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Demands
Published in Prem K. Kythe, Elements of Concave Analysis and Applications, 2018
Whereas the Hicksian demand comes from the expenditure minimization problem, the Marshallian demand comes from the utility maximization problem. Thus, the two problems are mathematical duos, and hence the duality theorem provides a method of proving the above relationships.
Teaching cost minimization – can a non-calculus approach help?
Published in International Journal of Mathematical Education in Science and Technology, 2021
Vedran Kojić, Mira Krpan, Zrinka Lukač
Some of the standard cost minimization problems include minimization of average costs, finding economic order quantity (EOQ problem), perimeter and area problems, surface area and volume problems, expenditure minimization problem, and firm’s cost minimization problem. The common approach to solving the first four problems is to use single variable calculus, while the last two are solved by applying multivariate differential calculus. The expenditure minimization problem and the firm’s cost minimization problem are important parts of the consumer theory and the theory of the firm, which are also taught by using calculus (see for example Jehle & Reny, 2011). In this paper, we present a non-calculus approach to teaching and solving all of the problems mentioned above. The method is based on the use of the weighted inequality between the arithmetic mean and geometric mean (weighted AM-GM inequality or WAG) (see for example Cvetkovski, 2012; or Hung, 2007). We show that for these problems it is possible to derive necessary and sufficient conditions for the point of (global) minimum without using calculus. Instead of taking derivatives, the method requires only the definition of the (global) minimum and the ‘formula’ for the weighted AM-GM inequality. This solves the problem completely.