Pricing Assets with Complete
Markets
This handout follows Ljungqvist and Sargent, Ch 8 “Equilibrium with Com
plete Markets".
1
Environment
•
In
fi
nite Horizon, Stochastic Pure Exchange Environment
•
In each period, there is a realization of a stochastic event
s
t
∈
S
which is
observable at date
t
≥
0
.
•
The history of events up to (and including
t
)
is denoted
s
t
= (
s
0
, s
1
, ..., s
t
−
1
, s
t
)
∈
S
t
+1
which can be represented as a tree diagram. Such a representation
will be useful in understanding decentralization.
•
The unconditional probability of observing history
s
t
is denoted
π
t
(
s
t
)
and
the probability of observing
s
t
conditional upon realization
s
τ
for
t > τ
is
denoted
π
t
(
s
t

s
τ
)
.
•
Each agent
i
is endowed with
y
i
t
(
s
t
)
of the time
t
history
s
t
nonstoreable
consumption good. We will assume
y
i
t
(
s
t
)
>
0
.
•
Household
i
orders possibly history dependent consumption streams
U
(
c
i
) =
T
X
t
=0
X
s
t
∈
S
t
+1
β
t
π
t
(
s
t
)
u
(
c
i
t
(
s
t
))
(1)
which is an increasing, twice di
ff
erentiable strictly concave function which
satis
fi
es the Inada condition
lim
c
→
0
u
0
(
c
) =
∞
.
In most of what follows we
will take
T
=
∞
.
•
Note that while preferences are identical among agents, incomes are not
(so a planner or the market may treat people di
ff
erently).
2
Planner’s Problem
•
Before moving onto the decentralized competitive equilibrium, we solve
the planner’s problem for this economy.
•
Since not all agents are assumed to have identical income realization paths,
the planner solves
max
{
c
i
}
I
i
=1
I
X
i
=1
λ
i
U
(
c
i
)
1
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subject to the resource feasibility constraint
s.t.
X
i
c
i
t
(
s
t
)
≤
X
i
y
i
t
(
s
t
)
,
∀
t,
∀
s
t
.
(2)
•
Assigning multipliers
θ
t
(
s
t
)
for each resource constraint, the langrangian
for the plannner’s problem is
∞
X
t
=0
X
s
t
∈
S
t
+1
(
I
X
i
=1
λ
i
β
t
π
t
(
s
t
)
u
(
c
i
t
(
s
t
)) +
θ
t
(
s
t
)
I
X
i
=1
£
y
i
t
(
s
t
)
−
c
i
t
(
s
t
)
¤
)
•
The foc for any
c
i
t
(
s
t
)
is
β
t
π
t
(
s
t
)
u
0
(
c
i
t
(
s
t
)) =
θ
t
(
s
t
)
λ
i
,
∀
i,
∀
t,
∀
s
t
.
(3)
•
Taking ratios of (3) for consumers
i
and WLOG
1
gives
u
0
(
c
i
t
(
s
t
))
u
0
(
c
1
t
(
s
t
))
=
λ
1
λ
i
⇐⇒
c
i
t
(
s
t
) =
u
0
−
1
μ
λ
1
u
0
(
c
1
t
(
s
t
))
λ
i
¶
,
∀
t,
∀
s
t
.
(4)
•
Plugging this expression into the feasibility constraint (2) yields
X
i
u
0
−
1
μ
λ
1
u
0
(
c
1
t
(
s
t
))
λ
i
¶
=
X
i
y
i
t
(
s
t
)
≡
Y
(
s
t
)
,
∀
t,
∀
s
t
(5)
which is one equation in one unknown
c
1
t
(
s
t
)
for each
t.
•
For the example where
u
(
c
) = log(
c
)
,
then (4) is just
c
i
t
(
s
t
) =
λ
i
λ
1
c
1
t
(
s
t
)
so agent
i
’s consumption is linear in agent
1
’s and the more the planner
weights
i
, the more he gives
i.
Further (5) is just
c
1
t
(
s
t
) =
Y
(
s
t
)
³
1 +
P
i
λ
i
λ
1
´
.
Since all the variables on the rhs are numbers, the lhs is a number.
•
Prop CC1.
Given a realization of aggregate output
Y
(
s
t
)
,
every agent’s
consumption is a function of the aggregate endowment in an e
ﬃ
cient allo
cation and does not depend on the realization of individual endowments.
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 Spring '07
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