2.6 Leontief (Fixed Proportions) Production Functions

In many production processes, labor and capital are used in a “fixed proportion.” For example, a steam locomotive needs to be driven by two people, an engineer (to operate the train) and a fireman (to shovel coal); or a conveyor belt on an assembly line may require a specific number of workers to function. How do we model this kind of process?

Let’s return to our island, and suppose Chuck has only one way of cracking open a coconut: he needs to use a sharp rock (a form of capital). With a pile of rocks at his disposal, Chuck could crack 2 coconuts open per hour. Unfortunately, the rock itself is shattered in the production process, so he needs one rock for each coconut he cracks open. If he has $L$ hours of labor and $K$ rocks, how many coconuts can he crack open?

Since he has to use labor and capital together, one of the two inputs is going to create a capacity constraint. Suppose, for example, that he has 2 rocks; then he can crack open up to 2 coconuts, depending on how much time he spends. On the other hand, suppose he’s decided to devote 3 hours; then he can crack open up to 6 coconuts, depending on how many rocks he has.

In general, if he has less than twice as many rocks as hours of labor — that is, $K < 2L$ — then capital will be the constraining factor, and he’ll crack open $K$ coconuts. On the other hand, if he has at least twice as many rocks as hours — that is, $K > 2L$ — then labor will be the limiting factor, so he’ll crack open $2L$ coconuts. In other words, we can define this as a piecewise function, \(q = f(L,K) = \begin{cases}2L & \text{ if } & K > 2L \\K & \text{ if } & K < 2L \end{cases}\) Another way of thinking about this is that it’s a function that returns the lower value of $2L$ and $K$: that is, \(q = f(L,K) = \min\{2L, K\}\) (You may note that this corresponds to the problem you had for homework after the first lecture!)

This class of function is sometimes called a fixed proportions function, since the most efficient way to use them (i.e., with no resources left unused) is in a fixed proportion. It is also called a Leontief production function, after the influential Nobel laureate Wassily Leontief, who pioneered its use in input-output analysis.

Marginal products and the MRTS

What are the marginal products of labor and capital? That depends on whether $K$ is greater or less than $2L$: \(\begin{aligned} K > 2L & \Rightarrow f(L,K) = 2L & \Rightarrow MP_L = 2, MP_K = 0\\ K < 2L & \Rightarrow f(L,K) = K & \Rightarrow MP_L = 0, MP_K = 1 \end{aligned}\) How do we interpret this economically? Well, if $K > 2L$, then some capital is going to waste. For example, if $K = 12$ and $L = 2$, then Chuck is only using 4 of his 12 stones; he could produce 2 more coconuts if he spent a third hour of labor, so $MP_L = 2$. On the other hand, getting more capital wouldn’t boost his production at all if he kept $L = 2$. Likewise, if he has 2 rocks and 2 hours of labor, he can only produce 2 coconuts; spending more time would do him no good without more rocks, so $MP_L = 0$; and each additional rock would mean one additional coconut cracked open, so $MP_K = 1$.

What about his MRTS? Again, we have to define things piecewise: \(MRTS = {MP_L \over MP_K} = \begin{cases}{2 \over 0} = \infty & \text{ if } & K > 2L \\{0 \over 1} = 0 & \text{ if } & K < 2L \end{cases}\) To make sense of this, let’s plot Chuck’s isoquants.

Isoquants

To draw Chuck’s isoquants, let’s think about the various ways Chuck could produce $q$ coconuts:

• He could use $q/2$ hours of labor and at least $q$ rocks. For example, to crack open 6 coconuts, he could use 3 hours of labor and either 6, or 8, or 10 rocks; therefore $(3,6)$, $(3,8)$, and $(3, 10)$ lie along his isoquant for $q = 6$.
• He could use $q$ rocks and at least $q/2$ hours of labor. For example, to crack open 6 coconuts, he could use 6 rocks and either 3, or 6, or 10 hours of labor; so $(3,6)$, $(6,6)$, and $(10, 6)$ also lie along his isoquant for $q = 6$.

Putting these all together gives us an L-shaped isoquant map:

Let’s pause for a moment to understand this map:

• The base of each L-shaped isoquant occurs where $K = 2L$: that is, where Chuck has just the right proportions of capital to labor (2 rocks for every hour of labor). You can see this “ridge line” by clicking the first check box. Along this line, the MRTS not well defined; there’s a discontinuity in the slope of the isoquant.
• Below and to the right of that line, $K < 2L$, so capital is the constraining factor; therefore in this region $MP_L = 0$ and so $MRTS = 0$ as well. We can see that the isoquants in this region do in fact have a slope of 0.
• Above and to the left of the line, $K > 2L$, so labor is the contraining factor; therefore in this region $MP_K = 0$ and so $MRTS$ is infinitely large. We can see that the isoquants in this region are vertical, which we can interpret as having “infinite slope.”

You’ll spend a fair bit of time in the live lecture talking about this case, since it’s new to most students. Come prepared with questions!