Lectures on
General Equilibrium Theory
⋆ ⋆ ⋆
Demand aggregation and the structure of equilibrium
John Quah
[email protected]
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 1/20
Weak Axiom of Revealed Preference
l
A function F : R++
→ Rl satisfies the weak axiom of revealed
preference if at any p and p′ with F (p) 6= F (p′ ), the following holds:
p′ · F (p) ≤ p′ · F (p′ ) =⇒ p · F (p′ ) > p · F (p).
In the case where F (p) = x̄a (p, w) (the demand of agent a), WARP
says that at any p and p′ with x̄a (p, w) 6= x̄(p′ , w), the following holds:
p′ · x̄a (p, w) ≤ p′ · x̄a (p′ , w) = w =⇒ p · x̄a (p′ , w) > p · x̄a (p, w) = w.
Equivalently, x̄a (p, w) ∈ B(p′ , w) =⇒ x̄a (p′ , w′ ) ∈
/ B(p, w).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 2/20
Weak Axiom of Revealed Preference
l
A function F : R++
→ Rl satisfies the weak axiom of revealed
preference if at any p and p′ with F (p) 6= F (p′ ), the following holds:
p′ · F (p) ≤ p′ · F (p′ ) =⇒ p · F (p′ ) > p · F (p).
In the case where F (p) = x̄a (p, w) (the demand of agent a), WARP
says that at any p and p′ with x̄a (p, w) 6= x̄(p′ , w), the following holds:
p′ · x̄a (p, w) ≤ p′ · x̄a (p′ , w) = w =⇒ p · x̄a (p′ , w) > p · x̄a (p, w) = w.
Equivalently, x̄a (p, w) ∈ B(p′ , w) =⇒ x̄a (p′ , w′ ) ∈
/ B(p, w).
p
p
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 2/20
Weak Axiom of Revealed Preference
p
p
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 3/20
Weak Axiom of Revealed Preference
p
p
Proposition: Agent a’s demand function obeys the weak axiom of
revealed preference if a is utility-maximizing.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 3/20
Weak Axiom of Revealed Preference
p
p
Proposition: Agent a’s demand function obeys the weak axiom of
revealed preference if a is utility-maximizing.
Proof: If p′ · x̄a (p, w) ≤ w then xa (p, w) is in B(p′ , w). But x̄a (p, w) is not
the demand at (p′ , w) so
U (x̄a (p′ , w)) > U (x̄a (p, w)).
Thus x̄a (p′ , w) ∈
/ B(p, w); otherwise it would be chosen over x̄a (p, w).
QED
In other words, p · x̄a (p′ , w) > w.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 3/20
Weak Axiom of Revealed Preference
Corollary: Agent a’s excess demand function z a obeys the weak axiom
of revealed preference: at prices p and p′ , if z a (p) 6= z a (p′ ), then
p′ · z a (p) ≤ 0 =⇒ p · z a (p′ ) > 0.
p
p
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 4/20
Weak Axiom of Revealed Preference
Corollary: Agent a’s excess demand function z a obeys the weak axiom
of revealed preference: at prices p and p′ , if z a (p) 6= z a (p′ ), then
p′ · z a (p) ≤ 0 =⇒ p · z a (p′ ) > 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 5/20
Weak Axiom of Revealed Preference
Corollary: Agent a’s excess demand function z a obeys the weak axiom
of revealed preference: at prices p and p′ , if z a (p) 6= z a (p′ ), then
p′ · z a (p) ≤ 0 =⇒ p · z a (p′ ) > 0.
Proof: Choose λ > 0 such that p′′ = λp′ satisfies p · ω a = p′′ · ω a = w.
Since z a (p) = x̄a (p, p · ω a ) − ω a , we may re-write p′ · z a (p) ≤ 0 as
p′ · x̄a (p, p · ω a ) ≤ p′ · ω a ; equivalently,
p′′ · x̄a (p, w) ≤ p′′ · ω a = w.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 5/20
Weak Axiom of Revealed Preference
Corollary: Agent a’s excess demand function z a obeys the weak axiom
of revealed preference: at prices p and p′ , if z a (p) 6= z a (p′ ), then
p′ · z a (p) ≤ 0 =⇒ p · z a (p′ ) > 0.
Proof: Choose λ > 0 such that p′′ = λp′ satisfies p · ω a = p′′ · ω a = w.
Since z a (p) = x̄a (p, p · ω a ) − ω a , we may re-write p′ · z a (p) ≤ 0 as
p′ · x̄a (p, p · ω a ) ≤ p′ · ω a ; equivalently,
p′′ · x̄a (p, w) ≤ p′′ · ω a = w.
Since x̄a obeys the weak axiom,
p · x̄a (p′′ , w) > w = p · ω a .
Re-write this as p · z a (p′′ ) > 0. Since z a (p′′ ) = z a (p′ ), we obtain
p · z a (p′ ) > 0.
Lectures onGeneral Equilibrium Theory⋆
QED
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 5/20
Structure of excess demand function Z
What is the structure of Z(p) =
P
a∈A
z a (p)?
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 6/20
Structure of excess demand function Z
What is the structure of Z(p) =
Hicks’s example
P
a∈A
z a (p)?
p
a
b
p
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 6/20
Structure of excess demand function Z
What is the structure of Z(p) =
Hicks’s example
P
a∈A
z a (p)?
p
a
b
p
The aggregate excess demand function Z need not obey the weak
axiom.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 6/20
Structure of excess demand function Z
Recall (from earlier theorem) that the excess demand function
l
Z : R++
→ Rl of the economy has the following properties: it is
zero-homogenous, it obeys Walras’ Law, it is continuous, it satisfies the
boundary condition, and it is bounded below.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 7/20
Structure of excess demand function Z
Recall (from earlier theorem) that the excess demand function
l
Z : R++
→ Rl of the economy has the following properties: it is
zero-homogenous, it obeys Walras’ Law, it is continuous, it satisfies the
boundary condition, and it is bounded below.
Can anything more be said about Z?
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 7/20
Structure of excess demand function Z
Recall (from earlier theorem) that the excess demand function
l
Z : R++
→ Rl of the economy has the following properties: it is
zero-homogenous, it obeys Walras’ Law, it is continuous, it satisfies the
boundary condition, and it is bounded below.
Can anything more be said about Z?
Indeterminacy Theorem (Sonnenschein-Mantel-Debreu): Let P be a
l
compact set in R++
and let S : P → Rl be a function with the following
properties: it is zero-homogenous, it obeys Walras’ Law, and it is
continuous.
There is an exchange economy of agents with utility functions obeying
(P1), (P2), and (P3) such that its excess demand function
l
Z : R++
→ Rl satisfies
Z(p) = S(p) for all p ∈ P .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 7/20
Structure of excess demand function Z
Corollary: Multiple equilibria and unstable equilibria are possible.
(1,
Z (1, )
Z (1, )
p
2
Lectures onGeneral Equilibrium Theory⋆
p
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 8/20
2
Demand Aggregation
Maybe it isn’t so bad after all... there is structure to Z if stronger
restrictions are imposed on utility functions and endowments.
Two types of aggregate structure on Z widely studied:
gross substitutability and the weak axiom.
We shall examine simple conditions for the latter property.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 9/20
Homothetic Preferences
A preference is homothetic if x x′ implies that λx λx′ where
λ > 0.
A demand function x̄ is linear in income if x̄(p, λw) = λx̄(p, w) for any
λ > 0.
Proposition: A homothetic preference has a demand function that is
linear in income.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 10/20
Homothetic Preferences
A preference is homothetic if x x′ implies that λx λx′ where
λ > 0.
A demand function x̄ is linear in income if x̄(p, λw) = λx̄(p, w) for any
λ > 0.
Proposition: A homothetic preference has a demand function that is
linear in income.
Proof: Let y = x̄(p, w). If x′ ∈ B(p, λw) then x′ /λ ∈ B(p, w).
So y x′ /λ.
By the homotheticity of , we obtain λy x′ .
Therefore, λy = x̄(p, λw).
QED
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 10/20
Demand Aggregation: example 1
Theorem: Suppose that all agents in E have the same utility function
and thus the same demand function x̄, with x̄(p, w) = x̄(p, 1)w, i.e.,
demand is linear in income. Then Z obeys the weak axiom.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 11/20
Demand Aggregation: example 1
Theorem: Suppose that all agents in E have the same utility function
and thus the same demand function x̄, with x̄(p, w) = x̄(p, 1)w, i.e.,
demand is linear in income. Then Z obeys the weak axiom.
P
Proof: Denote the economy’s aggregate endowment a∈A ω a by ω̄.
P
Then X(p) = a∈A x̄(p, p · ω a )
P
= a∈A x̄(p, 1)(p · ω a )
P
a
= x̄(p, 1)
a∈A (p · ω )
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 11/20
Demand Aggregation: example 1
Theorem: Suppose that all agents in E have the same utility function
and thus the same demand function x̄, with x̄(p, w) = x̄(p, 1)w, i.e.,
demand is linear in income. Then Z obeys the weak axiom.
P
Proof: Denote the economy’s aggregate endowment a∈A ω a by ω̄.
P
Then X(p) = a∈A x̄(p, p · ω a )
P
= a∈A x̄(p, 1)(p · ω a )
P
a
= x̄(p, 1)
a∈A (p · ω )
P
a
= x̄(p, 1) p · ( a∈A ω )
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 11/20
Demand Aggregation: example 1
Theorem: Suppose that all agents in E have the same utility function
and thus the same demand function x̄, with x̄(p, w) = x̄(p, 1)w, i.e.,
demand is linear in income. Then Z obeys the weak axiom.
P
Proof: Denote the economy’s aggregate endowment a∈A ω a by ω̄.
P
Then X(p) = a∈A x̄(p, p · ω a )
P
= a∈A x̄(p, 1)(p · ω a )
P
a
= x̄(p, 1)
a∈A (p · ω )
P
a
= x̄(p, 1) p · ( a∈A ω )
P
= x̄(p, p · ( a∈A ω a ))
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 11/20
Demand Aggregation: example 1
Theorem: Suppose that all agents in E have the same utility function
and thus the same demand function x̄, with x̄(p, w) = x̄(p, 1)w, i.e.,
demand is linear in income. Then Z obeys the weak axiom.
P
Proof: Denote the economy’s aggregate endowment a∈A ω a by ω̄.
P
Then X(p) = a∈A x̄(p, p · ω a )
P
= a∈A x̄(p, 1)(p · ω a )
P
a
= x̄(p, 1)
a∈A (p · ω )
P
a
= x̄(p, 1) p · ( a∈A ω )
P
= x̄(p, p · ( a∈A ω a ))
= x̄(p, p · ω̄).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 11/20
Demand Aggregation: example 1
Theorem: Suppose that all agents in E have the same utility function
and thus the same demand function x̄, with x̄(p, w) = x̄(p, 1)w, i.e.,
demand is linear in income. Then Z obeys the weak axiom.
P
Proof: Denote the economy’s aggregate endowment a∈A ω a by ω̄.
P
Then X(p) = a∈A x̄(p, p · ω a )
P
= a∈A x̄(p, 1)(p · ω a )
P
a
= x̄(p, 1)
a∈A (p · ω )
P
a
= x̄(p, 1) p · ( a∈A ω )
P
= x̄(p, p · ( a∈A ω a ))
= x̄(p, p · ω̄).
so economy’s aggregate demand behaves like the demand of an agent
with endowment ω̄.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 11/20
Demand Aggregation: example 1
Theorem: Suppose that all agents in E have the same utility function
and thus the same demand function x̄, with x̄(p, w) = x̄(p, 1)w, i.e.,
demand is linear in income. Then Z obeys the weak axiom.
P
Proof: Denote the economy’s aggregate endowment a∈A ω a by ω̄.
P
Then X(p) = a∈A x̄(p, p · ω a )
P
= a∈A x̄(p, 1)(p · ω a )
P
a
= x̄(p, 1)
a∈A (p · ω )
P
a
= x̄(p, 1) p · ( a∈A ω )
P
= x̄(p, p · ( a∈A ω a ))
= x̄(p, p · ω̄).
so economy’s aggregate demand behaves like the demand of an agent
with endowment ω̄. Therefore, its excess demand function
QED
Z(p) = x̄(p, p · ω̄) − ω̄ obeys the weak axiom.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 11/20
The law of demand
Let O ⊆ Rl ; F : O → Rl is monotonic (obeys the law of demand) if
(p − p′ ) · (F (p) − F (p′ )) ≤ 0 for any p and p′ in O.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 12/20
The law of demand
Let O ⊆ Rl ; F : O → Rl is monotonic (obeys the law of demand) if
(p − p′ ) · (F (p) − F (p′ )) ≤ 0 for any p and p′ in O.
Suppose p = (p1 , p2 , ..., pl ) and p′ = (p′1 , p2 , ..., pl ), then if F obeys the
law of demand, we obtain
(p1 − p′1 )(F1 (p) − F1 (p′ )) ≤ 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 12/20
The law of demand
Let O ⊆ Rl ; F : O → Rl is monotonic (obeys the law of demand) if
(p − p′ ) · (F (p) − F (p′ )) ≤ 0 for any p and p′ in O.
Suppose p = (p1 , p2 , ..., pl ) and p′ = (p′1 , p2 , ..., pl ), then if F obeys the
law of demand, we obtain
(p1 − p′1 )(F1 (p) − F1 (p′ )) ≤ 0.
Proposition: Let O be an open and convex set in Rl and suppose that
F : O → Rl is a differentiable function defined on O . Then F obeys
the law of demand if and only if its derivative matrix ∂p F (p) is negative
semidefinite for all p ∈ O.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 12/20
The law of demand
Let O ⊆ Rl ; F : O → Rl is monotonic (obeys the law of demand) if
(p − p′ ) · (F (p) − F (p′ )) ≤ 0 for any p and p′ in O.
Suppose p = (p1 , p2 , ..., pl ) and p′ = (p′1 , p2 , ..., pl ), then if F obeys the
law of demand, we obtain
(p1 − p′1 )(F1 (p) − F1 (p′ )) ≤ 0.
Proposition: Let O be an open and convex set in Rl and suppose that
F : O → Rl is a differentiable function defined on O . Then F obeys
the law of demand if and only if its derivative matrix ∂p F (p) is negative
semidefinite for all p ∈ O.
The (Marshallian) demand function x̄ does not generally obey the law
of demand, but the law of demand generally holds for the Hicksian
demand function.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 12/20
The law of demand
Given a price vector p ≫ 0 and a utility function u, consider the
following problem:
(⋆) minimize p · x subject to u(x) ≥ û.
Suppose, that u is continuous, monotone, and strictly quasiconcave.
Then, so long as û is in the range of u, the problem (⋆) admits a unique
solution x∗ , with u(x∗ ) = û.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 13/20
The law of demand
Given a price vector p ≫ 0 and a utility function u, consider the
following problem:
(⋆) minimize p · x subject to u(x) ≥ û.
Suppose, that u is continuous, monotone, and strictly quasiconcave.
Then, so long as û is in the range of u, the problem (⋆) admits a unique
solution x∗ , with u(x∗ ) = û.
l
}.
Denote the range of u by U , i.e., U = {u(x) : x ∈ R+
l
For any (p, u) ∈ R++
× U , we denote the solution to (⋆) by h(p, u).
The function h is known as the Hicksian demand function.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 13/20
The law of demand
Given a price vector p ≫ 0 and a utility function u, consider the
following problem:
(⋆) minimize p · x subject to u(x) ≥ û.
Suppose, that u is continuous, monotone, and strictly quasiconcave.
Then, so long as û is in the range of u, the problem (⋆) admits a unique
solution x∗ , with u(x∗ ) = û.
l
}.
Denote the range of u by U , i.e., U = {u(x) : x ∈ R+
l
For any (p, u) ∈ R++
× U , we denote the solution to (⋆) by h(p, u).
The function h is known as the Hicksian demand function.
The map from (p, u) to e(p, w) = p · h(p, u) is called the expenditure
function.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 13/20
The law of demand
Proposition: h obeys the law of demand.
Proof: By definition of h, p′ · h(p′ , u) ≤ p′ · h(p, u) and
p · h(p, u) ≤ p · h(p′ , u).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 14/20
The law of demand
Proposition: h obeys the law of demand.
Proof: By definition of h, p′ · h(p′ , u) ≤ p′ · h(p, u) and
p · h(p, u) ≤ p · h(p′ , u). This may re-written as
−p′ · (h(p, u) − h(p′ , u)) ≤ 0 and
p · (h(p, u) − h(p′ , u)) ≤ 0.
Summing these inequalities we obtain
(p − p′ ) · (h(p, u) − h(p′ , u)) ≤ 0.
.
QED
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 14/20
The law of demand
Proposition: h obeys the law of demand.
Proof: By definition of h, p′ · h(p′ , u) ≤ p′ · h(p, u) and
p · h(p, u) ≤ p · h(p′ , u). This may re-written as
−p′ · (h(p, u) − h(p′ , u)) ≤ 0 and
p · (h(p, u) − h(p′ , u)) ≤ 0.
Summing these inequalities we obtain
(p − p′ ) · (h(p, u) − h(p′ , u)) ≤ 0.
.
QED
The Marshallian demand x̄ obeys the law of demand if
(p − p′ ) · (x̄(p, w) − x̄(p′ , w)) ≤ 0
at any two price vectors p and p′ , and at any fixed income w.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 14/20
The law of demand
The Marshallian demand function, x̄, obeys that law of demand if and
only if the derivative matrix ∂p x̄(p, w) is negative semidefinite.
By the Slutsky decomposition,
∂p x̄(p, w) = S(p, w) − I(p, w),
where S(p, w) and I(p, w) are the substitution and income effect
matrices.
Recall that S(p, w) = ∂p h(p, v(p, w)), where v(p, w) ≡ u(x̄(p, w)) is the
indirect utility function.
S is negative semidefinite since h obeys the law of demand.
But I may not be positive semidefinite; hence the negative
semidefiniteness of ∂p x̄(p, w) is not guaranteed.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 15/20
The law of demand
Proposition: I is positive semidefinite, and hence x̄ obeys the law of
demand, if x̄ is linear in income.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 16/20
The law of demand
Proposition: I is positive semidefinite, and hence x̄ obeys the law of
demand, if x̄ is linear in income.
Proof: The ij entry of the matrix I(p, w) is
∂ x̄j
(p, w) x̄i (p, w).
∂w
So I(p, w) = B T A, where B = (x̄1 (p, w), x̄2 (p, w), ..., x̄l (p, w)) and
∂ x̄2
∂ x̄l
∂ x̄1
(p, w),
(p, w), ...,
(p, w) .
A=
∂w
∂w
∂w
If x̄(p, w) = x̄(p, 1)w, then
v ∈ Rl ,
∂ x̄i
∂w (p, w)
= x̄i (p, 1). For any column vector
2
v · I(p, w)v = w (v · x̄(p, 1)) ≥ 0.
Since S(p, w) is negative semidefinite, we conclude that ∂p x̄(p, w),
which equals S(p, w) − I(p, w) is a negative semidefinite matrix. QED
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 16/20
The law of demand
Theorem: (Milleron, Mitjuschin-Polterovich) Suppose u is increasing,
strictly quasiconcave, and concave. Then u generates a demand
function x̄ obeying the law of demand if
x · ∂ 2 u(x)x
−
≤ 4.
∂u(x)x
If the inequality is strict, then we obtain the strict law of demand, i.e.,
(p − p′ ) · (x̄(p, w) − x̄(p, w)) < 0 whenever p 6= p′ .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 17/20
The law of demand
Theorem: (Milleron, Mitjuschin-Polterovich) Suppose u is increasing,
strictly quasiconcave, and concave. Then u generates a demand
function x̄ obeying the law of demand if
x · ∂ 2 u(x)x
−
≤ 4.
∂u(x)x
If the inequality is strict, then we obtain the strict law of demand, i.e.,
(p − p′ ) · (x̄(p, w) − x̄(p, w)) < 0 whenever p 6= p′ .
Pl
Corollary: Suppose u(x) = i=1 vi (xi ). Then its demand function
obeys the law of demand if, for all i,
tvi′′ (t)
≤ 4.
− ′
vi (t)
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 17/20
Demand Aggregation
Why the fuss about the law of demand?
Observation 1: If a function F obeys the strict law of demand then it
obeys the weak axiom.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 18/20
Demand Aggregation
Why the fuss about the law of demand?
Observation 1: If a function F obeys the strict law of demand then it
obeys the weak axiom.
Observation 2: The law of demand is preserved by aggregation.
Suppose
F (p) =
X
x̄a (p, wa ),
a∈A
i.e., F (p) is market demand at price p. Then if x̄a obeys the (strict) law
of demand for all a ∈ A, F also obeys the (strict) law of demand.
Hence F obeys the weak axiom if x̄a obeys the strict law of demand
(for all agents a).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 18/20
Demand Aggregation: example 2
Consider an exchange economy with collinear endowments, i..e,
P
a
a
a
ω = t ω̄, with t > 0 and a∈A ta = 1.
Suppose that x̄a obeys the (strict) law of demand. Consider two
distinct prices p and p′ with p · ω̄ = p′ · ω̄. Then
(p − p′ ) · [x̄a (p, ta p · ω̄) − x̄a (p′ , ta p′ · ω̄)] ≤ (<) 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 19/20
Demand Aggregation: example 2
Consider an exchange economy with collinear endowments, i..e,
P
a
a
a
ω = t ω̄, with t > 0 and a∈A ta = 1.
Suppose that x̄a obeys the (strict) law of demand. Consider two
distinct prices p and p′ with p · ω̄ = p′ · ω̄. Then
(p − p′ ) · [x̄a (p, ta p · ω̄) − x̄a (p′ , ta p′ · ω̄)] ≤ (<) 0.
Summing across all agents we obtain
(p − p′ ) · (X(p) − X(p′ )) ≤ (<) 0
and thus Z obeys the restricted (strict) law of demand:
(p − p′ ) · (Z(p) − Z(p′ )) ≤ (<) 0 if p 6= p′ and p · ω̄ = p′ · ω̄.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 19/20
Demand Aggregation: example 2
Proposition: Suppose Z obeys the restricted strict law of demand.
Then Z obeys the weak axiom.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 20/20
Demand Aggregation: example 2
Proposition: Suppose Z obeys the restricted strict law of demand.
Then Z obeys the weak axiom.
Proof: Recall that Z obeys the weak axiom if at any p and p′ , with
Z(p) 6= Z(p′ ),
p′ · Z(p) ≤ 0 =⇒ p · Z(p′ ) > 0.
Given p and p′ , choose λ such that p′′ = λp′ satisfies p · ω̄ = p′′ · ω̄.
Then, since Z obeys the restricted strict law of demand,
(p − p′′ ) · (Z(p) − Z(p′′ )) < 0.
This can re-written as −p′′ · Z(p) − p · Z(p′′ ) < 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 20/20
Demand Aggregation: example 2
Proposition: Suppose Z obeys the restricted strict law of demand.
Then Z obeys the weak axiom.
Proof: Recall that Z obeys the weak axiom if at any p and p′ , with
Z(p) 6= Z(p′ ),
p′ · Z(p) ≤ 0 =⇒ p · Z(p′ ) > 0.
Given p and p′ , choose λ such that p′′ = λp′ satisfies p · ω̄ = p′′ · ω̄.
Then, since Z obeys the restricted strict law of demand,
(p − p′′ ) · (Z(p) − Z(p′′ )) < 0.
This can re-written as −p′′ · Z(p) − p · Z(p′′ ) < 0.
Suppose p′ · Z(p) ≤ 0; then p′′ · Z(p) ≤ 0. This implies that
p · Z(p′′ ) > 0.
Since Z is zero-homogeneous, we obtain p · Z(p′ ) > 0.
Lectures onGeneral Equilibrium Theory⋆
QED
⋆ ⋆Demand aggregation and the structure of equilibrium – p. 20/20