Difference equations are formal models of dynamical systems in which time is assumed to evolve in discrete periods. Models of this kind are used in many areas of economic research, including macroeconomics, monetary economics, resource economics, game theory, etc. In this chapter, we introduce some basic concepts and illustrate them by a number of selected examples. Throughout the book, we restrict the presentation to deterministic systems, that is, we do not consider any models involving uncertainty.

One of the simplest types of difference equations arises through repeated iterations of one-dimensional maps. Because of their conceptual simplicity, we start our discussion of difference equations with these models, proceeding rather informally and without proving any theorems. Later in chapter 4, we shall see that even these simple difference equations can generate surprisingly rich dynamics. In section 1.2, we continue by introducing a general class of explicit difference equations and by extending the basic concepts to this framework. We also increase the level of rigor and formally prove several properties of the solutions of explicit difference equations including their existence and their uniqueness for a complete set of initial or boundary conditions. Finally, in section 1.3, we argue that economic models often take the form of implicit difference equations. Unfortunately, neither the existence nor the uniqueness of solutions to such equations can be ensured, a fact that we illustrate by means of a detailed economic example.

One-dimensional maps

Suppose that the economic system under consideration can be described by a single variable x ∈ X, where X ⊆ ℝ is a non-empty interval on the real line. Depending on the context, the variable x can measure the productive capital available in the economy, the price of a commodity, the stock of a resource, the fraction of the population with a certain characteristic, etc. We shall refer to x as the system variable and to X as the system domain. The system domain contains all possible values of the system variable.