MODELING NON-PRICE FACTORS IN THE DEMAND FUNCTIONS OF COMPUTERIZED BUSINESS SIMULATIONS Developments in Business Simulation & Experiential Exercises, Volume 11, 1984 240 MODELING NON-PRICE FACTORS IN THE DEMAND FUNCTIONS OF COMPUTERIZED BUSINESS SIMULATIONS Steven C. Gold, Rochester Institute of Technology Thomas F. Pray, Rochester Institute of Technology ABSTRACT This paper evaluates the way in which non-price factors of demand are modeled in computerized business simulations. It is found that several different functional forms are utilized. The properties of these functions are reviewed and compared to modern demand theory. A recommended functional form for modeling demand is then presented and illustrated with a numerical example. INTRODUCTION Non-price factors of demand, like advertising and promotion, are fundamental components of most computerized business simulations. Students by making these types of decisions in a simulation environment, are supposed to gain insights into business theories or concepts. Accordingly, it is necessary for the mathematical functions or algorithms embodied in the simulations to reflect the relationships described by business or economic theory. At the 1981 ABSEL conference a paper presented by Kenneth R. Goosen[6} emphasized the need to expand the research concerning the internal design or modeling of computerized business simulations. He noted that the current and past papers presented at ABSEL do not provide enough information to help designers develop simulations in an efficient and effective manner and concluded: The designing and developing of simulations... appears to be primarily an art form, a creative skill based on intuitive feel rather than acquired knowledge. This raises a number of interesting questions. How are non-price factors modeled in business simulations? Do the simulation functions have the properties described by conventional demand theory? What are the advantages and disadvantages of alternative modeling approaches? An understanding of these issues should enhance the development and effectiveness of future simulations. PURPOSE The purpose of this paper is fivefold: (1) to examine the different ways in which computerized business simulations have modeled non-price factors of demand; (2) to review the theory of demand, focusing on non- price factors and the characteristics of inflection points; (3) to specify a flexible and stable functional form that is consistent with the properties of modern demand theory; (4) to derive mathematical expressions for the elasticity and inflection point of the demand function; (5) to illustrate with a numerical example the procedure involved in determining the parameter values of the demand function after specifying the elasticity and inflection point. THE DATA: SELECTED SIMULATIONS Five commercially available business simulations of different vintages (publication dates) were selected for review and analysis. The selection was based on three criteria: (1) the utilization of a demand function; (2) the inclusion of non-price factors in the demand function; and (3) a published source listing of the computer software program. A list of the selected simulations is reported in Table 1. The vintages range from 1968 to 1983. This wide range provides a historical perspective of demand modeling. The complexity of the simulations is compared by measuring three factors: the number of decision variables, the number of products, and the maximum number of firms in the market. Based on this criteria, the least complex simulations were the earlier vintages, i.e. Integrated simulation and the Executive Game. The most complex simulation appears to be the Multinational Game with 18 decision variables and two market products. TABLE 1 CHARACTERISTICS OF SELECTED SIMULATIONS Title Vintage Number of (Publication Date) Decisions Products Firms (Max.) Integrated Simulation 1968 8 1 9 Executive Game 1972 8 1 9 Multinational Game 1980 18 2 9 DECIDE 1981 13 1 9 MICROSIM 1983 8 1 99 REVIEW OF SIMULATION DEMAND FUNCTIONS The selected simulations use a variety of different independent variables and functional forms. The precise firm level demand functions embodied in each simulation are reported in Table 2 with a list of variable definitions. Two simulations used non-linear functional forms: Integrated and the Multinational Game - Product A. All the remaining simulations used a common functional form referred to as log - linear. A paper by Pray and Gold[8] evaluated the advantages and disadvantages of these types of functions and concluded: (1) the non-linear demand functions permit variable elasticities, however, they tend to be "highly unstable and constraints on decision variables are needed.” (Constraints were placed on the simulations using this functional form.) Developments in Business Simulation & Experiential Exercises, Volume 11, 1984 241 (2) the log-linear demand functions constrain the elasticities to be constant, but the functions are stable. However, “at the firm level care must be taken to avoid zero level decision variables.” The primary focus in this study pertains to the way in which the marketing (M) and research and development (R) are modeled in the demand functions. Are the properties of the demand functions consistent with the propositions described in modern demand theory? Table 3 provides information on the non-price elasticities implied by the functional forms used in each of the simulations. TABLE 3 FIRM LEVEL NON-PRICE ELASTICITIES Simulation Marketing Research & Development Integrated 0.75 0.75 Executive 0.70 0.70 Multinational- Product A Product B 1.00 none 0.50 1.00 DECIDE 1.50 1.02 MICROSIM 0.60 0.50 Non-price elasticities measure the percentage change in the quantity demanded due to a percentage change in the non-price variable (in this case marketing or R & D). An elasticity less than 1 indicates diminishing returns to the non-price factor. An elasticity greater than 1 implies increasing returns to the non-price factor. Referring to Table 3, diminishing returns to marketing occurs in three of the five simulations. Only one simulation has increasing returns - DECIDE. The Multinational Game has constant returns to marketing expenditures for Product A, and constant returns to R & D expenditures for Product B. DEMAND THEORY: NON-PRICE FACTORS In theory, demand is a function of price, as well as a vector of non- price factors which includes: the prices of related goods, income, marketing, and product quality (R & D). Q=f(P, Ps, Pc, Y, M, R) where: P price of product Ps = price of substitute good Pc = price of complement good Y = income M = marketing R = research & development (quality measure) Apriori expectations as to the sign of the relationship between demand and the independent variables for a normal good are as follows: dQ/dP<0 ; dQ/dPs >0 dQ/dPc <0; dQ/dY>0 dQ/dM>0 ; dQ/dR>0 Since the focus of this study pertains to the non-price factors of marketing and R & D, the second order conditions for the variables are specified below: d2Q/dM2>0 ; d2Q/dR2>0 (increasing returns) d2Q/dN2=0 ; d2Q/dR2=0 (constant returns) d2Q/dM2<0 d2Q/dR2>0 (decreasing returns) This relationship for the marketing variable is illustrated graphically in Figure 1. FIGURE 1 RELATIONSHIP BETWEEN MARKETING AND DEMAND Increasing returns to marketing occur for expenditure levels between 0 and Me, i.e. the slope of the function rises. Expenditure level Me (point E)is the inflection point of the function or commonly referred to as the point of diminishing returns. After point E, additional expenditures on marketing realize decreasing returns, i.e. the slope of the function declines. Developments in Business Simulation & Experiential Exercises, Volume 11, 1984 242 COMPARISON OF SIMULATION DESIGN WITH DEMAND THEORY The demand functions embodied in the simulations (previously reviewed) did not possess the inflection point characterized by modern demand theory. The DECIDE simulation had increasing returns over all ranges of marketing and/or R & D expenditures. The Multinational Game had constant returns over all ranges of marketing (product A) and R & D (product B) expenditures. The remaining simulations possessed decreasing returns over all ranges of expenditure. The functional forms adopted in these simulations were not flexible enough to permit the modeling of an inflection point. The remainder of this paper presents and evaluates a more flexible functional form. A SUGGESTED DEMAND FUNCTION A functional form recommended for modeling demand in computerized business simulations was presented in a 1983 paper by Gold and Pray[5]. This function is given below: Q = ao p-(a1+ a2P) M+(a3 - a4M) R+(a5 -a6R) (1) where: Q = quantity P = price M = marketing R = R & D ai = parameter i The functional form is multiplicative but is not log- linear. The authors have shown this function to be stable while possessing the fundamental characteristics of demand theory. DERIVING THE INFLECTION POINT The inflection point (or point of diminishing returns) for the marketing variable will be derived. The derivation is general and holds for all non-price variables in this particular function. Using partial analysis (i.e. holding P and R constant) the demand function is reduced to: Q = kM(a3 – a4M) (2) where k = a constant (containing P and R) Taking the natural log of equation (2): ln Q = lnk + (a3-a4M) ln M (3) The derivative of the function yields: dQ/Q = (a3 - a4 M)dM/M -a4lnMdM (4) where: dQ, dM are partial derivatives Solving for the marketing elasticity (e) and simplifying e = a3 - a4 M(l + lnM) (5) where: e = (dQ/dM)(M/Q) elasticity The marginal impact of marketing (dQ/dM) may also be expressed from equation (4): dQ/dM = (a3 - a4M(1+1nM))(Q/M) (6) At the inflection point the second derivative of Q with respect to M is zero: d2Q/dM2 = 0 (7) Solving equation(7) by taking the derivative of equation (6) with respect to M and setting it equal to zero, yields: e2 = a3 + a4M (8) where: e = elasticity for marketing e2= (a3 - a4 M(l+lnM))2 Consequently, the inflection point is characterized by equation (8), that is, the marketing elasticity squared equals the sum a3 + a4 M at the inflection point. SOLVING THE PARAMETER VALUES: AN EXAMPLE An example of how to determine the parameter values of the demand functions after specifying the characteristics of the inflection point is presented to demonstrate the properties of the functional form and the ease in which the parameter values are solved. The simulation designer need only specify the level of marketing expenditures and the elasticity at the inflection point. Assume the following specification: oMarketing expenditures of $100,000 oMarketing elasticity of 2.0 Substituting the values for marketing expenditures and elasticity into equations (5) and (8) yield, respectively: 2.0 = a3 - a4100,000(1+1n100,000); from (9) equation (5) 4.0 = a3 + a4100,000; from equation(8) (10) Solving equations (9) and (10) simultaneously, the values for a3 and a4 are: a = 3.65256 a = 1.48007 x 10-6 Substituting the value of the parameters into equation (2) gives: Q = kM3.65256-0.00000148M (11) The parameter "k" is simply a scaling factor and may be arbitrarily assigned a value. (In this case the demand variables other than marketing are assumed to be held constant.) The relationship between marketing and demand portrayed by equation (11) is illustrated numerically in Table 4. (A value of k of 3.0008x10 was assumed.) The marginal impact of marketing is the change in quantity demanded divided by the change in marketing expenditures. Note that the marginal impact of marketing increased from 150 units per dollar to 182 units per dollar as marketing rises from $60,000 to $100,000. After marketing expenditures of $100,000 the marginal impact declines, indicating diminishing returns. The inflection point occurs, therefore, at the $100,000 level as initially specified by the simulation designer in the hypothetical example. Note Developments in Business Simulation & Experiential Exercises, Volume 11, 1984 243 that the shape of this function corresponds to the illustration in Figure 1. TABLE 4 THE MARGINAL IMPACT OF MARKETING EXPENDITURES Marketing Expenditures Quantity Demanded (units) Marginal Marketing Impact (units/$) 60,000 70,000 80,000 90,000 100,000 110,000 120,000 130,000 3,201,854 4,701,894 6,405,536 8,184,052 10,000,000 11,789,374 13,401,302 14,903,767 xxx 150 170 178 182 179 161 149 SUMMARY AND CONCLUSION The modeling of demand is an important component of most business simulations. The simulation designer, therefore, should be careful in selecting the functional form for demand. A review of several past simulations indicate a number of different functions were selected. However, none of these functions were flexible enough to embody the standard inflection point characterized by demand theory. A demand function which overcomes this problem was presented and evaluated. This function is flexible enough to possess the properties described by demand theory but is simple enough to be easily solved for the parameter values. The paper’s intent is not to criticize the sample simulations. The purpose is to encourage more open discussion pertaining to the modeling of simulations. This type of research and dialogue should enhance the effectiveness of simulations in the classroom and, perhaps, the future of experiential, learning via the computer. REFERENCES [1] Brooks, LeRoy D., Instructors Manual - The Financial Management Decision Game, (Richard D. Irwin, Inc., Homewood, ILL.), 1975. [2] Darden, William R., William H. Lucas, The Decision Making Game-An Integrated Operations Management Simulation, (Appleton-Century Crofts, New York), 1969. [3] Edge, Alfred G., Bernard Keys, William E. Reymus, Instructor’s Manual The Multinational Management Game, (Business Publications Inc., Dallas), 1980. [4] Gold, Steven C., Thomas F. Pray, Terry Dennis, MICROSIM - A Microeconomics Simulation, (MacMillan Publishing Co. copyright 1983). [5] Gold, Steven C. and Thomas F. Pray, “Simulation Market and Firm Level Demand - A Robust Demand System,” Developments in Business Simulation And Experiential Exercises, Vol. 10, 1983, pp. 101-106. [6] Goosen, K., “A Generalized Algorithm For Designing And Developing Business Simulations,” Developments In Business Simulation and Experiential Exercises, Vol. 8, 1981. [7] Henshaw, Richard C., James R. Jackson, The Executive Game, Rev. Ed., (Richard D. Irwin Inc., Homewood, ILL.), 1972. [8] Pray, T. and S. Gold, “Inside The Black Box – An Analysis of Underlying Demand Functions in Contemporary Business Simulations.” Developments In Business Simulation and Experiential Exercises, Vol. 9, 1982. pp. 110-115. [9] Pray, Thomas F.., Daniel R. Strang, Instructor’s Manual for DECIDE, (Random House, Inc., New York) 1981. [10] Smith W. Nye, Elmer Estey, Ellsworth Vines, Integrated Simulation (South-Western Publishing Co., Cincinnati) 1974. Table of Contents Volume 11, 1984 Simulation Gaming as a Means of Researching Substantive Issues: Another Look A Further Test of the Group Formation and its Impacts in a Simulated Business Environment Impact of Economic Patterns on Student Performance in Computer Business Simulation Games Majority Fallacy Game with Independent Student Simulation and a Case Introducing the Marketing Channel Laboratory A Comparative Evaluation of a Marketing Game A Study of Comparative Effectiveness of Problem-Solving Technologies The Impact of Hierarchical and Egalitarian Organization Structure on Group Decision Making and Attitudes Risk-Free Decision Making The EX-STRA Export Strategy Game Computer Education for Management Students Developing a Computer Game/Job Simulation to Teach Functional Literacy Skills Experiencing Socialization First Hand: An Experiential Exercise in Organizational Socialization Networking Distributive Versus Integrative Approaches to Negotiation: Experiential learning Through a Negotiation Simulation Managerial Education and the Real World: Foudations for Designing Educational Tools Diagnosing Group Climate to Improve Supervisory Effectiveness Student background as a Factor in Simulation Outcomes: The Collective bargaining Example The Use of Pre-Plays in Management Education Experiencing the Process Debrief: A Workshop ABSEL Megatrend Roots MEGATRENDS for Business Simulation and Experiential Learning The Effects and Consequences of the Megatrends on Simulation Gaming: One View Opportunities for the Future: ABSEL's Role Experiential Learning-Based Discussion vs. Lecture Based Discussion: A Comparative Analysis in a Classroom Setting An Evaluation of the Minitab Package in Teaching Business Statistics Concepts A Path Analytic Study of the Effects of Alternative Pedagogies Developing and Using Weighted Application Blanks: An Experiential Exercise Building Airplanes Individual vs. Group Grade: An Exercise in Decision making A Marketing Plan Exercise: Development of Interteam Cooperation Using a Coordinated Experiential Approach Using Student Experience as the Basis for a Consumer Behavior Learning Exercise Student Evaluations of Instructors: What do Students Believe? A Description of the SOFTCAT Computer Assisted Teaching System Comparisons of Practitioners' and Professors' Perceptions of Business Policy Content and Learning Methods The Perceived Relationship Between Pedagogies and Attaining Course Objectives in the Business Policy Course The Use of Simulation in the Teaching of Business Policy A Research Study on Strategic Decisions in a Business Simulation Strategic Management Decision Making Researched Via Simulation Gaming Using Simulation to Investigate Factors in Competitive Bidding Combining Experiential Learning and management Assistance A Model for Teaching Management Skills Putting Experience Back into Experiential Learning: A Demonstration The Teaching and Behavioral Measurement of Managerial/Organizational Competencies: Developing Experiential Exercises and Simulations A Simulation Game Model for Conglomerates QCLAB - A Microcomputer Laboratory in Quality Control CTSS: A Commodity Trading Simulation System Problem Solving: An Exercise on Learning, Coaching, and Operant Conditioning A Demonstration of the Effects of Feedback as a Category of Reinforcement The Assessment of Feedback and Disclosure in Interpersonal Relations: An Experiential Exercise A Study to Determine Whether the Teaching of Basic Grammar Skills in Business Communication Classes Improves Students' Business Letter Writing Corporate Maladies Through the Eyes of the Memo Writer: A Seldom Used Experiential Tool Executive Bailout at Shake & Spear, Inc. The H.E./L&P Merger Intercultural Nonverbal Communications: An Experiential Exercise The Evolving Business Policies Course - Is Management Gaming the Logical Pedagogy? The Use of Decision Simulations in Management Training Programs: Current Perspectives Humanizing the Business of Medicine: The Use of Simulated Patients to Train medical Students Systematic Integration of Simulation Methods in a Graduate Management Curriculum Modeling Non-Price Factors in the Demand Functions of Computerized Business Using Spacial Relationships to Estimate Demand in Business Simulations Two Algorithms For Redistribution Of Stockouts In Computerized Business Simulations Leadership And Strategic Behavior A Comparison Of Two Business Strategy Simulations For Microcomputers Incorporating Decision Support Systems Into Management Simulation Games: A Model And Methodology Using Micro-Computers To Support The Analysis Of Complex Cases: It's As Easy As 1-2-3 Strategic Formulation Consistent With Pims: A Micro-Computer Application