2
1.2. Slutsky equation
ˆ
ˆˆ
()
(,(,()
)
)
(,()
)
kk
k
j
j
jj
xh
x
pp
v
p
w
p
p
w
p
x
w
∂∂
∂
⎡
⎤
=−
−
ω
⎣
⎦
∂
%
,
j
,
k
= 1, 2,
(2)
where:
p
= (
p
1
,
p
2
),
,
h
11
2 2
ˆ
wp
p
p
≡ω
+ω
k
is the Hicksian demand function and
v
is the indirect
utility function.
1.3(a)
Without further information, what can we say about the sign of
112
1
ˆ
(, )
xpp
p
∂
∂
when the consumer is a borrower? A saver? Autarkic?
From (2)
[
111
ˆ
xhx
x
ppw
∂∂∂
]
=
−−
%
ω
.
(3)
By the Slutsky Theorem,
1
1
0
h
p
∂
≤
∂
(the substitution effect is nonpositive), and in fact
1
1
0
h
p
∂
<
∂
under
our assumptions (in particular, the utility function is differentiable with nonzero gradient).
Moreover, because good 1 is normal,
1
0
x
w
∂
>
∂
%
.
If the consumer is a borrower,
[ ]
0
x
−ω>
, hence the wealth effect
[]
1
x
x
w
∂
∂
%
ω
is
negative. The total effect is then the sum of two negative terms, and is therefore negative.
Intuitively, if you are a borrower you are a (net) buyer of present consumption (good 1). An
increase in the price of a good that you buy reduces your purchasing power, which tends to
decrease your demand for all normal goods. The wealth effect then reinforces the substitution
effect, and the demand for good 1 unambiguously decreases.
If the consumer is autarkic, then
[ ]
0
x
−ω=
, hence the wealth effect is zero. The total
effect is then equal to the substitution effect, negative.
If the consumer is a saver, then
[ ]
0
x
−ω<
, hence the wealth effect is positive: Now you
are selling good 1, and an increase in its price raises your purchasing power. The substitution
effect and the wealth effect move in opposite directions, and nothing can be said on the sign of
1
ˆ
p
∂
∂
in the absence of further information. Table 1 summarizes.