Advanced mathematical economics

Book Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 9
1.2 The space we work in......Page 12
1.3 Facts from real analysis......Page 13
1.4 Facts from linear algebra......Page 17
1.5 Facts from graph theory......Page 20
2.1 Fundamental theorem of linear algebra......Page 24
2.3 Non-negative solutions......Page 26
2.4 The general case......Page 30
2.5 Application: arbitrage......Page 31
2.6 Application: co-operative games......Page 35
2.7 Application: auctions......Page 36
3 Convex sets......Page 44
3.1 Separating hyperplane theorem......Page 45
3.2 Polyhedrons and polytopes......Page 51
3.3 Dimension of a set......Page 57
3.4 Properties of convex sets......Page 58
3.5 Application: linear production model......Page 60
4 Linear programming......Page 64
4.1 Basic solutions......Page 67
4.2 Duality......Page 71
4.4 Interpreting the dual......Page 75
4.5 Marginal value theorem......Page 79
4.6 Application: zero-sum games......Page 80
4.7 Application: Afriat's theorem......Page 83
4.8 Integer programming......Page 86
4.9 Application: efficient assignment......Page 89
4.10 Application: Arrows theorem......Page 91
5 Non-linear programming......Page 98
5.1 Necessary conditions for local optimality......Page 100
5.2 Sufficient conditions for optimality......Page 106
5.3 Envelope theorem......Page 111
5.4 An aside on utility functions......Page 114
5.5 Application: market games......Page 116
5.6 Application: principal-agent problem......Page 120
6.1 Banach fixed point theorem......Page 128
6.2 Brouwer fixed point theorem......Page 129
6.3 Application: Nash equilibrium......Page 136
6.4 Application: equilibrium in exchange economies......Page 138
6.5 Application: Hex......Page 144
6.6 Kakutani's fixed point theorem......Page 147
7 Lattices and supermodularity......Page 153
7.1 Abstract lattices......Page 160
7.2 Application: supermodular games......Page 162
7.3 Application: transportation problem......Page 163
7.4 Application: efficient assignment and the core......Page 165
7.5 Application: stable matchings......Page 170
8.1 Introduction......Page 174
8.2 Matroid optimization......Page 176
8.3 Rank functions......Page 177
8.4 Deletion and contraction......Page 180
8.5 Matroid intersection and partitioning......Page 181
8.6 Polymatroids......Page 186
8.7 Application: efficient allocation with indivisibilities......Page 190
8.8 Application: Shannon switching game......Page 197
Index......Page 202