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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

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