3.1 Production and Productivity
Firms are institutions whose purpose is to provide goods and services. Some firms can have very complex structures, defined by objectives, technology, hierarchies, labor contracts and much more. However, two basic traits characterize almost all firms: its technology, i.e. what and how the firm produces the goods or services it provides, and the objective of maximizing its profit. We will discuss another important issue, the contracts between firms and workers, in Chapters 12 and 13. Thus, in these chapters we will define a firm in the perhaps simplest possible way, that is, as a “black box” using inputs to produce output, with the goal of maximizing its profit.
As a first step, we must ask ourselves what are the production possibilities available to the firm, that is, the input-output combinations that the firm’s tecnology allows. For simplicity, we will assume that the firm’s output In reality, almost all firms provide several types of products. is a single good (e.g. pasta, measured in kilograms produced each week). We will also assume What we call labor therefore represents what in reality is a combination of factors: water, flour, and so on, besides labor itself. that production occurs in units (i.e. plants) that use a single input, namely labor, measured in days (abbreviated dd).
We must then ask ourselves what is the structure of the market that we are interested in analyzing. This chapter and the next deal with competitive markets, namely markets populated by a large number of small price-taking firms. Again for simplicity, “small” for us will mean “consisting of a single production unit”. A competitive firm can therefore choose how much labor to employ, i.e. labor is a variable input, but cannot choose the number (or the size) of its production units, i.e. its level of capital, which is a fixed input (fixed at a level equal to one).
The Firm's Production Function
To describe a firm’s efficient production frontier—that is, the only production possibilities the firm considers (the efficient ones)—we use a production function, that is, a function $Q=F(L)$ which associates to each quantity of labor $L$ the maximum amount of output $Q$ that can be obtained with that quantity of labor.
A type of production function very frequently used by economists (the only one we consider in detail in these notes) has the following form:
\(\begin{gathered} Q = A \sqrt{L} \end{gathered}\)
where $A>0$ is a productivity parameter that we will discuss later.
We have In Figure 3.1 the quantities of output indicated on the vertical axis are rounded. For example, with $L=2$ the quantity of output $Q=80\sqrt{2}=113.137..$ was rounded to 113. already seen this type of production function: setting $A=80$ we obtain the efficient frontier represented in Figure 3.1.
In the figure below, we can see how the shape of the production function changes as $A$ changes, under the assumption that labor is finely divisible.
Firm's Productivity
In brief, a firm’s productivity is given by the ratio between quantity of output and corresponding quantity of input (labor). Depending on what we mean exactly by “quantity of input” we obtain different measures of productivity. Two of these, average product and marginal product, we discuss next.
Average Product
Just as we can measure our average speed during a car trip by taking the ratio between distance covered and time elapsed, similarly we can measure a firm’s average productivity by taking the ratio between quantity of output produced and quantity of labor hired. And just like average speed can depend on how long we drive (in a longer trip we may feel tired and hence become more prudent and slower as time goes by), a firm’s average productivity depends, in general, on how much labor the firm hires. It is, in other words, a function of $L$. This function is called average product of labor. It measures, for each possible quantity $L$ of labor hired by the firm, the quantity of output produced, on average, by each unit of labor. The average product of labor is denoted by $AP_L$ and its formula is the following:
\(\begin{gathered} AP_L = \frac{Q}{L} = \frac{A\sqrt{L}}{L} = \frac{A}{\sqrt{L}} \end{gathered}\)
As Continuing the analogy with average speed, this means taking proportionally less time to make a short trip compared to a long one, for example due to less fatigue. can be seen from the formula (and in the next figure), the average product of labor is decreasing, meaning it becomes smaller as the quantity of labor $L$ used increases.
To better understand the concept of average product, let’s return to the example shown in Figure 3.1. We reproduce that example again, this time also showing the average product, in the following figure.
The figure below shows how output and average product vary with the parameter $A$ and with the quantity of labor used by the firm.
From the figure, we can see that, for any given amount of labor used, the quantity of output (and thus the average product) is higher when $A$ is larger. For example, a firm characterized by the production function $Q=10\sqrt{L}$ is more productive than a firm with the production function $Q=5\sqrt{L}$.
Marginal Product
To introduce the notion of marginal product, it is useful to go back to the analogy with speed. Say it is 7:45 PM and, while driving our car, we see the car’s speedometer indicate “72 km/h”. What does this number represent? It is a measure of the instantaneous speed of the car at 7:45 PM. In order to compute that number, the instrument performs two tasks. First, it measures the distance (in kilometers) covered in the fraction of time, say a tenth of a second, immediately before 7:45 PM. Second, given that speed is measures in kilometers per hour (km/h) and there are 36000 tenths of a second in one hour, it divides the distance by $1/36000$ (i.e. multiples it by 36000). In other words, the instrument computes the average speed during the extremely short trip that starts one tenth of a second before 7:45 PM, and ends at 7:45 PM. The indication “72 km/h” is apparently due to the fact that in that tenth of a second the car has covered 2 meters. Indeed, 2 meters are 1/500 of a kilometer, so the average speed during that tiny trip, i.e. the instantaneous speed at 7:45 PM, was 2 meters per tenth of a second, or $(1/500)/(1/36000)=72$ kilometers per hour.
Just like instantaneous speed is nothing else than average speed during the super-short trip that started a tenth of a second before glancing at the speedometer, similarly the marginal product of labor is just the average product of the last, minimal quantity $\Delta L$ of labor hired by the firm. Like average product, marginal product is also a function of the quantity $L$ of labor hired by the firm, and its formula is
\(\begin{gathered} MP_L = \frac{\Delta Q}{\Delta L} = \frac{F(L)-F(L-\Delta L)}{\Delta L} \end{gathered}\)
Given a production function of the form $Q = A\sqrt{L}$, assuming that labor is divisible, the marginal product is calculated as the derivative of the production function (the limit of the ratio $\Delta Q/\Delta L$ as $\Delta L$ tends to zero) and is therefore given by the formula
\(\begin{gathered} MP_L = \frac{A}{2\sqrt{L}}. \end{gathered}\)
Like average product, marginal product is decreasing. The larger the quantity of labor hired by the firm, the smaller is the additional output obtained from the last fraction of labor hired.
To understand better the notion of marginal product, let us return to the example illustrated in Figure 3.1.
The following figure shows how output, average product and marginal product of labor vary, as we change the parameter $A$ of the production function or the quantity of labor hired by the firm.
Average and marginal product of labor are closely related. As the figure shows, the two fuctions are both decreasing, but average product is higher than marginal product. If the marginal product were increasing (you can verify this by considering e.g. the production function $Q=L^2$), then average product would also be increasing, and it would be lower than marginal product. The reason for this is that marginal product is decreasing. Intuitively, think of what happens to your school grade average if during the year you get lower and lower grades. The average grade will also get lower and lower, but it will always be higher than the last grade you received.
Inverse Production Function
So far we have described the technology of a firm by means of a production function, that is, viewing the quantity of output produced as a function of the quantity of labor hired. For example, assuming $Q=80\sqrt{L}$ means that by hiring $L=1$ units of labor the firm obtains $Q=80\sqrt{1}=80$ units of output, by hiring $L=4$ units of labor the firm obtains $Q=80\sqrt{4}=160$ units of output, and so on.
In an equivalent way to describe technology, we can view labor hired as a function of output produced. Instead of asking how much output the firm obtains from each possible quantity of labor, we can ask: how much labor does the firm need, in order to obtain each possible quantity of output? Giving an answer to this question means specifying the inverse production function of the firm. Mathematically, this is simply the inverse function of the production function. For example, if the production function is $Q=80\sqrt{L}$, the corresponding inverse production function is $L=Q^2/6400$.
More generally, given any production of the form $Q=A\sqrt{L}$, it is easy to see that the corresponding inverse production function is given by
\(\begin{gathered} L = \frac{Q^2}{A^2} \end{gathered}\)
As we shall see in the next chapter, computing the inverse production function of the firm is important for the calculation of the firm’s costs.