1. Graded; worth 25 points. Consider the following planning economy problem: 00 10' (2 i' gt 3 1,131 ll Z 1 0' i=1} such that C: + I: = yr; y: = ztktai and kt-il =It+{1 _5lkt- In addition; lnzt = plnzhl + 5:. Above: c is aggregate consumption; 3:: is the aggregate capital stock; IE: is the ex pectation operator conditional on the information available up and through period t inclusive; .3 E (D; l) is the subjective parametric discount factor (,3 is the Greek letter \"beta\"}; 3; is aggregate output; I is aggregate investment; 5 > [I is the depreciation rate of capital (5 is the Greek letter \"delta\"); 2: is productivity; or E [I]; l} is a parame- ter (a is the Greek letter t1alpha\"); p E (0., 1) is the productivity process persistence parameter (p is the Greek letter Edrho\"); and {Z is disturbance term associated with the exogenous productivity process (g is the Greek letter ii\"). As usual; all nonprice variables are normalized by the aggregate population; which consists of a unit mass. [3) State the Lagrangian associated with this problem using a single constraint. (1)) State the rst order conditions associated with this problem. Please show;' explain your work thoroughly. (c) Derive the steady state values of c; I; k; and 2. Please show/explain your work thoroughly