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Must-Have Math Tools for Graduate Study in Economics

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Must-Have Math Tools for Graduate Study in Economics MUST-HAVE MATH TOOLS FOR GRADUATE STUDY IN ECONOMICS William Neilson Department of Economics University of Tennessee – Knoxville September 2009 © 2008-9 by William Neilson web.utk.edu/~wneilson/mathbook.pdf Acknowledgments Valentina Kozlova, Kelly Padden, and John Tilstra provided valuable proofreading assistance on the first version of this book, and I am grateful. Other mistakes were found by the students in my class. Of course, if they missed anything it is still my fault. Valentina and Bruno Wichmann have both suggested additions to the book, including the sections on stability of dynamic systems and order statistics. The cover picture was provided by my son, Henry, who also proofread parts of the book. I have always liked this picture, and I thank him for letting me use it. CONTENTS 1 Econ and math 1 1.1 Some important graphs . 2 1.2 Math, micro, and metrics . 4 I Optimization (Multivariate calculus) 6 2 Single variable optimization 7 2.1 A graphical approach . 8 2.2 Derivatives............................. 9 2.3 Uses of derivatives . 14 2.4 Maximum or minimum? . 16 2.5 Logarithms and the exponential function . 17 2.6 Problems.............................. 18 3 Optimization with several variables 21 3.1 A more complicated pro…t function . 21 3.2 Vectors and Euclidean space . 22 3.3 Partial derivatives . 24 3.4 Multidimensional optimization . 26 i ii 3.5 Comparative statics analysis . 29 3.5.1 An alternative approach (that I don’tlike) . 31 3.6 Problems.............................. 33 4 Constrained optimization 36 4.1 A graphical approach . 37 4.2 Lagrangians ............................ 39 4.3 A 2-dimensional example . 40 4.4 Interpreting the Lagrange multiplier . 42 4.5 A useful example - Cobb-Douglas . 43 4.6 Problems.............................. 48 5 Inequality constraints 52 5.1 Lame example - capacity constraints . 53 5.1.1 A binding constraint . 54 5.1.2 A nonbinding constraint . 55 5.2 Anewapproach.......................... 56 5.3 Multiple inequality constraints . 59 5.4 A linear programming example . 62 5.5 Kuhn-Tucker conditions . 64 5.6 Problems.............................. 67 II Solving systems of equations (Linear algebra) 71 6 Matrices 72 6.1 Matrixalgebra .......................... 72 6.2 Usesofmatrices.......................... 76 6.3 Determinants ........................... 77 6.4 Cramer’srule ........................... 79 6.5 Inverses of matrices . 81 6.6 Problems.............................. 83 7 Systems of equations 86 7.1 Identifying the number of solutions . 87 7.1.1 The inverse approach . 87 7.1.2 Row-echelon decomposition . 87 7.1.3 Graphing in (x,y) space . 89 iii 7.1.4 Graphing in column space . 89 7.2 Summary of results . 91 7.3 Problems.............................. 92 8 Using linear algebra in economics 95 8.1 IS-LManalysis .......................... 95 8.2 Econometrics ........................... 98 8.2.1 Least squares analysis . 98 8.2.2 A lame example . 99 8.2.3 Graphing in column space . 100 8.2.4 Interpreting some matrices . 101 8.3 Stability of dynamic systems . 102 8.3.1 Stability with a single variable . 102 8.3.2 Stability with two variables . 104 8.3.3 Eigenvalues and eigenvectors . 105 8.3.4 Back to the dynamic system . 108 8.4 Problems..............................109 9 Second-order conditions 114 9.1 Taylor approximations for R R ................114 ! 9.2 Second order conditions for R R ...............116 ! 9.3 Taylor approximations for Rm R ...............116 ! 9.4 Second order conditions for Rm R ..............118 9.5 Negative semide…nite matrices .! . 118 9.5.1 Application to second-order conditions . 119 9.5.2 Examples .........................120 9.6 Concave and convex functions . 120 9.7 Quasiconcave and quasiconvex functions . 124 9.8 Problems..............................128 III Econometrics (Probability and statistics) 130 10 Probability 131 10.1 Some de…nitions . 131 10.2 De…ning probability abstractly . 132 10.3 De…ning probabilities concretely . 134 10.4 Conditional probability . 136 iv 10.5Bayes’rule.............................137 10.6 Monty Hall problem . 139 10.7 Statistical independence . 140 10.8Problems..............................141 11 Random variables 143 11.1 Random variables . 143 11.2 Distribution functions . 144 11.3 Density functions . 144 11.4 Useful distributions . 145 11.4.1 Binomial (or Bernoulli) distribution . 145 11.4.2 Uniform distribution . 147 11.4.3 Normal (or Gaussian) distribution . 148 11.4.4 Exponential distribution . 149 11.4.5 Lognormal distribution . 151 11.4.6 Logistic distribution . 151 12 Integration 153 12.1 Interpreting integrals . 155 12.2 Integration by parts . 156 12.2.1 Application: Choice between lotteries . 157 12.3 Di¤erentiating integrals . 159 12.3.1 Application: Second-price auctions . 161 12.4Problems..............................163 13 Moments 164 13.1 Mathematical expectation . 164 13.2Themean .............................165 13.2.1 Uniform distribution . 165 13.2.2 Normal distribution . 165 13.3Variance ..............................166 13.3.1 Uniform distribution . 167 13.3.2 Normal distribution . 168 13.4 Application: Order statistics . 168 13.5Problems..............................173 v 14 Multivariate distributions 175 14.1 Bivariate distributions . 175 14.2 Marginal and conditional densities . 176 14.3 Expectations . 178 14.4 Conditional expectations . 181 14.4.1 Using conditional expectations - calculating the bene…t ofsearch..........................181 14.4.2 The Law of Iterated Expectations . 184 14.5Problems..............................185 15 Statistics 187 15.1 Some de…nitions . 187 15.2Samplemean ...........................188 15.3 Sample variance . 189 15.4 Convergence of random variables . 192 15.4.1 Law of Large Numbers . 192 15.4.2 Central Limit Theorem . 193 15.5Problems..............................193 16 Sampling distributions 194 16.1 Chi-square distribution . 194 16.2 Sampling from the normal distribution . 196 16.3 t and F distributions . 198 16.4 Sampling from the binomial distribution . 200 17 Hypothesis testing 201 17.1 Structure of hypothesis tests . 202 17.2 One-tailed and two-tailed tests . 205 17.3Examples .............................207 17.3.1 Example1.........................207 17.3.2 Example2.........................208 17.3.3 Example3.........................208 17.4Problems..............................209 18 Solutions to end-of-chapter problems 211 18.1 Solutions for Chapter 15 . 265 Index 267 CHAPTER 1 Econ and math Every academic discipline has its own standards by which it judges the merits of what researchers claim to be true. In the physical sciences this typically requires experimental veri…cation. In history it requires links to the original sources. In sociology one can often get by with anecdotal evidence, that is, with giving examples. In economics there are two primary ways one can justify an assertion, either using empirical evidence (econometrics or experimental work) or mathematical arguments. Both of these techniques require some math, and one purpose of this course is to provide you with the mathematical tools needed to make and understand economic arguments. A second goal, though, is to teach you to speak mathematics as a second language, that is, to make you comfortable talking about economics using the shorthand of mathematics. In undergrad- uate courses economic arguments are often made using graphs. In graduate courses we tend to use equations. But equations often have graphical coun- terparts and vice versa. Part of getting comfortable about using math to do economics is knowing how to go from graphs to the underlying equations, and part is going from equations to the appropriate graphs. 1 CHAPTER 1. ECON AND MATH 2 Figure 1.1: A constrained choice problem 1.1 Some important graphs One of the fundamental graphs is shown in Figure 1.1. The axes and curves are not labeled, but that just ampli…es its importance. If the axes are commodities, the line is a budget line, and the curve is an indi¤erence curve, the graph depicts the fundamental consumer choice problem. If the axes are inputs, the curve is an isoquant, and the line is an iso-cost line, the graph illustrates the …rm’scost-minimization problem. Figure 1.1 raises several issues. How do we write the equations for the line and the curve? The line and curve seem to be tangent. How do we characterize tangency? At an even more basic level, how do we …nd slopes of curves? How do we write conditions for the curve to be curved the way it is? And how do we do all of this with equations instead of a picture? Figure 1.2 depicts a di¤erent situation. If the upward-sloping line is a supply curve and the downward-sloping one is a demand curve, the graph shows how the market price is determined. If the upward-sloping line is marginal cost and the downward-sloping line is marginal bene…t, the …gure shows how an individual or …rm chooses an amount of some activity. The questions for Figure 1.2 are: How do we …nd the point where the two lines intersect? How do we …nd the change from one intersection point to another? And how do we know that two curves will intersect in the …rst place? Figure 1.3 is completely di¤erent. It shows a collection of points with a CHAPTER 1. ECON AND MATH 3 Figure 1.2: Solving simultaneous equations line …tting through them. How do we …t the best line through these points? This is the key to doing empirical work. For example, if the horizontal axis measures the quantity of a good and the vertical axis measures its price, the points could be observations of a demand curve. How do we …nd the demand curve that best …ts the data? These three graphs are fundamental to economics. There are more as well. All of them, though, require that we restrict attention to two di- mensions. For the …rst graph that means consumer choice with only two commodities, but we might want to talk about more. For the second graph it means supply and demand for one commodity, but we might want to consider several markets simultaneously.
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