Macroeconomics · A module in fourteen lessons
Economic growth: from Harrod–Domar to endogenous growth
Why are some countries rich, and why do their incomes change at such different speeds? Growth theory is the economist’s answer, built one model at a time, each fixing what the last got wrong. This module starts from the facts, builds every model the exams ask, and ends with the evidence and the history behind each idea. Every lesson can be read on its own, and every one ends on real past-paper questions: 135 so far.
Start at the beginning, or jump to the model you need.All 14 lessons are ready; the full course is about 11 hours. Each lesson opens with a short answer you can read in a minute, then builds the idea with something you can move, and ends with a real exam question solved and a graded set of past papers.
The path
Six parts, fourteen lessons
The order is the order the ideas arrived in, because each model is a response to the one before. Harrod and Domar found that growth could be a knife edge; Solow replaced the knife edge with a steady state; the steady state could not explain why growth goes on, so endogenous growth theory put technical progress inside the model. Read in order and the field is a story. Read singly and each lesson still teaches one model fully.
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- Part 1The facts and the language
What is measured, how growth compounds, how output is produced, and how much of growth the inputs explain.
- 1What is economic growth? The facts, the rule of 70 and Kaldor's factsLevels against rates, why growth compounds, the rule of 70, how the world's economies have actually grown, Kaldor's stylised facts, and growth against development.40 min · 9 past questions
- 2The production function: returns to scale, Cobb–Douglas and technical changeOutput from capital and labour, constant returns and why growth theory assumes them, Inada conditions, the intensive form, factor shares, and Hicks, Harrod and Solow neutral technical change.45 min · 10 past questions
- 3Growth accounting and the Solow residualSplitting growth into capital, labour and productivity, the residual as a measure of what we cannot explain, per-worker growth, and what the East Asian debate showed.40 min · 8 past questions
- Part 2The classics
Harrod and Domar's knife edge, Solow's steady state, the golden rule, and whether poor countries catch up.
- 4The Harrod–Domar model and the knife edgeGrowth as saving over the capital–output ratio, warranted, actual and natural rates, ICOR, capital widening, why the path is a knife edge, and what planners made of it.50 min · 24 past questions
- 5The Solow–Swan model: capital accumulation and the steady stateWhy diminishing returns bring a steady state, the capital accumulation equation, stability on the Solow diagram, and what grows at what rate once the economy settles.55 min · 15 past questions
- 6Solow in motion: saving, population and technologyLevel effects against growth effects, population and depreciation, technical progress, transitions and the speed of convergence.50 min · 3 past questions
- 7The golden rule: how much should an economy save?Consumption at the steady state, the golden-rule saving rate, dynamic inefficiency, and the cost of getting there.40 min · 4 past questions
- 8Convergence: do poor countries catch up?Absolute and conditional convergence, β and σ convergence, Mankiw–Romer–Weil, the Lucas paradox and convergence clubs.45 min · 2 past questions
- Part 3Choosing to save
Where the saving rate comes from: households who plan ahead.
- Part 4Endogenous growth
Why growth continues: constant returns to capital, human capital and ideas.
- 10The AK model and learning by doingWhy growth theory moved on from Solow, constant returns to broad capital, Arrow, Romer 1986, Lucas and human capital.45 min · 6 past questions
- 11Ideas and R&D: the Romer model, creative destruction and JonesNon-rival ideas, labour allocated to research, Schumpeterian growth and the scale-effect problem.50 min · 9 past questions
- Part 5Other traditions and the real world
Growth as distribution and class, and the models against East Asia, India and institutions.
- 12The Cambridge tradition: Kaldor, Robinson and PasinettiGrowth as a story about distribution and class, golden ages, who saves, and the capital controversy.45 min · 9 past questions
- 13Growth in practice: East Asia, India and institutionsThe models meet the evidence: East Asia, India's growth by period, institutions, and what each model says about raising growth.50 min · 1 past questions
- Part 6The hard end
The Delhi School's harder questions on growth.
Where to start
A two-minute check
One question from each lesson that is ready. Your first wrong answer tells you where to begin; get them all and go straight to the cheat sheet.
Question 1 of 14
A country's real GDP grows 6% a year and its population 2%. Real GDP per head grows by about
History
Seventy years of growth theory in one scroll
Each entry is the paper the idea first appeared in, with its year, and links to the lesson that teaches it once that lesson is published. Dates were checked against the papers’ own bibliographic records.
- 1928Frank Ramsey“A mathematical theory of saving”: how much a far-sighted society should save.Lesson 9: Optimal saving
- 1928Charles Cobb and Paul DouglasA production function fitted to US manufacturing, with labour about three-quarters and capital about one-quarter.Lesson 2: Production functions
- 1939Roy Harrod“An essay in dynamic theory”: the warranted rate of growth and its instability.Lesson 4: Harrod–Domar
- 1946Evsey Domar“Capital expansion, rate of growth, and employment”: investment both creates demand and adds capacity.Lesson 4: Harrod–Domar
- 1956Robert Solow and Trevor SwanThe neoclassical growth model: substitutable inputs, a steady state, growth per head from technology.Lesson 5: Solow model
- 1957Robert Solow“Technical change and the aggregate production function”: the residual.Lesson 3: Growth accounting
- 1957, 1961Nicholas KaldorA growth model with distribution, and the stylised facts.Lesson 1: The facts
- 1953P. C. Mahalanobis“Some observations on the process of growth of national income”: the two-sector model behind India's Second Plan.Lesson 12: Cambridge growth
- 1955–56Nicholas Kaldor“Alternative theories of distribution”: saving and investment fix the profit share.Lesson 12: Cambridge growth
- 1958Paul SamuelsonThe consumption-loan model: the first overlapping-generations economy.Lesson 9: Optimal saving
- 1961Edmund Phelps“The golden rule of accumulation”: the saving rate that maximises steady-state consumption.Lesson 7: Golden rule
- 1962Kenneth Arrow“The economic implications of learning by doing.”Lesson 10: AK and learning
- 1962Luigi Pasinetti“Rate of profit and income distribution in relation to the rate of economic growth”: the profit rate is g/s_c.Lesson 12: Cambridge growth
- 1965David Cass and Tjalling Koopmans; Peter DiamondOptimal saving in the Solow model, and overlapping generations.Lesson 9: Optimal saving
- 1986Paul Romer“Increasing returns and long-run growth”: the first modern endogenous growth model.Lesson 10: AK and learning
- 1988Robert Lucas“On the mechanics of economic development”: human capital.Lesson 10: AK and learning
- 1989Andrew Abel, N. Gregory Mankiw, Lawrence Summers and Richard Zeckhauser“Assessing dynamic efficiency”: a test from capital income and investment.Lesson 7: Golden rule
- 1990Paul Romer“Endogenous technological change”: ideas as non-rival goods.Lesson 11: Ideas and R&D
- 1990Robert Lucas“Why doesn't capital flow from rich to poor countries?”: the paradox of capital that stays put.Lesson 8: Convergence
- 1991Gene Grossman and Elhanan Helpman“Quality ladders in the theory of growth.”Lesson 11: Ideas and R&D
- 1992N. Gregory Mankiw, David Romer and David Weil“A contribution to the empirics of economic growth”: the Solow model with human capital against the data.Lesson 8: Convergence
- 1992Philippe Aghion and Peter Howitt“A model of growth through creative destruction.”Lesson 11: Ideas and R&D
- 1995Charles Jones“R&D-based models of economic growth”: ideas get harder to find, so policy changes levels, not the long-run growth rate.Lesson 11: Ideas and R&D
- 1995Alwyn Young“The tyranny of numbers”: East Asian growth read as accumulation.Lesson 13: In practice
- 2001Daron Acemoglu, Simon Johnson and James Robinson“The colonial origins of comparative development”: institutions as the fundamental cause.Lesson 13: In practice
- 2005Dani Rodrik and Arvind Subramanian“From ‘Hindu growth’ to productivity surge”: India's transition dated to about 1980.Lesson 13: In practice
The debates
The great arguments
Growth theory advanced by argument. Six of them, each with the lesson where it is settled, or left open.
Harrod's knife edge against Solow's steady stateFixed proportions make growth unstable; substitutable inputs make it settle. What changed was one assumption.Lesson 5Exogenous against endogenous growthIs technical progress a given, as in Solow, or the result of saving, schooling and research? And can policy change the long-run rate?Lesson 10Do poor countries catch up?Absolute against conditional convergence, and why capital does not flow to the poor.Lesson 8The Cambridge capital controversyCan capital be measured without knowing the rate of profit? Cambridge, England against Cambridge, Massachusetts.Lesson 12East Asia: accumulation or innovation?Perspiration against inspiration: was the miracle mostly more inputs?Lesson 13Institutions against geographyDo rules or climate and location explain who is rich?Lesson 13
Revise
Cheat sheet: every model
The formula, what it is for, and the trap examiners set on it. Each row links to the lesson that teaches it.
| Idea | Formula | Use it for | The trap |
|---|---|---|---|
| Growth rate | How fast real output is rising | Use real GDP. Per head: growth of output minus growth of population, approximately. | |
| Doubling time | How long steady growth takes to double a level | The rate goes in as a percentage: 70 ÷ 3, not 70 ÷ 0.03. | |
| Cobb–Douglas | The workhorse production function; exponents are factor shares | Constant returns only if the exponents sum to one. | |
| Intensive form | Output per worker as a function of capital per worker | Needs constant returns to scale. | |
| Growth accounting | The Solow residual: growth not explained by inputs | Shares weight the growth rates. The residual is a measure of ignorance, not of technology alone. | |
| Harrod–Domar | Warranted growth from saving and the capital–output ratio | The warranted path is a knife edge. Warranted ≠ actual ≠ natural. | |
| Solow steady state | Where capital per worker stops changing | Saving changes the level of income, not its long-run growth rate. | |
| Speed of convergence | How fast the gap to the steady state closes | Independent of the saving rate. About 5% a year in the basic model, about 2% in the data. | |
| Golden rule | The saving rate that maximises steady-state consumption | Above it the economy is dynamically inefficient; below it, reaching it costs consumption now. | |
| β-convergence | Regression slope of growth on initial log income, over T years | Absolute needs identical parameters; conditional needs controls. β is not σ. | |
| Euler equation | Optimal consumption growth in the Ramsey model | The steady state is f′(k) = δ + ρ, not the golden rule f′(k) = n + δ. | |
| AK model | Growth from inside the model, with constant returns to broad capital | No steady state and no convergence; saving changes the growth rate. | |
| Romer and Jones | Growth from the production of ideas | Only at φ = 1 can policy change the long-run growth rate. | |
| Kaldor and Pasinetti | Distribution from saving and growth | Workers' saving does not enter the Pasinetti profit rate. Golden age is not golden rule. | |
| Crude ICOR | Reading a country's growth through Harrod–Domar | A rough ratio: it mixes capital deepening, widening and productivity. | |
| Full depreciation | Delhi School one-period economies | Initial capital does not matter for the steady state. |
Real questions
Practise across the module
JNU22
Hyderabad20
UGC NET19
CUET-PG16
IIT JAM14
GATE8
Questions placed in the lessons so far, by exam.
Hyderabad and UGC NET ask which model says what: assumptions, warranted and natural rates, and the order of the theories. JNU and IIT JAM ask for steady-state algebra and growth accounting. CUET gives short calculations and matching lists. The Delhi School’s harder questions are collected in the last lesson, 35 of them grouped by technique.
Foundation
Definitions, one-step arithmetic and which-model questions. Most CUET and Hyderabad questions sit here.
27 questionsPractise →
Standard
Growth accounting, steady-state algebra, statement questions and matching, up to JNU, IIT JAM and UGC NET level.
73 questionsPractise →
Delhi School level
Every Delhi School entrance question above Easy that we could verify, from the steady state to labour-market growth models, grouped by technique in Lesson 14.
35 questionsPractise →
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 are the main theories of economic growth?
Classical growth theory (Smith, Ricardo, Malthus) stressed capital accumulation, the division of labour and diminishing returns. The Harrod–Domar model (1939, 1946) made growth depend on the saving rate and the capital–output ratio. The Solow–Swan neoclassical model (1956) added substitutable inputs and diminishing returns, so that long-run growth per head comes from technical progress. Endogenous growth theory (Romer, Lucas, Aghion and Howitt, from the mid-1980s) explains technical progress inside the model. The Cambridge school (Kaldor, Joan Robinson, Pasinetti) emphasises distribution between profits and wages.
What is the difference between the Harrod–Domar and Solow models?
Harrod–Domar uses fixed proportions: capital and labour are used in a fixed ratio, so growth is saving divided by the capital–output ratio, and the warranted growth path is unstable, a knife edge. Solow lets capital and labour substitute, with diminishing returns to capital, so the economy converges to a steady state in which capital per worker is constant. In Solow, a higher saving rate raises the level of income but not the long-run growth rate; in Harrod–Domar it raises growth.
What is the Solow residual?
The Solow residual is the part of output growth not accounted for by growth in measured inputs. It is calculated as output growth minus capital's share times capital growth minus labour's share times labour growth, and is usually read as growth in total factor productivity, which includes technical progress and everything else the inputs do not capture. Solow (1957) found it accounted for most of US growth per worker.
What is endogenous growth theory?
Endogenous growth theory explains long-run growth from inside the model instead of assuming technical progress arrives from outside. In the AK model, capital has constant returns so saving raises growth permanently. In Romer's models, ideas are non-rival and researchers choose to produce them, so growth comes from the allocation of resources to research. The key difference from Solow is that policy can change the long-run growth rate.
What does the golden rule saving rate mean?
It is the saving rate that maximises consumption per worker in the steady state. In the Cobb–Douglas Solow model it equals capital's share of income. Saving more than the golden-rule rate makes the economy dynamically inefficient: it could consume more today and in every future period by saving less.
Do poor countries grow faster than rich ones?
The Solow model predicts conditional convergence: an economy grows faster the further it is below its own steady state. Absolute convergence, in which all poor countries catch up, needs identical saving rates, population growth and technology. In data for 76 countries from 1990 to 2024 there is a weak negative relationship between initial income and growth. Controlling for saving, population growth and schooling, the speed of convergence is about 2 per cent a year (Barro; Mankiw, Romer and Weil), much slower than the basic Solow model's 5 per cent.
What is the Ramsey–Cass–Koopmans model?
It is the growth model in which households choose how much to save. An infinitely lived household maximises discounted lifetime utility, which gives the Euler equation ċ/c = (f′(k) − δ − ρ)/θ. Together with the capital accumulation equation it has a single saddle-point steady state, f′(k*) = δ + ρ, the modified golden rule, which lies below the golden rule because households are impatient.
Is each lesson readable without the others?
Yes. Every lesson starts with a short answer to the question its title asks, restates the few things it needs from earlier lessons in a primer, and ends with a quick-answers section. The lessons are in order because later models answer the weaknesses of earlier ones, so reading them in sequence gives the fullest picture.
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