On Dictatorship, Economic Development and Stability

2.3.2 First-order Kuhn-Tucker conditions

The first-order Kuhn-Tucker conditions are computed by forming and maximizing the

following Lagrangean with respect to τ_t, q_t and k_t₊₁:

max L =
τt, qt, kt+ι

X ( q_tτ_tAk? )¹^σ -

t⅛ ⁽¹^-^σ^{)(1 +} P)^t

XXX (T⅛^{^k^t+¹^-+⁽¹^-^τ^t⁾⁽¹^-^α)^Ak‘^}⁺

XX jɪ: {ln[(1 - Tt +1)] + ln[(1 - qt+1)Tt+ι Ak^ + l] - ln l}

∑ (T+>^{^q^t^-¹^}

The first-order Kuhn-Tucker conditions are :

∂ L

^τt^~
∂τt

(qtτtAk^α)¹^σ - λtτt {
μt-1^{(1 +} P) ^τt I ^-——
1 - τt

1+β⁽¹^- α) ^A^-. (1 - qt ) Akα ¾

(1 - qt)τtAkα +1 f

(16)

qt_7r = ( qtτtAkt )¹^-σ +

∂qt

⁺^q_l^μ_l^-¹⁽¹⁺^ρ ) ½ (1 - q^,AA k+1 ¾ - ^q_ti_t=^o (¹⁷)

∂ L

^dkt +1

= ^α⁽qt^τt+1 ^a)¹^σ^kt+1 ) ^-^λt^{(1 +} P^{) +}

⁺^λt+1 11 + β^α⁽¹^-^α⁾⁽¹^-^τt+1)Akt+x¹! ^-

(18)

⁽¹^-^qt+1)^τt+1 ^aAkt₊1 ¾

⁽¹^-^qt +1 ) ^τt +1 ^Akt+1 ⁺^l ʃ

=0

18

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