INTRODUCING CROSS-ELASTICITIES IN DEMAND ALGORITHMS Developments in Business Simulation & Experiential Learning, Volume 27, 2000 INTRODUCING CROSS-ELASTICITIES IN DEMAND ALGORITHMS Richard D. Teach, Georgia Institute of Technology Robert G. Schwartz, Mercer University ABSTRACT Demand for products is determined not only by the usual marketing variables of price, promo- tion, quality, etc., but also by the interrelation- ships among those variables. These interrela- tionships are referred to as cross-elasticities. To better model the actual market place, a simula- tion demand algorithm that allows not only for changing elasticities, but also for cross- elasticities, needs to be utilized. Building on Teach's Distance Model, this paper describes methodology to incorporate and control cross- elasticities in demand algorithms. INTRODUCTION Business literature has long recognized that de- mand elasticities are not constant over the range of the demand variables and are interrelated (Arora, 1979). Historically, as published in simulation and gaming journals, demand algo- rithms have generally not included cross- elasticity functions. There is at least one excep- tion, the Executive Game (Hindshaw and Jack- son, all editions), that included a rudimentary method to accommodate the cross-elasticity be- tween price and promotion, with no other de- mand affecting variables included. Further, in 1984, Teach (1984) introduced a Distance Model that included cross-elasticities, but this concept was not explored in his original papers (Teach, 1984, 1990), nor was a method developed to control them. This paper develops a methodology, based upon the use of the Dis- tance Model, to include the associated cross- elasticities in the demand algorithm. It further demonstrates how the game designer or admin- istrator may control the cross-elasticities and their impact on game outcomes. Only a two variable case is considered, but this work can be extended to the n variable case as well. BACKGROUND There are two important concepts in the devel- opment of demand algorithms. One concept in- volves the changes in demand elasticity over the range of the independent demand variables. The Gold and Pray (1983) model is an example of the use of this concept. The second concept is the use of the cross-elasticities of demand vari- ables and their subsequent effects on demand, i.e., the Distance Model. The two models are explained below. The Gold And Pray Model At the 1983 ABSEL meeting in Tulsa, Okla- homa, Gold and Pray (1983), introduced a very robust demand algorithm with a modified ver- sion subsequently published in Simulation & Gaming (Gold and Pray, 1984). This algorithm, using a set of exponential equations, was unique in that it utilized a methodology that allowed for changing elasticities of demand over the entire range of values for any set of independent vari- ables. However, the model did not allow for in- teraction effects among the demand variables. In the Gold and Pray model price elasticity (Ep) is defined as the first derivative of the demand function with respect to price (p) and is shown in Equation 1. Ep = k1 – [k2*p(1+ln(p))] (1) 125 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 K1 and k2 are constants that determine the de- gree of elasticity and are determined by either the game designer or the game administrator to best represent what "reality" the simulation re- sults should represent. Figure 2 Price Demand Schedule Promotion = $1,000,000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 250 275 300 325 350 375 400 425 450 475 500 Price U ni ts As seen in Equation 1, no variables other than price have any impact upon price elasticity, but having equal price elasticity, when price is held constant, but other demand variables change, is unrealistic. Figures 1 and 2 elucidate the point, i.e., when the amount of the promotional budget is set equal to either $100,000 or $1,000,000, the price elasticities are the same. Thus, the problem with this model is that each of the variables that affect demand are inde- pendent or orthogonal to all the others. The Distance Model Teach (1984) introduced a "spatial" model to incorporate non-economic variables in the de- mand algorithm. This model created a “product space” by using product attributes, measured in non-economic units, to determine part of the demand for the products in a simulated market place. The economic variables of price, promo- tion, quality, and size of the sales force were in- cluded in the algorithm by using the Gold and Pray model and combining the results into a hy- brid solution. Figure 1 Price Demand Schedule Promotion = $100,000 0 200 400 600 800 1000 1200 1400 1600 250 275 300 325 350 375 400 425 450 475 500 Price U ni ts In the Distance Model, as in most other models, the values used by the equations in the model are not the actual decision values, but instead are exponentially smoothed values. The use of these values prevents large changes in the deci- sion variables from causing wide variations in a simulation's results. CROSS-ELASTICITIES IN DEMAND ALGORITHMS In order to include not only the product attribute variables, but also the economic variables in the demand algorithm, the Distance Model needs to be further developed. First, for any economic variable demand calculation to be made, an "ideal" point for each of the variables needs to be located, i.e., differences between the value of the smoothed decision variables and their ideal points have to be determined. The axes' origin needs to be defined in order to perform these calculations and that process is "anchoring the scales." Modifications to the original distances have to be made to account for disproportionate influences of the demand variables. To do this, the scale ranges have to be appropriately modi- 126 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 fied so that the effect of the demand variables can be controlled by either the game designer or the game administrator. In a similar fashion, the cross-elasticities need to be controlled. To ef- fect this control, a transformation utilizing a non-orthogonal coordinate system is suggested. Finally, based upon the nature of the algorithms design and control, additional modifications may be necessary by the designer or administra- tor to move the economic "playing field" to the elastic portion of the demand curve. The meth- odologies are explained below. The Development of the Ideal Point In order to develop the ideal point, i.e., the start- ing point for calculating the allocation of de- mand across firms, the economic variables must be incorporated into the distance model. In the previous versions of the distance model, only discrete variable product attributes were in- cluded (Teach, 1984,1990). Locating the Ideal Point It is generally assumed that a lower price is al- ways better than a higher price, but if price is perceived to be too low, there are usually cus- tomer suspicions about quality or general dis- trust of the very low priced producer. As a re- sult, the Ideal Price Point for any period is lo- cated at the lowest marginal cost point in the in- dustry for that period and if any firm prices above or below this point, that firm’s demand is reduced– ceterus parabus. For all other market- ing variables, it is assumed that more is better and the ideal point is set at the periods' variable maximums. This is not unlike other models of demand in that higher budgets for advertising, quality, sales force, etc. and lower prices all re- sult in increasing the demand. By setting the ideal point for each market seg- ment at the maximum of every demand variable, except price, and price set as defined above, a “kink” in the demand curve is created at the cur- rent decision points. “Kinked” demand curves are a major economic phenomenon of oligopo- lies, the type of industries that are most often modeled in business simulations. Price A single vector represents the current prices charged by the firms for their products, and firms could have multiple products in the mar- ket at the same time. The price ideal point is lo- cated at the price that equals the marginal cost of the lowest cost producer. Lower prices are preferred to higher prices, the assumption of all games, except in this model, This model allows for customers to be leery of prices set below marginal cost. Customers prefer the lowest priced product, except if that product was priced below the marginal cost of the lowest cost pro- ducer. If a product were offered at a price be- low marginal cost, customers would be suspi- cious and have a tendency to purchase less. (This assumption can easily be altered so that customers always preferred the lowest priced product in the market every period. It is left to the game administrator to define the customers ideal points.) Promotion Like price, the promotional budgets for each product in the market can be represented as a vector (assuming that each product has its own promotion budget). If one assumes that promo- tion is the process that firms use to inform the marketplace about their product and that more promotion is better, the same assumption as all other demand models, then the ideal point would be located at the point on the vector that represents the maximum promotions budget. Anchoring the Scales The ideal point for the market segment becomes the anchor point for the scales. For explanation purposes, consider the case where the game has six competitors, each producing a single product and competing in a single market segment. The difference between the ideal price point and the exponentially smoothed price decisions by each 127 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 firm form a vector with six values. Similarly, the difference between the ideal (maximum) ad- vertising level and each firm’s smoothed deci- sions forms a vector with six vales. The same is true for each of the demand generating vari- ables. Each simulated firm can be represented by an n dimensional point in space, where n equals the number of demand generating vari- ables. Providing a two variable example might serve to clarify this issue. A firm’s exponentially smoothed decisions, the ideal points and the two space coordinate points and the calculations used to determine them are shown in Table 1. Table 1 Decision Versus Ideal Point Variables Variable Firm 1’s decisions Ideal point Firm 1’s two space coordinate points Price ($) 108 55 53 (109-55) Promotional budget ($) 250,000 450,000 -200,000 (250,000-450,000) The Distance From the Ideal Point The distance between any firm's decisions and the ideal point is calculated by the Pythagorean Theorem. In two dimensional space, the hy- potenuse (the distance between two points, Dis- tIj) is equal to the square root of the square of the difference between the two points on the first axis plus the square of the difference be- tween the two points on the second axis. The distance across n space is shown in Equation 2. DistIj = ((a1-a2)2 + (b1-b2) 2 + ··(n1-n2)2) 1/2 (2) Where I defines the ideal point and j defines the firm de- cisions and j indicates the firm which is being measured (j varies from 1 to the number of competing firms in the in- dustry) and n represents the number of demand affecting variables, both economic and product attribute variables. "a" and "b" are the variables for deriving demand where the maximum number of variables is n. Defining the Scale Ranges Using the above equation (2) with the exponen- tially smoothed decision values would result in the promotion budget, Table 1, dominating the calculated distance, simply because of the magni- tude of the numbers. To prevent the largest number in the decision set from dominating the distance, each firm’s smoothed decisions are normalized. That is, each smoothed decision, across all firms, is divided by a value serving to normalize the range of all the variables. One such value is the maximum value of each smoothed decision variable. This means that the greatest value for each of the demand generating variables is equal to one. Weighting the Relative Importance of the De- cision Variables The weighting process allows the game designer or the game administrator to differentially weight the impor-tance of each decision variable. Thus, the weighted distance (WDistij) equation is: WDistij = (w1*(a1-a2)2 + w2*(b1-b2) 2 + ··· wn*(n1-n2)2) 1/2 (3) Where wn is the assigned weight of the nth variable, and the sum of the w's equal n Demand is allocated to the competing firms in- versely proportional to their distances from the ideal point. As the marketing and product at- tribute variables become closer to the market segment’s ideal point, the better the product meets the needs of customers and the greater the firm’s market share. Market Share The market share for firmj (MSj) is defined as the inverse of the distance between firmj and the ideal point i (Distij), divided by the sum of the inverses across all n firms as described in Equa- tion 4. Since market share is a quadratic func- tion of distance across all smoothed decision variables greater than zero, then the derivative of the function has all the smoothed decision 128 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 variables included. Ergo, cross-elasticities for all possible combinations of variables greater than zero exist. A distance of zero simply means that the variable does not contribute to the calculation of demand. Finally, a distance between two points can not be negative. MSj = [(1/Distij) / (Sum(1/Distij)] for all ij (4) Cross-Elasticities The Pythagorean Theorem is the two- dimensional case of calculating the distance be- tween two points in an orthogonal space using Euclidean measures (taking the square root of a sum of squares based on coordinate points placed on axes set at 90 degrees). The fact that distance is a function of two axes causes an interaction ef- fect and produces the cross-elasticities of the economic variables. To investigate the cross-elasticity effect, a re- gression analysis was performed. Distance val- ues (the determinate of demand) were generated using 81 observations. The observations were developed using two variables, each having nine different values in a Latin Square design. The regression forced the intercept through zero. The distance between price and the ideal price as well as the distance between the promotional budgets and the ideal promotional budgets were rescaled to one to nine. The resulting Equation (5) is: DistIj = 1.239 * Mj + 1.239* Pj - 1.057 * [ (PjMj)1/2] (5) DistIj is the distance between the ideal point and the prod- uct offered by firm j. Mj is defined as the distance between the ideal marketing budget and the marketing budget of firm j. Pj is defined as the distance between the ideal price and the price set by firm j. PjMj is the cross product distances for firm j. The regression produced a Coefficient of De- termination (R2) of 0.9987. Thus, Equation 5 accounts for “almost” all of the variance in the distances between the firms’ products. One needs to remember that the greater the distance a product is away from the ideal point the less its sales. Thus, the negative value for the cross-product, PjMj, means the combination has positive cross-elasticity or greater sales than either price or promotion alone would indicate. Non-Orthogonal Space Now that cross-elasticity can be estimated, how can it be controlled for the purpose of game de- sign? The distance calculated in Equation 5 used Euclidean measures. But what if the assumption of 90-degree axes were to be abandoned? That is, a non-orthogonal space, where the angle be- tween the axes may be any angle between 00 and 1800 and not restricted to 900. Figure 3 illus- trates two products in an orthogonal space. Figure 3. Two products in Orthogonal Space M A R K E T I N G If a non-ortho of the straight nipulated in a cross-elasticit lustrates the s space. Ø 129 Product 1 PRICE gonal space is u line between p predictable way y can be contro ame problem in P Arrow is Distance Between 1 and 2 se, then the length roducts can be ma- , and the degree of lled. Figure 4 il- a non-orthogonal roduct 2 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 Figure 4. Two products in Non-Orthogonal Space PRICE Note that the distance between shortened and decreasing the reduced the cross-elasticity. To investigate the cross-elastic tive to a changing angle Theta, gressions were performed usi data points as before, but with for the angle Theta (Table 2). sion coefficients are standardiz directly compared, given an an at 15 degrees the cross-elasticity as important as the price and ables are, but at 165 degrees the as 13 percent. Thus by picking the cross-elasticities can be con Since the coefficients (beta) a they can be directly compared, Theta. Thus, at 15 degrees th effect is almost as important promotion effects, but at 165 de elasticity effect is only 13 per and promotion effects. Thus, angle Theta, the cross-elasticit trolled. Table 2 Regression Results with Varying Degrees for Theta Theta R2 beta1 Promotion beta2 Price beta3 Cross- elasticities 150 0.99 2.975 2.975 - 4.803 Ø Product 1 Arrow is Distance Between 1 and 2 the products has angle Theta has ity effects rela- a series of 11 re- ng the same 81 changing values Since the regres- ed, they can be gle Theta. Thus, effect is almost promotion vari- effect is as little an angle Theta, trolled. re standardized, given the angle e cross-elasticity as the price and grees, the cross- cent of the price by selecting an ies can be con- 300 0.99 2.298 2.298 - 3.436 450 0.98 1.736 1.736 - 2.328 600 0.99 1.337 1.337 - 1.551 750 0.99 1.064 1.064 - 1.023 900 0.99 0.876 0.876 - 0.664 1050 0.99 0.747 0.747 - 0.417 1200 0.98 0.658 0.658 - 0.246 1350 0.99 0.597 0.597 -0.131 1500 0.99 0.588 0.588 -0.056 1650 0.99 0.536 0.536 -0.140 THE OVERALL ELASTICITY OF DEMAND After the distances between the actual and ideal points are calculated, the overall elasticity of demand may need to be adjusted. Since demand is allocated by the inverse of the distances, its elasticity can be altered by raising each distance value to a constant power. If the power is greater than 1.0, the demand elasticity increases and if the power is less than 1.0, the demand elasticity decreases. Table 3 demonstrates this phenomenon for the power of 3.0 Table 3: Overall Demand Elasticity Dist Inverse Market Share Power 3 New Distance Inverse New Market Share 4.12 0.242 27.9% 70.09 0.0143 46.0% 5.10 0.196 22.6% 132.57 0.0075 24.3% 6.08 0.164 18.9% 225.06 0.0044 14.3% 7.07 0.141 16.3% 353.55 0.0028 9.1% 8.06 0.124 14.3% 524.05 0.0019 6.2% Sum 0.868 0.0310 Product 2 130 Developments in Business Simulation & Experiential Learning, Volume 27, 2000 THE DISTANCE MODEL Although not discussed in this paper, the origi- nal model (Teach, 1990) included two important concepts. First, in order to prevent a single product from capturing the entire market, the ideal point has one additional dimension that prevents a perfect match between any existing product and the ideal point. Mathematically this would result in a perfect match that would result in a number divided by zero or infinity. The second concept was that of a segment shadow. The shadow absorbed demand, when the prod- ucts offered in the market were deemed as “less desirable” by the customer. This feature was very important when non-economic aspects such as specific product attributes are included in the model. SUMMARY Market share allocations are based upon the relative distance each competing product is from the market segment’s ideal point; the greater the distance the less the market share. All decision variables are exponentially smoothed prior to use in the allocation of de- mand Equations. The Ideal-Point for price is set equal to the product with the lowest marginal cost. The Ideal-Point for all other economic variables are set equal to the greatest level of expenditures for each demand-generating variable across all firms. Each smoothed demand-generating variable is scaled in order that they all have a common range. They are then weighted to represent their desired relative importance to demand. The angle Theta between the axes is determined to set the desired cross-elasticities. If further adjustments to demand are necessary, then the distances between each product and the ideal point are raised by a power function to control the overall elasticity of demand. Finally, the harmonic value of these distances is derived and represents the market share allocated to each product. In equation form, this becomes: MSj = [(1/D'Ij) / Sum(1/D'Ij)] for all j (6) Where D'Ij = (DIj)P for all products in the market place. (7) Where D Ij = {[w1*(pj)2] + [w2*(mj)2 ] – 2*pj*mj*Cos (ø)}1/2 (8) Where: ø = angle between the axis of p (price) and m (marketing budgets controlling the Price and Marketing cross-elasticities. m j= the distance between the highest marketing budget and firmj’s marketing budget, nor- malized to a common range with price. pj = the distance between the lowest marginal cost product and the price set by firmj. w1 and w2 = weights designed to define the rela- tive importance of the demand generating variables D Ij = The total distance between the Ideal prod- uct and firmj’s product D'Ij = The total distance between the Ideal prod- uct and firmj’s product raised to a power to control the demand elasticities as a whole and MSj = the market share allocated to firmj’s product References Available Upon request 131 Table of Contents Volume 27, 2000 Internet International: A Simulation Exercise for Understanding Technological Innovation and customer Service In a Rapidly Growing Internet Server Company Simulations and Learning: Dialog and Directions Endnote Activity: A Tool for Integration of Course Content and Communication Skill Practice Incorporating Video as a Teaching Strategy in Interpersonal Communication Vision Quest: An Alternative Approach to Industry Analysis for MBA Courses in Strategic Strategic Management: An Evaluation of the Use of Three Learning Methods Trainer, Mentor, Educator: What Role for the College Business Instructor in the Next Century? Using the Internet and Shareware to Facilitate Computer Simulation in Distance Learning Classes Visual Modeling of Business Simulations Teaching about Information with Management Games A Self-Evaluation Based on the Discussion and Decision in Experts' Business Gaming The Restaurant Game Using Journals to Enhance Computer Simulation Based Learning Exercises to Facilitate Better Student Writing in the Undergraduate Strategy Class Identifying, Resolving, and Managing Common Ethical Dilemmas in the Workplace: An Experiential Approach Integrating the Digital Revolution into the Classroom The Wheel of Learning: An Integrative Business Curriculum Experiment The Changing Nature of Simulation Research: A Brief ABSEL History Perspectives on Simulation & Gaming's Review Process Experiential Learning Across Disciplines: Mixing International Business and Accounting Simulating Governmental Effects on Economic Development Internationalizing the Introduction to Business Course Using an International Text and Domestic Simulation with a Twist Using Stock Value as the Performance Measure in a Business Simulation Game Introducing Cross-Elasticities in Demand Algorithms Validating a Model of Currency Valuation An Exercise for Exploring the Relationship between Jungian Psychological Types and Organizational Politics Exercise: Preparing Financial Reports Using the Group Categorizing Technique Effect of Trust and Cultural Beliefs on Negotiation Processes: Data from an Experiential Role Play Experiential Learning Gets Stamp of Approval From the Boyer Commission Talent Search 2000 - An Experiential Activity to Help Strengthen Skills in Employee Recruitment and Selection Clemson University's Collaborative Learning Environment Initial Data on a Test Bank Assessing Total Enterprise Simulation Learning Changing the Assessment Paradigm: Using Student Portfolios To Assess Learning from Simulations How We Learn and Why We Don't: The Cognitive Profile Model: A Workshop in Teaching to Reach Your Students Knowing Thyself: A Portfolio Approach to Student Self-Assessment Collaborative Learning and Web-Based Instruction in a Cognitive Apprenticeship Model Teamwork Attributes in a Classroom Simulation Virtual Teams: Meeting the Next Challenge for Experiential Education New Age Learning: Nuance or Nonsense Developing Charisma: An Experiential Exercise in Leadership Problems and Solutions in Going Web-based with an Agribusiness Simulation Creating a Comprehensive Web-Enhanced Classroom Your Class is in Session, Now What? The Challenges of Teaching On-line An Application of Process Control Charts for Attributes as a Form of Classroom Assessment for Experiential Learning Work Goals and Life Aspirations: Do You Have What it Takes to be an Entrepreneur An Exercise to Develop Initiative: Possible Dream? The Ball Point Pen Assembly Company Management Game Review System Development Total Enterprise Simulations and Optimizing the Decision Set: Assessing Student Learning Across Decision Periods Facilitating Learning in the New Millennium with the Complete Online Decision Entry, System (CODES) The Marketing Management Experience The Right Venue for Your Simulation One More Time: Overall Dominance in Total Enterprise Simulation Performance A Profile of ABSEL Conference Attendees Learning Readiness: An Underappreciated Yet Vital Dimension in Experiential Learning Active Learning in a Professional Undergraduate Curriculum The Problem Is - They Think Differently! Cultures Integration in Mergers and Acquisitions: Putting Managers Together in a Business Simulation The Global Business Game: A Strategic Management and International Business Simulation