Harrod-Domar Equation Calculator
The Harrod-Domar model reduces economic growth to two numbers: how much of its income an economy saves, and how much capital it needs to produce one unit of output. Enter the savings rate as a decimal, the capital-output ratio and an initial output, and this calculator returns the warranted growth rate plus the investment and capital stock that sit behind it.
Enter as a decimal: 0.20 means a 20% savings rate. The live field shows a % sign, but the tool reads a fraction.
0.00%
Growth Rate (g)
The base Harrod-Domar model only: no depreciation, no population growth, no technology term. Those extensions are covered honestly in the sections below, not folded into the tool.
Quick Answer
With a savings rate of 0.20, a capital-output ratio of 3 and initial output of $1,000,000, the Harrod-Domar growth rate is 6.67% a year: g = s/v = 0.2/3 = 0.0667. The same run requires $200,000 of investment and a capital stock of $3,000,000. The calculator above is pre-filled with these exact numbers.
Key takeaways
- Growth in this model is a division: savings rate over capital-output ratio. Nothing else enters the growth rate, which is both the model's power and its main weakness.
- The savings rate goes in as a decimal (0.20), never as 20. The field caps at 1, so a percent entry cannot sneak through.
- Initial output does not change the growth rate at all. It sizes the two money outputs: investment (s × Y) and capital stock (v × Y).
- This is the base model: no depreciation, no population growth, no technology. A textbook extension subtracts depreciation from the result, and this tool does not.
- Quick Answer
- How to use the Harrod-Domar equation calculator
- The equation, and the two papers behind it
- Worked example: a 20% savings rate and a capital-output ratio of 3
- How sensitive the growth rate is to s and v
- What the model leaves out, stated plainly
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the Harrod-Domar equation calculator
Enter the two model parameters and a starting output, then click Calculate. Results stay hidden until you do.
- Savings rate (s): the share of national income that is saved and, in this model, fully invested. Enter it as a decimal fraction: 0.20 for 20%. For real-world values, the World Bank's gross savings as a share of GDP series is the standard source. Type carefully: the field clamps any entry above 1 back to 1 as you type, so "20" becomes "1" on the first keystroke, not a 2,000% savings rate.
- Capital-output ratio (v): the stock of capital behind one unit of annual output, K/Y. Textbook problems usually use 2 to 5. Some books write this ratio as k or as Harrod's Cr; this tool labels it v, and the formula readout uses v too.
- Initial output (Y₀): starting annual income or GDP in dollars. It scales the investment and capital stock outputs and leaves the growth rate untouched, so a student with no output figure can still read g straight from s and v.
- Read all four outputs: the growth rate, the investment the model requires, the capital stock that output implies, and the formula line showing your exact substitution.
The equation, and the two papers behind it
The warranted growth rate is g = s/v: the savings rate divided by the capital-output ratio. Roy Harrod set out the logic in An Essay in Dynamic Theory in the Economic Journal in 1939, and Evsey Domar reached the matching condition independently in Capital Expansion, Rate of Growth, and Employment in Econometrica in 1946.
Investment required I = s × Y
Capital stock K = v × Y
The derivation is four short steps. Saving is a fixed share of income, S = sY. The model assumes saving becomes investment one for one (S = I), and investment is the addition to the capital stock, ΔK = I. Because each unit of output needs v units of capital, output can grow by ΔY = ΔK ÷ v. Divide by Y and the growth rate falls out: g = ΔY/Y = s/v. The reference entry in The New Palgrave Dictionary of Economics walks the same steps with Harrod's original notation, where the ratio appears as Cr and the result is called the warranted rate: the one growth rate at which producers are content to keep expanding at the same pace.
Domar's route stressed the double life of investment: it is demand today (spending) and capacity tomorrow (new capital). Growth has to be fast enough for demand to absorb the capacity that yesterday's investment created. That is why the same equation keeps turning up in development planning: it converts a growth target directly into a savings and investment requirement.
Worked example: a 20% savings rate and a capital-output ratio of 3
The pre-filled run is the standard textbook setup: s = 0.20, v = 3, initial output $1,000,000. The tool returns a 6.67% growth rate, $200,000 of required investment and a $3,000,000 capital stock.
| Output | How the tool gets it | Worked example |
|---|---|---|
| Growth Rate (g) | (s ÷ v) × 100, two decimals | 6.67 |
| Investment Required (I) | s × Y₀ | 200,000 |
| Required Capital Stock (K) | v × Y₀ | 3,000,000 |
| Output Growth Formula | g = s/v with your numbers substituted, ratio at four decimals | g = s/v = 0.2/3 = 0.0667 |
Read the three money and rate outputs as one statement: an economy producing $1,000,000 a year with $3 of capital behind every $1 of output holds a $3,000,000 capital stock, and saving 20% of income puts $200,000 a year into adding to it, which is exactly enough to grow output by 6.67%. Change the initial output and the two dollar figures scale in proportion while the growth rate stands still, which you can confirm by running $2,000,000: investment doubles to $400,000, capital to $6,000,000, and g stays at 6.67%.
How sensitive the growth rate is to s and v
Because g is a ratio, equal proportional changes in s and v cancel out, and a one-point move in either input hits harder at low values of the other. The table below holds initial output at $1,000,000 and varies one parameter at a time; every cell is a separate run of the tool above.
| Run (Y₀ = $1,000,000) | Growth rate | Investment | Capital stock |
|---|---|---|---|
| s = 0.10, v = 3 | 3.33% | 100,000 | 3,000,000 |
| s = 0.20, v = 3 (worked example) | 6.67% | 200,000 | 3,000,000 |
| s = 0.30, v = 3 | 10.00% | 300,000 | 3,000,000 |
| s = 0.20, v = 2 | 10.00% | 200,000 | 2,000,000 |
| s = 0.20, v = 4 | 5.00% | 200,000 | 4,000,000 |
| s = 0.25, v = 5 | 5.00% | 250,000 | 5,000,000 |
Two planning lessons fall out of the table. First, efficiency is as powerful as thrift: cutting v from 4 to 2 doubles growth at the same savings rate, with no extra sacrifice of current consumption. Second, the required-savings form of the equation, s = g × v, is the version planners actually budget with: a 5% growth target at v = 4 demands a 20% savings rate, and at v = 5 it demands 25%. William Easterly's World Bank study The Ghost of Financing Gap documents what happened when that arithmetic was applied mechanically to aid and investment targets: the model's fixed ratio and one-for-one saving-to-investment assumption repeatedly over-promised growth, which is the honest reason to treat any single run of this calculator as a benchmark, not a forecast.
What the model leaves out, stated plainly
The tool computes the base 1939-1946 model and nothing more: a fixed capital-output ratio, saving that turns into investment automatically, and no other moving parts. Four omissions matter when you interpret a result.
- Depreciation. Many textbooks extend the equation to g = s/v − δ, subtracting the rate at which capital wears out. At s = 0.20, v = 3 and δ = 2%, that net rate is 4.67%, a full two points below this tool's 6.67%. This calculator has no depreciation input and does not subtract anything; if your course or plan uses the net form, subtract δ from the result yourself.
- Population and per-capita growth. The result is total output growth. Per-person growth is roughly g minus the population growth rate, and Harrod's own framework compares the warranted rate with a natural rate set by labour force growth plus productivity growth. Neither rate appears in this tool.
- A fixed ratio and no technology. The model holds v constant, so capital never runs into diminishing returns and technology never improves on its own. Robert Solow's 1956 contribution to growth theory was built largely to relax exactly that assumption, letting the ratio adjust and adding technological progress; the two models answer different questions.
- The knife-edge. The warranted rate is an unstable equilibrium in Harrod's telling: grow slightly faster and capacity shortages push the economy further off path, grow slightly slower and excess capacity does the same in reverse. The equation gives the rate an economy needs, with no promise that it will stay there.
Keep the neighbouring tools separate in your head. A GDP growth rate calculator measures growth that already happened, from recorded output in two periods; this page models the growth an economy's savings and capital efficiency can support. A compound annual growth calculator smooths a past series into an average rate, and a general growth rate calculator handles the plain percentage change between any two values. The perpetual-growth assumption valued in a growing perpetuity calculator is a finance cousin of the same idea: a constant g, carried forward indefinitely, which is exactly the assumption the knife-edge discussion warns about.
What competing calculators miss
Most pages ranking for this keyword stop at the single division and leave the two money outputs, and the model's limits, unexamined:
- CalculatorsHub's Harrod-Domar tool: computes g from s and v and adds a growth-outlook table, but returns no investment or capital stock figures, so the planning half of the model (how much investment a target requires) never appears.
- Calculator Ultra's version: solid history of Harrod and Domar and the same g-only calculation, with an example at v = 5. No money outputs, no sensitivity runs, and no mention of depreciation or the decimal-entry trap.
- Number Analytics' growth rate guide: the strongest theory write-up among the rivals, covering assumptions and the knife-edge, but it is a guide rather than a working tool with verifiable outputs.
- Exam-prep pages (Testbook, Prepp): teach the rearrangements (s = g × v, v = s ÷ g) and the depreciation variant well, because exams test them, but offer no calculator and blur which form is the base model.
- The gap this page fills: all four of the tool's outputs on one run, a sensitivity table built from verified runs of this exact tool, the depreciation and per-capita extensions named and quantified without being smuggled into the math, and the decimal-entry behaviour of the savings field explained before it can bite.
Frequently asked questions
It is a Keynesian growth model from Roy Harrod (1939) and Evsey Domar (1946) that ties an economy's growth rate to two quantities: the share of income saved and invested, and the amount of capital needed to produce one unit of output. Growth equals the savings rate divided by the capital-output ratio, g = s/v.
Because the equation works with the savings share as a fraction, so 20% enters as 0.20. The field accepts values from 0 to 1 and pulls anything larger back to 1 as you type, and the % sign shown beside the field describes the concept, not the entry format. Entering 20 would ask for a 2,000% savings rate, which the model does not allow.
It is the value of an economy's capital stock divided by its annual output, written v here and k or C with a subscript in some textbooks. A ratio of 3 means $3 of capital stands behind every $1 of yearly output. A lower ratio means capital produces output more efficiently, so the same savings rate buys faster growth.
Investment equals the savings rate times output, I = s × Y, because the model assumes all saving is invested. At a 0.20 savings rate and $1,000,000 of output, the tool returns $200,000 of investment, alongside the $3,000,000 capital stock implied by a ratio of 3 (K = v × Y).
Rearrange the equation to s = g × v. A 5% growth target with a capital-output ratio of 4 needs a savings rate of 0.05 × 4 = 0.20, or 20% of income. This calculator runs the equation in its standard direction, from savings to growth, so check a candidate savings rate by entering it and reading the resulting growth rate.
No. It computes the base model, g = s/v, exactly as Harrod and Domar set it out. Textbooks often extend it by subtracting a depreciation rate (g = s/v − δ) and by comparing total growth with population growth to get per-capita growth; neither adjustment is built into this tool, so apply them to the result yourself if your work calls for the extended forms.
Harrod showed that the warranted growth rate is an unstable balancing point. If actual growth runs slightly above it, capacity shortages encourage even faster expansion; slightly below it, unused capacity discourages investment and growth falls further. The economy balances on a knife-edge rather than settling back to the path on its own.
Solow (1956) kept the savings-driven engine but let producers substitute between capital and labour, so the capital-output ratio adjusts instead of staying fixed, and he added technological progress as a driver of long-run growth. Harrod-Domar's fixed ratio produces the knife-edge; Solow's flexible ratio produces convergence to a steady state.
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References
- Harrod, R. F. An Essay in Dynamic Theory, The Economic Journal, 49(193), 1939. Accessed October 22, 2026.
- Domar, E. D. Capital Expansion, Rate of Growth, and Employment, Econometrica, 14(2), 1946. Accessed October 22, 2026.
- Solow, R. M. A Contribution to the Theory of Economic Growth, The Quarterly Journal of Economics, 70(1), 1956. Accessed October 22, 2026.
- Hoover, K. D. Harrod-Domar Growth Model, in The New Palgrave Dictionary of Economics, Springer. Accessed October 22, 2026.
- Easterly, W. The Ghost of Financing Gap: How the Harrod-Domar Model Still Haunts Development Economics, World Bank Policy Research Working Paper 1807, 1997. Accessed October 22, 2026.
- Groningen Growth and Development Centre. Penn World Table, University of Groningen. Accessed October 22, 2026.
- World Bank, World Development Indicators. Gross Savings (% of GDP). Accessed October 22, 2026.
- World Bank, World Development Indicators. Gross Capital Formation (% of GDP). Accessed October 22, 2026.
- World Bank, World Development Indicators. Gross Fixed Capital Formation (% of GDP). Accessed October 22, 2026.
- World Bank, World Development Indicators. GDP Growth (annual %). Accessed October 22, 2026.