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