1
Investment
• Neo-Classical (Jorgenssen)
• irreversability (Bentolila-Bertola / Bertola-Caballero)
• Fixed costs (Caballero-Engle)
— firm level
— aggregation
— general equilibrium
2
User Cost
Value with complete markets
X
t,st
¡ ¢ ¡ ¢
qt0 st dt st
¡ ¢
¡ ¡
¢ ¢
¡ ¡
¢
¡ ¢
¢
¡ ¢ ¡ ¢
dt st = π kt st−1 , st − c kt st−1 , it st , st − pt st it st
¡ ¢
¡
¢
¡ ¢
kt+1 st = (1 − δ) kt st−1 + it st
subsituting
¡ ¢
¡ ¡
¢ ¢
¡ ¡
¢
¡ ¢
¡
¢
¢
dt st = π kt st−1 , st − c kt st−1 , kt+1 st − (1 − δ) kt st−1 , st
|
{z
}
“blat (st )”
• value
¡
¡ ¢
¡
¢¢
−p (st ) kt+1 st − (1 − δ) kt st−1
X
t,st
¡ ¢¡
¡ ¢
¡ ¢¡
¡ ¢
¡
¢¢¢
qt0 st blat st − pt st kt+1 st − (1 − δ) kt st−1
1
• if pt+1 (st+1 ) = pt (st ) then
¡ ¢
¡ ¢ X 0 ¡ t+1 ¢
¡ ¢
qt+1 s
(1 − δ) kt+1 st
qt0 st kt+1 st −
st+1
¶
1−δ
Rt,t+1 (
st )
¡ ¢¡ ¡ ¢
¢
q 0 (st )
= t t kt+1 st R st − 1 + δ
R (s )
X ¡ ¢
¡ ¢¡ ¡ ¢
¢
qt0 st kt+1 st R st − 1 + δ
=
µ
¡ t¢
¡ t¢
0
= qt s kt+1 s
1−
st+1
• where risk-free rate
• or at t + 1 simply
¡ ¢
R st = P
qt0 (st )
0
t+1 )
st+1 qt+1 (s
¡ ¢¡ ¡ ¢
¢
kt+1 st R st − 1 + δ
• user cost (rental rate)
¡ ¢
¡ ¢¡ ¡ ¢
¢
ν st ≡ pt st R st − (1 − δ)
• assume pt (st ) 6= pt+1 (st , st+1 ) = pt+1 (st , ŝt+1 )
i.e. prices are not constant but predictable
¡ ¢ ¡ ¢
¡ ¢ X
¡
¢
¡
¢
¡ ¢
pt st qt0 st kt+1 st −
pt+1 st+1 qt0+1 st+1 (1 − δ) kt+1 st
st+1
¶
µ
¡ t¢
¡ t¢
¡ t¢
¡ t+1 ¢ 1 − δ
0
= qt s kt+1 s
pt s − pt+1 s
R (st )
X
¡ t+1 ¢
¡ ¢¡ ¡ ¢ ¡ ¢
¡
¢¢
0
=
qt+1
s
kt+1 st R st pt st − (1 + δ) pt+1 st+1
st+1
• user cost (rental rate)
¶
µ
¡ t¢
¡ t¢
¡ t¢
pt+1 (st+1 )
ν s ≡ pt s
R s − (1 − δ)
pt (st )
→ physical and economic depreciation
2
• thus...
¡ ¢
¡ ¡
¢ ¢ ¡ ¡
¢
¡ ¢
¢ ¡ ¢¡
¡ ¢
¡
¢¢
dt st = π kt st−1 , st −c kt st−1 , it st , st −p st kt+1 st − (1 − δ) kt st−1
...equivalent to
¡ ¢
¡ ¡
¢ ¢
¡ ¡
¢
¡ ¢
¢
¡
¢
d˜t st = π kt st−1 , st − c kt st−1 , it st , st − ν (st ) kt s
t−1
3
Irreversability or Costly Reversability
Discrete example (see Dixit-Pindyck)
• three periods t = −1, 0, 1
t = −1 invest k0
t = 0 invest k1
• at t = 0 learn A1 ∈ {AH , AL }
• irreversability
k1 ≥ k0 (1 − δ)
• risk neutral pricing
¡ ¢
¡ ¢
qt0 st ≡ R−t Pr s
t
• constant user cost v
Firm problem
µ
µ
−1
p
max A0 F (k0 ) − vk0 + R
k0
max
k1 ≥k0 (1−δ)
equivalently
¡ H
¢
A F (k1 ) − vk1 + (1 − p)
max A0 F (k0 ) − vk0 + R−1 E V (k0 , A1 )
k0
V (k, A) =
max AF (k 0 ) − vk0
k0 ≥k(1−δ)
• unconstrained optimum
max
AF (k0 ) − vk0
0
k
¢
¡
⇒ AF 0 k# (A) − v = 0
3
max
k1 ≥k0 (1−δ)
¶¶
(AL F (k1 ) − vk1 )
• if k1∗ (AL ) = k0 (1 − δ)
AF 0 (k1 ) − v ≤ 0
⇒A0 F 0 (k0 ) − v +
1−p
(AL F 0 (k1 ) − v) (1 − δ) = 0
{z
}
R |
−
⇒
k0∗
<
k0#
(A0 )
• another way of seeing this:
A0 F 0 (k0 ) − v + E [Vk (k0 , A1 )] = 0
but Vk ≤ 0 and Vk < 0 in some state...
• irreversability
⇒ lower investment
• if kt ∈ {0, 1} ⇒ option value intuition [Dixit&Pyndick]
• Q: lower average capital?
A: NO
• investment rule
k0 ≥ k1∗ → don’t invest
k0 < k1∗ → invest until k1 = k1∗
→ barrier control
• costly reversability
capital sells at discount
→upper barrier control
• comparative statics
increase uncertainty
4
4
Bentolila and Bertola
• homogenous return
π (k, z) = k α z 1−α
• Bellman
V (k− , z− ) = E
½
¾
max [π (k, z) − rk + βV (k, z)]
k≥(1−δ)k−
• guess
µ
¶
µ ¶
k
k
V (k, z) = zV
, 1 = zv
z
z
½ µ µ
¶
¶
∙ µ
¶¸¾
µ
¶
k
k
k
k−
z π
,1 −r
+ βE zV
,z
z− V
,1 = E
max
z k
k
z−
z
z
z
≥(1−δ) − −
z
z z−
• normalized
v (k− ) = E ε max[π (k) − rk + βv (k)]
k
k ≥ ε−1 (1 − δ) k−
s.t.
• FOC
π 0 (k∗ ) = r − βV 0 (k∗ )
• lower investment
V 0 (k) ≤ 0 ⇒ k∗ < k#
• Example:
ε = { b , g }
prob. λ, 1 − λ
• no adjustment ⇒
k = (1 − δ) ε−1 k−
then
g b
(1 − δ)2 = 1 ⇒
g
=
−1
b
(1 − δ)−2
keeps us on a grid
k∗ , k∗
−1
b
(1 − δ)−1 , k ∗
k ∗ ( b (1 − δ))−n
5
−2
b
(1 − δ)−2 , ...
n = 0, 1, 2, ...
• invariant
pn = (1 − λ) pn+1 + λpn−1
p0 = (1 − λ) p1 + (1 − λ) p0
• solving
1 = (1 − λ)
n = 1, 2, ...
p1
p1
λ
+ (1 − λ) ⇒
=
p0
p0
1−λ
substituting
pn+1
pn−1
pn+1
λ
+λ
⇒
=
pn
pn
pn
1−λ
µ
¶n
λ
⇒ pn = p0
1−λ
1 = (1 − λ)
• costly reversability:
regulate k from above as well
• from Bertola-Bentolila
Figure removed due to copyright restrictions.
See Figure 1 on p. 388 in Bentolila, Samuel, and Giuseppe Bertola.
"Firing Costs and Labour Demand: How Bad is Eurosclerosis?"
Review of Economic Studies 57, no. 3 (1990): 381-402.
• Abel and Eberly (1999; JME)
6
5
Labor: Firing Costs
• effects of firing costs on labor demand?
• Bertola and Bertolila (1990)
• Hopenhayn and Rogerson (1993)
Figure removed due to copyright restrictions.
See Figure 2 on p. 392 in Bentolila, Samuel, and Giuseppe Bertola.
"Firing Costs and Labour Demand: How Bad is Eurosclerosis?"
Review of Economic Studies 57, no. 3 (1990): 381-402.
Figure removed due to copyright restrictions.
See Figure 3 on p. 392 in Bentolila, Samuel, and Giuseppe Bertola.
"Firing Costs and Labour Demand: How Bad is Eurosclerosis?"
Review of Economic Studies 57, no. 3 (1990): 381-402.
7
Higher uncertainty
Figure removed due to copyright restrictions.
See Figure 4 on p. 393 in Bentolila, Samuel, and Giuseppe Bertola.
"Firing Costs and Labour Demand: How Bad is Eurosclerosis?"
Review of Economic Studies 57, no. 3 (1990): 381-402.
• Hopenhayn-Rogerson → ergodic distribution of shocks
• entry and exit of firms
• ergodic distribution of labor demand
• compare steady states with and without
6
Aggregate Shocks
• concerted shocks in z
• distribution is state variable
• average reaction depends on distribution
7
Fixed Costs
• fixed cost if it > 0
8
• don’t invest unless its important
→ innaction (fixed cost is irreversable)
• avoid fixed costs:
lump investment (no small investments)
•
Ss rule
• other end: generalized Ss
8 Generalized Hazard
Caballero-Engle
• probability of investing
depends on distance from preferred point
• previous case: discontinuous hazard
• Calvo: constant hazard
• example 1
aggregate firms with different sS rules
• example 2
firms with random fixed cost
9 Aggregation
• state: distribution of firms
• evolution of distribution
aggregate vs. idiosyncratic shocks
• non-linear response
• history matters
9
10
General Equilibrium
Julia Thomas → small effects in GE
Bachman-Caballero-Engle → large in recalibrated model
10