2.9 Scaling Production in the Short Run

For the last part of this chapter, let’s consider what happens as Chuck “scales up” production: that is, increases his quantity of output.

A critical question to ask is whether he can scale all his inputs at a moment’s notice. Clearly, each morning he can decide how much time to devote to, say, fishing or cracking coconuts open; so his labor is completely flexible. But it may not be possible for him to scale up his capital stock: he starts the day with a certain number of sharp rocks and fishing nets, and probably (unless he wants to take the day making more) that’s the capital he’s stuck with.

To account for the fact that different inputs may take different amounts of time to scale, we’ll introduce the concept of the short run and the long run. In the short run, we’ll assume that labor can be freely scaled up or down, but that capital is fixed; in the long run, we’ll assume that both capital and labor can be changed.

These, of course, are oversimplifications: in the real world, there are many more than two inputs, and various time scales on which they may be changed. Many technology startups find that the hardest part about scaling up is finding top talent (i.e., hiring more labor quickly), while cloud computing allows them to scale up their capital stock quickly. But for now, we’ll use the metaphor of labor as a quickly changeable input, and capital as the resource which is fixed in the short run.

The short-run production function

If there are only two factors of production, and one of them is fixed, the short-run production function becomes a univariate function: that is, $f(L, \overline K)$ may be written as $f(L | K = \overline K)$. We can interpret this as the function defined by taking a vertical slice of the production function at $K = \overline K$. For example, if $f(L,K) = \sqrt{LK}$, then the short-run production function may be seen in the diagram below:

We might also just write $f(L)$ having substituted $\overline K$ in: for example, if $f(L,K) = \sqrt{LK}$, then if $\overline K = 100$ we could just write this as the univariate function \(f(L) = 10 \sqrt{L}\) This is shown in the right-hand graph above.

Scaling and the marginal product of labor

A critical attribute of a production function is how the marginal product changes as inputs are added. In this case, we can ask how adding labor affects the $MP_L$. One can imagine that splitting coconuts open with a rock is tiring work, and that Chuck could produce less and less coconut milk for each additional hour of labor. This is reflected in a decreasing marginal product of labor, as shown in the diagram below:

On the other hand, a different production technology might exhibit different characteristics. The company Silk, for example, makes coconut milk and sells it in grocery stores. They have giant machines that can work for hours at a time extracting coconut milk from coconuts. The workers feeding coconuts into those machines don’t get nearly as tired as Chuck, so the number of coconuts they can produce each hour is fairly constant — say, 100 coconuts per hour. This production process would be described by the linear production function $f(L) = 100L$, with $MP_L = 100$.