Economic growth · Lesson 4 of 14

The Harrod–Domar model and the knife edge

Q: What is the Harrod–Domar model?

The Harrod–Domar model is an early growth model in which the growth rate of output equals the saving rate divided by the capital–output ratio, g = s/v. It starts from the point that investment creates demand now and capacity later, so steady growth needs demand to keep pace with capacity. It was developed independently by Roy Harrod (1939) and Evsey Domar (1946).

Q: What is the formula for the Harrod–Domar growth rate?

g = s/v, where s is the saving rate and v is the capital–output ratio. With depreciation at rate δ the condition is s/v = g + δ, so g = s/v − δ. The incremental capital–output ratio (ICOR) is ΔK/ΔY, which equals s/g on the warranted path.

Q: What is the ICOR and how do you calculate it?

The incremental capital–output ratio is the extra capital needed to produce an extra unit of output: ΔK/ΔY. Since investment is the change in capital, ICOR = investment ÷ change in output, and when saving equals investment this is the saving (investment) rate divided by the growth rate. A saving rate of 40% and growth of 8% give an ICOR of 5. To grow at g you need an investment rate of ICOR × g.

Q: What is the difference between the warranted, actual and natural rates of growth?

The warranted rate (Gw = s/Cr) is the growth rate at which firms are content with the capital they have, so planned saving and investment match capacity. The actual rate (G) is what the economy achieves in a period. The natural rate (Gn) is the maximum sustainable growth, the growth of the labour force plus the growth of labour productivity. Full employment of both labour and capital needs G = Gw = Gn.

Q: Why is the Harrod–Domar growth path called a knife edge?

Harrod argued that if actual growth ever departs from the warranted rate, the departure feeds on itself. Growth above the warranted rate means capital is scarcer than firms want, so they invest more and growth rises further; growth below it leaves spare capacity, so they invest less and growth falls further. The economy is balanced on an edge, with no force pulling it back. Solow's later model removed this instability.

Q: What are the assumptions of the Harrod–Domar model?

A fixed capital–output ratio (fixed proportions of capital and labour, with no substitution), a constant saving rate, no technical progress in the simplest version, a closed economy without a government, and no depreciation in the basic form. Diminishing returns to inputs is not an assumption: the model has none.

Q: What does it mean if the warranted rate exceeds the natural rate?

Harrod argued that if the warranted rate is above the natural rate, capital accumulates faster than the labour force plus productivity can use, so growth cannot be sustained at the warranted rate and the tendency is towards stagnation and unemployment. If the warranted rate is below the natural rate, growth is held back by saving and investment, and the tendency is towards a boom and a growing pool of unemployed labour.

Keynes explained why spending can fall short of output at a moment in time. Roy Harrod and Evsey Domar asked what that means over many years of growth: if investment both creates demand and adds capacity, what must happen for the two to stay in step? The answer was one line of arithmetic and one uncomfortable conclusion, that steady growth is a knife edge.

← Economic growth moduleLesson 4 of 14 · about 50 minutes

The short answer

What is the Harrod–Domar model?

The Harrod–Domar model says an economy’s growth rate equals its saving rate divided by its capital–output ratio. It is the simplest link between saving and growth, it is still the arithmetic behind questions such as “what investment rate does an 8% target need?”, and its conclusion that the growth path is unstable is the reason the Solow model was built.

g_{w} = \frac{s}{v} ICOR = \frac{Δ K}{Δ Y} = \frac{s}{g} full employment needs G = G_{w} = G_{n}

Growth = saving rate ÷ capital–output ratio. A saving rate of 20% and a capital–output ratio of 4 give 5% a year.
Investment has two sides: it creates demand (through the multiplier) and it adds capacity. Steady growth needs the two to match.
Three rates: actual (G), warranted (G_w = s/v, where firms are content) and natural (G_n, labour force plus productivity growth). They coincide only by luck.
A knife edge: growth above or below the warranted rate feeds on itself. Solow’s model (Lesson 5) removes this by letting capital and labour substitute.

A question to hold on to

Suppose the GDP growth rate is 8% and the rate of saving in the economy is 40%. The incremental capital output ratio (ICOR) is ________.

IIT JAM 2025Solved at the end ↓

New to this? The few things you need first

Saving equals investment: In a closed economy without a government, whatever households do not consume is saved, and in equilibrium saving equals investment: S = I.
The multiplier: A rise in investment ΔI raises income by ΔI ÷ s, because each round of income is partly spent. The multiplier is 1/s.
Capital–output ratio: How much capital it takes to produce one unit of output: v = K/Y. The incremental version, ΔK/ΔY, is the extra capital needed for extra output.

The idea

Investment has two sides

Keynes had stressed one side of investment: it is spending. A rise in investment raises demand, and through the multiplier raises income by $Δ I / s$ . Domar saw the other side: today’s investment is tomorrow’s capital, and new capital can produce more output. Investment is, in his phrase, a “double-edged sword”. It makes demand now and supply later.

Suppose each unit of capital can produce $σ$ units of output (so the capital–output ratio is v = 1/σ). Then investment I adds $σ I$ to the economy’s capacity. And the same investment, through the multiplier, adds $Δ I / s$ to demand when it changes. For demand to keep pace with capacity:

Domar's condition

\frac{Δ I}{s} = σ I ⟹ \frac{Δ I}{I} = s σ = \frac{s}{v}

Investment must grow at the rate s/v for demand to keep absorbing the extra capacity. With a constant saving rate, output then grows at s/v too.

Harrod reached the same number from the other direction. Let firms require a certain amount of capital per unit of output, the required capital–output ratio $C_{r}$ . If saving is a fraction s of income and all of it is invested, the growth rate that leaves firms with exactly the capital they want is the warranted rate:

The Harrod–Domar equation

g_{w} = \frac{s}{v}

s: saving rate. v (or C_r): the capital–output ratio. With depreciation at rate δ: s/v = g + δ.

In words

An economy that saves a bigger share of income, or needs less capital per unit of output, can grow faster. And it has to grow at exactly s/v for demand and capacity to stay in balance.

The arithmetic

The arithmetic: g = s/v and the ICOR

Almost every numerical question on this model is a rearrangement of g = s/v. The incremental capital–output ratio is the capital that must be added to get one more unit of output, ΔK/ΔY. Since investment is the change in capital, ICOR = I/ΔY, and with saving equal to investment:

Three forms of one equation

g = \frac{s}{v} v = \frac{s}{g} s = g v

Use whichever has the unknown on its own: growth rate, ICOR, or the investment rate a target needs.

Given	Find	Working	Answer
s = 40%, g = 8%	ICOR	40 ÷ 8	5
Investment ratio 31.2%, g = 6%	Capital–output ratio	31.2 ÷ 6	5.2
s = 20%, g = 10%, δ = 0	Capital–output ratio	0.20 ÷ 0.10	2
s = 36%, ICOR = 4	Growth rate	36 ÷ 4	9%
s = 20%, ICOR = 5, GDP = 1,000	GDP next year	g = 4%, 1,000 × 1.04	1,040
Consumption share 0.72, ICOR = 4, population +3%	Growth of income per head	s = 0.28, g = 7%, 7 − 3	4%
Capital–output ratio 4, investment 10% of income	Growth rate	10 ÷ 4	2.5%

The rows come from IIT JAM 2025, CUET-PG 2023, GATE 2025, Hyderabad 2021, JNU 2016, Hyderabad 2011 and Hyderabad 2013. Notice the last-but-one: the model gives growth of total output, so for growth per head you subtract population growth, as in Lesson 1. When capital depreciates at rate δ, part of saving only replaces worn-out capital, and the condition becomes s/v = g + δ.

Capital widening and deepening

Investment does two jobs. Capital widening is the part of investment needed to equip new workers with the same capital per worker as before (and to replace depreciation). Capital deepening is investment beyond that, which raises capital per worker. CUET-PG 2020 asks it directly: capital widening is the part of investment needed to equip new units of effective workers at the same capital per unit of effective labour. In the Harrod–Domar model with a fixed capital–output ratio there is only widening; deepening matters from the Solow model on.

Pause and predict

An economy wants 6% growth with a capital–output ratio of 5. What share of income must it invest?

Interactive

Warranted, actual and natural: the knife edge

Harrod distinguished three growth rates, and the model’s drama is that nothing makes them equal:

Rate	Symbol	What it is
Actual	$G = s / C$	What the economy achieves: saving divided by the capital actually added per unit of extra output
Warranted	$G_{w} = s / C_{r}$	The rate at which firms find they have exactly the capital they need, so they are content to go on investing at the same pace
Natural	$G_{n} = n + a$	The most the economy can grow without running out of workers: labour-force growth n plus growth of labour productivity a

Why the warranted path is a knife edge

Suppose actual growth G exceeds the warranted rate. Then the capital actually added per unit of extra output, C, is below what firms require, C_r: they are short of capital. Their response is to invest more, which raises growth further, which makes the shortage worse. If G is below G_w the reverse holds: spare capacity, cuts in investment, and still slower growth. Nothing pulls the economy back to the warranted path, so a small departure grows into a boom or a slump. This is Harrod’s instability principle. The lab below runs a simple version of that reaction; the response rule is a stylised one chosen to show the logic, not Harrod’s own equations.

Growth above the warranted rate means output outruns capacity. Firms read it as a shortage of capital and invest more, which pushes growth higher still: the path runs away upward. The line leaves the chart: in a real economy the labour force or a depression would stop it long before.

Warranted growth is s/v = 20% ÷ 4 = 5%; natural growth is 3%. Warranted exceeds natural: capital piles up faster than labour can use it, so Harrod expects chronic depression and unemployment.

Warranted against natural

Even a perfectly balanced economy can fail on a second front. For capital and labour to be fully employed together you need G = G_w = G_n, a coincidence of three independently determined numbers.

G_w > G_n: capital piles up faster than workers and productivity can use it. Growth cannot stay at the warranted rate, and Harrod expected chronic depression and unemployed capital (Hyderabad 2020’s key).
G_w < G_n: saving and investment limit growth, and a pool of unemployed labour builds up: a tendency towards a boom, with inflationary pressure.

JNU 2010 asks it with numbers: a saving propensity of 14%, an ICOR of 4, population growth of 3%, constant returns and no technical progress. G_w = 14 ÷ 4 = 3.5% and G_n = 3%, so the warranted rate exceeds the natural rate. And UGC NET 2025 asks which statement is wrong, with “actual growth can exceed the natural rate in the long run” being the wrong one, since the labour force sets the ceiling.

Pause and predict

Harrod's warranted growth rate is above the natural rate. The model's prediction is

History

Harrod and Domar: what each added

The two models are often taught as one, but they were written independently and differ in emphasis. Both reach g = s/v, both depend on a fixed capital–output ratio, and both conclude that steady growth requires a particular growth rate to be hit and held. The differences are what exam questions pick on:

	Harrod (1939)	Domar (1946)
Starting point	The growth rate at which firms are satisfied with their capital	The growth rate of investment needed for demand to absorb new capacity
Key mechanism	The accelerator: investment depends on expected output growth, with capital requirement C_r	The multiplier against the capacity effect; no accelerator
Distinctive idea	Three rates, and the instability principle	Investment as a double-edged sword

That is why UGC NET 2026 asks which assumption is not common to both: the principle of the accelerator belongs to Harrod’s model, and Domar’s does not use it. Both accept the Keynesian condition that saving equals investment, and the knife-edge conclusion is in both.

Joan Robinson used the term golden age for a path of steady growth with full employment, and her point was that it is mythical: nothing in an economy guarantees it will be found (UGC NET 2023). The term is the subject of Lesson 12.

One more thing to keep straight: Domar’s name is also on a condition from public finance, the Domar condition for debt sustainability, which says a government’s debt ratio is kept in check when the interest rate it pays is below the economy’s growth rate. It is a different result from his growth model and appears in different parts of the syllabus.

Critique

Assumptions, limits and the road to Solow

The model rests on assumptions that its critics, and its successors, went after:

Fixed proportions. Capital and labour are used in fixed ratio: no substitution, so the capital–output ratio v is a constant, not something the economy can adjust. JNU asks which is not an assumption: “diminishing returns to input”. There are none.
A constant saving rate, and no technical progress, in the basic version.
No price or interest adjustments. If capital became scarce, its price would rise and firms would economise on it. The model has no mechanism for that.

The decisive criticism was the knife edge itself. If the economy were really unstable, we should see booms and slumps growing without limit, not decades of steady growth. In 1956 Solow and Swan asked what changes if capital and labour can substitute, so that the capital–output ratio adjusts. The answer is a stable steady state, which is the next lesson.

The two models also disagree about saving. In Harrod–Domar, a higher saving rate raises the growth rate for good: capital accumulation is the engine of long-run growth. In Solow it raises the level of output but not long-run growth, which depends on technical progress (JNU 2016 tests exactly this contrast).

Watch out for

The mistakes examiners are fishing for

ICOR is incremental. It is ΔK/ΔY, the capital for extra output, not the total capital stock over total output, though the two are equal when v is constant.
The model gives growth of total output. For income per head subtract population growth.
Calling diminishing returns an assumption. The model has fixed proportions and no diminishing returns.
Believing the knife edge means growth is impossible. It means it is unstable if disturbed. The warranted path can be followed, but nothing keeps the economy on it.
Mixing the two Domars. The growth model and the debt-sustainability condition are different results.

Solving the opening question

The opening question, solved

IIT JAM 2025 · standard

GDP growth is 8% and the saving rate is 40%. The ICOR is…

Write the equation.
$g = \frac{s}{v}$
Solve for v, the ICOR.
$v = \frac{s}{g} = \frac{0.40}{0.08}$
Compute.
v = 5.

Answer

5. Each extra rupee of annual output needs five rupees of capital. Check: s ÷ v = 0.40 ÷ 5 = 8%.

Revise

The lesson on one screen

Core equation: g_w = s/v. With depreciation: s/v = g + δ.
ICOR: ΔK/ΔY = s/g. Investment rate needed for growth g: s = g·v.
Two sides of investment: Demand (the multiplier, 1/s) and capacity (adds σI). Domar: investment must grow at s/v.
Three rates: Actual G; warranted G_w = s/C_r; natural G_n = n + a. Full employment needs all three equal.
Instability: G > G_w feeds on itself upward; G < G_w downward: a knife edge.
G_w against G_n: G_w > G_n: stagnation and unemployment. G_w < G_n: boom and a labour surplus.
Assumptions: Fixed proportions, constant saving rate, no diminishing returns, no technical progress (basic version).

Practice

Check yourself

0 of 9 correct so far

A. Numbers

Problem 1

A country saves 24% of income and has a capital–output ratio of 3. Find the warranted growth rate, in per cent.

g (%)

Problem 2

A government targets growth of 7% a year. The ICOR is 4. What share of income must be invested, in per cent?

Investment rate (%)

Problem 3

s = 30%, v = 3, depreciation δ = 4% of capital. Find the growth rate, in per cent.

g (%)

Problem 4

s = 20%, v = 4, labour force growth 2% and productivity growth 2%. Find G_w and G_n, in per cent. (Then say which is larger.)

Warranted (%)Natural (%)

B. Concepts

1Which is NOT an assumption of the Harrod–Domar model?

2Actual growth is above the warranted rate. In Harrod's argument, firms

3The natural rate of growth is

4In the Harrod–Domar model a rise in the saving rate

Real questions

Past papers, by level

The model has been a favourite with Hyderabad, JNU and UGC NET for years: assumptions, the equation, ICOR, and warranted against natural.

Foundation

Definitions, one-step arithmetic and which-model questions. Most CUET and Hyderabad questions sit here.

9 questionsPractise →

Standard

Growth accounting, steady-state algebra, statement questions and matching, up to JNU, IIT JAM and UGC NET level.

15 questionsPractise →

All 24 past-paper questions for this lesson, easiest first: practise the full set →

These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

Quick answers

Frequently asked questions

What is the Harrod–Domar model?

The Harrod–Domar model is an early growth model in which the growth rate of output equals the saving rate divided by the capital–output ratio, g = s/v. It starts from the point that investment creates demand now and capacity later, so steady growth needs demand to keep pace with capacity. It was developed independently by Roy Harrod (1939) and Evsey Domar (1946).

What is the formula for the Harrod–Domar growth rate?

g = s/v, where s is the saving rate and v is the capital–output ratio. With depreciation at rate δ the condition is s/v = g + δ, so g = s/v − δ. The incremental capital–output ratio (ICOR) is ΔK/ΔY, which equals s/g on the warranted path.

What is the ICOR and how do you calculate it?

The incremental capital–output ratio is the extra capital needed to produce an extra unit of output: ΔK/ΔY. Since investment is the change in capital, ICOR = investment ÷ change in output, and when saving equals investment this is the saving (investment) rate divided by the growth rate. A saving rate of 40% and growth of 8% give an ICOR of 5. To grow at g you need an investment rate of ICOR × g.

What is the difference between the warranted, actual and natural rates of growth?

The warranted rate (Gw = s/Cr) is the growth rate at which firms are content with the capital they have, so planned saving and investment match capacity. The actual rate (G) is what the economy achieves in a period. The natural rate (Gn) is the maximum sustainable growth, the growth of the labour force plus the growth of labour productivity. Full employment of both labour and capital needs G = Gw = Gn.

Why is the Harrod–Domar growth path called a knife edge?

Harrod argued that if actual growth ever departs from the warranted rate, the departure feeds on itself. Growth above the warranted rate means capital is scarcer than firms want, so they invest more and growth rises further; growth below it leaves spare capacity, so they invest less and growth falls further. The economy is balanced on an edge, with no force pulling it back. Solow's later model removed this instability.

What are the assumptions of the Harrod–Domar model?

A fixed capital–output ratio (fixed proportions of capital and labour, with no substitution), a constant saving rate, no technical progress in the simplest version, a closed economy without a government, and no depreciation in the basic form. Diminishing returns to inputs is not an assumption: the model has none.

What does it mean if the warranted rate exceeds the natural rate?

Harrod argued that if the warranted rate is above the natural rate, capital accumulates faster than the labour force plus productivity can use, so growth cannot be sustained at the warranted rate and the tendency is towards stagnation and unemployment. If the warranted rate is below the natural rate, growth is held back by saving and investment, and the tendency is towards a boom and a growing pool of unemployed labour.

← Lesson 3Growth accounting and the Solow residual Lesson 5 →The Solow–Swan model: capital accumulation and the steady state

The Harrod–Domar model and the knife edge

Investment has two sides

The arithmetic: g = s/v and the ICOR

Capital widening and deepening

Warranted, actual and natural: the knife edge

Why the warranted path is a knife edge

Warranted against natural

Harrod and Domar: what each added

Assumptions, limits and the road to Solow

The mistakes examiners are fishing for

The opening question, solved

The lesson on one screen

Check yourself

A. Numbers

B. Concepts

Past papers, by level

Foundation

Standard

Frequently asked questions

Read next

The Harrod–Domar model and the knife edge

Investment has two sides

The arithmetic: g = s/v and the ICOR

Capital widening and deepening

Warranted, actual and natural: the knife edge

Why the warranted path is a knife edge

Warranted against natural

Harrod and Domar: what each added

Assumptions, limits and the road to Solow

The mistakes examiners are fishing for

The opening question, solved

The lesson on one screen

Check yourself

A. Numbers

B. Concepts

Past papers, by level

Foundation

Standard

Frequently asked questions

Read next