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Lesson M18.L04: The Golden Rule of Capital Accumulation

Module: Economic Growth Part I Level: intermediate Duration: 30 minutes Learning Objective: Derive the golden rule saving rate that maximises steady-state consumption per effective worker. Data as of: 2024 Provenance: ABS National Accounts โ€” Saving Rate | RBA Research

Explanation

The Solow model shows that higher saving raises k and y. But more saving also means less consumption today and in steady state โ€” households sacrifice current consumption by saving. The Golden Rule asks: what saving rate maximises consumption per effective worker in steady state?

Steady-state consumption per effective worker:

In steady state, output equals consumption plus investment. Investment = break-even investment = (ฮด+n+g)k*. 

c* = y* โˆ’ (ฮด + n + g)k*
c* = k*แต… โˆ’ (ฮด + n + g)k*

Notation: - c = steady-state consumption per effective worker - k_GR = golden rule capital per effective worker (maximises c*) - s_GR = golden rule saving rate - MPK = marginal product of capital = โˆ‚y/โˆ‚k = ฮฑk^(ฮฑโˆ’1) - All other notation as in M18.L02

Maximisation: Treat k as the choice variable (since k is determined by s, adjusting s adjusts k*). 

dc*/dk* = ฮฑk*^(ฮฑโˆ’1) โˆ’ (ฮด + n + g) = 0
MPK = ฮด + n + g    at the golden rule

Solving for k*_GR: 

ฮฑยทk*_GR^(ฮฑโˆ’1) = ฮด + n + g
k*_GR^(ฮฑโˆ’1) = (ฮด + n + g)/ฮฑ
k*_GR = [(ฮด + n + g)/ฮฑ]^(1/(ฮฑโˆ’1))

Equivalently (taking the positive root explicitly): 

k*_GR = [ฮฑ / (ฮด + n + g)]^(1/(1โˆ’ฮฑ))

Golden Rule saving rate โ€” Cobb-Douglas result:

At k_GR, the required saving rate satisfies s_GRยทf(k_GR)/k*_GR = (ฮด+n+g), which for Cobb-Douglas simplifies to: 

s_GR = ฮฑ

This is the elegant Cobb-Douglas result: the golden rule saving rate equals the capital income share ฮฑ. For Australia, ฮฑ โ‰ˆ 0.35, so s_GR = 35%.

Interpretation: - If s < s_GR (= ฮฑ): economy is dynamically efficient but under-saving โ€” raising s increases both k and c (moving toward the golden rule from below) - If s > s_GR (= ฮฑ): economy is dynamically inefficient โ€” over-saving (consuming less than possible). Reducing s would raise c* (moving toward the golden rule from above) - Australia: s โ‰ˆ 24% < s_GR โ‰ˆ 35% โ†’ Australia is below the golden rule, suggesting potential welfare gains from higher saving

Second-best caveat: The golden rule applies to the steady state. Getting from s = 0.24 to s = 0.35 requires sacrificing consumption today (current generations save more) to benefit future generations (higher c*). The appropriate saving rate depends on inter-generational welfare weighting โ€” a normative question beyond the Solow model.

Worked Example

Australian parameters: ฮฑ = 0.35, ฮด = 0.05, n = 0.015, g = 0.015

Step 1 โ€” Compute ฮด + n + g: 

ฮด + n + g = 0.05 + 0.015 + 0.015 = 0.080

Step 2 โ€” Derive k*_GR (golden rule capital): 

k*_GR = [ฮฑ / (ฮด + n + g)]^(1/(1โˆ’ฮฑ))
k*_GR = [0.35 / 0.080]^(1/0.65)
k*_GR = [4.375]^(1.5385)

ln(k*_GR) = 1.5385 ร— ln(4.375)
ln(4.375) = 1.4759
ln(k*_GR) = 1.5385 ร— 1.4759 = 2.2707
k*_GR = e^2.2707 = 9.684

Step 3 โ€” Compute golden rule output y*_GR: 

y*_GR = k*_GR^ฮฑ = 9.684^0.35
ln(y*_GR) = 0.35 ร— ln(9.684) = 0.35 ร— 2.2707 = 0.7948
y*_GR = e^0.7948 = 2.214

Step 4 โ€” Compute golden rule consumption c*_GR: 

c*_GR = y*_GR โˆ’ (ฮด+n+g)ยทk*_GR
c*_GR = 2.214 โˆ’ 0.080 ร— 9.684
c*_GR = 2.214 โˆ’ 0.775 = 1.439

Step 5 โ€” Verify golden rule saving rate: 

s_GR = (ฮด+n+g)ยทk*_GR / y*_GR = 0.775 / 2.214 = 0.350 = 35% = ฮฑ  โœ“

Step 6 โ€” Compare to Australia's actual steady state (s = 0.25):

From M18.L02: k = 5.771, y = 1.847 

c*_actual = y* โˆ’ (ฮด+n+g)ยทk* = 1.847 โˆ’ 0.080 ร— 5.771 = 1.847 โˆ’ 0.462 = 1.385

Comparison:

Variable Actual (s=0.25) Golden Rule (s=0.35) Difference
k* 5.771 9.684 +67.8%
y* 1.847 2.214 +19.9%
c* 1.385 1.439 +3.9%

Raising saving from 0.25 to 0.35 would increase steady-state consumption per effective worker by ~3.9%. This is the welfare gain from reaching the golden rule โ€” but it comes at the cost of lower consumption during the transition period.

Common Misconception

Misconception: "The golden rule saving rate maximises output, so we should always aim for it."

Correction: The golden rule maximises consumption, not output. At s > s_GR (over-saving), extra output is ploughed back into capital but less is consumed โ€” a welfare-reducing outcome despite higher output. Maximising output would mean s โ†’ 1 (saving 100%), which is clearly not the right goal. Furthermore, the golden rule is a steady-state concept. Getting there requires a costly transition. For a below-golden-rule economy like Australia (s < s_GR), raising s boosts future consumption but requires current sacrifice. Whether this is desirable depends on how much weight the social planner places on current vs. future generations โ€” a normative judgement beyond Solow's positive model.

Practice Prompts

  1. Conceptual: Prove algebraically that for a Cobb-Douglas production function, the golden rule saving rate equals the capital income share ฮฑ. โ†’ Answer:
    At the golden rule: MPK = ฮด + n + g
    MPK = ฮฑk^(ฮฑโˆ’1) = ฮฑยทy/k (since y = k^ฮฑ, so ฮฑk^(ฮฑโˆ’1) = ฮฑ(k^ฮฑ)/k = ฮฑy/k)
    Therefore: ฮฑยทy/k = ฮด + n + g
    In steady state: sยทy = (ฮด+n+g)ยทk (kฬ‡ = 0)
    So (ฮด+n+g)/y = s/k... rearranging:
    (ฮด+n+g) = sยท(y/k)
    Substituting the golden rule condition: ฮฑยท(y/k) = sยท(y/k)
    Dividing both sides by (y/k) > 0: s_GR = ฮฑ โœ“

  2. Numerical: For an economy with ฮฑ = 0.40, ฮด = 0.06, n = 0.02, g = 0.02, and actual saving rate s = 0.50: Is this economy above or below the golden rule? What would happen to steady-state consumption if s were reduced to s_GR? โ†’ Answer:
    s_GR = ฮฑ = 0.40. Since s = 0.50 > s_GR = 0.40, the economy is above the golden rule (dynamically inefficient / over-saving).
    At s = 0.50: ฮด+n+g = 0.10; k = (0.50/0.10)^(1/0.6) = 5^(1.667)
    ln(k    _high) = 1.667 ร— ln(5) = 1.667 ร— 1.6094 = 2.6829; k_high = e^2.6829 = 14.62
    y    _high = 14.62^0.4; ln = 0.4 ร— 2.6829 = 1.0732; y_high = e^1.0732 = 2.924
    c    _high = 2.924 โˆ’ 0.10 ร— 14.62 = 2.924 โˆ’ 1.462 = 1.462
    At s_GR = 0.40: k_GR = (0.40/0.10)^(1/0.6) = 4^1.667; ln = 1.667 ร— 1.3863 = 2.3106; k_GR = e^2.3106 = 10.08
    y_GR = 10.08^0.4; ln = 0.4 ร— 2.3106 = 0.9242; y_GR = e^0.9242 = 2.520
    c_GR = 2.520 โˆ’ 0.10 ร— 10.08 = 2.520 โˆ’ 1.008 = 1.512
    Reducing s from 0.50 to 0.40     raises c from 1.462 to 1.512 (+3.4%) โ€” confirming the over-saving result.

  3. Application: Australia's household saving rate fell from ~20% in the early 1980s to ~3% by 2020, recovering to ~11โ€“14% during COVID (ABS data). The national saving rate (including government and business) is approximately 24%. Given s_GR = 35% for Australia, what are the welfare implications of this shortfall? โ†’ Answer: Australia's s โ‰ˆ 24% is well below s_GR โ‰ˆ 35%, placing it in the under-saving range. In the Solow model, this means the economy has less capital than the golden rule โ€” a higher saving rate would increase both steady-state capital and consumption per effective worker. Welfare implications: future generations will have lower consumption per effective worker than they would under golden rule saving. Policy options include superannuation (compulsory retirement saving โ€” Australia's ~10.5% super rate boosts national saving), infrastructure investment (public capital formation), and fiscal surplus policies. However, the welfare cost of the transition (current generations must consume less) must be weighed against the benefit to future generations โ€” a classic inter-generational equity problem.

Further Resources