International macroe.. - Free

9.1. THE REDUX MODEL 271consumption and taxes (P t (C t +T t )). Wealth is accumulated in a similarfashion by the foreign agent. The budget constraint for home andforeign agents areP t B t +M t =(1+r t−1 )P t B t−1 +M t−1 +p t (z)y t (z)−P t C t −P t T t , (9.22)Pt ∗ B∗ t + M t∗ =(1+r t−1 )Pt ∗ B∗ t−1 + M t−1 ∗ + p∗ t (z∗ )yt ∗ (z∗ ) − Pt ∗ C∗ t − P t ∗ T t ∗ .(9.23)We can simplify the budget constraints by eliminating p(z) andp ∗ (z ∗ ).Because output is demand determined, re-arrange (9.20) to getp t (z)y t (z) =P t y t (z) θ−1θ [Ct w +G w t ] 1 θ , and substitute the result into (9.22).Do the same for the foreign household’s budget constraint using the zeronet supply constraint on bonds (9.15) to eliminate B ∗ to getC t = (1+r t−1 )B t−1 − B t − M t − M t−1− T tP t+y t (z) θ−1θ [Ct w + G w t ] 1 θ , (9.24)Ct ∗ = (1+r t−1 ) −nB t−11 − n + nB t1 − n − M t ∗ − Mt−1∗Pt∗ − Tt∗+yt ∗ (z∗ ) θ−1θ [Ct w + G w t ] 1 θ . (9.25)⇐(157)Euler Equations. C t ,M t , and B t are the choice variables for the domesticagent and Ct ∗ ,Mt ∗ , and Bt∗ are the choice variables for the foreignagent. For the domestic household, substitute the budget constraint(9.22) into the lifetime utility function (9.2) to transform the probleminto an unconstrained dynamic optimization problem. Do the samefor the foreign household. The Euler-equations associated with bondholding choice are the familiar intertemporal optimality conditionsC t+1 = β(1 + r t )C t , (9.26)Ct+1 ∗ = β(1 + r t )Ct ∗ . (9.27)The Euler-equations associated with optimal cash holdings are themoney demand functionsM tP t=" # 1γ(1 + it²)C t , (9.28)i t