In his work Input-Output Economics, Leontief provides an...

Name: Problem 19, Chapter 2: Finite Mathematics and Calculus with Applications
Uploaded: 2020-10-07T15:33:22-08:00
Duration: 3 min 18 s
Channel: Jacquelyn Trost
Description: In his work Input-Output Economics, Leontief provides an example of a simplified economy with just three sectors: agriculture, manufacturing, and households (i.e., the sector of the economy that produces labor). It has the following input-out- put matrix: Source: Input-Output Economics. TABLE CANNOT COPY Find the amount of each commodity that should be produced.

In his work Input-Output Economics, Leontief provides an example of a simplified economy with just three sectors: agriculture, manufacturing, and households (i.e., the sector of the economy that produces labor). It has the following input-out- put matrix: Source: Input-Output Economics. TABLE CANNOT COPY Find the amount of each commodity that should be produced.

In his work Input-Output Economics, Leontief provides an example of a simplified economy with just three sectors: agriculture, manufacturing, and households (i.e., the sector of the economy that produces labor). It has the following input-out- put matrix: Source: Input-Output Economics.
TABLE CANNOT COPY
Find the amount of each commodity that should be produced.

Key Concepts

Matrix Inversion and the Leontief Inverse

Once the system of linear equations is established in the form (I - A)x = d, where A is the input coefficients matrix, I is the identity matrix, x is the production vector, and d is the final demand vector, the solution involves finding the inverse of (I - A). This inverse, known as the Leontief inverse, is crucial as it provides a direct method for computing the vector of required production levels.

Production Vector and Final Demand

The production vector represents the total outputs of all sectors required to meet both inter-sectoral demands and final consumer demand. By incorporating the final demand into the input?output model, one can determine the necessary production levels needed across all sectors to maintain economic balance.

Leontief Input?Output Model

This model represents an economy as a system of interdependent sectors, where each sector's output is used partly as an input by other sectors. The model is constructed using a matrix that encapsulates the technical coefficients and inter-sectoral relationships, allowing analysts to study how perturbations in one sector can affect the whole economy.

System of Linear Equations

In the context of input?output analysis, the economic equilibrium is determined by setting up a system of simultaneous linear equations. Each equation represents the balance between the production of a sector and its distribution among other sectors as intermediate consumption and final demand. Solving these equations yields the production levels that satisfy overall demand.

Transcript

00:01 Okay, in this problem, we have been given an input -output matrix showing agriculture, manufacturing, and households, and a demand matrix d, showing how much of everything we need to have left over after our manufacturing process.

00:18 As you can see, i've already entered those matrices into a desmos application here on my computer.

00:25 You can tell from the icon this is a computer or graphing calculator problem.

00:31 So you can use whatever tool you are most comfortable with, either a calculator or a computer.

00:36 This is just one that i like to use.

00:38 I think it works very nicely.

00:40 Okay, so i have these matrices, but the real question is what do i do with them? so let's review what these matrices mean.

00:50 If i have a demand matrix d, that means that's the demand i have.

00:54 After i've manufactured whatever it is i'm doing, this is what i want to have as my final output.

01:01 Now, when i produce something, that's my matrix x.

01:05 And if manufacturing didn't cost anything, then what i produced would go to cover my demand, and d would equal x.

01:13 But that's not the case.

01:16 Manufacturing things costs something.

01:18 There is a cost involved.

01:22 And that cost is represented by ax.

01:24 A is how much manufacturing each of these things costs, how many units it costs, x is how many things are manufacturing.

01:33 So ax tells me my total cost involved in manufacturing x amount of each of these items.

01:40 So when i have my production matrix x, i subtract the cost it requires to make these things.

01:49 What i have left, that difference needs to be enough to cover my demand.

01:54 Okay...

Please give Ace some feedback