Hedging Strategy and Electricity Contract Engineering - IFOR

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198 Proofs
again be solved for by using the active water constraint.
Corollary 6.8
The pumping decision is related to the production decision by the marginal
value of water and
(i) Assume production at price S i , then the marginal value of water is u i .
Pumping occurs below price S j G j X iH , where the marginal value of
water u j
=
u i u j4 1.
(ii) Assume pumping at price S j with marginal value of water u j . Then production
occurs at price S i G j X iH where
=
u i u j
=
u i4 1.
Proof Observe that u j
1
i u j and that i 6 G such jH that u i u j u j4 6 1
1
and consequently u i i u
= j u j4 1 6 u i u j 6 u j4 1 . By definition
aWi
u i is the marginal value of water in the 6 interval b4i
b4i 1
and
aWiS 17 bWiS 17
from
bWi
7
the
= aWj
assumption about i 6 G , u jH i , which is the marginal value of
j bW
17 aWjS bW 17 jS
water in the interval b4
j 6 b4
j 7
1 .
Proposition 6.11
The optimal value function zG bH is piecewise linear and concave.
Proof The optimal value function can through the strong duality be written as
zG bH max c T x Ax b min b T y A T y c 6 y 0
8
(A.3)
Trough projection (A.3) can written as the minimum of all dual basis solutions
zG bH min b T y k k 1 6 88J8 6 q 6 b such that x Ax b 6 (A.4)
which proves that zG bH is piecewise linear.
To prove concavity we need to show that