What is Generalized Growth Theory?

 

Generalized Growth Causal Model

Generalized Growth Theory is a synthesis of Unified Growth Theory and the Neoclassical Growth Model. Unified Growth Theory is a synthesis of the  Malthusian Growth Model and the Neoclassical Solow-Swan Growth Model. Generalized Growth Theory derives a model from the Kaya Identity (an identity is an equation that is true by definition) that can (1) be test under all conditions where basic data (Population and Production) are available (from Ancient Civilizations to the Present) and (2) can be extended and tested in data rich environments with data generated by Complex Systems. This post derives a Generalized Growth Model starting with the Kaya Identity.

Kaya Identity

Malthusian Model

Unified Growth Model

Generalized Growth Model

Notes

Kalman, R. E. (1982) Dynamic Econometric Models: A System-Theoretic Critique

Wallerstein, Immanuel (1974). The Modern World-System. New York: Academic Press

Annales (2015) Debating the long Duree

Pasdirtz, G.W. (1981) Instability and Late Nineteenth Century German Development UW-Madison Thesis.

Higgins, B. (1968,1959) Principles of Economic Development

Wikipedia Links

• Unified Growth Theory Unified growth theory, founded by Oded Galor, explores the evolution of societies across the entire span of human history. The theory suggests that forces operating in the distant past have shaped the growth process and determined the differential timing of transitions from stagnation to growth across the globe.
• Malthusian Growth Model sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert Malthus, who wrote An Essay on the Principle of Population (1798), one of the earliest and most influential books on population.
• Solow-Swan Growth Model or exogenous growth model is an economic model of long-run economic growth. It attempts to explain long-run economic growth by looking at capital accumulation, labor or population growth, and increases in productivity largely driven by technological progress. At its core, it is an aggregate production function, often specified to be of Cobb–Douglas type, which enables the model "to make contact with microeconomics". The model was developed independently by Robert Solow and Trevor Swan in 1956, and superseded the Keynesian Harrod–Domar model.
• Kaya Identity a mathematical identity that disaggregates the total emission level of the greenhouse gas carbon dioxide so that it can be expressed as the product of four calculable factors: human population, GDP per capita, energy intensity (per unit of GDP), and carbon intensity (emissions per unit of energy consumed). It is a concrete form of the more general I = PAT equation relating factors that determine the level of human impact on climate. Although the terms in the Kaya identity would in theory cancel out, it is useful in practice to calculate emissions in terms of more readily available data, namely population, GDP per capita, energy per unit GDP, and emissions per unit energy. It furthermore highlights the elements of the global economy on which one could act to reduce emissions, notably the energy intensity per unit GDP and the emissions per unit energy.
• Ancient Civilizations The earliest human civilizations developed independently of each other and a cradle of civilization is a location and a culture from which one of them emerged. A civilization is any complex society characterized by the development of a state, social stratification, urbanization, and symbolic systems of communication beyond signed or spoken languages (namely, writing systems and graphic arts).
• Complex Systems a system composed of many components that interact with one another. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as a power grid, transportation or communication systems, complex software and electronic systems, social and economic organizations (like cities), an ecosystem, a living cell, and, ultimately, for some authors, the entire universe.

Examples

Malthusian Model

Solow-Swan Model

Unified Growth Model

Extensions

A list of models that you can run using the R Programming language on my Google site.

• The Limits to Growth  Jay Wright Forrester's World 1 model that started the The Limits to Growth (LTG) movement.  The 1972 report discussed the possibility of exponential economic and population growth with a finite supply of resources, studied by computer simulation. Classical Economists understood "limits to growth" but the idea was lost under Neoclassical Economics in the 20th Century.
• The Malthusian Model A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert Malthus, who wrote An Essay on the Principle of Population (1798), one of the earliest and most influential books on population.
• The Leibenstein Malthusian Model Introduced the concept of Historical Controllers.
• The Leibenstein Generalized Model Generalizing the concept of Historical Controllers.
• The Leibenstein Controller Model Controllers and Control Theory are central concepts in Systems Theory.
• The Adam Smith Model and the Smith Controller.
• The Adam Smith Small Country Model. with no State Variables!
• The Adam Smith Exponential Growth Model
• The Schumpeter Model Cyclical Creative Destruction.
• The Marxian Steady-State Model What I think Marx envisioned as Communism.
• The Marxian Collapse Model What happens in the Capitalist Future
• The Marxian Cyclical Model Capitalist Crises.
• Karl Marx's Philosophy  In 1845, Karl Marx declared: “philosophers have only interpreted the world in various ways; the point is to change it”.
• The David Ricardo Cyclical Model A strong influence on Marx.
• IPCC Kaya Scenarios For Complex Integrated Assessment Models.
• Kaya Growth State Space Models Deriving Generalized Growth Theory.

 

Historical Controller Models

Controllers that can be tested where data is available (not always general across time).

Classic References

These works are available on line:

• Adam Smith Wealth of Nations
• Malthus An Essay on the Principal of Population
• Malthus Principles of Political Economy
• Ricardo Collected Works
• Marx Collected Works
• JS Mill Collected Works
• JM Keynes The Collected Writings
  
• Milton Friedman Major Works
• Robert Solow A Contribution to the Theory of Economic Growth
• Oded Galor Unified Growth Theory

How do we Generalize from Ancient and Historical Societies?

 

We have to first estimate general systems models on whatever data and theory are available. Then we can compare the models.

Societal Complexity and Unstable Growth

 

A System-Theoretic, Causal Interpretation of Unified Growth Theory

 

Notes

• Galor, Oded; Weil, David N. (2000). "Population, Technology, and Growth: From Malthusian Stagnation to the Demographic Transition and Beyond" (PDF). American Economic Review. 90 (4): 806–828. doi:10.1257/aer.90.4.806.
• Unified Growth Theory  Forces operating in the distant past have shaped the growth process and determined the differential timing of transitions from stagnation to growth across the globe. For most of human history, technological progress expanded population size but had little effect on living standards. As technological progress gradually gathered momentum, through the mutually reinforcing interaction between population growth, adaptation, and innovation, the increasing importance of human capital in coping with a rapidly changing technological environment prompted parents to prioritize child quality over quantity, triggering the onset of the demographic transition. The decline in fertility freed the growth process from the counterbalancing effects of population growth and, along with advancements in knowledge and education, laid the foundation for sustained economic expansion and modern prosperity.

The Malthusian Model

 

This Page is UNDER CONSTRUCTION but you can watch the video above until it is complete.

Notes

Malthusian theory is surprisingly accurate as a description of preindustrial population dynamics, but much less accurate as a prediction of the modern world.

Thomas Malthus's central idea was that population tends to grow faster than the food supply, so that population is repeatedly pushed back toward the limits imposed by resources. When population exceeds those limits, mortality rises through famine, disease, war, and other forms of "positive checks."

How well did it work?

Malthusian proposition	Accuracy	Assessment
Population tends to grow rapidly when resources permit	★★★★★	Very strong
Preindustrial societies were constrained by food/resources	★★★★★	Very strong
Population growth tends to reduce living standards when technology is fixed	★★★★★	Strong
High population density produces greater mortality/famine risk	★★★★☆	Generally strong
War and disease can act as population checks	★★★★☆	Strong
Population eventually encounters diminishing returns to land	★★★★★	Strong
Population will continually outrun food production	★★☆☆☆	Too deterministic
Technological progress cannot permanently escape resource constraints	★★☆☆☆	Incorrect in its strong form
Long-run living standards necessarily return to subsistence	★☆☆☆☆	Clearly contradicted by modern experience

The key insight: the Malthusian trap

The theory works particularly well for much of human history before industrialization.

A simplified model is:

Population ↑ → labor/land ratio ↑ → wages ↓ → nutrition ↓ → mortality ↑ → population ↓

But there was an important feedback loop:

Population ↑ → labor supply ↑ → innovation/land intensification ↑ → food production ↑

Malthus recognized technological improvement, but he underestimated how dramatically technology could alter the relationship between population and resources.

For thousands of years, societies repeatedly experienced something resembling:

economic growth → population growth → resource pressure → declining income per person

This is why economic historians often describe the preindustrial world as Malthusian.

Where Malthus was remarkably right

The strongest evidence comes from historical economic data.

Before roughly 1800, increases in productivity often produced more people rather than permanently higher living standards.

For example:

Agricultural productivity ↑
↓
food becomes cheaper / mortality falls
↓
population ↑
↓
land becomes relatively scarce
↓
wages and consumption per person fall back toward subsistence

This helps explain why living standards in many agrarian societies could remain relatively stagnant for centuries despite substantial technological improvements.

There is also a strong empirical relationship between population density, agricultural productivity, fertility and mortality in preindustrial societies.

---

But then something extraordinary happened

The Industrial Revolution broke the classic Malthusian mechanism.

Instead of:

technology → more people

the modern economy increasingly produced:

technology → more output per person

Mechanization, fossil fuels, fertilizers, transportation, sanitation, medicine and eventually digital technology dramatically expanded the effective resource base.

The relationship became approximately:

Technology ↑↑ → productivity ↑↑ → income/person ↑

while eventually:

income ↑ → fertility ↓

This produced the Demographic Transition.

That is the fundamental reason Malthus's prediction failed.

The irony

Malthus assumed something like:

More prosperity → more population → eventual impoverishment.

Modern societies increasingly exhibit:

More prosperity → lower fertility → slower population growth → higher living standards.

That is almost the reverse of the classical Malthusian mechanism.

---

A better way to think about Malthus

I would give Malthus a 7/10 overall, but with an enormous difference between historical periods.

Period	Malthusian theory
Hunter-gatherer societies	8/10
Early agricultural societies	9/10
Ancient civilizations	9/10
Medieval Europe	9/10
Early modern Europe	8/10
1800–1900	5/10
1900–1950	3/10
1950–2000	2/10
2000–2026	2/10

The important qualification is that Malthusian mechanisms have not disappeared. They have been displaced.

We can still encounter resource constraints through:

• climate change
• water scarcity
• soil degradation
• ecological overshoot
• energy constraints
• biodiversity loss
• fisheries depletion
• food-system vulnerability

So the modern question is not really:

"Will population outrun food?"

It is:

"Can technological and institutional innovation expand humanity's effective carrying capacity faster than population and consumption expand?"

That is a much more sophisticated version of the Malthusian problem.

The deepest lesson

Malthus was wrong about the permanent equilibrium, but right about the existence of constraints.

His great mistake was treating the carrying capacity of a society as relatively fixed.

Modern economics and ecological science suggest that carrying capacity is partly endogenous:

Population + technology + institutions + resources + environment → effective carrying capacity

That makes Malthus an important precursor to modern ecological economics, population economics, resource economics and systems dynamics, even though his original prediction of inevitable subsistence-level living standards did not survive the Industrial Revolution.

Malthusian Theory: Historical Accuracy Scorecard

Scale: 0 = essentially wrong; 10 = remarkably accurate.

#	Malthusian proposition	Score	Assessment
1	Population has a natural tendency to grow exponentially	9/10	Very strong. Reproductive potential makes rapid population growth the default when constraints are relaxed.
2	Population growth eventually encounters resource constraints	9/10	Strongly supported by ecological and historical evidence.
3	Food production historically constrained population	9/10	Extremely accurate for agrarian societies.
4	Population pressure reduces wages/living standards	9/10	One of Malthus's strongest insights; particularly evident in preindustrial economies.
5	Diminishing returns to land limit agricultural output	9/10	Strongly supported, although technology can shift the production frontier.
6	Population growth can absorb technological improvements	9/10	Very important. Preindustrial productivity gains often produced population growth rather than permanently higher per-capita income.
7	Famine acts as a population check	8/10	Historically important, although famine became less common with modern food systems.
8	Disease acts as a population check	9/10	Exceptionally important before modern sanitation and medicine.
9	War can act as a population check	6/10	Sometimes true, but war is not primarily determined by population pressure.
10	High fertility is a response to economic conditions	8/10	Broadly supported, although the relationship is more complicated than Malthus assumed.
11	Population tends to return toward subsistence when resources are fixed	9/10	Excellent description of the classical Malthusian equilibrium.
12	Living standards cannot permanently rise much above subsistence	3/10	Major failure after industrialization.
13	Population will ultimately outrun food production	3/10	Too deterministic. Agricultural productivity has repeatedly outpaced population growth.
14	Technological progress cannot escape the Malthusian trap	2/10	Major failure. Industrialization radically changed the relationship.
15	Higher incomes inevitably produce more population	4/10	True in many preindustrial societies; reversed during the demographic transition.
16	Population pressure is the fundamental determinant of human welfare	5/10	Important, but institutions, technology, distribution and politics are equally important.
17	Resource scarcity necessarily produces catastrophe	4/10	Sometimes, but societies can respond through innovation, trade, institutions and substitution.
18	Humanity is ultimately constrained by ecological carrying capacity	8/10	Strong modern insight, although carrying capacity itself can change dramatically.

Overall score: 7.0/10

But that headline number hides something important.

Malthus's score by historical era

Era	Accuracy
10,000–3,000 BCE	8.5/10
Ancient civilizations	9/10
Medieval period	9/10
1500–1800	8.5/10
1800–1900	5.5/10
1900–1950	4/10
1950–2000	2.5/10
2000–2026	3/10

This produces an interesting historical pattern:

Malthus was increasingly accurate the further back we go.

---

The Five Most Successful Malthusian Predictions

1. Population has enormous reproductive potential — 9/10

This is probably his most fundamental insight.

If mortality falls while fertility remains high:

Population → exponential growth

This is basic population dynamics and remains true today.

---

2. Population pressure eventually encounters limits — 9/10

Malthus understood something that remains fundamental to ecology:

No population can grow indefinitely within a finite physical environment.

The mistake is not recognizing limits; it is assuming that the limits are fixed.

---

3. Preindustrial economies were substantially Malthusian — 9/10

This is probably the greatest vindication of Malthus.

For centuries:

Productivity ↑ → population ↑ → land scarcity ↑ → wages ↓

rather than:

Productivity ↑ → permanent improvement in living standards

Economic historians have found substantial evidence for this mechanism.

---

4. Technological improvement can be consumed by population growth — 9/10

This is an especially powerful insight.

Suppose agricultural productivity rises 20%.

In a rapidly growing population:

Food/person may increase only temporarily.

Population subsequently expands, absorbing much of the gain.

This mechanism helps explain why ancient societies could experience technological advances without sustained increases in per-capita consumption.

---

5. Ecological constraints matter — 8/10

This is perhaps the aspect of Malthus that has become more relevant again.

Modern civilization has not abolished resource constraints.

It has shifted them.

Instead of simply:

food → population

we now have:

energy → industry → food → population + consumption → environmental impact

That brings Malthus surprisingly close to modern ecological economics.

---

Where Malthus Was Wrong

The biggest error was not recognizing that technology could change the carrying capacity itself.

His implicit model was approximately:

Population ↑ → resource scarcity ↑ → mortality ↑

Modern history added another variable:

Population ↑ → innovation ↑ → productivity ↑ → carrying capacity ↑

And eventually another:

Income ↑ → education ↑ → fertility ↓

That creates a completely different system.

---

The Great Historical Break

We can represent the difference very simply.

Classical Malthusian system

Population ↑
↓
Land/person ↓
↓
Food/person ↓
↓
Wages ↓
↓
Mortality ↑ / fertility remains high
↓
Population ↓

Modern system

Technology ↑
↓
Productivity ↑
↓
Income ↑
↓
Education/urbanization ↑
↓
Fertility ↓
↓
Population growth ↓
↓
Income/person remains high

This is the Demographic Transition that Malthus did not anticipate.

---

Malthus vs. Marx

This is where your previous Marx scorecard becomes particularly interesting.

Dimension	Malthus	Marx
Population dynamics	9/10	5/10
Resource constraints	9/10	7/10
Technological change	4/10	8/10
Class conflict	2/10	9/10
Capital accumulation	5/10	9/10
Inequality	5/10	9/10
Economic crises	4/10	8/10
Long-run living standards	3/10	5/10
Ecological constraints	8/10	6/10
Demographic transition	2/10	2/10
Political institutions	3/10	8/10
Overall	7/10	~7/10

The striking result is that Malthus and Marx were almost mirror images.

Malthus tended to see the fundamental problem as:

Too many people relative to resources.

Marx tended to see the fundamental problem as:

The organization of production and ownership.

Modern history suggests that both identified something real but each made his mechanism too dominant.

---

A More Modern Malthusian Model

I think the most defensible version of Malthus today would be:

$$\text{Population} + \text{Consumption} > \text{Technology-adjusted ecological capacity}$$

creates increasing resource and environmental pressure.

But:

$$\text{Innovation} + \text{Institutions} + \text{Substitution}$$

can expand that effective capacity.

And:

$$\text{Higher income} \rightarrow \text{lower fertility}$$

can eventually stabilize population.

So the modern Malthusian proposition becomes:

Human societies cannot escape physical constraints; they can, however, continually redefine their effective carrying capacity through technology, institutions, trade and changes in consumption and fertility.

That is considerably more defensible than Malthus's original prediction of inevitable subsistence.

Bottom line

Malthus was wrong about the destination but remarkably right about the mechanism.

He correctly identified the population-resource feedback loop. What he failed to anticipate was that industrialization would introduce two enormous countervailing forces:

technological acceleration and fertility decline.

That is why I would characterize Malthus as one of the most historically successful preindustrial social theorists, but a poor prophet of the industrial and post-industrial world.

And there is a particularly interesting connection to your recent work on World Dynamics and systems theory: the modern Limits to Growth tradition can be viewed as a technologically sophisticated neo-Malthusian model, in which population is only one of several interacting stocks—alongside capital, resources, pollution and food.

Notes