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# Consider the management of a productive asset that generates a return stream of S, where dS = dt + dW is negative, implying a tendency for the asset

Consider the management of a productive asset that generates a return
stream of S, where
dS = ¹dt + ¾dW
¹ is negative, implying a tendency for the asset to generate lower (pos-
sibly negative) returns over time. The asset can be replaced at a ¯xed
cost of K, with the new asset generating a return of Sn. Suppose you
want to maximize the discounted expected °ow of returns over time,
using a discount rate or r.
The value of the discounted expected °ow of returns, V (S) satis¯es the
complementarity conditions
rV (S) ¸ S + LV (S)
V (S) ¸ V (Sn) ¡ K
with one of these two condition satis¯ed with equality at each value
of S. When the ¯rst condition is satis¯ed with equality it is optimal
to keep the current asset; when the second condition is satis¯ed with
equality it is optimal to replace the asset.
Suppose that r = 0:05, ¹ = ¡0:03, ¾ = 0:1, K = 5 and Sn = 1. Ap-
proximate V using Á(S)c for some choice of Á and solve the associated
complementarity problem. Determine the optimal rule for replacing
the asset. You should ¯nd that it is optimal to replace the asset when
S falls below a certain value. Determine this value.
Be sure to experiment with the number of nodes and the approximation
interval to be sure that you solution is not adversely a®ected by these choices.

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