Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-02T20:21:21.152Z Has data issue: false hasContentIssue false

12 - Revealed Preference and Systems of Polynomial Inequalities

Published online by Cambridge University Press:  05 January 2016

Christopher P. Chambers
Affiliation:
University of California, San Diego
Federico Echenique
Affiliation:
California Institute of Technology
Get access

Summary

It should be apparent by now that systems of linear inequalities emerge naturally in revealed preference theory. They constitute the essence of Afriat's Theorem, for example; and we formulated revealed preference problems using systems of linear inequalities in Chapters 3, 6, 7, and 8. In this chapter, we describe how revealed preference problems can generally be understood as a system of inequalities. From a purely computational perspective, one can very often solve a revealed preference problem by algorithmically solving the corresponding system of inequalities.

When the system of inequalities is linear, the problem is easy to solve both computationally and analytically. Here we develop a GARP-like acyclicity test (similar to the ones in Chapters 2 and 3). The test will follow from the linearity of the system of inequalities embodied in the revealed preference question.

We shall discuss an extension of the linear theory to systems of polynomial inequalities. The theory of polynomial inequalities will be seen to be very relevant for revealed preference theory (but harder to work with compared to the theory of linear inequalities).

LINEAR INEQUALITIES: THE THEOREM OF THE ALTERNATIVE AND REVEALED PREFERENCE

We start by revisiting the Theorem of the Alternative, or Farkas’ Lemma from Chapter 1. It is easy to see why it is useful in revealed preference theory. We then discuss some sources of linear systems for popular models in economics. The following is a bit weaker than Lemma 1.13. It is written so as to emphasize that the lemma can be used to “remove existential quantifiers;” it states that an existential statement (one that start with “there is… ”) is equivalent to a universal statement (one that starts with “for all… ”). Note that the discussion of the Tarski–Seidenberg Theorem in Chapter 9 is also about removing existential quantifiers.

Lemma 1.13'(Integer–Real Farkas) LetAii=1M be a finite collection of vectors inQK. The following statements are equivalent:

  1. I) There exists yRK such that for all i = 1, , M, Ai. y > 0.

  2. II) For all zZ+M \ ﹛0﹜, it holds that.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2016

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×