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0 i

x (c) =

F (θ )b2xj F (θ )b2xi

)1 b(F (θ )1 b(F (θ

kc(c) )1 kc(c) )1

c EQ,j

f (θ

c EQ,j

c EQ,j

c EQ,i

f (θ

c EQ,i

c EQ,i

kc(c))

0 j

x (c) =

kc(c))

exist. The first order conditions for the optimal investment decision in an asymmetric equilibrium are the following system of differential equations:

(39)

The proof is in two parts:

(40)

( i ) T a k e a p o t e n t i a l e q u i l i b r i u m c a n d i d a t e x 0 a n d d e n o t e b y c 0 i t h e t e c h n o l o g y w h e r e firm i starts to have positive investment. Wlg sort firms such that firm 1 is the one investing into the lowest technology according to the candidate equilibrium x0 (i.e. c 0 1 = m i n [ c 0 i ] ) .

W e fi r s t s h o w n o w t h a t x 0 c a n n o t b e a n e q u i l i b r i u m i f c 0 1 < c : N o t i c e t h a t x 0 1 ( c ) > 0

f o r a l l c > c 0 1 c a n o n l y b e a s o l u t i o n i f :

1

F ( 0 + c 0 1 ) + k c ( c 0 1 ) = 0

(41)

Since the above equation is solved by cand since kcc(c) > f(c) for all c (see assump-

tion 2 (iii)) we necessarily have 1

F ( 0 + c 0 1 ) + k c ( c 0 1 ) < 0 , c o n t r a d i c t i n g ( 4 1 ) .

Since furthermore xi(c) cannot become negative, we can conclude that all (potentially

asymmetric) equilibrium candidates pass through (0, c).

(ii) We now derive the slopes of the solutions of the above equation system (given by expressions (39) and (39)) at the point (0, c). Since the right hand sides of the a b o v e e q u a t i o n s y s t e m y i e l d 0 0 , t h e s l o p e s c a n b e d e t e r m i n e d b y a p p l y i n g t h e r u l e o f l H o p i t a l , t h e r e s u l t i s g i v e n i n e x p r e s s i o n ( 4 3 ) , a n d w e o b t a i n x 0 i ( c ) = x 0 j ( c ) .

Now suppose there exists an asymmetric equilibrium xi(c) and xj(c). As shown in part (i), both necessarily pass through (0, c). Then for any c > c, such that xi(c) < xj(c) the equations (39) and (39) imply that xi(c)0 < xj(c)0. However this is inconsistent , with the above statement (i.e. xi(c) < xj(c)).

We can thus conclude that an asymmetric equilibrium cannot exist.

47

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