# MRTS question involving production function [closed]

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...

Thanks

## closed as off-topic by Giskard, user20105, Bayesian, Herr K., E. SommerMay 24 at 10:46

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – Giskard, user20105, Bayesian, Herr K., E. Sommer

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.

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