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APPENDIX

21 3sh (85) Dx5 = β 21β

21 3sm x2 + β 21β

4 21 x3 + β 21β

x1 β 21x5

31 3sh (86) Dx6 = β 31β

31 3sm x2 + β 31β

4 31 x3 + β 31β

x1 β 31x6

41 3sh (87) Dx7 = β 41β

41 3sm x2 + β 41β

4 41 x3 + β 41β

x1 β 61x7

Equations (78)-(80) and (83)-(87) form the autonomous system of linear differential equations of degree one we use to study the dynamics of the model in the case in which steady state r.o.g.’s of patents flows are equal. The stability propriety of such a system may be studied by considering the characteristic equation of the following matrix and applying the Routh-Hurwitz necessary and sufficient conditions

x8 )

µ p1 =

Pat T * 1

* i1

Pati1 µ p1 = 1 T

µ p1

(83)

x1 β11x4

( 8 2 ) x x e i

8

(1+ xi x8 )

1 1 4 1 1 3 x β β +

Pat * 1 T

* i1

Pat * 1 T

* i1

i

(xi+2

( 8 4 ) 1 1 3 1 1 4 D x s h β β =

1 1 3 1 1 2 x s m β β +

i

Pat T * 1

* i1

Dx8 =

x

i

i+2

i e Pat T T * * i1 xi + 2 x8 1 * µ p1 x8 µ p1 1 * i1 Pat µ p1

(88)

α1 γ sh1γ γ sm1γ

1 sh1

1 sm1

α 1α γ 1sh γ 1sm

1 sh2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

i P a t P a t * 1 * 2 1

i P a t P a t * 1 * 3 1

i U S P a t P a t * 1 * 1

2 1 s 1 m α α

µ p1

i

µ p1

i

µ p1

i

1 3 1 γ sh1γ γ sm1γ α α 1 2sh 1 2sm

0

Pat Pat * 11 * i1

µ p1

i

4 11 β 11β

11 3sh β 11β

11 3sm β 11β

4 21 β 21β

21 3sh β 21β

21 3sm β 21β

4 31 β 31β

31 3sh β 31β

31 3sm β 31β

4 US1 β US1β

US 3sh β US1β

US1 3sm β US1β

0

β 11

0

0

0

0

0

β 21

0

0

0

0

0

β 31

0

0

0

0

0

β US1

Such conditions are usually difficult to interpret from an economic point of view when the corresponding differential equation is of degree greater than three (here is seven). Hence the analysis is strictly linked to the numerical values of the parameters of the model. Specifically: a) some elasticities may be close to 1 or 0 thus simplifying the characteristic equation, b) we can check the system convergence through a numerical solution, c) the final solution depends on the constrains during estimation.

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