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July 20th, 2010, 10:04 PM   #1
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Gradient and linear approximation

Assume output, Q, is a function of total capital, K, and labour, L, employed given by Q(K, L) = 10?K?L
(a) Find Q(10000, 25).
(b) Find dQ/dK and dQ/dL at the point in (a).
(c) Use (b) to estimate the increase in production should capital increase to 11000.
(d) Estimate Q(9000,30).

I already found (a).
For b, I found dQ/dK and dQ/dL. But it says at the point in (a). So am I supposed to find it at point Q(10000, 25) or the result from a (which is 5000)?
And I have no idea about (c) and (d). Could anyone please solve and explain it?

Thank you very much!
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July 21st, 2010, 05:24 AM   #2
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Re: Gradient and linear approximation

Quote:
Originally Posted by dk1702
For b, I found dQ/dK and dQ/dL. But it says at the point in (a). So am I supposed to find it at point Q(10000, 25) or the result from a (which is 5000)?
dQ/dK will be a function of K and L. Substitute 10000 for K and 25 for L into dQ/dK. Do the same for dQ/dL.

Quote:
Originally Posted by dk1702
(c) Use (b) to estimate the increase in production should capital increase to 11000.
From the point you already calculated in (a), assume (for the sake of estimation) that Q will increase by dQ/dK times the increase in capital -- that is, Q is a linear function of K. Over large ranges this is a bad assumption, but for small changes it's better than you might assume.

Quote:
Originally Posted by dk1702
(d) Estimate Q(9000,30).
Just like in (c), but assume it's linear in K then take that point and assume it's linear in L from there. Remember that the movement in K is negative.
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