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.
$$A=\left[\begin{array}{ll}{0.2} & {0.04} \\ {0.6} & {0.05}\end{array}\right], \quad D=\left[\begin{array}{r}{3} \\ {10}\end{array}\right]$$

Key Concepts

Input-Output Analysis

Input-output analysis is an economic framework that represents the interdependencies between different sectors of an economy. It uses matrices to model the flow of goods and services among industries, thus allowing analysts to understand how changes in demand for one sector’s output can affect production levels in others.

Open Model

An open model in input-output analysis refers to an approach that incorporates exogenous final (or external) demand into the system. In this model, the production levels are determined not only by the internal inter-industry transactions but also by the additional demand placed on the economy, which is used to compute the necessary output to meet both intermediary and final demands.

Production Matrix

The production matrix, often denoted by a vector or matrix of total outputs, represents the quantity of production required from each sector to satisfy a given level of final demand. In the context of the open model, it is computed by considering both the internal consumption specified by the technical coefficients and the external demand, reflecting the overall output needed by an economy.

Leontief Inverse

The Leontief inverse is a fundamental concept in input-output analysis and is derived from the identity matrix and the technical coefficient matrix (i.e., I ? A). It encapsulates the direct and indirect production requirements; by applying this inverse to the final demand vector, one obtains the total production vector necessary to satisfy that demand, taking into account the multiplier effects throughout the economic sectors.

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