4.5 Linear Production Functions
Suppose Chuck has two ways of fishing: he can try to catch fish in his bare hands, or by setting a net out:
- If he collects fish with his hands, he can catch 2 fish per hour.
- If he puts the net out, he can collect 4 fish without using any of his time.
Therefore, if Chuck spends $L$ hours fishing and puts out $K$ nets, the total number of fish caught is \(q = f(L,K) = 2L + 4K\)
Because this is a linear function, we call this a linear production technology.
Marginal products and the MRTS
The marginal products of labor and capital for this production function are constants: \(\begin{aligned} MP_L &= {df \over dL} = 2 {\text{fish} \over \text{hour}}\\ \\ MP_K &= {df \over dK} = 4 {\text{fish} \over \text{net}} \end{aligned}\) Notice that this linear function doesn’t exhibit increasing or diminishing marginal returns to either input: another hour of labor produces 2 additional fish, and another net produces 4 additional fish, regardless of how many fish have already been caught. We can also notice that the MRTS is constant: along any isoquant, at any point, the MRTS is exactly the same.
Visual representation: surface plot and isoquants
The isoquant for $q = 20$ fish is given by \(2L + 4K = 20\) or \(K = 5 - \tfrac{1}{2}L\) Note that the slope of each isoquant is $MRTS = {1 \over 2}$ regardless of the values of $L$ and $K$. On Friday we’ll see why this is the case.
The three-dimensional plot of this production function is familiar to us from the introduction to multivariate calculus last class: indeed, this was one of the functional forms you analyzed in section last week:
[ See interactive graph online at https://www.econgraphs.org/graphs/firm/technology/linear_isoquants ]
When do we use this production function?
We largely use a linear production function for highly automated processes, where each input is directly related to a specific amount of additional output. In fact, it wouldn’t be particularly realistic to say that no matter how may hours he had already been fishing, Chuck catches another 2 fish per each additional hour. Chuck surely gets tired! So let’s look at a different type of function that can capture diminishing marginal products of labor and capital.