A much-simplified version of Leontief's 42 -sector analysis...

A much-simplified version of Leontief's 42 -sector analysis of the 1947 American economy has the following input-output matrix. Source: Input-Output Economics.
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Find the amount of each commodity that should be produced.

Key Concepts

Matrix Inversion in Economic Models

Solving a Leontief input-output model typically involves matrix algebra, especially the process of inverting the matrix (I - A), where A is the matrix of technical coefficients and I is the identity matrix. The inverse, known as the Leontief inverse, directly gives the relationship between final demand and total production required.

Production-Demand Equilibrium

This key concept relates to balancing the supply produced with the demand faced by each sector, including both intermediate inputs and final consumption. The equilibrium ensures that the calculated production levels are sufficient to meet both the direct demands of consumers and the ancillary demands of other industries.

Input-Output Analysis

This concept involves studying the economic system by using a matrix to represent the interdependencies between different sectors of an economy. Each cell in the matrix shows how much output from one sector is required by another, allowing analysts to determine the overall production levels needed to fulfill both final and intermediate demands.

Leontief Model

Named after Wassily Leontief, this model is a formal representation of the inter-industry relationships in an economy. It uses the input-output matrix to link the production of various sectors through simultaneous equations, which can be solved to find the total output needed in each sector when final demand is specified.

Transcript

00:01 For this problem, we need to find the production matrix for a given input -output matrix looking at agriculture, manufacturing, and households or labor.

00:13 Now, as you can see from the icon next to this problem, it is a technology problem.

00:17 So you can use either a graphic calculator or some kind of application on your computer.

00:22 I'm using the desmos application on my computer.

00:25 I find this one to work very nicely.

00:27 You can use whichever one you are most comfortable with.

00:30 Now, as you can see from here, i have already entered our input -output matrix a and our demand matrix d along with the identity matrix b here.

00:40 But what do i actually do with these matrices now that they've been given to us? now what? well, let's review how these work.

00:50 D is my demand matrix.

00:52 That says after my whole manufacturing process is over, this is what i need to have as a result.

00:57 Now, if manufacturing didn't cost anything, d would equal x.

01:04 What i produce would go toward demand.

01:07 But that's not exactly the way it works.

01:08 There is a cost involved.

01:10 Some of the things that i produce have to be put back into the system as input into the system.

01:17 It costs something in order to produce something.

01:23 So i have to be able to account for that cost.

01:26 So my production doesn't just equal demand...

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