sign in sign up
<<
Home , Cost curve, Returns to scale

Returns to Scale and Size in Agricultural

John W. McClelland, Michael E. Wetzstein, and Wesley N. Musser

Differences between the concepts of returns to size and are systematically reexamined in this paper. Specifically, the relationship between returns to scale and size are examined through the use of the envelope theorem. A major conclusion of the paper is that the level of abstraction in applying a function derived from a homothetic technology within a relevant range of the expansion path may not be severe when compared to the theoretical, estimative, and computational advantages of these technologies.

Key words: of scale, envelope theorem, returns to size.

Agricultural economists investigating long-run scale, Euler's theorem, and the form of pro- relationships among levels of inputs, outputs, duction functions. However, the direct link be- and generally use concepts of returns to tween returns to size and scale is not devel- scale and size. Econometric studies which uti- oped. Hanoch investigates the elasticity of scale lize production or functions commonly and size in terms of variation with output and are concerned with returns to scale (de Janvry; illustrates that only at the cost-minimizing in- Lau and Yotopoulos), while synthetic studies put combinations are the two concepts equiv- of relationships between output and cost uti- alent. Hanoch further provides a proposition lize returns to size (Hall and LaVeen; Rich- stating that Frisch's "Regular Ulta-Passum ardson and Condra). However, as noted by Law" is neither necessary nor sufficient for a Bassett, the word scale is sometimes also used production technology to be associated with to mean size (of a plant). Thus, increasing (de- U-shaped curves. Unfortunately, creasing) returns to scale are used to refer to Hanoch's article does not consider specific economies (diseconomies) of size, and vice production technologies such as homothetic or versa. For example, Raup uses scale and size ray-homogenous technologies and does not ex- interchangeably in discussing economies and plicitly establish the relationships between re- diseconomies of "large-scale" agricultural pro- turns to scale and size. In an effort to clarify duction. Feder, and Gardner and Pope also the concepts of size and scale economies, invoke the scale concept, but the bulk of their Chambers, in a 1984 proceedings paper on discussions center on the farm size issue. Such economies-of-size studies, presents the two use of these concepts presents an interpretation concepts and discusses their interrelation- problem for researchers and students in agri- ships. Chambers presents a sound discussion . of the concepts blending the works of previous Previous research addressing this problem efforts within this area. However, despite generally has not been definitive. McElroy es- Chambers' and other past researchers' efforts tablishes the relationship between returns to Ito delineate the similarities and differences be- tween returns to scale and size, confusion still John W. McClelland is an agricultural economist, USDA, ERS, exists in the profession. As noted in a recently NRED, Washington, D.C. Michael E. Wetzstein is an associate published textbook by Beattie and Taylor, the professor, Department of at the University of Georgia. Wesley N. Musser is an associate professor, Depart- distinction between returns to scale and size is ment of Agricultural and Resource Economics at Oregon State not always clear, and the terminology is at University. times, inappropriately, employed interchange- Incisive comments by Donald Keenan, Paul Wenston, and Rod Ziemer on earlier drafts contributed a great deal to the final prod- ably. uct. The purpose of this paper is to reexamine

Western Journal of Agricultural Economics, 11(2): 129-133 Copyright 1986 Western Agricultural Economics Association 130 December 1986 Western Journal of Agricultural Economics systematically the differences between the con- (3b) J(kX) = Fkr h(X)], cepts of returns to scale and size. Specifically, a homothetic technology, where h(X) = the relation between returns to scale and size F-'[(X)] and F is a monotonic transform of is examined through the use of the envelope the technology; theorem. In the first section, the production and cost relationships between size and scale (3c) (kX) = kH(xlxl).j(X), are investigated, and based on these results, a ray-homogenous technology, where H(X/ an envelope relation between the two concepts IXI) is a strictly positive and bounded func- is developed. The final section concludes with tion; and some methodological views on the distinctions between these concepts. (3d) f(kX) = F[kH(XX./ l h(X)], a ray-homothetic technology (Chambers 1985). Returns to size, alternatively called returns Production and Cost Relationships to outlay, can be defined as the proportional between Size and Scale change in output associated with a propor- tional change in cost or outlay. Specifically, The function coefficient e is the most common elasticity of size is denoted means of discriminating scale economies. This - coefficient is known under various names, such aln y/dln c = 7 1, as elasticity of scale, local returns to scale, elas- where n is defined as the elasticity of cost (Var- ticity of production, and passus coefficient. ian, Chambers 1984). Thus, based on the prop- Specifically, e is defined as the proportional erties of elasticity, elasticity of size is equal to change in output resulting from a unit pro- the ratio of average cost to and portional change in all inputs. Mathematically, is directly associated with the average cost as noted by McElroy, curve. (1) Et(, X/IXI) = (VfX)/y, As implied by McElroy, ray production functions process the following relation be- where Vf is derived from y = f(X), a produc- tween elasticity of scale and size tion function which is assumed to be regular, monotone, and convex (Varian). Finally, as (4) r?- =E. noted by Chambers (1985), IXI is the Euclid- The conditions establishing (4) differ for the ean norm of the original input vector X and four technologies represented in (3). From (3a) X/IXI characterizes a ray from the origin in it can be demonstrated that e = r, and thus (4) Euclidean N space. does not depend on the input or output levels. A general representation of variations in The cost function associated with (3a) is mul- output associated with a proportional increase tiplicatively separable in output y and input in inputs (k) is vector w, c = yl/rA(w). Thus, n- 1 = r. For (2) (kX) = G[k, X/IXI, f(X)]. a homogenous technology, e and nr1 are con- stant in the isoquant space and exhibit con- Equation (2) corresponds to a class of produc- stant proportional returns. Equation (4) then tion functions with expansion paths which are holds at every point within the isoquant space. linear from the origin. This class of functions Every point in the isoquant space is a sufficient may be called ray production functions and but not necessary condition for (4) to hold. can be simplified as follows (McElroy): For a homothetic technology (3b), scale elas- (3a) (kX) = kr-j(X), ticity is defined as e = (Vf X)/J(X) = h(y)/ [h'(y)y], where h(y) = F- 1(y) and h'(y) = dh(y)/ a homogenous technology with r indicating the dy (Chambers 1985). In this case e is a function degree of homogeneity; of output and exhibits variable proportional returns. Denoting marginal cost as X, size and ' The norm in Euclidean N space, R", is defined as IX = (X scale elasticity for a homothetic technology may X)% and is interpreted as the distance in R" from X to the origin. X/IXI is a vector of trigonometric functions defining an angle be related with the aid of the following prop- associated with a ray through the origin Rn. In R2, X/IXl = [sin osition. 0, cos 0], where 0 is an angle between 0 and -r/2. Taking the inner 2 2 PROPOSITION 1. c = X[h(y)/h'(y)], if and only product of [sin 0, cos 0] with itself yields sin 0 + cos 0 = 1. In this manner, all points can be defined on the unit circle and the level if (X) is homothetic. of output is dependent on the angle 0. Proof.Consider the cost function c = w X(w, McClelland, Wetzstein, and Musser Returns to Scale and Size 131

Cost y). All X on the expansion path are character- Per Unit ized by Output (5) w = F'(h)VJ, E>1 where h(y) = J*(X) is a linear homogenous AS' function. Substitute (5) into the cost function and using Euler's theorem, c = XF'(h)h(y). Not- ing that F is monotonic, (6) c = x[h(y)/h'(y)]. Q.E.D. Dividing both sides of (6) by y and X estab- n<1 lishes the relation between size and scale elas- ticity

E = [h(y)/h'(y)y] = c/Xy= r-. I · _ For a homothetic technology, (4) is true when 0 y y associated with e corresponds to the cost- Figure 1. Average cost and scale average cost minimizing level of output. A necessary con- curves dition for (4) to be true is for e to be associated with all points on the isoquant corresponding curves associated with output along a rayfrom to the cost-minimizing output level. A suffi- the origin called scale average cost curves. cient condition is a point on the expansion Proof. Consider the following scale-cost path. Although (4) still holds at every point in function, c(y, X(y)), where X(y) is the vector of the isoquant space, the of the equality cost-minimizing inputs on a scale expansion changes with output. path. Let y* be the level of output at which the For a ray-homogenous technology, (3c), actual cost function, c(y) is equivalent to the equation (4) no longer holds for all points in scale-cost function, the isoquant space. For this technology, a point on the expansion path is a necessary and suf- c(y*) = c(y*, X(y*)). ficient condition for (4) to be true. Size and Let X* = X(y*) be the associated cost-mini- scale elasticities are defined as e = kH(x/lXl) and ). mizing inputs for output level y*, such that 7-1 = kH(X/'X\ Only when input ratios are the c(y*) = p X*. At any output level, the scale same for e and 7 will (4) hold. Thus, when all cost function must be at least as great as the inputs are increased by a fixed proportion, being ordinary cost function. Thus, the scale cost on the expansion path is a necessary and suf- curve and ordinary must be tangent ficient condition for (4) to be true. The same at y*, which is a geometrical statement of the correspondence holds when the production envelope theorem (Varian). The slope of the function is generalized to a ray-homothetic cost curves at y* is technology and to nonray classes. dc(y*)/dy = c(y*, X(y*))/ay + [c(y*, X(y*))/Xl(oX/8y), but Size as the Envelope of Scale ac[y*, X(y*)]/8X = 0. Q.E.D. Hanoch illustrates for nonray production func- Proposition 2 ties two important relation- tions that elasticity of scale and size are equiv- ships of elasticities of scale and size together. alent only at points on the expansion path. He First, Hanoch demonstrated that the change concludes that the behavior of the elasticity of in the two elasticities with respect to a change scale does not yield conclusive results in de- in y are not equivalent unless a ray production termining the shape of average costs. Further technology exists. This result is apparent from investigation of this issue reveals an important Proposition 2 and is illustrated in figure 1. property between the two elasticities. Elasticity of size and scale are equal at point PROPOSITION 2. Elasticity of size is the en- A associated with scale average cost curve AS velope of elasticity of scale, or the long-run av- and output yo. However, for alternative output erage cost curve is the envelope of average cost levels other than y°, ]- 1 and e differ. Second, 132 December 1986 Western Journal of Agricultural Economics

Frisch's "Regular Ulta-Passum Law" states assumption also does not appear to be dam- that for an outward movement along an ar- aging to agricultural economics. The wide- bitrary curve in an input space, the function spread use of enterprise budgets to estimate coefficient will decrease from a value in excess costs of production, the common practice of of one to values less than zero. This law es- including only one production activity for each tablishes the U-shaped scale average cost curves enterprise in mathematical programming in figure 1. Thus, Hanoch's concept and Frisch's models, and the estimation of homogenous law are directly related through proposition 2. production functions all, at least implicitly, as- Further, Hanoch's proposition 3, stating that sume a ray production technology. This view Frisch's law is neither necessary nor sufficient is at odds with the conventional beliefs of many for "classical" U-shaped average cost curves, agricultural economists. For example, Madden follows directly from proposition 2. Elasticity recently noted that is an of size, as the envelope of scale and Frisch's "empty box" which should be discarded as Law, implies that e may range from -oo to oo, unrealistic. The continued parallel use of scale within the range of 7 > 1, figure 1. with size concepts in agricultural economics suggests that this recommendation is not likely to be followed. In large part, the persistence of Conclusion scale is related to its theoretical value in in- terrelating cost functions and production tech- Basic concepts such as economies of scale and nology. At this point in the development of size can be very confusing when a clear dis- agricultural economics, the standard that the- tinction is not presented. Economies of scale oretical concepts must be completely realistic is solely related to the particular production surely is not taken seriously. Most widely used technology and does not require economic ef- economic concepts are unrealistic and could ficiency as one of its determinants. Economies also be considered empty boxes. func- of size requires the firm to be operating on its tions and the perfectly competitive model are input efficiency locus and thus is an economic two examples. Furthermore, Boggess recently efficiency concept. In the cost space, this cor- reminded us that the production function is responds to long-run average cost. Declining another such abstraction. This paper suggests long-run average cost indicates increasing re- only that the same methodological standards turns to size. Similarly, rising average cost im- be applied in the size versus scale issue as in plies decreasing returns to size. A rigorous def- other theoretical issues in agricultural econom- inition of the size and scale relationship reveals ics. that elasticity of size is the envelope of elas- [Received September 1985; final revision ticity of scale. Within the neighborhood about received May 1986.] any point on the expansion path where e = r -1, elasticity of scale can approximate elasticity of size and allows for the direct derivation of a cost function based on an associated ray pro- References duction technology, such as the homogenous technology. Thus, the level of abstraction in Bassett, L. "Returns to Scale and Cost Curves." S. Econ. applying a cost function derived from a ho- J. 36(1969): 189-90. mogenous production function within a rele- Beattie, B. R., and C. R. Taylor. The Economics of Pro- vant range of the expansion path may not be duction. New York: John Wiley & Sons, 1985. severe when compared to the theoretical es- Boggess, W. G. "Discussion: Use of Biophysical Simu- timative and computational advantages of ho- lation in Production Economics." S. J. Agr. Econ. mogenous functions. For example, Hoch re- 16(1984):87-89. ported Cobb-Douglas functions were superior Chambers, R. Applied Production Analysis. Cambridge: to quadratic functions; however, returns to Cambridge University Press, 1985. . "Some Conceptual Issues in Measuring Econ- scale varied among regions which he attributed omies of Size." Economics of Size Studies. Ames: to different sized production units. Center for Agricultural and Rural Development, Iowa McElroy concluded that in view of the con- State University, 1984. venient properties of ray production functions, de Janvry, A. "The Generalized Power Production Func- it is useful in many general economic appli- tion." Amer. J. Agr. Econ. 54(1972):234-37. cations to assume this class of functions. This Feder, G. "Adoption of Interrelated Agricultural Inno- McClelland, Wetzstein, and Musser Returns to Scale and Size 133

vations: Complementarity and the Impacts of Risk, Factor Demand Functions." Amer. J. Agr. Econ. Scale, and Credit.' Amer. J. Agr. Econ. 64(1982):94- 54(1972):11-19. 101. Madden, J. P. "Discussion: What to Do with Those Emp- Frisch, R. Theory of Production. Chicago: Rand-Mc- ty Economic Boxes-Fill or Decorate Them." Econ- Nally, 1965. omies-of-Size Studies. Ames: Center for Agricultural Gardner, B. D., and R. D. Pope. "How is Scale and Struc- and Rural Development, Iowa State University, 1984. ture Determined in Agriculture." Amer. J. Agr. Econ. McElroy, F. W. "Returns to Scale, Euler's Theorem, and 60(1978):295-302. the Form of Production." Econometrica 37(1969): Hall, B., and E. P. LaVeen. "Farm Size and Economic 275-79. Efficiency: The California Case." Amer. J. Agr. Econ. Raup, P. "Economics and Diseconomies of Large-Scale 60(1978):589-600. Agriculture." Amer. J. Agr. Econ. 51(1969):1274-85. Hanoch, G. "The Elasticity of Scale and the Shape of Richardson, J., and G. D. Condra. "Farm Size Evaluation Average Costs." Amer. Econ. Rev. 65(1975):492-97. in the El Paso Valley: A Survival/Success Approach." Hoch, I. "Returns to Scale in Farming: Further Evi- Amer. J. Agr. Econ. 63(1981):430-37. dence." Amer. J. Agr. Econ. 58(1976):745-49. Varian, H. R. Microeconomic Analysis. New York: W. Lau, L. J., and P. A. Yotopoulos. "Profit, , and W. Norton & Company, 1978.