a ξ(1 —a)
c = ξ-α (ad∖a a ξ-a E ^—^ L a⅛-a (/( (1 - 1 - βS) ∖ ∙
ξ ∖ + J к (1+ β - 2βδ)2 J
a(ξ+θ) — ξ aη a(ξ+η + θ) — ξ
• (α (ξ + θ) - )) a-ξ (dη)a-ξ (α (ξ + η + θ) - )) ξ-a
Finally from (28) we obtain:
ɪ (c - whT)
r
a a ɪ
d αd∖ ξ-a 1 ξ
∖~ξ ) aξ-α
ξ(1 —a)
ξ(1-a) ξ(1-a) {β (1 - δ) (1 - βδ)∖ ξ-a
E (1—δ)(ξ-a) L ■ ■ _2------LS----__
к (1 + β - 2βδ)2 J
a(ξ + θ) — ξ aη a(ξ+η + θ) — ξ
• (α () + θ) - )) a—ξ (αη)a—ξ (d () + η + θ) - )) ξ' +
. . fdd∖ — „ î a(1—ξ) a(1—ξ) ∕β (1 - δ) (1 - βδ) ∖ ~
- (1 - d) — A■a E (1—*)«—a) L ξ a ------LS---J
к ) J к (1+ β - 2 2 J
ξ — a(ξ + θ) aη a(ξ+η + θ) — ξ
• (d () + θ) - )) î—a (dη)a—ξ (d () + η + θ) - )) ξ a •
pμτ∙ β (1 - δ) 1 - βδ 1
• Ej 1 O •
1 + β - 2βδ 1 + в - 2вд_
a ξ(1
в )ζ - dτ 1 A fdd∖ ¢^^ ,, î(1— a) . a(1—¢) ββ (1 - δ)(1 - βδ)λ ~
a =1-e ∖iτ - + dJuJ î—a (1")(î~a) î—a к (1 + - - 2βδ)2 J
ξ —a(ξ + θ) aη a(ξ+η + θ) —ξ
• (d () + θ) - )) î—a (dη)a—ξ (d () + η + θ) - )) î—a
7.3 Comparative statics in the decentralized economy
The comparative statics of the model in the decentralized economy can be obtained
from equations (15)-(22). In particular, from (15) we have:
dr -β - 1 + β 1
dβ = β2 = - β2
From (16) we then have:
∂T _ -δ (1 + β - 2βδ) - (1 - βδ)(1 - 2δ) _ -δ - βδ + 2βδ2 - 1 + 2δ + βδ - 2βδ2
∂β = (1 + β - 2βδ)2 = (1 + β - 2βδ)2
_ δ - 1
= (1 + β - 2βδ)2
and also:
∂T _ -β (1 + β - 2βδ) + 2β (1 - βδ) _ -β - β2 + 2β2δ + 2β - 2β2δ
∂δ = (1 + β - 2βδ)2 = (1 + β - 2βδ)2
= β - β2 = β (1 - β)
(1 + β - 2βδ)2 (1 + β - 2βδ)2
32