Systems of equations, differentials and derivatives
In many economic problems, functions of more than one variable arise natu-
rally. Preferences are represented by a utility function that depends on goods con-
sumed, u(x1,x2, . . . ,xn), output in production is represented as a function of inputs,
f (k1,k2, . . . ,kn), and so on. As with the single-variable case, differential tech-
niques 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 k1 is given by [ f (k1+ ∆,k2, . . . ,kn)− f (k1,k2, . . . ,kn)]/∆, 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 tech-
niques leads to a number of applications and provides the necessary tools to de-
velop some important results – in particular the implicit function theorem.