Consumption Demand

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 45 in Conjugate Duality in Economic Analysis, 2026, pp 339-349 from Springer

Abstract: Abstract The fundamental research of Hildenbrand and Kannai justifies the use of concave utility, but what economists have not done is to develop consumer demand theory via concave utility. Because indifference curves are ordinal, the concept of the concave conjugate of utility is not part of economic thinking. Concave utility allows analysis by duality, which makes possible the derivation of results and the discovery of relationships by algebraic calculation. We specify the properties of the utility function of a “classical consumer.” We contrast expenditure minimization and utility maximization via the directional derivative and the subdifferential of utility. Consider consumption x and price v*≻0. Utility is $$f\left ( \boldsymbol {x}\right )$$ , and income/expenditure is $$\left \langle \boldsymbol {v}^{\ast },\boldsymbol {x}\right \rangle$$ . We study whether x is compensated demand: does it minimize expenditure among all consumption having the same utility or higher? We study whether x is consumer demand: does it maximize utility among all consumption costing the same or less? Axis reversal explains the situation. If income/expenditure is positive, then expenditure is minimized if and only if utility is maximized. If income/expenditure is zero, then necessarily expenditure is minimized, because it cannot be nonnegative. However perhaps utility is not maximized: to consume more free goods perhaps raises utility.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_45

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DOI: 10.1007/978-3-032-21396-9_45

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