Harrod-Domar Growth Model Explained: Formula, Derivation & Limitations

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Theories of Economic Growth

Theories of economic growth are the frameworks economists use to explain how and why national income, output, and productive capacity expand over time. They try to identify the key drivers behind a rising standard of living—such as capital accumulation, labor, technology, and institutions—and to show how these drivers interact to produce a sustained increase in an economy’s output.

Economic growth theory broadly divides into two families of models: exogenous growth models and endogenous growth models. Exogenous growth models, such as the Solow-Swan model, treat technological progress as an outside factor that is not explained within the model itself; growth in the long run is driven by forces assumed to occur independently of the economy’s own saving and investment decisions.

Endogenous growth models, by contrast, argue that growth is generated from within the economic system itself, through factors such as investment in human capital, innovation, and knowledge, which are treated as internal (endogenous) variables that the model actively explains rather than assumes (Romer, 1986; Lucas, 1988).

Before either of these traditions took shape, one of the earliest and most influential attempts to formally model economic growth was the Harrod-Domar Growth Model.

Background of the Harrod-Domar Model

The Harrod-Domar model is a classical economic growth model that explains the relationship between economic growth, capital accumulation, and savings. It states that the growth rate of GDP ( $\Delta Y/Y$ ) is jointly determined by the net saving ratio and the capital-output ratio ( $c$ ).

It was developed independently by economists Roy Harrod and Evsey Domar during the 1930s and 1940s, in the aftermath of the Great Depression, as a way of understanding the conditions required for a steady rate of economic growth (Harrod, 1939; Domar, 1946).

The basic idea of the Harrod-Domar model is that economic growth depends on two things:

The net savings ratio
The capital-output ratio

1. Net Saving Ratio

The net savings ratio can be defined as the proportion of national income that is saved over some period of time. It is assumed to be fixed in the model.

Thus,

$S = sY$

2. Capital-Output Ratio

The capital-output ratio shows the amount of capital required to produce one unit of output. It is denoted by $c$ .

Thus,

$c = \dfrac{K}{Y}$

Deriving the Harrod-Domar Growth Equation

Because $S = sY$ and $c = K/Y$ , so:

$c = \dfrac{\Delta K}{\Delta Y}$

Or:

$\Delta K = c\Delta Y$

Net Investment

Net investment ( $I$ ) can be defined as gross investment ( $I^{g}$ ) minus the depreciation of capital. If net investment is positive, then capital stock will increase. Thus, net investment is the change in capital stock, K, and can be expressed as: $I = \Delta K$

Because net national saving is equal to net investment, such as

$S = I$

$I = \Delta K = c\Delta Y$

So, the identity of saving and investment can be written as

$S = sY = \Delta K = c\Delta Y = I$

Or simply

$sY = c\Delta Y$

Dividing both sides by $Y$ , we get:

$s = c\dfrac{\Delta Y}{Y}$

Now dividing both sides by $c$ , we get:

$g = \dfrac{\Delta Y}{Y} = \dfrac{s}{c}$

This is the famous equation of the Harrod-Domar Growth Model. It states that the growth rate of GDP ( $\Delta Y/Y$ ) is jointly determined by the net saving ratio and the capital-output ratio ( $c$ ).

The GDP growth rate depends positively on the net saving ratio—the higher the saving, the higher the investment and the higher the capital stock, and hence the faster the growth, because higher investment today increases the production capacity of the economy in the future.

The GDP growth rate depends inversely on the capital-output ratio—the higher the $c$ , the lower the GDP growth rate.

Accounting for depreciation, we can write the equation as

$\dfrac{\Delta Y}{Y} = \dfrac{s^{G}}{c} - \delta$

Where $\delta$ is the rate of depreciation and $s^{G}$ is gross saving.

Efficiency of Capital Utilisation

How much additional output can be obtained from an additional unit of investment can be measured by the inverse of the capital-output ratio, $c$ , which is simply $1/c$ , the output-capital ratio or output-investment ratio. It is also known as the efficiency of capital utilisation.

The efficiency of capital utilisation ( $1/c$ ) shows the rate at which GDP will increase with an additional unit of investment.

A Numerical Example

For example, if we assume that the national capital-output ratio in a less-developed country is, say, 3, and the aggregate net saving ratio is 6% of GDP, then this country can grow at a rate of 2% per year, because

$\dfrac{\Delta Y}{Y} = \dfrac{s}{c} = \dfrac{6\%}{3} = 2\%$

Now, if the national net savings rate increased from 6% to 15%—through increased taxes, foreign aid, and general consumption sacrifices—GDP growth can be increased from 2% to 5%.

$\dfrac{\Delta Y}{Y} = \dfrac{s}{c} = \dfrac{15\%}{3} = 5\%$

Not only do national saving and investment cause the growth rate of GDP to rise, but also by improving investment efficiency. For example, if $c$ can be lowered from 3 to 2, then, other things equal, the model predicts that growth would increase.

If net savings were 6%, then the model predicts a GDP growth rate of 3%. Likewise, if net savings were 15%, then growth rises from 5% to 7.5%.

Finally, the model predicts that if depreciation can be decreased by 1%, then growth would rise by 1% (assuming $s = 6\%$ and $c = 3$ ).

Constraint to Development

The main obstacle to, or constraint on, development, according to this theory, is the relatively low level of new capital formation in most poor countries.

If, for example, a country wanted to grow at the rate of 7% per annum, it must generate saving and investment at the rate of 21% of national income (assuming $c = 3$ ).

But if it could only manage to save 15%, then the saving gap of 6% must be filled by either foreign aid or private foreign investment.

Conclusion of the Model

Rostow and others defined the “takeoff” stage in precisely this way. Countries that were able to save 15% to 20% of GDP could grow (“develop”) at a much faster rate than those that saved less (Rostow, 1960).

Moreover, this growth would then be self-sustaining. The mechanism of economic growth and development, therefore, would simply be a matter of increasing national savings and investment.

Limitations of the Harrod-Domar Model

It is difficult to increase the savings ratio in low-income countries. Savings propensities are low in many developing countries, as extra income is frequently spent rather than saved.
Many developing countries lack an efficient financial system; therefore, savings are not translated into investment.
Due to a low level of human capital, capital-output ratios are higher in these countries.
R&D required to enhance the capital-output ratio is frequently underfunded.
The model ignores factors such as labour productivity and technological innovation.

Necessary vs Sufficient Conditions

The mechanism of development explained by the theory of stages of growth does not always hold. This is because, although saving and investment are necessary conditions for growth, they are not sufficient to guarantee economic growth on their own. Economic growth requires more than investment and saving—these are the preconditions or assumptions in the Harrod-Domar model that must be fulfilled.

The Marshall Plan succeeded because the European countries receiving aid possessed the necessary structural, institutional, and attitudinal conditions, such as efficient commodity and money markets, highly developed transport facilities, a well-trained and educated workforce, motivation to succeed, and an efficient government system (Todaro & Smith, 2015).

However, underdeveloped countries often lack these conditions. As a result, simply increasing investment—as suggested by models like Rostow’s stages of growth or the Harrod-Domar model—does not automatically lead to growth in these nations.

Thus, saving and investment are necessary conditions for economic growth, but not sufficient conditions. For developing and underdeveloped countries to achieve economic growth, they must also focus on managerial skills, a skilled labor force, and an efficient government system to plan and administer development projects.

Moreover, these countries also need to focus on reducing the capital-output ratio, i.e., increasing the efficiency with which investment generates extra output.

To summarize the two concepts:

Necessary condition: A condition that must be present, although not in itself sufficient, for an event to occur. For example, capital formation may be a necessary condition for sustained economic growth but does not guarantee it.
Sufficient condition: A condition that, when present, guarantees that an event will occur. For example, in an economic model, if all the required assumptions are met, the result must hold—such as when the necessary social, institutional, and attitudinal changes have also occurred.

Final Thoughts

The Harrod-Domar model remains a foundational tool in development economics for illustrating how saving, investment, and capital efficiency interact to determine an economy’s growth rate. While its assumptions are simplistic compared to later exogenous and endogenous growth models, it correctly highlights that capital accumulation alone cannot guarantee sustained development—institutional capacity, human capital, and technological progress are equally indispensable.

References

Domar, E. D. (1946). Capital Expansion, Rate of Growth, and Employment. Econometrica, 14(2), 137–147.
Harrod, R. F. (1939). An Essay in Dynamic Theory. The Economic Journal, 49(193), 14–33.
Lucas, R. E. (1988). On the Mechanics of Economic Development. Journal of Monetary Economics, 22(1), 3–42.
Romer, P. M. (1986). Increasing Returns and Long-Run Growth. Journal of Political Economy, 94(5), 1002–1037.
Rostow, W. W. (1960). The Stages of Economic Growth: A Non-Communist Manifesto. Cambridge University Press.
Solow, R. M. (1956). A Contribution to the Theory of Economic Growth. The Quarterly Journal of Economics, 70(1), 65–94.
Todaro, M. P., & Smith, S. C. (2015). Economic Development (12th ed.). Pearson Education.

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About the author

Muhammad Minhaj Akhtar

Muhammad Minhaj Akhtar is a Lecturer in Economics at Government Graduate College Jauharabad, Pakistan. He holds an M.Phil. in Economics from Quaid-i-Azam University, Islamabad, and an MSc in Economics from the University of Sargodha, where he earned a Silver Medal. His academic passion lies in Econometrics, with a strong focus on applying empirical methods to real-world economic issues. Through MinhajMetrixHub, he shares learning resources, research guidance, and practical econometric insights for students and researchers.