Lesson M18.L02: The Solow Model: Full Mathematical Treatment

Module: Economic Growth Part I Level: intermediate Duration: 30 minutes Learning Objective: Derive the steady-state capital per effective worker from the fundamental Solow equation. Data as of: 2024 Provenance: MIT OCW 14.02 | ABS National Accounts

Explanation

The Solow-Swan model (Solow 1956, Swan 1956) is the foundation of modern growth theory. It explains long-run GDP per worker through capital accumulation and exogenous technological progress.

Production Function (aggregate):

Y = F(K, AL) = Kᵅ (AL)^(1−α)

Notation: - Y = real GDP; K = capital stock; L = labour (workers); A = level of technology (labour-augmenting) - AL = "effective labour" (efficiency units of labour) - α ∈ (0,1) = capital elasticity of output = capital income share (by competitive factor pricing); α ≈ 0.35 for Australia - 1−α = labour income share ≈ 0.65 - n = growth rate of L (population/labour force growth); n ≈ 0.015 for Australia - g = growth rate of A (technological progress); g ≈ 0.015 for Australia - δ = depreciation rate of capital; δ ≈ 0.05 for Australia - s = gross saving/investment rate; s ≈ 0.25 for Australia (rounded from ~24%; used throughout examples)

Intensive form (per effective worker): Define:

k = K / (AL)   [capital per effective worker]
y = Y / (AL)   [output per effective worker]

Substituting into the production function:

y = Y/(AL) = Kᵅ(AL)^(1−α) / (AL) = (K/(AL))ᵅ = kᵅ

Capital accumulation equation: Gross investment = sY. Capital per effective worker changes as:

k̇ = dk/dt = s·y − (δ + n + g)·k = s·kᵅ − (δ + n + g)·k

The term (δ + n + g)k is "break-even investment" — the investment needed to: - Replace depreciated capital (δk) - Equip new workers with existing capital per worker (nk) - Maintain k as A grows (gk, since more efficient workers "dilute" existing capital per effective worker)

Steady State: Set k̇ = 0:

s·k*ᵅ = (δ + n + g)·k*
s·k*^(α−1) = (δ + n + g)
k*^(1−α) = s / (δ + n + g)
k* = [s / (δ + n + g)]^(1/(1−α))

Steady-state output per effective worker:

y* = k*ᵅ = [s / (δ + n + g)]^(α/(1−α))

Key comparative statics: - Higher s → higher k and y (but not higher long-run growth rate — only level effect) - Higher δ, n, or g → lower k and y (more break-even investment required) - Higher α → more sensitive k* to changes in s

Worked Example

Australian parameters: s = 0.25, δ = 0.05, n = 0.015, g = 0.015, α = 0.35

Step 1 — Compute break-even investment rate:

δ + n + g = 0.05 + 0.015 + 0.015 = 0.080

Step 2 — Compute s/(δ+n+g):

s / (δ + n + g) = 0.25 / 0.080 = 3.125

Step 3 — Compute the exponent:

1/(1−α) = 1/(1 − 0.35) = 1/0.65 = 1.5385

Step 4 — Compute k*:

k* = 3.125^(1.5385)

Taking logs: ln(k*) = 1.5385 × ln(3.125)
ln(3.125) = 1.1394
ln(k*) = 1.5385 × 1.1394 = 1.7529
k* = e^1.7529 = 5.771

Step 5 — Compute y*:

y* = k*ᵅ = 5.771^0.35

Taking logs: ln(y*) = 0.35 × ln(5.771) = 0.35 × 1.7529 = 0.6135
y* = e^0.6135 = 1.847

Step 6 — Verify steady state (check k̇ = 0):

Investment: s·y* = 0.25 × 1.847 = 0.4618
Break-even: (δ+n+g)·k* = 0.080 × 5.771 = 0.4617  ✓  (small rounding difference)

Interpretation: In steady state, an economy with Australian parameters has k ≈ 5.77 units of capital per effective worker and produces y ≈ 1.85 units of output per effective worker. 25% of this output (≈ 0.46) is invested, exactly replacing depreciation and equipping new effective workers.

Step 7 — Output per actual worker (grows at rate g in steady state):

Y/L = y × A = k*ᵅ × A
Since A grows at rate g = 1.5%, Y/L grows at 1.5% per year in steady state.

This matches Australia's long-run GDP per capita growth of ~1.5–2%.

Common Misconception

Misconception: "A higher saving rate leads to permanently faster economic growth in the Solow model."

Correction: In the Solow model, a higher saving rate raises the level of k and y (a level effect), but not the long-run growth rate. In steady state, output per effective worker y is constant regardless of s. Output per actual worker Y/L grows at rate g (technological progress), independent of s. The saving rate determines where* on the growth path the economy sits, not how fast it travels along it. Permanent growth effects require endogenous growth models (Romer, Lucas) where knowledge/technology is itself produced by saving and investment.

Practice Prompts

1. Conceptual: Why must break-even investment include a term for technological progress (gk) in the capital accumulation equation? → Answer: Capital per effective worker k = K/(AL). As A grows at rate g, the denominator (AL) grows, so k would fall even if K were constant. To maintain k at its current level, investment must also cover this "dilution" effect. More precisely: d(K/AL)/dt = (K̇·AL − K·(Ȧ·L + A·L̇)) / (AL)² = K̇/(AL) − K·(n+g)/(AL) = (K̇/AL) − (n+g)k. Setting this equal to sk^α − (n+g)k gives the full equation. The gk term represents the investment needed to equip each unit of new technological capability with the existing capital-per-effective-worker ratio.

2. Numerical: An economy has s = 0.30, δ = 0.06, n = 0.02, g = 0.02, α = 0.40. Derive k and y. → Answer:
   δ + n + g = 0.06 + 0.02 + 0.02 = 0.10
   s/(δ+n+g) = 0.30 / 0.10 = 3.0
   1/(1−α) = 1/(1−0.4) = 1/0.6 = 1.6667
   k = 3.0^(1.6667)
   ln(k) = 1.6667 × ln(3.0) = 1.6667 × 1.0986 = 1.8310
   k* = e^1.8310 = 6.238
   y = k^0.4 = 6.238^0.4
   ln(y) = 0.4 × 1.8310 = 0.7324
   y = e^0.7324 = 2.080
   Verify: s·y = 0.30 × 2.080 = 0.624; (δ+n+g)·k = 0.10 × 6.238 = 0.624 ✓

3. Application: If Australia's saving rate rose from 0.25 to 0.30 (a 5 pp increase), by what percentage would the steady-state capital per effective worker (k) increase? Use α = 0.35, δ = 0.05, n = 0.015, g = 0.015. → Answer:
   k(s=0.30) = [0.30/0.080]^(1/0.65) = [3.75]^1.5385
   ln(k_new) = 1.5385 × ln(3.75) = 1.5385 × 1.3218 = 2.0337
   k_new = e^2.0337 = 7.637
   From worked example: k_old = 5.771
   Percentage increase = (7.637 − 5.771)/5.771 = 1.866/5.771 = 0.323 = 32.3%
   A 20% rise in the saving rate (from 0.25 to 0.30) raises k by approximately 32%, illustrating how the level effect is amplified through the exponent 1/(1−α) = 1.54.

Visual — Solow Mathematics in Phase-Diagram Form

Figure: The Solow steady state k is where the saving curve sk^α intersects the break-even line (δ+n+g)k. Left of k, capital deepening occurs (k̇ > 0); right of k, capital per effective worker shrinks (k̇ < 0).*

Further Resources

• 📺 Solow Growth Model Part 1: Model Intro & Solution — Intermediate Macro Series (18 min)
• 📺 Solow model #3 — Steady State (algebra) — Macro Algebra Series (12 min)
• 📚 MIT OCW 14.02 Lecture Notes on Growth — Full mathematical treatment of the Solow model