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## Due to assumption 1(ii) the integrand is negative. Notice: Since the integrand of the first

o r d e r c o n d i t i o n i s c o n t i n u o u s a t θ c F B

(equals zero), the derivative with respect to this lower

limit drops out according to Leibnitz rule.

# Proof of Lemma 5 Preliminaries: Properties of the Spot–market Equilibrium

In order to proof lemma 5, we need to precisely characterize the Cournot-spot market equilibrium and it’s reaction to changed investment of firms. In section 5 we characterized the spot market equilibrium somehow unusually without explicitly making use of marginal c o s t f u n c t i o n s C j q ( q ) b u t o n l y i n t e r m s o f t h e i n v e s t m e n t s x j ( c ) m a d e b y e a c h fi r m . O n throughout appendix 8 we will make use of the usual notation in terms of marginal cost l y C j q ( q ) , a s a l r e a d y e m p h a s i z e d m a r g i n a l c o s t C j q ( q ) a r e j u s t t h e i n v e r s e o f t h e i n v e s t m e n t

f u n c t i o n x j ( c ) . F o r q 0 = x j ( c 0 ) w e o b t a i n t h u s t h e w e l l k n o w n r e l a t i o n s h i p : x 0 j ( c 0 ) =

C j q 1 q ( q 0 )

.

(i) Properties of the spot market equilibrium For xed θ:

Derive the reaction of the spot market equilibrium for fixed values θ to a change in investment level of firm i at some specific marginal cost c (denoted by dx(c)). The spot m a r k e t e q u i l i b r i u m f o r g i v e n m a r g i n a l c o s t f u n c t i o n s C j q ( q j ) , f o r j = 1 , . . . , n i s c h a r a c t e r i by the usual equilibrium conditions for an asymmetric Cournot-equilibrium: z e d

P (QEQ

q , θ) + P (QEQ

j EQ , θ)q

P (QEQ

q , θ) + P (QEQ

i EQ , θ)q

j: i:

x c i )

)

= C j q ( q E Q j = C i q ( q E Q i

(25)

T h u s i n v e s t m e n t o f t h e a m o u n t x c i w i l l a l l o w fi r m i t o p r o d u c e n o t a t C i q ( q ) b u t a t l o w e r

m a r g i n a l c o s t g i v e n b y C i q ( q

x c i ) ( w h e r e x c i i s s m a l l , w i t h x c i & 0 ) . D i ff e r e n t i a t i o n o f

expression (25) with respect to dx(c)

yields:37

j: i:

( P q ( Q , θ ) + P q q ( Q , θ ) q j ) ( P q ( Q , θ ) + P q q ( Q , θ ) q i )

dQ

c) dx dQ

dx

c)

+ P q ( Q , θ ) d q j + P q ( Q , θ dx ) d q i d c) x c)

= C j q q ( q j ) d q j = C i q q ( q i dx ) d q i d c) x c)

C i q q ( q i )

Solving fo function, and

r

dx dx d q i d q c) j c)

, the reaction of spot market output of firm i to it’s change in the cost the reaction of spot market output of the other firms j:

j:

dq dx

j c)

=

i:

dq dx

i c)

=

³ ³ P q ( Q , ) + P q q ( Q , ) q j C j q q ( q j ) P q ( Q , ) ´ ( P q ( Q , ) + P q q ( Q , ) q i ) C i q q ( q i ) P q ( Q , )

dQ dx c)

´

dQ dx c)

+

C i q q ( q i ) C i q q ( q i ) P q ( Q , )

37From here on we drop the superscript to equilibrium outputs of stage 2.

EQ, in order to save notation. In what follows we always refer

36

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