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09Appendix-A-1

# 09Appendix-A-1 - 2September2009 APPENDIX A MATHEMATICAL...

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2 September 2009 APPENDIX A: MATHEMATICAL FOUNDATIONS A.1 IS IT REALLY TRUE? 1 Methods of proof A.2 MAPPINGS OF A SINGLE VARIABLE 6 Neighborhood and limit point Continuous Function A.3 DERIVATIVES AND INTEGRALS 12 Rules of Differentiation Integral of a function A.4 OPTIMIZATION 23 First Order Condition Second Order Condition A.5 SUFFICIENT CONDITIONS FOR A MAXIMUM 33 Concavity and Convexity Quasi-concave Function 43 pages

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Appendix A page 1 A.1 IS IT REALLY TRUE? Key ideas: direct proof, proof by induction, proving the contrapositive, proof by contradiction The whole point of building a mathematical model is to be able to use the power of mathematical analysis to draw conclusions about economic phenomena. Thus it is essential to have on hand a good toolkit of mathematical propositions. For example suppose that you wish to analyze the choice of a profit maximizing firm that sells in a market where the price of the commodity q is p . For any quantity, price pair ( , ) q p there is some profit ( , ) f q p . To work with this model it is useful to have in the toolkit a clear notion of the rate of change in the function as output changes. Two possible functions are depicted below. Clearly it is easier to characterize the profit maximizing output in the first case, since the top of the profit hill is the unique point where the slope is zero. In the second case there are lots of outputs where the slope is zero but not all of them yield the profit maximum. Thus, to simplify the model, it is useful to seek plausible conditions under which the profit curve has a shape that is simple to analyze. This is relatively easy if the firm is choosing a single output. But what if q represents a vector of outputs of different commodities? Taking the example a little further, suppose that ( ) q p is a solution to the maximization problem for each output price p . To ( , ) f q p q q ( , ) f q p q Fig. A.1-1: The profit of a firm
Appendix A page 2 analyze the firm’s behavior it is simpler if this output varies smoothly with the output price. But is this a reasonable assumption? One answer to such questions might be that the key is to put together a tool-kit of mathematical methods and results that is large enough to deal with all the basics of economic modeling. However, while this approach works quite well at an introductory level, it is ultimately inadequate for someone wanting a deeper understanding of economic models.. Without a clear appreciation of why results hold and how techniques work it is extremely difficult deciding how to approach a problem. For this reason understanding the derivation of results and how techniques are developed is very important. How does one go about proving some proposition, that is, demonstrate convincingly that the proposition is true? First there must be some common set of principles or results that are sufficiently familiar that they do not need reiterating or reproving. Mathematicians then employ several approaches. We discuss each in turn.

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09Appendix-A-1 - 2September2009 APPENDIX A MATHEMATICAL...

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