Production functions and marginal product

Section 3.1 Production functions and marginal product

We begin our discussion of firm theory with a model of firm production. The activity of most firms can be captured by considering how firms transform some combination of inputs - factors which contribute to their production - into output - a good or service that is then sold to a buyer. Firms typically use many inputs in the production of its output. If an auto manufacturer produces a pickup truck, it needs to synthesize:

steel; glass; paint; leather (for seats); bulbs (for headlights and interior lights); engineers (to design the truck); workers in assembly plants; assembly machinery; the assembly plant itself; and, much more!

For a restaurant who produces a salad, it needs to incorporate:

ingredients; bowls; utensils; napkins; a chef; a server; aprons for the chef and servers; a manager; a restaurant; tables; chairs; advertising; refrigerator, and, probably even more!

These lists are not exhaustive, but we can quickly gain an appreciation for how lengthy a list of inputs can be. The analysis of production focuses on how economists model the transformation of inputs into output. Often, this transformation is called the firm’s production technology.

Modeling production is no easy task. Take the restaurant’s salad: how many units of each input is really needed to produce one salad? What are the relationships between inputs? Are some substitutable for one another (like plates for bowls, or romaine lettuce for iceberg lettuce)? How does the number of aprons needed relate to the number of servers? In the abstract, we can model the production process with a production function. If $q$ gives the number of units of output, and each input is numbered $x_1\text{,}$ $x_2\text{,}$ $\ldots x_n\text{,}$ then the production function would be the mathematical function $f\text{:}$

\begin{equation*} q = f(x_1, x_2, _3, \ldots x_n) \end{equation*}

In practice, $n$ would be the number of inputs, which could be rather large. So, let’s look for a way to simplify our analysis. First, we split our inputs into two very broad categories, labor and capital. Labor will serve as our representative for inputs of human effort: the server, the chef, the manager. One “unit” of labor may vary from context to context, whether it is an hour of labor, or a day of labor, or an employee as a whole. While advanced analyses often consider further subdivisions (skilled labor versus unskilled labor, for example), it will suffice to generalize all units as labor, denoted by $L\text{.}$

The second category is capital, which we denote by $K\text{.}$ Capital refers to physical objects used in the production process. In the restaurant example, capital covers everything from the candlestick holders and the chef’s mixing bowls to the refrigerator and the restaurant itself. With this simplification from lots of inputs to two, we typically expression the firm’s production function as

\begin{equation*} q = f(K, L) \end{equation*}

Our model of inputs is critical for analyzing firm decision-making. To see why, consider how many salads the restaurant wishes to serve on any given day. What if the restaurant decided it wanted to produce and sell 100 more salads on Tuesday than it did the day before? On Tuesday, there are inputs the restaurant would need more of: more ingredients, more bowls, maybe more chefs or more servers, perhaps another refrigerator or more tables. Now, consider which of those inputs would be fairly easy for the restaurant to acquire on short notice. More ingredients could be purchased at a local supermarket, and new servers could be hired.

What about other inputs? If the restaurant needed a new refrigerator to keep those extra ingredients fresh, it might be more difficult to get an industrial-size fridge very quickly. What about the tables? Even if the restaurant could acquire the physical tables on short notice, the restaurant might not have the physical space to accommodate the new tables. The restaurant is bound to its lease, and couldn’t relocate for some time - maybe a year. So, decision making over a short time horizon is bound by some constraints: the inputs that the firm cannot change.

Now, if the restaurant were considering an expansion for the distant future, it surely could consider how to equip its new kitchen with enough refrigerator space, or how to plan for a larger space for more customers. The decision makers running the restaurant have far fewer constraints on their decision making when they consider the longer time horizon.

The lesson? Some inputs are easier to change than others. This leads us to distinguish between two time horizons: the short run and the long run. In short,

the short run is any period of time short enough that at least one input cannot be changed;
the long run is any period of time long enough that any input can be changed.

If the decision maker is stuck with an input that cannot be changed, the decision maker is considering a short-run decision, such as changing output from day to day. If a decision maker plans for a context where any input is able to be changed, he is considering a long-run decision. One helpful way to think of this is to identify the one input most difficult to change. For the restaurant, maybe this is the physical space itself, which the restaurant rents according to a yearly lease. In this example then,

any decision making less than a year ahead would be a short-run decision, since the restaurant will be constrained to their current lease and unable to change it;
any decision making over a year ahead would be a long-run decision, since the restaurant can consider changing any of its inputs, including its physical space.¹

Short-run versus long-run time horizons allow us to then categorize all of our inputs based on their ease of change. An input in production is a fixed input if it cannot be changed in the short run (like the restaurant itself.). An input is a variable input if it can be changed in either the short run or the long run (like the ingredients for the salad, or the servers.) If we reconsider the long list of inputs for the restaurant, some are clearly fixed inputs while others are variable inputs. But once again, it will benefit us to simplify from many inputs to two. Labor ($L$) will be our variable input, since hiring (and firing) workers is generally simpler. Capital ($K$) will be our stand in for a fixed input. Recall that the concept of physical capital covers a very wide range of objects, from forks and tablecloths (easy to change) to heavy appliances and the physical space itself (not so easy to change.) With $K$ as the fixed input, this means we will interpret capital more as the heavy machinery and physical spaces required for the production process.

As with any relationship, we can express the relationship between inputs and output in a table or with a mathematical function. We will start with a table, and begin by considering short-run production. This will simplify the model a bit, because one input ($K$) is fixed. This allows us to represent production as

\begin{equation*} q = f(\bar{K}, L) \end{equation*}

where the notation $\bar{K}$ is used to indicate that the level of capital is fixed and cannot be changed.² Here, a unit of labor can be interpreted in several ways: one unit could mean “one employee” or “one hour of labor from an employee.” In a table, we can express the short-run production relationship between labor $L$ and output $q$ since capital is fixed:

labor ($L$)	output ($q$)

0	0
1	10
2	18
3	24
4	26

An example of a short-run production function in tabular form.

One critical measure of production is the firm’s marginal product of labor. Marginal product of labor gives the added output produced from an additional unit of labor, and can be expressed as

\begin{equation*} MP_L = \frac{\Delta q}{\Delta L} \end{equation*}

Marginal product of labor is a rate of change, much in the same way marginal utility is a rate of change.³ Through the marginal product of labor, we can examine the differing impact of hiring each worker as production expands. To measure the $MP_L$ in the table above, we calculate the change in output from row to row, dividing by the change in labor from row to row. Since labor changes by one unit each row, the denominator of $MP_L$ will always be 1:

labor ($L$)	output ($q$)	$MP_L = \frac{\Delta q}{\Delta L}$

0	0	-
1	10	$\frac{10}{1} = 10$
2	18	$\frac{8}{1} = 8$
3	24	6
4	26	2

An example of a short-run production function including marginal product of labor.

We can visualize the table above by plotting points on a graph. The dependent variable $q$ will go on the y-axis, while labor $L$ goes on the x-axis. This allows us to see a pair of important assumptions economists often make about production:

Figure 3.1.1. Short-run production function as given above.

Assumption P1 (labor has positive productivity): The production function is increasing in all inputs.

Assumption P1 indicates that adding more units of an input should lead to more output. Buying more ingredients, hiring more wait staff, and expanding restaurant space should allow a restaurant to produce more meals. In the model of production, this assumption gives a positive relationship between $L$ and $q\text{,}$ making the production function slope upward.⁴ Additionally, since the marginal product of labor is the rate of change between $L$ and $q\text{,}$ we can interpret $MP_L$ as the slope of the production function. Under P1, the marginal product of labor is positive, since the production function under $P1$ has a positive slope. This should make sense: as $L$ increases, $q$ should increase as well, giving $MP_L = \frac{\Delta q}{\Delta L} = \frac{+}{+} = +\text{.}$

Figure 3.1.2. $MP_L$ is the slope of the short-run production function.

The second assumption we will want to consider is the most critical assumption economists make regarding production:

Assumption P2 (diminishing marginal product of labor): As $L$ increases, $q$ increases at a decreasing rate.⁵

Diminishing marginal product of labor (sometimes called diminishing marginal returns) says that as more units of labor are added to the production process, additional units add fewer and fewer units of output. Hiring the first unit of labor increases output by a lot, hiring the second worker increases output but not by as much, and hiring the 10th worker increases output but not by very much, and so on. But what is the intuition behind this assumption?

This assumption is really about the concept of crowding out. To see why, consider Diversions Cafe, the coffee shop on campus. In the short run, capital at the coffee shop - the size of the Cafe, the number of espresso machines, the number of refrigerators - is fixed. But Diversions can hire however many baristas (labor) in needs to take orders, make drinks, serve tasty treats, and otherwise operate the cafe. Consider the added production from hiring each new barista. When Diversions goes from having 0 baristas working to having 1 baristas working, its output will experience a big jump. There’s actually someone there! This “big jump” means the marginal product of the first worker is very high. Now, consider when Diversions has 5 baristas working already. What is the added output having a 6th worker would bring? Well, a 6th barista could fill in for someone who is on break, or maybe help grab pastries out of the case, but with 5 other baristas already taking orders and making drinks, the added amount of output worker 6 would bring is low. The marginal product of the 6th worker is small, because the production process is already crowded! This captures the essence of diminishing $MP_L\text{:}$ when other inputs ($K$) are fixed, crowding out occurs and added units of labor get less and less productive.

Diminishing marginal product of labor translates to the concave shape of the production function, since as more $L$ is added, output continues to increase but it does so at a decreasing rate. This can be expressed equivalently by saying that as $L$ increases, the marginal product of labor decreases - or diminishes, as the name suggests!

Figure 3.1.3. Short-run production function and diminishing $MP_L\text{.}$ For an identical increase in labor, the change in output is larger with few workers than with many workers. $MP_L$ gets lower as the number of workers increases.

Crowding out represents a fundamental narrative about production processes, and assumption P2 is essential to our models of production. But there is an important twist to the narrative we should consider. Let’s go back to Diversions.

Consider what happens when Diversions hires its first barista. This first barista is likely to have a high marginal product of labor, as the output increases from 0 cups of coffee to more than zero. But think about this lone barista, who must take orders, make drinks, serve pastries, and keep things organized - all by herself! When the cafe hires a second barista, the two baristas could work more efficiently by specializing across tasks: one barista could handle taking orders and serving pastries, while the other made drinks. The added specialization from hiring the second barista can actually make the first barista more productive! That is, the cafe could experience increasing marginal product of labor if specialization occurs: added workers increase output at an increasing rate!

Now, specialization cannot occur forever. A third barista could specialize by focusing on pastries, leaving one for orders and one to make drinks. And perhaps a fourth barista could specialize in another task. But eventually, opportunities for specialization are exhausted because there just aren’t enough tasks to warrant that much specialization. When this occurs, crowding out drives production: with no task to specialize in, added workers are less and less productive, and diminishing marginal product arises. This leaves us with an alternate assumption ...

Assumption P2B (eventually diminishing marginal product of labor): There is some number of units of labor past which as $L$ increases, $q$ increases at a decreasing rate.⁶

Figure 3.1.4. Short-run production function with eventually diminishing $MP_L\text{.}$ To the left of the inflection point, specialization. To the right, crowding out.

Assumption P2B states that for the first several units of labor, specialization occurs and production exhibits increasing diminishing marginal product of labor; however, eventually, there is a point where crowding out occurs and production begins to exhibit diminishing marginal product of labor. Since $MP_L$ is about the slope of production, assumption P2B states that for low levels of labor, production is convex (increasing at an increasing rate), while past some critical number of workers, production is concave: increasing at a decreasing rate. The point where the switch in shape occurs is an inflection point on the curve.

Figure 3.1.5. Short-run production function and eventually diminishing $MP_L\text{.}$ To the left of the inflection point, $MP_L$ is increasing, since $\Delta q$ is getting larger; to the right, $MP_L$ is diminishing, since $\Delta q$ is getting smaller.

	Assumption P1	Assumption P2	Assumption P2B

Production $f(L)$ is ...	increasing	concave	convex then concave
Marginal product of labor ($MP_L$) is ...	positive	decreasing	increasing then decreasing

Economists often simplify models of production by using a production function that is an actual mathematical function. Just as we have seen with demand functions or utility functions, a mathematical function can provide precision to the relationship between variables, here input and output. In this section, we will focus on short-run production functions. We have seen that in general, they take the form $q = f(\bar{K}, L)\text{,}$ since $K$ is fixed in the short run. Since short-run production functions have only one variable - the variable input - they can be a bit easier to work with.

The exact function we want to use depends on the assumptions we make:

Under assumption P1, the production function should be increasing (have positive slope);
Under assumption P2, the production function should be concave (have a decreasing slope);
Under assumption P2B, the production function should be convex, then have an inflection point, beyond which the function is concave.

So, for example, if we assume there is diminishing marginal product of labor (P2), then an easy and common production function is a square root function:

\begin{equation*} q = \sqrt{L} \end{equation*}

Notice that even if this production process includes capital, since capital is fixed in the short run, economists typically omit it from the function.⁷ With this production function, we can compute the exact number of units of output which result from any value of $L\text{:}$

$L$	$q = \sqrt{L}$	$MP_L$

0	0
1	$\sqrt{1} = 1$	1
2	$\sqrt{2} = 1.41$	0.41
3	$\sqrt{3} = 1.73$	0.32
4	$\sqrt{4} = 2$	0.27

The short-run production function $q = \sqrt{L}$ in tabular form.

Figure 3.1.6. Short-run production function $q = \sqrt{L}\text{.}$

Under assumption P2B, a more complex mathematical function would have to be used, since the function would need to capture both the specialization (convex) and the crowding out (concave) stages of production. A function like

\begin{equation*} q = \sqrt[3]{L - 1} + 1 \end{equation*}

would do the trick, as shown in the graph below.

Figure 3.1.7. Short-run production function $q = \sqrt[3]{L - 1} + 1\text{.}$ Notice the inflection point at $L = 1$ and $q = 1\text{.}$

In Depth 3.1.8. Diminishing $MP_L$? Or eventually diminishing $MP_L$?

We have reached a critical modeling question for economists: which assumption should an economist make? P2 or P2B? There is certainly a tradeoff. On one hand, assumption P2B is admittedly more realistic. The theory of specialization and division of labor seems more than reasonable for many production processes, especially if the variable input is labor. On the other hand, working with eventually diminishing marginal product of labor production functions is more mathematically complex, since the functions themselves are less straightforward. As a result, any economist choosing to model production must choose which assumption to make, and how to trade the model’s realism with its simplicity.

In most applications, economists tend to choose the simpler P2 assumption and its associated concave-everywhere functions. The rationale for this choice may become clearer later in this section, and certainly becomes crystal clear in later economics courses. But, in essence, the choice comes down to optimal decision making for the firm. Since firms achieve incredible efficiency during the specialization section of the production function, it will never be optimal for firms to stop hiring workers during the specialization phase of production. If a firm continues to experience increasing $MP_L\text{,}$ it should continue hiring to maximize its profit.⁸ Therefore, if the objective of the model is to analyze optimal decision making, and an optimal decision will never be made during the specialization phase, then omitting this phase from the model - while technically less realistic - does not detract from the explanatory power of the model.

Omitting the specialization phase: a firm will only optimally choose to the right of the inflection point.

(long run - bonus section)

Key terms in this section:

production
input
output
production function
short run
long run
variable input
fixed input
marginal product of labor
diminishing marginal product of labor
crowding out
eventually diminishing marginal product of labor
specialization

Prev Top Next

labor (\(L\))	output (\(q\))	\(MP_L = \frac{\Delta q}{\Delta L}\)

0	0	-
1	10	\(\frac{10}{1} = 10\)
2	18	\(\frac{8}{1} = 8\)
3	24	6
4	26	2

\(L\)	\(q = \sqrt{L}\)	\(MP_L\)

0	0
1	\(\sqrt{1} = 1\)	1
2	\(\sqrt{2} = 1.41\)	0.41
3	\(\sqrt{3} = 1.73\)	0.32
4	\(\sqrt{4} = 2\)	0.27