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Lesson M18.L05: Population Growth and Technological Progress in the Solow Model

Module: Economic Growth Part I Level: intermediate Duration: 30 minutes Learning Objective: Derive the balanced growth path of the Solow model when population growth and technological progress are incorporated. Data as of: 2024 Provenance: ABS Population Projections | World Bank Total Factor Productivity Data

Explanation

The basic Solow model (no technology, no population growth) predicts that growth must eventually stop โ€” capital accumulation faces diminishing returns and reaches a steady state with zero growth. To explain sustained growth in GDP per worker, the model must incorporate labour-augmenting technological progress (g) and population growth (n).

Notation: - A(t) = technology level at time t; grows at exogenous rate g: ศฆ/A = g - L(t) = labour force at time t; grows at rate n: Lฬ‡/L = n - AL = effective labour (efficiency units); grows at rate n + g - k = K/(AL) = capital per effective worker - y = Y/(AL) = output per effective worker - ฮด = capital depreciation rate - n + g = "effective labour growth" โ€” the rate at which effective labour expands

Why does AL grow at n + g? 

d(AL)/dt = ศฆยทL + AยทLฬ‡ = (gยทA)ยทL + Aยท(nยทL) = (g + n)ยทAL
Growth rate of AL = (n + g)

Full capital accumulation equation (per effective worker): 

kฬ‡ = sy^ฮฑ โˆ’ (ฮด + n + g)k = sk^ฮฑ โˆ’ (ฮด + n + g)k

Break-even investment: (ฮด + n + g)k

  • ฮดk: replaces depreciated capital
  • nk: equips new workers (population growth) with existing k ratio
  • gk: offsets dilution of k as technology A improves (more effective workers per actual worker)

Steady state (kฬ‡ = 0): 

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

Growth on the balanced growth path (BGP):

In steady state, k = K/(AL) is constant. Therefore: - K grows at rate (n + g) - Y grows at rate (n + g) - Y/L grows at rate g (output per actual worker grows at rate of technology) - Y/L does NOT grow due to capital accumulation โ€” only technology drives long-run per-capita growth - K/Y is constant (consistent with Kaldor Fact 3) - Rate of return to capital r = ฮฑY/K = constant (consistent with Kaldor Fact 4)

Effects on k* (comparative statics):

Parameter change Effect on k* Intuition
Higher s โ†‘ k* More saving, more capital
Higher n โ†“ k* More workers dilute capital
Higher g โ†“ k* Technology progress requires more break-even investment
Higher ฮด โ†“ k* More depreciation reduces net accumulation

Australian parameters (2024): - n โ‰ˆ 0.015 (1.5% p.a. โ€” combination of natural increase ~0.5% and net migration ~1%) - g โ‰ˆ 0.015 (1.5% p.a. TFP growth, long-run estimate) - ฮด โ‰ˆ 0.05 (5% p.a.) - s โ‰ˆ 0.24โ€“0.25

The balanced growth path predicts Y/L grows at g โ‰ˆ 1.5% p.a. โ€” consistent with observed Australian GDP per capita growth of ~1.5โ€“2%.

Connecting to Kaldor's Facts (M18.L01):

The Solow model with technology explicitly generates all six Kaldor facts on the balanced growth path:

  1. Y/L grows at constant rate g โœ“
  2. K/L grows at constant rate g โœ“ (since k = K/(AL) constant โ†’ K/L = kยทA โ†’ grows at g)
  3. K/Y = k/y* constant โœ“
  4. r = MPK = ฮฑy/k constant โœ“
  5. Capital share = ฮฑk^(ฮฑโˆ’1)ยทk/y* = ฮฑ constant โœ“ (Cobb-Douglas)
  6. Different countries have different steady states โ†’ different growth rates โœ“

Worked Example

Australian parameters: s = 0.25, ฮด = 0.05, n = 0.015, g = 0.015, ฮฑ = 0.35

Step 1 โ€” Baseline steady state (replicates M18.L02): 

ฮด + n + g = 0.05 + 0.015 + 0.015 = 0.080
k* = (0.25/0.080)^(1/0.65) = 3.125^1.5385 = 5.771
y* = 5.771^0.35 = 1.847

Step 2 โ€” Effect of higher population growth (n rises from 0.015 to 0.025): 

ฮด + n + g (new) = 0.05 + 0.025 + 0.015 = 0.090
k*_new = (0.25/0.090)^(1/0.65) = 2.778^1.5385

ln(k*_new) = 1.5385 ร— ln(2.778) = 1.5385 ร— 1.0217 = 1.5720
k*_new = e^1.5720 = 4.816
y*_new = 4.816^0.35

ln(y*_new) = 0.35 ร— 1.5720 = 0.5502
y*_new = e^0.5502 = 1.733

Change in k*: (4.816 โˆ’ 5.771)/5.771 = โˆ’0.166 = โˆ’16.6% Change in y*: (1.733 โˆ’ 1.847)/1.847 = โˆ’0.062 = โˆ’6.2%

Higher population growth reduces capital per effective worker by 16.6% and output per effective worker by 6.2% in steady state.

Step 3 โ€” Effect of faster technological progress (g rises from 0.015 to 0.025): 

ฮด + n + g (new) = 0.05 + 0.015 + 0.025 = 0.090  (same as above)
k*_new = 4.816, y*_new = 1.733  (same calculation)

Faster technology growth also reduces k* โ€” same direction as higher n. But there is a crucial difference:

Long-run growth rate of Y/L: - Under g = 0.015: Y/L grows at 1.5% p.a. - Under g = 0.025: Y/L grows at 2.5% p.a.

Higher g means lower k and Y in effective-worker units, but higher growth of actual living standards Y/L. The level effect (lower k*) is dominated in welfare terms by the faster growth rate. This is one reason why productivity-enhancing policies (education, R&D, innovation) are the primary long-run development strategy.

Step 4 โ€” GDP per worker growth on BGP:

On the balanced growth path, Y/L grows at rate g = 1.5%. Cumulative effect over 40 years: 

Y/L(t+40) / Y/L(t) = e^(gร—40) = e^(0.015ร—40) = e^0.6 = 1.822
GDP per worker is 82.2% higher after 40 years due to technological progress alone. This is the Solow model's explanation for why living standards have roughly doubled every 35โ€“40 years in developed economies.

Common Misconception

Misconception: "Faster population growth always raises total GDP, so it is unambiguously good for an economy."

Correction: Higher n raises total GDP (more workers) but reduces GDP per worker (Y/L) in the Solow model. Higher n increases the break-even investment requirement (nk), lowering k and hence y. Output per effective worker falls. Living standards โ€” measured by Y/L, not total Y โ€” are lower. This is why countries that have achieved high living standards generally have lower population growth rates (demographic transition). The Solow model does not say population growth is "bad," but it clearly implies a trade-off: more people to share a smaller capital stock per person, unless saving rates rise to compensate. Australia's high immigration intake (~1% p.a.) implies downward pressure on k*, partially offset by immigrants' own savings and capital brought with them.

Practice Prompts

  1. Conceptual: Why is the term gk included in break-even investment, even though technological progress does not depreciate the capital stock? โ†’ Answer: k = K/(AL). Even if K is constant (no depreciation), as A grows, each unit of capital is paired with more "effective labour" โ€” so k = K/(AL) falls. To maintain k at a constant level, investment must offset this dilution. Formally: if Kฬ‡ = I (gross investment), then kฬ‡ = Kฬ‡/(AL) โˆ’ k(n+g) = I/(AL) โˆ’ (n+g)k. Setting kฬ‡ = 0 requires I/(AL) = (n+g)k, i.e., break-even investment per effective worker must cover both population growth (nk) and technology dilution (gk). Think of it as: more efficient workers "use up" the existing capital-per-effective-worker ratio unless investment replenishes it.

  2. Numerical: An economy has s = 0.20, ฮด = 0.04, n = 0.03, g = 0.02, ฮฑ = 0.30. Find k and y. Then calculate: what is the steady-state growth rate of output per actual worker (Y/L)? โ†’ Answer:
    ฮด + n + g = 0.04 + 0.03 + 0.02 = 0.09
    s/(ฮด+n+g) = 0.20/0.09 = 2.222
    1/(1โˆ’ฮฑ) = 1/0.7 = 1.4286
    k = 2.222^1.4286
    ln(k    ) = 1.4286 ร— ln(2.222) = 1.4286 ร— 0.7985 = 1.1407
    k* = e^1.1407 = 3.129
    y = 3.129^0.30
    ln(y    ) = 0.30 ร— 1.1407 = 0.3422
    y* = e^0.3422 = 1.408
    Steady-state growth rate of Y/L = g = 2% per year
    (In steady state, k = K/(AL) is constant, so Y/L = y*ยทA grows only at rate g = 2%)

  3. Application: Australia's population growth rate has been significantly boosted by net immigration (~1% p.a. average, rising to ~2% in 2022โ€“23). Using the Solow model, explain the predicted effect on GDP per capita in the steady state. What offsetting policies could preserve GDP per capita? โ†’ Answer: In the Solow model, higher n increases break-even investment (ฮด+n+g)k. For given s, this lowers k = [s/(ฮด+n+g)]^(1/(1-ฮฑ)). Lower k โ†’ lower y โ†’ lower GDP per effective worker. GDP per actual worker (Y/L) on the balanced growth path equals yยทA, and while A keeps growing at g, the level of Y/L at any point in time is lower due to the reduced k. Offsetting policies: (1) Raise s (e.g., increase superannuation rate from 10.5% to higher) โ€” counteracts higher n in the denominator; (2) Boost g through R&D subsidies, education investment, skilled migration (bringing higher-productivity workers with above-average human capital โ€” effectively raising A directly); (3) Public infrastructure investment (government capital formation) โ€” raises effective K, partially offsetting the dilution of k. Australia's high skilled-migration intake partially addresses this by selecting immigrants who raise labour productivity (boosting g), which dampens the k*-reducing effect of higher n.

Visual โ€” Higher Population Growth Lowers the Level of Y/L, Not Its Long-Run Growth Rate

Left: higher n shifts the break-even line up sf(k) (ฮด+n_old+g)k (ฮด+n_new+g)k k*_old k*_new Right: Y/L keeps growing at g, but from a lower level Old balanced growth path After permanent rise in n Population growth rises Time Y/L Same slope = g

Figure: Higher population growth raises break-even investment and lowers the steady-state capital stock per effective worker. In levels, GDP per worker shifts onto a lower balanced-growth path โ€” but the long-run growth rate of Y/L still equals technological progress g.

Further Resources