6.4 Application II: Deriving the Slope of a PPF (MRT)

We can use the same approach when developing an expression for the MRT. In general, the equation for the PPF will be something like \(L(x_1,x_2) = \overline L\) where $L(x_1,x_2)$ is the labor required to produce $x_1$ units of good 1 and $x_2$ units of good 2, and $\overline L$ is the total amount of labor available to produce these two goods.

If we just apply the formula of the implicit function theorem, therefore, the slope of the PPF (i.e. the MRT) is given by \(MRT = \left|-\frac{\partial L/\partial x_1}{\partial L/\partial x_2}\right|\) For example, in the case of a linear PPF with the equation \(\tfrac{1}{3}x_1 + \tfrac{1}{2}x_2 = 150\) the MRT is \(MRT = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}\) In the case of the curved PPF with the equation \(\frac{x_1^2}{100} + \frac{x_2^2}{36} = 100\) the MRT is \(MRT = \frac{\frac{x_1}{50}}{\frac{x_2}{18}} = \frac{9x_1}{25x_2}\) Note that as one moves to the right along the PPF (increasing $x_1$ and decreasing $x_2$), the MRT increases, indicating a steeper PPF.

Interpreting the MRT as the ratio of resource requirements

Recall that the marginal product of labor is defined as the amount of output produced by another unit of labor: \(MP_L = {\Delta x \over \Delta L}\) The inverse of the marginal product of labor is the amount of labor needed to produce another unit of output: \({1 \over MP_L} = {\Delta L \over \Delta x}\) For example, think back to how we calculated the MRT in the linear example in Wednesday’s lecture: we were given \(\begin{aligned} x_1 &= 3L_1\\ x_2 &= 2L_2 \end{aligned}\) and we argued:

What is Chuck’s opportunity cost of producing another fish, therefore? Well, it will take 20 minutes to produce another fish. In that time, he could produce 2/3 of a coconut (since an additional coconut would take him 30 minutes). Therefore his MRT is 2/3 coconuts per fish.

The “20 minutes to produce another fish” is 1/3 of an hour, which is the inverse of the MPL for good 1: \({1 \over MP_{L1}} = {1 \over 3 \text{ fish per hour}} = {1 \over 3} \text{ hours/fish = 20 minutes/fish}\) Likewise, the “30 minutes to produce a coconut” is 1/2 of an hour, which is the inverse of the MPL for good 2: \({1 \over MP_{L2}} = {1 \over 2 \text{ coconuts per hour}} = {1 \over 2} \text{ hours/coconut = 30 minutes/coconut}\) Therefore the MRT can be thought of as the ratio of the inverses of the MPL’s: \(\begin{aligned} MRT &= {\partial L/\partial x_1 \over \partial L/ \partial x_2}\\ &= {1/MP_{L1} \over 1/MP_{L2} }\\ &= \frac{ {1 \over 3} \text{ hours/fish} }{ {1 \over 2} \text{ hours/coconut} }\\ &= {2 \text{ coconuts} \over 3 \text{ fish} }\end{aligned}\) or, more intuitively, \(MRT = \frac{\text{Resources required to produce another unit of good 1} }{\text{Resources required to produce another unit of good 2} }\) The more resources are required to produce another unit of good 1, relative to good 2, the steeper the MRT, and the more good 2 must be given up to get another unit of good 1; and vice versa.

A simpler formula, and some danger

Note that if you simplify the above equation, you get \(MRT = {1/MP_{L1} \over 1/MP_{L2} } = {MP_{L2} \over MP_{L1}}\) This is nicely intuitive formula: if you spend one less hour making good 2 and one more making good 1, you give up (approximately) $MP_L2$ units of good 2, and gain (approximately) $MP_L1$ units of good 1. However, my experience is many students get tripped up by this formula, because in all our other formulas using the implicit function theorem (the MRTS for production functions, and MRS for utility functions, which we’ll see next week), it’s a derivative with respect to good 1 in the numerator and a derivative with respect to good 2 in the denominator. So, feel free to use this formula – but make sure you apply it correctly, and understand the distinction between it and the others.