Differential Calculus and Smooth Optimisation

Norman Schofield
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Norman Schofield: Washington University in Saint Louis

Chapter 4 in Mathematical Methods in Economics and Social Choice, 2014, pp 135-187 from Springer

Abstract: Abstract In this chapter we develop the ideas of the differential calculus. Under certain conditions a continuous function f:ℜ n →ℜ m can be approximated at each point x in ℜ n by a linear function df(x):ℜ n →ℜ m , known as the differential of f at x. In the same way the differential df may be approximated by a bilinear map d 2 f(x). When all differentials are continuous then f is called smooth. For a smooth function f, Taylor’s Theorem gives a relationship between the differentials at a point x and the value of f in a neighbourhood of a point. This in turn allows us to characterise maximum points of the function by features of the first and second differential. For a real-valued function whose preference correspondence is convex we can virtually identify critical points (where df(x)=0) with the maxima of the function. We use calculus to derive important results in economic theory, namely conditions for existence of a price equilibrium for an economy, and the Welfare Theorem for an exchange economy.

Keywords: Global Maximum; Walrasian Equilibrium; Degenerate Critical Point; Welfare Theorem; Competitive Allocation (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sptchp:978-3-642-39818-6_4

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DOI: 10.1007/978-3-642-39818-6_4

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