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152 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESCobb—Douglas production functions for the home and foreign counties,with normalized labor input N = N ∗ = 1. (5.30) and (5.31) are thedomestic and foreign capital accumulation equations, and (5.31) is thenew form of the resource constraint. Both countries have the sametechnology but are subject to heterogeneous transient shocks to totalproductivity according to" # " # " #" # " #At 1 − ρ − δ ρ δ At−1 ²t=++ , (5.33)1 − ρ − δ δ ρA ∗ tA ∗ t−1where (² t ,² ∗ t ) 0 iid∼ N(0, Σ). We set ρ =0.906, δ =0.088, Σ 11 = Σ 22 =2.40e−4, and Σ 12 = Σ 21 =6.17e−5. The contemporaneous correlationof the innovations is 0.26.Apart from the objective function, the main difference between thetwo-county and one-country models is the resource constraint (5.32).World output can either be consumed or saved but a country’s net saving,which is the current account balance, can be non—zero(y t − c t − i t = −(y ∗ t − c ∗ t − i ∗ t ) 6= 0).Let λ t =(k t+1 ,k ∗ t+1,k t ,k ∗ t ,A t ,A ∗ t ,c ∗ t ) be the state vector, and indicatethe dependence of consumption on the state by c t = g(λ t ), andc ∗ t = h(λ t )(whichequalsc ∗ t trivially). Substitute (5.28)—(5.31) into(5.32) and re-arrange to getc t = g(λ t )=f(A t ,k t )+f(A ∗ t ,kt ∗ ) − γ(k t+1 + kt+1),∗+(1 − δ)(k t + kt ∗ ) − c ∗ t (5.34)c ∗ t = h(λ t )=c ∗ t . (5.35)For future reference, the derivatives of g and h are,² ∗ tg 1 = g 2 = −γ,g 3 = f k (A, k)+(1− δ),g 4 = f k (A ∗ ,k ∗ )+(1− δ),g 5 = f(A, k)/A,g 6 = f(A ∗ ,k ∗ )/A ∗ ,g 7 = −1,h 1 = h 2 = ···= h 6 =0,h 7 =1.