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Section 2.6: Finding the Input-Output Matrix
This document covers a part of the material of Section 2.6 on
Input-Output Models, andis not intended to serve as a replacement
for the actual section 2.6 of the textbook. Input-
Output models are concerned with the production and flow of
goods or services.
Definition. In an economy with n basic commodities, or sectors,
the production of eachcommodity uses some (or all) of the
commodities in the economy as inputs. The amountsof each commodity
used in the production of one unit of each commodity can be
writtenas an n n matrix A, called the technological matrix or
input-output matrix of theeconomy.
Example (1). Suppose a simplified economy involves just three
commodities categories:agriculture, manufacturing, and
transportation, all in appropriate units. Productionof 1 unit of
agriculture requires 1/2 unit of manufacturing and 1/4 unit of
transportation;production of 1 unit of manufacturing requires 1/4
unit of agriculture and 1/4 unitof transportation; and production
of 1 unit of transportation requires 1/3 unit ofagriculture and 1/4
unit of manufacturing. Give the input-output matrix for this
economy.
Solution. The input-output matrix is
A =
0 1/2 1312
0 14
14
14
0
The first column of the input-output matrix represents the
amount of each of the threecommodities consumed in the production
of 1 unit of agriculture. The second column givesthe amounts
required to produce 1 unit of manufacturing, and the last column
gives theamounts required to produce 1 unit of transportation.Note:
Notice that for each commodity produced, the various units needed
are put in acolumn.
Definition. The matrix giving the amount of each commodity
produced is called the pro-duction matrix, or the matrix of gross
output. In an economy producing n commodities,the production matrix
is an n 1 matrix, commonly denoted by X.Example (2). In example 1,
suppose the production matrix is
X =
605248
Then 60 units of agriculture, 52 units of manufacturing, and 48
units of transportation areproduced. Because 1/4 unit of
agriculture is used for each unit of manufacturing produced,
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Math 018 Elementary Linear Algebra Spring 2011
1/452 = 13 units of agriculture must be used in the production
of manufacturing. Similarly,1/3 48 = 16 units of agriculture will
be used in the production of transportation. Thus,13 + 16 = 29
units of agriculture are used for production in the economy.
Thus since, X gives the number of units of each commodity
produced and A gives theamount (in units) of each commodity used to
produce 1 unit of each of the various commodi-ties, the matrix
product AX gives the amount of each commodity used in the
productionprocess.
AX =
0 14 1312
0 14
14
14
0
605248
=2942
28
From this result, 29 units of agriculture, 42 units of
manufacturing, and 28 units of trans-portation are used to produce
60 units of agriculture, 52 units of manufacturing, and 48units of
transportation.
The matrix AX represents the amount of each commodity used in
the production process.The remainder (if any) must be enough to
satisfy the demand for the various commoditiesfrom outside the
production process.
Definition. In an n-commodity economy, this demand can be
represented by an n 1matrix, called the demand matrix, commonly
denoted by D.
The difference between the production matrix X and the amount AX
used in the pro-duction process must equal the demand D, or
D = X AX.
In example 2,
D =
605248
2942
28
=3110
20
,so production of 60 units of agriculture, 52 units of
manufacturing, and 48 units of trans-portation would satisfy a
demand of 31, 10, and 20 units of each commodity respectively.
In practice, A and D are known and X must be found. That is, we
need to decide whatamounts of production are needed to satisfy the
required demands.
Using matrix algebra, we have
X = (I A)1D,
if the matrix (I A) has an inverse, where I denotes the identity
matrix of the same size asA.
Example (3). As described in class.Example (4). An economy has
two basic products, wheat and oil. To produce 1 unit ofwheat
requires 0.25 units of wheat and 0.33 units of oil. Production of 1
unit of oil consumes0.08 units of wheat and 0.11 units of oil.
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Math 018 Elementary Linear Algebra Spring 2011
(a) Find the production that will satisfy a demand for 500 units
of wheat and 1000 unitsof oil.
Solution. The input-output matrix is
A =
[0.25 0.080.33 0.11
]Then
I A =[1 00 1
][0.25 0.080.33 0.11
]=
[0.75 0.080.33 0.89
].
Next, calculate (I A)1.
(I A)1 =[1.3882 0.12480.5147 1.1699
].
To find the production matrix X, use the equation X = (I A)1D,
with
D =
[5001000
].
So, the production matrix is
X =
[1.3882 0.12480.5147 1.1699
] [5001000
][
8191427
]This means production of 819 units of wheat and 1427 units of
oil is required to satisfythe indicated demand.
(b) As done in class.
Remark. The entries in the matrix (I A)1 are often called
multipliers, and they haveimportant economic interpretations.
Definition. The input-output models discussed above are called
open models, since theyallow for a surplus from the production
equal to D. In the closed model, all the productionis consumed
internally in the production process, so that X = AX.
In this case, the sum of each column of the input-output matrix
equals 1. To solve aclosed model, we need to solve the system of
equations corresponding to the matrix equation
(I A)X = O,
where O is the n 1 zero-matrix. The system of equations does not
have a unique solution,but it can be solved in terms of a parameter
using the Gauss-Jordan method or the echelonmethod.
Example (5). As done in class.Penn State University, University
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Math 018 Elementary Linear Algebra Spring 2011
Summary
Finding the production matrix
To find the production matrix, X, for an open input-output
model, follow these steps:
1. Form the n n input-output matrix, A, by placing in each
column the amount of thevarious commodities required to produce 1
unit of a particular commodity.
2. Calculate (I A), where I is the n n identity matrix.3. Find
the inverse, (I A)1.4. Multiply the inverse on the right by the
demand matrix, D, to obtain
X = (I A)1D.
To obtain a production matrix, X, for a closed input-output
model, solve the system
(I A)X = O,
where O is the n 1 zero-matrix.
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