Productivity Measures: Total, Marginal, and Average Product

Introduction

Welcome back! In our last lesson, we established the crucial distinction between the short run and the long run in production. We learned that in the short run, a firm operates with at least one fixed input (like capital) and can only change its output by varying its variable inputs (like labor).

Today, we will zoom in on that short-run production process. We're going to answer the question: "As a firm hires more workers, what happens to its total output?" This will lead us to three fundamental concepts in the theory of the firm: total product, marginal product, and average product.

Your learning outcome for this lesson is to be able to calculate and graph total product, marginal product, and average product of labor. We'll start with a practical example to build intuition, then formalize the concepts with calculations and graphs, and finally, we'll touch on the mathematical representation that connects to your analytical background.

1. From Inputs to Output: A Factory Simulation

To understand these concepts, let's start with a tangible example. Imagine a small factory that produces paper chains. The factory has a fixed amount of capital (scissors, tape dispensers) and adds workers (labor), one by one, to see how output (length of the chain) changes.

The following video brilliantly simulates this scenario and walks through the process of collecting data, calculating our key metrics, and plotting them. It’s an excellent way to see these abstract concepts in action.

2. Defining and Calculating the Three Measures of Product

As you saw in the video, we can describe a firm's production in the short run using three related metrics. Let's formally define them. We'll use $L$ for the quantity of labor (our variable input) and $Q$ for the quantity of output (our total product).

• Total Product (TP or $Q$): This is the simplest concept. It's the total amount of output produced with a given amount of labor and a fixed amount of capital. It's the "total length of the paper chain" from the video.

• Average Product (AP): This measures the output per worker. It gives us a sense of the average productivity of the workforce.

  $$\text{Average Product (AP)}=\frac{\text{Total Product}}{\text{Quantity of Labor}}=\frac{Q}{L}$$

• Marginal Product (MP): This is one of the most important concepts in microeconomics. It measures the additional output produced by hiring one additional unit of labor.

  $$\text{Marginal Product (MP)}=\frac{\text{Change in Total Product}}{\text{Change in Quantity of Labor}}=\frac{\Delta Q}{\Delta L}$$

  The "marginal" value tells you the contribution of the last worker hired.

Let's look at another numerical example to solidify your understanding of the calculations.

3. The Relationships Between the Curves

Now that you can define, calculate, and plot these three curves, the most important step is to understand how they relate to each other. The shapes of these curves are not random; they follow a predictable pattern dictated by economic principles.

The following segments of the first video you watched provide an excellent analysis of these relationships.

Here is a summary of those key graphical relationships, which you can see illustrated in the image below.

1. When MP > AP, the AP curve is rising. If the last worker you hired is more productive than the average of all previous workers, they pull the average up.
2. When MP < AP, the AP curve is falling. If the last worker is less productive than the average, they drag the average down.
3. The MP curve intersects the AP curve at the maximum point of AP. This is the logical result of the first two points. The average can only be at its peak when the marginal contribution is exactly equal to the average.
4. When TP is increasing at an increasing rate, MP is rising.
5. When TP is increasing at a decreasing rate, MP is falling (but still positive). The point where TP's slope changes from increasing to decreasing is where MP is at its maximum.
6. When TP reaches its maximum, MP is zero. If adding another worker doesn't change total output at all, their marginal product is zero.
7. When TP begins to fall, MP is negative. If adding another worker causes total output to decrease (perhaps due to overcrowding), their marginal contribution is negative.

4. A More Formal Approach: Calculus and the Production Function

Given your BTech background, we can express these relationships with more mathematical precision. Recall the short-run production function, $Q=f(\bar{K},L)$. Since $K$ is fixed, we can simply write output as a function of labor: $Q=f(L)$.

• Average Product is, as defined before, the function divided by the variable:

  $$A{P}_L=\frac{Q}{L}=\frac{f(L)}{L}$$

  Graphically, this is the slope of a line (a ray) from the origin to any point on the Total Product curve.

• Marginal Product is the rate of change of the Total Product. In calculus terms, it is the first derivative of the production function with respect to labor.

  $$M{P}_L=\frac{dQ}{dL}=f^{\prime }(L)$$

  This confirms our earlier observation: the marginal product is precisely the slope of the Total Product curve at any given point.

To see how this works, let's explore a resource that formally defines these concepts using calculus.

This mathematical perspective confirms the graphical relationships we observed. For example, the fact that Total Product is maximized when Marginal Product is zero is a direct application of the first-derivative test for finding maxima of a function, which you may be familiar with.

Conclusion

In this lesson, we dissected the short-run production process by quantifying the relationship between a variable input (labor) and output. We've moved from simple data tables to graphical representations and even to the underlying calculus.

Here are your key takeaways:

• Total Product (TP) is the total output produced.
• Average Product (AP) is the output per unit of labor ($Q/L$).
• Marginal Product (MP) is the additional output from one more unit of labor ($\Delta Q/\Delta L$), and is also the slope of the TP curve ($dQ/dL$).
• The MP curve intersects the AP curve at its maximum point. This is a crucial geometric relationship.
• The TP curve reaches its maximum when the MP curve is at zero.

You are now equipped with the fundamental tools for analyzing a firm's production capabilities. In our next lesson, we will use these concepts to explain a cornerstone principle of microeconomics: the law of diminishing marginal returns. We have seen this law in action in the shapes of our curves today; next time, we will explore it in detail.