The Leontief Input-Output Model

This model typically requires solving large systems of linear equations which can be done efficiently using Wolfram Language

Consider an economy with

sectors that produce goods and services. Also, assume that there is an open sector that does not produce any goods or services but only consumes them.

Each producing sector creates an intermediate demand by consuming goods and services produced by the other sectors and itself. The open sector creates a demand for goods and services from all producing sectors, called the final demand. According to the Leontief model, the production equation is

production=intermediatedemand+finaldemand

Let

be the production vector,

the final demand vector and

the matrix whose columns represent the units of each good consumed by a particular sector called the consumption matrix for the economy. The production equation can now be written in matrix form as:

x=Cx+d

Assuming

(

-C)

is invertible, the production vector for the Leontief model is

-1

(

-C)

Consider an economy with three sectors: manufacturing, agriculture, and utilities. Suppose that for each unit of output, manufacturing requires

0.4

units from manufacturing,

0.3

units from agriculture, and

0.2

units from utilities. For each unit of output, agriculture requires

0.2

units from manufacturing,

0.4

units from agriculture, and

0.1

units from utilities. For each unit of output, utilities requires

0.1

units from manufacturing,

0.1

units from agriculture, and

0.5

units from utilities. Determine the production levels necessary for a final demand of

units for manufacturing,

units of agriculture and

units of utilities:

The consumption matrix for the economy is:

In[1]:=

(

={{0.4,0.3,0.2},{0.2,0.4,0.1},{0.1,0.1,0.5}})//MatrixForm

Out[1]//MatrixForm=

0.4	0.3	0.2
0.2	0.4	0.1
0.1	0.1	0.5

The final demand vector for the economy is:

In[2]:=

(d={70,50,20})//MatrixForm

Out[2]//MatrixForm=

Then, the production vector for this economy is:

In[3]:=

Inverse[IdentityMatrix[3]-

].d//MatrixForm

Out[3]//MatrixForm=

254.4

189.6

128.8

Round the answers to the nearest whole unit:

In[4]:=

Round[%]//MatrixForm

Out[4]//MatrixForm=

254

190

129

Hence, manufacturing, agriculture and utilities sectors must produce 254, 190 and 129 units respectively.

The original Leontief input-output model required solving a linear system of

500

equations in

500

unknowns. Leontief reduced the problem to solving a system of

equations in

unknowns. The reduced system could be solved in

hours using the computing technology available at the time. Today, large systems of linear equations can be solved quickly using Wolfram Language.

Define a random

500500

consumption matrix and a random final demand vector for an economy:

In[5]:=

SeedRandom[1234];

=RandomReal[{0,1},{500,500}];d=RandomReal[{0,100},500];

Find the production vector for this economy and check how long it took for the system to be solved:

In[6]:=

(x=Inverse[IdentityMatrix[500]-

].d;)AbsoluteTiming[x=Inverse[IdentityMatrix[500]-

].d]〚1〛

Out[6]=

0.0078583

Verify that the solution satisfies the system of equations:

In[7]:=

.x-d//Chop//Flatten//Union

Out[7]=

{0}

External Links

Wolfram U Introduction to Linear Algebra

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Contributed by: Juan Ortiz

The Leontief Input-Output Model

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