In a modern industrialized country, there are hundreds of different industries that supply goods and services needed for production. These industries are often mutually dependent.
The agricultural industry, for instance, requires farm machinery to plant and harvest crops, whereas the makers of farm machinery need food produced by the agricultural industry. Because of this interdependency, events in one industry, such as a strike by factory workers, can significantly affect many other industries. To better understand these complex interactions, economic planners use mathematical models of the economy, the most important of which was developed by the Russian-born economist Wassily Leontief.
While a student in Berlin in the 1920s, Leontief developed a mathematical model, called theinput–output model, for analyzing an economy. After arriving in the United States in 1931 to be a professor of economics at Harvard University, Leontief began to collect the data that would enable him to implement his ideas. Finally, after the end of World War II, he succeeded in extracting from government statistics the data necessary to create a model of the U.S. economy. This model proved to be highly accurate in predicting the behavior of the postwar U.S. economy and earned Leontief the 1973 Nobel Prize for Economics.
Leontief’s model of the U.S. economy combined approximately 500 industries into 42 sectors that provide products and services, such as the electrical machinery sector.
To illustrate Leontief’s theory, which can be applied to the economy of any country or region, we next show how to construct a general input–output model. Suppose that an economy is divided into n sectors and that sector i produces some commodity or service Si (i =1, 2,. . .,n). Usually, we measure amounts of commodities and services in common monetary units and hold costs fixed so that we can compare diverse sectors. For example, the output of the steel industry could be measured in millions of dollars worth of steel produced.
For eachi andj, letcij denote the amount ofSi needed to produce one unit ofSj. Then then×n matrixC whose (i,j)-entry iscij is called theinput–output matrix (or theconsumption matrix) for the economy.
To illustrate these ideas with a very simple example, consider an economy that is divided into three sectors: agriculture, manufacturing, and services. (Of course, a model of any real economy, such as Leontief’s original model, will involve many more sectors and much larger matrices.) Suppose that each dollar’s worth of agri- cultural output requires inputs of $0.10 from the agricultural sector, $0.20 from the
∗This section can be omitted without loss of continuity.
manufacturing sector, and $0.30 from the services sector; each dollar’s worth of man- ufacturing output requires inputs of $0.20 from the agricultural sector, $0.40 from the manufacturing sector, and $0.10 from the services sector; and each dollar’s worth of services output requires inputs of $0.10 from the agricultural sector, $0.20 from the manufacturing sector, and $0.10 from the services sector.
From this information, we can form the following input–output matrix:
C =
Ag. Man. Svcs.
.1 .2 .1
.2 .4 .2
.3 .1 .1
Agriculture Manufacturing Services
Note that the (i,j)-entry of the matrix represents the amount of input from sector i needed to produce a dollar’s worth of output from sector j. Now let x1, x2, and x3 denote the total output of the agriculture, manufacturing, and services sectors, respectively. Sincex1 dollar’s worth of agricultural products are being produced, the first column of the input-output matrix shows that an input of .1x1 is required from the agriculture sector, an input of.2x1 is required from the manufacturing sector, and an input of.3x1 is required from the services sector. Similar statements apply to the manufacturing and services sectors. Figure 1.20 shows the total amount of money flowing among the three sectors.
Note that in Figure 1.20 the three arcs leaving the agriculture sector give the total amount of agricultural output that is used as inputs for all three sectors. The sum of the labels on the three arcs,.1x1+.2x2+.1x3, represents the amount of agricultural output that is consumed during the production process. Similar statements apply to the other two sectors. So the vector
.1x1+.2x2+.1x3
.2x1+.4x2+.2x3
.3x1+.1x2+.1x3
Agriculture
Services Manufacturing
.1x2 .1x1
.2x3 .2x1
.2x2
.1x3
.3x1
.1x3 .4x2
Figure 1.20 The flow of money among the sectors
gives the amount of the total output of the economy that is consumed during the production process. This vector is just the matrix–vector productCx, where xis the gross production vector
x=
x1
x2
x3
.
For an economy with input–output matrix C and gross production vector x, the total output of the economy that is consumed during the production process is Cx.
Example 1 Suppose that in the economy previously described, the total outputs of the agriculture, manufacturing, and services sectors are $100 million, $150 million, and $80 million, respectively. Then
Cx=
.1 .2 .1
.2 .4 .2
.3 .1 .1
100 150 80
=
48 96 53
,
and so the portion of the gross production that is consumed during the production process is $48 million of agriculture, $96 million of manufacturing, and $53 million of services.
Since, in Example 1, the amount of the gross production consumed during the production process is
Cx=
48 96 53
,
the amount of the gross production that is not consumed during the production pro- cess is
x−Cx=
100 150 80
−
48 96 53
=
52 54 27
.
Thusx−Cx is thenet production (or surplus) vector; its components indicate the amounts of output from each sector that remain after production. These amounts are available for sale within the economy or for export outside the economy.
Suppose now that we want to determine the amount of gross production for each sector that is necessary to yield a specific net production. For example, we might want to set production goals for the various sectors so that we have specific quantities available for export. Let d denote the demand vector, whose components are the quantities required from each sector. In order to have exactly these amounts available after the production process is completed, the demand vector must equal the net production vector; that is,d=x−Cx. Using the algebra of matrices and vectors and the 3×3 identity matrixI3, we can rewrite this equation as follows:
x−Cx=d I3x−Cx=d (I3−C)x=d
Thus the required gross production is a solution of the equation (I3−C)x=d.
For an economy withn×n input–output matrixC, the gross production neces- sary to satisfy exactly a demanddis a solution of (In −C)x=d.
Example 2 For the economy in Example 1, determine the gross production needed to meet a consumer demand for $90 million of agriculture, $80 million of manufacturing, and
$60 million of services.
Solution We must solve the matrix equation (I3−C)x=d, where C is the input–output matrix and
d=
90 80 60
is the demand vector. Since I3−C =
1 0 0 0 1 0 0 0 1
−
.1 .2 .1
.2 .4 .2
.3 .1 .1
=
.9 −.2 −.1
−.2 .6 −.2
−.3 −.1 .9
,
the augmented matrix of the system to be solved is
.9 −.2 −.1 90
−.2 .6 −.2 80
−.3 −.1 .9 60
.
Thus the solution of (I3−C)x=dis
170 240 150
,
so the gross production needed to meet the demand is $170 million of agriculture,
$240 million of manufacturing, and $150 million of services.
Practice Problem 1 䉴 An island’s economy is divided into three sectors—tourism, transportation, and ser- vices. Suppose that each dollar’s worth of tourism output requires inputs of $0.30 from the tourism sector, $0.10 from the transportation sector, and $0.30 from the services sector; each dollar’s worth of transportation output requires inputs of $0.20 from the tourism sector, $0.40 from the transportation sector, and $0.20 from the ser- vices sector; and each dollar’s worth of services output requires inputs of $0.05 from the tourism sector, $0.05 from the transportation sector, and $0.15 from the services sector.
(a) Write the input–output matrix for this economy.
(b) If the gross production for this economy is $10 million of tourism, $15 million of transportation, and $20 million of services, how much input from the tourism sector is required by the services sector?
(c) If the gross production for this economy is $10 million of tourism, $15 million of transportation, and $20 million of services, what is the total value of the inputs consumed by each sector during the production process?
(d) If the total outputs of the tourism, transportation, and services sectors are $70 million, $50 million, and $60 million, respectively, what is the net production of each sector?
(e) What gross production is required to satisfy exactly a demand for $30 million of tourism, $50 million of transportation, and $40 million of services? 䉴