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 differentiable^{19 }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:

5

equilibrium of are satisfied^{20}.

the overall Each firm

x ∗ E Q

(c) =

0

(x, c) :

# X:

x^{0}(c) with

=

^{F}(

c EQ

)

1 k_{c}(c)

f(

c EQ

)(n 1)b^{2}x b(F (

c EQ

)

θ c E Q

= c + b(n + 1)x

n

θ^{∗ }

(n + 1)bx = c^{∗ }

o

1 k_{c}(c))

c < c^{∗ }

c^{∗ }≤ c ≤ c^{∗ }

c^{∗ }< c.

(20)

# Proof

Proof see appendix 8.

dQ

i

dx

i c)

^{19}This 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). ^{20}As 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:

x^{0}(c)

with

=

¢ 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

x^{2 }

1 ¢

k_{c}(c)

+ P q

³

F

³

θ c E Q

´

1

k_{c}(c)

´

19