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Proof of Lemma 6

F i r s t d e t e r m i n e x 0 E Q

( c ) : S i n c e x E Q

to

zero

at

c.

In

order

to

determine

x (c) 0 E Q = 0 both numerator and denominator of (37) equal (c) we apply the rule of l’Hopital and obtain:

ccl i m x 0 E Q

(c) =

lim

cc

f θ c E Q

¢

(n

1 ) ³ P 2 q x E Q

F θ c E Q + P q P q q x ¢ 1 E Q

¢2

kc(c) ´ + P q F θ c E Q

¢

1

kc(c)

¢

Differentiation of numerator and denominator with respect to c yields40:

x 0 E Q

(c) =

f (c) (n

f ( c ) θ 0 E Q 1 ) 0 P 2 q x E Q + P (c) q

k c c ( c ) f ( c ) θ 0 E Q

(c)

kcc(c)

¢

(42)

W h e r e a c c o r d i n g t o e x p r e s s i o n ( 3 7 ) , 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 c E Q

is given by:

θ c E Q

( c ) = B ( n x E Q

)

P q ( n x E Q

) x E Q

+c

a n d d i ff e r e n t i a t i o n w r t c y i e l d s θ 0 E Q ( c ) = x 0 E Q

P q ( n + 1 )

P q q n x E Q

¢ + 1. We can thus

r e p l a c e θ following 0 E Q i n ( 4 2 ) a n d t h e n s o l v e t h e r e s u l t i n g q u a d r a t i c f o r m f o r x unique positive solution: 0 E Q

(c). This yields the

x

0 EQ

(c) =

k c c ( c )

(n + 2)f(c) +

q

(kcc(c)

P q

(n + 2)f(c))2 + 8f(c)(kcc(c)

(0)

f(c)

4

f(c))

(43)

I n o r d e r t o p r o o f t h e l e m m a w e n o w c o m p a r e x 0 E Q

( c ) t o x 0 F B

(c). Remember in section

3 w e o b t a i n : x 0 F B

(c) =

1 n

k c c ( c ) f ( c ) P q ( 0 ) f ( c )

. Direct comparison of both results reveals now:

x µ 0 EQ (c) kcc(c)(n

x

0 FB

4)

(c) =

P 1 q ( 0 ) f ( c ) 4 n

(n(n + 2)

4)f(c) + n

q

(kcc(c)

(n + 2)f(c))2 + 8f(c)(kcc(c)

f(c))

  • >

    0?

(kcc(c)

f(c))

µ

k c c ( c )

2 f(c)

n n

1 2

  • >

    0?

Since by assumption 2 (iii) we have kcc(c)

f(c) > 0, we observe over–investment

(with respect to first best investment) in efficient production technologies if and only if

k c c ( c ) > 2 f ( c ) n 1 n2

,

which proofs the lemma

.

40Notice as c c, we obtain x plugged in.

EQ

0 a n d θ c E Q

c, after di erentiation these values can directly be

48

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