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Lesson M19.L02: Endogenous Growth Theory: The AK Model

Module: Economic Growth Part II Level: intermediate Duration: 30 minutes Learning Objective: Derive sustained per-capita growth from the AK model without diminishing returns to capital. Data as of: 2024 Provenance: Marginal Revolution University | MIT OCW 14.02

Explanation

The Solow model generates diminishing returns to capital: as K rises, each additional unit of capital adds less to output (โˆ‚ยฒY/โˆ‚Kยฒ < 0). This is why growth eventually stops at a steady state. But real-world economies have sustained per-capita growth for centuries. Where does it come from?

The AK model provides the simplest endogenous growth framework. Replace the Cobb-Douglas production function with:

\[Y = AK\]

where A is a constant (technology parameter) and K is broadly defined capital โ€” encompassing physical capital, human capital, and knowledge. Because ฮฑ = 1 (constant returns to K), there are no diminishing returns and growth can persist indefinitely.

Deriving the growth rate. Capital accumulates as:

\[\dot{K} = sY - \delta K = sAK - \delta K\]

Dividing both sides by K:

\[\frac{\dot{K}}{K} = sA - \delta\]

Since Y = AK, the growth rate of output equals the growth rate of capital:

\[g_Y = sA - \delta\]

Key implication โ€” growth effect vs. level effect. In the Solow model, a rise in s raises the level of steady-state income permanently but does not change the long-run growth rate (which is tied to exogenous technology growth g). In the AK model, a rise in s raises the permanent growth rate g_Y = sA โˆ’ ฮด. This is a growth effect, not just a level effect.

Intuition. Broadly defined capital (physical + human + knowledge) does not suffer diminishing returns because: (i) human capital complements physical capital; (ii) knowledge is non-rival and generates spillovers. Even as any single firm's physical capital depreciates, the aggregate stock of ideas keeps expanding, preventing the MPK from falling to zero.

Australian policy relevance. If the AK model is correct, policies that increase s (compulsory superannuation, infrastructure investment) or A (R&D subsidies, education quality) have permanent growth effects โ€” not just temporary boosts. This motivates Australia's R&D tax incentives and Productivity Commission recommendations on human capital investment.

Notation: Y = output; K = broadly defined capital; A = productivity parameter; s = savings/investment rate; ฮด = depreciation rate; g_Y = per-capita output growth rate; dot notation (แบŠ) = dX/dt.

Worked Example

Question: Calculate the steady-state growth rate g_Y for two scenarios: (a) s = 0.30, A = 0.15, ฮด = 0.05; (b) s = 0.40, A = 0.15, ฮด = 0.05.

Scenario (a):

\[g_Y = sA - \delta$$ $$g_Y = 0.30 \times 0.15 - 0.05$$ $$g_Y = 0.045 - 0.05$$ $$g_Y = -0.005 = -0.5\%\]

A negative growth rate means capital and output are shrinking over time. With s = 0.30 and A = 0.15, investment (sAK = 0.045K) is insufficient to cover depreciation (ฮดK = 0.05K). The economy contracts.

Scenario (b):

\[g_Y = sA - \delta$$ $$g_Y = 0.40 \times 0.15 - 0.05$$ $$g_Y = 0.060 - 0.05$$ $$g_Y = 0.010 = 1.0\%\]

By raising s from 0.30 to 0.40, the economy now achieves 1% sustained per-capita growth. The extra savings fund enough investment (0.06K) to cover depreciation (0.05K) and net capital accumulation.

Comparison with Solow. In a Solow model calibrated with these same parameters, a rise in s from 0.30 to 0.40 would raise the steady-state level of output per worker but eventually return the growth rate to the exogenous rate g (e.g., 2%). In the AK model, the savings rate determines the growth rate permanently.

Break-even savings rate (where g_Y = 0): s = ฮด/A = 0.05/0.15 = 0.333 = 33.3%. Below this, the economy contracts; above it, it grows sustainably.

Common Misconception

Misconception: "In the AK model, higher saving always leads to faster growth, so countries should save as much as possible."

Correction: Higher s raises g_Y = sA โˆ’ ฮด, but there are diminishing returns to saving at the household level (less current consumption). The AK model does not imply saving 100% is optimal. Optimal growth theory (Ramsey model) balances current utility against future consumption โ€” the "Golden Rule" savings rate maximises long-run consumption, not output growth. Policy should also target A (technology, institutions), not just s.

Practice Prompts

  1. Conceptual: Explain the difference between a "level effect" and a "growth effect" of an increase in the savings rate. Which does the Solow model predict, and which does the AK model predict?

โ†’ Answer: A level effect means the economy reaches a permanently higher level of income per capita but then grows at the same long-run rate as before. A growth effect means the long-run rate of growth itself increases permanently. The Solow model predicts only a level effect: more savings โ†’ higher k* โ†’ higher y*, but the growth rate returns to g (exogenous). The AK model predicts a growth effect: more savings โ†’ higher g_Y = sA โˆ’ ฮด, sustained forever.

  1. Numerical: An economy has A = 0.20 and ฮด = 0.04. What savings rate s is needed to achieve exactly 2% per-capita growth? Show full algebra.

โ†’ Answer: - Set g_Y = 0.02: - g_Y = sA โˆ’ ฮด - 0.02 = s ร— 0.20 โˆ’ 0.04 - s ร— 0.20 = 0.02 + 0.04 = 0.06 - s = 0.06 / 0.20 = 0.30 = 30%

A savings/investment rate of 30% is required to generate 2% sustained per-capita growth under these parameters.

  1. Application: Australia's compulsory superannuation rate rose from 9% (2013) to 11.5% (2024). Using the AK model logic, explain why proponents argue this boosts long-run GDP growth. What are the limitations of this argument?

โ†’ Answer: In the AK model, higher s permanently raises g_Y = sA โˆ’ ฮด. Super contributions increase the national savings/investment rate, funding more capital accumulation (broadly defined). Proponents argue this raises long-run growth, not just a temporary level boost. Limitations: (i) superannuation may partially crowd out private saving (Ricardian effect); (ii) actual returns depend on where super is invested โ€” domestic vs. offshore; (iii) the AK model abstracts from diminishing returns, which likely exist in physical capital; (iv) if A (technology) is the binding constraint rather than s, super rate rises have smaller effects.

Further Resources