Lisø PhD Dissertation Manuscript - NTNU

Building economics of climate change
For simplicity, we include maintenance in the conversion
cost function whenever a conversion takes place.
s s
s
V ( M) = v ( W, z ) −c( W,
z ) −m
3
d 3
Mi
+ T all s∈S 2
d
2 2 2 2 2 2
(9b)
V 2s (T)=T 2s all s ∈ S Ti (9c)
The V 2 -functions above are a kind of indirect utility
functions and their values depend on the z and W values
that are chosen in period 1. The choice of action in
period 2 can be seen as a three-step procedure. First,
the state of period 2 is observed. Next, the optimal effort
under the M-strategy and the conversion activity under
the C-strategy are calculated. In the last step, the value
of the building under each of the three strategies are
compared, and the strategy yielding the highest value is
chosen.
Similarly, the owner has three different possible
actions in period 1. The object function for period 1
choice will be somewhat more complex than the object
functions for period 2. There are two reasons for this:
(1) When the choices of period 1 are made, the
future impact of climate change on building
enclosure performance is not known. The value
of the building under each of the possible
actions in period 2 is affected by the choices of
period 1.
(2) The choices of period 1 will partly determine
which actions are optimal in period 2. Hence the
sets S Mi , S Ti and S Ci are affected by choices made
in period 1.
The period 1 value of the building under each of the
strategies can be written as Equations 10a–c. The value
of the building (NV 1 ) in period 1 is given in Equation
10d.
NV( M) = v ( Wˆ, z ) - c ( Wˆ,z)
−m
+ p dV ( C)
1 1 1 1 1
Â
2s 2 2s
CM
s∈S 2s 2 2s 2s 2 2s
 Â
+ p dV ( M) + p dT (10a)
MM TM
s∈S s∈S 1 1 1 1
NV( C) = v ( W, z ) −c( W, z ) −C(
Wˆ, W)
+
2s 2 2s p dV ( C) +
2s 2 2s
p dV ( M)
+
 Â
CC MC
s∈S s∈S Â
2s 2 2s
p dT
(10b)
TC
s∈S NV(T) = T 1 (10c)
NV 1 = max(NV(T), NV(C), NV(M)) (10d)
771
Equations 9a–10c constitute a complex dynamic
stochastic optimization problem. Instead of spelling out
complete and general solutions to this optimization
we will characterize some important properties of the
solution.
The paper is primarily addressing the uncertainty of
future climate affects and the behaviour of owners of
buildings. For this reason, the analysis in the remaining
parts will focus on the choices made in the first period of
the model. Period 2 choices are treated as they are
nested within the period 1 choices.
The maintenance strategy
Previously in the paper we analysed the choice of effort
(m) in the case where no conversions and no termination
took place in the first two periods. The conclusion
from this analysis is altered when taking into consideration
that the owner knows that for some climate
change scenarios she will terminate the building and
for other scenarios, she will choose to rehabilitate or
convert the building.
The first order condition for the choice of effort in
this more general case will be:
1
∂
( − ) (
1
∂
∂ ∂
∂
+
−
∂ ∂
∂
∂
v c z
∂
z z m ∂ ∂
d
1 1
2s
2s
1
2 2sv
c z
1 1 Â p ) a
2s
2s
1
z z m
= 1
s∈S M
s
(11)
As long as the sets S C and S T are non-empty, the
expected marginal return on period 1 effort, for any
level of m 1 , is lower than it is when conversions and
‘early scrapping’ is not a part of the action set. Hence, in
the presence of climate uncertainty a lower level of
effort is put into the maintenance of a building.
Furthermore, the probability of scrapping the building
or converting it in period 2 will be higher if a given
climate change scenario was anticipated than if it comes
as a surprise. If the climate change scenario is not anticipated,
the effort in period 1 will be chosen without
taking the possibility of scrapping or conversion in order
to adapt to a changed climate into consideration. Consequently,
a higher level of effort is chosen, and the state
of the building, measured by z, will be better than if
effort is chosen according to Equation 11. This will
increase the value of V 2 (M) for any s. As a result, for
some s where V 2 (C) or V 2 (T) gives the maximum of
(V 2 (C), V 2 (T), V 2 (M)) when m 1 is chosen according to
Equation 11, V 2 (M) will give the maximum when the
owner did not take climate change into consideration.
In order to enhance the understanding of the consequences
of choosing the M-strategy in period 1,
Equation 10a is rewritten.