1. Input Output Analysis Dr. Shaveta Kohli Assistant Professor
Department of Economics
Central University of Jammu
Samba

2. Introduction
Input Output analysis is an important application of static general equilibrium approach to the empirical analysis of production. Rationale behind this analysis is that output of every industry is either used as an intermediary input for other industries or as an direct consumption. On the basis of this we will find how much we will have to produce or whether it is viable or not ( if D< S, Viable & if D> S then not Viable).

3. Open Model
If besides the n industries the model contains an open sector which exogenously determines a final demand (non-input demand for the product of each industry & which supplies a primary input (say labour service) not produced by n industries themselves, the model is an open model.

4. Viable Requirements
If we get a situation to find a combination which can satisfy C1 & C2 then situation is viable.

5. Feasible Requirement
Viability requirement find combination of X1 & X2 to satisfy C1 & C2. Once viable requirement is satisfied then we check whether it is possible for economy to produce that product

6. Types of Goods
There are two types of goods i.e. Producing & Non-Producing goods. For producing goods there is no problem if system is viable but problem arises for non-producing goods

7. Assumptions
Each industry produces only one homogeneous commodity.
Each industry uses a fixed input ratio for production of its output.
Production in every industry is subject to constant return to scale.
Only one method is viable to produce the commodity
No joint production takes place means single process can produce either X1 and X2

8. Transaction Matrix X1 X2 Final Demand Total Output X11 X12 C1 X21 X22
Here X1 & X2 means total output produced
X11 means first commodity required to produce X1 commodity
Row stands for distribution function
Column stands for production function
Final Demand i.e. non-input demand is C1 & C2

9. Solution of Input-Output Model
Rows can be written in equation form as:
X11+X12+C1=X1
X21+X22+C2=X2
X01+X02=X0
X11, X21,X01 are inputs required to produce first commodity. If we divide X11/X1, X21/X1, X01/X1 then we get a11, a21,a01 i.e. per unit requirement of X11 i.e. then by having this we will get another matrix i.e. input coefficient matrix.

10. a11 amount of first commodity required to produce 1 unit of first commodity
a21 amount of second commodity required to produce 1 unit of first commodity
a01 amount of non producing good (NPG) required to produce 1 unit of first commodity
Similarly for the second commodity i.e. a21 , a22 , a01

11. Input Co-efficient Matrix
Inter Industry Requirement
a11
a12 c1
a21
a22 c2
a01
a02
Input Coefficient of NPG

12. a11 , a12 , a21 , a22 are input coefficients, we assume it to be remain fixed.
Equation form of input coefficients is as follows:
a11 X11 + a12 X C1=X1
a21 X21 + a22 X C2 =X2
a01 X01 + a02 X02 =X ( )
Inter Industry requirement

13. If all the input coefficients and final demand i. e
If all the input coefficients and final demand i.e. C1 & C2 are known then we have 3 equations of function X1, X2, X0 . Before solving 3 equations for these unknown we have to check viable condition for this .

14. For the viability condition we proceed like this C1 = X1 – a11 X1 – a12 X2 (i) Subtracting unit utilised from total output so remaining part will satisfy final demand. C2 = X2 – a21 X1 – a22 X2 (ii) From (i) and (ii) (1- a11) X1 - a12 X2 = C1 - a21 X1 + (1-a22)X2= C2

15. In Matrix form 1-a11 -a12 X1 C1 -a21 (1-a22) X2 C2 If we subtract input coefficient matrix from I ( Identity Matrix), we get, 1 0 a11 a a21 a22
I-A

16. a01 X01 + a02 X 02 = X0 (I- A) X = C (I-A) ( I-A) X (I-A) C C From ( )
Demand for Labour
Supply of Labour

17. If Demand for Labour < Supply of Labour Then it is feasible
If Demand for Labour < Supply of Labour Then it is feasible. It will be possible to find the utilisation of X1 & X2 only if system is viable. X = (I-A) C X1 B11 B12 C1 X2 B21 B22 C2
(I-A)

18. The values of X1 and X2 can be written as: X1 = B11C1 + B12 C2
Value of X0 is given , thus on the basis of that value, it may be possible to produce X1 & X2. Supply of Labour is known & we have to find demand for labour. If supply is less than demand then it is not feasible to produce X1 & X2 and also not possible to satisfy C1 & C2 and Vice-Versa.

19. Hawkins Simon Viability Condition
System is viable if and only if it is possible to find optimum levels which can satisfy the final demand requirement. Thus, for viability condition we have, I-A > 0

20. References
Chiang, A. C. (1992). Elements of dynamic optimization. McGraw-Hill, New York.
Chiang and Wainwright(2005), Fundamental Methods of Mathematical Economics, Mc. Graw Hill, New York
Miller and Blair (2009), Input Output Analysis: Foundations and Extensions, Cambridge University Press, UK.
Raa, Thijs Ten ( 2005), The Economics of Input Output Analysis, Cambridge University Press, UK.