Behavior and Choice: EC 102

To build a model of how a firm optimally chooses its output and maximizes its profit, we will construct a general framework for this type of optimal decision making. This framework, marginal analysis, can be used to study profit maximization, as well as many other optimal decision making problems as we will see. Marginal analysis works by weighing the costs and benefits of taking an action, such as studying for an exam or producing output as a firm. Marginal analysis can help identify the optimal decision when faced with the following types of questions:

In the context of studying for an exam, a decision maker might consider: “how much should I study for this exam?” She may weigh the costs and benefits or studying, not wanting to study too little but also not wanting to study too much, and ultimately determine how much time she should spend studying. In the context of profit maximization, we will see how firms balance the costs of production with the benefits in order to find the optimal quantity of output the firm should produce to maximize its profit.

Let’s use the example of studying for an exam to motivate the marginal analysis framework. Consider our question:

Even if your professor hopes your answer is “as many hours as possible,” realistic decision making should weigh the benefits and the costs of studying. But how exactly can they be used to highlight what an optimal decision looks like?

First, consider the benefits to studying. You are likely to do better on your exam. Studying can increase your self-esteem. Plus, studying can help you learn and retain the content! There are lots of reasons why studying generates total benefit to you. For every minute you study, the total amount of benefit you receive from studying increases. More studying leads to more benefit.

As we can see in both the table and graph below, if the total benefit to studying is increasing and concave,¹ then the marginal benefit to studying is positive and decreasing. Much like other marginals, your marginal benefit to studying another minute is the slope of your total benefit curve.

An example of total benefit and marginal benefit from studying for an exam. Benefit is measured in units of utility for the decision maker.

What about the costs to studying? Studying takes effort, which costs you energy and makes you hungry and tired. There is also the opportunity cost of studying: all of the other activities you give up when you spend time studying. Naturally, the more you study, the more your total cost increases: you get hungrier, more tired, and give up more alternative activities the longer you study.

This is analogous to the marginal cost as defined for firms. If marginal cost is increasing, then the total cost curve is increasing and convex - like the total cost curve for firms under assumption P2.

An example of total cost and marginal cost from studying for an exam. Here, cost is measured in terms of units of utility for the decision maker.

Marginal analysis is a framework for optimal decision that weighs the costs and benefits of engaging in an activity to determine the optimal level of that activity. As the name implies, the key factor in determining, say, the optimal number of minutes to study, is to weigh the marginal benefit and marginal cost of studying. Since marginals are about the added benefit or cost to studying a little bit more, marginal analysis asks, "Based on how much I have studied so far, should I study one more minute?"

The intuition is as follows: if additional studying is going to add more benefit than it will add cost, then the minute is worth studying! If that additional studying is going to cost you more than it will benefit you, then the additional studying is not worth it! Why would I study when it will cost me 40 utility to gain a benefit of 25 utility? A rational decision maker ³ will only study minutes for which the marginal benefit exceeds or is equal to the marginal cost.

The first two rules capture the intuition from earlier: if studying will add more benefit than it will cost, it is rational to do! The third rule can be used to identify the exact optimal number of hours to study. To explore the first two rules in action, let’s combine our example from above and put benefits and costs in one table. From here, we will approach the table row by row, comparing \(MB\) and \(MC\text{,}\) starting at the top.

Marginal analysis compares marginal benefit and marginal cost.

First, when studying 0 minutes, you obviously receive 0 total benefit and 0 total cost. What about studying for the first 30 minutes? Well, the first rule above states that since \(MB = 3\) and \(MC = \frac{1}{6}\text{,}\) then \(MB \gt MC\) and it is rational to study at least 30 minutes. Notice that this is confirmed if we compare the changes to total benefit and total cost: the first 30 minutes will benefit you 90 utility, and will only cost you 5 utility. If we move to the second row and examine studying another 30 minutes (to 60 total), we see that \(MB = 1 \gt \frac{1}{3} = MC\text{,}\) so studying at least an hour is rational. And once again, the next 30 minutes of studying will add 30 benefit while only adding 10 cost. The next 30 minutes are rational to study as well: \(MB = \frac{2}{3} \gt \frac{1}{2} = MC\text{.}\) Finally, notice how diminishing marginal benefit and increasing marginal cost are at work here. As you study more, the added benefits start shrinking, while the added costs grow.

Now, look at studying 120 minutes. The \(MB\) of studying that 30 extra minutes is \(\frac{1}{2}\text{,}\) while the \(MC\) of studying that extra minutes is 1: the second rule above states that it isn’t rational to study that next 30 minutes all the way up to 120. This is confirmed by the totals: the next 30 minutes would add 15 benefit but add 30 cost! Moreover, we do not need to keep checking rows past this point. Since \(MB\) continues to diminish, and \(MC\) keeps growing, no number of minutes beyond 120 can be optimal either. If it wasn’t optimal to study 120 minutes, I definitely won’t find it optimal to study 150 minutes.

In this example, then, the optimal number of minutes to study is 90 minutes. As a general rule when working with a table like this one, any row where \(MB \gt MC\) is a row worth doing; once you hit a row where \(MC \gt MB\text{,}\) this is the first row not worth doing - and no row beyond it will be worth doing either.

The optimal number of minutes to study, \(x^* = 90\text{,}\) can be found by continuing to study as long as \(MB \gt MC\text{.}\) Once \(MC \gt MB\text{,}\) this quantity and all quantities beyond it are not rational to study.

Notice in the table that by following the rules outlined above, the choice \(x^* = 90\) does lead to the highest net benefit, which is equal to 110. This provides some added intuition to the marginal analysis framework. Even though studying more than 90 minutes would raise total benefit, doing so would raise total cost by more, and the net benefit would fall.

The optimal number of minutes to study, \(x^* = 90\text{,}\) will maximize the decision maker’s net benefit.

What about the third rule? This one stated that at the optimal number of hours, \(MB = MC\) should be true. But this doesn’t happen in the table above! This is an important nuance in our framework that we should be wary of, a nuance that comes down to a modeling choice. Like we have seen with demand functions and consumption bundles, there are often an infinite number of variables to model: infinite possible prices, infinite quantities of apples to purchase, and so on. So, often, we represent these infinite possibilities in a table to simplify our analysis, while acknowledging there are possibilities left out.

Our table to analyze optimal studying is the same! Certainly, it is possible to study 75 minutes, or 91 and a half minutes, yet these options are omitted from the table. You can imagine a scenario, then, where the variable for minutes studying, \(x\text{,}\) is continuous⁴, and can take any numerical value, from 0 to infinity. With a continuous \(x\text{,}\) it is possible to pinpoint the optimal quantity with more precision. The graph below shows how, if the table above allowed for a continuous \(x\text{,}\) the optimal number of minutes to study actually lies somewhere between 90 and 120.⁵

The intuition here remains the same. If, at some point in studying, \(MB \gt MC\text{,}\) then continuing to study will increase net benefit because additional studying will add more benefit than cost. Any point where \(MB \gt MC\) is a point where there is still an opportunity for the decision maker to improve by studying more. If, at some other point, \(MC \gt MB\text{,}\) then continuing to study will decrease net benefit by adding more cost than benefit. Another way to interpret this is to say that when \(MC \gt MB\text{,}\) reducing your studying will increase net benefit by reducing cost more than it reduces benefit.⁶ That is, any point where \(MC \gt MB\) is a point where there is still an opportunity for the decision maker to improve by studying less. And, of course, at \(MB = MC\text{,}\) the decision maker has both exhausted all opportunities to improve from studying more, and avoided overstudying - reaching the optimal number of minutes to study.

This same concept can be visualized on the graphs of total benefit and total cost. The table shows how total benefit is increasing and concave, while total cost is increasing and convex. We can represent these values on a graph with an approximation as if \(x\) were continuous. By doing so, we can also visualize net benefit: since it is defined as the difference between \(TB\) and \(TC\text{,}\) it can be seen on the graph as the vertical distance between the \(TB\) and \(TC\) curves on the graph.

Since the optimal number of minutes to study \(x^{*}\) occurs where net benefit is largest, this can be identified on the graph as the peak of the net benefit curve. By definition, this is also where the vertical gap between \(TB\) and \(TC\) is largest.

Notice that geometrically, at the optimal quantity \(x^{*}\text{,}\) where the distance between the two curves is the greatest, the curves have the same slope, represented in the graph below by drawing the lines tangent to each curve. But the slope of total benefit is marginal benefit, and the slope of total cost is marginal cost, which means that at \(x^{*}\text{,}\) \(MB = MC\text{.}\) This is just our third rule from above!⁷