1. Chapter 8 Slutsky Equation
Key Concept: How to decompose total effect into substitution effect and income effect. How to sign them?
Slutsky substitution effect and Hicks substitution effect
A Giffen goods must be inferior and the income effect has to be strong enough.

2. Chapter 8 Slutsky Equation
total effect (TE)
=substitution effect (SE) +
income effect (IE)

3. When p1 decreases, p1/ p2 decreases, consume more x1 because good 1 becomes cheaper (SE).
At the same time, the purchasing power of income increases, consume more (less) x1 if good 1 is normal (inferior) (IE).
Sign for normal goods seems right!

4. Draw a general case to illustrate the “pivot and shift” decomposition.
When we pivot, we want to make the original optimum just affordable as a control of purchasing power.
Slutksy substitution effect

5. Fig. 8.1

6. Fig. 8.2

7. (p1, p2, m) (x1, x2)
→ (p1’, p2, m’) (控制購買力不動)
→ (p1’, p2, m) (購買力改變)

8. At (p1’, p2, m’), (x1, x2) can still be bought, hence
p1 x1 +p2 x2 =m and
p1’x1 +p2 x2 =m’.
So m’- m=(p1’- p1)x1.
∆x1(SE) = x1(p1’, p2, m’) - x1(p1, p2, m)
∆x1(IE) = x1(p1’, p2, m) - x1(p1’, p2, m’)

9. An example: x1=10+m/10p1
p1=3, m=120, p1’=2, hence TE=10+120/20-(10+120/30)=16-14=2.
To calculate SE
m’- m=(p1’- p1)x1=(2-3)14
m’=120-14=106
SE=10+106/20-(10+120/30)=1.3
IE=TE-SE=2-1.3=0.7

10. How to sign the SE?
Suppose p1>p1’, then x1(p1’, p2, m’)≥ x1(p1, p2, m).
So SE is weakly negative, i.e., moving in the opposite direction as the price change.

11. Fig. 8.2

12. The Slutsky identity
x1 (p1’, p2, m)-x1 (p1, p2, m)
=x1 (p1’, p2, m’)-x1 (p1, p2, m)
+x1 (p1’, p2, m)-x1 (p1’, p2, m’)
TE=SE+IE

13. Suppose p1>p1’, m>m’
SE makes consumption of 1 increase
IE (restoring income from m’ to m) makes consumption of 1 increase if 1 is normal.
Hence for normal good, we will have a downward sloping demand.

14. To get an upward sloping demand, IE must dominate SE.
Giffen good must not only be inferior but IE has to be strong enough to dominate SE.

15. Fig. 8.3

16. TE/(p1’-p1)=[x1(p1’, m)-x1(p1, m)] /(p1’-p1)
Suppress the dependence on p2
TE/(p1’-p1)=[x1(p1’, m)-x1(p1, m)] /(p1’-p1)
SE/(p1’-p1)=[x1(p1’, m’)-x1(p1, m)] /(p1’-p1)
(-)
IE/(p1’-p1)
=[x1(p1’, m)-x1(p1’, m’)] /(p1’-p1)
=[(x1(p1’, m)-x1(p1’, m’)) /(m’-m)] x1(p1, m)) Recall that m’- m=(p1’- p1)x1
=-[(x1(p1’, m)-x1(p1’, m’)) /(m-m’)] x1(p1, m)) (-)

17. ∆x1 / ∆p1 = ∆xs1 / ∆p1 – (∆xm1 / ∆m )x1
(-) (+)

18. Some examples
p1’< p1
Perfect complement
zero SE

19. Fig. 8.4

20. p1’<p1
Perfect substitute
zero IE

21. Fig. 8.5

22. p1’<p1
Quasilinear
after some point, zero IE

23. Fig. 8.6

24. 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+R
For an average consumer, R=tx’, hence px’+y’=m
worse off
consume less of x, a pure SE

25. Fig. 8.7

27. Hicks SE: the original bundle may no longer be affordable but will be able to purchase a bundle just indifferent to the original bundle.
Consider again the case where p1>p1’.

28. Fig. 8.9

29. Neither can be revealed preferred to the other.
Some ideas of revealed preference
at (p1, p2) (x1, x2) is bought
at (p1’, p2) (y1, y2) is chosen
(x1, x2) i (y1, y2) by Hicks SE
Neither can be revealed preferred to the other.
Hence p1x1+p2 x2≤ p1y1+p2 y2 and
p1’y1+p2 y2≤ p1’x1+p2 x2
→ (p1’-p1) (y1-x1)≤0
So Hicks SE must be negative too.

30. So Hicks IE depends similarly on whether a good is normal or inferior.
p1’y1+p2 y2≤ p1’x1+p2 x2≤ p1x1+p2 x2
So Hicks IE depends similarly on whether a good is normal or inferior.

31. Chapter 8 Slutsky Equation
Key Concept: How to decompose total effect into substitution effect and income effect. How to sign them?
Slutsky substitution effect and Hicks substitution effect
A Giffen goods must be inferior and the income effect has to be strong enough.