Section 3.4 Profit maximization
With marginal analysis in the toolkit, we are equipped to tackle the firm’s profit maximization problem. We define a firm’s
profit as the difference between the firm’s
total revenue and the firm’s
total cost. Profit is almost always denoted by the symbol (not the number) pi (
\(\pi\)). Total revenue (
\(TR\)) is a measure of the money the firm brings in from sales of its output; this is often calculated by multiplying the number of units sold by the price at which the firm sells each unit
1 . Typically, this is expressed as
\(TR = p\times q\text{.}\) In general,
\(TR\) is a function of
\(q\text{:}\) the amount of revenue a firm makes depends on how many units it sells. Total cost (
\(TC\)), as we have seen, is the cost of the firm’s production. Therefore, we can express the firm’s profit as
\begin{equation*}
\pi = TR - TC
\end{equation*}
or
\begin{equation*}
\pi (q) = TR(q) - TC(q)
\end{equation*}
The second expression above emphasizes that the amount of profit the firm makes depends on its level of output \(q\text{,}\) in particular because \(q\) impacts both total revenue and total cost.
The optimization question we ask, then, is: how much output \(q\) should the firm choose to maximize its profit? This model fits perfectly over the model of marginal analysis. The firm’s total benefit from producing is its total revenue, since taking money in is the benefit of producing, while its total cost is its, well, total cost. Notice then that profit here will play the role of net benefit in our marginal analysis: it is the difference between the firm’s total benefit and the firm’s total cost.
But marginal analysis is about comparing marginal benefit and marginal cost to find an optimal quantity. In the context of profit maximization, then, the marginal benefit to the firm is the firm’s marginal revenue, the additional revenue the firm generates from an increase in output. This allows the firm to address the added value (value is in terms of revenue dollars) selling another unit will generate. Marginal revenue is defined just as any other “marginal”:
\begin{equation*}
MR = \frac{\Delta TR}{\Delta q}
\end{equation*}
As previously defined, we will use the firm’s marginal cost as the added cost the firm incurs from selling another unit. Now, let’s apply the rules of optimal decision making from marginal analysis dictate that a firm should choose the level of output where marginal benefit is equal to marginal cost. But here, since benefit \(=\) revenue, and cost \(=\) cost, we can express the rules for profit maximization for a firm:
If, for some level of output \(q\text{,}\) \(MR > MC\text{,}\) then it is optimal to produce more output than \(q\).
If, for some level of output \(q\text{,}\) \(MC > MR\text{,}\) then it is optimal to produce fewer than \(q\) output.
If, for some level of output \(q\text{,}\) \(MR = MC\text{,}\) then \(q^*\) is the exact optimal number of units of output to produce!
The intuition behind these conditions is analogous to the general marginal analysis intuition. If producing another unit of output adds more revenue for the firm than it adds in cost, then that unit of output is worth producing! The firm will earn more profit by producing a unit which costs the firm $8 to produce and can be sold for $10. If producing another unit of output costs more to produce than it will sell for, then it is never rational for the firm to produce that unit. Why spend $40 to make a unit of output that will only sell for $25? As long as added units generate more revenue than cost, the firm should continue to produce those units, right up until the point where \(MR = MC\text{.}\)
|
marginal analysis |
marginal analysis |
|
in general |
profit maximization |
|
|
|
| choice variable |
quantity of hours studied \(x\)
|
quantity of output \(q\)
|
| benefit |
total benefit |
total revenue |
| cost |
total cost |
total cost |
| optimality condition |
\(MB = MC\) |
\(MR = MC\) |
|
|
|
| objective |
maximize net benefit: |
maximize profit: |
|
\(NB = TB - TC\) |
\(\pi = TR - TC\) |
Comparison between the general framework for marginal analysis and its application to profit maximization in firm theory.
The profit maximization model is a nearly direct application of the marginal analysis framework. The general optimality condition for firms - profit maximization where marginal revenue equals marginal cost - is a powerful result which will provide us deep insight into firm decision making.
You may be wondering why there are no specific examples presented in a graph or table for profit maximization just yet. As we will see, while the foundation of profit maximization model is identical for any firm, the shape of the problem is greatly impacted by the type of firm being modeled. For example, a farmer who competes in a perfectly competitive market, an airline, and Google may all seek to maximize profit, but several factors will influence exactly how the firms go about maximizing profit. Our next step, then, is to answer the questions about market structure and the nature of competition that shape profit maximization. How many competitors do the firms have? Can the firms influence the price they charge for their product?
In Depth 3.4.1. Descriptive analysis? Or prescriptive analysis?
As we continue to see, a major point of focus in this course is on models of decision making. Since markets are critically influenced by consumers’ decision, sellers’ decisions, and policy makers’ decisions, it is important to understand those decisions. So, to better understand those decisions, models of optimal decision making play a key role. But what is the objective of these models?
One purpose of decision-making models is prescriptive. These models prescribe or offer what the optimal decision should be in a particular scenario. A model like this might suggest that, given a level of income, prices for goods, and preference, that a consumer would maximize her utility by making a particular decision or purchasing a particular optimal bundle \((x^*, y^*)\text{.}\) A prescriptive model of decision making can provide useful guidance for what rational decision making should look like, and can help real-world decision makers better understand the incentives, costs, and benefits at play when making a decision.
A decision-making model can also be descriptive. A descriptive model focuses less on what decisions individuals should make, and instead attempts to address how individuals actually make decisions. Here, the objective is to figure out how the model can best reflect real-world behavior, and the priority is realism of the predicted outcome. A descriptive model of consumer choice, for example, might need to reflect many additional elements of consumers’ decisions, such as inconsistencies in consumer preferences or impulse purchases in the checkout line.
Each type of model has a role to play in understanding behavior. Prescriptive models can highlight the role of assumptions in modeling, provide guidance on the core incentives facing decision makers, and help guide real-world decision makers on what an optimal decision should be. In that way, they can teach decision makers valuable lessons on how to make better decisions in practice. Descriptive models provide value by formalizing the many complex and at times unpredictable nature of real-world decision making. Prescriptive and descriptive models complement one another in the theorist’s toolkit.
Of course, models can be both prescriptive and descriptive! The profit maximization model is an oft-used example of a model that both tells us how private firms should make decisions and reasonably approximates actual decision making. Importantly, navigating this gap can lead to important insights on decision making. If a prescriptive model does not describe actual decision making, why not? What assumption is off-base? What aspect of reality is absent from the model? In short, why doesn’t the decision maker choose in reality what the model predicts is optimal?
Behavioral economics is a subfield of economics which focuses on developing a suite of decision making models that are more descriptive than many standard models in economics. Models in behavioral economics often build on existing models to increase their ability to capture real-world decision-making, focusing on discrepancies between the predictions of prescriptive models and observed behavior. By studying that behavior - and incorporating elements of psychology - behavioral economics has helped economists better understand where and why real-world decisions diverge from models.
Key terms in this section:
profit
total revenue
marginal revenue
total cost
marginal cost
marginal profit
