Lectures on
General Equilibrium Theory
⋆ ⋆ ⋆
The existence of equilibrium
John Quah
[email protected]
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 1/21
Notation
The set
Rl = {x = (x1 , x2 , ..., xl ) : xi ∈ R}
is known as the Euclidean space. We denote
l
R+
=
{x ∈ Rl : xi ≥ 0 ∀ i}
l
R++
=
{x ∈ Rl : xi > 0 ∀ i}
l
as the positive orthant.
We sometimes refer to R+
For vectors x and y in Rl , we say
x ≥ y if xi ≥ yi for all i and
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 2/21
Notation
The set
Rl = {x = (x1 , x2 , ..., xl ) : xi ∈ R}
is known as the Euclidean space. We denote
l
R+
=
{x ∈ Rl : xi ≥ 0 ∀ i}
l
R++
=
{x ∈ Rl : xi > 0 ∀ i}
l
as the positive orthant.
We sometimes refer to R+
For vectors x and y in Rl , we say
x ≥ y if xi ≥ yi for all i and x > y if xi ≥ yi for all i and x 6= y.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 2/21
Notation
The set
Rl = {x = (x1 , x2 , ..., xl ) : xi ∈ R}
is known as the Euclidean space. We denote
l
R+
=
{x ∈ Rl : xi ≥ 0 ∀ i}
l
R++
=
{x ∈ Rl : xi > 0 ∀ i}
l
as the positive orthant.
We sometimes refer to R+
For vectors x and y in Rl , we say
x ≥ y if xi ≥ yi for all i and x > y if xi ≥ yi for all i and x 6= y.
Finally, x ≫ y if xi > yi for all i.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 2/21
Notation
The set
Rl = {x = (x1 , x2 , ..., xl ) : xi ∈ R}
is known as the Euclidean space. We denote
l
R+
=
{x ∈ Rl : xi ≥ 0 ∀ i}
l
R++
=
{x ∈ Rl : xi > 0 ∀ i}
l
as the positive orthant.
We sometimes refer to R+
For vectors x and y in Rl , we say
x ≥ y if xi ≥ yi for all i and x > y if xi ≥ yi for all i and x 6= y.
Finally, x ≫ y if xi > yi for all i.
Examples: (1, 3, 3) > (1, 2, 3) and (1, 3, 3) ≫ (0, 2, 1).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 2/21
Demand Theory
Assume that there are l commodities and that the consumption space
l
is R+
(the positive orthant).
l
Agent has the utility function U : R+
→ R.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 3/21
Demand Theory
Assume that there are l commodities and that the consumption space
l
is R+
(the positive orthant).
l
Agent has the utility function U : R+
→ R.
Recall the following:
U is said to be increasing if x′ ≫ x implies U (x′ ) > U (x).
U is strictly increasing if x′ > x implies U (x′ ) > U (x).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 3/21
Demand Theory
Assume that there are l commodities and that the consumption space
l
is R+
(the positive orthant).
l
Agent has the utility function U : R+
→ R.
Recall the following:
U is said to be increasing if x′ ≫ x implies U (x′ ) > U (x).
U is strictly increasing if x′ > x implies U (x′ ) > U (x).
l
: U (x) ≥ α} is a
U is quasiconcave if, for any α, the set {x ∈ R+
convex set. Equivalently, whenever U (x′ ) ≥ α and U (x) ≥ α, then
U (tx′ + (1 − t)x) ≥ α for any t ∈ [0, 1].
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 3/21
Demand Theory
Assume that there are l commodities and that the consumption space
l
is R+
(the positive orthant).
l
Agent has the utility function U : R+
→ R.
Recall the following:
U is said to be increasing if x′ ≫ x implies U (x′ ) > U (x).
U is strictly increasing if x′ > x implies U (x′ ) > U (x).
l
: U (x) ≥ α} is a
U is quasiconcave if, for any α, the set {x ∈ R+
convex set. Equivalently, whenever U (x′ ) ≥ α and U (x) ≥ α, then
U (tx′ + (1 − t)x) ≥ α for any t ∈ [0, 1].
U is strictly quasiconcave if, whenever U (x′ ) ≥ α and U (x) ≥ α, then
U (tx′ + (1 − t)x) > α for any t ∈ (0, 1).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 3/21
Demand Theory
l
At price (vector) p = (p1 , p2 , ..., pl ) in R++
(we also write p ≫ 0) and
income w > 0, the agent’s budget set is
l
B(p, w) = {x ∈ R+
: p · x ≤ w}.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 4/21
Demand Theory
l
At price (vector) p = (p1 , p2 , ..., pl ) in R++
(we also write p ≫ 0) and
income w > 0, the agent’s budget set is
l
B(p, w) = {x ∈ R+
: p · x ≤ w}.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 4/21
Demand Theory
l
At price (vector) p = (p1 , p2 , ..., pl ) in R++
(we also write p ≫ 0) and
income w > 0, the agent’s budget set is
l
B(p, w) = {x ∈ R+
: p · x ≤ w}.
A commodity bundle x∗ is a demand bundle of the agent at (p, w) if x∗
maximizes utility in B(p, w); formally,
x∗ ∈ argmaxx∈B(p,w) U (x).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 4/21
Demand Theory
l
At price (vector) p = (p1 , p2 , ..., pl ) in R++
(we also write p ≫ 0) and
income w > 0, the agent’s budget set is
l
B(p, w) = {x ∈ R+
: p · x ≤ w}.
A commodity bundle x∗ is a demand bundle of the agent at (p, w) if x∗
maximizes utility in B(p, w); formally,
x∗ ∈ argmaxx∈B(p,w) U (x).
Note that B(tp, tw) = B(p, w) for any t > 0, so
argmaxx∈B(p,w) U (x) = argmaxx∈B(tp,tw) U (x).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 4/21
Demand Theory
Let x∗ ∈ argmaxx∈B(p,w) U (x).
Quite easy to see that
(A) if U is increasing, then p · x∗ = w. In this case, we say that the
agent’s demand obeys the budget identity.
(B) if U is strictly quasiconcave, then argmaxx∈B(p,w) U (x) has at most
one element.
(C) argmaxx∈B(p,w) U (x) is nonempty if U is a continuous function.
(This is a straightforward consequence of Weierstrass Theorem.a )
a
Weierstrass Theorem: Suppose that
function. Then arg maxx∈K
K is a compact set in Rl and F : K → R a continuous
F (x) is nonempty.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 5/21
Demand Theory
Result below summarizes (A)-(C) and more.
l
Proposition: Suppose that the utility function U : R+
→ R is
(P1) continuous, (P2) strictly increasing, and (P3) strictly
quasiconcave.
l
× R++ , there exists a unique element x∗ in
Then for any (p, w) in R++
argmaxx∈B(p,w) U (x).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 6/21
Demand Theory
Result below summarizes (A)-(C) and more.
l
Proposition: Suppose that the utility function U : R+
→ R is
(P1) continuous, (P2) strictly increasing, and (P3) strictly
quasiconcave.
l
× R++ , there exists a unique element x∗ in
Then for any (p, w) in R++
argmaxx∈B(p,w) U (x).
l
l
The function x̄ : R++
× R++ → R+
mapping
(p, w) to x̄(p, w) = argmaxx∈B(p,w) U (x) has the following properties:
(a) it is continuous,
(b) it obeys the budget identity [i.e., p · x̄(p, w) = w],
(c) it is zero-homogeneous, [i.e. x̄(tp, tw) = x̄(p, w) for any t > 0] and
(d) it obeys this boundary condition: if (pn , wn ) → (p̄, w̄) such that
w̄ > 0 and I = {i : p̄i = 0} is nonempty, then
X
x̄ai (pn , wn ) → ∞.
i∈I
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 6/21
Exchange Economy
Assume that there is a finite set A of agents.
l
Agent a ∈ A has the utility function U a : R+
→ R and endowment
l
.
ω a = (ω1a , ω2a , ..., ωla ) in R+
We require U a to obey (P1), (P2), and (P3), and that aggregate
endowment
X
ω̄ =
ω a ≫ 0.
a∈A
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 7/21
Exchange Economy
Assume that there is a finite set A of agents.
l
Agent a ∈ A has the utility function U a : R+
→ R and endowment
l
.
ω a = (ω1a , ω2a , ..., ωla ) in R+
We require U a to obey (P1), (P2), and (P3), and that aggregate
endowment
X
ω̄ =
ω a ≫ 0.
a∈A
Agent a’s demand function is x̄a .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 7/21
Exchange Economy
Assume that there is a finite set A of agents.
l
Agent a ∈ A has the utility function U a : R+
→ R and endowment
l
.
ω a = (ω1a , ω2a , ..., ωla ) in R+
We require U a to obey (P1), (P2), and (P3), and that aggregate
endowment
X
ω̄ =
ω a ≫ 0.
a∈A
Agent a’s demand function is x̄a .
In an exchange economy, income is determined by the prevailing price
p. With an endowment of ω a , agent a’s income wa = p · ω a .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 7/21
Exchange Economy
Assume that there is a finite set A of agents.
l
Agent a ∈ A has the utility function U a : R+
→ R and endowment
l
.
ω a = (ω1a , ω2a , ..., ωla ) in R+
We require U a to obey (P1), (P2), and (P3), and that aggregate
endowment
X
ω̄ =
ω a ≫ 0.
a∈A
Agent a’s demand function is x̄a .
In an exchange economy, income is determined by the prevailing price
p. With an endowment of ω a , agent a’s income wa = p · ω a .
His demand at price p is x̄a (p, p · ω a ).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 7/21
Exchange Economy
Assume that there is a finite set A of agents.
l
Agent a ∈ A has the utility function U a : R+
→ R and endowment
l
.
ω a = (ω1a , ω2a , ..., ωla ) in R+
We require U a to obey (P1), (P2), and (P3), and that aggregate
endowment
X
ω̄ =
ω a ≫ 0.
a∈A
Agent a’s demand function is x̄a .
In an exchange economy, income is determined by the prevailing price
p. With an endowment of ω a , agent a’s income wa = p · ω a .
His demand at price p is x̄a (p, p · ω a ).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 7/21
Exchange Economy
l
l
Define x̂ a : R++
→ R+
by x̂ a (p) = x̄a (p, p · ω a ).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 8/21
Exchange Economy
l
l
Define x̂ a : R++
→ R+
by x̂ a (p) = x̄a (p, p · ω a ).
Agent a’s excess demand function is z a (p) = x̂a (p) − ω a .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 8/21
Exchange Economy
l
l
Define x̂ a : R++
→ R+
by x̂ a (p) = x̄a (p, p · ω a ).
Agent a’s excess demand function is z a (p) = x̂a (p) − ω a .
Claim: z a is zero-homogeneous, i.e., z a (λp) = z a (p) for any scalar
λ > 0, and p · z a (p) = 0 for all p.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 8/21
Exchange Economy
l
l
Define x̂ a : R++
→ R+
by x̂ a (p) = x̄a (p, p · ω a ).
Agent a’s excess demand function is z a (p) = x̂a (p) − ω a .
Claim: z a is zero-homogeneous, i.e., z a (λp) = z a (p) for any scalar
λ > 0, and p · z a (p) = 0 for all p.
Proof: x̄a (p, w) = argmaxx∈B(p,w) U a (x) is zero-homogeneous so
x̂a (λp) = x̄a (λp, (λp) · ω a ) = x̄a (p, p · ω a ) = x̂a (p)
which in turn guarantees that z a (λp) = z a (p).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 8/21
Exchange Economy
l
l
Define x̂ a : R++
→ R+
by x̂ a (p) = x̄a (p, p · ω a ).
Agent a’s excess demand function is z a (p) = x̂a (p) − ω a .
Claim: z a is zero-homogeneous, i.e., z a (λp) = z a (p) for any scalar
λ > 0, and p · z a (p) = 0 for all p.
Proof: x̄a (p, w) = argmaxx∈B(p,w) U a (x) is zero-homogeneous so
x̂a (λp) = x̄a (λp, (λp) · ω a ) = x̄a (p, p · ω a ) = x̂a (p)
which in turn guarantees that z a (λp) = z a (p).
Since p · x̄a (p, p · ω a ) = p · ω a , we have
p · [x̄a (p, p · ω a ) − ω a ] = 0.
So p · z a (p) = 0.
QED
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 8/21
Exchange Economy
Aggregate (or market) demand at price p is
X
X(p) =
x̂a (p).
a∈A
l
Aggregate excess demand function Z : R++
→ Rl is
Z(p) = X(p) − ω̄.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 9/21
Exchange Economy
Aggregate (or market) demand at price p is
X
X(p) =
x̂a (p).
a∈A
l
Aggregate excess demand function Z : R++
→ Rl is
Z(p) = X(p) − ω̄.
Z is zero-homogeneous and obeys Walras’ Law, p · Z(p) = 0 for all p.
Both inherited from z a , obviously.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 9/21
Exchange Economy
Aggregate (or market) demand at price p is
X
X(p) =
x̂a (p).
a∈A
l
Aggregate excess demand function Z : R++
→ Rl is
Z(p) = X(p) − ω̄.
Z is zero-homogeneous and obeys Walras’ Law, p · Z(p) = 0 for all p.
Both inherited from z a , obviously.
Fundamental Question: What conditions guarantee that there is
p∗ ≫ 0 such that Z(p∗ ) = 0?
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 9/21
Exchange Economy
Aggregate (or market) demand at price p is
X
X(p) =
x̂a (p).
a∈A
l
Aggregate excess demand function Z : R++
→ Rl is
Z(p) = X(p) − ω̄.
Z is zero-homogeneous and obeys Walras’ Law, p · Z(p) = 0 for all p.
Both inherited from z a , obviously.
Fundamental Question: What conditions guarantee that there is
p∗ ≫ 0 such that Z(p∗ ) = 0?
Note that, since Z is zero-homogeneous, if p∗ is an equilibrium price
so is λp∗ for any λ > 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 9/21
Exchange Economy
With two agents A and B and two goods:
Agent A’s utility function U A (x1 , x2 ) = ln x1 + 2 ln x2 .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 10/21
Exchange Economy
With two agents A and B and two goods:
Agent A’s utility function U A (x1 , x2 ) = ln x1 + 2 ln x2 .
Demand function x̄A (p, w) =
2w
w
,
3p1 3p2
.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 10/21
Exchange Economy
With two agents A and B and two goods:
Agent A’s utility function U A (x1 , x2 ) = ln x1 + 2 ln x2 .
Demand function x̄A (p, w) =
2w
w
,
3p1 3p2
.
Endowment ω A = (1, 0), so wA = p1 , and
1 2p1
A
x̂ (p) =
,
.
3 3p2
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 10/21
Exchange Economy
With two agents A and B and two goods:
Agent A’s utility function U A (x1 , x2 ) = ln x1 + 2 ln x2 .
Demand function x̄A (p, w) =
2w
w
,
3p1 3p2
.
Endowment ω A = (1, 0), so wA = p1 , and
1 2p1
A
x̂ (p) =
,
.
3 3p2
Assume that agent B’s utility function is U B (x1 , x2 ) = 2 ln x1 + ln x2
and that his endowment is (0, 1).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 10/21
Exchange Economy
Exercise: show that
Z(p) =
2 2p2 2p1
2
− +
,
−
3 3p1 3p2
3
.
Setting Z1 (p) = 0 we obtain p1 = p2 .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 11/21
Exchange Economy
Exercise: show that
Z(p) =
2 2p2 2p1
2
− +
,
−
3 3p1 3p2
3
.
Setting Z1 (p) = 0 we obtain p1 = p2 .
Setting Z2 (p) = 0 we obtain p1 = p2 .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 11/21
Exchange Economy
Exercise: show that
Z(p) =
2 2p2 2p1
2
− +
,
−
3 3p1 3p2
3
.
Setting Z1 (p) = 0 we obtain p1 = p2 .
Setting Z2 (p) = 0 we obtain p1 = p2 .
Equilibrium price is (λ, λ) for any λ > 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 11/21
Exchange Economy
Exercise: show that
Z(p) =
2 2p2 2p1
2
− +
,
−
3 3p1 3p2
3
.
Setting Z1 (p) = 0 we obtain p1 = p2 .
Setting Z2 (p) = 0 we obtain p1 = p2 .
Equilibrium price is (λ, λ) for any λ > 0.
Note: if at p∗ , we have Z1 (p∗ ) = 0 then Z2 (p∗ ) = 0. This follows from
Walras’ Law, which says that p∗1 Z1 (p∗ ) + p∗2 Z2 (p∗ ) = 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 11/21
Exchange Economy
Exercise: show that
Z(p) =
2 2p2 2p1
2
− +
,
−
3 3p1 3p2
3
.
Setting Z1 (p) = 0 we obtain p1 = p2 .
Setting Z2 (p) = 0 we obtain p1 = p2 .
Equilibrium price is (λ, λ) for any λ > 0.
Note: if at p∗ , we have Z1 (p∗ ) = 0 then Z2 (p∗ ) = 0. This follows from
Walras’ Law, which says that p∗1 Z1 (p∗ ) + p∗2 Z2 (p∗ ) = 0.
(1,
Z (1, )
p2
1
2
3
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 11/21
Exchange Economy
Clear from picture that the existence of a solution to Z(p) = 0 requires
the continuity of Z and also the right boundary condition ...
Continuity of Z is guaranteed if x̄a is continuous for all a. This is in turn
guaranteed by (P1) (the continuity of U a for all agents a).
Recall the boundary property satisfied by x̄a : if (pn , wn ) → (p̄, w̄) such
that w̄ > 0 and I = {i : p̄i = 0} is nonempty, then
P
a n
n
i∈I x̄i (p , w ) → ∞.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 12/21
Exchange Economy
Clear from picture that the existence of a solution to Z(p) = 0 requires
the continuity of Z and also the right boundary condition ...
Continuity of Z is guaranteed if x̄a is continuous for all a. This is in turn
guaranteed by (P1) (the continuity of U a for all agents a).
Recall the boundary property satisfied by x̄a : if (pn , wn ) → (p̄, w̄) such
that w̄ > 0 and I = {i : p̄i = 0} is nonempty, then
P
a n
n
i∈I x̄i (p , w ) → ∞.
Example: Cobb-Douglas demand for good j is
infinity if pj → 0 and wn → w̄, with w̄ > 0.
wn
αj pj .
Clearly tends to
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 12/21
Exchange Economy
Clear from picture that the existence of a solution to Z(p) = 0 requires
the continuity of Z and also the right boundary condition ...
Continuity of Z is guaranteed if x̄a is continuous for all a. This is in turn
guaranteed by (P1) (the continuity of U a for all agents a).
Recall the boundary property satisfied by x̄a : if (pn , wn ) → (p̄, w̄) such
that w̄ > 0 and I = {i : p̄i = 0} is nonempty, then
P
a n
n
i∈I x̄i (p , w ) → ∞.
Example: Cobb-Douglas demand for good j is
infinity if pj → 0 and wn → w̄, with w̄ > 0.
wn
αj pj .
Clearly tends to
The requirement that w̄ > 0 is crucial. Compare pn = (1, n1 ) with
ω = (1, 0) and ω = (0, 1); wn tends to 1 in the first case and 0 in the
second...
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 12/21
Exchange Economy
Corollary: In economy E, suppose pn tends to p̄ 6= 0, such that
P
I = {i : p̄i = 0} is nonempty. Then i∈I Zi (pn ) → ∞.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 13/21
Exchange Economy
Corollary: In economy E, suppose pn tends to p̄ 6= 0, such that
P
I = {i : p̄i = 0} is nonempty. Then i∈I Zi (pn ) → ∞.
Proof: Suppose that for good k, p̄k > 0. Since ω̄ = a∈A ω a ≫ 0, there
is ã with ωkã > 0. (In other words, there is some agent ã who has a
strictly positive endowment of good k.) Then pn · ω ã tends to
w̄ã = p̄ · ω ã > 0.
P
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 13/21
Exchange Economy
Corollary: In economy E, suppose pn tends to p̄ 6= 0, such that
P
I = {i : p̄i = 0} is nonempty. Then i∈I Zi (pn ) → ∞.
Proof: Suppose that for good k, p̄k > 0. Since ω̄ = a∈A ω a ≫ 0, there
is ã with ωkã > 0. (In other words, there is some agent ã who has a
strictly positive endowment of good k.) Then pn · ω ã tends to
w̄ã = p̄ · ω ã > 0.
P
So
P
ã n
i∈I x̂i (p )
X
=
P
n
Zi (p )
i∈I
x̄ãi (pn , pn · ω ã ) → ∞, which implies that
=
X
Xi (p ) −
≥
X
x̄ãi (pn , pn
i∈I
n
i∈I
i∈I
X
ω̄i
i∈I
ã
·ω )−
X
ω̄i → ∞.
i∈I
QED
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 13/21
Exchange Economy
l
Theorem: The excess demand function Z : R++
→ Rl of the economy
E has the following properties:
•
it is zero-homogenous,
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 14/21
Exchange Economy
l
Theorem: The excess demand function Z : R++
→ Rl of the economy
E has the following properties:
•
it is zero-homogenous,
•
it obeys Walras’ Law,
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 14/21
Exchange Economy
l
Theorem: The excess demand function Z : R++
→ Rl of the economy
E has the following properties:
•
it is zero-homogenous,
•
it obeys Walras’ Law,
•
it is continuous,
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 14/21
Exchange Economy
l
Theorem: The excess demand function Z : R++
→ Rl of the economy
E has the following properties:
•
it is zero-homogenous,
•
it obeys Walras’ Law,
•
it is continuous,
•
it satisfies the boundary condition,
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 14/21
Exchange Economy
l
Theorem: The excess demand function Z : R++
→ Rl of the economy
E has the following properties:
•
it is zero-homogenous,
•
it obeys Walras’ Law,
•
it is continuous,
•
it satisfies the boundary condition,
•
it is bounded below.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 14/21
Exchange Economy
l
Theorem: The excess demand function Z : R++
→ Rl of the economy
E has the following properties:
•
it is zero-homogenous,
•
it obeys Walras’ Law,
•
it is continuous,
•
it satisfies the boundary condition,
•
it is bounded below.
Note: Clear that Z is bounded below since
Z(p) = X(p) − ω̄ > −ω̄.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 14/21
Exchange Economy
l
Theorem: The excess demand function Z : R++
→ Rl of the economy
E has the following properties:
•
it is zero-homogenous,
•
it obeys Walras’ Law,
•
it is continuous,
•
it satisfies the boundary condition,
•
it is bounded below.
Note: Clear that Z is bounded below since
Z(p) = X(p) − ω̄ > −ω̄.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 14/21
Exchange Economy
Theorem (Arrow and Debreu; McKenzie): Suppose Z satisfies
properties (1) to (5). Then there is p∗ such that Z(p∗ ) = 0.
Proof uses Kakutani’s fixed point theorem, which generalizes
Brouwer’s fixed point theorem to correspondences.
Brouwer’s fixed point theorem is a (far-reaching) generalization of the
intermediate value theorem.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 15/21
Exchange Economy
Theorem (Arrow and Debreu; McKenzie): Suppose Z satisfies
properties (1) to (5). Then there is p∗ such that Z(p∗ ) = 0.
Proof uses Kakutani’s fixed point theorem, which generalizes
Brouwer’s fixed point theorem to correspondences.
Brouwer’s fixed point theorem is a (far-reaching) generalization of the
intermediate value theorem.
Intermediate value theorem Let f be a continuous function defined on
some interval [a, b]. If f (a) and f (b) are of different signs, then there is
c ∈ [a, b] such that f (c) = 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 15/21
Exchange Economy
When the economy has just two goods, existence can be proved using
the intermediate value theorem.
Consider the function Z2 (1, ·) : R++ → R.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 16/21
Exchange Economy
When the economy has just two goods, existence can be proved using
the intermediate value theorem.
Consider the function Z2 (1, ·) : R++ → R.
First note that as p → 0, we have Z2 (1, p) → ∞, so there is p′ such that
Z2 (1, p′ ) > 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 16/21
Exchange Economy
When the economy has just two goods, existence can be proved using
the intermediate value theorem.
Consider the function Z2 (1, ·) : R++ → R.
First note that as p → 0, we have Z2 (1, p) → ∞, so there is p′ such that
Z2 (1, p′ ) > 0.
Now consider p → ∞.
Then Z1 (1, p) = Z1 ( p1 , 1) → ∞. In particular, there is p′′ such that
Z1 (1, p′′ ) > 0. This implies that Z2 (1, p′′ ) < 0 (by Walras’ Law).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 16/21
Exchange Economy
When the economy has just two goods, existence can be proved using
the intermediate value theorem.
Consider the function Z2 (1, ·) : R++ → R.
First note that as p → 0, we have Z2 (1, p) → ∞, so there is p′ such that
Z2 (1, p′ ) > 0.
Now consider p → ∞.
Then Z1 (1, p) = Z1 ( p1 , 1) → ∞. In particular, there is p′′ such that
Z1 (1, p′′ ) > 0. This implies that Z2 (1, p′′ ) < 0 (by Walras’ Law).
So there is p′ and p′′ such that Z2 (1, p′ ) > 0 and Z2 (1, p′′ ) < 0.By IVT,
there is p∗ such that Z2 (1, p∗ ) = 0.
QED
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 16/21
Brouwer’s fixed point theorem
Intermediate value theorem can be re-stated as a fixed point theorem.
Theorem Let K be a compact (i.e., closed and bounded) interval and
suppose that φ : K → K is a continous function. Then there is x∗ ∈ K
such that φ(x∗ ) = x∗ .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 17/21
Brouwer’s fixed point theorem
Intermediate value theorem can be re-stated as a fixed point theorem.
Theorem Let K be a compact (i.e., closed and bounded) interval and
suppose that φ : K → K is a continous function. Then there is x∗ ∈ K
such that φ(x∗ ) = x∗ .
Brouwer’s fixed point theorem Let K be a compact and convex set in
Rl and suppose that the function φ : K → K is continuous. Then there
is x∗ such that φ(x∗ ) = x∗ .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 17/21
Proof of equilibrium existence
We present a simple proof of equilibrium existence in the case where
ωa ≫ 0 for all a.
Define B̃(p, a) = B(p, p · ωa ) ∩ {x ≤ 2ω̄}.
This is a truncated budget set for agent a.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 18/21
Proof of equilibrium existence
We present a simple proof of equilibrium existence in the case where
ωa ≫ 0 for all a.
Define B̃(p, a) = B(p, p · ωa ) ∩ {x ≤ 2ω̄}.
This is a truncated budget set for agent a.
Assuming (P1), (P2), and (P3), then argmaxx∈B̃(p,a) U a (x) exists and is
Pl
unique for all p ∈ ∆ = {p > 0 : i=1 pi = 1}.
We denote this (modified demand) by x̃a (p).
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 18/21
Proof of equilibrium existence
We present a simple proof of equilibrium existence in the case where
ωa ≫ 0 for all a.
Define B̃(p, a) = B(p, p · ωa ) ∩ {x ≤ 2ω̄}.
This is a truncated budget set for agent a.
Assuming (P1), (P2), and (P3), then argmaxx∈B̃(p,a) U a (x) exists and is
Pl
unique for all p ∈ ∆ = {p > 0 : i=1 pi = 1}.
We denote this (modified demand) by x̃a (p).
The crucial feature of x̃a that makes it useful is that it is also defined on
the boundary of ∆ (the unit simplex), unlike x̂a which is defined only in
the interior of ∆.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 18/21
Proof of equilibrium existence
We present a simple proof of equilibrium existence in the case where
ωa ≫ 0 for all a.
Define B̃(p, a) = B(p, p · ωa ) ∩ {x ≤ 2ω̄}.
This is a truncated budget set for agent a.
Assuming (P1), (P2), and (P3), then argmaxx∈B̃(p,a) U a (x) exists and is
Pl
unique for all p ∈ ∆ = {p > 0 : i=1 pi = 1}.
We denote this (modified demand) by x̃a (p).
The crucial feature of x̃a that makes it useful is that it is also defined on
the boundary of ∆ (the unit simplex), unlike x̂a which is defined only in
the interior of ∆.
Note that x̃a satisfies p · x̃a (p) = p · ω a . Furthermore, x̃a is continuous
in p. (This property relies crucially on ω a ≫ 0 – can you see why?)
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 18/21
Proof of equilibrium existence
Therefore, map Z̃ : ∆ → Rl defined by
X
Z̃(p) =
[x̃a (p) − ω a ]
a∈A
is continuous and obeys Walras’ Law.
Lemma 1 There is p∗ ≫ 0 such that Z̃(p∗ ) = 0.
Lemma 2 If there is p∗ ≫ 0 such that Z̃(p∗ ) = 0, then Z(p∗ ) = 0.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 19/21
Proof of equilibrium existence
Therefore, map Z̃ : ∆ → Rl defined by
X
Z̃(p) =
[x̃a (p) − ω a ]
a∈A
is continuous and obeys Walras’ Law.
Lemma 1 There is p∗ ≫ 0 such that Z̃(p∗ ) = 0.
Lemma 2 If there is p∗ ≫ 0 such that Z̃(p∗ ) = 0, then Z(p∗ ) = 0.
Proof of Lemma 1: Define ψ : ∆ → ∆ by
pj + max{Z̃j (p), 0}
ψj (p) =
for all j.
Pl
1 + i=1 max{Z̃i (p), 0}
∆ is compact and convex set and this map is continuous. Brouwer’s
theorem guarantees that there is p∗ such that ψ(p∗ ) = p∗ .
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 19/21
Proof of equilibrium existence
Therefore, map Z̃ : ∆ → Rl defined by
X
Z̃(p) =
[x̃a (p) − ω a ]
a∈A
is continuous and obeys Walras’ Law.
Lemma 1 There is p∗ ≫ 0 such that Z̃(p∗ ) = 0.
Lemma 2 If there is p∗ ≫ 0 such that Z̃(p∗ ) = 0, then Z(p∗ ) = 0.
Proof of Lemma 1: Define ψ : ∆ → ∆ by
pj + max{Z̃j (p), 0}
ψj (p) =
for all j.
Pl
1 + i=1 max{Z̃i (p), 0}
∆ is compact and convex set and this map is continuous. Brouwer’s
theorem guarantees that there is p∗ such that ψ(p∗ ) = p∗ .
If p∗k = 0 for some k, then max{Z̃k (p∗ ), 0} > 0 and so ψk (p∗ ) 6= p∗k = 0 –
contradiction.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 19/21
Proof of equilibrium existence
So p∗ ≫ 0. By Walras’ Law, there is h such that max{Z̃h (p∗ ), 0} = 0.
Since ψ(p∗ ) = p∗ , in particular,
p∗h
p∗h + 0
∗
= ψh (p ) =
This gives
l
X
1+
Pl
i=1
max{Z̃i (p), 0}
.
max{Z̃i (p), 0} = 0,
i=1
so Z̃i (p∗ ) ≤ 0 for all i. By Walras’ Law and the fact that p∗ ≫ 0, we
QED
have Z̃i∗ (p∗ ) = 0 for all i.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 20/21
Proof of equilibrium existence
Lemma 2 If there is p∗ ≫ 0 such that Z̃(p∗ ) = 0, then Z(p∗ ) = 0.
Proof: We claim that x̂a (p∗ ) = x̃a (p∗ ) for all agents. Clearly, this implies
that Z(p∗ ) = Z̃(p∗ ) = 0.
Suppose, to the contrary, that for some agent b, x̂b (p∗ ) 6= x̃b (p∗ ), which
means that x̂b (p∗ ) is not less than 2ω̄ and U b (x̂b (p∗ )) > U b (x̃b (p∗ )).
Since Z̃(p∗ ) = 0, it must be the case that x̃b (p∗ ) ≪ 2ω̄.
Lectures onGeneral Equilibrium Theory⋆
⋆ ⋆The existence of equilibrium – p. 21/21
Proof of equilibrium existence
Lemma 2 If there is p∗ ≫ 0 such that Z̃(p∗ ) = 0, then Z(p∗ ) = 0.
Proof: We claim that x̂a (p∗ ) = x̃a (p∗ ) for all agents. Clearly, this implies
that Z(p∗ ) = Z̃(p∗ ) = 0.
Suppose, to the contrary, that for some agent b, x̂b (p∗ ) 6= x̃b (p∗ ), which
means that x̂b (p∗ ) is not less than 2ω̄ and U b (x̂b (p∗ )) > U b (x̃b (p∗ )).
Since Z̃(p∗ ) = 0, it must be the case that x̃b (p∗ ) ≪ 2ω̄.
Choose t ∈ (0, 1) such that x = tx̃b (p∗ ) + (1 − t)x̂b (p∗ ) satisfies x ≪ 2ω̄.
Note that U b (x) > U b (x̃b (p∗ )) (by the strict quasiconcavity of U b ) and
that x ∈ B̃(p∗ , b).
This contradicts the optimality of x̃b (p∗ ) in B̃(p∗ , b).
Lectures onGeneral Equilibrium Theory⋆
QED
⋆ ⋆The existence of equilibrium – p. 21/21