Introduction

In this chapter we begin to explore the fundamental structure of the input–output model, the assumptions behind it, and also some of the simplest kinds of problems to which it is applied. Later chapters will examine the special features that are associated with regional models and some of the extensions that are necessary for particular kinds of problems – for example, in energy or environmental studies or as part of a broader system of social accounts.

The mathematical structure of an input–output system consists of a set of n linear equations with n unknowns; therefore, matrix representations can readily be used. In this chapter we will start with more detailed algebraic statements of the fundamental relationships and then go on to use matrix notation and manipulations more and more frequently. Appendix A contains a review of matrix algebra definitions and operations that are essential for input–output models. While solutions to the input–output equation system, via an inverse matrix, are straightforward mathematically, we will discover that there are interesting economic interpretations to some of the algebraic results.

Notation and Fundamental Relationships

An input–output model is constructed from observed data for a particular economic area – a nation, a region (however defined), a state, etc. In the beginning, we will assume (for reasons that will become clear in the next chapter) that the economic area is a country. The economic activity in the area must be able to be separated into a number of segments or producing sectors. These may be industries in the usual sense (e.g., steel) or they may be much smaller categories (e.g., steel nails and spikes) or much larger ones (e.g., manufacturing).