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
This formula drops a β from the exponent on A. For the production function shown earlier, the marginal product of capital is proportional to A^(1−α−β), not A^(1−α).
Full reasoning
Earlier in the same section, the article gives the augmented Solow production function as
[
Y(t)=K(t)^\alpha H(t)^\beta (A(t)L(t))^{1-\alpha-\beta}.
]
Differentiating that with respect to physical capital (K) gives
[
\frac{\partial Y}{\partial K}=\alpha K^{\alpha-1}H^\beta (AL)^{1-\alpha-\beta}.
]
Rewriting in terms of (H/L) and (K/L), this becomes
[
MPK = \alpha A^{1-\alpha-\beta}(H/L)^\beta /(K/L)^{1-\alpha}.
]
So the exponent on (A) should be (1-\alpha-\beta). The displayed formula in the article instead uses (1-\alpha), which is missing the (-\beta) term.
A convenient cross-check is the standard Cobb–Douglas result (MPK = \alpha Y/K). Substituting the article’s own production function into (\alpha Y/K) also yields (\alpha A^{1-\alpha-\beta}(H/L)^\beta /(K/L)^{1-\alpha}), again showing that the published formula’s exponent on (A) is wrong.
2 sources
- Mankiw, Romer, and Weil (1992), A Contribution to the Empirics of Economic Growth
"Let the production function be Y(t) = K(t)^α H(t)^β (A(t)L(t))^(1−α−β)"
- Mankiw Solutions Manual excerpt on Cobb–Douglas marginal products
"The text showed that the marginal products for the Cobb–Douglas production function are: MPL = (1 − α)Y/L. MPK = αY/K."