Leontief Production Equation Calculator
This calculator solves the Leontief input-output model, a fundamental tool in economics for analyzing the relationships between different sectors of an economy.
Enter the Technology Matrix (A), which describes the internal "recipes" of your economy, and the Final Demand (d), which represents the "shopping list" for consumers. The calculator will determine the Total Output (x) each industry must produce to satisfy all demands.
Enter Economic Data
Understanding the Leontief Production Equation
What is the Leontief Input-Output Model?
Developed by Nobel laureate Wassily Leontief, this model shows how industries in an economy are interconnected. It demonstrates that to produce goods for final consumption (e.g., cars for people), industries also need to produce goods for each other (e.g., steel for car factories). The model calculates the total production required to support a given level of final demand.
Core Formula: The Leontief Production Equation
The central equation that this calculator solves is:
x = (I - A)⁻¹ d
Where:
- x (Total Output): The result we want. A vector listing the total value of production required from each industry.
- I (Identity Matrix): A square matrix with ones on the main diagonal and zeros elsewhere. It's the matrix equivalent of the number "1".
- A (Technology Matrix): Your input. It describes the input requirements for each industry.
- (I - A) (The Leontief Matrix): This represents the net output of the economy after internal consumption is accounted for.
- (I - A)⁻¹ (The Leontief Inverse): The most important part. Its elements show how much total output is needed across all industries to produce one unit of a specific industry's product for final demand.
- d (Final Demand): Your input. A vector listing the value of goods demanded by final consumers (not by other industries).
Practical Examples of Input-Output Analysis
Click on an example to see how different economic scenarios are calculated.
Example 1: Basic 2-Industry Economy
Scenario: A simple economy with Agriculture (Industry 1) and Manufacturing (Industry 2).
1. Technology Matrix (A):0.1 0.3
0.2 0.05
2. Final Demand (d):100
200
3. Interpretation: Consumers want $100 of Agriculture and $200 of Manufacturing. To produce $1 of its own goods, Agriculture needs $0.10 from itself and $0.20 from Manufacturing, etc.
4. Result (Total Output x): Agriculture must produce ~$185 and Manufacturing must produce ~$261. The extra production is consumed internally by the industries themselves.
Example 2: 3-Industry Economy
Scenario: An economy with Energy, Manufacturing, and Services.
1. Technology Matrix (A):0.2 0.3 0.1
0.4 0.1 0.2
0.1 0.3 0.1
2. Final Demand (d):1000
5000
8000
3. Result: The calculator finds the total output for all three sectors, showing the significant ripple effects of the large final demand in Manufacturing and Services.
Example 3: Highly Interdependent Sectors
Scenario: Energy and Heavy Industry rely heavily on each other.
1. Technology Matrix (A):0.4 0.5
0.5 0.2
2. Final Demand (d):50
50
3. Result: Total output will be very high relative to the small final demand, illustrating the large amount of internal trade required to keep the system running.
Example 4: A Mostly Independent Sector
Scenario: A traditional sector (Industry 1) and a modern Tech Services sector (Industry 2) that requires very few physical inputs.
1. Technology Matrix (A):0.3 0.05
0.4 0.01
2. Final Demand (d):100
500
3. Result: The total output for Tech Services will be only slightly higher than its final demand of 500, as it consumes very little from other industries.
Example 5: Zero Final Demand for One Product
Scenario: Consumers want a product from Industry 1, but no one wants the product from Industry 2 directly. However, Industry 1 needs Industry 2's product to function.
1. Technology Matrix (A):0.2 0.4
0.3 0.1
2. Final Demand (d):200
0
3. Result: Industry 2 will still have a positive total output. It must produce goods not for consumers, but as a necessary input for Industry 1.
Example 6: Single-Industry Economy (Sanity Check)
Scenario: An economy with only one industry that consumes part of its own output (e.g., an electric company using electricity to run its plants).
1. Technology Matrix (A):0.25
2. Final Demand (d):150
3. Result: Total output will be 200. Calculation: x = 150 / (1 - 0.25) = 200. For every dollar of output, $0.25 is used internally, leaving $0.75 for final sale.
Example 7: Economy with Significant Imports
Scenario: An economy where industries rely more on imported goods than domestic ones, resulting in low inter-industry coefficients.
1. Technology Matrix (A):0.05 0.02
0.08 0.03
2. Final Demand (d):10000
25000
3. Result: Total output will be very close to final demand because the "multiplier effect" of internal trade is small.
Example 8: Error - Non-Square Matrix
Scenario: User enters a technology matrix with a different number of rows and columns.
1. Technology Matrix (A):0.1 0.2 0.3
0.4 0.1 0.1
2. Final Demand (d):100
100
3. Result: An error message: "Technology Matrix (A) must be square."
Example 9: Error - Mismatched Dimensions
Scenario: The number of industries in the matrix does not match the number of industries in the demand vector.
1. Technology Matrix (A):0.1 0.2
0.3 0.4
2. Final Demand (d):100
200
300
3. Result: An error message: "The number of rows in Matrix A must match the number of rows in Demand Vector d."
Example 10: Error - Singular Matrix (Non-viable Economy)
Scenario: A flawed economic model where an industry uses up all of its own product, leaving none for others or final sale.
1. Technology Matrix (A):1 0.5
0.2 0.3
2. Final Demand (d):100
100
3. Result: An error message: "Matrix (I - A) is singular and cannot be inverted..." This indicates a fundamental problem with the economic structure itself.
Frequently Asked Questions (FAQs)
1. What is the Technology Matrix (A)?
It's the "recipe" for the economy. Each column represents an industry. The numbers in that column show how many dollars' worth of inputs are needed from other industries (represented by the rows) to produce one dollar's worth of that industry's output.
2. What is the Final Demand (d)?
This is the "shopping list." It represents the value of goods and services that final consumers (households, government, foreign exports) want to buy. It does not include the goods that industries buy from each other to produce things.
3. What does the Total Output (x) represent?
This is the calculator's main result. It's the total production level each industry must achieve to satisfy both the final consumer demand (d) and the internal demand from other industries (as defined by matrix A).
4. Why is the Total Output (x) always higher than the Final Demand (d)?
Because just producing the "shopping list" isn't enough. Industries must also produce extra goods to supply each other. For example, to produce $100 worth of cars for consumers, the manufacturing industry needs steel, which the steel industry must produce. The Total Output includes both the final cars and the intermediate steel.
5. What does a "singular matrix" error mean?
This is a critical economic indicator. It means the system of equations has no unique solution, which usually implies the economy is not viable. A common cause is a technology coefficient of 1 or more (e.g., it takes $1 of electricity to produce $1 of electricity), creating an infinite loop of demand that can never be met.
6. How do I format the inputs correctly?
For the Technology Matrix (A): Each row of the matrix should be on a new line. The numbers within a row should be separated by one or more spaces. The matrix must be "square" (same number of rows and columns).
For the Final Demand (d): Each industry's demand value should be on a new line. The number of rows must match the number of industries in your matrix A.
7. Can I use this for an economy with more than 3 industries?
Yes. The calculator is designed to work with any number of industries. Simply add more rows and columns to your Technology Matrix (A) and more rows to your Final Demand vector (d), following the formatting rules. For a 5-industry economy, A would be a 5x5 matrix and d would have 5 rows.
8. What are the main limitations of this simple Leontief model?
The model assumes fixed "recipes" (technology coefficients don't change), no constraints on production (you can produce any amount), and a linear relationship between inputs and outputs (doubling inputs doubles outputs). Real-world economies are far more complex.
9. What if a number in my Technology Matrix is negative or greater than 1?
A negative number is economically meaningless (an industry can't be paid to take an input). A number greater than or equal to 1 for a single industry's own input means an industry uses more than $1 of an input to create $1 of its own output, which is an unsustainable and non-viable model that will cause an error.
10. How can this model be used in practice?
Governments and economists use it for economic planning, to understand the impact of a shock in one industry (e.g., a steel shortage) on the entire economy, or to predict the total economic activity needed to achieve a certain goal (e.g., building a new high-speed rail network).
