MODIFICATION3 0? YR! COBB-WUGLAS :‘r’UNCTION f0 DESTROY CONSTANT ELASTIC”? AND SYMAETR‘.’ Thesis {at the Wm 55 M. S. MECHMAN STATE UNNERSITY Ham“ 0. Caz-hr {955 ___—-— MODIFICATIONS OF THE COBB-DOUGLAS FUNCTION To Destroy Constant Elasticity and Symmetry By HAROLD o. cum AN ABSTRACT Submitted to the College of Agriculture of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Economics 1955 Harold 0. Carter ABSTRACT The objective of this study was to develop and appraise methods of modifying the Cobb-Douglas function to destroy characteristics of constant elasticity and symmetry. In past studies these characteristics 'were suspected of forcing certain undesirable restrictions on the fitted function. The procedure was to make a transformation in the N-dimensional input space by replacing certain independent variables with dummy vari- ables. conceptually, these dummy variables introduced "ridge lines" on the production surface which imposed limits on the substitutability of x1 for 13. The dummy variables were formulated mathematically into what is referred to as modification I and II as follows: zj = Pxi (l-R x3) (1) .32. “23 = PXi (1-3 1‘1) (2) The "P" represents the ridge line proportion of In to X1. The "R” is a ratio of a decreasing geometric series, the terms of which are the re- spective increments in the dummy variable 23 due to sucessive unit in- creases in the independent variable X3. The prominent characteristics of both modification I and II were ascertained and their economic implications determined. Modification I was fitted twice statistically using different estimates of the parameter "P" with the method of least squares. ii Harold 0. Carter Modification II was also fit by least squares to the same data that had been previously fitted by vanes Wagley, using the unmodified Cobb—Douglas function. Comparisons were made using various statistical measures. The functions investigated did not appear to give a better fit statistically than the standard Cobb-Douglas function for these parti- cular data. However, the modified function did indicate certain eco- nomic advantages over the unmodified function. These advantages permit the fitted function to show non-constant elasticity, conditions of symmetry more in agreement with empirical findings, and more realistic marginal value productivity estimates. In addition, modification 11 in the labor input dimension gives strong indication of deveIOping, with further work, into a usable pro- duction function that will show three stages of production simultaneously. Such preliminary findings suggest that research in this area should be extended. iii Pm‘fz.‘ l \ H‘U~.'"‘“' ‘ - x" " .' C ‘c 131-. ‘V i ‘ .°‘ 61%;." " MODIFICATIONS OF THE COBB-DOUGIAS FUNCTICN To Destroy Constant Elasticity and Symmetry By HAROLD o. CARTER A THEIS submitted to the College of Agriculture of Michigan State University of Agriculture and Applied Science in partial fulfillment of the 'requirements for the degree of MASTER OF SCIENCE Department of Agricultural Economics 1955 ACKNOWLFHI‘: MEETS The author wishes to extend his appreciation to the people who contributed to this study. Sincere graditude is expressed to Dr» G. L. Johnson for his inspiration and guidance throughout the study. The author extends his thanks to Dr. L. L. Bcger, Head of the Department of Agricultural Economics, for the financial aid in the form of a graduate research assistantship. Thanks are also due Dr. L. H. Brown for his helpful suggestions. Helpful criticisms and suggestions were gratefully received fran fellow graduate students Albert Halter, Christoph Beringer, Bert Sundquist and Jack Knetseh. The author wishes to express his special thanks to his wife, Janet, who typed the manuscript and gave encouragement when it was needed most.' ‘ The author assumes responsibility for any errors or omissions in this manuscript. V 881866 TABLE OF CONTENTS CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . Organisation of Thesis. . . . . . . . . . . . . . . II. THEOAETICAL BACKGROUND . . . . . . . . . . . . . . . . . History and Origin of Cobb-Douglas Function . . . . Application of the Cobb-Douglas Function in Agri- culture . . . . . . . . . . . . . . . . . . . . . Relevant Characteristics of the Function. . . . . . Returnat°3oal.easeeeeeeeeesoo Smetryesoosss Origin always at the Y Opoint....... Need for Modification . . . III. POSSIBLE MODIFICATIONS OF THE COBB-DOUGLAS “Odificationlsoassesassso Estimating the parameters. . . . Some relevant characteristics of cationlsoaseeoess Disadvantages of modification 1. Advantages of modification I . . MOdification 11 s s a a a s s s e e 0 Estimating the parameters. . . . vi 0,0000, xn : FUNCTION. . s modifi- ll 13 l4 l5 17 18 21 22 22 23 CHAPTER PAGE Relevant characteristics for modification II. . 33 Advantages Of mOdifioation II a e e e s s e e s 26 Disadvantages of modification II. . . . . . . . 27 IV. STATISTICAL EVALUATION AND COMPARISON . . . . . . . . . . 28 Evaluation of the Unmodified Cobb-Douglas Fit. . . . 30 Evaluation of Modification I . . . . . . . . . . . . 52 Evaluati on or “Odi fi oati on I-b a s s e e o e e s s s 40 Evaluation Of MOdifioati on 11 a a e s s e e e s s e s 51 Va SWMANDCONCLUSIONSOOesesassesses... 6]. Summary . . . . . . . . . . . . . . . . . e . . . . 61 Modification I. . . . . . . . . . . . . . . . . 61 Modification II . . . . . . . . . . . . . . . . 62 Evaluation and comparison . . . . . . . . . . . 63 Conclusions. . . . . . . . . . . . . . . . . . . . . 66 BIBLIOGRAPHY...................'.... 69 LIST OF TABLES TABLE PME I. Estimated Regression Coefficients, Standard Errors, and Marginal Value Products at the Geometric Means, Using the Unmodified Cobb-Douglas Function, for Selected InghamCountyDairyFarms, 1952 . . . . . . . . . . . . 31 II. Estimated Regression Coefficients and Standard Errors Using the Modification I and the Standard Cobb- Douglas Fit for Selected Ingham County Dairy Farms, 1952......................... 35 III. Estimated Marginal ValueiProducts Computed from Standard Cobb-Douglas and Modification I at the Geometric Mean for Selected Ingham County Dairy Farms, 1952. . . 36 IV. Estimated Marginal Value Products Using the Modification I-b and the Standard Cobb-Douglas at the Geometric Mean Quantities for Selected Ingham County Farms, 1952......................... 45 V. Estimated Marginal Value Products Using the Modification II and the Standard Cobb-Douglas at the Geometric Mean Quantities for Selected Ingham County Farms, 19:32 0 O O O O O O O O O O O O O O O O O O O O O O O O 56 viii LIST OF FIGURES FFIURE 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. An.Ekamp1e Showing Increasing Returns at a Decreasing late . . . . . . . . . . . . . . . .,. . . . . . . . An Example Showing Constant Returns to Scale. . . . . . An Example of Increasing Returns to Scale . . . . . . . An Example Showing Contour Lines with Characteristics of Symmetry and a Scale line with a Constant SloPe . An Example Showing the Discrepancy That May Exist by Fitting a Cobb-Douglas Function to Fertilizer Input-output Data. . . . . . . . . . . . . . . . . . An Iso-product Line from Modification I Equation in the xixj Dimension . . . . . . . . . . . . . . . . . The Marginal value Productivity of Labor, Modification I Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 . . . . . . . . . The Marginal value Productivity of Livestock-Forage, Modification I Compared with the Standard Cobb- Douglas, on Selected Ingham.County Farms, 1952 . . . Graph Showing Range of Data for Labor and Machinery . . Graph Showing Range of Data for Labor and Livestock . . The Marginal Value Productivity of Livestock-Forage, Modification I-b Compared with Standard Cobb- Douglas, on Selected Ingham County Farms, 1952 . . . ix 10 11 ll 14 21 37 39 42 95 46 SURE 12. 13. 14. 15. 16. The Marginal Value Productivity of Machinery, Modifi- cation I-b Compared with the Standard Cobb-Douglas, on Selected lngham.County Fanms, 1952. . . . . . . . The Marginal Value Productivity of Labor, Modification I-b Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 . . . . . . . . . The Marginal Value Productivity of Machinery, Modifi- cation II Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952. . . . . . . . The Marginal Value Productivity of Livestock-Forage, Modification 11 Compared with the Standard Cobb- Douglas on Selected Ingham County Farms, 1952. . . . The Marginal Value Productivity of Labor, Modification II Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 . PAGE 47 48 53 54 55 CHAPTER I INTRODUCTION Many of the problems in agricultural production economic re- search center around selecting and using appropriate equations to des- cribe basic input-output relationships. It is difficult to devise equations that will express the very complicated ”true relationships” found in the physical and social spheres of the agricultural sciences. Much of the difficulty is created by the various uncontrolled biologi- cal, climatic, and sociological variables which tend to obscure the ”true relationships.” The difficulty is aggravated still further by the inability of finite hunan.minds to understand fully the nature and causality of certain relationships. Oftentines, the best that can be done is to recognise that a functional relation does exist between certain variables. Experience and insight along with observation of data at hand help the researcher determine what type of equation is needed to express the relationship. In productivity studies in agriculture one of the more widely used equations is a power function of the general fans: I = ax1b1,....,x1b1,....,xnbn (1.1) This equation, as it is used in agricultural economics, is referred to as a Cobb-Douglas function. Empirical research using this particular equation indicates that :it has certain major advantages over other functions. Not least among 2 these advantages is the fact that the function can be transfonmed into logarithmic foam, and the parameters estimated by the very sbmple and expedient method of least squares. However, as in most situations, simplicity and expedienmy have a ”price." In this case the "price" is certain shortcomings inherent within the mathematical form.of the Cobb- Douglas function. It will be the purpose of this study to investigate methodically certain modifications or alterations of the Cobb-Douglas equation capa- ble of alleviating or lessening these shortcomings. Among the more prominent of these shortcomings are: (I) constant elasticity not only with respect to the specific 11's but also in respect to all the variables x1,...,xn collectively. (2) intersection of Y and the Yx1,...,Yxn planes at Y = O. (3) inability of the function to describe, simultane- ously, any two relationships such as increasing positive, decreasing positive or negative marginal returns. Organization of Thesis The historical and theoretical background, to be presented in Chapter II, will place the problem in its prOper context. It consists of a brief discussion of the origin and use of the Cobb-Douglas function ‘with special reference to its use in agriculture. The discussion is ;followed by an analysis of the function, both mathematical and economi- cal, with particular attention to the relevant characteristics. Chapter III wdll deal mainly with a methodical investigation of taro possible modifications in the Cobb—Douglas function. The character- istics of each modification are ascertained and their economic impli- cations determined. 3 Chapter IV will present canparative applications and evaluations of the two modifications with the unmodified Cobb-Douglas function after each has been fitted to the same empirical data. The sunsnary and conclusions of the study will be presented in Chapter V. CHAPTER II THEORETICAL BACKGROUND gistory and Origin of Cobb-Douglas Function In 1928, Professor Paul Douglas,1 while at the University of Chicago, computed indexes of labor and capital for American manufactur- ing industries.2 With the help of a mthanatician from Amherst College, Charles W. Cobb, he developed a formula which would measure the relative effects of labor and capital upon productivity during this period.3 The following equation is what is now referred to as the origi- nal Cobb-Douglas function. P = bchl’k (2.1) The dependent variable P represents the value of the total production of the industry. The C stands for the total fixed capital available for production, L represents labor used in production, and b is a con- stant. The exponents k and l-k are coefficients of elasticity for P in respect to the independent variables labor and capital. The sum of the exponents‘were made equal to one‘which implies the assumption of con- stant returns to scale. It may be possible that Cobb's familiarity 1 Now U. S. Senator Douglas from Illinois. 2 Paul H. Douglas, Theory;of wages, (New York: The Macmillan COo, 1934)e 3 Paul H. Douglas and Charles W. Cobb, "A Theory of Production," American Economic Review, XVIII, Supplement, (March, 1928), pp. 139-165. 5 ‘with Euler's theoremfi affected his decision to force the exponents equal to one. This function is linear in logarithmic form, and the values of the b and k can be estimated by a modified5 method of least squares. This equation was later modified on the recommendation of David Durand,6 so that the sum.of the exponents need not equal one. The resulting equation is what Professor Douglas used in his many manufacturing studies. P = bLij (2.2) The exponents k and j are the co-efficients of elasticity of P with respect to ldbor (L) and capital (C) while b is a constant. The function is linear in logarithmic foam and the values of b, k and j can‘be esti- mated by the method of least squares. _Application of the Cobb—Douglas Function in Agriculture Some of the first applications of this type of function in agri- culture were made at Iowa State College by Tintner, Brownlee, and Heady. Tintner used farm.business records from.609 Iowa farms for the year 1942 to derive productivity estimates of various input categories.7 4'Cf., R. G. D. Allen, Mathematical Analysis for Economists, (LOMOns “Quillan and 00e, Ltde. Ste Martin's Ste, 1945’, fil. pe 317s Briefly Euler's theorem states that if each input is attributed its marginal product, the total product, under specified conditions, will be exhausted. 5Gerhart Tintner, "A Note on the Derivation of the Production Functions from Farm Records," Econometrica, XII, No. 1, (January, 1944), ppe 310 6 David Durand, "Some Thoughts on Marginal Productivity with Special Reference to Professor Douglas' Analysis," Journal Political Economics, XLV, (December, 1937), pp. 740-758. ' 7 Tintner, op. cit., pp. 26-34. 6 Tintner and Brownlee made similar estimates for 468 Iowa farms for the year 1939.8 Heady studied a randm sample of 738 Iowa farms.9 Fienup, at Montana State College, also used a random sampling procedure to study productivity on Montana dry land crop farms.10 Johnson applied the Cobb-Douglas analysis in studies of farms in the Purchase Area and western Kentucky.“ He used what he refers to as a "purposive" sample ”a in all of his studies. This means selecting sample farms that are not in scale line adjustment; thus reducing the intercorrelation among input categories and thereby increasim the reliability of the estimated regression coefficients. A similar study was made by Toon at Kentucky, also using a ”purposive” sample.” Similar to Tintner and Brownlee, Drake at Michigan State College used farm account records to gain esti- mates of the marginal productivity of inputs, as well as study some of 8 Tintner and O. H. Brownlee, ”Production Functions Derived from Farm Records,“ Journal of Farm Economics, XXVI, (August, 1944), pp. 566-571. 9 Earl 0. Ready, ”Production Functions from a Endom Sample of Farms," J_ournal of Fara Economics, XXVIII, No. 4, (November, 1946), pp. 989-1004. 1° Darrell F. Fienup, Resource Productivity on Montana Dry Land Crgp Fame, Mimeographed Circularjé'éc, (Roseman: Montana State College Agricuftural Experiment Station, 1952). 11 Glenn L. Johnson, Sources of Income on Upland Marshall County Fame, Progress Report No. l, and Sources of Income on Upland McCracken m Farms, Progress Report No. 2, (Lexington: Kentufiy Agricultural fiperiment Station, 1953). 12 Thomas G. Toon, The Earning Power of Inputs; Investment and Expenditures on Upl_and Grayson County Farms During 1951, Progress Report No. 7, (Lexington: Kentucky Agricultural Experiment Station, 1953). 7 the problems encountered in this approach.15 ‘Wagley, at Mdchigan State College, used a ”purposive" sample in deriving the earning power of selected input categories on thirty-three Ingham County dairy farms.14 This study, as well as Toon's, presents an excellent ”cook-book" des- cription of the computations involved in applying the Cobb-Douglas function to empirical data. Trant, also at Michigan State College, doe rived a.method of adjusting marginal value productivity estimates for changing prices of‘both inputs and outputs.15 Other applications of Cobb-Douglas productivity functions have determined earning powers of certain inputs for individual enterprises. Heady used this approach in fitting power functions with pork dependent on both corn and protein.16 P. R. Johnson, at North Carolina State College, fitted the Cobb-Douglas and other algebraic functions to fertilizerbyield data.17 13 Louis Schneider Drake, "Problems and Results in the Use of Farm Account Records to Berive Cobb—Douglas Value Productivity Functions," Unpublished Ph.D. Dissertation, Department of Agricultural Economics, Michigan State College, 1952. 14 Robert Vance Wagley, "Marginal Productivity of Investments and Expenditures, Selected Ingham County Farms, 1952," Unpublished M. S. Thesis, Department of Agricultural Ebonomics, Michigan State College, 1953. 15 Gerald Ion Trant, ”A Technique of Adjusting Marginal Value Productivity Estimates for'Changing Prices." Unpublished M. S. Thesis, Department of Agricultural Economics, Michigan State College, 1954. 16 Earl 0. Handy, Roger C. Whodworth, Damon Catron, and<30rdon C. Ashton, "An Experiment to Derive Productivity and Substitution Co- efficients in Pork Production," Journal of Farm Economics, XXIV, (August, 1953), pp. 341-354. 17 Paul R. Johnson, "Alternative Functions for Analyzing A ZFertiliser-Yield Relationship," Journal of Farm Economics, XAAV, (November, 1953), pp. 519-529. 8 More recently C. Beringer at Michigan State University developed concepts and methods of utilizing the Cobb-Douglas analysis to estimate the marginal value productivities of input categories in separate enter- prises of multiple enterprise farms. Relevant Characteristics of the Function Returns to Scale Consider the general equation in the following fonm: Y = n1b1,...,x1b1 x3b3,...,xnbn (2.3) Y is the output or dependent variable, the A is a constant and the 11's are independent variables. The bi's are constant coefficients of elasticity for the Y in respect to the 11's.18 The sum of the regression n coefficients (1:13 ) indicates the nature of the returns to scale. Since the coefficients of elasticity are constant over the entire range of the ._—‘ _‘_ _—_ ‘— __ 18 The equation for elasticity is: s =§_¥..-££ (2.4) 3X1 Y Taking the partial derivative of'Y with respect to X1 of (2.3), the resulting equation is: as = b1 E (Y) en *1?“ (M) Solving (2.5) for the regression coefficient gives; BY 11 b =-~—-—-° which is identically equal to the elasticity equation (2.4). 9 function, the elasticities of the dependent variable in respect to the independent variables are necessarily constant. That is, the function can show decreasing positive, increasing positive, constant, and nega- tive marginal returns, singularly but not simultaneously. Decreasing returns.--If the sum of the elasticity coefficients n is greater than zero and less than one (0 <121b1< l), the function 3 indicates total returns which increase at a decreasing rate. This case is illustrated in.Figure 1. Y Total physical product Marginal physical product A xl,eeeeexn (used in any constant set of preportions) Figure 1 An Example Showing Increasing Returns at a Decreasing Rate 'While the total product always increases at a decreasing rate, the mar- ginal product decreases becoming asymptotic to the horizontal axis. This shortcoming cannot be considered too serious from the standpoint of economic analysis since it is irrational for an entrepreneur to Operate ‘within the range of increasing returns or negative marginal returns. Thus, it is not difficult to believe that most fans units are Operating :in the area of decreasing marginal returns to individual inputs. Constant returns to scale.--If the sum of the elasticity co- efficients is one (iglbi = 1), there exists constant returns to scale. 10 Such an equation is linearly homogeneous in the first degree,19 and is illustrated in Figure 2. The total product in this case will go to Aotal physical product Y. Marginal physical product g ' x1,eeee,xn (used in any constant set of proportions) Figure 2 An Example Showing Constant Returns to Scale infinity and the marginal product will be constant at a certain level. This is to say that, if the use of 11 is increased by a given percentage, output increases by the same percentage. This, in view of the law of diminishing returns, implies control over all measurable variables and that unneasurable variables such as management and weather are randomly and normally distributed. Increasing returns to scale.--If the sum of the elasticity co- n 2 i=1 returns which increase at an increasing rate. This case is illustrated efficients is greater than one ( b1>1), the function indicates total in Figure 3. As the use of the 11 is expanded, the total physical product will go to infinity and the marginal physical product will always 19 on, a. s. D. Allen, op. cit., p. 515. Briefly Z=f(x,y) 1. a linear hanogeneous function if f(/\x, Ay)=-.Af(x, y,) for any points (x, y,) and for any value ofA whatever. ll increase. Total physical product ‘fl”’//////////fiarginal physical product x1,0000,xn (used in any constant set of proportions) Figure 3 An Example of Increasizg Returns to Scale Symmetry The summetry of the function is illustrated in Figure 4, which shows that contour lines20 in an xix, plane become asymptotic to both the vertical and horizontal axes. This implies that there is an 11 ./ Scale line iso-product lines 3‘3 Figure 4 An Example Showing Contour Lines with Characteristics of Symmetry and a Scale Line with a Constant Slope 20 Contour lines or ice-product lines show all combinations of two inputs which will product a given output. 12 unlimited range in which the proportions of any two inputs could be varied to produce a given level of output}?1 The symmetry characteristic does not always correspond to reali- ty. For example, output often can be produced with one input, i.e. a dairy cow may produce milk at a low level, utilising only forage but no grain with other inputs constant. The other illustration in Figure 4 shows the contour lines to have equal slope when 11 and 11 are used in the same proportion at successive levels of output. This results in straight scale lines in all subspaces of the ll dimensional input space, but not in spaces in- volving output unless Zbi's in the subspace equals one.22 21 Referring back to equation (2.3) let the dependent variable Y = c (constant level of output) as follows: C = ulbl,eee'11b1 X3b3,...,ann (2s?) Solving (2.7) for 11 in terms of X3, holding other inputs constant, yields: c T Pl 13133.45 n Ifli =0, thenxi :00 or if Xi =0, thenxj =00 . 22 The proof of this in two dinensional subspaces is shown by first considering the general equation of the scale line: MPPXi (Y) =PX1 “1?ij (Y ) Px1L (2.9) This relationship in terms of the Cobb-Douglas equation is secured by deriving the MPP of 11 and K using equation (2.5) and setting the re- sults in a proportion equal to the price ratio Pxi/ij as follows: 13 Origin Always at the Y==O, X1 = o,....,xn = 0 Point The Cobb-Douglas possesses the characteristic of always origi- ating at Y =:x1 =.0. In addition, if any.11 = 0, then Y = 0. These characteristics are closely related to symmetry which shows output being produced with a combination of inputs but never in the absence of a single input.23 Shortcomings of this nature lhmit the applicability of the function since empirical data often reflects input-output relation- ships which: (1) originate somewhere on the Y axis, (2) indicate output even in the absence of certain inputs. .An example of (l) is applying fertiliser to a crcp; the output even at the hero application.may be high due to the natural fertility present in the soil, as shown in Figure 5. An example of (2) exists on farms that produce siseable incomes without livestock investments, since their labor can be marketed in the form of cash crops. b1 E‘Y) X1 - P11 133 Ely) - Fig (2.10) 13 Solving (2.10) for 13 in terms of xi, the equation for the scale line in the Xixd plane is secured: 2: b: PXi . x3 b1 n3 xi (2°11) b Rx As 41—33: K (a constant), X3 = 1011 (2.12) hi PXJ Thus, the scale line is straight in the two dimensional input space and by analogous reasoning the scale line could be shown to be straight in the N-dimensional input space. 23 Refer back to Figure 4 and equation (3.8). l4 Applied Fertilizer) Figure 5 An Example Showing the Discrepancy That May Exist by Fitting a Cobb-Douglas Function to Fertiliser Input-output Data Need for Modification The review of the relevant characteristics of the unmodified Cobb-Douglas equation demonstrates, among other things, that its applic- ation is limited to data which reflect certain kinds of relationships. In order to broaden the use of such a convenient productivity function, alterations of the function to overcome these limitations are needed. Mathematically there are several methods by which this power function could be altered to permit it to reflect certain desirable characteristics. However, any modification must be considered on the basis of (1) whether certain mathematical and economic specifications are met and (a) whether the function retains the important properties of expediency and simplicity in estimating the parameters. The following chapter will consider some possible modifications. CHAPTER.III POSSIBLE MODIFICATIONS OF THE COBB-DOUGLAS FUNCTION Though the need for this study has long been apparent, hmpetus for it was given in the form.of criticism by Professor L. H. Brown1 of certain value productivity estimates in the'wagley Cobb-Douglas study.2 The estimates under criticism.bad the following characteristics: 1. The iso-product lines in the labor-livestock plane possessed the usual Cobb-Douglas characteristics of symmetry where there is an unlimited range in which the absolute amounts of labor and forage- livestock investment can be varied to produce a given level of output. 2. The iso-product lines in the laborqmachinery plane possessed the some conditions of'symmmtry. These characteristics indicate that a farmer with a fixed amount of labor available can increase the use of capital in the form of machinery and livestock indefinitely and, according to the function, continue to increase gross income. However, as Brown points out, the physical ca- pacity of a.man limits the amount of machinery and/or the volume of livestock he can handle and that after this capacity is reached it is illogical to assume the marginal value product of capital investments 1 Professor L. H. Brown is an extension specialist in agricultur— al economics at Michigan State University. 2 R. v. Wagley, op. cit. 16 to be anything but zero or negative. Thus, it was thought the'Wagley estimates might over-estimate the MVP's of machinery and forage-livestock investments and underestimate the MVP of relatively small amounts of labor. These criticisms appeared justifiable as the Cobb-Douglas function does assume symmetry and constant elasticity which.might force these undesirable restrictions on the fitted function in'flagley's study. The current study shows that the shortcomings of constant elasticity and symmetry can be reduced by introducing a "ridge line" in factor-factor dimensions to correspond more closely to the actual pro- duction response. The ridge line can be visualized as a "crease" on the production surface which imposes limits on the substitutability between two inputs at a given level of output. The ridge line also can be considered as the loci of points where additional amounts of’ma- chinery and/or livestock used in conjunction with various fixed levels of labor result in zero marginal returns as a result of ever-present physical limitations. Visualization of a new concept is an important step in research. However, conversion of a concept into mathematical or quantitative form is an equally important step for empirical application. First,oonsider the general form of the standard Cobb-Douglas equation: Y .-.axlbl“...,X1bi,....,X3b3,....ann (301) Y is the dependent variable, the Xi's the independent variables, the bi's the coefficients of elasticity of Y in respect to the Xi's, and the "a" is constant. To place the modification in proper perSpective, ll the assumption is made that all input-output relationships possess the usual Cobb-Douglas form, except Xi and 13 for which empirical evidence shows there exists a limited range within which the proportions of X1 and 13 can be varied in producing a given output. The extreme limit of the range is the "ridge line" where additional 13 results in little change in output. The mathematical procedure followed is to replace Is by a dummy variable, 23, which is a function of both.X1 and 13. In algebraic toms 23 = f(Ii,Xj). Modification I The first modification defines the dummy variable as: 23 = Pxi(l-Rx3) (3.2) The P represents the ridge line proportion of.Xj to x1. R is the ratio of a decreasing geometric series, the terms of which are the respective increments in the dummy variable 23 due to sucessive unit increases in In. X1 and X3 are the independent variables whose iso-product lines are under modification. For computational purposes, it is convenient to measure X1 and X3 in units such that P = l. The dummy variable 23 approaches 2x1, the total amount of X3 that can be associated with a given amount of Xi and still yield MPij(Y):>0 as the input of the variable factor X3 becomes infinitely large. Estimating the Parameters Estimation of the "P".--It is difficult to estimate the "ridge line" proportion of X3 to 11 because other inputs vary which influence 18 the substitutability of xi and X3, thus creating the problem of isolating the studied variables in order to determine estimates of "P." The estimate "P" is also affected considerably by the organizational set-up of the farms involved. A farm, for instance, with a well-arranged milking parlor can obviously handle a greater capital investment in live- stock with a given supply of labor than a poorly organised famm. These disturbances must be taken into consideration by the researcher in evaluating past input-output studies to gain estimates of "P." In addition, consultation with farm.managmment men provides reliable sources of such estimates. Estimation of the "R."--The estimate of the constant "R" is limited to the area between zero and one so as to realize diminishing returns to the dummy variable 23. Between these limits,the value of the ”R" depends on (1) the size of the units in which the x3 is measured, and (2) the nature of the input-output relationship between X3 and Y. A suggested method of determining the appropriate "R” is to make several reasonable estimates of "R" and then plot the 23 function varying only 13. The apprOpriate "R"'will be determined by the curve which yields the closest approximation to the relationship shown in the data. Some Relevant Characteristics of Modification I Egaeticity.--After modification I, the elasticity of the Y with respect to the input X3 is no longer constant, which is shown algebrai- cally as follows: The equation for elasticity is: =_?I.O_xl 3.3 a My ( ) 19 The modification I equation is written as: Y = ulb1,...,xibl [P11(l-Rxd)] bj,...,xnbn (3.4) The partial derivative of Y with respect to 13 in (3.4) is: air -bJRlenR . 8(Y) .. (3.5) 3x3 l-RXJ Substituting (3.5) in (3.3) the elasticity of Y with respect to xi becomes: b x3 .- R O n .X 1-Ex3 As I = -bj ln R,(a constant), equation (3.6) simplifies to: I E = FLEX-L (3e?) l-nxd Equation (3.6) shows that as 13 increases, the elasticity of Y with respect to 13 is no longer constant but increases in.a constant ratio. Scale line of modification I.--The scale line in the 11 and x3 plane of the modified function I is no longer a straight line. This is demonstrated algebraically by the following derivation of the scale line. The equation of a scale line in the 1113 dimension is: MPij (Y) PXj (3.8) The marginal physical product (MPPXi) of x1 for equation (3.4) is: BY =(b;+ bi) E(Y) 8X1 11 (3.9) For the marginal physical product of X3 in equation (3.4), refer to (3.5). The scale line of the modified I equation (3.4) is derived by dividing equation (3.9) by (3.5) and setting the result equal to the 20 price ratio as follows: (bl+bj) E(Y) 2; 5’5. - 3 ln Ron 1:, my) P13 (3-10) l-R x3 Solving (3.10) for 11 gives: ( +b ) PX 41-213) 11 .-. 3—4—1 i (3.11) P11 (-bj 1n R) 313 A: (b1+b3) P13 (-b3'ln B)PX.1 equation for (3.3) in the 1113 dimension becomes: - x3 Equation (3.12) shows that the slope of the scale line in the X113 = K, -a constant, the final simplified scale line plane does not remain constant, but increases with each added increment of Ijo Mw-In Chapter II symmetry was defined as the phenomenon of the contour lines becaning asymptotic to the X1 and 13 axes. In modified form I, each contour line because asymptotic to the X1 axes and to X, = a constant in the x3 dimension. This can be considered mathanatically with the derivation of the contour line in the xix: plane holding other inputs constant. Referring back to the modified form I (3.4), let Y = c (level of output) and P = l. The result is: C = ulb1,eeee.X1bi [H1(1-R x33bj'eeee.xnbn (3.13) Solving (3.13) for xi in terms of x3 yields: “‘ 21 l x g c A A b1"'+"'b3‘ 1 “lulueeee,(l’¥j)vj,eeee,xn1n (3.14) In (3.14), as X3 increases, the denominator approaches a constant and X1 approaches a limit, which is shown in Figure 6. The equation for the line representing the limiting amount of x1 is: l j‘fi 3 c (3 l4a) dF,eeee,an ° 11: 11 Isa-product line for 0 level of output Figure 6 An Isa-product Line from Modification 1 Equation in the 1113 Dimension Disadvantages of Modification I Difficulty in estimating parameters "P" and "R".--Estimatlng the "ridge line” proportion ”P" is difficult, as stated, because organi- sations of farms vary so greatly within a given sample and this greatly affects the proportions that ii to X3 can be varied. Estimating "R“ is also difficult in spite of the fact that the range is narrowed to 0< R(l in order to realise diminishing marginal returns for the dummy variable. Increasingly complex.--The complexities of deriving marginal value product estimtes, iso-product lines, elasticities, and scale line 22 relationships, are increased as shown by equations (3.5), (3.8), (3.11) and (3.13). Advantages of Modification I Blasticity.--The elasticity of the 13 is no longer constant. Scale lines.--The marginal rates of substitution between the 11 and 13 are not constant for successively higher output levels using the same proportions of 11 and 13, and thus the scale lines are no longer straight. §ZEEEEEZ!"Th° characteristics of symmetry no longer exist to such a degree in the In plane which.means that after the ridge line proportion is reached in varying 13 relative to fixed amount of I'D-3% is no longer greater than zero. Modification II A second modification was investigated which in many respects is similar to modification I. It expresses mathematically the “ridge line" condition in the X1X3 dimension anxj 3:) = PK: 0-3 fl.) ‘ (3.15) The 23, as in the first modification, represents a dummy variable which replaces 13 in equation (3.1). The "P” is the "ridge line" proportion of Xj to x1. The R is the ratio of a decreasing geometric series, the terms of which are the respective increments in the dummy variable Z3 due to successive unit increases in X3. The 13 andxi for computational advantages are measured in units such that P = l. The essential differ- ence between this second modification and the first is that R, as shown 23 in equation (3.15), is raised to a power which is the ratio of the relative amounts of X3 and Xi used. This results in the magnitude of the 23 being more dependent on the variable Xi than in modification I. Thus, the elasticity of Y with respect to X3 increases as X3 is expanded but at a smaller constant ratio than modification I. Estimating the Parameter for Modification II Estimation of the "P".--The problens of estimating the parameter "P” are the same as discussed in conjunction with modification 1. Estimation of the "R“.--As stated in modification I , the value of the ”8" depends on (1) the size of the units in which X3 is measured, (2) the nature of the input-output relation between X3 and Y. In modification II the variable factor is measured as a ratio of the rela- tive amounts of Xj and X1, resulting in smaller units and thus a smaller value of "R" . Relevant Characteristics of Modification II Elasticity.” After modification II, the elasticity of the product with respect to the input X3 is no longer constant. This is shown mathematically as follows: The equation for elasticity is: “-181 E;-— . (3.1s) an Y The modification II equation is: x3 Y = AX1b1,...,X1b1,... [PXfil-R KM] ,uxnbn (3.17) Taking the partial derivative of Y with reapect to X3 in (3.17) yields: 24 gflolnR-inEU) a 3 13 (3.18) X, (14:11?) f9. QY 1 Substituting (3.18) in (3.16), the elasticity of Y with respect to 5:3 becomes : x3 if Eg-bj'lnn‘a (3.19) x3 1-31; As K = b3 In R (constant), equation (3.19) simplifies to: non-17 E: i=3 l-R (3.20) As the 13 increases in (3.20), holding Xi constant, the numerator tends to zero, the denominator tends to one, and the elasticity increases to infinity. Comparing the elasticity equations of modification I (3.7) and (3.20) for any given level of X3, holding X1 and K the same, the elasticity of Y with respect to 13 will be greater in (3.7) than (3.20). Mafia-In modification form II each contour line beoanes asymptotic to the Xi axes and to X1 = a constant in the 13 dimension. Derivation of the contour lines in the xixj space follows: Referring back to (3.17), let Y = a (constant level of output). X b' ' a. X J as. x bn 302]. C = Axlbl,eeee,x1b1 [1711(1‘3 1)] ’ ' n ( ) If P = 1, the equation simplifies to: ' "',_,..,“,. ; y 25 1 \T x 1 - A _° 3 = i1 u151,..xibi'53..,x,;bn R (3'22) d Then, putting (3.22) in logarithnic form and solving for 13 yields: e 1 _ c ‘5" 1°5 It‘ll—11’1"». may". 13) 3 8ng (3.23) log R Equation (3.23) shows that X3 ~90, when 11—? co and the XL approaches a constant when Xj --? on , given all other inputs at a constant level other than zero. goale line.--The scale line for modification 11 in the 1113 space no lozger has a constant slope as shown by the unmodified Cobb- Douglas. The scale line equation is represented by (3.8). Derivation of (3.8) in terms of the modified function II follows: The marginal product of X1 for equation (3.17) is: X3 3Y:(b1+b1) 200+ 5 1n n - 311x: 3(1) E1 11 X3 (3.24) I 112(1-3 X?) For the marginal product of X3 refer back to equation (3.18). Dividing (3.24) by (3.18) and setting the results equal to the price ratio, Pxi/ij, defines the scale line in the X113 space. X 3 (b1+b3)(1-nr{) 1 :Pxi x: “if" Ff; (3.25) b3 1n 1:: 311 26 As K = Pij3 ln R, C = PXibj In R and A = PXj(bj+b1), all constants, (3.25) simplifies to: 1‘1 _ K R x1 X1 - 7:3 ““1" (5.26) 1(1-11 Rho-Rf} This means that the marginal product of.11 decreases even more as the "ridge line" proportion of X3 to Xi is approached, and that the use of 11 is expanded proportionally more than X3. Advantages of Modification II The advantages of modification II are much the same as with modification I. These are: Elasticity.--The elasticity of the product with respect to X3 is no longer constant. However, as X3 increases, the elasticity of the modification I increases at a faster rate than the modification 11 equation. This may, or may not be an advantage, depending on the input- output relationship of the data under study. §gale lines.--The marginal rates of substitution between.X3 and X1 are no longer constant for successively higher output levels using the same proportions of X1 and X3. This results in scale lines which do not have a constant slepe. However, the s10pe of the scale line for the modified form 11 is slightly less at most points than for the modified form I because the elasticity of Y with respect to Xj is less. §ymmg_sz.--The contour lines in the X1X3 space become asymptotic to the X1 axes and asymptotic to X1 =:C, a constant, in the X3 space. 21 This partially eliminates the characteristic of symmetry present in the unmodified function. Disadvantages of Modification lI Difficulty in estimating parameter "P" and "R".--The same diffi- culty exists in estimating the parameters as was discussed in modifi- cation 1. ComplexityLe-The complexity of estimating marginal value pro- ductivities, ice-product lines, and scale lines is increased over and above that for'modification I as shown by equations (3.18), (3.20), and (3.26). CHAPTER.IV STATISTICAL EVALUATION AND CGKPABISOH It was shown in Chapter III that characteristics of symmetry and constant elasticity, possessed by the Cobb-Douglas equation, could have forced certain undesirable restrictions on‘Wagley's input-output esti- mates. A conceptual modification'was introduced into the Cobb-Douglas function which can be visualized as a "ridge line" in the factor-factor dimension. This conceptual modification was formulated mathanatically into what is referred to as modification I and II. In this chapter, the problem.must be faced of which of the alternative functions, the origi- nal Cobb-Douglas, the modified Cobb-Douglas 12 or 11 best describes the data. The procedure followed is to fit, by least squares using a loga- rithmic transformation,1 the (1) unmodified Cobb-Douglas function, (2) modification I and (3) modification Il,to a common set of data and evalu-' ate the "goodness of fit" by various statistical measures. 1 Direct statistical tests are available to determine whether a significant reduction in variance is obtained when using the modified function compared to the original Cobb-Douglas, e.g., Fbtest on the sum 1 Cf. Gerhart Tintner, "A.Note on the Derivation of Production Functions from Fanm Records," op. cit., p. 27. Briefly, Tintner states that if the errors in the data are small and nonmally distributed, a loga- rithmic transformation of the variables will preserve the normality to a substantial degree. But even if the errors are not normally distributed and not independent, Tintner further states that we shall still get the best linear estimation by the application of the method of least squares. However, tests of significance are no longer valid. 29 of the squares of the residual quantities. Other.methods to determine the best "fit" are in terms of such statistics as standard error of esti- mate, coefficient of determination, and standard error of regression co- efficients. The standard errors of estimate indicate the closeness with which new estimated values may be expected to approximate the true but unknown values. The coefficient of determination measures the percentage of the variance in Y which is associated with.Xl,....,Xn. The standard error of the regression coefficients measures the accuracy of the esti- mated regression coefficients. These statistical measures have a pre- cise and definite meaning when the assumptions of (l) normality and (2) randomness or at least independence with respect to the Xi's are met. The conditions to be met are: E(u|Xi) = E(u) = 0 (4.1) This states that the unexplained residual (u) given any value of any independent variable (X1) is equal to the expected value of the unex- plained residual (u) which is in turn equal to zero. The additional condition to insure normality and randomness or independence is: 0-“ x1 = 1 2 an (402) The a“, is computed from a normal probability distribution. Equation (4.2) states that the standard error of the residuals given any value of any independent variable is equal to one if the residuals are randomly and nonmally distributed about the regression line. However, there is an inconsistency which exists when testing several functions fitted to the same data. These assumptions of normality and randomness or inde- pendence are made for each regression line. That is, the assumptions of 30 E (ulxi) = 0 and c’u X1 = 1 cannot hold for an unmodified Cobb-Douglas fitted as well as for a modified Cobb-Douglas line fitted to the same data. Thus, if these assumptions are not met for both regression lines, and obviously they are not, statistical tests of comparison based on these assumptions become less meaningful and require more careful inter- pretation. Evaluation of the Unmodified Cobb-Douglas Fit The input-output data used in the evaluation were taken from a study of thirty-three purposively sampled Ingham.County dairy farms located mainly on Miami, Hillsdale and Conover soils for the year 1952.2 Purposive sampling is selecting farms that are not in scale line adjust- ment so as to (I) reduce the inter-correlation among input categories, (2) increase the variance of individual inputs categories and thereby decrease the standard error of the regression coefficients which in— creases their reliability. In lhgley's study inputs were grouped into five categories with gross income (X1) as follows: X2, land, (acres) X3, labor, (months) X4, productive expenses, (dollars) 15, livestock-forage investment, (dollars) X6, machinery investment, (dollars) The data were fitted by the Doolittle method of least squares3 Vance R. Wagley, op. cit., pp. 19-27. 3 Mordecai Ezekiel, Methods of Correlation Analysis, 2nd Ed., (New York: John Wiley and Sons, Inc., 1949): pp. 455-485. 31 using the Cobb-Douglas equation which is linear in logarithmic form as follows: (4.3) log X1 = log a+bz log X2+b3 log X3+b4 log X4+b5 log X5+b6 log X6 The regression coefficients with their standard errors and the marginal value product at the geometric mean quantities are shown in Tab 1 O 1 0 TABLE I 83131112131) 33112353101: COEFFICIENTS, STANDAEJ ERRORS, AND mum. VALUE Pmnuc'rs AT THE csomsrmc MEANS, USING THE mmonirisn coca-comm memos, FOR 331me 11401111: comm DAIRY rims, 1952 W : G t . a : Regression : Marginal Input Category : Men?” ’1; . Coefficients and . Value ; Quan y 3 Standard Error 3 Products : : : 12. Land : 130 Acres . .211c7 7. .09868 . 816.56 : : : X3, Labor : 14 Months : .04166 3|- .13083b : 30.19 : : : x4, EXPOneee : 33,348 : .25001 3 .11432 . .76 : : : 15, LiV'estock-F‘orage . : - : ' 7 6 . 82 .0839 .64 InVrestment : 3 '12 : 44 1 + 4 : x Ma- 8 8 3 5' Gleanery : , : . .. b : In‘resmem : $6,803 3 .12556 + .10929 : .18 : : : \ ‘ Cf. F. E. Croxton and D. J. Cowden, Applied General Statistics, (New York: Prentice-Hall, Inc., 70 Fifth Avenue), p. 221. The geometric me?“ 18 defined as the Nth root of the product of N items which is Written symbolically as: ‘ N ‘7 x11. x12 . 113,00ee,xiN The <3°!nputation is usually carried out by means of logarithmic thus, (4.4) 10; G : log X11 + log X12 + log X13 + ,eeee, + log Xiu (4.5) N by ,, . b Not significantly different from zero at 95 percent confidence ' test. 32 The coefficient of determination was found to be .92, indicating that ninety-two percent of the variance in the dependent variable gross income was associated with variations in the independent variables. The other eight percent may be associated with such unmeasurable factors as mmnagement and weather. “ The standard error of estimate of the dependent variable (E) was found to be .09028 which indicates that for sixty-seven out of one hundred farms randomly sampled from the same population, given 1952 conditions, € gross income would be expected to fall between the fiducial limits of g 8,287 and 12,570 dollars. Modification I The first modification was adapted to Wagley's data by the following procedure: 1. The input categories of land (X2) and expenses (X4) were used in the usual Cobb-Douglas form. 2. The input categories of livestock-forage (X5) and.machinery (Kg) were considered as capital investments whose earning power is limited, as pointed out in Chapter III, by the physical capacity of the labor (X3). Therefore, in modification 1, introduction of the "ridge line" on the production surface'which imposes limits on the substitut- ability of X5 for X3, and.Xg for 13, was accomplished by replacing x5 and X5 by the dummy variables Z5 and 25. The dummy variables are defined 8.88 25 .—. P113 (l-Rx5) (4-6) 36 Ze = Pm (1-2“) (4.7) In estimating the "P's“, the problem'was to determine: 1. The "ridge line" preportion of dollars invested in livestock- forage to labor. 2. The "ridge line” proportion of dollars invested in machinery to labor. The following P's were secured in consultation with.Professor L. H. Brown4 who based his estimates on the observed operating ratios of capital to labor for numerous Michigan farms. 1. Thirteen hundred dollars of livestock-forage investment per month of labor. 2. One thousand dollars of machinery investment per month of labor. In order to make the Z5 and Zg easily computable, the livestock- forage investment was measured in thirteen hundred dollar units and the machinery investment in one thousand dollar units. This made P’: l in 25 and 26. The constant R was made equal to nine-tenths for both the 25 and 25 after considering the graphical relationship between x5 and Z5 and X5 and 26 using different values of R. No statistical tests were applied. The modified equation 1' is linear in logarithmic form and was fitted by the Doolittle method of least squares.5 4 Professor L. B. Brown is an extension Specialist in Agri- cultural Economics, Michigan State UniverSity. 5 Ezekiel, op. cit., pp. 455-485. M-~ V" m B... - .rex (54 log XI = log a+b2 log X2+b3 log 13+b4 log X4+b5 log P1X3(l-Rx5) + b6 log rzx3(1-836) (4'8) In modification I, the regression coefficients and the standard errors were computed. These are canpared with the regression coefficients of the standard Cobb-Douglas fit in Table II. A canparison of the elasticities for the input categories of land (12), labor (13)6 and expenses (14) for the modified equation I with the standard Cobb-Douglas shows no significant difference at the 95 ”merit level. However, a comparison of the regression coefficients b5 “‘1 he would be meaningless because in the modified function the bi's rePresent the elasticities of the dummy variables 25 and 26. A more meaningm1 ounparison between the "fits" would be to compare the esti- mat°d marginal value products as shown in Table III. Table III shows no significant difference between the MW of land f°r the modification I and the unmodified function. However, the marginal "1‘10 product of expenses for the modified form is greater than for the standard Cobb-Douglas estimate. A normal return on expenses is a dollar f°r an dollar. The estimated MVP of labor for all quantities of labor is greater for the modified form I than the standard Cobb-Douglas fimction as shown in I"igure 7 (Cf. modification I-b and 11, discussions on labor). The marginal value product of livestock-forage, at the geometric mean amount, for the modified function is greater than the unmodified \ 5 See footnote b, Table II. 35 TABLE II ESTIMATE) REGRESSION COEFFICIENTS AND STANDARD ERR01§ USIN} THE MODIFICATION I AND THE STANDARD COBB-DOWLAS FIT FOR SELECTED 1181M COUNTY DAIRY FAEMS, 1952 8 8 8 : : Regression“ : Regression Input : teifimi’m. 3 Coefficients for . Coefficients for ...__ : : Modification I : Standard Cobb-Douglas 8 8 8 18nd x2 : 130 Acres 3 .20321 3; .11416 . .21105 1; .09868 8 8 8 Labor x3 , 14 Months 3 -.52093 ; .19956b : .04166 1 .13083 8 8 8 Expenses x4. $3,348 : .36461 3; .12425 : .25001 1 .11452 L1 8 8 8 Orage x5 : $7,126 3 .66225 + .10855 , .44821 + .08394 Ma 8 8 8 o - tangy ' 36,603 : «08649 ; .07545° : .12556 t; .10929 ‘ The regression coefficient for the livestock-forage and machinery indicate the elasticities of II with respect to the dummy variables 25 and Ze. b The negative regression coefficient b3 does not indicate a :38 ative elasticity as shown by the following proof. According to °r1nition the equation for the elasticity of X1 with respect to 13 is: 9X1 X3 33s I: “'9’ The partial derivative of the modified equation with respect to X3 is: 311 = (bs+b5+b6)j(xil 9X3 13 Suba‘tituting (4.10) back into (4.9) and simplifying yields: EXS = b3+b5+b6 (4.11) (4.10) :heh, substituting the values canputed into the equation reveals that he elasticity of X1 with respect to X3 is: 1913 = -.52093+.66225+(-.06649) = .05483 (4.12) f ° The "t" test indicated that be was not significantly different Pom zero at the 95 percent level. 36 TABLE III ETIMATED MAEINAL VALUE PRODUCTS COMPUTED FROM STANDARD COBB-DOWLAS AND MODIFICATION I AT THE GEONETRIC MEAN FOR SELETED INGRAM COUNTY DAIRY FANS, 1952 Standard Cobb-Douglas‘s Modification Ib : Input Categories *_ ._ 8 8 8 8 land, (12) : 16.56 : 15.38 8 8 Labor, (is) 8 30.19 8 39.26 8 8 kpeneee, (x4) : .76 : 1.10 8 8 Livestock-Forage, (X5) : .64 : .79 8 8 l“schemata-y, (x6) : .18 : -.10° 8 —— 8 ‘ The marginal value products for the standard Cobb-Douglas is °°ulputed from the following form: MVPX1 =‘b1 E(X1) (4.13 X1 ) b When using the modified equation (4.8), the problem of com- Puting the MVP's becomes slightly more canplex. The general procedure 13 to take a partial derivative of the dependent variable in respect to the independent variable. 9X1 __ (b3+b5+b5) E(X1) " (4.14) 15 ' 9x1 __ -b5R in R-E(X1) , _ ' (4.15 ) 9115 (1-315) 9x1 4:63“ in sosul) = . (5.16) 3 X6 (1-Rxb) :29 Tnethcd of deriving the MVP's for the x2 and X4 is the same as for atlclard Cobb-Douglas and is shown in footnote above. 95 ° The "t" test indicated no significant difference from zero at Percent level. 200 Modi ficati on I / 160 120 80 Standard /‘7 Cobb-Douglas // 5 q 10 15 20 25 Labor Months Figure 7 The Marginal Value Productivity of Labor, Modification I Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 37 ?{~,g“"“ 38 Cobb-Douglas. However, this situation does not hold true when the amount of livestock—forage investment exceeds the "ridge line proportion." That is at the "ridge line" proportion of eighteen thousand dollars, (labor fixed at fourteen months and thirteen hundred dollars the esti- mated maximum amount of livestock-forage investment per month of labor) to ,-._._..s’ the IV? of livestock-forage in the unmodified Cobb-Douglas exceeds the MVP for the modified form I as shown in Figure 8. The negative marginal value product of machinery for the modi- .\ . N “9-.5_._‘ ‘g: m.-. o . fied function does not appear too meaningful in view of the author's knowledge of the sample farms. Assuming the normality of the residuals, the standard error of estimate (S) for the modification I was computed and found to be .09137. This indicates that of sixty-seven out of one hundred farms randomly sampled, from the same population conditions, gross insane would be expected to fall between 8,267 and 12,590 dollars. The (S) of the un- revised Cobb-Douglas fit is .09028 indicating that sixty-seven percent of farms randomly sampled from the same population, given 1952 conditions, would expect to have a gross income between 8,287 and 12,560 dollars. It is apparent that there is no significant difference between the stan- dard error of estimates of the modified form I and the standard Cobb- Douglas, at the ninety-five percent level. The coefficient of determination (R2) was computed to be .88 for the modified equation 1 which indicates that eighty-eight percent of the variance in gross income (X1) is associated with variations in the in- dependent variables. 'l‘he coefficient of determination for the standard 39 MVP 3.4 2.0 6______Standard Cobb-Douglas 1.6 1.2 .8 Modification I .4 4 8 12 16 20 Livestock-Forage Investment (Thousands of Dollars) Figure 8 The Marginal Value Productivity of Livestock-Forage, Modification 1 Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 4O Cobb-Douglas was computed to be .92 and there appears to be no signifi- cant difference between these estimates. Another method used to evaluate "goodness of fit” among the 7 on the variances of the computed functions was to apply the F-test residuals, i.e. the difference between the estimated gross income and the actual gross income. The standard error of the residual quantities for the modification I was found to be 1,777 dollars compared to 2,490 dollars for the standard Cobb-Douglas estimate. The reduction in the variance for the modification I proved to be significant at the ninety- five percent level. The final conclusions that the "fit" of the modified function I is not superior to the standard Cobb-Douglas, in spite of the fact that there is a significant reduction in variance of the residuals, was drawn because: 1. It is meaningless to assume a negative or zero marginal value product for machinery in view of the farms studied. 2. The standard error of estimate and coefficient of determi- nation between.modification I and the unmodified function were not significantly different, at the ninety-five percent level. Evaluation of Modification l-b An iteration of the first modification was made, which_is re- ferred to as modification l-b, because only a few of the surveyed fanms 7 George H. Snedecor, Statistical Methods, 4th Ed., (Ames: Iowa State College Press, 1946), pp.38‘: 41 had more than thirteen hundred dollars invested in livestock-forage per month of labor and/or one thousand dollars invested in machinery per month of labor. For modification l-b a new ridge line proportion ”P" was esti- mated of (1) seven hundred dollars investment in livestock-forage per month of labor, and (2) seven hundred dollars investment in machinery } 3 per month of labor. These new proportions included a larger share of the sample farms whose capital-labor ratio exceeded the estimated "ridge line" proportion as shown in Figures 9 and 10. Thus, it was expected V'm.--.-.‘ 4...; 'L- that the estimated MVP of livestock-forage and machinery would decrease and the MVP of labor increase. For computational simplicity the machinery and livestock invest- ments were measured in seven hundred dollar units. Thus the P in both dummy variables, was made equal to one. 26 =.P2x3 (l-hx5) (4.18) The value of the constant R was left at nine-tenths. With these changes the modification I-b was fitted in logarithmic form by the Doolittle method of least squares.8 The regression coefficients were computed along with the standard errors and found to be: be = .12602 1 .11335 b3 2-.82817 ; .26248 8 Ezekiel, 0p. cit., pp. 455-485 9 The negative b3 for the modification l-b does not indicate a negative elasticity with respect to labor (X3) as shown in footnote b, Table II. The elasticity of gross income with respects to X3 would be equal to the sum of b5, b5, and be which is .12047. 3O 28 26 24 22 20 18 16 14 12 10 LABOR'MOHTHS 42 *- 1:700 line + / I I , T // i- . O O f’ 131000 line 2/, b 1’1 / 2 4 6 8 10 12 14 16 18 20 22 Machinery Investment (Thousands of Dollars) Figure 9 (Eraph Showing Range of Data for Labor and Machinery 30 28’ 26 24 20 18 LABORlIUNTHS 43 1:700 line 121300 line Livestock Investment (Thousands of Dollars) Figure 10 Graph Showing Range of Data for Labor and Livestock ' V Cl .30.??’ "‘1? sub. p.__. . ' . . A- h 44 .38438 3 .12531 as N b5 = .64612 I .13286 b6 = .30252 1 .17344 The constant "a" was computed to be 2.45972. With these coefficients, the marginal value products were esti-. mated and compared with the Cobb-Douglas marginal value products in Table IV. ..-~..«-+~ "'1- 11; The estimated marginal value products of livestock-forage for the modified equation I-b and the unrevised Cobb-Douglas are shown in "7?"? V. 1 Figure 11. Figure ll indicates that at very low rates of investment of livestock-forage, the modified function I-b yields a higher marginal value product to this investment than the unmodified Cobb-Douglas. However, at higher rates of investment, which exceed the "ridge line" proportion, the marginal value product of the modified function I-b is less than the unmodified Cobb-Douglas estimates. In Figure 12, the same type of comparison is made between the marginal value products of machinery investment, holding other inputs constant at the geometric mean. Figure 12 indicates that at low rates of investment of machinery, the modified function I-b yields a higher MVP than the standard Cobb-Douglas estimates. However as the invest- ment of‘machinery exceeds the "ridge line" proportion, the.MVP of the modified functions decreases at a faster rate than the modified function. The estimated MVP of labor for the modified function I-b and the unrevised Cobb-Douglas are shown in Figure 13. Figure 13 indicates TABLE IV 45 ESTIMATED “AEGINAL VALUE PRODUCTS USING THE.MODIFICATION I-b AND THE STANDARD COBB-DOUGLAS AT THE GEOMEI’RIE MEAN QUANTITIE FOR SELECTED INGRAM COUNTY FARMS, 1952 Input Category G ecmet ric Mean Marginal Value‘ Product--S tandard Marginal Valueb Product-- : z and Unit : Quantities : Cobb-Douglas : Modification I-b Land, (x2) : 130 Acres : 16.56 1 9.54 Labor, (x5) : 14 Months : 30.19 : 86.27 Expenses, (X4) : 3,348 : .76 i 1.16 2 I 2 $332215) : 7,126 i .64 z .50 llachinery, (is): 6,803 : .18 i .23 8 s x __ ‘ For method of derivation refer back to footnote a, Table III. b For method of derivation refer back to footnote b, Table III. . ‘_..,‘"m. .o‘ufi’. l 332:"? '. a J Yolk, g' ' 46 km? 2.E3 2. 4 §____. Modification I-b 2.() 1.2 Standard Cobb-Douglas / 4 8 12 16 20 Livestock-Forage Investment (Thousands of Dollars) Figure 11 The Marginal Value Productivity of Livestock-Forage, Modification I-b Compared With Standard Cobb-Douglas, on Selected lngham County Farms, l9b2 .\ “D _ . '27 . .. 'l-"q [9-h- am.--.-.. _._. -. . . . . I o '- 3"; : — . 47 MVP 2..4 2 . O ¢____l Modification I-b 1..<3 1 . 2 1 i \ . \ -_8 \ o 4 I Standard Cobb-Douglas__~, 4 8 12 16 20 Machinery Investment (Thousands of Dollars) Figure 12 The Marginal Value Productivity of Machinery, Modification I-b Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 MVP 240 200 1160 JJBO 80 40 Modification I-b Standard /n Cobb-Douglas/” 5 10 15 20 25 Labor Months Figure 13 The Marginal value Productivity of Labor, Modification I-b Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 46 j Inn 1" ‘Ll‘l-r'l.b(.fipfl 1 .. . I V: [HUS-L49 ‘I’I—d . ._ l '- n . - a a 1' I I , '. E4 w 49 that the MVP of labor for the modified function l-b is considerably higher, for all quantities of labor, than the unmodified Cobb—Douglas estimate. These results, which show high relative MVP's for labor and low relative MVP's for livestock-forage and machinery for the modified function I-b, can be related back to Brown's original criticism of certain estimates in lhgley's Cobanouglas study. These criticisms were concerned with estimates that showed relatively low earning power for labor and relatively high earning power for livestock-forage and machinery when in fact the capital-labor operating ratio may have exceeded the physical capabilities of the operator. These estimates of modification I-b indicate that the Cobb-Douglas function can be altered to reflect more closely these types of relationships. However, the earning powers reflected by the modified function I-b do not give a significantly better statistical fit than the unmodified Cobb-Douglas for these particular data. Using the F-test on the sum of the squares of the residuals to evaluate the ”goodness of fit," the reduction in the sum of the squares for the modification I-b compared to the unmodified function did not prove to be significant at the ninety-five percent level.10 The standard error of the computed residuals for the modification I was found to be 1,981 dollars compared to 2,490 dollars for the standard Cobb-Douglas, and 1,777 dollars for the modification I estimate. 10 Snedecor, op. cit., p. 380. if?!" } VP” “BL“.-. 50 When applying the F-test on the residual variance quantities of the farms which had more than seven hundred dollars investment in machinery per month of labor and/or more than seven hundred dollars of livestock-forage per month of labor,11 the reduction in the sum of the squares for the modification I-b as compared to the unmodified function did not prove to be significant at the ninety-five percent level. The standard error of estimate (S) about the regression line for the modified equation I-b was found to be .09468 in logarithns as compared to .09028 for the standard Cobb-Douglas, and there was no significant difference between these estimates at the ninety-five per- cent level. The (S) for modification I was found to be .09137‘which‘was not significantly different fran (S) of I-b, at the ninety-five percmt level. The coefficient of determination (F?) was .89 for the modified function I-b compared to .92 for standard Cobb-Douglas. This was not significantly different at the ninety-five percent level. The co- efficient of determination for modification I was .88 which was not significantly different from.the modification I-b estimate. The conclusion that the statistical fit of modification I-b is not superior to the standard Cobb-Douglas was drawn because: (I) the standard error of estimates (S) were not significantly different, (2) the reduction in the variance of the residuals from all the farms using the modified equation I-b was slight, but not signifi- cant, and (3) the reduction in variances, of the farms whose capital 11 Refer back to Figures 9 and 10. Irv-s'rr-"u . '1 51 investments in livestock-forage and/or machinery exceeded the estimated ”ridge line" proportion, was not significant. Conclusions were drawn concerning the comparative fits of modifi- cation I-b and modification I as follows: 1. The standard error of estimate at the geometric mean showed no significant difference between the two fits. 2. The variance of the computed residual quantities was smaller, but not significantly, for modification I as compared to modification I-b. 3. The computed coefficients of determination were not signifi— cantly different between the fits. Evaluation of.Modification II The same input-output data were fitted, by the Doolittle method of least squares,12 using modification 11 which is linear in logarithmic fonm, as follows: 10g X1 = log a + 6210512 + balogX3 + b4logx4 + 312 32.5. (4.16) bslog P1X3(l-R X3} + b6log P2X3(l-R x3) The estimates of the "ridge line" proportion of machinery invest- ment to labor months, and livestock-forage investment to labor months, were identical to those in modification l-b. These estimates of "P" were seven hundred dollars machinery investment per month of labor and seven hundred dollars livestock-forage investment per month of labor. 12 Ezekiel, Op. Olte, ppe 405-4850 52 These ”ridge line" proportions were used in preference to the modifi- cation 1 "P‘s” because they appeared more reasonable in View of the operating ratio of capital to labor for farms in the study. For compu- tational purposes, the machinery and livestock-forage investments are measured in seven hundred dollar units maxing the "P's" equal to one. f~+ The constant B was made equal to three-tenths. This is smaller than or r.r -..‘ .I’ previous estimates of R because the variable factor, capital, is measured in units relative to labor months and not in absolute terms (Cf. 4.16). The marginal value products for the modified equation 11 and the pm4~wrn, unmodified Cobb-Douglas were computed at the geometric mean organi- zation and are shown in Table V. The marginal value products at the geometric mean for modifi- cation 11 and the unmodified function do not appear to be significantly different for land, machinery, and livestock-forage respectively. A comparison between the MVPs, of the modified equation II and the stan- dard Cobb-Douglas, with varying amounts of (l) livestock-forage, (2) machinery investment, and (3) labor months shows relationships as re- presented in Figures 14, 15, and 16. Figure 14 shows (1) the marginal value product for machinery in- vestment with the modified equation 11 to be slightly greater than the Cobb-Douglas when using amounts of machinery investment less than the "ridge line" proportion and (2) that the marginal value product tends to decrease more than the unmodified Cobb—Douglas estimate when using amounts greater than the ridge line proportion. 3%. _‘ 53 MVP 2.0 4‘} E 1.6 (____ Modification II werr+~ . 1'. 1.2 .8 .4 Standard Cobb-Douglas 4 8 12 16 20 Machinery Investment (Thousands of Dollars) Figure 14 The Marginal Value Productivity of Machinery, Modification 11 Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 g_____.Modification II 2.0 t——-— Standard Cobb—Douglas \ 4 8 12 16 20 Livestock-Forage Investment (Thousands of Dollars) Figure 15 The Marginal value Productivity of Livestock-Forage, Modification II Compared with the Standard Cobb-Douglas on Selected Ingham County Farms, 1952 54 55 MVP 200 ,___ Modification 11 160 120 80 40 é’,,Standard Cobb-Douglas O i_J- -40 -80 5 10 15 20 25 Labor Months Figure 16 The Marginal Value Productivity of Labor, Modification 11 Compared with the Standard Cobb-Douglas, on Selected Ingham County Farms, 1952 11—2.“...— ‘. . i- ‘ “*7 ~ ~, “—1er“ 5V ' 'u.“" 56 TABLE V ESTIMATE) WINAL VALUE PmDUCTS USING THE MODIFICATION 11 AND THE STANDARD COBB-DOIBLAS AT THE GEOMETRIC MEAN QUANTITIES FOR SELETE) INGHAM COUNTY FANS, 1952 V—__ Marginal Value‘ b Product-~Standard Marginal Value G M ecmetric can Product-- Input Category 2 2 2 2 2 2 t t‘ and "n” : Q‘m‘ 1 1°” : Cobb-Douglas : Modification 11 2 2 2 Land, (x2) x 130 Acres : 16.56 : 13.67 2 2 2 Labor, (13) : 14 Months : 30.19 : 23.78 2 2 2 Expenses, (X4) : 3,348 : .76 : .93 2 2 2 Livestock- : : : Fons” (x5) 3 7,126 , .64 x .68 2 2 2 Machinery, (X6): 6,803 : .18 : .22 2 2 2 9‘ The MV'P's were canputed from the equation: MVPxi : b1 E(Xl) Xi b The MVP's are computed by taking a partial derivative of the dependent variable with respect to the individual inputs or independent nn‘bloae x6 X6 X1 = szbzxsb'gxcb‘ [hitch-11753)] b5 [P2X3(1-R1'5)] ”3 (4.20) (4.19) The MVP or partial derivation for the 12 and X4 variable is computed form (4.19). The MVP's for inputs X3, X5 and X5 are more complex as follows: X5 X6 911 =(b3+b5+bg)E(xl) + b5'R 3'3 ln R-xsmxl) + 6611 15 1n Roxsmxl) x 1: X5 X6 9 3 3 X32(1~R TS) x32(l-R I5) (4'21) x q is 3.1 = “’5“ X3 1“ “(W (4.22) 81 X5 16 X = - ‘x3 'E.X 9...}. 1’53 1" R < 1’ (4.23) 3x6 X6 13(l-R XE ) . m! . a. sfifi " pan-7.-..- 1. .ln. ’ - d '— «In: «:6 1.. .Ih. 57 Figure 15 shows that the marginal value product of livestock- forage with modified equation 11 to be smaller at.all points compared to the standard Cobb-Douglas. Furthermore, the divergence increases with the larger livestock-forage investment. Figure 16 shows labor, using the modification II equation, to * 1 have an extremely high earning power of 504 dollars at five months and L 1' negative 94 dollars at the other extreme of eighteen months. The MVP E" I! of 504 dollars at five months is not so strange considering the capital- : ' .1. labor ratio at this point exceeds the estimated "ridge line" propor- ii tion,13 and according to the nature of the modified function 11 the m of labor would be relatively high. No statistical significance was attached to the negative MVP because there were no farms available with fifteen or more months of labor and geometric mean preportions of other inputs, and thus extrapolating beyond the range of the data is meaning- less. These results suggest that it is possible to reflect more than one stage of production by a transformation of the independent variables. The standard error of estimate of the modification II was found to be .06382 compared to .09028 for the unmodified Cobb-Douglas esti- mate. This indicates that for sixty-seven out of one hundred randomly sampled farms from the same pepulation, given 1952 conditions, the esti- mated gross income would fall within the fiducial limits of 8,808 dollars and 11,817 dollars using the modified 11 regression equation. 13 The livestock-forage and machinery investments are held constant at the geometric mean amount of 7,126 dollars and 6,803 dollars, respectively, while the estimated "ridge line" proportion of capital for five months of labor is 3,500 dollars. b8 This is compared to the fiducial limits of 8,227 dollars and 12,560 dollars using the standard Cobb-Douglas estimating equation. The dif- ference in the (8) was not significant at the ninety-five percent level. The coefficient of determination (E?) for the modification II was found to be .89 compared with .92 for the standard Cobb-Douglas. The ‘ ‘— E] T difference was not significant at the ninety-five percent level. The F-test was applied to the variance of the computed residuals. The standard error of the residuals for the.modification II was found : to be 1,730 dollars compared to 2,490 dollars for the standard Cobb- L-§ Douglas. The reduction in the sum of the squares for modification 11 proved to be significant at the ninety-five percent level. The same F-test was applied on the variance of the residuals for the specific farms having more than seven hundred dollars worth of machinery investment per month of labor and/or seven hundred dollars 'worth of livestock-forage investment per month of labor. The reduction in the variance was not significant at the ninety-five percent level. Conclusions were drawn concerning the comparative statistical fits of modification 11 and the unmodified function as follows: 1. The standard error of esttmate (§) was not found to be significantly different for the modification 11 as compared to the un- znodified function. 2. The variance of the computed residual quantities were significantly smaller with.the modification 11 as compared with the Lnnnodified function. 3. The computed coefficients of determination were not signifi- cantly different between the fits. 59 4. Estimated earning power of machinery and livestock-forage for modification 11 decreased relatively more than estimates for the un- modified function when using amounts of investments greater than "ridge line" proportions. 5. The MVP of labor for the modification 11 showed decreasing positive, and decreasing negative marginal returns. The estimates were not statistically significant but the transformation suggests greater possibilities with more applicable data. Conclusions were drawn concerning the comparative statistical fits of modification II and modification I-b as follows: 1. The standard error of estimate (§) was not significantly different for the modification 11 as compared to the modified function I-b. 2. The variance of the computed residual quantities was slightly smaller, but not significantly so, for the modification II compared with modified function I-b. 3. The computed coefficients of determination were not signifi- cantly different between the fits. 4. The variance of the computed residual quantities, for the farms whose capital-labor ratio exceeded the estimated "P," was slightly smaller but not significantly so, for the modification 11 as compared to I-b. Conclusions were drawn concerning the comparative statistical fits of modification 11 and modification I as follows: 1. The standard error of estimate (E) at the geometric mean a...‘ '0‘. -r«; r e ' E-m\# 60 'was not significantly different for the modification II compared to modification 1. 2. The variance of the computed residual quantities were not significantly different for the two fits. 3. The computed coefficients of determination were not considered significantly different between the two fits. R . I W' h:4 a!" a, ’4‘“ CHAPTER V SUMMARY AND CONCLUSIONS Summagy This study has considered a method of destroying the character- . . ' .maamw _,___:=r" istics of symetry and constant elasticity, inherent within the Cobb- Dctglas function, which may force certain undesirable restrictions on .1. .. _- 'ah. ~ ‘ 'W’."’i*. the fitted function. A conceptual modification was introduced in the form of a ”ridge line" in the factor-factor dimension to correspond more realistically to the actual production responses. The modifications were formulated mathematically by replacing the 13 with a dumny variable 23 in the unnodified Cobb-Douglas equation. The 23 = £(xi,x3). Hodificati on I The first modification defined the dummy variable as: z, a non-a 13) (5.1) The P represents the ridge line proportion of X3 to Xi. The R is the ratio of a decreasing geometric series, the terms of which are the re- spective increments in the dummy variable Z due to successive unit increases in the independent variables Xj. The prominent characteristics of modification 1 are: 1. The elasticity of Y with respect to 13 is no longer constant. 2. The slope of the scale line in the 11x3 plane varies with the amount of X3 present. 62 3. The contour in Xixj space becomes asymptotic to the X1 axes and to x, =.constant in thexj dimension. Modification II The second modification defined the dummy variables as: In 23. = min-a XI) (5.2) . The parameter P is the ridge line proportion of.xj to Xi, the R.is a constant which is a ratio of a decreasing geometric series, the terms of which are the respective increments in the dummy variable Zj, due to successive unit increases in the independent variable In. The difference between this second modification and the first is that R is raised to a power which is the ratio of the relative amounts of x. J and I used. This results in the magnitude of the Z. being more do- i J pendent on the variable X, than in modification I which causes the elasticity of Y with respect tc.x3 to increase more slowly than modification 1. The characteristics of modification II are: 1. The elasticity of the Y with respect to the Xj varies with the amount of x3 present. Using modification I-b estimates of the pa- rameters "B" and "R", the elasticity of Xj'will be less than.modifi- cation 1 estimates. This may or may not be an advantage depending on the nature of input-output data. 2. The scale lines no longer have constant slope which indicates variable rates of substitution between Xi andxj when expanding the use of resources in the same proportion. 'J‘PI-‘lm A u... “as . .i. w..- .- . - dflfiw, ‘21:: v I “ 9a: 63 3. The contour lines as in modification l, become asymptotic to thex1 axes and to X1 = constant in the.Xj space. 4. The Xi variable indicates both positive decreasing and nega- tive marginal returns. Evaluation and Comparison The problem was to determine which of the alternative functions, ‘u 1 ‘.'-.‘.~ the unmodified Cobb-Douglas,modification I, or modification 11 best describe the data. The procedure was to fit by least squares the three functions and evaluate the 'gcodness of fit" by various statistical measures. Input-output data used in the evaluation were taken from a study by Vance Wagley cf thirty-three purposively sampled lngham County farms, using six input categories meaningful with the dependent variable gross income. Evaluation of.modificaticn l.--The "ridge line” proportion "P" ‘was estimated at (1) thirteen hundred dollars livestock-forage invest- ment per'mcnth of labor, and (2) one thousand dollars machinery invest- ment per.mcnth of labor. The equation was fitted by least squares and the results indi- cated: l. A negative MVP for machinery which is meaningless in view of the farms studied. 2. A significant reduction in the variance of the residual quantities compared to the unmodified function. 3. The standard error of estimate was not significantly ’- .65”! _- - a ultra-f“ “-i - - -. C e."g'iS-s'ih. _. . rurav' 64 different from.the unmodified Cobb-Douglas. 4. The coefficient of determination was not significantly different from the unmodified Cobb-Douglas. 5. The MVP of labor was relatively higher when using amounts less than the estimated "ridge line" proportion for modification I as compared to the unmodified function. 6. The MVP of livestock-forage was relatively less for amounts greater than the estimated "ridge line” proportion when using the modification 1 as compared to the unmodified function. The conclusions were that the modification I equation did not yield a superior statistical fit to the unmodified Cobb-Douglas for these particular data. However, the function did permit a higher MVP for labor and a relatively low.MVP for livestockvfcrage when using amounts greater than the ridge line prcpcrticn. Evaluation of modification I-b.--An iteration of the first modification was made using a new "ridge line“ proportion of (1) seven hundred dollars investment in livestock-forage per month of labor, and (2) seven hundred dollars investment in machinery per month of labor. The equation was fitted by least squares and the results indicated: 1. The reduction of the residual variance quantities was slight but not significant when using the modification I-b compared to the un- modified Cobb-Douglas. 2. The reduction in variance of the residual quantities of the fanms whose capital investment in livestock-forage and/or machinery exceeded the estimated "ridge line" pr0pcrticn was not significant when I. , _rr 65 using the modification I-b as compared to the unmodified function. 3. The standard error of estimate was not significantly different from the unmodified Cobb-Douglas estimate of (g). 4. The coefficient of determination was not significantly dif- ferent for the modification I-b and the unmodified function. E 5. The MVP of labor for modification l-b indicated the same re- i l t lationship .. did modification 1. 6. The MVP cf livestock-forage for modification I-b indicated iron—- m; . 1.1; the same relationship as did modification I. 7. The MVP of machinery was relatively lower for amounts greater than the estimated ”ridg: line" proporticn‘when using modification I-b as compared to the standard Cobb-Douglas function. The conclusions were that modification I-b did not dancnstrate a superior statistical fit for these particular data but it did show certain economic advantages. These advantages were in the fcrm of non- constant elasticity for the dependent variable in respect to the modified independent variables, and further it permitted the fitted function to show more economically realistic isc-prcduct relationships in the xixj dimension. Evaluation of modification lI.--The same estimates of the "ridge line" proportion of’machinery investment to labor months, and livestock- fcrage investment to labor months, were used as in modification I-b. The equation was fitted by least squares and the results indicated: 1. The standard error of estimate (5) was not significantly dif- ferent from the unmodified Cobb-Douglas estimate. 66 2. The reduction in the residual variance quantities was signifi- cant when using the modification II as compared to the unmodified Cobb- Douglas estimate. 3. The reduction in the variance of the residuals of the farms whose capital investments in livestock-forage and/or machinery exceeded f‘ifi the estimated ridge line prcpcrticn was not significant when using the g 2:1 modification 11 function as compared to the unncdified Cobb-Douglas : equation. g .a§ 4. The MVP of livestock-forage and.machinery, for modification i;j 11, showed essentially the same relationships as did modification I and I'be 5. The MVr of labor for the modification 11 was decreasing positive, and negative. Conclusions This study demonstrated that alterations on the Cobb-Douglas function can be realized by introducing various transformations on the independent variables. These transformations take the form of dummy variables in the N-dimensicnal input space. The functions investigated did not appear to give a better fit statistically than the standard Cobb-Douglas function for these particular data. However, the modified functions did appear to have certain eco- nomic advantages over the unmodified function. These modifications per- mit the fitted function to show non-constant elasticity, conditions of Symmetry more in agreement with empirical findings, and more economically realistic marginal value productivity estimates. 67 More specifically, the results of modification I indicate that the MVP estimates for labor at the geometric mean are not significantly different from the estimate derived by using the standard Cobb-Douglas. However, the apparent economic advantage of modification I is based on "‘_-.(.h .i the high MVP estimates for labor when using small amounts in relation "- "l 4L a? ._iss.- tc the amount of investment in livestock-forage and machinery. It seems more reasonable economically for labor to reflect high earning power ‘ "J‘ T?“ . .. when there is a small supply being used concurrently with an abundance of livestock-forage and machinery. This is in opposition to the rela- vr'f - ‘Eq-fl;_ tively low earning power of labor shown by past Cobb-Douglas studies. In addition, modification 1 showed mVP estimates for livestock-forage which were relatively low when using a large amount of livestock-forage investment with a small supply of labor. Estimates which reflect low earning powers of livestock-forage when there is an abundance of it in relation to small amounts of labor also seem more reasonable economically. The MVP of machinery for modification I was negative which is meaning- less for the farms studied. Modification l-b demonstrated that the modified function I can be made to show different relationships by adjusting the parameters in the dummy variables. The results indicated that even higher MVP esti- mates of labor can be realized, if the data warrants it, by lowering the "P" or ridge line proportion of combining labor to capital. Modification II shows the most promise for additional research. The preliminary findings strongly indicate the possibility of developing, with further work, a usable production function that will reflect three 68 stages of production, simultaneously. Such preliminary results as these functions have shown suggest that more intensified research in this area should be forthcoming. kv"-ll I. a er‘rgr 69 BIBLIOGRAPHY Allen, R. G. D., Mathematical Analysis for Economists, Vol. XII, Macmillan & Co.,Itd., St. Martin's Street, London, 1947. Cobb, Charles W. and Paul H. Douglas, "A Theory of Production," The American Economic Review, vol. XVII, Supplement, (March, I928}. Croxton, Frederick E. and Dudley J. Cowden, Applied General Statistics, Prentice-Hall, Inc., 1939. Douglas, Paul 3., Theory of Wages, The Macmillan Company, 1934. 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Ashton, "An Experiment to Derive Productivity and Substitution Coefficients in Pork Production,” Journal of Farm Economics, Vol. XXIV, (August, 1953). Johnson, Glenn L., Sources of Income on Upland Marshall County Farms, Progress Report—No. l;—Kentucky AgriculturaI'Experiment Station, Lexington, 1955. , Sources of Income on Upland McCracken County Farms, Progress Report No. 2,‘Rentu3ky Agricultural Experiment Station, Lexington, 1953. Johnson, Paul R., "Alternative Functions for Analyzing a Fertilizer-Yield Relationship," Journal of Farm Economics, Vol. XXXV, (November, 1953). . I:.!II- 145% II. me. . A fit P ... Slim 70 Snedecor, George'W., Statistical methods, 4th Ed., Iowa State College Press, 1946. Tintner, Gerhard, "A Note on the Derivation of Production Functions from Farm.Records,' geonometrica, Vol. XII, No. l, Tintner, Gerhard, and D. H. Brownlee, "Production Functions Derived from Fans Records,"ngurnal of Farm Economics, Vol. XXVI, (August, 1944). as?! was Toon, Thomas G., The Earning_Power of Inputs, Investments and Expendi- tures on Upland Grayson County Farms During 1951, Progress 9 Report no. 7, Kentucky Agricultural Experiment Station, ' Lexington, 1953. Trant, Gerald Ion, “A Technique of Adjusting Marginal value Productivity J ; Estimates for Changing Prices," Unpublished M. S. Thesis, De- L j partment of Agricultural Economics, miohigan State College, 93—? 1954. ‘flagley, R. vanes, ”Marginal Productivity of Investments and Expenditures Selected Ingham.County Fauna, 1952,” Unpublished M. S. Thesis, Department of Agricultural Economics, Michigan State College, 1953. _ -, _--’—._.‘ n game {if} BNL‘! h 3 03082 6956 Wfiu‘flflufiyijlmmiimmmmmmmuwuxmm