Application Leontief input-output analysis

Leontief Input-Output Analysis

Developed by Wassily Leontief (Nobel Prize, 1973), input-output analysis uses linear algebra to model the interdependencies between different sectors of an economy.

The Model

Consider an economy with n industries. Let:

• x = total output vector (n x 1)
• M = technology/consumption matrix (n x n), where m_ij is the amount of output from industry i needed to produce one unit of output from industry j
• d = final demand vector (n x 1), representing external demand

The fundamental equation is:

x = Mx + d

which rearranges to:

(I - M)x = d

Solution

If (I - M) is invertible, the production levels needed to satisfy demand are:

x = (I - M)^(-1) d

The matrix (I - M)^(-1) is called the Leontief inverse.

Example: Two-Industry Economy

Suppose an economy has energy (E) and water (W) sectors:

M = [[0.2, 0.1],   d = [10]
     [0.3, 0.2]]        [20]

Then (I - M)x = d gives:

[[0.8, -0.1],  [x1]   [10]
 [-0.3, 0.8]]  [x2] = [20]

Solving: x1 approx 16.0, x2 approx 31.0 units of output needed.

Applications

• National economic planning
• Environmental impact analysis
• Supply chain analysis
• Regional economics