Globalization, Divergence and Stagnation

.l∕<ι.n,_φ)∣ U + θY_s

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 {∕_o^{1 [a (i) φ}s^{(i)]<7 di}} ^{σ [a (i) φ}s^{(i)]<7-1 φ}s⁽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 w_s, 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»V^{11 σ}'d,     1^l'''" = 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 d}7    = ^Г

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