Leontief Production Equation Calculator
Every sector of an economy buys from the others, so meeting final demand takes more production than the demand itself: steel needs coal, coal needs steel, and the chain only settles when you solve the Leontief production equation, x = (I − A)⁻¹ × d. Pick 2 to 5 sectors below, enter the technology matrix A and final demand d, and this calculator returns the Leontief inverse, total output per sector, and the output multipliers.
The tool checks only that inputs are non-negative and (I − A) invertible. It does not check productivity, so an over-consuming matrix yields negative outputs, not a warning. See the limits section.
Quick Answer
With the pre-filled two-sector economy (0.1 on A's diagonal, 0.05 off it, final demand 100 per sector), the Leontief inverse is [[1.1146, 0.0619], [0.0619, 1.1146]] so each sector must produce 117.65 units: 235.29 units total to deliver 200 of final demand. Multipliers are 1.176 per sector: one extra unit of demand sets 1.176 units of production in motion.
Key takeaways
- The answer is always bigger than the demand: total output x covers final demand plus every intermediate round behind it.
- Read A by columns: column j is sector j's recipe, the inputs it consumes per unit of its own output. Swapping rows and columns is the classic way to feed this model garbage.
- The tool inverts at full precision. Rounding the inverse to two decimals before multiplying, as an earlier version of this page did in its worked example, moves the answer visibly.
- Nothing here validates your economy: coefficients for sectors consuming more than they produce return negative outputs, no warning.
- Quick Answer
- How to use the Leontief production equation calculator
- The equation: x = (I − A)⁻¹ × d explained
- Worked example A: the pre-filled two-sector economy
- Worked example B: steel and coal, and the rounding trap
- Reading the production multipliers
- Where real coefficients come from
- The honest limits: what this tool does not check
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the Leontief production equation calculator
Set the number of sectors first, fill in A and d, then click Calculate. Results stay hidden until you click.
- Set the matrix size (2 to 5): the grid rebuilds the moment the size changes, resetting every cell to the defaults, so choose the size before typing anything.
- Enter the technology matrix A: aij is how much of sector i's product one unit of sector j's output consumes. Real coefficients come from an input-output table: flows into sector j divided by its total output.
- Enter final demand d: one figure per sector, same order as the matrix.
- Read all five outputs: the inverse, the output vector x, total production, the multipliers, and the equation summary. The inversion is a Gauss-Jordan elimination; the inverse matrix calculator shows that operation worked entry by entry for a 2×2 case.
- Probe the model: change one demand figure and rerun; the change in x is the inverse column at work.
The equation: x = (I − A)⁻¹ × d explained
Total output x has to cover two claims: the intermediate demand Ax that sectors consume while producing, and the final demand d. Setting supply equal to both gives x = Ax + d, and solving for x gives the Leontief production equation.
(I − A) x = d (gather the x terms)
x = (I − A)⁻¹ × d (multiply by the Leontief inverse)
| Symbol | Meaning |
|---|---|
| x | Total output vector: what each sector must produce for the economy to deliver d |
| A | Technology matrix: aij = units of sector i used per unit of sector j produced |
| I | Identity matrix, ones on the diagonal, zeros elsewhere, same size as A |
| (I − A)⁻¹ | The Leontief inverse: entry ij is sector i's total output, direct and indirect, per unit of final demand for sector j |
| d | Final demand vector: what households, government, investors and exporters take from each sector |
The model is Wassily Leontief's, built in the 1930s. The Nobel Prize biography of Wassily Leontief records his 1973 award "for the development of the input-output method and for its application to important economic problems": the analysis finds how much production must rise everywhere to meet a given rise in final demand.
Worked example A: the pre-filled two-sector economy
The calculator loads with A = [[0.1, 0.05], [0.05, 0.1]] and d = [100, 100]. Here is the run, step by step, every figure taken from the tool.
| Output | Value from the tool |
|---|---|
| Leontief inverse (I − A)⁻¹ | [[1.1146, 0.0619], [0.0619, 1.1146]] |
| Total output x | [117.65, 117.65] |
| Total production required | 235.29 units |
| Production multipliers | [1.18, 1.18]; 1.176 each in the equation summary |
Read the inverse before the answer: delivering one unit of sector 1 output to final users takes 1.1146 units of sector 1 production in total and 0.0619 units of sector 2. Multiply the inverse by d = [100, 100] and each sector must produce 117.65. The extra 35.29 units above the 200 demanded are the intermediate rounds.
Worked example B: steel and coal, and the rounding trap
Set A = [[0.2, 0.1], [0.4, 0.2]] and d = [100, 50]: steel uses 0.2 of its own output and 0.4 of coal's per unit; coal uses 0.1 of steel and 0.2 of its own:
| Output | Value from the tool |
|---|---|
| Leontief inverse (I − A)⁻¹ | [[1.3333, 0.1667], [0.6667, 1.3333]] |
| Total output x | [141.67, 133.33]: steel 141.67, coal 133.33 |
| Total production required | 275.00 units |
| Production multipliers | [2.00, 1.50]; 2.000 and 1.500 in the equation summary |
An earlier version of this page worked this same example by hand and printed 141.5 and 133.5. That arithmetic rounded the inverse to [[1.33, 0.17], [0.67, 1.33]] before multiplying, and the rounding alone moved steel by 0.17 units. The tool inverts at full precision and rounds only for display, so its figures (141.67 and 133.33) are the ones to quote. Checking a Leontief result by hand, carry at least four decimals through the inverse.
Reading the production multipliers
A multiplier is a column sum of the Leontief inverse: the total production across every sector that one extra unit of final demand in that column's sector sets in motion.
That reading matters because the multiplier is an economy-wide figure, not a sector figure. In example B, the steel multiplier of 2.000 does not mean steel produces 2 units; one more unit of steel demanded by final users forces 2.000 units of production somewhere in the economy (1.3333 of steel and 0.6667 of coal, the first column of the inverse). The same column-sum idea applied to jobs instead of output is an employment multiplier.
Where real coefficients come from
Real A matrices come from published input-output tables: an agency surveys who sells what to whom, then divides each column of flows by that sector's total output.
In the United States that agency is the Bureau of Economic Analysis. The BEA input-output accounts data are updated each year for 71 industry categories, with a detailed benchmark roughly every five years running to about 402 industries; the historical benchmark input-output tables reach back to 1947. The BEA's total requirements tables are, in effect, published Leontief inverses of the whole US economy, inside the same national accounts system as the GDP figures tracked by the GDP growth rate calculator. This page's five-sector cap is a teaching scale.
The honest limits: what this tool does not check
The tool solves the algebra you give it, and only the algebra. Four limits worth knowing before you quote its output:
- No productivity check: a workable economy must satisfy the Hawkins-Simon condition (roughly: sectors cannot consume more, directly and indirectly, than they produce). This tool never tests it. Feed it A = [[0.9, 0.9], [0.9, 0.9]] with d = [100, 100] and it returns x = [−125.00, −125.00], total −250.00 units. Negative output is the model saying this economy cannot work; the tool will not say so. Quick screen: every column of A should sum to less than 1.
- Singular matrices stop the run: if (I − A) has no inverse, for instance when A is the identity matrix, the tool reports "Error in calculation: Matrix is singular and cannot be inverted" and shows nothing else.
- Fixed coefficients, no prices: the model assumes sector j always needs the same inputs per unit, whatever the scale or prices: a real limitation for long-horizon forecasting.
- Housekeeping rules: blank cells and demand entries count as 0, negatives are rejected with a message, and resizing wipes your entries back to the defaults.
What competing calculators miss
The rival field for this keyword is thin, and the one working tool gets its own example wrong:
- CalculatorsHub's Leontief model calculator: a real tool, but its worked example treats the matrices as single numbers: it sets "Identity Matrix (I) = 2" and "Input-Output Coefficient Matrix (A) = 3" and computes Y = (2 − 3)⁻¹ × 5 = −5, presenting a negative scalar as the output of a matrix model. The example was plainly never run through the model it teaches.
- TrueGeometry's Leontief model page: states the coefficient convention correctly, but the tool sits behind an API-style explorer that is hard to drive, with no clean worked answer to check against.
- Fiveable's mathematical economics notes: solid theory (the X = AX + Y setup, the inverse, ΔX for demand changes) with no calculator at all.
- Bohrium's Leontief inverse explainer and NumberAnalytics' linear algebra treatment: a good narrative example in one, eigenvalue context in the other, but both are essays: nothing to compute with, and their figures cannot be re-run or varied.
- The gap this page fills: a working 2 to 5 sector solver showing the full inverse, total output and correctly labelled multipliers, worked examples at full precision, and the failure modes stated on the page.
Frequently asked questions
It calculates the total output x each sector must produce so that, after sectors have supplied each other, the final demand d is met. The equation is x = (I - A) inverse times d, where A is the technology matrix of input coefficients. Because sectors consume each other's output, x is always at least as large as d.
Entry aij is the amount of sector i's output needed to produce one unit of sector j's output. Read down a column to get a sector's recipe: column j lists every input sector j consumes per unit it makes. Rows and columns must use the same sector order; swapping the convention transposes the whole model and is the most common input error.
The Leontief inverse is the matrix (I - A) raised to the power -1. Its entry ij gives the total output of sector i required, directly and through every indirect round of intermediate demand, to deliver one unit of sector j's output to final demand. The calculator displays the full inverse to four decimal places.
Each multiplier is a column sum of the Leontief inverse: the total production across all sectors that one extra unit of final demand in that sector sets in motion. In the pre-filled economy both multipliers are 1.176. A multiplier is an economy total, not the output of the named sector alone.
Between 2 and 5 sectors. Set the size before entering data, because changing the size rebuilds the grid with the default coefficients (0.1 on the diagonal, 0.05 elsewhere) and a demand of 100 per sector, discarding your entries. Real statistical tables run far larger: the BEA's annual US accounts use 71 industries.
That message means (I - A) has no inverse, so the equation has no solution of this form. The clearest case is A equal to the identity matrix, which makes I - A all zeros. Check the coefficients for duplicated rows or a sector consuming a full unit of output per unit produced.
Negative outputs mean the economy you described is not productive: somewhere in the chain, sectors consume more than a unit of output per unit produced, so no production level can satisfy final demand. The calculator does not test the Hawkins-Simon productivity condition, so it reports the algebra instead of warning you. Setting every coefficient to 0.9 returns outputs of -125.00 per sector. Quick screen: every column of A should sum to less than 1.
From a published input-output table. Take the table's flows into sector j and divide each one by sector j's total output; the results are the coefficients for column j. In the United States the Bureau of Economic Analysis publishes the tables (71 industries annually, about 402 in the five-year benchmark).
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References
- Nobel Prize Outreach. Wassily Leontief: Facts and the 1973 Prize in Economic Sciences. Accessed October 22, 2026.
- U.S. Bureau of Economic Analysis. Input-Output Accounts Data. Accessed October 22, 2026.
- U.S. Bureau of Economic Analysis. Historical Benchmark Input-Output Tables. Accessed October 22, 2026.
- U.S. Bureau of Economic Analysis (Karen J. Horowitz and Mark A. Planting). Concepts and Methods of the U.S. Input-Output Accounts (BEA Primer). Accessed October 22, 2026.
- U.S. Bureau of Economic Analysis. Benchmark Input-Output Accounts of the U.S. Economy, 1997. Accessed October 22, 2026.