Linear Algebra and Economics

Steve Levandusky

Input Coefficient Matrix

We start by defining an input coefficient matrix

 

 

The elements, aij, indicate the amount needed of the good i to produce the good j. The sum of the elements in the jth column represents the cost of producing one unit of good j, and is assumed to be less than 1 because when we combine all the inputs there is only one product. Since each good can be an input for the other goods, including itself, A is considered to be a square matrix.

Now, let aijxj be the portion demanded of good i for the production of good j. Also there exists a demand for the final good i by the consumer denoted di. For the market to be in equilibrium, the total supply xi must be equal to the total demand for good i,

 

 

for i = 1, 2, , n. And expressed as a matrix:

 

 

 

Or x = Ax + d, which can be rewritten as

(I - A)x = d. The matrix (I-A) is called the technology matrix and if non-singular there exists a solution for the total output x needed for the final demand d, x = (I - A)-1d. [1]

 

Example

 

 

 

To contact:

E-mail: steve.levandusky@bucknell.edu