In the DS system, the Marshallian income slope is 'x3(p8,M8)/'M8 = "3, so that
each good has a separate income effect ("3), unlike the semilog demand system, where all
income effects must be the same. The income elasticity for good i is, then,
_ 'X3(p8,M8) M8
%3Q ´ 'M8 ■ x3
Al
X3
-p
!3
(6)
where !3 ´ p3x3∕M is the Marshallian budget share of good i. Each good has an
independent income effect, unlike the semilog system, where all income effects must be
equal.
The Marshallian own- and cross-price elasticities %33 and %34 are, respectively,
´ 'x3(p8M8) . p8
%33 - 'p8 x3
œ #3p8 1 - "!p8
(7)
and
= ⅜(p8,M8) . P8
34 ´ 'p8 x3
œ#4 p48 ’1 -
"P
!3
(8)
where !3 ´ p8x3/M8 is the budget share of good i. Noting, from (6), that "3p8/!3 is the
income elasticity for good i, (7) and (8) can also be written as