The Foundations of the Revealed Preference Theory - Assignment Example

 The Foundations of the Revealed Preference Theory There is growing tendency of greater rigor in modern economic theory in: 1) Formulation of its assumptions 2) Standards of proof on which assumptions imply The use of rigor is commendable because it: 1) Assures that conclusions are incurred by hypotheses 2) Spares us with unrealistic economic assumptions (perfect competition, divisibility, and restrictive mathematical conditions) 3) Leads to more precise theory and greater realism However, its dedication to rigor can lead to indiscriminate application, formalism, and a level of abstraction that makes it difficult for economists. However, this unfamiliarity does not validate the objection to its implementation. Newman (1955) represents theory of revealed preference in rigorous terms. By careful formulation of usual assumptions, many unresolved issues can be sorted. In particular, vexed problem of integrability can be assigned in a better way (Newman, 1995). 2) According to the usual ordinal preference theory, consumer can compare all possible commodity bundles with an arbitrary bundle x where: a) X is preferred to some bundles b) Some bundles are preferred to x c) Some bundles are indifferent to x 4) Conditions for a complete weak ordering are not complicated. When we consider all commodity bundles as X, and denote the relation (preferred or indifferent to) by R, the necessary postulates are: AXIOM I: For all x1, x2 in X, either x 1Rx2 or x 2Rx1 AXIOM II: For all x1, x2, x3 in X, such that x 1Rx2 and x 2Rx3, we have x 1Rx3 The first axiom specifies that all points in the 'commodity space' X are ordered by relation R while second stresses the transitivity of this relation (Newman, 1995). 5) Slope of an indifferent curve at a certain point x may be denoted by simple type of differential equation. In some mathematical conditions, this differential equation can be integrated to solve it as 'indifference' curve through x which is called integrability problem (Newman, 1995). 6) Georgescu-Roegen proved that if we consider two goods where differential equation defining an 'indifferent element' is always integrable, it is essential to consider that indifference is transitive relation that denotes that integral curves are indifferent curves. Georgecu proved that transitivity condition guarantees integrable differential equations in all dimensions. Therefore, integrability conditions are unnecessary. In economic terms, integral curves are meaningless unless Axiom I and II are adopted (Newman, 1995). II. Some Current Approaches (8, 9) By following Sauelson's techniques, Herman Wold claimed that demand-function approach is rationally equal to the ordinal preference approach. He asserted that Houthakker's theorem is same as his method of proof. However, an analysis of Wold's proof revealed his implicit assumption that his 'revealed preference' is a transitive relation. Wold proved his point by using certain assumptions of perfect competition; however, it is impossible to do because Wold didn't define indifference (Newman, 1995). (10) Wold's integrability assumptions are obsolete because existence of a convex boundary curve comes from this implicit assumption of the transitivity of preference not from explicit assumptions of integrability. (11) Ville deduced the existence of a utility function through excellent analytic evidence. A reasonable interpretation of his axiom can lead to a relation that brings a complete weak ordering on the commodity space. However, his criterion for consumer as "worse off" is insufficient (Newman, 1995). (12) Integrability problem suddenly appears when commodity space has three goods while transitivity problem arises when three situations are considered. Therefore, it arises even when there is two-goods case. Little's treatment assumes that revealed preference is a transitive relation but it failed to produce a utility function since he does not show the points around boundary curve which are equivalent to each other (Newman, 1995). III. The Nature of Revealed Preference The major issue is to create directly a complete ordering of commodity space from the assumptions of revealed preference. Utility function is selected for representation once the ordering is obtained. (21) Samuelson defined first consistency condition, the 'weak axiom' as: If p0. (x1-x0) ≤ 0, then not p1 .(x0-x1) ≤ 0, If p0(x1-x0) ≤ 0, then p1 .(x1-x0) < 0. Newman considers Samuelson's axiom too restrictive and offers some modifications in Samuelson's axiom: If x0 Rx1,then not x1Px0 i.e. if p0 .(x1-x0) ≤ 0, then not p1 .x1 >p1.x0 if p0(x1-x0) ≤ 0, then p1 .(x1-x0) ≤ 0. When compared with Samuelson's weak axiom, it appears to be the identical save for the equal sign in second inequality. IV. Two Theorems in Revealed Preference Theory With first consistency condition, Newman established the new theorem: Theorem 1: For three distinct points x0, x1, x2 such that, not x0Px1 and not x 0Px2 In this case, for no point x3 on the convex combination of x1 and x2 can we have x 0Px3. The theorem is trivially true if x1=x2 or x2=x3 so we may consider 0< α Read More