Find the production matrix for the following input-output...

Find the production matrix for the following input-output and demand matrices using the open model.
$$A=\left[\begin{array}{ll}{0.1} & {0.03} \\ {0.07} & {0.6}\end{array}\right], \quad D=\left[\begin{array}{r}{5} \\ {10}\end{array}\right]$$

Key Concepts

Production Matrix (or Production Vector)

In the open model, the production matrix, often computed as X = (I-A)?¹D, represents the total output required from each industry to satisfy a given external demand. It links the final demand with the necessary gross outputs by accounting for the inter-industry relationships detailed in the technical coefficient matrix.

Identity Matrix and Leontief Inverse

The identity matrix (I) is used to form the difference (I-A), which is then inverted to obtain the Leontief inverse. This inverse plays a central role in input-output models, as it quantifies how changes in final demand affect the overall production levels across the economy.

Technical Coefficient Matrix

Often denoted as matrix A, the technical coefficient matrix contains the input requirements per unit of output for each industry. Each element signifies the proportion of inputs from one sector needed to produce a unit of output in another, providing essential details for how production is interlinked.

Input-Output Analysis

This concept involves studying the interdependencies between different sectors of an economy. It examines how the output of one industry can become the input of another, forming a chain of economic activities. The analysis is typically represented in matrix form, capturing the flows of goods and services between industries.

Open Model

The open model in economic analysis considers not only interactions within the production sectors but also includes external demand as a factor driving production. In this approach, in addition to inter-industry transactions, the final demand (or exogenous demand) of products is taken into account to determine the necessary level of production.

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