Monetary Policy Design in a DSGE Model
1. A simple model 1.1 Households The utility function of a representative household is
∞ ⎛ ξ C1−σ L1+η ⎞ Et ∑ β s ⎜ t + s t + s − t + s ⎟ 1+η ⎠ s =0 ⎝ 1−σ

(1)

The dynamics of the demand shock is

ζ t = ζ t −1 + eζ ,t , where ξt = eζ . The consumption is t θ −1 ⎡ 1 ⎤ θ −1 composed of lots of goods. It is composed in a CES function Ct = ⎢ ∫ Ct (i ) θ di ⎥ . The i =0 ⎣ ⎦
1 1−θ 1−θ consumer price index is P = ⎡ ∫ P (i ) di ⎤ . Then the demand function of each good is t t ⎢ i =0 ⎥ ⎣ ⎦ 1

θ

⎡ P (i ) ⎤ Ct (i ) = ⎢ t ⎥ Ct ⎣ Pt ⎦
The budget constraint is

−θ

PCt + Bt = Wt Lt + Π t + Rt −1 Bt −1 t
The first order condition for Ct , Lt , and Bt are

Ct−σ = λt Pt Lη = λtWt t

λt = Rt λt +1|t
After some calculation, we have the Euler equation and labor supply equation

ξt +1Ct−+σ / Pt +1 1 β Rt Et =1 −σ ξt Ct / Pt ξt Ct−σ
Pt
1.2 Firms

(2)

=

Lη t Wt

(3)

Assume there is a type of price stickiness in the economy, that each firm has a probability

φ

that cannot change its price, and fixed the level as the last time. And it has a probability 1 − φ that can re-optimal its price. The problem of a firm which can re-optimal its price in time t is

1

Et ∑ (βφ ) s Λ t + s [ Pt* (i )Yt (i ) − Wt Lt (i )] s =0

∞

⎡ Pt * ⎤ s.t.Yt + s (i ) = ⎢ ⎥ Ct and Yt (i ) = At Lt (i ) ⎣ Pt + s ⎦
The dynamics of the technology shock is at = ρ a at −1 + ea ,t , where at = ln( At ) . The first order condition is
∞ ⎡ θ Wt + s ⎤ Et ∑ (βφ ) s Λ t + sYt (i ) ⎢ Pt * (i ) − ⎥=0 θ − 1 At + s ⎦ s =0 ⎣

−θ

(4)

1.3 Log Linear Form of the Model Log linear form of Euler equation is

rt − σ (ct +1|t − ct ) − π t +1|t + ζ t +1|t − ζ t = 0
Impose the market clear condition, i.e. consumption is equal to output, we have

yt = yt +1|t −

1

σ

(rt − π t +1|t ) + ζ t − ζ t +1|t

This function is the