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Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 1
Welfare function
Second-order approximation of welfare The second-order Taylor approximation of a real function
yt = 𝑓 𝑥𝑡 around a point (e.g. the equilibrium) 𝑥 is given by:
The instantaneous utility of the representative household is given by • the utility function is additively separable in consumption and
hours, i.e. 𝑈𝑐𝑐 = 0 • (the marginal utility of 𝐶 is independent of 𝑁, and the marginal
utility of 𝑁 is independent of 𝐶)
( ) ( ) ( )
( ) ( )( ) ( )( )
( )
0
2
!12!
nn
t t tn
x t xx t
f xy f x x x
n
f x f x x x f x x x
∞
=
= = − ≈
≈ + − + −
∑
( )1 1
,1 1
t tt t
C NU C N
σ ϕ
σ ϕ
− + = − − +
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 2
Welfare function
Second-order approximation of welfare Second-order approximation with respect to 𝐶 around the
steady state
Second-order approximation with respect to 𝑁 around the efficient (flex-price) equilibrium
Putting both together (additive separability)
( ) ( ) ( ) ( )1
211 2
tt c t cc t
CU C U C U C C U C Cσ
σ
−
= ≈ + − + −−
( ) ( ) ( )
( ) ( ) ( )
1 1
2 2
, ,1 1
1 12 2
t tt t c t
n t cc t nn t
C NU C N U C N U C C
U N N U C C U N N
σ ϕ
σ ϕ
− + = − ≈ + − + − +
+ − + − + −
( ) ( ) ( ) ( )1
211 2
tt n t nn t
NU N U N U N N U N Nϕ
ϕ
+
= − ≈ + − + −+
first derivative of the utility function evaluated at the steady state
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 3
Welfare function
Second-order approximation of welfare Rearranging terms
( ) ( )
2 22 2
, ,
1 12 2
t t t
t tc n
t tcc nn
U C N U C N U U
C C N NU C U NC N
C C N NU C U NC N
− = − ≈
− − ≈ + +
− − + +
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 4
Welfare function
Second-order approximation of welfare Then, define the inverse intertemporal elasticity of
substitution 𝜎 as
and the inverse elasticity of labor supply (elasticity of marginal disutility of labor) 𝜑 as
1cc
c
U CC CU C
σ
σσσ σ
− −
−
−≡ − = − =
1nn
n
U NN NU N
ϕ
ϕϕϕ ϕ
−−≡ = =
−
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 5
Welfare function
Second-order approximation of welfare Using the definitions for 𝜎 and 𝜂 to replace second-order
derivatives, the second-order Taylor expansion of 𝑈𝑡 can be formulated as
Let 𝑥�𝑡 = log (𝑋𝑡 𝑋⁄ ) denote the log deviation from steady state and calculate a second-order approximation of 𝑋𝑡−𝑋
𝑋= 𝑋𝑡
𝑋−
1 = exp 𝑥�𝑡 − 1 = ∑ 1𝑐!𝑥�𝑡𝑐∞
𝑐=0 − 1 ≈ 𝑥�𝑡 + 12𝑥�𝑡2
1 12 2
t
t t t tc n
U U
C C C C N N N NU C U NC C N N
σ ϕ
− =
− − − − = − + +
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 6
Welfare function
Second-order approximation of welfare Using 𝑋𝑡−𝑋
𝑋≈ 𝑥�𝑡 + 1
2𝑥�𝑡2 one finally gets
2 2
2 2
2 2 3 4
2 2 3 4
1 1ˆ ˆ ˆ ˆ12 2 2
1 1ˆ ˆ ˆ ˆ12 2 2
1 1ˆ ˆ ˆ ˆ ˆ2 2 4
1 1ˆ ˆ ˆ ˆ ˆ2 2 41ˆ
t c t t t t
n t t t t
c t t t t t
n t t t t t
c t
U U U C c c c c
U N n n n n
U C c c c c c
U N n n n n n
U C c
σ
ϕ
σ
ϕ
− ≈ + − + +
+ + + + = = + − + + +
+ + + + + ≈
−
≈ + 2 21ˆ ˆ ˆ2 2t n t tc U N n nσ ϕ+ + +
terms of an order higher than 2 are so small so that we can ignore them
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 7
Welfare function
Second-order approximation of welfare Using the market clearing condition �̂�𝑡 = 𝑦�𝑡 for all 𝑡 gives
When the optimal subsidy is in place, then the steady state is efficient
2 21 1ˆ ˆ ˆ ˆ2 2t c t t n t tU U U C y y U N n nσ ϕ− + − ≈ + + +
( ) ( ) ( )111 1 1
n
c
U WMRSU P
N YMPN AN ANNN
αα α
αα α α−
− −−
− = = =
= = − = − = −
1n
c cU N U Y U C
α⇔ = − = −
−
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 8
Welfare function
Second-order approximation of welfare The second-order approximation of 𝑈𝑡 around the steady
state then reads
( ) ( ) ( ) ( ){ }
2 2
2 2
1 1ˆ ˆ ˆ ˆ2 2
1ˆ ˆ ˆ ˆ1 1 1 12
t c t t n t t
c t t t t
U U U C y y U N n n
U C y n y n
σ ϕ
σ ϕα α
− + − ≈ + + + =
= − − + − − − +
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 9
Welfare function
Second-order approximation of welfare In the next step we rewrite 𝑛�𝑡 in terms of the output gap
• each firm produces according to 𝑌𝑡 𝑖 = 𝐴𝑡𝑁𝑡(𝑖)1−𝛼
• market clearing in the labor market requires 𝑁𝑡 = ∫ 𝑁𝑡 𝑖 𝑑𝑖10
• using the production function 𝑁𝑡 = ∫ 𝑌𝑡(𝑖)𝐴𝑡
11−𝛼 𝑑𝑖1
0
• and using the households’ demand equation 𝑌𝑡 𝑖 = 𝑃𝑡(𝑖)𝑃𝑡
−𝜀𝑌𝑡
• aggregate labor can be expressed as 𝑁𝑡 = 𝑌𝑡𝐴𝑡
11−𝛼 ∫ 𝑃𝑡(𝑖)
𝑃𝑡
−𝜀1−𝛼 𝑑𝑖1
0
• or after taking logs 1 − 𝛼 𝑛𝑡 = 𝑦𝑡 − 𝑎𝑡 + 𝑑𝑡
• where 𝑑𝑡 = 1 − 𝛼 log∫ 𝑃𝑡(𝑖)𝑃𝑡
− 𝜀1−𝛼 𝑑𝑖1
0
• subtracting the steady state (𝑎𝑡 = 0) on both sides of the equation 1 − 𝛼 𝑛𝑡 − 1 − 𝛼 𝑛 = 𝑦𝑡 − 𝑦 − 𝑎𝑡 + 𝑑𝑡
• finally gives 1 − 𝛼 𝑛�𝑡 = 𝑦�𝑡 − 𝑎𝑡 + 𝑑𝑡
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 10
Welfare function
Second-order approximation of welfare Using this to replace the linear terms in the second-order
approximation of 𝑈𝑡 around the steady state
where t.i.p. denotes (exogenous) terms independent of policy (here 𝑡. 𝑖.𝑝. = 𝑎𝑡)
ignoring these terms is based on the implication of the New Keynesian model that the steady state is independent of monetary policy
( )
( ) ( ) ( ){ }2 2
ˆ ˆ1 . . .
1 ˆ ˆ2 1 1 1 . . .2
t t t t t
tt t t
c
y n a d d t i p
U U d y n t i pU C
ϕασ
α− − = − ≈ − +
−⇒ ≈ − − − + − + +
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 11
Welfare function
Second-order approximation of welfare Replacing the quadratic labor term in the second-order
approximation of 𝑈𝑡 around the steady state
( )( ) ( )
( ) ( )
( )
( ) ( )
2
2
2
22
2
2
2
ˆ ˆ1
ˆ ˆ1
ˆ ˆ2
ˆ
1 1ˆ ˆ2 1 . . .2 1
t t t t
t t t t
t t t t t
t t
tt t t t
c
t
n y a d
n y a d
y a d d y a
y a
U U d y y a t i pU C
ϕ
α
σ
α
α
− = − +
− = − + =
= − + + − ≈
≈ −
− + ⇒ ≈ − − − + − + −
terms of higher than second order
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 12
Welfare function
Second-order approximation of welfare The last step follows from the presumption that 𝑑𝑡 is a term of
second order • 𝑑𝑡2 and 𝑑𝑡(𝑦�𝑡 − 𝑎𝑡) will be of higher than second order
Second-order approximation of 𝑑𝑡 in the neighborhood of the symmetric flex-price equilibrium
𝑑𝑡 = 1 − 𝛼 log∫ 𝑃𝑡(𝑖)𝑃𝑡
− 𝜀1−𝛼 𝑑𝑖1
0 ≈ 𝜀2Θ𝑣𝑎𝑟𝑖(𝑝𝑖 𝑡 ) where
Θ = 1−𝛼1−𝛼+𝛼𝜀
the proof of this approximation will be presented in the Tutorial
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 13
Welfare function
Second-order approximation of welfare The second-order approximation of 𝑈𝑡 around the steady
state then reads
price dispersion, as measured by 𝑣𝑎𝑟𝑖(𝑝𝑖 𝑡 ), leads to deviations of utility from steady-state utility (and by this to welfare losses)
( )( ) ( ) ( )2 21 1ˆ ˆvar 1 . . .2 1
ti t t t t
c
U U p i y y a t i pU C
ε ϕσα
− + ≈ − − − + − + Θ −
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 14
Welfare function
Second-order approximation of welfare Some further manipulations
( )( ) ( ) ( )
( )( ) ( ) ( )
( )( )
2
2 2 2
2
2 2
. .
1 1ˆ ˆvar 1 . . .2 1
1 1ˆ ˆ ˆvar 1 2 . . .2 11 1 1 1ˆ ˆvar 2 . . .2 1 1 2 1
ti t t t t
c
i t t t t t t
i t t t t t
t i p
U U p i y y a t i pU C
p i y y y a a t i p
p i y y a t i p a
α
α
αα
ε ϕσ
ε ϕ
ϕα α
σ
ε ϕ ϕσ
− + ≈ − − − + − + = Θ − + = − − − + − + + = Θ −
+ + + = − + + − + − Θ − − − .
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 15
Welfare function
Second-order approximation of welfare Then replace 𝑎𝑡 by 𝑦�𝑡𝑐
The second-order approximation of 𝑈𝑡 around the steady state then reads
( ) ( )( ) ( )
( )( )
1 ln 1 11 1
1ˆ1
1ˆ
1
t
nt
t
t
nt
ny a
y a
a y
α α ϕσ α α ϕ σ α α ϕ
ϕσ α α ϕ
σ α α ϕϕ
− − += +
− + + − + +
+=
− + +
− + +⇒ =
+
( )( ) ( )21 ˆ ˆ ˆvar 2 . . .2 1
nti t t t t
c
U U p i y y y t i pU C
αα
ε ϕσ− + ≈ − + + − + Θ −
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 16
Welfare function
Second-order approximation of welfare Finally replace 𝑦�𝑡 and 𝑦�𝑡𝑐 by 𝑦�𝑡
The second-order approximation of 𝑈𝑡 around the steady
state then reads
( ) ( ) ( )22 2 22 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 2 . . .n n n n n
t t t t t t t t t t ty y y y y y y y y y y y t i p= − = − = − + = − +
( ) ( )( ) 211 var . . .2 1 1
ti t t
c
U U p i y t i pU C
αε α αεαϕσ
α − +− + ≈ − + + + − −
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 17
Welfare function
Second-order approximation of welfare Accordingly, we can write a second-order approximation to
the consumer‘s welfare losses (resulting from deviations from the efficient allocation) as a fraction of steady-state consumption (and up to additive terms independent of policy)
( ) ( )( )
00
20
0
11 var2 1 1
nt t t
t c
ti t t
t
U UW EU C
E p i y
β
ε α αε ααϕβ σ
α
∞
=
∞
=
−= ≈
− + + ≈ − + + − −
∑
∑
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 18
Welfare function
Second-order approximation of welfare The final step consists in rewriting the terms involving the price
dispersion variable 𝑣𝑎𝑟𝑖 𝑝𝑡 𝑖 as a function of inflation. Up to a first order approximation 𝑝𝑡 = 𝐸𝑖𝑝𝑡(𝑖) and
𝜋𝑡 = 1 − 𝜃 (𝑝𝑡∗ − 𝑝𝑡−1) Using these, we can write the cross-sectional price dispersion as
( )( ) [ ] [ ][ ][ ]
[ ]
( )( )
*
1
2 2
21
221 1 1
22 2 2
1 1
2
var ( ) ( ) ( )
( )
( ) (1
( )
)
1
var1
i t i t i t t
i t t t
i
i t
t
i
t t t i t t
i t t t t
tt
p i E p i E p i E p i p
E p i p
E p i p E p p
E p i p
p i
π
θ π θ π
θθ θπ πθ
θθ πθ
−
− − −
−
− −
= − = − =
= − − =
= − − + − − − =
= − − + =−
= +−
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 19
Welfare function
Second-order approximation of welfare Taking the discounted sum yields
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( ) ( )
10 0
0
2
0
0
2
0
2
0
0
var var1
var var1
var1 1
t t tti t i t
t t
i t i tt
t
t t tt
t
t tt
t
t
i tt
p i p i
p i p i
p i
θβ θ β β πθ
θβ θβ β β πθ
θβ β πθ θβ
∞ ∞
−= =
∞ ∞
= =
∞
=
∞
=
∞
=
∞
=
= +−
⇔ = +−
⇔ =− −
∑ ∑
∑ ∑∑
∑
∑
∑
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 20
Welfare function
Second-order approximation of welfare
Using the loss function reads ( )( )1 1 11
θ βθ αλθ α αε
− − −=
− +
( )
2 20
0
2 2 2 20 0
0 0
12 11
2 2
tt t
t
t tt t t t
t t
W E y
E y E y
ε ϕβ π σλ
ε κβ επ κ β πλ λ ε
αα
∞
=
∞ ∞
= =
+ = − + + = − = − + = − +
∑
∑ ∑
Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 21
Welfare function
Second-order approximation of welfare which is equal to the expected sum of discounted current and
future period losses
By scaling the intertemporal loss function by a factor (1 − 𝛽) it can be shown that when 𝛽 → 1, the scaled intertemporal loss approaches the weighted sum of the unconditional variances of the output gap and inflation
The average welfare loss is a linear combination of the variances of the output gap and inflation.
( ) ( )1
11lim 1 var var2 1 t tW y
β
ϕ σ α σ εβ πα λ→
+ + −− = + −
00
12
tt
tW E Lβ
∞
=
= − ∑