I- A. Consider a Money-in-the-Utility model of the kind we studied in the class with no...

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I- A. Consider a Money-in-the-Utility model of the kind we studied in the class with no population growth, i.e., n = 0: A-1) Assume that the aggregate production function is Yt = F(Kt-1,Nt) =Kt-1N-; K-1 is total capital stock at the beginning of t carried over from t-1; N is total physical labor; and 0 =0 to maximize lifetime discounted utility given by (1) Et-o ' u (ct, mt) subject to the following per capita budget constraint (2) f(kt-1) + (1-8)kt-1 + [(1+it-1) bt-1]/(1+) + tt + m-1/(1+) = c + k + b + mt for all t >=0 where individual subjective discount rate, it-1= nominal interest rate at time t-1, 8-a constant capital depreciation rate, and inflation rate The first order conditions of the optimization problems are given by the following equations: (3) u (c,,m) = uc (C+m+) (f' (k)+1-8) (1 + i) (1++1) (4) u (c,,m) = uc (C+19 m +1) (5) u (c,,m) = um (c,m) + (uc (C+1, m+1)) (1++1) for all t >=0 for all t >=0 for all t >=0 QI-A-1) Show that the above production function is a constant return to scale production function jointly with respect to capital (= K-1) and labor (Nt), and derive the per capita output yt = (Y+/Nt) in terms of kt-1-(Kt-1/Nt-1) and n (population or labor growth rate) using the constant return to scale property of the production function. (7pts) Q I-A-2) Explain clearly what the right hand side of each equation (3), (4), and (5) represents. (12pts) I- A. Consider a Money-in-the-Utility model of the kind we studied in the class with no population growth, i.e., n = 0: A-1) Assume that the aggregate production function is Yt = F(Kt-1,Nt) =Kt-1N-; K-1 is total capital stock at the beginning of t carried over from t-1; N is total physical labor; and 0 =0 to maximize lifetime discounted utility given by (1) Et-o ' u (ct, mt) subject to the following per capita budget constraint (2) f(kt-1) + (1-8)kt-1 + [(1+it-1) bt-1]/(1+) + tt + m-1/(1+) = c + k + b + mt for all t >=0 where individual subjective discount rate, it-1= nominal interest rate at time t-1, 8-a constant capital depreciation rate, and inflation rate The first order conditions of the optimization problems are given by the following equations: (3) u (c,,m) = uc (C+m+) (f' (k)+1-8) (1 + i) (1++1) (4) u (c,,m) = uc (C+19 m +1) (5) u (c,,m) = um (c,m) + (uc (C+1, m+1)) (1++1) for all t >=0 for all t >=0 for all t >=0 QI-A-1) Show that the above production function is a constant return to scale production function jointly with respect to capital (= K-1) and labor (Nt), and derive the per capita output yt = (Y+/Nt) in terms of kt-1-(Kt-1/Nt-1) and n (population or labor growth rate) using the constant return to scale property of the production function. (7pts) Q I-A-2) Explain clearly what the right hand side of each equation (3), (4), and (5) represents. (12pts)