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Note: as an inter-disciplinary question, it has its twin on Mathematics Exchange: https://math.stackexchange.com/questions/4193896/why-the-definition-of-productive-economy-in-leontief-open-model-is-such . There you can find some potentially useful insights.

The Leontief Open (Production) Model is a simplified economic model for an economy in which consumption equals production, or input equals output.

In this model, the entries in the consumption matrix represent non-negative monetary values (in some arbitrary monetary unit) of various types of outputs (such as commodities or services). Every $(i,j)$th entry in the consumption matrix is the monetary value of the output of $i$th industry needed by the $j$th industry to produce one monetary unit of its own output. It means that arbitrary consumption matrix, commonly denoted $C$, can be represented in the form of an input-output table: enter image description here

For example, in order to obtain a unit of output from industry 3 it takes $c_{13}$ units of output from industry 1, as well as $c_{23}$ of output from industry 2, and $c_{33}$ of output from industry 3 (i.e. itself).

The open sector, in this context, refers to the external consumer whose demands are required to be satisfied and its demands are represented in the model by special kind of column vector where each $i$th entry corresponds to the non-negative monetary value of the output required by the external consumer from the $i$th industry. Such column vectors are called demand vectors and such individual vectors are often denoted $D$ (capital "d") or $\textbf{d}$ (boldface lower case "d").

Total production associated with a particular consumption matrix $C$ and some demand vector $D$ in this model is represented by a column vector where every $i$th column is a non-negative monetary value of the output of $i$th industry required to be produced to satisfy the needs of the external consumer. Such column vectors are called production vectors and are commonly denoted $X$ or $\textbf{x}$.

These triplets of associated matrices (production vector $X$, consumption matrix $C$, and demand vector $D$), based on the assumption that production equals to the sum of internal and external demand/consumption, are related by the matrix equation \begin{equation} X = CX + D \end{equation} where $CX$ is the internal consumption.

In this model, an economy is productive by definition if for its consumption matrix $C$ exists Leontief inverse $(I-C)^{-1}$ and $(I-C)^{-1} \geq 0$.

Here comes the weird part.

A consumption matrix C is said to be productive if $(I − C)^{−1}$ exists and $(I − C)^{−1}$ ≥ 0

The matrix $(I-C)^{-1}$ even has its own name, Leontief inverse, yet it is still unclear to me why it is even needed to exist.

According to Derrick Chung,

An economy is productive if it can meet any external demand. In other words, for any D, there is an X such that X − CX = D with X ≥ 0 (i.e. X contains only non-negative entries)

and right below there is an analogous definition with Leontief inverse. Why can't an economy be considered productive if $X(I − C) = D$ (without requiring matrix $(I-C)$ to be invertible)? Or, equivalently, why should the relation between $X$ and $D$ be bijective? Or does it have something to do with the eigen-values?

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  • $\begingroup$ I am not sure if I understand the question, thats just terminology, to me it sounds like asking why rocks are called rocks and not something else. In Leontief model economy is only productive if all sectors are profitable if the matrix is not invertible, they are not. Its just terminology used in this context, in wider macro productivity is actually even defined differently (i.e. outputs/inputs). The Leontief model has its own taxonomy, I would not read too much into it. $\endgroup$
    – 1muflon1
    Jul 11 at 22:33
  • $\begingroup$ @1muflon1 That's precisely the answer that I wanted! I still have to figure out why the singularity of $(I-C)$ implies that there is at least one unprofitable sector, but that is much more intuitive than existence of positive Leontief inverse. $\endgroup$ Jul 11 at 23:06
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This is a specific terminology employed in the literature on Leontief model. Productive here means that all sectors must be profitable (do not confuse it with notions of productivity used elsewhere in the economics, where productivity refers generally to outputs over inputs).

Profitability requires that:

$$X-CX>0$$

Actually, in order to arrive at sensible solution the Leontief matrix must be invertible with non-negative inverse (see Zalai 1983), since you are trying to solve for $X$ which should be positive (negative production does not make sense, how do you produce negative number of PCs?):

$$X(I−C)=D \implies X = (I-C)^{-1}D$$

As far as I understand unless all sectors are profitable its not possible to get non-negative inverse of $I-C$.

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