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This may well be a stupid question. I'm currently trying to find out whether a production function I have has convex isoquants.

I'm aware I can find the derivative $\frac{dL}{dK}$ by using the rule of implicit differentiation:

$$\frac{dL}{dK} = -\frac{∂Q}{∂K}/\frac{∂Q}{∂L} $$

Where Q is the production function.

Is it possible to find the second derivative in a similar way? i.e.

$$\frac{d^2L}{dK^2} = -\frac{∂^2Q}{∂K^2}/\frac{∂^2Q}{∂L^2} $$

Nik-D
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    take a look at something similar http://math.stackexchange.com/questions/1072869/why-doesnt-it-work-when-i-calculate-the-second-order-derivative/1072948#1072948 – janmarqz Sep 21 '15 at 11:57

1 Answers1

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Hint: The equation is not the result of implicit differentiation. It is the result of total differentiation:

$$dQ(K,L)=\frac{\partial Q}{\partial L}\cdot dL+\frac{\partial Q}{\partial K}\cdot dK=0$$

$$\frac{\partial Q}{\partial L}\cdot dL=-\frac{\partial Q}{\partial K}\cdot dK$$

Dividing the equation by $\frac{\partial Q}{\partial L}$ and $dK$ gives the final result.

callculus42
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