> |
MATH 244 -- Linear Algebra
A Leontief Model
February 7, 2007
Say we have an economy with three sectors: manufacturing, agriculture,
services.
Say to produce 1 unit of its output, the manufacturing sector will
use .3 units input from its own sector, .3 units input from agriculture,
and .2 units from the service sector. Say to produce 1 unit of its
output, the agricultural sector will use .4 units from manufacturing,
.1 units from agriculture, and .2 units from service. Finally, suppose
that for each 1 unit of its output, the service sector uses .3 units
from manufacturing, .2 units from agriculture, and .1 units from
service.
This information specifies the Leontief consumption matrix C:
> |
> |
> |
The Leontief model says: In order to meet a final demand vector d,
the three sectors must set their production vector x so that
or (if I - C is invertible)
> |
> |
> |
Say we want to meet a final demand of
> |
> |
> |
> |
This says: the manufacturing sector must produce approx. 376 units, the
agricultural sector must produce approx. 277 units, and the service
sector must produce approx. 179 units. We could also solve the model
by computing:
> |
> |