & px=sxÞ is equal to ex, px þ 1. Again, interpret this finding with a few numerical examples.

c. Use your results from part

(b) to show that the ‘‘expenditure elasticity’’ of good x with respect to its own price

½ex&px , px ¼ @ð px & xÞ=@px & 1=x* is also equal to ex, px þ 1.

d. Show that the elasticity of a good’s budget share with respect to a change in the price of some other good

ðesx , py ¼ @sx=@py & py=sxÞ is equal to ex, py .

e. In the Extensions to Chapter 4 we showed that with a CES utility function, the share of income devoted to good x is given by sx ¼ 1=ð1 þ pk yp'k x Þ, where k ¼ d/(d – 1) ¼ 1 – s. Use this share equation to prove Equation 5.56: exc, px ¼ 'ð1 ' sxÞr.

Hint: This problem can be simplified by assuming px ¼ py, in which case sx ¼ 0.5.