Solow Model – Notes Part 1: The Solow Growth Model without Technological Change

Solow Model – Notes Part 1

Introduction to the Solow Growth Model

The Solow Growth Model is a foundational framework in macroeconomics for understanding long-run economic growth, capital accumulation, and the role of savings and population growth. This set of notes focuses on the Solow model without technological change, analyzing how economies reach a steady state in terms of capital per worker and output per worker.

Capital Accumulation Equation

Defining the Capital Accumulation Equation

In the Solow model, the capital accumulation equation describes how the stock of capital per worker evolves over time. The equation is:

• Capital accumulation equation:

$\Delta k = s f(k) - (\delta + n)k$

• Variables:

  • $k$ = capital per worker

  • $s$ = saving rate

  • $f(k)$ = output per worker (production function)

  • $\delta$ = depreciation rate

  • $n$ = population growth rate

• Interpretation: The change in capital per worker depends on new investment (savings) and the loss of capital due to depreciation and population growth.

• Steady State: The steady state is reached when $\Delta k = 0$, meaning capital per worker remains constant over time.

Example: If the saving rate increases, more output is devoted to investment, raising the steady-state level of capital per worker.

Investment and Depreciation

Investment Function and the Steady State

Investment per worker is a fixed proportion of output per worker, given by $s f(k)$. Depreciation and population growth together reduce capital per worker at a rate $(\delta + n)k$.

• Key Points:

  • Actual investment is not always linear; it depends on the shape of the production function.

  • At the steady state, investment per worker equals depreciation plus population growth.

  • If capital per worker is below the steady state, investment exceeds depreciation and population growth, so capital per worker rises.

  • If capital per worker is above the steady state, depreciation and population growth exceed investment, so capital per worker falls.

• Example: If a country increases its saving rate, the steady-state level of capital per worker will rise, leading to higher output per worker in the long run.

Growth Rates of Capital and Output per Worker

Deriving the Growth Rate Equations

The growth rate of capital per worker is derived from the capital accumulation equation:

• Growth rate of capital per worker:

$\text{growth rate of capital per worker} = \frac{\Delta k}{k} = \frac{s f(k)}{k} - (\delta + n)$

• Growth rate of output per worker: Since output per worker is a function of capital per worker, its growth rate depends on the growth rate of capital per worker.

• Graphical Representation: The difference between the investment curve and the depreciation curve determines whether capital per worker increases or decreases.

Example: If the depreciation rate increases, the steady-state level of capital per worker will decrease, reducing output per worker.

The Cobb-Douglas Production Function and Steady State

Applying the Cobb-Douglas Function

The Cobb-Douglas production function is commonly used in the Solow model:

$Y = K^{\alpha} L^{1-\alpha}$

• Per Worker Form: Divide both sides by $L$ to get output per worker:

$y = k^{\alpha}$

• Marginal Product of Capital (MPK): The extra output produced by an additional unit of capital per worker.

$MPK = \alpha k^{\alpha - 1}$

• Steady-State Capital per Worker: At steady state, capital per worker is constant, so:

$k^* = \left( \frac{s}{\delta + n} \right)^{\frac{1}{1-\alpha}}$

• Steady-State Output per Worker:

$y^* = \left( \frac{s}{\delta + n} \right)^{\frac{\alpha}{1-\alpha}}$

Example: If the saving rate $s$ increases, both steady-state capital per worker and output per worker increase.

Golden Rule Level of Capital

Maximizing Consumption per Worker

The Golden Rule level of capital is the steady-state level of capital per worker that maximizes consumption per worker.

• Steady-State Consumption per Worker:

$c^* = f(k^*) - (\delta + n)k^*$

• Golden Rule Condition: The Golden Rule is achieved when the marginal product of capital equals the depreciation plus population growth rate:

$MPK = \delta + n$

• Implication: If the economy saves too much, capital per worker is too high and consumption is lower than optimal. If it saves too little, capital per worker is too low and consumption is also suboptimal.

Example: Policymakers can use the Golden Rule to guide optimal saving rates for maximum long-run consumption.

Summary Table: Key Equations in the Solow Model

Concept	Equation (LaTeX)	Description
Capital Accumulation	$\Delta k = s f(k) - (\delta + n)k$	Change in capital per worker
Growth Rate of Capital	$\frac{\Delta k}{k} = \frac{s f(k)}{k} - (\delta + n)$	Growth rate of capital per worker
Cobb-Douglas Output	$y = k^{\alpha}$	Output per worker
Steady-State Capital	$k^* = \left( \frac{s}{\delta + n} \right)^{\frac{1}{1-\alpha}}$	Steady-state capital per worker
Steady-State Output	$y^* = \left( \frac{s}{\delta + n} \right)^{\frac{\alpha}{1-\alpha}}$	Steady-state output per worker
Steady-State Consumption	$c^* = f(k^*) - (\delta + n)k^*$	Steady-state consumption per worker
Golden Rule Condition	$MPK = \delta + n$	Maximizes consumption per worker

Conclusion

The Solow model provides a powerful framework for understanding the determinants of long-run economic growth, the role of savings, and the impact of population growth and depreciation. By analyzing the steady state and the Golden Rule, students can appreciate how policy choices affect the long-run prosperity of an economy.