This is to certify that the dissertation entitled THE ROLE OF FLEXIBILITY IN THE DESIGN, ACQUISITION, AND USE OF DURABLE INPUTS presented by Larry Lev » has been accepted towards fulfillment ofthe requirements for A ricultyral Ph.D. degree in conomics M' professor Lindon Rdbison Date July 31, 1984 . AMSU 15 an Affirmative Action/Equal Opportunity lnslirutiun 0-12771 MSU LIBRARIES .—:— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. THE ROLE OF FLEXIBILITY IN THE DESIGN, ACQUISITION, AND USE OF DURABLE INPUTS By Larry Lev A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Economics 1984 5 WW7 © Copyright by Larry Lev I984 ii ABSTRACT THE ROLE OF FLEXIBILITY IN THE DESIGN, ACQUISITION, AND USE OF DURABLE INPUTS By Larry Lev While flexibility, or the ability to respond to changing circum- stances, is discussed in sequential decision making models, this con- cept has not been integrated in economic models used to study the deci- sion process. This study first surveys factors which influence the firm's desire for flexibility and illustrates the role of flexibility in the design, acquisition, and use of durable inputs. Distinction be- tween use flexibility, the ability to operate in different modes, and adjustment flexibility, the ability to define different resource con- figurations, is made. The characteristics of strategies designed to increase use flexibility are compared with those which are risk reducing. An ex ante uncertainty model and a flexibility model in which the firm is permitted to separate its investment decision from its subse- quent production decisions are compared. Comparisons of average input ’ and output levels are ambiguous between the two models; the average level of investment in the flexibility model is greater than the amount acquired and used, on average, in the ex.agtg_model. Subsequent models examine how use flexibility affects the selection of nonhomogeneous durable inputs which produce homogeneous units of ser- vice. Initially, only durables with fixed temporal lifetimes are Larry Lev considered and relative use flexibility is measured as a function of the variable input costs associated with the use of each durable. The specification of a model which forces the firm to optimally set three design parameters within an explicit budget constraint adds richness lacking in previous models which are inattentive to the trade-offs a firm must make between specialization and flexibility. Another model examines use flexibility in terms of the costs as- sociated with using up the durable's own capacity to provide services. A new vocabulary of the physical and monetary costs of durable use and a new definition of use flexibility are developed. The analytical results reveal that greater flexibility is desired as the range of demands increases in an exogenous quantity mode. Deter- minite results can only be obtained if a lower bound is placed on the price since specialization protects the firm from the hazards of low output prices. Neither model yields determinate results which link the desire for flexibility to the degree of risk aversion of the decision maker. ACKNOWLEDGMENTS Lindon Robison, my acting major professor and thesis supervisor, deserves much of the credit for guiding me through the long and tortur— ous research process. Without his time, his effort, and, perhaps most importantly, his good humor, this dissertation would not have been possible. Each of the other members of my Guidance Committee played a special role in the research. David Campbell brought a fresh perspective to the major research issues. Lester Manderscheid aided me throughout my grad- uate career and gave detailed comments at my defense. J. Roy Black helped me to figure out how it all fits together. From my earlier years of graduate school I wish to thank Tom Zalla and Carl Eicher who provided me with research opportunities. My major professor, Eric Crawford, provided guidance up until his departure for an overseas assignment. A number of graduate students provided intellectual and emotional support. Special mention must be given to Marion Gold, David Trechter, and Kristen Allen. Beverly Fleisher provided esSential last minute editorial assistance. Although the funding for the actual research period was provided by the Michigan Agricultural Experiment Station, earlier portions of my graduate career were funded by the United States Agency for Internation- al Development. I express gratitude to both. I would also like to thank Cindy Spiegel and Nancy Creed who pro- vided excellent typing services. Finally, I wish to express my appreciation to my wife, Ann Shriver, who kept my spirits up throughout. iv TABLE OF CONTENTS Page ACKNOWLEDGMENTS ......................... iii LIST OF FIGURES ......................... vii CHAPTER 1. INTRODUCTION ........................ 1 Problem Setting ..................... l Flexibility in Economics ................. 4 Research Procedures and Methods ............. 6 Research Objectives ................... 6 Plan for the Discussion ................. 7 2. REVIEW OF THE LITERATURE .................. 9 A Comparison of Static and Sequential Decision Problems .................. . ...... ll A Review of Flexibility Literature ............ l5 Summary ......................... 29 3. AN EXAMINATION OF INVESTMENT AND PRODUCTION DECISIONS UNDER CONDITIONS OF UNCERTAINTY AND FLEXIBILITY ...... 30 Introduction ....................... 30 Background and Review .................. 31 A Restricted Flexibility Model .............. 4O Introducing Risk Aversion ................ 46 Summary ......................... 47 4. USE FLEXIBILITY MODELS FOR DURABLES WITH FIXED TEMPORAL LIFETIMES ..................... 49 Variable vs. Durable Input Models ............ 53 Why Firms Own Durables .................. 56 Review of Use Flexibility Models ............. 59 Derivation of a New Use Flexibility Model ........ 69 The Relationship Between Flexibility and Risk Aversion ......... . ............... 76 Summary ......................... 79 5. AN EXTENSION OF THE CONCEPT OF USE FLEXIBILITY ....... Bl Some Cost and Benefit Issues in the Use of Durable Inputs ...................... 83 Types of Durables .................... 86 Use Flexibility in an Exogenous Demand Model ....... 89 Use Flexibility in an Exogenous Price Model ....... 101 Summary ......................... l05 V Page CHAPTER 6. CONCLUSIONS ........................ 108 Summary of Problem Setting and Background ........ 108 Summary of the Analytic Results ............. 112 Implications for Future Research ............. 117 APPENDIX A ............................ 119 APPENDIX B ............................ 121 APPENDIX c ............... ' ............. 124 APPENDIX D ............................ 126 APPENDIX E ............................ 128 BIBLIOGRAPHY ........................... 129 vi LIST OF FIGURES £192 A Static Decision Process ................. 12 A Sequential Decision Process ............... 16 A Comparison of an §x_Ante Uncertainty and a Certainty Model ..................... 33 Turnovsky's Flexibility Model ............... 38 A Comparison of an Ex Ante Uncertainty Model and a Restricted Flexibility Model .............. 45 Two Durables Which Vary in Flexibility .......... 61 A Second Comparison of Durables Which Vary in Use Flexibility. . . . .................. 65 The Design Parameters h, c, and 2 for a Representative Durable .................. 71 TSE Curves for "Fixed Lifetime" and "Completely Flexible" Durables .................... 92 Time of Operation for "Fixed Lifetime" and "Completely Flexible" Durables .............. 94 TSE Curves for Two Flexible Durables ........... 97 Physical "Cost" Curves for a Flexible Durable ....... 104 vii CHAPTER I INTRODUCTION Problem Setting Firm decision makers recognize that the majority of their decisions have an impact which extends into the future. Therefore, instead of simply selecting the action choice which appears optimal given the cur- rent circumstances, they also evaluate the performance of alternative choices in permitting the firm to respond to changing circumstances in later periods. Flexibility is the term which is designated to represent that characteristic of the initial choice which measures, on a relative basis, "...the [resulting] ability to respond or conform to new or'chang- ing conditions“ (Webster's New Collegiate Dietionany, 1977). Flexible initial action choices are valued because they permit the firm to process new information as it becomes available and thereby make more enlightened decisions at later points in time. In the absence of flexibility, when the course of action is completely and unalterably set early on, the firm is unable to respond to factors such as changes in tastes and preferences or the resolution of stochastic events. In everyday discussions of decision making the desire for flexi- bility is commonly cited as a partial justification for a particular action choice. For example, a doctor living in a mountainous and snowy region may select a four-wheel drive vehicle in order to provide a capa- bility of handling a wider range of future weather conditions. 2 Acquisition of a two-wheel drive vehicle would provide less flexibility. Similarly, a freshman at college may choose a general rather than specialized course of study so as to leave open a variety of options for future years. Decision makers, however, realize that flexibility is an intermedi- ate or instrumental goal of the firm and not a final objective. This implies that flexibility will only be Valued to the extent that it aids the firm in achieving final objectives be they profit maximization, utility maximization, or some other criterion. Since the maintenance of flexibility results in costs as well as benefits, in most instances the firm would not be well served by a policy of simply seeking to maxi- mize the degree of flexibility retained. In general terms, the flexibility of an action choice can be viewed as synonymous with its adaptability to a wide range of conditions. This generalization holds true for both adjustment flexibility, which is a measure of how easily the quantity and quality of resources under the control of the firm can be varied, as well as u§g_flexibility, which represents the ease with which the intensity and/or manner of resource use is varied. Actions which are widely adaptable are, however, less well adapted to a particular situation. Thus, in opting for greater flexibility of any sort, the firm must sacrifice either current income or the potential to perform extremely well under specific future condi- tions. Some decision makers for reasons of either myopia, a focus solely on the near term, or tunnel vision, a focus on a single final objective at an early point in the decision process, may choose to ignore flexi- bility considerations. Most, however, will seek to retain some 3 flexibility when they are confronted by a sequential decision problem characterized by variable and/or unknown future circumstances. Their initial choices will thus be situated somewhere on a continuum ranging from extreme specialization to complete flexibility with the exact loca- tion being determined by the risk attitudes of the decision maker as well as her/his understanding of the probabilities associated with oc- currence of events in the future. Although flexibility is often regarded as a risk reducing strat- egy, it shares only some characteristics with the more familiar of such strategies. In both instances the firm is solely interested in the final results rather than the strategy selected per se. Aikey differ- ence, however, exists between the two--while the firm's motives for selecting a specific degree of diversification can be directly inferred, such is not the case with respect to flexibility since the firm's final decisions are yet to be made. Stated in a different manner, the deci- sion to diversify to a specific degree represents the selection of a particular outcome distribution;1 the degree of flexibility which is picked, in contrast, must be viewed as the selection of a specific dis- tribution of outcomes for the short-run and a distribution of distribu- tions for the long-run. It is only after these future decisions are made that the firm's underlying motivations can be clarified. Thus, firms with divergent risk preferences may select identical initial ac- tions since each for its own reasons wishes to maintain a sufficient 1If the decision maker has more than a one-period time horizon, the choice of a risk reducing strategy may in fact be motivated by the de- sire to maintain options into the future. This would then be a flexi- bility strategy. 4 amount of flexibility to follow its own preferred course of action in later periods. This point can be illustrated by observing that frequently the most risk averse as well as the most risk preferring investors maintain the largest percentage of their resources in liquid assets. The risk averse individuals wish to be prepared for emergencies, while the risk prefer- ring investors want to retain the possibility of taking advantage of attractive new opportunities. Both classes of decision makers are willing to sacrifice short-term earnings in order to maintain these future options. This characteristic of strategies motivated by flexibility concerns makes them particularly difficult for outside analysts to interpret since the observation of initial choices does not reveal the ultimate goals of the firm. As a result, in cases where a sequential model is a more appropriate representation of the decision maker's view of the world than is a static model, care must be taken to either extend the model over a sufficiently long period of observation, or to supplement the observation of initial choices with participant explanations of longer-run objectives.2 Flexibility in Economics This research is motivated by recognition that although flexibility issues are of great concern in lay discussions of decision making and in many applied disciplines they are rarely explicitly treated in formal 2Farming systems research represents an example of a research process which requires the participation of the actual decision makers in order to form a view of what strategies are being pursued and why those strategies are chosen. 5 economic models of firm behavior. The argument has been made that economists need not devote a great deal of attention to these issues since well developed solution techniques for sequential problems are already available in other disciplines. This argument fails to recog- nize that while these techniques provide a method of deriving an optimal solution to specific problems, they do not seek to fully explore the underlying nature of the general problem situation. Economists, with their predilection for tracing through the influence of individual elements in an overall problem, can play a useful role in generalizing the relationships which are derived. A more fundamental reason for the lack of interest which econo- mists have shown in flexibility issues has to do with the difficulty of deriving unambiguous results from what turn out to be quite complex multiperiod models. Economists, on the whole, have tended to devote their energies to subjects which are more likely to yield determinate solutions. The results of this research concur with many previous efforts in finding that unambiguous results are only derived from severely re- stricted models. Nonetheless, including the concept of flexibility permits a better understanding of how the component parts of the deci- sion problem interrelate. This is achieved, and the focus of the dis- sertation is retained within reasonable bounds, by deriving a series of simple deductive models which examine the implications of different types of flexibility on the decisions to design, acquire, and use dur- able inputs. Although there are many other applications of flexibility concepts, the models deduced herein serve to highlight the trade-offs 6 and complementarities which the firm must consider when confronted by sequential decision problems. Research Procedures and Methods This research is disciplinary in nature and of known relevance. Because flexibility has attracted relatively little attention in eco- nomics, one of the objectives is to bring the research conducted in other fields such as farm management, planning, and architecture to the attention of economists. Often it is useful to begin a research effort by translating results from other areas. The main thrust of the research is to set up and solve deductive economic models. These models permit an investigation of the implica- tions of introducing flexibility into models of the firm and lead to an understanding of when and why flexibility is valued by decision makers. The usefulness of these models is judged in terms of their internal logic and their external correspondence to what is either known or ex- pected to take place. The empirical testing of the theoretical results derived, however, is not carried out in the course of this research. Research Objectives This research is undertaken with a full understanding of "the Billings Phenomenon" which indicates that: The conclusions of most good research are obvious once the research is completed. They may have been much less obvious before the research was undertaken. There is no expectation of developing startling new conclusions. In- stead, the goal is to draw together existing research and highlight the general conclusions which can be drawn. 7 More specifically, the objectives of the dissertation are: 1. To compare and contrast the structure of static versus sequential decision models. 2. To review previous attempts to formally define and use the concept of flexibility in economics as well as in other disciplines. 3. To analyze the implications of introducing adjustment flex- ibility in its simplest form to a model which determines the level of durable input acquisition and use. 4. To explore the importance of use flexibility issues in the design, selection, and use of different classes of durable inputs. 5. To investigate the relationship between flexibility and risk attitudes. Plan for the Dissertation Chapter Two introduces a framework for distinguishing sequential decision problems from static decision problems. Within that framework, a wide range of previous examinations of flexibility issues are reviewed. Chapters Three through Five build upon the previous work through the deveIOpment of simple deductive models. The goal in these chapters is to investigate specific topics rather than cover the broad and often confusing spectrum of flexibility issues. Chapter Three considers the impact of introducing a restricted form of adjustment flexibility into a simple durable input production model. This chapter fulfills two major objectives. First, it illustrates a solution process required to analyze a two-period model under uncertainty. 8 Second, interesting analytic results are developed which demonstrate how decision makers in a multi-period model can be expected to alter their actions in the face of uncertainty. Chapters Four and Five redirect attention to use flexibility issues as they investigate the trade-offs which firms consider in designing the optimal durable for different distributions of use conditions over time. Chapter Four examines the design problem under the assumption that all durables have a fixed temporal lifetime. Chapter Five removes that as- sumption and thus begins with a discussion of how the variable use costs of durable inputs should be assessed. In each chapter both variable quantity and price models are considered and the influence of the risk attitudes of the decision maker on the degree of use flexibility of the optimal durable is examined. Chapter Six summarizes the main conclusions derived and indicates profitable areas for future research. CHAPTER 2 REVIEW OF THE LITERATURE Concern with flexibility, the ability to respond or conform to new or changing situations, arises in any problem which can be modeled as a decision tree. When facing problems of this type, the firm must concern itself with both the short and long range consequences of its actions. But despite the pervasiveness of problems which involve a concern with flexibility, the concept has not received much attention within the mainstream of economic theory. The lay concept of flexibility has gained greater acceptance and application within applied disciplines such as farm management, planning, and architecture. This may result from the fact that researchers in applied fields must work closely with real world decision makers and their immediate problems, which often include concerns about flexibility. Nevertheless, even in these dis- ciplines, flexibility remains a fairly vague descriptive term rather than a precise measurable attribute of action choices. In disciplines such as statistics (Wald, 1947) and operations re- search (Bellman, 1957) solution techniques have been derived for sequen- tial decision problems which include concerns about flexibility. Cocks (1968) and Rae (1971a, 1971b) have progressed the farthest in the development and application of discrete stochastic programming to derive optimal solutions in specific problem situations. Their general 10 conclusion has been that the computational complexity of such problems can rapidly exceed reasonable bounds. The applied problem solving research and that done in quantitative modeling techniques comprise two distinct paths, the former using the concept of flexibility quite casually in the context of qualitative models of decision making while the other presents formal models of flexibility type problems without explicitly considering the concept of flexibility. The majority of research reviewed in this chapter falls into a middle ground, seeking to find an explicit role for the concept of flexibility within formal sequential decision making models. Although this third research path would appear to have a natural home within the discipline of economics, this has been precluded by the preponderance of comparative statics in microeconomics research. Flex- ibility issues have no relevance in these static models. Economists recognize that inclusion of a temporal dimension greatly increases the complexity of decision problems and thereby reduces the likelihood of deriving unambiguous results. Therefore, the tendency has been to maintain the use of a discount rate as the sole time related element in economic models. This chapter reviews the efforts of those few economists and re- searchers in related disciplines who have explicitly confronted the intertemporal issues of flexibility. These researchers have recognized the complementarity of simple deductive economic models which include flexibility and the qualitative and quantitative discussions of sequen- tial problems which appear elsewhere. Chapters Three, Four, and Five build upon this body of research and consider, from the economist's per- spective, the relationship between the key elements, including ll flexibility, which influence decisions regarding the design, acquisi- tion, and use of durable inputs. A Comparison of Static and Sequential Decision Problems In order to focus on the new elements which the concept of flex- ibility permits the firm to consider, a sequential decision process is compared and contrasted to a static decision process. The argument, in- troduced in Chapter One, that flexibility represents an instrumental strategy called upon by firms to deal with sequential decisions in much the same way that strategies such as diversification are used by firms in a static context is developed further and demonstrated. Figure 2.1 illustrates the major components of the static decision problem. The uppermost box, "the state of the system," reflects the environment within which the decision maker operates. An omniscient observer or an all-knowing decision maker could evaluate the state of the system as well as the factors under the control of the firm and define a "choice set" consisting of all of the technically feasible alternative actions available to the firm.1 If the assumption of per- fect information is not made, there is no a priori reason to believe that the decision maker understands the implications, or indeed is even aware, of all the potential alternatives. The "learning/evaluation" stage is included to indicate the role which the decision maker plays in defining the problem context. Decision makers with the same "objec- tive” circumstances may thus develop completely different views of the problem situation. In the static model, learning and evaluation are not 1Inaction is one member of this set, but this alternative is of more relevance in the discussion of the sequential problem below. 12 STATE OF SYSTEM __,L__ CHOICE SET . __—._¥_.——— LEARNING/ EVALUATION L PREFERENCE OUTCOME (IMMEDIATE) Figure 2.1 A Static Decision Process EXOGENOUS FACTORS 13 endogenous activities; since there will be no opportunity to apply any knowledge gained there can be no value associated with acquiring it. Under conditions of certainty and perfect information, the decision maker can calculate the exact outcome of each alternative and thus select the choice which maximizes the utility of the firm. Under cer- tainty, utility maximization is assumed to be equivalent to profit maxi- mization. Under uncertainty, an intervening stage is inserted between the derivation of the effective choice set and the actual selection of the preferred choice. When more than one outcome can occur for individ- ual action choices, the firm must evaluate the range of outcomes and their likelihood of occurrence. The expected utility hypothesis de- scribed in Hey (1979) represents one means of carrying out this process. The assignment of utilities to individual probabilistic outcomes occurs in the "preference" stage in Figure 2.1. As in the learning/evaluation stage, there is no expectation that individual decision makers will re- veal the same preferences from what may or may not be the same effective choice set. It is at this stage of actually selecting an action choice that it is instructive to detail how and why specific types of strategies which are largely ignored in the certainty problem come to the forefront under uncertainty. As will be seen below, other types of strategies come to the fore when the problem is transformed from a static to a sequential problem. The introduction of uncertainty brings forth a group of "risk re- ducing" strategies. Among these are: l. The selection of action choices with lower variances. 2. The purchase of insurance. l4 3. The hedging of input or output prices. 4. The diversification of enterprises. This last strategy will be examined more closely so that it can later be compared and contrasted with flexibility related choices. Eco- nomists from the time of Adam Smith have emphasized the scale economies to be reaped through specialization in production activities. Under un- certainty, however, increasing specialization may bring attendant dis- advantages. In particular, the firm may be left susceptible to wide fluctuations in earnings on a period by period basis. The sage use of a diversified mixture of enterprises may result in a marked decrease in the degree of income fluctuation. The firm will continue to diversify until the utility gained by reducing the variations of profits just off- sets the loss in expected profits. Since more risk averse individuals suffer greater utility losses from variations in profits, they would chose more diversified investment packages. It is, however, incorrect to regard diversification uniquely in the context of risk reduction. In many instances an increase in specializa- tion will not result in an increase in expected profits since the orig- inal impetus for developing multiple enterprises grew out of a recogni- tion of their complementarity in the use of available resources. This relationship between expected profits and diversification may not be obvious to analysts more familiar with the relationship between diversi- fication and risk aversion. Norman (1973) and Baker and McCarl (1982) both report examples of this sort of confusion on the part of research- ers conducting farm level studies in developing and developed economies. In both instances, diversification was ultimately identified primarily as a profit increasing rather than a risk reducing strategy. Making 15 reference to Figure 2.1, it becomes clear that it is essential to trace through all of the stages of the decision process in seeking an explana- tion as to why a particular action choice gains favor and not arbitrarity focus on a single stage such as risk preferences. Sequential decision models require modifications of Figure 2.1. At a minimum, the consequences of the prospective action choices must be traced through the requisite number of periods before their merits are assessed. In Figure 2.2 the inclusion of three feedback loops highlights the various ways in which actions in different periods become inter- dependent. As a result, the decision maker in this model must be con- cerned not only with the immediate outcome of any action choice con- sidered, but also the implications of such a choice for the situation u: be faced in the future. The "flexibility" loop which links action choices in earlier periods and choice sets in subsequent periods is complemented by the "endogenous learning" loop which links initial actions and outcomes to the evalua- tion process in later periods and the "resiliency" loop which links the outcome of one period to the state of the system in subsequent periods. While the interdependence of flexibility and learning is often noted in the flexibility literature, the issue of resiliency, which is treated in greater detail later in this chapter, is rarely examined. A Review of Flexibility Literature This discussion begins with the flexibility loop which can be view- ed as a technical measure of how action choices are linked over time. An examination of the flexibility literature reveals a distinction be- tween two basic types of flexibility. The first, tactical or use 16 I ENDOGENOUS LEARNING STATE OF SYSTEM CHOICE SET RESILIENCY LEARNING/ EVALUATION I FLEXIBILITY PREFERENCE ( OUTCOME IMMEDIATE) / Figure 2.2 A Sequential Decision Process EXOGENOUS FACTORS l7 flexibility can be defined as "the [resulting] ability of the system in its configuration at any point in time to operate in a number of dif- ferent modes" (Rosenhead, 1981). Tisdell classifies this as flexibility with static decision making since any and all flexibility must be a "built-in" characteristic of the resources when they are purchased and is simply used to advantage at later points in time. Heady (1952) iden- tifies two subcategories of use flexibility as gggt flexibility and product flexibility. In both instances the resources are fixed to the firm, but can be used in a variety of ways. Rosenhead cites the construction of a hospital as an issue which can be examined from a use flexibility perspective. If the external structure of the hospital is considered fixed once it is erected, then use flexibility is derived through the possibilities of subsequently altering the internal structure of the building. Compared to an inflex- ible or specialized hospital, a flexible hospital would permit rela- tively inexpensive responses to changes in the nature or volume of demand. Heady's (1952) example of a livestock barn which is moderately efficient for a variety of livestock enterprises can be explained in similar terms. Chapters Four and Five will present a more detailed re- view of the use flexibility literature and will develop simple use flexibility models. Adjustment or strategic flexibility represents the second major flexibility classification. Adjustment flexibility can be defined as "the resulting ability of the firm to define different resource con- figurations for the system as a whole" (Rosenhead, 1981). Thus, adjust- ment flexibility depends upon re-entering the marketplace to buy or sell resources or engaging in totally new physical production processes. 18 Rosenhead refers again to the hospital example and notes that adjustment flexibility is maintained through the construction of an external struc- ture which can be easily modified in the future. Adjustment flexibility in this case implies far more dramatic changes. In this context it is significant to note that the two basic types of flexibility, use and adjustment, need not be mutually reinforcing and may very well be con- tradictory (that is to say that a hospital planning committee may be forced to trade-off use versus adjustment flexibility concerns). Often use flexibility can be increased by increasing the level of investment, but this, of course, decreases the firm's adjustment flexibility. As with diversification, flexibility can have important implica- tions even under certainty conditions. Hart's influential 1942 article "Risk, Uncertainty, and the Unprofitability of Compounding Probabili- ties" demonstrated the essential difference between sequential and static problems by showing that even risk neutral firms are obliged to concern themselves with the higher moments of the individual period out- come distributions rather than simply the means in seeking to define an optimal path in a sequential decision model. Hart's work can be seen as a criticism of tunnel vision approaches to decision making in which the firm needlessly commits itself to the achievement of what appears at an early stage to be the preferred final goal. He thus emphasizes that the maintenance of options is a valuable attribute which should not be lightly sacrificed. Koopman (1964) presents similar arguments in con- sumer theory based upon a preference for waiting before final decisions need to be made. Whereas under certainty it is not difficult to calculate the opti- mal decision path to take, under uncertainty it is necessary to 19 carefully calculate the costs and benefits of maintaining flexibility. The degree of flexibility maintained can be viewed as inversely related to the commitment of the firm to a specific resource configuration or plan. Thus, the chief opportunity cost of maintaining flexibility can be evaluated as the benefits foregone by not committing the firm to a particular course of action at an earlier point in time. Examples of the potential for gain through early commitment or specialization in- clude a producer who sees great promise in a new product requiring new and specialized equipment, a politician who becomes an early backer of a dark horse, but potentially successful presidential candidate, and a young couple who are quickly convinced that they have found true love. In each case, great benefits can be derived if the course of events develops in the expected way. The maintenance of flexibility through the minimization of early commitments also has potential benefits. This is because once a com- mitment has been made, it may be difficult to respond to changing condi- tions. There are two factors which define the degree of commitment: the ease with which resources can be transformed back into money (liquid- ity), and the extent to which physical processes restrict future op- tions (reversibility). Resources which can be easily and costlessly transformed back into money are defined as "liquid" assets. Many specialized resources are not easily traded and hence are "illiquid." Hirschliefer (1972) notes that the great advantage of liquid resources is that they “facilitate the utilization of new information as it becomes available over time." Clearly, illiquid resources must possess a return advantage if they are to be preferred. Jones and Ostroy (1984) demonstrate for a two-period 20 model that liquid assets may be held even if other assets have a return advantage which is known with certainty in individual periods because the liquid asset permits intertemporal combinations not otherwise avail- able. Goldman (1978) extends the liquidity discussion to the selection of portfolios rather than individual investments. He found that since firms are not required to make "all or'nothing" decisions with respect to the portfolio, it is not possible to develop meaningful liquidity or flexibility rankings for entire portfolios. The firm expects to keep some assets and liquidate others. Goldman thus demonstrates that under uncertainty conditions it is possible to justify a risk neutral firm's holding of virtually any combination of assets of different liquidities. The only deterministic result which he obtains indicates that as the model approaches certainty, the firm tends to hold portfolios including only a small presence of assets of middle degrees of flexibility.2 The second factor which restricts the adjustment flexibility of the firm is the physical requirements of the production process. As an ex- ample, paving over a field is less flexible than farming it because it reduces the economically viable uses of the plot of land in future periods. In the flexibility literature this concern with the physical availability of options in later periods has been referred to as the "reversibility" issue. Arrow and Fisher (1974) as well as Henry (1974) demonstrate that, under conditions of uncertainty, “irreversible" action choices lose preference to reversible action choices even under the 2In other words, the firm would divide its portfolio between highly liquid, but low yielding assets which it expects to sell and illiquid, high yield assets which it expects to keep. 21 assumption of risk neutrality. They indicate that this provides the "feel" of risk aversion to these choices, since the decision maker appears to sacrifice profits. Their findings are but an extension of Hart's general result reported above.3 In response to the specific flexibility results derived above, attempts have been made to formalize a general definition of flexibil- ity. Rosenhead (1978) in the planning literature expanded upon Hart's views on the relationship between flexibility and expected profit. He developed flexibility or "robustness analysis" as an alternative to "optimal" planning based upon the assumption of no learning or other changes. As a case study he considered the situation of a 14 year old British school girl who must imnediately make educational choices which will influence her future educational and career choices. Over the relevant time frame of the decision problem the girl can be expected to obtain additionaL and one assumes more accurate, information with re- spect to her own interests and abilities as well as the job market. Rosenhead argues that it would be foolish to base current actions on what appeared to be the "optimal" final objective given her current situation and suggested instead an approach which would maximize the likelihood of obtaining an acceptable final state (career path). This is accomplished by first outlining all potential final states which at present hold some interest. The cut-off point is arbitrarily fixed. Next a calculation is made to determine how many of these final states remain attainable if each of the possible current actions is selected. 3This reversibility proposition is also demonstrated in Epstein (1978) and Jones and Ostroy (1984). As will be shown below, the op- posite results (the "feel" of risk preferring behavior) can also occur based upon similar reasoning. 22 Finally, a "robustness index" is calculated by dividing the number of acceptable final states achievable when a given action is taken by the total number of such states. Rosenhead defines the initial action with the highest score in this index as the most flexible initial action and indicates that this action would have some support as a good initial action in cases where initial information is poor. While Rosenhead offered his discussion as a nonstatistical alterna- tive to optimization analysis, a careful examination of his procedures reveals that he does carry out a statistical analysis using specific assumptions. In particular, all of the acceptable final states are established as having an equal value to the firm and as being equally likely to occur. Based on these two assumptions, Rosenhead calculates what others would simply term a dynamic optimal solution. Dynamic opti- mization, depending as it does on assumptions about utilities, is quite different than flexibility which should be regarded as merely a measure of technical possibilities. Increased flexibility, since it requires trade-offs, should at some point result in a reduction in the utility provided to the firm. The economists who have attempted to generalize the definition of flexibility have sought to utilize this concept to predict firm response to known changes rather than in simply defining the optimal choice. Marschak and Nelson (1962) seek to discover a means of integrating the concept of flexibility into microeconomics through the presentation of three definitions. Their goal is to establish a characteristic of action choices which can be used as a proxy for optimality as condi- tions change in known directions. Their first definition simply re- states the intuitive notion that the flexibility of an action choice 23 can be measured as a function of the size of the choice set of subse- quent options. Their second and third definitions differentiate between actions in terms of the average cost of each in meeting varying levels of demand in subsequent periods. They argue that Action 2 (A2) can be considered to be more flexible than Action 1 (A1) if and only if the amount by which the second period cost associated with A1 is less than the cost associated with A2 is bounded for all levels of demand, and the amount by which the cost associated with A2 is less than the cost associated with Al is unbounded for at least one value. This defini- tion of flexibility yields relatively modest predictive gains to the' firm since it merely indicates that as conditions become extremely vari- able the more flexible action choice will eventually gain favor. Jones and Ostroy define flexibility in the context of a multi- period profit function which depends upon the action, a, taken in period 1, the action, b, taken in period 2, and the actual state, 5, which is not known until period 3. Between periods 1 and 2 the firm acquires additional information about 5 which may cause it to decide to change its preferred action in period 2. The overall profit function is thus: f(a.b.5) = r(a.S) + u (b.S) - C(a.b.5) returns in periods 1 and 2, respectively; n, U C II the cost of switching from a given period 1 action to a 0 ll given period 2 action. The switching function which is used to develop a flexibility rank- ing is defined as: G (61.5.00 5 [b.C(a.b.S) 5a] 24 In words, G(a,s,a) represents the set of second period positions at- tainable from initial action a at a cost that does not exceed a in state s. The mapping G is used to define a partial ordering on A, the set of all possible first period actions. Position a is more flexible than position a', denoted azfa', when for all 0:30 and 555. G (a,s,a) > G (a',s,a) with the exception of g (a'). This represents the statement that ¢13_fcz' if the set of positions obtainable from a always contains the set obtainable from af,excluding what Jones and Ostroy term the "zero cost option," [9 (a') ]. This is the option of maintaining the same path in the second period and thus avoiding the switching cost. The Jones/Ostroy definition thus succeeds in establishing the sense of the term flexibility without having to resort to unbounded costs. According to this definition of flexibility, the more flexible initial action will yield a greater second period profit except in the case where the zero cost option is preferred by the firm. A problem which arises with this definition is that the allocation of costs between the switching function and the second period cost function is arbitrary. One is forced to consider the range of subsequent actions which should fall in the category of the zero (or low) cost option. Jones and Ostroy's work focuses on the examination of the relation- ship between learning and flexibility. Although they succeed in deriv- ing the reversibility results cited above, they demonstrate that it is not possible to make statements at the most general level about the re- lationship between preference for flexibility and changes in the firm's prospects for learning or what Jones and Ostroy term reductions in the 25 variability of beliefs. Thus, for example, it is not possible to derive deterministic results based upon a rank ordering of partially reversible action choices. In later chapters of this research this matter will be dealt with by simplifying assumptions about the learning process. In most in- stances, the attention here will be restricted to the type of model examined by Baumol and discussed in Chapter Four in which the firm moves from gx_ggtg_uncertainty to gx_gg§t_certainty. In those models the flexibility/endogenous learning link is quite clear.4 The relationship between changes in gx_ggtg uncertainty and flex- ibility is, of course, quite different from the relationship between changes in Ex Egg: uncertainty and the desire for flexibility. Tisdell considers a model in which the firm has constant ex_ggtg uncertainty about future conditions, but also recognizes that its gxupggt uncer- tainty (i.e., uncertainty at the time of the second decision) will in- crease. In this case, Tisdell argues that a less flexible choice will be selected in the new situation because the firm will not have the necessary information to complement a flexible initial choice and will therefore be better off not passing up opportunities in order to main- tain options. Heiner (1983) and Harris (1974) both use real world examples to demonstrate that as problems become increasingly complex, decision makers will turn to more flexible alternatives. In Figure 2.2 this can 4Since the endogenous learning aspect of the decision problem is a whole separate research topic, the reader is referred to Gould (1974), Jones and Ostroy (1984), and Merkhofer (1977) for further discussion. Also see Robison and Fleisher (1983) for a discussion of why learning should not be used synonymously with a reduction in uncertainty. 26 be seen as a breakdown at the learning/evaluation stage. When informa- tion cannot be processed, for whatever reason, the firm reaps no bene- fits from maintaining flexibility and therefore should not take on the costs associated with more flexible choices. The reaction of farmers to increasingly unpredictable prices represents an example of such behavior. If farmers will not have an opportunity to modify their pro- duction or marketing plans after the discovery of prices, they may react to increasing uncertainty through the selection of a less flexible action choice such as the hedging of output prices. As is discussed below, farmers and others who are near some critical income level may also react to the increase in gx_ggtg_uncertainty by moving towards more rigid, but "safer" production patterns. In summary, the attractiveness of more flexible action choices is not directly related to the degree of uncertainty which exists. Rather the firm's desire for flexibility grows as the amount of uncertainty which will be resolved before subsequent decisions must be made is per- ceived to increase. Thus, two contrasts emerge between flexibility and static risk strategies. First, the desire for flexibility cannot be directly re- lated back to the risk preferences of the firm. Firms with all risk at- titudes may be viewed as selecting their initial choices at least par- tially on the basis of relative flexibility measures. Secondly, the value of flexibility can only be assessed in terms of what additional insight the firm will have at the point in the future when subsequent decisions are to be made. The third linkage in Figure 2.2, "resiliency,“ has not been treated in the flexibility literature because of the assumption that 27 the options which will be available to the firm in subsequent periods are known with certainty and it is only the relative attractiveness of these options under the stochastic states of the world which is uncer- tain. Once the options themselves are made to depend upon the outcomes which occur in previous periods the "resiliency“ of initial actions assumes importance in determining "the ability of the firm to respond or conform to new or changing circumstances" (the original flexibility definition). As a result, there would appear to be a need to explore the relationship of resiliency to the overall concept of flexibility. The standard definition of flexibility has evolved around the idea of preserving at least one more option for future periods when compared to other alternatives. It is argued here that when the options them- selves are probabilistic, this definition can be modified to state that flexibility (or resiliency) is reflected in increasing the probability of the preservation of at least one option. A concern with resiliency is generally found in approaches grouped under the heading of safety-first models. These are models which stress the achievement of a critical threshold outcome. Although formally the models tend to be static, many of the proponents of safety-first models clearly have a longer time horizon in mind. In an early safety-first model, Roy (1952) used the threshold con- cept to define a model in which investors have in mind some disaster level of returns, yd, and behave so as to minimize the probability of returns below that level. The criterion is expressed as: Minimize Pr (yt_<_yd) Following this criterion, a decision maker faced with a choice between two distributions F and G would order the two based on the difference 28 F (yd) - G (yd). F is preferred to G when the difference is negative. This is equivalent to the first-degree stochastic dominance criterion at a single point. The inattention to outcomes above yd was corrected in later safety-first models by Telser (1956) and Kataoka (1968). They in- corporate a constraint which recognizes the importance of avoiding the outcome yd, but beyond that maximizes either expected profits or ex- pected utility. In Telser's model the objective function is: Maximize expected yt subject to Pr (ytiyd) < a. In Kataoka's model, the goal is: Maximize expected utility of yt subject to Pr (ytiyd) < a; where a is the acceptable level of probability of obtaining outcomes of yd or less. If a longer time horizon is adopted, the failure to attain the minimum threshold in a given period obviously influences income earned in future periods.5 Thus, even a risk neutral individual might adopt a seemingly very risk averse strategy in order to protect the income earn- ing potential of the firm. As Robison and Lev (1984) demonstrate in the single period context, the reverse is also true. Consider, for example, a young farmer who faces cash flow requirements which cannot be met given the prices avail- able for hedging. The farmer's decision to leave the crops unhedged represents the only opportunity to preserve the firm intact for future periods. Similarly, a farmer living on the edge of subsistence may 5The distinction which Robison and Lev (1984) make between initial and final outcome variables is useful in this regard. 29 choose a highly risky production package in order to make a "big push" away from the minimum survival level and therefore increase the firm's long-run survival potential. In sum, the concern with obtaining a threshold level outcome in a given period can take on a different meaning when interpreted over a longer time horizon. In many instances this threshold level has sig- nificance in terms of the likelihood of maintaining options into the future. Summar This review of the literature demonstrates that although flexi- bility is widely used by both lay decision makers and researchers in applied disciplines, it has not become well accepted in the mainstream of economics due to the difficulty of deriving unambigous analytic re- sults. Still, the concept is quite useful in highlighting the differ- ence between static and sequential models and in assessing how the dif- ferent components of the decision problem are related. Finally, it is argued that the concept of resiliency or the probabilistic maintenance of some Options for future periods should be built into the notion of flexibility. Although in some respects flexibility and resiliency may appear to be opposites, when considered over a longer time horizon, they reflect the same concerns on the part of the firm. CHAPTER 3 AN EXAMINATION OF INVESTMENT AND PRODUCTION DECISI NS UNDER CONDITIONS OF UNCERTAINTY AND FLEXIBILITY Introduction Chapter Two presented a broad perspective on the selection of the optimal action choice in the context of a sequential decision problem. This chapter addresses the specific question of how the analytic results derived from more familiar single period uncertainty models compare to the results of uncertainty models in which the decision maker has the option of altering or adjusting initial decisions at some specified 2 point before production decisions are finalized. In a sense, this chapter seeks to determine how a change in the rules of the game influ- ences decisions taken by the firm.3 Flexibility, or the ability to ad- just at a later point in time, is introduced by allowing the decision maker to separate the initial investment decision, in this case the ac- quisition of a durable input, from the decision which determines the use of the durable. The first section of the chapter provides a brief background and reviews similar studies of this type of model. The second section 1This chapter is based upon a paper co-authored with L.J. Robison and L. Sonneborn. 2In this paper we will restrict ourselves to a consideration of output price uncertainty. 3Note that in contrast to subsequent chapters the degree of flexi- bility desired by decision makers is not a focus here. 30 31 develops an analytic model of a risk neutral firm which makes decisions under the conditions of flexibility outlined above. The results de- rived from this model are compared to the results derived under static uncertainty. The third section generalizes the model further by allow- ing the firm to be either risk averse or risk loving. Background and Review Models which examine firm behavior under price uncertainty vary greatly in their assumptions about the firm's potential responses to un- certainty, the types of inputs employed and the timing of firm decisions. This chapter focuses on the latter element. Sandmo (1971) examined a firm production model in which simultaneous acquisition and use deci- sions for nondurable inputs were made under uncertainty. Sandmo showed that, compared to the output of a certainty model with a known output price equal to the mean of the uncertain price, a competitive firm's output is greater than, equal to, or less than in the certainty case depending on whether the firm is risk preferring, risk neutral, or risk averse. This result implies a direct link between the risk attitudes of decision makers and the firm's output level. Since this model forms the basis for future comparisons it will be formalized and graphed for the simple case in which the stochastic ele- ment has two possible values e2>e1 with respective probabilities 9(52) and g(e]). Representing expected profits by EU], the profit equation is written as: E[n] = [(Pteilfixl-PXXIMEI; i=1.2 (3.1) -‘M2 The condition for expected profit maximization is: 32 mix" = tf'-Px19(€1’ <3 2) + [(p+ez)f' (XI-px] 9(62) = 0 where p+e1 and p+e2 are the output prices, and x and px are the input and its price, respectively. The solution to equation (3.2) is repre- sented graphically in Figure 3.1 by drawing two marginal value product curves, (p+e])f'(x) and (p+ez)f'(x), and the input price line, px. Forced to make an gx,ggtg decision under uncertainty, the decision maker seeks to balance the probability weighted difference between the marginal value product and the marginal factor cost from ordering either too much or not enough x. If g(e]) equals 9(52), then the vertical distances [(p+e2)f(x)-px] and [px-(p+e])f'(x)] (or the distances ab and cd above and below px) in Figure 3.1 are equal. Thus, the El ggtg profit maxi- mizing choice of x without nggg§t_flexibility (or the ability to choose the input level after 6 is known) is 2. Gilbert et al. (1978) and Shalit, Schmitz and Zilberman (1982) modi- fied the Sandmo model by comparing the amount produced under uncertainty (i) to the amount produced under what they termed "price instability" (and certainty).4 Both sets of authors concluded that the average level of production under price instability, R, would be greater than, less than, or equal to the amount produced under uncertainty as a function of whether the third derivative of the cost function is less than, equal to, 5 or greater than zero. Thus, in contrast to the Sandmo model, the shape 4Price instability merely implies that the price varies, but the decision maker discovers the true price before the production decision is made. 5The average amount must be used as a basis for comparison since the actual amount produced would depend upon the individual draw from the probability distribution. 33 2X pone: xymwwpcwu a new xucmmpemucs muc< xm cm to comwemgeou < _.m mL=m_L AxV.LANu+aV .m>z 34 of the cost function determined how the firm's output under gx_ggtg_un- certainty compared to the average certainty output under price in- stability.6 Since the prices are known before the input choice is made, the average profit equation for this model is written: it = ['(p+61)f(x1)-pxx1] 9(61) (3.3) + [(p+€2)f(X2)-PXX2] 9(82) and the marginal conditions are: an __ _ 5;; - (p+epf(x,)-px - o (3.4) an _ = 5;; - (p+ez)f(x2)-px 0 (3-5) Graphically, the solution to (3.4) and (3.5) above occurs at either input level x1 or x2 in Figure 3.1 depending upon which price occurs. The average input level equals: SE= gx2 (3.6) Both Gilbert and Shalit demonstrated the ambiguity of x relative to R as well as f(x) relative to the sum of g(e])f(x])-+g(e2)f(x2). Johnson (1972) introduced a useful addition to firm level produc— tion theory under uncertainty by specifying that outputs are produced using both durable and nondurable inputs (i.e., some inputs provide services which last for more than one period). This is important, he argued, because market conditions may occur in which the acquisition and salvage prices of durable inputs differ. Therefore, firms should not be 6These comparisons are made for risk neutral decision makers. 35 expected to react in the same fashion to shortages and surpluses of these inputs. Although Johnson does not specify the source of uncertainty in his model, assuming output price uncertainty sets it in the same framework as the previous models and facilitates comparison. The starting point of the Johnson model is the assumption that, as a result of past errors, the firm may enter the current period with virtually any inventory level (I) of the durable input. After output price discovery in each period, the firm Optimizes using either the acquisition or the salvage price of the input as the appropriate opportunity cost. When the firm is faced with a "shortage" of any input in inventory, it is permitted to purchase more at price px. The condition for profit maximization in this case is: ~3- = (p+ellf'IX)-px=0. or 3—;;- = (p+ezlf'(X)-px=0 (3.7) When the amount in inventory is too large, the firm will wish to sell off some of that inventory. In that instance the salvage price of the input, px-r, where r is the per unit difference between the acquisi- tion and salvage price of the input x, becomes the relevant opportunity cost. Thus, the condition for profit maximization becomes: 31_ _ 8x 3T1 a .. 5-,; (p+e2)f (X)-px+r-0 (p+e])f'(X)-px+r=0. or (3.8) In Figure 3.1 the firm will end up producing at one of four input levels, x1, x2, x3, or x4 , depending upon the starting inventory level and the output price which occurs. It is clear that if on at least some occasions the firm finds itself with too much inventory on hand, greater input use and output production will take place in this model compared to the price instability model since x3 is greater than x1 and x4 is 36 greater than x2. Johnson termed this "overproduction" since, on aver- age, the acquisition price of the durable input is not covered by the returns to production. The challenge which this chapter addresses is the incorporation of Johnson's work on the establishment of the proper opportunity cost for the input use decision into a model which integrates acquisition and use decisions taken at different points in time. In Johnson's work all of the conclusions are based upon continuous and random commission of errors by the firm. That is to say the firm apparently makes between period inventory adjustments as if operating in an environment of cer- tainty and does not react to past mistakes by becoming more cautious in its investment decisions in the future.7 The class of firm production models which are termed flexibility models are designed to show how the initial acquisition decision is in- fluenced by the consideration of use decisions to be made at future points in time. In all of the flexibility models the firm is forced to make some production or investment decisions before the output price is known. However, the firm retains some gx_pg§t_ability to make adjust- ments away from the initially selected production levels at a later point in time when more information is known.8 These adjustments are never costless; before making an initial decision in a flexibility model, 7It may be argued that the iso-marginal value product (MVP) curves in the "overproduction trap" are adjusted for risk. In that case, these discounted MVP curves cannot be compared to output levels under certainty --nor could one claim they resulted in “overproduction" levels of out- put. 8The complex relationship between flexibility and imperfect informa- tion discussed in the previous chapter is relevant here. The assumption is made that the firm acquires perfect information gx_gost. 37 the firm must consider the range of potential consequences it will have on subsequent decisions. Flexibility models are differentiated by the nature as well as the cost of making gx.gg§t_adjustments. The general approach is well rep- resented by Turnovsky's (1973) flexibility model which permits the firm to increase or decrease its gx_ggtg production plan after the price of the output becomes known. In order to do so, however, the firm is re- quired to operate on a higher average cost curve. This relationship is demonstrated in Figure 3.2. Ex ggtg the firm moves along the C(Y,O) cost curve, but gxupgst, after the price has been revealed, the firm is restricted by adjustment costs to the C(y,z) cost curve. Turnovsky's conclusion was that it could not be determined whether input/output levels would be higher or lower when risk neutral firms are faced with certainty as compared to uncertain, flexible conditions. Although they have the same basic form as Turnovsky's model, the other flexibility models are more specific with respect to the factors which limit gxflgggt adjustments. Smith (1970) presented a flexibility model in which both the levels of the variable inputs and the utiliza- tion rate of capital stock are chosen after the output price becomes known. The amount of the capital stock in his model, however, is fixed before the price is revealed. The cost of varying output from the initially planned production level is incorporated through the assump- tion of a quadratic capital utilization rate function. Under these as- sumptions the capacity use rate was lower and for most production func- tions the optimal capital stock was greater than in the certainty model. Although Smith did not solve for the average level of capital services used, the above two changes imply an ambiguous impact on the level of 38 Total Costs C(Y,O) Output Figure 3.2 Turnovsky's Flexibility Model 39 capital services used and, by extension, an ambiguous effect on the output produced. Hartman (1976) constructed a similar model in which capital is selected gxflggtg_and labor is chosen gx_gg§t.9 He showed that the aver- age level of output produced by a risk neutral firm can be greater than, less than, or equal to the amount produced under certainty; the result is dependent upon production and cost relationships. He further demon- strated that even a risk averse decision maker can end up producing more under uncertainty than under certainty. Zylberberg (1981) presented a flexibility model in which a single production input, labor, is available in two different forms. "In- flexible" or "stable" labor is characterized by a lower average cost, but must be retained beyond the subsequent production period if hired. In contrast, temporary labor is more costly to the firm, but can be easily varied in response to changing market conditions. Zylberberg con- cluded that no clear relationship could be derived between the amount of inputs used and outputs produced in certain and uncertain situations for models with and without flexibility. In summary, the models do not yield unambiguous predictions of the influence of flexibility possibilities on total input use and output production. They do, however, help explain such observed behavior as the maintenance of capital stocks and the use of temporary and permanent labor in the same production process. As such, the formulation of these models represents an important step in understanding real world phenomena which are difficult to explain in the context of more familiar models. ngstein (1978) derived largely the same results from a similar model. 40 A Restricted Flexibility Model In many instances, such as the gx_ggtg_decision to purchase a dur- able (e.g., a tractor) and the gx_gg§t decision to determine its rate of use (i.e., the number of hours of operation),the gx_ggtg_choice places an absolute bound or limitation on the gxflgggt use decision. Such a limitation is not present in the flexibility models discussed above. This assumption is particularly well suited to models involving large transactions costs. In such a model, reorders are not considered be- cause they are too costly. Also, this model applies to physical pro- cesses such as the growing of crops which cannot be increased after the planting season is past. To develop a model which incorporates this constraint, assume the firm faces an uncertain output price (p+e) described earlier and an input/output relationship described by: y = f(x|XF) (3.9) where XF is a vector of fixed factors. For notational convenience, the XF vector is suppressed and the first and second derivative of i’ are written as f'(x) > O and f"(x) < 0, respectively. The amount of x the firm employs is limited by an inventory of the input equal to I, the restriction being: x §_I (3.10) The decision problem is to select I gx_ggtg_while e is unknown, so that an optimal selection of x can be made gx_gg§t_when e is revealed. The selection of I is made interesting by the inclusion of a per unit holding cost, r, imposed on all inventory I acquired, but not used in a given period. Alternatively i: may be thought of as either an option price (the price one pays for the right to purchase a specified quantity 41 of the input x at price px at some point in the future) or as the dif- ference between the acquisition and salvage prices of the input (assum- ing the unused inventory is liquidated at the end of the period). With r set equal to zero this flexibility model will collapse into the price instability model considered earlier, since the firm would choose to costlessly maintain infinite inventories as it waited to discover the price. At very high levels of r the model collapses to Sandmo's pure exuggtg model since holding inventories is never a profitable activity under these conditions. The flexibility model is formalized by first assuming that the firm maximizes expected profits. Representing profits by n, the profit equa- tion is written as: n = (p+e)f(x)-pxx-r(I-x), where e~(0,oz) (3.11) subject to: x 5_1 (3.12) The dynamic programming principle of backward induction (Bellman, 1957) suggests that the expression (3.11) be maximized §x_gg§t_for optimal values of x. The optimal values for x derived from the first order con- ditions of the profit function can be written as: PX-r r0 fOTEETr-(O-y-p - px-r P -r P -r x = i’ f' up“) forfim-pf<1)- pxru 9(a) de 2 2 where 9(a) is the probability density function for e. The first order conditions for I satisfy the expression below: Z2 I (-r) 9(a) de +27 [(p+e)f'(1) -pxlg(e)de=0 (3.15) -m 2 Having established the first order conditions for the model's solu- tion, a series of questions may be posed: (1) What is the effect on I of an increase in r? (2) How does the level of inventory, I, acquired in the flexibility model compare with i, the amount of input acquired and used in the gxhggtg uncertainty model? (3) How does the average level of input and output in the restricted flexibility model compare with the level of input used and output produced in the g§_ggtg_uncer- tainty model? The first question is easy to answer. It requires the differentia- tion of (3.15) with respect to I and r. 12 }' 9(€)Ck: 10 ;: §_O (3.16) f [(p+8)f"(1)] 9(€)de Z2 1: dr -r 10Equals zero only in the case that g(1r) =0 for 54.413 - p. 43 As expected, the higher cost of acquiring an inventory and not using it encourages the firm to be more cautious in its acquisition decisions. To answer the second question, recall that the gxnggtg choice of x without gxflpggt flexibility was shown earlier to be 2. Next assume that I is selected gxuggtg and that x is selected §x_gg§t_after e is known to be either e1 or £2. The mathematical results of a comparison between I and I are presented in Appendix A. An intuitive solution is presented next using the simple model described in Figure 3.1. Suppose the firm begins the period with inventory levels of its choosing--that is to say, the inputs on hand have been adjusted through use or salvage to a desired level. What determines the optimal gx_ggtg_ holding of I? On the one hand, .an inadequate inventory of I results in foregone income by not being able to produce at the desired level. This loss would be p(e2)f'(I)-px. On the other hand, when output price (p+e]) occurs, the firm holds more of the input than it desires and suffers the smaller of either a per unit production loss of (p+e])f(x)--px or the holding cost, r. If r < px-(p+e])f'(x), the firm chooses to hold I-x in inventory. A reversal of the inequality suggests the firm minimizes the cost of its overacquisition by producing rather than holding idle inventory. The possibility of holding an inventory partially shields the firm from the effects of the adverse price compared to the situation where the firm is forced to use all of its inputs. Thus, the acquisition level I of the input in the flexibility model will equal or exceed the acquisition and use level of I in the g; ggtg_uncertainty model. Intui- tively then, I 3,2. 44 This argument is presented graphically in Figure 3.3. Starting from the point i which was derived in Figure 3.1 (the gxuggtg acquisi- tion and use of i) the firm decision maker asks what are the potential gains and losses from acquiring one more unit of the input gx_ggtg? The potential gain for an additional unit of x remains unchanged. It is still the difference between (p+e2)f'(x) and px (the distance ab in Figure 3.3). The potential loss, however, has been reduced from px - (p+e])f'(x) to the holding cost, r. In the case where the probabilities are equal, the firm therefore acquires additional inventories until the potential gains of (p+ez)f'(I)--px is set equal to the potential losses of r. This occurs at the point marked I in Figure 3.3. The graphical description of the problem associated with the choice of I has some interesting implications. First, x2, the input level which satisfies the first order condition for the higher price when there is no holding cost, represents the upper bound on I since the firm would never choose to stockpile a greater input quantity. Second, I varies inversely with r and will descend towards i as r increases. Once r is greater than px-(p+e])f'(x), the firm pays a greater cost for storing than for using the input. Thus, at such a level of r, storage is no longer considered and the model's solution is the solution to the strict gx.ggtg_model. The bounds on I can be written as i 5_I < x2. The next question is how much x will be used and how much output will result from the production process in the restricted flexibility model? When the higher price occurs, the firm would prefer to use the input to the level x2, but the gx_ggtg_ordering constraints, I, will be binding. When the lower price occurs, the firm recognizes that the salvage price of the input (px-r) is the relevant opportunity cost and 45 Eva: 33.5.58: 233.58”. a 23 Eva: 353325 35 mm cm .8 5.0.2328 < ex fl v «x v mx H ANNIE + 9253 u ex «x H mx _.x m.m mesmwm 7f m _ _ _ _ Axv.mfipw+qv 3 Egg a .a>z 46 chooses to put x3 of the input in production. Thus, x*, the average amount used in the flexibility model, is a weighted mean derived from the two levels x3 and I. The relationship between x* and i is ambiguous. Moreover, the average amount produced in the restricted flexibility model g(e])f(x3)-+ g(ez)f(I) can be less than, equal to, or greater than the amount produced under gx_ante uncertainty.n In Appendix B, an example using a Cobb- Douglas production function shows that it is possible for the restricted flexibility model to result in lower average input use and lower average production, higher average input use and lower average production, or higher average input use and higher average production than in the gx ggtg undertainty model. All of these changes occur as a function of the elasticity of output. As the elasticity of output approaches 1, in- creased production ("overproduction") occurs in the flexibility model. Introducing Risk Aversion Next the question of what effect the introduction of risk aversion has on the restricted flexibility model will be examined. Instead of expected profit maximization, assume the decision maker maximizes the expected utility of profits, U(n), in accordance with the expected util- ity hypothesis. The objective function is: max EU(n) (3-17) where, lJ'(n) > O and U"(n) < 0. 11The reader may note that these results for I and x* relative to x hold for any spreading of the stochastic element 8. As the stochastic element in price varies more, I, the amount in inventory, will in- crease while the actual quantity of the input used (as well as output produced) will remain ambiguous. 47 Sandmo demonstrated that f(x), the expected output level (and i the expected input level) in the gx_ggtg_model, would be less under uncer- tainty and risk aversion than under uncertainty and risk neutrality which is equivalent to expected profit maximization. The reduction oc- curred because the utility function caused a concave transformation of the profit function. Thus, the input level is reduced as a result of the introduction of a concave utility function. In the flexibility model the gxupggt "use" decision is made after 5 is known. Since cer- tainty prevails, the decision is made so as to maximize profits, i.e., the gxnpggt choice of x is unaffected by the risk preferences of the firm. The ex.ggtg choice of the inventory level, I, in our model is made under uncertainty and thus is affected and reduced as the firm maximizes expected utility rather than expected profits. Hence, intro- ducing risk aversion to the strict gxflggtg and flexibility models re- sults in a reduction in the output and average output respectively com- pared to these models under risk neutrality. But unlike the strict gx 3333 case, it still remains impossible to make comparisons between the flexibility model under risk aversion and a stable certainty model since the risk aversion effect need not outweigh the previous potentiality of) increased production. Thus, once again, ambiguity results. Summary Two important and related firm decisions are the decision to acguire inventory of an input, and the decision to Egg the input. Be- cause these decisions can, in most instances, be separated in time, they may add flexibility to the firm's production decisions. When the cost of holding an inventory is excessive, the possibility of placing an in- put in inventory adds little or no flexibility. But when storage costs 48 are less than the amount lost if the input were used, the separation of the acquisition and restricted use decision increases the firm's ability to respond to uncertainty. In the flexibility model introduced here, it was demonstrated that this response would take the form of acquiring greater levels of inputs than would be used on average if the firm lacked flexibility. The amount of the input used and the output pro- duced, on the other hand, was ambiguous relative to the inputs and out- puts of the firm facing uncertainty and lacking flexibility. Obviously, these results apply to a relatively simple deductive model. They should also be examined in more complicated models. The basic model in this chapter is next modified by permitting decision makers to purchase inputs which offer varying levels of flexibility. CHAPTER 4 USE FLEXIBILITY MODELS FOR DURABLES WITH FIXED TEMPORAL LIFETIMES The next two chapters present a consideration of the role that the economist can play in designing and/or selecting optimal durable inputs. In particular, the discussion will focus on whether or not the concept of use flexibility can aid in predicting changes in optimal durable choices as either the situation or the risk preferences of the decision maker undergo known transformations. In the real world, a durable acquisition decision would require the consideration of both adjustment and use flexibility issues. The topic of these two chapters is restricted to use flexibility considerations by making a number of simplifying assumptions. These include fixing the time and price of the acquisition decision and ruling out any resale possibilities for owned durables. Consequently, any flexibility which the firm wishes to maintain must be "built in" to the nature of the dur- able input itself. Use flexibility is a matter of interest to the firm because, by definition, a durable input provides services over a multiple period time horizon and, hence, will influence the firm's production possibili- ties and costs in the future. Since market conditions may vary within that time frame, the firm will be concerned with the durable's peform- ance over the entire range of potential circumstances. 49 5O Returning to the flexibility definitions proposed in Chapter Two, more flexible durables can be defined simply as those durables which 1 The perform relatively better as the range of use conditions widens. analytical problem therefore is to develop a rigorous definition which will never provide incorrect predictions. A large part of the analysis in both this and the next chapter will be carried out under the assumption of perfect information in order to take advantage of the greater simplicity gained. The very important linkage between flexibility and learning will be mentioned. but not dealt with in detail. The inclusion of the relationship between the relative demand for use flexibility and the risk preferences of the decision maker represents an extension to the topics previously treated in this literature. The durable input investment problem, while clearly of concern to firm managers, has elicited only sporadic attention from economists. Most of that literature has focused on the optimal quantity of invest- ment and has done so in models with fairly rigid assumptions (Smith, 1957; Lutz and Lutz, 1951). Others have also considered disinvestment and optimal use rates (Edwards, 1958; Baquet, 1978). The research de- veloped here follows a somewhat different strand by focusing on the choice among nonhomogeneous durable inputs which produce homogeneous services to the firm.2 1The range of conditions considered in these two chapters will al- ways refer to the intentisy of use in a given time period. Use flexi- bility also refers to the product flexibility issues raised by Heidy and others, but those problems are less easily handled either mathematically or graphically. 2Much of the literature which is cited actually refers to choices made among techniques which produce the same outputs rather than dur- ables which produce the same services. 51 The concept of use flexibility which is considered in this chapter is further restricted through the simplifying assumption that the produc- tive lifetime of all competing durables is known (in terms of units of time) and is not influenced by the output production rate in individual periods. Since only durables of equal purchase prices will be compared, these durables will be differentiated solely in terms of the cost sched- ules of the variable inputs associated with their use.3 Chapter Five relaxes the assumption that durables can or should be characterized by fixed service lifetimes. Each durable in that chapter will be distinguished by its own variable per period loss of lifetime capactiy function. Although the two chapters are quite similar in for- mat, an examination of the variable costs associated with the durable's own productive capacity is regarded as sufficiently unique to merit separate treatment. In both chapters the objective will be to derive models which focus on the trade-off between one characteristic termed efficiency or special- ization and another referred to as flexibility. In the context of specific models a workable definition of flexibility is proposed and illustrated. It should be recognized that each chapter is restricted to a single aspect of the overall flexibility issue. The following discussion demonstrates that, although economists have often ignored these design issues, they do have the ability to pro- vide helpful insights in this area. Working in conjunction with engi- neers, they can provide assistance to real world decision makers. 3This avoids the introduction of adjustment flexibility issues. In general, higher purchase prices imply reduced adjustment flexibility, but may result in increased use flexibility. 52 In this chapter the durable input problem is first examined through comparison with the variable input problem. Although the introduction of a multiperiod framework creates certain new analytical complications,4 it is demonstrated that in many respects the selection of the optimal durable for a given set of circumstances parallels the optimal variable input choice under conditions of static uncertainty. After the introduction of the problem context, and before a de- tailed review of past models, a brief digression examines in general terms the costs and benefits of durable ownership. This section exam— ines why firms choose to own durables and will lead naturally into a consideration of models to determine the optimal durable for specific circumstances. The review of the literature section highlights the models pre- sented by Stigler and Tisdell. While both authors succeed in deriving a relationship between the relative flexibility of the optimal durable and the variability of demand conditions, each requires specific re- strictive assumptions to do so. Furthermore, neither adequately assesses the trade-offs involved in opting for increased flexibility. In order to answer questions left unresolved by these earlier models, a more detailed and specific flexibility model is derived which makes explicit the trade-offs between design parameters. Although the results of this model (derived for a case in which output demands are exogenous and required) can be generalized only with great caution, they 4One key complication which arises is the need to discount costs and returns which occur in different time periods. Due to the nature of the models considered, however, the discount rate will not influence the results and, hence, will be ignored. 53 are still indicative of how the design process takes place. The implica- tions of this model are also briefly discussed. Variable vs. Durable Input Models In economics, perfect information or certainty models are generally solved under the assumption that the firm wishes to maximize profits. The simplest models considered in the theory of the firm are single period one nondurable input/one output models. Since both the input and the output are assumed to be homogeneous and perfectly divisible, the only interesting question is "How much of the input should be acquired and used?" Representing profits by n, the output price by p, the output level by x, and the cost function by C(x),the profit equation can be written: n = px - C(x) (4.1) Assuming that C(x)' > O and C"(x) < O, the condition for profit maxi- mization is: '91 3x = p - C' (x) = 0 (4.2) All beginning students in economics learn that this marginal condi- tion indicates that additional units of output are produced until the marginal revenue gained just equals the marginal cost of production. There is no need in this model to distinguish between the acquisition and use decision since they are simultaneous. As was demonstrated in Chapter Three, the introduction of uncer- tainty complicates matters. Whereas under certainty all decision makers who prefer more to less will make the same action choice (i.e., purchase the same level of variable input), this need not be true under under- tainty since in this case the decision makers must choose among 54 alternative probability distributions of outcomes. In order to assess the competing probability distributions, each decision maker develops a weighting scheme (commonly termed a utility function). Since these schemes can assign weights to the outcomes in different ways, the rank orderings of action choices (here input levels) can also differ. The simplest uncertainty models are once again static single input/ single output models in which no distinction is made between acquisition and use decisions. Uncertainty is introduced by replacing the known output price with a stochastic probability distribution of prices with the mean of the distribution equal to the certainty price: w = p*x - C(x) (4.3) where, p* is a stochastic distribution of output prices. Under the assumption that decision makers are expected utility maximizers, Sandmo, among others, demonstrated that while risk neutral decision makers would be unaffected by the introduction of risk, risk averse (loving) decision makers will respond by reducing (increasing) the amount of variable input used and output produced under conditions of uncertainty. The marginal condition for utility maximization for risk averters is: E [U'(1r)] [C' (X) - Ep*] <‘ 0 (4.4) which, since U' > 0 by assumption, requires that C'(x) < Ep* The reduction (increase) in production occurs because the utility function causes a concave (convex) transformation of the profit func- tion. Under uncertainty each decision maker evaluates the full range of potential outcomes associated with an action choice rather than a sum- mary measure . 55 The inclusion of durable inputs results in some similar concerns in the derivation of optimal acquisition rules.5 Durable inputs, in contrast to variable inputs, are lumpy in acquisition. The purchase decision taken at time ID will have consequences for the firm until the durable's services are exhausted at time Tn“ As a result, this longer time horizon must be taken into account when the acquisition decision is made. Over the longer time horizon, market and/or other conditions may vary. The optimal durable choice (whether under certainty or uncer- tainty) must, in a manner similar to the optimal variable input choice under uncertainty, be able to perform well over the range of conditions. This involves an ability to perform well on average rather than simply performing well under average conditions. The profit equation can be written as: N j , 11 = {3 [pi x1. - C (xi)] (4.5) 1 Depending upon how the model is constructed, there may or may not be marginal conditions which are relevant for the individual period use decisions. The choice variable of interest, the durable input, is represented by the superscript on the cost function. In what follows it will be argued that decision makers must take into account design trade-offs and possibilities in selecting the ap- propriate durable input. Competing durables with equal acquisition prices can be ranked on a "specialization or flexibility scale" ranging from durables which are very efficient under specific conditions to 5The reader is reminded that there is the assumption of no resale market. 56 general or flexible durables which gain favor as there is a broadening of demand conditions. It is essential that engineers, economists, and decision makers clearly understand what trade-offs are involved in mov- ing from one end of the durable scale to the other. Both flexibility and specialization should be seen as desirable characteristics which, at the same time, impose penalty costs in terms of other characteristics sacrificed. Why Firms Own Durables Before examining the question of which durable to select, the prior question of why firms choose to employ durable inputs deserves some con- sideration. A simple answer to the above question is that firms choose to own durables when the benefits exceed the costs. In order to gain a better understanding of the durable problem it is useful to examine more closely what these costs and benefits are. Consider the example of a consumer who requires both city and high- way transportation services. In a world made up solely of variable in- puts the consumer would be able to buy city and highway transportation services at the times and in the quantities desired. In the real world, however, the consumer may be faced with a choice among vehicles designed specifically for one or the other type of transportation or a "hybrid" vehicle which performs moderately well under both conditions. The dis- cussion below indicates how the consumer should evaluate these choices in order to gain maximum utility. Whereas it was argued in the previous section that the purchase of a unit of variable inputs enables a firm to directly transform available cash or credit reserves into a factor of production, the relationship 57 between the acquisition and use decisions for durable inputs is a more complicated one. Specifically, the longer time horizon associated with the use of a durable requires the inclusion of two types of opportunity costs which are not a matter of concern in the purchase of variable in- puts (at least under certainty conditions).6 The first is the cost of tying up the firm's equity or credit reserves in maintaining an in- ventory of services acquired, but not used up in the initial (as well as subsequent) production period. The funds that are tied up in this man- ner cannot be used by the firm for other purposes. This is the main reason why the option of buying two separate vehicles to specialize in the different types of transportation services is not an attractive one to the consumer since such a course of action would more or less double the holding charges the consumer would pay (and thereby greatly increase the average per unit cost of transportation services). A second type of opportunity cost which must be considered is the opportunity cost of not having a more appropriate durable in place in any given period. In the car example, the opportunity cost associated with either of the specialized cars might be excessive when the Mother“ type of transportation services is required and, hence, encourage the consumer to opt for the hybrid car. In sum, the combination of these two factors might encourage the consumer to acquire a car which is not optimal for either of the proposed use situations considered individually. In principle, the existence of a well organized and complete rental market might permit the firm to make use of the optimal durable for each 6There are, of course, other costs associated with durable owner- ship such as maintenance costs which are, for simplicity's sake, not treated. See Robison (1982) for details. 58 individual situation.7 In many cases, however, such a market does not exist. General Motors, for example, does not have the option of renting fully operational factories on a year-to-year or period-by-period basis. Even in cases in which rental markets exist, there are a number of reasons why firms prefer to own durables rather than rent durable ser- vices. One reason is that through acquisition the firm can greatly re- duce the average cost paid for durable services. In effect, the firm does so by buying in bulk. In general, the decision to own rather than rent will hinge on the amount of services which the firm plans to use. For example, while it is clearly less expensive to rent a car or hire a taxi for infrequent service demands, as the quantity of transportation services demanded increases their average cost can be much reduced by owning a car. A second reason why firms prefer to own durables is that they there- by ensure a readily available stock of those services when needed. A farmer who elects to rely upon harvesting services rather than purchase a combine may not always find a timely supply of those services. Finally, in addition to providing production services, durable ownership may serve the investment and/or speculative objectives of the firm. Thus, the decision to purchase a durable such as land or housing may be based upon a combination of motives. In conclusion, although the selection of the "best" durable for a given distribution of uses may not provide as neat a fit as does the selection of variable inputs on a period-by-period basis, firms still 7The consumer would not be able to totally avoid the holding costs associated with durable ownership since one can assume these would be passed along by the rental agent. 59 regard the ownership of durable inputs in many instances to be superior to all other options. The next objective of the research, then, is to determine which durable will yield the greatest net benefit in different situations and what, if any, relationships can be derived between the characteristics of the optimal durable and the changing nature of the production circumstances (or risk attitudes of the decision maker). Review of Use Flexibility Models Although the general issue of flexibility was raised in the 19205, Stigler's 1939 article "Production and Distribution in the Short-Run" is commonly cited as the most influential early discussion which specif- ically focused on the question of use flexibility. In this article Stigler examines how a firm facing certain but variable demands for its future production would select the optimal factory design. He thus recognizes that the initial construction (or acquisition) decision must be made in the context of the future production decisions which will be taken. Stigler's model differs from the standard economic model in that the quantity rather than the price of output is assumed to be variable. As a result, the firm does not optimize on a period-by-period basis since it is required to meet an exogenously determined demand level. His re- sulting objective function of minimizing total cost over time is the type of problem which public utilities and other regulated industries often face. In addition, many multicomponent production processes may also result in an overall model quite similar to what Stigler presents. Consider, for example, a truck driver making a coast-to-coast run. Even if the driver knows that the average variable cost of producing a unit of transportation services is significantly lower at highway speeds, 60 the truck must, for certain stretches of the journey, be operated at city speeds in order to get from one highway to another. In effect, the producer is not permitted to set the optimal production rate for each period due to the constraints of the overall production process. As a result, the firm may be forced to operate at production levels for which the average variable cost is declining and/or less than the output price (if one can be identified). Neither of these situations would occur in a standard production model since the firm would respond by either mov- ing to a more profitable level of production or shutting down entirely. In this type of model the firm carefully scrutinizes the "package deal" offered by each durable at the time of the purchase and then merely meets demand levels in each period. Stigler based his discussion of the implications of flexibility for the selection of the optimal durable on a version of the following graph (Figure 4.1). He defines durable flexibility in terms of the relative 8 In a convexity (the second derivative) of the average cost curves. paired comparison of durables, a more flexible durable is one which has everywhere a smaller second derivative of average cost. This definition resembles Marschak and Nelson's definition in its generality and suffers from the same difficulty in ordering pairs of durables. Before discussing those shortcomings, Stigler's discussion of use flexibility under conditions of certainty will be summarized. The basic 8As is discussed below, it makes more sense to use the second derivative of the total cost curve as the flexibility measure, but Stigler's definition is convenient for graphical analysis. 61 Durable I AVC I | I I I | Durable F I l I I I I 1 l 1 I I l I I I I I I I I I I I I I 0 J J I I l D B A c E OUFPUt Figure 4.1 Two Durables Which Vary in Flexibility 62 assumptions of the Stigler model are:9 l. The firm wishes to maximize expected profits. 2. The output requirements are known with certainty in advance and must be met. 3. The marginal and average cost functions for each technique do not vary from period to period. 4. The two techniques have the same length of life of N periods. 5. All N periods are of the same length. 6. A single durable must be chosen for the N periods. In Figure 4.1 durable F is more flexible than durable I by Stigler's definition since its average cost curve has everywhere a smaller second derivative than durable I. As a first point, consider which of the two durables would be selected by the firm if a stable out- put of A units per period were required. By inspection, durable I pro- duces that output for a total cost of y2* 0A less per period and thus would be selected. The greater flexibility of durable F is a needless extravagance in producing a stable level of output. An inspection of the graph reveals that durable I is preferred for any and all combinations of output levels between B and C since the aver- age cost curve for I lies below the average cost curve for F. Once the output requirements spread beyond that range (on either side), it be- comes necessary to know the distribution of output requirements in order to calculate which durable will provide the output for the minimum total 9A key unmentioned assumption in Stigler's model is that all dur- ables considered attain their minimum average cost at the same output level. This assumption will not be enforced in the model derived in this chapter. 63 cost. The solution for the preferred durable from a paired comparison given known output requirements is: N N Min [i TCi, i TCE] (4.6) where: TC: = total variable cost of operating durable I at output level i; To: = total variable cost of operating durable F at output level i; N = number of periods of production. The flexibility ranking which Stigler proposes is not intended to aid in determining which durable will be preferred in any given situa- tion. Its role, rather, is to predict what types of durables may gain favor in response to a known transformation of the output demand condi- tions. In particular, Stigler's hypothesis is that more flexible dur- ables (durables with smaller second derivatives of average cost) will replace less flexible durables as the range of output requirements widens.10 Stigler examines this hypothesis graphically rather than mathemati- cally. A convenient way of widening the output range is through a mean preserving spread of the output requirements. Once again, using Figure 4.1, consider which durable will now be preferred if, as a result of a spread in the distribution,the firm must produce 50 percent of the time each at output levels 0 and E. Although the mean production remains A, 10The reader should note that in order to rank durables, the second derivative of average cost must everywhere be smaller. Many pairs of durables thus cannot be ranked. 64 the firm now minimizes its cost by selecting durable F. This example demonstrates the difference between performing well at the average out- put level and performing well on average (or overall) at the actual out- put levels. It can be intuitively seen in this example that as the range of re- quired outputs widens a change in preference is only possible from the less to the more flexible durable since durable F's average cost of production varies less from its minimum than does durable I's average cost. A second conclusion is that if the range widens enough, durable F will become preferred since, as for Marschak and Nelson's model, the total cost of the less flexible durable approaches infinity more quickly. A fundamental problem with Stigler's model is that there is no jus- tification or explanation given as to the choice set from which alterna- tive durables must be drawn. The seriousness of this shortcoming can be illustrated by examining a comparison of the two durables presented in Figure 4.2. In contrast to Figure 4.1, the two durables here attain their minimum average costs at very different output levels. Assume that initially the firm wishes to select a durable to produce a constant output level of A for each period. At this level the firm is indif- ferent between the two durables. If the distribution is then spread so that for one-half of the periods the production level is A—a and for the other half the level is A+a, it becomes unclear which durable will pro- vide the required outputs at minimum cost. The possibility that what Stigler defines as a less flexible durable can gain favor as the range of required outputs widens arises because the first derivatives of the average cost curves have opposite signs over part of the expanded range. AVC 65 Durable I Durable F \ | I l l | I | I I J I I A-o, A A+o OUtPUt Figure 4.2 A Second Comparison of Durables Which Vary in Use Flexibility 66 This result demonstrates that an ordering of durables in terms of the relative convexity of the second derivatives of their average cost curves is not a sufficient condition to predict how the sum of total costs will vary as the range of required output levels is spread. Some- how the full sense of the word flexibility is not being captured. In order to more effectively address the flexibility issue it will be necessary to more precisely specify the-required trade-offs. All of the other use flexibility models suffer from this same lack of attention to the actual benefits and costs of increasing flexibility. Nevertheless, they do present other results which deserve mention. Baumol (1959) extended Stigler's framework of analysis to a situation in which the firm faces a probability distribution of output require- ments in future time periods. Before each individual production period decision, however, that uncertainty is resolved. The results of this uncertainty model replicate the results of the original Stigler model. Tisdell (1968) and Marschak and Nelson (1962) examine the value of flexibility in models for which output price rather than quantity is the variable element. Tisdell defines a more flexible durable as one which has everywhere a smaller second derivative of total cost (i.e., a smaller slope of the marginal cost curve). While this flexibility measure is defined on the cost side, dur- able selection in this case must be viewed in terms of profit because having different durables in place will result in different levels of production. Since the decision maker in this model will carry out a period-by-period process of optimization, the preferred durable will be the one which results in the greatest profit over the lifetime of the durable [see equation (4.5)]. 67 Tisdell's certainty model is based upon many of the same assump- tions as Stigler's model, but does not assume that output requirements are known with certainty in advance. In addition, Tisdell assumes that: 7. After reaching their minimum, the marginal cost functions increase monotonically. 8. Output price always exceeds the maximum of the two minimum average variable costs (hence, there is no shutdown for either durable). 9. The output decisions which are made at different points of time are independent. He demonstrates that, under certainty with the output price vari- able, the durable with a smaller second derivative of total cost will always gain favor as the variance of the output prices increases if the output price never falls below the maximum of the minimum average costs. The mathematics of this proof are provided in Appendix C. Intuitively, the model functions in a similar fashion to Stigler's model. Tisdell's model clearly does not suffer from the problem outlined in Figure 4.2 since production would not occur at rates of output to the left of the minimum average variable cost level. Another problem, how~ ever, arises in that assumption #8 restricts the number of demand situa- tions which can be evaluated. Why should there not be some instances where one durable stays in production and the other is forced to remain idle? Should not the ability to remain in production be related to flexibility? The Tisdell model does not address this issue. The main focus of Tisdell's work revolves around a demonstration that as uncertainty increases the firm's preferences will be redirected towards durables characterized by less rather than more use flexibility. 68 Although initially this result seems counter-intuitive, it becomes comprehensible once one realizes that Tisdell introduces uncertainty in a quite different fashion that does Baumol. For Tisdell, increased un- certainty is measured at the time of the individual period production decisions rather than at the time of the durable acquisition decision. Thus, gxwgg§t_uncertainty reduces the potential benefits which the firm can gain from having a more flexible durable in place since without adequate information the firm is relegated to producing in the expecta- tion of the average price rather than having the opportunity to await the discovery of the actual price. As is graphed in Figure 4.1, the less flexible durable can produce more efficiently at the average level. The necessity of having adequate information in order to make it profit- able to maintain flexibility is the same point made previously by Hart and others and was highlighted in the discussion of Figure 2.2. A summary of this research reveals that under certainty conditions the derivation of the sought-after positive relationship between the degree of durable use of flexibility desired and the variability of demand conditions (either exogenous quantity demands or output prices) is dependent in Stigler's model on the assumption that all durables are centered around the same output level and in Tisdell's model on the assumption that prices never fall below the maximum of the minimum aver- age variable costs. Baumol extends the Stigler model to include ex ggte uncertainty and derives the same results with respect to the role of flexibility. In contrast, Tisdell demonstrates that for increasing ex ‘pggt uncertainty the firm reacts by decreasing its demand for durable use flexibility. 69 None of the models reviewed adequately address the issue of specifying the choice set from which the optimal durable is selected. Consequently, these models do not make clear what options are available to the firm as different demand conditions occur. This question can be most usefully clarified and considered in the context of the underlying design problem. The following section derives these results through the examination of a specific design problem. For the sake of brevity, the analyses in this chapter are restricted to the exogenous quantity model. Both quantity and price models will be examined in Chapter Five. Derivation of a New Use Flexibility Model The chief problem cited with the previous use flexibility models is the lack of attention paid to the trade-offs required in order to in- crease the flexibility of a durable. This shortcoming can best be over- come by moving away from the general models cited above to a very specif- ic example derived below. The durable design problem which is intro- duced and solved highlights the process of integrating the design in- formation provided by the engineers with the known demand information in order to form an optimization model. A definition of flexibility is derived in accordance with the design constraints. This definition succeeds in predicting the change in the preferred durable design in response to changes in demand conditions or risk preferences. The central assumptions of the model are: 1. All competing durables have the same purchase price. 2. The purchase of the durable is a “once and for all decision" (i.e., no resale possibilities). 7O 3. The firm wishes to maximize profits (or expected utility) over the entire time horizon of the problem and not over individual subperiods. 4. Demand conditions in each period are set exogenously and must be met by the firm. The problem is made tractable by stipulating that the family of durables is defined by average cost curves of the form: AC = c+£(h-x)2 (4.7) subject to, c, h,£>0 There also is an overall cost constraint, k, on the durable: k = h-c-Jl (4.8) An underlying assumption of the model is that the three design parameters, h,<:, and K, have been identified by the engineers and are under the control of the firm decision maker. Figure 4.3 illustrates these design parameters for a representative durable. All of the dur- ables which are defined by equation (4.7) will have the familiar U- shaped average cost curve. The first design parameter, c, is calculated as the height to the minimum point on the average cost curve. As such, c can be regarded as measuring the specialization of the durable in producing at a particular output level. The lower c is (at any output level), the more special- ized the durable is. From equation (4.8) it is clear that specializa- tion is inversely related to the cost of the durable; a decrease in c, holding other design parameters constant, can only be obtained through an increase in the acquisition price of the durable. In terms of a common example, a car which gets 40 miles per gallon has a higher price AVC 71 241, i h 5- Output Figure 4.3 The Design Parameters h, c, and L for a Representative Durable 72 than one, with otherwise identical characteristics, which gets 20 miles per gallon. A second factor, 2, influences the steepness of the average cost curve. As E increases, the average per unit production cost increases more rapidly at points other than the durable's most efficient produc- tion level. This design parameter closely resembles the factor which Stigler and others have identified as representing flexibility (i.e., a durable with a smaller 2 parameter will have a less convex average cost function). The constraint equation specifies an inverse relationship between k and 2 which indicates again that a decrease in C can only be achieved by increasing k. This ensures that the firm will have to pay more for a durable which has relatively moderate cost increases away from its most efficient use rate. Both of the above design constraints have implicitly been included in previous models. The third design parameter, h, has been the ne- glected element. Whereas c locates the curve vertically, h fixes its location horizontally. In all of the other studies the horizontal loca- tion of all competing average cost curves is apparently fixed by assump- tion and the firm is only permitted to trade off c for 2. In this model the constraint equation permits a three-way trade-off. Once again, from the constraint equation the partial derivative of k with respect to h is positive which indicates that it is more costly to purchase a durable which attains its most efficient production at a higher output level. Thus, a car which can achieve 40 miles per gallon at 75 miles per hour will have a higher price than one which attains the same fuel efficiency at 55 miles per hour. 73 Since in this model the output demanded is both exogenous and re- quired, the total revenues earned by the firm in a given period will be identical no matter which durable is selected. Thus, the revenue side can be ignored. This is convenient for analytical purposes. In designing the optimal durable the firm will seek to minimize its total cost over the production life of the durable. The total cost function can be calculated to be: TC = cx+-i€x(x-h)2 (4.9) This is a cubic function with regard to the output rate x and has the expected characteristics of a total cost function (the first derivative is always positive while the second derivative is first negative and then positive). As a first question one might wish to know how equation (4.9) would be used to design the optimal durable to produce at a gigglg out- put rate. This equation should not, however, be used in that context because the 2 parameter would have no usefulness and thus the con- straint would not operate as one would wish. In particular, the firm would continually select a higher and higher 2 in order to reduce c. For cases in which the firm is required to produce over a range of output levels, equation (4.9) can be used to derive the optimal values of the three design parameters. As an example, this will be derived for the case of a uniform and certain distribution of output require- ments. The mean of the distribution is defined as p and the distance between the mean and each end of the distribution is defined as plus or minus d. The objective function of the firm is: Min 233 [$3 cx+£x (x-h)2 dx 74 Integrating and then solving the appropriate first order conditions results in the following reduced form equations for each of the design parameters: O I w ‘C+ C. I _a 03 O. 'N N “C N I N N I X- Initial examination of these reduced form equations reveals that h, the centering point for average cost curves, does not fall at the mean of the output distribution. Furthermore, the firm will clearly choose to adjust h, and the other two design parameters, in response to changes in the distribution. Thus, the assumed placement of h by Stigler and others does not appear to be the correct approach. Given the design constraint and knowledge of different possible output distributions, it is possible to derive an isocost frontier of the preferred designs at a given k level. The direction of the change in each of the optimal design parameters can be obtained by differenti- ating the reduced form equations by either p, the mean, or d, the value equal to one-half the range. All three of the partial derivatives with respect to a change in u are positive. This indicates that the firm would respond to a higher 11As an additional restriction on the model, 6d2-2u2 must be less than or equal to 18 or the solution will be an irrational number. 75 mean by moving the entire average cost curve to the right and accepting a higher c and a higher 2. None of the other studies reviewed examine such a change. Thus, when faced with a production process with higher average demands the firm would sacrifice other design characteristics in order to select a durable which performs well at these higher output levels. The firm's response to a widening of the range, holding the mean constant, can now be derived. This requires differentiating the three reduced form equations with respect to d. Once again, unambiguous re- 82 SEI< This indicates that in response to increased output variability the sults are revealed for all three equations with gg-and %§-> O and 0. firm chooses to increase h and decrease 2. As a consequence, it is forced to accept an increase in c. In previous research the reduction in the E parameter was denoted as a movement towards greater flexibil- ity. In light of the above results 'it seems appropriate to describe more flexible durables as those which trade an increase in c for changes in both 2 and h. Thus, flexibility, or the ability to produce effi- ciently over a wider range, is simply seen as the inverse of a special- ization ranking, with lower c values indicating greater specialization. The advantage of this definition is that it captures the influence that the location of the average cost curve has on the ability of the firm to produce efficiently for different ranges of output. This result represents an important break with the existing litera- ture. Flexibility has frequently been contrasted with specialization, but it has generally been defined independently of specialization. This model suggests that it may be more appropriate to define flexibility in terms of an inverse ordering of the specialization index. Using this 76 new definition precise predictions of the direction of changes in the optimal durable input can be derived. The Relationship Between Flexibility and Risk Aversion A subject which has received little analytical consideration is the relationship between the risk attitude of the decision maker and the flexibility ranking of the durable input which is preferred. This ques- tion can be addressed by assuming that the decision maker's utility function is represented by a negative exponential of the form: EU = I-eA (TC) g(x) dx (4.10) An increase in total costs reduces the firm's expected utility. The risk preferences are denoted here by X which is the average risk aver- sion coefficient. As I increases, the firm becomes more risk averse. It is now possible to determine the changes in h, c, and 2 as X is increased. Due to the complexity of the problem, it only proves pos- sible to vary the design parameters in pairs holding the third parameter constant. The first step is to use the constraint equation (4.8) to substitute for one of the design parameters in equation (4.10). Next the first order condition is calculated for one of the remaining two parameters. The first order condition is then totally differentiated with respect to the selected design parameter and A while the third design parameter is held constant. In effect, this permits a calcula— tion of the partial derivative of the selected parameter with respect to X. By implication the sign of the partial derivative of the parameter substituted for in the first stage of this process is given as well, since the two changes must offset each other. 77 This process is detailed for the following example. Beginning with equation (4.10) and substituting for c results in: 2 Max EU(£,h) - fe)‘[(h'1'k)X+£x(x'h) Jg(x) dx Taking the derivative with respect to h results in the following first order condition: dEUdfi’h = f X [x- 211x (x-h)] ech] g(x) dx= O Fixing K and totally differentiating with respect to h and X yields: [.th - f {[x-2£x(x-h)] emch [Xx-A2£x(x-h)]e)‘[TC]TC} g(x) dx dX The second derivative with respect to h,[-], is negative by as- sumption of the second order condition for a maximum. The next step is to calculate the sign of the term associated with dX. The first term in the brace is identical to the first order condition divided by X and, hence, it is equal to zero. Thus, all that is necessary is to sign the second term in the brace. This term is identical to the first order condition with the addition of a variable weighting factor in the form of the total cost function. The sign of this term can be calculated from the formula for the product of two random variables: E(xy) = ExEy + COV (xy) (4.11) Since the expected value of the first order condition is zero, Ex* Ey must be equal to zero. Thus, it is only necessary to sign the co- variance. The first term in the product will be positive for small values of x and negative for large ones. Conversely the total cost function always increases with x and thus the covariance between the two is negative. Taking into account the negative sign outside of the 78 dh brace the entire coefficient of dX is positive and, hence,-ax is posi- tive. The other parameter which has been left free to vary is c, so-%§ must be positive as well in order to satisfy the design constraint. This result can be interpreted to indicate that as the average risk aversion coefficient increases, a firm which is permitted to trade off the minimum height for the location of the average cost curve will chose to move the average cost curve to the right. In the terminology of this chapter this is a movement towards a less specialized durable. This turns out, however, to be the only unambiguous result of these paired examinations of potential trade-offs. As is detailed in Appendix 0, when the firm is restricted to design changes between h and L or 2 and c 'the direction of the change cannot be determined. The overall implication of these three paired comparisons is that no predictions can be made as to what type of durable would gain favor as the firm became more risk averse. I These results conflict with the intuitive notion that the 'steep- ness" design parameter, 2, should gain in preference as risk aversion increases. This intuition stems, perhaps, from thinking about the problem in terms of the graph in average cost space. In looking at a graph of alternative durables in total cost space it becomes clear why an increase in 2 has not been revealed to be a clearly risk averse strategy. Whereas a reduction in 2 reduces the variance of average cost it does not necessarily reduce the variance of the total cost and, thus, has no obvious advantage for the expected utility maximizer. This result supports the discussion in Chapter Two which indicates that flex- ibility should not be viewed as a characteristic of an initial decision which decision makers value as a direct function of their own risk 79 attitudes. Rather, flexibility is sought as a means of responding to current uncertainty in cases in which the firm expects that some of the initial uncertainty will be resolved before subsequent decisions are made. Summary This chapter examines whether or not the concept of use flexibility can aid decision makers in determining the appropriate durable for their circumstances. The models reviewed, when suitably restricted, agree with the intuitive idea that flexibility is an increasingly valued char- acteristic as the demands on the durable become more varied. The shortcoming of these models is that they do not direct adequate attention to the trade-offs required in order to achieve increased use flexibility. A model is developed which explicitly solves for the opti- mal durable under different required output demand conditions. This model allows the firm to vary important design parameters along a given isocost line. In the context of this particular model, flexibility is best defined as the inverse of specialization. For changes in the mean of the output requirement when the total range is constant, the firm's preferred durable is shown to be "centered" farther to the right. This results in a higher minimum average cost as well as a larger penalty associated with output levels away from the minimum. According to the definition used here, this is a more flexible durable since c, which in- dicates movement away from specialization, increases. For an increase in the range around a given mean, the new preferred durable was shown to have lower 2 and h values and once again a higher c value. The new curve was thus located to the right, was flatter, and higher up than the previous curve. 80 As a final topic, the relationship between the average risk aver- sion coefficient of the decision maker and the optimal values of the three design parameters is examined. Since, of the three paired compari- sons, only the possibility of trading off h for c yielded a determinate result, it remains unclear how the design of the durable would change as a result of an increase in the average risk aversion coefficient. Thus, flexibility, even in this restricted model, cannot be related back to risk preferences. All of these flexibility relationships were derived only for an exogenous demand model. In Chapter Five the analysis will also include an attempt to use this flexibility definition in a price variability model. CHAPTER 5 AN EXTENSION OF THE CONCEPT OF USE FLEXIBILITY The use flexibility models reviewed and developed in the previous chapter shared the basic assumption that all durable inputs are charac- terized by fixed temporal lifetimes. As a result, the analysis in Chapter Four was limited to a comparison of the characteristics of the schedule of variable input costs associated with alternative durable in- puts. This chapter extends the discussion of the use flexibility of durables through a relaxation of the fixed lifetime assumption. In- stead, the durables considered here are differentiated in terms of their ability to produce units of service at different per period rates. Thus, the focus of this chapter is the derivation and comparison of the vari- able costs associated with the durable's own loss of productive capacity. In more familiar terms. this chapter examines the process of selecting the optimal durable input from a set of durables which differ in terms of their schedule of physical depreciation. This chapter represents an important addition to the use flexibil- ity literature because variable cost durables are more widespread in real world applications than are fixed lifetime durables. Models based upon the fixed lifetime hypothesis ignore important cost elements and thus (k) not perform well in evaluating the relative merit of competing durables for specific demand and/or risk preference situations. 81 82 The importance of variable use costs associated with the extrac- tion of durable services is recognized in the literature. Baquet (1978) examined the earlier work of Neal (1942) and Lewis (1949) and proposes a marginal user cost definition which consists of three parts: 1. The marginal cost of acquiring nondurable inputs used in the production of services. 2. The difference between the ending salvage value with and without use in a specific period. 3. The marginal opportunity cost suffered as a result of producing in later rather than earlier periods. Baquet compares the marginal user cost to the marginal benefits gained from the use of the durable's services and thereby internalizes the process of selecting the optimal durable service extraction rate. Robison (1982) motivated the present work with his study of the costs and benefits associated with durable use. Many of his cost categories are adopted (although often in modified form) in this chapter. Robison carried his analysis one step beyond Baquet's work on optimal service extraction rates to look at the problem of optimally designing a durable in order to meet a distribution of service demands or market prices. This chapter continues the thrust of Robison's work by examining, for a specific design constraint, the usefulness of the concept of flexibility in assessing and/or predicting which durable will be optimal. As in previous chapters, the focus on flexibility forces a consideration of the trade-offs which the decision maker must weigh in order to best achieve the firm's goals. The chapter is organized in four sections. The first section dis— cusses the important cost issues in analyzing this class of durable 83 inputs. This is followed by a section which derives the same basic optimality results as were derived in Chapter Four. The analysis is extended in the third section by allowing the output price rather than quantity to vary. This last model requires a more in-depth considera- tion of the cost issues raised in the earlier section. A final section summarizes the main points and conclusions of the chapter. Some Cost and Benefit Issues in the Use of Durable Inputs Many, if not most, production processes require the use of both variable and fixed inputs. While the procedures to evaluate the eco- nomic benefits and costs of the variable inputs are well known, the calculation of these same measures with respect to the fixed or durable inputs poses a somewhat greater problem. The consideration of two examples will illustrate the difficulties of establishing a single evaluation method. Consider first a hay stor- age process which requires the combination of variable inputs, bales of hay and labor, with a fixed input such as a barn in order to produce an output defined as x units of storage services per period. This will be contrasted with a transportation production process which requires the variable inputs gasoline and labor to be used in conjunction with an automobile, a durable input, in order to produce units of transportation services per period. In both cases, the calculation of the gross benefits earned simply requires multiplying the total quantity of output produced times the per unit market price. For convenience, the assumption is made that the output price is constant. Consequently, the benefits derived in a given period are completely divisible in the sense that each additional bale of hay stored or mile traveled can be assigned a value. 4 84 The cost side poses a greater analytical challenge. The costs as- sociated with the variable inputs are easily accounted for by simply keeping track of how many units of these inputs disappear in a given production period and valuing these units at the appropriate market price or opportunity cost. The relationship between the units of output produced and the variable input costs expended trace out the familiar cost functions. An analysis of the cost of the durable inputs consumed in a par- ticular period is not as easily accomplished. These costs can be divided into two categories: those that are related to the durable's physical capacity to provide production services, and those that are calculated as the opportunity costs of tying up the firm's resources. Since the main focus is on the former category, the latter category of costs will be considered first. As we discussed in earlier chapters, the maintenance of a stock of durable services results in a holding or inventory cost which must be charged to the durable whether the firm purchases the input using bor- rowed or debt capital. While inventory costs were previously treated as fixed costs, this assumption does not hold for durables with variable capacity use rates. A profit maximizing firm can respond to variations in the opportunity cost of capital by modifying durable use decisions, thereby altering the level and expense of carrying inventories into future periods. The holding of inventories of durables may cause the firm to suffer additional costs if changes in market conditions over time cause a de- valuation of the durable's resale value for reasons other than the direct 85 loss in the physical capacity of the durable to provide services. Fol- lowing Robison (1982) these losses are termed "time depreciation costsfl Replacement opportunity costs represent a final category of mone-. tary costs to the firm. This cost equals the opportunities which the firm loses by failing to replace an existing durable with a new and/or improved one. These last two cost categories are of limited interest in the cur- rent work due to the assumption that the acquisition of the durable is a "once and for all decision" and, hence, the firm cannot resell an owned durable. However, both are very important in the discussion of adjust- ment flexibility issues. For example, a farmer considering purchase of either a gasoline or a diesel powered tractor must consider the effect that possible future changes in the relative price of the two fuels will have on both production costs and resale value of the tractor. Purchase of a diesel powered tractor and a subsequent increase in diesel prices relative to gasoline prices would result not only in increased direct production costs, but also losses in the resale market due to the effect of time depreciation costs. The effect of replacement opportunity costs are best illustrated with respect to durables, such as personal computers, which are under- going rapid improvement. Because of the speed at which the technology is developing, a farmer may wish to place restrictions on the level of investment in an initial system under the expectation that it will soon need to be replaced by a more advanced model. The second major cost category consists of those costs which result in losses in the physical capacity of the durable to produce services. One subcategory of these losses, "capacity time costs," is associated 86 with the passage of time alone. Transportation services from tires, for example, decline even without use. Roofs on barns, exterior paint, and the human body are other examples of durables whose service capacity is reduced by time.1 Similar to inventory costs, capacity time costs need not be strict- ly fixed. For example, if capacity time costs each period represent a percentage of units held in capacity at the beginning of the period, then the decision to increase current period service extraction can de- crease future period capacity time costs. For simplicity's sake, it will be assumed here that capacity time costs are on a per period basis rather than a percentage basis and, hence, need not be included in the modeling which follows. The category of greatest interest to the present research is direct user costs which resemble most closely the costs attributed to variable inputs. The two models which follow examine the design of durables to optimize physical loss of capacity function. Types of Durables Before introducing these models it is useful to examine the types of durables which exist in the real world and the manner in which they can be represented in economic models. The most common approach to modeling durables has been to assume that they have fixed temporal life- times. This assumption has several important implications for the pro- duction model derived. First, it means that the durable use cost can be simply regarded as a fixed charge for each period. Beyond that it 1These were the only physical capacity losses considered in Chapter Four. 87 implies that the marginal cost of extracting an additional unit of ser- vices in any given time period is zero. This further implies that cur- rent production does not influence future production possibilities. The storage barn cited above is one of the few durables which is fairly well depicted by assumption of a fixed temporal lifetime. To a large extent, the barn loses its capacity to produce storage services at the same rate whether it is empty or filled to any point below its ca- pacity. Since the marginal cost of durable use is near zero, the deci- sion maker would only consider the relationship between the marginal benefits and marginal costs of the variable inputs when setting the optimal production level. In making long-run acquisition decisions, the firm might have the choice between durables with different inflexible per period costs. In Chapter Four, the simplifying assumption that the purchase price and number of periods of service are the same for all members of this class of durables was made. Thus, attention was direct- ed solely to the cost schedules of the associated variable inputs. While this is a convenient view of the world, a more realistic per- spective on this issue reveals that most durables do have positive mar- ginal costs which are associated with their use. This is quite obvious- ly recognized by automobile manufacturers who commonly provide warran- ties on their products which are valid for a given length of time or quantity of use. The calculation of durable costs which is implied by this second approach is only slightly more complicated than the previous one. Under the fixed service lifetime assumption, durable costs can be calculated on a per unit of service extracted basis which each addi- tional unit of service produced viewed as having a constant marginal 88 2 These durables are termed "completely flexible" durables since cost. if there are zero time costs, they will always produce the same total quantity of services over their lifetime regardless of the individual per period production rates. This second approach appears to be a far superior means of model- ing the costs associated with a durable such as an automobile which clearly is influenced by the quantity of services extracted. The essen- tial distinction between the two types of durables and, hence, the motivation for deriving different analytical approaches for the calcula- tion of durable use costs can be documented through reference to the market for used durables. Whereas a prospective used barn purchaser may not be interested in whether the barn had stored two or four thousand bales of hay in the past, a potential used car buyer will be quite con- cerned with the odometer readings of the cars examined. The difference between these two cases is the degree to which the past quantity of use affects the remaining capacity to provide services. Although the second approach represents an improvement over the zero marginal cost assumption, the question arises whether further im- provements are possible. Since the marginal costs of variable inputs are rarely regarded as constant, it is unclear why the use costs as- sociated with durables should be expected to be different. The argument developed in this chapter is based on the contention that durable inputs generally are characterized by variable marginal costs. While this third approach is analytically more complex, it also addresses a number of key issues not treated if the other assumptions are used. 2This is only partially true since there are intertemporal cost con- siderations (termed here "indirect" costs) which influence the actual marginal costs. 89 The assertion that durable costs are influenced by factors beyond the simple lifetime quantity of units of service extracted is supported by referring back to the used car example and noting that not only did the proverbial "little old man or lady" who supplied so many cars to so many used car dealers drive sparingly, he/she also extracted those ser- vices gently. Somehow this quality of driving gently must be formalized. Use Flexibility in an Exogenous Demand Model Although there are a variety of factors which influence the level of marginal (and hence average) costs, the focus of this chapter is limited to examination of the influence of per period use rates on production costs of the firm. Other issues will be raised and reserved for further research. As in Chapter Four, the goal of the firm is to select or design the durable which best permits the achievement of the firm's goals. The objective of the research is to trace the relationship between the flex- ibility characteristics of the optimal durable and the changing nature of demand conditions or the risk preferences of the firm. The models derived are based upon a production process which trans- forms a single durable characterized by variable marginal capacity losses and a vector of "fixed" durable inputs into a single output. The production relationship for a single period can be described as: s = f(x|XF) (5.1) where: s = the output of the production process (the units of service produced per period); x = the flow measure of the input (the units of durable capacity used up in a production period); 90 XF = the inflexible durable inputs which are part of the production process. Note that the absence of variable inputs eliminates the first element in Baquet's user cost formula. The treatment of the units of durable service, 5, as the final out- put with a per unit market price avoids the need to include an addi- tional transformation function. This greatly facilitates the analysis and does not change the main conclusions derived. The modeling of the benefits obtained from a durable is quite ob- viously a simplification since the services extracted from most durables have multiple dimensions. For instance, the services derived from an automobile can be measured in terms of the distance covered, the weight transported, and the comfort provided. Here it will be assumed that all of these various benefits have been reduced to a single common denomina- tor. For an automobile, that might be measured in miles driven. All units of service, 5, extracted in a single period will have the same price, ps. The gross benefit or revenue received by the firm in a single period is: TR = Ps*s (5.2) Whereas the objective of the firm facing exogenous and required demands in Chapter Four was to minimize the sum of the variable input costs, here the objective of the firm will be to purchase the durable which will produce the greatest quantity of services (an output measure) over its lifetime. In order to achieve a better understanding of these new concepts associated with durable use~ it is instructive to first examine how they apply to the two classes of durables which have pre- viously been defined as fixed lifetime and completely flexible durables. 91 Unlike variable input production and cost schedules which are reported on a period-by-period basis, the production and costs associated with durable use are calculated through a process of long-term experimenta- tion on a lifetime basis.3 Figure 5.1 represents a "lifetime output function" which differs from a production function in that the horizon- tal axis is not an input measure. The vertical axis in this figure represents the total quantity of services derived from the durable, re- ferred to as the total services extracted or TSE. In this model this is an output measure. The horizontal axis indicates the per period service extraction rate, SER, at which the durable is being operated. The units of the horizontal axis are thus 5 or equivalently f(x). The individual points on the graph are traced out by assuming that over time all ser— vices are extracted at a single continuous rate. Thus, when operated at an SER of sj, both durables in Figure 5.1 produce a TSE of C. When subjected to this process of experimentation, all fixed life- time durables trace out curves such as 0A. Since inflexible durables lose their service producing capabilities only as a function of the passage of time, their TSE curves are simply lines out of the origin with each having a slope equal to the number of periods that the specif- ic durable will provide services. The maximum TSE for an inflexible durable will always occur at the maximum per period SER. In terms of the earlier example, the barn's maximum production of storage services would occur when it is filled to capacity each period. In contrast to the durable AB, the completely flexible durable CD is characterized by zero time costs and constant direct user costs. The 3Lawless (1982) presents examples of experiments of this nature carried out by engineers. 92 TSE O SER Figure 5.1 TSE Curves for "Fixed Lifetime" and "Completely Flexible" Durables 93 absence of time costs implies that this and other completely flexible durables will never wear out from the passage of time alone. The con- stant marginal use cost ensures that the durable CD will always provide the same quantity of services over its lifetime irrespective of what SER or combination of SERs is chosen. Thus, the TSE curve for this durable is a horizontal line which stops just short of intersecting the vertical axis. Other completely flexible durables would also trace out horizon- tal TSE curves at different heights. An equivalent, and sometimes useful, means of observing or assess- ing a durable's capacity to provide services at different extraction rates is to simply record the number of periods of service that are pro- vided at specific SERs until the durable wears out. In Figure 5.2 the durables specified above are shown on a graph in which the vertical axis is defined at T, or the number of periods of operation. T can be cal- culated as TSE : SER. In this space the curve 0A is a horizontal line since T is a constant while the curve CD is a rectangular hyperbola since the product 1 *SER is a constant for completely flexible durables. The durables of interest in this chapter are neither as insensitive to use conditions as durable DA or as flexible as durable CD. They possess both positive time and marginal use costs. As a result, each of these durables will attain a maximum TSE at a unique SER. That max- imum TSE is designated as the lifetime capacity, LTC, of the durable and will prove important in calculating the opportunity cost of operating the durable at rates which result in smaller TSEs. The particular SER which yields the LTC is given a special name, "the efficient service extraction rate" or ESER. As a class, these durables will be termed "flexible" durables. 94 SER Figure 5.2 Time of Operation for "Fixed Lifetime" (and "Completely Flexible" Durables 95 As in the previous chapter, the trade-offs of concern to econo- mists and firm decision makers are made concrete by assuming that the engineers have identified the key design parameters which influence the performance of the durables in different situations. The job of the economist is to optimally adjust these design parameters within a cost constraint (i.e., along an isocost curve). To keep the analysis in this chapter as similar as possible to the discussion in Chapter Four, the family of durables under consideration is defined as:4 TSE = c -£(s-h)2 (5.3) where: c,£,h,s>0. In order for the above equation to make sense, it must be con- strained as follows: c =£h2 (5.4) This forces all of the curves to start at the origin and thus makes the necessary provision that at a constant SER of zero the TSE will be zero as well. The inclusion of the constraint in equation (5.4) and a cost con- straint which will be specified later restrict the permitted design trade-offs to a greater degree than was done in the previous chapter. An alternative form of equation (5.3) which provides a far greater de- gree of free movement was also examined: 4For these curves, c represents the maximum height, h represents the point on the horizontal axis around which each curve is symetric and 2 is the parameter which influences the steepness of the slope. 96 TSE = c [1 - (31:2) 2"1 h 9%) S.t. g)'(n) > 0, S'én) < 1 (5.5) c,h, n,s>0 Although this formulation yielded trade-offs similar to those found in the previous chapter, it was ultimately rejected because it proved ana- lytically impossible to solve for the optimal design parameters in specific use situations. This illustrates the necessity of sacrificing elegance in formulation for the possibility of achieving a solution. Figure 5.3 presents a comparison of two flexible durables. While the two share certain general features, they differ in other respects. All durables in this class can provide services in a range from O to Zhi' Thus, once a specific h value is set for a distribution, the capability of a particular durable to provide services at all of the required SERs can be determined. All flexible durables achieve their maximum LTCs and ESERs at the point where Si hi' From equation (5.3) it is evident that the LTC is calculated to be equal to c. The final and most im- portant characteristic that the two durables share is that, by assump- tion, the area underneath each of the curves is equal. The actual area constraint, A, can be calculated by integrating equation (5.3) over the range from O-to 2h which yields: A=%ch (am By a further assumption, the acquisition price of the durable (k) is set as a function of A: 5Dr. Lee Sonneborn provided this equation. 97 TSE l I I 1 1A h h* 2h* Figure 5.3 TSE Curves for Two Flexible Durables 98 k = f(A); k' > 0 (5.7) Since, as in Chapter Four, only durables with a common acquisition price will be considered, the firm can be seen as maximizing along a single isocost line. In summary, the assumptions of the basic model are: 1. The firm wishes to maximize expected benefits over the life of the durables which is equivalent to maximizing the total quantity of s extracted. 2. The service requirements are known with certainty in ad- vance, must be met, and are in the form of a uniform distribution. 3. The total service extraction functions are of the form derived in (5.3) subject to the constraints in (5.4) and (5.5). 4. All production periods are of the same length. 5. Only a single durable may be selected and no resale is possible. In order to solve for the optimal values of the design parameters for a specific distribution of output demands, equation (5.3) is first integrated over 5. Next, the first order conditions are taken and finally the individual parameters are solved for directly. When the three design parameters are optimally selected the largest lifetime TSE for a particular distribution is achieved. Carrying out the integration with respect to s over a uniform distribution with a mean of p and a range 2d yields an equation which can be solved for the optimal design parameters. The optimal values are: 99 d2+3u2 4n h = (5.8a) 3A 4n 3A - 4(d2‘,3112)3 (5 8C) u N I It is useful to note that if d is set equal to p, creating a uni- form distribution from O to 2d, the h parameter for the optimal durable will also be equal to p. In Chapter Four the average cost functions were found to be "off center" due to the effect of a weighting factor. Here, however, the boundary conditions ensure that the optimal durable will be the durable which is centered at h. Durables centered to the left of h will be unable to meet some of the service demands and, thus, are excluded, while durables centered to the right of h will yield a lower average TSE over the range 0 to 2d. If the three reduced form equations are differentiated with respect to u, the mean, it is possible to determine how the optimal design para- meters will change in response to an increase in the mean of the service requirements. These results are: at an gt Bu gt an The firm responds to a shift of the distribution to the right, yielding the same length of range, but a higher mean, by selecting a durable with a smaller LTC (a smaller c value), a larger ESER (a larger h value), and 100 a less steeply sloped curve (a smaller 2). In the previous chapter the value of c is established as the ranking mechanism for a specialization/ flexibility scale. The model clearly predicts that an increase in the mean required output will result in a demand for a more flexible durable and not merely the exchange of equally specialized durables over dif- ferent ranges. This occurs because a durable of equal specialization would be beyond the budget constraint over the new higher range. As an example, a taxi company will switch towards a vehicle which has superior characteristics at higher speeds as the range of service demands moves towards more highway driving. In order to attain these new characteristics, the firm must sacrifice its previous level of ef- ficiency at lower service extraction rates. Differentiating with respect to d measures the effect of spreading the output distribution. The partial derivatives which are calculated are identical to those reported above. Thus, as the distribution spreads, the relatively more flexible durables along the isocost fron- tier will gain in preference. This indicates that the "hybrid" car of the previous chapter gains favor over a specialized vehicle as the range of service demands around a given mean widens. In sum, the shape and location of the TSE curves for the optimal durable can be calculated as a function of the required use distribu- tions. As these use distributions are altered, the changes in the de- sign parameters which define the TSE curve can also be calculated. Under conditions of ex ggtg_uncertainty (i.e., uncertainty at the time of the acquisition decision) this model can be used to explore the relationship between flexibility and risk preferences. As in Chapter 101 Four, the utility function of the firm is assumed to be a negative exponential function. The utility function is:6 -X[c-£(s-h)2] EU = f - e g(s)ds (5.9) After making the necessary substitutions, this is rewritten as: 3A 3A 2 EU = f - e ")‘[(4h)' (21’ H3) (S'h) ]g(s)ds (5.10) Taking the first order condition with respect to h yields: , f «((52%) - (144%) (Caz—(gag) (s-hu (5.11) [-e 4['11 g(S)ds = 0 gg-is positive. This and the restrictions of the model ensure that g; and %§-are both negative. Thus, previously ambiguous relationships have been successfully signed and, as the firm becomes more risk averse, it will select a flatter TSE curve centered farther to the right. In view of the constraints imposed on the model, however, these results should not be overemphasized. Use Flexibility in an Exogenous Price Model If the output price, rather than the output quantity, is exogenous, the firm will be able to maximize its profits through the enforcement of marginal conditions on a period-by-period basis. The results obtained above must be manipulated further to determine costs which can then be compared to the benefits which the firm may receive. A simplified ver- sion of these marginal costs is calculated by employing a number of pre- liminary assumptions. First, it will be assumed that firms do not have the opportunity to sell or replace an owned durable before it is 6Since the exponent is a benefit rather than a cost, there is a change in sign from the previous chapter. - 102 depreciated through use to a capacity of zero service units. There- fore, replacement opportunity costs and time depreciation costs may be ignored. Time capacity costs need not be considered because they are assumed to be fixed for each period. However, holding or inventory costs will influence the per period use decision. Specifically, as the holding costs increase, the firm will seek to increase its current use of the durable.7 Since the costs are calculated as a fixed percentage of the value of units held in storage, they will shift all cost curves by an equal amount and thus will not influence the choice between durables. This leaves the consideration of the direct user costs. A normal cost curve graphs monetary costs versus different output levels. Direct user costs must therefore be calculated in monetary terms on an in- dividual period basis. The derivation of a monetary cost level for a given quantity of production services in a single period can be cal- culated for each durable as part of a two-step exercise. The first step requires the calculation of the physical cost of operating the durable. This physical loss of productive capacity can be calculated by dividing each service extraction rate, SER,, by its corresponding total services extracted, TSEi, at that rate. This calculation yields the percentage of the total service capacity which is used up in a single production period at each Si’ This will be termed the percentage total loss in capacity, %TLC. SER; = i (5.12) TSE; Ti %TLCi = 7This point was demonstrated in Chapter Three. Note that an in- crease in the discount rate will have the same effect. 103 Graphing the calculated %'HJ: against the allowable range of SERs yields the curve shown in Figure 5.4 which resembles a cost function. It differs from a normal cost function in that the vertical axis is in physical terms. Since only durables with equal acquisition prices are considered, the vertical axis can be directly transformed into an axis defined in monetary terms by multiplying each point on each curve by k, the acquisi- tion price. This yields a total cost curve defined in monetary terms: TCi = k * %TLC.i If durables with different acquisition prices were being consider- ed, the curves would shift in relative terms in moving from the physical cost measure to the monetary cost measure. If Tisdell's assumption that the output price never falls below the maximum of the minimum average cost level is once again enforced, it is a simple matter to show that more flexible durables will gain favor as the range of prices widens since Tisdell's proof (in Appendix C) still holds. By inspection of Figure 5.4. it is clear that durables with higher c values, which are less flexible, will have larger second deriv- atives of total cost. As in Chapter Four, however, if the output price falls below the maximum of the minimum average cost level, this relationship need not hold since the more specialized durable will be able to stay in produc- tion at lower output prices. Furthermore, if the price were to fall low enough so that both durables were forced to shut down for some produc- tion periods, the "fixed“ costs (which can be defined on a per period basis as l/TO) would be lower for the more specialized durable. Thus, once again, the notion of more flexible durables being able to handle 104 % TLC Figure 5.4 Physical "Cost" Curves for a Flexible Durable 105 more diverse demands is called into question when the rules of the game permit the firm to set its own production level. Similar problems arise when an attempt is made to derive a rela- tionship between the relative flexibility of preferred durables and the risk attitudes of the decision maker. Whereas in the quantity model a risk averse decision maker would be concerned with meeting required quantities at either end of the distribution, in the price model the decision maker only has low prices to fear since high prices always re- sult in high profits. Consequently, an extremely risk averse decision maker such as a "maxi-miner" would select a very specialized durable in order to protect against losses from relatively low prices. At the op- posite extreme, an extremely risk preferring decision maker would select a very flexible durable in order to be in a position to take advantage of high prices. In between these two extremes it is not clear how a marginal increase in the risk aversion of the decision maker would in- fluence the relative flexibility of the desired durable. Thus, although the risk attitudes of the decision maker continue to influence the degree of flexibility in the price model, the interaction between the two is no longer of a known direction. Summar This chapter builds upon previous work in the area of user cost and develops several simple models which examine how optimal durable design is a function of how they will be used and the risk attitudes of the owners. Once again, the process of maximizing profits within a design constraint proves to be a useful way of forcing the engineers, econo- mists, and decision makers to work together. 106 The first part of the chapter discusses the question of the divisi- bility of the costs and benefits associated with durable use. Whereas the benefits for durables are, in most instances, completely divisible, this is not always true for the costs. "Fixed lifetime" durables are identified as durables with no variable use costs. In contrast, the "completely flexible" durable has only variable and no fixed physical use costs. The durable considered in this chapter, the "flexible" dur- able,is characterized by both positive fixed and marginal use costs. The chapter also proposes using a process of experimentation to determine the physical costs associated with the use of flexible dur- able inputs. These new cost concepts are used to optimally solve a con- strained durable design problem for the case in which the firm faces exogenous and required outputs. Solution of the more complex exogenous price case requires intro- duction of additional cost concepts. Since the firm in this model is permitted to optimize on a period-by-period basis, the solution to the model requires that the direct user costs for the durable be calculated on a monetary basis for individual production periods. This is accom- plished through a process of two transformations. First, the lifetime total services extracted calculations are transformed to provide per period physical use costs. Next, a shadow price for a unit of capacity is calculated and used to transform the physical costs into monetary ones. The results of the analysis of the relationship between flexibility and increases in either the variability of demand or changes in risk attitudes are mixed. In examining changes in variability, unambiguous flexibility results are only obtained for the exogenous quantity model 107 and for the price model with additional restrictions placed on how low prices can fall. In the more general price case, more specialized dur- ables are shown to have an advantage at low prices. Consequently, these durables might gain in preference as the variation in prices increases. This calls into question the usefulness of a technical measure such as flexibility in predicting the direction of changes in economic models. The prediction problem is, of course, caused by the rules of the game which allow the firm to shut down in order to minimize costs. If the) firm were forced to operate at all price levels, the more flexible dur- able would unambiguously gain in preference as compared to a less flex- ible durable. This result is based upon Tisdell's proof presented in Appendix C. Finally, the relationship between risk attitudes and the relative amount of flexibility desired is examined. While in the exogenous quan- tity model the relative flexibility of the preferred durable (using the flexibility definition derived here) clearly increased as the firm be- came more risk averse, such is not the case in the price model; In the price model the only definitive results are that extremely risk averse decision makers would select very specialized (inflexible) durables while extremely risk preferring decision makers would select very flex- ible durables. CHAPTER 6 CONCLUSIONS Summary of Problem Setting and Background Decision makers in the real world must concern themselves with the long-run, as well as the immediate, consequences of their action choices. This research is motivated by the recognition that the static models commonly used by economists provide little insight into the nature of these complex, intertemporal problems. The primary goals of the research are disciplinary in nature and focus on the potential for integrating the concept of flexibility or "the ability to respond or conform to new or changing circumstances" into the mainstream of microeconomics. Because of the disciplinary focus, no specific problems or groups of decision makers are addressed. The lack of attention to flexibility issues within economics is highlighted by contrasting it to the well developed literature on flex- ibility in other disciplines. This literature can be divided between the qualitative treatments which contain explicit references to flex- ibility (Heady, 1952; Rosenhead, 1981) and the more formal development of solution techniques for sequential problems (Wald, 1947; Bellman, 1957; Cocks, 1968; Rae, 1971a, 1971b) which implicitly include flex- ibility concerns. The rigorous examination of flexibility within simple deduction models which is presented here is of a complementary nature to both of these existing orientations. Clearly, the work here remains 108 109 at a much more abstract level than is found in either of the two other literatures. Although the literature reviewed in the dissertation covers the broad spectrum of flexibility issues, the analytical work executed is restricted to the consideration of the role of flexibility in deriving optimal rules for the design, acquisition, and use of durable inputs. This limitation of the focus of the research permits the develOpment of more detailed case studies. In order to set the stage for the subsequent discussion, Chapter Two begins with the identification of those characteristics which dis- tinguish sequential decision problems from the more familiar static problems. Three types of linkages are defined for sequential models: "flexibility," "exogenous learning," and "resiliency." The first, flex- ibility, represents a relative measure of the range of options which are maintained for any initial action choice. Flexibility therefore should be viewed as a purely technical measure of the future opportunity set. Endogenous learning denotes the change in the decision maker's percep- tions of the probabilities of the outcomes associated with the different action choices available in future periods. Finally, resiliency refers to the linkage between the actual outcome of initial action choices and the overall production environment in subsequent periods. Within the flexibility category, two subcategories, use flexibility and adjustment flexibility, are identified. Use flexibility is defined as the firm's ability to vary the intensity or manner in which already controlled resources are put to use. The degree of use flexibility can therefore be measured as a "built-in" characteristic of these resources at the time that the acquisition decision (the initial action choice) is 110 made. In the literature, use flexibility has been examined through a comparison of alternative cost functions. This approach is used in Chapters Four and Five. In contrast, adjustment flexibility refers to the ease with which the firm can vary the resource under its control or its general plan of action. Two factors influence the adjustment flexibility which charac- terizes alternative action choices. The first, liquidity, is defined as the ease with which owned resources can be transformed back into money. The second, reversibility, measures the extent to which the physical production process which is initiated restricts the range of options available to the firm in future periods. The literature reviewed in Chapter Two is used to illustrate the key relationships which typify flexibility problems. The first of these is the recognition that the firm views flexibility as an intermediate rather than a final goal. The importance of this intermediate goal stems from the assistance that it can provide the firm in overcoming the twin problems of myopia (an overconcern with short-run outcomes) and tunnel vision (the selection too early in the decision process of a single final objective). The firm does not merely seek to maximize the flexibility it retains because there are costs as well as benefits as- sociated with its maintenance. The principal costs suffered by the firm are the opportunities foregone by not committing early on to a preferred course of action. The required complementarity between flexibility and learning rep- resents a second major theme in the literature. This point can be re- stated as indicating that the firm cannot expect to reap benefits from maintaining options if no additional information is gained, or if 111 greater knowledge of the situation is acquired, but the opportunity to act upon that new information does not exist. The analysis of the interaction between the firm's desire for flex- ibility and the degree of uncertainty inherent in the problem can be best understood as an outgrowth of this relationship between flexibility and learning. In general, the relative amount of flexibility desired by the firm increases as the amount of gx_ggte (or prior) uncertainty increases if the ex gggt uncertainty is held constant. In contrast, the attractiveness of more flexible initial action choices decreases as the amount of ex_pg§t uncertainty increases, holding ex_ggtg uncertainty constant, since the firm will be less capable of deriving benefits from the flexibility retained. These results further imply that flexibility should not be classi- fied as a strategy which parallels the risk reducing strategies which are utilized in static uncertainty problems. Instead, flexibility should be viewed as a means of maintaining options which can be employed equally well by either risk averse or risk preferring individuals. As a final topic, Chapter Two considers the merits of including the concept of resiliency within the general rubric of flexibility issues. The traditional flexibility literature begins with the assumption that the availability of options in future periods is known with certainty and that only the relative attractiveness of these options depend upon the stochastic states of the world. If, however, the future existence of the options themselves becomes stochastic, then the resiliency of alternative action choices becomes a relevant characteristic for assess- ing how the firm will be able to respond or conform to changing circum- stances. Although the inclusion of resiliency considerations introduces 112 yet further cemplications to the analysis of sequential decision prob- lems, it also highlights rather than hides the trade-offs which decision makers must weigh. Summary of the Analytic Results The complexity of sequential models and the difficulty of obtain- ing unambiguous results from all but the most restricted models are two of the prominent characteristics of the literature reviewed in Chapter Two. Chapters Three through Five focus exclusively on the development of deductive models specifically dealing with the use of durable inputs by the firm. Chapter Three examines the implications of introducing adjustment possibilities to a production process involving durable inputs. The model which is developed permits the firm to separate the durable in- vestment decision from the subsequent production decision. Specifi- cally, in this model, the ex ggte acquisition decision places an abso- lute bound on the e; EQSE use decision. This assumption is particularly well suited to models characterized by large transaction costs. In such situations, reorders are not viable because they are too costly. Addi- tionally, this model can be applied to physical processes such as crop production which cannot be greatly increased after the planting season is past. Given these conditions, the addition of flexibility to the firm's production plan, via separation of the two decisions, is dependent upon the cost of maintaining an inventory. If the inventory is excessive, the stocking of inputs will not be a viable economic option and no flex- ibility will be gained. However, if storage costs are less than the 113 amount the firm would lose if inputs were forced into production in spite of a low output price, the separation of the acquisition and re- stricted use decision increases the firm's ability to respond to un- certainty. In this model, as well as in several others reviewed, this takes the form of acquiring greater levels of the inputs than would be used on average if the firm lacked flexibility. The average amount of inputs used and outputs produced under these conditions share an ambiguous relationship to the average input and out- put levels for firms facing uncertainty and lacking flexibility. This ambiguity remains even with the introduction of increasing risk aver- sion on the part of the decision makers. Chapters Four and Five examine the problem of optimal design of nonhomogeneous durable inputs for the production of homogeneous out- puts. Because these durables may be called upon to produce over a range of market conditions, a concern with flexibility influences the design process. Although both use and adjustment flexibility are relevant to the design and/or selection of the optimal durable, the focus of both chapters is restricted to use flexibility issues by the assumption that owned durables are fixed to the firm. Chapter Four begins with a review of the use flexibility literature. In keeping with past work, only durables with fixed temporal lifetimes and, hence, no variable use cost; are considered. Consequently, these durables are differentiated by the cost schedules of their associated variable inputs. Although this literature has produced success in de- riving a relationship between the optimal amount of use flexibility and variability of demand conditions for both exogenous output (Stigler) 114 and stochastic price models (Tisdell), these models are criticized for not adequately defining the choice set and, hence, the trade-offs in- herent in the durable design problem. As a consequence, these models can only be solved if additional assumptions are made. In order to focus on the trade-offs which the firm must weigh, a specific and restricted production model is formulated and analyzed in Chapter Four. Whereas the standard formulation implicitly'permits the firm to select among durables differentiated with respect to two char- acteristics of their cost functions, this model allows the firm to de- sign the optimal durable as a function of three parameters and an ex- plicit budget constraint. In the use flexibility literature the shape and the minimum height of the average cost curve are the only parameters specified. The location of the average cost curve is added in this chapter as a third factor which the firm may vary. The previous literature defines use flexibility as a function of the shape of the cost curve and assumes a trade-off with the minimum height or specialization of the durable. In the three-way trade-off considered in Chapter Four, results are improved when flexibility is redefined in terms of an inverse ranking of the degree of specializa- tion. This permits the firm to obtain greater flexibility and, hence, an improved ability to produce over a wider range, through changes in .gither the shape of location of the average cost curve. As a result, an isocost curve is traced out which shows the optimal combinations of these three parameters for different demand distributions. As a final topic, the relationship between the average risk aver- sion coefficient of the decision maker andthe optimal values of the three design parameters is examined. No determinate results are 115 derived, confirming the previous consensus that flexibility should not be viewed as a risk reducing strategy. Chapter Five presents models of durable inputs which do not fit within the fixed temporal lifetime assumption of Chapter Four. The class of "flexible" durables considered in Chapter Five are character- ized by positive marginal and fixed use costs. Whereas in the previous chapter comparisons between durables are made solely in terms of the schedules of variable input coSts, in Chapter Five the focus is on the variable costs associated with the loss of the durable's own productive Capacity. A considerable portion of the chapter is devoted to the considera- tion of how different types of durables lose their ability to provide services. For "flexible" durables, two broad categories of costs are identified. The first category, the monetary or opportunity costs, in- cludes the holding costs of tying up resources in stocks of durable capacity, the replacement opportunity cost of having an outmoded durable in place, and the time depreciation cost which is defined as changes in the salvage value of a durable resulting from factors other than use. The second category of physical use costs consists of the direct user costs associated with the per period rate of production and the capacity time cost which is solely a function of-the passage of time. The modeling in the chapter is restricted to consideration of direct user costs through the enforcement of a series of assumptions. The central operational problem which must be overcome is the determina- tion of the actual decline in the durable's productive capacity which results from a particular rate of service extraction. This difficulty 116 does not arise in variable input production models since the market value of the inputs used can always be calculated. The solution which is proposed is the organization of a process of long-run experimentation. Durables are run continuously at a set ser- vice extraction rate and the number of production periods as well as total number of units of service produced are recorded. From this ex- perimental data it is possible to work backwards to the calculation of single period use costs defined in either physical or monetary units. As in Chapter Four, flexibility issues are examined in the context of rigidly defined production models. In this chapter, the curves traced out by the experimental data are defined in terms of three design parameters and a budget constraint. A positive relationship is found between the firm's desire for flexibility and the variability of demands in an exogenous quantity model. This conforms with the results found in Chapter Four. Addi- tionally, the previously ambiguous relationship between risk preferences of the firm and desire for flexibility can be solved unambiguously in the model developed. However, the results are not given much weight since the model in this chapter is more heavily restricted than the model presented in Chapter Four. When a stochastic price model is considered, however, an unambigu- ous relationship between flexibility and the variability of demand can only be derived if the output price remains above the maximum of the minimum average costs. Furthermore, the only definitive results relat- ing risk aversion to the desired degree of flexibility for the price model reveal that extremely risk averse decision makers would prefer very specialized (inflexible) durables in order to avoid the most 117 negative outcomes while extremely risk preferring decision makers would select very flexible durables in order to maintain the potential to derive maximum benefits from favorable prices. Implications for Future Research Further research in the role of flexibility remains feasible. A likely next step would be to derive a durable acquisition model in which use and adjustment flexibility are allowed to vary simultaneously rather than individually. This subsequent modeling would have to continue to follow the pattern presented here, highlighting a restricted aspect of the overall problem and greatly simplifying all other aspects. The dissemination of the results of flexibility models Within the discipline will continue to be restricted by the difficulty of deriving unambiguous relationships. Simply stated, flexibility models remain too complex to be easily incorporated in introductory presentations of eco- nomic theory. The research presented here also has implications beyond its narrow disciplinary focus. Researchers from other disciplines who investigate flexibility issues in the context of qualitative or formal technical models may profit from the insights gained from these deductive models. Gains can be made through increased interaction between the abstract theorists who look very carefully at narrow issues and the applied re- searchers who tackle real world problems. An important advance in applied research can be made through the recognition of the limitations present when economists work with their comfortable set of Static tools when they step outside their own dis- cipline. Although a concern with flexibility does not, as of yet, 118 provide a complete set of tools, it does provide a degree of insight into what should be regarded as the important components of these com- plicated applied problems. Farming systems research represents but one fruitful area in which a sensitivity to flexibility issues provides a checklist of issues which the research team should consider.1 1This topic is examined in a forthcoming article by Lev and Campbell entitled "The Temporal Dimension in Farming Systems Research: The Impor- tance of Flexibility and Resiliency Under Conditions of Uncertainty." APPENDICES APPENDIX A Define ; as the input level x used in the production process f(x) which maximizes the ex ante model of expected profits E(n): em = ,{ [(p+e)f (x) - DXXI g(eldx (An) In the model above (p+e) and px are output and input prices, respec- tively, and e is a random variable with probability distribution g(e) with expected value 0. It can be easily shown that in such a model: f'(x) =35 or x = f"1(Ep)-(') (A°2) It is now shown that the inventory I acquired in the ex_ggte/§x BREE model is greater than or equal to i. The first order conditions for I in the ex ggtg/ex_gg§t model were given in equation (3.13); expected profit defined over the control vari- able I in that model was expressed in equation (3.14) while the first order condition for the choice of I was given in (3.15). Suppose I equals i or that f'(I) equals px/p. If this is true, we can substitute for the limits of integration and in the integrand of (3.15) to obtain: :2 .0 p 139: f" (-r)g(e)d+___£B :25 9(8)“ (A.3) -oo px In the first integral above it is clear that: E: > fl or -p_X_€- < '1” (A04) 119 120 Recall that the first order condition for i could be written as: 1531? = 7 [(p+e)f'_§ (A.7) APPENDIX B Consider a profit maximizing firm facing the following production conditions: The expected output price (p) is 35. A low output price (q]) of 30 occurs q or 50 percent of the time and a high output price (qZ) of 40 occurs (l-q) or 50 percent of the time. There is a storage cost (r) of l. The price of the input (px) is 10. The production function is y==xa. a is less than 1. A comparison of the expected input use, E(x), in the flexibility model minus the input use, x, in the ex ante uncertainty case can be written: E(x*) -3? or q 1 1 l/(l-a) p-qq] l/(l-a) “116 [Q(px-r ) '+(1'Q)(ll-q) px+qr (8.1) - (pl) ”("91 X Since (11%3: is positive when a < 1, this term can be ignored. The first two terms represent the flexibility input use and the last term represents the gx ante uncertainty input use. If the whole bracketed term is positive, the average input use is greater under flexibility 121 122 than under 25 pptp uncertainty (the converse being true if the term is negative). In order to get a comparison of outputs, it is necessary to sub- stitute the input levels into the production function y==xa. Once again, this permits a comparison between the average output in the flexibility model and the output in the ex gptp uncertainty model. Three examples follow in which a, the elasticity of output, is varied from .25 to .95. Example 1 Given a = .25, the input equation taken from (B.l) after ignoring the first term, is: (.5) (%gT)1/(1-.25),(.SHMUfl/(inzs) -%%I/(I"25) = 5.282 - 5.312 [negative] Result 1a: average input use is less in the flexibility model. To compare outputs, the input levels are substituted into the production function. The output difference equation is obtained by raising (8.1) to the a power. .25/(1-.25) .25/(1-.25) (.5) (3.333) +~(.5) (3.636) .5 .25/(1-.25) -3 = 1.516 - 1.518 [negative] Result lb: average production is also less under flexibility. Example 2 Given a = .85, the input equation is: (.5) (3.333) '85/(I"85) +(.5) (3.636) '85’("°85) - (3.5)1/(I'°85)= 4.263 - 4.239 [positive] 123 Result 2a: at this a level, greater average input use occurs in the flexibility model as compared to the ex_ante uncertainty model. For the production levels substituting again: (.5) (3.333) “BS/(1"85)+ (.5) (3.636) '85/(I'-85I .85/(1-.85) -(3.5) = 1210.755 - 1211.2495 [negative] Result 2b: the average output produced under flexibility is less than production in the E! ante uncertainty model. Example 3 Given a = .95, the input equation is: (.5) (3.333) ”(1"95) + (,5) (3.535) 1/(I'-95) - (3.5)I/II'°95I = 9.587E10-7.61OE10 [positive] Result 3a: there is greater average input use in the flexibility model. The output equation is: .95/(1-.05) .95/(1-.05) (.5)(3.333) i-(.5)(3.636) - (3.5) '95/(“°°5) = 26.724E9 - 21.74259 [positive] Result 3b: there is greater average production for the flexibility model. The results of these three examples can be summarized as follows: Average Average .g Input Use Production Example 1 .25 F* < _e_x_ _a_n_t_e__** F gpflg F<_e_x_an£ Example 3 .9 F > ex_a_n_te F>§§flljg *F is the flexibility model. *ffixpgpte is the ex_ante uncertainty model. APPENDIX C Tisdell compares two production techniques, indicated by the sub- scripts 1 and 2, and finds that the maximum profit function for each technique is constant up to the level of production at which variable average cost is at a minimum and then increases at an increasing rate. The maximum profit function for technique 1 is: w1(p) = 991(91- C (91011) for p 3. min AVC (C.l) w1 (p) =-k1 for p < min AVC The first and second order conditions are: '( ) - _,I__ f ' AV c 2 wlp-%u) wp>mm C (-I W" = n > 0 f . o 1 (p) C‘(x) or p > min AVC (C 3) The difference between the profit earned using the two techniques can be defined by an excess maximum profit function, W(p): “(19) = w1(p) - wz (p) ((3.4) For prices above the minimum of the maximum average variable cost, W" can be defined as: W" (p) = w; - w; or (C.5) 1 _ 1 C; (X) C2“) 124 125 Since by the second order conditions for maximization, C?(x), 02 > 0: > W(p) < 0 accordingly as: (C.6) I! < n C] (X) > C2 (X) Therefore, under conditions of certainty, as price instability in- creases, the profit associated with the technique with the smaller second derivative of costs will increase at a more rapid rate and, hence, will gain favor. : APPENDIX D This appendix continues the examination begun in the text of the effect that a change in the average risk aversion coefficient, X, has on pairs of the optimal design parameters. Due to the complexity of the problem, only two of the design parameters are allowed to vary at a time. The discussion here will be greatly shortened by presenting only the equation which is derived as the total differential of a first order condition of equation (4.10). The equations presented here for the two remaining pairs of design parameters are the equivalent of the third equation on page 77. In examining the trade-off between h and 2, the following equation is calculated: [-] dh - f {[(x-h)2 - 2 (h-c-k) (x-h)] eAUC] (D 1) + X[(x-h)2 - 2 (h-c-k) (x-h)] em” TC} g(x) dx dX Using the same approach as in Chapter Four it becomes clear that the sign of the term associated with dX cannot be determined since the covariance of the two final terms in the brace is ambiguous. This is also true for the pair of C and 2. That equation is: [.] dc - f {[x-x (x-h)2] e)‘[TC:| + X [x-x (x-h)2] (0.2) e)‘[Tc] [TC]} g(x) dx d). Since two out of the three pairs in the paired comparisons are ambiguous, it is not possible to determine the sign of the change 126 127 for any of the parameters if all three are allowed to vary simultaneously. APPENDIX E In light of the solutions worked out in Appendix D, the solution here can be considerably simplified. As before, the relationship be- tween X, the average risk aversion, and the design parameters is sought. All the component parts of the total differential equation are the same, so once again, the deciding factor turns out to be the covariance be- tween the expression in (5.11) and the exponent in (5.10). Since the exponent in (5.10) is a TSE, its first derivative is positive and then negative. The expression in (5.11), when all the necessary sign changes are accounted, has the same sign. Hence, the term associated with dX must be positive and gg-must be positive as well. The other relation-' ships follow from the constraints imposed on this model. 128 BIBLIOGRAPHY BIBLIOGRAPHY Anderson, Jock R., John L. Dillon and J. Brian Hardaker. 1977. Agri- cultural Decision AnaLysis. Ames, Iowa: Iowa State University Press. Arrow, Kenneth J. and Anthony Fisher. 1974. "Environmental Preserva- tion, Uncertainty, and Irreversibility." Quarterly Journal of Economics, 88:312-319. Baker, Timothy G. and Bruce A. McCarl. 1982. "Representing Farm Re- source Availability Over Time in Linear Programs: A Case Study." North Central Journal of Agricultural Economics, 4:59-67. Baquet, Alan. 1978. "A Theory of Investment and Disinvestment Includ- ing Optimal Lives, Maintenance, and Usage Rates for Durables.“ Unpublished Ph.D. dissertation, Michigan State University. Baumol, William J. 1959. Economic Dynamics. New York: MacMillan Company. Bellman, R. 1957. Dynamic Programming. Princeton, New Jersey: Princeton University Press. Bergman, Lars and Karl-Goran Maler. 1981. "The Efficiency-Flexibility Trade-Off and the Cost of Unexpected Oil Price Increases." Scandinavian Journal of Economics, 83:253-268. Binswanger, H.P. 1980. "Attitudes Toward Risk: Experimental Measures in Rural India." American Journal of Agricultural Economics, 62: 395-407. Bradford, Lawrence and Glenn L. Johnson. 1953. Farm Management Analy- §j§, New York: John Wiley and Sons, Inc. Brink, Lars and Bruce McCarl. 1979. "The Adequacy of a Crop Planning Model for Determining Income, Income Change, and Crop Mix." Canadian Journal of Agricultural Economics, 27:13-75. Cocks, K.D. 1968. "Discrete Stochastic Programming." Management Science, 15:72-79. Crawford, Eric. 1982. A Simulation Study of the Growth of Small Farms in Northern Nigeria. MSU International Development Paper No. 2, Department of Agricultural Economics, Michigan State University, East Lansing, Michigan. 129 130 Cukierman, Alex. 1980. "The Effects of Uncertainty on Investment Under Risk Neutrality With Endogenous Information." Journal of Political Economy, 88:462-475. Day, Richard H., Dennis J. Aigner and Kenneth R. Smith. 1971. "Safety Margins and Profit Maximization in the Theory of the Firm." Journal of Political Economy, 79:1293-1300. Edwards, Clark. 1959. "Resource Fixity and Farm Organization." Journal of Farm Economics, 42:755-766. Epstein, Larry. 1978. "Production Flexibility and the Behavior of the Competitive Firm Under Price Flexibilityfl' Review of Economic Studies, 45:251-262. Epstein, Larry G. 1980. "Decision Making and the Temporal Resolution of Uncertainty." International Economic Review, 21:269-283. Friedman, Milton. 1962. Price Theory: A Provisional Text. Chicago: Aldine Publishing Company. Gilbert, Richard J., David M.G. Newberry and Joseph E. Stiglitz. 1978. “An Overview of the Economic Theory of Uncertainty and Its Implica- tions for Energy Supply." Electric Power Research Institute Special Report FA-586-SR, Palo Alto, California. Golbe, Devra. 1981. "The Effects of Imminent Bankruptcy on Stockholder Risk Preferences and Behavior." Bell Journal of Economics, 12: 321-328. Goldman, Steven. 1974. "Flexibility and the Demand for Money." Journal of Economic Theory, 9:203-222. Goldman, Steven. 1978. "Portfolio Choice and Flexibility: The Pre- cautionary Motive." Journal of Monetary Economics, 4:263-279. Gould, John P. 1974. "Risk Stochastic Preference and the Value of In- formation." Journal of Economic Theory, 8:64-84. Greenbaum, Stuart. 1971. ”Liquidity and Reversibility." Southern Economic Journal, 38:83-85. Hadar, J. and W.E. Russell. 1969. "Rules for Ordering Uncertain Pros- pects." American Economic Review, 59:25-34. Hanoch, G. and H. Levy. 1969. "Efficiency Analysis of Choices Involv- ing Risk." Review of Economic Studies, 36:335-345. Harris, Marvin. 1978. Cows, Pigs, Wars, and Witches. New York: Vintage Books. Hart, Albert Gailord. 1940. "Anticipations, Uncertainty, and Dynamic Planning." Studies in Business Administration, Vol. XI, Chicago: University of Chicago Press. 131 Hart, Albert Gailord. 1942. "Risk, Uncertainty, and the Unprofitabil- ity of Compounding Probabilities." Studies in Mathematical Eco- nomics and Econometrics, O. Lange, F. McIntyre and F. Yntema, editors, Chicago: University of Chicago Press, pp. 110-118. Hartman, Richard. 1976. "Factor Demand With Output Price Uncertainty." American Economic Review, 66:675-681. Heiner, Ronald. 1983. "The Origin of Predictable Behavior." American Economic Review, 73:560-595. Henry, Claude. 1974. "Investment Decisions Under Uncertainty: The Irreversibility Effect.“ American Economic Review, 64:1006- 1012. Heady, Earl O. 1952. Economics of Agricultural Production and Resource ygg. Englewood Cliffs, New Jersey: Prentice-Hall, Inc. Hey, John D. 1979. Uncertainty in Microeconomics. New York: New York University Press. Hirschleifer, Jack. 1972. "Liquidity, Uncertainty, and the Accumula- tion of Information." Uncertainty and Expectations in Economics, C.F. Carter and J.L. Ford, editors, Oxford: Basil Blackford. Hirschman, Albert 0. and Charles E. Lindbloom. 1969. "Economic Devel- opment, Research and Development, Policy Making: Some Converging Views." Systems Thinking, F.E. Enery, editor, Wfiddlesex, England: Penguin Books. Johnson, Glenn L. 1972. The Overproduction Trap in U.S. Agriculture. Baltimore: Johns Hopkin University Press. Johnson, D. Gale. 1947. Forward Pricing in Agriculture. Chicago: University of Chicago Press. Jones, Robert A. and Joseph M. Ostroy. 1984. "Flexibility and Un- certainty." Review of Economic Studies, 51:13-32. Kataoka, S. 1968. "A Stochastic Programming Model." Econometrica, 31:181-186. Koopmans, T. Jalling. 1964. "On Flexibility of Future Preferences." Human Judgments and Optimality, Maynord W. Shelly II and Glenn L. Bryon, editors, New York: John Wiley and Sons, Inc., pp. 243-254. Lawless, J.F. 1982. Statistical Models on a Methods for Lifetime Data. New York: John Wiley and Sons, Inc. Lavington, F. 1921. The English Capital Market. London: Methuen and Company, Ltd. 132 Lev, Larry, Lindon Robison and Lee Sonneborn. 1984. "An Examination of Investment and Production Decisions Under Conditions of Uncertainty and Flexibility." Mimeograph. Lewis, W.A. 1949. Overhead Costs. London: Allen and Urwin. Love, Ross 0. and Lindon J. Robison. 1982. "An Empirical Analysis of the Intertemporal Stability of Risk Preferences." Department of Agricultural Economics Staff Paper No. 82-24, Michigan State University. Marschak, Thomas and Richard Nelson. 1962. "Flexibility, Uncertainty, and Economic Theory." Metroeconomica, 14:42-58. Masson, R.T. 1974. "Utility Functions with Jump Discontinuities: Some Evidence and Implications from Peasant Agriculture." Economic Inguiry, 12:559-566. Merkhofer, M.W. 1977. "Value of Information Given Decision Flexibil- ity." Management Science, 23:716-727. Neal, Alfred C. 1942. "Marginal Cost and Dynamic Equilibrium." Journal of Political Economy, 50:45-62. Nickell, S.J. 1978. The Investment Decisions of Firms. Oxford: University Press. Norman, David. 1973. Methodology and Problems of Farm Management In- vestigations: Experience From Northern Nigeria. African Rural Employment Paper No. 8, Department of Agricultural Economics, Michigan State University. Officer, R. and A. Halter. 1968. “Utility Analysis in a Practical Setting." American Journal of Agricultural Economics, 50:257-277. Pyndyck, Robert. 1982. "Adjustment Costs, Uncertainty, and the Be- havior of the Firm." American Economic Review, 72:415-427. Rae, A.N. 1971a. "Stochastic Programming, Utility, and Sequential Decision Problems in Farm Management." American Journal of Agri- cultural Economics, 53:448-460. Rae, A.N. 1971b. "An Empirical Application and Evaluation of Discrete Stochastic Programming in Farm Management.“ American Journal of Agricultural Economics, 53:625-638. Robison, Lindon J. 1983. "Uncertainty in Economics." Unpublished class notes, Department of Agricultural Economics, Michigan State University. 133 Robison, Lindon J. 1981. “Investment/Disinvestment and Use of Our- ables: An Analytic Framework." In "Theoretical and Practical Models for Investment and Disinvestment Decision Making Under Un- certainty in the Energy Supply Industryf Lindon J. Robison and Michael H. Abkin, editors, Michigan State University Agricultural Economics Report No. 390. Robison, Lindon J. and Peter J. Barry. 1978. "Risk Efficiency Using Stochastic Dominance and Expected Gain-Confidence Limits." Journal of Finance, 33:244-249. Robison, Lindon J. and Beverly Fleisher. 1983. "Risk: Can We Model What We Can't Define or Measure?" Risk Management Strategies for Agricultural Production Firms; proceedings of a seminar sponsored by the Southern Regional Research Project S-180, San Antonio, Texas, March 28-30, pp. 34-51. Robison, Lindon J. and Larry Lev. 1983. "Distinguishing Between Ini- tial and Final Impact Variables to Predict Choices Under Risk or Why Woody Chip Went to the Air." Mimeograph. Rosenhead, Jonathan. 1978. "An Education in Robustness." Journal of the Operational Research Society, 29:105-111. Rosenhead, Jonathan. 1980. "Planning Under Uncertainty: A Methodology for Robustness Analysis." Journal of Operational Research Society, 31:331-341. Rosenhead, Jonathan. 1980. "Planning Under Uncertainty: The Inflexi- bility of Methodologies.“ Journal of Operational Research Society, 31:209-216. Rothschild, M. and J.E. Stiglitz. 1970. "Increasing Risk: 1. A Def- inition." Journal of Economic Theory, 2:225-243. Roy, A.D. 1952. "Safety-First and the Holding of Assets." Econo- metrica, 20:431-449. Sandmo, Agnar. 1971. "On the Theory of the Competitive Firm Under Price Uncertainty." American Economic Review, 61:65-73. Shalit, Haim, Andrew Schnitz and David Zilberman. 1982. "Uncertainty, Instability and the Competitive Firm." California Agricultural Experiment Station Working Paper No. 234. ‘ Shrieves, Ronald E. 1981. "Uncertainty, the Theory of Production and Optimal Operating Leverage." Southern Economic Journal, 47:690-702. Smith, Kenneth R. 1970. "Risk and the Optimal Utilization of Capital." Review of Economic Studies, 37:153-159. Stigler, George. 1939. "Production and Distribution in the Short-Run." Journal of Political Economy, 47:305-327. 134 Telser, L. 1955-56. "Safety-First and Hedging." Review of Economic Studies, 23:1-16. Theil, Henry. 1961. Economic Forecasts and Policy. Amsterdam: North Holland Publishing Company. Tisdell, Clement Allan. 1968. The Theory of Price Uncertainty, Produc- tion, and Profit. Princeton, New Jersey: Princeton University Press. Turnovsky, Stephen J. 1973. "Production Flexibility, Price Uncertainty and the Behavior of the Competitive Firm." International Eco- nomics Review, 14:395-412. Wald, Abraham. 1947. Sequential Analysis. New York: John Wiley and Sons, Inc. Young, D., et a1. 1979. "Risk Preferences of Agricultural Producers: Their Measurement and Use." Proceedings of the 1979 Annual Meeting of the Western Regional Research Project W-149, pp. 1-28. Zylberberg, André. 1981. "Flexibilite, Incertain et Theorie de la Demande de Travail." Annales De L'INSEE, 42:31-51.