For the production function P = 3(L)0.4 (K)0.6, find the marginal productivities of labour (L) and capital (K) when L = 10 and K = 6. [use: (0.6)0.6 = 0.736, (1.67)0.4 = 1.2267]
Solution
Given that P = 3(L)0.4 (K)0.6…….(1)
Differentiating partially with respect to L we get,
`(del"P")/(del"L") = 3("K")^0.6 (del/(del"L") ("L")^0.4)`
`= 3("K")^0.6 (0.4)("L"^(0.4-1))`
`= 1.2 ("K"^0.6)("L"^-0.6)`
`= 1.2 ("K"^0.6) 1/"L"^0.6`
`= 1.2 ("K"/"L")^0.6`
When L = 10, k = 6, `(del"P")/(del"L") = 1.2(6/10)^0.6`
= 1.2(0.6)0.6
= 1.2(0.736)
i.e., the marginal productivity of labour = 0.8832
Again differentiating partially with respect to ‘k’ we get,
Marginal productivity of labour when L = 10, K = 6 is
`(del"P")/(del"k") = 3("L")^0.4 (del/(del"k") ("k")^0.6)`
`= 3("L")^0.4 (0.6) k^(0.6 - 1)`
`= 1.8 ("L"^0.4) "k"^(-0.4)`
`= 1.8 ("L"^0.4) (1/k^0.4)`
`= 1.8 ("L"/"K")^0.4`
Marginal productivity of capital when k = 10, k = 6
`= 1.8(10/6)^0.4`
= 1.8(1.66666)0.4
= 1.8(1.67)0.4
= 1.8 × 1.2267
= 2.2081
