In a linear algebra textbook I came across the following question (not included in the answer key):

Consider an open economy with a consumption matrix

\begin{equation} C = \begin{pmatrix} 0.5 & 0.25 & 0.25 \\ 0.5 & 0.125 & 0.25 \\ 0.5 & 0.25 & 0.125 \\ \end{pmatrix} \end{equation} If the open sector demands the same dollar value from each product-producing sector, which such sector must produce the greatest dollar value to meet the demand? Is the economy productive?

This textbook defines productive as: "Economies for which $(I − C)^{−1}$ has nonnegative entries are said to be productive.".

Below is a theorem from the textbook.

If $C$ is the consumption matrix for an open economy, and if all of the column sums are less than $1$, then the matrix $I − C$ is invertible, the entries of $(I − C)^{−1}$ are nonnegative, and the economy is productive.

Immediately I notice that the column sum of $C$'s first column is greater than $1$, and from what the textbook says that means the given sector requires more inputs than its output, thus it's not profitable. However, it turns out that this given matrix is invertible and its inverse has nonnegative entries.

I understand mathematically in the cited theorem that the column sums being less than $1$ is not an if and only if condition and thus exceptional circumstances are possible.

However, in economic terms the question seems odd, in other words, asking whether an economy with an unprofitable sector is productive.

If the question is not faulty: how would you explain this in economic terms? Or if it is, what is wrong with the question?

  • 3
    $\begingroup$ In the future, please consider using MathJax to format the mathematical expressions. $\endgroup$
    – Herr K.
    Commented Oct 24, 2020 at 4:46

2 Answers 2


In terms of if and only if statements according to Peterson & Olinick (1982);

A substochastic matrix A is productive if and only if $I-A$ is nonsingular.

In substochastic matrix the sum of entries by row or columns will not be greater than 1 so it is part of the condition but in addition the matrix $I-C$ should also be nonsingular. The nonsingularity is important for the invertibility of the matrix.

Consequently, I do not think that there are exceptions where the industries are unprofitable but the matrix is still productive. Rather there are exceptions where just because the entries sum to less to 1 and matrix is non-negative but it is singular and so we cannot invert it, in which case it would not satisfy conditions to be productive.


You are right when saying that mathematically in the cited theorem the condition of column sums being less than 1 is not an "if and only if" condition and thus exceptional circumstances are possible. In fact, this is just an "if" condition.

There are more general theorems that do not assume that all column sums are less than 1 and resort to another more general condition, as for example to the value of the greatest eigenvalue of the matrix.

This is one criterion that is given involving the eigenvalues of $C$. The Perron-Frobenius theorem states that if $C$ is a nonnegative matrix, then there is a real eigenvalue $V(pf)$ of $C$ such that $V(pf) > 0$ and $|V| < V(pf)$ for all other eigenvalues $V$ of $C$. We call $V(pf)$ the maximal eigenvalue of $C$. Then the following result holds.

Theorem. A nonnegative matrix $C$ is productive if and only if the maximal eigenvalue $V(pf)$ of $C$ satisfies $V(pf) < 1$.


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