The Closed and Open Leontief Models
The standard input-output analysis is based upon a number of simplifying assumptions with regard to the productive structure of an economy. It is assumed that the economy can be divided into n sectors of production, each of which is uniquely associated to a specific product (for example, the steel producing sector).
In addition, all production processes are characterized by constant returns to scale, and all production functions are of the fixed proportions type. This means that it is possible to describe the relations between the sectors of production by means of a square matrix of technical coefficients, habitually designated by A. The j-th column of A (/ = 1, 2,..., n) describes the amounts of goods 1, 2,..., n required as inputs to produce one unit of good / as output.In the so-called closed Leontief model the inputs are defined in such a way that they include the consumption of goods by those who have contributed to production. This implies that there is no net output: on the level of the economy as a whole, the inputs are exactly equal to the outputs. In mathematical terms, this can be expressed as:
where x represents the activity vector. In terms of value, the counterpart of (1) is: 
where p stands for the vector of prices. (By convention we use the term “vector” to denote column vectors. A prime is used to denote row vectors.) The implication of (1) and (2) is that matrix (I - A) is a singular matrix, and hence there is no unique solution of x orp.
In the open Leontief model a separate final demand vector is introduced. Consumption is no longer integrated into the input coefficients. Instead of (1) we now have:
where d stands for the final demand vector.
Provided matrix (I - A) is regular, the activity levels which sustain the final demand are given by the solution:
Under plausible conditions matrix (I - A)-1 exists and is semi-positive. This implies that any positive demand vector d can be satisfied by positive activity levels. Matrix (I - A)-1, which plays a crucial role in input-output analysis, is called the Leontief inverse. It also occurs when we look at the value side. The counterpart of (2) can be written as:
where r stands for the value-added vector. The value-added represents what is paid to the primary factors such as labour and capital in the form of wages, profits, and so on. This leads to the following relation between value-added and prices:
These relationships can be used to estimate the effects of exogenous changes in the economic system. For instance, let us suppose that the export of a particular good increases as a result of an increase in world demand. If the new final demand vector is
demand.