3.3 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. When there’s only one input — labor — this means shifting a single hour of labor from producing good 2 to producing good 1. Since the $MP_L$’s of the production functions measures the amount of output produced by the last hour of labor, it follows that spending one less hour producing good 2 means we give up approximately $MP_{L2}$ units of good 2; and likewise, when we spend one more hour producing good 1, we gain $MP_{L1}$ more units of good 1. Therefore \(MRT = \frac{\Delta x_2}{\Delta x_1} = \frac{MP_{L2}}{MP_{L1}}\) Let’s look at the MRT for the two examples we derived in the previous section.
Example 1: MRT with linear technologies
In our first example we had \(x_1 = f_1(L_1) = 3L_1\) \(x_2 = f_2(L_2) = 2L_2\) therefore \(MP_{L1} = df_1(L_1)/dL_1 = 3\) \(MP_{L2} = df_2(L_2)/dl_2 = 2\) Hence \(MRT = \frac{MP_{L2}}{MP_{L1}} = \frac{2}{3}\) Indeed, if you look at the graph of the PPF on the previous page, you can see that it has a constant slope of $-\frac{2}{3}$. In other words, when the marginal products are constant, the opportunity cost of producing your first unit of good 1 is the same as the opportunity cost of producing the last (or any other); so the MRT is constant, and the PPF is a straight line.
Example 2: MRT with diminishing $MP_L$
In our second example we had the production functions \(x_1 = f_1(L_1) = 10\sqrt L_1\) \(x_2 = f_2(L_2) = 6\sqrt L_2\) Therefore the marginal products of labor are \(MP_{L1} = f^\prime_1(L_1) = \frac{5}{\sqrt L_1}\) \(MP_{L2} = f^\prime_2(L_2) = \frac{3}{\sqrt L_2}\) so the MRT is \(MRT = \frac{MP_{L2}}{MP_{L1}} = \frac{3 \sqrt L_1}{5 \sqrt L_2}\) This is an OK start, but we want to express the slope of the PPF in terms of the quantity of the two goods, not the quantity of labor. To do this, we invert the production functions (i.e. solve them for $L$ as a function of $q$) to get \(L_1 = (x_1/10)^2\) \(L_2 = (x_2/6)^2\) Substituting this into the equation for the MRT gives us \(MRT = \frac{3 \times x_1/10}{5 \times x_2/6} = \frac{9x_1}{25x_2}\) This is increasing as we move along the PPF to the right — that is, as $x_1$ increases and $x_2$ decreases, the slope of the PPF gets steeper.
Let’s think about why the MRT increases as you move to the right along the PPF for these production functions. Recall that in this example, the production functions exhibit diminishing marginal products of labor. That means that 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.
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 $MRT$: