Ensaios Econômicos - Sistema de Bibliotecas da FGV - Fundação ...

Using (2.11), with CES utility and recalling that the conditional variance of logged observed
consumption is ! t 2 tP
1
= 11 t +
2 22 j + 2 tP
1
12 j, we can compute the welfare cost of business
j=0
cycle when 12 6= 0. The analogous formula to (2.13) is:
8
>< exp ( (2e 12 + e 22 ) =2) 1 for 6= 1,
P =
>:
exp ((2e 12 + e 22 ) =2) 1 for = 1.
j=0
: (2.22)
Similarly, using (2.14), we can compute the welfare cost of economic-growth variation. The analogous
formula to (2.15) is:
8
><
T =
>:
(1 (1+1 ) (1 ) exp( (1 ) 11 =2))exp( (1 )e 12 =2)
(1 (1+ 1 ) (1 ) )
exp
11 +2(1 )e 12
2(1 )
1
(1 )
1 for = 1.
1 for 6= 1,
Finally, using (2.17), we can compute the welfare cost of all macroeconomic uncertainty.
analogous formula to (2.18) is:
8
><
D =
>:
exp((1 )(2e 12 +e 22 )=2)(1 (1+ 1 ) (1 ) exp( (1 ) 11 =2))
(1 (1+ 1 ) (1 ) )
exp
11 +(1 )(2e 12 +e 22 )
2(1 )
1
(1 )
1 for = 1.
1 for 6= 1,
(2.23)
The
(2.24)
The key di¤erence between (2.13), (2.15), and (2.18), and their respective counterparts in the
1P
previous section is their dependence on e 12 = 12 j, the unconditional counterpart of 12 .
j=0
Finally, it is straightforward to compute the marginal welfare costs of business cycles, economicgrowth
variation, and all macroeconomic uncertainty. They are, respectively:
8
>< exp ( (2e 12 + e 22 )) 1 for 6= 1
0 P (0) =
(2.25)
>:
exp (2e 12 + e 22 ) 1 for = 1
8
exp(2e 12 ) 1 (1+ 1 ) (1 ) (1 )11
exp
2
><
0 1 (1+ 1 ) (1 ) (1+)11
1 for 6= 1
exp
2
T (0) =
; and, (2.26)
>:
exp(2e 12 )(1 )
1 exp( 11 )
1 for = 1
8
exp((2e 12 +e 22 )) 1 (1+ 1 ) (1 ) (1 )11
exp
2
><
0 1 (1+ 1 ) (1 ) (1+)11
1 for 6= 1
exp
2
D (0) =
: (2.27)
>:
exp(2e 12 +e 22 )(1 )
1 exp( 11 )
1 for = 1
14