Leontief Input-Output Matrix Calculator
Analyze economic relationships between industries and calculate the total production required to satisfy final demand. Enter a technical coefficient matrix and a final demand vector to calculate the Leontief inverse, gross output, intermediate demand and output multipliers.
Input Data
Choose the number of economic sectors, enter the technical coefficients and specify final demand for each sector.
Enter aij as the amount of input supplied by sector i that is required to produce one unit of output in sector j. Rows represent supplying sectors and columns represent producing sectors. Use non-negative technical coefficients.
Technical Coefficient Matrix (A)
Each cell represents the input from one sector required per unit of output from another sector.
Final Demand Vector (d)
Enter final demand for each sector, using consistent units such as millions of currency units.
Leontief Input-Output Model Formula
The Leontief input-output model describes how different sectors of an economy depend on one another. Industries purchase inputs from other industries to produce their own goods and services.
Total output equals intermediate demand plus final demand.
Rearranging the equation gives:
What do the variables mean?
- A: Technical coefficient matrix describing the inputs required by each sector.
- I: Identity matrix with the same dimensions as A.
- d: Final demand vector, representing demand outside the production system.
- x: Gross output vector required to satisfy final demand.
- (I − A)−1: Leontief inverse, representing direct and indirect production requirements.
Understanding the output multipliers
The sum of each column of the Leontief inverse is the output multiplier for the corresponding sector. A larger multiplier indicates that an increase in final demand for that sector generates more total production across the economy.
Value-added coefficients
In a simplified input-output model, the value-added coefficient for a producing sector is calculated as one minus the sum of its technical input coefficients. Multiplying this coefficient by gross output gives an estimate of value added for that sector.
This calculator assumes fixed technical coefficients, constant returns to scale and non-negative intermediate input requirements. It does not automatically model price changes, capacity constraints, imports, taxes, inventory changes or substitution between inputs. The value-added calculation is a simplified estimate and should be adapted when using a more detailed national accounting framework.
Frequently Asked Questions
What is a Leontief input-output matrix?
It is a mathematical representation of the production relationships between sectors in an economy. Its coefficients describe how much input each sector needs from other sectors.
What is the Leontief inverse?
The Leontief inverse is (I − A)−1. It measures the total direct and indirect production needed throughout the economy to satisfy a given final demand vector.
Why must the matrix I − A be invertible?
The inverse is required to calculate a unique gross output vector using the standard Leontief solution. If I − A is singular, this calculator cannot produce that solution.
What is a technical coefficient?
A technical coefficient describes the quantity of an input from one sector required to produce one unit of output in another sector.
Can I use this calculator for multiple industries?
Yes. You can calculate models with 2 to 8 sectors. Each sector is represented by one row and one column in the technical coefficient matrix.
What is an output multiplier?
An output multiplier measures the total production generated across all sectors by one additional unit of final demand in a particular sector. It is calculated as a column sum of the Leontief inverse.
Does this calculator include imports?
No. This is a simplified domestic input-output model. Imported inputs and other adjustments must be incorporated separately when working with an actual national input-output table.
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