MUST-HAVE MATH TOOLS FOR GRADUATE STUDY IN …

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't like) . . . . . . . 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 A new approach . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2 Uses of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4 Cramer's rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

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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-LM analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 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

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