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Revealed Preference
Hal R. Varian∗ January 2005 Revised: September 20, 2006

Abstract This is a survey of revealed preference analysis focusing on the period since Samuelson’s seminal development of the topic with emphasis on empirical applications. It was prepared for Samuelsonian Economics and the 21st Century, edited by Michael Szenberg.

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Introduction

In January 2005 I conducted asearch of JSTOR business and economics journals for the phrase “revealed preference” and found 997 articles. A search of Google scholar returned 3,600 works that contained the same phrase. Surely, revealed preference must count as one of the most influential ideas in economics. At the time of its introduction it was a major contribution to the pure theory of consumer behavior, and the basic ideahas been applied in a number of other areas of economics. In this essay I will briefly describe of the history of revealed preference, starting first descriptions of the concept in Samuelson’s papers. These papers subsequently stimulated a substantial amount of work devoted to refinements


Email contact: hal@sims.berkeley.edu

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and extension of Samuelson’s ideas. This theoretical works, inturn, led to a literature on the use of revealed preference analysis for empirical work that is still growing rapidly.

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The pure theory of revealed preference

Samuelson [1938] contains the first description of the concept he later called “revealed preference.” The initial terminology was “selected over.”1 In this paper, Samuelson stated what has since become known as the “Weak Axiom ofRevealed Preference” by saying “. . . if an individual selects batch one over batch two, he does not at the same time select two over one.” Let us state Samuelson’s definitions a bit more formally. Definition 1 (Revealed Preference) Given some vectors of prices and chosen bundles (pt , xt ) for t = 1, . . . , T , we say xt is directly revealed preferred to a bundle x (written xt RD x) if pt xt ≥ ptx. We say xt is revealed preferred to x (written xt Rx) if there is some sequence r, s, t, . . . , u, v such that pr xr ≥ pr xs , ps xs ≥ ps xt , · · · , pu xu ≥ pu x. In this case, we say the relation R is the transitive closure of the relation RD . Definition 2 (Weak Axiom of Revealed Preference) If xt RD xs then it is not the case that xs RD xt . Algebraically, pt xt ≥ pt xs implies ps xs < ps xt. Subsequently, building on the work of Little [1949], Samuelson [1948] sketched out an argument describing how one could use the revealed preference relation to construct a set of indifference curves. This proof was for two goods only, and was primarily graphical. Samuelson recognized that a general proof for multiple goods was necessary, and left this as an open question.
As Richter [1966] haspointed out, “selected over” has the advantage over “revealed preference” in that it avoids confusion about circular definition of “preference.” Unfortunately, the original terminology didn’t catch on.
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Houthakker [1950] provided the missing proof for the general case. As Samuelson [1950] put it, “He has given us the long-sought test for integrability that can be formed in finiteindex-number terms, without need to estimate partial derivatives.” Houthakker’s contribution was to recognize that one needed to extend the “direct” revealed preference relation to what he called the “indirect” revealed preference relation or, for simplicity, what we call the “revealed preference” relation. Houthakker’s condition can be stated as: Definition 3 (Strong Axiom of Revealed Preference (SARP))If xt Rxs then it is not the case that xs Rxt . Algebraically, SARP says xt Rxs implies ps xs < ps xt . Rose [1958] later offered a formal argument that the Strong Axiom and the Weak Axiom were equivalent in two dimensions, providing a rigorous, algebraic foundation for Samuelson’s earlier graphical exposition. (See Afriat [1965] for a different proof.) Samuelson [1953], stimulated by Hicks [1939],...
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