380 L2-Optimization

380 L2-Optimization - Lecture 2Optimization-Getting Started...

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Lecture 2- Optimization-Getting Started R. Pope Introduction Economics is about optimizing behavior which can be stated as maximization (or minimization) subject to constraint. Somewhere in your calculus class, you were asked to find a maximum or minimum of a function y=f(x). In economics, f(x) is called the objective function and x is the control or decision variable. That is, what value of x corresponds to an extremum (minimum or maximum). Calculus is an amazing useful tool for economic theory, applied economics, and econometrics. One can study the theory of optimization one’s whole life and many people do. Here only a few simple notions will get us started. Approximation of Functions It is useful to remember a result from calculus. A function f(x) can be approximated to second order via a Taylor’s series at a point x 0 as: 2 0 00 0 0 ''( ) () ( ) ' ( ) ( ) 2 fx f xf x f x x x x x  A first order approximation would only include the two terms on the right where means approximately because there will be an error or remainder. Example: y = 3ln(x) approximated at x=x 0 =10 2 2 33 ( ) 3ln(10) ( 10) ( 10) 10 (10 )2 x x approximates the function to the second order. That is, Taylor’s series approximate functions with polynomials. Univariate Optimization Throughout, maximum problems are considered. Maximizing a function f(x) is equivalent to minimizing –f(x). If a maximum occurs on an open set, the maximum will occur where '( ) 0 . Call that choice x*. Thus, at a maximum (or a minimum) Result 1: The first order condition for a maximum is 1) '( *) 0 . Example: let y=ax+bx 2 a > 0, b < 0. The maximum will occur at a+2bx=0 if it has a maximum or x* = -a/(2b). Why the restriction on an open set and how can we be assured there is a maximum. To make things more concrete, let a=2 and b = -.1. If there is a maximum on (0, ), it will occur at x*=10 or (-a/2b). Suppose that I optimize this function over a closed set [0,5], what value of x maximizes this function? The answer is x=5 because it is increasing over that interval. What if the closed interval where [5,15]. We would find the same answer as before with an open set at x*=10. Thus, maximizing over open or closed sets can make a difference. Note that many if not most economic variables are presumed to be positive so does not cause any issues to maximize a function f(x) on [0, ). To illustrate, consider the function y=16-(x+2) 2 . The unconstrained optimum obtained by differentiation is at x = -2 (the derivative is -2(x+2) and setting to zero gives x = -2). At that point the function is 16. It declines for increasing x. For example for x=0, the function is 12. If the maximum is constrained over the set [0, ), we
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note that the maximum occurs where the derivative is negative. To verify, at x=0 the maximum, the slope is -4. This is an example of what are called the Kuhn-Tucker conditions which will be discussed later.
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This note was uploaded on 04/07/2011 for the course ECON 380 taught by Professor Showalter,m during the Winter '08 term at BYU.

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380 L2-Optimization - Lecture 2Optimization-Getting Started...

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