Lisø PhD Dissertation Manuscript - NTNU

Building economics of climate change
operating costs to vary between states. In the first-order
conditions in Equations 6 and 7, a definition of a state-
s dependent survival rate, , of the variable component is
inserted: a s
s
z
=
z
∂
2
1 .
∂
1 1 1
2 2 1
∂
2 2
( − ) ( )
1 1 1
2 2 1
∂
∂ ∂
∂
+
−
∂ ∂
∂
∂
v c z
∂
z z m ∂ ∂
d
S
s s
s v c z s
Âp
a = 1 s s
(6)
z z m
s=
1
( ) 1 s 1,2,3, ... 2 2 2
∂
− ,S
2 2 2
∂
∂
s s s
v c ∂z
= =
s s
(7)
z ∂z
∂m
This gives S + 1 equations determining the S + 1 levels
of effort.
In each period 2 states, effort is simply chosen so that
the marginal contribution to the net value of the services
produced by the building of the last money unit spent
on effort equals one. The period 1 condition has a similar
interpretation. However, the return of effort made in
period 1 is a probability weighted aggregate over periods
and states, i.e. it is the expected marginal contribution.
2 2
∂
By solving Equation 7 for ( − )
2 2
∂
∂
s s
v c
s s and inserting
z ∂z
into Equation 6, one gets another interpretation of how
period 1 effort is chosen:
1
1 1 1
∂
1
2 2
( − ) 1
1 1 1
2
∂
1
2
∂
∂z
s
S a
v c ∂z
s
= −d
∂m
 p s
(6’)
z ∂z
∂m
s=
∂z
∂m
1
∂z
s
a 1
∂m
2s
The expression ∂z
is the ratio of the marginal tech-
2
∂m
nical efficiency in producing z2 by the alternative factors
m1 and m2 . Measured at the optimal values of (m1 , m2 )
the ratio is state dependent. When the ratio is multiplied
by the discounting factor it can be interpreted as an economic
marginal efficiency, it captures both the technical
efficiency and the fact that discounting makes the price
of effort made early (in period 1) more expensive than
effort made later (i.e. in period 2).
Effort in period 1 is increasing in the expected
economic efficiency of m1 in producing z2 . Hence, it is
increasing in the expected survival rate of investments in
z made in period 1 and in the discounting factor. This
implies that period 1 effort is decreasing in the interest
rate used in the discounting. Note that d2 = (1 + i) −1 ,
where i is the relevant interest rate.
The relation between first and second period effort at
optimum depends strongly on the substitutability pattern
between z1 and m2 in the gs-functions. Two stylized
assumptions can be made:
769
(1) The ‘period 2-state s’ marginal product of effort
(at optimum) depends on the levels z 2s , irrespective
of the amount of effort undertaken in ‘state s
of period 2’.
(2) The ‘period 2-state s’ marginal product of effort
(in optimum) depends on the amount of effort
undertaken in ‘state s of period 2’, and not on
how much of the factor z is brought forward
from the first period.
To see the substantial content of these technical
assumptions consider one high and one low period 1
level of z: (z 1L , z 1H ). These levels taken forward to any
state in period 2 yield alternative starting values of z:
(a s z 1L , a s z 1H ). By starting values it is simply meant the z 2s ,
that will prevail for m 2s = 0.
Define m 2# as the effort needed to increase the level of
z 2s from a s z 1L to a s z 1H .
z 2s = a s z 1H = g s (a s z 1L ,m 2# )
Then consider a zˆ 2s > a s z 1H , and let m 2## be the effort
needed to increase the level of z 2s from a s z 1H to zˆ 2s .
Under assumption (1), the costs of increasing the
level of z 2s from a s z 1L to zˆ 2s will be equal to the sum
m 2# + m 2## . Hence, the marginal efficiency of m 2s is, at
optimum, independent of the starting values.
When the period 2 choice of effort is analysed and
described, the distinction between (1) and (2) is not
very interesting. Both of them produce a concave relation
between effort and z 2s . The effect of period 1 effort
on the period 2 choices does, however, differ strongly
between these two cases. In case two, the effort made in
period 1 does not only determine the amount of z that is
brought forward from period 1 to period 2, but also the
production technology of period 2. We proceed the
paper using assumption (1).
The optimal state of the building in each state in
period 2 will, under assumption (1), be independent of
the state of the building in period 1, and consequently of
effort in period 1. The optimal state of the building in
period 2, however, will affect the effort of period 1. The
optimal effort in period 1 will exceed the effort made
under a myopic optimization in period 1.
Think of a regional-level climate change scenario with
higher probabilities for states with a harsh climate. Furthermore,
assume that the survival rate of z is lower
under harsh climates. Then the expected technical and
economic efficiency of period 1 effort (m 1 ) in producing
z 2 will shift down as a response to climate uncertainty.
Consequently, period 1 effort will be reduced as a result
of (increased) uncertainty in the regional-level climate
change scenarios.
How can these algebraic exercises be used to define
and say something about the building economics of