International macroe.. - Free

9.2. PRICING TO MARKET 295ˆv t (z) = θ[Ŝt + ˆP t ∗ t(z)] + Ĉ∗ t , (9.148)ˆx ∗ t (z ∗ ) = θ[ ˆP t ∗ − ˆp ∗ t (z ∗ )] + Ĉ∗ t , (9.149)ˆv t ∗ (z ∗ ) = θ[ ˆP t − Ŝt − ˆp ∗ t (z ∗ )] + Ĉt. (9.150)Log-linearizing the labor supply rules (9.116) and (9.117) and using theprice markup rules (9.124)—(9.125) to eliminate the wage yieldsŷ t (z) = ˆp t (z) − ˆP t − Ĉt, (9.151)ŷt ∗ (z ∗ ) = ˆp ∗ t (z ∗ ) − ˆP t ∗ − Ĉ∗ t . (9.152)Log-linearizing the intertemporal Euler equations (9.112) and (9.113)givesˆP t + Ĉt = ˆδ t + Ĉt+1 + ˆP t+1 , (9.153)ˆP t ∗ t = ˆδ t + Ĉ∗ t+1 + ˆP t+1 ∗ − Ŝt. (9.154)Long-Run ResponseThe log-linearized equations hold for arbitrary t andalsoholdinthenew steady state. By the intertemporal optimality condition (9.112),δ = β in the new steady state which implies ˆδ =0. Notingthatthenominal exchange rate is constant in the new steady state, it followsfrom (9.141) and (9.142)ˆM − ˆP = 1 Ĉ,²(9.155)ˆM ∗ − ˆP ∗ = 1 ² Ĉ∗ . (9.156)By the law-of-one price ˆp(z) = q∗ ˆ (z)+Ŝ. (9.143) and (9.144) become ⇐(194-196)Ĉ = ˆp(z)+ŷ(z) − ˆP − βˆb, (9.157)" #Ĉ ∗ = ˆp ∗ (z ∗ )+ŷ ∗ (z ∗ ) − ˆP nβ∗ + ˆb. (9.158)1 − nTaking a weighted average of the log-linearized budget constraints (9.157)and (9.158) givesĈ w = n[ˆp(z) − ˆP +ŷ(z)] + (1 − n)[ˆp ∗ (z ∗ ) − ˆP ∗ +ŷ ∗ (z ∗ )]. (9.159)