0 i

x (c) =

F (θ )b^{2}x_{j }F (θ )b^{2}x_{i }

)1 b(F (θ )1 b(F (θ

k_{c}(c) )1 k_{c}(c) )1

c EQ,j

f (θ

c EQ,j

c EQ,j

c EQ,i

f (θ

c EQ,i

c EQ,i

k_{c}(c))

0 j

x (c) =

k_{c}(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 x_{0 }(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 c^{∗ }and since k_{cc}(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 x_{i}(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 x_{i}(c) and x_{j}(c). As shown in part (i), both necessarily pass through (0, c^{∗}). Then for any c > c^{∗}, such that x_{i}(c) < x_{j}(c) the equations (39) and (39) imply that x_{i}(c)^{0 }< x_{j}(c)^{0}. However this is inconsistent , with the above statement (i.e. x_{i}(c) < x_{j}(c)).

We can thus conclude that an asymmetric equilibrium cannot exist.

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