$4\mathrm{.}18$ Willig $(1976)$ has shown that when income elasticity of demand is independent of price, so that

$\frac{\text{d}\text{e}\text{l}\text{q}(p,y)}{\text{d}\text{e}\text{l}\text{y}}\frac{y}{q(p,y)}-=(y)$

for all $p$ and $y$ in the relevant region, then for base price $p^0$ and income $y^0,CS$ and $CV$ are related, exactly, as follows:

$-CS={\displaystyle \underset{y^0}{\overset{CV+y^0}{}}}$exp$(-{\displaystyle \underset{y^0}{\overset{}{}}}\frac{()}{}d)d.$

$($a$)$ Show that when income elasticity is constant but not equal to unity,

$CV=y^0[\frac{-CS}{y^0}(1-)+1{]}^{\frac{1}{1-}}-y^0.$

$($b$)$ Use this to show that when demand is independent of income, $\frac{-}{{?}_{\frac{?}{?}}}CS=CV,$ so consumer surplus can then be used to obtain an exact measure of the welfare impact of a price change.

$($c$)$ Derive the relation between $CV$ and $\frac{?}{{?}_{\frac{?}{?}}CS}$ when income elasticity is unity.

$($d$)$ Finally, we can use the result in 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 when income elasticity is constant. Show that when income elasticity is constant and not equal to unity, we have $|)CS|$~~$\frac{|CS|}{2}{y}^0|.$