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APPENDIX

(71) x2 = log

S h e S * 1 h1 µSh1

t

(72) x3 = log

S m e S * 1 m1 µSm1

t

(73) x8 = log

e T T * 1 1 µ p1t

  • (74)

    x4 = log

  • (75)

    x5 = log

e P a t Pat * 1 1 e Pat P a t * 2 1 11 µ p11 21 µ p 21 t t

(76) x6 = log

Pat e P a t * 3 1 31 µ p 31

t

(77) x7 = log

Pat U S e P a t * 1 US1 µ pUS1

t

By substituting the steady state values in endogenous variables eqs. (5), (8), (8’), (11), (12), (11), (20) and (21) and subtracting them from the same equations expressed in terms of actual values we obtain

i =1, 2, 3, 4 where 4 stands for the U.S.

( 7 8 ) 2 1 2 1 2 1 1 1 1 x x D x s m α α α α α + 1 + =

1 1 3 x α α 1 +

x8

(81) 1 * 1 * 3 1 * 1 * 2 1 * (79) Dx2 = −γ sh1x2 +γ 1shγ (80) Dx3 = −γ 1sm x3 +γ 1smγ 1 * + 1 1 1 1 Pat11 1 * 1 1 8 1 1 1 l o g l o g T e T e P a t T e P a t T e 1 sh1 1 1sm + P a t T T T T x1 +γ 1shγ x1 +γ 1smγ D T D D x t µ p1t 1 sh2 1 2sm + + x8 t Pat x8 Pat21 t p * US1 * p + p µ µ µ = Pat31 =

+

PatUS1 1 T

by linearization around the steady state (that is possible in the case of autonomous systems as the conditions of the Poincarè-Liapunov-Perron theorem are automatically satisfied) we can write

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