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F i n a l l y u n d e r t h e a s s u m p t i o n f 0 0 ( θ ) < 0 , w h e n e v e r f 0 ( θ ) > 0 , a s s t a t e d i n l e m m a 5

condition (b) we obtain

f( f0(

c,EQ,i c , ) E Q , i )

> θ c , E Q , i

and thus (35) is satisfied. Concavity of profits for

the interior case can consequently be guaranteed, provided conditions (a), (b), and (c) as specified in lemma 5 are satisfied.

We now show that also for the boundary case second derivatives are negative, i.e. < 0. In order to do so we need to differentiate expression (19) with respect EQ) d 2 π i ( x , q 2 c 0 , c 0 0 d(x ) t o d x ( c ) i , which yields: )

d π 2 i ( x , q E Q d ( x ( c i ) )2

)

=

Z ci E Q , i

d d x ( c i )

Ã

P QEQ

, θ ¢ + P q

QEQ

¢

x à 1 +

d Q E Q d x ( c i i )

!

c i !

dF (θ) < 0 ?

Notice that according to Leibnitz rule we need to consider only the derivative of the inte- grand, the derivative wrt to the lower border cancels out, since the integrand evaluated at

θ c E Q , i

equals zero.

The derivative of the integrand obtains as follows:

Ã

1+

d Q E Q d x ( c i i )

2 P q Q E Q

¢

+ P q q

QEQ

¢

x à 1 +

d Q E Q d x ( c i i )

!!

+ P q

QEQ

¢

x

Ã

d 2 Q d (x 2 (ci) E Q i )

!

<0 ?

When analyzing the sign of

d 2 Q E Q d ( x c i ) ) i

2

, again assumption (c) as stated in the lemma is crucial:

the steeper the marginal cost functions of the other firms j, the less pronounced are their r e a c t i o n s d Q E Q i d x c i ) t o c h a n g e d c o s t f u n c t i o n o f fi r m i . I n t h e l i m i t , w h e n e v e r fi r m j constrained (marginal cost vertical) it will not react at all to changed cost functions of firm i s c a p a c i t y

i.

We thus obtain

d 2 Q d(x E Q c i ) ) i

2

  • >

    0.

Furthermore due to assumption 1, also the first summand

is negative (both for the case of linear and nonlinear demand). Concavity of profits for the

boundary case can consequently be guaranteed, provided conditions (a), and (c) as specified in lemma 5 are satisfied.

42

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