Leontief Input Output Model

Simplified examples using the same idea:

I A group is baking chocolate chip cookies and fudge. They eat .12 dozen cookies for each dozen cookies baked. The fudge makers eat .06 lbs of fudge for each lb of fudge made. If they must have 7 dozen cookies and 4 lb of fudge, how much of each must they bake?

One way to solve this is to work it out for the cookies and the fudge separately.

If they bake x dozen cookies, they eat 0.12x dozen and must have 7 left. We have

So x = 7 + 0.12x Then x - 0.12x = 7 or (1-0.12)x = 7 Then x = 7/(1-0.12)=7.95 rounded.

Similarly if they make y pounds of fudge, they must produce enough so that

y = 4 + 0.06y Then (1-0.06)y = 4 so y = 4.3 rounded.

We can do both at once in matrix form.

Then we have a matrix equation .

II As above but suppose the chocolate chip cookie bakers eat .05 doz. cookies and also eat .03 lb of fudge. the cookies and 3% of the fudge per dozen cookies baked. The fudge makers eat .07 dozen cookies and .04 lb of fudge per lb of fudge made. Then the equation looks like:

Notice the setup of the matrix .

Column 1 represents what portion of cookies and what portion of fudge is used up by the cookie bakers. Column 2 represents what portion of cookies and fudge are used by the fudge makers.

shows the amount of cookies used up (top entry) and the amount of fudge used up in the production process.

To set up the input-output matrix A : Enter the portions of each product used up by production of a product in the columns.

Column 1 of example II is the amounts of cookies and fudge used up in the production of a dozen cookies.

Column 2 is the amounts of cookies and fudge used up in the production of a pound of fudge.

The solution of how many units of each product should be produced is

Calculator instructions for computing

Enter the matrix D. Then type:(Matrix > Math 5 (column length of A)-A) [xinverse] D

Question: How much of each product is used in the production process?

Answer: We produced X and have D left so X-D

Example from Finite Math With Calculus by Lial, Greenwell and Ritchey

A simple economy is based on 3 sectors; agriculture, manufacturing and households (which produce labor) .

Production of a unit of agricultural products uses 0.25 unit of ag. products, 0.14 unit of manufacturing products and 0.80 unit of labor.

Production of a unit of manufacturing uses 0.4 unit of ag., 0.14 unit of manufacturing and 3.6 units of labor.

Production of a unit of labor uses 0.133 unit of ag., 0.1 unit of manufacturing and 0.133 unit of labor.

How much of each sector must be produced to meet a demand for 35 units of ag., 38 units of manufacturing and 40 units of labor.

Set up A:

Set up D:

Enter A and D in the calculator.

Find

How much of each sector is used in the production process:

They should produce 1018.307 units of ag., 627.594 units of manufacturing, and 3591.677 units of labor.

This will use up 983.307 units of ag, 589.594 units of manufacturing and 3551.677 units of labor.