12  Differentiated Bertrand Competition

12.1 Introduction

Differentiated Bertrand competition is arguably the most prevalent form of oligopoly in real-world markets. It’s characterized by product differentiation and strategic pricing—a scenario familiar in industries like cola, desktop computers, automobiles, airlines, and ski resorts. The defining feature here, in contrast to undifferentiated Bertrand competition, is the unique characteristics of each product. To be part of this competitive landscape, the market must exhibit the following criteria:

  • A Few, Influential Firms: These firms have the power to shape market equilibrium and influence their competitors’ strategies.
  • Product Differentiation: Products are distinct in the eyes of consumers, who may prefer one over another based on these differences.
  • Entry Barriers: These maintain the concentration of key players in the industry.
  • Numerous, Individual Customers: A large customer base with limited individual market influence.
  • Perfect Information: All parties are well-informed about product prices and availability.
  • Price Competition: Despite product differentiation, pricing remains a crucial competitive tool.

12.2 Differentiation and the Dynamics of Pricing

Differentiated Bertrand markets allow firms to command higher, more sustainable prices, thanks to the unique appeal of their products. For instance, consider two firms offering similar but distinct products, each enjoying its own loyal customer base. A scenario where one firm offers its product for free doesn’t automatically mean the other loses all its customers. Brand loyalty and product preferences ensure that some customers remain willing to pay a premium.

In this market, firms lose the typical price war incentives seen in undifferentiated scenarios. Unique product attributes allow firms to adjust their pricing strategies flexibly. If one firm raises its price, its competitor might also increase its price, potentially adding a premium for its distinct features. This pricing interplay continues until reaching a threshold where customers no longer perceive additional value in paying a higher price.

12.3 Differentiated Bertrand Equilibrium

In differentiated Bertrand oligopolies, products are distinct yet substitutable, catering to the diverse needs of heterogeneous customers. This scenario is typical in industries where strategic pricing and product uniqueness play pivotal roles.

Demand Curves for Differentiated Products

Here, the customer’s perception of products directly influences demand. The general demand equations for two competing firms in such a market can be expressed as:

  • Firm 1: \(\quad \mathsf{Q_1 = a_1 + b_1 P_1 + b_{12}P_2}\)

  • Firm 2: \(\quad \mathsf{Q_2 = a_2 + b_2 P_2 + b_{21}P_1}\)

In these equations, \(\mathsf{Q_1}\) and \(\mathsf{Q_2}\) denote the demand for Firm 1 and Firm 2, respectively, while \(\mathsf{P_1}\) and \(\mathsf{P_2}\) are their corresponding prices. The coefficients \(\mathsf{a_1}\), \(\mathsf{b_1}\), and \(\mathsf{b_{12}}\) (and their Firm 2 counterparts) represent the demand intercept, slope, and cross-price effect, illustrating how each firm’s demand is affected by both its own and its competitor’s pricing.

Maximizing Profits with Differentiation

Profit maximization for each firm involves finding the optimal price that balances revenue and costs. The profit functions are:

  • Firm 1: \(\quad \mathsf{\pi_{1} = P_{1}Q_{1} - f_{1} - c_{1}Q_{1}}\)

  • Firm 2: \(\quad \mathsf{\pi_{2} = P_{2}Q_{2} - f_{2} - c_{2}Q_{2}}\)

Where \(\mathsf{\pi_{1}}\) and \(\mathsf{\pi_{2}}\) represent the profits, incorporating both fixed (\(\mathsf{f}\)) and variable unit costs (\(\mathsf{c}\)). Substituting the demand equations into the profit functions, we get:

  • Firm 1: \(\quad \mathsf{\pi_1 = P_{1}(a_{1} - b_{1}P_{1}+b_{12}P_{2}) - f_{1} - c_{1}(a_{1} - b_{1}P_{1}+b_{12}P_{2})}\).

  • Firm 2: \(\quad \mathsf{\pi_2 = P_{2}(a_{2} - b_{2}P_{2}+b_{21}P_{1}) - f_{2} - c_{2}(a_{2} - b_{2}P_{2}+b_{21}P_{1}) }\).

Reaction Functions

Each differentiated Bertrand oligopolist’s pricing decision is intertwined with its competitor’s pricing strategy. The relationship between the company’s optimal price and the competitor’s price is captured in reaction functions which map out the optimal pricing response to every possible competitor price.

Firm 1’s Reaction: Illustrated in Fig. R1, showing how its optimal price adjusts in response to Firm 2’s pricing. Firm 2’s Reaction: Similarly, Firm 2’s optimal pricing, in response to Firm 1’s strategy, is also defined by a reaction function.

Company 1’s reaction function showing how its optimal price adjusts in response to Firm 2’s pricing is illustrated in Figure 12.1. The intercept shows that if Company 2 sets price at zero, Company 1 would still set its price above zero. The slope shows that Company 1 will optimally raise its price when Company 2 raises its price. In short, this reaction function demonstrates that each firm’s optimal price is a strategic response to its competitor’s pricing, leading to a state of mutual dependency. For Company 1, the only remaining question is “what price will Company 2 choose?”

Figure 12.1: Reaction function for Company 1 – Company 1’s optimal prices as its best response to every possible price from Company 2

You might already be anticipating that Company 2, much like Company 1, has its own reaction function, dependent on the pricing decisions of Company 1. This interdependence raises an intriguing question: “What price will Company 1 choose?” Logically, we are left with a strategic stalemate with both companies waiting for their competitor act so they can respond with their best price. Fortunately, there is a simple solution to the mutually dependent pricing conundrum.

Figure 12.2 introduces the reaction function of Company 2 alongside that of Company 1, providing a comprehensive view of their interactive pricing strategies. Observing Company 2’s reaction function, we notice that if Company 1 were to set its price at zero, Company 2 would still opt for a price above zero, as indicated by its reaction function’s intercept on the x-axis. Moreover, the positive slope of Company 2’s reaction function suggests a strategic response pattern: as Company 1 increases its price, Company 2 will also elevate its price proportionally.

This addition of Company 2’s reaction function to the plot not only enriches our understanding of the market dynamics but also sets the stage for identifying the equilibrium point where both companies’ strategies converge.

Figure 12.2: Reaction functions for Company 1 and Company 2

Figure 12.2 not only illustrates the individual reaction functions of Companies 1 and 2 but also reveals a critical intersection point between them. This intersection signifies where each company’s optimal pricing strategy, or best response, aligns with the other’s. In seeking to optimize, both companies naturally gravitate towards setting their prices on their reaction functions. The convergence of these functions at a single point highlights the only scenario where both firms can simultaneously choose their best responses. This point of intersection defines the optimal, equilibrium prices in our differentiated Bertrand oligopoly, as depicted in Figure 12.3.

The emergence of this equilibrium, however, raises a pertinent question: How do Bertrand competitors, without any coordination, arrive at the equilibrium prices where their reaction functions intersect? While it’s conceivable that both firms could analytically deduce and simultaneously choose the exact equilibrium prices, real-world evidence suggests this is not commonly the case.

In practice, firms often reach the Bertrand equilibrium through a process of observation and reaction. Take, for instance, Company 2 setting its initial price at zero as depicted in Figure 12.3. In response, Company 1 would set its price at the intercept of its reaction function. Company 2, observing this, adjusts its price upward to align with its reaction function, as indicated by the bottom arrow in Figure 12.3. This iterative process, characterized by a series of adjustments and counter-adjustments, eventually leads the firms to converge at the equilibrium. The key is consistent observation and responsive action, ensuring that each firm continually adjusts its pricing in line with the evolving market dynamics.

Figure 12.3: Reaction functions for Company 1 and Company 2
Bertrand-Nash Equilibrium

This logic of both companies choosing their best response to their competitor is known as a Nash equilibrium.1 Since we are seeking the equilibrium prices in a differentiated Bertrand oligopoly, this equilibrium is known as a Bertrand-Nash equilibrium or the Bertrand equilibrium for short.

12.4 Quantitative Example: Coke v. Pepsi in Bertrand Competition

Coca-Cola and Pepsi, two of the most iconic soft drink brands, exemplify the dynamics of differentiated Bertrand competition. Utilizing demand and cost curves estimated from historical data,1 we can delve into their competitive strategies and profitability within the Bertrand framework.

Let’s designate Coca-Cola as Company \(\mathsf{c}\) and Pepsi as Company \(\mathsf{p}\). Their respective demand curves have been estimated as2

  • Coke’s demand curve: \(\mathsf{\quad Q_c = 63.42 - 3.98\ P_c + 2.25\ P_p}\)

  • Pepsi’s demand curve: \(\mathsf{\quad Q_p = 49.52 - 5.48\ P_p + 1.40\ P_c}\)

These demand curves suggest a scenario where a price increase by Coke (Company \(\mathsf{c}\)) relative to Pepsi (Company \(\mathsf{p}\)) would reduce Coke’s sales, but not all Coke consumers would switch to Pepsi. This reflects the brand loyalty found among consumers of differentiated products.

The estimated variable unit costs for Coke and Pepsi are:

  • Coke’s variable unit cost: \(\mathsf{\quad c_c = \$4.96}\)

  • Pepsi’s variable unit cost: \(\mathsf{\quad c_p = \$3.96}\)

Incorporating these costs into the demand functions, we derive the profit functions for each company:

  • Coke’s profit: \(\mathsf{ \quad \pi_c = (P_c - c_c) (63.42 - 3.98\ P_c + 2.25\ P_p) }\)

  • Pepsi’s profit: \(\mathsf{\quad \pi_p = (P_p - c_p) (49.52 - 5.48\ P_p + 1.40\ P_c)}\)

Next, we determine the reaction functions for each company:

  • Coke’s reaction function: \(\mathsf{\quad P_c = 10.45 + 0.2827\ P_p}\)

  • Pepsi’s reaction function: \(\mathsf{\quad P_p = 6.498 + 0.1277\ P_c}\)

Figure 12.4 illustrates these reaction functions and the Bertrand equilibrium optimal prices for Coke and Pepsi. At this equilibrium, each company selects its optimal pricing strategy in response to the other’s pricing, balancing competitive pressure with profit maximization.

Figure 12.4: Differentiated Bertrand reaction functions for Coke and Pepsi. The optimal price for Coke is $12.74 and for Pepsi is $8.12. Coke earns $242.71 in economic profit and Pepsi earns $95.08.

Now that we know the optimal prices for Coke and Pepsi, we can calculate their profits.

  • Coke’s profits: \[\begin{align} \mathsf{\pi_c} &= \mathsf{ (P_c - c_c) Q_c} \\ &= \mathsf{ (12.74 - 4.96) (63.42 - 3.98\cdot 12.74 + 2.25\cdot 8.12) }\\ &= \mathsf{\$242.71} \end{align}\]

  • Pepsi’s profits: \[\begin{align} \mathsf{\pi_p} &= \mathsf{ (P_p - c_p) Q_p} \\ &= \mathsf{ (8.12 - 3.96) (49.52 - 5.48\cdot 8.12 + 1.40\cdot 12.74) }\\ &= \mathsf{\$95.08} \end{align}\]


12.5 Conclusion

As we conclude our exploration of Bertrand oligopolies and their implications for entrepreneurial ventures, several critical lessons emerge. These lessons are not just theoretical musings but practical guidelines for navigating the complex landscape of market competition.

Exercise Pricing Discipline

One of the most important takeaways is the concept of pricing mutual forbearance. In the relentless quest to attract customers, the temptation to engage in aggressive price cutting can be strong. However, our analysis underscores the peril of this approach, particularly in undifferentiated markets. Instead, firms must practice restraint and avoid triggering destructive price wars that erode profitability for all players involved. Recognizing and respecting the pricing strategies of competitors can lead to a more stable and sustainable competitive environment.

The Significance of Product Differentiation

Differentiated Bertrand competition highlights the value of product differentiation and the discipline it brings to pricing strategies. Differentiation allows firms to escape the trap of competing solely on price, enabling them to command premium prices for unique product features that resonate with specific customer segments. This approach not only helps in maintaining healthy profit margins but also fosters innovation and brand loyalty.

Measure Competitiveness for Evidence-Based Decisions

Finally, the ability to compete effectively is not a matter of guesswork. It requires a rigorous measurement of expected profits and a deep understanding of one’s competitive position. In the realm of entrepreneurship, where resources are often limited and the cost of missteps high, making evidence-based decisions is crucial. By quantitatively assessing how well a venture can compete against its rivals, entrepreneurs can make informed strategic choices, from pricing to product development.

In conclusion, the journey through the nuances of Bertrand oligopolies reveals much about the strategic underpinnings of effective competition. For entrepreneurs, the lessons learned here are invaluable. They provide a framework for understanding market dynamics, a guide for strategic decision-making, and a reminder of the importance of innovation and customer focus. By applying these insights, entrepreneurs can navigate the competitive landscape with greater confidence and precision, turning challenges into opportunities for growth and success.


12.6 Workout Problems

Dorsal Packs

A group of entrepreneurs, concerned about pick-pocketing in tourist destinations, created the Dorsal travel pack. This stylish, modern backpack has zippers facing inward, toward the back of the wearer, for security. For added security, it is made of slash-resistant fabric in nondescript colors to avoid the attention that more colorful, fashion-oriented brands draw.

The team validated the problem and solution, refined the concept, and confirmed direct sales through their website. They conducted a profitability analysis with 107 potential customers, who rated the innovation, suggested improvements, stated their maximum willingness to pay, and provided travel and demographic information.

The team recognizes that not all travelers care more strongly about security than other features the bag may have. They have seen competitors in the market that focus on the versatility in of a travel pack, emphasizing pockets, packing, and compatability with airline luggage restrictions, in addition to branded logos and colors. The VersaPack travel pack is the dominant competitor in the travel pack market, primarily targeting travelers who value versatility by designing and placing pockets in convenient places, enhancing packability, and developing TSA-friendly features. While the VersaPack is less secure from pick-pocketers than the Dorsal pack, it has many features that travelers value.

How effectively can Dorsal pack expect to compete with VersaPack. What is the impact on Dorsal pack’s predicted profits when we add a competitor to the analytics?

Dorsal Pack data

The Dorsal pack team reached out to travelers who have either traveled internationally recently or are about to travel internationally. They showed the travelers both the Dorsal pack and the VersaPack and gathered data about their impressions and willingness to pay. They were able to get 97 responses from potential customers.

Demand and Revenue

The respondents were shown the Dorsal backpack and its features, emphasizing the physical design that enhances security without adding mechanical locks. They were asked

  1. To rate how strongly they care about the security features of a travel backpack (scale 1-10);
  2. To rate the innovation and offer suggestions to improve it (scale 1-10).

The respondents were also shown the VersaPack travel pack. They were asked

  1. “What is the most you would be willing to pay for the Dorsal travel pack?”
  2. “What is the most you would be willing to pay for the VersaPack travel pack?”

Finally, the respondents provided information about their travel patterns as well as demographics.

The responses are stored in a tibble named dp (Dorsal Pack).

dp <- tibble(security_preference = c(8, 9, 8, 8, 8, 9, 7, 10, 8, 8, 7, 8, 9, 10, 7, 9, 9, 7, 10, 0, 0, 4, 2, 1, 6, 1, 0, 5, 2, 3, 5, 0, 2, 6, 4, 3, 6, 1, 4, 6, 0, 0, 1, 6, 2, 3, 4, 6, 4, 2, 5, 0, 1, 4, 4, 3, 6, 4, 1, 0, 0, 2, 0, 5, 4, 6, 0, 6, 1, 3, 6, 3, 5, 5, 2, 5, 5, 6, 6, 0, 5, 1, 0, 1, 3, 4, 4, 5, 6, 2, 0, 3, 5, 0, 5, 0, 6),
             wtp_Dorsal = c(255, 310, 229, 272, 262, 284, 248, 290, 253, 240, 219, 266, 289, 301, 258, 321, 270, 194, 320, 86, 86, 201, 134, 96, 224, 117, 100, 208, 133, 173, 196, 107, 162, 229, 173, 183, 240, 131, 185, 207, 127, 88, 164, 251, 135, 139, 166, 225, 175, 133, 181, 99, 104, 147, 172, 178, 208, 192, 88, 99, 110, 146, 102, 187, 163, 200, 102, 201, 110, 155, 257, 147, 205, 202, 121, 199, 229, 229, 221, 92, 159, 143, 71, 135, 198, 151, 194, 195, 189, 110, 68, 149, 171, 114, 242, 74, 236),
             wtp_Versa = c(139, 118, 95, 117, 113, 124, 121, 124, 111, 146, 104, 88, 191, 90, 137, 126, 98, 143, 109, 195, 202, 159, 233, 171, 113, 191, 208, 161, 169, 143, 182, 191, 158, 134, 155, 198, 142, 209, 148, 145, 192, 202, 168, 107, 230, 185, 129, 125, 130, 235, 183, 193, 204, 150, 148, 150, 125, 201, 189, 203, 206, 211, 187, 125, 202, 129, 182, 109, 158, 156, 155, 198, 168, 141, 181, 132, 132, 162, 115, 249, 148, 195, 182, 176, 137, 155, 170, 158, 120, 160, 187, 207, 122, 196, 198, 197, 106)
)

# inspect the data
glimpse(dp)
Rows: 97
Columns: 3
$ security_preference <dbl> 8, 9, 8, 8, 8, 9, 7, 10, 8, 8, 7, 8, 9, 10, 7, 9, …
$ wtp_Dorsal          <dbl> 255, 310, 229, 272, 262, 284, 248, 290, 253, 240, …
$ wtp_Versa           <dbl> 139, 118, 95, 117, 113, 124, 121, 124, 111, 146, 1…
Cost Structure

Dorsal plans to have the pack manufactured by a contract manufacturer who charges $40 per backpack to be manufactured, packaged, and delivered to the startup’s offices. The startup is also spending $200 per month for web hosting and $5500 per month for contracted web development. Their total cost curve is linear: \(\mathsf{C = cQ + f}\) where \(\mathsf{c}\) is the variable cost per unit and \(\mathsf{f}\) is fixed cost per month.

Rescaling

To conduct profitability analytics, it’s crucial to align the scale of sample demand with the fixed costs required to serve the market. For this analysis, we will rescale the demand up to the market level. Concerning market size:

  • International Travelers from the U.S.: According to the U.S. National Travel and Tourism Office, approximately 93 million international trips were made by U.S. citizens in 2019. We focus on 2019 data because international travel dynamics significantly changed due to the COVID-19 pandemic, and recent recovery data is not yet fully available.

    • We estimate that the average international traveler makes about 1.5 trips per year, accounting for both frequent travelers and those who rarely travel abroad.
    • It is generally estimated that 70 to 80 percent of these trips are for leisure purposes.
    • Combining these factors, we assume there were approximately 43.4 million unique international travelers in 2019.

  • Percentage of Young Travelers Likely to Use a Backpack: While detailed data is unavailable, a conservative estimate suggests that 25% of these travelers are in the 20-39 age group, a demographic typically more active in international travel. This estimation brings our target population down to 10.85 million.

  • Market share in a crowded backpack market: Recognizing the diverse needs and preferences of travelers, as well as the crowded nature of the backpack market, achieving a 1-5% market share would be substantial. Opting for a midpoint estimate of 3%, we approximate the target market to be around 65,100 potential customers.

In summary, our assumption for the target market is about 565,500 potential international travelers, some of whom care more about security and others who care more about versatility.

Questions

  1. What is the rescaling factor \(\mathsf{m}\) based on the calculation of the target population?

  2. How many Dorsal travel packs would you expect to sell to the sample of 97 respondents if the price of a Dorsal pack is $200 and the price of a VersaPack is $150?

  3. How many VersaPack travel packs would you expect to sell to the sample of 97 respondents if the price of a VersaPack is $200 and the price of a Dorsal travel pack is $175?

  4. How many Dorsal travel packs would you expect to sell to the complete market if the price of a Dorsal pack is $250 and the price of a VersaPack is $200?

  5. How many VersaPack travel packs would you expect to sell to the complete market if the price of a VersaPack is $220 and the price of a Dorsal travel pack is $150?

  6. Using an iterating function or manual iteration, what is the differentiated Bertrand equilibrium price for a Dorsal pack? Use the rescaled market demand curve for your calculation and continue to use the market demand for the rest of the questions.

  7. What is the differentiated Bertrand equilibrium price for a VersaPack travel pack?

  8. How many Dorsal travel packs would you expect to sell to the complete market at the Bertrand equilibrium prices?

  9. How many VersaPack travel packs would you expect to sell to the complete market at the Bertrand equilibrium prices?

  10. How much profit does Dorsal pack expect to earn per year in competition with VersaPack?

  11. How much profit does VersaPack expect to earn per year in competition with Dorsal pack?

  12. Does Dorsal pack expect to earn more profit or less in competition with VersaPack compared to what it expected to earn under the assumption it was a monopolist?

  13. Which market assumption is more appropriate (monopoly or Bertrand competition)?

  14. How does Dorsal pack’s expected profit change when considering only the responses of customers who value security highly (security preference score of 7 or higher)?

  15. Is Dorsal a stronger competitor (higher expected profit) with customers who care about more about security or with all travelers?

  1. This equilibrium is named after the Princeton mathematician John Nash who won a Nobel prize in economics for his contributions to game theory. His journey to this discovery and eventually receiving the Nobel prize was portrayed in the film A Beautiful Mind.↩︎

  2. Gasmi, Vuong, and Laffont (1992) used data to estimate the demand and cost curves and then analyze the Bertrand competition between the soft drink behemoths. We will reproduce their analysis here.↩︎