Hedging Strategy and Electricity Contract Engineering - IFOR

Ax 3 V AxK1 G 1 V H AxK2 V b 1 G 1 V H b 2 b 3
8
b
b
z V end
] 7
z V end \ ]
b
b 7 \
b
_
199
be optimal and hence
b V zG 3H b G 1H V H zG 1 b 6
zG 2H
where b 3 V b 1 V H 1 G b 2 0 6 1 . Let xK1 and 2
andV
feasible solutions to max c T x Ax b 1 and max c T x Ax b 2 respectively.
A feasible solution to max c T x Ax b 3 is given by x xK 3 G xK1
V H 1 V xK2
since
By definition we have that
zG 1H V H 2H G 1 V zG V G V H
and zG since c not b _
3H
b 1 b c T xK1 c T xK2 c T x 3
T x 3 is necessarily optimal, c T x 3 .
Corollary 6.12
The marginal value of water,
increasing in V end .
is piecewise constant and
Proof The marginal value z b z b 7 \ ] 4¡b ] is given by the gradient of \ the optimal
bH value function , which according to Proposition 6.11 is piecewise linear
zG
and concave. Hence z b z b 7 \ ] 4¡b ] will be piecewise constant \ and decreasing
in b and the marginal value of water z V end
] 7
z V end \ ] will, because of the
b 7 \
opposite inequality sign ( instead of ) consequently be piecewise constant
and increasing in V end .
Corollary 6.13
The efficient frontier zG CH is piecewise linear and concave.
Proof Observe that zG CH is the one-dimensional version of the optimal value
function zG bH . The results then immediately follows from Proposition 6.11. _