3-period consumption-savings problem (30%). This question extends the 2 -period model we discussed n class to 3-periods. Agents live for 3-periods and maximize their discounted intertemporal utility subject o their resource constraint. s.t.max[ln(c1)+ln(c2)+2ln(c3)]ct+kt+1(1+r)kt00given. (a) Write down the Lagrangian for this optimization problem and solve for the optimal c1,c2,c3,k2, k3 and k4. (7 marks) (b) Write the problem in recursive form and solve for the optimal c1,c2,c3,k2,k3 and k4. (10 marks) (c) Compare your answers from part (b) with part (a), are they the same? (7 marks) (d) Suppose (1+r)=1, what are the optimal consumption levels in periods 1,2 , and 3 ? (6 marks) 3-period consumption-savings problem (30%). This question extends the 2 -period model we discussed n class to 3-periods. Agents live for 3-periods and maximize their discounted intertemporal utility subject o their resource constraint. s.t.max[ln(c1)+ln(c2)+2ln(c3)]ct+kt+1(1+r)kt00given. (a) Write down the Lagrangian for this optimization problem and solve for the optimal c1,c2,c3,k2, k3 and k4. (7 marks) (b) Write the problem in recursive form and solve for the optimal c1,c2,c3,k2,k3 and k4. (10 marks) (c) Compare your answers from part (b) with part (a), are they the same? (7 marks) (d) Suppose (1+r)=1, what are the optimal consumption levels in periods 1,2 , and 3 ? (6 marks)