1. Ohio Wesleyan University
Goran Skosples
12. Input-Output Example

2. The Economy
Owustan is a labor-abundant socialist economy and produces two types of goods: agricultural (1) and industrial (2). Moreover, the only factor Owustan uses is labor (no capital).
Currently Owustan produces 5 million tons of agricultural goods (X1) and 10 million tons of industrial goods (X2).
However, some of agricultural and some of industrial goods not available for final consumption as they are used as intermediate inputs in production of agricultural and industrial goods.

3. The Economy Specifically,
to produce 5 million tons of X1 (agricultural goods), 0.5 million tons of x1 and 1 million tons of x2 need to be used.
to produce 10 million tons of X2 (industrial goods), 0.5 million tons of x1 and 3 million tons of x2 need to be used.
After the revolution of the proletariat, we need to set up a plan that will replace the market and allow central control of resource allocation, production and distribution. How would we go about it?

4. The Economy
We need input coefficients (aij), where i is input and j is output (aij=units of i needed to produce one unit of j):
to produce one unit of X1 we need 0.1 units of X1 and 0.2 units of X2 (a11=x1/X1=0.5/5=0.1; a21=x2/X1=1/5=0.2)
to produce one unit of X2 we need 0.05 units of X1 and 0.3 units of X2 (a12=x1/X2=0.5/10=0.05; a22=x2/X2=3/10=0.3)
This gives us input coefficients:
Outputs
Agricultural goods
Industrial goods
Inputs
a11=
a12=
a21=
a22=

5. Input-Output Relationships
X1 ≥ a11X1 + a12X2 + Y1, or
X1 ≥ 0.1X X2 + Y1
X2 ≥ a21X1 + a22X2 + Y2=, or
X2 ≥ 0.2X X2 + Y2
In matrix notation, this becomes:
X=AX+Y  X-AX=Y  (I-A)X=Y  X=[I-A]-1Y
Let’s graph these 2 inequalities:
1 X2 ≤ [(1-a11)/a12]X1 - Y1/a12, or
X2 ≤ 18X1 - 20Y (1)
2  X2 ≥ [a21/(1-a22)]X1 + Y2/(1-a22), or
X2 ≤ 8X1/7 + Y2/ (2)

6. Feasibility X2
X2 ≤18X1-20Y1
X2 ≤8X1/7 + Y2/0.7
Y2/0.7 X1
10/9Y1

7. Constraint
Recall that we need to use labor to produce either agricultural or industrial goods
Suppose we have 40 million workers (L=40)
To produce 1 ton of agricultural goods, we need 4 workers and to produce 1 ton of industrial goods, we need 2 workers. Thus;
aL1 = 4
aL2 = 2
and L ≥ aL1X1 + aL2X2, which becomes
40 ≥ 4X1 + 2X2  X2 ≤ X1

8. Feasibility X2 ≤ 20 - 2X1 X2 X2 ≤18X1-20Y1 X2 ≤8X1/7 + Y2/0.7 Y2/0.7

9. Owustan Given our economy, we have:
20 million workers in the agricultural sector
5 million tons of agricultural output
4 million tons for final consumption
20 million workers in the industrial sector
10 million tons of industrial output
6 million tons for final consumption
all the workers are employed
Let’s look at the input-output table of our economy

10. Owustan Using Sectors Producing Total Inter-Industry Uses Final
Using Sectors
Producing
Total
Inter-Industry Uses
Final
Sectors
Output
Agricultural
Industrial 5
0.5 4
(=0.1xAG)
(=0.05xIND) 10 1 3 6
(=0.2xAG)
(=0.3xIND)
Labor
 40 20
(=4xAG)
(=2xIND)

11. Problem
You need to increase production of agricultural products so that 6 million tons are available for final consumption and that all the workers are employed.
We need to increase X1, but in order to do so, we need to reduce production of X2 to free up some workers.
How to solve it?
try playing around with numbers until you figure it out
you can use principles of linear algebra
Excel may be of some assistance

12. Owustan Using Sectors Producing Total Inter-Industry Uses Final
Using Sectors
Producing
Total
Inter-Industry Uses
Final
Sectors
Output
Agricultural
Industrial
(=0.1xAG)
(=0.05xIND)
(=0.2xAG)
(=0.3xIND)
Labor
(=4xAG)
(=2xIND)
We need 6 units of agricultural output for final consumption