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Unformatted text preview: The Simple Mathematics of Optimization L A T E X file: Optimization-nb-all Daniel A. Graham, June 14, 2005 There are two main approaches to solving the optimization problems that arise in Economics: the method of Substitution and the method of Lagrangian Multipliers . The method of Substitution is stressed in this class. While this handout illustrates the substitution method in the context of the consumer choice problem, the method will be applicable to optimization problems throughout the course. The Standard Consumer Choice Problem The standard optimization problem is to maximize a utility function subject to a budget constraint. More precisely, the problem is to choose quantities of two commodities denoted by x 1 and x 2 , respec- tively, in order to maximize a funtion, u( x 1 , x 2 ) subject to the requirement that the chosen quantities satisffy the budget constraint. In: budget = p 1 x 1 + p 2 x 2 == m ; The values of x 1 and x 2 are to be chosen to solve the maximization problem. Their values will thus be determined " inside the model " and they are, accordingly, called endogenous variables. The prices of the commodities, p 1 and p 2 , and the persons income, m, are determined " outside the model " and are called exogenous variables. The exogenous variables can be thought of as constants. The Substitution Method This approach first asks a question and then, depending upon the answer, proceeds through a series of specific steps. Question: Will the budget constraint hold as an equality in the optimal solution? The answer to this is " yes " either if the partial derivative of the utility function wrt x 1 or if the partial derivative of the utility function wrt x 2 is is positve. Why? If either of these marginal utilities is positive then the optimal choice could not lie below the budget constraint since utility could be made greater by increasing the consumption of which ever commodity had a positive marginal utility. And, as noted by Sherlock Holmes, " When you have eliminated the impossible, whatever remains must be the truth. " Suppose, for example, that the utility function is In: utility = x 1 x 2 ; Then since the partial derivatives In: D[ utility , x 1 ] Out: x 2 and In: D[ utility , x 2 ] Page 1 of 7 Out: x 1 are both positive when...
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