en.wikipedia.org/wiki/Solow%E2%80%93Swan_model
1 correction found
M P K = ∂ Y ∂ K = α A 1 − α ( H / L ) β ( K / L ) 1 − α {\\displaystyle MPK={\\frac {\\partial Y}{\\partial K}}={\\frac {\\alpha A^{1-\\alpha }(H/L)^{\\beta }}{(K/L)^{1-\\alpha }}}}
The displayed MPK formula is missing a β in the exponent on A. For the stated production function, the correct marginal product is proportional to A^(1−α−β), not A^(1−α).
Full reasoning
A few paragraphs earlier, the page defines the human-capital-augmented production function as
[
Y = K^{\alpha} H^{\beta} (AL)^{1-\alpha-\beta}.
]
Differentiating that function with respect to physical capital (K) gives
[
\frac{\partial Y}{\partial K} = \alpha K^{\alpha-1} H^{\beta} (AL)^{1-\alpha-\beta}.
]
Rewriting in per-worker terms,
[
\frac{\partial Y}{\partial K} = \alpha A^{1-\alpha-\beta} \frac{(H/L)^{\beta}}{(K/L)^{1-\alpha}}.
]
So the exponent on (A) should be (1-\alpha-\beta). The article’s displayed formula instead uses (1-\alpha), which is algebraically inconsistent with the production function stated on the same page.
Independent sources on the augmented Solow/MRW model give the corrected expression with (A^{1-\alpha-\beta}).
2 sources
- The Quality vs. The Quantity of Schooling: What Drives Economic Growth?
Equation (5): MPK_i = δY_i/δK_i = α (K/L)_i^(α−1) (H/L)_i^β A_i^(1−α−β) = α (K/Y)_i.
- Human Capital and Growth in Japan: Converging Perspectives
Equation (10): MPK = ∂Y/∂K = α (K/L)^(α−1) (H/L)^β A^(1−α−β).