# 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...

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 '19 at 9:31