Finding the Minimum Using Newton’s Method

1. Finding the Minimum Using Newton’s Method Lecture I

2. Many problems encountered in agricultural and applied economics are posed as optimization problems. • The basic economic problem implies optimization: Economics is the study of the allocation of scarce resources across unlimited and competing human wants and desires.

3. We generally assume that individuals allocate their income to maximize utility while firms allocate inputs to produce output in a way that maximized profit. • At a somewhat more applied level most of our econometric tools also imply some form of optimization of parameters based on an objective function

4. Closed-Form Solutions • In the classroom many of our optimization problems yield closed-form solutions. • If the primal production function can be expressed as a quadratic function of inputs, the optimizing behavior can be determined as linear functions of the relative input prices.

5. Closed Form Solution of the Primal Production Problem

6. However, the set of all functions with closed form solutions is a subset of feasible functions. • Thus, limiting the set of functions to those functions with closed form solutions may unduly limit the production functions considered.

7. Fourier Production Function

8. My contention is that a simple closed form as presented in Equation 1 no longer exists. • The question then becomes: Can we use this specification to derive useful information about economic optimizing behavior? • The answer is obviously yes (you are leading the witness).

9. Let us start by positing that a series of steps could be used to solve the general optimization problem

11. Finding a Univariate Minimum • Finding the Univariate Minimum (Algorithm1.ma) • Finding a minimum of a simple univariate quadratic is trivial given the conditions we developed in the previous section

13. In addition, straightforward transformations add little additional complexity

14. One possibility for a more complex function is

17. Thus, these functions have discontinuities at some points in their range. If we restrict the search to points where x is greater than -10.0, then the function is defined at all points

19. The method of bisection involves shrinking an interval known to contain the zero of the derivative. • Starting with an interval of [-8, 20]. The bisection of that interval is 6. At 6 the value of the derivative is 2.882. • Thus, the subinterval [6,20] can be excluded from the search and the new interval becomes [-8,6].

20. Solving Using Bisection

21. Newton’s Method • Newton’s Method • The theoretical foundation of Newton’s method involves inscribing triangles inside the function. Mathematically

22. f(x) x

25. Finding Multivariate Maximum • Finding the Multivariate Maximum (Algorithm2.ma) • The basic difference between univariate and multivariate optimization is that we want to solve for the x that simultaneously make a system of equations equal to zero.

26. Appealing again to the second order Taylor series expansion

27. A simple problem with a Cobb-Douglas utility function

28. Starting with x=(1,1,1)