Consumer "Habit Persistence" problem (35\%). Consider the following infinite horizon utility maximization problem. Notice that utility depends on both current consumption and previous period consumption. The habit persistence parameter shows how much of current consumption is influenced by previous consumption patterns. max[t=0t(lnct+lnct1)]s.t.ct+kt+1AktA>000k0>0,andc1given. (a) Write down the Lagrangian for this optimization problem. (You do not have to solve it (yet), just write down the Lagrangian equation.) (5 marks) (b) State the control and state variables. (5 marks) (c) Write down the Bellman equation. Follow the convention to drop the time subscripts - that is express ct=c,ct+1=c,ct1=c1 and so on. (5 marks) (d) Derive the Euler equation. (8 marks) (e) Show that the solution of the Bellman equation has the form E+Flnk+Glnc1 where E,F and G are constants. Express these constants in terms of parameters A,, and . (This is the method of undetermined coefficients or the so-called "guess and verify method".) (6 marks) (f) What is the role of habit persistence in the solution to this model? Does current consumption c depend on c1, based on consumer habit, given this setup? What is the intuition behind this result? Hint: Show that the optimal policy is of the form c=Ak where is a constant. (6 marks) Consumer "Habit Persistence" problem (35\%). Consider the following infinite horizon utility maximization problem. Notice that utility depends on both current consumption and previous period consumption. The habit persistence parameter shows how much of current consumption is influenced by previous consumption patterns. max[t=0t(lnct+lnct1)]s.t.ct+kt+1AktA>000k0>0,andc1given. (a) Write down the Lagrangian for this optimization problem. (You do not have to solve it (yet), just write down the Lagrangian equation.) (5 marks) (b) State the control and state variables. (5 marks) (c) Write down the Bellman equation. Follow the convention to drop the time subscripts - that is express ct=c,ct+1=c,ct1=c1 and so on. (5 marks) (d) Derive the Euler equation. (8 marks) (e) Show that the solution of the Bellman equation has the form E+Flnk+Glnc1 where E,F and G are constants. Express these constants in terms of parameters A,, and . (This is the method of undetermined coefficients or the so-called "guess and verify method".) (6 marks) (f) What is the role of habit persistence in the solution to this model? Does current consumption c depend on c1, based on consumer habit, given this setup? What is the intuition behind this result? Hint: Show that the optimal policy is of the form c=Ak where is a constant. (6 marks)