Math Eco Study Guide.pdf - Mathematical economics M Bray R...

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Undergraduate study in Economics, Management, Finance and the Social Sciences Mathematical economics M. Bray, R. Razin, A. Sarychev EC3 120 2014 This subject guide is for a 300 course offered as part of the University of London International Programmes in Economics, Management, Finance and the Social Sciences. This is equivalent to Level 6 within the Framework for Higher Education Qualifications in England, Wales and Northern Ireland (FHEQ). For more information about the University of London International Programmes undergraduate study in Economics, Management, Finance and the Social Sciences, see:
This guide was prepared for the University of London International Programmes by: Dr Margaret Bray, Dr Ronny Razin, Dr Andrei Sarychec, Department of Economics, The London School of Economics and Political Science. With typesetting and proof-reading provided by: James S. Abdey, BA (Hons), MSc, PGCertHE, PhD, Department of Statistics, London School of Economics and Political Science. This is one of a series of subject guides published by the University. We regret that due to pressure of work the authors are unable to enter into any correspondence relating to, or arising from, the guide. If you have any comments on this subject guide, favourable or unfavourable, please use the form at the back of this guide. University of London International Programmes Publications Office Stewart House 32 Russell Square London WC1B 5DN United Kingdom Published by: University of London © University of London 2007 Reprinted with minor revisions 2014 The University of London asserts copyright over all material in this subject guide except where otherwise indicated. All rights reserved. No part of this work may be reproduced in any form, or by any means, without permission in writing from the publisher. We make every effort to respect copyright. If you think we have inadvertently used your copyright material, please let us know.
Contents Contents 1 Introduction 1 1.1 The structure of the course . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.5 Reading advice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.5.1 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.6 Online study resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6.1 The VLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6.2 Making use of the Online Library . . . . . . . . . . . . . . . . . . 5 1.7 Using the subject guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.8 Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Constrained optimisation: tools 9 2.1 Aim of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 The constrained optimisation problem . . . . . . . . . . . . . . . . . . . 10 2.7 Maximum value functions . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.8 The Lagrange sufficiency theorem . . . . . . . . . . . . . . . . . . . . . . 14 2.9 Concavity and convexity and the Lagrange Necessity Theorem . . . . . . 16 2.10 The Lagrangian necessity theorem . . . . . . . . . . . . . . . . . . . . . . 19 2.11 First order conditions: when can we use them? . . . . . . . . . . . . . . . 19 2.11.1 Necessity of first order conditions . . . . . . . . . . . . . . . . . . 20 2.11.2 Sufficiency of first order conditions . . . . . . . . . . . . . . . . . 21 2.11.3 Checking for concavity and convexity . . . . . . . . . . . . . . . . 22 2.11.4 Definiteness of quadratic forms . . . . . . . . . . . . . . . . . . . 23 i
Contents 2.11.5 Testing the definiteness of a matrix . . . . . . . . . . . . . . . . . 23 2.11.6 Back to concavity and convexity . . . . . . . . . . . . . . . . . . . 26 2.12 The Kuhn-Tucker Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.13 The Lagrange multipliers and the Envelope Theorem . . . . . . . . . . . 29 2.13.1 Maximum value functions . . . . . . . . . . . . . . . . . . . . . . 29 2.14 The Envelope Theorem

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