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El-Hodiri, Mohamed

Journal of Political Economy

English

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MiscellanyComparative Statics When the ObjectiveFunction Is Concave: Old Winein Old Bottles?The purpose of this note is to provide an alternative to the harassmentthat students are subjected to at the hands of Hessians, bordered or other-wise. The idea is to utilize the fact that differentiate concave function liesbelow the tangent plane to derive qualitative relations between two optimacorresponding to two values of the parameters.^ We shall use consumer'sbehavior theory as an illustration. Let .r"̂  be the bundle which maximizesthe consumer's utility when income is m and when prices are p-^, where .r"*and p"^ are w-vectors, subject to the budc^et constraint: p-^ x^ m and toA: ^ 0, Let x^ be the optimal bundle when income is m and prices are p^.Assuming that the utility function, denoted by U{x), is concave anddJfferentiable,' it is necessary and sufficient for .v̂  to be optimal that:with equality whenXi^ > 0,where^ 0, X^m = K^{p^ • x"^). (1)Now let .r^' be the optimal bundle under p^ when income is changed to w®',so that the consumer gets to the old indifference curve, that is, U^' = U"^,where U^' denotes U{x^') and U"^ denotes f/(x^). The vector .r"' — x"^ isthen the compensated change in consumption and is denoted by A'a:, LetA/> denote p^ — p^. That ,r^' is an optimum is equivalent to:1 See, for example, Fleming 1965.2 Admittedly, concave functions do not stay concave as they undergo monotonetransformation, but that should not be interpreted as lack of invariance on their partif invariance means getting the same demand functions. Still, assuming concavity isworse than assuming quasi-concavity.1320MISCELLANYUan —f̂  Pi ,with equality whenx^' > 0, X '̂ ̂  0, X^'m^' = l^'p^x^'. i 2)In view of compensation and concavity of U we have:0=U^' -U^^ t / / ix^' -x^) = f//A'jc. (3)From (1 ) and (3 ) we have :l^ip^ • W) ^ 0 (4)A s s u m i n g the consumer not to be s a t i a t e d at any p o i n t , i t fo l lows^ f r o m ( 1 )that l^ > 0. Thus:p^ ' A 'x^O. (5)Again, by concavity, and compensation we have:O=U^-U^'^ ^ / ' • ix^ - xn = U,^' • i-^'x). (6)By (2) and (6) we have-l^' p^ • A'x^O. (7)By nonsatiation, concavity and (3), l^' > 0. Thus:—p^ ' A ' x ^ O . (8)Adding (5) and (8) we have:(pA _ pB>^ . A ' . r^O. (9)Noting that p^ — p^ — — A/>, (9) implies:A/>-A 'x^O. (10)Relation (1) could be used to show that the own substitution term is non-positive, if all prices except one are unchanged A/> will have zero com-ponents except one (say the /th) and then (10) implies A/>, A'x,-^ 0, or toshow that the Slutsky matrix is negative semidefinite. To accomplish thislast task we follow Samuelson's discussion.'* We could write (10) as:dp • dx^O. ( 1 1 )Assuming the demand function to be differentiable we get, by compensationand definition of the Slutsky terms (denoted by ^,7):(12)"̂  By contradiction, since if X^. [^ not positive then, as }A "^ 0,}A -=0. By (1), theni/j.^^ ^ 0, with equality if a:-̂  > 0, for z = 1, . . . , n. This, by concavity, implies thatU has an absolute maximum at x^ subject to x ^ 0 which means that the consumeris satiated at x^.* See Samuelson 1947.1322 JOURNAL OF POLITICAL ECONOMYMultiplying both sides of (12) by dpi and summing over i we get(13)By (11) and (12) the Slutsky matrix is negative semidefinite.MOHAMED E L - H O D I R IUniversity of KansasReferencesFleming, \V. Functions of Several Variables. Reading, Mass.: Addison-Wesley,1965^Samuelson, Paul A. Foundations of Economic Analysis. Cambridge, Mass.: Har-vard Univ. Press, 1947.

Comparative Statics When the Objective Function Is Concave: Old Wine in Old Bottles?

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Comparative Statics When the Objective Function Is Concave: Old Wine in Old Bottles?

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