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Proof of Theorem 3

The proof of the theorem is in three parts. First we derive the symmetric equilibrium candidate, second we show that deviation from that candidate is not profitable and third we prove that an asymmetric equilibrium cannot exist.

Part I) Derive the Symmetric equilibrium candidate: A symmetric equilibrium candidate needs to satisfy the first order conditions given in lemma 5. Since firms can choose to invest at any level of marginal cost c 0, we directly consider the integrand of expression (18), which yields for the case of symmetry:

1

F θ c E Q

¢

+ k c ( c ) + f θ c E Q

¢

(n

1)

( 1 P q + P q q x ) x 0 P q x 0

P q x = 0

(36)

For x(c) close to zero, the second summand drops out and we simply obtain39 1 F (c)+kc(c). Equilibrium investment can thus not be desirable whenever the first derivative is negative, which is always the case for c small enough (compare assumption 2 (i)). Thus as in the case of perfect competition and monopoly investment is profitable only whenever c > c, where chas been characterized in lemma 3.

Whenever c > c, the interior solution has to satisfy the following differential equation which is obtained directly from (36):

x0(c) =

f θ c E Q

¢

(n

¢ F θ c E Q

1 ) P 2 q x + P q P q q x 2

1 ¢

+ kc(c) P q F θ c E Q

¢

1

kc(c)

¢

(37)

with

θ c E Q

= c + B ( n x ) + P q x

For

the

linear

case

where

P q q

=

  • 0

    and

P q

=

b this yields expression (20) of theorem 3.

Solutions of the differential equation (37) are illustrated as dotted lines in figure 9. The l o c u s o f a l l p a i r s ( x , c ) w h e r e ( 3 7 ) e q u a l s 0 i s d e n o t e d b y x 0 ( c ) , a l l s o l u t i o n s o f d i ff e r e n t i a l e q u a t i o n ( 3 7 ) p a s s t h r o u g h x 0 ( c ) w i t h s l o p e 0 . x 0 ( c ) i s g i v e n b y :

F θ c E Q

¢

1

kc(c) = 0

with

θ c E Q

= c + B ( n x ) + P q x

F o r m a l l y t h e s o l u t i o n s o f t h e d i ff e r e n t i a l e q u a t i o n s e x h i b i t n e g a t i v e s l o p e a b o v e t h e x 0 ( c locus, since firms cannot make negative investment, however, they find it optimal to make ) -

n o f u r t h e r i n v e s t m e n t a b o v e t h e x 0 ( c ) - l o c u s

. I n o t h e r w o r d s , f o r a l l p a i r s ( x , c ) b e l o w

3 9 N o t i c e f o r x = 0 w e o b t a i n θ c E Q

= c, compare expression (17), since B(0)

P q

0 = 0.

43

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