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# The derivative of the Integrand reads as follows:

X Ã Ã j=6 i

¦ j dR

d x(c)

X Ã Ã j=6 i

R j!

!

h

P q q E Q i

i

+

( P q ) q E Q i

( θ c E Q , i

## Ã jX=6 i

R j !

) f 0 ( θ c E Q , i

)

( P q )

# !

f ( θ c E Q , i

)+

f ( θ c E Q , i

)

!

d θ dx(c) c E Q , i

< 0?

For the case of linear demand (condition (a) of the lemma) and concave marginal cost

f u n c t i o n s o f t h e r i v a l s , i . e . x 0 0 j ( c ) < 0 ( c o n d i t i o n ( b ) o f t h e l e m m a ) , w e o b t a i n

d R dx ¦ j c)

0 as

derived in appendix 8 expression (30).

# Thus whenever conditions (a) and (c) of lemma 5

are satisfied, this term is negative and can be omitted.

We now check only the remaining terms.

After plugging in for

d

c EQ,i

dy

c i

=

P q ³ 2 +

Pj=6

i

R j ´ , a s d e r i v e d i n a p p e n d i x 8 e x p r e s s i o n ( 2 9 ) w e o b t a i n :

f ( θ c E Q , i

) " ( P q )

f 0 ( θ c E Q , i

) P 2 q q E Q i

## Ã jX=6 i

R j !

Ã X j 6 = i

R j ! Ã

( P q )

Ã

2+

Ã X j 6 = i

Ã 2+ X j 6 = i R j ! ! <0 ? # R j ! !

+

# and after slight simplification:

Ã f ( θ Ã jX=6 i c E Q X , i ) 2 P q + f 0 ( θ c E Q , i ) P 2 q q E Q i R j ! Ã R <0 ? j ! 2+ ! j =6 i N o t i c e t h a t t h i s i s a l w a y s s a t i s fi e d f o r f 0 ( θ ) < 0 . H o w e v e r , w h e n e v e r f 0 ( θ ) > 0 w e c o u l d g e t p r o b l e m s . R e a r r a n g i n g u n d e r t h e a s s u m p t i o n f 0 > 0 t h e f o l l o w i n g s h o u l d h o l d t r u e :

f ( θ c E Q , i f 0 ( θ c E Q , i ) )

>

³

Pj=6

2

i

R j ´

( P q )

Ã

2+

# X

j =6 i

R j !

q E Q i

?

(35)

I n a p p e n d i x 8 e x p r e s s i o n ( 3 3 ) w e o b t a i n a n u p p e r b o u n d f o r q E Q i

. By making use of

(33)38

we can find an upper bound on the right hand side of (35) and obtain:

³

Pj=6

2

i

R j ´

( P q )

Ã

2+

j =6 i

R j !

q E Q i

³

Pj=6

2

i

R j ´

θ c E Q , i

c Ã 1 +

# X

j =6 i

R j ! !

## 38This is given as follows:

q E Q i

θ c E Q , i c ³ 1 + R j ´ ( P q ) Pj=6 Pj=6 i i ³ 2 + R j ´

41

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