International macroe.. - Free

120 CHAPTER 4. THE LUCAS MODELn t = Pt ∗ c yt, (4.43)which you can use to eliminate m t and n t from the allocation of currentperiod wealth to rewrite (4.41) asW t = c xt + S tPt∗P tc yt| {z }Goods+ ω xt e t + ω yt e ∗ t| {z }Equity+ ψ Mt r t + ψ Nt rt∗ | {z }. (4.44)Money transfersThe consolidated budget constraint of the home individual is thereforec xt + S tPt∗P t+ S tP ∗c yt + ω xt e t + ω yt e ∗ t + ψ Mtr t + ψ Nt rt ∗ = P t−1ω xt−1 x t−1P tt−1ω yt−1 y t−1 + ψ Mt−1∆M t+ ψ Nt−1S t ∆N tP tP tP t+ω xt−1 e t + ω yt−1 e ∗ t + ψ xt−1r t + ψ yt−1 rt ∗ . (4.45)The domestic household’s problem is to maximize⎛∞XE t⎝j=0⎞β j u(c xt+j ,c yt+j ) ⎠ (4.46)(88-92)⇒subject to (4.45). The associated Euler equations areS t Pt∗c yt : u 1 (c xt ,c yt )=u 2 (c xt ,c yt ), (4.47)P tà !#Ptω xt : e t u 1 (c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) x t + e t+1 , (4.48)P t+1ÃSt+1 P ∗tω yt : e ∗ t u 1(c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) y t + e ∗ t+1 , (4.49)P t+1à !#∆Mt+1ψ Mt : r t u 1 (c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) + r t+1 , (4.50)P t+1à !#ψ Nt : rt ∗ u ∆Nt+1 S t+11(c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 )+ rt+1∗ .(4.51)P t+1The foreign agent solves the analogous problem which generate a set ofsymmetric Euler equations, do not need to be stated here.!#