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Chapter 4 / Monday, September 30 | Production Functions

4.6 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:

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

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

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!

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