L. Karp
Notes for Dynamics
VII. Limit Cycles in Intertemporal Adjustment Models
1. Describe basic models of convex adjustment, with one state variable.
2. Review basic fish problem with no costs of adjusting control.
3. Add adjustment costs to basic resource (e.g. fishing) problem, to get twostate variable
problem.
1) What is an "adjustment model"?
(Recall example of Krugman’s paper in first set of notes.) Standard example, costly adjustment
of capital.
π
(
K
) = restricted profit function
C
(
I
) = investment cost
C
′
,
C
′′
> 0 e.g.
C
wI
γ
I
2
2
First let’s consider the problem when adjustment costs are linear. In that case adjustment to a
steady state is instantaneous. If current level of capital is K
0
, the cost of increasing stock to K
∞
and then keeping it there (i.e, buying
δ
K
∞
each unit of time) is w[K
∞
K
0
+
δ
K
∞
/r, so the
marginal cost of an additional unit of capital is just w(1 +
δ
/r). The marginal benefit of an extra
unit of capital is
π′
/r. Setting marginal benefit equal to marginal cost gives
π′
= w(r+
δ
). This
equation determines the equilibrium capital stock in the absence of adjustment costs.
The
dynamics in this problem were trivial, because it is optimal to jump immediately to a steady
state.
2) Now consider control problem with convex adjustment costs, where it is optimal to approach
steady state gradually. The control problem is
(1)
max
I
⌡
⌠
∞
0
e
rt
π
(
K
)
C
(
I
)
dt
K
˙
=
I
–
δ
K
δ
= depreciation rate
H
=
π
(
K
)–
C
(
I
)+
λ
(I –
λ
K
)
FOC
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C
′
(
I
)+
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 Fall '06
 KARP
 Economics, Derivative, L. Karp, SS r

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