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142 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESmoving steady state which is inconvenient. To Þx this,youcanÞrsttransform the model to get a Þxed steady state by normalizing all thevariables by labor efficiency units. Let lower case letters denote thesenormalized valuesy t = Y t, k t = K t, i t = I t, c t = C t.X t X t X t X tDividing (5.2) by X t gives y t = A t kt α. Dividing (5.3) by X t givesγk t+1 = i t +(1− δ)k t . To normalize lifetime utility (5.1), note thatP ∞j=0β j ln X t+j = P ∞j=0 β j ln(γ j X t )=ln(X t )/(1 − β)+ln(γ) P ∞j=0 jβ j(ch.5-1)⇒= ln(X t )/(1 − β) +ln(γ)β/(1 − β) 2 < ∞. Using this fact, addingand subtracting P ∞j=0 β j Pln(X t+j ) to (5.1) gives E ∞j=0t β j U(C t+j )=PΩ t +E ∞j=0t β j U(c t+j )whereU(c) =ln(c) andΩ t =ln(X t )/(1 − β)+β ln(γ)/(1 − β) 2 .SinceΩ t is exogenous, we can ignore it when solvingthe planner’s problem. We will call the transformed problem, problem2. This is the one we will solve.Problem 2. It will be useful to use the notation f(A t ,k t )=A t k α .SinceΩ is a constant, the social planner’s growth problem normalized bylabor efficiency units is to maximizeE t ∞ Xj=0β j U(c t+j ), (5.6)subject to y t = f(A t ,k t )=A t kt α , (5.7)γk t+1 = i t +(1− δ)k t , (5.8)y t = c t + i t , (5.9)U(c) =ln(c). (5.10)It will be useful to compactify the notation. Let λ t =(k t+1 ,k t ,A t ) 0 andcombine the constraints (5.7)—(5.9) to form the consolidated budgetconstraintc t = g(λ t ) = f(A t ,k t ) − γk t+1 +(1− δ)k t= A t k α t − γk t+1 +(1− δ)k t . (5.11)Under Cobb-Douglas production and log utility, you havef k = αyk , f kk = α(α − 1) y k 2 , u c = 1 c , u cc = −1c 2 .