If no industry (sector) draws its own output as input, then the principle diagonal element in the technological coefficient matrix will be

Question

Prepp Team · Accepted Answer

Understanding the Technological Coefficient Matrix

The question asks about the value of the principal diagonal elements in a technological coefficient matrix under a specific condition. Let's first understand what a technological coefficient matrix represents in economics, particularly in the context of an input-output model.

A technological coefficient matrix, often denoted by $A$, is a fundamental component of the Leontief input-output model. This matrix shows the direct input requirements of each industry (sector) from other industries (sectors) to produce one unit of its own output.

The elements of this matrix are typically represented as $a_{ij}$, where:

$a_{ij}$ is the amount of input from industry $i$ required by industry $j$ to produce one unit of output of industry $j$.

This matrix captures the technology or production methods used by different sectors in an economy.

Principal Diagonal Elements Explained

In the technological coefficient matrix $A$, the elements on the principal diagonal are $a_{11}, a_{22}, a_{33}, \dots, a_{nn}$, where $n$ is the number of industries. These elements have a specific meaning:

$a_{11}$ represents the amount of output from industry 1 required as input by industry 1 itself to produce one unit of output of industry 1.
$a_{22}$ represents the amount of output from industry 2 required as input by industry 2 itself to produce one unit of output of industry 2.
And so on for all $a_{ii}$.

In essence, the principal diagonal element $a_{ii}$ indicates the amount of a sector's own output that is used as input for its own production process, per unit of its output.

Condition: No Industry Uses Own Output as Input

The question states, "If no industry (sector) draws its own output as input...". This is a crucial condition that directly relates to the principal diagonal elements $a_{ii}$.

If an industry $i$ does not use any of its own output as input for its production, it means that the requirement $a_{ii}$ is zero for that industry. The condition states that this is true for every industry.

Conclusion on Principal Diagonal Elements

Given that the condition applies to all industries, if no industry $i$ uses its own output as input, then for every $i$, the value $a_{ii}$ must be equal to zero.

Therefore, all the elements on the principal diagonal of the technological coefficient matrix will be zero.

Let's visualize a simple $2 \times 2$ technological coefficient matrix where industries 1 and 2 exist:

	To Industry 1 (per unit output)	To Industry 2 (per unit output)
From Industry 1	$a_{11}$	$a_{12}$
From Industry 2	$a_{21}$	$a_{22}$

According to the condition, industry 1 does not draw its own output as input ($a_{11}=0$), and industry 2 does not draw its own output as input ($a_{22}=0$). The matrix would look like this:

	To Industry 1 (per unit output)	To Industry 2 (per unit output)
From Industry 1	$0$	$a_{12}$
From Industry 2	$a_{21}$	$0$

The principal diagonal elements ($a_{11}$ and $a_{22}$) are indeed zero.

Based on this analysis, if no industry draws its own output as input, the principal diagonal elements in the technological coefficient matrix will be Zero.

Revision Table: Key Concepts Reviewed

Concept	Explanation	Relevance to Question
Technological Coefficient Matrix ($A$)	Shows input requirements from industry $i$ to industry $j$ per unit of $j$'s output ($a_{ij}$).	The matrix being discussed.
Principal Diagonal Elements ($a_{ii}$)	Amount of industry $i$'s output used as input by industry $i$ itself per unit of $i$'s output.	The specific elements the question asks about.
Condition in Question	No industry uses its own output as input.	Directly implies $a_{ii} = 0$ for all $i$.
Result	Principal diagonal elements are zero.	The conclusion derived from the condition.

Additional Information: Input-Output Model Basics

The input-output model, developed by Wassily Leontief, is a quantitative economic technique that represents the interdependencies between different branches of a national economy or different regional economies.

The model uses a table, called the input-output table, to show how the output of one industry is used as input by other industries, as well as for final demand (consumption, investment, government spending, exports).

The technological coefficient matrix ($A$) is derived from the input-output table. It is calculated by dividing the input from industry $i$ to industry $j$ by the total output of industry $j$.

The core equation of the Leontief model is $(I - A)X = D$, where:

$I$ is the identity matrix.
$A$ is the technological coefficient matrix.
$X$ is the vector of total output for each industry.
$D$ is the vector of final demand for the output of each industry.

Solving this equation for $X$ (i.e., $X = (I - A)^{-1}D$) allows economists to determine the total output required from each industry to satisfy a given level of final demand. The matrix $(I - A)^{-1}$ is known as the Leontief inverse matrix.

Understanding the properties of the technological coefficient matrix, such as the values on its principal diagonal, is essential for working with input-output models.