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My work out shows constant MRTS and also increasing returns to scale. I thought the answer was C as I only found increasing marginal products of labour and capital. I really don't see how the answer is d.

I have double checked my working also, I don't think I made any mistakes...

Thanksenter image description here

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This function is a Cobb-Douglas type production function. This family of functions has the general form $ Q = f(x_1,x_2) = A \cdot x_1^a \cdot x_2^b $. For your specific question, $ A = 1, x_1 = l, x_2 = k, a = 1, b = 0.5 $.

These functions have an interesting property, which is handy to keep in mind (you can show this using the same logic you used in your answer)

i) if $ a+b > 1 $, then the functions exhibits increasing returns to scale; ii) if $ a+b = 1 $, then the functions exhibits constant returns to scale; iii) if $a+b > 1 $, then the functions exhibits decreasing returns to scale.

Therefore, the function you presented has increasing returns to scale. Now, in regards to the marginal productivity of the factors, we have:

i) $ MP_L = \frac{\partial Q}{\partial L} = K^{0.5} $. But we want to check whether $ MP_L $ is increasing or decreasing, so we take the second derivative of Q with respect to L: $ \frac{\partial^2 Q}{\partial L^2} = \frac{\partial MP_L}{\partial L} = 0 $, thus L has constant marginal returns;

ii) Doing the same for K, we arrive at $ \frac{\partial MP_K}{\partial K} < 0 $, which means K has decreasing marginal returns.

This alone would eliminate all alternatives but d.

However, for the sake of completeness:

$ MRTS_{K,L} = \frac{MP_L}{MP_K} $

but we showed that $ MP_L $ is constant and $ MP_K $ is decreasing, thus $ MRTS_{K,L} $ is also decreasing.

If you still have any doubts, feel free to ask.

Kind regards, Pedro.

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    $\begingroup$ Thank you so much for this explanation, I really appreciate the help. $\endgroup$ – onetwothree May 21 at 9:31

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