
44 
APPENDIX
21 3sh (85) Dx_{5 }= β ^{21}β
21 3sm x_{2 }+ β ^{21}β
4 21 x_{3 }+ β ^{21}β
x_{1 }− β ^{21}x_{5 }
31 3sh (86) Dx_{6 }= β ^{31}β
31 3sm x_{2 }+ β ^{31}β
4 31 x_{3 }+ β ^{31}β
x_{1 }− β ^{31}x_{6 }
41 3sh (87) Dx_{7 }= β ^{41}β
41 3sm x_{2 }+ β ^{41}β
4 41 x_{3 }+ β ^{41}β
x_{1 }− β ^{61}x_{7 }
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 RouthHurwitz necessary and sufficient conditions
− x_{8 })
µ _{p1 }=
Pat T * 1
* i1
Pat_{i1 }µ _{p1 }= 1 T
µ _{p1 }
(83)
x_{1 }− β^{11}x_{4 }
( 8 2 ) x x e i −
8
≅ (1+ x_{i }− x_{8 })
1 1 4 1 1 3 x β β +
Pat * 1 T
* i1
Pat * 1 T
* i1
^{≅}∑
i
(x_{i+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
Dx_{8 }=
∑^{x }
i
i+2
i ∑^{e }Pat T T * * i1 x_{i + 2 }− x_{8 }1 * µ _{p1 }− x_{8 }µ _{p1 }1 * i1 Pat µ _{p1 }−
(88)
−α^{1 }γ ^{sh1}γ γ ^{sm1}γ
1 sh1
1 sm1
α ^{1}α −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 ^{sh1}γ γ ^{sm1}γ α α 1 2sh 1 2sm
0
Pat Pat * 11 * i1
µ _{p1 }
∑
i
4 11 β ^{11}β
11 3sh β ^{11}β
11 3sm β ^{11}β
4 21 β ^{21}β
21 3sh β ^{21}β
21 3sm β ^{21}β
4 31 β ^{31}β
31 3sh β ^{31}β
31 3sm β ^{31}β
4 US1 β ^{US1}β
US 3sh β ^{US1}β
US1 3sm β ^{US1}β
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.