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Given the company's production function $f(L,K)=L^{1/3}K^{3/4}$, find slope of the isoquant passing through $(L,K)=(20,40)$ is equal to $-4/5$ (K is on the vertical axis).

I need to state whether the statement above is true or false. In my opinion that is true because i need to calculate $MRTS_{LK}$ which is obviously equal to $-4/5$, but the answer in my textbook is: $-8/9$. Thus my question is am i doing something wrong or just the answer in my textbook is wrong?

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The book is right. The proper way to go about this is of course to work out the theory and the math before plugging in any specific numbers. Fixing the quantity produced, we have $$ \bar Q = L^bK^a \implies K = (\bar Q)^{1/a}L^{-b/a}$$

Then, since capital is on the vertical axis

$$\frac {dK}{dL} = -\frac{b}{a}(\bar Q)^{1/a}L^{(-b/a)-1} $$

Note that we do not differentiate $\bar Q$ since we want it to be fixed, for different combinations of inputs.

Inserting the production function expression

$$\frac {dK}{dL}= -\frac{b}{a}[L^bK^a]^{1/a}L^{(-b/a)-1}$$

$$\implies \frac {dK}{dL}= -\frac{b}{a}\frac{K}{L}$$

Then, plug in the specific numbers: $b=1/3, a=3/4, K = 40, L = 20$ so

$$\frac {dK}{dL} = -\frac{1/3}{3/4}\frac{40}{20} = -\frac{4}{9}\cdot 2 = -\frac{8}{9}$$

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