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Unformatted text preview: Math Guide for Econ 11 University of California, Los Angeles These notes are intended as a reference guide for the mathematical tools of calculus that are used in Econ 11. The following pages provide all the mathematical results, along with brief explanations, illustrative examples of their use and additional exercises. You should have seen most of this material in your calculus course, especially the parts related to single-variable calculus. Any new materials will be introduced as well in lecture. 1 Introduction Consider the following two examples: 1. A &rm must decide how much to produce of a certain good. The market price for the &rms product is p , while it costs the &rm C ( q ) = 1 2 q 2 to produce a quantity q of the good. The &rm wants to maximize its pro&ts, which are de&ned as sales revenue less production costs, and can choose the quantity q that it produces. Thus we de&ne the &rms pro&ts as F ( q ) = pq & 1 2 q 2 , and the &rms decision problem consists in &nding the quantity q that maximizes F ( q ) . 2. A student must prepare for an exam the next day. She has 12 hours remaining. For each additional hour of studying, she expects to raise her grade by 2 points out of 100, but her grade also depends on how much she sleeps. If she sleeps 7 hours, she is able to work with full concentration. If she sleeps less, her grade will su/er. WIthout any further studying, the student expects a grade of 11 + y (14 & y ) where y is the number of hours she sleeps. Therefore, if she studies x hours and sleeps y hours, the students expected grade is F ( x; y ) = 2 x +11+ y (14 & y ) , and the students decision problem consists in &nding x and y that maximize F ( x; y ) subject to the constraint that x + y 12 . These two examples are representative of the decision problems that are studied in microeconomics. Formally, we consider problems in which a decision- maker , (i.e. a consumer, a household, a &rm, etc.) has to maximize an objective function F ( x ) with respect to some choice variable x . In the &rst example above, the decision-maker was a &rm, its objective was to maximize pro&ts and its choice variable was the quantity produced. In the second example, the decision-maker was the student, her objective was to maximize her grade, and her choice variable consisted of two components: study and sleep. As well, the students decision problem is subject to a constraint, that the decision- makers choice of hours studied, x; and hours slept, y , must satisfy the additional constraint that g ( x; y ) = 12 & x & y . The advantage of formulating individual decision problems in this mathe- matical format is that we can then use the mathematical tools of optimization 1 to analyze them. The &rms problem is one of unconstrained optimization , generally represented as max x F ( x ) while the students problem is one of constrained optimization , generally repre- sented as max x F ( x ) subject to g ( x ) & : In both cases, x can consist of one or multiple variables, and both the objec-...
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This note was uploaded on 09/23/2008 for the course ECON 11 taught by Professor Cunningham during the Summer '08 term at UCLA.
- Summer '08