Macroeconomics: An Overview of Long-Run Economic Growth

Introduction to Economic Growth and Global Welfare

• Current Global Realities and Challenges: A specific set of harsh living conditions is used to frame the necessity of economic growth. Characteristics of a developing nation might include:

  • Life expectancy of less than $50$ years.

  • Infant mortality rate where $1$ in $10$ infants dies before their first birthday.

  • Lack of basic infrastructure: More than $90\%$ of households have no electricity, refrigerator, telephone, or car.

  • Educational attainment: Fewer than $10\%$ of adults have completed high school.

  • Options presented for this specific scenario: Yemen, Ethiopia, Haiti, or "None of the above."

• Transformation of the United States: The U.S. serves as a primary example of how sustained growth improves welfare:

  • Life Expectancy: In 1900, life expectancy was approximately $50$ years; today, it is approximately $78$ years.

  • Medical Advancements: Nathan Rothschild, the mid-1800s great European financier and then-richest man in the world, died from an infection that could be cured today by antibiotics costing only $\$10$.

Motivation and the Importance of Small Growth Differences

• Economic Growth as a Driver of Living Standards: The study of growth explains:

  • Differences between countries (e.g., why per capita GDP is higher in the U.S. than in Bangladesh).

  • Differences over time (e.g., why standards of living in the U.S. increase despite periodic recessions).

  • Crucial Insight: High-GDP countries maintain higher standards of living even during recessions compared to low-GDP countries; growth is not a short-term result.

• Small Rate Differences (Hypothetical GDP Scenario): Consider a 100-year period where Country A and B both start at $\$100$ billion GDP.

  • Country A: Grows at $1\%$ annually. After 100 years, GDP reaches approximately $\$270$ billion.

  • Country B: Grows at $0.25\%$ annually. After 100 years, GDP reaches approximately $\$130$ billion.

  • Conclusion: A seemingly small $1\%$ change in annual growth leads to massive differences in total output and welfare over time.

Growth Over the Very Long Run: The Great Divergence

• Historical Timeline: Sustained increases in living standards are a modern phenomenon.

  • First Agricultural Revolution: Approximately $10,000$ years ago.

  • Modern Economic Growth: Only appeared in the last $200$ to $300$ years.

• Global Inequality and Per Capita GDP: The varying times at which growth emerged led to the "Great Divergence":

  • Some countries have per capita GDP close to zero.

  • Others have one-fifteenth of U.S. GDP.

  • Others have one-third, or less, of U.S. GDP.

  • Wealthy nations have three-fourths or more of U.S. GDP.

• Case Study: Great Britain (1200–2000): Historically, English real wages and population were stagnant for centuries. Significant rising trends in the real wage index (normalized to $1913 = 100$) and population only began around the Industrial Revolution (late 1700s to 1800s).

Defining and Calculating Economic Growth

• Definition: Economic growth is the exact rate of change of per capita GDP.

• The Percentage Change Formula: To calculate the percentage change in GDP between period $t$ and $t+1$:     $g = \frac{y_{t+1} - y_t}{y_t}$     where $y_t$ is the value in the initial period.

• Relation to Next Period Value:     $y_{t+1} = y_t(1 + g)$     (Where $\bar{g}$ denotes a constant growth rate).

• Numerical Examples (UK Real GDP Per Capita):

  • Between 2014 ($y_{2014} = 37411$) and 2015 ($y_{2015} = 37936$):         $g = \frac{37936 - 37411}{37411} = 0.014$

  • Between 2019 ($y_{2019} = 40123$) and 2020 ($y_{2020} = 36031$):         $g = \frac{36031 - 40123}{40123} = -0.102$

The Constant Growth Rule and Method of Forward Iteration

• Evolution of Variables: Tomorrow's value (population or GDP) depends on today's value multiplied by the growth factor $(1 + g)$.

• Forward Iteration Process:

  1. $L_1 = L_0(1 + \bar{n})$

  2. $L_2 = L_1(1 + \bar{n})$

  3. Substituting step 1 into step 2: $L_2 = [L_0(1 + \bar{n})](1 + \bar{n}) = L_0(1 + \bar{n})^2$

• The Constant Growth Rule Formula:     $y_t = y_0(1 + \bar{g})^t$

  • $t$: Time period.

  • $y_t$: Value at time $t$.

  • $y_0$: Initial value at period $0$.

  • $\bar{g}$: Constant growth rate.

• Application (World Population Projection):

  • If $L_0 = 6$ billion and $\bar{n} = 0.02$ (2% growth):

  • $L_{100} = 6 \times (1 + 0.02)^{100} \approx 6 \times 7.24 \approx 43.5$ billion.

  • Caveat: Demographers expect population growth rates to fall toward zero or become negative in the coming century; thus, constant growth rates over 100 years are used as a tool for understanding, not literal prediction.

The Rule of 70

• Definition: The Rule of 70 is used to estimate how many years it takes for a variable growing at a rate of $g\%$ to double.     $\text{Years to double} \approx \frac{70}{g}$

• Key Principles:

  • Doubling time depends only on the growth rate, not the initial value.

  • If $g = 1\%$, doubling takes $70$ years.

  • If $g = 2\%$, doubling takes $35$ years.

• Mathematical Derivation:     $y_t = 2 \times y_0$     $y_0(1 + \bar{g})^t = 2 \times y_0$     $(1 + \bar{g})^t = 2$     $t \times \ln(1 + \bar{g}) = \ln(2)$     $t \times \bar{g} \approx \ln(2) \approx 0.7$     $t = \frac{70}{100 \times \bar{g}}$

Graphical Representation: Standard vs. Ratio Scales

• The Ratio (Logarithmic) Scale: A plot where equally spaced tick marks on the vertical axis represent a constant ratio (e.g., doubling: $1, 2, 4, 8$ or $6, 12, 24, 48$).

  • Growth Interpretation: A variable growing at a constant rate appears as a straight line on a ratio scale.

  • Slope Interpretation:

    • Straight line: Constant growth rate.

    • Increasing slope: Increasing growth rate.

    • Decreasing slope: Decreasing growth rate.

• US and UK GDP Examples:

  • US per capita GDP has grown at approximately $2\%$ annually over the past 150 years ($1880$-$2022$).

  • UK per capita GDP exhibits similar long-run linearity on a ratio scale, showing growth rates around $1.0\%$ to $2.0\%$ depending on the era.

Calculating Yielded Growth Rates Over Time

• To find the average growth rate between two distant time points ($y_0$ and $y_t$):     $y_t = y_0(1 + \bar{g})^t$     $\frac{y_t}{y_0} = (1 + \bar{g})^t$     $(1 + \bar{g}) = \left(\frac{y_t}{y_0}\right)^{\frac{1}{t}}$     $\bar{g} = \left(\frac{y_t}{y_0}\right)^{\frac{1}{t}} - 1$

• United States Example (1870–2022):

  • $y_{1870} = \$4,200$

  • $y_{2022} = \$76,000$

  • $t = 152$ years.

  • $\bar{g} = \left(\frac{76000}{4200}\right)^{\frac{1}{152}} - 1 \approx (18.095)^{0.00657} - 1 \approx 1.0192 - 1 \approx 0.02$ (or $2\%$).

Global Trends: Modern Growth and Convergence

• Post-WWII Acceleration: Germany and Japan saw rapid growth acceleration after World War II, which eventually slowed as they approached higher income levels.

• Convergence: The principle that poorer countries grow faster to "catch up" to the income levels of richer countries.

  • Argentina: Showed accelerated growth until 1980, then slowed significantly.

  • China and India: Showed the reverse; slow initial growth followed by significant acceleration. Because these two nations account for $40\%$ of the world population, their growth has significantly reduced the global fraction of people living in poverty.

• Global Distribution (1960–2019):

  • Most countries grow at roughly $2\%$.

  • Some countries (e.g., Ethiopia, Botswana, South Korea, China) have sustained growth near $7\%$.

  • Some countries (e.g., D.R. Congo, Central African Republic) have exhibited negative growth rates.

Mathematical Properties of Growth Rates

• Growth rates follow several approximation rules that simplify operations:

  • Product Rule: If $z = x \times y$, then $g_z \approx g_x + g_y$.

  • Ratio Rule: If $z = \frac{x}{y}$, then $g_z \approx g_x - g_y$.

  • Power Rule: If $z = x^a$, then $g_z \approx a \times g_x$.

• Justification for Product Rule Approximation:     $1 + g_z = (1 + g_x)(1 + g_y) = 1 + g_x + g_y + (g_x \times g_y)$     Because $g_x \times g_y$ is extremely small (e.g., $0.02 \times 0.02 = 0.0004$), it can be ignored.

• Application: Cobb-Douglas Production Function:

  • Original: $Y = KA^{\alpha}L^{1-\alpha}$

  • Growth rate form: $g_Y = g_K + \alpha \times g_A + (1 - \alpha) \times g_L$

The Benefits and Costs of Economic Growth

• Benefits:

  • Improvements in health and life expectancy.

  • Higher incomes and consumption power.

  • Increased variety and quality of goods and services.

• Costs:

  • Environmental degradation.

  • Increased income inequality both within and across countries.

  • Economic disruption and the loss of specific job sectors.

• Consensus: Most economists agree that the benefits of growth outweigh the costs.

Long-Run Roadmap for Future Lectures

• Lecture 3: Exploiting the Production Model to explain income level differences.

• Lecture 4: The Solow Growth Model.

• Lecture 5: Knowledge as the primary driver of growth.

• Lecture 6: Labor markets, wages, and unemployment in the long run.

• Lecture 7: Determinants of long-run inflation.