a) The marginal rate of technical substitution is
MRTS = ΔK/ΔL
If labor input, L, is decreased by one unit, by how much must one increase capital, K, to hold production constant? The answer is given by MRTS.
b) To show the relation with the marginal products, we start by the following reasoning: Suppose we increase the input of labor, ΔL, by one unit. Depending on how productive the labor is, this will increase production. The total increase will then be MPL*ΔL = Δq.
Suppose we then decrease the input of capital, ΔK, by an amount that makes the associated decrease in production just as large as the previous increase. We then get that MPK*ΔK = -Δq. Summing these two together must give a total change of zero:
MPL*ΔL = Δq
MPK*ΔK = -Δq
MPL*ΔL+MPK*ΔK = 0
Now, rearrange the last expression: Move MPK*ΔK over to the right-hand side, and then divide both sides by MPK and by -ΔL. You will then have an expression for MRTS:
This is the relation we asked for.