5.3 The Marginal Rate of Transformation: the Slope of the PPF

Anywhere along the PPF, Chuck cannot make more of both goods: if he wants to produce more fish, he needs to produce fewer coconuts, and vice versa. The slope of the PPF measures the rate at which his available technology allows him to trade off between two goods. In particular, it represents the opportunity cost of producing an additional unit of good 1, in terms of units of good 2 given up. We call this slope the marginal rate of transformation, or MRT.

We’ll generally just use the absolute value of the MRT, since we know it represents a tradeoff and will (nearly) always be negative.

The curvature of the MRT is clearly related to the nature of the production functions for the two goods. To see how, think of what happens as we move to the right along the PPF.

Case 1: Constant Marginal Products of Labor

Let’s think first of the linear case, in which \(\begin{aligned}x_1 &= f_1(L_1) = 3L_1\\ x_2 &= f_2(L_2) = 2L_2\end{aligned}\) Intuitively, we can say that these mean that Chuck can produce 3 fish per hour, or 2 coconuts per hour.

When we inverted these, we got the labor requirement functions \(\begin{aligned}L_1(x_1) &= \tfrac{1}{3}x_1\\ L_2(x_2) &= \tfrac{1}{2}x_2\end{aligned}\) Intuitively, these mean that producing another fish takes one-third of an hour, or 20 minutes; and producing another coconut takes half an hour, or 30 minutes.

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 \(MRT = {2 \over 3}\text{ coconuts per fish}\) And indeed, we can see that the slope of the PPF is indeed $-2/3$:

[ See interactive graph online at https://www.econgraphs.org/graphs/scarcity/ppf/resource_constraint_ppf ]

Case 2: Diminishing Marginal Products of Labor

Now let’s think about why the MRT increases as you move to the right along the PPF for the production functions with diminishing marginal products of labor. We’ll do the math on this on Friday, but for now we can consider what’s happening intuitively. Because the production functions exhibit diminishing marginal products of labor, as Chuck spends more time fishing and less time on coconuts, his $MP_{L1}$ is decreasing and $MP_{L2}$ is increasing. Intuitively, each hour of additional fish production produces less and less additional fish, while each worker he subtracts from coconuts production would have produced an increasing number of coconuts. Therefore, his opportunity cost of producing an additional fish is increasing, causing the MRT to increase.

Visually, you can see this in the following set of diagrams. The large square diagram on the left shows the PPF; the two smaller diagrams show the production functions for fish and coconuts. Try shifting labor by moving the points in the smaller diagrams left or right; and see what happens to the marginal products of labor and the $MRT$:

[ See interactive graph online at https://www.econgraphs.org/graphs/scarcity/ppf/mpls_and_mrt ]

Again, we’ll get to the mathematical derivation of the MRT’s on Friday; for now, focus on the relationship between the verbal description of the diminishing marginal products of labor represented by the decreasing slopes of the production function, and the increasing opportunity cost represented by the increasingly negative slope of the PPF.