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274CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSTogether, these relations tell us that 0-steady-state output at home andabroad are equal to consumption" # 1/2θ − 1y 0 (z) =y0(z ∗ ∗ )== C 0 = C0 ∗ = C0 w . (9.37)ρθNominal and real interest rates in the 0-steady state are equalizedwith (1+i 0 )/i 0 =1/(1−β). By (9.28) and (9.29), 0-steady state moneydemand isM 0P 0= M ∗ 0P ∗ 0=" # 1/²γy0 (z). (9.38)1 − βFinally by (9.38) and PPP, it follows that the 0-steady-state nominalexchange rate isS 0 = M 0M0∗ . (9.39)(9.39) looks pretty much like the Lucas-model solution (4.55).Log-Linear Approximation About the 0-Steady StateWe denote the approximate log deviation from the 0-steady state witha ‘hat’ so that for any variable ˆX t =(X t − X 0 )/X 0 ' ln(X t /X 0 ). Theconsolidated budget constraints (9.32) and (9.33) with B t−1 = B 0 =0becomeC t = p t(z)y t (z) − B t − G t , (9.40)P tCt ∗ = p∗ t (z ∗ )y ∗Pt∗ t (z∗ )+µ nBt1 − n− G ∗ t . (9.41)Multiply (9.40) by n and (9.41) by 1 − n and add together to get theconsolidated world budget constraintC w t= nà !pt (z)y t (z)+(1− n)P tÃp∗t (z ∗ )Log-linearizing (9.42) about the 0-steady state yieldsP ∗t!y ∗ t (z∗ ) − G w t . (9.42)Ĉ w t= n[ˆp t (z)+ŷ t (z) − ˆP t ]+(1− n)[ˆp ∗ t (z∗ )+ŷ ∗ t (z∗ ) − ˆP ∗t ] − ĝw t , (9.43)