realizations and the realized market prices will change as investment xi(c) of firm i is
Lemma 1 First Derivative
(i) Interior Case: Consider a variation of investment xi(c), a ecting all technologies c ∈ [c0, c00], where c0 < c00 ≤ ci. This leads to the following change in firm i’s profits:
dπi(x, q) dx c0,c00)
Z c c 0
F ( θ c i ) + k c ( c ) + f ( θ c i )
d θ dc c i
P (Q, θ)
dP (Q, θ) dx c0,c00)
(ii) Boundary case: Consider a variation of investment xi(c), a ecting all technologies c ≥ ci. This leads to the following change in firm i’s profits:
dπi(x, q) dx ci)
Z θ θ c i i
P (Q, θ)
dP (Q, θ) dx ci)
Proof see Appendix 8.
Lemma 1 gives the impact of a variation of firm i’s investment on it’s profits both for the interior and for the boundary case14. The analysis up to now does not yet specify the type of (strategic) behavior of firms both at the spot markets and the investment stage. In order to solve for the cases of Perfect competition, Monopoly and of strategic interaction in the subsequent section, these first derivatives of lemma 1 provide a valuable starting point.
Perfect Competition – Welfare Maximization
As a Benchmark we determine the case of spot markets and the investment stage. competition, only industry investment X(c) 15
perfectly competitive behavior both at the In our framework, for the case of perfect
xj(c) and industry output Q(θ) =
q j ( θ ) m a t t e r . I n o r d e r t o c o m p a r e l a t e r o n t o t h e c a s e o f s t r a t e g i c b e h a v i o r , h o w e v e r ,
we will explicitly consider n firms which equally share investment and output among each other. 14A change in total investment by dx which a ects all c ≥ c0 can be determined by setting c00 = ci and
just summing then over (3) and (4).
15This has already been analyzed in the peak load pricing literature, the first best solution of our frame- work, however, will serve as a valuable benchmark in order to compare to the case of strategic interaction of firms as analyzed in section 5. Furthermore in the framework chosen, we obtain a smooth solution, which makes its characterization as given in theorem 1 rather short, in contrast to previous contributions on that topic.