The main object of microeconomics is the analysis of individual decisions concerning goods and services, with the aim of deriving general laws that can later be tested against real-world data. In many situations, the problems encountered can be solved graphically by invoking the concept of tangency, in various forms. Three classic microeconomic examples illustrate this: basic producer or consumer theory involves lines tangent to curves, the Edgeworth box uses curves tangent to one another, and a firm's long-run average cost is graphically expressed as the envelope of a family of curves.
Lines tangent to curves -------------------------------
The notion of a line tangent to a curve is often used in microeconomics. From a mathematical standpoint, certain economic problems amount to finding the optimum of an objective function of several variables, possibly subject to constraints. Let's consider the (simple) case of finding a maximum of a function of two variables, assumed to be non-negative, satisfying a genuine constraint given by a first-degree inequality. The basic problems in production theory and consumption theory are of this type. Formally, they are analogous in every respect; only the notation and vocabulary used differ, as shown in the equivalence table below.
The notion of a level curve deserves closer attention. An isoquant is the curve taking into account the capital and labor variables that make it possible to reach an equal degree of production level. An indifference curve gathers all the baskets offering the same satisfaction to the consumer. Every level curve takes the form of a decreasing function curving upward. The optimal choice for reaching the sought maximum lies at a point where a level curve of the objective is tangent to the line whose equation is obtained by converting the genuine constraint's inequality into an equality. At this point of tangency, the slope of this line therefore coincides with what is called the marginal rate of substitution, which measures the rate at which one variable must be substituted for the other in order to maintain the same optimal level of the objective. The curve linking the optimal points, as the line defined by the constraint moves parallel to itself, is of particular interest to economists. This is referred to as the expansion path (or expansion locus).