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5.2. CALIBRATING A TWO-COUNTRY MODEL 157You can write second—order stochastic difference equation (5.51) as(1 − m 1 L − m 2 L 2 )ˆk t+1 w = Wt w . The roots of the polynomial(1 − m 1 z − m 2 z 2 )=(1− ω 1 L)(1 − ω 2 L) satisfy m 1 = ω 1 + ω 2 and m 2 =−ω 1 ω 2 . Under the parameter values used to calibrate the model and usingtheqquadratic formula, the roots are, z 1 = (1/ω 1 ) =[−m 1 − m 2 1 +4m 2 ]/(2m 2 ) ' 1.17, and z 2 = (1/ω 2 ) =q[−m 1 + m 2 1 +4m 2 ]/(2m 2 ) ' 0.84. The stable root |z 1 | > 1 lies outsidethe unit circle, and the unstable root |z 2 | < 1 lies inside the unit circle.From the law of motion governing the technology shocks (5.33), youhaveà w t+1 =(ρ + δ)Ãw t + ² w t , (5.52)where ² w t = ² t + ² ∗ t . Now E t W t+k = m 3 à w t+1 + m 4 à w t + m 5 =[m 3 (ρ + δ) +m 4 ](ρ + δ) k à w t + m 5 . As in the one-country model, usethese forecasting formulae to solve the unstable root forwards and thestable root backwards. The solution for the world capital stock is˜k t+1 w = ω 1˜k t w (ρ + δ) ³− [m3 (ρ + δ)+m 4ω 2 − (ρ + δ)t + m 5´. (5.53)Now you need to recover the domestic and foreign components ofthe world capital stock. Subtract (5.49) from (5.48) to get!Øb7˜k t+1 − ˜k t+1 ∗ = − ã 7ã 3 − ã 4!Øb8 − ã 8à t+1 + à ∗ t+1ã 3 − ã 4 (5.54)Add (5.53) to (5.54) to get˜k t+1 = 1 2 [˜k t+1 w +(˜k t+1 − ˜k t+1 ∗ )]. (5.55)The date t+1 world capital stock is predetermined at date t. Howthatcapital is allocated between the home and foreign country depends onthe realization of the idiosyncratic shocks Ãt+1 and Ã∗ t+1.Given ˜k t ,and˜k ∗ t , it follows from the production functions that theoutputs areỹ t = f A à t + f k˜kt = yÃt + α y k ˜k t , (5.56)ỹ ∗ t = f ∗ AÃ∗ t + f ∗ k ˜k ∗ t = y ∗ à ∗ t + α y∗k ∗ ˜k ∗ t , (5.57)