Homework

1. Program the solution of unconstraint minimization for \(T=80\). How many variables we need to solve the problem?

2. Program the solution of constraint minimization for \(T=80\). How many equations, variables and constraint we need?

3. Program the solution of Euler equation system solving by \(c_t\) for \(T=80\). How many equations and variables we need?

4. Program the solution of Euler equation system solving by \(a_t\) for \(T=80\). How many equations and variables we need?

5. Program the solution using bisection method (we review using Solow) for \(T=80\). Explain the implementation.

6. Plot the solution of (1), (2), (3), (4 )and (5) for \(c_t\) and \(a_t\) in a subplot.

7. Plot the time of solution by T and by method.

8. Analize the role of \(\gamma\) in the utility function. How change utility, the dynamics of \(c_t\) and \(a_t\) for different \(\gamma\) values. Analize theoretically and graphically.

9. Present the Lagrangean, and the FOC and the Euler equations for the consumption-saving model.

10. Using your fast method from 1-5, solve the consumption-saving model using a CARA utility function \(u(c)=\frac{1-e^{ac}}{a}\) for \(a\neq 0\). Plot the solution for \(c_t\) and \(a_t\) and compare with the solution of the CRRA utility.