# 16.5 The MRT and Marginal Cost

## Case 1: MRT with only labor

In Chapter 3 we showed that if there was only one resource in an economy, labor, then the MRT was the ratio of the marginal products of labor: \(MRT = {MP_{L2} \over MP_{L1}}\) In Chapter 13, we showed that marginal cost may be written \(MC = {w \over MP_L}\) That is, the marginal cost of producing another unit of output is the wage rate times the amount of labor that it would take to produce another unit of output, which is $1/MP_L$. If we cross multiply, we get \(MP_L = {w \over MC}\) This is true for firms that produce good 1, as well as firms that produce good 2; so we have \(MP_{L1} = {w \over MC_1}\) \(MP_{L2} = {w \over MC_2}\) Hence we may rewrite the MRT as \(MRT = {MP_{L2} \over MP_{L1}} = \frac{w \over MC_2}{w \over MC_1}={MC_1 \over MC_2}\)

## Case 2: MRT with more than one input

Most production functions involve a lot more than one input. There is another way of thinking about the MRT: instead of thinking about moving one worker from producing good 2 to producing good 1, we can think about moving $$1$ of resources from good 2 to good 1.

Put another way: instead of thinking about the total amount of labor in the economy $\overline L$, let’s think about the total *market value of all resources*, $\overline C$. For example, if we have $\overline L$ units of labor and $\overline K$ units of capital in an economy, then $\overline C = w\overline L + r\overline K$.

Using our normal methods for finding cost functions, we can then say that the total cost of producing $y_1$ units of good 1 and $y_2$ units of good 2 must be the total value of all resources. This is just another way of writing the PPF: \(C_1(Y_1)+C_2(Y_2) = \overline C\) By the implicit function theorem, the MRT must be \(MRT = \frac{MC_1}{MC_2}\) In other words, if the economy is producing at a point along the PPF, the ratio of the marginal cost of the last unit of good 1 produced to the marginal cost of the last unit of good 2 produced will be equal to the slope of the PPF.