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# APPENDIX

(64)

) ( ) [ ] 1 4 1 1 1 1 4 1 1 1 1 1 1 1 1 4 1 3 * 1 1 0 1 4 1 1 1 4 1 3 * 1 1 2 1 0 1 4 * 1 1 1 0 1 1 4 0 1 1 3 1 0 +γ * 1 1 / / * l o 1sm 2 g / * l o g l o g l o g l o g l o g + + + + + + + + + + + = s m s 1 m s m s h s h s h y s m m S s m s m s m s m s h S h s h s h s h s h s h I C T S T R T I C T S T R T T L K Y α α γ α α γ α µ γ µ γ γ γ α γ µ γ γ γ γ α α α α α

×

( 6 5 ) h S * 1 l o g γ =

1sh 0

+γ

1sh 1

Y * 1 l o g γ +

1sh 2

T * 1 l o g γ +

1sh 3

log STR1 +γ

1sh 4

s h S h I C T 1 1 * 1 / l o g γ µ

(65’) log S

* m1

=γ

1sm 0

+γ

1sm 1

Y * 1 l o g γ +

1sm 2

T * 1 l o g γ +

1sm 3

logSTR1 +γ

1sm 4

s m S m I C T 1 1 * 1 / l o g γ µ

# The solution for initial levels of patents is rather complex as it depends also on the initial level of technology which, in turn, depends on the aggregation of initial level of patents (eqs. (46)- (48)). A complication lies in the fact that we find a solution in logs of variables which depends on the sum of the variables themselves. However, although numerical solutions are always possible, we need a closed form solution to be used for economic analysis (an appealing application is comparative dynamics). To find this we need the sum of the patents flows:

(66)

[ [ 3 1 3 3 1 3 log Pat 3 * 1 3 3 1 3 5 1 1sm 0 * i1 1sh 0 2 1 1 0 l o i1 g i1 l o g i1 l o g γ s m γ 1sm 1 1sh 1 i1 s m s m i s m i1 s h s i1 h s h i s h i i T F A T F A H K a d i s t β β β γ β β β β γ β β β β β * + + 0 1 i1 + + + i1 + + + + + γ γ 1sh 2 i1 = 1sm 2

3 log HKR * 1 l o g s h 0 1 i1 T β

µSs1 /γ 1sh

]+

3 * 1 l o g s m i1 T β

µ Sm1

/γ 1sm

]+

1 1 1 4 * 1 1 4 / l o g i p i i A T F β µ β β +

) [ 1 1 1 4 1 3 1 0 1 4 1 3 0 1 1 4 0 1 1sh 0 1 3 1 0 1 /γ 1sh / * l 1 4 Sh o g l o g + + + + + + + y s m s γ m 1 s m S h s h S T R I C T S 1sm A T R = ICT * µ Sm1 L K γ α µ γ γ γ µ α γ γ 4 Sm 1sh α α α α ]1 1sh 1 α 1Sm 1 sh 4 1sm 1 1 4 sm α α γ α

/ γ 1sh

) +

## F

=

(α

1 1

+α

1 4 sh

γ 1shγ

1sh 2

+α

1 4 sm

γ 1smγ

1 2 sm

)(1 γ

1sh 1

α 1shα

1 sh 4

γ

1sm 1

α 1Sm

α

1 4 sm

)1

(67)

[ ] [ ] { } * 1 1 4 1 3 1 3 1 3 1 3 1 1 4 1 1 3 1 1 1 3 1 0 1 3 1 1 3 1 3 1 3 0 1 1 5 0 1 1 i1 2 1 1 1 1 0 * 1 l o g ) ( 1sh 2 0 e x p / 1sh l o g l o g e x p T F F F A A A H K R H K a d i s t P a t i i s m i s m i s h 1 i s h i 1sh p S m i s m s m i s m s m i s m S s i s h i s h i s h i i i i i i β γ β β γ β β β µ β µ β γ β 1sm 2 γ β µ β γ β γ β β β β β + + + + + + + + + + + + =

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