A TECHNIQUE OF ADJUSTING MARGINAL VALUE PRODUCTIVITY. ESTIMATES FOR CHANGING PRICES Thesis for the Degree of M. S. MICHIGAN STATE COLLEGE Gerald Ion Trim? 1954 ...... This is to certify that the thesis entitled A Technique of Adjusting Marginal Value Productivity Estimates for Changing Prices presented by Gerald Ion Trent has been accepted towards fulfillment of the requirements for _MS_ degree in We Ma' rofessor Date October 7, 195).; 0-169 ; _ , ll mum; 1112"!"me Lm I II 1| In I'll M I _ _ . A TECHNIQUE OF ADJUSTING MARGINAL VALUE PRODUCTIVITY ESTIMATES FOR CHANGING PRICES AN ABSTRACT Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Economics by Gerald Ion Trant 19Sb Approved by Gerald Ion Trant ABSTRACT A Technique of Adjusting Marginal Value Productivity Estimates for Changing Prices The purpose of this study was to evolve a technique of adjusting estimates of margizal value productivity, of the various input and investment categories of farm businesses, for conditions of changing price. I It was believed that such a technique would.be useful to research nd extension workers in farm management since it would obviate in part the necessity of costly annual surveys to obtain current estimates of fanm business resource productivity. Marginal value productivity estimates were derived by fitting a b2 b b "Cobb-Douglas" function of the form X1 ' aX2 3 7 13 . . . . X7 . This equation which is linear in logarithims was fitted in that form by the method of least squares regression to derive estimates of the exponents or elasticities of the input categories with respect to gross income. Estimatesof marginal value productivity were derived from the rela- tionship MVP = bi :(Y) where E(y) is the geometric mean of the anti- i log of gross income from the predicting equation ii is the geometric mean of input category Xi. The resultant estimates were adjusted by applying price indices of the relevant factor and product categories to the estimated MVP'S, so that for inputs measured in physical terms the adjusted MVP of h Gerald Ion Trant i aX-' = . Xi = IY(t=n) - bi 'jfLLEng and for inputs measured in dollars the i(t=O) bi adjusted MVP of Xi = IY(t=n) ° abl X(tzoll_ where IY is a x1 (t=0) - Ixi(t=n) Ph n F- (P n \] price index of the form 42:. 0‘03 and IXi is a similar price 1 =1 191: 1-?er 3] 1 index. Data were collected from 65 purposively selected farms in Northern Lower Michigan for the calendar year 1952. The input categories, their estimated marginal value productivities during 1952, and their estimated marginal value productivities for 1953 were: Marginal Value A‘Marginal Value Prdduct Input Category Product 1952 Adjusted for 1953 ‘_Price Conditions X2 Land 8 9.05 3 7.68 X3 Labor 15.11 12.81 Xh Machinery .h029 .3390 X5 Livestock-Forage .1?15 .1217 X6 Expenses .8593 .7292 X7 Animal Units of Housing 68.6b 58.?1 Capacity Tentative conclusions were that labor, expenses, and livestock- forage investments were yielding low returns in both years, and that since increasing returns to scale were indicated by a sum of exponents greater than one proportionate increases in the other input categories are suggested. Gerald Ion Trent Adjusting estimates of marginal value productivity for price changes by means of price indices appears to have the following characteristics. The metlmd is easily understood and fairly inexpensive. djustments between categories are indicated. However, it should be noted that intra category adjustrents due to price changes are obscured and that adjustments due to changes in such factors as technology and institutions are disregarded. A TECHNIQUE OF ADJUSTING MIRSINAL VALUE PRODUCTIVITY ESTIIATES FOR CHANGING PRIJSS A Thesis Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Economics by Gerald Ion Trant 19Sb ‘n ACKNOWLEDthNTS The author wishes to express his obligations to the many people who assisted in the development and construction of this piece of work. Particular thanks are due Dr. G. L. Johnson for his continuous inspiration and incisive criticisms in the preparation of this thesis. The constructive criticism and suggestions of the following members of the Department of Agricultural Economics, Dr. Boger, tr. Brown, Dr. French, Dr. Sorensen and Dr. Wright were eSpecially useful. Thanks are also due Mrs. Agnes Cepus and Mrs. Carol 1220 of the secretarial staff for typing the original manuscript and to Mrs. Dorothy Norford for typing the final copy. Finally, the author wishes to thank his wife, Norma, for fitting the several regreSsion equations and providing encouragement when it was most needed. 343493 ' CHAPTER I. INTRODUCTION . . . . . . Organization of The Thesis . . . . II. THEORETICAL BACKGROUND . Orientation to Relevant Static Economic Theory Classification of Economic Theory. . . . . . . Static Factor-Product and Factor-Factor Analysis PPOduc‘toion MCI/5.0118 0 0 . 0 0 0 0 . . Value Productivity and Profit. . . . . The Categorization of Multiple Inputs. The Pricing of Inputs. . . . . . . . . Factor-Factor Analysis and the Index Number PTOblemee....00.0000000 Changing Marginal Value Productivities Due To Conditions of Changing Price. . . . . . . III. THE SAMPLING TECHNIQUE AND THE Requirements of the Sampling Empirical Techniques Used to Conditions. . The Area and The Sample. Soils. . . . . . . . . Climatic Factors . . . Present Land Use . . . The Sample Farms . . . IV. AN EVALUATION OF THE STATISTICAL THE FUNCTION . . . SAMPI‘E O C O O 0 Technique . . . . . . Approach the Required RESULTS AFTER FITTING 0.0.0....00.0.00.. Description of Technique . . . . . . . . . . . . . . t” 11" 11’ {\D \n 10 16 18 19 23 27 27 28 30 3O 32 33 3h 36 36 C HAL-’TER Rejection of the First Set of Coefficients . . . The Third Function, Appraisal and Rejection. . . The Statistical Results on Fitting the Fourth FuDCtion. O O O O 0 O O O O O O 0 O O O O O A Comparison of Traditional and "Cobb—Douglas" Estimates of Value Productivity . . . . . . Usual Organization, and Statistical Tests of Significancee00000.00.00.... v. THE ADJUSTMENT OF THE 1952 MARGINAL VALUE PRODUCTIVITY ESTIMATES FOR 1953 PRICE COI’JDITIONS AND GROSS INcoms . IndiCQSRequiI‘ed............. Computation of the Indices . . . . . . . . The Adjusted.largina1 Value Productivities Interpretation of Adjusted Marginal Value Productivities. . . Prediction of Gross Income VI. SUMMARY AND CONCLUSIONS. . Summary. . . . . . . . . Conclusions. . . . . . . APPENDIX A . . . . . . . . Supplementary Tables . . APPENDIX B. Calculation of APPENDIX C. Calculation of Price Indices . Adjusted Marginal PTOdUCtiVity Estimates 0 O O O O O O O O O O O 0 Value B IBLImRAPHY O O O O O O O O O O O O O O O O O O 0 PAGE 37 38 39 in 143 ha he 50 52 52 53 55 55 S7 60 61 7O 72 7h _-fl TASE: II. III. IV. V. VI. VII. VIII. IX. LIST OF TABLES PAGE Comparison of Actual Estimates of bi's and the bi's Required to Yield the Esti.ated Minimum Marginal Value Products . . . . . . . . . . . . . . . . . . . . hh 65.27 Percent Confidence Limits for Marginal Value Productivity Estimates . . . . . . . . . . . . . hS Comparison of Marginal Value Productivity Estimates for 1952 and the Adjusted Estimates for 1953 Price Conditions . . . . . . . . . . . . . . . . . . . . . . 52 Value Weights, Classes of Items in Machinery Investment Category, 30 Selected Farms, Northern Michigan, 1952 . 61 Value weights, Classes of Items in LivestockéForage Investment Category, 30 Selected Farms, Northern Michigan, 1952 . . . . . . . . . . . . . . .b. . . . . 62 Value weights, Classes of Items in Cash Expense Input Category, 30 Selected Farms, Northern Michigan, 1952 . 63 Value Weights, Classes of Items in Cross Receipts, 30 Selected Farms, Northern Michigan, 1952. . . . . . . . 6h Price Relatives, and weighted Price Relatives of the Classes of Items in the Machinery Investment Category, 30 Selected Farms, Northern Michigan, 1952 . . . . . . 65 Price Relatives, and Weighted Price Relatives of the Classes of Items in Livestock—Forage Investment Category, 30 Selected Farms, Northern Michigan, 1952 . . . . . . 67 TA DIE PAGE X. Price Relatives, and Weighted Price Relatives of the Classes of Items in the Cash Expense Input Category, 30 Selected Farms, Northern Michigan, 1952 . . . . . . 68 XI. Price Relatives, and weighted Price Relatives of the Classes of Items in Gross Income Category, 30 Selected Farms, Northern Michigan, 1952 . . . . . . . . 69 FIGURE 2. 3. 14. LIST OF FIGURES Total Physical Product, All Inputs Variable. . . . . . The Three Stages of a Production Function. . . . . . . Location of the High Profit Point Using X1, With X2, . . . . , Kn Fixed . . . . . . . . . . . . . . . ISO-Product Contours With Superimposed Iso-Cost Line . ISO-Value Product Contours With Superimposed ISO-Cost Lines. . . . . . . . . . . . . . . . . . . . . . . . Outline Map of Michigan Showing Counties From Which SMPICWESDram..0.0......0....0. 11 1h 20 31 GENERI INTRODUCTION There has been considerable work done in the field of agricultural economics to derive estimates of marginal value productivities, for various input categories of farm businesses.1 Although there have been several mathematical functionsavailable to research‘workers in this field of endeavor, most of the literature has dealt with applications of specific type of exponential2 equation known as the "Cobb-Douglas" function to farm management data. For descriptions of the technique of applying this function the theses done by Toon3 and WSgleyh are good examples. 1 For additional information on the subject see: Bradford, L. A. and Johnson, G. L., Farm Management Analysis, John Wiley and Sons, New York, New York, 1953, chapters 16, 17, 18, 19; Ready, E. 0., Economics of Aggigultural Production and Resource Use, Prentice Hall, New York, New ork, 1952, Chapters 3, h, 5, 6; Jensen, 3., et al, Input-Output Relationshi s in Milk Production, Technical Bulletin Number 1 , United States Department of Agriculture, lashington, D. C., 19h33 Spillman, I; J., Dec of the §§ponentia1 Yield Curve in Fertilizer Egperiments, Technic Bulletin Number 3&8, U.§.D.A., Washington, D. C., 1933; Tintner, G. H., ”A Note on the Derivation of Production Functions from Farm Records," Econometrics Vol. 12 number 1, pp. 26-3h. Tintner, G. H., and Brownlee, O. H., "PFoduction Functions Derived from Records," Journal of Farm Economics, Vol. 28,19hh, pp. 566-571; Fineup, D. F., Resource Productivity on Montana Dzijand Crop Farms, Mimeo. Circular number 66, Montana State College, Agricultural Experiment Station, Bozeman, Mentana, 1952. 2 It could probably be equally well described as a power function see: Griffin, F. L., Introduction to Mathematical Analysis, Houghton Mifflin 00., Boston, 1935. p. 33. 3 Toon, T. 6., Marginal Value Productivities of In uts Investments and Expenditures on Upland Grayson County Farms Dur 19 unpublished M.S. Thesis, university of Kentucky, Department of Agri— cultural Economics, 1952. h Whgley, R. V., Marginal Productivities of Investments and Expenditures, Selected Ingham Count! Farms, 1952 unpublished M.S. Thesis, Michigan State College Department of Agricultural Economics, 1953. In his thesis Toon outlined some of the advantages and disadvantages of using this particular function in making marginal value product estimates. One disadvantage of which he makes no mention has been the high cost of the estimates. Most of the estimates have been derived for a one year period, in the sense that the data collected have only covered that time Span. Consequently, the estimates have tended to be valid only in the year covered by the data. As a result, eXpensive year to year surveys have been required to keep the estimates current. It is the purpose of this study to extend the applicability of marginal value productivity estimates. This is done by applying price indices for different years to the marginal value productivity estimates derived for the year for which the data were collected. Such a procedure reduces costs by making it unnecessary to conduct expensive annual surveys in order to have annual estimates. Organization of the Thesis The relevant economic theory, its relation to the problem at hand, and the theoretical conditions for adjusting marginal value productivity estimates for price changes by means of price indices will be developed in Chapter II. Some of the more important empirical requirements for deriving estimates of marginal value productivity from farm business data by means of a "Cobb—Dbuglas" production function.nill be discussed in Chapter III. In the same chapter the actual empirical techniques used in this study will be outlined. Chapter IV will be an appraisal of the statistical results on fitting the function to the datr. The technicue of adjusting the estimates of marginal value productivity and the resultant values of the adjusted estimates will be delineated in Chapter V. A summary and general conclusions will be presented in Chapter VI. CHAPTER II THEORETICAL BACKGRQUND Orientation to Relevant Static Economic Theory Classification of Economic Theory. A useful classification which covers most of existing economic theory divides it into four broad categories: Micro static economic theory lacro static economic theory Micro dynamic economic theory Macro dynamic economic theory There have been various levels of development and integration achieved within and between these various categories. Both micro static, and macro static economic theories have been rather well deve10ped and integrated. On the other hand the dynamic systems of theory, although approaching more closely the real world in some of their assumptions than the static system, have yet to achieve a comparable development and integration. A.major difficulty associated with static economic theory has been the nature of the assumptions made. By assuming as fixed.most of the changes or variables associated with the real world, the system has been simplified at the expense of reality. Thus technology, wants and preferences, asset distribution, and institutions, have been posited as immutables. At the same time individuals have been assumed both rational and omniscent. Dynamic systems of economic thought have arisen when one or more of the assumptions of the static system have been relaxed. Although the dynamic systems may be more realistic in their assumptions, the S attendent complexities have inhibited their usefulness to a large extent. Despite the fact that the static system yields an admittedly unrealistic picture of the real world it has given valuable insights into the work- ings of particular parts of the economic system. One of these areas is factor-factor and factor—product analysis. This area of the static economic system is concerned with problems of resource combination and allocation in the production of a single product. In this thesis, static factor-factor and factor-product theory is applied to an empirical problem by the joint use of mathematics (as represented by the "Cobb-Douglas" function) and statistics. Static Factor-Product and Factor-Factor Analysis The problems of static factor-factor and factor-product analysis are respectively; 923, the determination of proportions in which inputs should be combined to yield a maximum.of a product for a given outlay or inputs. And, 332, the detenmination of how much of the various inputs, combined in Optimum proportions are required to yield a maximum profit. Factorqfactor and product-factor analysis does not deal with the problem of proportions in which various products should be combined to yield maximum economic efficiency. In its present stage of development, the "Cobb-Douglas Function" deals with questions of factor-factor combinations and factor-product problems. There has been no system devised as yet which enables the function to handle the problem of multiple enterprise1 combinations. 1 This problem is the subject of a doctoral dissertation presently being prepared by Christoph Beringer, a graduate student in Agricultural Economics at Michigan State College. Production Functions The general form of a production function (which is a mathematical statement of the way in which output or product depends upon inputs) has been expressed in mathematical symbolism in the following generalized 2 ram Y - f (X1,X2,X3,Xh, e e e .,Xn) Its meaning has been expressed verbally as output (y) depends upon or is a function of (X1, . . . . , In). This relationship may also be expressed in geometrical form. The geometrical interpretation appears in Figure 1. If nothing has been fixed i.e. if all the variable inputs can be controlled, then as they are increased in constant proportions, output (y) increases at a constant rate. The locus of relationship is then a straight line passing through the origin and the slope of the line is dependent upon the proportions among the inputs (ignoring the (o, ...., 0) point) and the scales selected. Although a universally applicable mathematical production function has not been devised, the concept is useful since all production functions which have been developed are an approximation of subfunctions of the more general one. In distinguishing between a subfunction and more general form, the essential difference is than in a subfunction, more variables have been fixed. A special case of the general function exists when only one input is required to produce output. In this case the graphic representation made in Figure 1 is again applicable. 2 Allen, R. G. D., Mathematical Analysis for Economists, Macmillan and Co., London, 19h7, pp. 235-551. (x1, x2, X3, 0 o 0 0 , Xn) FIGURE 1. TOTAL PHYSICAL PROLUCT, ALL INPUTS VARIABLE Mbst functions which have been derived empirically correspond to a situation in which one or more inputs determine output with one or more of the variables fixed and usually some uncontrolled. Such a function may be written in mathematical symbols as I - f (X1, . . . . ,Xg, ' xEfiJ-' . . . .,Xn) (Xn+l,. . . . ,Xt) where I is output, X1, . . . . ,Xg are variable inputs Xg+1, . . . .,Xn are held constant, and Xn+1, . . . .,Xt are uncontrolled variables or inputs influencing the relationship between Y and X1, . . . ., XE. When some variables are held constant and others are allowed to vary a rather special relationship exists between product increases and input increases: This relationship has been termed the Flaw of diminishing returns"3 or the "law of variable preportions,' and may be stated as follows: ‘when output depends upon both variable and fixed inputs, as the amount of the variable inputs is increased from zero the total physical producth first increases at an increasing rate, then increases at a decreasing rate, and finally tends to decrease. A subfunction such as Y . f (X1, X2, . . . . , Xn) has been represented graphically5 in Figure 2. The second stage is the only rational one to produce6 in if it pays to produce at all. Any good 3 Stigler, G. J., The Theory of Pricg, Macmillan Co., New'Iork, N.Y., 1950, pp. 116-fe h This relationship also holds for the marginal and average physical products see leintraub, 8., Price Theory, Pitman Publishing 00., New York, N. L, 19119, pp. 78 1'. 5 Stigler, G. J., Qp.'§it. p. 1235 Knight, F. H., Risk Uncertaint and Profit, Houghton flifflin 00., Boston and New Ybrk, 1921, p. loo. 6 Stigler, G. J°!.92'.9$E°’ pp. 12h-125. Stage I Stage II Stage III TPP (x1| x2, x3, . . . . , xn) FIGURE 2. THE THREE STAGES OF A PRODUCTION FUNCTION 10 text on economic theory will provide a definition of the marginal, and average physical product. The "Cobb-Douglas" function is capable of reflecting any one stage of production, but for any input or the sum of any combination of inputs it can reflect only one stage at a time. However it can reflect different stages for different inputs at the same time. In most farm management work, it has been assumed that stage II7 is the one in which farms are being operated. (This is not unrealistic if a purposive sample has been drawn).8 Thus, only diminishing returns are usually reflected for any input category in the coefficients usually9 obtained. (Ealue Productivity and Profit The physical relationships discussed so far may be converted to value relationships by multiplying the marginal, average, and total physical products by the price of the product. Thus the value of the marginal product, the value of the total product, and the value of the average product may be obtained. Under conditions of perfect competition asswming homogeneity and divisibility of factors and products the following relationships are also true VMP - MVP, TVP - VTP and VAP s AVP. The value productivity curves are shown in Figure 3. The determination of the maximum profit point, when the price of the product and of the variable inputs is known can be accomplished either graphically 7 Bradford, L. A. and Johnson, G. L., Farm Management Analysis, John Wiley & Sons, New York, N. Y., 1953, p. 1kg? 8 The method of drawing a purposive sample is discussed in a later section. 9 A possible exception might occur if a deficiency of a minor element were discovered and remedied. Value . . . . X xll X2, X3, : n FIGURE 3. LOCATION OF THE HIGH PROFIT POINT USING X1, With x2, 0 e e e , Xn FIXED 11 1? as in Figure 3 as the point at which the le line intersects the MVP line, or mathematically as follows. The profit equation is: Profit - TVP - X1 Pg - FC x1(y) 1 where TVP is total value product. Px is price of X1 and F0 is fixed 1 cost. The maximum profit point of the equation may be determined by taking the first derivative of the profit equation with respect to the variable input X1. The derivative is d1T * MVP . If this - P --- x1( x1 dxl Y) derivative is set equal to zero the maximum point on the profit curve is defined if the law of diminishing returns holds, as dfiT . 0 “h At this point the following relationship is true: “vpk1(y) - le or MVPx1(Zz ' 1 2x1 Only the single variable input has been discussed up to this point but it can also be proven10 for multi—inputs that the condition for maximum profit is: MVP P MVP .1111). . ”szo) =- WPXam . . . . .= xno) PX]. X2 PX3 13:11 For example in the case where I has depended upon 11 and x2 “3 variable inputs, X3, . . . . ,Xn being fixed, the equation for profit 10 Bradford, L. A. and Johnson, G. L., onygiio: P- 132’ 13 determination becomes: profit = TVP - P X - P X -- FC 3 X1X2 (y) X1 1 X Tr 2 2 If the partial derivatives of the profit equation are taken with respect to X1 and X2 while the other is held constant, the results are as follows: an“ - MVP -- P and at” =- uvp —— P 5X1 1(y) 1 5X2 2 x2 If the two partial derivatives are set equal to zero, thus defining the maximum point, mm ) MIN 1 T“!- v.1). "1 1‘2 A problem which has not been considered but which was implied in the last profit equation was that of the optimum preportion or combination of inputs to use in the production of I. What is sought is that combina— tion which yields the greatest amount of Y for a given.money outlay on X1 and X2. The prodUCtion function I . {(X1, X2, X3, 0 o e O , Xn) in the 11 X1, X2, plane may be represented graphically as in Figure h. The curves labeled Y1, . . . . , 25 are isoquants or isoproduct lines. Each line represents a constant and specific amount of I which is considered to be measured vertically from the surface of the paper. Until contour line I5 is reached higher outputs of Y are to the * v—v 11 Wientraub, 22: 931'" pp. 56-59. FIGURE b. ISO-PRODUCT cowrouus WITH SUPERIMPOSED ISO-COST LINE "North East." The dotted lires are "ridge lines"12 and enclose the rational area of proouction since in that area all positive of Y as dictated by the function I ' f(X], X2, X3, . . . . , Xn) are in this area. These quantities of I also exist beyond the ridge lines but production outside of this area requires for the same output greater amountsof either X1 and or X2 than the area enclosed by the ridge lines. If an isocost line such as I, I' is superimposed on the diagram, it can be seen that the greatest output of Y for a given number of dollars occurs at that point on the isocost line just tangent to an isoproduct contour line. At the point of tangency, the s10pes of the two lines are equal. The slape of the isoproduct contour line is the marginal rate of sub— stitution of X2 for II which is MPP and the slope of the isocost or X2 X1 isooutlay line is 3x2 . Therefore MPPXQ . PxZ P MPP: PR1 x1 1 at the point of tangency.13 This is the optimum combination of resources. If a series of these points are joined together the resultant line is termed the expansion path or scale line. 12 For a complete discussion of any of the terms used herein see an advanced text on economics theory such as: Boulding, K. E., Economic Anal sis, Harper and Brothers, New York, N. Y., 19h8, p. 669-ff3 Weintraub,‘22:.git., pp. 52—62. 13 WGintraub, m 933%" p. 60. For more than two inputs, combined in scale line proportions KPP IPPx MPPx P P "1 x2 Xn which defines the optimum combination of inputs. The optimum?”1 amount to produce occurs when MVP MVP MVP x1 ‘9 12(22 ' cos. 3 xn(!2 2” 1 Ex P P x x 1 2 n The marginal physical productivity of X1 or 12 is the slope of the product contour in the X1 and 12 dimension. The magnitude of the marginal physical product of 11 or X2 is dependent upon the amount of both x1 and x2. One advantage of the "Cobb-Douglas" function is that it can reflect the effect of the quantities of associated inputs on the marginal value productivity of a single input. The Categorization of Multiplgrlgputs Although the "Cobb-Douglas" function is simple to manipulate when compared to some others15 it is costly to use for a large number of variables. Consequently it has been usually fitted to categories16 of inputs rather than to individual inputs in farm management work. This categorization has a sound theoretical background since many inputs 1 . L Ibid 5 For example, the Spillman function. 16 Bradford, L. A. and Johnson, G. L., 92: Cit., pp. lb3-f. 17 used in agriculture production have been either close complements or close substitutes. Both of these general categories may be handled as single inputs. In the case of close complements, the problem created by the influence of relative prices on the proportions used is not important since they must be combined in fixed proportions. An important problem arises in "Cobb-Douglas" analysis when good complements are placed in different categories. The resultant rij's are large. Consequently the standard errors of estimate of the regression coefficients have tended to be high.17 When inputs are close substitutes, one for another, their relative prices are very important in determining the proportion in which to combine them to yield maximum output for a given expenditure. A further consideration is that the better one input substitutes for the other, the less likely they are to be considered as separate inputs for if factor A is contained in both input B and input C, then as source of factor A, both B and C are substitutes. Near perfect substitutes may be considered to exist only when both contain a factor capable of performing a very similar function, hence the problem is one of measuring the common element or factor in both inputs. Near perfect substitutes may be grouped in one category and measured in terms of their common factor. In any one input category there can be subcategories of inputs such that two subcategories might contain close complements or substitutes and the relation between the two subcategories might have been either 1? Ezekiel, M., Methods of Correlation Analysjg, John Wiley and Sons, Inc., NEW‘York, New York, 19h9, p. 502. 15 as close complements or as close substitutes. The important consideration however is that close complementarity should not have existed between categories. The problem of defining categories has often been a vital one in "Cobb-Douglas" analysis. This problem has been especially knotty when differentiating between cash expense items and the various investment categories, with the result that faulty estimates of marginal value productivity may occur. In some instances the demarcation has been fairly clear but in fringe cases it has not been so lucid. The policy in such cases has been to make an arbitrary choice, indicate what it was, and abide by it consistently. The Pricing_g£_lppgtg The question of pricing variable inputs has been fairly adequately resolved by using market prices, but for fixed assets the problem has been a more complex one. The value in use of a fixed asset has been considered to lie between its salvage value and its replacement cost. When a good or service is divisible, homogenous, and frequently traded on the open market, no great difference between replacement and salvage value exists, but when a good is not homogenous, or not divisible, or if it is infrequently exchanged, the difference between replacement and salvage price may be considerable. Many of the fixed assets in farming have fallen into this latter grouping and it has been a problem of importance to determine their actual value. In many cases, the problem has been solved by measuring fixed assets in physical units, but some categories 19 do not lend themselves to such measurement. At present, there has not been an adequate "a priori" method of making such a determiration without CXCBSSlVE costs. lfiaptor—Factor Analysis and thewgpdex Number Problem If isoproduct contours similar to those drawn in Figure b have been constructed to indicate levels of gross income rather than levels of output, iso-value contours such as are presented in Figure 5 are the result. YO to Yb represent successively higher values or levels of income. X1 and X2 are inputs used in the production of Y. POPl, and P1P2 represent isoexpenditure combinations for different relative prices of 11 and X2. If X1 and X2 are input categories, the combinations of X1 and X2 which can be purchased in period zero for a given amount of money are represented by line PbBl. At the prices which obtained in period zero, it was possible to achieve a level of income between Y0 and Yi. If in period one the price of X1 declines as indicated by line P1P2, a new level of income Ii is made possible with the same dollar outlay. Thus, aside from changes in the production function, gross income Can vary because of: l. A relative change in the prices of 11 and X2. 2. A change in the price of Y. The index number problem as it has related to this study has been to estimate the new levels of income and earning power associated with price changes by means of index numbers. A price index is essentially a comparison of the price of a good or group of goods at one time or place with the price of the same goods 20 P2 // f PO \ 1' \ \ \ \\ .\ \ \ \\\ \\\\ r 0 P1 x? FIGURE 5. ISO-VALUE PRODUCT commas wm—r SUPERWPOSED ISO—COST LINES 21 at a different time or place. This comparison usually takes the form of a ratio.18 In some instances, an average of several periods may be used as a base; in other cases, an actual time period may be used. If several prices are involved in the comparison, each price is expressed as a ratio of the price which was obtained in the base period. If the prices of several commodities are being compared from year to year, it is also usual to weight each commodity according to its relative importance in one or more of the time periods. In instances where the quantity has fluctuated widely from one period to another, the problem of which period or combination of periods to use as a base for weighting become important. For indexes which have extended over a considerable period of time varying weights are often used. There are many types of index number formulae in existencelg most of which have been designed to meet a specific situation. The present discussion will be confined to the type selected as most suitable for inflating or deflating marginal value productivity estimates under the circumstances peculiar to the present problem. The method chosen was Laspeyre's. Essentially, it involves using base year quantity weights. In mathematical terms it has been 8 Pearson, F. A. and Bennett, K. 3., Statistical Methods, JOhn Wiley and Sons, Inc., New York, New'YOrk,'l9h9, pp. 53+?. 19 Pearson, F. A., and Bennett, K. R.,.QE; Cit., Po b5 — p. 75: Croxton, F. E., and Cowden, D. J., A lizafiGeneral Statistics Prentice - Hall Inc., New Ybrk, New gork, 1557, Chapter XXI. 2? T Pn‘io T expressed as i '_7?TEETU.' , where i is the index, p 8 price, Q = quantity, and O and n refer to the base period and the period being compared to the base period, respectively. There are disadvantages to most index numbers and the Laspeyre's method is no exception. For one thing it tends to be biased upward when it is applied to the factors of production. Since a firm would tend to purchase less of an input which has risen relatively in price and more of an input which has declined relatively in price. On the other hand when products are being considered, the same index has a.bias in the opposite direction. For if the price of a product tends to rise relative to the price of other products produced by a firm the tendency will be to produce more of that product and less of those which have declined relatively in price. It would appear from the foregoing discussion of biases that the method chosen would tend to have a downward bias on the marginal value product estimates if it were applied to both gross income estimates and For if marginal value productivity is 20 . E determined by the formula MVP - bi 1.19.). , and bi is constant then input category estimates. a downward bias in E(y) and an upward bias in 1i will both tend toward a downward bias of the marginal.value productivity estimate. Even though the foregoing biases were known to be present in Laspeyre's method it was selected for the following reasons. —‘ 20 T0011, To Go, $.- 93$” p. 80 I)‘ ¢'. ) 1. If other than base year quantities were required the purpose of the study would have been defeated. t3indicated in Chapter I, Laspeyre's method has the advantage of requiring only base period weights. 2. The method is easily understood and its interpretation is not difficult. 3. By relatively simple manipulation either base period quantities or values may be used for weights. Changing Marginal Value Productivities Due to Conditions of Changing_?rice If the price of an input varies, other things being equal, the marginal value productivity of a physical unit of the input does not change. Hewever, if the input has been measured inznonetary terms the marginal value productivity of a dollar's worth of the input does change. If the price of the product varies the marginal value productivity of both physical and monetary units of a specific input changes. There are certain conditions which price indices have to meet if they are to be used to accurately estimate the new marginal value productivities resulting from price changes. 21 P P .2 FCC ( n QC) P .( O ) since the price in the ITE?Z;;V- 0 (P0 00) base period multiplied by the price relative of the period being compared, and the base period is equal to the price in the period being compared to the base period. The general form of a Cobb-Douglas function is k y ‘ u so my 1 _ 1A . Y = a X1 *9 . . . . Xn where Y is output, ”a" is a constant, X1, . . . . , Xn are inputs, and bl, . . . . , bn are the eXponents associated with the inputs. If a simplified example such bl as Y = aXl is used the marginal value product of X1 in the production of Y may be determined by taking the first derivative of Y with respect to X1, thus b1 ' dz = b1 aXl “ or multiplying the result by'.§l the fonn Xm x1 b1 bl .§§%i_. emerges. If a marginal value productivity estimate is required for 11 when the price of Y changes between two periods such as a base period (T=O) and a year being compared with the base period b1 (T-N) then Y(T=O)o IY(T=n) = Iy(T=n)' aX1(T=O) _ Price of Y in eriod (T=n) h _ - P i__ W ere ;y(T-n) Price of Y in period (T553 If the first derivative of I(T=O)° IY(T=n) is taken with reapect to xl(T=0)’ the result is b I dY(r:gl_o Iy(T=n) - Iy(T‘n) . b1 8X1(T202 dx1(T=o) X1(T=o) which is the marginal value productivity of input xl(T=O) in the production of Y(T=n)’ since Y(T=O) . Iy(T=n) = Y(T=n)' Thus the marginal value productivity of xl(T=O) can be determined for a change in the price of Y. P.) KIT If the price of X1 fluctuates between the two periods while the price of Y is constant then b1 x1 b1 _ (T= ) ,YW‘O) g Y(T=n) ’ ax1(T=0) ' ‘I(T=n)n the marginal value product of xl('1‘=n) under these circumstances is then dy [x 1 131—1 X g 2 X1('r---'n) b1 eQual to (Ten) - 3 b1 WET-n) 0 ii“) = :1 b1 [TE—“u“- n M --—-,::::> -n but since ‘;E£I:El ' xl(T=O) then (T-n) - b1 H dVg'ra-n) a b1 aha-02 dx1(T’n) X1(T.0) . IXi(T=n) If the price of both X1 and Y vary from one period to another then I xl(T=n) (T=0) ° Iy('r=o) ‘ Y(T=n) 'Iy(T=n) - a m) The marginal value product of x1('1‘=n) then becomes b1 CIY(Tam) x1(T-O) #4; III ......_.._.. - z . 3 .cf dx(T=n) Iy(T n) b1 x1('1“0) ' IX1(T=n) Verbally the last mathematical expression means that the marginal value productivity of a monetary unit of input 11 in the production of y in period 11 is equal to the IV? of a monetary unit of Kl in the 26 production of Y in period zero times the index of the price of Y in period one over the index of price of X] in period one. For those inputs which are measured in physical terms, equation I is the relevant one to consider. For inputs measured in monetary torus equation III is the appropriate one to use. Equation III may be extended to include more variable inputs as follows: MVPX1(T=n) = Iy(T=n) ° 23. E(y)(T=Ql xi(T=O) ' IXi(Tun) bi bj ' a: x. z where E(Y) _ a a .EE£E_22 . , , , _J(T.n) (T‘O) Ix (r~ ) IXJ(T ) i -n =n and Xi (Tgo) 1' 3.1.9.22 Ixi(T==n) The four equations which have been developed in the latter section of this chapter are basic to the problem of this thesis. They establish the way in which the various price indices will be applied to marginal value productivity estimates in adjusting them for changes in prices. As previously noted, in "Cobb-Douglas" analysis of farm business data, categories of inputs rather than single inputs are considered as the independent variables. Therefore, price indices of categories rather than of separate inputs are used to adjust estimates of marginal value Preductivity. Such indices disregard any internal adjustments which may occur in the categories as a result of price changes. I s; ":.~'~i :'.—"- ‘I CHAPTER III THE SfiMPLING TECHNIQUE AND THE SAMPLE Requirements of the Sampling Technique In most empirical‘work in agricultural economics the technique of the controlled experiment is precluded because of the very high costs involved. Consequently a large part of the burden of controlling variables falls on the sampling procedure. A useful classification of the variables involved in most empirical work is: A. Controlled 1. Independent studied nonrandom 2. Independent fixed nonrandom B. Uncontrolled l. Dependent and studied 2. Independent not studied nonrandom; 3. Independent random Control over the independent nonrandom variables, both studied and fixed, is usually accomplished by the sampling technique. The errors due to random variables are "averaged out" by statistical processes,'while those due to nonrandom, unstudied variables can.be handled.mainly on an "insight“ basis. Consequently it is very important that the number of uncontrolled nonrandom variables be mini- mized by studying and fixing as many as feasible. 5 would all be included as studied variables, however in practice limitations such as, difglgifigi of measurement, limited resources, and loss of degrees 0 r have to be balanced against their inclusion. 1 Conceptually nonrandom'variable 28 To control independent nonrandom variables is an important objective in sampling technique when only non experimental data are available. The methods used depend upon the type of study being done. In this study the sampling problem was to control independent nonrandom variables in data collected from a group of farms in such a manner that satisfactory estimates of the marginal value productivities of input and investment categories could be derived from the data when W fitted to a "Cobb-Douglas" function. To insure a reasonable attainment of this objective the following i conditions should be met. 1. A similar level of technology should exist among the farms, and they should be using the same inputs. 2. Farms included in the sample should.be producing the same product, or if multiple enterprises do exist they should be complementary (joint products may be considered as a single product). 3. The soils should be of the same inherent productive capacity. h. Inputs within each category should be in optimum combination. These four conditions insure that the farms are on the same production function.2 Empirical Techniques Used to Approach the Required Conditions In the present study it was sought to implement the foregoing conditions in the following manner. 2 Pr duction T°°n:.22; Cit., pp. 20-ff; See also Bronfenbrenner, M., 0 Functions Cobb-Douglas, Interfirm, Intrafirm,‘§conometrigg, Vol. 12 No. 1, January 19hh. Some legree of CJLtrO] over the problem of multiple enterprises was achieved by selecting only farms which had dairy sales as their major single source of cash receipts. However, since the agriculture of the area was diversified it was not possible to obtain the desired number of farm records from dairy specialists, i.e., seventy yercent or more of cash receipts from dairy sales. To insure that some homogeneity as to technical development among sample farms existed, a farm was not included unless all six input and investment categories were present. Sich a procedure also insured that farms were using the same input categories. By restricting the sample to two fairly similar soil associations in the same geographic area, and collecting records for the year 1952, it was possible to avoid wide variations in productive capacity. However, it should be noted that there was a good deal of variation within the area as to growing season and rainfall.3 Year to year variations in prices paid and received were eliminated by collecting all records for the same year. Variation within the area during the year was considered to be randomly distributed. An intra-firm rather than an inter—firmh function "83 sought by purposely selecting a wide range of quantities and combinations of quantities in the various input and investment categories. It is this deliberate seldction of range and the attempt to minimize the correla- tions among input categories that gives the name of purposive sampling 3 Hill, E. Bo, Q. 9-1-32.) p. 90 s . Bronfenbrenner, M., 22. Cit. 30 technique. 5 The standard error of the regression coefficients was held down by selecting farms so that the correlation between input categories was low and the variance of the quantities of each input category was high. The following variables were not controlled but were considered to be randomly distributed. Within area variations in price and weather, management, institutional arrangements, asset control, and variations of productivity within soil association. The Area And The Sample The data for this study were collected in the area shown in Figure 6, for the year 1952. If a line were drawn joining the southern boundaries of Benzie, and Alcondécounties, that portion of the southern peninsula of Michigan to the north of the line would include the area from which farm records were taken for this study. Only farms on either of two soil associations were selected. Consequently although there are fifteen counties in the area delineated, two, Benzie and Crawford were not included since neither contained the required soil associations. Soils The two soil associations from which farms were selected were nonbers eight and sixteen.7 Soil association eight has a terrain'which 5 Ezekiel, m., Methods of Correlation Analysis, (2nd ed.) New York, N. L: John Wiley and Sons, Inc., 19149, p. 302. 6 See Figure 6. 7 For the location and complete description of these soil association as they occur in Michigan see: Veatch, J. 0., Soils and Land of Michi an, The'Michigan State College Press, East Lansin M1 5933, pp. 69-73- g, Chigan, MICHIGAN FIGURE 6. OUTLINE MAP OF MICHIGAN SHOWING COUNTIES FROM.WHICH SAMPLE WAS DRAWN 31 ranges from smooth to strongly rolling. Its subsurface is typically a thin reddish brown friable clay. Veatch8 rates the upland soil of this association very highly 311d considers that it is Well suited to the staple crOps grown in the area. Where the climate is satisfactory, as in those portions in close proximity to the moderating influence of Lake Michigan, it is highly'valued as orchard land. Soil association sixteen which occupys a larger portion of the area than eight is comparable to the latter in several respects. The soil types in both associations are rather alike, the important difference being that of tapography. Association sixteen is rolling and hilly and does not have as high a productive capacity as association eight. Veatch8 has made the following comment with regard the economic use of soil association sixteen, "the place of most of the land is not firmly established in the economy." Climatic Factore9 Rainfall is not regarded as a limiting factor in crop production in the area, since the annual precipitation is usually in excess of twenty-eight inches. Temperature, on the other hand, limits not only yields but also the types and.varieties of crops which may be grown. The western.portion of the area bordering on.Lake Michigan.has an average temperature some ten degrees higher than the eastern part. A variation of nearly fifty days exists between the growing season.ot 8 Ibid. 9 Hill, E. 8., Types of'Farming in Michigan, Special Bulletin 206 Michigan State College,:§3st.Lansing, Michigan, 1939. ’ _- ‘W Le fanau county and portions of Otsego and Montmorency counties, the former having a growing season of one hundred and fifty days whereas parts of the latter counties can expect only about one hundred days of growing weather. With such a range in.growing season it is not surprising to find a wide range of cropping enterprises. Present Land Qge Inversification of agricultural production is one of the main characteristics of the area. However, there is considerable Speciali- i zation in fruit production in Leelanau and Grand Traverse counties. l Milk sales represent the largest single source of cash farm income. I Potatoes, cattle, and some cash crOps such as wheat and pulses account for most of the remainder. A good deal of the land is unsuited to agriculture. As much as eighty percent of type of farming area 1310 is not good farm land. One important factor which has inhibited farm development in this area has been the great distance from markets. For example although much of the land is suited to dairying only a small portion goes to the major Michigan markets as fluid milk, as a consequence, cream and some local f1uid.milk are the main sources of dairy income in the area. In total sixty-five individual farm records were obtained by personal interview from farms whose major source of cash receipts came from the sale of dairy products. Thirty records were taken from farms on soil association eight and thirtybfive from farms on soil associ- ation sixteen. Two of the records from the latter group were discarded Ibid.» pp. 2h. 3!. as they were incomplete. The following11 information was obtained for each farm. X1 Gross income measured in dollars. Land measured in tillable acres. X Labor measured in months. Xh Machinery investment in dollars. XS ForagenLivestock investment in dollars. X6 Productive cash expenses. . . . 1 X Buildings measured in terms of animal.unit housing capac1ty. The Sample Farms Some idea of the nature of the sample farms may be derived by showing the range in the data for the several categories of variables 13 studied and the "usual" farm organization. For.farms located on soil association sixteen the usual organization was Land 97.h acres tillable Labor 1h.2 months Machinery Investment 33619 Livestock-Forage Investment $h733 Cash Expenses $1761 Animal Units of Housing Capacity 2h Units 11 For a complete discussion of the method of handling and the items included in the various categories see Toon, T. G".QE:.EEE" pp. 29-ff; and wagiey, R. V°:.QE:.EEE': pp. 37-ff. 12 . Wagley, R. v'9‘92;.9££'3 p. hS. 13 The usual organization is considered to be at the geometric means of the various input categories. This combination of resources yielded a gross income of $9597. Gross income varied from a low of $1587 to a high of R2h,529. The range in tillable acres was also considerable, with a minimum of 36 acres and a maximum of 270 acres. The smallest amount of labor used for one farm was two months as compared to the largest of 30 months. Machinery investment ranged from $518 to $19,058, livestock forage investment from $850 to $10,837, expenses from $21h to $9,013, and building animal housing units from 11 to 59 units. The usual organization for farms on soil association eight was: Land 105.23 acres Labor 16.L2 months machineny Investment 8h80h. livestock-Forage Investment $572b. Cash Expenses $2550. Animal Unitsof Housing Capacity' 28.8 Units Gross income ranged from $3056 to 81b,30h; land from 51 to 279 tillable acres; labor from 12 to 3b months; machinery from $1,217 to 317,319; livestock—forage investment from $2,910 to $19,969; cash expenses from $3h8 to $12,2h8; and animal units of housing capacity ranged from 15.7 to 56.29 units. 36 CHAPTER IV AN EVALUATION OF THE STATISTICAL RESULTS AFTER FITTING THE FUNCTION Egsgripticn of Techgiggg After the data were collected and checked for internal consistency, the quantities in the various input and investment categories and gross income were converted to logarithms. Using the Doolittle1 method of multiple correlation analysis, two least squares regression equations were fitted to the logarithms of the data2, one for the thirty farms on soil association eight and one for the thirty-three farms on soil association sixteen. The resultant coefficients (elasticities)3 and associated standard errors for the two functions were as follows: For soil association eight. Land p2 - -.232073 3: .235095 Labor b3 - «130581; _4_- .225911 Machinery bh - .26hb27 ; .1560h1 Forage-Livestock bS - .558385 3; .188853 Expenses b6 - .397683 3; .127227 Animal Units or b7 - .ohh379 _+_ .23173h Housing Capacity For soil association sixteen. Land b2 - .285962 3 .15683803 Labor b3 - -.1738h555 .3 .1h170100 n 1 Ezekiel, 15., 93. 933., Appendix I 2 Toon, 22. £13., p. 3b, 3 Toon, k. Cite, pp. 8-ffo 37 Machinery bh - .Ohh63333 ‘: .12h52925 Forage-Livestock b5 = .26693169 ‘: .lSthESO Expenses b6 8 .39250352 .I .096th66 Animal Units of Housing Capacity b7 = .25278055 ‘: .13301986 Rejection of the. First Set of Coefficients It will be noted that negative elasticities were obtained for land and labor on soil type eight, and for land on soil type sixteen. While it is possible that increased quantities of land or labor might decrease gross income it was not believed probable that they would do so. Furthermore according to Tintner and Brownleo,h "negative elasti— cities, within the range of inputs on most farms are meaningless." The procedures of least squares regression analysis provide three types of information about regression coefficients which are: the amount of change, the proportionate importance, and the accuracy of the estimate. The amount of change is indicated by the value derived for the regression coefficient, the proportionate importance by the correlation, and the accuracy of the estimate by the standard error. These are all useful types of information and should all be considered in evaluating estimates. Although necessary, they are not a sufficient basis for conclusions unless the original data is of an experimental nature. When data is of a nonexperimental nature, Wo1d5 takes the position that "in regression analysis the formal tests of significance, m h Tintnar, Bromlee, Q. gigs, p. 5680 S wold, H., Demand Anal sis, John Wiley and Sons, Inc., New York, No Yo, 1953, p. EB. is 0‘) however refined, carry little weight as compared with the non-formal and non-quantative significance that is embodied in results derived from independent sources, provided these results support one another and form an organic whole." Although statistical manipulations had provided negative coeffi- cients, economic theory and "common sense" indicated that although negative coefficients had been determined they were in the nature of 6 a nonsense result. The Third Functiop, Appraisal and Rejection As previously7 indicated, the possibility of competing multiple enterprises existed in the sample. Furthermore, product prices had shown a marked variation, with prices received for milk ranging from three to five dollars per hundred pounds. Several of the farms on soil association sixteen received a large portion of their gross receipts from the sale of cherries and on one farm cherry sales represented a major source of cash receipts. Both the cash sales of cherries and the associated inputs (as near as they could be determined) were removed from the data'by accounting procedures and a new function was fitted to the farms on soil associ- ation sixteen. The resultant coefficients were: Land b2 - .057035 3; .200659 Labor ' b3 =- -.262270 1 .158770 Machinery bh s .013775 3; .125018 6 That is it indicates that stage III has been entered. 7 Infra p. Ch. 3 and Ch. L . 39 Forage-Livestock b; = .625661 ‘1 .l9h989 EXpenses b6 = .h86678 ‘: .100236 Animal Units of b7 3 .090170 ‘I .197100 Housing Capacity Even after the removal of the cherry enterprises negative coefficients were again obtained. However, since large unexplained variances in the data indicated that hetrogeneity of product was a likely cause of the negative coefficients, the sixty—three records were sorted according to the proportion of cash receipts from milk sales and a function was fitted to the thirty records meeting the condition that over forty percent of their cash receipts be from the sale of milk. If dairy cattle sales are considered as a joint product with milk sales then the average dairy income per farm in the new grouping was close to eighty percent. _The Statistical Results on Fitting the Fourth Functign The coefficients obtained by fitting a Cobb-Douglas function to the group of farms deriving forty percent or more of their total cash receipts from milk sales were found to be: Land b2 = .1h812b 3; .1h5787 Labor b3 - .035117 1 .182655 Machinery bu - .27h29h 1 .13202h Forage-Livestock b5 = .10889h ‘1 .lSthB Expenses b6 8 .2850bh .1 .097680 Animal Units of b7 - .3277140 _+_ .12383b Housing CapaCity The sum.of the regression coefficients was 1.18. Since this sum is greater than one, increasing returns to scale are indicated on the thirty farms. In natural numbers the fitted regression equation was: .1h812h7 .o351169h .27b39h5'0 .10889h17 .2850hh37 X1 2 he22hex2 0 x3 0 Kb 0 le 0 X6 . .3277L036 R7 The multiple correlation coefficient or R was .8889h8. Under conditions of random sampling with.six independent and one dependent variables, a multiple correlation coefficient this high would be expected in one sample out of 20 on the average if the true (R) were .75.8 Consequently the degree of correlation is significant. Since extreme values were selected in the sample, (R) is higher than would be expected from a random sample of farms in the universe studied but not necessarily different than for a similarly drawn sample from the same universe. The coefficient of determination or R? was .790292 which may‘be interpreted as meaning that seventy—nine percent of the variation in gross income (X1) is associated with variations in the input and investment categories. The remaining twentybone percent of the variation in X1 may be due to such factors as variations in product price, and soil association, variations in management, and other non studied variables which were assumed to be randomly distributed as discussed in Chapter III. 8 Ezekiel, M0, fl. Ell-0’ pp. 508“5090 Lu Correlation between input categories was not a serious problem. The highest rjj for any two input or investment categories was between forage—livestock investment and buildings measured in animal units of housing capacity. The value of r5.7 was only .568558. Once the elasticity coefficients of the various input and investment categories are determined it is fairly simple to derive estimates of marginal value9 productivity from the relationships MVF = bi __.LE(Y x1 where bi is the elasticity of the input with respect gross income, E(Y) is predicted gross income and Xi is the quantity of the input. Both E(Y) and X1 are measured in natural numbers. For this study the MVP estimates for the various input categories calculated at their geometric means were: Land 89.05 per acre Labor 15.11 per month Machinery .hO per dollar invested Forage-Livestock .12 per dollar invested Productive Cash Expenses .86 per dollar spent Animal Units of Housing 68.6h per unit Capacity A Comparison of Traditional and "Cobb-Douglas" Estimates of veins Productivipy When such measurements are compared with those Obtained by the "traditional method" of farm management there appears, at first, to be a marked discrepancy. For example, the earning power of the last month of labor calculated at the geometric mean was $154Yl per month, while _h 9 Toon, To Go, E:- Efie, p. 8 the gross income per man day calculated for farms operating in a similar area10 for the same year ranged between $12.9h and $13.28 per day. The discrepancy, however, is more apparent than real. Before such measures may be logically compared they must be brought to a comparable basis. This was accomplished in the following manner. First, the total derivative of the fitted Cobb-Douglas function was taken with respect to labor at the geometric mean for labor with the other inputs at geometric mean.levels and changing with one month of labor in geometric mean proportions. This may be represented symbolically as follows: 7 11 dxl - MVP x ‘+E:: nan: dxi + MVP x dx2 3 'What has been derived thus far has been the marginal value product of one month of labor in the usual or geometric mean organization plus the sum of the marginal value productivities of the other input and investment categories usually associated with one month of labor. If the result is divided by thirty (the approximate number of days in a month) a rough estimate of the earning power of a day of labor and the associated inputs is derived. For this study the estimate was $16.97. This is a little higher than the estimates obtained by traditional 10 See: Area Reports 12 and lb of Farm Business Anal sis 20,Michigan State College COOperative Extension Service, Department of Agricultural Economics, East Lansing, Michigan. 11 In lieu of using scale line proportions of the inputs the geometric mean proportions were used, consequently dXi as used for computational 33' anti log of the mean of the_log of Xi ant—flogme mean 0? the log of 1., ° purposes is equal to b3 methods but when it is remembered that "products for home use" are included in.gross income in these computations, whereas they were excluded from the traditional calculation,the difference is small. Usual Organization, and Statistical Tests of Significance The usual organization of the thirty farms obtaining forty percent or more of their cash receipts from the sale of milk was: X2 Land 101.hl tillable acres X3' Labor lb.hl months Kb Machinery Investment 3&220. X5 Livestock-Forage Investment $5557. X6 Cash Expenses $2056- X7 Animal Units of Housing 29.6 units Capacity which.yielded a usual gross income of $6199. The standard error of estimate of the logarithm of gross income was found to be .08990. Under conditions of random sampling, given 1952 conditions, two thirds of the actual logarithms of gross income would be expected to fall between the logarithm of gross income at the geometric mean plus .08990 and the logarithm of gross income at the geometric mean minus .08990 or in dollars between $5051 and $7609. Instead of testing the elasticities against a null hypothesis, it was decided that a more meaningful test would result if they were tested against the elasticity which would be required to yield (at the geometric mean levels of inputs and investments) a reservation price for the input or investment associated with the elasticity. The following earnings or reservation prices were considered to be reasonable minima to expect. f) Land $5.00 per acrelc Labor $100.00 per month13 Machinery 20% of investment Cash Eernses $1.00 per dollar expended Forage-Livestock hO% of investment Animal Units of Housing $b0.00 per unit Capacity The resultant coefficients which would be required to yield the reservation prices appear in Table I. Table I Comparison of Actual Estimates of bi's and the bi'S Required to Yield the Estimated Minimwm Marginal Value Products A WM‘ A . actual bi to yield b1 bi's minimum return Differgnce 4:: b2 .1l;812 .08179 .0663 3 b3 .03512 .22810 -.l?298 bb .27h29 .13612 .1381? b5 .10889 .35857 -.2);968 b6 .2850b .33166 -.oh622 _— It was hypothesized for each bi that the actual predicted bi was equal 1 9 to the bi required. The hypothesis was tested by means of a "t test"*“ 12 Based on 5% interest rate with land valued at $100 per acre. 13 Karl A. Vary, "wage Rates Reported by Farmers", Michigan Farm Economics (East Lansing Co-operative Extension) 0‘ b; Where t = bi (required). See Ezekiel,.M.,.Qp..Qi1o, p. 506. MS and it was concluded that there was no reason for rejecti:g it at the five percent level of significance. Since the marginal value productivity estimates are closely related to the values obtained for the regression coefficients, some ideas of the variance in the earning power of the several input and investment categories may be obtained by computing marginal value productivities, based on the regression coefficients plus or minus their respective standard errors. Under conditions of random sampling, . . 1% given 1952 price and weather conditions, Sixty-seven ' percent of the 1argina1 value productivities obtained should fall between the limits obtained in this manner. Table II shows the marginal value product estimates thus obtained. Table II 63.2 Percent Confidence Limits* for Marginal Value Productivity Estimates ____ 4 _— bi !._ Lower Limit Upper Limit ‘_ b2 Land, in acres $1.5 $18. b3 Labor, in months -6h. 9h. bh machinery, in dollars .19 .60 b5 Livestock-Forage, in dollars .051 .29 b6 Expenses, in dollars .56 1.2 D? Animal Units of Housing Capacity b3. 95. A- * . . . Includes only variation due to variance of bi's If two input categories are highly correlated and the regression lg ' This does not take into account the variation in the means of the X13, just the variation in the bi's. he coefficient of one of the categories is biased upward the regression . . . . . 16 . . coeffic1ent of the other W111 be biased downward. In this particular study the unexplained variance was rather large (.21). Thus the rjj'S are low. As previously noted the highest ri- J forage—livestock investment. Thus it is possible that they may be was between buildings and fairly highly correlated. If livestock-forage returns are as high as the most probable marginal value productivity estimates would indicate, then it would seem reasonable to expect a decline in dairy cow numbers in the area. Figures from "Michigan Agricultural Statistics" indicate a decline of fourteen percent in dairy cattle numbers in the area between 19h? and 1952 as compared.with a one half of one percent decline in dairy cattle numbers for the whole state. If the earnings of farm buildings are high as their estimated marginal value productivity would indicate, there is some reason.for believing that farms in the area have too feW'buildings relative to optimum requirements. Vincent17 et. a1. indicate that the building investment per animal unit in the area is between $128 and $155 as compared with $212 for the state average. These external evidences would tend to indicate that dairy cow earnings are low and building investment earnings in the area are high. In view of the small sample, and the large variance obtained, it is not possible to state on a statistical basis alone that the earnings of 6 Wagley, R. V., Op. Cit., pp. S9-ff. 17 Vincent, W. H., et. a1., Michigan Farm Business Report for 1952 (Reprinted from the qwerterly Bulletin, of t§54¥ichigan.Agricultural _ggperiment Station, Michigan State College, East Lansing, Michigan:— Vbl. 36, No. 2, pp. 202-211). h? livestock—forage investments or buildings are significantly different from what they should be to maximize profits. External information, however tends to lend significance to differences between the estimates and standards of marginal value productivity necessary to maximize profits. In evaluating the results after fitting the function the following conclusions appear to hold. (1) The accuracy of the marginal value productivity estimates was considerably impaired by the small size of the sample and the large variances obtained for the regression coefficients. Hence, any recommendations made on the basis of these estimates have to be carefully qualified and interpreted in view of all available information external to the study. (2) From a methodological standpoint the estimates can still be used to illustrate the technique of adjusting marginal value producti- vities for price changes. b8 CHAPTER V THE ADJUSTMENT OF THE 1952 MfiRGINAL VALUE PRcIUCTIVITY ESTILMTES FOR 1953 PRICE ccrmI'rIcrs AND GROSS INCOME The main purpose of this thesis was to evolve a technique for adjusting estimates of marginal value productivity for conditions of changing price by the application of price indices to gross income and the various input and investment categories of farm businesses. Indices Required In Chapter II, four equations were developed which indicated the conditions which should be met by the price indices. It was noted there that equation I was the relevant one to consider for inputs which are measured in physical terms. Equation I requires only an index of product prices. Equations III or IV which involve two indices, one for product price and one for input price, are required for inputs measured in dollars. Since three input categories, Xh, X5, and X6 were measured in dollars, a total of four price indices are required, one for X1 or gross income, and one for each of: machinery, livestock-forage, and cash expenses. The base period was the calendar year 1952 and the year being compared with the base period was 1953. In Laspeyre's method of calculating index numbers the general formula is N 31 (P ) I(T=H)- -N nQO~_ E31 (POQO) to compute such an index, quantity weights in the base period (QC) are required, and prices in the base period (Pb), and in the period being compared with the base period (Pn) are also necessary. It was found in collecting the records that accurate estimates of physical quantities of sales and purchases were not readily procured, the estimate often being little better than a guess. On the other hand, there was usually a record of the value of sales and expenses giving an accurate estimate of the year's transactions. It was decided therefore, that base year values would provide a better weighting system than base year quantities. Such a weighting system necessitates using a:modified.but theoretically equivalent form of Laspeyre‘s method. The equivalence was demonstrated in a footnote in Chapter II. The modified form of Laspeyre's method is N [P l P I(T=n) ‘ 1'1 ?% (POQO where .3 is a price relative, m 1’0 £21 i.e., the ratio of the price of an item in the period.being compared with the base period to the price of the item in the base period. Although this form is a theoretical equivalent of the general formula, in.this study the following inequality existed: The prices used in computing the price relatives were obtained from unpublished data collected by the Crop Reporting Service1 of the State of Michigan and represented state averages rather than averages for the area encompassed.by this study. On the other hand, the base period values (PbQO) include actual prices received by farmers in the area rather than state average prices. Since the base period price in the 1 Infra, p. 50. 50 price relative is not the same as the base period price in the value weight, the equivalence of the two forms of the index number formula is impaired. It should be noted that so long as prices are obtained from two sources, an inaccuracy in the index number will result whether the general or modified form of the index is used. The data used in this study to compute price relatives were obtained from the Cooperative CrOp Reporting Service of the Michigan.Department of Agriculture, Lansing, Michigan, and represent state average prices. Although these price data are collected by the CrOp Reporting Service, they are not published on a state basis. Local prices were not used since in the opinion of Mr. Blood2 the "Crop Reporting" sample in the area of this study was not adequate for the purposes of this thesis. Computation o£_the Indices The following procedure was used to compute the four price indices. The 1952 values of the several classes of items within each input category were summed to provide base period value weights.3 Thus, in the machinery investment category the value of all tractors was used as the base period weight for the tractor classification, and the value of all hay balers as the base period weight fer that class I.‘ 2 Statement by J. C. Blood, Statistician, Michigan Department of Agriculture, Lansing, Mdchigan. 3 In instances where no state price estimates were available for a class or item, that class was given a zero weighting in both numerator and denominator. All such omissions are shown in the Appendix Tables IV, V, VI, VII. It is realized that such omissions would bias the price index but it was not believed that the omissions in this study would seriously distort the resultant indices. of item, etc. The sum of all such classes within a category provided N the base period value, i.e., E: (POQO) which is the denominator of the modified Laspeyre's index. i=1 Price relatives for the individual classes of items within a category were obtained by dividing the 1953 state average price for the specific class of item by the 1952 state average price. The resultant relatives were then multiplied by the base period.weights for their appropriate class and the weighted relatives summed. Divid- ing the numerator thus obtained by the total base period value of the category, yielded the price indexh for the category. The following price indices expressing weighted averages of 1953 prices as ratios of similarly weighted 1952 prices were obtained.5 X1 Gross income prices index ‘ .8b938 Xh Machinery price index - 1.0080 X5 Livestockbforage price index - .8h809 X5 Cash expense price index = .999h Both livestock—forage and gross income prices showed about a 15 percent decline from 1952 to 1953‘while machinery and cash expense prices remained relatively constant. See appendix Tables IV, V, VI, VII, VIII, IX, X, XI for price relatives and.value weights used in Computing indices. 5 See appendixIBforcomputation of price indices. The Adjusted Magginal Value Productivities Using formula 1,111, andIVf as developed in Chapter II, the adjusted.marginal value productivities for the input and investment categories were obtained. They appear in Table III along with the 1952 marginal value productivity estimates. Table III Comparison of Marginal Value Productivity Estimates for 1952 and the Adjusted Estimates for 1953 Price Conditions* 1952 MVP Eitifiate Input or Investment 1952 MVP Adjusted for 1953 Li Categogy Estimate Price Conditions V X2 Land (per acre) $9.05 $7.68 X3 Labor (per month) 15.11 12.81 Xh Machhnery (per dollar) .b029 .3390 X5 Livestock-forage (per dollar) .1215 .1217 X6 Cash expenses (per dollar) .8593 .7292 17 Building animal units (per unit) 68.6h 58.21 'fgfill MVP estimates computed at Efiz”géoms£rie mean. Interprgtation of Adjusted.Marginal Value Productivities Some consideration should be given the meaning of the adjusted marginal value product estimates which have just been derived. Primarily they are estimates of the earning power of the various input and investment categories of the thirty farms deriving more than forty “ 6 See Appendix C for computations of adjusted MVP estimates. 53 percent of their cash receipts from milk sales, as they were physically organized in 1952, but under 1953 price conditions. To the extent that weather, institutions, management, technology, human relations, etc., remain fixed, or change but little, the estimates indicate the earning power of the various input and investment categories during 1953. Since monetary units were used to measure some of the input and investment categories, a decline in the price of items included in those categories would increase the physical amount which could be purchased for a dollar thereby increasing the marginal physical product of a dollar of the category. One disadvantage of applying price indices to the input and investment categories is that unless prices within the categories move by the same relative amount and in the same direction, adjustment between classes of items within a category are obscured. The grouping of substitute, and complement within categories, however, tends to offset this disadvantage. Prgdiction of Gross Incomg If it were desired to predict the 1953 gross income of a farm similar to those in the sample, and the 1953 dollar values and physical amounts in the various input categories on that farm were known, the procedure would be essentially as follows. First the input and investment categories measured in 1953 dollars would have to be divided by their appropriate price indices. Next the income which would have resulted in 1952 for the farm as organized in 1953 could be predicted by substituting the adjusted.va1ues of the input and investment categories into the estimating equation presented in Sh Chapter IV. 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