Economic Growth, Underdevelopment and Economic Development

Lessons

Senior 6 · Economics

Economic Growth, Underdevelopment and Economic Development

5 lessons

Lesson 3 of 5

Theories of economic growth

25 min

Case: how much must a country save?

NST2 sets two targets side by side. Real GDP is to grow by an average of 9.3% a year from 2024 to 2029, and total investment is to reach 32.1% of GDP by 2029, while national saving rises from 12.4% of GDP in 2023 to 25.9% (MINECOFIN, NST2, pp. 9–10).

Think: why would planners link a growth target to an investment target? And if Rwandans save only 25.9% of GDP but invest 32.1%, where does the rest come from? The oldest model in development economics answers both questions.

Definition

Capital–output ratio ( $k$ ): the amount of capital needed to produce one unit of output a year, $k = \frac{K}{Y}$ . The incremental capital–output ratio (ICOR) is the extra capital needed for one extra unit of output, $\frac{\Delta K}{\Delta Y}$ .

Example: if a Frw 400 million maize mill produces Frw 100 million of flour a year, $k = \frac{400}{100} = 4$ .

Definition

Savings ratio ( $s$ ): the fraction of national income that is saved, $s = \frac{S}{Y}$ .

Savings gap: the difference between the investment needed to reach a target growth rate and the saving a country provides itself. It must be filled by foreign saving (aid, foreign direct investment or borrowing) or the target is missed.

The Harrod–Domar model

Roy Harrod (1939) and Evsey Domar (1946) built the model separately; development planners in the 1950s and 1960s used it to work out how much investment poor countries needed. It rests on these assumptions:

Output is proportional to the capital stock, with a fixed capital–output ratio $k$ .
A fixed fraction $s$ of income is saved, and all saving is invested: $S = I$ .
Investment adds directly to the capital stock: $\Delta K = I$ (ignoring depreciation for now).
Labour is plentiful, so capital is the only constraint on output.
Prices are constant, there is no technical progress, and the economy is closed.

Derivation

From assumption 1, $Y = \frac{K}{k}$ , so a change in capital changes output by

\Delta Y = \frac{\Delta K}{k}

From assumptions 2 and 3, $\Delta K = I = S = sY$ . Substitute:

\Delta Y = \frac{sY}{k}

Divide both sides by $Y$ to get the growth rate:

g = \frac{\Delta Y}{Y} = \frac{s}{k}

If a fraction $\delta$ of capital wears out each year, the growth rate is $g = \frac{s}{k} - \delta$ , and growth of income per head is roughly $\frac{s}{k} - n$ , where $n$ is the population growth rate.

The model's message is simple: growth rises when a country saves and invests more (higher $s$ ) or uses its capital more productively (lower $k$ ). Read the other way, a growth target fixes how much investment is required: $s = g \times k$ .

Worked example

Worked example 1: the growth rate (illustrative figures)

A country saves 24% of its income, its capital–output ratio is 4, and its population grows by 2.3% a year. Find the growth rate of output and, approximately, of output per head.

g = \frac{s}{k} = \frac{0.24}{4} = 0.06 = 6\%

g_{pc} \approx 6 - 2.3 = 3.7\%

If 2% of the capital stock wears out each year, growth falls to $6 - 2 = 4\%$ and growth per head to about 1.7%.

Worked example 2: what NST2's targets imply

Use NST2's figures: investment of 32.1% of GDP and growth of 9.3% a year. (a) What capital–output ratio do these imply? (b) What share of GDP must come from foreign saving if national saving is 25.9%? (c) What growth would the same investment give if $k$ were 4.5?

\text{(a)}\quad k = \frac{s}{g} = \frac{0.321}{0.093} = 3.45

\text{(b)}\quad 32.1 - 25.9 = 6.2\% \text{ of GDP}

\text{(c)}\quad g = \frac{0.321}{4.5} = 0.071 = 7.1\%

So the growth target depends as much on using capital efficiently (a low $k$ ) as on the amount invested. This is a rough check, not NST2's own method: the targets are for different years, and the model ignores depreciation.

Warning

Keep $s$ and $k$ in the right places. Growth rises with $s$ and falls with $k$ . A higher capital–output ratio means capital is less productive, so the same saving buys less growth. And write rates as decimals or percentages consistently: $\frac{24}{4} = 6$ gives 6%, but $\frac{0.24}{4} = 0.06$ also means 6%.

Limitations of the Harrod–Domar model

$k$ is not fixed. It differs between sectors and rises as more capital is added (diminishing returns), so the same saving gives less and less growth.
Saving is not automatically invested. In poor countries weak banks and small capital markets mean saving may be held as cash, cattle or land, or sent abroad.
It ignores technology, skills, institutions and entrepreneurship. Capital without trained workers, reliable power and honest administration produces little.
Labour is not always plentiful in the right form. Shortages of skilled workers limit how fast capital can be used.
Investment can be wasted on prestige projects, or lost to corruption.
Poor countries find it hardest to raise $s$ , because most income goes on necessities; this is one strand of the vicious circle of poverty in lesson 4.
Filling the savings gap with foreign money has costs: debt repayments, profits sent abroad, and dependence on donors (Unit 8).
It was designed for other questions. Harrod and Domar were studying the stability of growth in industrial economies, not development.

The neoclassical answer: Solow

Robert Solow (1956) let the capital–output ratio vary. Because each extra unit of capital adds less output than the one before, investment alone cannot raise the growth rate for ever: the economy approaches a steady state where saving only replaces worn-out capital. In the long run, growth of income per head comes from technical progress, the total factor productivity of lesson 2. The model also predicts that poorer countries, with little capital, can grow faster than rich ones while they catch up, but only if they have similar saving, population growth and technology.

Definition

Rostow's stages of economic growth: W. W. Rostow's theory (The Stages of Economic Growth, 1960) that every country passes through five stages on the way to a modern economy, with a short "take-off" into self-sustaining growth at its centre.

Rostow's five stages

Stage	Main features
1. Traditional society	Mostly subsistence farming with limited technology; output per head barely rises
2. Preconditions for take-off	Education spreads, banks and entrepreneurs appear, transport improves, and an effective national state emerges
3. Take-off	Investment rises from about 5% to over 10% of national income; one or more "leading sectors" grow fast; political and social institutions support growth
4. Drive to maturity	Growth spreads to many industries; technology is widely used; the economy diversifies
5. Age of high mass consumption	Output shifts to durable consumer goods and services; welfare spending rises

Critique of Rostow

History does not fit neatly. The stages overlap, and some countries have skipped or reversed them.
Take-off is hard to date, and later studies (for example by Simon Kuznets) found that investment did not jump as Rostow claimed.
It is based on the history of Britain, western Europe and the United States, which grew in a world without powerful earlier industrialisers to compete with.
It ignores external constraints such as colonial history, falling terms of trade and protection in rich countries.
It says what happens, not how: it does not tell a government which policies bring about the preconditions.
High mass consumption may not be the goal of every society, and it has environmental costs.

Tip

Three models, three messages. Harrod–Domar: save and invest more. Solow: investment runs into diminishing returns; technology is the long-run engine. Rostow: growth comes in stages, with a take-off. For any essay, state each model's message, then its key limitation.

Important

Discuss. Worked example 2 shows that NST2's growth target needs both much more saving and efficient use of capital. Which of the Harrod–Domar model's limitations do you think is the biggest risk for Rwanda, and why?

Klaus Prettner, "Harrod-Domar Growth Model": a university economics lecture on the model. Compare his notation with this lesson's g = s ÷ k; many books write the capital–output ratio as v.

✦

Practice

State four assumptions of the Harrod–Domar model.
Derive the Harrod–Domar growth equation $g = \frac{s}{k}$ .
A country saves 18% of its income and its capital–output ratio is 3. Calculate its growth rate. If its population grows by 2.5%, what is the approximate growth of output per head?
A government wants 8% growth. The capital–output ratio is 3.5 and domestic saving is 20% of GDP. Calculate the savings gap as a percentage of GDP.
Explain why a lower capital–output ratio raises the growth rate.
Explain four limitations of the Harrod–Domar model as a guide to development in Africa.
Describe the take-off stage in Rostow's theory, and give two criticisms of the theory.
"Poor countries grow slowly because they do not save enough." Evaluate this statement, using the Harrod–Domar and Solow models.

Show answersHide answers

Any four: fixed capital–output ratio; a fixed savings ratio with all saving invested; investment adds to the capital stock; labour is plentiful; constant prices, no technical progress, closed economy.
$Y = \frac{K}{k}$ , so $\Delta Y = \frac{\Delta K}{k}$ . Since $\Delta K = I = S = sY$ , $\Delta Y = \frac{sY}{k}$ ; dividing by $Y$ , $g = \frac{s}{k}$ .
$g = \frac{0.18}{3} = 0.06 = 6\%$ ; output per head grows by about $6 - 2.5 = 3.5\%$ .
Required saving $s = g \times k = 0.08 \times 3.5 = 0.28$ , that is 28%. Gap: $28 - 20 = 8\%$ of GDP.
A lower $k$ means each franc of capital produces more output, so a given amount of investment adds more to output: $\Delta Y = \frac{I}{k}$ is larger.
Any four explained: $k$ is not fixed (diminishing returns); saving is not automatically invested; it ignores technology, skills and institutions; skilled labour can be scarce; investment may be wasted; foreign finance brings debt and dependence.
Take-off: a short period in which investment rises from about 5% to over 10% of national income, leading sectors expand fast and institutions support continued growth. Criticisms, any two: stages overlap or are skipped; take-off is hard to date and the investment jump is not observed; based on Western history; ignores external constraints; does not say which policies create the preconditions.
Harrod–Domar supports the statement: low $s$ gives low $g$ , and poor people cannot save much. But saving is not the only constraint: Solow shows diminishing returns to capital and makes technology the long-run source of growth; capital is often used inefficiently (high $k$ ); and skills, institutions, infrastructure and markets matter. Foreign saving can fill part of the gap. Judgement: low saving is one cause, but productivity and institutions usually matter more.

Sources and credits

Growth, investment and saving targets: National Strategy for Transformation (NST2) 2024–2029, abridged version, MINECOFIN, pages 9–10.
Theories summarised from the original works: R. F. Harrod, "An Essay in Dynamic Theory", Economic Journal, 1939; E. D. Domar, "Capital Expansion, Rate of Growth, and Employment", Econometrica, 1946; R. M. Solow, "A Contribution to the Theory of Economic Growth", Quarterly Journal of Economics, 1956; W. W. Rostow, The Stages of Economic Growth: A Non-Communist Manifesto, Cambridge University Press, 1960.
Video: Harrod-Domar Growth Model, Klaus Prettner, embedded from YouTube.

Written by Ganzaa: the text, the Harrod–Domar cycle and the Rostow staircase diagrams, the tables, the worked examples and the practice items. The figures in worked example 1 and the practice items are illustrative.

1Case: how much must a country save?

2The Harrod–Domar model

3Derivation

4Limitations of the Harrod–Domar model

5The neoclassical answer: Solow

6Rostow's five stages

7Critique of Rostow

Case: how much must a country save?

The Harrod–Domar model

Derivation

Limitations of the Harrod–Domar model

The neoclassical answer: Solow

Rostow's five stages

Critique of Rostow