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This is to certify that the thesis entitled A THEORY OF INVESTMENT AND DISINVESTMENT INCLUDING OPTIMAL LIVES, MAINTENANCE AND USAGE RATES FOR DURABLES presented by Alan E. Baquet has been accepted towards fulfillment of the requirements for Doctoral Agricultural Economics degree in gm kinda.» thyfessor Date 2 l1 7 I 7 g v—v'v 0-7639 , it; a 4L A THEORY OF INVESTMENT AND DISINVESTMENT INCLUDING OPTIMAL LIVES, MAINTENANCE AND USAGE RATES FOR DURABLES By Alan E. Baquet A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Economics 1978 ABSTRACT A THEORY OF INVESTMENT AND DISINVESTMENT INCLUDING OPTIMAL LIVES, MAINTENANCE AND USAGE RATES FOR DURABLES BY Alan E. Baquet The acquisition, use and disposal of durable assets has long been an important aspect of firm management. Managers are faced with two inter- related decisions concerning durable assets. The first decision concerns investments and/or disinvestments in the stock of a durable asset while the second concerns the rate at which to extract or generate services from a given stock of durable assets. A further dimension of the durable asset decision process concerns the economic life of the stock. The primary purpose of this study was to develop a model which per- mits the simultaneous determination of optimal investment, disinvestment and use decisions for durable assets. In the model, the firm's pro- duction process involves both the stock of durable assets and the flow of services from the stock. We conceived of the production process as vertically integrated with the flow of services from durables being gen- erated or produced from the stock at one level and subsequently fed into the production of the final output at a second level. When the services produced from durable assets are allowed to vary, the economic life of the durable asset is also variable. we allowed for Alan Eugene Baquet an aggregated maintenance input and specified a physical relationship among maintenance performed in each time period, services generated in each time period and the physical life of the durable. The economic life for each durable is determined internally as part of the optimal dis- investment criteria. The objective function for firms engaged in intertemporal alloca- tion decisions has received considerable attention in the literature. We assumed that the appropriate objective in each time period was to maximize current net revenue plus the gain achievable by reorganizing the initial endowment of durable assets. Reorganization can be achieved either by acquiring additional units of the durable (investment) or by disposing of currently held units of the durable (disinvestment). Optimal decision rules for the production activities and the invest— ment/disinvestment activities of the firm were derived by maximizing the objective function subject to the constraints imposed by the production functions, the service generation functions, and the length of life functions. The interactions and interdependencies between the production activities and the investment/disinvestment activities were reflected in the optimal decision rules. The research in this dissertation has advanced that branch of dy- namic production economies which is concerned with specifying the physical production process and the interactions among the production, investment and disinvestment process. we have specified the stock-flow conversion process more completely; we have accounted for the interaction among maintenance, use and the length of life of durable assets. Finally, we have specified in greater detail the interaction among Alan Eugene Baquet production, investment and disinvestment activities and have derived optimal decision rules which reflect this interaction. To the memory of my father, Eugene C. Baquet. His courageous fight against adversity serves as a desideratum. And to my mother, Gladys M. Baquet. Her unwaivering faith has always been a source of strength. ii ACKNOWLEDGEMENTS I wish to express my appreciation to Dr. Glenn L. Johnson for his assistance and guidance on this dissertation and throughout my graduate program. His ability to teach both in and out of the classroom greatly enhanced my entire graduate program. Appreciation is also extended to the members of my guidance committee, Dr. Norman Obst, Dr. Thomas Manetsch, Dr. Robert Barr and Dr. Robert Gustafson. Among my graduate student colleagues who contributed to my education Timothy Baker stands out as deserving a special thank you. His keen mind contributed much to my education while his quick wit contributed to my enjoyment of the educational process. I wish to thank the Department of Agricultural Economics at Michigan State University and the Electrical Power Research Institute for providing financial assistance. Finally, I wish to express my deepest thanks to my wife, Janie, and our daughter, Kristen. They were both instrumental in lightening the burden of graduate school. iii TABLE OF CONTENTS Chapter I O INTRODU CT I on O O O O O O O O O I O O O O O O O O O O I 0 Lack of Theoretical Foundation . . . . . . . . . . . Problem Statement . . . . . . . . . . . . . . . . . From Statics to Dynamics . . . . . . . . . . . . . . Research Methodology . . . . . . . . . . . . . AsSumptions on Which the Theory is Based . . . . . . The Production Process . . . . . . . . . . . . Motivational Assumptions . . . . . . . . . . . Degree of Knowledge . . . . . . . . . . . . . . The Managerial Process . . . . . . . . . . . . Other Assumptions . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . Dissertation Format . . . . . . . . . . . . . . . . II. REVIEW OF PART OF THE LITERATURE OF DYNAMIC PRODUCTION ECONom C S O O C O O O O O O O O O I O O O O I O O O O 0 Introduction . . . . . . . . . . . . . Historical Development of Dynamic Production Economics . . . . . . . . . . . . . . . . . . . . Summary of Historical Development . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . III. A MODEL OF PRODUCTION, INVESTMENT AND DISINVESTMENT . . . IntrOduction . . . . . . . . . . . . . . . . . . . . The Production, Investment and Disinvestment Model . The Physical Production Process . . . . . . . . The Economizing Process . . . . . . . . . . . . Production Decisions . . . . . . . . . . . . . Nondurable Inputs in the Final Production Process . . . . . . . . . . . . . . . . Generation of Services from Durables . . . Maintenance Activities . . . . . . . . . . . . Investment and Disinvestment Principles . . . . Length of Use Principles . . . . . . . . . . . Formal Derivation of Necessary Conditions . . . . . Production Activities . . . . . . . . . . . . . iv Page 38 38 38 38 42 45 46 46 51 51 52 53 Chapter Page Investment and Disinvestment Activities . . . . . . 60 Feasibility of Empirical Specification . . . . . . . . . 69 Summary . . . . . . . . . . . . . . . . . . . . . . . . 70 IV. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . 72 Summary . . . . . . . . . . . . . . . . . . . . . . . . 72 Summary of Theoretical Advances . . . . . . . . . . . . 82 Areas for Future Research . . . . . . . . . . . . . . . 84 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 LIST OF FIGURES Figure 1. Two-tiered vertical production process 2. Two-tiered vertical production process vi CHAPTER I INTRODUCTION Managers of agricultural firms are faced with a variety of de- cisions concerning the manner in which to organize and use various types of inputs in the production of alternative products. One kind of input, durable assets, are becoming increasingly important in agri- cultural production. Agricultural managers are faced with two interrelated decisions concerning durable assets. The first decision concerns investments and/ or disinvestments in the stock of a durable asset, while the second con- cerns the rate at which to extract or generate services from a given stock of durable assets. A further dimension of the durable asset decision process concerns the economic life of the stock. When the amount of services extractable in a particular production period is variable, the economic life of the durable is also variable. This becomes even more significant when we consider the maintenance activities of the firm. With perfect mainten- ance, it is theoretically possible to extend the life of an asset indefinitely. Ricardian land is a durable asset with an infinite life- time and perfect maintenance at zero cost. In actual practice, we observe less than perfect maintenance for most assets; hence, they de— preciate and are subsequently scrapped. There are two types of depreciation. One type is associated with using the durable assets. This type can be referred to as use depreci— ation as it represents the loss in the value of the durable as a result 1 of using it. This loss in value has been referred to as user cost by some economists. The second type of depreciation is associated with owning the asset over time. This type of depreciation can be referred to as time depreciation. It is this type of depreciation which is generally discussed in investment/disinvestment literature. As an illustration of the types of decisions discussed above, con- sider the following situation. A farm manager is considering investing in a tractor. The tractor can be used, say, in the production of both wheat and corn. If the manager is interested in making an optimal investment decision with respect to each tractor he buys, he needs to weigh the acquisition cost of the tractor against the values to be generated from its use in the production of wheat and corn. These values would be derived from the services generated during each pro- duction period. The number of production periods in which the tractor is to be used must also be determined. If the present value of the future uses of the tractor exceeds its acquisition cost, the manager can advantageously invest in the tractor. The disinvestment decision for each tractor on hand can be stated in an analagous manner. In this case, the manager weighs the present value of the tractor in future use against its present salvage value. The value in use is based on the services to be extracted in the re- maining production periods. The number of remaining production periods is also variable and depends, among other things, on the amount of ser- vice extracted and the maintenance performed. We need to consider investment separately from disinvestment de- cisions because the acquisition prices paid for durables usually exceed the salvage prices received for the identical durables because of 3 transfer costs and because durables are "lumpy" with an additional tractor to be purchased playing a different role than an existing tractor to be sold. Acquisition prices exceed salvage prices for var- ious reasons. Some of the reasons are (1) transportation costs and (2) transactions costs. Lack of a Theoretical Foundation The static theory of production economics is well documented (Ferguson, Heady). Most static theories of production economics treat the services of durable assets as flow variables and do not consider the economics of generating service flows from stocks of durable assets. For this dissertation, flow inputs are defined to be "one-use" services which are repeated through time, e.g. tractor services per production period. Stock variables are defined at a point in time, e.g. tractors at time to. Some static theories have recognized this stock-flow conversion problem, but have assumed it away by assuming a constant rate for con- verting the stock variable to a flow variable and vice versa (Edwards, 1958, 1959). The theory of investment with fixed extraction rates and hence fixed lives of durables is well documented, also (Smith, Vernon; Lutz and Lutz). The theory of disinvestment is not as well documented (Edwards, 1958, 1959; Paris, 1960; warren and Weintraub). There have been relatively few attempts to combine the theories of production, investment and disinvestment. Theories which have combined these areas have made the simplifying assumption that durable assets generate a prespecified fixed amount of services per unit of time, and that, hence, the assets also have prespecified fixed lives (Edwards, 1958, 1959). To our knowledge, no theory has been developed which considers the simultaneous decisions involved in production, investment and disinvest- ment when the rate at which services can be extracted from durable assets is allowed to vary. Allowing the extraction rate to vary also allows the lifetime of the durable assets to V3fY€ll Problem Statement The intent of this research is to develop and extend those parts of production and investment theory which explicitly recognize the simultaneous decisions in production, investment and disinvestment when the rate of extracting services from durables is varied optimally. The affect that varying the extraction rate of services has on the economic life of the durables is also to be explicitly examined. Examining these issues requires the development of a theory of dynamic production economics. It is beyond the scOpe of this disserta- tion to deve10p a completely dynamic theory, rather we will be concerned with analyzing the theory of production, investment and disinvestment in the context of limited dynamics. Our theory does not deal with uncer- tainty, nor does it include the managerial process. we are concerned with specifying the physical production relationships among durable in- puts, nondurable inputs and outputs. Our analysis will focus on the optimizing decisions of managers concerning the level and usage of 1-/Francis 8. Idachaba in an unpublished manuscript considered the effects of a variable extraction rate for services on the production activities and investment/disinvestment activities. However, he did not consider the effect of the variable extraction rate for services on the life of the durables. dual 0 aetai bi“. saUL r.) r o. n durable assets. The remainder of this chapter specifies in greater detail the objectives, boundaries and limitations of this research. From Statics to Dynamics When the production, investment and/or disinvestment decisions of a firm are considered simultaneously, we move from the realm of static economics to dynamic economics. An analysis which relies on the as- sumptions of static economics is not concerned with (l) the effects of imperfect knowledge, (2) the effects of changes in technology, (3) the effects of management, (4) the effects of changes in the institu- tional structure, (5) the effects of changes over time, (6) the effects 0f changes in the behavior of humans. Investment and disinvestment decisions require knowledge of future PrOduction activities as investment/disinvestment decisions are based on the present value of future services generated from durable assets. A completely (dynamic theory would consider the uncertainty associated with these future uses. It would also consider the management activi- ties of the firm. These involve knowledge acquisition and use, decision making, execution and responsibility bearing. To deal with all aspects of dynamic production economics would be beyond the scope of this dissertation. We are concerned with developing a theory with a limited dynamic nature. The limitations and their con- sequences are discussed in that part of a subsequent section which deals th assumptions . as [:11 Research Methodology This research is disciplinary and of known relevance. Our major tests for validity will be internal and external consistency. We re- quire our theory to be internally consistent or coherent, i.e. the propositions must follow logically one from another and the conclusions must follow from the propositions. We also require the theory to be externally consistent, i.e. we require the theory to be consistent with previous empirical research on and experience with production, in- vestment and disinvestment. We also require our theory to pass the test of clarity, i.e. we require the theory to be presented in such a way that it is unambiguous and, hence, testable as to its internal and external consistency as well as transmissible from one person to another. Assumptions on Which the Theory is Based Several assumptions are needed to reduce the problem stated above to a manageable size. Some of these assumptions limit the dynamic nature of our theory, while others limit the effort to reasonable bounds and, at the same time, define those bounds. Thewduction Process Nearly all previous research on production economics has modeled the Production process as a flow relationship or has assumed that stocks are Converted to flows at some prespecified rate (Edwards, 1958, 1959; YotoPoulas). In our model, we do not assume that the production process is 8ttictly a flow relationship. Furthermore,‘we do not prespecify the re te at which services are extracted from the stock of durables. Rather, We determine the optimal extraction rate internally. 6 deg PI' 0U pr m M- [M "J We conceive production to be a vertically integrated process. The degree to which the production process is disaggregated in the vertical dimension depends upon the problem being addressed.-2- For our theoretical model, we could use a multi-tiered vertical process, but for the sake of simplicity (and, hence, clarity) we limit ourselves to a two-tiered vertical production process. On one level we treat the services of durable assets as being produced from the stock of durable assets, Dt’ by using one nondurable production input, x2t' These services thus produced are an input in the second or upper level production process. At this second level, the final outputs, Yt’ are produced from the services from the durable assets and one nondurable input, x1t° This second level in the production process is strictly a relationship while the first level involves both stocks and flow flows .2, In our model we consider two durable assets and two final Ontputs. Figure 1 is a diagrammatic representation of our vertical Preduction process . t and D2t are used along with , to produce services, alt and 92: Figure 1 shows that the durables D1 the aggregated nondurable input, x2t at One level. At the upper level, the services produced are used along t andY With the nondurable, xlt’ to produce the final outputs Y1 2t° In limiting our production process to two outputs, two durables and two nondurables, we are following the aggregation rules discussed \ Pr 1a é[There is an economics of aggregation which determines the appro- Pri te level of aggregation for a particular research effort. The basic addnciple is to disaggregate both outputs and inputs sufficiently to 1:388 the problem at hand (Heady, Johnson and Hardin). 3 (111111 ‘IWe assume that each production function conforms to the law of no nIlshing returns and that inputs are neither perfect substitutes Perfect complements. I Y1: 1th Production X Production function It function 61: 62:: Production X Production function 2t function Q @ Figure l. Two-tiered vertical production process. in footnote two. No durables permit us to analyze the interactions between the durables' services. More durables would not add signifi- cantly to the analysis. In considering only one nondurable input in eBCh level of the vertical process, we cannot analyze the interactions between nondurables at each level; however, the economics of these interactions has been studied extensively elsewhere and is easily accessible (Ferguson). The interaction between nondurables, durables and the services of durables, which we can analyze, is more central to our theory. Thus, we consider nondurables, X and X2t’ to be aggre- 1t gates of all the nondurable inputs used in each production processfi/ I n considering two outputs, we are allowing for the analysis of alter- n atiVe uses for the services from durables, as well as for the \ 4 ‘IWe recognize the possibility of an index number problem in de- Ping these aggregates, but will not let it concern us in our Oretical presentation. Velo the IL: '1‘, nondurable, X Having more than two outputs would not add signifi- lt' cantly to the analysis. In limiting our consideration of the production process to the dimensions defined above, we have selected a level of complexity which permits us to analyze significant interactions in the production, in- vestment and disinvestment process. At the same time, we have confined ourselves to a manageable domain which can be discussed without failing the test of clarity. Recognizing both stocks and flows in our production process builds upon and extends the earlier work done by Georgescu-Roegen (1971a, 1971b) and Kenneth Smith (1968, 1969). While both these men recognized the importance of the stock-flow conversion process for durable assets, neither conceived of production processes involving durable assets as Vert 1cally integrated. .l‘fitivational Assumptions The usual motivational assumption in static production economics is Profit maximization. In dynamic production economics when uncertainty is cI-‘-Ons:ldered, managers are assumed to maximize expected profit or the exPefited utility of profit. In the traditional treatment of investment decislons, managers are assumed to maximize the net present value of the investment or some variant of net present value. These basic motivational assumptions have been modified by various researchers. Clark Edwards (1958, 1959) amplified the meaning of pro- fit maximization. He assumed that managers of existing firms maximize the gain achievable from reorganizing an initial endowment of resources. TypiQal profit functions can be viewed as gain functions with a zero 10 initial endowment. There is a considerable body of literature on portfolio theory which suggests that managers of a portfolio of assets should maximize the return or the value of the entire portfolio (Van Horn). In our theory, managers are concerned with both production de- cisions and investment/disinvestment decisions. Thus, we need an objective function which will take both aspects into account. In the early 19508, Boulding (1950) suggested that managers of firms which are not required to sell all their output in the period it is produced should maximize the current net returns plus the gain achievable from selling in a future production period. His objective function can be modified to apply to our production, investment, disinvestment theory. Although we are not directly concerned with intertemporal mar- keting strategies, Boulding's basic objective function can be used. Our primary concern is with the intertemporal usage of durable assets and the effect of this usage on the value of the firm's portfolio of durable assets. For our theory, we assume that in each period the firm Seeks to maximize current profit from its production activities plus the 3315 in the net present value of its portfolio of durable assets. This gain is achievable by either investing in additional assets, disin- vesting in currently held assets, or by reorganizing the pattern of use for the initial endowment of durable assets. In making this motivational assumption, we are extending Boulding's earlier work by applying it to durable assets. We are also combining it w ith the work of Edwards by considering the gain in the value of the (1 “table assets. 11 Degree of Knowledge For firm managers to make optimal investment and disinvestment de- cisions, they need to compare current costs and returns with future costs and returns. In actuality, the future costs and returns are not known with certainty. To be completely accurate in this regard, our theory should account for the uncertainty of future events. However, adding this element to our theory would unduly complicate it at this stage. Thus, we will leave this refinement for the future. Even though we assume perfect knowledge of the future, we are con- cerned with the consequences of previous decisions which have turned out ex post to be nonoptimal. If we assumed all past decisions were made with perfect knowledge, the firm would not need to adjust its initial endowments. However, since decisions in past periods were not made with Perfect knowledge and foresight, mistakes may have been made. Our con- cern is in analyzing the consequences of previous mistakes under the assumption of perfect future knowledge. Following Knight, this elimin- ates the need at this point in the development of theory to consider managerial processes, insurance, gambling, etc. EWrial Process Management is a complex process which is only beginning to be under- stood and carefully analyzed. The study of management is a part of dynamic economics. The relationships between the managerial process and the Other aspects of dynamic economies are still being established. As indicated above, our theory will not address the economics of the mana- Serial process. We are concerned only with the consequences of k1mizing decisions assuming perfect knowledge of a changing future. 12 We recognize the importance of studying the managerial process but feel it is necessary to develop the basic theory of production, invest- ment and disinvestment separately. At some future date, emerging theories of the managerial process will need to be merged with theories of production, investment and disinvestment.2/ We further recognize that the implications and results of our theory would be modified if the managerial process were included. Other Assumptions The income tax consequences of investments and disinvestments in durable assets are often significant. Investment tax credits received for investments in durables significantly reduce the actual price of the durable while some taxes may increase costs. We have not incorpor- ated tax considerations in our theoretical model. It is a refinement Whitch could be added, but we chose not to since it would not add sig- nificantly to the theory being developed and would encumber the Presentation. Analyzing production, investment and disinvestment decisions 811mlltaneously requires the specification of a planning horizon for the firm, The economic life of each type of durable asset is determined within the model. The planning horizon for the firm must transcend the life of all but perfectly maintained durable assets. Thus, we arbi- trarily prespecify the planning horizon for the firm as the point in time beyond which the present value of the costs and returns would be e seentially zero for any positive interest rate. \ 5 "1‘1 ‘lEmerging theories of managerial processes can be found in the tinge of Knight, Dillon, Stiglitz and Glenn L. Johnson (1960). Summary Prior to this effort, there was no solid theoretical definition of how to simultaneously optimize investments and/or disinvestments in durable assets and the rate at which to use the services of the durable assets in production. In this research we develop a theoretical model for making such optimizing decisions simultaneously. To reduce the theoretical problem to manageable proportions, we made several as- sumptions which limit the dynamics of our theory, simplify the analysis and specify its bounds. Our primary focus is on developing the optimizing conditions for the simultaneous production, investment and disinvestment decisions. In deve10ping these conditions, we limit the firm's production to a two- tiered vertically integrated process which uses one nondurable at each level- The firm has two durable assets which produce services. We aBSUIne each durable's acquisition price exceeds its salvage price be- cause of transfer and transactions costs. The firm produces two outPI-Its for sale. We assume perfect knowledge of a changing future. We do not include the managerial process in our theory. We prespecify the planning horizon for the firm but determine the optimal life for each type Of durable internally. Dissertation Format The following chapter traces the historical development of the parts of dynamic economies which are relevant to our theory. This per- “111:3 us to place our theory in the proper historical perspective. The mathematical specification of our theoretical model is pre- 8e IIted in Chapter III. Optimizing conditions are derived. Since our 13 14 theory is of known relevance to practical problems, we discuss the feasibility of empirically specifying our theoretical model. Chapter IV contains our summary and conclusions. We also specify in Chapter IV areas for future research. CHAPTER II REVIEW OF PART OF THE LITERATURE OF DYNAMIC PRODUCTION ECONOMICS Introduction In the previous chapter we identified some aspects of a more com- plete theory of dynamic production economies which should be addressed. Alternative theories of dynamic economics have concentrated on particu- lar aspects of dynamics. As yet, the separate theories have not been merged into a complete theory of dynamic production economics. There are two main tracks being followed in the development of the theory of dynamic economics. One track is concerned primarily with risk and uncertainty in production. Followers of this track are also con- cerned with the role of management in production. Much of the study of management and uncertainty in production involves the learning aspects of management. This track has its origins in Knight's writings. Other contributors to this track are Hart, wald, Bayes and Johnson, et a1. (1961). Information theorists, cyberneticists and some system scien- tists have also contributed to this track. The focus of the second main track is on specifying the physical production process and relating the production activities of firms to their investment and disinvestment activities. Followers of this track have been concerned with the manner in which durable assets enter the PrOduction process. Writings on this particular issue have not ade- quately handled the stock/flow conversion problem. More recently, the foCus has been on the divergence between acquisition and salvage prices 15 16 for inputs in the production process, and the implications of this for supply responses. George warren was an early contributor to this track. More recently, Neal, weintraub, Johnson (1958), Edwards, Georgescu- Roegen and Kenneth Smith have made contributions to this track. Also, Hirschleifer discusses some consequences of salvage/acquisition price differentials in his text on price theory. We recognize that there are important contributions to be made by merging both tracks. However, there are also advancements to be made within each track. We leave the merging of the two tracks for the future. The contributions of this dissertation will be to the second track. Our concern is in specifying the optimal manner for durable assets to enter and leave the production processes of a firm. we are also con— cerned with the timing of the use of durable assets and the affect the time pattern of use has on the value of the durable assets. In con- sidering the use pattern through time for durable assets, we are in effect linking production, investment and disinvestment. The remainder of this chapter traces the historical development of part of the theory of dynamic production economics. The main focus is on what was identified above as the second track of dynamic production economics. Historical Development of Dynamic Production Economics John Bates Clark laid the groundwork for that part of dynamic pro— Cthtion economics, on which this dissertation is based, in 1899 in his 1)001:: titled The Distribution of Wealth. He made the following observa- tic>118. a5: 11‘4“: 1%? Cc “Oak 11 l7 1. Realism is the striking trait of dynamic theory. This quest for realism underlies nearly all the writings on dynamic economics. 2. A theoretical dynamic world is exactly like the actual world if the theory that constructs it is a valid and complete one. (Clark failed to recognize that the cost of completeness is infinite and that, thus, it is not possible to have the com- plete dynamic theory specified.) 3. Historical economics records and measures differences while the theory of dynamics accounts for them. 4. The mere theory of economic dynamics will enlarge by many fold the scope of political economy; it will lift theory to a new plane. Clark wrote in vague generalities about economic dynamics--more about the virtues of having such a theory than about the theory itself. Few changes or additions to J. B. Clark's formulation were made until Frank H. Knight's Ph.D. dissertation in 1916. This later appeared as his now famous book entitled Risk, Uncertainty and Profit. This book laid the foundation for much subsequent work on uncertainty and the decision process and marked the beginning of the separation of dynamic production economics into two main tracks previously delineated. A contemporary of Knight, George F. warren, wrote perhaps the earliest statement concerning the divergence between acquisition value and value in use. He wrote, "If hay is worth $15 a ton at the railroad 8tation, it is usually not worth more than $12.50 on the farm because the cost of baling and hauling to the station must be deducted. Live- Stock need only return $12.50 for hay to make it pay to feed rather 18 than to sell" (warren, p. 208). Defining fixed assets has been a central theme in the second track. warren provided a partial definition. He omitted the part of the definition which states that livestock need re- turn at least $15 plus transport costs for hay to make it pay to buy more from the railroad station. The 19203 saw the publication of a book by Jacob H. Holland en- titled Economic Essays in Honor of John Bates Clark. While not adding anything new to J. B. Clark's formulation, it does clarify some of his writings. Two essays in this book addressed issues relevant to dynamic econ- omics. One was by John M. Clark, J. B. Clark's son. In an essay titled "The Relation Between Statics and Dynamics" he expanded the ideas his father conceived. He was more definite on how a dynamic theory might evolve. His main points were: 1. Static assumptions should be replaced by dynamic assumptions. The dynamic theory should be built on these rather than starting with static conclusions and adding dynamic elements. 2. Dynamic economics represents a shift from searching for equi- librium levels to the studying of the processes involved in achieving levels. 3. He emphasized the importance of studying errors. It is im- portant to ask how they are corrected, if they are and what happens if not. (Note the implication of less than perfect knowledge and foresight, ex post of least.) 4. Definitions of basic concepts change when we go from static to dynamics. Capital is one example. 5. Dynamics should be an explanation rather than a mere description of economic behavior. 19 A second article of relevance was by Frank A. Fetter entitled "Clark's Reformulation of the Capital Concept." According to Fetter, Clark was one of the earliest economists to recognize that capital was an economic factor of production in addition to being the fruit of past labor used to further production. (Note the similarity between this statement and Marshall's concept of capital.) We can summarize the pre-1930 position of dynamic production economics by saying it was in its infancy. J. M. Clark became more specific about how the theory ought to develop but did not develop a rigorous statement of dynamic production theory. With the writings of Frank Knight and George warren (who was not a theoretician but recognized the practical implications of price divergence), we see the theoretical developments beginning to split into two tracks which are still followed. Interest in the development of dynamic economics continued in the 19303. Several articles were written on various aspects. Economic theory texts published in this decade included sections on dynamic economics. we will key on a few of the major articles and texts which represent the general development of the theories of economic dynamics. Nicholas Kaldor, writing in the March 1934 issue of the Economic Journal, argues that the concept of a firm is essentially a dynamic one. In a static theory characterized by equilibrium points, the supply function of the firm is indeterminate in the long-run. It is dynamics and possible imperfections in competition which lends indeterminateness t0 the firm. It was during this decade that Keynes published his General Theory. Although the major emphasis in his book was on macro or aggregate 20 economics, some of the concepts he developed are applicable to micro- economics. The most noteworthy development of interest to our theory is his concept of user cost. He was the first to attempt to identify the cost of using durable assets in the production process. His purpose for considering user cost was to define income precisely and to explain changes in production not due to investment and disinvestment. He was not directly interested in the costs of production except as a means of defining income. To Keynes, user cost was the sacrifice in value in- volved in the production of output. It represented the current disinvestment involved in using durable assets. Later writers added some clarification to user cost, but its role in determining production levels is as yet unclear. Sune Carlson in his book A Study on the Pure Theory of Production makes reference to Keynes' concept of user cost, but he does not extend or clarify it. He does have a chapter on "Poly-Periodic Production Theory" in which he discusses some features and problems of poly-periodic production. The main features according to Carlson are: l. The time period interdependencies--i.e. production this period affects production in future periods. 2. The distinctions between fixed and variable services and between durable and nondurable resources. Resources which render ser- vices for more than one period are durable. Services for which the cost remains unchanged as the quantity of output changes are considered fixed. A different treatment of dynamic economic theory is contained in J? IR, Hick's Value and Capital, published in 1939. He contributed the following advancements . 21 1. With economic dynamics, he restressed that we must date every quantity. 2. The concept of capital is only relevant in dynamic economics. 3. The objective function should be to maximize the capitalized value of the stream of surpluses--surplus - value of output - costs of production. This is formally identical to maxi— mizing surpluses of receipts over costs in static problems as outputs of different dates are regarded as different outputs. 4. The conditions for equilibrium are: (a) the marginal rate of substitution between any two outputs of any two dates must be equal to the ratio of discounted prices; (b) the marginal rate of substitution between any two inputs for any two dates must be equal to the ratio of discounted prices; (c) the marginal rate of transformation between inputs and outputs must be equal to the ratio of prices. 5. Hicks was the first to distinguish between the decision process for an existing business and the decision process for starting a new business and the alterations in the firm's objective function that this distinction requires. Hicks made significant contributions. He is the first to discuss the appropriate objective function. This allowed him to consider the conditions necessary for an optimum. His conception of the differences between analyzing existing firms and beginning firms is very important. Neither Carlson nor Hicks was concerned with the affects of uncer- tainty or the role of management in production. Thus, they were both tleking about a specific type of dynamics. Their writings are contribu- t1(ms to the second track of dynamic production economics. 22 At about the time Hicks was developing a theoretical framework for economic dynamics, T. w. Schultz in an article appearing in the 1939 Journal of Farm Economics was pointing out the need for developing such a theoretical framework. Schultz's interest in such a theory was spurred in part by his criticisms of current farm management studies. The criticisms at that time were: 1. The research results do not provide a basis for guiding entrepreneurial decisions when economic change confronts the farmer. 2. They afford no way of relating the actions taken within the farm to that of the economy as a whole. Schultz devotes most of the article to the first criticism. He discusses why the criticism is valid and what should be done to make farm management research more relevant when conditions change. Basically Schultz argued that the reason farm management studies had not been relevant to changing conditions was because they had been done in a framework of economic statics. He argued that the firm exists in only a dynamic setting and, hence, must be examined in terms of dynamics. The remainder of the paper is devoted to "... pointing out some of the more fundamental rules which underlie the trial and error procedures which are followed by the entrepreneur as he seeks to keep his firm in equilibrium as the operations of the firm are adjusted to the changes which arise" (Schultz, p. 586). His analysis is based both on Hicks and Knight. At this stage the theory is far from complete, but it has progressed from J. B. Clark's early conception to a point where specific aspects of the second track can be identified and worked on separately. The specific areas that Schultz identified were: 23 1. Price and technological expectations--any time we consider more than one time period, we must consider expectations. 2. Production plan--Schu1tz has in mind the method of producing output as well as which output to produce. 3. Time span of the production plan--Schultz means here the planning horizon over which the production plan will hold. The following can be added to Schultz's list. 4. The actual adjustment mechanism between time periods--it is one thing to say what the input-output combinations should be in each time period and quite another thing to say how the changes will be made that will result in the appropriate in- puts for subsequent time periods. 5. The role of investment and disinvestment--to this point the whole area of durable assets has not been discussed in con- nection with dynamic economics. Some authors note that capital is a dynamic concept, but the idea of investing or disinvesting in capital is not mentioned except in a macro context. 6. The notion of varying the rate of use on capital is not brought out explicitly. Keynes introduces the concept of user cost, but it is unclear from his discussion if changes in user cost occur from changes in the use rate or changes in the stock level at a constant use rate or both. We should note that Schultz also called for a theory of production Q'Qonomics which is completely dynamic, i.e. one that would span both of it:t1!e separate tracks being developed. The Schultzian demands go beyond I‘Llcks' theory by relying on Knight's work on risk and uncertainty and 233: art: 24 his own theories of capital rationing (Schultz). Such a theory has im- portant implications for studying the supply responsiveness of firms due to changes in their economic environment. However, both tracks continued to be developed individually in economics, and after Schultz's article, in agricultural economics. Alfred C. Neal made a significant contribution to the theory of dynamic economics in his article "Marginal Cost and Dynamic Equilibrium of the Firm" which appeared in the Journal of Political Economy in 1942. Neal makes explicit reference to an aspect of dynamics that was pre- viously included only implicitly at best. This aspect deals with the notion of "clock-time" as distinctly different from short, intermediate, and long-run time concepts. In addition to his discussion of clock-time and the changes which take place within clock-time, Neal recognizes, as did Schultz, that it is not only current costs and current revenues which determine decisions, but also expected future costs and revenues. Neal recognizes that the production plan includes more than the determination of input-output combinations and levels. He says the plan 8.1 so includes: 1. Additions to or alterations of the plant, i.e. investments or disinvestment in durable assets. 2. Market maneuvering. This is of course relevant not only for firms in non-competitive positions but also for those in competitive positions . 3. Research. This can be considered as investing in knowledge. Neal's discussion of the objective function and the consequences of '31) timizing it is particularly insightful. He assumes the appropriate ijective is to maximize the present value of the difference between the rec: 25 total value of the products and the total costs over the planning horizon under consideration. He further points out that because of the ‘variety of time shapes for the receipts and cost streams, there is no Immessity that the adoption of a plan which.maximizes the present value of net receipts over the entire planning horizon would involve at the same time maximizing net money receipts for any current time period. 'This implicit recognition of the opportunity costs between time periods (is particularly insightful. The trade-off between current production and future production is an important aspect of dynamic economics. Neal's arguments and presentation are the second stream of dy- Ilaumics and are similar to Frank Knight's work. Neal's work does not parallel Knight's in the first stream; however, his recognition of the trade—off mentioned above is very similar to Knight's definition of the Iralte of capitalization: "the .... ratio of conversion of present goods into future income" (Knight, p. 166). Neal does not pursue this idea; instead he shifts his attention to '61!) elaborate discussion of the costs associated with production in a ‘ilvnamic setting. He begins by identifying those costs which would be 3lecurred if no production takes place in the next production period. fJTlhese costs include: 1. Depreciation due to deterioration and obsolescence. 2. Interest on actual value of investment. 3. Maintenance. 4. Taxes. 5. Insurance for idle equipment and plants. 6. Costs due to retaining personnel and maintaining contact with markets. ope: labs Q q, 5'"! :a: Sea. c113 ta! at ‘1 Mr 26 The magnitude of these "supplementary" costs varies with the expected length of zero production. The shorter the idleness, the larger the preportion of supplemental cost to total cost. The second cost category set up by Neal is labeled prime cost. These are the costs associated with production. He defines them to be total cost less supplementary costs. Prime costs include more than what is traditionally included in the variable cost category of the fixed/ variable cost dichotomy. The prime cost category is further broken down into factor costs--the cost of factors of production associated with operating as opposed to not operating; purchases from other entrepreneurs, labeled M by Neal; and U, user cost--the change in the value of facil- ities due to operating as opposed to not operating. The word "facilities" according to Neal covers all types of durable assets. Neal's definition of user cost is different than Keynes' in that pur- chases of goods and services from other entrepreneurs (Keynes' A, Neal's M) are excluded from Neal's user cost. Also Neal includes maintenance in user cost. Perhaps Keynes was considering maintenance to be perfect in which case depreciation would be zero. To illustrate Neal's user cost definition, let E0 be the value of " facilities" at the beginning of the production period; let S be the Expected change in their value at the end of the production period if 2110 production took place; and let El be the expected value of the facilities at the end of the production period if production was under- taken. User cost is then (Eo - S) - E1. This is the change in value as a result of using the facilities. From the definition of user cost, it is obvious that more than Qturrent costs influence current production. Incorporating the notion 6.5. 27 that future costs and returns affect current production greatly compli- cates the usual static analysis. Basically, with this type of analysis, we must consider the production affects of a change in any one of the following: 1. Current costs. 2. Current revenues. 3. Future costs. 4. Future revenues. Neal goes through some examples of the effect that changes in one (If these four areas would have upon current production. He makes the following points . l. The current cost of using facilities beyond a certain level is the foregone future use with the highest present value sacri- ficed. Future opportunity cost--i.e. future "within-firm" demand. 2. Because of (l) the true present marginal cost of production (including user cost) involved in using these facilities will almost always exceed the marginal cost calculated solely from the costs of the so—called variable inputs. Tlfhe Neal article, although incomplete, is probably the best single refer- Gence on user cost in the literature. His article verbalizes much of what ithe second track in a theory of dynamic economics must contain. He is Specific about the various components that would compose such a partial ‘theory of economic dynamics. It was during the 19403 that Abraham wald (1947) contributed to the -first track of dynamic economics initiated by Knight's writings. wald ilaid the groundwork for analyzing the learning process of management (I) (- an '———4 PETE 28 with his work on sequential analysis. Earlier A. G. Hart (1946) inter- preted Knight's work trying to clarify the distinction between risk and uncertainty. While both of these men made important contributions to the theory of dynamic economics, their contributions are not in the portion of dynamic economics which we are concerned with in this disser- tation. Friedman and Savage (1948) and Von Neumann and Morgenstern (1947) also contributed to the first track during the 19403. These writers were concerned with the utility analysis of decisions under uncertainty. Their writings were original contributions in the area of decision making under uncertainty . D. Gale Johnson's book, Forward Prices of Agriculture, was pub- lished in 1947. He applied Knight's concepts to the public sector. The Friedman/ Savage, Von Neumann/Morgenstern and D. Gale Johnson contributions were to the first track of dynamic production economics, not the track followed in this dissertation. Samuelson's Foundations of Economic Analysis appeared in the late J~9403. Several ’chapters in this text are devoted to economic dynamics. 4 During the remainder of the 19403, no major advances were made in tI121croeconomic dynamics. Some things were written which concerned macro— Q(:onomic dynamics. Examples of the writings in macroeconomics can be found in the works of Paul Samuelson and R. F. Harrod. The 19503 saw an interest in the investment/disinvestment activities Of the firm. Freidrich and Vera Lutz wrote a book on the firm's invest- ment decisions. Vernon L. Smith wrote articles and a book on equipment replacement. These writers gave lucid accounts of one of the problems faced by firms, namely investing or disinvesting in durable assets. . . 'c' "I“ 31C!e ‘ .gnfiv“ "~va 0 for se fiat t 1 l ‘l 4:!”- a..&34 ‘ :. luv-‘5‘ ‘ Q 1 n. .7, Lil... I Q-O‘IV asks! a “7" as f 2f us .1“ fin f) 29 However, they did not link the investment/disinvestment decisions to production decisions, nor did they consider varying the extraction rate for services from durable assets. Thus, these writings are incomplete even though they add to the body of knowledge concerning investment decision making. Kenneth Boulding contributed to both tracks during the 19503. His book, A Reconstruction of Economics, presented an objective function Which permitted the simultaneous consideration of current production activities and the affect of future production activities on the current Value of the firm's portfolio of assets. Later, in 1956, his book, The $33, contributed to the first track of dynamic economics. The concept of user cost introduced by Keynes and extended by Neal Was further extended by A. D. Scott in 1953. He discussed four aspects of user cost. 1. Keynesian user cost. 2. User cost as a determinant of current output. 3. Calculation of user cost. 4. Differences between user cost and "retainer cost." 4 Scott points out that Keynes developed the concept of user cost so he could better define aggregate income and explain changes in output 11:; 1: due to investment or disinvestment. The Keynesian conception of user Q“ 8!: should be modified if it is to be used as a determinant of micro production. Scott uses a modification developed by A. P. Lerner (1943), namely that of prime user cost. Prime user cost is the difference be- Ween the value of the asset after production and what it would have been without production. It is a measurement of wear and tear during use to the extent it is not made good by maintenance. There is no IE'FEHUQ £58! CC I 's. a :3..ch ever ti: n'... C' .46 gl 30 simple algebraic function relating Lerner's prime user cost to output alone, for it depends upon maintenance and replacement decisions. Scott presents a diagrammatic view of how user cost is used to determine current output. He establishes a user cost curve and a net revenue curve. The appropriate output level is the one where marginal user cost equals marginal net revenue (user cost is not deducted from net revenue). Although Scott discusses user cost, he does not explicitly consider varying the rate of use of durable assets. Thus, he does not specify the Optimal rate of service extraction. Scott illustrates the calculation of user cost by using a simplified model of a firm's plans Over time. He illustrates the three user costs of using equipment in the first period. These are the three costs that w. A. Lewis (Lewis, P- 10) identified in his book, Overhead Costs. These are: l. The salvage value of the asset if sold now and not used less the discounted value of future yields and scrap value. 2. The difference between scrap value now and scrap value next year if used this year. The yield of the extra future year for which the asset is avail- able if it is neither used nor scrapped in this year. The highest of these three costs is the relevant user cost for the Q‘-‘-l'rent production period. Scott recognizes the interdependencies between current production and future production. "... for each subsequent rate of output, which helps determine today's output, is itself influenced by today's decision" (Scott, p. 378). All current and future outputs need to be determined Q:'~111t.11taneously. Such problems amount to finding a maximum integral of d1Scounted revenues over time and can be solved by calculus of variations. . 0‘." u .‘v . flit p s . . . .M 2. .5 up“ ... u ~ v?‘ .u an . . Wu» . v. 4k h A . :v- . 31 Scott distinguishes between prime user cost and retainer cost. The value of an asset falls as a result of use (user cost) and also as a result of retaining it over time (retainer cost). Scott says only user cost affects output decisions. He does not make the analogous case that retainer costs affect investment decisions. Scott's article suffers from the lack of recognition of the inter- dependence of‘ investment and production decisions. Even with this shortcoming, the article is good as far as it goes. Scott had a good grasp of the complexities of the problem of production over time. He Points to how it can be solved for the simplified model he used. There were related developments in the 19503 which apply to both Static and dynamic production economics. One of the most important of these was the recognition by Glenn Johnson, Joseph Willet, Lowell Hardin, and others that an asset can and generally does have different acquisi- tion and salvage values. The recognition of this acquisition-salvage Price differential led to a new definition of resource fixity. This new definition was an economic rather than a technical or physical one. A I T and T . H-—- D1 D2 hk(°)-tfunctional relationship among services, maintenance and the physical life of the durable. It has the following properties: “'1. “‘1. __.__(0,.__._ 36kt akat > 0. his wrl inc the : :aim 9'5; WC, 42¢ a; 42 The inputs X , and X t are all aggregates. Treating the in- lt’ x2: 3 puts as aggregates masks a lot of the interaction which occurs in real world production processes. The theory of combining nondurable inputs in optimal proportions is well documented (Ferguson). Our primary inter- est is in examining the interactions between nondurables, durables, and the role of nondurables in generating services as well as the role of maintenance in extending the life of durables. The preceding six equations model the production processes con- sidered in this dissertation. It is the form of these production relationships which guarantees that the appropriate second order condi- tions will be met in the Optimizing calculations which follow our discussion of the economic processes involved in production, investment and disinvestment. The Economizing Process The preceding section specified our model of the firm's production processes. In this section we discuss the optimizing decisions of the firm. The firm must determine the optimal levels for the durable in- puts, the optimal amount of services to extract from the durable, and the optimal quantities of the nondurable to use in its production pro- cesses. All these decisions need to be made simultaneously since the optimal level for each variable depends on the levels of the other variables. An important component for making such decisions is an appropriate objective function. Much has been written about what the appropriate ob- jective for the firm should be when intertemporal decisions are being made. Maximizing the present value of disposable income (Boehlje and 43 White), maximizing net worth (Cocks and Carter), and maximizing the net present value of the firm (Van Horne) have all been proposed as appro- priate objectives for firms making intertemporal decisions. In addition to the time dimension, there is a second aspect which needs to be accounted for in the objective function. The firm operates over time with durable assets which are not entirely used up in each production period. Thus, at the beginning of each production period, the firm has an initial endowment of durables. In other words, the firm does not start from scratch each production period. The optimal organi— zation for the firm is conditioned by this initial endowment of resources. This has long been recognized by practioners in farm management, if not by "zero-base" planners in government. Boulding, writing in 1950, recognized this possibility on the output side. He was concerned with situations where not all the output of the firm is sold in the same per- iod that it is produced. In these cases he said simple profit maximization is not appropriate, rather the firm should maximize net returns plus the net gain achievable from selling output in future time periods (Boulding, p. 99-101). 1 Edwards considered the gain achievable from reorganizing an initial endowment of resources by either buying additional units or by selling units from the initial endowment (Edwards, 1958, 1959). we will assume that the firm operates in each time period to maxi- mize current profits plus the changes in the net present value of the durable assets. This objective function is consistent with the Edwardian gain function and Boulding's writings. Equation 7 is our expression for the firm's objective function. 44 (7) Gt 8 PchYlt + PYZtYZt " let(xllt + x12t) ’ Px2t(x21t + x22t) _ _ _ O + Px3t(x31t + X329 TUCN1(91::) 1:11chng Fclt _ O _ O _ 0 Pen + Cl1mm Dlt) + cl2(DZt D2t) where PY = the price of output Y. in time period t. j = l, 2. 3t 3 PK = price of nondurable input X1 in time period t. i = 1, 2. it TUCNk(ekt) = total user cost of extracting serv1ces ekt in time period t k = 1, 2 Fckt = fixed costs associated with durable k in time period t.§/ k = 1, 2. “R = gain in the net present value of a unit of the durable Dk' k = 1, 2. for Dk > Dfl a a - NRD (6*, T* ) - P k k k Dk Dkt 0 for Dk < DR S k k k Dk Dkt B O for Dk DR 2 * * ck NRDk(ek’ TDk where P; = acquisition price for Dk' k = 1, 2. kt P; = salvage price for Dk. k = 1, 2. kt 8/ -The "°" notation refers to initial levels, while the "*" notation denotes optimal levels. 443 In determining a k , T* 7' 9* Dk klt * * I: ' NRDk(ek’ TDk) :51 MVPe deklt klt ek1t'o * * 6k2: 9k: + MVP d9 — MFC d6 ek2t k2t ekt kt ek2t=0 °kt=° 1 * —-—-— * + Sk(TDk) (1+rk) TDk k 1’ 2 (1+rk)t 45 * * NRDk(ek’ TDk) = net return to durable k. k = l, 2. MVPekjt = marginal value product of services generated from durable k used to produce product Y.. MFCekt - marginal factor cost of services generated from Dk' * Sk(TDk) - salvage value of durable asset k. k = 1, 2. rk = discount rate for durable Dk' k = 1, 2. kt fixed cost associated with Dk. k = 1, 2. '11 C II Before formally deriving the necessary conditions which determine the optimal value of seven, we will discuss the underlying economic principles which correspond to the mathematical conditions. Though the production decisions and the investment/disinvestment decisions have to be made simultaneously, we discuss them spearately for ease of presenta- tion. Production Decisions In this section, we will discuss the economizing principles which apply to the production activities. Our main concern is with the princi- ples which specify the optimal levels for the inputs in the production processes. ,Our specification of the production processes involves two types of inputs. The first category consists of the nondurable inputs used in producing the final products. The second category consists of the services generated from the durable assets. Specifying the econo- mizing principle for the generation of services involves specifying optimizing conditions for the use of the nondurable inputs in the gen- eration of services. We will discuss the economizing principles for each category separately. Nondurable Inputs in the Final Production Process The basic economizing principle is to match added costs with added returns under appropriate second order conditions to locate the maximum difference between total returns and total costs. For the nondurable input X . lt’ the cost of an additional unit is the price paid for it, P The added return is the addition to the total value of the product X1: produced. Since X1t can be used to produce both Y needs to equate the price of X It and Y2t’ the firm 1t with the added returns from using X in either production process. Equation 8 expresses the economizing 1t principle for xlt' aY aY (3) Px = PY ax 1t 8 FY ax 2t 1t 1t llt 2t 12t Equation 8 indicates that the firm should use X lt until the added returns in each production process are equal to each other and equal to the price of xlt' The second order conditions are guaranteed by the form of Equations 1 through 6. Generation of Services from Durables The basic economizing principle for generating services from dur- ables is to match the marginal value of the services generated with the marginal cost of generating those services. The marginal value of the services generated is their marginal value in producing either Y r 1t° 2t° 1t or th times the marginal physical product of services used in either production process. Y This is calculated as the price of either Y There are several components of the marginal cost of generating services. The first component concerns the cost associated with using nondurable inputs in the generation of services. There are economizing 46 47 'principles which apply to the usage ofnondurable inputs in the genera- tion of services. The principle is to use the input until the value of the marginal unit equals the cost of the marginal unit. The cost of the imarginal unit is simply its price. The value of the marginal unit is the ‘marginal value of using the nondurable in the generation of services for use in producing the final product. This is an instrumental marginal value product and is calculated as the price of the output times the marginal physical product of the services in producing the output times the marginal physical product of the nondurable in generating the ser- vices. Equating the price of the nondurable with its instrumental marginal value product will determine the appropriate quantity to use and will specify the first component in the marginal cost of generating services. The second category to be considered will be labeled user costs. These costs are the opportunity costs associated with the extraction of services during a particular time period. The opportunity cost of extracting services in a particular time period is the present value of those services in a future time period which must be foregone. Some previous authors have recognized this cost category (Lewis, Neal, Keynes); but to our knowledge, none have included it as a component in the produc- tion, investment, disinvestment process. User cost was defined in 1942 (Neal) as the change in the value of the asset as a result of using it during a particular production period. Assuming Neal meant the difference between ending salvage without use and ending salvage value with use, we can express this as Equation 9. (9) TUCN(et) = smilet = 0) - 8(t. at) 48 The marginal user cost associated with this definition would be given as the derivative of TUCN(6) with respect to St. Equation 10 expresses this. 3TUCN(6t) aet aS(t, 9t) aet A broader definition of user cost was presented in 1949 (Lewis). (10) MUCN(et) = Three alternative user costs were identified. Actual user cost would be the maximum of the three alternatives. The first alternative measures user cost as the difference between current salvage value and its value in use. The second possibility is the difference between the current salvage value and the salvage value one period hence if the asset is used during that period. The final possibility is the differ- ence between the value of the services this time period and their value in some future time period which is excluded by using the asset during the current time period. These three alternatives can be expressed as follows. (11) TUCl(6t) = S(to, 6t) - NPV (12) TUC2(9t) = S(to, 6t) - S(tl, 6t) (13) TUC3(6t) - NPVT+dt The following observations can be made concerning these four user cost concepts. First, we should note that neither Lewis nor Neal made the variable rate of use explicit in their presentations. Their presen- tations were in terms of either using the asset or not using the asset. Thus, the explicit inclusion of a variable rate of use represented here as at, the variable services from the asset, is an extension of their user cost concepts. 49 The second point to note is that Lewis' second user cost, TUC2(8t). contains more than the opportunity cost associated with the use of the asset. .It also contains a time component in that it compares current salvage without use with a future salvage with use. To more accurately specify the costs associated only with use, we should compare the sal- vage values at the same points in time. Neal's specification does this, the time variations eliminated by comparing the salvage values with and without use at the same point in time. Thus, Neal's conception of user cost is preferred to Lewis' second alternative. The final point to note is that these alternative formulations can be phrased in terms of being either a "within firm" opportunity cost or an "off-firm" opportunity cost. For an asset, which is specialized as to product, the within firm user cost is the opportunity cost associated with the time dimension, i.e. the user cost of current use is the value _ of future use foregone. Lewis captures this in his third formulation for user cost. The off-firm user cost is the opportunity cost in the use dimension at a particular point in time, i.e. the user cost associated with using an asset within the firm is its value in off-firm use, namely the salvage value. Neal's definition of user cost focuses on this dimen- sion. It is this aspect of user cost which becomes part of the economizing principle for extracting services. The other aspect of user cost will be discussed in the following section on the economics of determining the length of run. A third component of the marginal cost of extracting services deals with the affect that extracting services in any production period has on the economic life of the asset. Extracting an additional unit of service 50 in time t shortens the life of the asset by some amount. With perfect maintenance, there is no shortening of the life of the asset; thus, there is an economics of maintenance and use or service generation which will be discussed in the following section. However, in stating the econo- mizing principle for service generation, we need to account for the marginal effects of service generation on asset life. We label this -cost the opportunity cost of time since it represents the value foregone by current service generation and is similar to Lewis' third user cost. With these three components of the marginal cost of generating ser- vices in mind, we can state the economizing principle for service generation as equate the marginal value product of services with the marginal cost of acquiring nondurable inputs in the production of ser- vices plus the marginal user cost as defined by Neal plus the marginal opportunity cost of time (Lewis' third definition of user cost). Maintenance Activities The basic principle for determining optimal maintenance activities is to equate the marginal factor cost of a maintenance variable with its marginal value product in terms of maintenance. The marginal factor cost is simply the price of the maintenance input. The marginal value of maintenance is the marginal value of the services which the durable asset can render as a result of performing maintenance. This completes our discussion of the economizing principles for the production activities. The following section considers the economizing principles for investment/disinvestment decisions. 51 Investment and Disinvestment Principles The economizing principles for either investing or disinvesting in durable assets can be stated quite briefly. However, the implicit calculations necessary are quite complicated. In this section we shall be concerned with stating the principles. The formal deviation of the necessary conditions will illustrate the calculations involved in applying the principles. For investing in additional units of a durable asset, the firm should match the value of the additional unit of the durable with its acquisition price. The value of the durable is de- rived from the services it would generate over its lifetime within the firm. Both the services generated:h1any time period and the number of time periods are variables determined endogenously. The exact form for the value of the durable is specified when we formally derive the neces- sary conditions. For disinvestments, the firm matches the present value in use with the salvage price of the durable. Again, the present value in use is derived from the services generated in each time period where the optional number of time periods is also endogenously determined. For both investments and disinvestments, the economizing principle relies on the present value of the services generated from the durable as well as on the life of the durable. In the next section we discuss the economic principles involved in determining the optimal life for durable assets. Length of Use Principles When the firm has control over both the amount of services to gener- ate from a durable asset in each time period and the amount of maintenance 52 to perform in each time period, it can control the physical life of the durable. Determining the optimal life for a durable involves matching the present value of using the asset an additional time period with the present value of the cost of using the asset that additional time period. The value of the asset is derived from the services it would render in that additional time period. The.cost of using the asset is the cost of generating the services which includes the change in the salvage value of the asset as a result of use. The firm also needs to account for the change in the salvage value with respect to time. Our formal derivation of the necessary conditions indicates the exact manner in which these factors interact in determining the optimal life for durable assets. Formal Derivation of Necessary Conditions The preceding sections have specified the physical production pro— cesses for the firm we are modeling. The decision making process was also discussed. An objective function was specified and the economizing principles were presented. In this section, we derive the necessary conditions which must hold for the firm to maximize its objective func- tion subject to the physical production relationships. The second order conditions are insured by the form of our physical production relation- ships. For ease of presentation, we separate the production activities from the investment/disinvestment activities. Even though we present the pro- duction and investment/disinvestment decisions separately, the maximization of the firms objective function requires that the necessary conditions for both sets of decisions be solved simultaneously. 53 Production Activities The production activities of the firm affect both the current profit and the investment/disinvestment activities of the firm. The levels of the production activities determine the current profit of the firm di- rectly. The production activities enter the second portion of the objective function indirectly. A durable's value in use is determined from the services generated in each production period over the life of the durable. In this section we specify the necessary conditions which must hold for the firm's production activities to be optimally organized. We will present the conditions which must hold in each time period. A subsequent section will relate the production activities to the invest- ment/disinvestment activities and indicate the simultaneity between the production and the investment and disinvestment activities of the firm. In optimizing its production activities, the firm is concerned with acquiring the optimal quantity of nondurable inputs as well as with ex- tracting the optimal amount of services from the stock of durable assets. In specifying optimal production activities, we are in essence assuming . a fixed level for the durables D t and D For the firm to maximize its 1 2t. objective function as specified in Equation 7, it must optimize its pro- duction and investment/disinvestment activities simultaneously. We separate the activities for ease of presentation only. To maximize the production activities, we form the following Lagrangian which is composed of the current return portion of (7) plus the Lagrangian multipliers times the physical production constraints.2/ 9/ -The theory of constrained optimization has been presented else- where (Beveridge and Schechter). The form of our production relationships guarantee the satisfaction of the second order conditions. 54 (14). L E P Y + P Y - P [X + X ] Y1t 1t Y2t 2t x1t llt 12c - P [x + x ] - P [x + x ] X2t 21t 22t x3t 31t 32c ”TUCN1(°1c) ' TUCN2(62t) ‘ Fclt ' F02: - A - f . 9 ltlylt 1(xllt’ 611: 21c)] ' A2c[Y2t ' f2(x12c’ 912c’ 622x” ‘ ¢1t[°1t ' gl(x21tID1t)] ‘ ¢2t[62t - 32(X22cID2c)] _ yltlTD1 - h1(611, ..., alt, ..., 61TH, x311, ..., X31c’ ..., x31TH)] - 72t[TD - h2(621, ..., 92:, ..., 92TH, x321. ..., 2 x32c’ "" x32TH)] + Blttelt - 911: - 612t] + BZtIGZt - 921: ' 622:] L is maximized by equating its partial derivatives with respect to the variable factors of production and the Lagrangian multipliers with zero. This yields a set of simultaneous equations. The solution of which will yield the optimal quantities for the control variables and Lagrangian multipliers. The following conditions must hold at each point in time for the firm's production activities to be optimally organized within each time period. (15) 3%i;-- PYlt - Alt = 0 (l6) 3%i;-- PY2t - Azt = 0 3f ‘1” 33L. ' ‘qu * “til—1'; ' ° 3f (18) 31%;; - -let + AZtaxl:t o a (19) '3%§IZ" -Px2c + °1£§i§i:" 0 a (20) 33:2: . -Px2t + ¢2t3i§§2 I O (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) _a_L__ ax31c 3L ax32c L 3911: BL aeth 3L 3621: 22c -P x3: Px3t a 1:39 a 2:39 a ltDO a 2t89 A A A A -MUC N1 "MUCN2 - f - 81(X + Y1t + Y2c f1 _ 11: f2 _ 12: f1 21: f2: 22t ( 61:) (e2t f2(X g2(x22t h (e )- 1(xllt Bhl ax31c Bhl 3X 32t 9 91 lDz: 55 «121: + Y21:36 1t 12t’ 012t’ 622t) 21:;‘01:J = ) l 11’ '°°’ h ( e _ llt 621: 2 e21’ - 6 e12:; 22t .., 6 . 9 0 0 3h 1 - ¢1t + Yltae + 611: - 0 1t 3h .— 2t I O + B2t - th) g 0 = 0 1T ’ X211’ .., X ) - H 21'1‘H , X , ..., X ) - 2TH 221 21TH 0 Each of the Lagrangian multipliers in Equations 14 to 36 has an economic significance which we should note. The Langrangian kit is the value of a marginal change in the production of the final output, Y it. 56 Equations 15 and 16 indicate that the optimal organization involves equating the price of the nondurable input X t to the value of its mar— l ginal product. The Langrangian B is the opportunity cost of using the jt services from the durable Dkt in the production of either final output. For Bit’ Equations 23 and 24 indicate that the optimal organization involves having the opportunity cost of using services 6 in the pro- it duction of either output equal to the marginal value product of those services in the production of the other output. Equations 25 and 26 indicate the analogous case for th. The remaining two Lagrangians ¢it and Yit are the opportunity cost of generating services and the current value of altering the physical life of the durable. The generation of services in any time period affects the life of the durable asset; thus, the optimal levels for these Lagrangians are interrelated as indicated by Equation 27 for ¢lt and Ylt and Equation 28 for ¢2t and Y2t' From Equation 19 we determine that the optimal organization will involve having ¢lt equal to the ratio of the price of th to its marginal physi- cal product in the production of 8 This ratio must also equal the 1t' net marginal value of the services generated in each time period as indicated in Equation 27. Upon transposing 27, we get the following expression for ¢ 1t' ' Bfl Px3t ahl (37) q = P + , - MUC (e ) 1t Y1c3811c ahl Belt N1 1t _ 3X3. _ The optimal organization will involve having the above condition hold. The term in the square brackets is our expression for Ylt° It repre- sents the opportunity cost of changing the length of life of the durable Dlt' It is the ratio of the price of the maintenance input to its 57 marginal physical product in changing TDl. If we again look at Equation 27, we can transpose it so as to isolate Ylt° In doing this we substi- tute the expression for blt derived in 19. We see from the following that the optimal organization involves equating the present value of services, Ylt’ with the net value of current services. MUCNl(elt) + (38) Ylt = 3h Similar expressions can be derived for b and Y . 2t 2t . From the above set of necessary conditions, we have a set of 22 equations which must be solved simultaneously to determine optimal levels for the firm's production activities within each time period. we will have a similar set of equations for each time period within the firm's planning horizon, T H For the firm's production activities to be optimally organized over its entire planning horizon, a set of 22 TH first order conditions would need to be solved simultaneously. First order conditions for future time periods would be discounted to the present when solving these 22 TH equations. Our presentation here will focus on the within time period pro- duction activities. These results can be generalized to cover future as well as present time periods by including the appropriate discount factor for future periods and requiring the appropriately discounted marginal conditions to be equal across time periods. By making the appropriate substitutions, the set of 22 equations for each time period can be reduced to the following 8 necessary conditions. For the nondurable xlt’ we have the following. 58 A ail (39) P = P x1: Y1:3X11: sz (40) P = P x1: Y2:3x12: Equations 39 and 40 indicate that the firm should use the quantity of X in the production of Y 1t t that equates the price of X 1: and Y2 1t with the value of its marginal product in the production of each pro- duct. For the nondurable th used to generate services, the following conditions must hold. _ asfi_ 3911: ah1 aeljt 361 (41) Px ' FY 39 ax + Y1:36 ax ' MUCN (61:) ax 2: jt ljt 21: ljt 21: 1 21: j = 1, 2 (42) Px = PY 32F1 ::21: + Y2:3:h2 2:21: ’ MUCH (92:) 3:62 2t jt 23: 22: 2jt 22t 2 22t j = 1, 2. Equations 41 and 42 give the conditions for the optimal quantities of X2t to use in each time period in the generation of services 61t and 9 For X used in producing services 9 t for use in producing Y 2:' 2t 1 lt’ the firm would equate the price of X2t to the instrumental opportunity cost of altering the life of the durable by extracting current services minus the instrumental marginal user cost of generating the services. Similar conditions hold for the use of X t to produce 6 2 l in producing Y2t' Equation 42 gives the similar conditions for x2t to for use I: produce 62t for use in producing either Y1t or Y2t° The optimal maintenance to perform on the durables is given by Equations 43 and 44. 59 P x2: 8g 8f MUC (e ) + 1 - P -——¥L- N1 1: ex . Y 36 . 3h 21: 1: 13: 1 - (43) Px ’ ah ax j ‘ 1’ 2' 3: 1 31: as _. P _ x2: 8g 8f MUC (e ) +--—-3—-- P -——41— N2 2: ax Y 36 ab _ 22: j: 2jt 2 - (44) Px — 3h 3X j - l, 2 3: 2 32: L. aGZt d Equation 43 indicates that the optimal maintenance to perform on durable asset Dlt will equate the marginal factor cost of a maintenance input with the marginal physical product of a maintenance input times the net value of a unit of maintenance. Once the optimal level for the input th is known, the firm can de- termine the optimal quantity of services to generate by using the service generation function. An alternative way of determining the opti- mal quantity of services to generate in each time period is given by the following. 3f Px 3h (45) FY 39 ' MUCN1(91:) + 32t' ' Y1:39 1 j 3 1’ 2‘ j: ljt 81 1: ax21: a: Px2t ahz (46) P =MUC (9 )+-_-Y j=l,2 thaezjt N2 2: aeg2 2:392t ax22: Equation 45 indicates that the optimal organization for the firm will involve having the marginal value product for services 91: used in the production of either Y t or Y2t equal to the marginal cost of l 60 generating those services. The marginal cost consists of the marginal user cost plus the marginal cost of the nondurable th used to generate services less the opportunity cost of the change in the life of the durable. Equation 46 specifies the analogous conditions for services 62t' Equations 39 to 46 specify the conditions which must hold at an optimum. The simultaneous solution of these equations will determine the optimal quantities for the nondurable inputs X and x2t' the optimal 1t quantity of maintenance X and the optimal quantity of services to gen- 3t erate in each time period. The following section relates the optimal production activities to the investment/disinvestment activities. Investment and Disinvestment Activities In the previous section we specified the optimizing conditions for the production activities. In this section we specify the conditions which must be met for the firm to make optimal adjustments to its initial quantities of durable assets. The simultaneous satisfaction of the condi- tions specified in this section with the conditions specified above for the production activities will optimize the firm's gain function speci- fied in Equation 7. In making adjustments to its initial quantities, the firm will want to acquire units of a durable when its value in use exceeds its acqui- sition price. It will want to dispose of units of a durable when its value in use is less than its salvage price. The firm's objective is to maximize the net change in the value of the durable assets as a result of either investments and/or disinvestments. Since we are considering 61 the durables to be discrete units, the optimality conditions cannot be derived via ordinary calculus. We first consider the investment decision. The value of a durable in use is determined by the net present value of the services generated over the life of the durable plus the ending salvage value of the durable. Both the quantity of services to extract in each time period and the num- ber of time periods are endogenous variables. The determination of the maximum value in use for the durable in- volves determining the optimal quantity of services to extract in each time period and the optimal number of time periods to use the durable. The determination of the optimal quantity of services to extract was specified in the previous section. The determination of the optimal length of time to use the durable will be considered in this section. Even though we present these decision processes separately, they are actually simultaneous decisions. We separate them for ease of presenta- tion only. To determine the maximum value in use for the initial set of dur- ables, the following equation is maximized for each type of durable. Equation 47 is a restatement of the final portion of Equation 7. T 9* D klt (47)NRD(6,T)= ks _ MVP de k k Dk :51 6P1: 0 6k1: klt 5%: + _ MVP :19 let 0 Bth k2t eé- P ah kt x2t th 0 RN kt agk kt36k ax2k: 1 l T (1+rk) k (1+rk) Dk 62 where NRDk(6k, T ) = net return to the durable as a function of services and length of use. k = l, 2. MVPe = marginal value product of 6k used in producing Yj kjt in time t. k = l, 2; j = l, 2; t = l, ..., T . Dk MUCkN(ekt) = marginal user cost of generating services ekt. k = 1’ 2' 6k: = 9k1: + 6P2: All other notations retain their earlier meaning. The maximum value determined for Equation 47 will give the value in use for the durable asset. The maximum value is found by determining the optimal amount of services to extract in each time period (eat) and the optimal number of time periods to use the durable (Tgk). The determina- tion of the optimal number of time periods to use the asset depends on the net value of the services generated and the salvage value of the durable. Determining the final time in which to use the durable, in essence, determines the point in time when the firm should disinvest in the durable. It will be advantageous for the firm to disinvest in a durable when its current salvage value exceeds the present value of the future ser- vices of the durable. To apply this disinvestment decision rule to determine the optimal length of time to use a durable asset, we need to determine the optimal T in Equation 47. A method for doing this is Dk to compare the changes in the salvage value of the durable with the net value of the services generated in each time period. This method is a version of the marginal conditions for disinvesting derived by Perrin. He considered the continuous time-perfectly divisible units case. Ours 63 is the discrete time-indivisible units case; thus, we cannot take deriva- tives with respect to either time or the durable asset. Since the firm knows the optimal quantity of services to generate in each time period, the net present value of the services can be ex- pressed in general as a function of the final time period only, i.e. we can express the summation portion of (47) as a function of the final time period. Denoting this as PVSk(T ), we can rewrite (47) as: D k * = (48) NRDk(ek, TDk) vak(TDk) + Sk(TDk) The marginal rule for disinvesting (determining TD ) is to equate k the additions to PVS (T ) with the reductions in S (T ).lg/ If (48) k Dk k Dk were a continuous function of time, we could (as Perrin did) equate the derivative of (48) with respect to time with zero to determine T5 .- How- k ever, our model treats time in discrete units; thus, we cannot take derivatives and can only specify the method for determining Ts . In our k discrete time model, it is unlikely that the additions to vak(TD ) at k T* will just equal the reductions in S (T ) at T* , so we state the Dk k Dk Dk marginal rule as: Use the durable Dk until the additions to PVSk(TD ) k are less than the reduction in Sk(TD ). In other words, the value in k use at T* is greater than the salvage value at T*.; but at (T* + l), Dk Bk ”1: the salvage value exceeds the value in use. The above method determines the optimal length of time to use the durable asset. This optimal length, Tgk, is then used in Equation 47 to determine NRD*, the maximum value in use for the durable. Value in use thus determined is compared with acquisition and salvage value. If NRD; for the initial set of durables (Dkt) exceeds the acquisition cost -lQ/Since the additions to PVSk(T ) and the reductions in S (TD ) are both at time T and both would k be discounted at the same k rate, we need not k consider the discount factor in our marginal rule. 64 for the durable, the firm will consider investing in an additional unit of the durable. To determine if it will pay to invest in the additional unit, the firm recalculates its optimal production activities assuming it owns the additional unit. It then recalculates the value in use of the new quantity of durable, i.e. it recalculates NRDk(6*, T3 ) for the k durable under consideration. We label this recalculated value as ' * * ' * t - * * NRDk(ek’ TD ). If NRDk(6k, TDk) NRDk(6 , TD ) is greater than the acquisitionkcost of the new durable, the firm Sill acquire the additional unit of the durable. The firm repeats the above calculations for each type of durable whose initial level, Dkt’ is such that the value in use exceeds the ac- quisition cost. After completing these calculations, the firm will be optimally organized with respect to acquisitions of durable assets. For firms with more than one type of durable asset, the order in which the durables are considered for investment and disinvestment decisions will have an effect on the optimal investments and disinvest- ments. For example, in our firm with two durables, determining the optimal level of D t given D = D° its initial level, may yield dif— 1 2: 2:’ ferent results than determining the optimal level of D2t given D1t = Dlt or given D equals its optimal level. lt One technique which deals with discrete variables is integer pro- gramming. It appears that most integer programming methods involve some type of complete enumeration of all possible combinations of the discrete variables in seeking the optimal combination (Hillier and Lieberman). For durable assets whose initial levels are such that the value in use is less than their salvage prices, the firm will disinvest in units of the durable until the value in use of the last unit disinvested in 65 equals the salvage value. The firm's disinvestment activities are con- strained by the size of its initial inventory of durables. It cannot dispose of more than it has in its initial inventory. For disinvestments, the firm recalculates its optimal production activities assuming it has one less unit of the durable. It then recal- culates NRDk(ek’ TB ). we label this recalculated value NRD£'(9:, T* ). D k k The firm disinvests in the unit of the durable if NRDk(e*, TD ) k - NRD£'(6*, T3 ) is less than the salvage value of the durable. The firm k repeats these disinvestment calculations for each type of durable whose initial level Dkt is such that the value in use is less than its salvage value. The firm will then be optimally organized with respect to dis- investments in durable assets. Durable asset replacement has received considerable attention in economic literature. Smith has written on the general theory and appli- cations. Faris and Windner and Trant debated the appropriate replacement strategy for orchard trees, while more recently Perrin has presented a theory which focuses on the optimal replacement age. The assumption underlying nearly all replacement literature is that the assets gener- ate services which are perfect substitutes in production. Our more detailed specification of the role of durable assets in production treats durable assets which generate perfectly substitutable services as the same asset. This together with our assumption that acquisition prices exceed salvage prices implies that the firm will never replace an asset with another identical asset. Thus, we do not consider replacement de— cisions in the traditional manner. When a firm replaces a durable with another durable capable of generating closely substitutable services, 66 the firm needs to make both a disinvestment decision for the current durable and an investment decision for the new durable. Both decisions follow the methods presented above. To illustrate the calculations discussed in this section, consider the following situation for our theoretical firm. The firm has an in- itial endowment of both tractors and harvestors. Let us illustrate how the firm would determine an optimal quantity of tractors. The firm has already determined the optimal amount of services to generate from its initial endowment of tractors. These calculations were performed to satisfy Equations 39 through 46. Given these calculations, it remains for the firm to determine the optimal length of time to use the initial endowment of tractors. Tsk is calculated so as to maximize Equation 48. This requires the firm to compare the net value of the tractor services generated in each time period with the change in salvage value as a result of using the tractor in each time period. In early time periods, the net values of the tractor services will exceed the changes in the tractor's salvage value. In subsequent time periods the net value of the tractor's services will decrease and the tractor's sal- vage value will also decrease. Eventually, there will be a point in time such that further use of the tractor will not offset the change in the tractor's salvage value. This point will be T6 and will maximize k (42). The maximum value of the initial endowment is determined as NRDi(ef, T3 ). The firm compares this maximum value with the acquisition 1 . price for another tractor and salvage price for the initial endowment of tractors. If NRDf(6i. Ts ) is less than the acquisition price and l 67 greater than the salvage price, the initial endowment is optimal and the firm will neither invest nor disinvest in tractors. If, however, NRDi(ei, T3 ) exceeds the acquisition price of 1t another tractor, the firm will consider investing in another tractor unit. The additional value generated by the additional tractor is calculated as the difference between NRD*(6*, T* ) and NRD'*(6'*, T'* ) where l . D1t 1 l Dlt NRD'*(9'*, T* ) is calculated in a manner similar to NRD*(T* ) but 1 1 D1t 1 D1': assumes the firm owns the additional tractor. If this difference ex— ceeds the acquisition price of the tractor, the firm will purchase the tractor. This process will be repeated for additional units until the value generated by an additional tractor does not cover its acquisition price. D1!: is less than the salvage value of tractors, the firm would consider dis- If the initial endowment of tractors is such that NRDf(Gi, T* ) posing of units of the tractors. It would recalculate the optimal production activities assuming it had one less tractor to use. It would then recalculate the value in use of this reduced quantity of tractors. The difference between the value of the reduced quantity and the value of the initial quantity is compared to the salvage price of tractors. If the salvage price exceeds the value differential, the firm disposes of a unit of the tractors. This process is repeated until the salvage value does not cover the value of the unit being disposed of or the initial endowment is entirely disposed of. In this manner, the firm determines its optimal organization of tractors. It would carry out a similar process for its second type of durables. As indicated above, the order in which these investment and dis- investment decisions are carried out may affect the final result. For 68 two durables, it would be possible to consider both orderings, i.e. , given D = D° , and then consider 1t 2t 2t , to determine a "global" optimum. When consider altering the level of D altering D2t’ given D t = D 1 It the number of durables is large, the number of combinations which would be required is also large. One operational method for considering the order of investment decisions would be to consider those durables which have the largest divergence between acquisition and salvage values first and then consider durables with successively smaller divergence between acquisition and salvage values. It appears that this type of operationalization could be combined with an integer programming technique to limit the number of enumerations that would be required by the integer program alone. Our theory does not specify an optimal order for altering durable assets. In this section we presented a method whereby the firm would adjust its initial endowment of durable assets. We showed how these investment/ disinvestment decisions are dependent on the related production activ- ities of the firm. The general rule for investments is to invest in additional units of the durable when the value of the additional unit in use exceeds its acquisition price. Calculating the value in use was shown to depend on both the optimal amount of services to extract in each time period as well as the optimal length of time to use the durable. Determining the optimal length of life for the durable in- volved specified a disinvestment decision rule. The disinvestment decision rule which we used was to disinvest when the value added by using the durable an additional time period did not offset the reduc- tion in salvage value. This decision rule corresponds to Lewis' first definition of user cost presented above. we presented it here as a 69 disinvestment decision rule, while Lewis discussed it as a cost of production. In either case, the value of the asset in use must cover its salvage price. we indicated that replacement decisions as they are traditionally formulated are not included in our theoretical model. Rather, the firm makes an investment and a disinvestment decision whenever it replaces one type of durable with another type which generates closely substitu- table services. Finally, we illustrated the calculations that our hypothetical firm would carry out with respect to optimally reorganizing its initial en- dowment of tractors. This example showed how the production and investment and disinvestment activities are interrelated. We noted that the order in which we consider the investment and/or disinvestment activ- ities is important. In the following section, we discuss the feasibility of empirically specifying the model developed in this research. Feasibility of Empirical Specification Applying the model developed in this research to a real world firm would involve specifying empirically the following relationships. 1. The productiOn function for each enterprise the firm is engaged in. 2. The service generation function for each durable asset the firm uses. 3. The relationship between use and maintenance and the physical life for each durable. 70 In addition to specifying the above physical relationships, we would need to know certain economic relationships and values. An important part of determining the optimal amount of services to extract in each time period involves knowing the user cost of those services. Thus, we would need to know the relationship between use and salvage value for the durables. we would also need to know input and output prices for each time period over the firm's planning horizon. We also need to know the acquisition and salvage prices for the durable assets for each time period. Given the above information, we can specify and optimize the firm's objective function. Optimizing the objective function subject to the physical constraints would involve some type of numerical search routine. Several such routines are available (Kuester and Mize). The major problem in specifying empirically our model would center around the lack of data for the service generation function and the physi- cal 1ife-service-maintenance function. The lack of data may necessitate some type of approximation for these functions. Previous researchers, some unknowingly, have approximated the service generation function by assuming a one-to-one correspondence between services generated and the stock value of the durable. Summary In this chapter we developed a model which treated the generation of services from durable assets as an explicit part of the firm's pro- duction process. Our treatment of the production process as a vertically integrated process permitted us to specify the physical pro— cess in more detail than previous writings. In doing this we have 7l moved beyond the writings of Idachaba. The explicit treatment of the services from durable assets allowed us to link the firm's production activities with its investments and disinvestments in durables. We also determined, endogenously, the optimal length of life for the durable assets, as well as the optimal maintenance to perform. We relied on writings by Boulding and Edwards to specify the firm's objective function. We assumed the firm.maximizes the current profit plus the net changes in the value of its durable assets. Maximizing this objective function involved the simultaneous determination of the optimal production activities and the optimal investment/disinvestment activities. The final section of this chapter discussed the feasibility of spec- ifying our model empirically. It was pointed out that the major problem would be concerned with the lack of empirical observations on the rela— tionships between the services generated from the durable and the inputs used to generate those services. The relationship between the physical life of the durable and the services extracted and the maintenance per- formed may also be difficult to estimate empirically. CHAPTER IV SUMMARY AND CONCLUSIONS Summary Managers of agricultural firms are faced with two interrelated de- cisions concerning durable assets. They must decide about the optimal amount of services to extract in each production period. They also make decisions about the optimal stock of durable assets. For an existing firm, these latter decisions involve both additions to the stock of dur- able assets and decreases in the initial stock of durable assets. An important aspect of these decisions involves determining the optimal maintenance for a durable asset and, hence, an optimal life for the durable. Previous attempts to model and analyze theoretically the production, investment and disinvestment decisions of firms have not satisfactorily handled the simultaneous decisions concerning durable assets. Our literature review presented in the second chapter indicated that those people who have recognized the simultaneity of production, investment and disinvestment have simplified their models by assuming a constant extraction rate for services from the durables. This also implies a fixed life for the durables. The objective of this dissertation was to develop and extend that part of the theory of production economies which explicitly recognizes the simultaneous nature of decisions concerning the acquisition, use and disposal of durable assets. In our theory, we do not make the simplifying 72 ‘4. 73 assumption that the services from durable assets are extracted at an arbitrarily fixed rate; rather we determine the optimal quantity of ser- vices internally in our theoretic model. Allowing the extraction rate to vary implies that the economic lives of the durables are not fixed but are endogenously determined. In developing our theory, we moved beyond the static theory of pro- duction economics to a partially dynamic theory of production economics. To aid the presentation of our theory and to permit us to focus directly on the problem at hand, we made some assumptions which limit the dynamic nature of our theory. These assumptions concerned: 1. The degree of knowledge held by decision makers-ewe assumed that decision makers have perfect knowledge about the future. We did not assume that they have had perfect knowledge in previous time periods, i.e. they could have made decisions in the past, the results of which are currently nonoptimal. 2. The role of management in production--we did not include manage- ment in our theory. Our concern was in developing a theory which will aid the decision making process of management. We were not concerned with developing a theory of management. 3. The firm's planning horizon was prespecified-dwe specified the planning horizon for the firm to be the point in time beyond which costs and returns would be discounted to near zero for any relevant positive discount rate. Our literature review revealed that a total dynamics would include imperfect knowledge and the theory of management. We recognize the need to develOp this broader dynamics. That development will include the partial dynamics considered herein. When this development occurs, the 74 results and implications derived from our theory will be modified. How- ever, we leave the consequent modifications of our theory for the future. In addition to the assumptions which limit the dynamics in our theory, we made the following assumptions to simplify the presentation. 1. The production process of the firm was limited to two outputs Y1 and Y , two durable inputs D , their services t 2t t and D 1 2t 61: and 62:, two aggregated nondurable "production" inputs X and x2t’ and an aggregated maintenance input X lt The diagram- 3:' matic representation of the vertical production process is presented in Figure 2. 2. We did not consider tax implications in our theory. The in- clusion of taxes is a modification which can easily be made but would encumber the current presentation. While these latter assumptions limit the scope of our theory, re- laxing them would not alter the basic implications of our theory. Y1: 1Y2: Production Production function 1t function F 191: [ I 62: Production X Production function 2t function Figure 2. Two-tiered vertical production process. ll:- PC- 75 The literature review in our second chapter traced the development of part of the theory of dynamic economics. We concentrated on the branch of dynamic economics concerned with specifying the production, investment and disinvestment process assuming perfect knowledge of the future. An important part of this branch has been concerned with the theory of fixed assets. In our review we discovered several writings on either the theory of production or investment theory. There were fewer writings on the theory of disinvestment. Edwards (1959) and Glenn L. Johnson (1958, 1972) have both written on the theory of production, in- vestment and disinvestment. However, they assumed a fixed extraction rate for services from durables. The theoretical contributions made in the present dissertation build on and extend the work done by Carlson, Neal, Lewis, Georgescu-Roegen, Kenneth Smith and Idachaba. The mathematical specification of our model was presented in Chapter III. we modeled the physical production process for our theoretical firm as a vertically integrated system. The services from each durable are produced from the stock of the durable by using one nondurable input. This production relationship is characterized by the following equation. (49) okt = 8kt(x2k:|Dkt) ' for k = 1, 2. The services from the durables are inputs along with a nondurable input in the production of the final output. These production relationships are characterized by Equation 50. (50) Y (X for j = l, 2. jt ' f1 ljt’ 6ljt’ 623:) In addition to the production function specified above, we modeled the physical relationships among the life of a durable, the services ex- tracted and maintenance performed. we characterized this relationship by the following. 76 (51) TDk = hk(ak1, ..., ekt, ..., ekTH, x3k1, ..., x3kt, ..., x3kTH) for k = l, 2. The relationships in Equations 49 to 51 are our model of the firm's physical processes. Equations 49 and 50 form the vertical production process, while Equation 51 accounts for the physical depreciation or appreciation process for the durable assets. Our basic motivational assumption was that the firm seeks to maxi- mize an objective function subject to the physical constraints imposed by the production process. We did not consider financial constraints such as a limit on credit availability. The objective function we specified for the firm reflects both the current production decisions and the affect of current and future pro- duction decisions on the current value of the durable assets. Our objective function is a modification of both an objective function specified by Boulding in his book, A Reconstruction of Economics, and of the objective function Edwards used. The objective for our firm is specified in Equation 7 and restated here as Equation 52. (52) G: 3 13131: + P3:232: ’ P2:1:(x11: + x12:) " P2:2:0‘21: + x22:) - - _ _ o P3:3:0‘31: + x329 TUCN1(°1:) TUCN2(62t) Fclt - O _ O _ o FCZ: + “1(D1: D1:) + “2(D2: D2:) There are two things to note about our objective function. First, it measures the gain in the net present value of the firm in each time period as the sum of the net receipts from the current production activities plus the net gain in the value of the durable assets. The second thing to note is that in calculating the net receipts from the current production activities, we have included a component in the cost of current production which is generally overlooked. This cost is the 77 user cost of the durable asset represented in (52) by TUCNk(ekt)’ k 8 l, 2. One aspect of user cost is the use depreciation of the durable asset. The use depreciation may be partially or totally offset by the maintenance activities of the firm. The maintenance costs are given as an index of price times the aggregated maintenance input. The user cost in our objective function represents the change in the ending salvage value of the durable as a result of using the durable during the period. This cost was formalized by Neal (1942) after Keynes conceived of it in 1936. There is a second user cost involved in extracting services in the current time period. This cost is the highest discounted value of the future services foregone by current use of the durable. This cost com- ponent was identified by Lewis (1949) and is a portion of our disinvest- ment criteria for the firm. The gain in the net present value of the firm is achievable by in- vesting in additional units of the durable, disinvesting in some or all of the units currently held, or by reorganizing the usage pattern for the currently held durables. When investing in durables, the gain is the difference between the current acquisition price and the net present value of the future services generated from the durable. When disinvest- ing in durables, the gain is the difference between the current salvage price and the net present value of the future services generated from the durable. When reorganizing the usage pattern of the durables, the gain is the change in the net present value of the future services gener- ated from the durables. The variables a1 and 02 in Equation 47 take on the appropriate values depending on the investment/disinvestment activities of the firm. 78 The optimization of (47) as presented in Chapter III involves the simultaneous determination of the optimal production activities and the optimal investment/disinvestment activities. we separated, for dis- cussion purposes only, the derivation of the optimality conditions for the production activities from the derivation of the optimality condi- tions for the investment/disinvestment activities. In the theory they are simultaneous. Our derivation of the optimality conditions for the production activities indicates that the following conditions must hold at an optimum. 1. For the nondurable inputs, the marginal cost of the input must equal its marginal value in use. The marginal value in use of the nondurable in the final production process is its marginal value product, while its marginal cost is simply its price. Equating these two determines the optimal quantity. The mar— ginal value in use of the nondurable used in the generation of services is determined as an instrumental marginal value pro- duct. An instrumental marginal value product is measured as the marginal physical product of the input used in the production of an output times the marginal physical product of that output used as an input in a higher level production process times the price of the higher level output. This corresponds to the concept of an input's value in a vertically integrated pro— duction process. The marginal cost of using the nondurable involves three components. The first component is the price of the nondurable. The second component is the instrumental mar- ginal user cost of generating services. The third component is 2. 79 the opportunity cost of economically altering the physical life. This final component is essentially an instrumental maintenance cost. The optimal quantity of the nondurable is determined by equating its instrumental marginal value product with the sum of the three marginal cost components. The optimal quantity of maintenance to perform is determined by equating the cost of a maintenance input with its marginal value product. The marginal value product of maintenance is given as the marginal physical product times the marginal value of main- tenance. The value of maintenance is the value of the services which can be generated from the durable as a result of perform— ing maintenance. The optimal quantity of services to generate in each production period is determined by equating the marginal value product of the services in producing the final outputs with the marginal cost of generating those services. The marginal cost is com- posed of three elements. The first element is the marginal user cost incurred by using the durable to generate services. The second element is the net cost of using the nondurable input to generate services. This component recognizes the aggregative nature of the nondurable input by taking the price of the input net of the opportunity cost of adjusting the life of the durable. The final element reflects the opportunity cost of economically reducing the physical life of the durable by generating current services, i.e. maintenance costs. The marginal cost of gener- ating services is the sum of all three elements. 80 Equations 39 through 46 in Chapter III contain the formal mathemat- ical expressions for the optimality conditions which must hold at each point in time for the production activities. This results in a set of 8ffiisimultaneous equations. Because the production, investment and dis- investment activities are simultaneous, the levels at which the above optimality conditions will be met will be influenced by the investment and disinvestment activities of the firm. The optimality conditions for the investment/disinvestment decisions of the firm derived in Chapter III can be summarized here as follows. 1. The firm optimally organized with respect to investments in durable assets when the net present value of the last unit of durable invested in exceeds its acquisition price while the net present value of the next unit of durable, if it were acquired, would not cover its acquisition cost. The net present value of the durable asset is derived from the services generated in each production period over the economic life of the durable plus the final salvage value. 2. The firm is optimally organized with respect to disinvestments in currently held durables when the net present value of the last unit of durable disposed of is less than its salvage value, while the net present value of the next unit to be dis- posed of exceeds its salvage value. In meeting these conditions, the firm cannot dispose of more durables than it has in its initial endowment. Our seemingly complicated optimality conditions for investments and disinvestments were necessitated by our assumption that the durables are "lumpy" and by our treatment of time as a discrete rather than a 81 continuous variable; thus, we could not derive marginal conditions in the usual sense. We assumed that each unit of durable is accompanied by an inventory statement which indicates its physical condition. We noted the importance of the order in which the investments and disinvestments are made and the influence of this order on the optimal investment and dis- investment activities. We did not specify the optimal order in which to consider investment and disinvestment decisions. The net present value of the durables used in the investment/disin- vestment decisions is derived from the net value of the services generated by the durable over its economic life. In calculating the maximum value in use, the firm controls both the amount of services to generate in each time period and the number of periods or the economic life of the durable. The simultaneous determination of the optimal amounts of services to generate from each durable in each period was specified in the discussion of the production activities. The determination of the optimal number of periods to use a durable asset was specified in Chapter III. To summarize that presentation, the firm will use the durable until the change in the salvage value from one time period to the next is greater than the value of the services to be generated during that time period. This determines the point in time be- yond which it is not advantageous for the firm to use the durable asset. Thus, determining the maximum value in use for the durable assets was shown to depend on both the optimal amount of services generated in each production period and the optimal number of production periods. The optimal organization for the durable assets will depend upon the order which they are altered. We did not determine the optimal order for altering durable assets. 82 Our more detailed theory of the production, investment and disin- vestment process permitted us to treat the durable asset replacement decision as a combined investment and disinvestment decision. Summary of Theoretical Advances The additions to the theory of production, investment and disinvest- ment made in this dissertation build upon and extend previous work done by Edwards and Glenn L. Johnson in the area of investment and disinvest- ment with fixed extraction rates, Idachaba in the area of production with variable extraction rates, Georgescu-Roegen in the area of durable assets in production and Perrin in the area of investments in durables with variable lifetimes. In our theory we have relaxed Edwards' fixed extraction rate; we have specified, in greater detail, the production process considered by Idachaba. This permits us to link the production process with the invest- ment/disinvestment process-~something neither Georgescu-Roegen nor Perrin did explicitly. By specifying the production process in greater detail, we were able to identify more precisely the manner in which durable assets enter the production process. Our conception of the production process as being vertically integrated permitted us to move beyond Idachaba's work. we considered only the services from the durables to be inputs in the pro- duction of the final outputs whereas Idachaba considered both the stock of the durable and the services from the durable to be inputs in the final production processes. As a consequence of considering the production process in this manner, we were able to identify a cost of production and its composition 83 which is usually overlooked--namely, the user cost of generating services from durable assets. This has important implications for firms which practice marginal cost pricing. Previous analyses would indicate a lower marginal cost than an analysis based on our theory. A further consequence of our vertically integrated production pro- cess is in the area of supply response. Our analysis indicates that firms may either expand or contract their supply by using their durable assets either more or less intensely rather than by investing or dis- investing in durable assets. An explanation of the perceived lack of a supply response by pro- ducers to changes in output prices was offered by Edwards (1959) in his theory which incorporated divergent acquisition and salvage prices for inputs. Keynes (Chapter 6) suggested that aggregate output could be varied without a corresponding change in the levels of productive inputs. Our theory suggests that with variable extraction rates for services from durables, producers may respond to changes in output price by altering supplies even with divergent acquisition and salvage prices for the durable inputs. Producers, in our theory, could alter the amount of services extracted from durable assets to either increase or decrease quantities supplied in response to a price change. Our theory provides the micro foundation for Keynes' aggregate response. In addition, we have extended the Edwardian analysis. In muCh of Georgescu-Roegen's (1972, a,b) recent work, he has recog- nized the difference between the durable stock or fund and the flow of services which can be derived from the stock. In this dissertation we have specified, in greater detail, the actual process whereby services are extracted from the stock of durable assets. we conceived of the 84 services as being produced from the stock by using nondurable inputs. Our modeling of the service generation process in this manner permitted us to link investments and disinvestments in durable assets with the pro- duction activities of the firm. Our more detailed specification of the service generation process extends Georgescu-Roegen's earlier writings on the stock/flow conversion problem. Our determination of the optimal investment and disinvestment strategies for the firm relied on comparing an asset's value in use with its acquisition and salvage prices. In computing the asset's value in use, we considered the optimal amount of services to generate in each time period, the optimal maintenance to perform and, hence, the optimal number of time periods to use the asset. In developing our criteria for determining the optimal economic life for durable assets, we extended Perrin's (1972) work to consider the "lumpy" durable-~discrete time case. Our basic criteria is the same as his: Continue to use the durable until the change in its salvage value offsets the change in its value in use. However, our results appear to be more complex because we did not con- sider continuous time nor perfectly divisible durable assets. Our determination of the asset's value in use was specified in more detail than Perrin did in his work. We left the issue of the optimal order in which to alter durable assets unresolved. Areas for Future Research While the research in this dissertation contributes in its own right to part of the theory of production economics, it is hoped that it will also provide the basis for continued development of the theory of dynamic production economics. It is also hoped that the current research will 85 provide the groundwork for better applications of production economics to the problems faced by agricultural producers. The major areas for future disciplinary research would involve re- laxing the assumptions which restricted the dynamic nature of the current research. These assumptions involved risk and uncertainty and the man- agerial process. Relaxing our assumption of perfect knowledge would be a natural ex- tension of this research. The optimality conditions derived in this research would be affected by the relaxation of this assumption. Thus, it would be necessary to derive a set of optimality conditions for the firm operating under uncertainty. Our assumption concerning the managerial process restricted our re— search to a particular branch of dynamic economics. Research is needed in both developing the theory of the managerial process and in merging that branch with the branch being followed in this dissertation. One aspect of optimal investment/disinvestment decision making con- cerns the order in which durables are altered. We did not specify an optimal order in this research; thus, further work is needed to develop a criterion by which the optimal order for altering durable assets can be specified. Empirical research which would examine the supply response implica- tions of our theory is needed. We indicated in the previous section the nature of the supply response that our theory implies. 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