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 - .04P = D

.96P = D

.96(200) = $192 = D

P: Total Production D: Demand

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:

B F.03

.02

.04

.05

.03 .04

.02 .05

B F

B

From

To

F

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

.03 .04

.02 .05

- =300

400

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

.03 .04

.02 .05

- =300

400

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 1st row, 1st 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)-1D

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)-1D

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)-1D

We finally get the equation we really want:

P = (I – C)-1D

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)-1D

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)-1D

I = because of the two sectors.

C = D =

1 0

0 1

.03 .04

.02 .05

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)-1D

1 0

0 1

.03 .04

.02 .05

200

700

( )-=

-1

=236.78

741.83

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)-1D

So the company must produce $236.78 worth of bolts and $741.83 worth of office furniture.

=236.78

741.83