input output matrix

Suppose that the economy of a certain region depends on three industries: service, electricity and oil production. Monitoring the operations of these three industries over a period of one year, we were able to come up with the following observations:

1. To produce 1 unit worth of service, the service industry must consume 0.3 units of its own production, 0.3 units of electricity and 0.3 units of oil to run its operations.

2. To produce 1 unit of electricity, the power-generating plant must buy 0.4 units of service, 0.1 units of its own production, and 0.5 units of oil.

3. Finally, the oil production company requires 0.3 units of service, 0.6 units of electricity and 0.2 units of its own production to produce 1 unit of oil.

Find the production level of each of these industries in order to satisfy the external and the internal demands assuming that the above model is closed, that is, no goods leave or enter the system.

Solution Consider the following variables:

1. p₁= production level for the service industry

2. p₂= production level for the power-generating plant (electricity)

3. p₃= production level for the oil production company

Since the model is closed, the total consumption of each industry must equal its total production. This gives the following linear system:

The input-output matrix is

and the above system can be written as (A-I)P=0. Note that this homogeneous system has infinitely many solutions (and consequently a nontrivial solution) since each column in the coefficient matrix sums to 1. The augmented matrix of this homogeneous system is

which can be reduced to

.

To solve the system, we let p₃=t (a parameter), then the general solution is

and as we mentioned above, the values of the variables in this system must be nonnegative in order for the model to make sense; in other words, t≥0. Taking t=100 for example would give the solution

Laura Geagea