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d X F B




k c c ( c ) f( FB

f( (c))( FB P (c)) q )

  • >

    0 due to assumption 2(iii), there k(c) is assumed to be convex

enough, which ensures that at the solution all technologies c [c, c] indeed are active.

At this point it is important to point out that the analysis of the first best case has already been dealt with extensively in the peak load pricing literature. However, that literature focuses on discrete instead of continuous technology choices. Here, we (re–)solve the first best case within our continuous model since it later serves as a benchmark for the case of imperfect competition, which is the main focus of our analysis. It is still worthwhile to mention, however, that the framework of continuous technology sets not only seems to make it easier to gather some intuition, but also crucially simplifies the mathematical exposition of the results.17


Pro t Maximization – Monopoly Outcome

Next we consider the case that where firms’ total profits are maximized, i.e. the case of monopoly (or collusion in case several firms are active in the market). The solution

is obtained analogously to the solution of the first best case.

maximization again only industry investment X(c) =


n j=1



In order to compare to the case of strategic behavior

Under firms joint Profit and industry output Q(θ) of firms in section 5, we

explicitly n firms which equally share investment and output.

T h e p r o fi t m a x i m i z i n g S p o t m a r k e t s o l u t i o n f o r g i v e n i n v e s t m e n t X 0 ( c ) a t e a c h d e m a n realization θ solves the standard condition of ”Marginal Revenue equals marginal cost”, i.e. d P ( X ( c ) , θ ) + P q ( X ( c ) ) X ( c ) = c θ . A g a i n w e d e fi n e t h e c r i t i c a l d e m a n d r e a l i z a t i o n θ c M , w h i c h m a k e s fi r m p r o d u c e a t m a r g i n a l c o s t g i v e n t h e y h a v e i n v e s t e d X 0 ( c ) a n d m a x profits at the spot market. The critical demand realization and the corresponding spot- i m i z e

market output QM (θ) are given by:

c θ M


QM (θ)



{(Q, θ) :

P q ( X ( c ) ) X ( c ) + c c ,

{ Q = X ( c ) , θ = θ c M

c}} .

(10) (11)

Having specified the outcomes at the spot markets under profit maximization for given

17Just to give an example: In their more than 50 page survey on the theory on peak load pricing in Crew and Kleindorfer (1986) exact characterizations of quantities invested and exact production of each technology is omitted: ”While the lemma indicates which plants are used it does not indicate the amounts of capacity ql and amounts produced by each plant in each period qli. To derive this is complicated and calls for a lengthy theorem which we do not state here (see Crew and Kleindorfer 1979a, pp-42–50, 63-65).”, see Crew and Kleindorfer (1986), p 45.


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