Price setting for non-tradable goods. All non-tradable goods firms that are able to reset their prices, which happens with the probability 1 - ξN, choose the same price widetilde P NT,t. Firms that do not reset their prices index their prices according to Pt = (ΠNT,t-1)χ_ N Π 1-χ_ N Pt-1, i.e., to a geometric average of past sector-specific inflation, ΠNT,t-1 ≡ PNT,t-1 / PNT,t-2 and the (constant) inflation target, \Pi. The parameter χN measures the degree of indexation. Reoptimising firm maximizes the discounted sum of its expected nominal profits subject to the price-indexation scheme and taking as given the demand for its brand. The marginal costs MCN,t are symmetric across producers. Λi,t,t+k is the stochastic discount factor of i-type households, who own firms.
The implied first-order condition is:
Et[sum from k = 0 to ∞ of (ξN)kΛi,t,t+k(product from s = 1 to k of ΠNT,t+s-1χ_ N Π 1-χ_ Nwidetilde P NT,t-(θN)/(θN-1)MCN,t+k)NTt+k] = 0
(43)
Firms whose price contracts are re-optimized set prices to equate the discounted sum of expected revenues to the discounted sum of expected marginal costs.
With price setting as described above, the sector-specific price index PNT,t is:
PNT,t = [ξN(ΠNT,t-1χ_ N Π 1-χ_ NPNT,t-1)1-θ_ N+(1-ξN)(widetilde P NT,t)1-θ_ N]1/(1-θN)
PHT,t = [ξH(ΠHT,t-1χ_ H Π 1-χ_ HPHT,t-1)1-θ_ T+(1-ξH)(widetilde P HT,t)1-θ_ T]1/(1-θT)
(44)
Price setting for home tradable goods. Price setting for home tradable goods is analogous to that of non-tradable goods. The probability of resetting the price is (1 - ξH). The sector-specific price index PHT,t evolves according to:
(45)
Price setting for export goods. Firms discriminates across countries, by invoicing and setting the price in the currency of the generic destination market CO (local currency pricing assumption). The probability of optimally resetting prices (1 - ξX), and all firms that reset their prices choose the same price, widetilde P X,tH,CO, while the others index their prices to PX,tCO = (product from X,t-1 to H,CO of)χX Π 1-χX PX,t-1CO, where ΠX,t-1H,CO ≡ PX,t-1H,CO / PX,t-2H,CO is the sector-specific inflation. Foreign inflation target is time invariant and equal to the Home inflation target, Π CO = Π.
The bilateral exports price index (of country H to the generic country CO) is:
PX,tH,CO = [ξX((ΠX,t-1H,CO)χ_ X Π 1-χ_ XPX,t-1H,CO)1-θ_ T+(1-ξX)(widetilde P X,tH,CO)1-θ_ T]1/(1-θT)
(46)
This appendix if based on, and follows closely the model description in Gomes et al. (2023), and the reader should refer to this paper for all the details. The labour market structure with search frictions follows the standard search model of Mortensen and Pissarides (1999), but adds sticky wages by means of staggered wage setting, and a