Economic Systems Ohio Wesleyan University Goran Skosples 14. Input-Output Example

1 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 (X 1 ) and 10 million tons of industrial goods (X 2 ).  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.

2 The Economy  Specifically, to produce 5 million tons of X 1 (agricultural goods), 0.5 million tons of x 1 and 1 million tons of x 2 need to be used. to produce 10 million tons of X 2 (industrial goods), 0.5 million tons of x 1 and 3 million tons of x 2 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?

3 The Economy  We need input coefficients (a ij ), where i is input and j is output (a ij =units of i needed to produce one unit of j): to produce one unit of X 1 we need 0.1 units of X 1 and 0.2 units of X 2 (a 11 =x 1 /X 1 =0.5/5=0.1; a 21 =x 2 /X 1 =1/5=0.2) to produce one unit of X 2 we need 0.05 units of X 1 and 0.3 units of X 2 (a 12 =x 1 /X 2 =0.5/10=0.05; a 22 =x 2 /X 2 =3/10=0.3)  This gives us input coefficients: Outputs Agricultural goods Industrial goods Inputs Agricultural goods a 11 =a 12 = Industrial goods a 21 =a 22 =

4 Input-Output Relationships 1. X 1 ≥ a 11 X 1 + a 12 X 2 + Y 1, or X 1 ≥ 0.1X X 2 + Y 1 2. X 2 ≥ a 21 X 1 + a 22 X 2 + Y 2 =, or X 2 ≥ 0.2X X 2 + Y 2  In matrix notation, this becomes: X=AX+Y  X-AX=Y  (I-A)X=Y  X=[I-A] -1 Y  Let’s graph these 2 inequalities: 1  X 2 ≤ [(1-a 11 )/a 12 ]X 1 - Y 1 /a 12, or X 2 ≤ 18X Y 1 (1) 2  X 2 ≥ [a 21 /(1-a 22 )]X 1 + Y 2 /(1-a 22 ), or X 2 ≤ 8X 1 /7 + Y 2 /0.7 (2)

5 Feasibility X1X1 X2X2 X 2 ≤18X 1 -20Y 1 X 2 ≤8X 1 /7 + Y 2 /0.7 Y 2 /0.7 10/9Y 1

6 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; a L1 = 4 a L2 = 2 and L ≥ a L1 X 1 + a L2 X 2, which becomes 40 ≥ 4X 1 + 2X 2  X 2 ≤ X 1

7 Feasibility X1X1 X2X2 X 2 ≤18X 1 -20Y 1 X 2 ≤8X 1 /7 + Y 2 /0.7 Y 2 /0.7 10/9Y 1 X 2 ≤ X 1

8 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

9 Owustan Using Sectors Producing TotalInter-Industry UsesFinal SectorsOutputAgriculturalIndustrialOutput Agricultural (=0.1xAG)(=0.05xIND) Industrial10136 (=0.2xAG)(=0.3xIND) Labor 4020 (=4xAG)(=2xIND)

10 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 X 1, but in order to do so, we need to reduce production of X 2 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

11 Owustan Using Sectors Producing TotalInter-Industry UsesFinal SectorsOutputAgriculturalIndustrialOutput Agricultural (=0.1xAG)(=0.05xIND) Industrial (=0.2xAG)(=0.3xIND) Labor (=4xAG)(=2xIND) We need 6 units of agricultural output for final consumption