Consider the dynamics programming problem we discussed in the class:

1. Solve the above problem using brute-force grid search method using only loops (do not use any matrix algebra). Plot the theoretical and numerical value and policy functions

2. Solve the above problem using brute-force grid search method by exploiting monotonicity of the policy function. Plot the theoretical and numerical value and policy functions

3. Solve the above problem using brute-force grid search method by exploiting the concavity of the objective function. Plot the theoretical and numerical value and policy functions

4. Solve the above problem using brute-force grid search method and use matrix algebra as much as you can to avoid the loops. Plot the theoretical and numerical value and policy functions

5. Prepare a table comparing the speed of the code in each four cases above.

6. Using the policy function simulate two economies for 100 periods: (1) starting from initial capital stock k0 = ¯ k (2) starting from initial capital stock k0 = k. Plot the evolution of these two economies on the same plot (The figure will have time in the x-axis and capital stock in the y-axis).