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Returns to Scale and Size in Agricultural Economics

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Returns to Scale and Size in Agricultural Economics Returns to Scale and Size in Agricultural Economics John W. McClelland, Michael E. Wetzstein, and Wesley N. Musser Differences between the concepts of returns to size and returns to scale 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 cost 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: elasticity 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 costs 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 profit 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 average cost 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 cultural economics. 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 Agricultural Economics 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 marginal cost 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 price 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 value of the equality cost-minimizing inputs on a scale expansion changes with output.
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