5.2 The Production Possibilities Frontier (PPF)

Given his resource constraints, Chuck is going to be limited in the number of fish he can catch, and the number of coconuts he can break open. Some “bundles,” or combinations of outputs, will be possible for Chuck to produce; others will not. We will call Chuck’s “feasible set” of output bundles his production possibilities set. The boundary between that set and the set of bundles he cannot produce we’ll call his production possibilities frontier.

If Chuck is not using all of his resources – or is using them inefficiently – he will produce at a point strictly within his production possibilities set. At such a point, he can produce more fish without necessarily producing fewer coconuts, and vice versa. Along the PPF, however, he faces a tradeoff: if he wants to produce more fish, he needs to produce fewer coconuts.

A PPF might look something like this:

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

Chuck’s PPF is related to his resource constraints via his production functions: depending on how he can transform labor and capital into fish and coconuts, different combinations of output will be available to him.

For simplicity, let’s first examine the case in which time (i..e hours of labor) is the only resource Chuck can shift. One way of thinking about this is that he can produce fish and coconuts using only labor, so that his production functions are one-variable functions, of the form $f(L)$. Another way of thinking about this is that his production functions are still functions of labor and capital, but that we’re thinking about the “short run” production functions in which capital is fixed: that is, $f(L,\overline K)$ for some fixed $\overline K$. In the last part of the chapter we’ll examine the “long run” problem in which Chuck can shift both labor and capital, and see how the short-run and long-run production sets relate to one another.

A bit of notation: we’ll call fish “good 1:” we’ll write the quantity of fish as $x_1$, the amount of labor devoted to fish $L_1$ ,and the production function producing fish $f_1(L_1)$; likewise, we’ll call coconuts “good 2” and write the quantity of coconuts as $x_2 = f_2(L_2)$.

Example 1: A PPF with linear technologies

Let’s start with a simple case of linear technologies: assume that with each hour of labor, Chuck can produce 3 fish or 2 coconuts. This simple setup is really saying that there is an economy with a single resource (labor) which can be used in one of two production functions: the quantity of fish, $x_1$, is given by \(x_1 = f_1(L_1) = 3L_1\) and the quantity of coconuts, $x_2$, is given by \(x_2 = f_2(L_2) = 2L_2\) For any division of labor $(L_1,L_2)$, the production functions tell us how many fish and coconuts Chuck will produce. Therefore the set of feasible production possibilities depends on the set of feasible resource allocations.

We can illustrate this by viewing the resource constraint and the PPF side-by-side, as shown below. Drag the orange dot all the way to the right: we can see that if Chuck devotes all 150 hours of labor to producing fish, he can produce $150 \times 3 = 450$ fish. Likewise, if you move the orange dot all the way up and to the left, you can see that if Chuck devotes all 150 hours to coconuts, he can produce $150 \times 2 = 300$ coconuts.

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

How do we derive the equation of this PPF? Note that the PPF is defined by three equations:

• The production functions relate $x_1$ to $L_1$ and $x_2$ to $L_2$.
• The resource constraint relates $L_1$ and $L_2$ to the total supply of labor, $\overline L$.

If we invert the production functions — that is, solve each for $L$ as a function of $x$, instead of $x$ as a function of $L$ — we get the amount of labor required to produce $x_1$ and $x_2$ units of output: \(\begin{aligned} L_1 &= \tfrac{1}{3}x_1\\ L_2 &= \tfrac{1}{2}x_2 \end{aligned}\) If we plug this into the resource constraint, we get the equation of the PPF: \(\begin{aligned} \tfrac{1}{3}x_1+\tfrac{1}{2}x_2 &=150 \end{aligned}\) This is the equation of the red line in the right-hand panel above.

One way of thinking about it is that the left-hand side of the equation represents “the total labor required to produce the bundle $(x_1,x_2)$,” which we could write as $L(x_1,x_2)$; and the right-hand side represents the total amount of labor we have, $\overline L$. Thus another way of thinking about the equation fo the PPF is as \(L(x_1,x_2) = \overline L\)

Example 2: A PPF with diminishing $MP_L$

Let’s now think of a production function in which the marginal product of labor is diminishing. In particular, suppose \(x_1 = f_1(L_1) = 10\sqrt L_1\) \(x_2 = f_2(L_2) = 6\sqrt L_2\) and there is a total of $\overline L = 100$ units of labor to be devoted to the production of fish (good 1) and coconuts (good 2).

Again we can illustrate this by viewing the resource constraint and the PPF side-by-side, as shown below. Now if you drag the orange dot all the way to the right, we can see that if the economy devotes all 100 hours of labor to producing fish, it can produce $10\sqrt{100} = 100$ fish. Likewise, if you move the orange dot all the way up and to the left, we can see that if the economy devotes all 100 hours to coconuts, it can produce $6\sqrt{100} = 60$ coconuts.

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

Following the same method as in the linear case, we invert the production functions to get the labor requirements: \(\begin{aligned} L_1 &= \frac{x_1^2}{100}\\ L_2 &= \frac{x_2^2}{36} \end{aligned}\) Plugging this into the resource constraint gives us the equation of the PPF, \(\begin{aligned} \frac{x_1^2}{100} + \frac{x_2^2}{36} = 100 \end{aligned}\) This is the equation of the red curve in the right-hand diagram above.

We see that the slope of the PPF increases as we move to the right, whereas in the previous example of the linear PPF it was constant. What does this slope represent, and what does this slope have to do with whether the $MP_L$ of the production functions are constant or decreasing?