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Sum

If the production of a firm is given by P = 4LK – L^{2} + K^{2}, L > 0, K > 0, Prove that L `(del"P")/(del"L") + "K"(del"P")/(del"K")` = 2P.

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#### Solution

P = 4LK – L^{2} + K^{2}

P(K, L) = 4LK – L^{2} + K^{2}

P(tK, tL) = 4(tL) (tK) – t^{2}L^{2} + t^{2}K^{2}

= t^{2}(4LK – L^{2} + K^{2})

= t^{2}P

∴ P is a homogeneous function in L and K of degree 2.

∴ By Euler’s theorem, L `(del"P")/(del"L") + "K"(del"P")/(del"K")` = 2P.

Concept: Applications of Partial Derivatives

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