One Perfect Price: Profit Maximization for the Single Price Firm
Ultimately, the purpose of this textbook is to develop innovative pricing strategies designed to capture consumer surplus. But prior to devising such strategies, we must first derive the profit-maximizing strategy for the single-price firm so we have a basis for comparison.
Let’s assume Wendy has created her own interior design business. To start her business, she purchased interior design software for $1000 that would allow prospective clients to visualize her ideas. She also bought an assortment of wallpaper, carpet, and paint samples for $500. Of course, each individual job will incur additional costs. Although the cost and price of each job would vary in real life, we will assume the costs associated with each job are equal to $550 and she will charge a flat fee to each client.
The costs of the software and samples are considered to be fixed costs because they will not vary with the number of jobs Wendy takes on. Moreover, if she has already purchased the software and samples and cannot return them for a partial refund, they constitute sunk costs, meaning they cannot be recovered. Wendy’s first decision is to determine whether she could take on her first job. If this were to be her one and only job, she could scour the marketplace and find the individual is willing to pay the most for her services. As not all interior designers are alike, Wendy faces competition but retains some flexibility to set her own price.
Suppose Wendy finds a customer who is willing to pay $1300. Should Wendy take on the job? At first glance, one may say “no.” After all, she has already expended $1500 on the samples and software, and the job will cost an additional $550. But this is where the significance of sunk costs comes into play. Assuming she cannot return the samples or re-sell the software, if she turns down the job, her business will suffer a $1500 loss. But if she takes the job, she expends an additional $550, collects an additional $1300, and her overall loss falls from $1500 to $750. By agreeing to the job, her losses decrease by $750. Economists argue that because sunk costs cannot be recovered, they should be ignored in decision-making. In Wendy’s case, the only factors that are relevant to the decision are the additional costs the job will incur and the additional revenues it will generate. The $550 cost of completing the job is called a variable cost because it represents an expenditure that varies with the number of jobs completed. Alternatively, economists label the additional cost of each additional job as its marginal cost. Clearly, Wendy should agree to the job if she can get a price that exceeds $550.1 Therefore, because the client is willing to pay $1300 for the job, Wendy will profit by $750 by agreeing to the work.
The law of demand suggests that the more jobs Wendy wants to take on, the lower the price she will have to charge. Again, for the time being, we will assume that Wendy plans to charge all of her customers the same price. If Wendy wants to perform two jobs, she will have to lower her price for both clients. Suppose she decides she can attract two clients if she charges a price of $1200. Is it worth her while? At face value, the answer is “yes”. The second job will cost $550, so if she can get a price of $1200, she adds $650 to her overall profits. But is this really the case? Let’s do some simple math: if she only performs one job, her revenues will equal $1300 and her variable costs will total $550, leaving a profit contribution of $750. If she decides to do two jobs, her revenues will be $2400 and her variable costs will sum to $1100. Note that the second job increased her overall profits by only $550, not $750.
Why did her profits rise by only $550? To understand, let’s examine the full ramifications of her decision making. If she only agrees to one job, she can charge her client $1300. If she decides to do two jobs (and under the assumption that all of her clients will be charged the same price), she will charge $1200 for each job. Although she gains an extra $1200 in revenue from the second job, the revenue she could have earned from the first job falls from $1300 to $1200. In understanding the ramifications from the revenue side, the additional $1200 is referred to as the output effect. It represents the revenue she will receive from the second job. The $100 price cut on the first job is referred to as the price effect. Economists refer to the additional revenue generated from an additional unit (or job, in this case) as the marginal revenue. The marginal revenue from performing two jobs instead of one is equal to the sum of the output and price effects. Put another way, although the second job is priced at $1200, if Wendy decides to do it, her revenues only increase by $1100 ($2400 from two jobs less the $1300 she could have earned from one job). Thus, the marginal revenue from the second job is $1100.
In essence, when considering whether to perform two jobs instead of one, she need only ask two questions: by how much will her costs rise if she performs the job ( marginal cost), and by how much will her revenues rise if she performs the extra job ( marginal revenue)? In this case, if she decides to do two jobs, the additional (marginal) cost is $550 and the additional (marginal) revenue is $1100. Hence, her profits will rise by $550 if she does the second job. (In this case, because her business is not yet profitable if she only agrees to two jobs, her losses will decline by $550).
Table 4.1 summarizes the potential jobs Wendy can take on. As the table illustrates, each job can be evaluated by comparing its marginal revenue and marginal cost.
Table 4.1. Finding the Profit-Maximizing Price and Quantity Using Marginal Revenue and Marginal Cost
As the table indicates, the marginal revenue from each of the first four jobs exceeds the marginal cost. Wendy would not be willing to take on five jobs because the marginal cost ($550) exceeds the marginal revenue ($500).
We can also use the data in Table 4.1 to calculate the consumer surplus. Wendy is charging her clients the profit-maximizing price of $1000. However, we know that one of her clients was willing to pay $1300, one was willing to pay $1200, and a third was willing to pay $1100. By charging all four clients $1000, Wendy is missing out on $600 in consumer surplus.
Figure 4.1 illustrates the single price strategy. Due to the price effect, the marginal revenue from an additional job is less than the price of the job (except for the initial one). Therefore, the marginal revenue curve lies below the demand curve. Because we assumed the marginal cost of each job to be constant, the marginal cost curve is a horizontal line.
Figure 4.1. Graphical representation of the profit-maximizing price and quantity.
Figure 4.1 shows the marginal revenue for each of the first four jobs exceeds the marginal cost. Beginning with the fifth job, marginal cost is greater than marginal revenue. The demand curve indicates that the four jobs can be sold at a price of $1000. Figure 4.2 focuses on consumer surplus. As the graph indicates, the consumer surplus on the first job is $300, the consumer surplus on the second job is $200, and the consumer surplus on the third job is $100. The client for the fourth job retains no consumer surplus.
Figure 4.2. The profit-maximizing quantity and consumer surplus.
Figure 4.3 combines Figures 4.1 and 4.2 into a single generic graph. The firm has an incentive to produce until the point where the marginal revenue is equal to the marginal cost (i.e. it produces every unit for which marginal revenue exceeds marginal cost, but produces no unit whose marginal cost is greater than its marginal revenue. The corresponding profit-maximizing price (P*) is indicated by the demand curve: it shows the highest price that would allow the firm to sell the profit-maximizing quantity (Q*). The shaded area illustrates the consumer surplus and the striped area is the firm’s profit contribution.
Figure 4.3. Graphical representation of profit contribution and consumer surplus at the profit-maximizing quantity.
Summary
• Marginal revenue is the change in revenue associated with a given change in output. Marginal revenue has two components: the output effect, which is the revenue generated by the additional units of output; and the price effect, which is the revenue lost on previous units due to the price reductions necessary to sell more units.
• If the revenue generated by the additional units (i.e. output effect) exceeds the lost revenue on the other units due to the price reduction (i.e. price effect), marginal revenue is greater than zero. Increased production will lead to more revenue.
• If the revenue generated by the additional units (i.e. output effect) is less than the lost revenue on the other units due to the price reduction (i.e. price effect), marginal revenue is less than zero. Increased production will cause total revenues to fall.
• Marginal cost refers to any change in costs that are associated with an increase in production. Marginal costs are, by definition, variable costs.
• If the marginal revenue associated with an increase in production exceeds its marginal costs, the firm’s profits will rise (or its losses will become smaller).
• If the marginal cost associated with an increase in production exceeds its marginal revenue, the firm’s profits will fall (or its losses will rise).
• Under a single price strategy, uncaptured consumer surplus will exist.