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Question
Find the marginal productivities of capital (K) and labour (L) if P = 8L – 2K + 3K2 – 2L2 + 7KL when K = 3 and L = 1.
Solution
P = 8L – 2K + 3K2 – 2L2 + 7KL
Marginal productivity of labour, `(del"P")/(del"L")` = 8 – 0 + 0 – 2(2L) + 7K(1)
= 8 – 4L + 7K
Marginal productivity of labour when K = 3 and L = 1 is
`((del"P")/(del"L"))_((3;1))` = 8 – 4 + 21
= 29 – 4
= 25
Marginal productivity of capital, `(del"P")/(del"K")` = 0 – 2(1) + 3(2K) – 0 + 7L(1)
= -2 + 6K + 7L
Marginal productivity of capital when K = 3 and L = 1 is
`((del"P")/(del"K"))_((3;1))`
= -2 + 18 + 7
= -2 + 25
= 23
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