MODELING COST FUNCTIONS 1N COMPUTERIZED BUSINESS SIMULATION: AN APPLICATION OF DUALITY THEORY AND SHEPPARD’S LEMMA Developments In Business Simulation & Experiential Exercises, Volume 17, 1990 70 MODELING COST FUNCTIONS 1N COMPUTERIZED BUSINESS SIMULATION: AN APPLICATION OF DUALITY THEORY AND SHEPPARD’S LEMMA Dr. Steven C. Gold, Rochester Institute of Technology ABSTRACT The paper develops an algorithm to model jointly, cost and production functions in computerized business simulations. The algorithm utilizes the concepts of duality theory to derive a generalized cost function, where costs depend both on the level of input prices and the production rate. Sheppard’s Lemma is applied to derive the cost minimizing input demand levels based on the characteristics of the cost function. The application of Sheppard’s Lemma is shown to help avoid ¶inconsistencies between the cost structure and production technology of the firm. A recommended set of equations is presented and discussed to simulate the cost function of the firm. A numerical example is given to illustrate how the function may be used to demonstrate the stability of the system. INTRODUCTION In 1982 Kenneth R. Goosen presented paper at the ABSEL Conference identifying the need to expand research pertaining to the internal design of computerized business simulations. A review of the literature by Goosen (1982) showed that past research on simulation design was simply not extensive enough to assist in a meaningful way in the development of new and better simulations; and he concluded: “The designing and developing of simulations, appears to be an art form, a creative skill based on intuitive feel rather than acquired knowledge.” Although research in the area of simulation design has increased, a study by Goosen in 1966 raised a concern about the nature of the research relating to mathematical modeling of functional relationships, specifically he stated: “Very little research has been published concerning the development of functional equations for business games... Satisfactory mathematical equations that have inflection points or maximum and minimum values at the desired points over a desired range of values are difficult to develop. In many cases equations that appear suitable only give desired results over a limited range of values.” The focus of this paper is to address this concern with respect to the modeling of cost functions in computerized business simulations. A review of a number of contemporary business simulations by Gold and Pray (1989) identified a problem in the design of cost functions. Almost all simulations reviewed displayed a Linear relationship between production and costs in both the short-run and long run, implying constant returns to the variable inputs and no economies-of-scare in the cost structure. Economists have published a great many studies on the cost structure of firms and industries, utilizing wide variety of statistical, engineering, and accounting methods. Although there were some disagreements in the result’s of these studies, a comprehensive review of the literature by Walters (1963), showed that economies-of -scale were pervasive and existed to some degree In almost all industries. Despite these findings, the cost structure embodied in most computerized business simulations appear to be linear, as indicated by Gold and Pray (1989), who concluded: "Most simulations that permit capital expansion have fixed dollar ratios for plant expansion. This fixed ratio approach imposes constant returns to scale on au firms over any time horizon.” (p.26) Gold and Pray (1989) also noted that some designers modeled economies of scale by changing input prices while keeping productivity constant. While this approach may seem adequate it creates an inconsistency between the cost function and the production function. In this case productivity is constant but average costs are declining. Although this result is possible, it is not the general scenario. Duality theory argues that economies of scare are derived, more generally, from increasing returns in the production process and then manifest themselves in the cost Structure. PURPOSE AND PROCEDURE The paper will proceed to develop an algorithm for the modeling of cost functions in computerized business simulations. The model allows the designer to develop the cost relationship in the simulation first and then derive, jointly, the levels of input usage and the production function implied by the cost structure. The advantages of this approach are threefold. First, it guarantees that the behavior of the production function will be consistent with the cost structure of the firm. Second, the cost information needed to model "real world” firms is more accessible in published sources than production data. Since the approach in this paper uses cost information to develop the cost function first, ans then derives the implied production technology, it s easier to simulate. Third, the impact of costs in the simulation on financial performance are more direct. Costs impact profits directly, whereas productivity changes first impact costs. The paper proceeds in the following manner. (1) Summarizing the theoretical properties of cost functions that are most important in the design of computerized business simulations; especially the properties of duality between cost and production implied y Sheppard’s Lemma. 2) Developing a stable and flexible system of cost equations that encompass the key theoretical properties implied by duality theory and empirical research. The recommended system of equations will permit the designer to specify, simultaneously, the degree of economies of scale in the cost structure and returns to scale in production. (3) Presenting a procedure to derive the parameters of the cost system based on the appropriate specifications of the designer. A numerical example is given to illustrate the procedure. (4) Deriving the input (or factor) demand equations from the parameters of the cost function by applying Sheppard’s Lemma. (5) Simulating the cost system and the derived production function given the parameters derived in the numerical example to demonstrate and discuss how the system functions. DUALITY AND COST THEORY Duality theory states that the cost function and the production function are associated and must behave in a consistent manner. There is a direct relationship between costs and production, such that n increase in production efficiency will decrease costs or, conversely, a decrease in production efficiency will increase costs. The dual’ relationship between cost and production may be illustrated, succinctly, by first assuming there are only two inputs, Labor (L) and material (M); however, these results may be generalized easily to "n" inputs. The total variable costs (TVC) are then: TVC (PL) +(Pm)M Where: PL = price of labor ($/hr) Pm + price of Mat’l ($/lb) In equation I the decision variables are the hours of labor and pounds of material’ to be used by the firm. The exogenous parameters facing the firm are the price of labor and materials. It follows that the production function is also dependent on the same two inputs: Dividing both sides of equation I by the quantity produced 0, we get the average variable cost equation (AVC): Equation 3 shows average variable costs (AVC) are inversely related to the average products of ‘labor and materials. As the average product of Labor (API) or material (APm) increase the AVC will decrease. The marginal cost equation may be derived by taking the derivative of equation 1 with respect to Q: Equation 4 shows marginal costs (MC) are inversely related to the marginal’ products of the variable inputs. As the marginal product of labor or materials increase, the MC will decline. DERIVING THE GENERALIZED COST FUNCTION Applying the approach presented by Sheppard (1970) it may be shown that costs can be expressed as a function of the Level of input prices and production. The first step is to formulate the lagrangian equation for minimizing total variable costs, equation 1, subject to the production constraint, equation 2: The cost minimizing input usage may then be obtained by setting the partial derivatives of the lagrangian equation Z with respect to the inputs equal to zero, giving us the following first order conditions Solving the equation set simultaneously, the cost minimizing levels of labor and capital are a function of PL, Pm and Q. Substituting into equation 1, the generalized cost function may be written as: Developments In Business Simulation & Experiential Exercises, Volume 17, 1990 71 Equation is significant for the purpose of modeling cost functions in business simulations. Equation 6 shows that total variable Costs may be expressed as ~ function of input prices and production levels, without directly specifying the level of input use. The level of input use may be derived by applying Sheppard’s Lemma. SHEPPARD’S LEMMA AND THE DERIVED DEMAND FOR INPUTS Sheppard <1970) proved that the demand for inputs may be obtained by differentiating the cost function with respect to the variable input prices. Given the generalized Cost function, equation ~, and applying Sheppard’s Lemma we get: Equation 7 specifies that the quantity of labor used by the firm may be determined through the cost function by taking the derivative of total variable costs with respect to the price of ‘labor. Similarly, equation B specifies the quantity of material used by the firm is the derivative of total variable costs with respect to the price of materials. Sheppard’s Lemma is a powerful theoretical tool for the design of cost and production functions. Once the cost function is specified, the demand for inputs (labor and materials) may be ascertained in a manner consistent with duality theory. In this case, increases in average variable costs or marginal costs would imply decreases in average products or marginal projects of the variable inputs (as described by equations 3 & 4). SHAPE OF THE COST FUNCTION A cost function is an expression relating the costs of doing business to the level of production. The general shapes of the total, average, and marginal variable costs are illustrated in Figures 1 & 2. Figure 1 shows total variable Costs are generally “S’ shaped. Initially, variable Costs rise at a decreasing rate with respect to output. After point ‘A’ variable costs rise at an increasing rate. Point “A” is referred to as the point of diminishing returns, and indicates that the productivity of the variable input starts to decline after this point. In the short-run this is due to fixed factors of production and capacity constraints. In the long run, this Is due to decreasing returns to scale in production. Decreasing returns to scale implies diseconomies of scale ~r rising average costs (given fixed factor prices). Figure 2 shows the marginal and average cost Curves are “U” shaped. Initially the marginal cost declines up to point ‘A’ and then begins to increase. Average variable costs continue to decline with increases in output until point ‘B” and then begin to increase. At point “B” MC is equal to AVC. After point “B’ MC exceeds AVC. Duality between production and cost flies marginal products of the variable inputs (MP) would rise until point “A’ and then decline; whereas the average products of the variable inputs (AP) would continue to decline until “B” and there rise. At point “B’, the MP should also equate the AP. After point ‘9” MP is Less than Al’. Economies of scale exist in the long run if AVC exceeds MC. Economies of scale implies increasing returns to scale in production given fixed factor prices. Constant economies of scale occurs when MC exceeds AVC Diseconomies of scale implies decreasing returns to scale in production given fixed factor prices. Constant economies of scale occur when AVC = MC. The degree of economies of scale (E) is measured by the ratio of AVC to MC, such that E= AVC/MC If E > 1 then economies of scale exist; if E < 1 then diseconomies of scale exist; and if E 1 then there are constant economies. Generally, it is expected that at low levels of output the firm would be able to achieve economies of scale, and after some point would only be able to obtain constant economies; and eventually diseconomies of scale would occur. A RECOMMENDED COST SYSTEM A recommended system of cost equations for modeling business simulations is presented that is consistent with the theory of cost and duality. The cost function is multiplicative in nature and is flexible enough to model Increasing and decreasing returns to the variable input; is well as economies and diseconomies of scale. For clarity of exposition, the two input cases will be illustrated, but the function is easily generalized to any number of arguments. The parameter c1 is simply a scaling factor to obtain the desired level of cost. Parameters c2 and c3 are the input price elasticities and show the proportion of input costs to total variable costs. The exponent term (c4 + c5q) allows for variable cost elasticities with respect to output, 0. Variable cost elasticity is necessary to model Increasing and decreasing returns and economies and diseconomies of Scale. Economies of scale (E) may be derived from equation 10 since E AVC/MC = (TVC/Q)/(dTVC/dQ). Economies of Scale E=1 / (c4+c5 Q(1.0 +1nQ) Where E = economies of scale 1nQ = natural log of Q The resulting input demand equations derived from the cost function using sheppard’s lemma are: INPUT DEMAND EQUATIONS L = c2 (TVC)/P1 M = c3 (TVC)/Pm A well-behaved cost function also requires the following restrictions: HOMOGENEITY RESTRICTION C2 +c3 =1.0 This restriction guarantees that the cost function is homogenous of degree one. This simply means that if all variable input prices increase by some proportion, say 10~, then total variable costs will increase 10k, given a constant production level. This relationship holds by definition, refer to equation 1. NUMERICAL EXAMPLE: DESIGNING THE COST FUNCTION AND DETERMINING THE PARAMETERS OF THE SYSTEM Although the equations appear complex, it is relatively easy to design a cost function and solve for the parameters of the system. To illustrate the ease of application, a numerical example will be given. Suppose a simulation designer wants to model a cost function that possesses increasing returns or economies of scale at an initial output level of 1000 units, and constant returns at 1400 units of output. Of course any Scenarios consistent with standard cost behavior could be evaluated. To summarize, we have the following: Assuming a two input Case (labor and materials), the designer needs to specify the proportion of variable costs that are attributed to labor (c2) and the proportion of variable costs attributed to materials (C3). The sin of c2 plus c3 must equal 1.0. Lets suppose we specify Developments In Business Simulation & Experiential Exercises, Volume 17, 1990 72 finally, the designer needs to specify the total variable costs corresponding to the initial output level of 1000 units; and the input prices: Given the above data, the first step is to solve for the parameters C4. and C5 by using equation 11 and substituting in the values for E and 0 given above. The second step is to solve for the parameter C1. Simply substitute all known parameters and variable’s into equation 10 and solve. At this point all parameters and variables are known except for C1: Note that C1 is a scaling factor and does not affect the shape or properties of the cost function. The final total variable cost equation becomes: Once the parameters of the cost function are known, the input levels for labor and materials may be derived by substituting into equations 12 and 13 the values for TVC given any level of 0, P1, and Pm. The input levels derived in this manner are consistent with the “dual” relationship between cost and production functions. SIMULATING THE COST FUNCTION The cost function in the numerical example will be simulated along with the dual production function to illustrate its behavior and the relationship between production and total variable costs. The output level was varied between 900 units and 1600 units, given fixed input prices of $25 and $1O for labor and materials respectively. Table I summarizes the results. Average variable costs are “U” shaped with the minimum level occurring at the production rate of 1400 units. After 1400 units of output, AVC begins to rise. Economies of scale behave as modeled by the designer in the example. Returns to scale start at 2.64 and gradually decline. Constant economies of 1.00 correspond to the minimum AVC, which is consistent with the theory of cost. After an output rate 0f 1400 diseconomies of scale are exhibited and the scale coefficient, E, drops below 1.00. The “dual production function can be determined through the cost function by using the derived input demand equations (estimated in the numerical example). Given the same output levels, Table 2 summarizes the results, focusing on the labor input. The average product of Labor, AP1, is the dual of the average variable cost. Average product of labor begins at relatively Low Level, 0.295 units per hour, and gradually rises. In tandem, average variable costs fall in response to the increased productivity of labor. The maximum average project of labor occurs at an output rate of 1400 units, which corresponds to the minimum point on the average variable cost function. The second input, materials, behaves in a consistent fashion but the results are not displayed. SUMMARY AND CONCLUSIONS The properties of duality theory need to be addressed when designing cost and production functions. The characteristics of the cost structure embodied in a business simulation imply certain over a wide range of values. (2) The cost function is restricted to be homogeneous of degree one, a requirement of a well-behaved cost structure. (3) The Levels of inputs and the production function is derived directly from the cost function, guaranteeing the properties of duality are maintained. As a consequence, only cost information i~ characteristics relating to the production technology. If these relationships are not carefully modeled, inconsistencies between production and cost may develop. A review of the Literature has indicated that there are some common problems in the way in which contemporary business simulations have designed there cost and production relationships, especially pertaining to economies of scale. The system of equations developed in this paper to model the cost structure of the firm possesses a number of desirable properties: (1) A multiplicative functional form that relates total variable costs directly to input prices and the level of production. The functional form is flexible and allows for variable elasticities, increasing and decreasing returns, and economies and diseconomies of scale. The function also appears to be stable needed to model and simulate both the cost and production functions. (4) The parameters of the cost system may be easily calculated with only limited data requirements. Only information pertaining to input proportions, total variable costs at the starting point of the simulation and economies of scale coefficients for two discrete points are needed. REFERENCES Gold, S. and Pray, T., (1989) “The Production Frontier: Modeling Production in Computerized business Simulations,” Simulation and Games, forthcoming September 1989. Goosen, K.R., (1982) “A Generalized Algorithm for Designing and Developing Business Simulations, ABSEL Proceedings, Vol. 8. Goosen, K.R., (1986) ‘An Interpolation Approach to Developing Mathematical Functions for Business Simulations, Developments in Business Simulations and Experiential Learning, vol. 13, ~956, pp. 248-255. Sheppard RU, (1970) Theory of Cost and Production Functions (Princeton: Princeton University press, 1970) Walters, AA., (1963) “Production and Cost Functions,” Econometrica, Vol. 31, no. 1 (January), pp. 1-66. Table of Contents Volume 17, 1990 The Impact of Decision Support Systems on the Effectiveness of Small Group Decisions - Revisited The Relationship Between Financial Performance and Other Measures of Learning on a Simulation Exercise Use and Effectiveness of an Analogy-Based Expert System Suggestions for Computerized Business Authors Dealing with Power: An Experiential Exercise Using Movie and Personal Diary Analysis Techniques A Model for Developing Student Skills and Assessing Outcomes Through Outdoor Training Computer-Aided Exercises Versus Workbook Exercises as Learning Facilitator in the Principles of Marketing Course An Exposition of Guilford's Si Model as a Means of Diagnosing and Generating Pedagogical Strategies in Collegiate Business Education Formal Planning and Simulation Team Performance: A Cross Sectional Approach Cases: Real Organizations in Real Time in the Classroom An Empirical Investigation of the Internal Validity of Marketing Simulation Game An Empirical Evaluation of the Pedagogical Value of Playing a Simulation Game in a Principles of Marketing Course Factors Affecting Effective Teaching of Strategic Planning: Some Preliminary Evidence An Experiential Exercise for Learning About the Relationship Between Organizational Form & the Project Management Process Accounting Communication Skills can be Taught in the Auditing Course Modeling Cost Functions in Computerized Business Simulation: An Application of Duality Theory and Sheppard's Lemma A Life Cycle Analysis of Decision Making for a Strategic Management Team What's the Problem? A Dynamic Model for Teaching Problem Solving Skills Experientially International Currency Fluctuations: Money$im, A Simulation Superstores: A Specialized Retailing Simulation Within a Specialized Marketing Curriculum Factors Affecting Student Perceptions of Learning in a Business Policy Game VC + EL = VL The Name Game: An Experiential Exercise in Intergroup Relations The Effects of Experiential Accounting Work Experience on Student Performance in Intermediate Accounting Courses The Results of Using the Experiential Activity Group Performance Evaluation in a Business Policy Setting Using a Legal Database to Describe the Legal Environment of Marketing (and Business) Matching Environmental Uncertainty and Organizational Configuration An Instructional Computer Simulation of Tampering in QC Executive Evaluation of Student Learning in the Looking Glass Simulation Group Personality Composition and Total Enterprise Simulation Performance An Expert System for Selecting Analytical Techniques for Analyzing Marketing Research Data Effects of Cognitive Styles on Responses in an In-Basket Simulation A Psychometric Analysis of Kolb's Revised Learning Style-Inventory Selecting and Developing Experiential Exercises Using Movies Application of a Real-World Strategic Management Model in the Classroom Demand Equations which Include Product Attributes Consumption as the Objective in Computer-Scored Total Enterprise Simulations The Effects of Decision Format and Evaluation on Simulation Performance, Decision Time, and Team Cohesion The Effects of Computer Related Assignments on Student Performance in Business Administration The Money Game: A Dynamic Simulation Including Random Shocks for Money and Banking Courses Methods for Evaluating Performance on Business Simulations: A Survey The Effects of Synergogy on the Policy Course: Significant Improvements in Student Learning and Teacher Evaluation Conditions and Outcomes of Trust in a Two-Person Bargaining Exercise Bankgame Enhancing Computer Business Simulation with the Use of VGA Graphics An Experiential Approach to Entrepreneurship An Advanced Simulation Method (ASM) for Multiple Objective Problems The Influence of Experiential learning Techniques on Student Recognition of Non-Primary Learning Styles Negotiating Mergers and Acquisitions: A Cocktail Napkin Approach Identification of Unintended Effects in Experiential Laboratory Exercises An Experimental Comparison of Paper and Pencil and Computer Aided Decision Support Tools Porting a Simulation from the IBM World to the Macintosh World A Transaction Cost Analysis of Experiential Learning The Development of Experiential Exercises for Courses in Entrepreneurship and Small Business Management The Assessment Center as and Experiential Classroom Exercise Pricing Strategy Algorithms for Playing Business Simulations Organizational Structures for International Operations: An Experiential Activity Simulation Emphasis in the Business School Capstone Course An Integrated Approach to Computerizing the Business Curriculum Cross-Cultural Business Negotiations Exercise Organizational Socialization and Gender Differences in Students at Work Understanding Student Work Experience: A Content-Analytic Approach Introducing Executive MBA Programs with Management Games An Analysis of Improvement in Business Decision Outcome with Sequential Use of Two Simulation Games Teaching Business Policy Utilizing Mass Lecture and Individual Case Labs Potholes Along the Road to Evaluating Learning Outcomes: The Case of Outdoor Management Training Experiential Learning for Interior Design Students: Using CADD, Lotus 1-2-3, and Wordperfect Cognitive Learning Using a Computer-Based, Qualitative Interactive Business Simulation Sex Discrimination: Does the Woman get the Job or Does the Best Man Win? A Search for Visual Aids to Support Experiential Learning Through the 1990's An Experimental Analysis fo the Effectiveness of Student Role-Playing in Sales Training How to Have Students Learn from their Term Projects Self-Evaluation Exercise (SEE): An Assessment of Class Contribution A Study of the Influence of Team Formation on Attitudes and Performance in Management Games A Hardware Based CIM Simulation Laboratory Model Test Substance Abuse in Organizations Micro Computer Training Models Teaching Forecasting, Cash Budgeting and Inventory Model Building Using SBTools A Comparison of the Effects of Experiential Learning Activities and Traditional Lecture Classes An Adjunct Writing Instruction Assistant, The Computer; With an Illustration Classroom Software for ABC Analysis Utilizing Information Processing Technology to Enhance the Business Policy Simulation Experience Progressive Cases Realistic Job Previews Vs. Traditional Job Previews: Experiencing the Differences and Understanding the Consequences Investment Analysis Using the Pragmatic Multiplier Approach A Computer Simulation Interface for Competitive and Firm Analysis Self-Assessment of Ethical Decision Making Predispositions Preparing Managers for Overseas Assignments An Inquiry into Japanese Marketing: Workshop on Teaching Japanese Marketing The Performance Appraisal Feedback Interview: A Role Play for Human Resources Management Teaching Counselor Selling techniques Using Experiential Techniques to Teach International Topics International Management Simulation Gaming: Current Status and Future Developments A Realism Comparison of Simulation Technologies/Methodologies A Time-Efficient Game to Illustrate Concepts Taught in Management Courses and Management Development Programs Teaching the Management of Technology The Concept of Face and the Applicability of Experiential Exercises in an Oriental Culture's