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 in this case are constants: \(\begin{aligned} MP_L &= {df \over dL} = 2 {\text{fish} \over \text{hour}}\\ \\ MP_K &= {df \over dK} = 4 {\text{fish} \over \text{net}} \end{aligned}\) Therefore the MRTS is constant as well: \(MRTS = \frac{MP_L}{MP_K}= \frac{2 \text{ fish/hour}}{4 \text{ fish/net}} = \frac{1 \text{ net}}{2 \text{ hours}}\) Because the MRTS is constant for all values of $L$ and $K$, we sometimes call this a perfect substitutes production technology, because one net is a “perfect substitute” for two hours of labor. Indeed, Chuck could produce fish using only labor or only capital if he wanted to: there’s no need to use them in conjunction with one another.

Visual representation: surface plot and isoquants

The isoquant for $q = 20$ fish is given by \(2L + 4K = 20\) or \(K = 5 - \tfrac{1}{2}L\) Again, we can see from this expression that the slope of each isoquant is $MRTS = {1 \over 2}$ regardless of the values of $L$ and $K$.

The three-dimensional plot of this production function is familiar to us from the math review:

Next, let’s consider the opposite kind of production function: one in which labor and capital have to be used together in a fixed proportion.