Notes on Durable Consumption
Russell Cooper
Outline Basic Optimization Model: Continuous Choice Dynamic Discrete Choice: Simple Dynamic Discrete Choice: Complex
Outline
1
Basic Optimization Model: Continuous Choice
Outline Basic Optimization Model: Continuous Choice Dynamic Discrete Choice: Simple Dynamic Discrete Choice: Complex
Outline
1
Basic Optimization Model: Continuous Choice
Outline
1
Basic Optimization Model: Continuous Choice
2
Dynamic Discrete Choice: Simple
Outline
1
Basic Optimization Model: Continuous Choice
2
Dynamic Discrete Choice: Simple
V
(
A
,
D
,
y
,
p
) = max
D
0,
A
0u
(
c
,
D
) +
β
E
y
0,
p
0|
y
,
p
V
(
A
0
,
D
0
,
y
0
,
p
0
)
(1)
for all (
A
,
D
,
y
,
p
) with
c
=
A
+
y
−
(
A
0
/
R
)
−
p
(
D
0
−
(1
−
δ
)
D
)
(2)
A
0
=
R
(
A
+
y
−
c
−
pe
)
.
(3)
FOC and Euler
u
c
(
c
,
D
) =
β
RE
y
0,
p
0|
y
,
p
V
A
(
A
0
,
D
0
,
y
0
)
(5)
and
u
c
(
c
,
D
)
p
=
β
E
y
0,
p
0|
y
,
p
V
D
(
A
0
,
D
0
,
y
0
)
.
(6)
u
c
(
c
,
D
) =
β
RE
y
0|
y
u
c
(
c
0
,
D
0
)
(7)
and
No Time to Build
V
(
A
,
D
,
y
,
p
) = max
D
0,
A
0u
(
c
,
D
0
) +
β
E
y
0|
y
V
(
A
0
,
D
0
,
y
0
,
p
0
)
(9)
Implies
pu
c
(
c
,
D
0
) = [
u
D
(
c
,
D
0
) +
β
E
y
0,
p
0|
y
,
p
p
0
(1
−
δ
)
u
c
(
c
0
,
D
00)]
(10)
If prices are constant:
u
D
(
c
,
D
0
) =
β
RE
y
0|
y
u
D
(
c
0
,
D
00Mankiw
β
R
= 1
separable quadratic utility
implications
E
t
D
t
+1
=
D
t
e
t
+1
=
a0
+
a1e
t
+
ε
t
+1
−
(1
−
δ
)
ε
t
empirical evidence implies
δ
= 1
Alternatives:
adjustment costs
Outline
1
Basic Optimization Model: Continuous Choice
2
Dynamic Discrete Choice: Simple
Simple Car Replacement
V
i
= max[
V
i
k
,
V
i
r
]
V
i
k
=
u
(
s
i
,
y
) +
β
V
i
+1
(12)
and
V
i
r
=
u
(
s
1
,
y
−
p
+
π
) +
β
V
2
where
β
∈
(0
,
1). Here
y
is income,
p
is the price of a car and
π
is
Car age (i)
values
V
irV
ikkeep
scrap
Going to the Data: Micro
the optimal scrapping time is a critical age,
i
∗
this age depends on the vector of parameters, Θ:
i
∗
(Θ)
Observations on car ownership would then determine the
optimal scrapping time
Θ would not be identified
Going to the Data: Macro
Aggregate Car Sales depends on initial distribution
smooth if initial distribution is uniform
deterministic aggregate cycles if distribution is degenerate
intermediate possibilities
evolution of cross sectional distribution
car age
i
period
t
f
i
+1
(
t
+ 1) =
f
i
(
t
)
Outline
1
Basic Optimization Model: Continuous Choice
2
Dynamic Discrete Choice: Simple
More Complete Model
V
i
(
z
,
Z
) = max[
V
i
k
(
z
,
Z
)
,
V
i
r
(
z
,
Z
)] where
V
i
k
(
z
,
Z
) =
u
(
s
i
,
y
+
Y
, ε
) +
(13)
β
(1
−
δ
)
EV
i
+1
(
z
0
,
Z
0
) +
βδ
EV
1
b
(
z
0
,
Z
0
)
and
V
i
r
(
z
,
Z
) =
u
(
s
1
,
y
+
Y
−
p
+
π, ε
) +
(14)
β
(1
−
δ
)
EV
2
(
z
0
,
Z
0
) +
βδ
EV
1
b
(
z
0
,
Z
0
)
and
V
1
b
(
z
,
Z
) =
u
(
s
1
,
y
+
Y
−
p
+
π, ε
) +
(15)
β
(1
−
δ
)
EV
2
(
z
0
,
Z
0
) +
βδ
EV
1
b
(
z
0
,
Z
0
)
u
(
s
i
,
c
) =
h
i
−
γ
+
ε
(
c
1
/λ
−
)
ξ
1−ξAggregate Dynamics
Aggregate Hazard
H
k
(
Z
t
, θ
) =
Z
h
k
(
z
t
,
Z
t
, θ
)
φ
(
z
t
)
dz
t
(16)
where
h
k
(
z
t
,
Z
t
, θ
) is individual hazard
Sales
S
t
(
Z
t
, θ
) =
X
k
H
k
(
Z
t
, θ
)
f
t
(
k
)
(17)
Distribution
Key Points
Sales driven by the interaction of the evolution of the
distribution and the hazard
Aggregate Shocks bunch the distribution and leads to echo
effects
Car age (i)
1
0 δ
Hazard
function
Estimation
estimate exogenous processes in the first-stage
estimate structural parameters using SMM and non-linear
least squares in the second stage
L
N
(
θ
) =
α
L
N
1
(
θ
) +
L
2
N
(
θ
)
L
1
N
(
θ
) =
1
T
P
T
t
=1
h
(
S
t
−
S
¯
t
(
θ
))
2
−
N
(
N
1
−
1)
P
N
n
=1
(
S
tn
(
θ
)
−
S
¯
t
(
θ
))
2
i
L
2
N
(
θ
) =
P
First-Stage Estimation
Y
t
=
µ
Y
+
ρ
YY
Y
t
−
1
+
ρ
Yp
p
t
−
1
+
u
Yt
p
t
=
µ
p
+
ρ
pY
Y
t
−
1
+
ρ
pp
p
t
−
1
+
u
pt
ε
t
=
µ
ε
+
ρ
ε
Y
Y
t
−
1
+
ρ
ε
p
p
t
−
1
+
u
ε
t
The covariance matrix of
the innovations
u
=
{
u
Yt
,
u
pt
,
u
ε
t
}
is
Ω =
ω
Y
ω
Yp
0
ω
pY
ω
p
0
0
0
ω
ε
Policy: Adda-Cooper
use estimated parameters from pre-policy period
simulate state dependent scrapping subsidies
