Modelling the Steady State Prices of an Economy Using the Leontieff Model

The due date is November 22.

You will need to be familiar with material in sections 9.1-9.8 of "Introduction to Maple 8" by D. I. Schwartz.

The economy of a country or a group of countries can be modelled by thinking of each sector of the economy as a producing and consuming unit. This leads to the idea an input and output matrix which was first studied by Vasily Leontieff in the 1930's.

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: In other words, there are no products leaving or entering the system, and suppose that all the goods produced are consumed during the period under consideration, say one year. Then, since the total production of is 1,



It turns out that it is possible to predict the prices of goods in this economy assuming stability of prices; in other words, assuming no producer is forced to raise prices because production costs are higher than income. 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. (This is what is known as a zero-sum game.

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 oorder to agree with our formulation you will need the transpose of the matrix. The stable prices correspond to a non-trivial eignenvector of the exchange matrix. There are two ways to find such an eigenvector. One is to use use Maple's LinearAlgebra package. The other is to start with a potential price vector an iterate applying the exchange matrix to this vector. Why should this yield an eignevector? Do the two method yield the same result. If not, why not?