Second-order approximation of welfare · Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 1 Welfare function Second-order approximation of welfare The second-order

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

− + = − − +


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


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

− = − ≈

− − ≈ + +

− − + +


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

−−≡ = =

−


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

− =

− − − − = − + +


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


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⇔ = − = −

−


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

− + − ≈ + + + =

= − − + − − − +


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


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− − = − ≈ − +

−⇒ ≈ − − − + − + +


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


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


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 −


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 − − − .


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 −


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 − +− + ≈ − + + + − −


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

∞

=

∞

=

−= ≈

− + + ≈ − + + − −

∑

∑


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

−

− − −

−

− −

= − = − =

= − − =

= − − + − − − =

= − − + =−

= +−


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

∞ ∞

−= =

∞ ∞

= =

∞

=

∞

=

∞

=

∞

=

= +−

⇔ = +−

⇔ =− −

∑ ∑

∑ ∑∑

∑

∑

∑


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

∞

=

∞ ∞

= =

+ = − + + = − = − + = − +

∑

∑ ∑


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

∞

=

= − ∑

Documents

Second-order approximation of welfare · Carstensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 1 Welfare function Second-order approximation of welfare The second-order