outputs Q i, since more aggressive cost functions will give a more advantageous position for the spot market competition.In principle the same argument holds true for the boundary c a s e ( a g a i n w e o b s e r v e t h e t e r m d Q i d x c i ) ) ) , h o w e v e r s i n c e t h e r e e x i s t o n l y s y m m e t r i c e q u of the overall game as we will show later on, the reaction of firms output choice will be i l i b r i a
irrelevant since firms anyhow are capacity constrained for those demand realizations.
Most importantly it remains to notice that the first order conditions of firm i do only i n v o l v e l e v e l s o f i n v e s t m e n t c h o i c e x i ( c ) , b u t n o t o n i t ’ s s l o p e x 0 i ( c ) . W h e n c h e c k i n g f potential deviations from a given equilibrium candidate with fixed investment X i(c), firm i faces a standard maximization problem can be solved by the point-wise first order conditions (point-wise, since all cross derivatives are zero). Derivation of entire equilibrium candidates o r
(e.g. symmetric candidate) on the other hand will involve the solution of a differential equation. We thus restrict attention to differentiable19 equilibrium candidates x∗(c), but
allow deviations to non–differentiable investment f u n c t i o n s x ∗ i ( c ) . W e n o w p r o v i d e c h a r a c t e r i z a t i o n ∗ o f e a c h fi r m ’ s i n v e s t m e n t c h o i c e x E Q (c) under strategic interaction of
a full firms:
Theorem 3 Strategic Behavior) There exists a unique
game, if the second c h o o s e s t o i n v e s t x ∗ E Q
order conditions established in lemma (c), which is uniquely characterized by:
equilibrium of are satisfied20.
the overall Each firm
x ∗ E Q
(x, c) :
)(n 1)b2x b(F (
θ c E Q
= c + b(n + 1)x
(n + 1)bx = c∗
c < c∗
c∗ ≤ c ≤ c∗
c∗ < c.
Proof see appendix 8.
19This ensures that
) is well defined. Similar restriction to di erentiable functions are found in
many contributions of the literature, compare for example the article on supply function competition by
Klemperer and Meyer (1989). 20As stated in lemma 5, the second order conditions are always satisfied if conditions (a), (b) and (c) are
satisfied. Especially the assumption of linear demand was made mainly in order to limit the computational burden when determining second order conditions. The symmetric candidate solution for the nonlinear case however would only change slightly and is given by the following di erential equation:
¢ F θ c E Q
f ³ θ c E Q 1 ) P 2 ´ (n q x + P q P q q θ c E Q = c + B ( n x ) + P q x
+ P q
θ c E Q