Leontief Input-Output Model and Functions of One Variable: Study Notes for College Algebra

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Leontief Input-Output Model

Introduction to the Leontief Model

The Leontief Input-Output Model is a mathematical framework used to analyze the interdependencies between different sectors of an economy. Each sector's output serves as input for other sectors, and the model helps determine the production levels required to satisfy both internal and external demands.

Internal demand: Inputs required by all sectors, including the sector itself.
External (final) demand: Demand from outside the system, such as exports or consumer needs.
All quantities are measured in monetary units.

Mathematical Formulation

Let n be the number of sectors. For sector i:

= total production required from sector i
= output of sector i consumed by sector j
= final demand for sector i

The basic equation:

Define , the proportion of sector i's output used by sector j to produce one unit of output in sector j.

The system of equations:

Matrix Representation

The system can be written in matrix form:

Where:

= Leontief matrix (technical coefficients matrix)
= production vector
= final demand vector

Solving the Leontief Model

To find the production vector :

If the final demand changes by , the change in production is:

Economic Interpretation

: Proportion of sector i's output required by sector j to produce one unit.
for each sector j (total input required is less than one unit).
: Amount sector i supplies to sector j.
External supply:

Worked Example: Three-Sector Economy

Given input-output data (in million dollars):

Output	Input 1	Input 2	Input 3	Final Demand
1	20	60	10	50
2	50	10	80	10
3	40	30	20	40

Production required: , ,
Leontief matrix (rounded):
- ... (other entries similarly calculated)

Solving for Production Levels

Given and , use:

Methods include matrix inversion or Cramer's rule for linear systems.

Input-Output Matrix Construction

For each sector:

Output	Input 1	Input 2	Input 3	Final Demand
1
2
3

Example values (rounded):

Output	Input 1	Input 2	Input 3	Final Demand
1	4.97	6.2	3.67	10
2	9.94	2.06	3.67	5
3	2.49	6.2	3.67	6

Effects of Changes in Final Demand

If final demand changes, use:

Example: If , then is calculated using the inverse matrix.

Special Cases and Parameterization

Technical coefficients can be parameterized (e.g., in ).
Total input required by a sector:

Two-Sector Example

For two sectors (agriculture and milling):

Technical coefficients: , , ,
Final demand: 300 units (agriculture), 500 units (milling)
Set up and solve

Functions of One Variable

Definition of a Function

A function is a rule that assigns to each element in the domain exactly one element in the codomain , denoted . The range of is .

Domain: Set of all possible input values ().
Range: Set of all possible output values ().
Graph: Set of ordered pairs .

Types of Fundamental Functions

Power Functions:
Exponential Functions:
Logarithmic Functions:
Trigonometric Functions: , , , , ,
Inverse Trigonometric Functions: , , ,

Domains and ranges for inverse trigonometric functions:

(, )
(, )
(, )
(, )

Limits and Continuity

The limit of a function as approaches is if can be made arbitrarily close to by taking sufficiently close to (but not equal to ):

Alternative notation: as
Left-hand limit:
Right-hand limit:

Theorem: if and only if

Properties of Limits

If and exist, then:
(for any )
(if )

Continuity

A function is continuous at if:

is defined
exists

If any of these conditions fail, is discontinuous at .

Example: Continuity at a Point

Investigate the continuity of at .

Check if is defined.
Compute left and right limits as .
Compare and .

Example: does not exist, as left and right limits differ.

Additional info: The Leontief model is a classic application of systems of linear equations and matrices, which are core topics in College Algebra. Functions, limits, and continuity are foundational concepts for calculus and advanced algebra.