To introduce price stickiness, we assume that there is a probability of 1 - ξ at each
period that the price of each intermediate good m is set optimally to Pt0(m). If the price
is not re-optimized, then it is held fixed.5 For each intermediate producer m, the objective
is at time t to choose {Pt0(m)} to maximize discounted profits
∞
Et X ξk Dt,t+k Yt+k ( m ) £ Pt ( m ) - Pt+k MCt+k ] (21)
k=0
subject to (18), where Dt,t+k is now the nominal stochastic discount factor over the interval
[t, t + k]. The solution to this is
Et X∞ ξkDt,t+kYt+k(m)
k=0
In (22) we have introduced a mark-up shock MSt to the steady state mark-up (1 J /ζ). By
the law of large numbers, the evolution of the price index is given by
Pt1+-1ζ = ξPt1-ζ + (1 - ξ)(Pt0+1)1-ζ (23)
P0(m) — и l1∕P Pt+kMCt+kMSt+k
(1 - 1/z)
=0
(22)
In setting up the model for simulation and estimation, it is useful to represent the
price dynamics as difference equations. Using the fact that for any summation St ≡
Pk∞=0 βkXt+k , we can write
St = Xt+X∞ βkXt+k = Xt + X∞ βk0+1Xt+k0+1 puttingk0=k+1
k=1 k0=0
= Xt + βSt+1 (24)
and defining here the nominal discount factor by Dt,t+k ≡ βλCjt+k∕ptt+k■ inflation dynamics
are given by
Ht-ξβEt[Πtζ+-11Ht+1]
YtΛC,t
(ɪɪ) YtΛc,tMCtMSt
ξ πh 1+(1—ξ ) μ h )1 ^ζ
(25)
(26)
(27)
Jt-ξβEt[Πtζ+1Jt+1]
1
Real marginal costs are no longer fixed and are given by
PW
MCt = t. (28)
Pt
5Thus we can interpret ι^ as the average duration for which prices are left unchanged.
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