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Unformatted text preview: 3 The Utility Maximization Problem We have now discussed how to describe preferences in terms of utility functions and how to formulate simple budget sets. The rational choice assumption, that consumers pick the best a f ordable bundle, can then be described as an optimization problem . The problem is to f nd a bundle ( x ∗ 1 , x ∗ 2 ) , which is in the budget set, meaning that p 1 x ∗ 1 + p 2 x ∗ 2 ≤ m and x ∗ 1 ≥ , x ∗ 2 ≥ , which is such that u ( x ∗ 1 , x ∗ 2 ) ≥ u ( x 1 , x 2 ) for all ( x 1 , x 2 ) in the budget set. It is convenient to introduce some notation for this type of problems. Using rather standard conventions I will write this as max x 1 ,x 2 u ( x 1 , x 2 ) subj. to p 1 x 1 + p 2 x 2 ≤ m x 1 ≥ x 2 ≥ , which is an example of a problem of constrained optimization. A common tendency of students is to skip the step where the problem is written down. This is a bad idea. The reason is that we will often study variants of optimization problems that di f er in what seems to be small “details”. Indeed, often times the di ﬃ cult step when thinking about a problem is to formulate the right optimization problem. For this reason I want you to: 1. Write out the “ max ” in front of the utility function (the maximand , or, objective func tion ). This clari f es that the consumer is supposed to solve an optimization problem. 2. Below the max, it is a good idea to indicate what the choices variables are for the consumer ( x 1 and x 2 in this example). This is to clarify the di f erence between the variables that are under control of the decision maker and variables that the decision maker has no control over, which are referred to as parameters . In the application above p 1 , p 2 and m are parameters. 38 3. Finally, it is important that it is clear what the constraints to the problem are. I good habit is to write “subject to” or, more concisely, s.t. and then list whatever constraints there are, as in the problem above. 3.1 Basic Math Review Read this only if you fell that you don’t quite know how to deal with simple optimization problems. 3.1.1 Derivatives 6 ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ x x 00 ax + b ax 00 + b b − b a x y Figure 1: A Linear Function Consider a linear function y = h ( x ) = ax + b as the one depicted in Figure 1. Recall that the slope of a linear function is slope = change in y change in x = f ( x 00 ) z } { ax 00 + b − f ( x ) ( z } { ax + b ) x 00 − x = a. Now, let’s think about a nonlinear function y = f ( x ) . If we just do the same thing as with the linear function and think of the “slope” as the ratio of the change in the value of the function to the change in the value of the argument....
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 Spring '10
 CarlosSerrano
 Microeconomics, Derivative, Optimization, Utility, X1

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