FIRST MOVER ADVANTAGE IN SEQUENTIAL BERTRAND MARKETS: AN EXPERIMENTAL APPROACH A THESIS Presented to The Faculty of the Department of Economics and Business The Colorado College In Partial Fulfillment of the Requirements for the Degree Bachelor of Arts By Perry Fitz April 2017
FIRST MOVER ADVANTAGE IN SEQUENTIAL BERTRAND MARKETS: AN EXPERIMENTAL APPROACH Perry Fitz April 2017 Mathematical Economics Abstract Literature and applied experimental evidence has established the consensus that firms competing with price competition in sequential markets have a second mover advantage. A high proportion of literature assumes firms have symmetric costs, while in real markets firms tend to have asymmetrical costs. In this paper, I use current literature to define various profit assumptions that yields a theoretical first mover advantage for a low-cost firm in a differentiated-product Bertrand-duopoly. I report on the findings of an experimental 30 round sequential game, where firms have varying levels of cost. The results show that a low-cost firm will not necessarily always have a first mover advantage against a high-cost competitor. KEYWORDS: Experimental Economics, Stackelberg, Bertrand, Asymmetric Firms JEL CODES: C71, C02, C70 ii
Acknowledgements I would like to thank my mentor, Rich Fullerton, for the amount of dedication and unconditional support he put throughout my thesis. Without his help from the beginning, I would be lost. Thank you to Kevin Rask and Kathryn Bryant for continuous support and input. iii
ON MY HONOR, I HAVE NEITHER GIVEN NOR RECEIVED UNAUTHORIZED AID ON THIS THESIS Signature iv
TABLE OF CONTENTS ABSTRACT ACKNOWLEDGEMENTS ii iii 1. INTRODUCTION... 1 2. MODEL... 6 3. METHOLOGY... 8 4. THEORITCAL BACKGROUND AND EQUILIBRIA... 10 5. HYPOTHESES... 15 6. EXPERIMENTAL RESULTS... 15 TABLE I.... 16 TABLE III.. FIGURE 3.. 17 23 7. DISCUSSION... 25 8. CONCLUSION... 26 FIGURE 2... 28 FIGURE 3... 29 APPENDIX... 30 9. REFERENCES... 31 v
1. Introduction From 2000 to 2016 the number of listed domestic companies in the United States decreased by 37% according to the World Bank s Federation of Exchanges database. In the same time frame, research and development expenditure towards technology has increased by 9% 1. As technology continues to grow in the United States, more firms are attempting to collect complete information of markets to stay ahead of their competition. Firms can compete against one another in a multitude of strategies; the most basic and popular being quantity or price competition. But having access to a growing amount of market information has added a third complexity: timing. Firms have the ability to use timing as an additional strategy with price or quantity. Firms can decide to compete with competitors simultaneously or sequentially. In sequential markets, companies select to be a leader or a follower, where both literature and experimental markets have supported the advantage of being a first-mover when using quantity competition. However, in price competition a consensus in literature has been gathered that the follower is at an advantage when symmetric firms compete against one another. Even so, real world markets rarely contain symmetric competitors in terms of cost. The main purpose of this paper is to apply experimental evidence to Amir and Stepanova s theory (2006) that a firm with a sufficient cost lead over its rival has a first 1 According to the World Bank s Research and Development Expenditure (% of GDP) database. vi
mover advantage in a differentiated-product Bertrand-duopoly with general demand and asymmetric linear costs. Asymmetric costs are more relevant in studying real world markets because it is unlikely that two competing firms will have identical cost functions for imperfect substitutes. Asymmetric costs are common in markets of large and small producers, where larger firms can achieve economies of scale (for example Barnes & Noble vs. an independent local bookstore) (Ledvina and Sircar, 2011). The experimental market in this study will be a two-firm duopoly with differentiated products. Differentiated products are goods that are not perfect substitutes, allowing each firm to receive demand regardless of setting the lowest price. For more complex study, the experiment will have three players : high-cost, medium-cost, and low-cost participants. Each participant will be randomly matched in the experiment to allow analysis of decision making based on the relative amount of cost a firm has. Finally, to see if a firm with a large cost advantage (Low-cost vs High-cost) has the same advantage as a firm with a modest cost advantage (Medium-cost vs High-cost). By replicating the assumptions used by Amir and Stepanova (2006), this paper focuses on Bertrand Stackelberg competition, which allows for a comparison to relevant literature on profit maximizing and first and second mover advantages. More importantly, this study will analyze the behavior of participants to understand if the Nash equilibrium and first mover advantage for the lower cost firm can be replicated. 1.1 Background Cournot (1929) and Bertrand (1883) competition are among the most frequently applied theories used to describe industry interaction. Firms using the Cournot model will compete with product output, while firms using the Bertrand model will compete on 2
prices. Cournot competition will result in the competitive price from market equilibrium, while Bertrand competition drives price to equal the marginal cost 2. In a duopolistic market with homogenous goods and equal marginal costs, Bertrand competition is more efficient than Cournot competition when demand is linearly structured 3 (Singh & Vives, 1984). However, firms need to determine the timing of their decision for quantity or price. Competing firms in any market will set their prices or quantities either simultaneously or sequentially. In a sequential duopolistic market, firms will act as either a leader or follower. The timing of the decision to be a leader or follower has a significant impact on the market outcome. Von Stackelberg (1934) argued that players in a market have a preference to which role (leader, follower) they hold 4. Both theory and applied experimental evidence 5 in quantity competition of homogenous duopolistic markets support that Stackelberg markets yield higher output, consumer rents, and higher welfare levels than Cournot markets 6. Higher levels of output and welfare levels in Stackelberg markets substantiate that sequential moves are more efficient than simultaneous timing in quantity competition 7. Theoretically, in quantity competition, the first mover in a homogenous duopolistic market has an advantage compared to the second mover as supported by Huck, etc. (2001). Price competition, however, varies to quantity 2 This will be true if the assumptions are set in a duopolistic situation with homogenous goods and equal marginal costs. 3 Regardless if goods are substitutes or complements. 4 In a duopoly both firms prefer the same role meaning that a stable equilibrium will not exist. When firms in a duopoly compete in quantity competition, the position of leader is most preferred, while in price competition the follower position is more advantageous as presented by Damme and Hurkens (2004). 5 See Huck, etc. (2001) and Kübler and Müller (2001). 6 Huck, etc. (2001) use experimental evidence to compare Stackelberg and Cournot duopolistic markets by focusing on homogenous products. Their results for Stackelberg markets show higher yields in output, higher consumer rents, and higher welfare levels than Cournot markets (Huck, 750). 7 This paper focuses on sequential timing because of the higher efficiency of sequential moves in markets. 3
competition as the advantage in Stackelberg markets is held by the follower 8. For example, in a homogenous duopolistic market where symmetric firms are competing in prices using sequential timing, the follower has the advantage because using perfect information to undercut the leader s price leads to the follower receiving higher demand. However, does this hold true when goods are heterogeneous? Kübler and Miller (2001) find when firms are symmetric competing in prices of heterogeneous duopolistic markets there is a significant first-mover disadvantage. Their experiment looks to study the difference between sequential and simultaneous movements. Our paper, however, will focus only on sequential movements in Bertrand markets, where firms are asymmetric in costs. There is a current lack of literature on asymmetric costs when duopolistic markets have differentiated products. Ledvina and Sircar (2011) conclude heterogeneous markets lead to inactive firms in equilibrium 9, however, do not focus on duopolistic markets. Alternatively, using literature on reaction functions, we can observe how asymmetric costs will impact first and second mover advantages. Gal-Or (1985) demonstrates that when two identical firms move sequentially, the first-mover earns lower profits than the follower, when the reaction functions of the firms are upward sloping 10. Amir and Stepanova (2006) use theoretical proofs to show that when both reaction functions 11 are downward sloping, both firms have a first-mover advantage. The same is true in a mixed case, where the firm with the downward sloping reaction function will have a first-mover advantage. In asymmetric cost functions, the 8 See Amir and Stepanova (2006) 9 Ledvina and Sircar (2011) study three-firm games before moving on to the general case of N-firm games, however, they do focus on heterogeneous markets with asymmetric cost firms. 10 Reaction functions are determined to be upward sloping (downward sloping) if they are twice differentiable and positive (negative). 11 Reaction functions of the two firms that are competing in a duopolistic market. 4
firm with a downward sloping reaction function will continue to have a first-mover advantage 12. However, this could be relative to a firm s timing, as the difference between costs of asymmetric firms would determine the range of profits for both firms in a duopolistic market. The purpose of this paper is to explore the theory that in price competition of duopolistic heterogeneous markets the first-mover has the advantage if their cost function is lower than their competitor. In a sequential move Bertrand-duopoly, firms will decide price inputs by committing to an action. The first-mover will engage in setting a price, which will be observed by the second-mover before making their own price decision, respectively. The experiment will allow for perfect information between randomly matched pairs 13, where players will have access to their opponent s price decision 14. Rebelein and Turkay (2016) create a classroom experiment to understand the first and second mover advantages in both price and quantity driven competition for sequential moving duopolistic markets. Our experiment is similar by creating a sequential Bertrand game by randomly matching students. However, their experiment fails to study asymmetric cost functions, games with perfect information, and analyze experimental evidence towards market efficiency. In this experiment, multiple asymmetric firms will be present. There are three different cost functions defined as Low-Cost, Medium-cost, and High-Cost firms. I aim to study how the effect of varying cost degrees between firms 12 This will be true if the relative firm operates with a lower cost function than their upward sloping reaction function competitor. 13 Huck, etc. (2001) report that in random matching Stackelberg markets report higher total yield quantities than theoretically predicted. Kübler and Müller (2001) also report that their theoretical predictions do better in treatments when random matching is present. 14 Only the follower will have access to their opponent s price decision, as the leader will move first. 5
affects first-mover advantage. The experiment will not provide firms the information as to which firm (Low-Cost, Medium-Cost, or High-Cost) they will be competing with. Although this brings up the issue that firms will have to make a guess as to which firm they are competing against, the three different cost firms have unique price equilibriums. I am looking to see if firms, depending on their opponent, can adapt in the latter half of the experiment to play the predicted equilibrium. 2. Model Observing a model of a duopolistic Bertrand market with differentiated goods, Firm i charges price p i, faces demand D i (p 1, p 2 ) and assumed to have linear production costs with asymmetric marginal cost c i, i = 1,2 15. The profit of firm i 16 is then given by: P i (p 1, p 2 ) = (p i c i ) D i (p 1, p 2 ) (1) The demand equation used by Amir and Stepanova (2006) that will be replicated in this experiment is the following: Equation (2) will use (1) for the profit of firm i. However, due to the fact that prices will need to be above marginal cost and that all quantities in the game will need to be positive 17, the following has to hold: i. C 1 > C 2, due to asymmetric costs. ii. The demand function (2) satisfies: 15 The standard model of duopolistic price competition with differentiated goods is defined by Amir and Stepanova (2006) on page 5. 16 This observation is taking from Amir and Stepanova (2006) on page 5, which is being used to replicate their assumptions. 17 This is taken from Amir and Stepanova (2006) on page 10 that is describing the assumptions needed for the demand equation. 6
iii. Let 18 iv. 0 < b < 1 Amir and Stepanova (2006) theorize that in order for either the low-cost or high-cost firm to have a first-mover advantage the final two assumptions have to hold: (Assumption 1) The demand equation D i (p 1, p 2 ) is twice continuously differentiable when > 0 and the following are held to be true 19 : (Assumption 2) D i (p 1, p 2 ) is strictly log-supermodular on P 1 x P 2, i = 1,2 20. This implies the following: (4) Concerning my experiment, I use the assumptions presented up to this point to observe the behavioral play of participants towards first-mover advantage. However, the demand model used for the experiment varies slightly compared to the one presented by Amir and Stepanova (2006). In order to make the experiment easier to follow when observing the optimal prices and profits, I adjusted the demand equation to decrease the 18 See 17. 19 This is assumption 1 from Amir and Stepanova (2006) that is being used for replication found on page 6. 20 This is assumption 2 also from Amir and Stepanova (2006) that is being used for replication found on page 7. 7
prices and profits 21. The demand equation allows for more comprehensible understanding as a participant, which is given by: 22 while the cost function for each firm is constant, yet asymmetric given by C a = 10 (6) C b = 7 (7) C c = 5 (8) Three firms are present with varying cost functions. By allowing varying costs when two of the three firms compete against each other, I hope to observe an effect of first and second-mover advantage based on the magnitude of costs 23. Finally, in regards to assumption i. that C 1 > C 2, C 1 can be either C a or C b, where C 2 can be either C b or C c. This possibly allows for observing decision making between firms that have varying percentage of higher costs of 40%, 43%, and 100% 24. 3 Methodology 3.1 Experimental Design The major setback in running a Stackelberg game at Colorado College was the lack of software that could replace the use of running the experiment with paper and pen. 21 Using the original demand equation (3), the prices were high around 60 units, with profits in the six-digit range. To allow more price inputs to be present on the payoff table, I decided to adapt the demand equation to give 5-digit profits and price inputs to be around 23 units. 22 a = 10 and b =.96. 23 In the experiment C 1 referred as the high-cost firm, C 2 is referred as the medium-cost firm, and C 3 is referred as the low-cost firm. 24 These percentages of cost differences are between C c and C b, C b and C a, C c and C a, respectively. 8
The use of software would allow for a major reduction in time and an increase in the rounds played. In order to run my experiment at Colorado College, it was first necessary to code my own software to allow for a high number of rounds to be played. This past summer I was taught Ruby on Rails at a ten-week intensive coding boot camp, where the last two weeks were spent on an individual project. The outcome of that time and the remainder of the summer was a real-time software that handled a 30-round Stackelberg Bertrand game. By building the game from scratch, I incorporated real-time connection to allow inputs to be sent back and forth immediately, and devise authentication with a random matching algorithm to allow for anonymity. 3.2 Subjects Twenty-four Colorado College undergraduate students took part in one session of sixty minutes. Fifty percent of the students were underclassmen (freshmen and sophomores), and forty-six percent of the students were female. Students were offered $5.00 for showing up to the experiment, $5.00 for finishing all thirty rounds, and told a bonus would be given of up to $18.00 depending on their average profit earned compared to cost similar competitors. All students had no prior knowledge of Stackelberg Bertrand competition and were selected from a wide variety of majors, including economics. Lastly, the issue of students knowing one another was not a concern because of the random and anonymous matching. 3.3 Procedure Students were instructed to show up to a testing lab furnished with 24 computers. Each computer was assigned either the role of the high-cost, medium-cost, or low-cost firm. Upon arrival, each participant was asked to draw a number for a random seat 9
assignment. When the participants sat down, the computer software instructed them to register using their name, student ID, and email address. They were then instructed to sign in using their student ID and a password they created. After signing in, each student was instantly inserted into the game lobby where they were anonymously matched with another student. On each screen, the participant saw that they were either the leader or the follower for that specific round. Before the experiment begun, I clarified the instructions and initiated a practice round. Participants were encouraged to ask questions during and after their practice round. The session of thirty rounds lasted approximately forty minutes. In each round two students were randomly matched together, with one being the leader and the other being the follower. The leader would initiate the game by inputting a price from their profit table which sent in real-time to their respective follower. The follower used the input from the leader to select a price from their profit table before recording their own profit and sending a response price to the leader. The leader would then record their profit for the round, click refresh and begin their next round with a new partner from the game lobby. Once the leader was matched, a click for new opponent button appeared on the follower s screen. The new pairings would randomly assign a leader and follower for that round, allowing each student to experience the role of leader and follower throughout the thirty rounds. Upon completion, each participant was paid $5.00 and allowed to leave. A few students were asked about their observations while playing. Two students responded that their strategy was to play the price input for the column that held their highest respective payoffs; however, they noted that somewhere in the middle of the session, 10
opponents started to send inputs that would award both parties lower profits, as if the leader was trying to punish both parties. 4. Theoretical Background and Equilibriums In this experiment we have three firms, all with three different cost functions as seen in equations (6), (7), and (8). Once assigned the role of either leader or follower, firms are able to choose their price for each round. This allows one to theorize and predict the strategy for each firm depending on their high, medium, or low cost opponent. Participants did not have access to profit maximization functions, since allowing them to try and solve for their best input would be too time consuming, and many of the participants lacked the proper mathematical background. Instead, each student was given a profit table depending on their cost function 25. Participants used the profit table to select their inputs, which I theorized would lead students to pick the price input 26 of the column with the largest profits. To optimize price inputs for every firm depending on their opponent, I used profit maximization through each firm s reaction function. Each firm has the following profit function: P i (p 1, p 2 ) = (p i c i ) D i (p 1, p 2 ), where c i is either C a, C b, C c. (9) In a Stackelberg-Bertrand equilibrium, I can solve for the optimal price input by calculating the reaction functions of both the leader and the follower. The reaction function of the follower can be substituted into the profit maximizing equation to solve 25 See appendix. 26 This input was $25 for the high-cost firm, $23.80 for the medium-cost firm, and $22.60 for the low-cost firm. 11
for the leader s ideal output, and then used to solve for the follower s best response output. Using the profit function, we can generalize each reaction function with the following 27 : P! = (%! '!) (10 %!.96%!%/ + 40%/) (10) 2P3 243 %! '! 10 %!.96%!%/ + 40%/ = 0 (11) %! = 567897:64;7.<=894; >75.<>4; (12) P/ = (%/ '/) (10 %/.96%/ 567897:64;7.<=894; >75.<>4; + 40 567897:64;7.<=894; ) (13) >75.<>4; For this experiment, there are six pairs of observations I plan to analyze. With three different asymmetric costs and the need to understand the conditions of the first or second mover advantage, the following pairs for equilibrium and respective profits need to be addressed: Low-Cost 28 L vs High-Cost F Low-Cost L vs Medium-Cost F Medium-Cost L vs High-Cost F Medium-Cost L vs Low-Cost F High-Cost L vs Medium-Cost F High-Cost L vs Low-Cost F 4.1 Game Equilibria Low-Cost L vs High-Cost F Equilibriu Using the equations (6), (10), (11), (12), and the following High-Cost reaction function: @! = >67:<.=4; >75.<>4; (14) yields the preceding result when combined with equations (14), (8), and (13): 27 In the following equations, p i is the price of the follower. 28 L and F stand for the respective leader or follower of the Stackelberg-Bertrand game. 12
@/ = 22.89 P/ = 7877.74 The strategy for the Low-Cost leader to play the ideal output of P j can be substituted into the reaction function of the follower to yield the following price and profit: @! = 25.14 P! = 7877.74 By following the same method for optimizing the maximizing price input and respective profit, I calculate the equilibriums for the remaining pairs. Low-Cost L vs Medium-Cost F Equilibrium The reaction function for the Medium-Cost follower: @! = 5E7:=.E>4; >75.<>4; (15) Yielding for the Low-Cost leader: Medium-Cost follower: @/ = 22.86 P/ = 7393.07 @! = 23.64 P! = 6355.91 Medium-Cost L vs High-Cost F Reaction function for the High-Cost follower: @! = >67:<.=4; >75.<>4; (16) Yielding for the Medium-Cost leader: High-cost follower: @/ = 23.86 P/ = 7020.61 13
@! = 25.17 P! = 5502.10 Medium-Cost L vs Low-Cost F Reaction function for the Low-Cost follower: @! = 5G7::.H4; >75.<>4; (17) Yielding for the Medium-Cost leader: Low-Cost follower: @/ = 23.80 P/ = 6300.60 @! = 22.67 P! = 7445.53 High-Cost L vs Medium-Cost F The High-Cost leader: Medium-Cost follower: @! = 5E7:=.E>4; >75.<>4; @/ = 23.86 P/ = 5465.70 @! = 23.71 P! = 7055.78 (18) High-Cost L vs Low-Cost F The High-Cost leader: Low-Cost follower: @! = 5G7::.H4; >75.<>4; @/ = 25.26 P/ = 5224.6 (19) 14
@! = 22.4 P! = 7915.27 5. Hypotheses Hypothesis 1. The average price input and profit for the low-cost, medium-cost, and high-cost leaders will be equal to the game equilibriums found in section 4.1. Hypothesis 2. The average price input and profit for the low-cost, medium-cost, and high-cost followers will be equal to the game equilibriums found in section 4.1. Hypothesis 3. Due to assumptions one and two from section 2, each firm will prefer to lead than to follow. Because there is no data directly asking each firm if they want to be the leader or the follower to reflect a preference, I will hypothesize if Õ Leader > Õ Follower for any respective cost firm, then each firm prefers to be the leader. Furthermore, firms will have a first-mover advantage where Õ Leader > Õ Follower. Hypothesis 4. In trying to understand a new game, I theorize that the subjects of the experiment will move towards the equilibrium of each pairing in the latter rounds of the experiment. 6. Experimental Results The results of the participants are organized into four subsections. Section 6.1 presents the aggregate results by analyzing the average profit per input and more importantly the number of cases when the equilibrium by each firm type was played. Section 6.2 displays price inputs and profits divided into three groups, where each group corresponds to a different set of rounds. Section 6.3 exhibits the observed average response price input to each Stackelberg-leader s price input. The response inputs are 15
compared to the calculated best response function. 6.3 also calculates the observed response function for each firm type. Finally, section 6.4 attempts at modeling the interaction between competitors using a system of differential equations. 6.1 Equilibrium Data Table I presents the collective number of cases where either the follower or the leader played the game equilibrium. One issue to note is the variability of N for each firm. This is due to two players of the twenty-four student sample group having to be dropped. The two players dropped were both low cost players and their respective observations were removed due to corrupt data 29. Table I shows the number of times a firm played the game equilibrium against the two other opponents. Inspection of table I reveals that Medium-Cost firms played the various games equilibriums the most often with 58% as a Stackelberg-leader and 69% as a Stackelberg-follower. High-Cost firms 29 The data was corrupt because both players failed to follow the instructions. They believed that they were leaders for all rounds when verbal instructions clarified that each round the leader and follower would be randomly selected and the role for that player would appear on the respective screen. 16
played the various games equilibriums the least with 34% as a Stackelberg-leader and 35% as a Stackelberg-follower. Table I also shows that the equilibrium was played, percentagewise, a higher number times when a firm was a follower compared to when that same firm was a leader. Behaviorally, this may make sense because observing the payoff tables for each firm, it is clear that a single column 30 in each table gives the highest payoff. Therefore, as a follower, it makes logical sense to always choose the highest payoff. This could be a possible explanation as to why followers played the game equilibrium a higher number of times. Table III displays the average price input for the leader against their respective opponent in the first section of the table, and the average price input for the follower 30 Appendix shows one column in each payoff table with a higher profit. 17
against their opponent in the second section of the table. To establish credibility of averages in Table III, all input pairs for each type of game were tested for normality. Table III also exhibits the equilibrium for the respective game being played. The average price inputs are broken into three groups: Rounds 1-10, Rounds 11-20, and Rounds 21-30. In breaking the average price inputs into these groups one can observe if firms moved closer or further from the game equilibrium as the number of rounds played for each firm grew. We note that there were three distinct instances when the average price input between the three groups approached the game equilibrium. This occurred for the games of Medium-Cost leader versus Low-Cost follower, High-Cost leader versus Medium-Cost follower, and High-Cost follower versus Low-Cost leader. Each type of game may not approach the game equilibrium throughout each grouping distinction because of a few players choosing inputs furthest from the equilibrium 31. 6.2 First and Second Mover Advantage In this subsection we analyze the average profit for each type of firm, displayed in figure 1. More specifically, we address the profit difference between being a leader or a follower against the same type of firm. Earlier in this paper we defined a first mover advantage when Õ Leader > Õ Follower. Following this definition of an advantage we analyze the profit difference for each firm by grouping the average profits in three distinct sections: Rounds 1-10, Rounds 11-20, and Rounds 21-30. In the graph of Low-Cost vs Medium-Cost, the profit as a leader outweighs the profit as a follower. In this case we observe that the Low-Cost firm holds a first mover advantage. However, the average 31 There are many cases when students played inputs far from the equilibrium, where their respective opponents capitalized and played non-equilibrium inputs yielding high profit. 18
profits displayed in the graph of Low-Cost vs High-Cost exhibits that the Low-Cost firm earns a higher average profit as the follower, depicting a second mover advantage. In the Medium-Cost vs Low-Cost graph we observe a partial first mover advantage due to the average profits of the Medium-Cost firm being higher in two of the three groupings. However, in the Medium-Cost vs High-Cost graph we observe the opposite, where the Medium-Cost firm has a second mover advantage in two of the three groupings. In the final two graphs we see a mix of advantages. In the High-Cost vs Low-Cost graph we note that the High-Cost firm has a second mover advantaged in two of the three groups, while having a first mover advantage in two of the three groupings in the remaining graph. Note the pattern of the Low-Cost and Medium-Cost firms; they have a first mover advantage when playing a firm with a relatively closer cost difference 32. The same pattern of high profits holds for the Low-Cost and Medium-Cost firms when they are matched with a firm with a greater cost difference, giving a second mover advantage. It is surprising that Low-Cost firms have a first mover advantage against firms with almost equal cost functions, yet a second mover advantage against firms with larger cost differences. Asymmetric firms will have a first mover advantage if their reaction function is downward sloping in a duopolistic market (Amir & Stepanova, 2006). The assumptions to our profit functions gave each firm the ability to have a first mover advantage, yet with larger cost discrepancies, firms preferred to play as the follower. 6.3 Observed Response and Best Response Figure 2 shows the calculated best responses as well as the observed responses from each game. The graphs in Figure 2 each display the collective average response 32 The costs for each firm are 5,7, and 10 for the Low-Cost, Medium-Cost, and High-Cost firms, respectively. 19
input by the follower for each price inputted by the respective Stackelberg-leader. The calculated best response functions all have slight positive slopes that are located near the equilibrium for the respective game played. The observed responses for each input vary greatly, however; each graph displays at least one point which falls either on or in close proximity to the calculated best response inputs. The responses of the Medium-Cost followers against High-Cost firms almost mimic the calculated best responses. Using the responses for each follower, the following observed reaction functions were calculated for each type of firm: Low-Cost Follower : @ I = 25.86.10@ J (20) Medium-Cost Follower : @ I = 24.10 +.02@ J (21) High-Cost Follower : @ I = 20.05 +.20@ J (22) Over time we observe that when the leader increases their prices, the Low-Cost follower will decrease their prices until zero, while Medium and High-Cost firms will increase their prices as P L increases. The observed reaction functions provide experimental understanding towards relevant literature. When two identical firms move sequentially, the leader earns a lower profit than the follower when the reaction functions are upward sloping (Gal-Or, 1985). Although the Medium and High-Cost firms are not identical, both observed reaction functions are upward sloping. Figure 1 of observed average profits displays Medium-Cost firms earning higher profits 33 as a follower against High-Cost firms. However, the opposite is true for the High-Cost firm against a Medium- Cost firm as inferred from Figure 2. Even though these firms are not identical, it is interesting to note the higher yield of profit for the Medium-Cost firm as a follower when playing an opponent with higher costs. Finally, when both reaction functions are 33 Medium-Cost firms yielded larger profits in 2/3 round groups, leading to the assumption that Medium- Cost firms have the first mover advantage. 20
downward sloping, both firms have a first mover advantage (Amir & Stepanova, 2006). Only the Low-Cost firm has an observed downward sloping reaction function, yet Figure 2 displays higher average profits as a follower against an opponent with larger cost differences. 6.4 Differential Equations Application Regardless of the type of cost-firm matchup, behavior in the experimental Stackelberg markets does not reach the predicted theoretical outcome. Instead, behavior by the varying cost firms fails to follow a common strategy. Using differential equations, we can attempt to model the long term behavior between two competing firms. Recall from Section 2 and Section 4.1 that each competing firm has varying costs and theoretically predicted profits. Further observation of Figure 1 shows that Low-Cost firms earn on average higher profits than medium and high cost firms. The same applies to Medium-Cost firms in respect to High-Cost firms. We will use the cost advantage between firms to model two competing firms in a predator-prey system. Predator-prey systems allow long term behavior to be analyzed; however, to apply this model to our competing cost firms, many assumptions need to be made. The following is a modified predator-prey system (Blanchard, 2012): 2; 2K = L/ 1 ; M N/! (23) 29 2K = O! + P/! (24) The first differential equation is a logistical growth model for the prey, where j represents the population, L represents the growth constant of j, N is the carrying capacity of j, and N is the proportion of interaction between i and j. The second 21
differential equation is the predator growth of model of population i by positively benefiting from the interaction with j, where O is the negative growth of i, and P is the positive proportion of interaction between i and j. To modify this equation to model interaction between two competing firms, we assume the prey is the high cost firm, and the predator the low cost firm. However, we need to assume that in our model only two firms compete in the experimental market. If one firm leaves the market, we assume that the remaining firm takes control of the monopolistic market. Furthermore, we assume that once a firm leaves the market, no other firms enter. Finally, the predator prey system measures the rate of profit instead of population growth, where j becomes P/, profit of the high cost firm, and i becomes P!, profit of the low cost firm. Therefore, both the predator and prey firms have logistical growth giving us the following system: 2P; 2K 2P9 2K = LP/ 1 P; M; P9 = QP! 1 M9 NP/P! (25) RP/P! (26) The second differential equation now has logistical growth where Q and R are the growth constant and the proportion of interaction, respectively. Due to P/ being the high cost firm, N > R is assumed due to the assumption that high cost is a disadvantage. Next, Q > L is inferred due to the assumption that in an asymmetric duopolistic market with heterogeneous goods, the lower cost firm will have a greater rate of profit growth over time. Finally, T! > T/ is assumed due to lower costs. Validating all of the assumptions made, the following system is created with numerical values: 2P; 2K = P/ 1 P; 566,666.2P/P! (27) 22
2P9 2K = 1.2P! 1 P9 5G6,666.1P/P! (28) The result of our system of differential equations is displayed in Figure 3. Figure 3 displays the phase portrait, which shows the directional field and various solutions curves for different initial conditions. The y-axis describes the profit behavior for the low cost firm, while the x-axis describes the profit behavior for the high cost firm. The solution curves for various initial conditions follow the directional field as time increases. The result is five distinct regions. If the initial condition for profit is in Region I, the long term behavior goes to the carrying capacity for the high cost firm, and zero for the low cost firm. This takes place only if the initial profit for the high cost firm is nearly three times the initial profit of the low cost firm, such as the point (30,10). Recall from Section 4 that the profit functions for a low cost and high cost firm are similar, where price inputs and cost vary. Therefore, 23
it is unlikely that long term profit will fall in Region I, as a high cost firm must earn a profit nearly three times greater than the low cost firm. However, this may explain the behavior of inputs away from the predicted Stackelberg equilibrium, as players may force favorable inputs for themselves as a leader, only to be punished by the follower. In Region II & Region III, we observe initial profit conditions that are positive for both firms and positive for the low cost and negative for the high cost, respectively. In both scenarios the long term behavior of profit approaches the carrying capacity for the low cost firm. Both these cases display that the low cost firm will drive the high cost firm out of the business. An interesting note is the directional vector in Region II, where positive initial conditions will decrease both firms profits. However, the high cost firm will continue to decrease towards zero, while the low cost firm will start to have logistical growth at the approximate line of y = 10. Region IV will be an initial condition of negative profit, and will continue to grow negative. Region V displays a positive initial condition for the high cost firm and a negative initial condition for the low cost firm. Due to the bounds of price in the experimental game, Regions III, IV, and V can be ignored as negative profit was unattainable. Focusing on Regions I & II, an equilibrium point to the system can be observed around the point (12,5). At these initial conditions for profit, both firms will be in equilibrium such that the growth of profit is zero. Lastly, the long term behavior in Regions I & II depicts the duopolistic market shifting into a monopolistic market. However, it is extremely difficult for the high cost firm to oust the low cost firm. The result is either the high cost firm earns profit until being removed from the market, or to play extreme price inputs so that profit is roughly three times greater than the low cost, 24
resulting in the behavior of Region I. However, landing in Region I by inputting extreme values will fail almost every time if the high cost firm is a market leader. This implies that a high cost firm is better off as a market follower. 7. Discussion In this section we address whether or not the various hypotheses stated in Section 6 hold. More importantly, we summarize our results by stating our own conclusions. Based upon our results, hypothesis 1 and 2 fail to fully hold as the highest percentage for the collective number of times an equilibrium was played was 69%. Hypothesis 3 holds for Low-Cost and Medium-Cost firms when the opponent is also either a Low-Cost or Medium-Cost firm, but fails when the Low-Cost or Medium-Cost firm is paired with the High-Cost firm. Hypothesis 4 holds true in three of the twelve potential firm pairings, which ultimately is a rejection of hypothesis 4. Result 1: Calculated Nash equilibriums are rarely played in Stackelberg games, as followers have the opportunity to capitalize on behavioral error, skewing the observed average inputs. Result 2: Subjects will have problems understanding the game leading to either removing observations or risk highly skewed data. Result 3: Various cost differences allow for long term profit advantages dependent on the initial starting profit of a firm. Result 4: A firm with a cost advantage will not always have a first mover advantage. If the cost advantage is much larger, a second-mover advantage is present. 25
Result 5: There appears to be a cost difference value when the lower cost firm switches from a first mover advantage to a second mover advantage. In spite of these results, it is difficult to predict the behavior of each various firm. It appears that subjects will vary in behavior which can be attributed to a multitude of reasons such as lack of knowledge, understanding or failure to follow directions. More importantly, we note that to make the game easier to comprehend the exact equilibrium solutions were not present on the profit tables provided. To account for the variability of these conditions it would be considerate to rerun this experiment with set roles and only two firms: low and high cost. Multiple sessions would be run, where the difference in cost between the low and high cost firms vary in each session. 8. Conclusion The theoretical results provided by Amir and Stepanova (2006) failed to hold in our experimental game, where a first mover advantage was the supposed outcome for low cost firms. If our game comprised of only two firms, one low cost and one high cost, then this hypothesis might have held. However, with the inclusion of a third firm, medium cost, our results show that there is a point between the differences in costs, which switch the low cost firm having a first mover advantage to having a second mover advantage. This allows for further study to identify the degree of cost differences that result in a first or second mover advantage. 26
Figure. 1. Average profit as leader and follower against the same type of firm. Low-Cost vs Medium-Cost Medium-Cost vs Low-Cost High-Cost vs Low-Cost Profit ($) 7800 7700 7600 7500 7400 7300 7200 7100 7000 Profit as Leader Profit as Follower Profit ($) 6800 6700 6600 6500 6400 6300 6200 6100 6000 5900 5800 Profit as Leader Profit as Follower Profit ($) 5400 5350 5300 5250 5200 5150 5100 Profit as Leader Profit as Follower Low-Cost vs High-Cost Medium-Cost vs High-Cost High-Cost vs Medium-Cost Profit ($) 8000 7900 7800 7700 7600 7500 7400 7300 7200 Profit as Leader Profit as Follower Profit ($) 7050 7000 6950 6900 6850 6800 6750 6700 Profit as Leader Profit as Follower Profit ($) 5750 5700 5650 5600 5550 5500 5450 5400 5350 5300 Profit as Leader Profit as Follower 27
Fig. 2. Observed response for each input compared to the theoretical best response. Observed Response for Low-Cost Follower vs Medium-Cost Leader Variable 27 Low4Response BestResponse4 26 Observed Response for Medium-Cost Follower vs Low-Cost Leader Variable 27.0 Medium2Response BestResponse2 26.5 Observed Response for High-Cost Follower vs Medium-Cost Leader Variable 27 High3Response BestResponse3 Price (Low-Cost Follower) 25 24 23 22 Price (Medium-Cost Follower 26.0 25.5 25.0 24.5 Price (High-Cost Follower) 26 25 24 24.0 21 21 22 23 24 25 26 27 Price (Medium-Cost Leader) 23.5 21 22 23 24 25 26 27 Price (Low-Cost Leader) 23 21 22 23 24 25 26 27 Price (Medium-Cost Leader) Price (Low-Cost Follower) Observed Response for Low-Cost Follower vs High-Cost Leader 27 26 25 24 23 Variable Low5Response BestResponse5 Price (Medium-Cost Follower) Observed Response for Medium-Cost Follower vs High-Cost Leader 26.5 Variable Medium6Response BestResponse6 26.0 25.5 25.0 24.5 24.0 Price (High-Cost Follower Observed Response for High-Cost Follower vs Low-Cost Leader Variable 27 High1Response BestResponse1 26 25 24 23 22 22 21 22 23 24 25 26 27 Price (High-Cost Leader) 23.5 21 22 23 24 25 26 27 Price (High-Cost Leader) 21 21 22 23 24 25 26 27 Price (Low-Cost Leader) 28
Appendix 29
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9. References Amir, R., & Stepanova, A. (2006). Second-mover advantage and price leadership in Bertrand duopoly. Games and Economic Behavior, 55(1), 1-20. Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. Boston, MA: Brooks/Cole, Cengage Learning, 2012. Print. Damme, E. V., & Hurkens, S. (2004). Endogenous price leadership. Games and Economic Behavior, 47(2), 404-420. Gal-Or, E. (1985). First Mover and Second Mover Advantages. International Economic Review, 26(3), 649. Huck, S., Muller, W., & Normann, H. (2001). Stackelberg Beats Cournot - On Collusion and Efficiency in Experimental Markets. The Economic Journal, 111(474), 749-765. Huck, S., Müller, W., & Normann, H. (2002). To Commit or Not to Commit: Endogenous Timing in Experimental Duopoly Markets. Games and Economic Behavior, 38(2), 240-264. Kaldor, N., & Stackelberg, H. V. (1934). Marktform und Gleichgewicht. Economica, 3(10), 227. Kübler, D., & Müller, W. (2002). Simultaneous and sequential price competition in heterogeneous duopoly markets: experimental evidence. International Journal of Industrial Organization, 20(10), 1437-1460. Ledvina, A. F., & Sircar, R. (2011). Bertrand and Cournot Competition Under Asymmetric Costs: Number of Active Firms in Equilibrium. SSRN Electronic Journal. Macgregor, D. H., Cournot, A., & Bacon, N. T. (1929). The Mathematical Principles of the Theory of Wealth, 1838. The Economic Journal, 39(153), 91. Rebelein, R., & Turkay, E. (2016). When do first-movers have an advantage? A Stackelberg classroom experiment. The Journal of Economic Education, 47(3), 226-240. Singh, N., & Vives, X. (1984). Price and Quantity Competition in a Differentiated Duopoly. The RAND Journal of Economics, 15(4), 546. 31