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 has the wrong exponent on A. Given the production function stated just above, the marginal product of physical capital is proportional to A^(1−α−β), not A^(1−α).
Full reasoning
The article defines the augmented Solow/Mankiw–Romer–Weil production function as:
- (Y = K^{\alpha} H^{\beta} (AL)^{1-\alpha-\beta})
Differentiating that with respect to physical capital (K) gives:
- (MPK = \partial Y/\partial K = \alpha K^{\alpha-1} H^{\beta} (AL)^{1-\alpha-\beta})
- Rewriting in per-worker terms: (MPK = \alpha A^{1-\alpha-\beta}(H/L)^{\beta}/(K/L)^{1-\alpha})
So the exponent on (A) must be (1-\alpha-\beta). The article’s displayed formula uses (A^{1-\alpha}) instead, which omits the (-\beta) term.
A university paper discussing this same augmented Cobb–Douglas setup gives the marginal product of physical capital as:
- (MPK = A^{1-\alpha-\beta},\alpha,(K/L)^{\alpha-1}(H/L)^{\beta})
That is algebraically equivalent to the corrected expression above and contradicts the formula shown in the article.
2 sources
- Mankiw, Romer and Weil (1992) — Econ 303a: Advanced Macroeconomics I
“Y(t) = K(t)^α H(t)^β (E(t)L(t))^{1−α−β}” — the standard augmented Solow/MRW production function used in the section.
- Universidad EAFIT repository PDF on human capital and national income
“MPK = ∂Y/∂K = A^{1-α-β} α (K/L)^{α-1} (H/L)^β” — showing the exponent on A is 1−α−β, not 1−α.