Ch 2
Consumer Choice
Consumer Choice
• 𝑙 = 1, 2, … . , L finite number of products
𝑥1
• 𝑥 = ⋮ , commodity vector
𝑥𝐿
• Let X⊂ ℝ𝐿 , a subset of ℝ𝐿 ⇒ consumption set
𝑝1
• 𝑝 = ⋮ be the price vector the consumer faces
𝑝𝐿
• ⇒ consumers are price takers (competitive buyers)
• ⇒ assume 𝑝 ≫ 0, i.e. 𝑝𝑙 , ∀𝑙, the price for each commodity
is positive.
Budget set
• Budget set:
– the set of affordable consumption bundles
– 𝐵𝑝,𝑤 = 𝑥 ∈ ℝ𝐿+ ∶ 𝑝𝑇 𝑥 ≤ 𝑤 (𝑜𝑟 𝑝 ∙ 𝑥 ≤ 𝑤) , where 𝑝𝑇 =
𝑝1 … 𝑝𝐿
• Remarks:
𝑥1
– 𝑝𝑇 𝑥 = 𝑝1 … 𝑝𝐿 ⋮ = 𝐿𝑙=1 𝑝𝑙 𝑥𝑙
𝑥𝐿
– Walrasian or Competitive budget set
Convexity
• Definition: The set 𝑋 ⊂
ℝ𝐿 is convex if for any
𝑥 ∈ 𝑋 and 𝑥′ ∈ 𝑋, we
have 𝑥 ′′ = 𝛼𝑥 +
1 − 𝛼 𝑥 ′ ∈ 𝑋, such
that x ′′ is also in the
set.
x‘’
x'
x
x
x‘’
x'
Convex Budget Set
• ⇒ 𝐵 𝑝,𝑤 is a convex set.
• (proof:)
– let 𝑥, 𝑥′ ∈ 𝐵 𝑝,𝑤
– By definition of 𝐵 𝑝,𝑤 , 𝑝𝑇 𝑥 ≤ 𝑤 & 𝑝𝑇 𝑥′ ≤ 𝑤
– let 𝑥′′ = 𝛼𝑥 + 1 − 𝛼 𝑥′
– 𝑝𝑇 𝑥 ′′ = 𝛼𝑝𝑇 𝑥 + 1 − 𝛼 𝑝𝑇 𝑥′ ≤ 𝑤
2.E Demand function & comparative Statics
• 𝑥 𝑝, 𝑤 : Walrasian demand correspondence
– The set of consumption bundles an individual
consumer likes most given 𝑝, 𝑤 .
– If single-valued → demand function
Homogeneous demand
• Definition: 𝑓 𝑡, 𝑎 is homogeneous of degree k in 𝑡, if
𝑓 𝛼𝑡, 𝑎 = 𝛼 𝑘 𝑓 𝑡, 𝑎 , ∀𝑡, 𝑎 & 𝛼 > 0
• Property: 𝑥 𝑝, 𝑤 is homogeneous of degree 0 in
𝑝 & 𝑤; that is, 𝑥 𝛼𝑝, 𝛼𝑤 = 𝛼 0 𝑥 𝑝, 𝑤 = 𝑥 𝑝, 𝑤 ,
∀𝑝, 𝑤 & 𝛼 > 0
– 𝐵𝑝,𝑤 : 𝑥: 𝑝𝑇 𝑥 ≤ 𝑤
– 𝐵𝛼𝑝,𝛼𝑤 : 𝑥: 𝛼𝑝 𝑇 𝑥 ≤ 𝛼𝑤 = 𝑥: 𝑝𝑇 𝑥 ≤ 𝑤 = 𝐵𝑝,𝑤
– the budget set remains the same if p & w experience the
same proportional change;
– 𝑥 𝛼𝑝, 𝛼𝑤 = 𝑥 𝑝, 𝑤 if 𝑥 𝑝, 𝑤 depends only on the
budget set.
Walras’ law
• Definition: 𝑥 𝑝, 𝑤 satisfies the Walras’ law if
for every 𝑝 ≫ 0 & 𝑤 > 0, 𝒑𝑻 𝒙 = 𝒘 , ∀𝑥 ∈
𝑥 𝑝, 𝑤 .
EXAMPLE
• 𝑥1 𝑝, 𝑤 =
• 𝑥2 𝑝, 𝑤 =
• 𝑥3 𝑝, 𝑤 =
𝑝2 𝑤
𝑝1 +𝑝2 +𝑝3 𝑝1
𝑝3 𝑤
𝑝1 +𝑝2 +𝑝3 𝑝2
𝛽𝑝1 𝑤
,
𝑝1 +𝑝2 +𝑝3 𝑝3
𝛽≤1
• 𝑥1 𝛼𝑝, 𝛼𝑤 = 𝑥1 𝑝, 𝑤
• 𝑥2 𝛼𝑝, 𝛼𝑤 = 𝑥2 𝑝, 𝑤
• 𝑥3 𝛼𝑝, 𝛼𝑤 = 𝑥3 𝑝, 𝑤 ,→ x(p,w) is h.d(0)
𝑝2 𝑤
𝑝1 +𝑝2 +𝑝3 𝑝1
𝑝1 +𝑝2 +𝛽𝑝3 𝑤
𝑝1 +𝑝2 +𝑝3
• 𝑝𝑇 𝑥 = 𝑝1
+ 𝑝2
𝑝3 𝑤
𝑝1 +𝑝2 +𝑝3 𝑝2
• x(p,w) satisfies W.L Iff 𝛽 = 1
+ 𝑝3
𝛽𝑝1 𝑤
𝑝1 +𝑝2 +𝑝3 𝑝3
=
Math front
• If 𝑓(𝑥): 𝑅𝐿 → 𝑅, is a
scalar function where
𝑥1
𝑥= ⋮ .
𝑥𝐿
𝜕𝑓(𝑥)
• 𝛻𝑓 𝑥 =
𝜕𝑥1
,
⋮
𝜕𝑓(𝑥)
𝜕𝑥𝐿
the gradient vector.
• EX: 𝑢 𝑥 = 3𝑥10.4 𝑥20.3
𝑥30.2 ,
– 𝛻𝑢 𝑥 =
1.2𝑥1−0.6 𝑥20.3 𝑥30.2
0.9𝑥10.4 𝑥2−0.7 𝑥30.2
0.6𝑥10.4 𝑥20.3 𝑥3−0.8
• EX: 𝐶 𝑥 = 𝑝 ∙ 𝑥
𝑝1
– 𝛻𝐶 𝑥 = 𝑝2
𝑝3
𝑓1 (𝑥)
⋮ : 𝑅𝐿 → 𝑅𝑀 , a
• 𝐼𝑓 𝑓 𝑥 =
𝑓𝑀 (𝑥)
vector function.
𝑥1 (𝑝, 𝑤)
⋮
– EX: 𝑥 𝑝, 𝑤 =
:
𝑥𝐿 (𝑝, 𝑤)
𝐿+1
𝐿
𝑅
→𝑅
𝜕𝑥1 (𝑝, 𝑤)
⋮
𝜕𝑥𝐿 (𝑝, 𝑤)
𝜕𝑥1 (𝑝,𝑤)
• 𝐷𝑥 𝑓(𝑥)=
𝜕𝑓1 (𝑥)
–
• EX:
• 𝐷𝑝 𝑥 𝑝, 𝑤 =
𝜕𝑥1 (𝑝, 𝑤)
𝜕𝑝1 ⋯
⋮
⋱
𝜕𝑥𝐿 (𝑝, 𝑤)
𝜕𝑝1 ⋯
𝜕𝑥1
⋮
𝜕𝑓𝑀 (𝑥)
𝜕𝑥1
⋯
⋱
⋯
𝜕𝑓1 (𝑥)
• 𝐷𝑤 𝑥 𝑝, 𝑤 =
𝜕𝑥𝐿
⋮
𝜕𝑓𝑀 (𝑥)
𝜕𝑥𝐿
𝜕𝑤
⋮
𝜕𝑥𝐿 (𝑝,𝑤)
𝜕𝑤
𝜕𝑝𝐿
𝜕𝑝𝐿
• If 𝑓(𝑥): 𝑅 𝐿 → 𝑅, the Hessian matrix 𝐷2 𝑓 𝑥 = 𝐷[𝛻𝑓 𝑥 ]
– EX: 𝑢 𝑥 = 3𝑥10.4 𝑥20.3 𝑥30.2
1.2𝑥1−0.6 𝑥20.3 𝑥30.2
– 𝛻𝑢 𝑥 = 0.9𝑥10.4 𝑥2−0.7 𝑥30.2
0.6𝑥10.4 𝑥20.3 𝑥3−0.8
– 𝐷2 𝑢 𝑥 = 𝐷 𝛻𝑢 𝑥 =
−0.72𝑥1−1.6 𝑥20.3 𝑥30.2 0.36𝑥1−0.6 𝑥2−0.7 𝑥30.2
0.36𝑥1−0.6 𝑥2−0.7 𝑥30.2 −0.63𝑥10.4 𝑥2−1.7 𝑥30.2
0.24𝑥1−0.6 𝑥20.3 𝑥3−0.8 0.18𝑥10.4 𝑥2−0.7 𝑥3−0.8
0.24𝑥1−0.6 𝑥20.3 𝑥3−0.8
0.18𝑥10.4 𝑥2−0.7 𝑥3−0.8
−0.48𝑥10.4 𝑥20.3 𝑥3−1.8
Comparative statics of x(p, w)
𝑥1 (𝑝, 𝑤)
⋮
• If demand function 𝑥 𝑝, 𝑤 =
is differentiable
𝑥𝐿 (𝑝, 𝑤)
𝜕𝑥1 𝑝, 𝑤 /𝜕𝑤
⋮
– 𝐷𝑤 𝑥 𝑝, 𝑤 =
, derivative matrix of
𝜕𝑥𝐿 𝑝, 𝑤 /𝜕𝑤
𝐿×1
𝑥 𝑝, 𝑤 with respect to w, the individual’s income.
– 𝐷𝑝 𝑥 𝑝, 𝑤 =
𝜕𝑥1
𝜕𝑝1
…
𝜕𝑥1
𝜕𝑝𝐿
, derivative matrix of 𝑥 𝑝, 𝑤
⋱
⋮
𝜕𝑥𝐿
𝜕𝑥𝐿
…
𝜕𝑝1
𝜕𝑝𝐿 𝐿×𝐿
with respect to p, the price vector.
⋮
• Chain Rule
• Suppose 𝑔(𝑥): 𝑅 𝐿 → 𝑅𝐾 and 𝑓(𝑥): 𝑅𝐾 → 𝑅𝑀 are
differentiable, and so is 𝑓(𝑔 𝑥 ). 𝐷𝑥 𝑓 𝑔(𝑥) =
𝐷𝑓 𝑔(𝑥 )𝐷𝑔 𝑥
– EX: 𝑓(𝛼𝑥)
• 𝐷𝛼 𝑓 𝛼𝑥 =𝐷𝛼𝑥 𝑓 𝛼𝑥 𝑥
𝑝
– EX: let 𝑦 =
, 𝑥(𝛼𝑝, 𝛼𝑤)=𝑥 𝛼𝑦 , (Note 𝑥 𝛼𝑦 and 𝑥 𝑦
𝑤
are of the same function, but evaluated at different value)
• 𝐷𝛼 𝑥 𝛼𝑦 =𝐷𝛼𝑦 𝑥 𝛼𝑦 𝑦 = 𝐷𝛼𝑝 𝑥(𝛼𝑝, 𝛼𝑤) 𝐷𝛼𝑤 𝑥(𝛼𝑝, 𝛼𝑤)
=𝐷𝛼𝑝 𝑥 𝛼𝑝, 𝛼𝑤 𝑝 + 𝐷𝛼𝑤 𝑥 𝛼𝑝, 𝛼𝑤 𝑤
𝑝
𝑤
• Product Rule
• Suppose 𝑓(𝑥): 𝑅𝐿 → 𝑅 with 𝑓 𝑥 = 𝑔 𝑥 ∙
ℎ 𝑥 , where 𝑔 ∙ and ℎ ∙ are both vectors.
• 𝐷𝑓 𝑥 = 𝑔 𝑥 ∙ 𝐷ℎ 𝑥 + ℎ 𝑥 ∙ 𝐷𝑔 𝑥
– EX: 𝑓 𝑝 = 𝑝 ∙ 𝑥 𝑝, 𝑤
– 𝐷𝑓 𝑝 = 𝑝 ∙ 𝐷𝑥 𝑝, 𝑤 + 𝑥 𝑝, 𝑤 ∙ 𝐷𝑝 =𝑝 ∙
𝐷𝑥 𝑝, 𝑤 + 𝑥 𝑝, 𝑤 ∙ 𝐼=𝑝 ∙ 𝐷𝑥 𝑝, 𝑤 + 𝑥 𝑝, 𝑤
Interchanging derivatives
• 𝛻𝑥 𝑓 𝑥 = 𝐷𝑥 𝑓(𝑥) 𝑇 if 𝑓(𝑥) is a real-valued
function. If 𝑓(𝑥) is a vector-valued function, note
that 𝐷𝑥 𝑓(𝑥) 𝑇 =𝛻𝑥 [𝑓 𝑥 𝑇 ]
• Chain Rule:
𝑇
– 𝛻𝑥 𝑓 𝑔(𝑥) = 𝐷𝑥 𝑓(𝑔(𝑥)) = 𝐷𝑔 𝑓(𝑔)𝐷𝑥 𝑔(𝑥)
𝑇
𝑇
𝐷𝑥 𝑔(𝑥)) 𝐷𝑔 𝑓(𝑔) =𝛻𝑥 𝑔 𝑥 𝛻𝑔 𝑓 𝑔
• Product Rule:
– 𝛻𝑥 𝑓 𝑥 𝑇 𝑔 𝑥 = 𝛻𝑥 [𝑔 𝑥 𝑇 𝑓 𝑥 ]𝑇 =
𝐷𝑥 [𝑔 𝑥 𝑇 𝑓 𝑥 ] 𝑇 = [𝑔 𝑥 𝑇 𝐷𝑥 𝑓 𝑥 +
𝑇
=
Implication of Walras’ law
• W. L. ⇒ 𝑝 ∙ 𝑥 𝑝, 𝑤 ≡ 𝑤,
• differentiating w.r.t. p
• 𝒑𝑻 𝑫𝒑 𝒙 𝒑, 𝒘 + 𝒙𝑻 𝒑, 𝒘 𝑰 = 𝟎𝑻 (2.E.5)
•
𝜕𝑥𝑙
𝐿
𝑙=1 𝑝𝑙 𝜕𝑝
𝑘
+ 𝑥𝑘 = 0, for 𝑘 = 1, … , 𝐿,
• Total expenditure cannot change in response
to a change in prices (Cournot aggregation).
(1) differentiate w.r.t. p
• 𝑝𝑇 𝐷𝑝 𝑥 𝑝, 𝑤 + 𝑥 𝑇 𝑝, 𝑤 = 0
• 𝑝1
𝑥1
• ⇒
…
𝑝𝐿
𝜕𝑥1
𝜕𝑝1
⋮
𝜕𝑥𝐿
𝜕𝑝1
…
⋱
…
… 𝑥𝐿 = 0
𝜕𝑥𝑙
𝐿
𝑙=1 𝑝𝑙 𝜕𝑝
𝑘
+ 𝑥𝑘 = 0
𝜕𝑥1
𝜕𝑝𝐿
+
⋮
𝜕𝑥𝐿
𝜕𝑝𝐿
∀ 𝑘 = 1,2, … , 𝐿
Implication of Walras’ law
(2) differentiate w.r.t. w
• 𝑝 ∙ 𝑥 𝑝, 𝑤 ≡ 𝑤, ⇒ 𝑝𝑇 𝐷𝑤 𝑥 𝑝, 𝑤 + 0 = 1
• 𝒑𝑻 𝑫𝒘 𝒙 𝒑, 𝒘 = 𝟏, (2.E.7)
•
𝑝1
… 𝑝𝐿
𝜕𝑥1 𝑝,𝑤
𝜕𝑤
⋮
𝜕𝑥𝐿 𝑝,𝑤
𝜕𝑤
=1⇒
𝜕𝑥𝑙 𝑝,𝑤
𝐿
𝑙=1 𝑝𝑙
𝜕𝑤
=1
• Engel aggregation, the total expenditure must change
by an amount equal to any income change
Property 𝑥 𝑝, 𝑤 → homogeneous of degree 0
•
•
•
•
𝑥 𝛼𝑝, 𝛼𝑤 ≡ 𝑥 𝑝, 𝑤
Differentiate w.r.t. 𝛼 ⇒ 𝐷𝑝 𝑥 𝛼𝑝, 𝛼𝑤 𝑝 + 𝐷𝑤 𝑥 𝛼𝑝, 𝛼𝑤 𝑤 = 0
Evaluate at 𝛼 = 1
⇒ 𝑫𝒑 𝒙 𝒑, 𝒘 𝒑 + 𝑫𝒘 𝒙 𝒑, 𝒘 𝒘 = 𝟎 (2.E.2)
𝜕𝑥1
𝜕𝑝1
•
⋮
𝜕𝑥𝐿
𝜕𝑝1
…
⋱
…
𝜕𝑥1
𝜕𝑝𝐿
⋮
𝜕𝑥𝐿
𝜕𝑝𝐿
𝐿×𝐿 × 𝐿×1
•
𝐿 𝜕𝑥𝑘
𝑙=1 𝜕𝑝 𝑝𝑙
𝑙
+
𝜕𝑥1
𝜕𝑤
𝑝1
⋮
𝑝𝐿 +
𝜕𝑥𝑘
𝑤
𝜕𝑤
⋮
𝜕𝑥𝐿
𝜕𝑤
𝑤=0
𝐿×1
=0
∀𝑘 = 1,2, … , 𝐿, (2.E.1)
• Price and wealth derivatives sum to zero when weighted by these
prices and wealth.
• If 𝑥 𝑝, 𝑤 is homogeneous of degree 0, then
for all p, w
• 𝐷𝑝 𝑥 𝑝, 𝑤 𝑝 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑤 = 0
•
•
𝜕𝑥𝑘 𝑤
𝐿 𝜕𝑥𝑘 𝑝𝑙
⇒ 𝑙=1
+
=0
𝜕𝑝𝑙 𝑥𝑘
𝜕𝑤 𝑥𝑘
𝜕𝑥𝑘 /𝑥𝑘
𝐿 𝜕𝑥𝑘 /𝑥𝑘
𝑙=1 𝜕𝑝 /𝑝 + 𝜕𝑤/𝑤 = 0
𝑙 𝑙
⇒ 𝐿𝑙=1 𝜀𝑘,𝑙 + 𝜀𝑘,𝑤 = 0 ∀𝑘
⇒
= 1,2, … , 𝐿,
Weak Axiom (WA)
• Definition: 𝑥 𝑝, 𝑤 satisfies the Weak Axiom of
revealed preference (WA) if the following holds
for any 𝑝′ , 𝑤 ′ & 𝑝′′ , 𝑤 ′′ :
• Suppose 𝒑′ ∙ 𝒙 𝒑′′ , 𝒘′′ ≤ 𝒘′ , 𝒂𝒏𝒅 𝒙 𝒑′ , 𝒘′ ≠
𝒙 𝒑′ ′, 𝒘′′ , then 𝒑′′ ∙ 𝒙 𝒑′ , 𝒘′ > 𝒘′.
• Suppose 𝑥 𝑝′ , 𝑤 ′ & 𝑥 𝑝′ ′, 𝑤 ′′ are both
affordable under 𝑝′ , 𝑤 ′ , 𝑏𝑢𝑡 𝑥 𝑝′ , 𝑤 ′ is chosen,
instead of 𝑥 𝑝′ ′, 𝑤 ′′ , then 𝑥 𝑝′ , 𝑤 ′ is not
affordable under 𝑝′′ , 𝑤′′ ; (If the consumer has a
stable preference)
𝒙𝟐
• Continuous on
definition of WA
• If 𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ ≤ 𝑤′
then 𝑝′′ ∙ 𝑥 𝑝′ , 𝑤 ′ >
𝑤′′
𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ ≤ 𝑤′
• ⇒ ′′
𝑝 ∙ 𝑥 𝑝′ , 𝑤 ′ ≤ 𝑤′′
will exist just one
𝒙(𝒑′′ , 𝒘′′ )
𝒙(𝒑′ , 𝒘′ )
𝒙𝟏
Property 2.F.1
• Suppose 𝑥 𝑝, 𝑤 is homogeneous of degree 0 & it satisfies
Walras’ Law, then 𝑥 𝑝, 𝑤 satisfies WA if and only if the
following holds:
• For any compensated price change from an initial situation
𝑝′ , 𝑤 ′ to 𝑝′′ , 𝑤 ′′ = 𝑝′′ , 𝑝′′ ∙ 𝑥 𝑝′ , 𝑤 ′ , we have
• 𝑝′′ − 𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ − 𝑥 𝑝′ , 𝑤 ′ ≤ 0 ⨂
– < 0 if 𝑥 𝑝′′ , 𝑤 ′′ ≠ 𝑥 𝑝′ , 𝑤 ′
• Compensated Law of Demand, (CLD)
– 𝑊𝐴 ↔ 𝐶𝐿𝐷
– The (uncompensated) law of demand (ULD) is a stronger
assumption!!!
Proof
1.
WA ⇒⨂
–
–
–
–
2.
𝑝′′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ − 𝑥 𝑝′ , 𝑤 ′ − 𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ − 𝑥 𝑝′ , 𝑤 ′
𝑝′′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ − 𝑝′′ ∙ 𝑥 𝑝′ , 𝑤 ′ − 𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ + 𝑝′ ∙ 𝑥 𝑝′ , 𝑤 ′ < 0
Because 𝑝′′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ = 𝑝′′ ∙ 𝑥 𝑝′ , 𝑤 ′ = 𝑤 ′′ by WL
𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ − 𝑝′ ∙ 𝑥 𝑝′ , 𝑤 ′ > 0 by WA
⨂⇒ WA
– First note WA holds for any change from (𝑝′ , 𝑤 ′ ) to (𝑝′′ , 𝑤 ′′ ) if and
only if WA holds for any compensated price change from (𝑝′ , 𝑤 ′ ) to
𝑝′′ , 𝑝′′ ∙ 𝑥 𝑝′ , 𝑤 ′ ,
– −𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ + 𝑝′ ∙ 𝑥 𝑝′ , 𝑤 ′ < 0 for a compensated price change
𝑝′′ ∙ 𝑥 𝑝′ , 𝑤 ′ = 𝑤 ′′ if 𝑥 𝑝′′ , 𝑤 ′′ ≠ 𝑥 𝑝′ , 𝑤 ′
– 𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ ≤ 𝑝′ ∙ 𝑥 𝑝′ , 𝑤 ′ = 𝑤 ′ if WA doesn’t hold
– It contradicts ⨂ by using 𝑝′′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ = 𝑝′′ ∙ 𝑥 𝑝′ , 𝑤 ′
CLD: a differentiable version
• For 𝑝′ → 𝑝′′ , 𝑤 ′′ = 𝑝′′ ∙ 𝑥 𝑝′ , 𝑤 ′
• CLD ⇒ 𝑝′′ − 𝑝′ ∙ 𝑥 𝑝′′ , 𝑤 ′′ − 𝑥 𝑝′ , 𝑤 ′ ≤ 0
• For differentiable 𝑥 𝑝, 𝑤 , and 𝑑𝑤 = 𝑥 𝑇 𝑑𝑝 ⇒ 𝑑𝑝𝑇 𝑑𝑥 ≤ 0
• 𝑥 ≡ 𝑥 𝑝, 𝑤
• take a total differentiation
• 𝑑𝑥 = 𝐷𝑝 𝑥 𝑝, 𝑤 𝑑𝑝 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑑𝑤 = 𝐷𝑝 𝑥 𝑝, 𝑤 𝑑𝑝 +
𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 𝑑𝑝 = 𝐷𝑝 𝑥 𝑝, 𝑤 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 𝑑𝑝
• CLD (or WA) ⇒ 𝑑𝑝𝑇 𝐷𝑝 𝑥 𝑝, 𝑤 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 𝑑𝑝 ≤ 0
• Let 𝑆 𝑝, 𝑤 = 𝐷𝑝 𝑥 𝑝, 𝑤 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 , Slutsky
(substitution) matrix
𝜕𝑥1
𝜕𝑝1
• 𝓁
⋮
𝜕𝑥𝐿
𝜕𝑝1
…
⋱
…
𝜕𝑥1
𝜕𝑝𝐿
𝜕𝑥1
𝜕𝑤
⋮
⋮
𝜕𝑥𝐿
𝜕𝑝𝐿
+
𝑥1
…
𝑥𝐿
𝜕𝑥𝐿
𝜕𝑤
𝓀
• 𝑆𝓁𝓀 =
𝜕𝑥𝓁
𝜕𝑝𝓀
+
𝜕𝑥𝓁
𝑥
𝜕𝑤 𝓀
• 𝑆 𝑝, 𝑤 : if WA (CLD) holds,
• ⇒ 𝑆 𝑝, 𝑤 is negative semidefinite
• ⇒ 𝑆𝓁𝓁 𝑝, 𝑤 ≤ 0 (𝑓11 < 0, 𝑓22 < 0, … )
Remarks:
•
𝜕𝑥𝑙
𝜕𝑝𝑙
+
– if
𝜕𝑥𝑙
𝑥𝑙
𝜕𝑤
𝜕𝑥𝑙
𝜕𝑝𝑙
≤0
≥ 0 (very unusual), then
𝜕𝑥𝑙
𝜕𝑤
≤0
– if good 𝑙 is a Giffen good, then 𝑙 must be an
inferior good
• ⇒ 𝑆 𝑝, 𝑤 is symmetric for L = 2, but not
for L ≥ 3
Property 2.F.3
• Suppose 𝑥 𝑝, 𝑤 is differentiable,
homogeneous of degree 0 in 𝑝, 𝑤 & it
satisfies Walras’ Law, then
• 1 𝑆 𝑝, 𝑤 𝑝 = 0
• 2 𝑝𝑇 𝑆 𝑝, 𝑤 = 0
Proof
• (1) 𝑥 𝑝, 𝑤 is homogeneous of degree 0
• 𝑥 𝛼𝑝, 𝛼𝑤 ≡ 𝑥 𝑝, 𝑤
• ⇒ 𝐷𝑝 𝑥 𝑝, 𝑤 𝑝 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑤 = 0 (by Property 2.E.1)
• ∴𝑆 𝑝, 𝑤 𝑝
• = 𝐷𝑝 𝑥 𝑝, 𝑤 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 𝑝
• = 𝐷𝑝 𝑥 𝑝, 𝑤 𝑝 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 𝑝
• = −𝐷𝑤 𝑥 𝑝, 𝑤 𝑤 + 𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 𝑝 = 0
• Note 𝑆 𝑝, 𝑤 𝑝 indeed is 2.E.2 if WL holds so that
𝑝𝑇 𝑥 𝑝, 𝑤 = 𝑤
Proof Cont.
•
•
•
•
•
•
•
•
(2) From WL, 𝑝𝑇 𝑥 𝑝, 𝑤 = 𝑤
Taking differentiation w.r.t. p
𝑝𝑇 𝐷𝑝 𝑥 𝑝, 𝑤 + 𝑥 𝑇 𝑝, 𝑤 = 0𝑇 (by Property 2.E.2)
∴𝑝𝑇 𝑆 𝑝, 𝑤 = 𝑝𝑇 𝐷𝑝 𝑥 𝑝, 𝑤 + 𝑝𝑇 𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 𝑝, 𝑤
= −𝑥 𝑇 𝑝, 𝑤 + 𝑝𝑇 𝐷𝑤 𝑥 𝑝, 𝑤 𝑥 𝑇 𝑝, 𝑤
Taking differentiation of 𝑝𝑇 𝑥 𝑝, 𝑤 = 𝑤 w.r.t. w
𝑝𝑇 𝐷𝑤 𝑥 𝑝, 𝑤 = 1
∴𝑝𝑇 𝑆 𝑝, 𝑤 = −𝑥 𝑇 𝑝, 𝑤 + 𝑥 𝑇 𝑝, 𝑤 = 0
Note 𝑝𝑇 𝑆 𝑝, 𝑤 is indeed a combination of 2.E.5 &
2.E.7