MRTS is equal to the ratio of marginal products of two inputs. I'm having a hard time understanding the intuition behind the formula.

For example, I am looking at the MRTS of good b for a. Thus, MRTS = marginal product of a / marginal product of b.

Let's say that good a is very productive with a high marginal product, while good b is unproductive and has a low marginal product. This means that the MRTS of good b for a is very high. Thus, a lot of input a must be reduced for a 1 unit increase in input b, to maintain the same level of output.

This conclusion does not make sense to me since I thought that input a was very productive. How come when we increase the unproductive input by 1 unit, we have to reduce a lot of the productive input in order to maintain the same level of output?

The question is based on the definition of the MRTS of B for A as the (the amount of input A)/(the amount of input B), which is flipping the numerator and the denominator of the correct definition, probably because of the definition of the MRTS using the marginal product of A and B.

The correct definition of the MRTS is (the amount of input B)/(the amount of input A). With this definition, the intuition behind the MRTS turns out to make sense. We want to know the meaning of the MRTS of input B for A: how much extra of input B we need if we are going to give up 1 unit of input A to produce the same amount of output.

We can simply say the level of output can be determined by (marginal product of an input)*(the amount of the input)=output level

With the equation above, in the MRTS context, we can mathematically express the "how much extra of input B (X units of input B) we need if we are going to give up 1 unit of input A to produce the same amount of output" part by the following equation.

(marginal product of A)*(1 unit of A)=(marginal product of B)*(X units of B) because we want to keep the amount of output the same with the changes in the amount of inputs.

Rearranging this equation above, we can get (marginal product of A)/(marginal product of B)=(X units of B)/(1 unit of A)

Then we can find the value of MRTS of B for A as "X", also as the value of the formula for the MRTS.

example)

MP of A = 2 (more productive)

MP of B = 1 (less productive)

then MRTS of B for A is 2, following the formula above, which means that we need 2 extra units of B when we lose 1 unit of input A to produce the same output because input A is twice as productive as input B.