Harrod-Domar Model: Formula, Examples and Limitations

In 30 seconds

The Harrod–Domar model says an economy’s growth rate depends on how much it saves and how productive its new capital is.
The core formula is $ g = s / v $: growth equals the savings ratio divided by the capital-output ratio.
Turned around, $ s = g \times v $ tells a government how much investment it needs to hit a target growth rate. Any shortfall in domestic saving is the “savings gap” or “financing gap”.
Aid agencies used this logic for decades, but the evidence that more aid or investment reliably raises growth is weak.
Exam point: always evaluate. The model ignores diminishing returns, technology, institutions and how well capital is used. The Solow model fixes several of these problems.

What is the Harrod–Domar model?

The Harrod–Domar model is a simple theory of economic growth. It says that the growth rate of national output depends on two things: the share of income a country saves and invests, and how much extra output each unit of new capital produces.

The model comes from two economists working separately. Roy Harrod, a British economist, published “An Essay in Dynamic Theory” in the Economic Journal in 1939^[1]. Evsey Domar, an American economist, reached a very similar result in “Capital Expansion, Rate of Growth, and Employment”, published in Econometrica in April 1946^[2]. Because their conclusions matched, textbooks joined their names. Harrod’s essay is often described as the beginning of modern growth theory^[10].

Their question was dynamic: if investment adds to demand and to productive capacity, how fast must it grow to keep the economy at full employment? The version taught today strips this down to one equation.

Key termHarrod–Domar model: a growth model in which the rate of growth of output equals the savings (investment) ratio divided by the capital-output ratio, $ g = s/v $.

The model rests on a short chain of logic that you can learn as a cycle:

Households and firms save part of their income.
Those savings are channelled into investment, such as factories, roads and machines.
Investment adds to the capital stock.
A larger capital stock produces more output, so national income rises.
Higher income allows more saving, and the cycle repeats.

The policy message is simple. If you want faster growth, raise saving and investment, or make capital more productive.

The Harrod–Domar growth model formula

You need three ideas to derive the formula.

Key termSavings ratio (s): the share of national income that is saved, $ s = S/Y $. In the model, all saving is invested, so $ S = I $.

Key termCapital-output ratio (v): the amount of capital needed to produce one unit of output, $ v = K/Y $. When measured on extra capital and extra output, it is called the incremental capital-output ratio (ICOR), $ v = \Delta K / \Delta Y $.

A high ICOR means capital is not very productive: you need a lot of new investment to get a little extra output. A low ICOR means each pound of investment buys a lot of growth.

Derivation. Start with the definition of the ICOR and rearrange it:

\[ v = \frac{\Delta K}{\Delta Y} \quad \Rightarrow \quad \Delta Y = \frac{\Delta K}{v} \]

Ignoring depreciation for now, the change in the capital stock equals investment, and investment equals saving:

\[ \Delta K = I = S = sY \]

Substitute this into the first equation:

\[ \Delta Y = \frac{sY}{v} \]

Divide both sides by $ Y $. The left-hand side becomes the growth rate of output, $ g = \Delta Y / Y $:

\[ g = \frac{\Delta Y}{Y} = \frac{s}{v} \]

That is the whole model. Growth is the savings ratio divided by the capital-output ratio.

Adding depreciation. Machines wear out. If a fraction $ \delta $ of the capital stock depreciates each year, only investment above that amount adds to capacity. A common extended version is:

\[ g = \frac{s}{v} – \delta \]

Per-person growth. Living standards depend on output per person. If the population grows at rate $ n $, then growth of output per head is approximately:

\[ g_{\text{per capita}} \approx \frac{s}{v} – n \]

Common mistakeStudents often write $ g = s \times v $. Check the logic: a higher capital-output ratio means capital is less productive, so it must lower growth. That is why $ v $ sits on the bottom of the fraction.

Worked examples: growth rates, required investment and the savings gap

All the numbers below are illustrative. Work through each one before reading the answer.

Worked example

Suppose an economy has national income of £100 billion and a capital stock of £400 billion. It saves 20% of income, and every extra £4 of capital produces £1 of extra output. Find the growth rate using the step-by-step logic, then check it with the formula.

Step 1Saving: $ S = sY = 0.20 \times 100 = £20\text{ bn} $

Step 2Investment equals saving, so the capital stock rises by $ \Delta K = £20\text{ bn} $

Step 3Extra output: $ \Delta Y = \Delta K / v = 20 / 4 = £5\text{ bn} $

Step 4Growth: $ g = 5/100 = 0.05 = 5\% $

Step 5Check: $ g = s/v = 0.20/4 = 0.05 = 5\% $

Saving one-fifth of income, with an ICOR of 4, lets the economy grow by 5% a year. Both routes give the same answer, which confirms the formula.

Worked example

Suppose a government wants GDP to grow by 7% a year. Its planners estimate the ICOR at 4. Domestic saving is 20% of GDP and GDP is $50 billion. How much investment is needed, and how large is the savings gap?

Step 1Rearrange the formula: $ s_{\text{required}} = g \times v $

Step 2$ s_{\text{required}} = 0.07 \times 4 = 0.28 $, so investment must be 28% of GDP

Step 3Savings gap as a share of GDP: $ 28\% – 20\% = 8\% $

Step 4In money terms: $ 0.08 \times \$50\text{ bn} = \$4\text{ bn} $ a year

The country can only afford investment of 20% of GDP from its own saving. To reach 7% growth, the model says it must find another $4 billion a year from abroad: foreign aid, borrowing or foreign direct investment. This is the “financing gap”.

Key termSavings gap (financing gap): the difference between the investment needed to reach a target growth rate and the saving a country can generate domestically. In the model it is $ (g^{*} \times v) – s $, where $ g^{*} $ is the target growth rate.

Worked example

Suppose a country saves 25% of GDP, has an ICOR of 5, loses 3% of its capital stock each year to depreciation, and its population grows by 1.5% a year. What happens to output per person?

Step 1Gross growth from new capital: $ s/v = 0.25/5 = 0.05 = 5\% $

Step 2Subtract depreciation: $ g = 5\% – 3\% = 2\% $

Step 3Subtract population growth: $ g_{\text{per capita}} \approx 2\% – 1.5\% = 0.5\% $

Most of the saving goes on replacing worn-out capital and equipping a growing population, so living standards rise by only about half a percent a year.

Exam tipIf a question gives you a target growth rate and asks “how much investment is needed?”, rearrange to $ s = g \times v $. Always state your answer as a percentage of GDP, then convert to money terms if GDP is given. Examiners reward a clear statement of what the number means.

Warranted growth, natural growth and the knife-edge

Harrod’s original essay was not only about how fast an economy can grow. It was about whether growth can be stable. He compared three growth rates^[1].

Key termActual growth rate (G): the rate at which output really grows in a given year.

Key termWarranted growth rate (G_w): the growth rate at which firms are satisfied that they invested the right amount. Saving is fully absorbed by the investment firms want, so $ G_w = s / v_r $, where $ v_r $ is the capital-output ratio firms require.

Key termNatural growth rate (G_n): the highest growth rate the economy can sustain in the long run, set by growth of the labour force plus growth of labour productivity from technical progress.

The knife-edge problem. Suppose actual growth runs above the warranted rate. Firms find demand stronger than expected and their capital stretched, so they invest more. Extra investment raises demand further, pushing actual growth even higher above the warranted rate. The reverse also holds: if growth falls short, firms have spare capacity, cut investment, and demand falls further. Small deviations do not correct themselves. They widen. That is why the balanced path is called a “knife-edge”. Harrod later highlighted this instability problem, which helped launch post-war research on growth^[10].

The second problem: G_w versus G_n. Even if the economy stays on its warranted path, there is no reason that path matches the natural rate. In the model, $ s $, $ v $, labour force growth and technical progress are all set independently.

If $ G_w > G_n $, capacity grows faster than the labour force can staff it. The economy keeps running into labour shortages and tends towards stagnation.
If $ G_w < G_n $, capital grows too slowly to employ every new worker. Unemployment rises over time.

Full employment with steady growth needs $ s/v = G_n $ exactly. Nothing in the model pushes the economy towards that point. This pessimistic result is what Robert Solow set out to challenge in 1956^[3].

DiagramDraw the savings ratio $ s $ on the horizontal axis (0% to 40%) and the growth rate on the vertical axis. The warranted growth line $ G_w = s/v $ is a straight line from the origin with slope $ 1/v $. With $ v = 4 $, it passes through (20%, 5%) and (40%, 10%). Draw a second, steeper line for $ v = 3 $ to show that more productive capital raises growth at every savings ratio. Add a horizontal line at the natural rate, say $ G_n = 6\% $. The $ v = 4 $ line crosses it at $ s^{*} = 24\% $, the only savings ratio that gives full-employment growth. Label the region to the right “$ G_w > G_n $: labour shortages” and to the left “$ G_w < G_n $: rising unemployment”. For A-Level, you may also draw the circular flow version: arrows from Savings → Investment → Capital stock → Output and income → back to Savings.

How the model was used in development planning and aid

After 1945, newly independent countries wanted rapid growth, and the Harrod–Domar model offered a clear recipe. Planners chose a target growth rate, estimated an ICOR, worked out the investment needed, and compared it with domestic saving. Western donors were then asked to fill the gap. A review of William Easterly’s book The Elusive Quest for Growth describes how this model, combined with W. W. Rostow’s 1960 idea of a “takeoff” into growth, produced the financing-gap approach to aid^[6].

The striking point is that the model’s co-founder rejected it as a long-run theory. Domar disavowed it in 1957, saying it “made no sense as a theory of long-run growth”^[6]. Yet the same review notes that it became “the most widely applied growth model in economic history”^[6].

Easterly, then a World Bank economist, tested the model directly. His 1997 World Bank working paper opens: “The Harrod-Domar growth model supposedly died long ago.” He explains that economists were still using it to estimate short-run investment needs for a target growth rate, calculate a financing gap and fill it with aid. He asks whether forty years of data justify this. “The answer is ‘no.'”^[5] The journal version appeared in the Journal of Development Economics in 1999 under the title “The ghost of financing gap: testing the growth model used in the international financial institutions”^[4].

The financing-gap approach needs two links to hold:

Aid must raise investment, ideally one-for-one, so that each dollar of aid becomes a dollar of capital.
Investment must raise growth in a stable, predictable way, with a constant ICOR.

Both links are weak in the data. Summarising Easterly’s later work, Berloffa, Folloni and Schnyder report that aid was followed by a positive, significant rise in investment of at least one-for-one in only 6 of 88 countries. In 36 countries the relationship was significantly negative^[7]. In other words, aid often replaced domestic investment rather than adding to it. Easterly’s broader explanation is about incentives. Aid did not change the incentive for people in recipient countries to invest^[6].

Real-world data: investment, growth and ICORs

You can estimate a country’s ICOR from published data. Take the average investment rate (gross capital formation as a share of GDP) and divide it by the average real GDP growth rate over the same years. The table uses World Bank data for 2015 to 2019, a period chosen to avoid the distortions of the COVID-19 shock in 2020^[8]^[9].

Country	Gross capital formation, % of GDP (avg 2015–19)	Real GDP growth, % (avg 2015–19)	Implied ICOR	Investment, % of GDP (2024)	GDP growth, % (2024)
China	42.7	6.7	6.4	40.5	5.0
Ethiopia	37.2	8.9	4.2	20.6	7.6
India	31.1	6.7	4.7	34.3	7.1
Bangladesh	30.8	7.1	4.3	30.7	4.2
Kenya	20.2	4.8	4.2	16.5	4.7
Pakistan	16.1	4.8	3.4	13.2	3.1

Sources: World Bank World Development Indicators, indicators NE.GDI.TOTL.ZS and NY.GDP.MKTP.KD.ZG^[8]^[9]. Implied ICOR is the author’s calculation: average investment share divided by average growth rate.

Investment and growth, 2015–2019 averages (%)

Gross capital formation (% of GDP)Real GDP growth (% a year)

Source: World Bank World Development Indicators, NE.GDI.TOTL.ZS and NY.GDP.MKTP.KD.ZG [8][9] Implied ICOR (investment share ÷ growth): China 6.4, Ethiopia 4.2, India 4.7, Bangladesh 4.3, Kenya 4.2, Pakistan 3.4. Five-year averages of annual World Bank figures.

Four lessons stand out.

Investment and growth are linked, but loosely. China invested over 42% of GDP and grew by 6.7% a year. India grew at the same average rate while investing about 31% of GDP. The same growth rate needed very different amounts of capital.
ICORs are not constant. China’s implied ICOR of 6.4 is the highest in the table. One reading is diminishing returns: after decades of heavy investment, each extra yuan of capital adds less output. The Harrod–Domar model, with its fixed $ v $, cannot explain this.
Low investment does not always mean low growth. Pakistan invested only about 16% of GDP yet grew by 4.8% a year, giving a low ICOR of 3.4. Ethiopia’s investment share fell from around 37% to 20.6% by 2024, while growth was still 7.6% that year. A planner using a fixed ICOR would have predicted much slower growth.
Short-run ICORs are meaningless in shock years. In 2020 India’s investment was still 28.9% of GDP, but GDP fell by 5.8%^[8]^[9]. The ratio comes out negative. In any single year, shocks move growth far more than the capital stock does.

Common mistakeA “measured” ICOR is not a technical constant. It is simply investment divided by growth, so it soaks up everything else that affects growth: technology, education, policy, weather and luck.

Limitations of the Harrod–Domar model

The model is a useful starting point, but it fails as a guide to policy in developing countries for several reasons.

Fixed capital-output ratio. The model assumes $ v $ is constant. In reality, capital runs into diminishing returns when it is added to a fixed amount of labour and land. The China data above show the ICOR drifting upwards.
No role for technology. Growth comes only from capital accumulation. Yet most long-run growth in living standards comes from better technology and better ways of organising production, which the model leaves out.
Labour is ignored. The model assumes there is always enough labour to work with new capital. Skills, health and education (human capital) matter as much as machines.
Saving does not automatically become productive investment. Weak banks, capital flight and corruption mean savings can leak away or fund “white elephant” projects.
Aid can crowd out domestic saving. If governments expect donors to fill the gap, they may save less themselves. The evidence that aid raises investment one-for-one is weak^[7].
Institutions and incentives are missing. Property rights, rule of law and political stability determine whether people want to invest at all. Easterly’s central argument is that people respond to incentives^[6].
Borrowing creates debt. If the gap is filled with loans rather than grants and the investment does not raise growth, debt burdens rise.

Evaluation. The model still has value. Savings and investment are necessary for growth: no country has grown rich without building up its capital stock. It also gives a quick consistency check: targeting 8% growth while investing 12% of GDP implies an ICOR of 1.5, far below the values above. The problem is treating a necessary condition as a sufficient one. More capital helps only if it is well chosen, well used and combined with skills, technology and sound institutions.

Harrod–Domar vs the Solow model

Robert Solow’s 1956 paper in the Quarterly Journal of Economics was written partly as a response to the knife-edge result^[3]. Solow dropped the assumption of a fixed capital-output ratio and allowed firms to substitute between capital and labour.

Feature	Harrod–Domar	Solow (1956)
Capital-output ratio	Fixed	Varies as capital per worker changes
Returns to capital	Constant	Diminishing
Effect of higher saving	Permanently higher growth rate	Higher level of income per head, but only a temporary rise in growth
Source of long-run growth per head	Saving and investment	Technical progress
Stability	Unstable knife-edge	Stable steady state
Policy message	Raise saving or fill the gap with aid	Capital deepening has limits; productivity matters most

The key difference is diminishing returns. In Solow’s model, extra capital per worker adds less and less output, so the economy settles at a steady state. After that, only technical progress raises output per head. This fits the China data better than Harrod–Domar does.

Exam tipGive a specific contrast: “Harrod–Domar predicts that doubling the savings ratio doubles growth forever; Solow predicts only a temporary rise because of diminishing returns” earns more credit than “Solow is better”.

Practice questions

EasyQ1. An economy saves 18% of its income and has a capital-output ratio of 3. Using the Harrod–Domar model, what is its growth rate? Show worked answer

$ g = s/v = 0.18/3 = 0.06 $, so the economy grows by 6% a year.

EasyQ2. Explain what a rise in the capital-output ratio from 3 to 5 means, and its effect on growth if the savings ratio stays at 15%. Show worked answer

A higher capital-output ratio means more capital is needed to produce each extra unit of output, so capital is less productive. Growth falls from $ 0.15/3 = 5\% $ to $ 0.15/5 = 3\% $.

MediumQ3. A country with GDP of $80 billion wants to grow by 6% a year. Its ICOR is 4.5 and domestic saving is 19% of GDP. Calculate the required investment ratio and the financing gap in dollars. Show worked answer

Required investment: $ s = g \times v = 0.06 \times 4.5 = 0.27 $, or 27% of GDP. Gap: $ 27\% – 19\% = 8\% $ of GDP. In dollars: $ 0.08 \times 80 = \$6.4 $ billion a year.

MediumQ4. A country saves 24% of GDP, has an ICOR of 4, and its population grows by 3% a year. Ignoring depreciation, estimate growth in GDP per head. What would the savings ratio need to be for GDP per head to grow by 4%? Show worked answer

GDP growth: $ 0.24/4 = 6\% $. Per head: $ 6\% – 3\% \approx 3\% $. For 4% growth per head, GDP must grow by about $ 4\% + 3\% = 7\% $, so $ s = 0.07 \times 4 = 0.28 $, or 28% of GDP.

ExamQ5. Evaluate the view that foreign aid to close the savings gap is the most effective way to raise economic growth in a developing country. [25 marks] Show worked answer

Knowledge and understanding (up to 5): Define the savings gap. State $ g = s/v $ and explain the cycle: saving, investment, capital stock, output, income.

Application (up to 5): Use a numerical example, for instance a 7% target with an ICOR of 4 needs 28% investment; with 20% saving, the gap is 8% of GDP. Use real data, such as Pakistan investing about 16% of GDP in 2015–19.

Analysis (up to 6): Explain how aid can raise investment, add to the capital stock and raise growth, and how it can also fund infrastructure that raises the productivity of private capital (lowering $ v $).

Evaluation (up to 9): The aid-to-investment link is weak: aid raised investment at least one-for-one in only 6 of 88 countries in Easterly’s analysis. ICORs vary over time and between countries, so the model’s predictions are unreliable. The model ignores diminishing returns (Solow), technology, human capital and institutions. Aid can crowd out domestic saving and create dependency; loans add debt. Alternatives may be more effective: improving governance, education, trade openness or domestic financial systems. Reach a supported judgement, for example that aid helps when it funds productive investment in countries with sound institutions, but closing a calculated “gap” is not by itself a reliable route to growth.

Frequently asked questions

What is the formula for the Harrod–Domar model?

The growth rate equals the savings ratio divided by the capital-output ratio: $ g = s/v $. With depreciation at rate $ \delta $, it becomes $ g = s/v – \delta $.

What is the capital-output ratio?

It is the amount of capital needed to produce one unit of output. The incremental version (ICOR) is the extra capital needed for one extra unit of output, $ \Delta K/\Delta Y $. A lower ratio means capital is more productive.

What is the savings gap in the Harrod–Domar model?

It is the difference between the investment needed for a target growth rate ($ g \times v $) and the saving a country can generate at home. Developing countries have often tried to fill it with aid, loans or foreign investment.

Why does the Harrod–Domar model fail for developing countries?

It assumes a fixed capital-output ratio and ignores technology, labour skills and institutions. In practice, aid does not reliably raise investment, and investment does not reliably raise growth, so plans based on the model often miss their targets.

References

Harrod, R. F. (1939). “An Essay in Dynamic Theory”. The Economic Journal, 49(193), 14–33. https://doi.org/10.2307/2225181
Domar, E. D. (1946). “Capital Expansion, Rate of Growth, and Employment”. Econometrica, 14(2), 137–147. https://doi.org/10.2307/1905364
Solow, R. M. (1956). “A Contribution to the Theory of Economic Growth”. The Quarterly Journal of Economics, 70(1), 65–94. https://doi.org/10.2307/1884513
Easterly, W. (1999). “The ghost of financing gap: testing the growth model used in the international financial institutions”. Journal of Development Economics, 60(2), 423–438. https://doi.org/10.1016/S0304-3878(99)00047-4
Easterly, W. (1997). “The ghost of financing gap: how the Harrod-Domar growth model still haunts development economics”. Policy Research Working Paper 1807. The World Bank. https://ideas.repec.org/p/wbk/wbrwps/1807.html
Federal Reserve Bank of Minneapolis (2003). “The Elusive Quest for Growth: Economists’ Adventures and Misadventures in the Tropics” (book review of Easterly, W. (2001), MIT Press). https://www.minneapolisfed.org/article/2003/the-elusive-quest-for-growth-economists-adventures-and-misadventures-in-the-tropics
Berloffa, G., Folloni, G. and Schnyder, I. (2009). “Development and economic growth: the effectiveness of traditional policies”. Working paper, University of Trento. https://www.economia.unitn.it/alfresco/download/workspace/SpacesStore/92177c6a-3e47-41a8-ad06-d64326bc7262
World Bank (2026). “Gross capital formation (% of GDP)” (NE.GDI.TOTL.ZS). World Development Indicators. https://data.worldbank.org/indicator/NE.GDI.TOTL.ZS
World Bank (2026). “GDP growth (annual %)” (NY.GDP.MKTP.KD.ZG). World Development Indicators. https://data.worldbank.org/indicator/NY.GDP.MKTP.KD.ZG
History of Economic Thought website (n.d.). “Roy F. Harrod, 1900–1978”: profile. https://hetwebsite.net/het/profiles/harrod.htm

The Harrod–Domar Model: Formula, Worked Examples and Why It Fails for Developing Countries

What is the Harrod–Domar model?

The Harrod–Domar growth model formula

Worked examples: growth rates, required investment and the savings gap

Warranted growth, natural growth and the knife-edge

How the model was used in development planning and aid

Real-world data: investment, growth and ICORs

Limitations of the Harrod–Domar model

Harrod–Domar vs the Solow model

Practice questions

Frequently asked questions

What is the formula for the Harrod–Domar model?

What is the capital-output ratio?

What is the savings gap in the Harrod–Domar model?

Why does the Harrod–Domar model fail for developing countries?

References

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