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( f^{0}(

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 ^{c}i E Q , i

d d x ( c i )

Ã

P Q^{EQ }

, θ ¢ + P q

Q^{EQ }

¢

x Ã 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 )

!Ã

2 P q Q E Q

¢

+ P q q

Q^{EQ }

¢

x Ã 1 +

d Q E Q d x ( c i i )

!!

+ P q

Q^{EQ }

¢

x

Ã

d 2 Q d (x 2 (c_{i}) 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.

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