Application of Linear Algebra to Economics
In Closing
Danna Yu
- Input-Output model is just one example of applied linear algebra in economics
- Most popular function of linear algebra in economics
Input-Output Model
basic assumption:
- hold prices of goods and services contant
- for each sector, there is a unit consumption vector c listing inputs needed per unit of output of sector
Leontief's question:
Is there a production level such that the total amount produced equals the total demand for production?
Is there an x such that x=intermediate demand+d?
Input-Output General Matrix
Questions?
Terminology
Example
n: number of sectors in the economy
x: production vector (output of each sector for a year)
d: final demand vector
(value of goods and services demanded from
sectors by non-production part of the economy)
intermediate demand: inputs producers need for
production
Wessily Leontief
Leontief Input-Output Model
- Divided U.S. economy into 500 sectors
- Wrote linear equation describing how sector distributes output to other sectors
Total intermediate demand:
x1c1+x2c2+...+xncn=Cx,
where C is consumprtion matrix C= [c1,c2...cn ]
Leontief's Question:
Is there an x such that x=Cx+d?
Alternatively, is there an x such that (In-C)x=d?
Intermediate Demand
Example