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2019 | OriginalPaper | Buchkapitel

9. Samuelson’s Approach to Revealed Preference Theory: Some Recent Advances

verfasst von : Thomas Demuynck, Per Hjertstrand

Erschienen in: Paul Samuelson

Verlag: Palgrave Macmillan UK

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Abstract

Since Paul Samuelson introduced the theory of revealed preference, it has become one of the most important concepts in economics. This chapter surveys some recent contributions in the revealed preference literature. We depart from Afriat’s theorem, which provides the conditions for a data set to be consistent with the utility maximization hypothesis. We provide and motivate a new condition, which we call the Varian inequalities. The advantage of the Varian inequalities is that they can be formulated as a set of mixed integer linear inequalities, which are linear in the quantity and price data. We show how the Varian inequalities can be used to derive revealed preference tests for weak separability and show how they can be used to formulate tests of the collective household model. Finally, we discuss measurement errors in the observed data and measures of goodness-of-fit, power, and predictive success.

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Vorheriges Kapitel The Samuelson Revolution in Australia

Nächstes Kapitel Paul Samuelson and the Economics of Pass-Through and the Envelope Theorem

The more refined search ‘revealed preference’ + samuelson returned approximately 4460 works on Google Scholar over the period from 2005 to 2018.

Other recent surveys or overviews of revealed preference, some with a different focus, are Cherchye et al. (2009a), Diewert (2012), Crawford and De Rock (2014), and Chambers and Echenique (2016).

By a bundle we mean an N-dimensional vector that gives the various quantities over the N goods that the consumer chooses.

By px, we mean the dot product \(\sum _{i\,=\,1}^{N} p_{i} x_{i}\).

A function \(u:\mathbb {R}^{N}_{+} \rightarrow \mathbb {R}\) is locally non-satiated if for all \(x \in \mathbb {R}^{N}_{+}\) there exists a neighborhood \(\mathcal{N}_{x}\) around x and a vector \(y \in \mathcal{N}_{x} \cap \mathbb {R}^{N}_{+}\) such that \(u(y) > u(x)\).

With nonlinear budgets, the property of quasiconcavity is testable (see Cherchye et al. 2014).

Varian’s algorithm is widely available in many programming languages. See, for example, www.revealedpreferences.org for codes.

A non-satiated and quasiconcave function is also called semistrict quasiconcave (see Ginsberg 1973).

Without loss of generality, the strict inequality on the right-hand side in the second row can be converted to a weak inequality by noticing that the inequalities are homogeneous of degree one in \(V^{t}\). Thus, the second row may be written as \(\text {if}\,p^{t}\left( x^{t} - x^{s}\right) >0\), then \(V^{t}-V^{s}\ge \varepsilon \), for any \(\varepsilon >0\) (the standard choice is to set \(\varepsilon = 1\)).

From a computational perspective, we have that LP problems are solvable in polynomial time, while MILP problems are in general NP-hard.

Cherchye et al. (2015a) showed that verifying Conditions (3) and (4) is an NP-hard problem, meaning that it is computationally intractable for very large datasets. See also Echenique (2014).

Of course, this disregards the fact that there are many other things in addition to the consumption of goods that influence the utility that partners derive from marriage.

Computing the maximum or minimum of the measure over all cycles can be done in a computationally efficient manner (i.e. in polynomial time).

It is straightforward to modify the exposition and instead allow for measurement error in prices (see, e.g., Jones and Edgerton 2009).

We use superscripts to refer to observations, while subscripts refer to the goods.

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Titel: Samuelson’s Approach to Revealed Preference Theory: Some Recent Advances
verfasst von: Thomas Demuynck
Per Hjertstrand
Verlag: Palgrave Macmillan UK
Buch: Paul Samuelson
Print ISBN: 978-1-137-56811-3

Electronic ISBN: 978-1-137-56812-0

Copyright-Jahr: 2019
DOI: https://doi.org/10.1057/978-1-137-56812-0_9