ABSTRACT
In many economic problems, functions of more than one variable arise naturally. Preferences are represented by a utility function that depends on goods consumed, u(x 1 x 2,…, xn ), output in production is represented as a function of inputs, f(k 1, k 2,…, kn ), and so on. As with the single-variable case, differential techniques play an important role in a variety of applications, such as comparative statics and optimization. For example, the impact of variation in one variable on the value of a function routinely arises in considering its marginal impact. In the production case, the marginal productivity of input k 1 is given by [f(k 1 + Δ, k 2,…, kn ) – f(k 1, k 2,…, kn )]/Δ, and measures the impact on output of a small change in input k1. Such a calculation is called a partial derivative, since only variation in one of a number of variables is considered. Developing these techniques leads to a number of applications and provides the necessary tools to develop some important results – in particular the implicit function theorem.
