When the income elasticity of demand is independent of price, so that ?q(p,y) ?y y q(p,y)
When the income elasticity of demand is independent of price, so that
y q(p,y) = ?(y)
for all p and y in the relevant region, then for the base price p0 and income y0, the consumer surplus, CS, and the compensating variation, CV , are related as follows: -?CS =Z CV +y0 y0 expZ ? y0 - ?(?) ? d?d?. a) Show that when the income elasticity is constant but not equal to unity, then CV = y0-?CS y0 (1-?) + 1 1 1-? -y0. b) Use this result to show the following: if demand is independent of income, i.e., -?CS = CV , then CS is an exact measure of the welfare impact of a price change.
c) Derive the relation between CV and ?CS when the income elasticity is unity.
d) We can use the result of part a) to establish a convenient rule of thumb that can be used to quickly gauge the approximate size of the deviation between the change in consumer surplus and the compensating variation for the case of a constant income elasticity. Show that (CV -|?CS|)/|?CS|˜ (?|?CS|)/(2y0) holds if the income elasticity is constant and not equal to unity.