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# The Leontief Input-Output Model - leontief matrix calculator

The Leontief Input-Output Model
Text Reference: Section 2.7, p. 152
The purpose of this set of exercises is to provide three more examples of the Leontief Input-
Output Model in action. We begin by reviewing the basic assumptions of the model and the
calculations involved. Refer to Section 2.7 of your text for more complete information.
Recall that the input-output model requires that the economy in question be divided into
sectors. Each sector produces goods or services except for the open sector, which only
consumes goods and services. A production vector x lists the output of each sector. A
final demand vector (or bill of final demands) d lists the values of the goods and services
demanded from the productive sectors by the open sector. As the sectors strive to produce
enough goods to meet the final demand vector, they make intermediate demands for the
products of each sector. These intermediate demands are described by the consumption
matrix. This matrix is constructed as follows.
We begin with a collection of data called an input-output table (or an exchange table)
for an economy. This table lists the value of the goods produced by each sector and how
much of that output is used by each sector. For example, the following table is derived from
the table Leontief created for the American economy in 1947. (See References 1 or 2 for
the complete table.) For purposes of this example we have collected the data from the 42
sectors into just 3: agriculture, manufacturing, and services. Of course, the open sector is
also present.
Agriculture Manufacturing Services Open Sector
Agriculture 34.69 4.92 5.62 39.24
Manufacturing 5.28 61.82 22.99 60.02
Services 10.45 25.95 42.03 130.65
Total Gross Output 84.56 163.43 219.03
Table 1: Exchange of Goods and Services in the U.S. for 1947 (in billions of 1947 dollars)
Reading the table is straightforward; for example, in 1947 the agriculture sector spent 84.56
billion dollars for the inputs it needed. These inputs were divided among the sectors as
follows: 34.69 billion dollars of agricultural output was consumed by the agriculture sector
itself, 5.28 billion dollars of manufacturing output was consumed by the agriculture sector,
etc.
To create the consumption matrix from the table, we divide each column of the 3 ? 3 table
by the Total Gross Output for that sector. The result is Table 2, which appears on the
following page.
The matrix with entries taken from this table is the consumption matrix C for the economy.
1
Agriculture Manufacturing Services
Agriculture .4102 .0301 .0257
Manufacturing .0624 .3783 .1050
Services .1236 .1588 .1919
Table 2: Inputs Consumed Per Unit of Sector Output

.4102 .0301 .0257
C = .0624 .3783 .1050
.1236 .1588 .1919
For the 1947 economy, the final demand vector d is the column of the table associated with
the open sector:

39.24
d = 60.02
130.65
We wish to find equilibrium levels of production for each sector; that is, production levels
which will just meet the intermediate demands of the sectors of the economy plus the final
demands of each sector. If x is the desired production vector, we know that x must satisfy
x = Cx + d
We may solve this equation for x to find that
x = (I - C)-1d
where I is the identity matrix.
In our example, we find that

1.7203 .1006 .0678
(I - C)-1 = .2245 1.6768 .2250
.3073 .3449 1.2921
and thus that
82.40
x = (I - C)-1d = 138.85
201.57
2
Question:
1. Suppose the bill of final demands is changed to

40.24
d = 60.02
130.65
What is the new eqilibrium production vector? Find the difference between this new
vector and the old equilibrium vector? How must extra production must each sector
provide?
Notice that in the above exercise the only difference in the old and new demand vectors is
the addition of one unit of demand to the agricultural sector. Also notice that the difference
in the old and new production vectors is just the first column of the matrix (I - C)-1. This
is a valuable interpretation of the entries of (I - C)-1:
Observation: The (i, j) entry in the matrix (I - C)-1 is the amount by which sector i must
change its production level to satisfy an increase of 1 unit in the final demand from sector j.
Question:
2. How much would the service production level need to increase if agricultural demand
for services increased by 1 unit? How much would the manufacturing production level
need to increase in this situation?
We will now consider a less abrupt consolidation of the 1947 economic data: we divide the
economy into 25 sectors. These sectors are:
1. Agriculture and Fisheries 13. Motor Vehicles
2. Food and Kindred Products 14. Other Transportation Equipment
3. Textiles and Apparel 15. Miscellaneous Manufacturing
4. Lumber, Wood, and Furniture 16. Coal, Gas, and Electric Power
5. Paper, Printing, and Publishing 17. Transportation Services
6. Chemicals, Petroleum Products, Rubber 18. Trade
7. Leather and Leather Products 19. Communications
8. Stone, Clay, and Glass Products 20. Finance, Insurance, Real Estate
9. Primary Metals 21. Business Services
10. Fabricated Metal Products 22. Personal and Repair Services
11. Machinery (non-electric) 23. Miscellaneous Services
12. Electrical Machinery 24. New Construction and Maintenance
25. Undistributed
The consumption matrix C1 and final demand vector d1 for this model accompany this
exercise set.
3

How do you find the Leontief inverse of a matrix? We can transform this equation as follows: InX AX = B (In A)X = B X = (In A) 1B if the inverse of the matrix In A exists. ((In A) 1 is then called the Leontief inverse.) For a given realistic economy, a solution obviously must exist.

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