*S. N. Afriat*

- Published in print:
- 1987
- Published Online:
- November 2003
- ISBN:
- 9780198284611
- eISBN:
- 9780191595844
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198284616.003.0024
- Subject:
- Economics and Finance, Microeconomics

This is the first of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It introduces the general ideas of optimal programming ...
More

This is the first of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It introduces the general ideas of optimal programming in economic terms, with reference to the programming problem of a firm. A theorem is proved that is basic to the entire subject, and requires no special assumptions about the programming functions. The ten sections of the chapter are: bounds, limits and maxima; programming problem of a firm; optimal programming theorem; input–output; output limit function; support gradients and marginal values; complementarity; shadow price decentralization; proof of the theorem; and Lagrange multipliers.Less

This is the first of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It introduces the general ideas of optimal programming in economic terms, with reference to the programming problem of a firm. A theorem is proved that is basic to the entire subject, and requires no special assumptions about the programming functions. The ten sections of the chapter are: bounds, limits and maxima; programming problem of a firm; optimal programming theorem; input–output; output limit function; support gradients and marginal values; complementarity; shadow price decentralization; proof of the theorem; and Lagrange multipliers.

*S. N. Afriat*

- Published in print:
- 1987
- Published Online:
- November 2003
- ISBN:
- 9780198284611
- eISBN:
- 9780191595844
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198284616.003.0026
- Subject:
- Economics and Finance, Microeconomics

This is the third of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It discusses linear programming, which might appear to ...
More

This is the third of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It discusses linear programming, which might appear to be a special case of convex programming, but is more substantial, and is really an embodiment of the theory of systems of linear inequalities (as reflected here). This chapter initiates the subject with reference to systems of linear inequalities and natural questions about them, and all LP (linear programming) theorems are encountered simply in pursuing those. Theorems about linear inequalities that have uses directly on their own are also derived (and are illustrated in many places in this book). The eight sections of the chapter are: linear inequalities; separation theorems; theorems of alternatives; polyhedra and polytopes; LP Duality Theorem; the pivot operation; the Simplex Algorithm; and BASIC program.Less

This is the third of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It discusses linear programming, which might appear to be a special case of convex programming, but is more substantial, and is really an embodiment of the theory of systems of linear inequalities (as reflected here). This chapter initiates the subject with reference to systems of linear inequalities and natural questions about them, and all LP (linear programming) theorems are encountered simply in pursuing those. Theorems about linear inequalities that have uses directly on their own are also derived (and are illustrated in many places in this book). The eight sections of the chapter are: linear inequalities; separation theorems; theorems of alternatives; polyhedra and polytopes; LP Duality Theorem; the pivot operation; the Simplex Algorithm; and BASIC program.

*S. N. Afriat*

- Published in print:
- 1987
- Published Online:
- November 2003
- ISBN:
- 9780198284611
- eISBN:
- 9780191595844
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198284616.003.0025
- Subject:
- Economics and Finance, Microeconomics

This is the second of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It introduces convexity conditions, and shows where ...
More

This is the second of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It introduces convexity conditions, and shows where they have effect, together with Slater's condition, in assuring the existence of a support to the limit function, so providing Lagrange multipliers, or shadow prices, of resources that have part in the optimality conditions. Then for the case of differentiable functions the Kuhn–Tucker conditions are obtained. The six sections of the chapter are: convexity; programming convexity theorem; Slater's condition; optimality theorem; non‐negative maxima; the Kuhn–Tucker conditions.Less

This is the second of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It introduces convexity conditions, and shows where they have effect, together with Slater's condition, in assuring the existence of a support to the limit function, so providing Lagrange multipliers, or shadow prices, of resources that have part in the optimality conditions. Then for the case of differentiable functions the Kuhn–Tucker conditions are obtained. The six sections of the chapter are: convexity; programming convexity theorem; Slater's condition; optimality theorem; non‐negative maxima; the Kuhn–Tucker conditions.