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demand realizations18

θ c E Q , i

a n d t h e c o r r e s p o n d i n g s p o t - m a r k e t o u t p u t s q E Q i

(θ):

θ c E Q , i

q E Q i

(θ)

= =

n ( θ , c ) : c = c E Q i

x i ( c E Q i

(θ))

θ, i

( θ ) θ o ,

(16)

I t i s w o r t h - w i l e t o n o t i c e t h a t t h e c r i t i c a l d e m a n d r e a l i z a t i o n θ market equilibrium has the following property: c E Q , i

of the Cournot spot

P (QEQ

, θ c E Q , i

) + P q ( Q E Q

) q E Q i

=c

θ c E Q , i

= B(QEQ

)

P q ( Q E Q

) q E Q i

+c

(17)

# Where the right expression just makes use of the initial separability assumption of demand

which was assumed to be given by P (Q, θ) = θ

B(Q).

Having solved for the outcomes at the spot market for fixed investment choice, we can now proceed and solve for the overall equilibrium with respect to firms investment choices. Again we make use of the general first order conditions derived in lemma 1, and derive the optimal capacity of firm i for fixed investment X i(c) of all other firms. The first order conditions of firm i for the case of strategic capacity choice are summarized in lemma 5.

Expression (18) is directly obtained from (3) This is due to the following two observa-

tions: first

dP (QEQ dx , ) c 0 , c 0 0 ) x i ( c ) = P q

µ

1+

d Q E Q ( ) d x c 0 i , c 0 0

x i ( c ) , a n d s e c o n d b y t h e d e fi n i t i o n o f θ c E Q

as

established in (16), we have P (QEQ

( θ c E Q

) , θ c E Q

)

c + P q ( Q E Q

( θ c E Q

))xi(c) = 0.

# obtain expression (19) from (4). We obtain the following optimality conditions for the case

of strategic interaction among firms:

Lemma 5 Optimality Conditions, Strategic Firms) .

(i) First order conditions: F o r a n i n t e r i o r c h a n g e a e c t i n g c [ c 0 , c 0 0 ] w h e r e ( c 0 < c 0 0 c i ) w e o b t a i n :

## 00

i(x, qEQ dx c0,c00)

)

=

c

c0

# Ã

1

F ( θ c E Q , i

) + k c ( c ) + f ( θ c E Q , i

)

# Ã

d θ dc c E Q , i

d Q dx c0,c00) E Q i

!

P q x i ( c )

!

dc = 0

(18)

nd

Ã

d θ dc c E Q , i

d Q dx c0,c00) E Q i

!

=

# X

j =6 i

³

P q

+ P 1 q q x j ( c E Q j x 0 j P q ) x 0 ´ j ( c E ) Q j

( c E Q j

)

1 8 R e m e m b e r : t h e c r i t i c a l d e m a n d r e a l i z a t i o n i s t h a t d e m a n d r e a l i z a t i o n θ c E Q , i

that will give rise to

production cost c for firm i in the Spot market Cournot equilibrium. In the present context this is just the

i n v e r s e o f c E Q i

(θ).

17

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