Economics 2 - Long-Run Economic Growth Study Guide

Historical Perspective on the World Distribution of Income

• The distribution of income per capita across the globe is characterized by significant regional variations observed in modern times (e.g., year 2000) and through a long-run historical lens.

• A lack of convergence was prominent from 1960 to 2000, where many countries failed to close the gap relative to the United States' GDP per worker.

• The Maddison Project: Economic historians utilize the Maddison project to track data from the year $1$ to $2010$. This data is primarily derived from administrative sources regarding historical taxes levied on populations across Western Europe, the United States ("Western offshoots"), China, Africa, and Latin America.

• Divergence Across Regions (1-2008):     - In the year $1$, the ratio of Rich to Poor regions was approximately $1.4$.     - By the year $2008$, this ratio expanded drastically to $16.9$.     - Specific income levels (Maddison estimates) for selected years:         - Western Offshoots: Year 1: $400$; Year 1000: $400$; Year 1820: $1202$; Year 2008: $30152$.         - Western Europe: Year 1: $576$; Year 1000: $427$; Year 1820: $1194$; Year 2008: $21672$.         - Latin America: Year 1: $400$; Year 1000: $400$; Year 1820: $691$; Year 2008: $6973$.         - Asia: Year 1: $456$; Year 1000: $470$; Year 1820: $581$; Year 2008: $5611$.         - Africa: Year 1: $472$; Year 1000: $425$; Year 1820: $420$; Year 2008: $1780$.

Stylized Facts: From Stagnation to the Great Divergence

• Stylized Fact 1 (Pre-1000 AD): In the year $1000$, European real income per capita was significantly lower than it had been $1000$ years prior in the same region. It was also lower than income in China and India at that time.     - From 400-700 AD: Europe experienced a contraction.     - From 700-1000 AD: Europe entered a long period of stagnation.     - Explanation (Maddison): The downfall of the Roman Empire led to the disaggregation of the global political system into fragmented, unstable administrations. This caused the near-disappearance of trade routes with North Africa and Asia and the replacement of great urban structures with small, self-sufficient rural communities.

• Stylized Fact 2 (Cumulative Growth): A cumulative growth process began around the year $1000$, though it was extremely slow and significantly stronger in Western Europe than elsewhere.

• Stylized Fact 3 (Dynamic Shift within Europe):     - Venezzia Republic (11th-15th century): Dominant economic power.     - Netherlands (1600-1820): Highest GDP per capita.     - United Kingdom (1700-1820): Strongest GDP per capita growth.

• Stylized Fact 4 (The Take-off):     - Around $1750$, the First Industrial Revolution caused a sudden acceleration.     - Around $1820$, a real economic take-off occurred.     - From 1820 to 2000, world average real income per capita was multiplied by $9$ ($6049 / 667 = 9$).     - Belgian GDP per capita was multiplied by $16$, while African GDP per capita was multiplied by only $3.5$.

• Stylized Fact 5 (The Great Divergence): Modern growth has increased world income inequality. The GDP per capita gap between Belgium and Africa grew from a factor of $3$ in $1820$ to a factor of $14$ in $2001$.

Phases of Economic Development and Regimes

• (1) The Malthusian Epoch:     - Timeline: Emergence of homo sapiens to $1750$ for developed countries; to $1900$ for Less Developed Countries (LDCs).     - Mechanism: Technological progress or land expansion leads to a temporary increase in income per capita, which subsequently causes the population size to increase. This population boom offsets the income gains, returning output per capita to a subsistence level in the long run.     - Land-rich or technologically advanced economies in this era display higher population density but similar long-run income levels.

• (2) The Post-Malthusian Regime:     - Timeline: $1750 - 1870$ for developed countries; $1900$ onwards for LDCs.     - Mechanism: Economies take off from Malthusian equilibrium. Population growth is still positively correlated with income, but technological progress accelerates faster than population growth, allowing income per capita to rise.

• (3) The Modern Growth Regime:     - Timeline: $1870$ onwards for developed countries.     - Mechanism: Technological progress accelerates further, but population growth declines due to the Demographic Transition. This allows for a sustainable increase in output per capita.

The Demographic Transition

• Definition: A rapid decline in fertility, mortality, and population growth marking the transition to modern growth.

• Reversal of Relationship: In the Malthusian regime, the relationship between income and population growth was positive; in the Modern Growth regime, it became negative.

• Theories of Demographic Transition:     1. Rise in income: Dismissed empirically.     2. Decline in child mortality: Dismissed empirically.     3. Decline in the gender wage gap: Increases the opportunity cost for women child-rearing; confirmed by data.     4. Human capital formation: Acceleration in technology necessitates higher human capital, leading to a "quantity-quality" trade-off; confirmed by data.

Identifying Potential Growth Factors: The Production Function

• Two categories increase an economy's ability to create income:     1. Production Factors: Productive capital ($K_t$), Labor ($L_t$), commodities, energy.     2. Productivity/Efficiency: Determined by technology ($A_t$) and factor combination efficiency (organization, institutions).

• The Neoclassical Aggregate Production Function:     - $Y_t = F(A_t, K_t, L_t)$     - $Y_t$: Final aggregate output or primary income.     - $K_t$: Productive capital (machines, buildings, infrastructure).     - $L_t$: Labor units (regardless of qualification).     - Technology Shifter ($A_t$): A non-rival, non-excludable shifter. It represents technological innovation, human capital, and organization. It has no natural unit.

• Properties of the Production Function:     - Constant Returns to Scale (CRS): $F(A, xK, xL) = xF(A, K, L)$. This implies firm size does not matter and justifies the aggregate approach.     - Positive and Decreasing Marginal Returns:         - \frac{\partial F}{\partial K} > 0, \frac{\partial F}{\partial L} > 0         - \frac{\partial^2 F}{\partial K^2} < 0, \frac{\partial^2 F}{\partial L^2} < 0

• Per-Capita/Per-Worker Transformation:     - Utilizing homogeneity of degree one: $y_t = \frac{Y_t}{L_t} = F(A_t, \frac{K_t}{L_t}, 1) = f(A_t, k_t)$.     - With exogenous technology, simplified to: $y_t = f(k_t)$.

• Cobb-Douglas Production Function:     - $Y_t = A_t K_t^{\alpha} L_t^{1-\alpha}$ where 0 < \alpha < 1.     - Per-capita form: $y_t = A_t k_t^{\alpha}$.

The Solow-Swan Model: Key Equations and Assumptions

• Environment: Closed economy, no government, discrete time $t = 0, 1, 2, \dots$

• National Accounting: $Y_t = C_t + I_t$.

• Behavioral Rule: Households save a constant fraction $s$ of their income.     - $S_t = sY_t$     - $C_t = (1 - s)Y_t$     - Investment equals savings: $I_t = S_t = sY_t$.

• Law of Motion of Capital:     - Capital depreciates at rate $\delta$.     - $K_{t+1} = (1 - \delta)K_t + I_t$     - Substituting for investment: $K_{t+1} = sF(A_t, K_t, L_t) + (1 - \delta)K_t$.

Long-Run Equilibrium without Population Growth

• If population is constant ($L_t = L$), the per-capita capital stock evolves as follows:     - $k_{t+1} = (1 - \delta)k_t + sf(k_t)$     - Variation in capital: $\Delta k_{t+1} = k_{t+1} - k_t = sf(k_t) - \delta k_t$

• Dynamics:     - If per-capita investment $sf(k_t)$ exceeds capital depreciation $\delta k_t$, $k_t$ increases (Capital accumulation).     - If sf(k_t) < \delta k_t, capital decreases.

• Steady State ($k^*$):     - Occurs when investment exactly compensates depreciation: $sf(k^*) = \delta k^*$.     - The steady state is unique (due to decreasing marginal returns) and stable (the economy converges to $k^*$ regardless of the initial value $k_0$).

• Observations:     - Income per capita increases higher for higher productivity ($A$), higher saving rates ($s$), or lower depreciation rates ($\delta$).     - Investment is necessary for growth (if $s=0$, there is no growth), but decreasing marginal returns make it insufficient to sustain long-run growth of income per capita alone.

Absolute vs. Conditional Convergence

• Absolute Convergence: All countries converge to a common long-run level regardless of initial conditions or structure. Divergence is seen as transitory.

• Conditional Convergence: Countries with similar structural characteristics (savings, depreciation, policy) converge to the same level. Poorer countries in the same structural group will grow faster (catch-up).

• Solow Model Prediction: It features conditional convergence, not absolute convergence. Economies with different structures will converge to different steady states.

Incorporating Population Growth and Technology

• Population Growth ($n$): $L_{t+1} = (1 + n)L_t$.     - Evolution of capital: $k_{t+1} = \frac{1}{1 + n} [(1 - \delta)k_t + sf(k_t)]$     - Change in capital: $\Delta k_{t+1} = sf(k_t) - \frac{(\delta + n)k_t}{1 + n}$     - Dilution Effect: Increased population requires more investment to maintain the capital-to-labor ratio because existing capital must be shared among more people.     - Steady State Effect: Higher demographic growth leads to a lower steady-state income per capita. At steady state, total output ($Y$) and total capital ($K$) grow at rate $n$, but income per capita ($y$) remains constant (stagnates).

• Technological Progress ($A_t$):     - A one-shot increase in $A_t$ shifts curves upward, leading to transitory growth and a higher new steady state ($k^*, y^*$).     - Perpetual Growth: For per-capita income to grow indefinitely, there must be regular, "perpetual" increases in the technology factor $A_t$.

Club Convergence and Poverty Traps

• Club Convergence: Shared structural characteristics lead to convergence only if initial conditions are similar. If countries belong to different "clubs" based on initial distance from a steady state, they will not converge to the same level.

• Poverty Trap: A situation featuring two steady states ($y^*_L$ and $y^*_H$).     - Potential causes: Increasing marginal returns at low income levels, savings rates that depend on income (unable to save at low income), or population growth rates that depend on income (high $n$ at low income).     - Under these conditions, INITIAL CONDITIONS MATTER for the long-run outcome.