Introductory Mathematics and Statistics for Islamic Finance, + Website

Chapter 8Difference Equations

In Islamic finance, many models have a dynamic feature; variables may be related to their past values and give rise to difference equations. In the cobweb model, a supplier's decision is based on lagged prices. High prices result in high output; the latter induces a fall in prices, which in turn induces lower output, followed by higher prices. At the corporate level, low levels of investment may cause higher investment spending in the future. A low performance may lead a firm to improve performance in the future. In econometrics, difference equations characterize the autoregressive models. A variable is influenced by its past levels. Models of economic growth use difference equations. In the same vein, macroeconomic models rely on difference equations. More specifically, in inflation models, inflationary expectations are determined by past price inflation rates.

In the forward-looking model, the expected future level of a variable may have an effect on the present level of the variable. For instance, if farmers anticipate a drought next year, they may decide to produce more this year. Likewise, if a car company anticipates high demand for cars in the future, it may decide to increase its present car production. If consumers anticipate a shortage of food, they may decide to store food now.¹

This chapter covers the theory of difference equations. In difference equations, time is discrete; variables are dated at equal time intervals (e.g., year, month, ...