# Leontief input output model with column sum greater than 1

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?

• In the future, please consider using MathJax to format the mathematical expressions. Oct 24 '20 at 4:46

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.