Social welfare function W : R
n
→ R gives social utility W(u
1
,u
2
,...,u
n
).Wis
strictly increasing.
x
?
is socially optimal if it solves:
max W[u
1
(x
1
),u
2
(x
2
),...,u
n
(x
n
)]
s.t.
n
[
i=1
x
i
≤
n
[
i=1
w
i
Proposition 1.29. If x
?
is SO, it is PO.
Proposition 1.30. Suppose that preferences are continuous, strictly monotonic, and
strictly convex. Then, for any PO allocation x
?
with x
?
i
>> 0, ? i, there exist a
i
>
0,i=1,...,n, s.t. x
?
solves
max W ≡
[
a
i
u
i
(x
i
)
s.t.
[
x
i
≤
[
x
?
i
,
where the weights a
i
are the reciprocals of the marginal utilities of income.
Note: competitive market favors individuals with large incomes.
Summary:
(1) Competitive equilibria and SO allocations are PO.
(2) PO allocations are competitive equilibria under convexity of preferences and endow-
ment redistribution.
(3) PO allocations are SO under convexity of preferences and a linear social welfare
function with special weights.
4.3. General Equilibrium with Production
m firms: j =1,2,...,m. Production possibilities: G
j
,j=1,2,...,m.
The firms are owned by n individuals i =1,2,...,n, with endowments w
i
,i=
1,2,...,n. Let α
ij
be the share of firm j owned by agent i, 0 ≤ α
ij
≤ 1,
S
i
α
ij
=
1, ? j.
1—21
Definition 1.5. An allocation (x, y),x≥ 0, is feasible if
n
[
i=1
x
i
≤
n
[
i=1
w
i
+
m
[
j=1
y
j
; G
j
(y
j
) ≤ 0, ? j.
Definition 1.6. (x
?
,y
?
,p
?
) is a Walrasian equilibrium if
(1) x
?
i
solves
max u
i
(x
i
)
s.t. p
?
·x
i
≤ p
?
· w
i
+
m
S
j=1
α
ij
p
?
· y
?
j
(2) y
?
j
solves
max p
?
·y
j
s.t. G
j
(y
j
) ≤ 0
(3) (x
?
,y
?
) is feasible:
n
[
i=1
x
?
i
≤
n
[
i=1
w
i
+
m
[
j=1
y
?
j
.
Example 1.19. Add a firm to the economy in Example 1.13. The firm inputs x to
produce y by production function y =
√
x. The two consumers share the firm equally.
Find the equilibrium price.
Example 1.20. Add a firm to the economy in Example 1.14. The firm inputs x to
produce y by production function y =
√
x. The two consumers share the firm equally.
Find the equilibrium price.
In equilibrium,
MRS
lh
i
(x
i
)=MRT
lh
j
(y
i
)=
p
l
p
h
, ? l, h, i, j.
Definition 1.7. A feasible allocation (x, y) is Pareto optimal if ? no feasible alloca-
tion (x
0
,y
0
) s.t. x
0
i
null
i
x
i
, ? i.
Proposition 1.31. Suppose that u
i
is di?erentiable, quasi-concave and D
x
u
i
(x) > 0,
for all i, and G
j
is di?erentiable, quasi-convex and D
y
G
j
(y
j
) > 0. Then, a feasible
allocation (x,y) is Pareto optimal i?
MRS
lh
1
(x
1
)=···= MRS
lh
n
(x
n
)=MRT
lh
1
(y
1
)=···= MRT
lh
m
(y
m
), ? l, h. (4.7)
1—22
Theorem 1.6. (The First Welfare Theorem). If (x
?
,y
?
,p
?
) is a Walrasian equi-
librium, (x
?
,y
?
) is PO.
Theorem 1.7. (The Second Welfare Theorem). Suppose preferences are contin-
uous, strictly monotonic, and strictly convex. Then, any PO allocation (x
?
,y
?
) is a
Walrasian equilibrium allocation with proper distributions of profit shares and endow-
ments.
The two results about social optimality also hold.
4.4. General Equilibrium with Uncertainty
Let T ≡ {all possible states} = {1,2,...,τ}. State t occurs with probability π
t
.
Individuals i =1,2,...,n, with endowment w
t
i
∈R
k
+
for t. Given t, i’s consumption
bundle is x
t
i
∈R
k
+
. In a plan, i’s consumption bundle is x
i
≡ (x
1
i
,x
2
i
,...,x
τ
i
) ∈R
kτ
+
with
u
i
(x
i
)=
τ
[
t=1
π
t
i
u
i
(x
t
i
).
m firms j =1,2,...,m, with production possibilities G
j
: R
kτ
→ R,j=1,2,...,m,
where y
j
≡ (y
1
j
,y
2
j
,...,y
τ
j
) ∈R
kτ
is the net output vector and p =(p
1
,p
2
,...,p
τ
) ∈R
kτ
+
is the price vector.
The firms are owned by individuals, where α
ij
is the share of firm j owned by agent
i, 0 ≤ α
ij
≤ 1,
S
i
α
ij
=1, ? j.
Contingent contracts economy: contracts on buying and selling are contingent on
all the states of the economy.
Definition 1.8. A allocation (x,y),x≥ 0, is feasible if
n
[
i=1
x
t
i
≤
n
[
i=1
w
t
i
+
m
[
j=1
y
t
j
, ? t; G
j
(y
j
) ≤ 0, ? j.
Definition 1.9. (x
?
,y
?
,p
?
) is a Walrasian equilibrium if
(1) x
?
i
∈R
kτ
+
solves
max u
i
(x
i
)
s.t.
τ
S
t=1
p
?t
· x
t
i
≤
τ
S
t=1
p
?t
· w
t
i
+
τ
S
t=1
m
S
j=1
α
ij
p
?t
· y
?t
j
1—23
(2) y
?
j
∈R
kτ
solves
max
τ
S
t=1
p
?t
· y
t
j
s.t. G
j
(y
1
j
,y
2
j
,...,y
τ
j
) ≤ 0
(3) (x
?
,y
?
) is feasible:
n
[
i=1
x
?t
i
≤
n
[
i=1
w
t
i
+
m
[
j=1
y
?t
j
,t=1,2,...,τ.
Definition 1.10. A feasible allocation (x,y) is Pareto optimal if ? no feasible allo-
cation (x
0
,y
0
) s.t. x
0
i
null
i
x
i
, ? i.
Since this economy is a special case of the previous one, the four welfare theorems still
hold.
This economy requires k ×τ markets – informationally ine?cient.
1—24