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Lesson M09.L04: Steady-State Capital and Output

Module: Savings, Capital Formation, and Economic Growth Level: intro Duration: 30 minutes Learning Objective: Compare steady-state capital and output levels across different saving rates using the Solow model. Data as of: 2024 Provenance: OpenStax Macro 3e | MIT OCW 14.02 | RBA Education

Explanation

In the Solow model, steady-state capital per worker (k*) is the unique level of capital at which investment exactly offsets depreciation. Finding k* — and understanding what changes it — is the central task of this lesson.

The steady-state formula (for a Cobb-Douglas production function y = Ak^Îą) is:

sf(k*) = Îīk* → k* = (sA/Îī)^(1/(1−α))

Key variables: - s (saving rate): higher saving → higher k* - A (total factor productivity, TFP): better technology → higher k* - Îī (depreciation rate): faster wear → lower k* - Îą (capital share in output, typically ~0.35 for Australia)

Marginal products at steady state:

  • MPK = Îą(Y/K) = Îą(y/k) — diminishing as k rises
  • MPL = (1−α)(Y/L) = (1−α)y — rises as productivity improves

At k*, the marginal product of capital equals the depreciation rate (in the basic model without population growth). This is a zero-profit condition for capital investment.

Comparing steady states: If country A has saving rate s_A = 0.20 and country B has s_B = 0.30, with otherwise identical technologies, country B will have a higher k* and higher y*. However, both settle at zero per-capita growth in the long run. Only TFP growth (A rising over time) generates permanent increases in living standards.

Australia's saving rate of roughly 22–23% of GNI (2020s) is moderate by OECD standards. Countries like South Korea and China (saving rates > 30%) have reached higher steady-state capital levels faster.

Worked Example

Two economies — Australia (A) and a high-saver (H) — same technology, different saving rates.

Both have: - Production function: y = k^0.35 (i.e., Îą = 0.35, A = 1) - Depreciation: Îī = 0.05

Economy A: s = 0.22 | Economy H: s = 0.35

Step 1 — Steady-state condition:

s × (k*)^0.35 = 0.05 × k* k* = (s/Îī)^(1/(1−0.35)) = (s/0.05)^(1/0.65)

Step 2 — Solve for k*_A (s = 0.22):

(s/Îī) = 0.22/0.05 = 4.40 k*_A = 4.40^(1/0.65) = 4.40^1.538

Calculate: ln(4.40) = 1.4816; × 1.538 = 2.279; e^2.279 = 9.77

k*_A ≈ 9.77 units per worker y*_A = (9.77)^0.35; ln(9.77) = 2.279; × 0.35 = 0.798; e^0.798 ≈ 2.22 units per worker

Step 3 — Solve for k*_H (s = 0.35):

(s/Îī) = 0.35/0.05 = 7.00 k*_H = 7.00^(1/0.65) = 7.00^1.538

ln(7.00) = 1.9459; × 1.538 = 2.994; e^2.994 = 19.96

k*_H ≈ 19.96 units per worker y*_H = (19.96)^0.35; ln(19.96) = 2.994; × 0.35 = 1.048; e^1.048 ≈ 2.85 units per worker

Step 4 — Calculate MPK at each steady state:

MPK_A = Îą × (y*/k*) = 0.35 × (2.22/9.77) = 0.35 × 0.227 = 0.0795 MPK_H = 0.35 × (2.85/19.96) = 0.35 × 0.143 = 0.0500

MPK is lower in the high-saving economy — more capital means diminishing returns have set in further.

Step 5 — Calculate MPL at each steady state (assume same L for both):

MPL = (1−α) × y* MPL_A = 0.65 × 2.22 = 1.443 MPL_H = 0.65 × 2.85 = 1.853

Workers are more productive in the high-saving economy — they have more capital to work with.

Common Misconception

Misconception: "The economy with the highest saving rate will have the highest living standards in the long run."

Correction: A higher saving rate raises k* and y*, but at the cost of consuming less today. There is an optimal saving rate — the Golden Rule saving rate — that maximises steady-state consumption per worker. Saving too much (above the Golden Rule) means high capital but low consumption; saving too little means high consumption now but a low capital stock. Japan and Germany save substantially more than Australia, but this does not automatically translate to higher welfare.

Practice Prompts

  1. What happens to steady-state capital k* if the depreciation rate Îī increases? Explain using the steady-state condition sf(k*) = Îīk*. → Answer: A higher Îī steepens the depreciation line in the Solow diagram. The intersection with the (unchanged) saving curve occurs at a lower k*. Intuitively, capital wears out faster, so the same amount of investment can only sustain a smaller capital stock. Output per worker y* also falls.

  2. NUMERICAL CALCULATION: Use y = k^0.4 (Îą = 0.4), s = 0.25, Îī = 0.05. (a) Find k* and y*. (b) Calculate MPK and MPL at steady state (assume y* and k* found in (a), with L normalised so y = Y/L). → Answer: (a) Steady state: 0.25(k*)^0.4 = 0.05k* (k*)^0.4 = 0.05/0.25 × k* / (k*)^0.4 → rearrange: (k*)^(1−0.4) = s/Îī = 0.25/0.05 = 5 (k*)^0.6 = 5 k* = 5^(1/0.6) = 5^1.667 ln(5) = 1.6094; × 1.667 = 2.683; e^2.683 = 14.62 k* ≈ 14.62 y* = (14.62)^0.4; ln(14.62) = 2.683; × 0.4 = 1.073; e^1.073 ≈ 2.92

(b) MPK = Îą(y*/k*) = 0.4 × (2.92/14.62) = 0.4 × 0.200 = 0.080 MPL = (1−α) × y* = 0.6 × 2.92 = 1.752

  1. Australia and South Korea have similar technologies but South Korea's saving rate is approximately 35% versus Australia's 22%. Using the Solow model, predict how their steady-state capital stocks and output per worker compare, and identify one real-world factor that might reduce the gap. → Answer: The Solow model predicts South Korea will have a higher k* and y* due to its higher saving rate. However, the gap may be reduced (or reversed) by: (1) Australia's stronger rule of law and property rights (higher effective A), (2) Australia's greater natural resource endowments boosting Y independently of K, (3) Australia's reliance on foreign capital inflows supplementing its lower domestic saving rate. In practice, Australia maintains high per-capita income through both productivity and resource wealth.

Visual — Higher Saving Raises k More Than y

Because f(k) is concave, output rises less than capital when saving increases Capital per worker, k Output / investment / depreciation k*_A k*_H y*_A y*_H f(k) s_A f(k) s_H f(k) Îīk Capital gap is large, output gap is smaller

Figure: Raising the saving rate from a low level (s_A) to a higher level (s_H) shifts the saving curve up and raises the steady state from k_A to k_H. But because the production function is concave, the increase in y is proportionally smaller than the increase in k.

Further Resources