Master Thesis - Humboldt-Universität zu Berlin

Û c t+1 − Û c t + ˆτ c t − ˆτ c t+1 − ˆπ C t+1 + ˆR t − ˆɛ b t = 0 (38)
Û c t = −σ c
1
1 − h (Ĉt − hĈt−1) + (σ c − 1)(¯l τ ) 1+σ l ˆLt (39)
In this equation, steady state labor supply of individual households ¯l τ is obtained
using the equation () for wage optimization, derived later in the paper
In additon, in steady state the entire income from labor is used for private
consumption, namely W τ l τ = P C C. To show this, insert TR from budget
constraint into the households budget constraint, evaluated at the aggregate
level, which implies that all the tax-terms cancel as they should (government
consumption is zero in the steady state). In addition, the return on capital
is used entirely for the investment to rebuild the steady state level of capital
affected by depreciation. These result in: (¯l τ ) 1+σ l
=
−1
gives us the law of motion for consumption:
1−τ l
(1+λ w)(1−h)
1+τ c . This
Ĉ t = 1
σ c − 1 1 − τ l
1 + h (Ĉt+1 − hĈt−1) +
σ c (1 + h)(1 + λ w ) 1 + τ (ˆL c t − ˆL t+1 )
−
1 − h
σ c (1 + h) ( ˆR t − ˆπ t+1 C − ɛ b t + ˜τ t c − ˜τ t+1) c (40)
We use again the definition of Q t to derive investment:
(
ɛ
Q I )
t(1−S t I t
)=Q ′( ɛ I )
I t−1 tS t I t ɛ I [
t I t
+1−E I t−1 I t β λ t+1
Q t−1 λ t+1 S ′( ɛ I )(
t+1 I t+1 ɛ
I ) ]
t+1
I t+1 I t+1
t I t I t I t
(41)
Rather than log-linearization for real consumption, we use a fraction of intertemporal
investment:
22