Explain why in the long run a firm that is cost minimizing will choose K and L where:

w/MPL = r/MPK
What does this tell you about the marginal cost of increasing output through hiring labor and the marginal cost of increasing output through adding capital?

This is just a simple rearrangement of the cost-minimization condition MRTS = w/r. Because MRTS = MPL/MPK, then rearranging gives the condition above. For a firm in the long run minimizing costs, if the firm wishes to increase output it can do so by adding labor or capital (or both). The condition above says the cost of an additional unit of output (Q) from increasing labor must equal the cost of increasing output from increasing capital. Intuitively, if this were not the case, the firm could reduce output by lowering the input corresponding to the higher marginal cost and replace that output by increasing the input correspond to the lower marginal cost. Doing so would reduce the overall cost without sacrificing output.