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(i) No technology comes for free, i.e. k(c) > 0 c. marginal cost is not available, i.e. k(0) = .


A technology which produces at zero

  • (ii)

    less efficient technologies (that is technologies with higher c) are less expensive,

  • i.

    e. kc(c) < 0 c.12

(iii) k(c) is sufficiently convex, i.e.

kcc(c) > f(c)


kcc(c) > f(F 1(kc(c) + 1)) c.


The situation we want to analyze is captured by the following two stage situation. At stage one firms determine their technology mix by choosing their investment function xi(c), we denote by x(c) = x1(c), . . . , xn(c) the vector of all investment choices and by

X(c) =


n j=1

xj(c) the industry investment. As illustrated in figure 2, the investment

choice xi(c) of firm i determines which output can be produced at Marginal cost c.13

At the second stage firms choose their output at the spot market after having observed both the investment choices of all firms and the realization of demand. We denote by q(θ) = (q1(x, θ), . . . , qn(x, θ)) the vector of outputs of the n firms in scenario θ, and let

Q(x, θ) =


n i=1

qi(θ) be total quantity produced in the market.

In the following we now determine profits of firm i for fixed cumulative investment x(c), and given Spot market outputs q(x, θ). In the subsequent sections we will be more specific on the precise characterization of Spot market competition, which will be given by perfect competition, monopoly and Cournot-Competition. For the framework chosen, Spot market outcomes are always nondecreasing in θ. We can thus characterize the demand realization θ c i w h i c h w i l l g i v e r i s e t o p r o d u c t i o n c o s t c :

θ (x, q) =

c i

{(θ, c) : qi(x, θ) = xi(c)}

i.e. for given (qi(x, θ), xi(c)), if θ

c0 i

occurs, then firm i will produce at marginal cost c0.

F u r t h e r m o r e f o r a l l θ < θ c 0 i fi r m i w i l l p r o d u c e a t m a r g i n a l c o s t b e l o w c 0 a n d f o r a l l θ > θ c 0 i firm i will produce at marginal cost above c0. This is illustrated in figure 2.

11This assumption is not crucial, whenever k(c) = 0 for some c < c, then we would just obtain a corner solution, where capacity could be infinite at that technology. For ease of exposition we exclude this corner

solution 12This is a natural observation which can already be found in the pioneering contributions on peak load

pricing, compare Boiteux (1948). 13In a sense it is just the inverse of the marginal cost function, however since firms choose their investment

in di erent technologies it is much more convenient to choose this formulation.


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