Phillips Policy Rules 3.1 A simple textbook macrodynamic model

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Anne Pope
7 years ago
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5 ( ) ( ) = ( ) + ( ) + ( ) ( ) ( ) ( ) [ ] &( ) = α ( ) ( ) α > 0 4

6 ( ) ( ) = ( ) 0 < < 1 ( ) = [ ] &( ) = α ( 1) ( ) + + ( ) 0 < < 1 + = 1 5

7 ( ) ( ) &( ) = β ( ( ) ( )) β > 0 ( ) ( ) ( ) β ( ) = β β 6

8 = γ > 0 * ( ) γ ( ) * γ * ( ) < ( ) > 0 * ( ) > ( ) < 0 ( ) < 0 & α( 1) α α = + * & βγ β βγ 1 + γ > 0, (ii) 1 + β α > 0 γ, β 0 < < 1 > 1 [ ] 2 4 αβγ > β α(1 ) ( ) * 1+ γ = 1 + γ * β, γ 7

9 , * * * * = (1 ) * * ( ) γ ( ) = * ( ) γ > 0 * ( ) = γ [ ( ) ] 0 & * ( ) = γ [ ( ) ] ( ) ( ) ( ) 8

10 & α( 1) 0 α α & * = γ γ & 0 β β 0 0 < < 1 [ ] β + α (1 ) (1 ) > γ β β 2 4 γ > α(1 ) * ( ) = γ &( ) γ > 0 γ * * ( ) = γ ( ( ) ) γ [ ( ) ] γ ( ) & γ 9

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12 &( ) = ( ) ( ) + ( ) ( ) ( ) ( ) 1 1 ( ) ( ) + [ ( ) ( ) ( ) ( ) ( ) ( )] ( ) ( ) ˆ( ) 1 ˆ( ) ( ) ( ) ( ) ( ) = ( ) & 1 ( ) = ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) = 11

13 ,,, 1 ˆ( ) ( ) = = 0 & 1 ( ) = [ ] ( ) 12

14 [ ] &( ) = α ( ) + ( ) ( ) ( ) = 1 ( ) 13

15 [ ( ) ( ) ] 0 α ( ) = ( ) α α = 0 γ > 0 α α 2 + (2 α + α σ ) = σ &( ) = α[ ( ) ( )] ( ) = ( ) + ( ) + ( ) ( ) = ζ [ ν &( ) ( )] ( ) ( ) 14

16 ( ), ( ) ( ) = θ ( ) + θ ( ) 1 2 θ1, θ 2 ( ) 2 ζ ( ) ( θ θ ζ ) θ ζ ( ) = + ( ) ( ) &( ) = α[ ( ) ( )] ( ) = ( ) [ ( ) π ( )] 1 2 ( ) = κ ( ) + π ( ) κ > 0 π& ( ) = ρ[ ( ) π ( )] ρ > 0 ( ) π ( ) ( ) ( ) = ( ) &( ) = ζ [ ( ) ( )] &( ) = ( ) 15

17 π ( ) ( π ) &( ) = α ( 1) ( ) [ ( ) ( )] 1 2 π& ( ) = ρκ ( ) ( ) ( ) = ϕ ( ) + ϕ π ( )

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19 α, γ 18

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21 γ [ ( ( ) ( ))] = = θ + ( ) (.) 1 1, (, + 1) 1 = [ ( ) 1 ] + ( ) = λ ( ) + [ ] 1 20

22 [ ( ( ) )] = (, + 1) ( ) = [ 1( )] [ ( + ) 1( + )] + (1 + θ ) (1 + θ ) (1 + θ ) = 1 ( 1) 1+ θ 1 1 = λ, λ = θ 1 + θ λ = θ

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24 2 2 β [ (1 ) ] (0) = 0 = (0)

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26 ( ) = θ + β ( ) 0 < β < (, + 1) ( 1, ) β + θ 25

27 β + θ < 1 = 2 θ ( β + θ ) 1 1( + ) + = 1 = µ 1 = µ 1 + µ β + θ > 1 26

28 0 1 [ 1 2 ] min ( ) ( ) ( ) ( ) ( ) ( ) &( ) = ( ) + ( ) + ( ) (0) 1 ( ) 2 ( ) ( ) 1( ) = 1 ( ), 2( ) = 2 ( ) 27

29 (, ) ( %%, ) (, ) ( %%, ) 28

30 1 1 (, ) (, ) + ( ) + ( )( ) + ( ) %% % % % % = ( ) = (, ) % ( %) + ( %) (, ) ( %%, ) + ( %%, ) ( + ) % + % = ( %) ( %) = ρ 29

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33 [ ' + ' ] 0 &= + = [ ] 1 2 = [ ] (0) = 0 (0) (0) = ( ) ( ) = 32

34 &( ) ( ) ( ) = &( ) ( ) ( ) = lim ( ) ( ) = 0 1 lim ( ) = lim ( ) = 0 [ 2]' [ 1]' & & = & & = = ( ) = 2( ) 1 ( ) [ 11 ( ) 12 2( ), 2( )] 1 [, ] [ ] & = [ ] &

35 (0) = 2 0 = = ( + ) ( ) 1 ( ) = 2( ) + ; ; ;

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