Macroeconomic Theory I: Growth I – Capital & Population (Solow Model and Golden Rule)

Growth Theory: Why It Matters

Importance of Economic Growth

Economic growth is a central concern in macroeconomics because it directly affects living standards, poverty rates, and social outcomes. Even small changes in the long-run growth rate can have profound effects over time.

• Infant Mortality Rates: Much higher in poorer countries (e.g., 20% in the poorest fifth vs. 0.4% in the richest fifth).

• Poverty: In some countries, a large proportion of the population lives on less than $2/day.

• Famines: One-fourth of the poorest countries have experienced famines in recent decades.

• Social Impact: Poverty is linked to oppression of women and minorities.

• Growth Raises Living Standards: Economic growth reduces poverty and improves quality of life.

Example: Impact of Growth Rates

Annual Growth Rate of Income per Capita	Increase in Standard of Living after 25 years	after 50 years	after 100 years
2.0%	64.0%	169.2%	624.5%
2.5%	85.4%	243.7%	1,081.4%

The Solow-Swan Model

Overview and Historical Context

The Solow-Swan model (or simply, the Solow model) is a foundational framework in growth theory, developed by Robert Solow and Trevor Swan in 1956. It is widely used in policy analysis and serves as a benchmark for more recent growth theories.

• Purpose: To analyze the determinants of long-run economic growth and living standards.

• Key Features: Focuses on capital accumulation, population growth, and saving rates.

Solow Model: Economy in the Long Run

The Solow model examines how the capital stock and labor force evolve over time, and how these changes affect output and consumption.

• Capital (K): Not fixed; grows via investment, shrinks via depreciation.

• Labor (L): Not fixed; grows with population.

• Consumption Function: Simplified for analysis.

• No Government (G) or Taxes (T): Excluded for simplicity, but fiscal policy can still be analyzed.

Production Function

Aggregate and Per Worker Terms

The production function describes how output is generated from inputs of capital and labor.

• Aggregate Form: $Y = F(K, L)$

• Per Worker Output: $y = Y/L$

• Capital per Worker: $k = K/L$

• Constant Returns to Scale: $zY = F(zK, zL)$ for any $z > 0$

• Per Worker Production Function: $y = f(k)$ where $f(k) = F(k, 1)$

Diminishing Marginal Product of Capital (MPK)

The production function typically exhibits diminishing returns to capital:

• Marginal Product of Capital: $MPK = f(k+1) - f(k)$

• As $k$ increases, $MPK$ decreases.

National Income Identity and Consumption

Income and Saving

In a closed economy without government or trade:

• National Income Identity: $Y = C + I$

• Per Worker Terms: $y = c + i$

• Saving Rate ($s$): Fraction of income saved (exogenous parameter).

• Consumption Function: $c = (1-s)y$

• Investment: $i = sy$

Capital Accumulation and Equation of Motion

Investment, Depreciation, and Capital Stock

Capital stock changes over time due to investment and depreciation:

• Depreciation Rate ($heta$): Fraction of capital that wears out each period.

• Change in Capital Stock: $\Delta k = i - \theta k$

• Equation of Motion: $\Delta k = s f(k) - \theta k$

This central equation determines the evolution of capital per worker and, consequently, output and consumption.

Steady State Analysis

Definition and Properties

The steady state is reached when capital per worker remains constant over time:

• Steady State Condition: $\Delta k = 0 \Rightarrow s f(k^*) = \theta k^*$

• At $k^*$, investment just covers depreciation.

• All endogenous variables (output, consumption, investment) depend on $k^*$.

Approaching the Steady State

• If $k < k^*$, investment exceeds depreciation, so $k$ grows toward $k^*$.

• If $k > k^*$, depreciation exceeds investment, so $k$ shrinks toward $k^*$.

Numerical Example: Cobb-Douglas Production Function

Formulation

Suppose the aggregate production function is Cobb-Douglas:

• $Y = K^{\alpha} L^{1-\alpha}$

• Per worker: $y = k^{\alpha}$

• Example parameters: $s = 0.3$, $\theta = 0.1$, initial $k = 4.0$

Use the equation of motion to solve for steady-state values.

The Golden Rule of Capital Accumulation

Definition and Purpose

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

• Consumption per Worker: $c^* = (1-s) f(k^*)$

• Increasing $s$ raises $k^*$ and $y^*$, but reduces $(1-s)$, so there is an optimal $s$.

• Golden Rule Condition: $MPK = \theta$ (marginal product of capital equals depreciation rate)

Finding the Golden Rule Steady State

• Express $c^*$ in terms of $k^*$: $c^* = f(k^*) - \theta k^*$

• Maximize $c^*$ with respect to $k^*$.

• At optimum: $MPK = \theta$

Transition to the Golden Rule

• If the economy has too much capital ($k > k_{GR}$), reducing $s$ increases consumption for all generations.

• If the economy has too little capital ($k < k_{GR}$), increasing $s$ raises future consumption but lowers current consumption.

Population Growth in the Solow Model

Incorporating Population Growth

Population and labor force grow at rate $n$ (exogenous):

• $\Delta L / L = n$

• Break-even investment: $(\theta + n)k$ (investment needed to keep $k$ constant)

• Equation of motion with population growth: $\Delta k = s f(k) - (\theta + n)k$

Impact of Population Growth

• Higher $n$ increases break-even investment, lowering steady-state $k^*$ and $y^*$.

• Prediction: Countries with higher population growth rates have lower capital and income per worker in the long run.

The Golden Rule with Population Growth

Modified Condition

• Golden Rule with population growth: $MPK = \theta + n$

• Maximizes steady-state consumption per worker.

Alternative Perspectives on Population Growth

Malthusian Model

• Predicts population growth will outpace food production, leading to poverty.

• Historical evidence: Living standards have risen despite population growth, due to technological progress.

Kremerian Model

• Suggests population growth can spur economic growth by increasing the number of innovators.

• Empirical evidence: Larger populations have historically experienced faster growth.

Summary of the Solow Growth Model

• Long-run living standards depend positively on the saving rate and negatively on the population growth rate.

• Increasing the saving rate raises output and capital in the long run, but does not affect the steady-state growth rate.

• Achieving the Golden Rule requires policy intervention to adjust the saving rate.

Key Equations

• Production function: $y = f(k)$

• Equation of motion: $\Delta k = s f(k) - (\theta + n)k$

• Steady state: $s f(k^*) = (\theta + n)k^*$

• Golden Rule: $MPK = \theta + n$

Example Application: Use the Solow model to analyze how changes in saving rates or population growth affect a country's long-run standard of living.

Additional info: The notes reference Mankiw's Macroeconomics, 9th edition, Chapter 8, for further reading and context.