OPTIMAL REPLACEMENT INTERVAL AND DEPRECIATION METHOD OF A COMBINE ON A REPRESENTATIVE DRYLAND GRAIN FARM IN NORTHCENTRAL MONTANA by Alfons John Weersink A thesis submitted in partial fulfillment of the requirements for the degree of Master of Siene in Applied Eonomis MONTANA STATE UNIVERSITY Bozeman, Montana Marh 1984
ii APPROVAL,, of a thesis submitted by Alfons John Weersink This thesis has been read by eah member of the thesis ommittee and has been found to be satisfatory regarding ontent, English usage, format, itation, bibliographi style, and onsisteny, and is ready for submission to the College of Graduate Studies. Date Chairperson, Graduate Committee Approved for the Major Department Date Head, Major Department Approved for the College of Graduate Studies Date Graduate Dean
iii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulflllment of the requirements for a master's degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without speial permission, provided that aurate aknowledgment of soure is made. Permission for extensive quotation from or reprodution of this thesis may be granted by my major professor, or in his absene, by the Dean of Libraries when, in the opinion of either, the proposed use of the material is for sholarly purposes. Any opying or use of the material in this thesis for finanial gain shall not be allowed without my permission. Signature----------------- Date
iv ACKNOWLEDGMENTS I wish to thank my major advisors, Dr. Daniel Dunn, for his time, enouragement and interest, and Dr. Steve Stauber, for his personal efforts and professional guidane. Thanks, also, to the remainder of my graduate ommittee: Drs. Myles Watts and Osar Burt. I would aiso like to extend my appreiation to Rotary International for providing the initial impetus to attend graduate shool. Speial thanks is due to my fellow graduate students whose friendship will always be remembered along with the good times they provided. Finally, I would espeially like to thank my best friend, my wife Maureen.
v TABLE OF CONTENTS Page APPROVAL........................................................ STATEMENT OF PERMISSION TO USE.................................. ACKNOWLEDGMENTS............................................... TABLE OF CONTENTS............................................... LIST OF TABLES.................................................... LIST OF FIGURES................................................... ABSTRACT... ii iii iv v vii ix x CHAPTER 1 INTRODUCTION.... Introdution.... I Purpose....... 3 ( 2 LITERATURE REVIEW.... 4 Literature Review-General Replaement Priniples... :.. 4 Dynami Programming Definitions and Conepts.... 10 Literature Review of DP Replaement Problems.... 13 3 FORMULATION AND IMPLEMENTATION OF EMPIRICAL MODEL.... 18 The General Deision Model.... 18 Representative Farm...,.... 21 The Empirial Problem.... Stages.... 26 26 States.... 26 Deision Alternatives...,.... Expeted Immediate Return.... 29 29 Disount Fator.... 38 Transitional Probabilities.... 38 Terminal Values.... 41
vi TABLE OF CONTENTS-Continued Page 4 RESULTS... 42 Results.................................................. 42 Cost of Capital............................................ 50 Cost of a Major Breakdown................................... 54 5 SUMMARY AND CONCLUSIONS................................ 58 Summary... 58 Conlusions............................................... 61 Limitations............................................... 63 REFERENCES...................................................... 65 APPENDIX... 69 (
vii LIST OF TABLES I. Depreiable Assets on the Farm Exluding the Combine.................. 24 2. Variable Operating Costs Per Are for a Representative Dry-land Grain Farm in Northentral Montana................................ 25 3. Deision Alternatives Available in DP Replaement Model................ 30 4. Probability of a Major Breakdown Ourring at Various Mahine Ages......................................................... 32 5. Remaining Market Value of Combine at Various Ages................... 34 6. Perentages for Investment Credit Reapture.......................... 36 7. Distribution of Random Prie Levels................................. 40 8. Optimal Poliy and Total Expeted Costs in Stage 30 for a Prie of$1.50... 44 9. Optimal Poliy and Total Expeted Costs in Stage 30 for a Prie of $3.50...................................................... 46 10. Optimal Poliy and Total Expeted Costs in Stage 30 for a Prie of $4.50....................................................... 48 11. Optimal Poliy and Total Expeted Costs in Stage 30 for a Prie of $6.50...................................................... 49 12. Optimal Replaement Age and Depreiation Shedule for Asset Presently Depreiated Under ACRS for Various Disount Rates............ 52 13. Optimal Replaement Age and Depreiation Shedule for Asset Presently Depreiated Under 5 Year Straight Line for Various Disount Rates................................................. 52 14. Optimal Replaement Age and Depreiation Shedule for Asset Presently Depreiated Under 12 Year Straight Line for Various Disount Rates................................................. 53 15. Optimal Replaement Age and Depreiation Shedule for Asset Presently Depreiated Under 25 Year Straight Line for Various Disount Rates................................................. 53
viii Tables Page 16. Optimal Replaement Age and Depreiation Shedule for Asset Presently Depreiated Under ACRS for Various Opportunity Costs of Breakdown.................................................. 56 17. Optimal Replaement Age and Depreiation Shedule for Asset Presently Depreiated Under 5 Year Straight Line for Various Opportunity Costs of Breakdown................................... 56 18. Optimal Replaement Age and Depreiation Shedule for Asset Presently Depreiated Under 12 Year Straight Line for Various Opportunity Costs of Breakdown................................... 57 19. Optimal Replaement Age and Depreiation Shedule for Asset Presently Depreiated Under 25 Year Straight Line for Various Opportunity Costs of Breakdown................................... 57
ix LIST OF FIGURES Figures Page 1. Relationship of hronologial time and stages in dynami programming................................................... 11 2. Probability of a major breakdown ourring at various mahine ages................................................... 32 3. Remaining market value of ombine at various ages..................... 34
X ABSTRACT Eonomi unertainty is one of the foremost problems in agriulture and introdues many omplexities into the deision making proess. To aount for these risks and unertainties in the replaement problem, a model is fommlated within a dynami programming framework and applied to a typial ash grain farm in northentral Montana. The deision riterion used under onditions of risk is the minimization of osts assoiated with eah asset through the firm's planning horizon. The asset under study is a ombine and the optimal replaement deision regarding this asset is based on the stohasti nature of winter wheat pries. Transition probabilities for prie hanges are alulated from a single equation prie predition model. The other state variables are deterministi and inlude fifteen asset ages and sixteen tax onditions. Together, they ompletely summarize the osts assoiated with the ombine. The optimal deision minimizes the expeted immediate osts and those from the n-1 stage proess whih are a funtion of the state variables and deision alternative seleted. Besides being able to keep or replae, the deision variable for replaement also inludes all the possible depreiation shedules and investment inentives whih an be used on the new asset. The optimal poliy seleted is dependent upon the state of the proess. The aelerated ost reovery system is used in high inome years after five years of servie and a longer reovery period when returns are very low. The evidene also indiates the value of investment tax redit. The pratial and wide ranging results obtained through the use of stohasti dynami programming ontributes to the body of theoretial knowledge on replaement analysis.
1 CHAPTER INTRODUCTION Introdution The tehnologial revolution in agriulture is a well-known phenomena whih has drastially hanged the struture of the setor. The impetus for adoption of the new hanges are provided for by the ability one has to expand output and lower prodution osts. Sine agriulture in both the U.S. and Canada developed under onditions of plentiful land and a sarity of labor, the innovations have onentrated in expanding the apaity of labor. Suh labor-saving tehnology is primarily of a mehanial nature rather than biohemial. Eonomies of sale in on-farm prodution are diretly related to mehanization and they an only be realized through farm enlargement and labor displaement. The result is an agriulture setor that is heavily dependent on mehanization to sustain its prodution. Other strutural hanges whih have aompanied this tehnologial revolution inlude growing apital and redit requirements and a rising ratio of farm prodution expenses to gross fam1 inome. With the trends expeted to ontinue and prodution to beome more heavily dependent on purhased inputs, greater emphasis will be plaed on finanial management. Among these apital outputs used by agriulture today, the average farm's annual equipment ost is mathed only by the harge for land use. The opportunity thus exists for an inrease in farm profits or, alternatively, finanial ruin depending upon how this setor of total farm expense is managed.
2 Proper investment planning of fann equipment onsists of analyzing two important problems. The first involves deiding if mahinery servies should be aquired through ownership, leasing or ustom hire. The latter two alternatives for ontrol are not onsidered here. Instead, this study fouses on the seond problem of asset replaement over time. To properly analyze the replaement problem, the investment deision should be ompared to others available to the finn. However, this depends on suh fators as the amount of apital aumulated and operator goals whih in turn transfonns the problem into one dealing with finn growth. Suh an analysis is beyond the sope of this study, so to keep the fous on asset replaement, only a partial analysis of the real problem an be onsidered. Sine the mehanization of agriulture is nearly omplete, the purhase of a new asset results from a need to sueed an older mahine whose servies must eventually be replaed if the prodution system is to ontinue. Besides being no longer reliable, the present asset may be replaed if it has beome obsolete or if its operating osts have beome exessive. Even though eonomi savings will result with replaement based on the above reasons, there still frequently exists a relutane on the part of managers to supplant physially satisfatory equipment. On the other hand, many fanns use the purhase of a new asset to try and elevate their omparative soial status despite the fat that the reasoning indues earlier replaement than is warranted. Letting suh intangible onsiderations get entangled in the final investment deision results in a replaement age different from the proper eonomi one. The optimal time between purhases is detennined by the basi marginal priniple of replaement theory whih ompares the gains from keeping the urrent asset for another period with opportunity gains whih ould be realized from a replaement asset during the same interval. From this deeptively simple riterion arises the real problem of speifying all the relevant ost elements. Traditionally, the rising variable osts of repair and main-
3 tenane were added with the delining fixed osts as determined by net investment to alulate aumulated osts. Reent works have added the important effet of inome taxes in deision analysis and parameters to aount for inflation and the asset's true remaining market value. While these are determinants of ost, their impat on the firm's investment deision is also greatly influened by the eonomi environment surrounding the firm. Due to the inherently unstable nature of the agriulture setor, unertainties with regard to new tehnology and risks with respet to returns must be reognized as important fators in analysis. These risks and unertainties introdue many omplexities into the deision making proess and are an important influene on replaement analysis. Purpose The purpose of this researh effort is to develop a deision making model whih will fous on the effets of eonomi unertainty in the evaluation of optimal farm equipment replaement deisions given the present tax laws and struture. The results should provide farmers with a profit maximizing deision riterion and may aid poliy makers in identifying the impat of various tax methods on replaement. In order to aomplish, this, the speifi objetives of this study will be to: 1. develop the general methodology for analyzing replaement deisions and then adapt a dynami programming proedure where the seletion of an optimal poliy is dependent on stohasti variables, and to 2. apply the model to a representative ash grain farm in northentral Montana where the asset is a ombine and the optimal replaement interval and depreiation shedule for this asset is based on the stohasti nature of winter wheat pries.
4 CHAPTER 2 LITERATURE REVIEW Literature Review-General Replaement Priniples Martin Faustman (1849) was the first to fully develop the onept of net present worth when disussing the forest management problems of rotation length and reation of a normal forest. Faustman used present disounted values to put a fair prie on forest land whih is omprised of both the land and of all inome and expenditures assoiated with the forest. This priniple of disounted ash flow has beome the basis for solving many investment deisions inluding optimal replaement patterns. Unfortunately it was not until Fisher's artile in 1906, that an eonomist put forth the idea of disounted revenues. With the delay, the first replaement artiles were not presented until 1923 by Taylor and 1925 by Hotel!ing. They determined the eonomi life of an asset with one yle by maximizing the present value of the output minus the operating ost of the asset, the interest on the salvage value and the assoiated rate of depreiation and dividing this sum by the mahine's rate of prodution. The minimum total unit ost of the produt defines the eonomi lifetime of the asset and this is found through substitution into the value funtion at time zero. The derivation is possible beause they assumed total dependene of operating ost on the value of the mahine. Preinreih (1940) was one of the first to deal speifially with replaement in eonomis sine most previous disussion of the topi was done in depreiation artiles. He feels that the Taylor-Hotelling riterion for eonomi replaement had severe limitations beause it did not onsider relevant dynamis. To orret this, Preinreih studies a number
5 of situations in whih an asset may be under three lassifiations; sope of replaement, input and output limitations and eonomi onditions. He onludes that replaement theory will have a separate solution for every kind of rigid sarity and for every volume of limited supply. In the ase of demand, the problem is simplified into making the ost per unit of outpl!t a minimum, whih is the Taylor Hotelling proposition. In all other ases, the entrepreneur should maximize profit per unit of input where the shortage is felt. When he ombines all sarities, Preinreih states "that exess profits must be made a maximum in terms of a omposite index of produtive ativity, not with referene to any single ingredient" (p. 36). In his 1937 artile, Samuelson shows that the value of apital invested in an asset will at all times be equal to the apitalization of the subsequent inome stream disounted at the market interest rate. As a result, the market prie of an asset is idential to its apitalized value. In addition, he dismisses Boulding's proposition (1935) that rational investors should maximize the internal rate of return over the whole period of an investment. Samuelson proves that given the market interest rate, an operator should hoose a replaement age that will maximize the present value of the assoiated inome stream. The result is true for varying rates too sine with "the time shape of interest being given an,d inome known, the apital invested up to any time is always equal to the value of the (investment) aount at that time, the value being a apitalization of subsequent inome" (p. 487). In one of the first artiles demonstrating the basi proedures involved in determining the optimum replaement pattern for agriultural assets, Faris uses three types of enterprises of a sequential nature in a 1960 JFE publiation. He follows the priniple that the "optimum time to replae is when the margirtal net revenue from the present enterprise is equal to the highest amortized present value of antiipated net revenue from the following enterprise" (pp. 761-762). For an operation that will be replaed several times a year suh
6 as attle finishing, he uses a disount rate of zero in whih ase the highest average net revenue is used as a basis for omparison rather than the amortized present value. In examining the longer prodution period enterprises, Faris inorporates the interest on the unpaid balane of the establishment osts in determining net revenue for operations in whih revenue was realized by the sale of the asset and for ones in whih there was a flow through the life of the asset. In both ases it was found that if marginal net revenue for the present asset was hanged, the amortized present value of the new asset would hange by the same amount thus having no effet on optimum replaement pattern. The impliation of this result is that fixed osts an be left out of suh alulations. In a subsequent omment on the preeding artile, Winder and Trant (1961) argue that the opportunity osts should not only inlude the usual elements whih Faris used. but also the foregone earnings of the time to apply the asset in onsideration. In their ritiism they use a situation with a zero disount rate and a seond with a positive rate of time preferene. They define opportunity osts as alternative inome possibilities and time preferene proper as the preferene for inome in one time period rather than another. They found in the no time preferene situation that equating the marginal net rate of profit per unit of time (marginal value produt) to the average net rate of profit pqr unit of time (marginal fator ost) will maximize profit per unit of time. When time preferene proper is onsidered, the optimum replaement age is where the marginal net rate of return per unit of time equals the average net rate of return per unit of time multiplied by the onstant (I +q)ln(l +q) l/ q where (l+q) = (! + r)n. With the time preferene disount rate (r) greater than zero, a shorter prodution (n) is implied than that of the first situation. Chisholm (1966) laims that the two previous artiles overlooked some of the elements of marginal ost with respet to time. There is agreement that the fixed and variable osts involved should be ompounded at an appropriate interest rate in order to ompare osts and returns inurring at differing points of time but Chisholm adds that money tied
7 up in the atual replaement asset under study is also part of the relevant opportunity osts. He suggests that the annual running ost, the interest on total revenue obtainable from sale of the asset and the amortized value of net returns from the next asset are elements to be inorporated in marginal ost. Optimum replaement age an then be seleted whih maximizes net present value of future profits for a perpetual sequene of prodution periods and not for just a single period. Perrin (1972) ties together past developments and presents a general model of asset replaement whih applies to both appreiating and depreiating assets in a number of different settings. With a single asset, he found that aquisition age is irrelevant and the optimum replaement age is that at whih the residual earnings plus hanges in asset value (marginal revenue) equals the interest whih ould be earned by selling the asset (marginal opportunity osts). If it is to be replaed by a series of idential assets, the opportunity ost of delaying the future earnings of these assets must be inluded. Replaement will then our when the net flow of benefits equals the flow whih ould be realized by immediate replaement. If the new assets are tehnologially improved, their higher apitalized value will indue earlier replaement than the previous senario. In reality, the relevant elements are disrete values rather than ontinuous and using the marginal riterion in a disrete world will often lead to a one year error in alulation of optimum replaement interval. In lieu of this, Perrin states that finding the present values for eah replaement year may be a better evaluating proedure. The operator must hoose the eonomi life whih will maximize these net present values of future inome streams from the asset. Perrin notes that this maximum will be zero due to the ation of market fores. If the value of the residual earnings is temporarily positive, input pries will be bid up and/or output pries will fall with expanded prodution
8 until the rent is eliminated. The effet of this proess on optimum replaement age will depend upon the elastiity of supply of those assets of various ages. Perrin also examined the theoretial impliations of hanging the disount rate on replaement. With appreiating assets suh as a forest, a higher rate will result in earlier replaement. However this general statement is not neessarily true for other assets and the effet will depend upon the shape of the earnings flow. The appropriate hoie for the disount rate depends on the irumstanes at hand. The ost of apital may be used as an indiation of the return on alternative investments if the owner faes a perfet apital market. If there is no suh market, then his personal preferene rate may be appropriate. A third alternative is the internal rate of return. Sine this value is determined by the market pries of the inputs, market fores will drive up the asset prie if the internal rate of return is above the market rate for ativities of similar risk. The latter rate an be viewed as the appropriate disount rate if equilibrium pries of all inputs are expeted to prevail by the first replaement date. Chisholm (1974) was one of the first to analyze the effets of inome tax poliy on the optimal timing of farm mahinery replaement. To do so, he develops a disrete time period model in whih the firms are assumed to minimize the present value osts 9f obtaining a onstant flow of idential mahinery servies over an infinite planning horizon. A firm will ontinue to maintain the urrent asset until the marginal ost of holding that mahine for another year exeeds its amortized ost. His results show that higher rates of disount are assoiated with longer replaement intervals and higher inome tax rates with shorter replaement intervals. Sine the annuity value of the tax saving from an investment allowane is a dereasing funtion of age, Chisholm onludes that suh a tax redit will signifiantly shorten replaement intervals. However suh deisions are only slightly influened by the method of depreiation used.
9 Kay and Rister (1976) extended Chisholm's work on tax poliies. Using a similar model but under United States rather than Australian tax regulations, they found that the after tax disount rate had the largest impat on replaement while the inome tax rate auses only slight differenes in optimal poliy. Like Chisholm, they onluded that the depreiation method had little effet. They also found that though the tax regulations have a small impat on replaement age, they do lower the present value of any poliy whih has enouraged the trend towards larger equipment. Kay and Rister listed some of the possible reasons why predited replaement age in their study and other previous ones is longer than that atually observed partiularly for farmers with a high disount rate. These inlude using the wrong pattern of repair osts or not adequately overing the ost assoiated with a loss in reliability as the mahine ages. A shorter replaement poliy may also be explained by ontinual tehnologial improvements and the farmer's desire for larger mahines. In their ontinuous time model, Bates, Rayner and Custane (1979) proved that the rate of inflation an have a signifiant impat on the optimal age of replaement. The inlusion of inflation is justified on the basis of two fats. First, sine taxes are based on histori osts, a signifiant level of inflation will redue the real value of dep!eiation allowanes. Seondly, the reeipts and benefits from tax allowanes are lagged and thus depreiated. In addition, resale pries for equipment will often be greater than the unexpired depreiation osts during inflationary times whih results in a gain in ordinary inome in the form of depreiation reapture and possibly apital gains. Bringing these fators into the model, they onlude that "the higher the rate of inflation, the greater the real value of osts and the higher the optimal replaement age; but in eah ase, the absolute differene made dereases as the rate of inflation beomes higber" (p. 333). The effet is greater, partiularly on osts, the higher the tax rate.
10 Reid and Bradford ( 1983) ontinue the improvement of the previous models by speifying a more generalized equation to estimate remaining market value whih along with tax inentives is the most important parameter influening agriultural replaement deisions. Using trators, they inlude more situation speifi explanatory variables suh as horsepower, realized new farm inome, the trator make and indexes for tehnologial hange. They use this remaining value equation in a disrete model similar to that of Kay and Rister but with additional terms for investment redit reapture and tax gains. This adjustment gave them results with a wider range of replaement ages than previous studies. As an example, they found that larger trators and ones of a ertain make have shorter replaement intervals beause they retain a higher market value relative to their initial osts than do smaller horsepower mahines and other manufatured models. They also examined the effets of the Eonomi Reovery At of 1981 (ERTA-81), detailed explanation of whih will be provided later. Replaement intervals are shorter with no expensing under ERTA-81 than with expensing emphasizing the value of investment tax redit. The ability to redue taxable inome with expensing does not offset the redued value of a lower investment tax redit. The replaement ages are shorter under ERTA-81 without expensing than under the pre-erta-81 onditions while the effet with the expensing option depends on the remaining value equation used and on the disount rate. They also found that under the new onditions, the after-tax ownership osts are higher beause the tax rate redution more than offsets the gain in the write off value of a more rapid depreiation. As a result, there is a smaller inentive to buy larger mahines though there are more funds available for reinvestment. Dynami Programming Definitions and Conepts The dynamis involved in the farm firm deision making proess must be inluded if the previous work on replaement is to be extended. To inorporate the effets of risk and
11 unertainty on future events, this paper uses dynami programming to analyze the replaement deision. Dynami programming is an optimization tehnique whih solves a multistage deision problem by onverting it into a problem requiring the solution of sequential single period problems rather than a programming algorithm that solves for a speifi type of problem (Dreyfus, p. 213 ). It is a bakward mathematial indution proess that seeks to find the sequene of deisions that will maximize, or in this ase minimize, the appropriately defined objetive funtion. The multistage deision proess is divided into time intervals or stages as shown by Figure 1 with a poliy deision required at eah one. Eah stage has a number of states assoia ted with it that desribe the urrent situation at any stage. The sum total of all relevant information about the proess at a given stage is defined by the magnitudes of the state variables. The poliy deision made at eah ontrols the state in whih the proess will be found in the following stage. The transition from one state to another an be made with ertainty or stohastially aording to a probability distribution. time (t) I 1 stage (n) n Figure 1. Relationship of hronologial time and stages in dynami programming.. In dynami programming or DP as it will be referred to in the rest of the study, the objetive funtion must be one of Markovian nature. Given the state of the proess at a given stage, the deision proess depends only on the state of the proess in that stage and not on the state at preeding stages. Thus, for DP to be appliable, the set of state variables must inlude all neessary information required to ensure that the optimal poliy depends only on the present stage and state and not upon how one got to that state. To satisfy the Markovian requirement, the researher must ahieve adequate realism of state desription whih will vary depending upon the depth of analysis.
12 Bellman is redited with the formal oneptualization of dynami programming in 1951 and his priniple of optimality lies behind the operation of the DP tehnique. An optimal poliy has the property that whatever the initial state and initial deision are, the remaining deisions must onstitute an optimal poliy with regard to the state resulting from the first deision (1961, p. 57). This priniple allows one to divide the total problem and solve the last deision stage, then work bakwards and solve the seond-to-last deision until the first deision is solved. The solution proedure moves bakward stage by stage through the use of a reursive relationship. It identifies the optimal poliy for eah state at the present stage, given the optimal poliy for eah in the future time period is available. If these optimal retums in the next stage are known, one would make the deision that maximizes (or minimizes) the total of the immediate return and the optimal return from the proess in the next time period starting in the new state. Solution of the following reurrene relation yields the sequene of deisions that optimizes the objetive funtion; where, = total value of a n-stage proess where an optimal poliy is used and the initial state of the proess is i Max = the maximum operator k = the set of deision deision alternatives the expeted immediate returns given the ith stage, kth deision altemative and the nth stage of proess B = the disount fator the transition probability for being in the jth state in stage (n-1) given the proess is in the ith state and the kth deision is made in stage (n) of proess = the total value of a (n-1) stage proess where an optimal poliy is used and the initial state of the proess is j.
13 Dynami programming provides a great omputational saving over exhaustive enumeration to find the optimal sequene of interrelated deisions, espeially for large problems. However, it does require formulating an appropriate reursive relationship for eah individual problem. DP is not desribed by a set of equations in a standardized format nor does a pre-programmed omputational algorithm exist. Instead, it is a general type of approah to problem solving that requires the development of equations fitted for eah distint situation. The literature review to follow will outline the various approahes different authors have used to examine the replaement problem with DP as the optimization tehnique. Literature Review of DP Replaement Problems Appropriately enough, it was Bellman (1955) who published the first paper using DP to determine the optimal replaement age of equipment. He did not use a speifi situation but did set up the following funtional equation; f(t) = Max[K:r(t) - u(t) + af(t + 1)] [P:s(t) - p + r(o) - u(o) + af(l)] With no tehnologial improvement in equipment or pratie, the only state variable is mahine age. The return assoiated with keeping the mahine for another time ppriod (K) is the output of the mahine (r) minus the upkeep for that year (u) plus the future disounted return (af(t + 1)). The deision to purhase a new mahine (P) involves the return linked to the new asset (r(o)- u(o)) and the disounted return when it is a year old (af(l)) plus the differene between the salvage value and the purhase prie (s(t)- p). It is assumed that the trade-in value and output of the mahine are dereasing funtions of age while its assoiated ost is inreasing over time. The optimal replaement poliy found by solving the above reurrene relation will maximize the overall return from the mahine. Bellman adds that if tehnologial improvements inrease the future returns from the same mahine, absolute time must be inluded as another state variable.
14 In his textbook, Howard ( 1960) onsiders an automobile replaement problem over a ten year planning horizon. The state variable is desribed by the age of the ar in three \ month periods and a replaement deision is made at eah of these intervals. The first deision alternative, k = 1, is to keep the present ar for another three months and the other, k > 1, is to buy a ar of age k- 2. The funtional equations are muh the same as Bellman's, however Howard has inluded the probability that a ar of a ertain age will survive to the next year without inurring a prohibitively expensive repair. A ar that suffers a major breakdown is sent diretly to state 40 indiating that it is worn out. The result is 40 states with 41 alternatives in eah and thus 40 to the 41st power possible replaement poliies. This example is presented in a textbook by Bellman and Dreyfus (1962) along with additional explanation of the original Bellman artile whih involved an infinite time prob-!em. In ontrast, they present a tehnique with an example to solve a finite duration proess by means of the iteration of a reurrene relation. This allows them to inlude ost variations as a funtion of real time as well as of age. They also desribe a variety of replaement problem formulations. For example, the purhase of a used mahine may be inluded as a third deision aiternative if one an define the appropriate ost funtion fo.r suh a transation. The DP replaement problem ould also be designed to ontain the posssibility of an overhaul with the inlusion of another state variable whih desribes the age of the asset at the last overhaul. In this problem, it must be assumed that the repairs will give the mahine harateristis of a younger asset depending upon the age and the effort devoted to the overhaul. Burt (1963) formulates the multistage deision proess of replaement in a different way. He defines the stage as the number of replaements yet to be made during the firm's planning horizon and the state variable as the number of years in that horizon. The age at whih to replae the equipment of the urrent stage beomes the deision variable. Using
15 this format, Burt finds that in the disrete ase, the optimal age to replae the urrent asset is where net marginal return of the next year is less than, and at the urrent age is greater than or equal to, the present value of returns under an optimal replaement poliy redued to a perpetual annuity. With a ontinuous model, optimal replaement age is where the marginal net returns are equal to the perpetual annuity equivalent of net returns. The net return funtion must be independent of the optimal replaement poliy for the model to be appliable. It was Burt, along with Allison (1963), who first indiated the potential appliation of dynami programming for farm management deisions. The use of DP was illustrated by examining the wheat-fallow deision on a dry land farm. The amount of soil moisture at seeding was defined as the state variable upon whih the deision to plant a rop or leave the land fallow was based. Though it is not a speifi replaement problem as suh, the artile does learly present the formulation and use of DP in agriulture. They also show how the optimal poliy onverges and how to derive long run expeted yields under a speifi poliy by obtaining the probabilities of being in a partiular state after a number of transitions. In another paper, Burt (1965) extended the analytial results of replaeme11t theory to the ase where the asset is subjet to involuntary replaement due to hane events. Age is again the state variable used to indiate the asset's expeted future eonomi produtivity. However Burt inludes both a voluntary replaement ost (prie of new asset minus terminal value of used one) and a ost for replaement aused by random fators. The latter reflets the salvage value under failure, the ost of a new mahine and the average proportion of periodi net revenues reeived under involuntary replaement. It may also be assumed that the gross returns from an asset are onstant, thus simplifying the problem to one in whih osts are only onsidered. In this model, Burt has an infinite planning horizon in whih the revenue, ost and probability parameters remain onstant. This implies
16 that the replaement age will be onstant for all mahines and is unaffeted by the age of the initial asset. As a result, the optimal poliy is one that maximizes the expeted value of returns from the first asset held and expeted present value of returns from all future assets. Using a marginal approah instead of the aforementioned disrete method, one should maintain the urrent asset until the expeted marginal net revenue minus expeted marginal ost of planned replaement is less than the weighted average net revenue from the potential replaement. The weights are produts of the disount fator and probability of survival for eah age whih is not aounted for in the measure of risk in the disount fator. Burt extends this general model to ases in whih the revenue assoiated with the first asset is different and for various probability distributions of asset failure. He also goes through the model when the maximum rate of return is the appropriate riterion for optimization rather than present value whih would our under onditions of apital rationing. The traditional replaement models examined so far have not aounted for the possible situation where the replae:nent age of the urrently held asset influenes the value of future assets. Burt aommodates this relationship in his 1971 artile on the optimal timing for learing brush and srub timber from pasture and range. As the length of time between pasture improvements inreases, the brush and timber ontinually deterio.rate the pasture and in the proess redue quasi-rents of the range in the renewal yle after their removal. With this senario, Burt formulates the model in a method similar to his 1963 artile. The stage of the proess is the number of pasture renewals yet to be made in the planning horizon rather than a disrete time period. One state variable is the number of years remaining in the planning horizon and the other is the length of the immediately preeding renewal period. With this fonnat, an optimal replaement age is one that maximizes the present value of all quasi-rents from the remainder of the planning horizon. Sine this time the use of DP as a useful analytial tehnique has grown. Textbooks suh as Dreyfus and Law (1977) even ontain a hapter devoted to replaement models,
17 yet there remains an apparent lak of popularity for DP among agriultural eonomists whih Burt (1982) has reently addressed. Using the past works ited as a basis for the methodology, this study will show the pratiality and flexibility of dynami programming when applied to the problem of optimal replaement in agriulture.
18 CHAPTER 3 FORMULATION AND IMPLEMENTATION OF EMPIRICAL MODEL The General Deision Model With the substitution of apital for labor projeted to ontinue in agriulture, greater emphasis will be plaed on replaement. Use of apital inputs require annual ash outflows whereas, to some degree at least, returns to farmer labor an be postponed in years of adversity. The result is that the farming setor is beoming inreasingly sensitive to flutuations in inome as the use of purhased inputs inrease. Mahinery represents the largest setor of apital inputs on the farm so it also has a large impat on the viability of individual enterprises. The aquisition of a major fann asset requires a substantial investment on the part of the owner and so is often purhased with the use of borrowed funds. A ash ommitment is neessitated regardless of the irumstanes surrounding the ability to pay whih explains in part the farming setor's. vulnerability to inome shortfalls. The farmer may delay purhases to avoid the above situation during low inome periods. However, if his returns are high, the ability to derease taxable inome through depreiation and investment inentives may offset the ash osts assoia ted with aquisition. Regardless of the level of returns, the impetus for replaement may be brought about by reliability loss and repair osts that are inreasing with the age of the asset. The farmer must take into onsideration all these fators and ost elements when ontemplating the replaement deision. Noting the inreased sensitivity of agriulture to inome flutuations due to apitalization and the inherently unstable nature of returns in farming, any study on optimal
19 replaement deisions in this setor of the eonomy must be onsidered in a stohasti framework. If there was no unertainty surrounding inome, the analysis would turn into a single-stage deterministi problem. However, the variability of returns requires the problem to be formulated as a sequene of annual deisions in whih the owner must deide whether to replae or keep his ombine for another time period. He is unsure of the possible prie levels in the next year but urrent onditions are an indiation if returns are assumed to be jointly distributed over time. The new information determines the relative value of tax dedutions whih the owner must weigh against purhase osts and inreasing repairs when making his replaement deision. The problem is thus properly viewed as a sequential deision proess. The proess is summarized at any point in time by the stohasti prie level, and the age of the asset and the depreiation shedule and inentives used. These state variables ompletely desribe the ombine and form the basis on whih the deision mle is made. The optimum replaement interval is then determined by solving the sequene of deisions whih will minimize the present value of all ash flows assoiated with the om bin e. Sine it is diffiult to distinguish whih returns are attributable to a partiular asset, the model is formulated so as to minimize these flows rather than as a profit maximizing problem. "When a fir;m's prie or output deisions are independent of its replaement deisions then ost minimization and profit maximization are ompletely separable" (Chisholm, p. 776). As Preinreih noted, the age annot be determined separately from the eonomi life of eah mahine to be used in the firm's planning horizon so the ri.terion seeks to minimize the osts assoiated with all assets during that time spetrum. The preeding desription is formulated in terms of a general model with the following notation and definitions. The model is represented in terms of disrete time variables and is evaluated by alulating the present value of all relevant osts assoiated with eah
20 deision alternative and for eah possible replaement year and depreiation shedule. All variables are on an annual basis and the stage of the proess is denoted by n where n = 0, I,...,N. S = set of all possible asset ages and tax alternatives (deision variables) at the present stage, u = the partiular deision variable seleted from the set S, s = the state variable whih designates status of ombine at the present stage in terms of age and depreiation method, p = set of expeted produt pries whih are state variables. The transition of the ombine status is detenninisti and does not involve the prie state variable and is denoted as follows; s(n-1) = h(u, s) The transition of the prie vetor is stohasti and does not involve the deision variable, u, or the present physial and finanial status of the ombine and is desribed mathematially as follows; 4 --+--+-+ p(n-1) = g (p, v) where, ~ ~ v = is the vetor of random variables where there is an element of v assoia.ted with ~ eah element of p. ~ ~..,. g = is the vetor of funtions assoiated with the elements of p and v. With these definitions, the reurrene of the dynami programming formulation for the replaement problem is as follows; where, 4 -+ -+-+-+ fn(s, p) = Min [R(u, s, p) +!lefn-1 ((h(u, s), g(p, v))] uz:s the expeted value of disounted osts from an-stage proess under an optimal replaement poliy when the initial state is desribed by s, the finanial and physial status of the ombine, and p, the vetor of prie state variables,
21 _,. R(u, s, p) = the expeted osts in stage n whih are a funtion of age, tax alternatives and the vetor of expeted pries, ~ = the appropriate disount fator (1/1+(1-t)r)) where r is the real rate of interest and tis the marginal tax rate, E = the expetation operator. Representative Farm The setting for the replaement model is a northentral Montana dry-land grain farm and the asset under onsideration is a self-propelled ombine harvestor. A representative farm has been onstruted for analysis rather than grouping results to avoid aggregation bias. While the firm struture for grain farms may be more standardized than for many fa1m types, there will still exist disrepanies between individual enterprises and the desribed representative farm. Despite this, it is felt that the assumptions and model oeffiients are very harateristi of this dry-land grain farming region. A ombine represents one of the major farm assets for this farm type, so proper replaement of this mahine is essential to the firm's viability. Historially on these farms, the owners hauled in and stored their grain. The protetion from the weather eliminated the timeliness fator involved in threshing, enabling the ommon pratie of joint ownership of threshing equipment. But with the shortage of labor brought about by World War II, farmers swithed inreasingly to threshing diretly in the field. The onern of losing a rop due to prolonged bad weather aused onflits among the o-operators of a threshing ring and resulted in a move towards individual ownership. The prosperous years following the war were marked by an expansion of farm size and a major wave of new farm mehanization. To own the mahines and/or to purhase bigger mahines, farmers had to expand their grain areages whih in turn required additional mahinery. This proess has slowed somewhat during the urrent period so tehnology in this study is assumed to be onstant through the firm's planning horizon. Thus. eah ombine of whih the farmer is the sole
22 owner is replaed by an idential mahine based on the urrent tehnology. With inflation assumed to be nonexistent, eah ombine arries a $80,000 prie tag and has a 160 horsepower engine that will handle a 24 foot grain header. The owner is assumed to be married with two hildren and neither his wife nor himself earn any supplemental inome from off-farm employment or from rents, royalties or trusts. Thus, their sole means of support is derived from growing grain on 2400 ares of rop land. The owner has a 90 perent equity in his land base whih is valued at $500 per are. Eah year, winter wheat will be sown on 1000 ares, barley on 500 ares and the remaining ground left as summer fallow. This typial ropping pattern is ommonly used in order to redue risk during planting time and to inrease soil moisture. The sequene is fixed as are the rop yields with wheat fields presumed to average 35 bushels per are and the barley rop 50 bushels per are. The stohasti nature of returns are thus aounted for solely by the prie level. Yield ould also be inluded as another state variable but there is no dynami trend assoiated with it. Sine the firrn operates in a perfetly ompetitive market with prie and output independent of one another, the inlusion of yield variability to enhane the authentiity of risks in returns is not signifiant enough to justify the addition of another state variable. While some of the ripple effet on returns will be missing, it is easier to assume average yields and then plug in different values later if neessary. Prie times the output determines gross farm inome for this study, and to simplify the omputations, barley prie is expressed in tenus of wheat prie equivalents through the following regression equation; BP =.72736 +.47822 (WP)' (.0512) (.04478) (!) 1 Annual pries for the last seventeen years were onverted into present day dollars. Soure: Montana Agriultural Statistis.
23 where BP is barley prie per bushel in urrent dollars and WP is winter wheat prie. The adjusted oeffiient of determination is.8828 and Durbin Watson statisti is 2.0323. The enterprise osts are assumed to be deterministi. The mahinery omplement and its usage per are are summarized in the following table for a si.rt:tilar size farm in the northentral region of Montana. 2 To obtain the ownership osts assoiated with the equipment, some arbitrary assumptions were made. First, the appraised value of the new assets were deflated by the pries paid index for trators and other farm mahinery to determine original purhase prie. 3 The seond assumption involved grouping these purhases into a restrited number of aquisition dates and depreiating the mahines purhased during the same ti.rt:te period together. These dedutions were determined by multiplying the basis or original investment ost by the perentages given under the present aelerated ost reovery system for the appropriate lassit1ation of3, 5 and 15 year property. The owner is assumed to have a 90 perent equity in his mahinery omplement similar to his land. The variable operating osts listed in the following table were generated on the basis of the ropping pratie assumed to be used in the region 4 The other expenses listed in the table that are neessary to alulate net farm profit are not well doumented. They were obtained through an interview with the operator of a farm omparable to, the one being studied. The amount of extra labor hired, utility bills and the building and liability insurane figures were values that this individual had experiened in the past and expets to fae again in the future. The remaining values in the table are itemized dedutions whih are needed to ompute taxable inome. They will ordinarily hange with inome levels as outlined by the Wall Street Journal. 5 However the small variation in their amount through 2 Data obtained from an unpublished Montana Agriultural Experiment Station Bulletin dealing with ost of prodution on Montana farms aording to region. 3 Indexes obtained from Inputs; Outlook and Situation. United States Department of Agriulture, Eonomi Researh Servie, June 1983, p. 17. 4 Costs are from the same unpublished Experi.rt:tent Station bulletin as above. 5 Figures obtained from the Wall Street Journal, 8 Deember 1982, p. I.