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.
A primitive society has three basic needs: food, shelter, and clothing. There are thus
three industries in the society—the farming, housing, and garment industries—that
produce these commodities. Each of these industries consumes a certain
proportion of the total output of each commodity according to the following
table.
Find the annual prices that each industry must charge for its income to equal its expenditures.
Let , , and be the prices charged per year by the farming, housing, and garment
industries, respectively, for their total output. To see how these prices are determined,
consider the farming industry. It receives for its production in any year. But it
consumes products from all these industries in the following amounts (from row 1 of
the table): of the food, of the housing, and of the clothing. Hence, the expenditures
of the farming industry are , so \begin{equation*} 0.4p_{1} + 0.2p_{2} + 0.3p_{3} = p_{1} \end{equation*}
A similar analysis of the other two industries leads to the following system of
equations. \begin{align*} 0.4p_{1} + 0.2p_{2} + 0.3p_{3} &= p_{1} \\ 0.2p_{1} + 0.6p_{2} + 0.4p_{3} &= p_{2} \\ 0.4p_{1} + 0.2p_{2} + 0.3p_{3} &= p_{3} \end{align*}
This has the matrix form , where \begin{equation*} E = \begin{bmatrix} 0.4 & 0.2 & 0.3 \\ 0.2 & 0.6 & 0.4 \\ 0.4 & 0.2 & 0.3 \end{bmatrix} \quad \mbox{ and } \quad{\bf p} = \left [ \begin{array}{c} p_{1} \\ p_{2} \\ p_{3} \end{array} \right ] \end{equation*} The equations can be written as the homogeneous system \begin{equation*} (I - E){\bf p} ={\bf 0} \end{equation*} where is the identity matrix, and the solutions are \begin{equation*}{\bf p} = \left [ \begin{array}{c} 2t \\ 3t \\ 2t \end{array} \right ] \end{equation*} where is a parameter. Thus, the pricing must be such that the total output of the farming industry has the same value as the total output of the garment industry, whereas the total value of the housing industry must be as much.
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, \begin{equation} \label{eq:IOcond1} 0 \leq e_{ij} \leq 1 \quad \mbox{for all } i \mbox{ and } j \end{equation} Moreover, all the output from industry is used by some industry (the model is closed), so \begin{equation} \label{eq:IOcond2} e_{1j} + e_{2j} + \cdots + e_{ij} = 1 \quad \mbox{for each } j \end{equation} 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 \begin{equation*} e_{1j}p_{1} + e_{2j}p_{2} + \cdots + e_{ij}p_{n} = p_{i} \quad \mbox{for each } i = 1, 2, \dots , n \end{equation*} If we write , these equations can be written as the matrix equation \begin{equation*} E{\bf p} ={\bf p} \end{equation*} This is called the equilibrium condition, and the solutions are called equilibrium price structures. The equilibrium condition can be written as \begin{equation*} (I - E){\bf p} ={\bf 0} \end{equation*} 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:
Let be any stochastic matrix. Then there is a nonzero vector with nonnegative
entries such that . If all the entries of are positive, the matrix can be chosen with all
entries positive.
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.
Find the equilibrium price structures for four industries if the input-output matrix is
\begin{equation*} E = \begin{bmatrix} 0.6 & 0.2 & 0.1 & 0.1 \\ 0.3 & 0.4 & 0.2 & 0 \\ 0.1 & 0.3 & 0.5 & 0.2 \\ 0 & 0.1 & 0.2 & 0.7 \end{bmatrix} \end{equation*}
Find the prices if the total value of business is .
If is the equilibrium price structure, then the equilibrium condition reads . When we
write this as , we obtain the following family of solutions \begin{equation*}{\bf p} = \begin{bmatrix} 44t \\ 39t \\ 51t \\ 47t \end{bmatrix} \end{equation*}
where is a parameter. If we insist that , then . Hence \begin{equation*}{\bf p} = \begin{bmatrix} 243.09 \\ 215.47 \\ 281.76 \\ 259.67 \end{bmatrix} \end{equation*}
to five figures.
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: \begin{equation*} p_{i} = (e_{i1}p_{1} + e_{i2}p_{2} + \cdots + e_{in}p_{n}) + d_{i} \quad \mbox{for each } i = 1, 2, \dots , n \end{equation*} The column is called the demand matrix, and this gives a matrix equation \begin{equation*}{\bf p} = E{\bf p} +{\bf d} \end{equation*} or \begin{equation} \label{eq:demandmatrix} (I - E){\bf p} ={\bf d} \end{equation} 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 \begin{equation*} (I - E)(I + E + E^2 + \cdots + E^{k-1}) = I - E^k \end{equation*} holds for all . If we can show that every entry of approaches as becomes large then, intuitively, the infinite matrix sum \begin{equation*} U = I + E + E^2 + \cdots \end{equation*} 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).
Let be a square matrix. In each case, is invertible and :
- (a)
- All row sums of are less than .
- (b)
- All column sums of are less than .
Practice Problems
Problems prob:i/o_1-prob:i/o_4
Find the possible equilibrium price structures for each given input-output matrix.
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.
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.
If is a matrix with entries between and , show that is invertible and if and only if
. Here, if , then and .
Problems prob:i/o_14-prob:i/o_17
In each case show that is invertible and .
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
2024-11-13 18:07:01