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A firm produces 231 doohickeys with 8.4 units of labour and 22.1 units of capital. the marginal product of labour is 18, the marginal product of capital is 20. Approximately how many doohickeys will it produce with 8.6 units of labour and 22 units of capital?

Is this just as simple as taking the difference between the new and original values and multiplying that by the marginal product?

i.e. (8.6-8.4) x 18 + (22-22.1) x 20 = 1.6

so they will produce approximately 232.6 doohickeys

Or am i supposed to take the integral of the marginal products and do something with those? i tried that but I couldnt get any where

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I think your solution is golden.

Taking the integral of the marginal products could only work if the marginal products were constant everywhere. This is not mentioned in the text, and in fact if you assume it to be constant then production given this level of $K$ and $L$ would not be 231.

Another interesting tidbit:
If you knew the marginal product function of $L$ and also that of $K$ taking the integrals and summing them up would only work if $MPL$ is constant in $K$ and vice-versa.

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