(Comment on notation: in model current production Yt depends on Kt-1. Uhlig’s In our earlier models Yt depended on Kt. So Kt-1 is the same as Kt Uhlig’s in our earlier models) - Productivity evolves in line with the following exogenous stochastic process: ln Zt = (1-) ln + ln Zt-1 + , N(0,) and 0<<1  t t is random here and so ln(Zt) is a random variable. ln(Zt) follows an AR1 t process that converges on ln . e.g. say start with: ln Zt= ln Zt-1= ln and then there is a positive shock t>0 lasting one period (so E(t)=0 thereafter). then in period t: ln Zt = ln + t period t+1: E(ln Zt+1)= (1-) ln + + = ln + (ln t) t period t+2: E(ln Zt+2)= (1-) ln + + = ln + (ln  t) 2t . period t+n E(ln Zt+n)= ln + nt - Making productivity a random variable makes the model stochastic. - This is new and says that the production function can shift in a random manner that shows some persistence (shock persists but gradually dies out). - Technology or productivity shocks are standard in RBC models. - Now lets go through five steps for this model. Uhlig’s Step 1: Find the Equations Characterizing the equilibrium - Planners problem: maximize the Lagrangian WRT Ct, Kt , t
(Comment on notation: in Uhlig’s model current production Yt depends on Kt-1. In our earlier models Yt depended on Kt. So Uhlig’s Kt-1 is the same as Kt in our earlier models) - Productivity evolves in line with the following exogenous stochastic process: ln Zt = (1-) ln +  ln Zt-1 + t , t N(0,) and 0<<1 t is random here and so ln(Zt) is a random variable. ln(Zt) follows an AR1 process that converges on ln . e.g. say start with: ln Zt= ln Zt-1= ln and then there is a positive shock t>0 lasting one period (so E(t)=0 thereafter). then in period t: ln Zt = ln + t period t+1: E(ln Zt+1)= (1-) ln + (ln + t) = ln + t period t+2: E(ln Zt+2)= (1-) ln + (ln +  t) = ln + 2t . period t+n E(ln Zt+n)= ln + nt - Making productivity a random variable makes the model stochastic. - This is new and says that the production function can shift in a random manner that shows some persistence (shock persists but gradually dies out). - Technology or productivity shocks are standard in RBC models. - Now lets go through Uhlig’s five steps for this model. Step 1: Find the Equations Characterizing the equilibrium - Planners problem: maximize the Lagrangian WRT Ct, Kt , t