Application to Input-Output Economic Models

In 1973 Wassily Leontief was awarded the Nobel prize in economics for his work on mathematical models. (See W. W. Leontief, “The world economy of the year 2000,” Scientific American, Sept. 1980.) Roughly speaking, an economic system in this model consists of several industries, each of which produces a product and each of which uses some of the production of the other industries. The following example is typical.

In general, suppose an economy has industries, each of which uses some (possibly none) of the production of every industry. We assume first that the economy is closed (that is, no product is exported or imported) and that all product is used. Given two industries and , let denote the proportion of the total annual output of industry that is consumed by industry . Then is called the input-output matrix. Clearly,

Moreover, all the output from industry is used by some industry (the model is closed), so This condition asserts that each column of sums to . Matrices satisfying conditions (eq:IOcond1) and (eq:IOcond2) are called stochastic matrices.

As in Example 006965, let denote the price of the total annual production of industry . Then is the annual revenue of industry . On the other hand, industry spends annually for the product it uses ( is the cost for product from industry ). The closed economic system is said to be in equilibrium if the annual expenditure equals the annual revenue for each industry—that is, if

If we write , these equations can be written as the matrix equation This is called the equilibrium condition, and the solutions are called equilibrium price structures. The equilibrium condition can be written as which is a system of homogeneous equations for . Moreover, there is always a nontrivial solution . Indeed, the column sums of are all (because is stochastic), so the row-echelon form of has a row of zeros. In fact, more is true:

Theorem 007013 guarantees the existence of an equilibrium price structure for any closed input-output system of the type discussed here. The proof is beyond the scope of this book.1

The Open Model

We now assume that there is a demand for products in the open sector of the economy, which is the part of the economy other than the producing industries (for example, consumers). Let denote the total value of the demand for product in the open sector. If and are as before, the value of the annual demand for product by the producing industries themselves is , so the total annual revenue of industry breaks down as follows:

The column is called the demand matrix, and this gives a matrix equation or This is a system of linear equations for , and we ask for a solution with every entry nonnegative. Note that every entry of is between and , but the column sums of need not equal as in the closed model.

Before proceeding, it is convenient to introduce a useful notation. If and are matrices of the same size, we write if for all and , and we write if for all and . Thus means that every entry of is nonnegative. Note that and implies that .

Now, given a demand matrix , we look for a production matrix satisfying equation (eq:demandmatrix). This certainly exists if is invertible and . On the other hand, the fact that means any solution to equation (eq:demandmatrix) satisfies . Hence, the following theorem is not too surprising.

Heuristic Proof

If , the existence of with is left as Practice Problem ex:ex2_8_11. Conversely, suppose such a column exists. Observe that

holds for all . If we can show that every entry of approaches as becomes large then, intuitively, the infinite matrix sum exists and . Since , this does it. To show that approaches , it suffices to show that for some number with (then for all by induction). The existence of is left as Practice Problem ex:ex2_8_12.

The condition in Theorem 007060 has a simple economic interpretation. If is a production matrix, entry of is the total value of all product used by industry in a year. Hence, the condition means that, for each , the value of product produced by industry exceeds the value of the product it uses. In other words, each industry runs at a profit.

If , the entries of are the row sums of . Hence holds if the row sums of are all less than . This proves the first of the following useful facts (the second is Practice Problem ex:ex2_8_10).

Practice Problems

Find the possible equilibrium price structures for each given input-output matrix.

(Use as the parameter.)

(Use as the parameter.)

Three industries , , and are such that all the output of is used by , all the output of is used by , and all the output of is used by . Find the possible equilibrium price structures.

(Use as the parameter.)

Find the possible equilibrium price structures for three industries where the input-output matrix is . Discuss why there are two parameters here.
Prove Theorem 007013 for a stochastic matrix by first writing it in the form , where and .
If is an stochastic matrix and is an matrix, show that the sum of the entries of equals the sum of the entries of the matrix .
Let . Let and denote matrices with nonnegative entries.
(a)
Show that is a stochastic matrix if and only if .
(b)
Use part (a.) to deduce that, if and are both stochastic matrices, then is also stochastic.
Find a matrix with entries between and such that:
(a)
has no inverse.
(b)
has an inverse but not all entries of are nonnegative.
If is a matrix with entries between and , show that is invertible and if and only if . Here, if , then and .
In each case show that is invertible and .
Use in Theorem 007060.
Use in Theorem 007060.
Prove that (1) implies (2) in the Corollary to Theorem 007060.
If , find such that .
If where and , find a number such that and .
If and , take any number where .

Text Source

This application was adapted from Section 2.8 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, p. 128