Demand Correspondence
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The demand correspondence vector
x
(
p
,
w
)
=
[
x
1
(
p
,
w
)
x
2
(
p
,
w
)
⋮
x
L
(
p
,
w
)
]
{\displaystyle x(p,w)={\begin{bmatrix}x_{1}(p,w)\\x_{2}(p,w)\\\vdots \\x_{L}(p,w)\end{bmatrix}}}
assigns a set of consumption bundles to each pair
(
p
,
w
)
{\displaystyle (p,w)}
. A single valued demand correspondence is a demand function .
Assumptions on demand correspondences
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Homogeneity of degree zero:
x
(
α
p
,
α
w
)
=
x
(
p
,
w
)
∀
p
,
w
,
α
>
0
{\displaystyle x(\alpha p,\alpha w)=x(p,w)\;\forall p,w,\alpha >0}
Walras's law:
p
⋅
x
=
w
∀
x
∈
x
(
p
,
w
)
,
p
>>
0
,
w
>
0
{\displaystyle p\cdot x=w\;\forall x\in x(p,w),\;p>>0,\;w>0}
Notice, the homogeneity assumption allows one argument of
x
(
p
,
w
)
{\displaystyle x(p,w)}
to be normalized.
Comparative Statics
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The Engel Function
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Holding the price vector constant, the demand correspondence
x
(
p
=
p
¯
,
w
)
{\displaystyle x(p={\bar {p}},w)}
is the Engel function . In
R
L
{\displaystyle \mathbb {R} ^{L}}
the Engel function is known as the wealth expansion path , illustrating changes in the demand correspondence at various levels of wealth. The first derivative of the Engel function with respect to wealth for good
l
{\displaystyle l}
∂
x
l
(
p
¯
)
∂
w
{\displaystyle {\frac {\partial x_{l}({\bar {p}})}{\partial w}}}
is the wealth effect.
for normal goods the wealth effect is nonnegative,
∂
x
l
(
p
¯
,
w
)
∂
w
≥
0
{\displaystyle {\frac {\partial x_{l}({\bar {p}},w)}{\partial w}}\geq 0}
for inferior goods the wealth effect is negative,
∂
x
l
(
p
¯
,
w
)
∂
w
<
0
{\displaystyle {\frac {\partial x_{l}({\bar {p}},w)}{\partial w}}<0}
Price Effects
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For any two goods
l
,
k
{\displaystyle l,k}
the representation of
x
l
(
p
k
,
w
¯
)
{\displaystyle x_{l}(p_{k},{\bar {w}})}
across all prices
p
k
{\displaystyle p_{k}}
is the offer curve .
Define the price effect of good
k
{\displaystyle k}
on good
l
{\displaystyle l}
,
∂
x
l
(
p
k
,
w
¯
)
∂
p
k
{\displaystyle {\frac {\partial x_{l}(p_{k},{\bar {w}})}{\partial p_{k}}}}
Aggregation
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Homogeneity Results
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Engel Aggregation
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Cornout Aggregation
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