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Pedagogy Using Mathematica Through the Web

by Sloan-C

ABSTRACT
This paper deals with two topics: (1) a web-based technology, called MathServ, that combines the computational engine of Mathematica (the Mathematica kernel) with web pages that are written in the HTML language; and (2) the use of this technology to teach complex economic models to students without requiring them to learn the tools necessary to build the models.. The marriage of Mathemtica with HTML creates a synergism that is a useful tool for teaching mathematics and mathematically-oriented topics.

I. INTRODUCTION

As the price of computing power has fallen, researchers have adopted computationally-intensive research tools that have allowed them to solve new problems with increasingly-complex and realistic models. In almost every branch of science, including the social sciences, new high-level languages such as Mathematica, Maple and MatLab have radically changed the way research is done.

Unfortunately, the potential of these tools to change teaching has gone unfulfilled. The reasons for this are two-fold: (1) the cost of the software and hardware necessary to use these programs is beyond the reach of most students; (2) the programs are often difficult to learn, particularly for entry-level or non technical students.

In response to these cost and pedagogic issues as well as looking for new instructional models, faculty from the Department of Mathematics and the Owen Graduate School of Management developed a new technology in 1995 called MathServ. The idea behind MathServ is to use existing Internet browsers such as Netscape or Internet Explorer as an interface to the Mathematica computational engine (the Mathematica kernel) running on a remote server. By putting a user-friendly web interface on Mathematica, instructors and students can utilize the computational power of the program for pedagogical applications at very low cost--where cost is measured not only in dollars, but in terms of the time expended by students. In short, the technology is very efficient. The web pages used in the system are typically focused on a narrow task that permits a greater degree of control over the learning process. The instructor designs tools or games, with an easy-to-use web interface, that illustrate one particular idea or concept. The student learns the ideas or models by playing the games or using the tools to answer questions.

The MathServ technology has many cost and pedagogical advantages:

• The system does not require the user to have the Mathematica software. The only software needed by the user is the Web browser software which is presently free. The only major cost is the Mathematica software for the server and a single server can service hundreds of users at a time. Furthermore, since Web browsers run on many platforms, the system is machine independent.

• Access to the system is available 24 hours per day, 7 days a week over the Internet.

• Each web page can be designed to do a specific calculation, thus focusing the user's attention on learning a particular model or concept. This is much easier than developing multipurpose stand-alone applications, e.g. Click&Learn Regression.

• The system is modular and is easily accessed by developers since all of the web pages and Mathematica script are stored on the server.

• The output created by the Mathematica kernel can be multimedia--text, graphics, sound, and/or movies.

• Data about who is using the system and what they are doing is easily collected.

• The system can be used to disseminate examples and applications of research which would be "beyond" the reach of most students using traditional pedagogy.

• The system permits relatively effortless communications between the student and the instructor using email through the web pages.

• MathServ is well suited for remote learning.

The MathServ technology has the potential to be used in a wide variety of courses. Presently, the Department of Mathematics at Vanderbilt uses it in their introductory courses for calculus and differential equations. For example, in the calculus sequence a toolkit has been developed that performs many of the basic mathematical operations that are taught in the first year of calculus (e.g., differentiation, integration, or series.) The Calculus Toolkit is used in conjunction with a laboratory as well as being available in and outside of the classroom. There is also a DE Toolkit. However in this article, the main focus is the use of MathServ for teaching MBA students some of the important ideas about competition that derive from complex oligopoly models. Students learn about competition by playing games that involve the loss of competition following a simulated merger between two competing firms.

II. MERGER SIMULATION GAMES

One of the first applications of the MathServ technology was to teach and disseminate a new methodology for merger enforcement. Using software developed by Professors Crooke, Froeb and Tschantz, and the Department of Justice Research Director Gregory Werden, both the Federal Trade Commission (FTC) and Department of Justice (DOJ) recently began using computer models to simulate the effects of mergers (see e.g. [1,2,3,4,5,6]). The use of the software in several high profile cases, such as the L'Oreal-Maybelline and General Mills-Chex mergers, led to rapid adoption of the technology. Writing in the Spring 1997 issue of Antitrust, a publication of the American Bar Association, one attorney called it "one of the most interesting and potentially useful recent developments in antitrust merger analysis."

To teach the methodology to attorneys as well as students, several interactive games were designed. The games are motivated by actual cases on which Professor Froeb was employed, either by the Justice Department or by one of the parties to the merger. This degree of realism heightens motivation to use the games, especially for MBA students who are ever wary of learning abstract models with little relevance to the business world. The games can be demonstrated in class, or assigned as homework. The games put the student into the role of economist, analyzing the effects of mergers or computing damages. The assignments ask questions that the student can answer only by playing the games. Below, some of these games are illustrated.

III. ORAL AUCTION GAME

This game illustrates the loss of competition in the setting of an oral auction [7]. In this game, the price of the good being sold is determined by the strength of the losing bidders. The stronger the losing bidders, the higher the winning bidder has to bid in order to win the auction. Mergers between the bidders will obviously have an effect on the winning price of the good. It is assumed that the value that each bidder places on a good comes from a probability distribution (with two free parameters: mean and standard deviation). The student chooses the means and standard deviations for each bidder in the auction and observes the outcome on the revenue and market share (percentage of time that bidder wins) that are generated by the auction. Below we have the web page that the student uses to enter the data for the simulation.

Figure 1. Simulation Inputs

If the Simulate Merger button is selected, then the mathematical model is executed giving information about the outcome of the merger. Figure 2 presents the typical output from the simulation.

Figure 2. Premerger and Postmerger Results

Figure 3. Graphs of Distributions: A B C Max(A,B)

In Figure 3, bidders A and B merge and become Max(A,B). The visual cue red+blue=purple helps students understand the rules of the game. After a merger, the only remaining competition is between C and Max(A,B). The price effects of the merger depend on how strong the bidders are (determined by their means) as well as the standard deviation. As in most of the games, students are asked to find anticompetitive and competitive mergers, and in doing so, develop an intuitive feel for how competition works in oral auctions.

IV. UNILATERAL EFFECTS MERGER GAME

The Unilateral Effects game illustrates the models used in the L'Oreal-Maybelline merger. There a merger was permitted by the Department of Justice, in part due to the low cross-price elasticity between the mascara products of the merging firms. L'Oreal produces a higher-end product than Maybelline (i.e., they were weak substitutes), so there was only a small degree of competition lost by the merger. The merger game asks the students to find anticompetitive mergers (those that raise price above 5%) and OK mergers (those that raise price by less than 2%) by changing the elasticity parameters. The output of the mathematical model show the effects of the merger on prices and market share for the merging and nonmerging companies. In answering the question, students develop an intuitive feel for the role that demand elasticities play in determining the degree to which substitute products compete with one another. Beyond its use as a merger model, the game can also be used to teach MBA students to price jointly-owned brands--be they substitutes or complements.

V. SPATIAL MERGER GAME

This game is based on a case reviewed by the Department of Justice.It involved predicting the loss of competition that resulted from mergers between timber mills in National Forests.

Students are put in the role of economist trying to predict the effects of a merger between several timber mills in a forest. They must find, by trial and error, the worst possible merger, and the best possible merger with respect to revenues that the government collects for leasing national forrests. The output of the simulation is a number, showing how much revenues decline, as well as a "topographical" map (see Figure 4) of the forest showing where the price rises are the largest. The darker colors indicate higher revenue loss and the dots indicate the position of the merging timber mills in the forest.

Figure 4. Loss of Competition in a Forest.

Students find that the worst merger (from the standpoint of the the government) is one where adjacent mills merge--and the best merger is one where distant mills merge. The analogy of a timber mill in a forest to positioning brands in product space is very easy to draw. Those consumers hurt most by the merger are analogous to forest tracts with strong preferences for the merging brands, i.e. those tracts close to the merging mills. The geographic map is analogous to "brand maps" that are used in marketing to analyze product positioning.

VI. CONCLUSION

While it seems clear to us that this pedagogical approach is working, there are still unanswered questions about the best way to use the technology in different settings. So far there seems to be a natural split between tool-based applications, such as the Calculus Toolkit, and the games approach. Tools allow users to harness the power of the computational algebra systems without having to learn the language, while games are designed to give users the intuition gleaned from complicated oligopoly models without incurring the high costs of learning how to build the models. Developing hypotheses about how students learn using this technology and testing the hypotheses in controlled settings seems like a natural next step.

VII. REFERENCES

1. Crooke, Philip, Luke Froeb & Steven Tschantz, "Simulate Mergers On-Line", Antitrust 11 (Spring, 1997) 29.
2. Froeb, Luke, Steven Tschantz, Philip Crooke, and Gregory Werden "The Effects of Assumed Demand Form on Simulated Post Merger Equilibria," Owen working paper (1997).
3. Froeb, Luke, and Gregory Werden, "Simulating Mergers among Noncooperative Oligopolists," in Computational Economics and Finance: Modeling and Analysis with Mathematica, edited by Hal Varian (TELOS, Springer-Verlag) 1996.
4. Werden, Gregory, and Luke Froeb, "Simulation as an Alternative to Structural Merger Policy in Differentiated Products Industries," chapter 4 in The Economics of the Antitrust Process, edited by Malcolm Coate and Andrew Kleit, Boston: Kluwer Academic Press, 1996.
5. Werden, Gregory, and Luke Froeb, "The Entry-Inducing Effects of Horizontal Mergers," Journal of Industrial Economics, forthcoming.
6. Werden, Gregory and Luke Froeb, "The Effects of Mergers in Differentiated Products Industries: Structural Merger Policy and the Logit Model," Journal of Law, Economics, & Organization, 10 (1994) pp. 407-426.
7. Froeb, Luke, Steven Tschantz & Philip Crooke, "Mergers Among Asymmetric Bidders: A Logit Second-price Auction Model," , Owen working paper (1997).