The welfare cost of macroeconomic uncertainty in the post-war period

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No ₆₂₄ _{ISSN 0104-8910}

The Welfare Cost of Macroeconomic

Uncertainty in the Post–War Period

Jo ˜ao Victor Issler, Afonso Arinos de Mello Franco, Osmani Teixeira de Carvalho Guill ´en

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Os artigos publicados são de inteira responsabilidade de seus autores. As opiniões

neles emitidas não exprimem, necessariamente, o ponto de vista da Fundação

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∗

†

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7 %

8 4 8

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4

7 5 ' ($$9%

7 7 : 3

7 3 7 3 % % ! /

6 7 % 5 0.9% 7 7 7 ;<"1)%$$=

5 3 %

∗

> 3 ' ? @ ! ' ? '

A ' & 5 % ' 8 ' % ' , ' / ' ' &

' - / / 7 7 % , %

> 6 B ! + ' &-6 C , D 77 % > 7 6E

' % 0 ' 8' 3 7 7 7 3 3 %

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(4)

? ;"#H1' *= . 7 3 B

5 I 5 3 B 5 / 7

B 3 % "#H* D ' 7 ' .

7 5 < H%)$ 7 7 % % ; 0.04% 7 7 7 7 ='

7 3 % 5 B 3 > / / 7 5

. 5 7 % . 7 ' ;"#H#= 3

7 > % -5 ;"##J=' 7;"##J=' 5 ;"##9=' 8 ;"##H=

;($$$= / 7 . .7 . + % , / +

;($$J= / . 3 > 7 7 5 ? 3

/ 5 ' 3 ' . ' 5 /

7 7 3 / . 5 5 / 7 7 3

%

3 7 5 7 / % ' 3 5

5 ' 7 3 5 7

% ' 5 B 7 7 ' 3 7 25%

7 7 7 ' 7 % , 5 - >;($$"=' K /

> 3 5 3 5 7 77 7

7 'L ' 3 7 5 '

: 3 %

7 . ? 5 / 3 I 3 %

' 3 > 7 7 . / B 5 - >% ' 3

5 3 7 : % 7 ! /

6 ;"#H"= 7 > 7 5 3 >"'

3 . F 5 / 7 ' 3 7 5

% , 3 7 5 /

/ ; ,&= %

7 5 I 3 77

7 ' % %' A ;"#1H=' 6 ;"#H(=' 75 ;"#H1=' 2

%;"##"=' ;"##J=' ;"##1=' ;($$"=' ;($$J='

5 ? ;"#H1' 77% (( (*=% 7 5

/ ; = 7 3 5 D ' 3 7 I

7 / 5 % , 5 -5 '

/ 5 7 > / 7 3 %

"

? ;"#H1' 77% (( (*' "= .7 7 5 7

6 ;"#H(=%

(5)

3 7 / 3 7 / / 5 '

3 / ,& % ' 77

3 7 7 3 %

,& 7 ' 3 77 8 7 ' 3 3

+ M 8 ' / ;($$)=%

( * 7 / 7 / 3 > 7 3

% J 7 7 ) %

? ;"#H1= 7 (ct) 5 5 4

ct=α (1 2α )texp −1

2σz zt,

3 ln (zt)∼N 0, σz % 7 D B {c∗t}∞t ' 3 c∗t =

E(ct) = α (1 2α )t% 6 ct 7 7 / 7 c∗t% ? 7 7

3 5 λ 4

E E

∞

t

βtu((1 2λ)ct) =

∞

t

βtu(c∗t)' ;(%"=

3 Et( ) .7 7 'β ' u( )

%

? 7 '

ln (ct) B / / 5 % I B /

' 5 7 ln (ct)% / '

E(ct) D ' 3 -5 ;"##J= 7 7 E ( ) D 3 4

E

∞

t

βtu((1 2λ)ct) =

∞

t

βtu(E (ct)). ;(%(=

6 3' λ 3 3 7 % ' 3

5 %

, / + ;($$J= 7 7 I / . 5 {c∗

t}∞t

{ct}∞t 4 (1−α)ct2αct∗' 3 c∗t = E (ct)% > 3 5

3 α'λ(α)' 3 / 4

E

∞

t

βtu((1 2λ(α))ct) =E

∞

t

βtu((1−α)ct2αc∗t)% ;(%*=

(6)

7 λ(0) = 0' λ' D 5 ? ' 5 λ = λ(1)% 5 λ(1)

D ' 5

I ;(%*= 3 7 α 4

λ′(0) = E

∞

t βtu′(ct)×E (ct)

E ∞_t βtu′₍_c t)×ct

−1% ;(%J=

I 7 ' 3 ? N 7

7 5 ' 3 / > / O φ4

u(ct) = c −φ t −1

1−φ % ;(%)=

, 3 ! / 6 ;"#H"=' / I 7 5 7

' 3 > ' ;ARMA7 =4

ln (ct) = ln (α ) 2 ln (1 2α ) t−ωt

2 2

t

i

ξi2 t−

j

bjζt−j ;(%9=

3 ln α (1 2α )t exp −ωt/2 / 7 ' ti ξi 7

3 > 7 ' t_j− bjζt−j M A(∞) 7 7

; =' ωt =σ t22 σ

t−

j

bj2σ t−

j

b_j / ln (ct)% 7

> 'ξt ζt 7 / ' 5 4

ξt

ζ_t ∼IN

0

0 ,

σ σ

σ σ ' ;(%1=

% %' > 6 7 5 5 7

σ = 0(%

β(1 2α ) −φexp − −φ φσ <1 β(1 2α ) −φ<1' 4

λ(β, φ) = exp φ(2σ 2σ )

2

  

1−β(1 2α ) −φexp − −φ φσ

1−β(1 2α ) −φ

  

/ −φ

−1, ;(%H=

3 7 σ t

−

j

bj σ

t−

j

b_j 5 7 / 7 *'σ =σ

∞

j

bj

σ =σ ∞

j

b_j% > 7 ' 3 3 λ(β, φ)

(

/ ! / 6 7 ξ ζ 7 ' 3

/ 3 > %

*

6 σ , σ <∞ ₃ _N ₇ _%

(7)

7 7 % ∂λ α,β,φ_∂α _α ≡λ′₍₀_{, β, φ}₎ ₄

λ′(0, β, φ) =

exp (φ(2σ 2σ )) 1−β(1 2α ) −φ exp −φ −φ σ

1−β(1 2α ) −φ exp φ φ σ −1; ;(%#=

77 3 φ= 1 5 %

7 / 7 3 % ' >

/ B ;(%H= ;(%#= 3 / I σ ;/

7 >= σ σ ;5 > =% ,

3 5 I 7 > ' 7

7 / 3 7 ' 3 7 7

. ' 5 3 3 /

%

!"

8 5 yt= (ln (ct),ln (It))′ 2×1/ 7 / 7

7 5 7 7 % 5 7 5

[−1,1]′yt 5 A 7 ; 75 ;"#H1==% ,

/ ;V ECM(p−1)= 4

∆yt= Γ ∆yt− 2 . . . 2 Γp− ∆yt−p 2γ[−1,1]′yt−p2εt. ;*%"=

;"##1= 3 3 . yt

7 7 % 7 ' ;*%"= 3 5 5 V ECM(1)' 3

7 4

∆yt =Zft ;*%(=

ft =T ft2Z′εt ' 3 '

ft =

  

∆yt

α′_y t−

 

, T =

  

Γ −γα′₋_γ

I 0 0

0 α′ ₁

 

, Z = [ ],

α / % ? 5 3 > 7

yt 7 / 5 !t ψt' ! / 6 ;"#H"= 4

ψt=−lim l→∞

l

i

Et[∆yt i] =−Z[I−T]− T ft' '

!t=yt−ψt.

(8)

,7 / ' / 7 ξt 7 [1,0] ×

×∆!t ×

'

5 3 >% / σ B V AR([1,0]×∆!t)% 6 4

ln (ct)−Et− (ln (ct)) = [1,0]×εt=ξt2ζt,

D ζt 7 / [1,0]×(εt−∆!t) =ζt' 3 3 7 σ

σ % , 77 3 7 σ σ 7 %

8 3 7 λ( )

λ′( ) ;(%H= ;(%#=% 77 ? ,& ; % %'

A ;"##J== 7 3 8 7 3

+ M 8 ' / ;($$)=%

#

$

, % % 7 5 / ' % % 6 ' % % 7 7 '

3 5 8& "#J1 ($$$% D 5 / ,& 7

% ? V AR(2)

M ;"##"=% 3 /

% ; = 3 / 3 /

7 λ % ' [−1,1]′

/ 7 / 0.1089% A ' 3 ,& 7

α= [−1,1]′%

3 7 5 "M

A > ;"##1= D . %

! / 6 7 5 0.9% 7 7 7 ' 3

$175.77 7 7 ($$$ <% , ($ 5 > /

5 ? ' / % 7 A >

;"##1= D ' 3 D ! / 6 7 7 3

5 A > D 7

5 5 ? %

5 ( 7 3 %

5 1.9% 7 7 7 ! / 6 7 P 3 5

3 % 5 7 5 , / + ;($$J=%

"#)J #1 7 ' D 5 0.20% 3 H 3 7 D . '

5 0.30% 3 D ' 5 0.77% 1.40% 3

D 7 / % , 3 / (' 3 7 3

7 / % , / +

(9)

5 B 7 %

3 7 5 " (' 3

3 + % , 3 > 3' D

7 % % %

- 5 7 7 7 7 7 7

7 7 3 % ' 7

5 " ( 3 3 / 5 5 7

7 % . 7 ' -5 ;"##J= ;($$$=

7 3 > 3 ' 3 8 ;"##H= 7 3 5

AR(1)7 %

B ;(%)=' -5 8

3 3 % 3 7 5

7 5 " (% 3 > -5

3 5

5 "% / ' ' I 5 3 ($

/ 3 ' > B / % 8 N AR(1)

' 3 5 ))Q 5 5 "% , ' I

B / P 5 ") / 3 % ,

7 3 7 5 (% -/ '

7 7 7

3 7 / %

F 3 3 5 / 3 % A ' 5

7 3 7 N %

5 3 3 / 7 -5 P 3

P 8 N ' 3 ! / 6 7 AR(1) 7 4

−φ

−φ(∆ lnct−!)' 3 φ D / O 7 3 !

% / 7 I

' 3 % ' 3 5 / 3

*(Q 3 )$Q 3 3 / 3 >

AR(1) 7 I % ! > ' / 7 7

;V ECM(1)= AR(1) 7 5 8 ' 3

.7 3 3 / 5 / 7 / 7 %

(10)

7 7 77

7 3 7 3 7 7

! / 6 ;"#H"= 7 % 7

/ B ' 3 3 5 7 / 5 0.9% 1.9%

7 % , 5 > / 7 5 ? 5 1/20

' 5 7 7 % 8 7

3 / ' 3 5 D %

3 3 / 7 7 3 5 7

% D 5 3

3 5 3 5 %

6 > > , / + ;($$J=%

7 7 + > ' 7 5 7

! / 6 7 7 > 7 % , 3

5 / ' 7 . ' 5 3 3 /

%

, / +' % ' %' ($$J' K , ! 'L

' ""(;9=' 77% "((* )9%

! / ' % 6 ' %&%' "#H"' K, 6 3 ,77 8 7

7 3 ,

R! N'L ' 1' ")" "1J%

75 ' % "#H1' K8 / , 7 8 ? 5 S , , /

A 7 'L ' / % ));9=' 77% "(J# 1*%

' %A%' "##J' K 7 6 > 'L

' *$' (J" (9)%

8 ' %' "##H' K& > ! L'

' "' 9J9 919%

8 ' ,% %' ' % %' / ' %' ($$J' K, !

, > 7 S'L %' ($$)' 3 5

74TT333% 7 % %5 T ($$)T T,$),$*"%7 %

(11)

A ' &% %' "#1H' K 7 ? A 7 4

/ 'L ' H9' 77% #1" #H1%

A ' "##J' K %L 4 / %

A >' &% % ' % %' "##1' K 3 % % ! 4 , 7 /

'L ! " " # $ ' (#' ""9%

' , ' "#H#' K ! / 5 ? B

L' ' #1 ;9= ' "*9J "*H*%

' % % ' %' ($$"' K 7 >

, 'L ' J1' JJ# J1)%

' %' "##"' K A 7

, L' ' / % )# 9' 77% "))" ")H$%

2 ' &% %' ' % %' >' %A% ' % %' "##"' K

L' ' H"' H"# HJ$%

? ' &%' "#H1' K " # ! 'L -. 4 ! >3 %

' %' ($$J' K& 5 , 7 8 'L

> 7 U 3"$("$4 6 ! & %

6 ' %&% ' %' "#H(' K & > 'L

' "$' "$J) "$99%

-5 ' %' "##J' K / & > 7 4 & 7 5

5 'L % ' *H' "J1" "JH9%

- >' %' ($$"' K- ! 'L

J1' 9" #(%

5 ' %' "##9' K 3 ' ' 3 6 .7

'L & ' )$' *H1 *#(%

' %' "##1' K 8 L''( " #

" )# ;*=' J$) J((%

%' %8%' ($$$' K& > / & ! L'

J)' )$1 )*(%

(12)

' % ' &% %' "##1' K 7 'L ' / % H$' 77% "## "("%

7' %' "##J' K & > L'

*J' "1) ($$%

5 "4 %

; = ? ! >

β B / E !

β = 0.950,0.971,0.985

φ= 1 φ= 5 φ= 10 φ= 20

0.008 0.042 0.08 0.17

;5= ! / 6 8 7 "#J1 ($$$

β B / E !

β = 0.950

β = 0.971

β = 0.985

φ= 1 φ= 5 φ= 10 φ= 20

0.45 0.76 0.79 0.74

(0.012) (0.020) (0.020) (0.019)

0.80 0.92 0.89 0.79

(0.022) (0.024) (0.023) (0.021)

1.59 1.06 0.96 0.83

(0.043) (0.028) (0.025) (0.022)

; = A > "#J1 ($$$

β B / E !

β = 0.950,0.971,0.985

φ= 1 φ= 5 φ= 10 φ= 20

0.01 0.04 0.08 0.16

(0.0002) (0.0011) (0.0022) (0.0043)

; = ? "#J1 ($$$

β B / E !

β = 0.950,0.971,0.985

φ= 1 φ= 5 φ= 10 φ= 20

0.05 0.27 0.54 1.08

(0.001) (0.007) (0.014) (0.029)

6 4 ;"= ; = . ? ;"#H1=% ;(= ;5= 77

B ;(%H= I 7 % ;*= ; = ; =

3 ct =α (1 2α )texp − σz zt' 3 7 c∗t =E(ct) =

α (1 2α )t' ln (zt)∼N 0, σz % ; ='ln (1 2α ) 3 A

D 7 % ; =ln (1 2α ) 3 7 %

' 3 7 B ;(%"= 7 3 %

(13)

5 (4 %

; = ? ! >

β B / E !

β = 0.950,0.971,0.985

φ= 1 φ= 5 φ= 10 φ= 20

0.008 0.042 0.08 0.17

;5= ! / 6 8 7 "#J1 ($$$

β B / E !

β = 0.950

β = 0.971

β = 0.985

φ= 1 φ= 5 φ= 10 φ= 20

0.91 1.58 1.70 1.75

(0.024) (0.042) (0.047) (0.055)

1.63 1.92 1.92 1.90

(0.044) (0.052) (0.054) (0.060)

3.26 2.22 2.08 2.00

(0.091) (0.061) (0.059) (0.064)

; = A > "#J1 ($$$

β B / E !

β= 0.950,0.971,0.985

φ= 1 φ= 5 φ= 10 φ= 20

0.02 0.08 0.16 0.32

(0.0004) (0.002) (0.004) (0.009)

; = ? "#J1 ($$$

β B / E !

β = 0.950,0.971,0.985

φ= 1 φ= 5 φ= 10 φ= 20

0.11 0.54 1.08 2.18

(0.003) (0.014) (0.029) (0.059)

6 4 ;"= ; = . ? ;"#H1=% ;(= ;5= 77

B ;(%#= I 7 % ;*= ; = ; =

3 ct =α (1 2α )texp − σz zt' 3 7 c∗t =E(ct) =

α (1 2α )t' ln (zt)∼N 0, σz % ; ='ln (1 2α ) 3 A

D 7 % ; =ln (1 2α ) 3 7 %

' 3 7 B ;(%J= 7 3 %

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