Returns to Scale in Welfare Economics

ROBERTE. KOHN*

I. Introduction

There are two different interpretations of returns to scale. This was first recognized by Edgeworth [1911, pp. 349-50], who distin- guished his own approach based on "the rate at which production inc reases . . . (with) equal doses of productive power" from the Marshal- lian interpretation based upon "proportionate changes." It was not until a half century later that the marginal and the proportional inter- pretations of returns to scale were formally distinguished by Levenson and Solon [1966] and by McElroy [1967]. The distinction is important in the case of production functions that exhibit increasing~ constant, and decreas- ing returns to scale because there is then a range of output over which there are increasing returns to scale according to the proportional definition but decreasing returns according to the marginal definition.

The proportional definition lends itself to empirical analysis and is consequently more widely used. For example, all of the economet- ric models in the survey article by Cowing and Smith [1978] use some form of the propor- tional definition of scale. In the context of wel- fare economics, however, the marginal interpre- tation is preferable because it alone explains for all possible technologies why a production possibility frontier may contain a segment that is convex to the origin when there are increas- ing returns to scale.

In this paper, the two definitions of returns to scale are contrasted and the case is noted in which there is an inconsistency. The relation- ship between returns to scale and the curvature of the production possibility frontier is ex- plained in the context of a simple model in

i

*Southern Illinois University at Edwardsville. I am grateful to Donald Aueamp, Stanford Levin, John Meisel, David Pines, and an unidentified referee for their helpful comments on this paper.

which other factors affecting curvature are absent. The range of output over which this frontier is convex to the origin is the range over which there are increasing returns ac- cording to the marginal definition rather than the proportional definition.

II. The Discrepancy in the Two Definitions

The proportional definition of returns to scale relates a proportional increase in output, Z~X/X, to the corresponding proportional in~ crease in scale, As/s. If the former is (a) greater than, (b) equal to, or (c) less than the latter, there are (a) increasing, (b) constant, or (c) decreasing returns to scale, respectively. This definition is sometimes construed as an elas- ticity of output with respect to scale. Given a production function, X = f(L, K), and measur- ing scale as the variable ratio, s, of the two in- puts to some base values (i.e., L = sL_ K = sK), the elasticity of output with respect to scale is (dX/X)/(ds/s). According to the proportional interpretation~ there are increasing returns when this elasticity exceeds unity and decreasing re- turns when it is less than unity. Because the elasticity of output with respect to scale can be estimated by summing the exponents of a Cobb-Doublas production function [Walter, 1963, p. 6], this interpretation of returns to scale is widely used in statistical analysis

The marginal definition is based on the rate at which the change in output with respect to scale (i.e., dX/ds) is changing with scale. When this rate of change, which is equivalent to the second derivative, d2X/ds 2, is (a) positive, (b) zero, or (c) negative, there are (a) increasing, (b) constant, or (c) decreasing returns to scale, respectively. In terms of an isoquant map, this is equivalent to (a) decreasing, (b) constant, and (c) increasing distances between successive

54

KOHN: WELFARE ECONOMICS 55

unit isoquants, along a ray from the origin. 1 The two definitions of returns to scale are

equivalent in the case of homogeneous produc- tion functions. In the case of production func- tions in which there are increasing and then de- creasing returns to scale, marginal returns to scale, dX/ds, increase with scale, attain a maxi- mum at some level, sl , and then decline. How- ever, average returns to scale, X/s, also rising, continue to increase beyond s~, and finally attain a maximum at some greater scale, s2, where dX/ds = X/s. It is in the range from sl to s2 that there are decreasing returns to scale according to the marginal definition (because dX/ds is declining with s and, hence, d2X/ds 2 is negative), but increasing returns to scale ac- cording to the proportional definition (because dX/ds exceeds X/s and, hence, the elasticity of output with respect to scale is greater than unity).

The relationship between average returns to scale and marginal returns to scale, which may also be called the marginal product of scale, is characterized at the bottom of Figure L (To demonstrate the main result of this paper, average and marginal returns to scale are plotted along the output axis. This preserves the basic relationship between the two curves because output increases monotonically with scale.) The quantity, X1, is the level of output at which the marginal product of scale, dX/ds, reaches a maximum, and the quantity, X2, is the level of output at which the marginal product of scale equals the average product of scale,X/s. Accord- ing to the marginal interpretation, there are in- creasing returns to scale between 0 and X1 and decreasing returns thereafter, whereas according to the proportional definition there are increas- ing returns to scale between 0 and X2. This is

i ii

The marginal definition of returns to scale is the analogue in long run analysis of marginal returns to a single variable factor in short run analysis. Both inter- pretations relate to distances between successive unit isoquants, the former along a diagonal ray and the latter along a line pependicular to the fixed-input axis. The U-shaped long run marginal cost curve turns upward at the onset of decreasing marginal returns to scale, just as the short run marginal cost curve turns upward at the onset of diminishing returns to the variable input. (See Kohn, 1981.)

demonstrated with a numerical example in the appendix.

The most common interpretation of a change in scale, which is the one used in the above definitions, is that of an equal proportional change in all inputs. However, economists do not limit their use of the scale concept to this special case. Rowe [1968], Geithman and Truett [1974], and Hanoch [1975] have demonstrated that the common practice of identifying changes in scale with movements along an expansion pathwhere input prices are constant is entirely appropriate.

A third interpretation, used in welfare eco- nomics, associates changes in scale with move- ments along the efficiency locus of an Edge- worth box for inputs. In that context, there are increasing (decreasing) returns to scale when the transfers of inputs between industries occur in regions of the respective production isoquants in which there are increasing (decreasing) re- turns along any ray through the corresponding origin, Although this third interpretation is most relevant to the present paper and permits a generalization of the result, the case that will be examined here is one in which all three defi- nitions of a change in scale are applicable.

III. Increasing Returns and the Product ion

Possibil i ty Frontier

The case in which an otherwise concave or linear production possibility frontier contains, because of increasing returns to scale, a seg- ment that is convex to the origin is familiar in welfare economics [Bator, 1957, pp. 50-3; Kindleberger, 1963, p. 102; Rader, 1970, pp. 820-21]. Such a production possibility fron- tier is depicted at the top of Figure I. To estab- lish the main result of this paper it must be demonstrated that this frontier, which is con- vex to the origin between 0 and X1, becomes concave for quantities greater than X1 because there is a shift at XI from increasing to decreas- ing returns to scale (in the production of good X) according to the marginal rather than the proportional definition.

The crucial geometric characteristic of the frontier in Figure I is that its slope, dY/dX, be- comes less steep as X increases from 0 to XI

56 ATLANTIC ECONOMIC JOURNAL

and more steep thereafter. It is well known that the slope of the frontier is equal to the ratio of the marginal product of any factor in the pro- duction of good Y to the marginal product of that same factor in the production of good X. These marginal products are a useful link to changes in scale.

There is a complication in relating the cur- vature of the frontier to returns to scale when factor proportions are also changing. For then, as Bator [1957, pp. 50-1] notes, the comple- mentary mesh of differing factor proportions along the technical efficiency locus can counter- act " . . . a little increasing returns to scale in both f u n c t i o n s . . . " Only if the locus coincides with the diagonal of the Edgeworth box will "increasing returns to s c a l e . . , necessarily im- ply an inward bending production possibility curve." Following Bator, the analysis is simpli- fied by assuming that the production functions for goods X and Y are homothetic, so that the technical efficiency locus lies along the diagonal of the Edgeworth box from which the produc- tion possibility frontier could be graphically determined. (This also provides the case in which all three interpretations of a change in scale coincide).

In this analysis it is assumed that good X is produced according to a homothetic function that exhibits increasing and then decreasing returns to scale. To avoid the expositional com- plication of increasing returns in industry X having to offset decreasing returns in industry Y, it is further assumed that good Y is pro- duced under constant returns to scale.

The model is one in which inputs are shifted from one industry to the other to generate the production possibility frontier. Because input proportions are constant, the marginal product of any input, say labor, is a constant multiple of the marginal product of scale in that same industry. As the scales of production of the two goods change, starting with all Y and no X, the marginal product of labor in the production of good X increases and then decreases at the same rate as the marginal product of scale in- creases and decreases.

The marginal product of labor in industry Y remains constant as the output of that good decreases. As long as the marginal product of labor in industry X is increasing, the slope of the frontier, which is equal to the ratio of the marginal product of labor in industry Y to that in industry X, becomes less and less steep and the frontier is convex to the origin. As soon as the marginal product of labor in industry X begins to decline, there is an inflection point on the production possibility frontier, and the lat- ter becomes concave (See appendix).

The slope of the production possibility fron- tier, which equals the ratio of the marginal products, also (in this simplified model) equals the ratio of the marginal products of scale. This tight relationship is characterized in Figure I, where the quantity X1, at which the inflection of the frontier occurs, corresponds to the maxi- mum marginal product of scale. The slope of the frontier is unrelated to average returns to scale and hence has no relationship to the proportional definition of returns to scale. The marginal definition of returns to scale alone explains the curvature of this production pos- sibility frontier.

IV. Concluding Remarks In the case of production functions that ex-

hibit increasing and then decreasing returns to scale, there is a discrepancy between the pro- portional and the marginal definitions of re- turns to scale. Although the proportional defi- nition is more useful in positive economics, the marginal definition has advantages in welfare economics for it alone explains the well known case in which an otherwise concave production possibility frontier contains a segment that is convex to the origin because of increasing re- turns to scale. Although this relationship has been demonstrated for the simple case in which one of the production functions is homothetic and the other linear homogeneous, it generalizes because the curvature of the production possi- bility frontier depends upon marginal rather than average products.

KOHN: WELFARE ECONOMICS

REFERENCES Francis M. Bator, "The Simple Analytics of Wel-

fare Maximization," The American Economic Review, Vol. 47, March 1957, pp. 22-59.

Thomas G. Cowing and V. Kerry Smith, "The Esti- mation of a Production Technology: A Survey of Econometric Analyses of Steam-Electric Generation." Land Economics, Vol. 54, May 1978, pp. 156-86.

F. Y. Edgeworth, "Contributions to the Theory of Railway Rates," The Economic Journal, Vol. 2I, 1911, pp. 346-70.

David T. Geithman and Dale B. Truett, "Long-Run Returns: An Ambiguity in the Theory of Production and Cost," Economia Internazionale, Vol. 27, May 1974, pp. 249-58.

Giora Hanoch, "The Elasticity of Scale and the Shape of Average Costs," The American Economic Review, Vol. 65, June 1975, pp. 492-97.

Charles P. Kindleberger, International Economics, Irwin, Homewood, 1963, p. 102.

Robert E. Kohn, Economic Efficiency: A Norma-

57

tire Approach to Microeconomics, duplicated, Ed- wardsville, Illinois~ 1981.

Albert M. Levenson and Babette Solon, "Returns to Scale and the Spacing of Isoquants," The American Economic Review, Vol. 56, June 1966, pp. 501-3.

F. W. McElroy, "Returns to Scale and the Spacing of Isoquants; Comment," The American Economic Review, Voi. 57, March 1967, pp. 223-24.

Alfred Marshall, Principles of Economics, Eighth Edition, Macmillan, New York, 1920, p. 150.

Trout Rader, "Resource Allocation with Increas- hag Returns to Scale," The American Economic Re- view, Vol. 60, December 1970, pp. 814-25.

John W. Rowe, Jr., "Returns to Scale and the Spacing of Isoquants: Comments," The American Economic Review, Vol. 58, June 1968, pp. 548-50.

A. A. Waiters, "Production and Cost Functions: An Econometric Survey," Econometrica, Vol. 31, January 1963, pp. 1-66.

APPENDIX

The product ion possibility frontier in Figure

I and the corresponding marginal and average product curves are derived from the following product ion functions and input constraints, to- gether with the marginal condit ion for equal marginal rates of technical substitution between

inputs:

X = 4 L K - L 2 K 2 / 1 6 , (1)

Y= 2 v ~ k , (2)

L + ~ = 12, and (3)

K + k = 3 . (4)

According to the marginal definition of re-

turns to scale, decreasing returns commence

when

d2X/ds 2 = 32 - 12s 2 = O, (8)

which occurs when s = V / 8/3 and X = 320/9. According to the proport ional definition, de-

creasing returns commence when

(dX/X)/(ds/s) = (32 - 4s2)/(16 - s2) = 1, (9)

which occurs when s = X/~-/3 and X = 512/9. Accordingly, the values of X1 and X2 in Figure I are 320/9 and 512/9, respectively.

The product ion possibility frontier,

Along the technical efficiency locus, the pro- duction function for good X in terms of scale,

s, is

X = 1 6 s 2 - s 4, (5)

the marginal product of scale is

dX/ds = 32s - 4s 3 , (6)

and the elasticity of scale is

(dX/X)/(ds/s) = (dX/ds)/(X/s) = (7)

(32 - 4s2)/(16 - s2).

X / 8 - X / - ~ - X + Y/4 = 3, (10)

is illustrated in Figure I. ( T h e X a n d Yintercepts are 63 and 12, respectively.)

The inflection point of this frontier occurs when the second derivative equals zero:

( l l j

( 3 / 4 ) V / ~ - X - 4 d2 Y/dX2 = [ ( 6 4 - X ) ( 8 - X/o4-~)'~'77-77""]3')2 ........ = 0

This occurs when X = 320/9, which is consistent with the marginal definition of returns to scale.