Mathematics for Economics — Matrices and Input-Output Analysis

Mathematics for Economics — Matrices Solve Inter-Industry Dependence in One Go, Not One Market

The integral of chapter 2 is the area under a single curve. When goods use one another as intermediates, adding those areas industry by industry does not give total output. This chapter reviews why $(I-A)^{-1}$ closes that loop.

1. Final demand and total output are not the same vector

The $(i,j)$ element of the intermediate input coefficient matrix $A$ is the amount of i needed to make one unit of j. If final demand is $d$ , total output $x$ must satisfy intermediate demand $Ax$ and final demand at the same time.

Leontief system

x = Ax + d \quad \Rightarrow \quad x = (I-A)^{-1}d

Each column of the inverse shows how much every industry must run to meet one unit of final demand for that industry.

With $A=[[0.2,0.3],[0.1,0.4]]$ and $d=(100,80)$ , $I-A=[[0.8,-0.3],[-0.1,0.6]]$ . The determinant is $0.48-0.03=0.45$ . The inverse is $(1/0.45)[[0.6,0.3],[0.1,0.8]]$ . Total output is $(84/0.45, 74/0.45)$ , roughly $(186.7, 164.4)$ units — more than the final demand of 100 and 80 units. The difference is the circulation of intermediates.

2. Having an inverse does not make a technology viable

What matrices do in economics
Object	Expression	When it fails
Input-output	(I-A)^{-1}d	Hawkins–Simon fails, spectral radius of 1 or more
Comparative statics with two goods	Signs of the Jacobian	Outside the local linearization
Portfolio	Covariance matrix	Singular matrix, perfect correlation

The Hawkins–Simon condition requires the leading principal minors of $(I-A)$ to be positive. Being able to plug in numbers is not the same as the technology being productive. The Jacobian of comparative statics is the multivariable version of the derivatives in chapter 1. The covariance matrix is what corporate finance reads as risk.

The Bank of Korea’s input-output tables extend this 2×2 to dozens of industries. The arithmetic is the same. If classifications or the treatment of imports change, $x$ changes even for the same $d$ .

Check your understanding

In the $A$ above, if only the second industry’s final demand rises by 20, from 80 to 100, multiply the second column of the inverse, $(1/0.45)(0.3, 0.8) \approx (0.667, 1.778)$ , by 20: total output rises by about 13.3 and 35.6 units. Only the second industry’s demand rose, yet the first industry’s output also rises by 13.3. Without intermediate links, the first would be 0.

References

Alpha Chiang and Kevin Wainwright, Fundamental Methods of Mathematical Economics, ch. 15–16
Wassily Leontief, Input-Output Economics
Bank of Korea, Input-output tables (Korean)