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140 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES0.150.1Investment0.05GDP0-0.05-0.1-0.15Consumption73 75 77 79 81 83 85 87 89 91 93 95Figure 5.2: Hodrick-Prescott Þltered cyclical observations.shock, A t . The production function isY t = A t K α t (NX t ) 1−α .α is capital’s share. Most estimates for the US place 0.33 ≤ α ≤ 0.40.Output can be consumed or saved. Savings (or investment I t )areused to replace worn capital and to augment the current capital stock.Capital depreciates at a rate δ and evolves according toK t+1 = I t +(1− δ)K t .There is no government and no foreign sector. There are also nomarket imperfections so we can work with Þctitious social planner’sproblem as we did with the Lucas model.Problem 1. The social planner wants to maximizeE t ∞ Xj=0β j U(C t+j ), (5.1)