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# Now we introduce the following notation for all j = 1, ..., n,

R j : =

Ã

P q ( Q , θ ) C j q q ( q j )

P q q ( Q , θ ) q j P q ( Q , θ ) !

.

(26)

N o t e t h a t R j c o r r e s p o n d s t o t h e s h a r e t h a t fi r m j h a s o n t h e t o t a l c h a n g e i n Q . W e c a n

now solve the equation system by summing up over all reactions

P

n j=1

d q dx j c)

, and obtain:

dQ dx(c)

=

C i q q

C i q q

P q

dQ dx(c)

Ã n X j = 1

R j !

dQ dx(c)

=

( C i q q

C i q q

P q ) ³ 1 +

# P

n j=1

R j ´

dQ

i

dx

c)

We finally obtain for

:

dQ dx i (c)

= ( 1)

Ã jX=6 i

R j !

dQ dx(c)

=

(

( C i q q

1 ) C i q q

³

Pj=6

i

P q ) ³ 1 +

P

R n j=1 j ´ R j ´ ( 1 , 0 ) .

(27)

( i i ) 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 t θ c E Q , i

:

# Now we derive the reaction of the spot market equilibrium at the critical demand real-

i z a t i o n θ c E Q , i

( x c i ) t o a c h a n g e i n i n v e s t m e n t l e v e l o f fi r m i a t s o m e s p e c i fi c m a r g i n a l c o s t c 0 .

T h e c r i t i c a l θ c E Q , i

is characterized through the following equation system:

i EQ +Q

c0 ,θ EQ,i

q j EQ )+P q

i EQ +Q

c0 ,θ EQ,i

i q EQ )+P q

q j j EQ C (q

) =0

c

=0

θ c E Q , i j : P ( q E Q i i : P ( q E Q i D i ff e r e n t i a t i o n w r t x c i y : i e l d s :

(28)

µ

1+

µ

1+

j

q :P

i:

q P

dQ dx

EQ i c)

dQ dx

EQ i c)

+ P q

d q dx EQ j c)

C j q q

+ P q +

d dx c EQ,i c)

d q dx EQ j c)

+

d dx c EQ,i c)

=0

=0

# Now we need to recover

d Q dx E Q i c)

from equations j =6

i. We obtain by summing up and

m a k i n g u s e o f R j a s d e fi n e d a b o v e :

d q dx(c) E Q j

=

R j

d Q dx(c) E Q i

R j

# Ã

1+

d dx c EQ,i c)

P q

!

Summing up over all equations:

d q dx EQ j c)

and solving for

d Q dx E Q i c)

yields the following system of two

j:

i:

c)

=

µ

1+

d Q i E Q dx

P q

( 1)(

## Pj=6

i

R j ) 1 +

c d EQ,i

c) dx

### Pq

d Q dx E Q i c)

1 + ( P j 6 = i R j ) + P q + d dx c EQ,i c)

=0

37

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