Modelling of the economy plays a very important role in the attempt to understand and manipulate the economy of a country or a region. The Leontief Closed Model is an early attempt in this direction. These problems are generally quite difficult, since they are not as well defined as problems in (for example) the physical sciences. In particular, it is not clear when enough data are known to approximate the economy, and what influence human nature has on the economy. The Leontief closed model was invented by Prof Wassily Leontief in the 1930's, and he was subsequently awarded the Nobel prize for his efforts (of course, Leontief's contributions goes beyond the model we will discuss here).

Consider the general problem where we have *n* manufacturers
making *n* products
(
makes ).
In one year we assume that
makes exactly one unit of ,
and that all of the
is consumed in that year in the manufacturing of the other products (including ).
Thus, let
be the amount of product
consumed in the manufacturing of .
Then .

Suppose that the system is **closed**: There are no products leaving
or entering the system, and suppose that all the goods are consumed at
the rate that they are made. Then, since the total production of
is 1,

The aim is understand the prices of the products. Let
be the price of product
(for one unit). Assuming that no manufacturer makes or loses any money,
what are the prices of the products? Since
pays
for product ,
the total cost in producing product
is

But 's
income must be ,
if there is no loss or profit. Thus, equating expenses with income, we
get

Let ,
and .
Then this equation reduces to

*A* is called the exchange matrix, since it describes the exchange
of goods between the manufacturers. We must find a solution
which satisfies this equation, and that will be the prices of the products.

One can argue that we should really require that no manufacturer is
making a loss: .
However, it can be proven that this implies that .
In other words, no manufacturer can make a profit without another making
a loss. The Leontief closed model is therefore called a *zero-sum*
game, describing a steady state system.

The following table describes the exchange of goods in a small economy
which includes a wheat farmer, a milk farmer, a wine producer, a tailor,
a cotton grower, a baker and a yoghurt maker.

Make a Leontief closed model from this data, and solve for the prices of the products using Maple. Observe that the columns do not add up to 1. Therefore, in order to use the assumption that the total produced is one unit, you will have to normalize the production; in other words, you will have to rescale to different units so that the sum of each column is equal to 1.