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Lesson M09.L03: The Solow Growth Model — Building Intuition

Module: Savings, Capital Formation, and Economic Growth Level: intro Duration: 30 minutes Learning Objective: Interpret the Solow diagram showing saving, depreciation, and steady-state capital graphically. Data as of: 2024 Provenance: OpenStax Macro 3e | MIT OCW 14.02 | RBA Education

Explanation

The Solow growth model (developed by Robert Solow in 1956) is the workhorse model of long-run economic growth. Its core insight is that capital accumulation alone cannot sustain rising living standards — eventually, diminishing returns bring growth to a halt.

The model uses a single diagram with capital per worker (k = K/L) on the horizontal axis and output per worker (y = Y/L) on the vertical axis. Two curves are drawn:

  1. Production function: y = f(k) — curves upward but flattens due to diminishing returns to capital. As k rises, each new unit of capital adds less output.
  2. Saving curve: sf(k) — a fixed fraction s of output is saved and invested. It is simply the production function scaled down by s.
  3. Depreciation line: Îīk — a straight line through the origin. For every unit of capital, fraction Îī wears out.

Steady state (k*) is where the saving curve intersects the depreciation line:

sf(k*) = Îīk*

At k*, investment exactly replaces depreciation — capital per worker stops changing.

  • To the left of k*: sf(k) > Îīk → investment exceeds depreciation → k is rising.
  • To the right of k*: sf(k) < Îīk → depreciation exceeds investment → k is falling.
  • At k*: sf(k) = Îīk → equilibrium.

This self-correcting mechanism means the economy always converges to k* from any starting point. Australia's strong investment and moderate saving rate places it somewhere on this convergence path, with foreign capital inflows supplementing domestic saving to push k toward (and beyond) what domestic saving alone would support.

Worked Example

Scenario: A stylised economy has: - Production function (per worker): y = k^0.5 (i.e., Y = K^0.5 L^0.5, Îą = 0.5) - Saving rate: s = 0.25 - Depreciation rate: Îī = 0.05

Step 1 — Write the steady-state condition:

sf(k*) = Îīk* 0.25 × (k*)^0.5 = 0.05 × k*

Step 2 — Solve for k*:

Divide both sides by (k*)^0.5:

0.25 = 0.05 × (k*)^0.5 (k*)^0.5 = 0.25 / 0.05 = 5 k* = 5Âē = 25 units of capital per worker

Step 3 — Find steady-state output per worker:

y* = (k*)^0.5 = 25^0.5 = 5 units of output per worker

Step 4 — Verify: saving = depreciation at k*:

sf(k*) = 0.25 × 5 = 1.25 Îīk* = 0.05 × 25 = 1.25 ✓

Step 5 — Check a starting point below k*: Suppose k = 16.

sf(16) = 0.25 × 16^0.5 = 0.25 × 4 = 1.0 Îī × 16 = 0.05 × 16 = 0.8 Since 1.0 > 0.8, Δk > 0 → capital is accumulating → economy moves toward k* = 25. ✓

Common Misconception

Misconception: "A country that saves more will always grow faster indefinitely."

Correction: A higher saving rate raises the saving curve, which shifts k* to a higher level — but growth is temporary. Once the economy reaches the new, higher k*, per-capita output stabilises again. The Solow model shows that saving rate increases produce a level effect (higher steady-state output), not a permanent growth rate effect. Sustained per-capita growth requires ongoing technological progress (TFP), not just higher saving.

Practice Prompts

  1. In the Solow diagram, what does it mean when the saving curve lies above the depreciation line at a given level of k? → Answer: When sf(k) > Îīk, investment exceeds depreciation — capital per worker is growing (Δk > 0). The economy is to the left of the steady state and is converging toward k*. Living standards (y) are rising.

  2. NUMERICAL CALCULATION: Use the same economy as the worked example (y = k^0.5, s = 0.25, Îī = 0.05) but now s rises to 0.30. Calculate the new steady-state k* and y*, and find the percentage increase in steady-state output per worker. → Answer: Steady-state condition: 0.30 × (k*)^0.5 = 0.05 × k* (k*)^0.5 = 0.30 / 0.05 = 6 k* = 6Âē = 36 y* = 36^0.5 = 6 Percentage increase in y*: (6 − 5) / 5 × 100 = +20% The saving rate rose by 20% (from 0.25 to 0.30), but steady-state output rose by only 20% in this example — confirming saving has a level effect, not a multiplier that keeps compounding.

  3. Australia opens its capital account wider, allowing more foreign investment inflows. Using the Solow diagram, explain the effect on k and y in the short and long run. → Answer: Foreign capital inflows effectively boost total investment above domestic saving (I > S domestically), shifting the effective investment curve upward. In the short run, k rises above its previous trajectory — capital per worker grows faster. In the long run, k converges to a higher steady state k*, with higher output per worker y*. This mirrors Australia's historical experience: persistent current account deficits (foreign borrowing) have supported a capital stock and productivity level above what domestic saving alone would fund.

Visual — The Solow Model and the Effect of a Higher Saving Rate

Savings shifts the steady state to the right in the Solow model Capital per worker, k Output / investment / depreciation f(k) sâ€ēf(k) sf(k) Îīk k* k*â€ē kĖ‡ > 0 kĖ‡ < 0

Figure: The steady state occurs where saving/investment equals depreciation. A higher saving rate shifts the saving curve upward, increasing the steady-state capital stock from k to kâ€ē and lifting long-run output per worker.

Further Resources