1. Lecture 4 Exposition of the Lagrange Method
This lecture is technical. Please read Chow, Dynamic Economics, chapters 1 and 2.

2. Brief explanation of the Lagrange method for dynamic optimization– 3 steps
1. Start with the constrained maximization problem max r(x,u) subject to x=f(u).
Set up the Lagrange expression
L = r(x,u) –λ[x-f(u)].
Differentiate L with respect to x, u and λ to obtain three first-order conditions.
Solve these equations for the three variables.

3. step 2 - Generalize above procedure to many periods
Objective function is a weighted sum of r(x(t),u(t)) over time t.
Constraints are x(t+1) = f(x(t),u(t)).
We call x the state variable and u the control variable.
Set up the Lagrange expression
L = Σt βt{r(x(t),u(t)) –λt+1[x(t+1)- f(x(t),u(t))]} and differentiate to obtain first-order conditions to solve for the u’s and x’s.

4. Step 3 - stochastic Model is xt+1 = f(xt, ut, εt), εt stochastic.
We now have an expectation operator in front of the previous objective function
L = E Σ1Tβt {r(xt ,ut) – λt+1[xt+1 - f(xt ,ut)]}
The first-order conditions can still be obtained by differentiation after the summation sign.
This summarizes all I know after 30 years of work on dynamic optimization.

5. Solution of a linear-quadratic optimal control problem

17. Problems
Do problems 1 and 3 of Chapter 2 of Chow, Dynamic Economics