Lesson M18.L03: Steady State, Convergence, and Transitional Dynamics

Module: Economic Growth Part I Level: intermediate Duration: 30 minutes Learning Objective: Explain conditional convergence using the Solow model phase diagram. Data as of: 2024 Provenance: World Bank Open Data | Marginal Revolution University Growth

Explanation

The Solow model makes a clear prediction about the path of capital accumulation: economies converge to their steady state from any initial capital level. The phase diagram is the central tool for understanding this transitional dynamics.

Notation (as in M18.L02): - k = capital per effective worker; k* = steady-state value - k̇ = dk/dt = rate of change of k - sk^α = gross investment per effective worker (savings curve) - (δ+n+g)k = break-even investment (straight line from origin)

Phase Diagram (k̇ equation):

k̇ = sk^α − (δ + n + g)k

Key regions: - If k < k: sk^α > (δ+n+g)k → k̇ > 0 → k is rising (economy accumulating capital) - If k > k: sk^α < (δ+n+g)k → k̇ < 0 → k is falling (economy decumulating toward k) - If k = k: k̇ = 0 → steady state (no change)

The steady state k is globally stable: from any initial k₀ > 0, the economy converges to k.

Speed of convergence: Near the steady state, the convergence rate is approximately:

speed ≈ (1−α)(δ + n + g)

Higher δ, n, g or lower α → faster convergence. For Australia: (1−0.35)(0.08) = 0.65 × 0.08 = 5.2% per year, meaning the gap between k and k* closes at ~5% per year.

Absolute convergence: The prediction that all countries converge to the same per capita income level — regardless of initial conditions. This requires all countries to have identical parameters (s, δ, n, g, α). Empirically, absolute convergence is not strongly supported across all countries.

Conditional convergence: Countries converge to their own steady state, determined by their own (s, δ, n, g, α). Given their structural parameters, poor countries should grow faster than rich ones conditional on those parameters. Mankiw, Romer, and Weil (1992) demonstrated empirically that conditional convergence holds strongly for OECD countries when controlling for saving rates and population growth. The regression:

growth = const − λ × ln(y₀) + controls for s, n

yielded λ ≈ 0.02 (2% per year convergence speed) — consistent with the Solow model prediction.

Asia-Pacific evidence: - South Korea and Taiwan: Both started the 1960s with low capital-per-worker but implemented high saving rates (s ≈ 0.35–0.40) and rapid human capital accumulation. Conditional on their parameters, their k is high — and they have been converging toward it rapidly (~5–7% p.a. growth). This is conditional convergence, not absolute. - Australia: Already close to its own k — slow, steady 1.5–2% growth reflects transitional dynamics near the steady state combined with slow growth in A. - Australian states: Similar parameters (same technology, institutions, policies) → should show absolute convergence. Evidence supports this: poorer states (NT, Tasmania) have shown higher growth rates, closing the gap with NSW/Victoria.

Worked Example

Setup: α = 0.35, s = 0.25, δ+n+g = 0.08, k* = 5.771 (from M18.L02)

Scenario A: Economy starts below steady state — k₀ = 2.0

Step 1 — Check k̇ sign at k₀ = 2:

Investment: s·k₀ᵅ = 0.25 × 2.0^0.35
ln(2.0^0.35) = 0.35 × 0.693 = 0.2426
2.0^0.35 = e^0.2426 = 1.2746
s·y₀ = 0.25 × 1.2746 = 0.3186

Break-even: (δ+n+g)·k₀ = 0.08 × 2.0 = 0.1600

k̇ = 0.3186 − 0.1600 = +0.1586 > 0

Capital is growing at rate 0.1586 units per year. The economy is below k* and accumulating capital rapidly.

Step 2 — Growth rate of k:

(k̇/k) = 0.1586 / 2.0 = 7.93% per year

Rapid growth because the economy is far below k*.

Scenario B: Economy starts above steady state — k₀ = 9.0

Step 1 — Check k̇ at k₀ = 9:

s·k₀ᵅ = 0.25 × 9.0^0.35
ln(9.0^0.35) = 0.35 × 2.197 = 0.7690
9.0^0.35 = e^0.7690 = 2.158
s·y₀ = 0.25 × 2.158 = 0.5394

Break-even: 0.08 × 9.0 = 0.7200

k̇ = 0.5394 − 0.7200 = −0.1806 < 0

Capital is declining. The economy is above k* and disinvesting (net capital per effective worker falls).

Step 2 — Growth rate of k:

(k̇/k) = −0.1806/9.0 = −2.01% per year

Summary:

Initial k₀	k̇	Growth of k	Direction
2.0	+0.159	+7.9%/yr	→ k* (rising)
5.771 (k*)	0	0%	Steady state
9.0	−0.181	−2.0%/yr	→ k* (falling)

Far below k: growth is fast. Near k: growth is slow. Above k: capital declines. This generates the convergence dynamic — faster growth for low-k economies conditional on having the same k.

Common Misconception

Misconception: "The Solow model predicts that poor countries should always grow faster than rich countries (absolute convergence)."

Correction: The Solow model predicts that countries grow faster when further below their own k. A poor country with low s and high n may have a low k — it converges quickly to its (low) steady state, then grows at rate g just like a rich country. There is no prediction that poor countries will catch up with rich ones unless they have the same structural parameters (s, δ, n, g, α). The failure of absolute convergence in sub-Saharan Africa (low s, high n, political instability reducing effective δ) is perfectly consistent with the Solow model: those countries have low k values and are already near them. Convergence is conditional*, not absolute.

Practice Prompts

1. Conceptual: What is the Mankiw-Romer-Weil (1992) finding, and why does it vindicate the Solow model? → Answer: MRW (1992) found that when growth regressions control for saving rates, population growth rates, and human capital (an augmented Solow model), the data exhibit strong conditional convergence — exactly as predicted. Countries with similar structural parameters converge to similar income levels, with poorer (lower-k) countries growing faster. The coefficient on initial income in the augmented growth regression is approximately −0.02, consistent with the Solow model's predicted convergence speed of about 2% per year. This vindicated the Solow model against critics who pointed to the failure of unconditional convergence.

2. Numerical: Using the Solow model with k = 5.771, α = 0.35, s = 0.25, δ+n+g = 0.08, calculate k̇ at k₀ = 4.0. Is the economy below or above steady state? What is the growth rate of k? → Answer:
   s·k₀ᵅ = 0.25 × 4.0^0.35
   ln(4.0^0.35) = 0.35 × ln(4.0) = 0.35 × 1.3863 = 0.4852
   4.0^0.35 = e^0.4852 = 1.6245
   s·y₀ = 0.25 × 1.6245 = 0.4061
   Break-even: 0.08 × 4.0 = 0.3200
   k̇ = 0.4061 − 0.3200 = +0.0861
   k₀ = 4.0 < k = 5.771: economy is below steady state.
   Growth rate of k = 0.0861/4.0 = 2.15% per year
   (Slower than at k₀=2.0 because now closer to k*)

3. Application: Taiwan's real GDP per worker in 1960 was approximately 15% of the US level. By 2023, it had risen to about 85% of the US level. Is this consistent with conditional or absolute convergence? What structural factors drove Taiwan's rapid convergence? → Answer: This is consistent with conditional convergence. Taiwan's structural parameters shifted dramatically: saving rates rose from ~10% to ~35%, education and human capital investment surged, institutions strengthened, and export-led growth raised TFP growth. These changes raised Taiwan's k substantially (high s → high k). Taiwan was far below its new, higher k, driving rapid capital accumulation and GDP growth. This is not absolute convergence (Taiwan converged to its own higher k, which happened to approach the US level) — it reflects the Solow model's prediction that countries with higher s and lower n converge to higher k and y.

Visual — Convergence in the Solow Model

Figure: The left panel shows why economies move toward k: below the steady state, saving exceeds break-even investment; above it, depreciation plus dilution dominates. The right panel shows the implied time paths of capital per effective worker converging toward the same long-run level from either side.*

Further Resources

• 📺 Absolute and Conditional Convergence: Solow Growth Model — Intermediate Macro Lectures (12 min)
• 📺 Conditional Convergence: Limits to Growth in the Solow Model — Marginal Revolution University (10 min)
• 📚 World Bank Open Data — GDP per capita growth — Country-level data for testing convergence claims