Master Thesis - Humboldt-Universität zu Berlin

5 Model analysis
We find it more convenient to treat the first order conditions, steady state
and log-linearized equations together. For the purpose of the paper, we need
to write the model in log-linearized form. This emerges from the first order
condition equations and linearizing the equations around the steady state.
5.1 Households: Optimization problem
Write the Lagrange function:
H = E 0
∞
∑
t=0
+ λ t ( 1 R e t
β t 1
[( (Ct τ − H t )) 1−σc · exp( σ c − 1
(lt τ ) 1+σ l
)
1 − σ c 1 + σ l
B τ t
P C
t
+ 1 Bt
τ∗
− Bτ t−1
Rt
e∗ Pt C S t Pt
C
− Bτ∗ t−1
P C
t S t
+ (1 + τ c t )C τ t + I τ t
− (1 − τt)w l t τ lt τ − ((1 − τt k )(rt k − δ)zt τ Kt−1) τ − δz t Kt−1 τ + Ψ(zt τ )Kt−1 τ − Divt τ − T Rt τ )
( ) ɛ
I
+ µ t (K t − K t−1 (1 − δ) − (1 − S t I t
)I t )]
I t−1
where λ t is the marginal value in utils of one real domestic currency’s worth
of bonds and µ t is the marginal value of a unit of capital in utils (as defined
in Kimball (1995)). The marginal value of capital Q t is:
Q t = µ t
λ t
(31)
Given the Ricardian equivalence implied by the model, it is not necessary
to keep track of the household stock of bonds or the government’s budget
constraint.
Deriving the Lagrangian function with respect to holdings of
domestic and foreign bonds, capital, investment, capital utilization, labor
yields the following FOCs:
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