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mathguide - Math Guide for Econ 11 These notes are intended...

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Math Guide for Econ 11 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 some, if not most of this material in your calculus course, especially the parts that are related to single-variable calculus. Some of the concepts may be new, but these will be discussed in detail, when they are introduced during the lectures; these notes are intended to serve you as a quick reference as you go along with the course. The f rst two TA sessions will review the material in this handout in detail. 1 Introduction Consider the following two examples: 1. A f rm that has to decide how much to produce of a certain good. The market price for the f rm’s product is p , and it costs the f rm C ( q )= 1 2 q 2 to produce a quantity q of the good. The f rm wants to maximize its pro f ts, i.e. sales revenue minus production costs, and can choose the quantity q that it produces. With a sales revenue pq ,wecanwr itethe f rm’s pro f ts as F ( q )= pq 1 2 q 2 ,andthe f rm’s decision problem consists in f nding the quantity q that maximizes F ( q ) . 2. A student has to prepare for an exam the next day. She has 12 hours remaining before the exam. For each additional hour of studying, she expects to raise her grade by 2 points (out of a 100), but she also realizes that her concentration level during the exam depends on how much she sleeps. If she sleeps 7 hours, she is able to work with full concentration. If she sleeps less than that, her concentration goes down. If she didn’t study any more, and slept exactly y hours before the exam, she expects a grade of 11 + y (14 y ) (for y=7, this gives a grade of 60; for y=6, this gives a grade of 59, for y=5, this gives a grade of 56, and so on). Therefore, if the student sleeps y hours and studies x hours until the exam, her grade (as a function of x and y )is F ( x, y )=11+ y (14 y )+2 x . The student wants to allocate her time optimally between sleeping and studying so as to maximize her expected grade. But her choices of x and y have to satisfy the additional constraint that 12 x + y ,i .e . the amount of time that the student studies plus the amount that she sleeps cannot exceed the 12 hours remaining before the exam. These two examples are representative of the decision problems that are studied in Mi- croeconomics, and that we are concerned with in much of this course. Formally, we consider problems in which a decision-maker (i.e. a consumer, a household, a f rm, etc. depending on the exact context), has to maximize an objective function F ( x ) with respect to some choice variable x .I nt h
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were pro f ts, and its choice variable was the quantity. In some of these problems, x may consist of multiple elements, or it may be subject to additional
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This note was uploaded on 04/05/2009 for the course ECON 11 taught by Professor Cunningham during the Spring '08 term at UCLA.

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mathguide - Math Guide for Econ 11 These notes are intended...

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