## 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