Question 2 (50 pts] Consider an economy in which households seek to maximize life- time utility 2:=oftu(ct) and the aggregate resource constraint is given by Ct +kt+1 = Yt, i.e. there is full depreciation of capital. 1 (a) Let u(c) = log c and yt k. Find the optimality conditions for social planners problem. Hint: Look at the hint above. (b) Derive the steady state values for all endogenous variables in this economy. (c) Find the optimal saving rate and show that is constant across periods. (d) Assume that u(c) = log c and yt Eztk where Zt is iid random variable with E[z] = 1. Show that optimal saving rate is constant again. (e) Assume that households have persistent habits, i.e. households don't like too much fluctuations in consumption. The preference of these households can be represented as Eo B'uct m x C41). t=0 Assume that there is full depreciation of capital, i.e. C+ +kt+1 = yt = ztk where Zt is an iid random variable and E[z] = 1. Derive the Euler equation of this model. Show the differences between this Euler equation and the usual one. Question 2 (50 pts] Consider an economy in which households seek to maximize life- time utility 2:=oftu(ct) and the aggregate resource constraint is given by Ct +kt+1 = Yt, i.e. there is full depreciation of capital. 1 (a) Let u(c) = log c and yt k. Find the optimality conditions for social planners problem. Hint: Look at the hint above. (b) Derive the steady state values for all endogenous variables in this economy. (c) Find the optimal saving rate and show that is constant across periods. (d) Assume that u(c) = log c and yt Eztk where Zt is iid random variable with E[z] = 1. Show that optimal saving rate is constant again. (e) Assume that households have persistent habits, i.e. households don't like too much fluctuations in consumption. The preference of these households can be represented as Eo B'uct m x C41). t=0 Assume that there is full depreciation of capital, i.e. C+ +kt+1 = yt = ztk where Zt is an iid random variable and E[z] = 1. Derive the Euler equation of this model. Show the differences between this Euler equation and the usual one