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AK Growth Model

Calculates the long-run growth rate of output per capita in an AK model.

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g = A δ - n

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Core idea

Overview

The AK growth model is a fundamental model in endogenous growth theory, explaining sustained economic growth without relying on exogenous technological progress. It posits that the aggregate production function exhibits constant returns to scale to capital, implying that capital accumulation alone can drive long-run growth. This formula determines the per capita growth rate based on the technology level, capital's productivity, and population growth.

When to use: Apply this equation when analyzing long-run economic growth in models where capital accumulation does not face diminishing returns. It is particularly relevant for understanding how policy interventions affecting technology (A) or capital's productivity ($\delta$) can influence sustained growth rates, or how population growth (n) impacts per capita growth.

Why it matters: The AK model is crucial because it provides an endogenous explanation for economic growth, unlike earlier models (e.g., Solow-Swan) that relied on exogenous technological progress. It highlights the importance of human capital, R&D, and infrastructure in fostering sustained development, influencing policy debates on innovation and investment.

Symbols

Variables

A = Technology Level, $δ$ = Capital Share/Productivity, n = Population Growth Rate, g = Growth Rate of Output per Capita

A

Technology Level

dimensionless

δ

Capital Share/Productivity

dimensionless

n

Population Growth Rate

% / year

g

Growth Rate of Output per Capita

% / year

Walkthrough

Derivation

Formula: AK Growth Model

The AK growth model describes the long-run economic growth rate as a function of technology, capital's productivity, and population growth, assuming constant returns to capital.

The aggregate production function is linear in capital: $Y = A K$ , where $Y$ is output, $A$ is a constant technology parameter, and $K$ is capital.
There are no diminishing returns to capital, allowing for sustained growth.
A constant fraction of output is saved and invested: $I = s Y$ , where $s$ is the savings rate.
Capital depreciates at a constant rate $δ_{K}$ .
Population grows at a constant rate $n$ .

Start with Capital Accumulation Equation:

The change in the capital stock ( $\dot{K}$ ) is equal to investment ( $s Y$ ) minus capital depreciation ( $δ_{K} K$ ). Here, $δ_{K}$ is the depreciation rate of capital.

\dot{K} = s Y - δ_{K} K

Substitute the AK Production Function:

Replace $Y$ with $A K$ from the AK production function. This highlights the constant returns to capital.

\dot{K} = s (A K) - δ_{K} K

Express in Terms of Capital Growth Rate:

Divide both sides by $K$ to get the growth rate of the total capital stock. This shows that the capital growth rate is constant.

\frac{K ˙}{K} = s A - δ_{K}

Derive Per Capita Output Growth Rate:

The growth rate of output per capita ( $g$ ) is approximately the growth rate of capital per capita ( $\dot{k} / k$ ), which is the growth rate of total capital ( $\dot{K} / K$ ) minus the population growth rate ( $n$ ).

g = \frac{y ˙}{y} = \frac{k ˙}{k} = \frac{K ˙}{K} - n

Note: In the AK model, output per capita ( $y$ ) and capital per capita ( $k$ ) grow at the same rate because $y = A k$ .

Final AK Growth Rate Formula:

Substitute the capital growth rate into the per capita growth rate equation. For simplicity, the term $s A - δ_{K}$ is often aggregated into a single parameter, say $A_{e f f}$ , or the capital share $δ$ in the prompt's formula is implicitly $s A - δ_{K}$ . If we interpret $A δ$ as the effective return to capital after depreciation, then $g = A δ - n$ is the final form.

g = s A - δ_{K} - n

Result

g = s A - δ_{K} - n

Source: Romer, D. - Advanced Macroeconomics, Chapter 2 (Endogenous Growth Theory)

Free formulas

Rearrangements

Solve for $A$

AK Growth Model: Make A the subject

A = \frac{g + n}{δ}

To make $A$ (Technology Level) the subject of the AK Growth Model formula, add the population growth rate ( $n$ ) to the per capita growth rate ( $g$ ) and then divide by the capital share ( $δ$ ).

Difficulty: 2/5

Solve for $δ$

AK Growth Model: Make $δ$ the subject

δ = \frac{g + n}{A}

To make $δ$ (Capital Share/Productivity) the subject of the AK Growth Model formula, add the population growth rate ( $n$ ) to the per capita growth rate ( $g$ ) and then divide by the technology level ( $A$ ).

Difficulty: 2/5

Solve for $n$

AK Growth Model: Make n the subject

n = A δ - g

To make $n$ (Population Growth Rate) the subject of the AK Growth Model formula, subtract the per capita growth rate ( $g$ ) from the product of the technology level ( $A$ ) and capital share ( $δ$ ).

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a linear function with a positive slope, meaning the growth rate of output per capita increases at a constant rate as the technology level improves. For an economics student, this relationship implies that even small improvements in technology lead to predictable, proportional gains in long-run growth, regardless of whether the initial technology level is high or low. The most important feature of this curve is its constant slope, which demonstrates that the impact of technological progress on economic growth remains uniform across all levels of development.

Graph type: linear

Why it behaves this way

Intuition

A self-reinforcing cycle where investments in capital (broadly defined to include human capital and technology) continuously generate returns without diminishing, propelling per capita output upward.

g

The long-run growth rate of output per capita.

How quickly the average person's economic output or income increases over time.

A

The level of technology or total factor productivity in the economy.

A higher 'A' means the economy can produce more output from a given amount of capital, reflecting better knowledge, innovation, or efficiency.

δ

A parameter representing the productivity of capital or the constant return on capital investment within the AK model.

How effectively each unit of capital contributes to output. A higher '

δ

' means capital is more productive.

n

The constant rate of population growth.

How quickly the number of people in the economy is increasing.

Signs and relationships

A δ: This product represents the contribution of capital accumulation and technology to the aggregate output growth rate. Both higher technology (A) and more productive capital ( $δ$ )
- n: Population growth dilutes the total output across more individuals. To maintain or increase per capita output, the aggregate economy must grow faster than the population; otherwise, per capita output declines.

Free study cues

Insight

Canonical usage

All terms in the equation represent rates and must have consistent units of inverse time (e.g., per year) and be used in decimal form for calculations.

Common confusion

A common mistake is using percentage values directly in the formula instead of converting them to decimal form, or using inconsistent time units (e.g., annual growth rate with quarterly population growth).

Unit systems

$g$ year^-1 - The long-run growth rate of output per capita. Often expressed as a percentage (e.g., 2% per year), but must be converted to decimal form (0.02 per year) for calculations.

$A$ dimensionless - The technology level or total factor productivity parameter. It is typically a dimensionless constant representing the efficiency of capital in generating output.

$δ$ year^-1 - Capital's productivity. This parameter represents a rate at which capital contributes to growth. It must be in consistent time units with 'g' and 'n'.

$n$ year^-1 - The population growth rate. Often expressed as a percentage (e.g., 1% per year), but must be converted to decimal form (0.01 per year) for calculations.

One free problem

Practice Problem

An economy following the AK growth model has a technology level (A) of 0.3, a capital share in production ( $δ$ ) of 0.2, and a population growth rate (n) of 0.01. Calculate the growth rate of output per capita (g).

Technology Level0.3 dimensionless

Capital Share/Productivity0.2 dimensionless

Population Growth Rate0.01 % / year

Solve for: result

Hint: Substitute the values directly into the formula g = A $δ$ - n.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Analyzing how investments in education and R&D (which increase 'A' or '$\delta$') can lead to sustained economic growth in developing nations.

Study smarter

Tips

Ensure 'A' (technology level) and ' $δ$ ' (capital share/productivity) are positive.
A higher 'A' or ' $δ$ ' leads to a higher growth rate.
A higher 'n' (population growth) reduces the per capita growth rate.
The model assumes constant returns to scale to capital, a key departure from neoclassical models.

Avoid these traps

Common Mistakes

Confusing the AK model with the Solow-Swan model, especially regarding returns to capital.
Incorrectly interpreting 'A' as total factor productivity without considering its broader role in endogenous growth.

Common questions

Frequently Asked Questions

The AK growth model describes the long-run economic growth rate as a function of technology, capital's productivity, and population growth, assuming constant returns to capital.

Apply this equation when analyzing long-run economic growth in models where capital accumulation does not face diminishing returns. It is particularly relevant for understanding how policy interventions affecting technology (A) or capital's productivity ($\delta$) can influence sustained growth rates, or how population growth (n) impacts per capita growth.

The AK model is crucial because it provides an endogenous explanation for economic growth, unlike earlier models (e.g., Solow-Swan) that relied on exogenous technological progress. It highlights the importance of human capital, R&D, and infrastructure in fostering sustained development, influencing policy debates on innovation and investment.

Confusing the AK model with the Solow-Swan model, especially regarding returns to capital. Incorrectly interpreting 'A' as total factor productivity without considering its broader role in endogenous growth.

Analyzing how investments in education and R&D (which increase 'A' or '$\delta$') can lead to sustained economic growth in developing nations.

Ensure 'A' (technology level) and '$\delta$' (capital share/productivity) are positive. A higher 'A' or '$\delta$' leads to a higher growth rate. A higher 'n' (population growth) reduces the per capita growth rate. The model assumes constant returns to scale to capital, a key departure from neoclassical models.

References

Sources

Economic Growth by David Romer, 4th Edition, W. W. Norton & Company
Macroeconomics by N. Gregory Mankiw, 10th Edition, Worth Publishers
Wikipedia: AK model
Romer, David. Advanced Macroeconomics. 5th ed. McGraw-Hill, 2018.
Mankiw, N. Gregory. Macroeconomics. 10th ed. Worth Publishers, 2019.
Barro, Robert J. Macroeconomics: A Modern Approach. 2nd ed. South-Western Cengage Learning, 2008.
Romer, D. - Advanced Macroeconomics, Chapter 2 (Endogenous Growth Theory)

AK Growth Model

Overview

Variables

Derivation

Start with Capital Accumulation Equation:

Substitute the AK Production Function:

Express in Terms of Capital Growth Rate:

Derive Per Capita Output Growth Rate:

Final AK Growth Rate Formula:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Walras' Law

Frequently Asked Questions

Sources