The Leontief Input-Output Method, Part 2
Example 1: Sunny Summer Beverages produces and bottles a variety of fruit juices.
For every dollar worth of juice it produces, it keeps $.04 worth of juice in house to help keep the workers hydrated and happy.
If the company produces $200 worth of juice, how much will be available for sale?

The Leontief Input-Output Method, Part 2
Example 1: Sunny Summer Beverages
Recall that we can calculate the Demand if we know the Production and the Consumption (.04 in this case):
P: Total Production P - .04P = D D: Demand
.96P = D
.96(200) = $192 = D

The Leontief Input-Output Method, Part 2
We can modify this equation slightly to determine the Demand for a 2-sector economy.
In this case, we’ll use the consumption matrix,
C:
P – CP = D

The Leontief Input-Output Method, Part 2
Example 2: ABC Furniture manufactures a variety of office furniture. It also manufactures bolts, some of which are used in its furniture.
Every dollar worth of bolts produced requires an input of $.03 worth of bolts and $.02 worth of office furniture. Each dollar worth of office furniture requires an input of $.04 worth of bolts and $.05 worth of office furniture.

The Leontief Input-Output Method, Part 2
Example 2: Recall our weighted digraph, as well as our consumption matrix:
.04
.05
.03
B .02
F
From
To
B F


.03 .04
.02 .05



The Leontief Input-Output Method, Part 2
Example 2: Suppose the company produces
$300 of bolts and $400 of office furniture.
How much of each will be available for sale?

The Leontief Input-Output Method, Part 2
Example 2: Suppose the company produces
$300 of bolts and $400 of office furniture.
How much of each will be available for sale?
P – CP = D

300

 400 



.03 .04
.02 .05




300

 400 
=

275



374



The Leontief Input-Output Method, Part 2
Example 2: Suppose the company produces
$300 of bolts and $400 of office furniture.
How much of each will be available for sale?
P – CP = D

300

 400 



.03 .04
.02 .05




300

 400 
=

275



374


So $275 of bolts and $374 of office furniture are available to sell.

The Leontief Input-Output Method, Part 2
Things get a little more interesting if we know the demand and need to determine the production.
Start with our previous equation:
P – CP = D

The Leontief Input-Output Method, Part 2
P – CP = D
We would like to factor out P on the left hand side of the equation, but it’s not quite as easy with a matrix as it is with a variable.
First, we have to multiply P by the Identity matrix, I.

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D
What is an Identity matrix? It assigns a coefficient of 1 to each variable. Then if you multiply I by any matrix, it returns the original matrix:
IP = P

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D
An Identity matrix is always a square matrix
(2x2, 3x3, 4x4, etc.). The diagonal starting in the 1 st row, 1 st column contains 1s, with all other entries being 0s.

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D
Because most of our examples will involve two sectors, I will normally be 2x2:

1 0

 0 1 

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D
Now we can factor out P:
(I – C)P = D

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D
(I – C)P = D
If (I – C) represented variables, we could simply divide each side of the equation by
(I – C) and be done. Because it is a matrix, however, we must multiply by the inverse matrix, (I – C) -1

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D
(I – C)P = D
(I – C) -1 (I – C)P = (I – C) -1 D

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D
(I – C)P = D
(I – C) -1 (I – C)P = (I – C) -1 D
Fortunately, (I – C) -1 and (I – C) are inverses, so when we multiply them, they essentially eliminate each other.

The Leontief Input-Output Method, Part 2
P – CP = D
IP – CP = D
(I – C)P = D
(I – C) -1 (I – C)P = (I – C) -1 D
We finally get the equation we really want:
P = (I – C) -1 D

The Leontief Input-Output Method, Part 2
Example 3: If ABC Furniture receives an order for $200 of bolts and $700 of office furniture, how much of each must it produce to fill the order?

The Leontief Input-Output Method, Part 2
Example 3: If ABC Furniture receives an order for $200 of bolts and $700 of office furniture, how much of each must it produce to fill the order?
We are trying to find the production, P, in a two-sector economy, so we will use
P = (I – C) -1 D

The Leontief Input-Output Method, Part 2
Example 3: If ABC Furniture receives an order for $200 of bolts and $700 of office furniture, how much of each must it produce to fill the order?
P = (I – C) -1 D

1 0

I = because of the two sectors.
0 1 


.03 .04



200

 700 

The Leontief Input-Output Method, Part 2
Example 3: If ABC Furniture receives an order for $200 of bolts and $700 of office furniture, how much of each must it produce to fill the order?
P = (I – C) -1 D
=
( 
1 0

 0 1 
-


.03 .04
.02 .05


)
-1
=


236.78
741.83



200

 700 

The Leontief Input-Output Method, Part 2
Example 3: If ABC Furniture receives an order for $200 of bolts and $700 of office furniture, how much of each must it produce to fill the order?
P = (I – C) -1 D
=


236.78
741.83


So the company must produce $236.78 worth of bolts and $741.83 worth of office furniture.