Differential Calculus and Smooth Optimisation

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.