Chapter 8 Slutsky Equation Decompose total effect (TE) into substitution effect (SE) and income effect (IE) When p 1 decreases, p 1 / p 2 decreases, you may want to consume more x 1 because good 1 becomes relatively cheaper (SE). At the same time, the purchasing power of income increases, you may want consume more (less) x 1 if good 1 is normal (inferior) (IE).

Draw a general case to illustrate the decomposition. (pivot and shift) (p 1, p 2, m) (x 1, x 2 ) → (p 1 ’, p 2, m’) → (p 1 ’, p 2, m) ( 控制購買力不動 )( 購買力改變 ) At (p 1 ’, p 2, m’), (x 1, x 2 ) can still be bought, hence, p 1 x 1 +p 2 x 2 =m and p 1 ’x 1 +p 2 x 2 =m’. So m’- m=(p 1 ’- p 1 )x 1. ∆x 1 (SE) = x 1 (p 1 ’, p 2, m’) - x 1 (p 1, p 2, m) ∆x 1 (IE) = x 1 (p 1 ’, p 2, m) - x 1 (p 1 ’, p 2, m’)

Fig. 8.1

Fig. 8.2

An example:p 1 =3, m=120, p 1 ’=2, x 1 =10+m/10p 1, hence TE=10+120/20- (10+120/30)=16-14=2. To calculate SE, m’- m=(p 1 ’- p 1 )x 1 =(2- 3)14, so m’=120-14=106, therefore SE=10+106/20-(10+120/30)=1.3 and IE=TE-SE=2-1.3=0.7 How to sign the SE? Suppose p 1 >p 1 ’, then x 1 (p 1 ’, p 2, m’)≥ x 1 (p 1, p 2, m). So SE is weakly negative.

The Slutsky identity: x 1 (p 1 ’, p 2, m)-x 1 (p 1, p 2, m)=(x 1 (p 1 ’, p 2, m’)-x 1 (p 1, p 2, m))+(x 1 (p 1 ’, p 2, m)-x 1 (p 1 ’, p 2, m’)) (TE=SE+IE) Suppose p 1 ’>p 1, m’>m, so SE makes consumption of 1 decrease and IE (restoring income from m’ to m) makes consumption of 1 decrease as well if 1 is normal. Hence for normal good, we will have a downward sloping demand.

To get an upward sloping demand, the IE effect must dominate SE. In other words, a Giffen good must not only be inferior but the IE has to be strong enough to dominate the SE.

Fig. 8.3

Suppress the dependence on p 2 : TE/(p 1 ’-p 1 )=(x 1 (p 1 ’, m)-x 1 (p 1, m)) /(p 1 ’-p 1 ) SE/(p 1 ’-p 1 )=(x 1 (p 1 ’, m’)-x 1 (p 1, m)) /(p 1 ’-p 1 ) IE/(p 1 ’-p 1 )=(x 1 (p 1 ’, m)-x 1 (p 1 ’, m’)) /(p 1 ’-p 1 ) =[(x 1 (p 1 ’, m)-x 1 (p 1 ’, m’)) /(m’-m)] x 1 (p 1, m)) (-)

Some examples: p 1 ’<p 1 Perfect complement (zero SE) Perfect substitute (zero IE) Quasilinear after some point (zero IE) Rebating a tax (ignore tax incidence) p’=p+t demand goes from x to x’ for an average consumer, so R=tx’ Old BC: px+y=m; new BC (p+t)x’+y’=m+tx’ or px’+y’=m (worse off and consume less of x, pure SE)

Fig. 8.4

Fig. 8.5

Fig. 8.6

Fig. 8.7

Hicks SE: the original bundle may no longer be affordable but will be able to purchase a bundle just indifferent to the original bundle Some ideas of revealed preference: at (p 1, p 2 ) (x 1, x 2 ) is bought, at (p 1 ’, p 2 ) (y 1, y 2 ) is chosen and (x 1, x 2 ) i (y 1, y 2 ) by Hicks SE. Neither can be revealed preferred to the other. Hence p 1 x 1 +p 2 x 2 ≤ p 1 y 1 +p 2 y 2 and p 1 ’y 1 +p 2 y 2 ≤ p 1 ’x 1 +p 2 x 2 → (p 1 ’-p 1 ) (y 1 -x 1 )≤0 So Hicks SE must be negative either.

Fig. 8.9

Ordinary demand (holding income fixed) (+ -) Slutsky demand (holding purchasing power fixed) (-) Hicks demand (holding utility fixed) (-) useful because utility is fixed