International macroe.. - Free

276CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSLong-Run ResponseThe economy starts out in the 0-steady state. We will solve for the newsteady-state following a permanent monetary or government spendingshock. For any variable X, let ˆX ≡ ln(X/X 0 ), where X is the new(forward-looking) steady state value. Since log-linearized equations(9.43)—(9.50) hold for arbitrary t, they also hold across steady statesand from (9.43), (9.47), (9.48), (9.49) and (9.50) you getĈ w = n[ˆp(z)+ŷ(z) − ˆP ]+(1− n)[ˆp ∗ (z ∗ )+ŷ ∗ (z ∗ ) − ˆP ∗ ] − ĝ w ,(9.55)ŷ(z) =θ[ ˆP − ˆp(z)] + Ĉw +ĝ w , (9.56)ŷ ∗ (z ∗ )=θ[ ˆP ∗ − ˆp ∗ (z ∗ )] + Ĉw +ĝ w , (9.57)(1 + θ)ŷ(z) =−θĈ + Ĉw +ĝ w , (9.58)(1 + θ)ŷ ∗ (z ∗ )=−θĈ∗ + Ĉw +ĝ w , (9.59)where ĝ = G/C0w and ĝ ∗ = G ∗ /C0 w . Log-linearizing the steady statebudget constraints (9.35) and (9.36) and letting ˆb = B/C0w yieldsĈ = rˆb +ˆp(z)+ŷ(z) − ˆP − ĝ, (9.60)µ nĈ ∗ = − rˆb +ˆp ∗ (z ∗ )+ŷ ∗ (z ∗ ) −1 − nˆP ∗ − ĝ ∗ . (9.61)Together, (9.55)—(9.61) comprise 7 equations in 7 unknowns(ŷ, ŷ ∗ , (ˆp(z) − ˆP ), (ˆp ∗ (z ∗ ) − ˆP ∗ ), Ĉ,Ĉ∗ , Ĉw ). There is no easy way tosolve this system. You must bite the bullet and do the tedious algebrato solve this system of equations. 6 The solution for the steady statechanges isĈ = 1 2θ [(1 + θ)rˆb +(1− n)ĝ ∗ − (1 − n + θ)ĝ], (9.62)Ĉ ∗ = 1 "#n(1 + θ)r−2θ (1 − n) ˆb + nĝ − (n + θ)ĝ ∗ , (9.63)Ĉ w = −ĝw 2 , (9.64)ŷ(z) = 1"ĝw1+θ 2 − θĈ#, (9.65)6 Or you can use a symbolic mathematics software such as Mathematica or Maple.I confess that I used Maple.