Economics at Zero Marginal Cost: Hawkins-Simon Conditions

In some cases, the input-output model equation solution may give numbers expressed by negative numbers. However, in the economic sense, the output cannot be negative since a firm may produce some amount of a product or no product at all. It cannot produce a negative amount of a product, say, it cannot produce negative 2 units of bread.

If our solution gives negative output, it means that more than one unit of a product is used up in the production of every one unit of that product; it's definitely an unrealistic situation. Such a system is not viable.

The solution of the Leontief input-output model equation will yield a non-negative output if and only if it satisfies certain conditions. These conditions are known as the Hawkins-Simon condition. It ensures that Leontief input-output model equation does not give negative numbers as a solution.

For n-industries case:

Given the Leontief matrix (I - A):

$\begin{bmatrix} 1-a_{11} & -a_{12} &...... &-a_{1n} \\ -a_{21} & 1-a_{22} &...... &-a_{2n} \\ . &. &...... & .\\ . &. &...... & .\\ -a_{n1} & -a_{n2} &...... &1-a_{nn} \\ \end{bmatrix}$

The two conditions are:

1. The diagonal elements $(1-a_{11}),(1-a_{22}) ,.....,(1-a_{nn})\\$ should all be positive. In other words, elements $a_{11},a_{22} ,.....,a_{nn}\\$ should all be less than 1. Thus, the production of one unit of output of any sector should use not more than one unit of its own output, and

2. The determinant of the matrix must always be positive

These two conditions are known as the Hawkins-Simon condition.

Economic Interpretation of Hawkins-Simon conditions:

Let us consider a two-industry case. The Leontief Leontief matrix (I - A) can be written as:

$\begin{bmatrix} 1-a_{11} & -a_{12} \\ -a_{21} & 1-a_{22} \\ \end{bmatrix}\\$

1. The first condition requires that $1-a_{11}$ and $1-a_{22}\\$ must be positive or $a_{11}$ and $a_{22}$ must be less than 1. Economically, this implies that the amount of a commodity used in the production of a hundred rupee's worth of that commodity must be less than a hundred rupee.

In the case of the 1st commodity, the amount of the 1st commodity used in the production of a hundred rupee's worth of the 1st commodity must be less than a hundred rupee.

In the case of the 2nd commodity, the amount of the 2nd commodity used in the production of a hundred rupee's worth of the 2nd commodity must be less than a hundred rupee.

2. The second condition implies that the determinant must be positive, i.e., D>0 implies that $\begin{vmatrix} 1-a_{11} & -a_{12} \\ -a_{21} & 1-a_{22} \\ \end{vmatrix}>0$ .

$\\or,\ (1-a_{11})(1-a_{22})-a_{12}a_{21}>0\\ \\or,\ 1-a_{22}-a_{11}+a_{11}a_{22}-a_{12}a_{21}>0\\ \\\mathrm{Transfering\ all\ the\ terms\ involving\ a_{ij}\ to\ the\ right\ hand\ side,\ we\ have}\\ \\or,\ 1>a_{22}+a_{11}-a_{11}a_{22}+a_{12}a_{21}\\ \\or,\ a_{22}+a_{11}-a_{11}a_{22}+a_{12}a_{21}<1\\ \\or,\ a_{22}-a_{11}a_{22}+a_{11}+a_{12}a_{21}<1\\ \\or,\ (1-a_{11})a_{22}+a_{11}+a_{12}a_{21}<1\\ \\\mathrm{Since,\ (1-a_{11})a_{22}\ is\ positive.\ According\ to\ the\ first\ condition\ both\ a_{11}\ and\ a_{22}\ are\ less\ than\ one. }\\ \\a_{11}+a_{12}a_{21}<1\\$

Economically, the first term $a_{11}$ shows the direct use of the 1st commodity in the production of the 1st commodity itself. The second term shows indirect use. $a_{12}a_{21}\\$ shows that output of the first industry is used as input in the production of the second commodity which is in turn used as input in the production of the 1st commodity. Economically, the second condition implies that the direct and indirect requirement of any commodity to produce one unit of that commodity must also be less than one.

Economics at Zero Marginal Cost

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Wednesday, July 28, 2021

Hawkins-Simon Conditions

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