Consider now the case of imperfect protection of IPRs in S, (θ = 0).
.l∕<ι.n,φ)∣ U + θYs
Ln {f [a (i) Φn(i)]σ di} + θLs ^J [a (i) φs(i)]σ di}
s.t.
J <'∣n (i) di
the FOCs for a maximum are, ∀i ∈ [0,1]:
, x 1-σ
Ln {ʃɪ [a (i) Φn(i)]σ di} σ [a (i) Φn(i)]σ-1 Φn(i)+
1 —σ
θls {∕o1 [a (i) φs(i)]<7 di} σ [a (i) φs(i)]<7-1 φs(i) = λ
where λ is the lagrange multiplier associated to the constraint. Using (9) and solving
for a (i):
a (i) =
/,■,'■,'■"/....■....'■....'
βλ
1∕(1-σ)
Comparing this condition with equation (16) in the text shows that the sectoral
distribution of the endogenous technology maximizes a weighted sum of Northern
and Southern aggregate output, with a weight of θ on the South. As Ln/ (θLs) → 0,
technologies maximize ws, whereas as Ln/ (θLs) → ∞ they maximize Wn∙
5.2 Properties of the wage ratio in autarky
To show that the North-South wage ratio in autarky is bounded by max Φn (i) ∕φs (i) =
Φn (0) φs (0), first note that ∂ω∕∂φN (i) > 0 and ∂ω∕∂φs (i) < 0. Therefore, by con-
struction:
Jo1 Φn (i)J/(1-a) di
Jo‘ Φn ' Φs (≈)σ di
!,; Φn»V11 σ'd, 1l'''" = Φn(0)
ʃ,1 Φn '' ' Φs (0)σ di] φs(0)
5.3 The growth rate under free-trade
Rewrite the marginal condition for buying innovation in a Northern sector as:
WnΦn (i) LnAn (i)σ 1 =
β Jo An (J)σ d7 = Г
34
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