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2. Dynamic Stochastic General Equilibrium (20pt for each) Consider an economy populated by many identical households who maximize their life time utility: U0=E0t=0tln(ctxnt), where ct and nt are the consumption and labor at period t=, and (0,1) is the discounting factor. lnx is the natural log function, and its first derivative is 1/x. Also, the production takes place using capital and labor inputs (kt and nt ) to the following technology at time t : yt=atktnt1 where yt and at denote the output and productivity (TFP). The productivity follows i.i.d. lognormal distribution: atLN(0.52,2). Note that E[x]=exp(+22/2) where xLN(,2). The capital accumulation is kt+1=it+(1)kt where we set to be one (full depreciation). (1) Set the maximization problem and Lagrangean equation. (2) Derive the intertemporal decision rule. Explain its economic meaning. (3) Derive the intratemporal decision rule. Explain its economic meaning. (4) Find the optimal comsumption. (You may need to "guess and verify".) 2. Dynamic Stochastic General Equilibrium (20pt for each) Consider an economy populated by many identical households who maximize their life time utility: U0=E0t=0tln(ctxnt), where ct and nt are the consumption and labor at period t=, and (0,1) is the discounting factor. lnx is the natural log function, and its first derivative is 1/x. Also, the production takes place using capital and labor inputs (kt and nt ) to the following technology at time t : yt=atktnt1 where yt and at denote the output and productivity (TFP). The productivity follows i.i.d. lognormal distribution: atLN(0.52,2). Note that E[x]=exp(+22/2) where xLN(,2). The capital accumulation is kt+1=it+(1)kt where we set to be one (full depreciation). (1) Set the maximization problem and Lagrangean equation. (2) Derive the intertemporal decision rule. Explain its economic meaning. (3) Derive the intratemporal decision rule. Explain its economic meaning. (4) Find the optimal comsumption. (You may need to "guess and verify".)