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Now plug in into the equation of firm i. It is important to notice that

d q dc EQ i

= x 0 i ( c ) =

C i q 1 q ( q EQ i

)

. In total we obtain:

d θ dc c E Q , i

d θ dc c E Q , i

d θ dc c E Q , i

EQ + d c EQ,i ( P dc 1 q P Pj=6 ) d q i j 6 = i R j + P q R i dc j R j + à 2 P q d 1=0q E Q i Pj=6 dc R 1+ j R j i à ! = 1 Pj=6 Pj=6 + ( P q ) d i i q E Q i R j 1 2 R Pj=6 Pj=6 dc j ! R j 1+ 1+ i i Ã1 + à ! ( Pj=6 P q 2+ ) C i 1 Pj=6 i q = 1+ Pj=6 q R j R j 1+ ! i i

d θ dc c E Q , i

= 1+

X

j =6 i

R j +

(

P q )

C i q q

Ã

2+

X

j =6 i

R j !

>0

It can be verified by simple algebraic manipulation that the above can also be rewritten:

d θ dc c E Q , i

=

C i q q

C i q q

P q

Ã

1+

X

j =6 i

R j + R i !

=

C i q q

C i q q

P q

Ã

1+

n

X

j=1

R j !

(32)

( i v ) P r o p e r t i e s o f t h e s p o t m a r k e t e q u i l i b r i u m , a p p r o x i m a t e q E Q i

( θ c E Q , i

)

L a t e r o n w e n e e d t o h a v e a n u p p e r b o u n d o n q E Q i

( θ c E Q , i

). This is obtained as follows.

Notice in the linear case the spot market equilibrium (characterized in (28)) can also be

written as follows:

j : θ c E Q , i i : θ c E Q , i

+ P q · ( q E Q i + Q E Q ) + P q q E Q j + P q · i ( 2 i q E Q i )c + Q E Q

C j q q q E Q j

  • +

    posj

=0 =0

Notice if Cqqq(q) > 0, then posj 0.

S o l v e f o r q E Q j

, t h e n s u m m i n g u p a n d s o l v i n g f o r Q E Q ( ( θ c E µ Q , i i + q E Q j = R + posj j Q E Q θ c i E Q , i ( P q ) y c 0 i )) yields:

Q E Q i

=

Pj=6

i

R j

1+

Pj=6

i

R j

µ

θ c E Q , i + posj ( P q )

y c 0 i

P l u g g i n g i n t o t h e fi r s t o r d e r c o n d i t i o n o f fi r m i a n d s o l v i n g f o r q E Q i yields: q c ³1 + E Q i P Pj=6 q θ c E Q Pj=6 , 1 i i R i j ´ R j pos = R j Pj=6 Pj=6 R j 2+ 2+ i i

(33)

39

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