International macroe.. - Free

116 CHAPTER 4. THE LUCAS MODELc yt : q t u 1 (c xt ,c yt )=u 2 (c xt ,c yt ), (4.31)Ã !#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 , (4.32)P t+1Ã !#ω yt : e ∗ t u Pt1(c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) q t y t + e ∗ t+1 .(4.33)P t+1The foreign household solves an analogous problem. Using the foreigncash-in-advance constraintm ∗ t = P t(c ∗ t + q tc ∗ yt ). (4.34)the consolidated budget constraint for the foreign household is(84-86)⇒c ∗ xt + q t c ∗ yt + ω ∗ xte t + ω ∗ yte ∗ t = P t−1P t[ω ∗ xt−1x t−1 + ω ∗ yt−1q t−1 y t−1 ]The job is to maximize+ ∆M t2P t+ ω ∗ xt−1e t + ω ∗ yt−1e ∗ t . (4.35)⎛∞XE t⎝j=0β j u(c ∗ xt+j,c ∗ yt+j)subject to (4.35).The foreign household’s problem generates a symmetric set of Eulerequationsc ∗ yt : q tu 1 (c ∗ xt ,c∗ yt )=u 2(c ∗ xt ,c∗ yt ),ω ∗ xt : e t u 1 (c ∗ xt,c ∗ yt) =βE t"u 1 (c ∗ xt+1,c ∗ yt+1)"⎞⎠ ,ÃPtP t+1x t + e t+1!#,Ãωyt ∗ : e∗ t u 1(c ∗ xt ,c∗ yt )=βE t u 1 (c ∗ xt+1 ,c∗ yt+1 ) Ptq t y t + e ∗ t+1P t+1The adding-up constraints that complete the model are1 = ω xt + ωxt,∗1 = ω yt + ωyt,∗M t = m t + m ∗ t ,x t = c xt + c ∗ xt ,y t = c yt + c ∗ yt.!#.