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Hidden Info with Persistent Shocks
•
A. Fernandez and C. Phelan (2000), “A Recursive Formulation for Re
peated Agency with History Dependence,”
Journal of Economic Theory
,
91, 22347.
1
Environment
•
Realizations of the nonstorable endowment take 2 values
h
t
∈
H
=
{
h
H
,h
L
}
where
h
H
>h
L
which follows a
f
rst order markov where
π
(
h
t
−
1
)denotes
the prob that
h
t
=
h
H
given
h
t
−
1
.
The initial seed is
h
−
1
assumed to be
public.
•
The conclusion considers what happens in the general
N
state Markov
process (which necessitates expanding the state space to
N
−
1threats)
.
•
Let
Π
(
h
t
+
j

h
t
) denote the probability of future history
{
h
t
+1
, ..., h
t
+
j
}
given
h
t
.
•
Endowment realizations are iid across agents.
•
All agents are also endowed with an initial entitlement to future utility
w
0
.
•
A planner can transfer resources across time at rate
q
∈
(0
,
1)
.
•
Agents are risk averse with preferences
U
(
c
)=
E
"
∞
X
t
=0
β
t
U
(
c
t
)
#
where
U
:
B
→
R
with
B
=[
b
,
b
] is cts, strictly increasing, strictly concave.
•
The implied values for feasible momentary utility is
D
d
,
d
]wh
e
r
e
d
=
U
(
b
)and
d
=
U
(
b
)
.
•
Info:
h
t
is priv. info to agent (except initial seed).
•
Timing: at the beginning of each period, agent observes
h
t
,
reports en
dowment shock, makes/receives transfer, consumes.
1.1
Sequence Representation
•
The only real decision for agents to make is about their reports.
•
Let
e
h
denote a
reporting strategy
given by the sequence
{
e
h
t
(
h
t
)
}
∞
t
=0
map
ping histories
h
t
=(
h
0
, ..., h
t
)
∈
H
t
into a report of the current endowment
e
h
t
.
1
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View Full Document•
De
f
ne a transfer system
τ
as a sequence of functions
{
τ
t
}
∞
t
=0
such that
τ
t
:
H
t
→
R
and
τ
t
(
h
t
)
≥−
h
t
for all
t
and
h
t
.
Notational note: the
dependence of
τ
t
(
h
t
) on the initial state (
h
−
1
,w
0
)isle
ftimp
l
ic
it(thatis
,
should really write
τ
t
(
h
t
;
h
−
1
0
)
.
•
The constraint does not rule out the possibility that
c
t
<b
.
This can be
taken care of many ways, but they simply assume that the agent cannot
claim a higher than actual endowment. See footnote 3 of FP.
•
If we let an agent’s current utility
b
u
t
after report contingent transfer be
denoted
b
u
t
=
U
(
h
t
+
τ
t
(
e
h
t
(
h
t
)))
and de
f
ne
C
:
D
→
B
by
C
(
b
u
t
)=
U
−
1
(
b
u
t
)
.
Since
C
is the inverse function of
U,
it is a uniquely de
f
ned strictly in
creasing and convex function.
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 Spring '07
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