Advanced Mathematical Economics

by Vohra; Rakesh V.

• ISBN13: 9780415700085
• ISBN10: 0415700086
• Format: Nonspecific Binding
• Copyright: 2005-01-10
• Publisher: Routledge

Summary

This concise textbook presents students with all they need for advancing in mathematical economics. Higher level undergrads as well as postgrad students in mathematical economics will find this book to be extremely useful in the course of their development as economists.

Table of Contents

Preface	viii
1 Things to know	1	(12)
1.1 Sets	1	(1)
1.2 The space we work in	1	(1)
1.3 Facts from real analysis	2	(4)
1.4 Facts from linear algebra	6	(3)
1.5 Facts from graph theory	9	(4)
2 Feasibility	13	(20)
2.1 Fundamental theorem of linear algebra	13	(2)
2.2 Linear inequalities	15	(1)
2.3 Non-negative solutions	15	(4)
2.4 The general case	19	(1)
2.5 Application: arbitrage	20	(4)
2.6 Application: co-operative games	24	(1)
2.7 Application: auctions	25	(8)
3 Convex sets	33	(20)
3.1 Separating hyperplane theorem	34	(6)
3.2 Polyhedrons and polytopes	40	(6)
3.3 Dimension of a set	46	(1)
3.4 Properties of convex sets	47	(2)
3.5 Application: linear production model	49	(4)
4 Linear programming	53	(34)
4.1 Basic solutions	56	(4)
4.2 Duality	60	(4)
4.3 Writing down the dual	64	(1)
4.4 Interpreting the dual	64	(4)
4.5 Marginal value theorem	68	(1)
4.6 Application: zero-sum games	69	(3)
4.7 Application: Africa's theorem	72	(3)
4.8 Integer programming	75	(3)
4.9 Application: efficient assignment	78	(2)
4.10 Application: Arrow's theorem	80	(7)
5 Non-linear programming	87	(30)
5.1 Necessary conditions for local optimality	89	(6)
5.2 Sufficient conditions for optimality	95	(5)
5.3 Envelope theorem	100	(3)
5.4 An aside on utility functions	103	(2)
5.5 Application: market games	105	(4)
5.6 Application: principal-agent problem	109	(8)
6 Fixed points	117	(25)
6.1 Banach fixed point theorem	117	(1)
6.2 Brouwer fixed point theorem	118	(7)
6.3 Application: Nash equilibrium	125	(2)
6.4 Application: equilibrium in exchange economies	127	(6)
6.5 Application: Hex	133	(3)
6.6 Kakutani 's fixed point theorem	136	(6)
7 Lattices and supermodularity	142	(21)
7.1 Abstract lattices	149	(2)
7.2 Application: supermodular games	151	(1)
7.3 Application: transportation problem	152	(2)
7.4 Application: efficient assignment and the core	154	(5)
7.5 Application: stable matchings	159	(4)
8 Matroids	163	(28)
8.1 Introduction	163	(2)
8.2 Matroid optimization	165	(1)
8.3 Rank. functions	166	(3)
8.4 Deletion and contraction	169	(1)
8.5 Matroid intersection and partitioning	170	(5)
8.6 Polymatroids	175	(4)
8.7 Application: efficient allocation with indivisibilities	179	(7)
8.8 Application: Shannon switching game	186	(5)
Index	191