2.3 Isoquants

One of the core questions we might consider regarding producing something is how to produce it. For example, suppose Chuck could produce 20 fish using many hours of labor and a flimsy spear, or just a few hours and a strong net, or any number of other combinations of labor and capital. The set of all such combinations that could produce 20 fish is of interest to Chuck, because it gives him a set of options to choose from.

For a production function with two inputs — such as labor ($L$) and capital ($K$) — the visual representation of this menu of options called an isoquant, from the Greek iso meaning “same,” and quant meaning “amount.” An isoquant shows the set of all combinations of labor and capital that could be used to produce $q$ units of output: \(\text{Isoquant for }q = \{(L,K)\ |\ f(L,K) = q\}\) Mathmatically, this is just the level set for quantity $q$ that we saw in the last chapter. And just as we could draw a contour map for a multivariable function, we can draw an “isoquant map” for a production function. For example, the following diagram shows the surface plot, isoquant, and isoquant map for $f(L,K) = 4L^{1 \over 2}K$: