International macroe.. - Free

156 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESAt this point, the marginal beneÞt from looking at analytic expressionsfor the coefficients is probably negative. For the speciÞc calibration ofthe model the numerical values of the coefficients are,ã 1 =0.105, ˜b1 =0.105,ã 2 =0.105, ˜b2 =0.105,ã 3 = −0.218, ˜b3 = −0.212,ã 4 = −0.212, ˜b4 = −0.218,ã 5 =0.107, ˜b5 =0.107,ã 6 =0.107, ˜b6 =0.107,ã 7 = −0.128, ˜b7 = −0.161,ã 8 = −0.159, ˜b8 = −0.130,ã 9 =0.158, ˜b9 =0.158,ã 10 =0.158, ˜b10 =0.158,ã 11 =0.007, ˜b11 =0.007.You can see that ã 3 +ã 4 = ˜b 3 +˜b 4 and ã 7 +˜b 7 =ã 8 +˜b 8 which meansthat there is a singularity in this system. To deal with this singularity,let Ãw t = Ãt + Ã∗ t denote the ‘world’ technology shock and add (5.48)to (5.49) to getã 1˜kw t+2 +ã3 +ã 42˜k t+1+ã w + ˜b 75˜kw t +ã7 Ã w2t+1+ã 9 Ã w + ˜b 11t +ã11 2=0. (5.50)(5.50) is a second—order stochastic difference equation in ˜k t w = ˜k t + ˜k t ∗ ,which can be rewritten compactly as 4˜k t+2 w − m 1˜k t+1 w − m 2˜k tw = Wt+1 w (5.51)where Wt+1 w = m 3 Ã w t+1 + m 4 Ã w t ,andm 1 = −(ã 3 +ã 4 )/(2ã 1 ),m 2 = −ã 5 /ã 1 ,m 3 = −(ã 7 + ˜b 7 )/(2ã 1 ),m 4 = −ã 9 /ã 1 ,m 5 = −ã11 + ˜b 112ã 11.4 Unlike the one-country model, we don’t want to write the model in logs becausewe have to be able to recover ˜k and ˜k ∗ separately.