E
CON
714: M
ACROECONOMIC
T
HEORY
II
TA: T
IM
L
EE
M
AY
1, 2010
Continuous Time Value Function with Poisson Arrivals
If
N
t
is a Poisson process with rate
λ
,
1.
P
(
N
t
=
0
) =
exp
(

λ
t
)
,
2. lim
t
→
0
P
(
N
t
=
1
)
t
=
λ
3. lim
t
→
0
P
(
N
t
>
1
)
t
=
0.
As in class, assume
 continuum 1 of agents
 interest rate
r
 arrival rate
α
∈
[
0,
∞
)
: NOT A PROBABILITY
 probability
x
of liking what the other guy produces

M
∈
[
0, 1
]
people hold money
Two value functions, one each of whether or not you have money.
Here’s a heuristic
argument of how we get
V
1
:
V
1
(
t
)
=
e

rdt
E
t
V
t
+
dt
(
1

e

rdt
)
V
1
(
t
)
=
e

rdt
E
t
[
V
1
(
t
+
dt
)

V
1
(
t
)]
.
Now note that the stochastic gain comes from a meeting, which happens with 1

e

α
dt
probability:
E
t
[
V
1
(
t
+
dt
)

V
1
(
t
)]

{z
}
expected instantaneous increase in
V
1

[
V
1
(
t
+
dt
)

V
1
(
t
)]

{z
}
deterministic instantaneous increase in
V
1
=
(
1

e

α
dt
)
x
(
1

M
)
π
[
u
+
V
0
(
t
+
dt
)

V
1
(
t
+
dt
)]
,
so
(
1

e

rdt
)
V
1

{z
}
cost of holding on to money
=
e

rdt
(
1

e

α
dt
)
x
(
1

M
)
π
(
u
+
V
0

V
1
)

{z
}
gain from a meeting
+
dV
1
{z}
deterministic gain
1
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Next divide by
dt
and take the limit as
dt
→
0,
lim
dt
→
0
1

e

rdt
dt
V
1
=
α
x
(
1

M
)
π
(
u
+
V
0

V
1
) +
˙
V
1
rV
1
=
α
x
(
1

M
)
π
(
u
+
V
0

V
1
) +
˙
V
1
.
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 Spring '08
 Staff
 Trigraph, TA, Keynesian economics, Xt, Φt

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