Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of...

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Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian Theory and Further Semiparametric Evidence Andrea Vaona WP Number: 9 May 2010 ISSN: 2036-2919 (paper), 2036-4679 (online)

Transcript of Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of...

Page 1: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

Working Paper SeriesDepartment of Economics

University of Verona

Inflation and Growth in the Long Run: A New Keynesian Theoryand Further Semiparametric Evidence

Andrea Vaona

WP Number: 9 May 2010

ISSN: 2036-2919 (paper), 2036-4679 (online)

Page 2: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

∀ΧΘ6Ι>∆Χ 6Χ9 Γ∆ΛΙ= >Χ Ι=: %∆Χ< +ϑΧ: A∋:Λ ∃:ΝΧ:Η>6Χ −=:∆ΓΝ 6Χ9 FϑΓΙ=:Γ

,:Β>Ε6Γ6Β:ΙΓ>8 EΚ>9:Χ8:

AΧ9Γ:6 /6∆Χ61

1.Χ>Κ:ΓΗ>ΙΝ ∆; /:Γ∆Χ6 (D:Ε6ΓΙΒ:ΧΙ ∆; E8∆Χ∆Β>8 ,8>:Χ8:Η) 6Χ9 ∃>:Α ∀ΧΗΙ>ΙϑΙ: ;∆Γ Ι=:0∆ΓΑ9 E8∆Χ∆ΒΝ, )6Α6ΟΟ>Χ6 32 ,8>:ΧΟ: E8∆Χ∆Β>8=: - :Μ C6Η:ΓΒ6 )6ΗΗ6Α68Φϑ6, />6Α:9:ΑΑΡ.Χ>Κ:ΓΗ>ΙΞ 4, 37129 /:Γ∆Χ6, ∀Ι6ΑΝ. E-Β6>Α: 6Χ9Γ:6.Κ6∆Χ6@ϑΧ>ΚΓ.>Ι. ∀ Λ∆ϑΑ9 Α>≅: Ι∆Ι=6Χ≅ ϑ>9∆ AΗ86Γ>, %ϑ><> B∆Χ6ΙΙ> 6Χ9 ΙΛ∆ 6Χ∆ΧΝΒ∆ϑΗ Γ:;:Γ::Η ;∆Γ >ΧΗ><=Ι;ϑΑ 8∆ΒΒ:ΧΙΗ.D:ΧΧ>Η ,Χ∆Λ:Γ Λ6Η :ΜΙΓ:Β:ΑΝ ΗϑΕΕ∆ΓΙ>Κ: 6Χ9 =:ΑΕ;ϑΑ Λ>Ι= 6 ΕΓ:Κ>∆ϑΗ Κ:ΓΗ>∆Χ ∆; Ι=>Η Ε6Ε:Γ.−=: ϑΗϑ6Α 9>Η8Α6>Β:Γ 6ΕΕΑ>:Η.

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∀ΧΘ6Ι>∆Χ 6Χ9 Γ∆ΛΙ= >Χ Ι=: %∆Χ< +ϑΧ: A∋:Λ ∃:ΝΧ:Η>6Χ −=:∆ΓΝ 6Χ9 FϑΓΙ=:Γ

,:Β>Ε6Γ6Β:ΙΓ>8 EΚ>9:Χ8:

Ab,−+ac−

−=>Η Ε6Ε:Γ :ΜΕΑ∆Γ:Η Ι=: >ΧΘϑ:Χ8: ∆; >ΧΘ6Ι>∆Χ ∆Χ :8∆Χ∆Β>8 <Γ∆ΛΙ= 7∆Ι=

Ι=:∆Γ:Ι>86ΑΑΝ 6Χ9 :ΒΕ>Γ>86ΑΑΝ. 0: ΕΓ∆Ε∆Η: Ι∆ Β:Γ<: 6Χ :Χ9∆<:Χ∆ϑΗ <Γ∆ΛΙ=

Β∆9:Α ∆; Α:6ΓΧ>Χ< 7Ν 9∆>Χ< Λ>Ι= 6 ∋:Λ ∃:ΝΧ:Η>6Χ ∆Χ: Λ>Ι= ΗΙ>8≅Ν Λ6<:Η.

0: Η=∆Λ Ι=6Ι Ι=: >ΧΙ:ΓΙ:ΒΕ∆Γ6Α :Α6ΗΙ>8>ΙΝ ∆; Ηϑ7ΗΙ>ΙϑΙ>∆Χ ∆; Λ∆Γ≅>Χ< Ι>Β:

>Η 6 ≅:Ν Ε6Γ6Β:Ι:Γ ;∆Γ Ι=: Η=6Ε: ∆; Ι=: >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ= Χ:ΜϑΗ. 0=:Χ >Ι

>Η Η:Ι :Φϑ6Α Ι∆ Ο:Γ∆, Ι=: >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ= Χ:ΜϑΗ >Η Λ:6≅ 6Χ9 =ϑΒΕ-Η=6Ε:9.

0=:Χ >Ι >Η <Γ:6Ι:Γ Ι=6Χ Ο:Γ∆, >ΧΘ6Ι>∆Χ =6Η 6 Η>Ο:67Α: 6Χ9 Χ:<6Ι>Κ: :Ω:8Ι ∆Χ

<Γ∆ΛΙ=. EΧ9∆<:Χ>Ο>Χ< Ι=: Α:Χ<Ι= ∆; Λ6<: 8∆ΧΙΓ68ΙΗ 9∆:Η Χ∆Ι Α:69 Ι∆ >ΧΘ6Ι>∆Χ

ΗϑΕ:ΓΧ:ϑΙΓ6Α>ΙΝ >Χ ΕΓ:Η:Χ8: ∆; 6 ΠΜ:9 8∆ΗΙ Ι∆ Λ6<: Γ:Η:ΙΙ>Χ<. (Χ8: 69∆ΕΙ>Χ<

Κ6Γ>∆ϑΗ Η:Β>Ε6Γ6Β:ΙΓ>8 6Χ9 >ΧΗΙΓϑΒ:ΧΙ6Α Κ6Γ>67Α: :ΗΙ>Β6Ι>∆Χ 6ΕΕΓ∆68=:Η ∆Χ

6 8Γ∆ΗΗ-8∆ϑΧΙΓΝ/Ι>Β:-Η:Γ>:Η 96Ι6Η:Ι, Λ: Η=∆Λ Ι=6Ι >Χ8Γ:6Η>Χ< >ΧΘ6Ι>∆Χ Γ:9ϑ8:Η

Γ:6Α :8∆Χ∆Β>8 <Γ∆ΛΙ=, 8∆ΧΗ>ΗΙ:ΧΙΑΝ Λ>Ι= ∆ϑΓ Ι=:∆Γ:Ι>86Α Β∆9:Α Λ>Ι= 6 Ε∆Η>Ι>Κ:

>ΧΙ:ΓΙ:ΒΕ∆Γ6Α :Α6ΗΙ>8>ΙΝ ∆; Ηϑ7ΗΙ>ΙϑΙ>∆Χ ∆; Λ∆Γ≅>Χ< Ι>Β:.

K!20)+ ,: >ΧΘ6Ι>∆Χ, <Γ∆ΛΙ=, Λ6<:-ΗΙ6<<:Γ>Χ<, Α:6ΓΧ>Χ<-7Ν-9∆>Χ<, Η:Β>-

Ε6Γ6Β:ΙΓ>8 :ΗΙ>Β6Ι∆Γ.

JEL C) !,: E31, E51, E52, (42, C14.

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1 I(−+) .c−%)(

−=>Η Ε6Ε:Γ ∆Ω:ΓΗ 6 Χ:Λ Ι=:∆Γ:Ι>86Α Β∆9:Α 6Χ9 Χ:Λ :ΒΕ>Γ>86Α :Κ>9:Χ8: ∆Χ Ι=:

8∆ΧΧ:8Ι>∆Χ 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9 <Γ∆ΛΙ=.

∀Χ Ι=: Ι=:∆Γ:Ι>86Α Ε6ΓΙ, Λ: Η=∆Λ Ι=6Ι 8=6Χ<:Η >Χ Ι=: >ΧΘ6Ι>∆Χ Γ6Ι: 86Χ ΕΓ∆-

9ϑ8: Ε:ΓΒ6Χ:ΧΙ 8=6Χ<:Η >Χ Ι=: <Γ∆ΛΙ= Γ6Ι: ∆; ∆ϑΙΕϑΙ - :Κ:Χ Ι=∆ϑ<= Ι=: Β∆9:Α

=:Γ: ΕΓ∆Ε∆Η:9 8∆ΧΙ6>ΧΗ Χ∆ Β∆Χ:Ν >ΑΑϑΗ>∆Χ, Χ∆ Ε:ΓΒ6Χ:ΧΙ Χ∆Β>Χ6Α Γ><>9>Ι>:Η,

6Χ9 Χ∆ 9:Ε6ΓΙϑΓ: ;Γ∆Β Γ6Ι>∆Χ6Α :ΜΕ:8Ι6Ι>∆ΧΗ. 0: 9∆ Η∆ 7Ν :ΜΙ:Χ9>Χ< Ι=:

Β∆9:Α 7Ν Γ6=6Β 6Χ9 ,Χ∆Λ:Γ (2004) ;Γ∆Β Ι=: >ΧΘ6Ι>∆Χ-∆ϑΙΕϑΙ Α:Κ:Α 9∆Β6>Χ

Ι∆ Ι=: >ΧΘ6Ι>∆Χ-∆ϑΙΕϑΙ <Γ∆ΛΙ= ∆Χ:. AΗ 6 8∆ΧΗ:Φϑ:Χ8:, Λ: Β:Γ<: 6Χ :Χ9∆<:-

Χ∆ϑΗ <Γ∆ΛΙ= Β∆9:Α ∆; Α:6ΓΧ>Χ<-7Ν-9∆>Χ< Λ>Ι= 6 ∋:Λ ∃:ΝΧ:Η>6Χ ∆Χ: Λ>Ι=

ΗΙ>8≅Ν Λ6<:Η 6ΑΑ∆Λ>Χ< Ι∆ :ΜΕΑ∆Γ: =∆Λ Ι=: :Ω:8Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ ∆ϑΙΕϑΙ <Γ∆ΛΙ=

>Η 8∆ΧΧ:8Ι:9 Χ∆Ι ∆ΧΑΝ Ι∆ Ι=: 9>Γ:8Ι :Ω:8ΙΗ ∆; >ΧΘ6Ι>∆Χ ∆Χ 86Ε>Ι6Α 688ϑΒϑΑ6Ι>∆Χ,

7ϑΙ 6ΑΗ∆ Ι∆ >Χ9>Γ:8Ι ∆Χ:Η Ε6ΗΗ>Χ< Ι=Γ∆ϑ<= Ι=: Α67∆ϑΓ Β6Γ≅:Ι.

∀Χ Ι=>Η 8∆ΧΙ:ΜΙ, Λ=:Χ Β∆Χ:Ν ΗϑΕΕΑΝ <Γ∆ΛΗ >Χ Ι=: ΕΓ:Η:Χ8: ∆; Ι:ΒΕ∆Γ6ΓΝ

Χ∆Β>Χ6Α ;Γ>8Ι>∆ΧΗ (>Χ Ι=: ;∆ΓΒ ∆; ΗΙ6<<:Γ:9 Χ∆Β>Χ6Α 8∆ΧΙΓ68ΙΗ), Χ∆Β>Χ6Α 69-

?ϑΗΙΒ:ΧΙΗ Χ:Κ:Γ =6Κ: 6 8=6Χ8: Ι∆ Λ∆Γ≅ Ι=:ΒΗ:ΑΚ:Η ∆ϑΙ ;ϑΑΑΝ. −=ϑΗ, >Χ 6Χ

:Χ9∆<:Χ∆ϑΗ <Γ∆ΛΙ= 8∆ΧΙ:ΜΙ, 8=6Χ<:Η >Χ Β∆Χ:Ν <Γ∆ΛΙ= 6Ω:8Ι Ι=: Β6Γ<>Χ6Α

ΕΓ∆9ϑ8Ι ∆; 86Ε>Ι6Α 6Χ9 Ι=:Γ:7Ν Ι=: Γ6Ι: ∆; :8∆Χ∆Β>8 <Γ∆ΛΙ=.

−=: 76Η>8 >ΧΙϑ>Ι>∆Χ ;∆Γ Ι=: Β∆9:Α =:Γ: ΕΓ∆Ε∆Η:9 86Χ 7: 9:Η8Γ>7:9 6Η ;∆Α-

Α∆ΛΗ.

0=:Χ Χ∆Β>Χ6Α Λ6<: 8∆ΧΙΓ68ΙΗ 6Γ: ΗΙ6<<:Γ:9, 8ϑΓΓ:ΧΙ 8∆ΧΙΓ68Ι Λ6<:Η

9:Ε:Χ9 ∆Χ Ι=: 8ϑΓΓ:ΧΙ 6Χ9 ;ϑΙϑΓ: ΕΓ>8:Η Ι=6Ι ΕΓ:Κ6>Α ∆Κ:Γ Ι=: 8∆ΧΙΓ68Ι

1

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Ε:Γ>∆9. −=: 8ϑΓΓ:ΧΙ 8∆ΧΙΓ68Ι Λ6<: >Η 6Ω:8Ι:9 Β∆Γ: 7Ν 8ϑΓΓ:ΧΙ ΕΓ>8:Η

Ι=6Χ 7Ν :ΜΕ:8Ι:9 ;ϑΙϑΓ: ΕΓ>8:Η, 9ϑ: Ι∆ Ι>Β: 9>Η8∆ϑΧΙ>Χ<.

−=: <Γ:6Ι:Γ >Η Ι=: Γ6Ι: ∆; Β∆Χ:Ν <Γ∆ΛΙ=, Ι=: ;6ΗΙ:Γ ΕΓ>8:Η Γ>Η:, 6Χ9 Ι=:

Β∆Γ: Ι=: 8∆ΧΙΓ68Ι Λ6<: Α6<Η 7:=>Χ9 Ι=: 8ϑΓΓ:ΧΙ ΕΓ>8: Α:Κ:Α. −=ϑΗ Ι=:

Α∆Λ:Γ >Η Ι=: 6Κ:Γ6<: Γ:6Α Λ6<: ∆Κ:Γ Ι=: 8∆ΧΙΓ68Ι Ε:Γ>∆9. C∆ΧΗ:Φϑ:ΧΙΑΝ,

Ι=: Β∆Γ: Α67∆Γ >Η 9:Β6Χ9:9 7Ν ΠΓΒΗ.

∀Χ ∆ϑΓ :Χ9∆<:Χ∆ϑΗ <Γ∆ΛΙ= Β∆9:Α, 6Χ >Χ8Γ:6Η: >Χ Ι=: Α67∆ϑΓ >ΧΕϑΙ Γ6>Η:Η

Ι=: Β6Γ<>Χ6Α ΕΓ∆9ϑ8Ι ∆; 86Ε>Ι6Α, Α:69>Χ< Ι∆ ;6ΗΙ:Γ :8∆Χ∆Β>8 <Γ∆ΛΙ=. −=:Γ: >Η

=∆Λ:Κ:Γ 6Χ >ΒΕ∆ΓΙ6ΧΙ 8∆ϑΧΙ:ΓΚ6>Α>Χ< :Ω:8Ι:

AΗ Β∆Χ:Ν <Γ∆ΛΙ= - 6Χ9 8∆ΧΗ:Φϑ:ΧΙΑΝ >ΧΘ6Ι>∆Χ - >Χ8Γ:6Η:Η, Γ:Α6Ι>Κ: ΕΓ>8:Η

7:8∆Β: Β∆Γ: Κ∆Α6Ι>Α:, >.:. Ι=: Γ:6Α Λ6<: Κ6Γ>:Η Β∆Γ: ∆Κ:Γ Ι=: 8∆ΧΙΓ68Ι

Ε:Γ>∆9, Η>Χ8: Ι=: Χ∆Β>Χ6Α Λ6<: ∆; :68= 8∆=∆ΓΙ >Η 8∆ΧΗΙ6ΧΙ ∆Κ:Γ Ι=: 8∆Χ-

ΙΓ68Ι Ε:Γ>∆9 Λ=:Γ:6Η Ι=: 6<<Γ:<6Ι: ΕΓ>8: Α:Κ:Α Γ>Η:Η <Γ69ϑ6ΑΑΝ Ι=Γ∆ϑ<=

Ι>Β:.

−=>Η Γ:6Α Λ6<: Κ∆Α6Ι>Α>ΙΝ >Χ9ϑ8:Η :ΒΕΑ∆ΝΒ:ΧΙ Θϑ8Ιϑ6Ι>∆ΧΗ, >.:. ":Β-

ΕΑ∆ΝΒ:ΧΙ 8Ν8Α>Χ<". ,>Χ8: Ι=:Γ: 6Γ: 9>Β>Χ>Η=>Χ< Γ:ΙϑΓΧΗ Ι∆ Α67∆Γ, Ι=>Η

>Η >Χ:ς 8>:ΧΙ Α:69>Χ< Ι∆ 6 Γ:9ϑ8Ι>∆Χ >Χ Ι=: Β6Γ<>Χ6Α ΕΓ∆9ϑ8Ι ∆; 86Ε>Ι6Α

6Χ9 Ι=:Γ:;∆Γ: >Χ ∆ϑΙΕϑΙ <Γ∆ΛΙ=.2

−=:Η: >ΧΘϑ:Χ8:Η >ΒΕΑΝ 6 Χ:<6Ι>Κ: Α∆Χ<-ΓϑΧ Γ:Α6Ι>∆Χ 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9

∆ϑΙΕϑΙ <Γ∆ΛΙ=.

2∋∆Ι: Ι=6Ι :ΒΕΑ∆ΝΒ:ΧΙ 8Ν8Α>Χ< Ι6≅:Η ΕΑ68: ∆Κ:Γ 6 Ι>Β: Ε:Γ>∆9 Η=∆ΓΙ :Χ∆ϑ<= (Ι=: 8∆Χ-ΙΓ68Ι Ε:Γ>∆9) ;∆Γ 9>Β>Χ>Η=>Χ< Γ:ΙϑΓΧΗ Ι∆ 7: Γ:Α:Κ6ΧΙ.

2

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0: Η=∆Λ, >Χ 6 ΧϑΒ:Γ>86Α :Μ6ΒΕΑ:, Ι=6Ι, Λ=:Χ Ι=: >ΧΙ:ΓΙ:ΒΕ∆Γ6Α :Α6ΗΙ>8>ΙΝ

∆; Ηϑ7ΗΙ>ΙϑΙ>∆Χ ∆; Λ∆Γ≅>Χ< Ι>Β: >Η Η:Ι Ι∆ Ο:Γ∆, Ι=: 9>Η8∆ϑΧΙ>Χ< :Ω:8Ι >Η 9∆Β-

>Χ6ΧΙ 6Ι Α∆Λ Β∆Χ:Ν <Γ∆ΛΙ= Γ6Ι:Η Λ=:Γ:6Η Ι=: :ΒΕΑ∆ΝΒ:ΧΙ 8Ν8Α>Χ< :Ω:8Ι >Η

9∆Β>Χ6ΧΙ 6Ι =><= Β∆Χ:Ν <Γ∆ΛΙ= Γ6Ι:Η. BΝ >ΒΕΑ>86Ι>∆Χ, Ι=: Α∆Χ<-ΓϑΧ Γ:Α6Ι>∆Χ

7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9 ∆ϑΙΕϑΙ <Γ∆ΛΙ= >Η 768≅Λ6Γ9 7:Χ9>Χ<: <Γ∆ΛΙ= Γ>Η:Η Λ>Ι=

>ΧΘ6Ι>∆Χ 6Ι Α∆Λ >ΧΘ6Ι>∆Χ Γ6Ι:Η, 7ϑΙ ;6ΑΑΗ Λ>Ι= >ΧΘ6Ι>∆Χ 6Ι =><= >ΧΘ6Ι>∆Χ Γ6Ι:Η.

!∆Λ:Κ:Γ, ∆Χ8: 6ΑΑ∆Λ>Χ< ;∆Γ 6 Ε∆Η>Ι>Κ: >ΧΙ:ΓΙ:ΒΕ∆Γ6Α :Α6ΗΙ>8>ΙΝ ∆; Ηϑ7ΗΙ>-

ΙϑΙ>∆Χ ∆; Λ∆Γ≅>Χ< Ι>Β:, >ΧΘ6Ι>∆Χ =6Η 6 Χ:<6Ι>Κ: >ΒΕ68Ι ∆Χ <Γ∆ΛΙ=, 7:86ϑΗ:

=∆ϑΗ:=∆Α9Η, =6Κ>Χ< 6 <Γ:6Ι:Γ ΕΓ:;:Γ:Χ8: ;∆Γ Α67∆ϑΓ ΗϑΕΕΑΝ ΗΒ∆∆Ι=>Χ<, Γ:ΗΕ∆Χ9

Ι∆ >Χ8Γ:6Η:Η >Χ Β∆Χ:Ν <Γ∆ΛΙ= - Λ=>8= :ΧΙ6>Α Β∆Γ: :ΒΕΑ∆ΝΒ:ΧΙ 8Ν8Α>Χ< - 7Ν

Γ6>Η>Χ< Ι=:>Γ Λ6<:, Γ:9ϑ8>Χ< :ΒΕΑ∆ΝΒ:ΧΙ 6Χ9 ∆ϑΙΕϑΙ <Γ∆ΛΙ=.

F>Χ6ΑΑΝ, :Χ9∆<:Χ>Ο>Χ< Ι=: Α:Χ<Ι= ∆; Ι=: Α67∆ϑΓ 8∆ΧΙΓ68Ι Γ:9ϑ8:Η Ι=: ΗΙΓ:Χ<Ι=

∆; Ι=: :Ω:8Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ <Γ∆ΛΙ=, 7ϑΙ, >Χ ΕΓ:Η:Χ8: ∆; 6 ΠΜ:9 8∆ΗΙ Ι∆ Λ6<:

Γ:Η:ΙΙ>Χ<, Ι=: Χ:<6Ι>Κ: :Ω:8Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ <Γ∆ΛΙ= 9∆:Η Χ∆Ι 9>Η6ΕΕ:6Γ. −=>Η

>Η 7:86ϑΗ: =∆ϑΗ:=∆Α9Η 86Χ Γ:9ϑ8: Α67∆ϑΓ 8Ν8Α>Χ< 7Ν Ε6Ν>Χ< Ι=: Λ6<: Γ:Η:ΙΙ>Χ<

8∆ΗΙ Β∆Γ: ∆;Ι:Χ. ,∆ Ι=:Ν 8=∆∆Η: Λ6<: Θ:Μ>7>Α>ΙΝ ∆ΧΑΝ ;∆Γ 6 Α∆Λ Α:Κ:Α ∆; Ι=>Η

8∆ΗΙ. −=: Λ:Α;6Γ: >ΒΕΑ>86Ι>∆ΧΗ ∆; Ι=:Η: 9>Ω:Γ:ΧΙ Β∆9:Α Ε6Γ6Β:ΙΓ>Ο6Ι>∆ΧΗ 6Γ:

6ΑΗ∆ 9>Η8ϑΗΗ:9 >Χ Ι=: Ε6Ε:Γ.

−=: :ΒΕ>Γ>86Α Ε6ΓΙ ∆; Ι=: Ε6Ε:Γ >Η Β6>ΧΑΝ 76Η:9 ∆Χ Η:Β>Ε6Γ6Β:ΙΓ>8 :ΗΙ>-

Β6Ι>∆Χ Β:Ι=∆9Η 6ΕΕΑ>:9 Ι∆ 6 8Γ∆ΗΗ-8∆ϑΧΙΓΝ/Ι>Β:-Η:Γ>:Η 96Ι6Η:Ι, Η>Β>Α6ΓΑΝ Ι∆

/6∆Χ6 6Χ9 ,8=>6Κ∆ (2007). !∆Λ:Κ:Γ, Λ: ;∆ΑΑ∆Λ 6 9>Ω:Γ:ΧΙ Γ:Η:6Γ8= ΗΙΓ6Ι:<Ν.

0: ΠΓΗΙ ϑΗ: Κ6Γ>∆ϑΗ Β∆9:Α Η:Α:8Ι>∆Χ 8Γ>Ι:Γ>6 ;∆Γ Η:Β>Ε6Γ6Β:ΙΓ>8 Ε∆Λ:Γ Η:Γ>:Η

:ΗΙ>Β6Ι>∆Χ 6ΕΕΓ∆68=:Η. −=:Χ, Λ: 8=:8≅ Λ=:Ι=:Γ ∆ϑΓ Γ:ΗϑΑΙΗ =6Κ: 6Χ :Χ9∆-

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<:Χ:Χ:>ΙΝ 7>6Η 6Χ9 Λ: 8∆Χ9ϑ8Ι Κ6Γ>∆ϑΗ Ηϑ7-Η6ΒΕΑ: ΗΙ67>Α>ΙΝ Ι:ΗΙΗ. F>Χ6ΑΑΝ,

>ΧΗΙ:69 ∆; 69∆ΕΙ>Χ< 6 ∋696Γ6Ν6-06ΙΗ∆Χ ∃:ΓΧ:Α :ΗΙ>Β6Ι∆Γ Λ: ϑΗ: 6 Α∆86Α Α>Χ-

:6Γ ∆Χ: Λ>Ι= 6 Κ6Γ>67Α: Λ>Χ9∆Λ Λ>9Ι=. F6Χ 6Χ9 >?7:ΑΗ (1992) Α>ΗΙ 6 ΧϑΒ7:Γ

∆; 69Κ6ΧΙ6<:Η ∆; Ι=: Α6ΙΙ:Γ :ΗΙ>Β6Ι∆Γ ∆Κ:Γ Ι=: ;∆ΓΒ:Γ ∆Χ:. ∀Χ Ε6ΓΙ>8ϑΑ6Γ, >Ι

9∆:Η Χ∆Ι Α∆Η: ΕΓ:8>Η>∆Χ Χ:6Γ Ι=: 7∆ϑΧ96ΓΝ ∆; Ι=: ∆7Η:ΓΚ6Ι>∆Χ >ΧΙ:ΓΚ6Α (7∆ϑΧ9-

6ΓΝ :Ω:8ΙΗ). 0: ΠΧ9 Ι=6Ι =><=:Γ >ΧΘ6Ι>∆Χ ?ϑΗΙ =6ΓΒΗ Γ:6Α :8∆Χ∆Β>8 <Γ∆ΛΙ=.

−=: :ΒΕ>Γ>86Α Γ:ΗϑΑΙΗ Λ: ∆7Ι6>Χ =:ΑΕ ϑΗ Ι∆ 8=∆∆Η: ∆ϑΓ ΕΓ:;:ΓΓ:9 86Α>7Γ6Ι>∆Χ

;∆Γ Ι=: Ι=:∆Γ:Ι>86Α Β∆9:Α =:Γ: ΕΓ∆Ε∆Η:9.

−=: Ε6Ε:Γ >Η ∆Γ<6Χ>Ο:9 6Η ;∆ΑΑ∆ΛΗ. ,:8Ι>∆Χ 2 Γ:Α6Ι:Η ∆ϑΓ 8∆ΧΙΓ>7ϑΙ>∆Χ Ι∆

Ι=: :Μ>ΗΙ>Χ< Α>Ι:Γ6ΙϑΓ:. ,:8Ι>∆Χ 3 ΕΓ:Η:ΧΙΗ Ι=: ϑΧ9:ΓΑΝ>Χ< Β∆9:Α. ,:8Ι>∆Χ 4

Η≅:Ι8=:Η Ι=: Β∆9:Α Η∆ΑϑΙ>∆Χ, Λ=>8= >Η >ΑΑϑΗΙΓ6Ι:9 >Χ 9:Ι6>Α >Χ AΕΕ:Χ9>8:Η A 6Χ9

B. ,:8Ι>∆Χ 5 8∆ΧΙ6>ΧΗ ∆ϑΓ Γ:ΗϑΑΙΗ Γ:<6Γ9>Χ< Ι=: Γ:Α6Ι>∆ΧΗ 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9

<Γ∆ΛΙ= 6Χ9 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9 Λ:Α;6Γ:. ,:8Ι>∆Χ 6 <:Χ:Γ6Α>Ο:Η =∆ϑΗ:=∆Α9ΗΡ

ΕΓ:;:Γ:Χ8:Η 6Χ9 :Χ9∆<:Χ>Ο:Η Ι=: ;Γ:Φϑ:Χ8Ν ∆; Χ∆Β>Χ6Α 69?ϑΗΙΒ:ΧΙΗ. ,:8Ι>∆Χ

7 Η=∆ΛΗ ∆ϑΓ :ΒΕ>Γ>86Α Γ:ΗϑΑΙΗ 6Χ9 ,:8Ι>∆Χ 8 8∆Χ8Αϑ9:Η.

2 R!&a−%)( −) −∃! L%−!+a−.+!

2.1 T∃!)+!−%ca& L%−!+a−.+!

0>Ι= Ι=: :Μ8:ΕΙ>∆Χ ∆; 6 ;:Λ Γ:8:ΧΙ ΗΙϑ9>:Η, Ι=: Ι=:∆Γ:Ι>86Α Α>Ι:Γ6ΙϑΓ: Β6>ΧΑΝ

ΕΓ∆9ϑ8:9 Β∆9:ΑΗ Λ=:Γ: >ΧΘ6Ι>∆Χ =6Η 6 Χ:<6Ι>Κ: >ΒΕ68Ι ∆Χ <Γ∆ΛΙ=.

(Χ: ∆; Ι=: ΠΓΗΙ Ι=:∆Γ:Ι>86Α ΗΙϑ9>:Η 8∆Χ8:ΓΧ>Χ< >ΧΘ6Ι>∆Χ 6Χ9 ∆ϑΙΕϑΙ >Η

−∆7>Χ (1965), 688∆Γ9>Χ< Ι∆ Λ=>8= >ΧΘ6Ι>∆Χ >Η 7:Χ:Π8>6Α Ι∆ Ι=: ∆ϑΙΕϑΙ Α:Κ:Α

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7:86ϑΗ: >Ι Α∆Λ:ΓΗ Ι=: >ΧΙ:Γ:ΗΙ Γ6Ι: 6Χ9 Ι=:Γ:;∆Γ: Ι=: ∆ΕΕ∆ΓΙϑΧ>ΙΝ 8∆ΗΙ Ι∆ >Χ-

Κ:ΗΙ. −=>Η >Χ8Γ:6Η:Η Ι=: 86Ε>Ι6Α-Α67∆ϑΓ Γ6Ι>∆ 6Χ9 Ι=:Γ:;∆Γ: ∆ϑΙΕϑΙ3. ,Ι∆8≅Β6Χ

(1981) Ε∆>ΧΙ:9 Ι∆ Ι=: Ε∆ΗΗ>7Α: :Μ>ΗΙ:Χ8: ∆; 6Χ >ΧΚ:ΓΗ: −∆7>Χ :Ω:8Ι, Λ=:Γ:7Ν

6Χ >Χ8Γ:6Η: >Χ Ι=: >ΧΘ6Ι>∆Χ Γ6Ι: 86ϑΗ:Η Ι=: 86Ε>Ι6Α ΗΙ∆8≅ Ι∆ 9:8Γ:6Η:, ∆Χ8:

ΗϑΕΕ∆Η>Χ< 6 86Η= >Χ 69Κ6Χ8: 8∆ΧΗΙΓ6>ΧΙ ;∆Γ 86Ε>Ι6Α 688ϑΒϑΑ6Ι>∆Χ 6Χ9 <>Κ:Χ

Ι=6Ι >ΧΘ6Ι>∆Χ Γ6>Η:Η Ι=: 8∆ΗΙ ∆; Β∆Χ:Ν =∆Α9>Χ<.

&∆Γ: Γ:8:ΧΙΑΝ, Ι=: Α>Ι:Γ6ΙϑΓ: =6Η Η=>;Ι:9 ;Γ∆Β Ι=: Α:Κ:Α ∆; ∆ϑΙΕϑΙ Ι∆ Ι=:

∆ϑΙΕϑΙ <Γ∆ΛΙ= Γ6Ι: 6Χ9 ;Γ∆Β ,∆Α∆Λ Β∆9:ΑΗ Ι∆ :Χ9∆<:Χ∆ϑΗ <Γ∆ΛΙ= ∆Χ:Η. >ΑΑ-

Β6Χ 6Χ9 ∃:?6≅ (20056) ΕΓ∆Ε∆Η:9 Ι∆ 9>ΗΙ>Χ<ϑ>Η= 7:ΙΛ::Χ Ε=ΝΗ>86Α 86Ε>Ι6Α

Β∆9:ΑΗ, Α67:ΑΑ:9 6Η A∃, =ϑΒ6Χ 86Ε>Ι6Α ∆Χ:Η, Α67:ΑΑ:9 6Η A∀, 6Χ9 8∆Β7>Χ:9

Β∆9:ΑΗ, Λ>Ι= 7∆Ι= =ϑΒ6Χ 6Χ9 Ε=ΝΗ>86Α 86Ε>Ι6Α. −=: Γ:ΗϑΑΙΗ =6Κ: 7::Χ Β>Μ:9.

,∆Β: 8∆ΧΙΓ>7ϑΙ>∆ΧΗ =6Κ: ΕΓ∆9ϑ8:9 >ΧΗ><Χ>Π86ΧΙ Α∆Χ<-ΓϑΧ >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ= :;-

;:8ΙΗ, Ηϑ8= 6Η Ι=: A∃ Β∆9:ΑΗ ∆; ∀Γ:Α6Χ9 (1994) 6Χ9 D∆ΙΗ:Ν 6Χ9 ,6ΓΙΓ: (2000)

6Χ9 Ι=: 8∆Β7>Χ:9 Β∆9:Α ∆; C=6Γ> e− a%. (1996). (Ι=:Γ 8∆ΧΙΓ>7ϑΙ>∆ΧΗ =6Κ:

ΕΓ∆9ϑ8:9 6 Χ:<6Ι>Κ: 6Χ9 Η><Χ>Π86ΧΙ >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ= :Ω:8Ι, Ηϑ8= 6Η Ι=: A∃

Β∆9:ΑΗ ∆; !6ΗΑ6< (1998) 6Χ9 >ΑΑΒ6Χ 6Χ9 ∃:?6≅ (2004), Ι=: A∀ Β∆9:Α ∆;

>ΑΑΒ6Χ :Ι 6Α. (1999), Ι=: 8∆Β7>Χ:9 Β∆9:ΑΗ ∆; ∆ΒΒ: (1993) 6Χ9 Ι=6Ι ∆;

>ΑΑΒ6Χ 6Χ9 ∃:?6≅ (2002, 20057). >ΑΑΒ6Χ 6Χ9 ∃:?6≅ (20056) ΕΓ∆Ε∆Η: 6

Β∆9:Α Ι=6Ι Χ:ΗΙΗ Β∆ΗΙ ∆; Ι=: Β∆9:ΑΗ 69Κ6Χ8:9 7:;∆Γ:. −=:>Γ Γ:ΗϑΑΙ >Η Ι=6Ι >Χ

Ι=: A∃ Β∆9:Α >ΧΘ6Ι>∆Χ Λ∆Γ≅Η 6Η 6 Ι6Μ ∆Χ Ε=ΝΗ>86Α 86Ε>Ι6Α, >ΒΕΑΝ>Χ< 6 Χ:<-

6Ι>Κ: −∆7>Χ (1965) :Ω:8Ι, Λ=:Γ:6Η >Χ Ι=: A∀ Β∆9:Α >ΧΘ6Ι>∆Χ Λ∆Γ≅Η 6Η 6 Ι6Μ

3F∆Γ Ι=>Η 9:ΠΧ>Ι>∆Χ ∆; −∆7>Χ (1965) :Ω:8Ι ∀ Γ:ΑΝ ∆Χ >ΑΑΒ6Χ 6Χ9 ∃:?6≅ (20056). ∀Ι>Η Λ∆ΓΙ= Χ∆Ι>Χ< Ι=6Ι 6 Ε∆Η>Ι>Κ: :Ω:8Ι ∆Χ >ΧΘ6Ι>∆Χ 6Χ9 Ι=: Α:Κ:Α ∆; ∆ϑΙΕϑΙ Β><=Ι Χ∆Ι >ΧΕΓ>Χ8>ΕΑ: :Μ8Αϑ9: 6 Χ:<6Ι>Κ: ∆Χ: ∆Χ Ι=: <Γ∆ΛΙ= Γ6Ι: ∆; ∆ϑΙΕϑΙ 6Χ9 Κ>8:Κ:ΓΗ6.

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∆Χ =ϑΒ6Χ 86Ε>Ι6Α, >ΒΕΑΝ>Χ< 6 Ε∆Η>Ι>Κ: −∆7>Χ (1965) :Ω:8Ι. F>Χ6ΑΑΝ, >Χ Ι=:

8∆Β7>Χ:9 Β∆9:Α, >ΧΘ6Ι>∆Χ Λ∆Γ≅Η Β∆Γ: Α>≅: 6 Ι6Μ ∆Χ =ϑΒ6Χ 86Ε>Ι6Α Ι=6Χ ∆Χ

Ε=ΝΗ>86Α 86Ε>Ι6Α >ΒΕΑΝ>Χ< 6 Ε∆Η>Ι>Κ: −∆7>Χ (1965) :Ω:8Ι.

0>Ι=>Χ 6 Ε:8ϑΧ>6ΓΝ ΙΓ6ΧΗ68Ι>∆Χ 8∆ΗΙΗ Β∆9:Α, 3=6Χ< (2000) Η=∆Λ:9 Ι=6Ι

6Χ >Χ8Γ:6Η: >Χ Β∆Χ:Ν <Γ∆ΛΙ= Α:69Η Ι∆ Α∆Λ:Γ ΗΙ:69Ν ΗΙ6Ι: ΕΓ∆9ϑ8Ι>∆Χ >ΧΕϑΙΗ,

8∆ΧΗϑΒΕΙ>∆Χ 6Χ9 Γ:6Α Β∆Χ:Ν 76Α6Χ8:Η.

F>Χ6ΑΑΝ, ΝΑ;6Η∆Χ 6Χ9 !:Γ7:ΓΙΗΗ∆Χ (2001) ΕΓ∆Ε∆Η:9 Ι∆ >ΧΗ:ΓΙ Γ:6Α Β∆Χ:Ν

76Α6Χ8:Η >ΧΙ∆ Ι=: ΕΓ∆9ϑ8Ι>∆Χ ;ϑΧ8Ι>∆Χ, 6Η 6 ΕΓ∆ΜΝ ∆; Ι=: :Ω:8Ι ∆; ΠΧ6Χ8>6Α

9:ΕΙ= ∆Χ ΕΓ∆9ϑ8Ι>∆Χ (67∆ϑΙ Λ=>8= Η:: ∃>Χ< 6Χ9 %:Κ>Χ:, 1993; %:Κ>Χ:, 1997;

ΝΑ;6Η∆Χ 6Χ9 3∆:<6, 2006), 6Χ9 Ι=:Ν ;∆ϑΧ9 6 Χ:<6Ι>Κ: :Ω:8Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ

<Γ∆ΛΙ= Ι=Γ∆ϑ<= Ι=Γ:: 8=6ΧΧ:ΑΗ: >Ι Α∆Λ:ΓΗ Ι=: Γ:6Α >ΧΙ:Γ:ΗΙ Γ6Ι:, 6Χ9 Ι=:Γ:;∆Γ:

Η6Κ>Χ<Η, >Ι Γ:9ϑ8:Η :ς 8>:Χ8Ν 7Ν 9Γ>Κ>Χ< 6 Λ:9<: 7:ΙΛ::Χ Ι=: Γ:ΙϑΓΧΗ Ι∆ Γ:6Α

6Χ9 ΠΧ6Χ8>6Α 86Ε>Ι6Α 6Χ9, ΠΧ6ΑΑΝ, >Ι Γ:9ϑ8:Η ΠΧ6Χ8>6Α 9:ΕΙ= =6ΓΒ>Χ< ∆ϑΙΕϑΙ.

∀Χ 06Χ< 6Χ9 2>Ε (1992) >ΧΘ6Ι>∆Χ >Η Χ:<6Ι>Κ:ΑΝ Γ:Α6Ι:9 Ι∆ <Γ∆ΛΙ=, 7:86ϑΗ:

6 Γ:9ϑ8Ι>∆Χ >Χ Γ:6Α 76Α6Χ8:Η 6Γ>Η>Χ< ;Γ∆Β 6Χ >Χ8Γ:6Η: >Χ Ι=: Γ6Ι: ∆; Β∆Χ:Ι6ΓΝ

<Γ∆ΛΙ= Γ6>Η:Η ΙΓ6ΧΗ68Ι>∆Χ Ι>Β: 6Χ9 Ι=:Γ:;∆Γ: ΙΓ6ΧΗ68Ι>∆Χ 8∆ΗΙΗ. (Χ Ι=: 8∆Χ-

ΙΓ6ΓΝ &>Χ∆ 6Χ9 ,=>76Ι6 (1995), >Χ 6Χ ∆Κ:ΓΑ6ΕΕ>Χ< <:Χ:Γ6Ι>∆Χ ;Γ6Β:Λ∆Γ≅,

Η=∆Λ Ι=6Ι >ΧΘ6Ι>∆Χ Β6Ν =6Κ: 6 Γ:9>ΗΙΓ>7ϑΙ>Κ: >ΒΕ68Ι ;Γ∆Β ∆Χ: <:Χ:Γ6Ι>∆Χ Ι∆

Ι=: ∆Ι=:Γ 6Χ9 ;∆ΗΙ:Γ 86Ε>Ι6Α 688ϑΒϑΑ6Ι>∆Χ. B∆Χ6ΙΙ> (20026, 7) 6Γ<ϑ: Ι=6Ι,

Λ=:Χ ΒϑΑΙ>ΕΑ: 76Α6Χ8:9 <Γ∆ΛΙ= Ε6Ι=Η :Μ>ΗΙ >Χ 6 Χ∆Χ-Β∆Χ:Ι6ΓΝ :8∆Χ∆ΒΝ, >Χ-

Θ6Ι>∆Χ Ι6Γ<:Ι>Χ< 86ΧΧ∆Ι Γ:Η∆ΑΚ: Ι=: Γ:ΗϑΑΙ>Χ< >Χ9:Ι:ΓΒ>Χ68Ν, Λ=:Γ:6Η 6 ΠΜ:9

Β∆Χ:Ι6ΓΝ <Γ∆ΛΙ= ΓϑΑ: 86Χ 9∆ >Ι 6Χ9 >Ι 6ΑΗ∆ 9:Ι:ΓΒ>Χ:Η Ι=: <Γ∆ΛΙ= Ε6Ι= ∆;

Ι=: :8∆Χ∆ΒΝ. FϑΓΙ=:ΓΒ∆Γ:, 6 Γ:ΗΙΓ>8Ι>Κ: Β∆Χ:Ι6ΓΝ Ε∆Α>8Ν Β6Ν Η:Α:8Ι 6 Α∆Λ:Γ

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<Γ∆ΛΙ= Ε6Ι= Ι=6Χ 6 Β∆Γ: :ΜΕ6ΧΗ>Κ: ∆Χ:.

∀Χ )66Α 6Χ9 ,Β>Ι= (2000), Ι=: Γ:Α6Ι>∆ΧΗ=>Ε 7:ΙΛ::Χ Β∆Χ:Ν <Γ∆ΛΙ= 6Χ9

Γ:6Α <Γ∆ΛΙ= >Η Η=∆Λ:9 Ι∆ 7: 8=6Γ68Ι:Γ>Ο:9 7Ν 6 Ι=Γ:Η=∆Α9. AΙ Α∆Λ Β∆Χ:Ν

<Γ∆ΛΙ= Γ6Ι:Η, 76Χ≅Η Ε:Γ8:>Κ: 6 ΗΒ6ΑΑ ∆ΕΕ∆ΓΙϑΧ>ΙΝ 8∆ΗΙ >Χ 9:Ι6>Χ>Χ< Γ:Η:ΓΚ:Η

>ΧΗΙ:69 ∆; Α:Χ9>Χ< ;ϑΧ9Η ;∆Γ >ΧΚ:ΗΙΒ:ΧΙΗ. AΗ Β∆Χ:Ν <Γ∆ΛΙ= Γ>Η:Η, Ι=: Χ∆Β>Χ6Α

>ΧΙ:Γ:ΗΙ Γ6Ι: Γ>Η:Η Ι∆∆ >Χ8Γ:6Η>Χ< Ι=: ∆ΕΕ∆ΓΙϑΧ>ΙΝ 8∆ΗΙ ∆; =∆Α9>Χ< Γ:Η:ΓΚ:Η 6Χ9

ΗΕϑΓΓ>Χ< Α:Χ9>Χ< 6Χ9 Ι=:Γ:;∆Γ: >ΧΚ:ΗΙΒ:ΧΙ 6Χ9 <Γ∆ΛΙ=. 0=:Χ Ι=: Χ∆Β>Χ6Α

>ΧΙ:Γ:ΗΙ Γ6Ι: <Γ∆ΛΗ 7:Ν∆Χ9 6 8:ΓΙ6>Χ Ι=Γ:Η=∆Α9 Α:Κ:Α 8Γ:9>Ι Γ6Ι>∆Χ>Χ< 769ΑΝ

6Ω:8ΙΗ Α:Χ9>Χ<, Γ:9ϑ8>Χ< 86Ε>Ι6Α 688ϑΒϑΑ6Ι>∆Χ 6Χ9 <Γ∆ΛΙ=. AΑΗ∆ B∆Η: (2002)

>Η 8∆Χ8:ΓΧ:9 Λ>Ι= Ι=: >ΒΕ68Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ <Γ∆ΛΙ= Ε6ΗΗ>Χ< Ι=Γ∆ϑ<= Ι=: 8Γ:9>Ι

Β6Γ≅:Ι. −=>Η Ε6Ε:Γ 6Γ<ϑ:Η Ι=6Ι, ϑΧ9:Γ 6ΗΝΒΒ:ΙΓ>8 >Χ;∆ΓΒ6Ι>∆Χ 7:ΙΛ::Χ

Α:Χ9:ΓΗ 6Χ9 7∆ΓΓ∆Λ:ΓΗ, Ι=:Γ: Β><=Ι 7: ΙΛ∆ Α:Χ9>Χ< Γ:<>Β:Η. ∀Χ Ι=: ;∆ΓΒ:Γ

∆Χ:, =><= 6Χ9 Α∆Λ Γ>Η≅ 7∆ΓΓ∆Λ:ΓΗ 6Γ: Η:Ε6Γ6Ι:9 7Ν 8Γ:9>Ι Γ6Ι>∆Χ>Χ< (Γ6Ι>∆Χ>Χ<

Γ:<>Β:). ∀Χ Ι=: Α6ΙΙ:Γ ∆Χ:, Η:Ε6Γ6Ι>∆Χ Ι6≅:Η ΕΑ68: Ι=Γ∆ϑ<= 8∆ΗΙΑΝ Η8Γ::Χ>Χ<

(Η8Γ::Χ>Χ< Γ:<>Β:). A Γ>Η: >Χ Ι=: >ΧΘ6Ι>∆Χ Γ6Ι: >Χ8Γ:6Η:Η Ι=: 8∆ΗΙ ∆; Η8Γ::Χ>Χ<

∆Γ Ι=: >Χ8>9:Χ8: ∆; 8Γ:9>Ι Γ6Ι>∆Χ>Χ< ∆Γ >Ι Β6Ν :Κ:Χ ΕΓ∆9ϑ8: 6 8=6Χ<: ;Γ∆Β

Ι=: Η8Γ::Χ>Χ< Ι∆ Ι=: Γ6Ι>∆Χ>Χ< Γ:<>Β:. AΑΑ Ι=:Η: Ι=Γ:: :Ω:8ΙΗ 6Γ: =6ΓΒ;ϑΑ Ι∆

<Γ∆ΛΙ=.

(Χ Ι=: ∆Ι=:Γ =6Χ9, FϑΧ≅ 6Χ9 ∃Γ∆Β:Χ (2005, 2006) >ΧΚ:ΗΙ><6Ι:9 Ι=: 8∆Χ-

Χ:8Ι>∆Χ 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9 <Γ∆ΛΙ= >Χ 6 ,8=ϑΒΕ:Ι:Γ>6Χ ;Γ6Β:Λ∆Γ≅ Λ>Ι=

Η=∆ΓΙ-ΓϑΧ ΕΓ>8: Γ><>9>ΙΝ. −=:Ν 6ΑΗ∆ ;∆ϑΧ9 6Χ =ϑΒΕ-Η=6Ε:9 >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ=

Α∆8ϑΗ 9ϑ: Ι∆ Ι=: 9>ΗΙ∆ΓΙ>∆Χ6ΓΝ :Ω:8ΙΗ ΕΓ∆9ϑ8:9 7Ν >ΧΘ6Ι>∆Χ ∆Χ Ι=: >Χ8:ΧΙ>Κ:

Ι∆ >ΧΧ∆Κ6Ι:.

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∀Ι >Η Ε∆ΗΗ>7Α: Ι∆ 8∆Χ8Αϑ9: Ι=6Ι Ι=: Ι=:∆Γ:Ι>86Α Α>Ι:Γ6ΙϑΓ: ;∆8ϑΗ:9 ∆Χ Ι=:

:Ω:8Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ <Γ∆ΛΙ= Ε6ΗΗ>Χ< Ι=Γ∆ϑ<= Ι=: 688ϑΒϑΑ6Ι>∆Χ ∆; :>Ι=:Γ =ϑ-

Β6Χ ∆Γ Ε=ΝΗ>86Α 86Ε>Ι6Α, Ι=Γ∆ϑ<= Ι=: 8Γ:9>Ι Β6Γ≅:Ι ∆Γ Ι=Γ∆ϑ<= Ι=: ΕΓ∆9ϑ8Ι

Β6Γ≅:Ι. −=>Η ΗΙϑ9Ν, >ΧΗΙ:69, 9:6ΑΗ Λ>Ι= Ι=: :Ω:8Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ Γ:6Α <Γ∆ΛΙ=

Ε6ΗΗ>Χ< Ι=Γ∆ϑ<= Ι=: Α67∆ϑΓ Β6Γ≅:Ι >Χ ΕΓ:Η:Χ8: ∆; Λ6<: ΗΙ6<<:Γ>Χ<. FϑΓΙ=:Γ-

Β∆Γ:, Ι=>Η ΗΙϑ9Ν <>Κ:Η 9>Ω:Γ:ΧΙ >ΧΗ><=ΙΗ >ΧΙ∆ Ι=: >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ= Χ:ΜϑΗ Ι=6Χ

FϑΧ≅ 6Χ9 ∃Γ∆Β:Χ (2005, 2006) 6Η Λ: ϑΗ: 6 Α:6ΓΧ>Χ< 7Ν 9∆>Χ< Β∆9:Α 6Χ9 Χ∆Ι

6 ,8=ϑΒΕ:Ι:Γ>6Χ ;Γ6Β:Λ∆Γ≅, Λ6<:-ΗΙ6<<:Γ>Χ< 6Χ9 Χ∆Ι ΕΓ>8:-ΗΙ6<<:Γ>Χ< 6Χ9,

Β∆Γ: >ΒΕ∆ΓΙ6ΧΙΑΝ, Α67∆ϑΓ ΗϑΕΕΑΝ >Η Χ∆Ι :Μ∆<:Χ∆ϑΗΑΝ <>Κ:Χ, 7ϑΙ 9:Ι:ΓΒ>Χ:9

7Ν Ι=: ∆ΕΙ>Β>Ο>Χ< 7:=6Κ>∆Γ ∆; :8∆Χ∆Β>8 6<:ΧΙΗ.

−:ΒΕΑ: (2000) ∆Ω:ΓΗ 6 7Γ∆69:Γ Γ:Κ>:Λ ∆; Ι=: Ι=:∆Γ:Ι>86Α Α>Ι:Γ6ΙϑΓ:.

2.2 E∋∗%+%ca& L%−!+a−.+!

−=: Α>Ι:Γ6ΙϑΓ: Γ:Κ>:Λ Ι=6Ι ;∆ΑΑ∆ΛΗ <>Κ:Η ΗΕ:8>6Α Λ:><=Ι Ι∆ Ι=: ΗΙϑ9>:Η ∆Χ Ι=:

>ΧΘ6Ι>∆Χ-<Γ∆ΛΙ= Χ:ΜϑΗ 6;Ι:Γ 2000, 6Η BϑΑΑ6Γ9 (1999) 6Χ9 −:ΒΕΑ: (2000) ∆;-

;:Γ:9 ΙΛ∆ ΗϑΓΚ:ΝΗ ∆; ΕΓ:Κ>∆ϑΗ Λ∆Γ≅. F∆Γ Ι=: ΗΙϑ9>:Η 7:;∆Γ: 2000, ∆ΧΑΝ Ι=: ≅:Ν

∆Χ:Η 6Γ: Β:ΧΙ>∆Χ:9 =:Γ:6;Ι:Γ.

+:<6Γ9>Χ< Ι=: Ι>Β:-Η:Γ>:Η Α>Ι:Γ6ΙϑΓ:, BϑΑΑ6Γ9 (1999) 6Χ9 −:ΒΕΑ: (2000)

6<Γ:: Ι=6Ι, 7:>Χ< Β6>ΧΑΝ 76Η:9 ∆Χ ϑΧ>Ι Γ∆∆Ι Ι:ΗΙ>Χ<, >ΙΗ Γ:ΗϑΑΙΗ Β><=Ι 7:

Φϑ:ΗΙ>∆Χ67Α: 9ϑ: Ι∆ Ι=: Α∆Λ Ε∆Λ:Γ ∆; Ι=:Η: Ι:ΗΙΗ >Χ ΠΧ>Ι: Η6ΒΕΑ:Η. (Χ: ∆; >ΙΗ

Β6>Χ Γ:ΗϑΑΙΗ >Η Ι=6Ι >ΧΘ6Ι>∆Χ 6Χ9 <Γ∆ΛΙ= 6Γ: Χ∆Ι 8∆ΧΧ:8Ι:9 >Χ Ι=: Α∆Χ<-ΓϑΧ,

<>Κ:Χ Ι=6Ι Ι=: ;∆ΓΒ:Γ Λ∆ϑΑ9 7: Χ∆Χ-ΗΙ6Ι>∆Χ6ΓΝ 6Χ9 Ι=: Α6ΙΙ:Γ ΗΙ6Ι>∆Χ6ΓΝ4.

4−=>Η 8∆Χ8ΑϑΗ>∆Χ >Η >Χ 8∆ΧΙΓ6ΗΙ Λ>Ι= C=Γ>ΗΙ∆Ε∆ϑΑ∆Η 6Χ9 −Η>∆Χ6Η (2005), Λ=∆, ϑΗ>Χ< Ε6Χ:Α

8

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−=: ΕΓ:Η:Χ8: ∆; 6 ϑΧ>Ι Γ∆∆Ι >Χ >ΧΘ6Ι>∆Χ Β><=Ι 7: Γ:-6ΗΗ:ΗΗ:9 ∆Χ8: 8∆ΧΗ>9:Γ>Χ<

Ι=6Ι Β6ΧΝ ΗΙϑ9>:Η ∆; Ι=: >ΧΘ6Ι>∆Χ Ε:ΓΗ>ΗΙ:Χ8: Χ:ΙΛ∆Γ≅ ∆; Ι=: ECB ;∆ϑΧ9 Ι=6Ι,

6ΑΑ∆Λ>Χ< ;∆Γ ΗΙΓϑ8ΙϑΓ6Α 7Γ:6≅Η >Χ Ι=: Β:6Χ ∆; Ι=: >ΧΘ6Ι>∆Χ ΕΓ∆8:ΗΗ, >ΧΘ6Ι>∆Χ

6ΕΕ:6ΓΗ Ι∆ 7: ΗΙ6Ι>∆Χ6ΓΝ 6Η Λ:ΑΑ (AΑΙ>ΗΗ>Β∆ :Ι 6Α., 2006).

&6ΧΝ 8Γ∆ΗΗ-Η:8Ι>∆Χ6Α 6Χ9 Ε6Χ:Α 96Ι6 ΗΙϑ9>:Η ;∆ϑΧ9 6 Χ∆Χ-Α>Χ:6Γ Γ:Α6Ι>∆Χ-

Η=>Ε 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9 Γ:6Α <Γ∆ΛΙ=. ΝΑ;6Η∆Χ 6Χ9 !:Γ7:ΓΙΗΗ∆Χ (2001)

Γ:;:Γ Ι∆ Κ6Γ>∆ϑΗ ΗΙϑ9>:Η Ε∆>ΧΙ>Χ< ∆ϑΙ Ι=6Ι 6Χ >Χ8Γ:6Η: >Χ >ΧΘ6Ι>∆Χ ;Γ∆Β 5% Ι∆

50% 9:8Γ:6Η:Η Ι=: Γ:6Α <Γ∆ΛΙ= Γ6Ι:. !∆Λ:Κ:Γ Ι=:Ν ;∆ϑΧ9 Ι=6Ι Ι=>Η :Ω:8Ι >Η

Χ∆Χ-Α>Χ:6Γ 6Χ9 Ι=6Ι >ΧΘ6Ι>∆Χ Γ6Ι:Η 7:Α∆Λ 10% 6Γ: Ε∆Η>Ι>Κ:ΑΝ 8∆ΓΓ:Α6Ι:9 Λ>Ι=

<Γ∆ΛΙ=, 7ϑΙ Ι=: ∆ΕΕ∆Η>Ι: =∆Α9Η ;∆Γ >ΧΘ6Ι>∆Χ Γ6Ι:Η 67∆Κ: 10%.

,>Β>Α6Γ Γ:ΗϑΑΙΗ Λ:Γ: ;∆ϑΧ9 7Ν C=6Γ> e− a%. (1996) 6Χ9 B6ΓΓ∆ (2001). FΓ∆Β

Ι=: ;∆ΓΒ:Γ ΗΙϑ9Ν Λ∆ϑΑ9 Γ:ΗϑΑΙ Ι=6Ι 6Χ >Χ8Γ:6Η: >Χ >ΧΘ6Ι>∆Χ ;Γ∆Β 10 Ι∆ 20%

Λ∆ϑΑ9 9:8Γ:6Η: <Γ∆ΛΙ= 7Ν 6Χ 6Β∆ϑΧΙ Γ6Χ<>Χ< ;Γ∆Β 0.2 Ι∆ 0.7%. ∃6=Χ 6Χ9

,:Χ=69?> (2001) ;∆ϑΧ9 Ι=: Ι=Γ:Η=∆Α9 Ι∆ 7: 6Γ∆ϑΧ9 1% ;∆Γ >Χ9ϑΗΙΓ>6Α>Ο:9 8∆ϑΧ-

ΙΓ>:Η 6Χ9 11% ;∆Γ 9:Κ:Α∆Ε>Χ< ∆Χ:Η. −=: Γ:ΗϑΑΙΗ ∆; =∆Η= 6Χ9 )=>ΑΑ>ΕΗ (1998)

Λ∆ϑΑ9 Ε∆>ΧΙ >Ι Ι∆ 7: 6Ι 2.5%, Λ=:Γ:6Η #ϑ9Η∆Χ 6Χ9 (ΓΕ=6Χ>9:Η (1999) 6Ι 10%

6Χ9 )∆ΑΑ>Χ 6Χ9 3=ϑ (2006) 6Ι 67∆ϑΙ 13-15%.

A Ι=Γ:Η=∆Α9 :Ω:8Ι Λ6Η 6ΑΗ∆ ;∆ϑΧ9 7Ν −=>ΓΑΛ6ΑΑ 6Χ9 B6ΓΙ∆Χ (1971) 6Ι 6Χ

6ΧΧϑ6Α >ΧΘ6Ι>∆Χ Γ6Ι: Γ6Χ<>Χ< ;Γ∆Β 8% Ι∆ 10%. A Η>Β>Α6Γ Κ6Αϑ: Λ6Η Ηϑ<<:ΗΙ:9

7Ν ,6Γ:Α (1996) 6Η Λ:ΑΑ. ΝΑ;6Η∆Χ (1991) ;∆ϑΧ9 Ι=6Ι :8∆Χ∆Β>:Η Λ>Ι= >ΧΘ6Ι>∆Χ

ϑΧ>Ι Γ∆∆Ι 6Χ9 8∆>ΧΙ:<Γ6Ι>∆Χ Ι:8=Χ>Φϑ:Η, ;∆ϑΧ9 7∆Ι= >ΧΘ6Ι>∆Χ 6Χ9 (ΕΓ∆9ϑ8Ι>Κ>ΙΝ) <Γ∆ΛΙ= Ι∆7: ∀(1), Ι=∆ϑ<= Ι=:Ν 8∆ϑΑ9 Χ∆Ι ΠΧ9 6 Χ:<6Ι>Κ: Α>Χ≅ 7:ΙΛ::Χ Ι=:Β 68Γ∆ΗΗ 6ΑΑ Ι=: 8∆ϑΧΙΓ>:Η>Χ8Αϑ9:9 >Χ Ι=:>Γ Η6ΒΕΑ:. −=:Ν Β:6ΗϑΓ:9 ΕΓ∆9ϑ8Ι>Κ>ΙΝ 6Η Ι=: Γ6Ι>∆ 7:ΙΛ::Χ Γ:6Α <Γ∆ΗΗ∆ϑΙΕϑΙ 6Χ9 :ΒΕΑ∆ΝΒ:ΧΙ 6Χ9 Χ∆Ι 6Η D) Ε:Γ 86Ε>Ι6.

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67∆Κ: 20% <Γ:Λ Α:ΗΗ Γ6Ε>9ΑΝ Ι=6Χ :8∆Χ∆Β>:Η Λ>Ι= >ΧΘ6Ι>∆Χ 7:Α∆Λ 5% 6 Ν:6Γ.

BΓϑΧ∆ 6Χ9 E6ΗΙ:ΓΑΝ (1998) Γ:Ε∆ΓΙ Ι=6Ι >ΧΘ6Ι>∆Χ Γ6Ι:Η 67∆Κ: 40 Ε:Γ8:ΧΙ 6

Ν:6Γ ;∆Γ 6Ι Α:6ΗΙ ΙΛ∆ Ν:6ΓΗ >Χ 6 Γ∆Λ 6Γ: <:Χ:Γ6ΑΑΝ =6ΓΒ;ϑΑ Ι∆ <Γ∆ΛΙ=. AΑΗ∆

F>Η8=:Γ (1993) Χ∆Ι:9 Ι=: :Μ>ΗΙ:Χ8: ∆; 6 Ε∆Η>Ι>Κ: Γ:Α6Ι>∆ΧΗ=>Ε 6Ι Α∆Λ >ΧΘ6Ι>∆Χ

Γ6Ι:Η 6Χ9 6 Χ:<6Ι>Κ: ∆Χ: 6Η >ΧΘ6Ι>∆Χ Γ>Η:Η. BϑΓ9:≅>Χ (2004) ;∆ϑΧ9 Ι=6Ι, ;∆Γ

>Χ9ϑΗΙΓ>6Α>Ο:9 8∆ϑΧΙΓ>:Η, >ΧΘ6Ι>∆Χ Γ6Ι:Η 7:Α∆Λ 8% =6Κ: 6Χ >ΧΗ><Χ>Π86ΧΙ :Ω:8Ι

∆Χ <Γ∆ΛΙ= 6Χ9 6 Χ:<6Ι>Κ: ∆Χ: 67∆Κ: Ι=6Ι Γ6Ι:. ∀Χ 9:Κ:Α∆Ε>Χ< 8∆ϑΧΙΓ>:Η,

>ΧΘ6Ι>∆Χ =6ΓΒΗ <Γ∆ΛΙ= 67∆Κ: 3%. !∆Λ:Κ:Γ, 67∆Κ: 50% Ι=: Β6Γ<>Χ6Α 8∆ΗΙ

∆; >ΧΘ6Ι>∆Χ 9:8Γ:6Η:Η Ηϑ7ΗΙ6ΧΙ>6ΑΑΝ. F>Χ6ΑΑΝ, /6∆Χ6 6Χ9 ,8=>6Κ∆ (2007) ϑΗ:9

7∆Ι= 6 Χ∆ΧΕ6Γ6Β:ΙΓ>8 6Χ9 6 Η:Β>Ε6Γ6Β:ΙΓ>8 >ΧΗΙΓϑΒ:ΧΙ6Α Κ6Γ>67Α: :ΗΙ>Β6Ι∆ΓΗ,

Λ=>8= =6Κ: Ι=: 69Κ6ΧΙ6<: ∆; Α:ΙΙ>Χ< 6Η Βϑ8= 6Η Ε∆ΗΗ>7Α: Ι=: 96Ι6 ΗΕ:6≅. −=:Ν

Η=∆Λ:9 Ι=6Ι, ;∆Γ 9:Κ:Α∆Ε:9 8∆ϑΧΙΓ>:Η, Α∆Λ >ΧΘ6Ι>∆Χ Γ6Ι:Η =6Κ: =6Γ9ΑΝ 6ΧΝ Γ:6Α

:Ω:8Ι 6Χ9 =><= >ΧΘ6Ι>∆Χ Γ6Ι:Η =6Κ: Χ:<6Ι>Κ: Γ:6Α :Ω:8ΙΗ. F∆Γ 9:Κ:Α∆Ε>Χ<

8∆ϑΧΙΓ>:Η Ι=:Ν ;∆ϑΧ9 Ι=6Ι Ι∆∆ =><= Κ6Γ>67>Α>ΙΝ 9∆:Η Χ∆Ι 6ΑΑ∆Λ Ι∆ Γ:68= 8Α:6Γ-

8ϑΙ Γ:ΗϑΑΙΗ.

ϑ:ΓΓ:Γ∆ (2006) ϑΗ:9 ΕΓ:Κ>∆ϑΗ =ΝΕ:Γ>ΧΘ6Ι>∆Χ6ΓΝ :ΜΕ:Γ>:Χ8: 6Η >ΧΗΙΓϑΒ:ΧΙ

;∆Γ >ΧΘ6Ι>∆Χ, ΠΧ9>Χ< Ι=6Ι >ΧΘ6Ι>∆Χ =6Η 6 Χ:<6Ι>Κ: >ΒΕ68Ι ∆Χ <Γ∆ΛΙ=. AΓ6> :Ι

6Α. (2004), ϑΗ>Χ< 9ΝΧ6Β>8 Ε6Χ:Α 96Ι6 Β:Ι=∆9Η ∆Χ 6 96Ι6Η:Ι ∆; 115 8∆ϑΧΙΓ>:Η

∆Κ:Γ Ι=: Ε:Γ>∆9 1960-1995, 9>9 Χ∆Ι ΠΧ9 6ΧΝ :Κ>9:Χ8: Ι=6Ι >ΧΘ6Ι>∆Χ >Η =6ΓΒ;ϑΑ

Ι∆ <Γ∆ΛΙ=. (Χ Ι=: 8∆ΧΙΓ6ΓΝ, Ι=: Χ:<6Ι>Κ: 8∆ΓΓ:Α6Ι>∆Χ 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9

<Γ∆ΛΙ= 86Χ 7: :ΜΕΑ6>Χ:9 7Ν ∆>Α ΕΓ>8: Η=∆8≅Η. !∆Λ:Κ:Γ, ∃>Β :Ι 6Α. (2000), ϑΗ-

>Χ< 8Γ∆ΗΗ-8∆ϑΧΙΓΝ/Ι>Β:-Η:Γ>:Η 96Ι6, ;∆ϑΧ9 Ι=6Ι Ι=: Χ:<6Ι>Κ: :Ω:8Ι ∆; >ΧΘ6Ι>∆Χ

∆Χ <Γ∆ΛΙ= >Η <Γ:6Ι:Γ >Χ 9:Κ:Α∆Ε:9 8∆ϑΧΙΓ>:Η Ι=6Χ >Χ 9:Κ:Α∆Ε>Χ< ∆Χ:Η 6Χ9 Ι=6Ι

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Ι=: >Χ8ΑϑΗ>∆Χ >Χ Ι=: Β∆9:Α ∆; ∆>Α ΗϑΕΕΑΝ Η=∆8≅Η Λ:6≅:ΧΗ Ι=: >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ=

Χ:ΜϑΗ, Λ=>8=, =∆Λ:Κ:Γ, 9∆:Η Χ∆Ι 9>Η6ΕΕ:6Γ.

B∆Ν9 :Ι 6Α. (2001), !6ΗΑ6< 6Χ9 ∃∆∆ (1999) 6Χ9 AΧ9ΓΨΗ :Ι 6Α. (2004)

;∆8ϑΗ:9 ∆Χ Ι=: >ΧΙ:Γ8∆ΧΧ:8Ι>∆ΧΗ 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ, ΠΧ6Χ8>6Α 9:Κ:Α∆ΕΒ:ΧΙ 6Χ9

<Γ∆ΛΙ=. −=: ΠΓΗΙ ΙΛ∆ ΗΙϑ9>:Η ΗϑΕΕ∆ΓΙ Ι=: Κ6Α>9>ΙΝ ∆; Ι=: <Γ∆ΛΙ=ΥΠΧ6Χ8>6Α

9:Κ:Α∆ΕΒ:ΧΙ Α>Χ≅ 7ϑΙ Χ∆Ι ∆; Ι=: >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ= ∆Χ:. (Χ Ι=: ∆Ι=:Γ =6Χ9,

AΧ9ΓΨΗ :Ι 6Α. (2004) 8=6ΑΑ:Χ<: Ι=: >ΒΕ∆ΓΙ6Χ8: ∆; ΠΧ6Χ8>6Α Β6Γ≅:ΙΗ >Χ ϑΧ9:Γ-

ΗΙ6Χ9>Χ< Ι=: >ΒΕ68Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ <Γ∆ΛΙ=.

AΑΑ >Χ 6ΑΑ, >Ι >Η Ε∆ΗΗ>7Α: Ι∆ ΗΙ6Ι: Ι=6Ι =><= >ΧΘ6Ι>∆Χ >Η 9:ΙΓ>Β:ΧΙ6Α Ι∆ <Γ∆ΛΙ=,

7ϑΙ Ι=:Γ: 6Γ: Η∆Β: 8=6Χ8:Η Ι=6Ι 6 Α∆Λ >ΧΘ6Ι>∆Χ Γ6Ι: Β><=Ι =6Κ: 6 Ε∆Η>Ι>Κ:

>ΒΕ68Ι ∆Χ <Γ∆ΛΙ=, Ι=∆ϑ<= =∆Λ Α∆Λ >Η 6 Φϑ:ΗΙ>∆Χ Ι=: Α>Ι:Γ6ΙϑΓ: =6Η =69 Β6ΧΝ

9>ς 8ϑΑΙ>:Η >Χ Γ:ΕΑΝ>Χ< Ι∆.

3 A M) !& )∀ N)∋%(a& R%#% %−%!, a( E(-

)#!()., G+)0−∃

∀Χ ∆ϑΓ 6Χ6ΑΝΗ>Η, Ι=: Χ∆Β>Χ6Α Γ><>9>ΙΝ Ι6≅:Η Ι=: ;∆ΓΒ ∆; ΗΙ6<<:Γ:9 −6ΝΑ∆Γ Λ6<:

8∆ΧΙΓ68ΙΗ5. (ϑΓ Β∆9:Α :8∆Χ∆ΒΝ 8∆ΧΙ6>ΧΗ 6 8∆ΧΙ>ΧϑϑΒ ∆; =∆ϑΗ:=∆Α9Η, ΗϑΕ-

5AΗ Χ∆Ι:9 7Ν Γ6=6Β 6Χ9 ,Χ∆Λ:Γ (2008), ;∆Γ "Ηϑς 8>:ΧΙΑΝ =><= Α:Κ:ΑΗ ∆; Β∆Χ:Ν <Γ∆ΛΙ=,C6ΑΚ∆ 8∆ΧΙΓ68ΙΗ 6Γ: Χ∆Ι 6ΕΕΓ∆ΕΓ>6Ι:. −=: Γ:6Η∆Χ >Η ΗΙΓ6><=Ι;∆ΓΛ6Γ9. 0>Ι= C6ΑΚ∆ 8∆ΧΙΓ68ΙΗ,Η∆Β: =∆ϑΗ:=∆Α9Η ≅::Ε Ι=:>Γ Χ∆Β>Χ6Α Λ6<: ϑΧ8=6Χ<:9 ;∆Γ 6 Κ:ΓΝ Α∆Χ< Ε:Γ>∆9 ∆; Ι>Β:, Λ=>8=Β:6ΧΗ Ι=6Ι, >Χ Ι=: ΕΓ:Η:Χ8: ∆; >ΧΘ6Ι>∆Χ, Ι=: Γ:6Α Κ6Αϑ: ∆; Ι=>Η Λ6<: 6ΕΕΓ∆68=:Η Ο:Γ∆. −=>Η>ΒΕΑ>:Η Ι=6Ι Ι=: ΠΓΒ Λ>ΑΑ Λ>Η= Ι∆ =>Γ: 6Η Βϑ8= ∆; Ι=: Α67∆Γ ∆; Ι=:Η: =∆ϑΗ:=∆Α9Η 6Η Ε∆ΗΗ>7Α:,6Χ9 6Η Α>ΙΙΑ: ∆; Ι=: ∆Ι=:Γ =∆ϑΗ:=∆Α9Η. −=>Η >Η Κ:ΓΝ >Χ:ς 8>:ΧΙ Η∆ ∆ϑΙΕϑΙ 6ΕΕΓ∆68=:Η Ο:Γ∆".−=>Η Η=∆ΓΙ8∆Β>Χ< ∆; C6ΑΚ∆ 8∆ΧΙΓ68ΙΗ 8∆ϑΑ9 7: ∆Κ:Γ8∆Β: 7Ν >ΧΙΓ∆9ϑ8>Χ< >ΧΘ6Ι>∆Χ >Χ9:Μ6Ι>∆Χ(∆Χ Ι=>Η >ΗΗϑ: Η::, ;∆Γ >ΧΗΙ6Χ8:, Ι=: Ε6Ε:ΓΗ Φϑ∆Ι:9 >Χ &>Χ;∆Γ9 6Χ9 )::Α, 2004). !∆Λ:Κ:Γ,Ηϑ8= 6 8∆ΧΙΓ68Ι>Χ< 9:Κ>8: >Η Χ∆Ι 6Η Λ>9:ΗΕΓ:69 6Η >Χ Ι=: Ε6ΗΙ >Χ 9:Κ:Α∆Ε:9 (c(∋−#∋.e,)

11

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ΕΑΝ>Χ< 9>Ω:Γ:ΧΙ>6Ι:9 Α67∆Γ, 6Χ9 6 Α6Γ<: ΧϑΒ7:Γ ∆; >9:ΧΙ>86Α ΠΓΒΗ, ΕΓ∆9ϑ8>Χ<

∆ϑΙΕϑΙ 7Ν Β:6ΧΗ ∆; 6ΑΑ Α67∆Γ ΙΝΕ:Η 6Χ9 86Ε>Ι6Α. −=: Α67∆Γ ΙΝΕ:Η 6Γ: >ΒΕ:Γ-

;:8Ι Ηϑ7ΗΙ>ΙϑΙ:Η >Χ Ι=: ΕΓ∆9ϑ8Ι>∆Χ ;ϑΧ8Ι>∆Χ, :Μ=>7>Ι>Χ< 9>Β>Χ>Η=>Χ< Γ:ΙϑΓΧΗ Ι∆

Α67∆Γ 7ϑΙ >Χ8Γ:6Η>Χ< Γ:ΙϑΓΧΗ Ι∆ Η86Α:. −=ϑΗ :68= =∆ϑΗ:=∆Α9 ;68:Η 6 9∆ΛΧΛ6Γ9-

ΗΑ∆Ε>Χ< Α67∆ϑΓ 9:Β6Χ9 8ϑΓΚ: >Χ Ι=: Η=∆ΓΙ ΓϑΧ. −=: <∆Κ:ΓΧΒ:ΧΙ ΕΓ>ΧΙΗ Β∆Χ:Ν

6Χ9 >Ι Γ:ΙϑΓΧΗ >ΙΗ Η:><Χ>∆Γ6<: ΕΓ∆8::9Η Ι∆ =∆ϑΗ:=∆Α9Η >Χ Ι=: ;∆ΓΒ ∆; ΑϑΒΕ ΗϑΒ

Ι6Μ Γ:76Ι:Η.

3.1 T∃! ,.∗∗&2 ,% ! )∀ −∃! !c)()∋2

AΗ ;6Γ 6Η Ι=: ΗϑΕΕΑΝ Η>9: ∆; Ι=: :8∆Χ∆ΒΝ >Η 8∆Χ8:ΓΧ:9, Λ: 6ΗΗϑΒ: Ι=: :Μ>ΗΙ:Χ8:

∆; ΙΛ∆ <∆∆9 Η:8Ι∆ΓΗ. ∀Χ Ι=: ΠΧ6Α <∆∆9 Η:8Ι∆Γ 6 Η:Ι ∆; Ε:Γ;:8ΙΑΝ 8∆ΒΕ:Ι>Ι>Κ:

ΠΓΒΗ ΙΓ6ΧΗ;∆ΓΒΗ 6 8∆ΧΙ>ΧϑϑΒ ∆; =∆Γ>Ο∆ΧΙ6ΑΑΝ 9>Ω:Γ:ΧΙ>6Ι:9 >ΧΕϑΙΗ >ΧΙ∆ 6Χ

=∆Β∆<:Χ:∆ϑΗ <∆∆9. ∀Χ Ι=: >ΧΙ:ΓΒ:9>6Ι: <∆∆9 Η:8Ι∆Γ 6 8∆ΧΙ>ΧϑϑΒ ∆; Β∆-

Χ∆Ε∆Α>ΗΙ>86ΑΑΝ 8∆ΒΕ:Ι>Ι>Κ: ΠΓΒΗ ΕΓ∆9ϑ8:Η 9>Ω:Γ:ΧΙ Κ6Γ>:Ι>:Η ∆; 6 =∆Γ>Ο∆ΧΙ6ΑΑΝ

9>Ω:Γ:ΧΙ>6Ι:9 <∆∆9 ϑΗ>Χ< 7∆Ι= Α67∆ϑΓ 6Χ9 86Ε>Ι6Α >ΧΕϑΙΗ. −=: 8∆ΧΙ>ΧϑϑΒ ∆;

Β∆Χ∆Ε∆Α>ΗΙ>86ΑΑΝ 8∆ΒΕ:Ι>Ι>Κ: ΠΓΒ, >Χ9:Μ:9 7Ν ∋ , >Η Χ∆ΓΒ6Α>Ο:9 ∆Χ Ι=: [0; 1]

>ΧΙ:ΓΚ6Α.

0: ;ϑΓΙ=:Γ ΗϑΕΕ∆Η: Ι=6Ι Ι=:Γ: :Μ>ΗΙ 6 86Ε>Ι6Α 6Χ9 6 Α67∆ϑΓ Β6Γ≅:Ι. C6Ε-

>Ι6Α >Η 6 =∆Β∆<:Χ:∆ϑΗ ΕΓ∆9ϑ8Ι>∆Χ ;68Ι∆Γ. ∀Χ Ι=: Α67∆ϑΓ Β6Γ≅:Ι, =∆ϑΗ:=∆Α9Η

7:Α∆Χ<>Χ< Ι∆ 9>Ω:Γ:ΧΙ 8∆=∆ΓΙΗ Η:Ι Ι=:>Γ Λ6<:Η >Χ 6 Β∆Χ∆Ε∆Α>ΗΙ>86ΑΑΝ 8∆ΒΕ:Ι>-

8∆ϑΧΙΓ>:Η. FϑΓΙ=:ΓΒ∆Γ:, >Χ C6ΑΚ∆ 8∆ΧΙΓ68ΙΗ, :8∆Χ∆Β>8 6<:ΧΙΗ =6Κ: >Χ :68= Ε:Γ>∆9 Ι=:Η6Β: ΕΓ∆767>Α>ΙΝ Ι∆ Γ:Η:Ι Λ6<:Η/ΕΓ>8:Η. −=>Η >Η Γ6Ι=:Γ ϑΧΓ:6Α>ΗΙ>8 Λ=:Χ 6ΕΕΑ>:9 Ι∆ Ι=:Α67∆ϑΓ Β6Γ≅:Ι, Λ=:Γ: 8∆ΧΙΓ68ΙΗ Λ>Ι= ΠΜ:9 9ϑΓ6Ι>∆ΧΗ 6Γ: Β∆Γ: Λ>9:ΗΕΓ:69 6Χ9 Ι=: Ι>Β>Χ<∆; 8∆ΧΙΓ68Ι Γ:Χ:Λ6ΑΗ >Η ϑΗϑ6ΑΑΝ 8=6Γ68Ι:Γ>Ο:9 7Ν 6 =><= 9:<Γ:: ∆; 8∆=∆Γ9>Χ6Ι>∆Χ. F∆Γ Ι=:Η:Γ:6Η∆ΧΗ Λ: ΕΓ:;:Γ Ι=: −6ΝΑ∆Γ ΗΕ:8>Π86Ι>∆Χ.

12

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Ι>Κ: :ΧΚ>Γ∆ΧΒ:ΧΙ 6Χ9 Η:ΑΑ Ι=:>Γ Α67∆ϑΓ Ι∆ Ι=: 8∆ΧΙ>ΧϑϑΒ ∆; Β∆Χ∆Ε∆Α>ΗΙ>86ΑΑΝ

8∆ΒΕ:Ι>Ι>Κ: ΠΓΒΗ ∆; Ι=: >ΧΙ:ΓΒ:9>6Ι: <∆∆9 Η:8Ι∆Γ. −=: 8∆ΧΙ>ΧϑϑΒ ∆; Β∆Χ∆Ε-

∆Α>ΗΙ>86ΑΑΝ 8∆ΒΕ:Ι>Ι>Κ: =∆ϑΗ:=∆Α9Η, >Χ9:Μ:9 7Ν (, >Η Χ∆ΓΒ6Α>Ο:9 ∆Χ Ι=: [0; 1]

>ΧΙ:ΓΚ6Α.

3.1.1 T∃! 4(a& #)) ,!c−)+

∀Χ Ι=: ΠΧ6Α <∆∆9 Η:8Ι∆Γ, 9ϑ: Ι∆ Ι=: :Μ>ΗΙ:Χ8: ∆; 6 Η:Ι ∆; 9>Ω:Γ:ΧΙ>6Ι:9 >Χ-

ΕϑΙΗ Λ>Ι= 6 8∆ΧΗΙ6ΧΙ :Α6ΗΙ>8>ΙΝ ∆; Ηϑ7ΗΙ>ΙϑΙ>∆Χ (p), Ι=: ΕΓ∆9ϑ8Ι>∆Χ ;ϑΧ8Ι>∆Χ

6ΗΗϑΒ:Η Ι=: ;∆ΓΒ ∆; 6 CE, 6<<Γ:<6Ι∆Γ 6t =

0

@1!

0

6p1

p

ft %∋

1

A

p

p1

, Χ6Β:ΑΝ >Χ

Ι=: ΠΧ6Α <∆∆9 Η:8Ι∆Γ ΠΓΒΗ ?ϑΗΙ ϑΗ: >ΧΙ:ΓΒ:9>6Ι: >ΧΕϑΙΗ, 6ft, Ι∆ ΕΓ∆9ϑ8: Ι=:>Γ

∆ϑΙΕϑΙ, 6t. F>ΓΒΗ Β6Μ>Β>Ο: ΕΓ∆ΠΙΗ Ηϑ7?:8Ι Ι∆ Ι=:>Γ ΕΓ∆9ϑ8Ι>∆Χ ;ϑΧ8Ι>∆Χ:

maxf ftg

0t6t

1!

0

0ft6ft%∋

2:3: 6t =

0

@1!

0

6p1

p

ft %∋

1

A

p

p1

Λ=:Γ: 0t >Η Ι=: 6<<Γ:<6Ι: ΕΓ>8: Α:Κ:Α 6Χ9 0ft >Η Ι=: ΕΓ>8: ∆; Ι=: ∋ -Ι= >ΧΙ:ΓΒ:-

9>6Ι: >ΧΕϑΙ. BΝ Η∆ΑΚ>Χ< Ι=>Η Β6Μ>Β>Ο6Ι>∆Χ ΕΓ∆7Α:Β >Ι >Η Ε∆ΗΗ>7Α: Ι∆ <:Ι Ι=:

9:Β6Χ9 ;ϑΧ8Ι>∆Χ ;∆Γ :68= <∆∆9 Κ6Γ>:ΙΝ:

6ft =

0ft0t

p6t (1)

13

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FϑΓΙ=:ΓΒ∆Γ:, 7Ν >ΒΕ∆Η>Χ< Ι=: Ο:Γ∆ ΕΓ∆ΠΙ 8∆Χ9>Ι>∆Χ 6Χ9 Ηϑ7ΗΙ>ΙϑΙ>Χ< (1)

>ΧΙ∆ Ι=: ΕΓ∆ΠΙ :Φϑ6Ι>∆Χ >Ι >Η Ε∆ΗΗ>7Α: Ι∆ ∆7Ι6>Χ Ι=: ΕΓ>8: >Χ9:Μ:

0t =

0

@1!

0

01pft %∋

1

A

11p

(2)

3.1.2 T∃! %(−!+∋! %a−! #)) ,!c−)+

∀Χ Ι=: >ΧΙ:ΓΒ:9>6Ι: <∆∆9 Η:8Ι∆Γ :68= ΠΓΒ ∋ 7ϑΝΗ 86Ε>Ι6Α 6Χ9 Α67∆ϑΓ Ι∆ ΕΓ∆-

9ϑ8: ∆Χ: Κ6Γ>:ΙΝ ∆; <∆∆9. ∀Χ Η∆ 9∆>Χ<, >Ι Β>Χ>Β>Ο:Η 8∆ΗΙΗ Ηϑ7?:8Ι Ι∆ Ι=:

8∆ΧΗΙΓ6>ΧΙ ∆; >ΙΗ ΕΓ∆9ϑ8Ι>∆Χ ;ϑΧ8Ι>∆Χ 6Χ9 >Ι Β6Μ>Β>Ο:Η Ι=: ΗΕΓ:69 7:ΙΛ::Χ

Ι=: ΕΓ>8: >Ι 8=6Γ<:Η 6Χ9 >ΙΗ Β6Γ<>Χ6Α 8∆ΗΙ.

>Κ:Χ Ι=6Ι Α67∆ϑΓ >Η =∆Γ>Ο∆ΧΙ6ΑΑΝ 9>Ω:Γ:ΧΙ>6Ι:9, Λ: 6ΕΕΑΝ 6 ΙΛ∆-ΗΙ6<:

7ϑ9<:Ι>Χ< 6ΕΕΓ∆68= 6;Ι:Γ C=6Β7:ΓΗ (1988, ΕΕ. 112-113) 6Χ9 !:>?9Γ6 6Χ9

/6Χ 9:Γ )Α∆:< (2002, ΕΕ. 360-363). F>ΓΗΙ, ΠΓΒΗ 8=∆∆Η: Ι=:>Γ 9:Η>Γ:9 Φϑ6ΧΙ>ΙΝ

∆; 86Ε>Ι6Α 6Χ9 ∆; 6 8∆ΒΕ∆Η>Ι: Α67∆ϑΓ >ΧΕϑΙ, .ft, Λ=>8= >Η 9:ΠΧ:9 6Η ;∆ΑΑ∆ΛΗ

.ft = N 1w1

8<

:

1!

0

[.t (()]w1w %(

9=

;

ww1

Λ=:Γ: ( >Η Ι=: =∆ϑΗ:=∆Α9 >Χ9:Μ 6Χ9 w >Η Ι=: :Α6ΗΙ>8>ΙΝ ∆; Ηϑ7ΗΙ>ΙϑΙ>∆Χ 7:ΙΛ::Χ

9>Ω:Γ:ΧΙ Α67∆ϑΓ ≅>Χ9Η. 0: ΗϑΕΕ∆Η: Ι=6Ι =∆ϑΗ:=∆Α9Η 6Γ: <Γ∆ϑΕ:9 >ΧΙ∆ N

8∆=∆ΓΙΗ 6Χ9 Ι=6Ι Ι=:Γ: 6Γ: 1N=∆ϑΗ:=∆Α9Η Ε:Γ 8∆=∆ΓΙ. ∀Ι >Η ΕΑ6ϑΗ>7Α: Ι∆ 6ΗΗϑΒ:

Ι=6Ι Ι=: ΕΓ:Η:Χ8: ∆; 6 Α6Γ<:Γ ∆Γ 6 ΗΒ6ΑΑ:Γ ΧϑΒ7:Γ ∆; 8∆=∆ΓΙΗ 9∆:Η Χ∆Ι =6Κ:

6Χ >ΒΕ68Ι ∆Χ Ι=: :ς 8>:Χ8Ν ∆; Ι=: ΕΓ∆9ϑ8Ι>∆Χ ΕΓ∆8:ΗΗ. −=:Γ:;∆Γ:, Ι=: Ι:ΓΒ

14

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N 1w1 6Κ∆>9Η :8∆Χ∆Β>:Η ∆; ΗΕ:8>6Α>Ο6Ι>∆Χ >Χ 9>Ω:Γ:ΧΙ Α67∆ϑΓ ≅>Χ9Η6, Λ=>8=

>Χ Ι=: ΕΓ:Η:ΧΙ Η:ΙΙ>Χ< 8∆>Χ8>9: Λ>Ι= 8∆=∆ΓΙΗ. !∆ϑΗ:=∆Α9Η Λ>Ι=>Χ 8∆=∆ΓΙΗ 6Γ:

ΗΝΒΒ:ΙΓ>8 6Χ9 Η∆ Ι=: 8∆ΒΕ∆Η>Ι: Α67∆ϑΓ >ΧΕϑΙ 86Χ 7: Γ:-ΛΓ>ΙΙ:Χ 6Η ;∆ΑΑ∆ΛΗ

.ft = N 1w1

"N1X

j=0

.j;tN

w1w

# ww1

(3)

Λ=:Γ: ∗ >Η Ι=: 8∆=∆ΓΙ >Χ9:Μ.

∀Χ Ι=: ΠΓΗΙ ΗΙ6<: ∆; 7ϑ9<:Ι>Χ<, ΠΓΒΗ Η∆ΑΚ:Η Ι=: ;∆ΑΑ∆Λ>Χ< ΕΓ∆7Α:Β:

minfnft;kftg

5t0t.ft + 1t+ft

2:3: 6ft = (.ft+t) (+ft)

1 (4)

Λ=:Γ: 5t >Η Ι=: Χ∆Β>Χ6Α 6<<Γ:<6Ι: Λ6<: Γ6Ι:, 1t >Η Ι=: Γ:ΒϑΧ:Γ6Ι>∆Χ ;∆Γ 86Ε>Ι6Α

Η:ΓΚ>8:Η, +ft >Η Ι=: 6Β∆ϑΧΙ ∆; 86Ε>Ι6Α ϑΗ:9 7Ν ΠΓΒ ∋ , +t >Η Ι=: 6<<Γ:<6Ι:

86Ε>Ι6Α ΗΙ∆8≅ 6Χ9 >Η 6 Ε6Γ6Β:Ι:Γ. (4) >Η 6 ΙΝΕ>86Α ΕΓ∆9ϑ8Ι>∆Χ ;ϑΧ8Ι>∆Χ Λ>Ι=

≅Χ∆ΛΑ:9<: :ΜΙ:ΓΧ6Α>Ι>:Η, Λ=:Γ:7Ν 6Χ >Χ8Γ:6Η: >Χ 6 ΠΓΒΡΗ 86Ε>Ι6Α ΗΙ∆8≅ Α:69Η Ι∆

6Χ >Χ8Γ:6Η: >Χ >ΙΗ ΗΙ∆8≅ ∆; ≅Χ∆ΛΑ:9<:. AΗΗϑΒ>Χ< Ι=6Ι :68= ΠΓΒΡΗ ≅Χ∆ΛΑ:9<:

>Η 6 Εϑ7Α>8 <∆∆9, Ι=: 6<<Γ:<6Ι: >Χ8Γ:6Η: >Χ ≅Χ∆ΛΑ:9<: >Η ΕΓ∆Ε∆ΓΙ>∆Χ6Α Ι∆ Ι=:

6<<Γ:<6Ι: 86Ε>Ι6Α ΗΙ∆8≅ (B6ΓΓ∆ 6Χ9 16Α6->-&6ΓΙ>Χ, 1995)7.

6,:: 6ΑΗ∆ !:>?9Γ6 6Χ9 /6Χ 9:Γ )Α∆:< (2002), Ε. 361.7∋∆Ι: Ι=6Ι (4) >Η Χ∆Ι 9:ΠΧ:9 ∆Κ:Γ Ι=: ΧϑΒ7:Γ ∆; :ΒΕΑ∆Ν::Η, Γ6Ι=:Γ ∆Κ:Γ 6 8∆ΒΕ∆Η>Ι:

>Χ9:Μ ∆; Λ∆Γ≅>Χ< =∆ϑΓΗ, Λ=∆Η: ΗΙ6Ι>∆Χ6Γ>ΙΝ >Η 8∆ΧΗ>ΗΙ:ΧΙ Λ>Ι= Ι=: ∃6Α9∆Γ ;68ΙΗ (C∆∆Α:Ν 6Χ9)Γ:Η8∆ΙΙ, 1995, Ε. 16). −=:Γ:;∆Γ:, ∆ϑΓ Β∆9:Α ∆; :Χ9∆<:Χ∆ϑΗ <Γ∆ΛΙ= >Η Χ∆Ι ϑΧ9:ΓΒ>Χ:97Ν Ι=: #∆Χ:Η (1995) 8Γ>Ι>Φϑ:, 688∆Γ9>Χ< Ι∆ Λ=>8= Ι=: Γ:6Α <Γ∆ΛΙ= Γ6Ι:, 7:>Χ< 6 ΗΙ6Ι>∆Χ6ΓΝΚ6Γ>67Α:, 86ΧΧ∆Ι 7: :ΜΕΑ6>Χ:9 7Ν Χ∆Χ-ΗΙ6Ι>∆Χ6ΓΝ Κ6Γ>67Α:Η, Α>≅:, >Χ =>Η 86Η:, Ι=: >ΧΚ:ΗΙΒ:ΧΙ

15

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−=: ΠΓΗΙ ∆Γ9:Γ 8∆Χ9>Ι>∆ΧΗ ;∆Γ Α67∆ϑΓ 6Χ9 86Ε>Ι6Α 6Γ::

5t0t

= −∃t (.ft+t)1 (+ft)

1 +t (5)

1t = −∃t (1 ) (.ft+t) (+ft)

(6)

Λ=:Γ: −∃t >Η Ι=: Γ:6Α Β6Γ<>Χ6Α 8∆ΗΙ.

!6Κ>Χ< 8=∆Η:Χ Ι=: 6Β∆ϑΧΙ ∆; Α67∆ϑΓ 6Χ9 86Ε>Ι6Α Ι=6Ι Β>Χ>Β>Ο: 8∆ΗΙΗ, ΠΓΒΗ

∆; Ι=: >ΧΙ:ΓΒ:9>6Ι: ∆ϑΙΕϑΙ Η:8Ι∆Γ Β6Μ>Β>Ο: ΕΓ∆ΠΙΗ 7Ν Β6Μ>Β>Ο>Χ< Ι=: ΗΕΓ:69

7:ΙΛ::Χ Ι=: ΕΓ>8: Ι=:Ν 8=6Γ<: 6Χ9 Ι=:>Γ Β6Γ<>Χ6Α 8∆ΗΙ ϑΧ9:Γ Ι=: 8∆ΧΗΙΓ6>ΧΙ

∆; Ι=: 9:Β6Χ9 ;∆Γ Ι=: ΗΕ:8>Π8 <∆∆9 Κ6Γ>:ΙΝ Ι=:Ν ΕΓ∆9ϑ8: (1):

maxfpftg

0ft −∃nt

0t6ft (7)

2:3: 6ft =

0ft0t

p6t (8)

Λ=:Γ: −∃nt >Η Ι=: Χ∆Β>Χ6Α Β6Γ<>Χ6Α 8∆ΗΙ.

BΝ Η∆ΑΚ>Χ< Ι=: ΕΓ∆7Α:Β (7)-(8), >Ι >Η Ε∆ΗΗ>7Α: Ι∆ Η=∆Λ Ι=6Ι Ι=: ΕΓ>8:

8=6Γ<:9 7Ν :68= ΠΓΒ >Η ?ϑΗΙ 6 Β6Γ≅-ϑΕ ∆Κ:Γ Ι=: Γ:6Α Β6Γ<>Χ6Α 8∆ΗΙ:

0ft0t=

pp 1

−∃t (9)

+:86ΑΑ>Χ< Ι=6Ι ΠΓΒΗ 6Γ: ΗΝΒΒ:ΙΓ>8 6Χ9 Ι=:>Γ ΧϑΒ7:Γ >Η Χ∆ΓΒ6Α>Ο:9 Ι∆

Γ6Ι: 6Χ9 +&D :ΒΕΑ∆ΝΒ:ΧΙ.

16

Page 20: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

∆Χ:, Ι=: ∋ >Χ9:Μ >Χ .ft 86Χ 7: 9Γ∆ΕΕ:9 6Η

1!

0

.ft%∋ = .t

∀Χ Ι=: Η:8∆Χ9 ΗΙ6<: ∆; 7ϑ9<:Ι>Χ<, ΠΓΒΗ "8∆ΧΗΙΓϑ8Ι" ∆ΕΙ>Β6ΑΑΝ Ι=: 8∆Β-

Ε∆Η>Ι: Α67∆ϑΓ >ΧΕϑΙ 7Ν 8=∆∆Η>Χ< Ι=: 6Β∆ϑΧΙ ∆; Λ∆Γ≅>Χ< Ι>Β: ∆; 8∆=∆ΓΙ ∗, .j;t,

Η∆ΑΚ>Χ< Ι=: ;∆ΑΑ∆Λ>Χ< ΕΓ∆7Α:Β

maxfnj;tg

N 1w1

"N1X

j=0

.j;tN

w1w

# ww1

(10)

2:3: 5t.ft

N1X

j=0

5j;tj

.j;tN

= 0 (11)

−=: Ι:ΓΒwj;tj>Η Ι=: Λ6<: ∆; 8∆=∆ΓΙ ∗ Ι=6Ι =6Η 7::Χ 9:ΙΓ:Χ9:9 7:86ϑΗ: Β∆Χ:Ν

>Η <Γ∆Λ>Χ< >Χ ΗΙ:69Ν ΗΙ6Ι: 7ϑΙ Ι=: Λ6<: ∆; :68= 8∆=∆ΓΙ >Η ΗΙ6<<:Γ:9.

,∆ΑΚ>Χ< Ι=: ΕΓ∆7Α:Β 67∆Κ: 6;Ι:Γ !:>?9Γ6 6Χ9 /6Χ 9:Γ )Α∆:< (2002, Ε. 363),

>Ι >Η Ε∆ΗΗ>7Α: Ι∆ ΠΧ9 Ι=: 9:Β6Χ9 ;ϑΧ8Ι>∆Χ ;∆Γ :68= 8∆=∆ΓΙΡΗ Α67∆ϑΓ Η:ΓΚ>8:Η

(19) 6Χ9 Ι=: 6<<Γ:<6Ι: Λ6<: >Χ9:Μ:

5t =

"1

N

N1X

j=0

5j;tj

1w#1

w1

(12)

3.2 T∃! ∃).,!∃)& 6, ∗+)b&!∋

0: 8∆ΧΗ>9:Γ 6 9:Ι:ΓΒ>Χ>ΗΙ>8 Η:ΙΙ>Χ<. ,ϑΕΕ∆Η: Ι=:Γ: 6Γ: N 8∆=∆ΓΙΗ ∆; =∆ϑΗ:-

=∆Α9Η. E68= =∆ϑΗ:=∆Α9 Β6Μ>Β>Ο:Η Ι=: ΕΓ:Η:ΧΙ Κ6Αϑ: ∆; >ΙΗ ϑΙ>Α>ΙΝ (U) Λ>Ι=

17

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Γ:ΗΕ:8Ι Ι∆ >ΙΗ 8∆ΧΗϑΒΕΙ>∆Χ (C), Λ6<: (W ), Γ:6Α Β∆Χ:Ν 76Α6Χ8:Η (M=P ), 86Ε-

>Ι6Α (K) 6Χ9 ΕΓ>Κ6Ι: 7∆Χ9 =∆Α9>Χ<Η (B), Ηϑ7?:8Ι Ι∆ >ΙΗ 7ϑ9<:Ι 8∆ΧΗΙΓ6>ΧΙ, >ΙΗ

Α67∆Γ 9:Β6Χ9 ;ϑΧ8Ι>∆Χ 6Χ9 Ι=: 86Ε>Ι6Α Α6Λ ∆; Β∆Ι>∆Χ. ∋∆Β>Χ6Α Λ6<:Η 6Γ: Η:Ι

;∆Γ N Ε:Γ>∆9Η 7:>Χ< =∆ϑΗ:=∆Α9Η <Γ∆ϑΕ:9 >ΧΙ∆ N Λ6<: Η:ΙΙ>Χ< 8∆=∆ΓΙΗ.

,Ε:8>Π86ΑΑΝ, Ι=: ΕΓ∆7Α:Β ∆; =∆ϑΗ:=∆Α9 ( >Η

maxfCt+j(h);Mt+j(h);Wt+jN (h);Kt+j+1(h);Bt+j(h)g

1X

j=0

(i+1)N1X

j=iN

jU

Ct+j(();

Mt+j(()

Pt+j; .t+j(()

(13)

2:3:

N1X

j=0

t;t+j

"

Ct+j(() +Mt+j(()

Pt+jMt+j1(()

Pt+j+

1

1 + 1bt+j+1

Bt+j(()

Pt+j

Bt+j1(()

Pt+j+ It+j(()

Tt+j(()

Pt+j

=

N1X

j=0

t;t+j

Wt(().t+j(()

Pt+j+

+t+j(() + 1t+jKt+j(()] (14)

.t+j(() =

Wt(()

Wt+j

w

.t+j (15)

Kt+j+1(() = It+j(() + (1 )Kt+j(() (16)

Λ=:Γ: >Η Ι=: 9>Η8∆ϑΧΙ ;68Ι∆Γ, I >Η >ΧΚ:ΗΙΒ:ΧΙ >Χ Ε=ΝΗ>86Α 86Ε>Ι6Α, T 6Γ: ΑϑΒΕ

ΗϑΒ ΙΓ6ΧΗ;:ΓΗ, >Η ΕΓ∆ΠΙ >Χ8∆Β:, 1b >Η Ι=: >ΧΙ:Γ:ΗΙ Γ6Ι: Ε6>9 Ι∆ 7∆Χ9 =∆Α9:ΓΗ,

1 >Η Ι=: ϑΗ:ΓΡΗ 8∆ΗΙ ∆; 86Ε>Ι6Α, t;t+j =

j

l=0

11+~rt+l

, Λ=:Γ: 1 >Η Ι=: Β6Γ≅:Ι >ΧΙ:Γ:ΗΙ

Γ6Ι:, 6Χ9 >Η Ι=: 9:ΕΓ:8>6Ι>∆Χ Γ6Ι:. −=: ΠΓΗΙ 8∆ΧΗΙΓ6>ΧΙ >Η Ι=: =∆ϑΗ:=∆Α9ΡΗ

7ϑ9<:Ι 8∆ΧΗΙΓ6>ΧΙ: Ι=: ΗϑΒ ∆; Ι=: ΕΓ:Η:ΧΙ Κ6Αϑ:Η ∆; Ι=: =∆ϑΗ:=∆Α9ΡΗ :ΜΕ:Χ9>-

ΙϑΓ:Η ∆Χ 8∆ΧΗϑΒΕΙ>∆Χ, Β∆Χ:Ν 76Α6Χ8: 688ϑΒϑΑ6Ι>∆Χ, 7∆Χ9 688ϑΒϑΑ6Ι>∆Χ 6Χ9

18

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>ΧΚ:ΗΙΒ:ΧΙ 6Γ: :Φϑ6Α Ι∆ 9>ΗΕ∆Η67Α: >Χ8∆Β:, Χ6Β:ΑΝ Ι=: ΗϑΒ ∆; Ι=: =∆ϑΗ:=∆Α9ΡΗ

Λ6<:, ΕΓ∆ΠΙ 6Χ9 86Ε>Ι6Α >Χ8∆Β:Η. −=: Η:8∆Χ9 8∆ΧΗΙΓ6>ΧΙ >Η Ι=: Α67∆Γ 9:Β6Χ9

;∆Γ Ι=: 8∆=∆ΓΙΡΗ Η:ΓΚ>8:Η (9:Γ>Κ:9 67∆Κ:), 6Χ9 Ι=: ΠΧ6Α 8∆ΧΗΙΓ6>ΧΙ >Η Ι=: Α6Λ ∆;

Β∆Ι>∆Χ ∆; 86Ε>Ι6Α. −=: ΙΓ6ΧΗΚ:ΓΗ6Α>ΙΝ 8∆Χ9>Ι>∆Χ >Η limt!1 tKt+1 = 0, Λ=:Γ:

t >Η Ι=: Χ∆Χ-9:ΙΓ:Χ9:9 %6<Γ6Χ<>6Χ ΒϑΑΙ>ΕΑ>:Γ.

A;Ι:Γ 9:ΙΓ:Χ9>Χ< Χ∆Β>Χ6Α Κ6Γ>67Α:Η ;∆Γ Χ∆Β>Χ6Α <Γ∆ΛΙ= () 6Χ9 Γ:6Α Κ6Γ>-

67Α:Η ;∆Γ Γ:6Α <Γ∆ΛΙ= ( ), >Ι >Η Ε∆ΗΗ>7Α: Ι∆ Γ:ΗΙ6Ι: ΕΓ∆7Α:Β (13) - (16) >Χ Ι=:

;∆ΑΑ∆Λ>Χ< ;∆ΓΒ:

maxfct+j(h);mt+j(h);wt+jN (h);kt+j+1(h);bt+j(h)g

1X

j=0

(i+1)N1X

j=iN

jU

∃t+j(();

−t+j(()

0t+j; .t+j(()

(17)

2:3:

N1X

j=0

t;t+j

"

∃t+j(() +−t+j(()

0t+j−t+j1(()

0t+j

1

+

1

1 + 1bt+j+1

#t+j(()

0t+j

#t+j1(()

0t+j

1

+ )t+j(()

3t+j(()

0t+j

=

N1X

j=0

t;t+j

5t(()

j0t+j.t+j(() + t+j(() + 1t+j+t+j(()

(18)

.t+j(() =

5t(()

5t+j

1

j

w.t+j (19)

+t+j+1(() = )t+j(() + (1 ) +t+j(() (20)

%∆Λ:Γ-86Η: Α:ΙΙ:ΓΗ 9:Χ∆Ι: 9:-ΙΓ:Χ9:9 Κ6Γ>67Α:Η.

−∆ ΗΙ6ΓΙ Λ>Ι=, ΕΓ:;:Γ:Χ8:Η 6Γ: 6ΗΗϑΒ:9 Ι∆ 7: 6Η ;∆ΑΑ∆ΛΗ:

19

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U =

1X

j=0

(i+1)N1X

j=iN

jlog ∃t+j(() &.t+j(() + V

−t+j(()

0t+j

(21)

Λ=:Γ: & >Η 6 Ε6Γ6Β:Ι:Γ 6Χ9 V [] Ι=: Ε6ΓΙ>6Α ϑΙ>Α>ΙΝ ;ϑΧ8Ι>∆Χ ∆; Γ:6Α Β∆Χ:Ν

76Α6Χ8:Η. −=: ΠΓΗΙ ∆Γ9:Γ 8∆Χ9>Ι>∆ΧΗ ;∆Γ 8∆ΧΗϑΒΕΙ>∆Χ, 86Ε>Ι6Α 6Χ9 :68= 8∆-

=∆ΓΙΡΗ Λ6<: 6Γ: ∆; Ε6ΓΙ>8ϑΑ6Γ >ΒΕ∆ΓΙ6Χ8: ;∆Γ Ι=: Η∆ΑϑΙ>∆Χ ∆; Ι=: Β∆9:Α. >Κ:Χ

Ι=6Ι Ι=:Γ: :Μ>ΗΙ 8∆ΒΕΑ:Ι: 6ΗΗ:Ι Β6Γ≅:ΙΗ, Ι=6Ι Ι=: ∆ΧΑΝ =:Ι:Γ∆<:Χ:>ΙΝ 6Β∆Χ<

=∆ϑΗ:=∆Α9Η >Η 9ϑ: Ι∆ Λ6<: ΗΙ6<<:Γ>Χ< 6Χ9 Ι=6Ι =∆ϑΗ:=∆Α9Η ;68: >9:ΧΙ>86Α N

Ηϑ7-ΕΓ∆7Α:ΒΗ, Ι=:Ν 86Χ 7: ΛΓ>ΙΙ:Χ 6Η ;∆ΑΑ∆ΛΗ:

j

∃t+j= t+j t;t+j (22)

t+j = t;t+1t+j+1(1 + 1t+j) (23)

N1X

j=0

.t+jj =

w 1

w&

N1X

j=0

t;t+j5j;t

0t+jj.t+j∃t+j

(24)

Λ=:Γ: t+j >Η Ι=: 9:-ΙΓ:Χ9:9 %6<Γ6Χ<: ΒϑΑΙ>ΕΑ>:Γ.

3.3 T∃! G)/!+(∋!(−

F∆Γ Η>ΒΕΑ>8>ΙΝ, Ι=: <∆Κ:ΓΧΒ:ΧΙ >Η 6ΗΗϑΒ:9 Ι∆ 9>ΗΙΓ>7ϑΙ: >ΙΗ Η:><Χ>∆Γ6<: >Χ Ι=:

;∆ΓΒ ∆; ΑϑΒΕ-ΗϑΒ ΙΓ6ΧΗ;:ΓΗ Ι∆ =∆ϑΗ:=∆Α9Η:

−t+j −t+j11

= 3t+j (25)

20

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4 T∃! M) !& S)&.−%)(

−=: Β68Γ∆:8∆Χ∆Β>8 Β∆9:Α >Η 8∆ΧΗΙ>ΙϑΙ:9 7Ν Ι=: :Φϑ6Ι>∆ΧΗ: (1), (2), (4), (5),

(6), (9), (12), (19), (22), (23), (24) 6Χ9 (25). −=: ΕΓ∆8:9ϑΓ: Ι∆ Η∆ΑΚ: Ι=:

Β∆9:Α >Η ∆ϑΙΑ>Χ:9 >Χ Β∆Γ: 9:Ι6>Α >Χ AΕΕ:Χ9>Μ A.

∋∆Ι: Ι=6Ι Γ6=6Β 6Χ9 ,Χ∆Λ:Γ (2004) Η=∆Λ:9 Ι=6Ι >Χ ΗΙ:69Ν ΗΙ6Ι: t;t+j =

j8. FϑΓΙ=:ΓΒ∆Γ: >Χ :Φϑ>Α>7Γ>ϑΒ Ι=:Γ: >Η Χ∆ 6Γ7>ΙΓ6<: 7:ΙΛ::Χ 6ΑΙ:ΓΧ6Ι>Κ:

6ΗΗ:ΙΗ 6Χ9 Η∆ 1 = 1b = 1 : ∀Ι >Η Ε∆ΗΗ>7Α: Ι∆ ∆7Ι6>Χ Ι=: Η∆ΑϑΙ>∆Χ ;∆Γ Ι=:

ΗΙ:69Ν ΗΙ6Ι: <Γ∆ΛΙ= Γ6Ι: ∆; Ι=: :8∆Χ∆ΒΝ, 7Ν ϑΗ>Χ< (4), (9), (6), (22) 6Χ9

(23):

=

1 +

p 1

p(1 ) (.)

(26)

−=: Χ∆Χ-9:ΙΓ:Χ9:9 Κ:ΓΗ>∆Χ ∆; (23) >ΒΕΑ>:Η Ι=6Ι Ι=: Α6Λ ∆; Β∆Ι>∆Χ ∆; Ι=:

Χ∆Χ-9:ΙΓ:Χ9:9 %6<Γ6Χ<>6Χ ΒϑΑΙ>ΕΑ>:Γ >Η

t+j =

1

t;t+1(1 + 1t+j)

jt

Λ=>8=, Ι∆<:Ι=:Γ Λ>Ι= Ι=: ΙΓ6ΧΗΚ:ΓΗ6Α>ΙΝ 8∆Χ9>Ι>∆Χ 6Χ9 Ι=: :Φϑ6Α>ΙΝ t;t+j = j,

>ΒΕ∆Η:Η Ι=: Γ:ΗΙΓ>8Ι>∆Χ ∆Χ Ι=: Ε:Γ8:ΧΙ6<: <Γ∆ΛΙ= Γ6Ι: Ι∆ 7: Χ∆Χ-Χ:<6Ι>Κ:.

AΙ Ι=>Η ΗΙ6<: >Ι >Η Χ:8:ΗΗ6ΓΝ Ι∆ ΠΧ9 Ι=: ΗΙ:69Ν ΗΙ6Ι: Α:Κ:Α ∆; Ι=: 8∆ΒΕ∆Η>Ι:

Α67∆ϑΓ >ΧΕϑΙ >Χ ∆Γ9:Γ Ι∆ Η∆ΑΚ: Ι=: Β∆9:Α. ∀Χ Ι=: 6ΕΕ:Χ9>Μ Λ: Η=∆Λ Ι=6Ι,

6ΗΗϑΒ>Χ< = 23, Ι=: 8∆ΒΕ∆Η>Ι: Α67∆ϑΓ >ΧΕϑΙ >Η <>Κ:Χ 7Ν Ι=: Γ∆∆ΙΗ ∆; Ι=:

8−=:>Γ 9:Β∆ΧΗΙΓ6Ι>∆Χ 6ΕΕΑ>:Η 6ΑΗ∆ =:Γ: 6Η ct+j >Η 9:ΙΓ:Χ9:9 8∆ΧΗϑΒΕΙ>∆Χ.

21

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;∆ΑΑ∆Λ>Χ< Ε∆ΑΝΧ∆Β>6Α

A3.3 3A2B.2 +3AB2 C3

.B3 = 0 (27)

Λ=:Γ: A, B 6Χ9 C 6Γ: 9:ΠΧ:9 6Η ;∆ΑΑ∆ΛΗ:

A : =

1

p 1

p(1 )

(28)

B : =

N1 (w1)

1 N(w1)p 1

p1 NN(w1)

1 (w1)

1 w

1 (w)Nw 1

w&

1

N

1 N(w1)

1 w1

ww1

(29)

C : = (1 ) ( 1) (30)

−=: Ε∆ΗΗ>7>Α>ΙΝ Ι∆ =6Κ: ΒϑΑΙ>ΕΑ: ΗΙ:69Ν ΗΙ6Ι:Η Λ>Ι=>Χ 6 Α:6ΓΧ>Χ<-7Ν-9∆>Χ<

Β∆9:Α >Η Λ:ΑΑ ≅Χ∆ΛΧ >Χ Ι=: Α>Ι:Γ6ΙϑΓ: (B:Χ=67>7 6Χ9 F6ΓΒ:Γ, 1994). 0:

86Α>7Γ6Ι: Ι=: Β∆9:Α ϑΗ>Χ< ΗΙ6Χ96Γ9 Ε6Γ6Β:Ι:Γ Κ6Αϑ:Η ( = 0:961N ; N =

52; = 10:921N ; p=10; =0:67, w=2) Η>Β>Α6ΓΑΝ Ι∆ AΗ86Γ> (2003), Γ6=6Β

6Χ9 ,Χ∆Λ:Γ (2004) 6Χ9 !ϑ6Χ< 6Χ9 %>ϑ (2002). & =6Η 7::Χ 69?ϑΗΙ:9 ;∆Γ Ι=:

Β∆9:Α Ι∆ ΕΓ∆9ϑ8: Γ:6Α>ΗΙ>8 <Γ∆ΛΙ= Γ6Ι:Η. .Χ9:Γ Ι=>Η 86Α>7Γ6Ι>∆Χ ΙΛ∆ ∆; Ι=:

Ι=Γ:: Γ∆∆ΙΗ ∆; (27) ΙϑΓΧ ∆ϑΙ Ι∆ 7: 8∆ΒΕΑ:Μ ΧϑΒ7:ΓΗ Λ=>8= 86Χ 7: ΓϑΑ:9 ∆ϑΙ

7:>Χ< Λ>Ι=∆ϑΙ :8∆Χ∆Β>8 Β:6Χ>Χ<. −=>Η Α:6Κ:Η ϑΗ Λ>Ι= 6 ϑΧ>Φϑ: Η∆ΑϑΙ>∆Χ ;∆Γ

Ι=: Β∆9:Α.

F><ϑΓ:Η 1 6Χ9 2 Η=∆Λ =∆Λ Ι=: Α∆Χ<-ΓϑΧ <Γ∆ΛΙ= Γ6Ι: ∆; Ι=: :8∆Χ∆ΒΝ

8=6Χ<:Η 6Η 6 ;ϑΧ8Ι>∆Χ ∆; p 6Χ9 w; ∆Χ8: Η:ΙΙ>Χ< = 1:02. AΧ >Χ8Γ:6Η:

>Χ p Γ:9ϑ8:Η Β∆Χ∆Ε∆Α>ΗΙ>8 Γ:ΧΙΗ >Χ Ι=: >ΧΙ:ΓΒ:9>6Ι: ΕΓ∆9ϑ8Ι Β6Γ≅:Ι, >ΒΕΑΝ-

22

Page 26: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

>Χ< 6 Ε∆Η>Ι>Κ: >Χ8∆Β: :Ω:8Ι 6Χ9 :Χ=6Χ8>Χ< :8∆Χ∆Β>8 <Γ∆ΛΙ=. −=>Η :Ω:8Ι >Η

86ΕΙϑΓ:9 7Ν Ι:ΓΒΗ 9:Ε:Χ9>Χ< ∆Χ p >Χ (26), (28) 6Χ9 (29): −=: Η6Β: =6ΕΕ:ΧΗ

;∆Γ w, :Κ:Χ Ι=∆ϑ<= >ΙΗ >ΒΕ68Ι ∆Χ Α∆Χ< ΓϑΧ <Γ∆ΛΙ= >Η Ι=: Γ:ΗϑΑΙ ∆; 6 ΧϑΒ7:Γ

∆; 9>Ω:Γ:ΧΙ Β:8=6Χ>ΗΒΗ. F>ΓΗΙ ∆; 6ΑΑ, 6Χ >Χ8Γ:6Η: >Χ w Γ:9ϑ8:Η Β∆Χ∆Ε∆Α>ΗΙ>8

Γ:ΧΙΗ ∆Χ Ι=: Α67∆ϑΓ Β6Γ≅:Ι, :ΧΙ6>Α>Χ< 6 <Γ:6Ι:Γ Α67∆ϑΓ ΗϑΕΕΑΝ. !∆Λ:Κ:Γ, 9ϑ:

Ι∆ Λ6<: ΗΙ6<<:Γ>Χ<, Ι=>Η >Η Χ∆Ι Ι=: 8∆ΒΕΑ:Ι: ΗΙ∆ΓΝ. C∆ΧΗ>9:Γ Ι=: Γ6Ι>∆ 7:ΙΛ::Χ

Ι=: Α67∆ϑΓ 9:Β6Χ9:9 Ι∆ 8∆=∆ΓΙ 0 6Χ9 8∆=∆ΓΙ ∗:

.0

.j= wj (31)

AΧ >Χ8Γ:6Η: >Χ w :ΧΙ6>ΑΗ Β∆Γ: Α67∆ϑΓ 8Ν8Α>Χ<, Λ=>8= ΕΓ∆9ϑ8:Η Β∆Γ: >Χ:ς 8>:Χ-

8>:Η. −=>Η Χ:<6Ι>Κ: :Ω:8Ι, =∆Λ:Κ:Γ, 9∆:Η Χ∆Ι ∆ΩΗ:Ι Ι=: 7:Χ:Π8>6Α Γ:9ϑ8Ι>∆Χ ∆;

Γ:ΧΙΗ Α:69>Χ< Ι∆ 6 Χ:Ι >Χ8Γ:6Η: >Χ Ι=: 8∆ΒΕ∆Η>Ι: Α67∆ϑΓ >ΧΕϑΙ 6Χ9 Ι=:Γ:;∆Γ:

>Χ :8∆Χ∆Β>8 <Γ∆ΛΙ=.

5 T∃! E7!c−, )∀ M)(!−a+2 P)&%c%!,

5.1 T∃! R!a& E7!c−, )∀ M)(!−a+2 ∗)&%c2

F><ϑΓ: 3 Η=∆ΛΗ Ι=6Ι Ι=: Γ:Α6Ι>∆ΧΗ=>Ε 7:ΙΛ::Χ Β∆Χ:Ν 6Χ9 Γ:6Α <Γ∆ΛΙ= >Η 8Α:6ΓΑΝ

Χ∆Χ-Α>Χ:6Γ. −=:Γ: >Η 6 Ι=Γ:Η=∆Α9: >Χ8Γ:6Η>Χ< Ι=: Β∆Χ:Ν <Γ∆ΛΙ= Γ6Ι: =6Η 6

Ι>ΧΝ Ε∆Η>Ι>Κ: >ΒΕ68Ι ∆Χ Γ:6Α <Γ∆ΛΙ= ϑΧΙ>Α 67∆ϑΙ 2% 6Χ9 6 Χ:<6Ι>Κ: ∆Χ: 67∆Κ:

Ι=>Η Κ6Αϑ:. AΗ Η=∆Λ:9 7Ν Γ6=6Β 6Χ9 ,Χ∆Λ:Γ (2004) 6Χ6ΑΝΗ>Χ< ΗΙΓϑ8ΙϑΓ6Α

:Φϑ6Ι∆ΧΗ, Ι=: 9>Η8∆ϑΧΙ>Χ< 6Χ9 Ι=: :ΒΕΑ∆ΝΒ:ΧΙ 8Ν8Α>Χ< :Ω:8ΙΗ ϑΧ9:ΓΑ>: Ι=:Η:

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Χ∆Χ-Α>Χ:6Γ>Ι>:Η. AΙ Α∆Λ >ΧΘ6Ι>∆Χ Γ6Ι:Η, Ι=: Ι>Β: 9>Η8∆ϑΧΙ>Χ< :Ω:8Ι ΕΓ:Κ6>ΑΗ

Α:69>Χ< Ι∆ 6 <Γ:6Ι:Γ Α67∆ϑΓ ΗϑΕΕΑΝ 6Χ9 Ι=:Γ:;∆Γ: Ι∆ ;6ΗΙ:Γ 86Ε>Ι6Α 688ϑΒϑΑ6-

Ι>∆Χ 6Χ9 <Γ∆ΛΙ=. (Χ Ι=: 8∆ΧΙΓ6ΓΝ, 6Ι =><= >ΧΘ6Ι>∆Χ Γ6Ι:Η Ι=: :ΒΕΑ∆ΝΒ:ΧΙ

8Ν8Α>Χ< :Ω:8Ι >Η ΗΙΓ∆Χ<:Γ Α:69>Χ< Ι∆ Α:ΗΗ Α67∆ϑΓ 9:Β6Χ9 6Χ9 Ι=:Γ:;∆Γ: Ι∆ ΗΑ∆Λ:Γ

<Γ∆ΛΙ=. −=: Α67∆ϑΓ 8Ν8Α>Χ< :Ω:8Ι >Η 9ϑ: Ι∆ Ι=: ;68Ι Ι=6Ι ΠΓΒΗ Ηϑ7ΗΙ>ΙϑΙ: 7:-

ΙΛ::Χ 9>Ω:Γ:ΧΙ ≅>Χ9Η ∆; Α67∆ϑΓ 7:86ϑΗ: 6<:ΧΙΗ 7:Α∆Χ<>Χ< Ι∆ 9>Ω:Γ:ΧΙ 8∆=∆ΓΙΗ

=6Κ: 9>Ω:Γ:ΧΙ Λ6<:Η, 7:>Χ< Η∆Β: ∆; Ι=:Β Α∆8≅:9 >Χ Ε6ΗΙ 8∆ΧΙΓ68ΙΗ.

−=: 6Χ6ΑΝΗ>Η 67∆Κ:, Γ:<6Γ9>Χ< =∆Λ Ι=: :Α6ΗΙ>8>ΙΝ ∆; Ηϑ7ΗΙ>ΙϑΙ>∆Χ 7:ΙΛ::Χ

9>Ω:Γ:ΧΙ <∆∆9Η 6Χ9 9>Ω:Γ:ΧΙ Α67∆ϑΓ ≅>Χ9Η 6Ω:8Ι Ι=: Γ:6Α <Γ∆ΛΙ= Γ6Ι:, 6Α-

Γ:69Ν <>Κ:Η Η∆Β: >ΧΗ><=ΙΗ ∆Χ =∆Λ Ι=:Ν 6Ω:8Ι Ι=: Α∆Χ<-ΓϑΧ Γ:Α6Ι>∆ΧΗ=>Ε 7:-

ΙΛ::Χ Β∆Χ:Ν <Γ∆ΛΙ= 6Χ9 Γ:6Α <Γ∆ΛΙ=. F><ϑΓ:Η 4 6Χ9 5 Η=∆Λ Ι=6Ι Ι=: Β∆Χ:Ν

<Γ∆ΛΙ=-Γ:6Α <Γ∆ΛΙ= Α∆8ϑΗ Β∆Κ:Η ϑΕΛ6Γ9 6Η :>Ι=:Γ p ∆Γ w >Χ8Γ:6Η:Η. .Χ9:Γ

Ι=: ΕΓ:Η:ΧΙ 86Α>7Γ6Ι>∆Χ, :8∆Χ∆Β>8 <Γ∆ΛΙ= 6ΕΕ:6ΓΗ Ι∆ 7: Βϑ8= Β∆Γ: Η:ΧΗ>Ι>Κ:

Ι∆ Ι=: ΗΙΓϑ8ΙϑΓ6Α Ε6Γ6Β:Ι:ΓΗ ∆; Ι=: Β∆9:Α Ι=6Χ Ι∆ Β∆Χ:Ν <Γ∆ΛΙ=, 6Η Ι=:

9>Η8∆ϑΧΙ>Χ< 6Χ9 Ι=: :ΒΕΑ∆ΝΒ:ΧΙ 8Ν8Α>Χ< :Ω:8ΙΗ Ι:Χ9 Ι∆ ∆ΩΗ:Ι ∆Χ: 6Χ∆Ι=:Γ.

5.2 O∗−%∋a& M)(!−a+2 P)&%c2

−∆ >Χ9>Κ>9ϑ6Ι: Ι=: ∆ΕΙ>Β6Α Γ6Ι: ∆; Β∆Χ:Ν <Γ∆ΛΙ= Λ: ϑΗ: 6 ΗΕ:8>Π86Ι>∆Χ ∆;

Ι=: Λ:Α;6Γ: ;ϑΧ8Ι>∆Χ Η>Β>Α6Γ Ι∆ 0∆∆9;∆Γ9 (1998), B:Χ><Χ∆ (2004) 6Χ9 A∆≅>

(2001) Λ=:Γ: Ι=: Λ:><=Ι ∆; Β∆Χ:Ν =∆Α9>Χ<Η >Η 6ΗΗϑΒ:9 Ι∆ 7: Η∆ ΗΒ6ΑΑ Ι∆ 7:

Χ:<Α><>7Α::

24

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W =

1X

j=0

j [logCt+j &.jt+j]

Ct+j >Η <Γ∆Λ>Χ< 6Ι Ι=: Γ6Ι: , Ι=:Γ:;∆Γ: Ct+j = ∃t j.

AΗ Η=∆Λ:9 >Χ AΕΕ:Χ9>Μ B, Β6Μ>Β>Ο>Χ< W >Η :Φϑ>Κ6Α:ΧΙ Ι∆ Β6Μ>Β>Ο>Χ<

Ι=: 8Γ>Ι:Γ>∆Χ

W 0 =1

1 N

1 N

(1 )2NN

1

log (32)

&.01

1 N1 NwN

1 w+

1

1 Nlog (. + 1 )

F><ϑΓ: 6 Η=∆ΛΗ Ι=6Ι Β∆Χ:Ν <Γ∆ΛΙ= 9∆:Η Χ∆Ι 6Ω:8Ι Λ:Α;6Γ: Ι∆ 6 <Γ:6Ι :Μ-

Ι:ΧΙ. !∆Λ:Κ:Γ, ϑΧ9:Γ Ι=: ΕΓ:Η:ΧΙ 86Α>7Γ6Ι>∆Χ Ι=: ∆ΕΙ>Β6Α >ΧΘ6Ι>∆Χ Γ6Ι: Λ∆ϑΑ9

7: 67∆Κ: 20%, 6 Γ:ΗϑΑΙ Ι=6Ι 9∆:Η Χ∆Ι =∆Α9 ;∆Γ Β∆Γ: Γ:6Α>ΗΙ>8 86Α>7Γ6Ι>∆ΧΗ 7:-

Α∆Λ. −=>Η =6ΕΕ:ΧΗ 7:86ϑΗ: Β∆Χ:Ν <Γ∆ΛΙ= 6Ω:8ΙΗ W 0 Ι=Γ∆ϑ<= ;∆ϑΓ 8=6ΧΧ:ΑΗ:

Ι=: Γ:6Α <Γ∆ΛΙ= Γ6Ι: ∆; Ι=: :8∆Χ∆ΒΝ, Ι=: Λ∆Γ≅>Χ< Ι>Β: ∆; 8∆=∆ΓΙ Ο:Γ∆, Ι=:

Ι:ΓΒ 1NwN

1w6Χ9 Ι=: >Χ>Ι>6Α Α:Κ:Α ∆; 8∆ΧΗϑΒΕΙ>∆Χ. −=: Η:8∆Χ9 8=6ΧΧ:Α 86Χ

7: Ι6≅:Χ Ι∆ Γ:ΕΓ:Η:ΧΙ Ι=: Α67∆ϑΓ 8Ν8Α>Χ< :Ω:8Ι9, Λ=>Α:, ;∆ΑΑ∆Λ>Χ< Γ6=6Β 6Χ9

,Χ∆Λ:Γ (2004, Ε. 18), Ι=: Ι=>Γ9 8=6ΧΧ:Α 86Χ 7: Ι6≅:Χ Ι∆ Γ:ΕΓ:Η:ΧΙ Ι=: Ι>Β:

9>Η8∆ϑΧΙ>Χ< :Ω:8Ι. −∆ ϑΧ9:ΓΗΙ6Χ9 Λ=>8= 8=6ΧΧ:Α 9∆Β>Χ6Ι:Η Λ: 9:8∆ΒΕ∆Η:

W 0 >Χ Ι=Γ:: Ι:ΓΒΗ

9−=: Β∆Γ: Ι=: Λ∆Γ≅>Χ< Ι>Β: ∆; 8∆=∆ΓΙ Ο:Γ∆ Η=Γ>Χ≅Η, Ι=: Β∆Γ: Α67∆ϑΓ 9:Β6Χ9 Λ>ΑΑ Η=>;ΙΙ∆ ∆Ι=:Γ 8∆=∆ΓΙΗ.

25

Page 29: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

W 01 =

1

1 N

1 N

(1 )2NN

1

log

W 02 = &.0

1

1 N1 NwN

1 w

W 03 =

1

1 Nlog (. + 1 )

6Χ9 Λ: 8∆ΒΕϑΙ: Η>Μ Ε:Γ8:ΧΙ6<: Κ6Γ>6Ι>∆ΧΗ%W 01a =

W 0

1maxW0

1ff

W 0

ff

100;%W 02a =

W 0

2maxW0

2ff

W 0

ff

100; %W 03a =

W 0

3maxW0

3ff

W 0

ff

100; %W 01b =

W1endW0

1max

W 0max

100;

%W 02b =

W 0

2endW0

2max

W 0max

100; %W 03b =

W 0

3endW0

3max

W 0max

100; Λ=:Γ: Ι=: Ηϑ7-

Η8Γ>ΕΙ "max" >Χ9>86Ι:Η Κ6Αϑ:Η Ι6≅:Χ 6Ι Ι=: Β6Μ>ΒϑΒ ∆; W 0, ∋∋ Ι=: Κ6Αϑ:Η

Ι6≅:Χ Λ>Ι= Θ:Μ>7Α: Λ6<:Η 6Χ9 &.% Ι=: Α6ΗΙ 8∆ΒΕϑΙ:9 Κ6Αϑ:Η. A;Ι:Γ ;∆Γ >Χ-

ΗΙ6Χ8: 0∆ΑΩ (2003), Λ: ;ϑΓΙ=:Γ 9:8∆ΒΕ∆Η: %W 02a 6Χ9 %W

02b Ι∆ >Η∆Α6Ι:

Ι=: :Ω:8Ι ∆; .0 ;Γ∆Β Ι=6Ι ∆;1NwN

1wϑΗ>Χ< Ι=: ;68Ι Ι=6Ι, ;∆Γ 6 <>Κ:Χ Κ6Γ>67Α:

7 = # ∋; ∆Χ: =6Η Ι=6Ι 72 71 = # (∋2 ∋1) + ∋

(#2 #1) Λ=:Γ: # = b2+b1

2

6Χ9 ∋ = f2+f12; Λ=:Γ: 1 6Χ9 2 >Χ9>86Ι: 9>Ω:Γ:ΧΙ Κ6Αϑ:Η ∆; Ι=: Κ6Γ>67Α:Η.

.Χ9:Γ Ι=: ΕΓ:Η:ΧΙ Ε6Γ6Β:ΙΓ>Ο6Ι>∆Χ, %W 01a = 0:0328%; %W 0

2a =

0:0715% 6Χ9 %W 03a = 0:0072%, Λ=>Α: %W

01b = 0:0263%, %W

02b =

0:0292%, %W 03b = 0:0057%: ,∆ Ι=: Η:8∆Χ9 Ι:ΓΒ >Χ (32) 6ΕΕ:6ΓΗ Ι∆

7: Ι=: Β∆ΗΙ Γ:ΗΕ∆ΧΗ>Κ: Ι∆ 8=6Χ<:Η >Χ Β∆Χ:Ν <Γ∆ΛΙ=. !∆Λ:Κ:Γ, %W 01a

6Χ9 %W 03a 86ΧΧ∆Ι 8∆ΒΕ:ΧΗ6Ι: ;∆Γ %W

02a, Ι=∆ϑ<= %W

01b 6Χ9 %W

03b

86Χ 8∆ΒΕ:ΧΗ6Ι: ;∆Γ %W 02b, :Κ:ΧΙϑ6ΑΑΝ Α:69>Χ< W

0 Ι∆ ;6ΑΑ. −=: 8=6Χ<:Η

>Χ %W 02a 6Χ9 %W

02b 6ΙΙΓ>7ϑΙ67Α: Ι∆ .0 6Χ9

1NwN

1w6Γ: 67∆ϑΙ ∆; Ι=:

26

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Η6Β: Β6<Χ>Ιϑ9:Η, Ι=∆ϑ<= Ι=: Α6ΙΙ:Γ Ι:Χ9Η Ι∆ 7: Η∆Β:Λ=6Ι <Γ:6Ι:Γ. ∀Χ9::9,

%W 02a = 0:0715% 86Χ 7: 9:8∆ΒΕ∆Η:9 >ΧΙ∆ ΙΛ∆ ;ϑΓΙ=:Γ Ε:Γ8:ΧΙ6<: Κ6Γ>6-

Ι>∆ΧΗ: ∆Χ:, 6ΙΙΓ>7ϑΙ67Α: Ι∆ .0; :Φϑ6ΑΗ 16:9854% 6Χ9 Ι=: ∆Ι=:Γ, 6ΙΙΓ>7ϑΙ67Α:

Ι∆ 1NwN

1w, :Φϑ6ΑΗ 16:9140%. F∆Γ %W 0

2b Ι=:Η: ΙΛ∆ 8=6Χ<:Η 6Γ: :Φϑ6Α Ι∆

5:3833% 6Χ9 5:3542% Γ:ΗΕ:8Ι>Κ:ΑΝ.

−∆ ΗϑΒ ϑΕ, ϑΧ9:Γ Ι=: ΕΓ:Η:ΧΙ Ε6Γ6Β:ΙΓ>Ο6Ι>∆Χ, Β∆Χ:Ν <Γ∆ΛΙ= =6Η 6 Ι>ΧΝ

Χ∆ΧΑ>Χ:6Γ :Ω:8Ι ∆Χ Γ:6Α <Γ∆ΛΙ= 6Χ9 Λ:Α;6Γ: 7:86ϑΗ: Ι=: Ι>Β: 9>Η8∆ϑΧΙ>Χ< 6Χ9

Α67∆ϑΓ 8Ν8Α>Χ< :Ω:8ΙΗ Ι:Χ9 Ι∆ 86Χ8:Α :68= ∆Ι=:Γ ∆ϑΙ. !∆Λ:Κ:Γ, Λ:Α;6Γ: Ι:Χ9Η

Ι∆ >Χ8Γ:6Η: ϑΕ Ι∆ Κ:ΓΝ =><= >ΧΘ6Ι>∆Χ Γ6Ι:Η 7:86ϑΗ: Ι=: :Ω:8ΙΗ ∆; Γ:6Α <Γ∆ΛΙ=,

Ι>Β: 9>Η8∆ϑΧΙ>Χ< 6Χ9 Ι=: >Χ>Ι>6Α Α:Κ:Α ∆; 8∆ΧΗϑΒΕΙ>∆Χ 86ΧΧ∆Ι 8∆ΒΕΑ:Ι:ΑΝ ∆ΩΗ:Ι

Ι=: Α67∆ϑΓ 8Ν8Α>Χ< :Ω:8Ι, Λ=>8= Α:69Η Ι∆ 6 Λ:Α;6Γ: :Χ=6Χ8>Χ< 9:68Γ:6Η: >Χ

Λ∆Γ≅>Χ< Ι>Β:. 0: Λ>ΑΑ Η:: 7:Α∆Λ Ι=6Ι Ι=>Η Γ:ΗϑΑΙ 9∆:Η Χ∆Ι =∆Α9 ;∆Γ Β∆Γ:

<:Χ:Γ6Α ΕΓ:;:Γ:Χ8:Η.

6 E1−!(,%)(,

6.1 G!(!+a&%3! P+!∀!+!(c!,

∀Χ Ι=>Η ,:8Ι>∆Χ Λ: Γ:Α6Μ Ι=: =ΝΕ∆Ι=:Η>Η ∆; Α>Χ:6Γ ΕΓ:;:Γ:Χ8:Η >Χ Α67∆ϑΓ Ι>Β:.

AΧ9 Λ: ΗϑΕΕ∆Η: Ι=6Ι Ι=: >ΧΙ:ΓΙ:ΒΕ∆Γ6Α :Α6ΗΙ>8>ΙΝ ∆; Ηϑ7ΗΙ>ΙϑΙ>∆Χ ∆; Λ∆Γ≅>Χ<

Ι>Β: >Η :Φϑ6Α Ι∆ :

U =

1X

j=0

j

(

log ∃t+j(() &.1+j;t+j(()

1 + + V

−t+j(()

0t+j

)

(33)

27

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C∆ΧΗ:Φϑ:ΧΙΑΝ Ι=: ΠΓΗΙ ∆Γ9:Γ 8∆Χ9>Ι>∆Χ ∆; Ι=: 8∆ΧΗϑΒ:ΓΡΗ Β6Μ>Β>Ο6Ι>∆Χ

ΕΓ∆7Α:Β 6Χ9 B Λ>ΑΑ Γ:ΗΕ:8Ι>Κ:ΑΝ 7: 6Η ;∆ΑΑ∆ΛΗ

N1X

j=0

.(1+)j;t+j

j =w 1

w&

N1X

j=0

t;t+j5j;t

0t+jj.j;t+j∃t+j

(34)

B :=

N1 (w1)

1 N(w1)p 1

p1 NN(w1)

1 (w1)

1 (1+)w

1 [(1+)w ]N

w 1

w&

1

N

1 N(w1)

1 w1

ww1

−=: Λ:Α;6Γ: 8Γ>Ι:Γ>∆Χ >Η Χ∆Λ

W 0 =1

1 N

1 N

(1 )2NN

1

log (35)

&.01

1 N1 Nw(1+)N

1 w(1+)+

1

1 Nlog (. + 1 )(36)

∀Χ ∆ϑΓ 7:Χ8=Β6Γ≅ Β∆9:Α, Λ: Η:Ι = 5; 8∆ΧΗ>ΗΙ:ΧΙΑΝ, ;∆Γ >ΧΗΙ6Χ8:, Λ>Ι=

!ϑ6Χ< 6Χ9 %>ϑ (2002), 7ϑΙ Λ: 6ΑΗ∆ 8=:8≅ Ι=: Η:ΧΗ>Ι>Κ>ΙΝ ∆; ∆ϑΓ Γ:ΗϑΑΙΗ ΠΜ>Χ<

Ι∆ 2:5. F><ϑΓ:Η 76 6Χ9 77 Η=∆Λ Ι=: Α∆Χ<-ΓϑΧ Γ:Α6Ι>∆ΧΗ=>Ε 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ

6Χ9 <Γ∆ΛΙ= 6Χ9 Ι=: Λ:Α;6Γ: Α:Κ:Α 6ΗΗ∆8>6Ι:9 Λ>Ι= 9>Ω:Γ:ΧΙ >ΧΘ6Ι>∆Χ Γ6Ι:Η ;∆Γ

Ι=>Η Χ:Λ ΗΕ:8>Π86Ι>∆Χ ∆; Ι=: Β∆9:Α. ,>Β>Α6ΓΑΝ Ι∆ Γ6=6Β 6Χ9 ,Χ∆Λ:Γ (2004)

;∆Γ Ι=: Γ:Α6Ι>∆ΧΗ=>Ε 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9 Ι=: Α:Κ:Α ∆; ∆ϑΙΕϑΙ, Ι=>Η >Η 9ϑ: Ι∆

Ι=: Α67∆ϑΓ ΗϑΕΕΑΝ ΗΒ∆∆Ι=>Χ< :Ω:8Ι. 0>Ι= > 0 =∆ϑΗ:=∆Α9Η =6Κ: 6 <Γ:6Ι:Γ

ΕΓ:;:Γ:Χ8: ;∆Γ ΗΒ∆∆Ι=>Χ< Ι=:>Γ Α67∆ϑΓ ΗϑΕΕΑΝ ∆Κ:Γ Ι=: 8∆ΧΙΓ68Ι Ε:Γ>∆9. AΗ

6 8∆ΧΗ:Φϑ:Χ8:, 7:>Χ< 69Κ:ΓΗ:ΑΝ 6Ω:8Ι:9 7Ν >Χ8Γ:6Η:Η >Χ Β∆Χ:Ν <Γ∆ΛΙ= Ι=6Ι

Α:69 Ι∆ Β∆Γ: :ΒΕΑ∆ΝΒ:ΧΙ 8Ν8Α>Χ< 9ϑ: Ι∆ Λ6<: ΗΙ6<<:Γ>Χ<, Ι=:Ν Γ6>Η: Ι=:>Γ

28

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Λ6<:, Γ:9ϑ8>Χ< Α67∆ϑΓ 9:Β6Χ9, Ι=: 6<<Γ:<6Ι: Φϑ6ΧΙ>ΙΝ ∆; Α67∆ϑΓ >ΧΕϑΙ 6Χ9,

Η∆, ∆ϑΙΕϑΙ <Γ∆ΛΙ=. −=: <Γ:6Ι:Γ >Η , Ι=: ΗΙ::Ε:Γ >Η Ι=: Γ:6Α <Γ∆ΛΙ=-Β∆Χ:Ν

<Γ∆ΛΙ= Α∆8ϑΗ.

−=: Β∆Χ:Ν <Γ∆ΛΙ=-Λ:Α;6Γ: Α∆8ϑΗ >Η Χ∆Λ 9∆ΛΧΛ6Γ9 ΗΑ∆Ε>Χ< 6Η Ι=: Γ:6Α

<Γ∆ΛΙ=, Ι=: Α67∆ϑΓ 8Ν8Α>Χ< 6Χ9 Ι=: >Χ>Ι>6Α 8∆ΧΗϑΒΕΙ>∆Χ :Ω:8Ι Β∆Γ: Ι=6Χ ∆ΩΗ:Ι

Ι=: Ι>Β: 9>Η8∆ϑΧΙ>Χ< :Ω:8Ι.

6.2 E( )#!(%3%(# −∃! &!(#−∃ )∀ −∃! &ab).+ c)(−+ac−

−=>Η ,:8Ι>∆Χ >Η 9:Κ∆Ι:9 Ι∆ :Χ9∆<:Χ>Ο>Χ< Ι=: ;Γ:Φϑ:Χ8Ν ∆; Χ∆Β>Χ6Α 69?ϑΗΙ-

Β:ΧΙΗ. AΗ >Χ Γ6=6Β 6Χ9 ,Χ∆Λ:Γ (2004), Λ: ΗϑΕΕ∆Η: Ι=6Ι N >Η Η:Ι >Χ ∆Γ9:Γ

Ι∆ Β6Μ>Β>Ο: =∆ϑΗ:=∆Α9ΗΡΛ:Α;6Γ::

maxN

1X

j=0

j

(

logCt+j &.1+j;t+j

1 + F

)

(37)

ϑΧ9:Γ (14), (15) 6Χ9 (16). F >Η 6 ΠΜ:9 8∆ΗΙ Ι∆ 8=6Χ<>Χ< Λ6<:Η.

+:6Η∆Χ>Χ< 6Α∆Χ< Ι=: Α>Χ:Η ∆; AΕΕ:Χ9>Μ B, (37) 86Χ 7: Γ:ΛΓ>ΙΙ:Χ 6Η

maxN

1X

j=0

N(j+1)1X

l=Nj

l

(

logCt+l &.1+l;t+l

1 + F

)

(38)

(38) >Χ ΗΙ:69Ν ΗΙ6Ι: >Η 6Χ >ΧΠΧ>Ι: ΗϑΒ ∆; N Ε:Γ>∆9Η ΕΓ∆7Α:ΒΗ. −=:Γ:;∆Γ:, >Ι

7:8∆Β:Η

29

Page 33: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

maxN

1

1 N

1 N

(1 )2NN

1

log +

1

1 Nlog (. + 1 )

1

1 N&.0

1

1 +

1(1+)w

N

1 (1+)w

1

1 NF

−=: ΕΓ∆7Α:Β 86Χ 7: Η∆ΑΚ:9 ΧϑΒ:Γ>86ΑΑΝ. A<6>Χ ;∆ΑΑ∆Λ>Χ< Γ6=6Β 6Χ9 ,Χ∆Λ:Γ

(2004), Λ: 6ΗΗϑΒ: Ι=6Ι, 6Ι 6 4% >ΧΘ6Ι>∆Χ Γ6Ι:, Λ6<:Η 6Γ: Η:Ι ;∆Γ ∆Χ: Ν:6Γ.

+:ΗϑΑΙΗ ;∆Γ 9>Ω:Γ:ΧΙ Κ6Αϑ:Η ∆; F 6Γ: Η=∆ΛΧ >Χ Π<ϑΓ:Η 86 6Χ9 87. F>ΓΗΙ, ϑΧ9:Γ

Ι=: 6ΗΗϑΒΕΙ>∆Χ F 9 0, :Χ9∆<:Χ>Ο>Χ< N 9∆:Η Χ∆Ι Α:69 Ι∆ Β∆Χ:Ν ΗϑΕ:ΓΧ:ϑ-

ΙΓ6Α>ΙΝ. −=>Η =6ΕΕ:ΧΗ 7:86ϑΗ: =∆ϑΗ:=∆Α9Η =6Κ: 6 ΕΓ:;:Γ:Χ8: 6<6>ΧΗΙ Α67∆ϑΓ

8Ν8Α>Χ< 6Χ9 Η∆ ;∆Γ Λ6<: Θ:Μ>7>Α>ΙΝ, 7ϑΙ Ι=:Ν 86Χ 68=>:Κ: >Ι ∆ΧΑΝ 7Ν Ε6Ν>Χ<

Β∆Γ: ;Γ:Φϑ:ΧΙΑΝ F . ,∆, >Χ Ι=: :Χ9, Ι=:Ν ;68: 6 ΙΓ69:-∆Ω 6Χ9 Ι=:Ν ΕΓ:;:Γ Χ∆Ι

Ι∆ Η:Ι N = 1: AΗ 6 Β6ΙΙ:Γ ∆; 8∆ΧΗ:Φϑ:Χ8:, ;∆Γ F ! 0, Ι=: Α:Χ<Ι= ∆; Ι=: Λ6<:

8∆ΧΙΓ68Ι 8∆ΧΚ:Γ<:Η Ι∆ 1 6Χ9 Β∆Χ:Ν 7:8∆Β:Η ΗϑΕ:Γ-Χ:ϑΙΓ6Α.

7 N!0 ,!∋%∗a+a∋!−+%c !/% !(c! )( −∃! %(5a−%)(-

#+)0−∃ (!1.,

−=: ΕΓ:Η:ΧΙ ,:8Ι>∆Χ ∆Ω:ΓΗ Χ:Λ :ΒΕ>Γ>86Α :Κ>9:Χ8: ∆Χ Ι=: >ΧΘ6Ι>∆Χ-<Γ∆ΛΙ=

Χ:ΜϑΗ ;∆ΑΑ∆Λ>Χ< Ι=: ;∆∆ΙΗΙ:ΕΗ ∆; /6∆Χ6 6Χ9 ,8=>6Κ∆ (2007). ∀Χ ∆Γ9:Γ Ι∆ ∆7Ι6>Χ

8∆ΒΕ6Γ67Α: Γ:ΗϑΑΙΗ Ι∆ Ι=:>ΓΗ Λ: ϑΗ: Ι=: Η6Β: Β∆9:Α ΗΕ:8>Π86Ι>∆ΧΗ 6Χ9 96Ι6Η:Ι,

Λ=>8=, ∆Χ Ι=:>Γ ∆ΛΧ, 7ϑ>Α9 ∆Χ Ι=: Λ∆Γ≅ ∆; ∃=6Χ 6Χ9 ,:Χ=69?> (2001).

30

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(ϑΓ 9:Ε:Χ9:ΧΙ Κ6Γ>67Α: >Η Ι=: <Γ∆ΛΙ= Γ6Ι: ∆; D) >Χ 8∆ΧΗΙ6ΧΙ Α∆86Α 8ϑΓ-

Γ:Χ8Ν ϑΧ>ΙΗ. −=: >Χ9:Ε:Χ9:ΧΙ Κ6Γ>67Α:Η ∆; ,Ε:8>Π86Ι>∆Χ ∀ 6Γ: Ι=: Α:Κ:Α ∆;

>ΧΘ6Ι>∆Χ, <Γ∆ΗΗ ΠΜ:9 86Ε>Ι6Α ;∆ΓΒ6Ι>∆Χ (∆Γ <Γ∆ΗΗ 86Ε>Ι6Α ;∆ΓΒ6Ι>∆Χ Λ=:Χ Ι=:

;∆ΓΒ:Γ >Η Χ∆Ι 6Κ6>Α67Α:) ∆Κ:Γ D), Ι=: Α∆< ∆; >Χ>Ι>6Α Ε:Γ 86Ε>Ι6 D) >Χ )))

69?ϑΗΙ:9 9∆ΑΑ6ΓΗ, Ε∆ΕϑΑ6Ι>∆Χ <Γ∆ΛΙ=, Ι=: 6Κ:Γ6<: Η8=∆∆Α>Χ< Ν:6ΓΗ >Χ Ι=: Ι∆-

Ι6Α Ε∆ΕϑΑ6Ι>∆Χ 6<:9 15 ∆Γ Β∆Γ:, 6Χ9 Ι=: Η=6Γ: ∆; <∆Κ:ΓΧΒ:ΧΙ :ΜΕ:Χ9>ΙϑΓ:

∆Κ:Γ D). ,Ε:8>Π86Ι>∆Χ ∀∀ 8∆ΧΙΓ∆ΑΗ 6ΑΗ∆ ;∆Γ Ι=: <Γ∆ΛΙ= Γ6Ι: ∆; Ι=: Ι:ΓΒΗ ∆;

ΙΓ69: 6Χ9 Ι=:>Γ 5-Ν:6Γ ΗΙ6Χ96Γ9 9:Κ>6Ι>∆Χ10. ∀Χ 7∆Ι= Ι=: ΗΕ:8>Π86Ι>∆ΧΗ Λ:

>Χ8Αϑ9: 6 Η:Ι ∆; 8∆ϑΧΙΓΝ 6Χ9 Ι>Β: 9ϑΒΒ>:Η Ι∆ 8∆ΧΙΓ∆Α ;∆Γ Ε∆ΗΗ>7Α: ϑΧ∆7Η:ΓΚ:9

=:Ι:Γ∆<:Χ:>ΙΝ.

0: Γ:Η∆ΓΙ Ι∆ 9>Ω:Γ:ΧΙ 96Ι6 Η∆ϑΓ8:Η. F∆Γ Ε:Γ 86Ε>Ι6 D) 6Χ9 Ι=: Η=6Γ:

∆; <∆Κ:ΓΧΒ:ΧΙ 8∆ΧΗϑΒΕΙ>∆Χ Λ: Γ:Κ:ΓΙ Ι∆ ):ΧΧ 0∆ΓΑ9 −67Α:Η ()0−). (ϑΓ

:9ϑ86Ι>∆Χ Β:6ΗϑΓ: >Η ∆7Ι6>Χ:9 ;Γ∆Β Ι=: B6ΓΓ∆ 6Χ9 %:: 96Ι6Η:Ι ∆Χ :9ϑ86Ι>∆Χ6Α

6ΙΙ6>ΧΒ:ΧΙ. 0: ϑΗ: Ι=: :ΜΕ∆ΓΙ 6Χ9 >ΒΕ∆ΓΙ ϑΧ>Ι Κ6Αϑ: Η:Γ>:Η ;Γ∆Β Ι=: ∀&FΡΗ

∀ΧΙ:ΓΧ6Ι>∆Χ6Α F>Χ6Χ8>6Α ,Ι6Ι>ΗΙ>8Η (∀F,) Ι∆ 7ϑ>Α9 Ι:ΓΒΗ ∆; ΙΓ69: 96Ι6.

(ϑΓ 96Ι6Η:Ι ΗΕ6ΧΗ ;Γ∆Β 1960 Ι∆ 1999 ∆Κ:Γ 167 8∆ϑΧΙΓ>:Η. A;Ι:Γ −:ΒΕΑ:

(2000), Λ: :ΜΕ:Γ>Β:ΧΙ Λ>Ι= 9>Ω:Γ:ΧΙ Ι>Β: ;Γ:Φϑ:Χ8>:Η 7ϑΙ ∆ϑΓ 76Η:Α>Χ: Γ:ΗϑΑΙΗ

6Γ: 76Η:9 ∆Χ 5-Ν:6ΓΗ Β:6ΧΗ11. &>ΗΗ>Χ< Κ6Αϑ:Η Γ:9ϑ8: Ι=: ΧϑΒ7:Γ ∆; 6Κ6>Α67Α:

∆7Η:ΓΚ6Ι>∆ΧΗ (:ΗΕ:8>6ΑΑΝ ;∆Γ 9:Κ:Α∆Ε>Χ< 8∆ϑΧΙΓ>:Η). 0: 6ΑΗ∆ 9Γ∆Ε ;Γ∆Β Ι=:

Η6ΒΕΑ: 6ΑΑ Ι=: ∆7Η:ΓΚ6Ι>∆ΧΗ Λ>Ι= 6Χ >ΧΘ6Ι>∆Χ Γ6Ι: 67∆Κ: 40% 6Η >Χ ∃6=Χ 6Χ9

10/6∆Χ6 6Χ9 ,8=>6Κ∆ (2007) Ε:Γ;∆ΓΒ:9 Γ∆7ϑΗΙΧ:ΗΗ 8=:8≅Η 6ΑΗ∆ >Χ8Αϑ9>Χ< ΠΧ6Χ8>6Α 9:Κ:Α-∆ΕΒ:ΧΙ >Χ9>86Ι∆ΓΗ. >Κ:Χ Ι=6Ι Ι=:>Γ Γ:ΗϑΑΙΗ Λ:Γ: ΗΙ67Α:, Ι=>Η Α>Χ: ∆; Γ:Η:6Γ8= >Η Χ∆Ι ΕϑΓΗϑ:96ΧΝ ;ϑΓΙ=:Γ =:Γ:.11.Η>Χ< Β:9>6ΧΗ >ΧΗΙ:69 ∆; Β:6ΧΗ Λ∆ϑΑ9 ΕΓ∆9ϑ8: Η>Β>Α6Γ Γ:ΗϑΑΙΗ Ι∆ Ι=∆Η: ΕΓ:Η:ΧΙ:9

7:Α∆Λ.

31

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,:Χ=69?> (2001)12. (ϑΓ ΠΧ6Α Η6ΒΕΑ: >Χ8Αϑ9:Η 85 8∆ϑΧΙΓ>:Η 6Χ9 421 ∆7Η:ΓΚ6-

Ι>∆ΧΗ. 119 ∆7Η:ΓΚ6Ι>∆ΧΗ 7:Α∆Χ< Ι∆ 19 >Χ9ϑΗΙΓ>6Α 8∆ϑΧΙΓ>:Η. C∆ΧΗ>9:Γ>Χ< Ι=:

Ι:ΓΒΗ ∆; ΙΓ69: Κ6Γ>67Α:Η Γ:9ϑ8:Η Ι=: ΧϑΒ7:Γ ∆; 9:Κ:Α∆Ε>Χ< 8∆ϑΧΙΓ>:Η ;Γ∆Β 66

Ι∆ 25, Α:6Κ>Χ< 6ΑΒ∆ΗΙ ϑΧ6Ω:8Ι:9 9:Κ:Α∆Ε:9 8∆ϑΧΙΓ>:Η.

0: ;∆ΑΑ∆Λ 6 9>Ω:Γ:ΧΙ Γ:Η:6Γ8= ΗΙΓ6Ι:<Ν Ι=6Χ /6∆Χ6 6Χ9 ,8=>6Κ∆ (2007).

F>ΓΗΙ, 6;Ι:Γ %> 6Χ9 +68>Χ: (2007, 8=. 15), Λ: ϑΗ: 6 Η:Β>Ε6Γ6Β:ΙΓ>8 Ε∆Λ:Γ

Η:Γ>:Η :ΗΙ>Β6Ι∆Γ 6Χ9 Λ: Α∆∆≅ ;∆Γ Ι=: 7:ΗΙ ΠΙΙ>Χ< Β∆9:Α ΗΕ:8>Π86Ι>∆Χ 7Ν ϑΗ>Χ<

Η:Κ:Γ6Α 8Γ>Ι:Γ>6. −=:Χ Λ: 8=:8≅ ;∆Γ :Χ9∆<:Χ:>ΙΝ 7>6Η 7Ν >ΒΕΑ:Β:ΧΙ>Χ< 6 ΙΛ∆

ΗΙ6<: Α:6ΗΙ ΗΦϑ6Γ: :ΗΙ>Β6Ι∆Γ (2,%,) 6Χ9 6 !6ϑΗΒ6Χ Ι:ΗΙ. F>Χ6ΑΑΝ Λ: Ε:Γ;∆ΓΒ

Κ6Γ>∆ϑΗ Γ∆7ϑΗΙΧ:ΗΗ 8∆ΧΙΓ∆ΑΗ ;∆Γ ∆ϑΓ Γ:ΗϑΑΙΗ 7Ν ΗΕΑ>ΙΙ>Χ< ∆ϑΓ Η6ΒΕΑ: 7:ΙΛ::Χ

9:Κ:Α∆Ε:9 6Χ9 9:Κ:Α∆Ε>Χ< 8∆ϑΧΙΓ>:Η, 7Ν ϑΗ>Χ< 8 Ν:6ΓΗ 6Κ:Γ6<:Η 6Χ9 7Ν Γ:Η∆ΓΙ-

>Χ< Ι∆ 6 Η:Β>Ε6Γ6Β:ΙΓ>8 Α∆86Α Α>Χ:6Γ :ΗΙ>Β6Ι∆Γ. −=>Η Γ:Η:6Γ8= ΗΙΓ6Ι:<Ν 6ΑΑ∆ΛΗ

ϑΗ Ι∆ Ι:ΗΙ Λ=:Ι=:Γ 6Χ >ΧΗΙΓϑΒ:ΧΙ6Α Κ6Γ>67Α: :ΗΙ>Β6Ι∆Γ >Η 6ΕΕΓ∆ΕΓ>6Ι:, Γ6Ι=:Γ

Ι=6Χ 6ΗΗϑΒ>Χ< >Ι Α>≅: >Χ /6∆Χ6 6Χ9 ,8=>6Κ∆ (2007).

C∆ΧΗ>9:Γ Ι=: ;∆ΑΑ∆Λ>Χ< :8∆Χ∆Β:ΙΓ>8 Β∆9:Α

!i = 01i + ( 2i) + 4i

!i >Η Ι=: 9:Ε:Χ9:ΧΙ Κ6Γ>67Α:, ji ;∆Γ ∗ = 1; 2 6Γ: ΙΛ∆ Η:ΙΗ ∆; >Χ9:Ε:Χ9:ΧΙ

Κ6Γ>67Α:Η, >Η 6 Κ:8Ι∆Γ ∆; 8∆:ς 8>:ΧΙΗ Ι∆ 7: :ΗΙ>Β6Ι:9, () >Η 6 Χ∆Χ-Α>Χ:6Γ

;ϑΧ8Ι>∆Χ ∆; ϑΧΗΕ:8>Π:9 ;∆ΓΒ, ) >Η Ι=: Ηϑ7-Η8Γ>ΕΙ ;∆Γ Ι=: )-Ι= ∆7Η:ΓΚ6Ι>∆Χ, 4i

>Η Ι=: 9>ΗΙϑΓ76Χ8:. 0: =:Γ: ΗϑΕΕ∆Η: Ι=6Ι 2i >Η 6 Η86Α6Γ. ∀Ι >Η Ε∆ΗΗ>7Α: Ι∆

12 >ΑΑΒ6Χ :Ι 6Α. (2004) Η=∆Λ Ι=6Ι ϑΗ>Χ< 9>Ω:Γ:ΧΙ ΙΓϑΧ86Ι>∆Χ Ε∆>ΧΙΗ <:Χ:Γ6Ι: Χ:<Α><>7Α:9>Ω:Γ:Χ8:Η >Χ Ι=: Γ:ΗϑΑΙΗ.

32

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6ΕΕΓ∆Μ>Β6Ι: () 7Ν ϑΗ>Χ< 6 Ε∆Λ:Γ Η:Γ>:Η ;ϑΧ8Ι>∆Χ

f0s ( 2i) = s2ig

K

s=1

!∆Λ:Κ:Γ, >Ι >Η Χ:8:ΗΗ6ΓΝ Ι∆ 8=∆∆Η: K. −=: Α>Ι:Γ6ΙϑΓ: =6Η ΕΓ∆Ε∆Η:9 Η:Κ-

:Γ6Α 8Γ>Ι:Γ>6 Ι∆ =:ΑΕ Ι=>Η 8=∆>8:. −Λ∆ ΕΓ∆Β>Χ:ΧΙ :Μ6ΒΕΑ:Η 6Γ: Ι=: A≅6>≅:

6Χ9 ,8=Λ6ΓΟ Ε6Γ6Β:ΙΓ>8 8Γ>Ι:Γ>6. 0: 6ΑΗ∆ 8∆ΧΗ>9:Γ Ι=Γ:: Χ∆ΧΕ6Γ6Β:ΙΓ>8 ΕΓ∆-

8:9ϑΓ:Η: &6ΑΑ∆ΛΗΡΗ CL, <:Χ:Γ6Α>Ο:9 8Γ∆ΗΗ-Κ6Α>96Ι>∆Χ 6Χ9 Α:6Κ:-∆Χ:-∆ϑΙ 8Γ∆ΗΗ-

Κ6Α>96Ι>∆Χ.

&6ΑΑ∆ΛΗΡΗ CL 8∆ΧΗ>ΗΙΗ >Χ Η:Α:8Ι>Χ< K Ι∆ Β>Χ>Β>Ο:

N1

NX

i

#!i

01i ( 2i)

∃2+ 22

K

N

Λ=:Γ: N >Η Ι=: ΧϑΒ7:Γ ∆; ∆7Η:ΓΚ6Ι>∆ΧΗ, 6Χ9 6Γ: Ι=: :ΗΙ>Β6Ι:9 8∆ϑΧΙ:Γ-

Ε6ΓΙΗ ∆; 6Χ9 , 2 =PNi u

2i

N6Χ9 4i = !i

01i ( 2i).

A88∆Γ9>Χ< Ι∆ <:Χ:Γ6Α>Ο:9 8Γ∆ΗΗ-Κ6Α>96Ι>∆Χ, ∆Χ: Η:Α:8ΙΗ K Ι∆ Β>Χ>Β>Ο:

PN

i

#!i

01i ( 2i)

∃2

1

Kn

2

F>Χ6ΑΑΝ, Ι=: Α:6Κ:-∆Χ:-∆ϑΙ 8Γ∆ΗΗ-Κ6Α>96Ι>∆Χ 8∆ΧΗ>ΗΙΗ ∆; Η:Α:8Ι>Χ< K Ι∆ Β>Χ-

>Β>Ο:

CVK =

NX

i

#!i

01ii i ( 2i)

∃2

Λ=:Γ: i6Χ9 i 6Γ: :ΗΙ>Β6Ι:9 Γ:Β∆Κ>Χ< (!i; 01i; 2i).

33

Page 37: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

0: 6ΕΕΑΝ Ι=: ΕΓ∆8:9ϑΓ:Η 67∆Κ: Ι∆ ∆ϑΓ 96Ι6 Η6ΒΕΑ: ;∆Γ 7∆Ι= ,Ε:8>Π86Ι>∆Χ

∀ 6Χ9 ∀∀ ΠΜ>Χ< Ι=: Β6Μ>ΒϑΒ ∆; K Ι∆ 4. AΗ Η=∆Λ:9 >Χ −67Α: 1, Ι=: Β6?∆Γ>ΙΝ

∆; Ι=: 8Γ>Ι:Γ>6 Ε∆>ΧΙ Ι∆ 6 ΠΓΗΙ ∆Γ9:Γ Ε∆ΑΝΧ∆Β>6Α 6Η Ι=: 7:ΗΙ ΠΙΙ>Χ< ∆ΕΙ>∆Χ, Η∆

Λ: ΗΙ>8≅ Ι∆ >Ι.

∋:ΜΙ Λ: 8∆ΒΕ6Γ: ∆ϑΓ (%, :ΗΙ>Β6Ι:Η Ι∆ Ι=: 2,%, ∆Χ:Η 7Ν Β:6ΧΗ ∆; 6

!6ϑΗΒ6Χ Ι:ΗΙ. AΗ >Χ B6ΓΓ∆ (1995) Λ: >ΧΗΙΓϑΒ:ΧΙ >ΧΘ6Ι>∆Χ 7Ν >ΙΗ ΠΓΗΙ Α6<.

.Η>Χ< Ι=>Η >ΧΗΙΓϑΒ:ΧΙ :ΧΙ6>ΑΗ 6ΗΗϑΒ>Χ< Ι=6Ι >Ι Η=∆ϑΑ9 Χ∆Ι :ΧΙ:Γ Ι=: :8∆Χ∆Β:Ι-

Γ>8 Β∆9:Α ;∆Γ <Γ∆ΛΙ=. −=>Η >Η 8∆ΧΗ>ΗΙ:ΧΙ Λ>Ι= ∆ϑΓ Ι=:∆Γ:Ι>86Α Β∆9:Α, Λ=:Γ:

Ι=:Γ: >Η Χ∆Ι 6ΧΝ 9:Α6Ν >Χ Ι=: >ΒΕ68Ι ∆; Β∆Χ:Ν <Γ∆ΛΙ= ∆Χ Γ:6Α <Γ∆ΛΙ=13. −67Α:

2 Η=∆ΛΗ ∆ϑΓ Γ:ΗϑΑΙΗ. +:<Γ:ΗΗ>Χ< ΕΓ:Η:ΧΙ >ΧΘ6Ι>∆Χ ∆Χ >ΙΗ ΠΓΗΙ Α6< (6Χ9 ∆Χ 6ΑΑ

Ι=: 8∆ΧΙΓ∆ΑΗ ∆; ∆ϑΓ Β∆9:Α) Γ:ΙϑΓΧΗ 6 Ε-Κ6Αϑ: ∆; 0.00 Λ>Ι= 6 Ι-ΗΙ6Ι>ΗΙ>8 ∆; 4.91

6Χ9 6Χ F-ΗΙ6Ι>ΗΙ>8 24.06 ;∆Γ ,Ε:8>Π86Ι>∆Χ ∀ 6Χ9 6 Ι-ΗΙ6Ι>ΗΙ>8 ∆; 4.65 6Χ9 6Χ

F-ΗΙ6Ι>ΗΙ>8 ∆; 21.61 ;∆Γ ,Ε:8>Π86Ι>∆Χ ∀∀. ,∆ Ι=: >ΧΗΙΓϑΒ:ΧΙ:9 Κ6Γ>67Α: >Η =><=ΑΝ

Η><Χ>Π86ΧΙΑΝ 8∆ΓΓ:Α6Ι:9 Λ>Ι= >ΙΗ >ΧΗΙΓϑΒ:ΧΙ 6Χ9 ∆ϑΓ Γ:ΗϑΑΙΗ Η=∆ϑΑ9 Χ∆Ι 7: 7>-

6Η:9 7Ν 6 Λ:6≅ >ΧΗΙΓϑΒ:ΧΙ ΕΓ∆7Α:Β. −=: !6ϑΗΒ6Χ Ι:ΗΙ Λ∆ϑΑ9 Χ∆Ι Γ:?:8Ι Ι=:

ΧϑΑΑ Ι=6Ι Ι=: (%, 6Χ9 2,%, :ΗΙ>Β6Ι∆ΓΗ 6Γ: :Φϑ6Α 6Χ9, >Χ9::9, ;∆Γ 7∆Ι= Ι=:

ΗΕ:8>Π86Ι>∆ΧΗ Ι=: >ΧΘ6Ι>∆Χ 8∆:ς 8>:ΧΙΗ 6Γ: Κ:ΓΝ 8Α∆Η:. ∀ΧΘ6Ι>∆Χ >Η Ι=:Γ:;∆Γ:

8∆ΧΠΓΒ:9 Ι∆ =6Κ: 6 Χ:<6Ι>Κ: Α>Χ:6Γ >ΒΕ68Ι ∆Χ <Γ∆ΛΙ=.

∀Χ ∆Γ9:Γ Ι∆ 8=:8≅ Ι=: Γ∆7ϑΗΙΧ:ΗΗ ∆; ∆ϑΓ Γ:ΗϑΑΙΗ, Λ: ΠΓΗΙ ΗΕΑ>Ι ∆ϑΓ Η6ΒΕΑ:

13&∆Γ: >Χ <:Χ:Γ6Α, Ι=>Η Β∆9:ΑΑ>Χ< 8=∆>8: >Η 8∆ΧΗ>ΗΙ:ΧΙ Λ>Ι= ΙΛ∆ ΕΑ6ϑΗ>7Α: Ι=∆ϑ<=ΙΗ. F>ΓΗΙ,Ε6ΗΙ >ΧΘ6Ι>∆Χ =6Η 6Χ >ΒΕ68Ι ∆Χ 8ϑΓΓ:ΧΙ <Γ∆ΛΙ= ∆ΧΑΝ 7:86ϑΗ: >Ι >Η 8∆ΓΓ:Α6Ι:9 Λ>Ι= 8ϑΓΓ:ΧΙ>ΧΘ6Ι>∆Χ 6Χ9, Η:8∆Χ9, 8ϑΓΓ:ΧΙ Γ:6Α <Γ∆ΛΙ= Β><=Ι =6Κ: 6 8∆ϑΧΙ:Γ:Ω:8Ι ∆Χ >ΧΘ6Ι>∆Χ 7Ν ∆Κ:Γ-=:6Ι>Χ< Ι=: :8∆Χ∆ΒΝ, 7ϑΙ Χ∆Ι ∆Χ Ε6ΗΙ >ΧΘ6Ι>∆Χ. >Κ:Χ Ι=6Ι Λ: 6Γ: ϑΗ>Χ< ΒϑΑΙ>-Ν:6Γ 6Κ:Γ6<:Η>Ι >Η >ΒΕΑ6ϑΗ>7Α: Ι∆ Ι=>Χ≅, ;∆Γ >ΧΗΙ6Χ8:, Ι=6Ι 8ϑΓΓ:ΧΙ >ΧΘ6Ι>∆Χ >Η =><= #∋ −∀e 4+,− ∀a% ( adecade 7:86ϑΗ: 6<:ΧΙΗ :ΜΕ:8Ι =><= <Γ∆ΛΙ= #∋ −∀e ,ec(∋d ∀a% ( −∀e decade Ι∆ :Μ68:Γ76Ι:>ΧΘ6Ι>∆Χ6ΓΝ 7∆ΙΙΑ:Χ:8≅Η.

34

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7:ΙΛ::Χ 9:Κ:Α∆Ε:9 6Χ9 9:Κ:Α∆Ε>Χ< 8∆ϑΧΙΓ>:Η (−67Α: 3). F∆Γ Ι=: Α6ΙΙ:Γ ∆Χ:Η,

6ΑΑ Ι=: 8Γ>Ι:Γ>6 Ε∆>ΧΙ Ι∆ 6 Α>Χ:6Γ Β∆9:Α 6Η Ι=: 7:ΗΙ ΠΙΙ>Χ< ∆Χ:. F∆Γ Ι=: ;∆ΓΒ:Γ

∆Χ:Η, >ΧΗΙ:69, Ι=:Γ: >Η 6 9>Η8Γ:Ε6Χ8Ν 7:ΙΛ::Χ ,Ε:8>Π86Ι>∆ΧΗ ∀ 6Χ9 ∀∀. ∀Χ Ι=:

ΠΓΗΙ 86Η: 6 ;∆ϑΓΙ= ∆Γ9:Γ Ε∆ΑΝΧ∆Β>6Α Λ∆ϑΑ9 7: Η:Α:8Ι:9 7Ν 6ΑΑ Ι=: 8Γ>Ι:Γ>6, 7ϑΙ

Ι=: ,8=Λ6ΓΟ ∆Χ:, Λ=>8= Λ∆ϑΑ9 Η:Α:8Ι 6 Η:8∆Χ9 ∆Γ9:Γ Ε∆ΑΝΧ∆Β>6Α. (Χ8: 699>Χ<

Ι∆ Ι=: Β∆9:Α Ι=: <Γ∆ΛΙ= Γ6Ι: ∆; Ι=: Ι:ΓΒΗ ∆; ΙΓ69: 6Χ9 Ι=:>Γ 5-Ν:6ΓΗ ΗΙ6Χ96Γ9

9:Κ>6Ι>∆Χ, Ι=: Β6?∆Γ>ΙΝ ∆; Ι=: 8Γ>Ι:Γ>6 Λ∆ϑΑ9 Η:Α:8Ι 6 Α>Χ:6Γ Β∆9:Α. ,ϑ8= 6

9>Η8Γ:Ε6Χ8Ν 8∆ϑΑ9 7: 86ϑΗ:9 7Ν Ι=: ∆Β>ΗΗ>∆Χ ∆; 6 Γ:Α:Κ6ΧΙ Κ6Γ>67Α:, <>Κ:Χ

Ι=6Ι Ι=: <Γ∆ΛΙ= Γ6Ι: ∆; Ι=: Ι:ΓΒΗ ∆; ΙΓ69: >Η =><=ΑΝ Η><Χ>Π86ΧΙΑΝ 8∆ΓΓ:Α6Ι:9

Λ>Ι= :8∆Χ∆Β>8 <Γ∆ΛΙ=14 6Χ9 Ι=6Ι Χ∆Χ-Α>Χ:6Γ Ι:ΓΒΗ 86Χ 7: ϑΗ:9 Ι∆ 9:Ι:8Ι Ι=:

∆Β>ΗΗ>∆Χ ∆; 6 Γ:Α:Κ6ΧΙ Κ6Γ>67Α: (Η:: ;∆Γ >ΧΗΙ6Χ8: +6ΒΗ:Ν, 1969).

F>Χ6ΑΑΝ, /6∆Χ6 6Χ9 ,8=>6Κ∆ (2007) ;∆ϑΧ9 Ι=6Ι 6 Β>Α9 Ε∆Η>Ι>Κ: :Ω:8Ι ∆;

>ΧΘ6Ι>∆Χ ∆Χ <Γ∆ΛΙ= 6ΕΕ:6ΓΗ 6Ι Α∆Λ >ΧΘ6Ι>∆Χ Γ6Ι:Η >Χ 9:Κ:Α∆Ε:9 8∆ϑΧΙΓ>:Η ;∆Γ

,Ε:8>Π86Ι>∆Χ ∀ Λ=:Χ 8∆ΧΗ>9:Γ>Χ< 8 Ν:6ΓΗ Β:6ΧΗ >ΧΗΙ:69 ∆; 5 Ν:6ΓΗ ∆Χ:Η. 0:

ΓϑΧ 6<6>Χ ∆ϑΓ Β∆9:Α Η:Α:8Ι>∆Χ 8Γ>Ι:Γ>6 ;∆Γ Ι=>Η 86Η:, 7ϑΙ Λ: 9∆ Χ∆Ι ΠΧ9 6ΧΝ

Χ∆Χ-Α>Χ:6Γ>ΙΝ >Χ Ι=: 96Ι6 6Η Η=∆Λ:9 >Χ −67Α: 4. A Α>Χ:6Γ Β∆9:Α 6ΕΕ:6ΓΗ Ι∆ 7:

Ι=: 7:ΗΙ ΠΙΙ>Χ< ∆Χ:.

(ϑΓ ΠΧ6Α Γ∆7ϑΗΙΧ:ΗΗ 8=:8≅ >Η Ι∆ 8∆ΒΕϑΙ: 6 Η:Β>Ε6Γ6Β:ΙΓ>8 Α∆86Α Α>Χ:6Γ

≅:ΓΧ:Α :ΗΙ>Β6Ι∆Γ, Λ=∆Η: Β6>Χ 69Κ6ΧΙ6<:, 8∆ΒΕ6Γ:9 Ι∆ Ι=: ∋696Γ6Ν6-06ΙΗ∆Χ

∆Χ:, >Η Χ∆Ι Ι∆ 7: Η∆ Η:ΧΗ>Ι>Κ: Ι∆ 7∆ϑΧ96ΓΝ ∆7Η:ΓΚ6Ι>∆ΧΗ. ∀Χ ∆ϑΓ 8∆ΧΙ:ΜΙ, Ι=>Η

69Κ6ΧΙ6<: >Η Κ:ΓΝ >ΒΕ∆ΓΙ6ΧΙ <>Κ:Χ Ι=6Ι Ι=: Ι=Γ:Η=∆Α9 Α:Κ:Α ∆; >ΧΘ6Ι>∆Χ =6Η

14(Χ8: Γ:ΗΙΓ>8Ι>Χ< Ι=: 6Χ6ΑΝΗ>Η Ι∆ 9:Κ:Α∆Ε:9 8∆ϑΧΙΓ>:Η >Ι =6Η 6 8∆:ς 8>:ΧΙ :Φϑ6Α Ι∆ 0.12Λ>Ι= 6 Ε-Κ6Αϑ: ∆; 0.006.

35

Page 39: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

∆;Ι:Χ 7::Χ ;∆ϑΧ9 Ι∆ 7: Γ6Ι=:Γ Α∆Λ. 0=:Χ ϑΗ>Χ< 6 Η:Β>Ε6Γ6Β:ΙΓ>8 ≅:ΓΧ:Α

:ΗΙ>Β6Ι∆Γ, ∆Χ: 86Χ ∆7Ι6>Χ 6Χ :ΗΙ>Β6Ι: ∆; 6Η ;∆ΑΑ∆ΛΗ:

=

"NX

i=1

( 1i −12i) ( 1i −12i)0

#1 " NX

i=1

( 1i −12i) (!i −2i)0

#

Λ=:Γ: −12i 6Χ9 −2i 6Γ: Ι=: Χ∆Χ-Ε6Γ6Β:ΙΓ>8 :ΗΙ>Β6Ι∆ΓΗ ∆; −12i = E ( 1ij 2i)

6Χ9 −2i = E (!ij 2i) ; Λ=:Γ: E >Η Ι=: :ΜΕ:8Ι6Ι>∆Χ ∆Ε:Γ6Ι∆Γ. .Η>Χ< 6 Α∆86Α

Α>Χ:6Γ :ΗΙ>Β6Ι∆Γ, ∆Χ: =6Η:

−2i = &01

.2 +

nX

i=1

∀iK

2i 2

( ( 2i)

∀0i

!1∀iK

2i 2

( ( 2i)

!i

Λ=:Γ: ∀0i = [1; ( 2i 2)], &1 >Η 6 8∆ΑϑΒΧ Κ:8Ι∆Γ ∆; Ι=: Η6Β: 9>Β:ΧΗ>∆Χ 6Η

∀i; Λ>Ι= ϑΧ>ΙΝ 6Η ΠΓΗΙ :Α:Β:ΧΙ 6Χ9 Ο:Γ∆ :ΑΗ:Λ=:Γ:, 6Χ9 K () >Η Ι=: ≅:ΓΧ:Α

;ϑΧ8Ι>∆Χ ()6<6Χ 6Χ9 .ΑΑ6=, 1999). !:Γ: Λ: ϑΗ: 6 6ϑΗΗ>6Χ ≅:ΓΧ:Α. A;Ι:Γ F6Χ

6Χ9 >?7:ΑΗ (1992), Λ: Λ∆Γ≅ Λ>Ι= 6 Κ6Γ>67Α: 76Χ9Λ>9Ι= ( ( 2i) = (∋ 15 ( 2i) ;

Λ=:Γ: ( Λ6Η 8=∆Η:Χ Ι∆ 7: :Φϑ6Α Ι∆ ,>ΑΚ:ΓΒ6ΧΡΗ ∆ΕΙ>Β6Α ΗΒ∆∆Ι=>Χ< 76Χ9Λ>9Ι=

6Χ9 ∋ ( 2i) >Η 6 ≅:ΓΧ:Α :ΗΙ>Β6Ι: ∆; Ι=: 9:ΧΗ>ΙΝ ∆; 2i: −12i 6ΗΗϑΒ:Η 6 Η>Β>Α6Γ

;∆ΓΒ Ι∆ −2i. +:86ΑΑ Ι=6Ι

( 2i) = E#!i

01ij 2i

>Κ:Χ ∆ϑΓ Γ:ΗϑΑΙΗ 67∆Κ:, Λ: Β6>ΧΑΝ ;∆8ϑΗ ∆Χ ,Ε:8>Π86Ι>∆Χ ∀∀ 6Η ,Ε:8>Π86-

36

Page 40: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

Ι>∆Χ ∀ ∆Β>ΙΗ 6 Γ:Α:Κ6ΧΙ Κ6Γ>67Α:, Ι=: <Γ∆ΛΙ= Γ6Ι: ∆; Ι:ΓΒΗ ∆; ΙΓ69:. )6Γ6Β:Ι:Γ

:ΗΙ>Β6Ι:Η ;∆Γ Ι=: 8∆ΧΙΓ∆Α Κ6Γ>67Α:Η 6Γ: Η:Ι ∆ϑΙ >Χ −67Α: 5. F><ϑΓ: 9 Η=∆ΛΗ Ι=:

:ΗΙ>Β6Ι:Η ∆; Ι=: Ε6ΓΙ>6Α :Ω:8Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ <Γ∆ΛΙ= ;∆Γ Ι=: Λ=∆Α: Η6ΒΕΑ:,

;∆Γ 9:Κ:Α∆Ε:9 8∆ϑΧΙΓ>:Η 6Χ9 ;∆Γ 9:Κ:Α∆Ε>Χ< ∆Χ:Η. (Χ8: 6<6>Χ Χ∆ Ι=Γ:Η=∆Α9

Α:Κ:Α ∆; >ΧΘ6Ι>∆Χ 86Χ 7: 9:Ι:8Ι:9, Ι=: :Ω:8Ι ∆; >ΧΘ6Ι>∆Χ ∆Χ <Γ∆ΛΙ= 6ΕΕ:6ΓΗ Ι∆

7: ?ϑΗΙ Χ:<6Ι>Κ: 6Χ9 Α>Χ:6Γ>ΙΝ >Η 6 <∆∆9 6ΕΕΓ∆Μ>Β6Ι>∆Χ. ,=>;Ι>Χ< Ι∆ 8 Ν:6ΓΗ

6Κ:Γ6<:Η ;∆Γ 9:Κ:Α∆Ε:9 8∆ϑΧΙΓ>:Η 6Χ9 Ι∆ ,Ε:8>Π86Ι>∆Χ ∀ Λ∆ϑΑ9 Χ∆Ι 8=6Χ<: ∆ϑΓ

Γ:ΗϑΑΙΗ.

(ϑΓ :ΒΕ>Γ>86Α Γ:ΗϑΑΙΗ 86Χ Η=:9 Α><=Ι ∆Χ Ι=: Ι=:∆Γ:Ι>86Α Γ:ΗϑΑΙΗ 67∆Κ:.

−=∆ϑ<= Λ: 86ΧΧ∆Ι ΕΓ∆Κ: Ι=6Ι Ι=: Α67∆ϑΓ Β6Γ≅:Ι 8=6ΧΧ:Α =><=Α><=Ι:9 7Ν ∆ϑΓ

Ι=:∆ΓΝ >Η Ι=: 9∆Β>Χ6ΧΙ ∆Χ: 8∆ΒΕ6Γ:9 Ι∆ ∆Ι=:Γ 8=6ΧΧ:ΑΗ 6ΑΓ:69Ν >ΑΑϑΗΙΓ6Ι:9 7Ν

Ι=: Α>Ι:Γ6ΙϑΓ:, Λ: Η=∆Λ Ι=6Ι, ∆Χ8: 6ΗΗϑΒ>Χ< 6 Χ∆Χ-ΧϑΑΑ >ΧΙ:ΓΙ:ΒΕ∆Γ6Α :Α6ΗΙ>8-

>ΙΝ ∆; Ηϑ7ΗΙ>ΙϑΙ>∆Χ ∆; Λ∆Γ≅>Χ< Ι>Β:, ∆ϑΓ Β∆9:Α 9∆:Η Χ∆Ι ΕΓ∆9ϑ8: ϑΧΓ:6Α>ΗΙ>8

Γ:ΗϑΑΙΗ. ,∆ ∆ϑΓ :ΒΕ>Γ>86Α Γ:ΗϑΑΙΗ =:ΑΕ ϑΗ Ι∆ Η:Α:8Ι Ι=: Β∆ΗΙ Ηϑ>Ι67Α: Ε6Γ6Β:-

ΙΓ>Ο6Ι>∆Χ ;∆Γ ∆ϑΓ Β∆9:Α.

8 C)(c&.,%)(,

−=>Η Ε6Ε:Γ :ΜΙ:Χ9Η Ι=: Γ:ΗϑΑΙΗ ∆; Ι=: ∋:Λ-∃:ΝΧ:Η>6Χ Α>Ι:Γ6ΙϑΓ: Λ>Ι= Λ6<:-

ΗΙ6<<:Γ>Χ< ;Γ∆Β Ι=: Γ:Α6Ι>∆Χ 7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9 Ι=: Α:Κ:Α ∆; ∆ϑΙΕϑΙ Ι∆ Ι=:

>ΧΘ6Ι>∆Χ-<Γ∆ΛΙ= Χ:ΜϑΗ. ∀Χ Ι=>Η Λ6Ν, >Ι >Η Ε∆ΗΗ>7Α: Ι∆ :ΜΕΑ∆Γ: =∆Λ >ΧΘ6Ι>∆Χ

6Ω:8ΙΗ <Γ∆ΛΙ= Ε6ΗΗ>Χ< Ι=Γ∆ϑ<= Ι=: Α67∆ϑΓ Β6Γ≅:Ι >ΧΗΙ:69 ∆; 86Ε>Ι6Α, 8Γ:9>Ι

∆Γ ΕΓ∆9ϑ8Ι Β6Γ≅:ΙΗ. (Χ8: 6ΑΑ∆Λ>Χ< ;∆Γ 6Χ >ΧΙ:ΓΙ:ΒΕ∆Γ6Α :Α6ΗΙ>8>ΙΝ ∆; Ηϑ7ΗΙ>-

37

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ΙϑΙ>∆Χ ∆; Λ∆Γ≅>Χ< Ι>Β: <Γ:6Ι:Γ Ι=6Χ Ο:Γ∆, >ΧΘ6Ι>∆Χ =6Η Η>Ο:67Α: Γ:6Α :Ω:8ΙΗ:

>Ι =6ΓΒΗ :8∆Χ∆Β>8 <Γ∆ΛΙ= 6Χ9 Γ:9ϑ8:Η :8∆Χ∆Β>8 6<:ΧΙΗΡΛ:Α;6Γ:. AΗΗϑΒ-

>Χ< 6Χ :Χ9∆<:Χ∆ϑΗ Α:Χ<Ι= ∆; Ι=: Α67∆ϑΓ 8∆ΧΙΓ68Ι 9∆:Η Χ∆Ι >Χ <:Χ:Γ6Α Α:69 Ι∆

Β∆Χ:Ν ΗϑΕ:ΓΧ:ϑΙΓ6Α>ΙΝ. ∀Χ ΕΓ:Η:Χ8: ∆; 6 ΠΜ:9 8∆ΗΙ Ι∆ 8=6Χ<>Χ< Λ6<:Η, 6<:ΧΙΗ

ΕΓ:;:Γ Λ6<: ΗΙ6<<:Γ>Χ< Ι∆ Λ6<: Θ:Μ>7>Α>ΙΝ ϑΧΑ:ΗΗ >ΧΘ6Ι>∆Χ >Η =><= - =∆Λ =><=

9:Ε:Χ9>Χ< ∆Χ Ι=: Β6<Χ>Ιϑ9: ∆; Ι=: ΠΜ:9 8∆ΗΙ.

(ϑΓ :ΒΕ>Γ>86Α Γ:ΗϑΑΙΗ 8∆ΧΠΓΒ Ι=6Ι >ΧΘ6Ι>∆Χ =6Η 6 Χ:<6Ι>Κ: >ΒΕ68Ι ∆Χ

<Γ∆ΛΙ= 6Χ9 Χ∆-Ι=Γ:Η=∆Α9 Α:Κ:Α ∆; >ΧΘ6Ι>∆Χ >Η ;∆ϑΧ9. 0: 6Γ<ϑ: Ι=6Ι Ι=>Η >Η

8∆ΧΗ>ΗΙ:ΧΙ Λ>Ι= ∆ϑΓ Ι=:∆Γ:Ι>86Α Β∆9:Α Λ>Ι= <:Χ:Γ6Α>Ο:9 ΕΓ:;:Γ:Χ8:Η.

R!∀!+!(c!,

415 AΑΙ>ΗΗ>Β∆, F., E=ΓΒ6ΧΧ, &. 6Χ9 F. ,Β:ΙΗ (2006). I∋5a−#(∋ Pe+,#,−e∋ce

a∋d P+#ce Se−−#∋! Be∀a/#(.+ #∋ −∀e E.+( a+ea. (886Η>∆Χ6Α )6Ε:Γ 46,

ECB, FΓ6Χ≅;ϑΓΙ.

425 AΧ9ΓΨΗ, #6Κ>:Γ, !:ΓΧ6Χ9∆, ∀<Χ68>∆, %∆Ε:Ο-,6Α>9∆, D6Κ>9 (2004). "−=:

Γ∆Α: ∆; Ι=: ΠΧ6Χ8>6Α ΗΝΗΙ:Β >Χ Ι=: <Γ∆ΛΙ=->ΧΘ6Ι>∆Χ Α>Χ≅", E.+()ea∋ J(.+-

∋a% ( P(%#−#ca% Ec(∋(&2, 20, 941-961.

435 A∆≅>, ∃∆Ηϑ≅: (2001). "(ΕΙ>Β6Α Ε∆Α>8Ν Γ:ΗΕ∆ΧΗ: Ι∆ Γ:Α6Ι>Κ: ΕΓ>8:

8=6Χ<:Η." J(.+∋a% ( M(∋e−a+2 Ec(∋(&#c, 48, 55-80.

445 AΓ6>, &6=Β∆∆9, ∃>ΧΧΛ6ΑΑ, &6ΙΗ, −=∆ϑΓΗ>: ):Ι:Γ ,≅∆<Β6Χ (2004),

ΣCΝ8Α>86Α 6Χ9 86ϑΗ6Α Ε6ΙΙ:ΓΧΗ ∆; >ΧΘ6Ι>∆Χ 6Χ9 D) <Γ∆ΛΙ=Τ, A))%#ed

38

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Ec(∋(&#c,, 1705-1715.

455 AΗ86Γ>, ϑ>9∆. (2003). ")Γ>8:/06<: ,Ι6<<:Γ>Χ< 6Χ9 ):ΓΗ>ΗΙ:Χ8:: A .Χ>-

;Ν>Χ< FΓ6Β:Λ∆Γ≅." J(.+∋a% ( Ec(∋(&#c S.+/e2, 17, 511-40.

465 B6ΓΓ∆, +∆7:ΓΙ #. (1995). "∀ΧΘ6Ι>∆Χ 6Χ9 E8∆Χ∆Β>8 Γ∆ΛΙ=". ∋BE+Λ∆Γ≅-

>Χ< Ε6Ε:Γ 5326.

475 B6ΓΓ∆, +∆7:ΓΙ #. (2001). "!ϑΒ6Χ C6Ε>Ι6Α 6Χ9 Γ∆ΛΙ=." A&e+#ca∋ Ec(-

∋(&#c Re/#e0 91, 12-17.

485 B6ΓΓ∆, +∆7:ΓΙ #. 6Χ9 16Α6->-&6ΓΙ>Χ, 16Κ>:Γ. (1995). E8∆Χ∆Β>8 Γ∆ΛΙ=.

∋2.: &8 Γ6Λ-!>ΑΑ.

495 B:Χ=67>7, #:ΗΗ 6Χ9 F6ΓΒ:Γ, +. E. A. (1994). "∀Χ9:Ι:ΓΒ>Χ68Ν 6Χ9 ∀Χ-

8Γ:6Η>Χ< +:ΙϑΓΧΗ." J(.+∋a% ( Ec(∋(&#c T∀e(+2 91, 12-17.

4105 B:Χ><Χ∆, )>:ΓΕ6∆Α∆. (2004). "(ΕΙ>Β6Α &∆Χ:Ι6ΓΝ )∆Α>8Ν >Χ 6 CϑΓΓ:Χ8Ν

AΓ:6." J(.+∋a% ( I∋−e+∋a−#(∋a% Ec(∋(&#c, 63, 293-320

4115 B∆Χ6ΙΙ>, %ϑ><>. (20026). "EΧ9∆<:Χ∆ϑΗ Γ∆ΛΙ=,

Α∆76Α ∀Χ9:Ι:ΓΒ>Χ68Ν 6Χ9 &∆Χ:Ι6ΓΝ )∆Α>8Ν." ΛΛΛ-

:8∆Χ∆.:8∆Χ∆Β>6.ϑΧ>ΙΧ.>Ι/Χ:Λ/Γ>8:Γ8=:/9>Ε6ΓΙ>Β/Ε6Ε:Γ/Ε6.Ε9; .

4125 B∆Χ6ΙΙ>, %ϑ><>. (20027). ",∆;Ι Γ∆ΛΙ= 6Χ9 Ι=: +∆Α: ∆; &∆Χ:Ι6ΓΝ )∆Α>8Ν

>Χ ,:Α:8Ι>Χ< Ι=: %∆Χ<-+ϑΧ EΦϑ>Α>7Γ>ϑΒ )6Ι=." Q.ade+∋# d# R#ce+ca de%

D#)a+−#&e∋−( d# Sc#e∋3e Ec(∋(&#c∀e "H2&a∋ P. M#∋,∃2", 6.

39

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4135 B∆Ν9, #., %:Κ>Χ:, +., ,Β>Ι=, B., (2001). "∀ΧΘ6Ι>∆Χ 6Χ9 ΠΧ6Χ8>6Α Β6Γ≅:Ι

Ε:Γ;∆ΓΒ6Χ8:". J(.+∋a% ( M(∋e−a+2 Ec(∋(&#c, 47, 221Υ248.

4145 B∆Η:, ∋>Α∆Ν, (2002), Σ∀ΧΘ6Ι>∆Χ, Ι=: 8Γ:9>Ι Β6Γ≅:Ι 6Χ9 :8∆Χ∆Β>8 <Γ∆ΛΙ=Τ,

O1 (+d Ec(∋(&#c Pa)e+,, 54 (2002), 412-434.

4155 BΓϑΧ∆, +. 6Χ9 E6ΗΙ:ΓΑΝ, 0. (1998). "∀ΧΘ6Ι>∆Χ CΓ>Η:Η 6Χ9 %∆Χ<-+ϑΧ

Γ∆ΛΙ=." J(.+∋a% ( M(∋e−a+2 Ec(∋(&#c, 41, 3-26.

4165 BϑΑΑ6Γ9, #. (1999). Σ−:ΗΙ>Χ< Α∆Χ<-ΓϑΧ Β∆Χ:Ι6ΓΝ Χ:ϑΙΓ6Α>ΙΝ ΕΓ∆Ε∆Η>Ι>∆ΧΗ:

Α:ΗΗ∆ΧΗ ;Γ∆Β Ι=: Γ:8:ΧΙ Γ:Η:6Γ8=Τ, Re/#e0, F:9:Γ6Α +:Η:ΓΚ: ∆; ,Ι.%∆ϑ>Η,

57-77.

4175 BϑΓ9:≅>Χ, +. C. ∃., A. −. D:ΧΟ6ϑ, &. 0. ∃:>Α, −. ,>ΙΙ=>Ν∆Ι, 0>ΑΑ:Ι

−. D. (2004). "0=:Χ 9∆:Η >ΧΘ6Ι>∆Χ =ϑΓΙ :8∆Χ∆Β>8 <Γ∆ΛΙ=? D>Ω:Γ:ΧΙ

Χ∆Χ-Α>Χ:6Γ>Ι>:Η ;∆Γ 9>Ω:Γ:ΧΙ 8∆ϑΧΙΓ>:Η." J(.+∋a% ( Mac+(ec(∋(&#c, 26,

519-532.

4185 C=6Β7:ΓΗ, +. . (1988). "AΕΕΑ>:9 )Γ∆9ϑ8Ι>∆Χ AΧ6ΑΝΗ>Η", C6Β7Γ>9<:

.Χ>Κ:ΓΗ>ΙΝ )Γ:ΗΗ: C6Β7Γ>9<:.

4195 C=6Γ>, /., #∆Χ:Η, %. E. 6Χ9 &6Χϑ:ΑΑ>, +. E. (1996). "∀ΧΘ6Ι>∆Χ, Γ∆ΛΙ=

6Χ9 F>Χ6Χ8>6Α ∀ΧΙ:ΓΒ:9>6Ι>∆Χ." Fede+a% Re,e+/e Ba∋∃ ( S−. L(.#, Re/#e0

78, 41-58.

4205 C=Γ>ΗΙ∆Ε∆ϑΑ∆Η, D>Β>ΙΓ> 6Χ9 −Η>∆Χ6Η, E;Ι=ΝΒ>∆Η . (2005). ")Γ∆9ϑ8Ι>Κ-

>ΙΝ <Γ∆ΛΙ= 6Χ9 >ΧΘ6Ι>∆Χ >Χ EϑΓ∆Ε:: EΚ>9:Χ8: ;Γ∆Β Ε6Χ:Α 8∆>ΧΙ:<Γ6Ι>∆Χ

Ι:ΗΙΗ". E&)#+#ca% Ec(∋(&#c, 30, 137-150.

40

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4215 C∆∆Α:Ν, −. F. 6Χ9 )Γ:Η8∆ΙΙ, E. (1995). "E8∆Χ∆Β>8 Γ∆ΛΙ= 6Χ9 BϑΗ>Χ:ΗΗ

CΝ8Α:Η". ∀Χ: C∆∆Α:Ν, −. F. (:9.), F+(∋−#e+, ( B.,#∋e,, C2c%e, Re,ea+c∀.

)Γ>Χ8:Ι∆Χ .Χ>Κ:ΓΗ>ΙΝ )Γ:ΗΗ, )Γ>Χ8:Ι∆Χ, ∋#.

4225 D∆ΙΗ:Ν, &. 6Χ9 ,6ΓΙΓ:, ). D. (2000). "∀ΧΘ6Ι>∆Χ, .Χ8:ΓΙ6>ΧΙΝ 6Χ9 Γ∆ΛΙ=

>Χ 6 C6Η=->Χ-A9Κ6Χ8: E8∆Χ∆ΒΝ." J(.+∋a% ( M(∋e−a+2 Ec(∋(&#c, 45,

631-655.

4235 F6Χ, #. 6Χ9 >?7:ΑΗ, ∀. (1992). "/6Γ>67Α: B6Χ9Λ>9Ι= 6Χ9 %∆86Α %>Χ:6Γ

+:<Γ:ΗΗ>∆Χ ,Β∆∆Ι=:ΓΗ", A∋∋a%, ( S−a−#,−#c,, 20. 2008-2036.

4245 F>Η8=:Γ, ,Ι6ΧΑ:Ν. (1993). "−=: +∆Α: ∆; &68Γ∆:8∆Χ∆Β>8 F68Ι∆ΓΗ >Χ

Γ∆ΛΙ=." J(.+∋a% ( M(∋e−a+2 Ec(∋(&#c, 41, 485-512.

4255 FϑΧ≅, ):Ι:Γ 6Χ9 ∃Γ∆Β:Χ, B:ΙΙ>Χ6. (2005). "∀ΧΘ6Ι>∆Χ 6Χ9 ∀ΧΧ∆Κ6Ι>∆Χ

DΓ>Κ:Χ Γ∆ΛΙ=." 0∆Γ≅>Χ< )6Ε:Γ ,:Γ>:Η >Χ E8∆Χ∆Β>8Η, .Χ>Κ:ΓΗ>ΙΝ ∆;

C∆Α∆<Χ:, 16.

4265 FϑΧ≅, ):Ι:Γ 6Χ9 ∃Γ∆Β:Χ, B:ΙΙ>Χ6. (2006). ",=∆ΓΙ-Ι:ΓΒ ΕΓ>8: Γ><>9>ΙΝ >Χ

6Χ :Χ9∆<:Χ∆ϑΗ <Γ∆ΛΙ= Β∆9:Α: Χ∆Χ-ΗϑΕ:ΓΧ:ϑΙΓ6Α>ΙΝ 6Χ9 6 Χ∆Χ-Κ:ΓΙ>86Α

Α∆Χ<-Ι:ΓΒ )=>ΑΑ>ΕΗ 8ϑΓΚ:." 0∆Γ≅>Χ< )6Ε:Γ ,:Γ>:Η >Χ E8∆Χ∆Β>8Η, .Χ>Κ:Γ-

Η>ΙΝ ∆; C∆Α∆<Χ:, 29.

4275 =∆Η=, A. 6Χ9 )=>ΑΑ>ΕΗ, ,. (1998). "∀ΧΘ6Ι>∆Χ, D>Η>ΧΘ6Ι>∆Χ 6Χ9 Γ∆ΛΙ=."

∀&F 0∆Γ≅>Χ< )6Ε:Γ 98/68, ∀ΧΙ:ΓΧ6Ι>∆Χ6Α &∆Χ:Ι6ΓΝ FϑΧ9.

4285 >ΑΑΒ6Χ, &., !6ΓΓ>Η, &. ∋., &6ΙΝ6Η, %., 2004. ∀ΧΘ6Ι>∆Χ 6Χ9 <Γ∆ΛΙ=:

:ΜΕΑ6>Χ>Χ< Ι=: Χ:<6Ι>Κ: :Ω:8Ι. E&)#+#ca% Ec(∋(&#c, 29, 149-167.

41

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4295 >ΑΑΒ6Χ, &. 6Χ9 ∃:?6≅, &. (2002). "&∆9:ΑΑ>Χ< Ι=: EΩ:8Ι ∆; ∀ΧΘ6Ι>∆Χ:

Γ∆ΛΙ=, %:Κ:ΑΗ 6Χ9 −∆7>Χ." >Χ )Γ∆8::9>Χ<Η ∆; Ι=: 2002 ∋∆ΓΙ= AΒ:Γ>86Χ

,ϑΒΒ:Γ &::Ι>Χ<Η ∆; Ι=: E8∆Χ∆Β:ΙΓ>8 ,∆8>:ΙΝ: &∆Χ:Ν, :9>Ι:9 7Ν D.

%:Κ>Χ: (=ΙΙΕ://ΛΛΛ.9≅Α:Κ>Χ:.8∆Β /ΕΓ∆8::9>Χ<Η/Β∆Χ:Ν.=ΙΒ).

4305 >ΑΑΒ6Χ, &. 6Χ9 ∃:?6≅, &. (2004). "−=: D:Β6Χ9 ;∆Γ B6Χ≅ +:Η:ΓΚ:Η

6Χ9 (Ι=:Γ &∆Χ:Ι6ΓΝ A<<Γ:<6Ι:Η." Ec(∋(&#c I∋∗.#+2 42, 518-533.

4315 >ΑΑΒ6Χ, &. 6Χ9 ∃:?6≅, &. (20056). "C∆ΧΙΓ6ΗΙ>Χ< &∆9:ΑΗ ∆; Ι=: EΩ:8Ι

∆; ∀ΧΘ6Ι>∆Χ ∆Χ Γ∆ΛΙ=." J(.+∋a% ( Ec(∋(&#c S.+/e2, 19, 113-136.

4325 >ΑΑΒ6Χ, &. 6Χ9 ∃:?6≅, &. (20057). "∀ΧΘ6Ι>∆Χ 6Χ9 B6Α6Χ8:9-)6Ι=

Γ∆ΛΙ= Λ>Ι= AΑΙ:ΓΧ6Ι>Κ: )6ΝΒ:ΧΙ &:8=6Χ>ΗΒΗ." Ec(∋(&#c J(.+∋a% 115,

1-25.

4335 >ΑΑΒ6Χ, &., ∃:?6≅, &. 6Χ9 /6Α:ΧΙ>ΧΝ>, A. (1999). "∀ΧΘ6Ι>∆Χ, Γ∆ΛΙ=,

6Χ9 CΓ:9>Ι ,:ΓΚ>8:Η" −Γ6ΧΗ>Ι>∆Χ E8∆Χ∆Β>8Η ,:Γ>:Η 13. />:ΧΧ6: ∀ΧΗΙ>ΙϑΙ:

∆; A9Κ6Χ8:9 ,Ιϑ9>:Η.

4345 ∆ΒΒ:, ). (1993). "&∆Χ:Ν 6Χ9 Γ∆ΛΙ=: +:Κ>Η>Ι:9." J(.+∋a% ( M(∋e-

−a+2 Ec(∋(&#c, 32, 51-77.

4355 Γ6=6Β, %>6Β 6Χ9 ,Χ∆Λ:Γ D:ΧΧ>Η. (2004). "−=: +:6Α EΩ:8ΙΗ ∆Χ &∆Χ:Ν

Γ∆ΛΙ= >Χ DΝΧ6Β>8 :Χ:Γ6Α EΦϑ>Α>7Γ>ϑΒ." ECB0∆Γ≅>Χ< )6Ε:Γ ∋∆ 412,

EϑΓ∆Ε:6Χ C:ΧΙΓ6Α B6Χ≅.

4365 Γ6=6Β, %>6Β 6Χ9 ,Χ∆Λ:Γ D:ΧΧ>Η. (2008). "!ΝΕ:Γ7∆Α>8 9>Η8∆ϑΧΙ>Χ< 6Χ9

Ι=: )=>ΑΑ>ΕΗ CϑΓΚ:", J(.+∋a% ( M(∋e2, C+ed#− a∋d Ba∋∃#∋!, 40, 427-448.

42

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4375 ϑ:ΓΓ:Γ∆, F:9:Γ>8∆ (2006), ΣD∆:Η >ΧΘ6Ι>∆Χ 86ϑΗ: Ε∆∆Γ Α∆Χ<-Ι:ΓΒ <Γ∆ΛΙ=

Ε:Γ;∆ΓΒ6Χ8:?Τ, Ja)a∋ a∋d −∀e W(+%d Ec(∋(&2, 18, 72-89.

4385 ΝΑ;6Η∆Χ, −. (1991). "∀ΧΘ6Ι>∆Χ, Γ∆ΛΙ= 6Χ9 EΜΙ:ΓΧ6Α D:7Ι: 6 />:Λ ∆;

Ι=: %6Χ9Η86Ε:." W(+%d Ec(∋(&2 14, 279-297.

4395 ΝΑ;6Η∆Χ, −. 6Χ9 !:Γ7:ΓΙΗΗ∆Χ, −. (2001). "D∆:Η >ΧΘ6Ι>∆Χ Β6ΙΙ:Γ ;∆Γ

<Γ∆ΛΙ=?" Ja)a∋ a∋d −∀e W(+%d Ec(∋(&2 13, 405 - 428.

4405 ΝΑ;6Η∆Χ, −. 6Χ9 3∆:<6, . (2006). "∋6ΙϑΓ6Α +:Η∆ϑΓ8:Η 6Χ9 E8∆Χ∆Β>8

Γ∆ΛΙ=: −=: +∆Α: ∆; ∀ΧΚ:ΗΙΒ:ΧΙ." W(+%d Ec(∋(&2 29, 1091-1115.

4415 !6ΗΑ6<, #. !. (1998). "&∆Χ:Ι6ΓΝ )∆Α>8Ν, B6Χ≅>Χ< 6Χ9 Γ∆ΛΙ=." Ec(-

∋(&#c I∋∗.#+2 36, 439-450.

4425 !6ΗΑ6<, #., ∃∆∆, #., (1999). F#∋a∋c#a% +e)+e,,#(∋, 4∋a∋c#a% de/e%()&e∋−

a∋d ec(∋(&#c !+(0−∀. F:9:Γ6Α +:Η:ΓΚ: B6Χ≅ ∆; AΙΑ6ΧΙ6 0∆Γ≅>Χ< )6Ε:Γ

99-02.

4435 !:>?9Γ6, B. #. 6Χ9 F. /6Χ 9:Γ )Α∆:< (2002). T∀e F(.∋da−#(∋, ( M(de+∋

Mac+(ec(∋(&#c,. (Μ;∆Γ9 .Χ>Κ:ΓΗ>ΙΝ )Γ:ΗΗ: (Μ;∆Γ9.

4445 !ϑ6Χ<, ∃. 1. D. 6Χ9 3. %>ϑ (2002). ",Ι6<<:Γ:9 ΕΓ>8:-Η:ΙΙ>Χ<, ΗΙ6<<:Γ:9

Λ6<:-Η:ΙΙ>Χ<, 6Χ9 7ϑΗ>Χ:ΗΗ 8Ν8Α: Ε:ΓΗ>ΗΙ:Χ8:." J(.+∋a% ( M(∋e−a+2 Ec(-

∋(&#c, 49, 405Υ433.

4455 ∀Γ:Α6Χ9, ). (1994). "&∆Χ:Ν 6Χ9 Γ∆ΛΙ=: 6Χ AΑΙ:ΓΧ6Ι>Κ: AΕΕΓ∆68=."

A&e+#ca∋ Ec(∋(&#c Re/#e0 55, 1-14.

43

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4465 #∆Χ:Η, C. ∀. (1995). "−>Β: Η:Γ>:Η Ι:ΗΙΗ ∆; :Χ9∆<:Χ∆ϑΗ <Γ∆ΛΙ= Β∆9:ΑΗ."

T∀e Q.a+−e+%2 J(.+∋a% ( Ec(∋(&#c, 110, 2, 495-525.

4475 #ϑ9Η∆Χ, +. 6Χ9 (ΓΕ=6Χ>9:Η, A. (1999). "∀ΧΘ6Ι>∆Χ, /∆Α6Ι>Α>ΙΝ 6Χ9

Γ∆ΛΙ=." I∋−e+∋a−#(∋a% F#∋a∋ce 2, 117-138.

4485 ∃6=Χ, ,. 6Χ9 ,:Χ=69?>, A. (2001). "−=Γ:Η=∆Α9 EΩ:8ΙΗ >Χ Ι=: +:Α6Ι>∆ΧΗ=>Ε

B:ΙΛ::Χ ∀ΧΘ6Ι>∆Χ 6Χ9 Γ∆ΛΙ=." IMF S−a6 Pa)e+, 48, 1-21.

4495 ∃>Β, ,ϑΧ< 6Χ9 0>ΑΑ:ΙΙ, −=∆Β6Η, D. (2000), Σ∀Η Ι=: Χ:<6Ι>Κ: 8∆ΓΓ:Α6Ι>∆Χ

7:ΙΛ::Χ >ΧΘ6Ι>∆Χ 6Χ9 <Γ∆ΛΙ= Γ:6Α? AΧ 6Χ6ΑΝΗ>Η ∆; Ι=: :Ω:8ΙΗ ∆; Ι=: ∆>Α

ΗϑΕΕΑΝ Η=∆8≅ΗΤ, AΕΕΑ>:9 E8∆Χ∆Β>8Η %:ΙΙ:ΓΗ, 7, 141-147.

4505 ∃>Χ<, +∆7:ΓΙ . 6Χ9 %:Κ>Χ:, +∆ΗΗ. (1993). "F>Χ6Χ8: 6Χ9 Γ∆ΛΙ=:

,8=ϑΒΕ:Ι:Γ &><=Ι B: +><=Ι." Q.a+−e+%2 J(.+∋a% ( Ec(∋(&#c, 108, 717-

737.

4515 %:Κ>Χ:, +. (1997). "F>Χ6Χ8>6Α D:Κ:Α∆ΕΒ:ΧΙ 6Χ9 E8∆Χ∆Β>8 Γ∆ΛΙ=:

/>:ΛΗ 6Χ9 A<:Χ96." J(.+∋a% ( Ec(∋(&#c L#−e+a−.+e 35, 688-726.

4525 %>, ∗> 6Χ9 +68>Χ:, #. ,. (2007). N(∋)a+a&e−+#c Ec(∋(&e−+#c,. )Γ>Χ8:Ι∆Χ

.Χ>Κ:ΓΗ>ΙΝ )Γ:ΗΗ: )Γ>Χ8:Ι∆Χ.

4535 &>Χ;∆Γ9, ). 6Χ9 )::Α, D. (2004). "C6ΑΚ∆ C∆ΧΙΓ68ΙΗ: A CΓ>Ι>Φϑ:" CE)+

D>Η8ϑΗΗ>∆Χ )6Ε:ΓΗ 4288.

4545 &>Χ∆, ∃. 6Χ9 A. ,=>76Ι6 (1995). "&∆Χ:Ι6ΓΝ )∆Α>8Ν, (Κ:ΓΑ6ΕΕ>Χ< :Χ-

:Γ6Ι>∆Χ, 6Χ9 )6ΙΙ:ΓΧΗ ∆; Γ∆ΛΙ=." Ec(∋(&#ca 62, 179-194.

44

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4555 )66Α, B:6ΙΓ>Μ 6Χ9 ,Β>Ι=, BΓϑ8:. (2000). "−=: ,ϑ7-(ΕΙ>Β6Α>ΙΝ ∆; Ι=:

FΓ>:9Β6Χ +ϑΑ: 6Χ9 Ι=: (ΕΙ>ΒϑΒ ∗ϑ6ΧΙ>ΙΝ ∆; &∆Χ:Ν." (Β6ΧϑΗ8Γ>ΕΙ,

,Ι6Χ;∆Γ9 .Χ>Κ:ΓΗ>ΙΝ).

4565 )6<6Χ, A. 6Χ9 .ΑΑ6=, A. (1999). "∋∆ΧΕ6Γ6Β:ΙΓ>8 E8∆Χ∆Β:ΙΓ>8Η". C6Β-

7Γ>9<: .Χ>Κ:ΓΗ>ΙΝ )Γ:ΗΗ.

4575 )6Γ≅, ,. (2003). ",:Β>Ε6Γ6Β:ΙΓ>8 >ΧΗΙΓϑΒ:ΧΙ6Α Κ6Γ>67Α: :ΗΙ>Β6Ι>∆Χ".

J(.+∋a% ( Ec(∋(&e−+#c, 112, 381-399.

4585 )∆ΑΑ>Χ, +∆7:ΓΙ 6Χ9 3=ϑ, AΧ9∆Χ< (2006). "∀ΧΘ6Ι>∆Χ 6Χ9 :8∆Χ∆Β>8 <Γ∆ΛΙ=:

6 8Γ∆ΗΗ-8∆ϑΧΙΓΝ Χ∆ΧΑ>Χ:6Γ 6Χ6ΑΝΗ>Η." J(.+∋a% ( P(,−-Ke2∋e,#a∋ Ec(-

∋(&#c,, 28, 593-614.

4595 +6ΒΗ:Ν # (1969) −:ΗΙ ;∆Γ ΗΕ:8>Π86Ι>∆Χ :ΓΓ∆ΓΗ >Χ 8Α6ΗΗ>86Α Α>Χ:6Γ Α:6ΗΙ

ΗΦϑ6Γ:Η Γ:<Γ:ΗΗ>∆Χ 6Χ6ΑΝΗ>Η. # +∆Ν6Α ,Ι6Ι ,∆8 (,:Γ>:Η B) 31: 350-371

4605 ,6Γ:Α, &. (1996). "∋∆ΧΑ>Χ:6Γ EΩ:8ΙΗ ∆; ∀ΧΘ6Ι>∆Χ ∆Χ E8∆Χ∆Β>8 Γ∆ΛΙ=."

IMF S−a6 Pa)e+, 43, 199-215.

4615 ,Ι∆8≅Β6Χ, A. (1981). "AΧΙ>8>Ε6Ι:9 ∀ΧΘ6Ι>∆Χ 6Χ9 −=: C6Ε>Ι6Α ,Ι∆8≅ >Χ

6 C6Η=->Χ-A9Κ6Χ8: E8∆Χ∆ΒΝ." J(.+∋a% ( M(∋e−a+2 Ec(∋(&#c, 8, 387-

393.

4625 −:ΒΕΑ:, #∆Χ6Ι=6Χ. (2000). "∀ΧΘ6Ι>∆Χ 6Χ9 Γ∆ΛΙ=: ,Ι∆Γ>:Η ,=∆ΓΙ 6Χ9

−6ΑΑ." J(.+∋a% ( Ec(∋(&#c S.+/e2, 14, 395-426.

45

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4635 −=>ΓΑΛ6ΑΑ, A. ). 6Χ9 B6ΓΙ∆Χ, C. A. (1971). "∀ΧΘ6Ι>∆Χ 6Χ9 Γ∆ΛΙ=: −=:

∀ΧΙ:ΓΧ6Ι>∆Χ6Α EΚ>9:Χ8:." Ba∋ca Na3#(∋a%e de% La/(+( Q.a+−e+%2 Re/#e0

98, 263-275.

4645 −∆7>Χ, #6Β:Η. (1965). "&∆Χ:Ν 6Χ9 E8∆Χ∆Β>8 Γ∆ΛΙ=." Ec(∋(&e−+#ca

33, 671-684.

4655 /6∆Χ6, AΧ9Γ:6 6Χ9 ,8=>6Κ∆ ,Ι:;6Χ∆. (2007). "∋∆Χ-)6Γ6Β:ΙΓ>8 6Χ9 ,:Β>-

Ε6Γ6Β:ΙΓ>8 EΚ>9:Χ8: ∆Χ Ι=: %∆Χ<-+ϑΧ EΩ:8ΙΗ ∆; ∀ΧΘ6Ι>∆Χ ∆Χ Γ∆ΛΙ=."

Ec(∋(&#c, Le−−e+, 94, 452-458.

4665 06Χ<, ). 6Χ9 2>Ε, C. ∃. (1992). "AΑΙ:ΓΧ6Ι>Κ: AΕΕΓ∆68=:Η Ι∆ &∆Χ:Ν 6Χ9

Γ∆ΛΙ=." J(.+∋a% ( M(∋e2, C+ed#− a∋d Ba∋∃#∋! 24, 553-562.

4675 0∆ΑΩ, E9Λ6Γ9 ∋. (2003), 0=6ΙΡΗ 7:=>Χ9 Ι=: Γ>Η: >Χ ΕΓ∆ΠΙ67>Α>ΙΝ >Χ Ι=:

., >Χ Ι=: 1980Η 6Χ9 >Χ Ι=: 1990Η?, C6Β7Γ>9<: #∆ϑΓΧ6Α ∆; E8∆Χ∆Β>8Η,

27, 479-499.

4685 0∆∆9;∆Γ9, &>8=6:Α (1998). "D∆>Χ< 0>Ι=∆ϑΙ &∆Χ:Ν: C∆ΧΙΓ∆ΑΑ>Χ< ∀ΧΘ6-

Ι>∆Χ >Χ 6 )∆ΗΙ-&∆Χ:Ι6ΓΝ 0∆ΓΑ9." Re/#e0 ( Ec(∋(&#c D2∋a&#c, 1, 173-

219.

4695 3=6Χ<, #ϑΧΜ>, (2000), Σ∀ΧΘ6Ι>∆Χ 6Χ9 Γ∆ΛΙ=: ):8ϑΧ>6ΓΝ −Γ6ΧΗ68Ι>∆ΧΗ

C∆ΗΙΗ 6Χ9 ∗ϑ6Α>Ι6Ι>Κ: EΦϑ>Κ6Α:Χ8:Τ, J(.+∋a% ( M(∋e2 C+ed#− a∋d Ba∋∃-

#∋!, 32, ΕΕ. 1-12.

46

Page 50: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

9 A∗∗!( %c!,

A T∃! S)&.−%)( )∀ T∃! M) !&

+:86ΑΑ Ι=6Ι Ι=: Β68Γ∆:8∆Χ∆Β>8 Β∆9:Α 8∆ΧΗ>ΗΙΗ ∆; Ι=: :Φϑ6Ι>∆ΧΗ: (1), (2), (4),

(5), (6), (9), (12), (19), (22), (23), (24), 6Χ9 (25).

%:Ι ϑΗ ΠΓΗΙ 8∆ΧΗ>9:Γ Ι=: ΕΓ>8: >Χ9:Μ (2) >Χ ΗΙ:69Ν ΗΙ6Ι:. >Κ:Χ Ι=: 67Η:Χ8:

∆; ΕΓ>8: ΗΙ6<<:Γ>Χ< 6Χ9 9ϑ: Ι∆ ΗΝΒΒ:ΙΓΝ 6ΑΑ Ι=: ΠΓΒΗ Λ>ΑΑ 8=6Γ<: Ι=: Η6Β:

ΕΓ>8: 0ft. ,∆ (2) Λ>ΑΑ 7:8∆Β::

0f0= 1

BΝ Ηϑ7ΗΙ>ΙϑΙ>Χ< Ι=: ΗΙ:69Ν ΗΙ6Ι: ΕΓ>8: >Χ9:Μ >ΧΙ∆ (9) >Ι >Η Ε∆ΗΗ>7Α: Ι∆ ∆7Ι6>Χ

Ι=6Ι >Χ ΗΙ:69Ν ΗΙ6Ι: Ι=: Β6Γ<>Χ6Α 8∆ΗΙ >Η Ι=: Η6Β: 68Γ∆ΗΗ 6ΑΑ Ι=: ΠΓΒΗ 6Χ9

Ι=6Ι >Χ >ΙΗ ΙϑΓΧ >Ι 86Χ 7: Ηϑ7ΗΙ>ΙϑΙ:9, Ι∆<:Ι=:Γ Λ>Ι= (8), >ΧΙ∆ Ι=: ΠΓΗΙ ∆Γ9:Γ

8∆Χ9>Ι>∆ΧΗ ;∆Γ 86Ε>Ι6Α 6Χ9 Ι=: 8∆ΒΕ∆Η>Ι: Α67∆ϑΓ >ΧΕϑΙ Ι∆ ∆7Ι6>Χ

5

0=

p 1

p6

.(39)

1 =p 1

p(1 )

6

+(40)

!6Κ>Χ< >Χ Β>Χ9 Ι=: :Φϑ6Ι>∆Χ ;∆Γ 86Ε>Ι6Α Γ:ΒϑΧ:Γ6Ι>∆Χ, >Ι >Η Ε∆ΗΗ>7Α: Ι∆

∆7Ι6>Χ Ι=: Η∆ΑϑΙ>∆Χ ;∆Γ Ι=: ΗΙ:69Ν ΗΙ6Ι: <Γ∆ΛΙ= Γ6Ι: ∆; Ι=: :8∆Χ∆ΒΝ, 7Ν ϑΗ>Χ<

(22) 6Χ9 (23):

47

Page 51: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

=

1 +

p 1

p(1 )

6

+

(41)

C∆ΧΗ>9:Γ Ι=: ΕΓ∆9ϑ8Ι>∆Χ ;ϑΧ8Ι>∆Χ >Χ ΗΙ:69Ν ΗΙ6Ι:, Ι=: ∆ϑΙΕϑΙ-86Ε>Ι6Α Γ6Ι>∆

>Η <>Κ:Χ 7Ν:

6

+= (.) (42)

Λ=:Γ: Ι=: ΠΓΒ Ηϑ7Η8Γ>ΕΙ Λ6Η 9Γ∆ΕΕ:9 7:86ϑΗ: 6ΑΑ Ι=: ΠΓΒΗ 6Γ: ΗΝΒΒ:ΙΓ>8.

,ϑ7ΗΙ>ΙϑΙ>Χ< (42) >ΧΙ∆ (41), >Ι >Η Ε∆ΗΗ>7Α: Ι∆ ΠΧ9 Ι=6Ι >Χ ΗΙ:69Ν ΗΙ6Ι: Ι=:

Γ:6Α <Γ∆ΛΙ= Γ6Ι: ∆; Ι=: :8∆Χ∆ΒΝ 9:Ε:Χ9Η ∆Χ Ι=: 8∆ΒΕ∆Η>Ι: Α67∆ϑΓ >ΧΕϑΙ:

=

1 +

p 1

p(1 ) .

(43)

−=:Γ:;∆Γ: >Ι >Η Χ:8:ΗΗ6ΓΝ Ι∆ ΠΧ9 Ι=: ΗΙ:69Ν ΗΙ6Ι: Α:Κ:Α ∆; Ι=: 8∆ΒΕ∆Η>Ι:

Α67∆ϑΓ >ΧΕϑΙ >Χ ∆Γ9:Γ Ι∆ Η∆ΑΚ: Ι=: Β∆9:Α. AΗ 6 ΠΓΗΙ ΗΙ:Ε Α:Ι ϑΗ ;∆8ϑΗ ∆Χ (24).

AΗ Η=∆Λ:9 >Χ Γ6=6Β 6Χ9 ,Χ∆Λ:Γ (2004), Ι=: Α67∆ϑΓ ΗϑΕΕΑΝ ∆; :68= 8∆=∆ΓΙ

86Χ 7: Γ:9ϑ8:9 Ι∆ Ι=: Α67∆ϑΓ ΗϑΕΕΑΝ ∆; 8∆=∆ΓΙ Ο:Γ∆ Ι>Β:Η 6 Ι:ΓΒ 688∆ϑΧΙ>Χ<

;∆Γ Β∆Χ:Ν <Γ∆ΛΙ=:

.0

.j= wj

−=:Γ:;∆Γ:, 8∆ΧΗ>9:Γ>Χ< Ι=6Ι t;t+j = j Ι=: Α:;Ι =6Χ9 Η>9: ∆; (24), 7:8∆Β:Η

>Χ ΗΙ:69Ν ΗΙ6Ι:N1X

j=0

.j;t+jj = .0

1 NNw

1 w(44)

48

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AΗ ;6Γ 6Η Ι=: Γ><=Ι =6Χ9 Η>9: ∆; (24) >Η 8∆Χ8:ΓΧ:9, 9>Κ>9: 7∆Ι= Ι=: ΧϑΒ:Γ-

6Ι∆Γ 6Χ9 Ι=: 9:Χ∆Β>Χ6Ι∆Γ 7Ν 6t+j:

w 1

w&

N1X

j=0

j5j;t0t+jj

.j;t+j∃t+j

=w 1

w&

N1X

j=0

jwj;tpt+jj

nj;t+j t+j

ct+j t+j

(45)

F∆8ϑΗ>Χ< ∆Χ Ι=: ΧϑΒ:Γ6Ι∆Γ, >Χ ΗΙ:69Ν ΗΙ6Ι:, ∆Χ: =6Η:

N1X

j=1

j5j;t0t+jj

.j;t+j6t+j

=5

0

.06

N1X

j=1

j(w1)j (46)

Λ=:Γ: 5 >Η Ι=: ΗΙ:69Ν ΗΙ6Ι: Χ∆Β>Χ6Α Λ6<: Γ6Ι: ∆; 8∆=∆ΓΙ Ο:Γ∆.

C∆ΧΗ>9:Γ Ι=6Ι, 9ϑ: Ι∆ (11) 6Χ9 Ι∆ (39) ∆Χ: =6Η:

5.

06=

N1X

j=0

wj;tj

nj;tN

06=5.006N

N1X

j=0

(w1)j =5.006N

1 N(w1)

1 (w1)=p 1

p

=:Χ8:

5.006

= N1 (w1)

1 N(w1)p 1

p

,∆ >Ι >Η Ε∆ΗΗ>7Α: Ι∆ Γ:ΛΓ>Ι: (46) 6Η ;∆ΑΑ∆ΛΗ

N1X

j=1

j5j;t0t+jj

.j;t+j6

= N

1 (w1)

1 N(w1)p 1

p1 NN(w1)

1 (w1)(47)

C∆ΧΗ>9:Γ>Χ< Ι=: 9:Χ∆Β>Χ6Ι∆Γ ∆; Ι=: Γ><=Ι =6Χ9 Η>9: ∆; (45), 7∆Ι= 8∆Χ-

49

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ΗϑΒΕΙ>∆Χ 6Χ9 ∆ϑΙΕϑΙ 6Γ: <Γ∆Λ>Χ< 6Ι Ι=: Η6Β: Γ6Ι: Η∆ Ι=:>Γ Γ6Ι>∆ >Η 6 8∆ΧΗΙ6ΧΙ

>Χ ΗΙ:69Ν ΗΙ6Ι: 6Χ9 ∆Χ: 86Χ :ΜΙΓ68Ι >Ι ;Γ∆Β Ι=: ΗϑΒ. FϑΓΙ=:ΓΒ∆Γ:, <>Κ:Χ Ι=:

6<<Γ:<6Ι: 7ϑ9<:Ι 8∆ΧΗΙΓ6>ΧΙ

6= 1

)

6(48)

C∆ΧΗ>9:Γ Ι=6Ι

)

6=)

+

+

6(49)

−=: 86Ε>Ι6Α-∆ϑΙΕϑΙ Γ6Ι>∆ 86Χ 7: Γ:8∆Κ:Γ:9 ;Γ∆Β Ι=: ΠΓΗΙ ∆Γ9:Γ 8∆Χ9>Ι>∆ΧΗ

;∆Γ 8∆ΗΙ Β>Χ>Β>Ο6Ι>∆Χ (1 = p1

p(1 )

k); Λ=:Γ:6Η Ι=: >ΧΚ:ΗΙΒ:ΧΙ-86Ε>Ι6Α

Γ6Ι>∆ ;Γ∆Β Ι=: Α6Λ ∆; Β∆Ι>∆Χ ∆; 86Ε>Ι6Α:

)

+= 1 +

,∆ 7Ν Ηϑ7ΗΙ>ΙϑΙ>Χ< Ι=: 86Ε>Ι6Α-∆ϑΙΕϑΙ Γ6Ι>∆ 6Χ9 Ι=: >ΧΚ:ΗΙΒ:ΧΙ-86Ε>Ι6Α

Γ6Ι>∆ >ΧΙ∆ (49) 6Χ9 Ι6≅>Χ< >ΧΙ∆ 8∆ΧΗ>9:Γ6Ι>∆Χ (40) 6Χ9 (41), >Ι >Η Ε∆ΗΗ>7Α: Ι∆

<:Ι:

)

6=)

+

+

6= [(1 ) + 1 + 1]

p 1

p(1 )

1

1

,>ΒΕΑ>;Ν>Χ< 6Χ9 Ηϑ7ΗΙ>ΙϑΙ>Χ< ;∆Γ 1 7Ν Ι6≅>Χ< >ΧΙ∆ 8∆ΧΗ>9:Γ6Ι>∆Χ (40) 6Χ9

50

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(42), >Ι >Η Ε∆ΗΗ>7Α: Ι∆ Ε:Γ;∆ΓΒ ΙΛ∆ ;ϑΓΙ=:Γ ΗΙ:ΕΗ

)

6= (1 ) ( 1)

p 1

p(1 )

1

1+

p 1

p(1 )

= (1 ) ( 1) (.) + p 1

p(1 ) (50)

−=:Γ:;∆Γ: Ι=: >ΧΚ:ΗΙΒ:ΧΙ-∆ϑΙΕϑΙ Γ6Ι>∆ 9:Ε:Χ9Η ∆Χ Α67∆Γ >Χ :ς 8>:Χ8Ν ϑΧ>ΙΗ.

∀Ι >Η Ι>Β: Ι∆ Ι6≅: ΗΙ∆8≅. BΝ 8∆ΧΗ>9:Γ>Χ< (44), (45), (47),(48) 6Χ9 (50), >Ι

>Η Ε∆ΗΗ>7Α: Ι∆ Γ:ΛΓ>Ι: (24) 6Η:

.0 =1 w

1 NNw

N 1(w1)

1N(w1)p1

p 1

NN(w1)

1(w1)

#1 (1 ) ( 1) (.) p1

p(1 )

∃ w 1w&

(51)

∀Ι >Η Λ∆ΓΙ= Χ∆Ι>Χ< Ι=6Ι Ι=: Α67∆ϑΓ ΗϑΕΕΑΝ ∆; 8∆=∆ΓΙ Ο:Γ∆ 9:Ε:Χ9Η ∆Χ Ι=:

8∆ΒΕ∆Η>Ι: Α67∆ϑΓ >ΧΕϑΙ Κ>6 Ι=: >ΧΚ:ΗΙΒ:ΧΙ Η=6Γ: ∆; ∆ϑΙΕϑΙ.

∀Χ ΗΙ:69Ν ΗΙ6Ι:, Ι=: 9:Β6Χ9 ;∆Γ Ι=: Α67∆ϑΓ >ΧΕϑΙ ∆; 8∆=∆ΓΙ Ο:Γ∆ >Η :Φϑ6Α

Ι∆:

.0 =

5

5

w. (52)

Λ=:Γ:, 8∆ΧΗ>9:Γ>Χ< (12),

5

5=

1

N

1 N(w1)

1 w1

1w1

BΝ Ηϑ7ΗΙ>ΙϑΙ>Χ< (51) >ΧΙ∆ (52), >Ι >Η Ε∆ΗΗ>7Α: Ι∆ ∆7Ι6>Χ:

51

Page 55: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

. =

N 1(w1)

1N(w1)p1

p1

NN(w1)

1(w1)

#1 (1 ) ( 1) (.) p1

p(1 )

∃ 1 w

1 (w)Nw 1

w&

1

N

1 N(w1)

1 w1

ww1

(53)

,>ΒΕΑ>;Ν>Χ<

.

1

p 1

p(1 )

(1 ) ( 1) .1 (54)

N1 (w1)

1 N(w1)p 1

p1 NN(w1)

1 (w1)

1 w

1 (w)Nw 1

w&

1

N

1 N(w1)

1 w1

ww1

= 0

(55)

,ϑΕΕ∆Η:, 6Η 8ϑΗΙ∆Β6ΓΝ, Ι=6Ι = 23:

.

1

p 1

p(1 )

(1 ) ( 1) .

13 (56)

N1 (w1)

1 N(w1)p 1

p1 NN(w1)

1 (w1)

1 w

1 (w)Nw 1

w&

1

N

1 N(w1)

1 w1

ww1

= 0

(57)

D:ΠΧ::

52

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A : =

1

p 1

p(1 )

C : = (1 ) ( 1)

B : =

N1 (w1)

1 N(w1)p 1

p1 NN(w1)

1 (w1)

1 w

1 (w)Nw 1

w&

1

N

1 N(w1)

1 w1

ww1

AΗ 6 8∆ΧΗ:Φϑ:Χ8: Ι=: Η∆ΑϑΙ>∆Χ ;∆Γ Α67∆ϑΓ >Χ :ς 8>:Χ8Ν ϑΧ>ΙΗ >Η <>Κ:Χ 7Ν

Ι=: Γ∆∆ΙΗ ∆; Ι=: ;∆ΑΑ∆Λ>Χ< Ε∆ΑΝΧ∆Β>6Α 15

A3.3 3A2B.2 +3AB2 C3

.B3 = 0

0>Ι= . >Χ Β>Χ9, ∆Χ: 86Χ :6Η>ΑΝ 8∆ΒΕϑΙ: ;Γ∆Β (43).

B O∗−%∋a& M)(!−a+2 P)&%c2

AΗ ΗΙ6Ι:9 67∆Κ: Λ:Α;6Γ: >Η <>Κ:Χ 7Ν:

W =1X

j=0

j (logCt+j &.jt+j)

∃::Ε>Χ< >Χ Β>Χ9 (31) 6Χ9 Ι=6Ι Ct+j = ∃t j; >Ι >Η Ε∆ΗΗ>7Α: Ι∆ ΛΓ>Ι: ;∆Γ Ι=:

15−∆ Η:: Ι=>Η, 8∆ΧΗ>9:Γ Ι=6Ι

nAB = Cn1

3

(nAB)3= C

3n

A;Ι:Γ :ΜΕ6Χ9>Χ< Ι=: Ε∆ΑΝΧ∆Β>6Α Ι∆ Ι=: Γ><=Ι ∆; Ι=: :Φϑ6Ι>∆Χ 6Χ9 Ηϑ7ΙΓ68Ι>Χ< ;Γ∆Β 7∆Ι=Ι:ΓΒΗ C3n, >Ι >Η Ε∆ΗΗ>7Α: Ι∆ ∆7Ι6>Χ (27) :

53

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ΗΙ:69Ν ΗΙ6Ι::

W =

1X

j=0

jlog j &.0

w(j) + log ∃:

(58)

>Κ:Χ Ι=: Γ:8ϑΓΗ>Κ: ΗΙΓϑ8ΙϑΓ: ∆; Α67∆ϑΓ 8∆ΧΙΓ68ΙΗ, ;∆ΑΑ∆Λ>Χ< Γ6=6Β 6Χ9

,Χ∆Λ:Γ (2004, Ε. 23), (58) 86Χ 7: ΛΓ>ΙΙ:Χ, 8∆ΧΗ>9:Γ>Χ< Ι=: ΗΙ:69Ν ΗΙ6Ι:, 6Η

W =

1X

j=0

N1X

l=0

l log l

1X

j=0

N1X

l=0

l&.0w(l) +

1X

j=0

N1X

l=0

l log ∃:

&6≅>Χ< :ΜΕΑ>8>Ι Ι=: Γ:Α6Ι>∆Χ ∆; Ι=: >Χ>Ι>6Α Α:Κ:Α ∆; 8∆ΧΗϑΒΕΙ>∆Χ Λ>Ι= Ι=:

>Χ>Ι>6Α Α:Κ:Α ∆; Ι=: 86Ε>Ι6Α ΗΙ∆8≅ ∆Χ: =6Η

∃ = 6 )

∃ = 6 ( 1 + ) +

∃ = (. + 1 ) +

B:>Χ< Ι=: Κ6Αϑ: ∆; + <>Κ:Χ 7Ν Ι=: >Χ>Ι>6Α 8∆Χ9>Ι>∆Χ, ∆Χ: ∆7Ι6>ΧΗ Λ>Ι=∆ϑΙ Α∆ΗΗ

∆; <:Χ:Γ6Α>ΙΝ:

W 0 =1

1 Nlog

N1X

l=0

l,&.01

1 N1 NwN

1 w+

1

1 Nlog (. + 1 )+∃/.23:

C∆ΧΗ>9:Γ Ι=6Ι

54

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(1 )

N1X

l=0

l, = + 2 + :::+ N1 + N NN

8∆ΧΗ:Φϑ:ΧΙΑΝ

N1X

j=0

l, = 1 N

(1 )2NN

1

6Χ9 Ι=:Γ:;∆Γ:

W 0 =1

1 N

1 N

(1 )2NN

1

log &.0

1

1 N1 NwN

1 w+

1

1 Nlog (. + 1 )

55

Page 59: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

Figure 1 – The effect of θp on real growth

6 7 8 9 10 11 12 13 14 15 161.5

2

2.5

3

3.5

4

4.5

5

θp

perc

enta

ge rea

l gro

wth

rat

e

Figure 2 – The effect of θw on real growth

2 3 4 5 6 7 8 9 10 11 125

6

7

8

9

10

11

12

θw

perc

enta

ge rea

l gro

wth

rat

e

Page 60: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

Figure 3 – The effect of money growth on real growth

0 5 10 15 20 25 304.36

4.37

4.38

4.39

4.4

4.41

4.42

4.43

4.44

percentage money growth rate

perc

enta

ge rea

l gro

wth

rat

e

Figure 4 – The effect of money growth on real growth for different values of θp

0 5 10 15 20 25 304.36

4.37

4.38

4.39

4.4

4.41

4.42

4.43

4.44

percentage money growth rate

perc

enta

ge rea

l gro

wth

rat

e

Benchmark modelAfter a 0.1% increase in θp

Page 61: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

Figure 5 – The effect of money growth on real growth for different values of θw

0 2 4 6 8 10 12

4.416

4.418

4.42

4.422

4.424

4.426

4.428

4.43

4.432

4.434

4.436

percentage money growth rate

perc

enta

ge rea

l gro

wth

rat

e

Benchmark modelAfter a 0.1% increase in θw

Figure 6 – The effect of money growth on welfare

0 5 10 15 20 25 300

0.005

0.01

0.015

0.02

0.025

0.03

0.035

percentage money growth rate

wel

fare

(% d

evia

tion

from

the

leve

l of w

elfa

re w

ith fl

exib

le p

rices

)

Page 62: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

Figure 7 – The effect of money growth on real growth and welfare for generalized preferences

(a)

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

4.5

percentage money growth rate

perc

enta

ge rea

l gro

wth

rat

e

Benchmark modelφ=2.5

(b)

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

percentage money growth rate

wel

fare

(% d

evia

tion

from

the

leve

l of w

elfa

re w

ith fl

exib

le p

rices

)

Benchmark modelφ=2.5

The intertemporal elasticity of substitution for working time is set equal to 5.

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Figure 8 – The effect of money growth on real growth and on wage contract length with endogenous labour contract duration

(a)

0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

percentage money growth rate

perc

enta

ge rea

l gro

wth

rat

e

F=0.1F=1

(b)

0 1 2 3 4 5 6 7 830

35

40

45

50

55

percentage money growth rate

leng

th o

f w

age

cont

ract

F=0.1F=1

The intertemporal elasticity of substitution for working time is set equal to 5.

Page 64: Inflation and Growth in the Long Run: A New Keynesian ... · Working Paper Series Department of Economics University of Verona Inflation and Growth in the Long Run: A New Keynesian

Table 1 – Model selection criteria Whole sample: specification I Whole sample: specification II Polynomial order Polynomial order

Criteria 1 2 3 4 1 2 3 4 Mallow’s

CL 4.0004 3.9954 3.9902 3.9936 2.7151 2.7023 2.7130 2.7265

Generalized cross-

validation 4.5821 4.5882 4.5942 4.6103 3.2551 3.2589 3.2914 3.3280

Leave-one-out cross-validation

1.2893 1.2911 1.2889 1.2901 428.8411 429.0025 431.2143 434.8971

Akaike Information

Criterion 1987.257 1987.294 1987.308 1988.337 929.1067 928.6779 930.2978 932.1711

Schwartz Information

Criterion 2413.22 2417.393 2421.543 2426.707 1140.4 1143.435 1148.519 1153.856

Table 2 - Coefficient estimates and t-statistics of the control variables

Specification I Specification II OLS 2SLS OLS 2SLS

Inflation -0.09* -0.09 -0.16* -0.23* t-statistics -4.76 -1.23 -4.51 -2.15 GDP per capita -4.74* -4.76* -6.46* -6.57* t-statistics -8.25 -6.50 -7.01 -6.95 Population Growth 1.50* 1.50* 0.64 0.70* t-statistics 9.12 8.93 1.92 2.02 Investment/GDP 0.20* 0.20* 0.30* 0.31* t-statistics 7.73 7.17 8.15 8.06 Government consumption/GDP -0.03 -0.03 -0.02 0.01 t-statistics -1.30 -1.00 -0.70 0.02 Education 0.19 0.19 0.11 0.08 t-statistics 0.96 0.85 0.46 0.33 Terms of trade growth - - 0.11* 0.11* t-statistics - - 3.07 3.11 Terms of trade standard deviation - - 0.89* 0.86* t-statistics - - 2.22 2.14 First-stage F-statistic (p-value) - 0.00 - 0.00 Hausman test (p-value) - 1.00 - 1.00 Adjusted R2 0.56 0.56 0.60 0.60 Observations 462 462 236 236

*: significant at the 5% level. The models also include a set of country and time-period dummies.

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Table 3 – Robustness checks for model selection criteria Developed countries: specification I Developed countries: specification II Polynomial order Polynomial order

Criteria 1 2 3 4 1 2 3 4 Mallow’s

CL 1.4604 1.4246 1.3982 1.3856 1.3996 1.4060 1.4038 1.3957

Generalized cross-

validation 1.7016 1.6739 1.6571 1.6568 1.7071 1.7339 1.7509 1.7611

Leave-one-out cross-validation

156.7271 153.7707 150.4885 149.8273 124.3379 126.8406 126.6606 127.6177

Akaike Information

Criterion 501.9681 498.8436 496.6573 495.9556 436.5977 437.9164 438.428 438.3876

Schwartz Information

Criterion 610.8278 610.7271 611.5647 613.8869 540.6503 544.8593 548.2613 551.1112

Developing countries: specification I Developing countries: specification II Polynomial order Polynomial order

Criteria 1 2 3 4 1 2 3 4 Mallow’s

CL 5.1022 5.1224 5.1116 5.1242 4.2152 4.2274 4.2635 4.2776

Generalized cross-

validation 6.0865 6.1377 6.1522 6.1953 6.0263 6.1554 6.3259 6.4710

Leave-one-out cross-validation

1060.6 1067.3 1063.4 1067.0 277.4053 281.1286 284.6060 287.3215

Akaike Information

Criterion 1415.244 1417.148 1417.180 1418.631 490.3006 491.4675 493.2615 932.1711

Schwartz Information

Criterion 1710.434 1716.074 1719.842 1725.030 598.3198 602.1872 606.6817 610.6177

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Table 4 – Model selection criteria using 8 years averages

Developed countries: specification I Polynomial order

Criteria 1 2 3 4 Mallow’s

CL 0.6674 0.6745 0.6748 0.6813

Generalized cross-

validation 0.9851 1.0202 1.0466 1.0849

Leave-one-out cross-validation

34.6133 35.7789 36.2510 37.1464

Akaike Information

Criterion 228.6061 230.3979 231.3121 1418.631

Schwartz Information

Criterion 309.9811 314.2387 317.6189 321.827

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Figure 9 – Semiparametric estimation of the effect of inflation on economic growth

Specification II: Whole Sample

0 5 10 15 20 25 3054

55

56

57

58

59

60

π

g(π)

Specification II: Developed countries

0 5 10 15 20 2544

45

46

47

48

49

50

51

π

g(π)

Specification II: Developing countries

0 5 10 15 20 25 3042

43

44

45

46

47

48

49

π

g(π)

Specification I: Developed countries 8 years averages

0 2 4 6 8 10 12 14 16 18 2041

42

43

44

45

46

47

48

π

g(π)

Note: π is the inflation rate, g(π) is the partial effect of inflation on real economic growth. The other regressors included in the estimated model are showed in Table 5. Dotted lines mark 95% confidence intervals.

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Table 5 – Coefficient estimates and t-statistics of the control variables of the semi-parametric estimation of the inflation-growth nexus

Specification II Specification I

All Developed Developing Developed / 8 years averages

GDP per cap -6.44* -5.54* -6.02* -4.96* t-statistics -7.07 -4.27 -4.34 -3.92 Population Growth 0.65 1.09* 0.71 0.94 t-statistics 1.97 2.44 1.30 1.93 Investment/GDP 0.31* 0.19* 0.34* 0.17* t-statistics 8.33 2.90 5.20 2.58 Government consumption/GDP -0.01 -0.10* 0.05 -0.08* t-statistics -0.35 -2.61 0.69 -2.54 Education 0.10 0.45* -0.32 0.36 t-statistics 0.45 2.07 -0.64 1.85 Terms of trade growth 0.10* 0.12* 0.10 - t-statistics 3.04 2.81 1.73 - Terms of trade standard deviation 0.86* -5.22 0.83 - t-statistics 2.15 -1.41 1.37 -

*: significant at the 5% level. The models also include a set of country and time-period dummies.