(A) Understanding Returns to Scale (i) Classification Returns to scale are technical properties of the production function, y = ¦ (x1, x2, ..., xn). If we increase the quantity of all factors employed by the same (proportional) amount, output will increase. The question of interest is whether the resulting output will increase by the same proportion, more than proportionally, or less than proportionally. In other words, when we double all inputs, does output double, more than double or less than double? These three basic outcomes can be identified respectively as increasing returns to scale(doubling inputs more than doubles output), constant returns to scale (doubling inputs doubles output) and decreasing returns to scale (doubling inputs less than doubles output). The concept of returns to scale are as old as economics itself, although they remained anecdotal and were not carefully defined until perhaps Alfred Marshall (1890: Bk. IV, Chs. 9-13). Marshall used the concept of returns to scale to capture the idea that firms may alternatively face "economies of scale" (i.e. advantages to size) or "diseconomies of scale" (i.e. disadvantages to size). Marshall's presented an assortment of rationales for why firms may face changing returns to scale and the rationales he offered up were sometimes technical (and thus applicable in general), sometimes due to changing prices (thus only applicable to situations of imperfect competition). As we are focusing on technical aspects of production, we shall postpone the latter for our discussion of the Marshallian firm. The definition of the concept of returns in to scale in a technological sense was further discussed and clarified by Knut Wicksell (1900, 1901, 1902), P.H. Wicksteed (1910), Piero Sraffa (1926), Austin Robinson (1932) and John Hicks (1932, 1936). Although any particular production function can exhibit increasing, constant or diminishing returns throughout, it used to be a common proposition that a single production function would have different returns to scale at different levels of output (a proposition that can be traced back at least to Knut Wicksell (1901, 1902)). Specifically, it was natural to assume that when a firm is producing at a very small scale, it often faces increasing returns because by increasing its size, it can make more efficient use of resources by division of labor and specialization of skills. However, if a firm is already producing at a very large scale, it will face decreasing returns because it is already quite unwieldy for the entrepreneur to manage properly, thus any increase in size will probably make his job even more complicated. The movement from increasing returns to scale to decreasing returns to scale as output increases is referred to by Frisch (1965: p.120) as the ultra-passum lawof production. We can conceive of different returns to scale diagramatically in the simplest case of a one-input/one-output production function y = ¦ (x) as in Figure 3.1 (note: this is not a total product curve!). As all our inputs (in this case, the only input, x) increase, output (y) increases, but at different rates. At low levels of output (around y1), the production function y = ¦ (x) is convex, thus it exhibits increasing returns to scale (doubling inputs more than doubles output). At high levels of output (around y3), the production function y = ¦ (x) is concave, thus it exhibits decreasing returns to scale (doubling inputs less than doubles output). [Note: the relationship between convexity and concave production functions and returns to scale can be violated unless the ¦ (0) = 0 assumption is imposed. Heuristically, a function exhibits decreasing returns if every ray from the origin cuts the graph of the production function from below. A production function which is strictly concave but intersects the horizontal axis at a positive level (thus ¦ (0) < 0) will not exhibit decreasing returns to scale. Similarly, a non-concave production function which intersects the vertical axis at a postitive amount (thus ¦ (0) > 0) will exhibit decreasing returns to scale.] Figure 3.1 - Returns to Scale for One-Output/One-Input Production Function (ii) Justification The economic justification for these different returns to scale turns out to be far from simple. At the most naive level, we justify increasing returns to scale by appealing to some "division of labor" argument. A single man and a single machine may be able to produce a handful of cars a year, but we will have to have a very amply skilled worker and very flexible machine, able to singlehandedly build every component of a car. Now, as Adam Smith (1776) famously documented, if we add more labor and more machines, each laborer and machine can specialize in a particular sub-task in the car production process, doing so with greater precision in less time so that more cars get built per year than before. The ability to divide tasks, of course, is not available to the single man and single machine. Specialization reflects, then, the advantage of large scale production over small scale. In Figure 3.1, assume we increase all factor inputs from x1 to x2, reflecting, say, the movement from a single man-and-machine to fifteen men-with-machines. The total output increases, of course, but so does the productivity of each man-and-machine since fifteen men-and-machines can divide tasks and specialize. So increasing factors fifteen-fold, increases output more than fifteenfold. In effect, we have increasing returns to scale. We should note that by justifying increasing returns by specialization implies that increasing returns is necessarily associated with a change of method. But this implies there are indivisibilities in production. In other words, the specialized tasks available at large scale are not available at the smaller scale; consequently, as the scale of production increases, these indivisibilities are overcome and thus methods not previously available become available. (K. Wicksell, 1900, 1901; F.Y. Edgeworth, 1911; N. Kaldor, 1934; A.P. Lerner, 1944; cf. E.H. Chamberlin, 1948). Nonetheless, we should note that there are direct examples of pure increasing returns to scale. For instance, consider a cylinder such as an oil pipeline and the mathematical relationship between the steel it contains (= 2p rl where r is the radius and l the length of the pipe) and the volume of oil it can carry (= p r2l). If one adds sufficient steel to the cylinder to double its circumference, one will be more than doubling its volume. Thus, doubling inputs (steel in pipeline) more than doubles output (flow of oil). In this example, increasing returns does notinvolve changes in technique. However, these pure examples are rare and the rationale for increasing returns is usually given by specialization. Thus, we can say equivalently that increasing scale captures the idea that there is technical progress with increasing scale. This is how we find it explicitly expressed in the work of Allyn A. Young (1928) and Nicholas Kaldor (1966) and, indeed, modern Neoclassical endogenous growth theory. As such, as it is generally discussed, increasing returns is more than a "pure scale" matter; it is about emerging techniques and changes in technique, of which we shall have more to say later. Decreasing returns to scale are more difficult to justify. We see that, in Figure 3.1, moving from x2 to x3, the production function is concave, so that by doubling inputs we less than double output. The naive justification is that the size of production has overstretched itself. The advantages of specialization are being outweighed by the disadvantages of, say, managerial coordination of an enterprise of such great scale. However often employed (e.g. Marshall, 1890: Ch. 12; Hicks, 1939: p.83; Kaldor, 1934), this "managerial breakdown" explanation is not really legimitate. This is because "returns to scale" requires that we double allinputs, yet we have not increased one of the factors: namely, the managers themselves. In the managerial breakdown argument, the manager implicitly remains as a "fixed factor", thus we are no longer talking of "decreasing returns to scale" in its pure technical sense but rather of diminishing marginal productivity, which is quite a different concept. In principle, then, decreasing returns to scale is hard to justify technically because every element in production can always be identically replicated (i.e. all inputs increase). To take another common but misleading example, suppose we increase the number of fishing boats in the North Sea. In this case, we would expect each boat to catch relatively less fish. Similarly, taking Pareto's (1896, II: 714) example, doubling the number of train lines from Paris will lead us to expect that each train will carry less passengers. But these examples are not examples of decreasing returns to scale be cause we have not, appropriately speaking, doubled all inputs: we have kept the North Sea and Paris constant. In other words, we have changed factor proportions: we have more fishing boats per square mile of North Sea and more trains per Parisian passenger. The accurate exercise for decreasing returns to scale in the first case is to double the number of boats anddouble the size of the North Sea (and thus double the number of fish). Similarly, we would need to double the number of trains and double the number of Parisians. In other words, one needs to replicate the North Sea and Paris completely, is entirely possible. The only way one might obtain decreasing returns to scale in these circumstances is if there were externalities of some sort, e.g. the existence of second replica implicates the operation of the first ("there is only one Paris...", etc.). Yet, barring this, two identical North Seas or two identical cities of Paris should not interfere with each other. Thus, decreasing returns simply do not make technical sense since replication does not complicate things. Another reason for doubting the existence of decreasing returns to scale is more empirical. Specifically, it would not be "rational" for an enterprise to ever produce in such a situation. To see this, suppose there is an entrepreneur who has a given set of laborers and machines willing to work for him. He can either put all these factors into a single factory, or just construct a series of smaller, but identical factories. Obviously, if he is facing decreasing returns to scale, then organizing them into several, decentralized, separate factoriesis better than throwing them all together into a single, centralized factory. Consequently, one of the justifications sometimes found for arguing for decreasing returns to scale is that production faces indivisibilities so that "dividing" a factory into several factories is simply not possible. Technically speaking, then, only constant and increasing returns can make sense; decreasing returns are harder to accept. The asymmetric nature of different returns to scale was explicitly admitted by Alfred Marshall in a footnote, "the forces which make for Increasing Return are not of the same order as those that make for Diminishing Return: and there are undoubtedly cases in which it is better to emphasize this difference by describing causes rather than results." (Marshall, 1890: p.266, fn.1). However, the problematic nature of this asymmetry of causes for a competitive economy, as we shall discuss later, were fully uncovered by Piero Sraffa (1925, 1926). It will be noticed that although most textbooks since have continued to refer to the possibility of decreasing returns to scale, they also often add parenthetically that they are assuming a fixed factor, or indivisibilities or some other imperfection that violates somewhat its pure definition. Be that as it may, it is important to note that decreasing returns to scale, in its proper symmetric definition, is rarely held among modern economists. In contrast, increasing returns, as noted, have become irretrievably associated with technical progress. (iii) Characterization We can characterize the "returns to scale" properties of a production function via the homogeneity properties of the production function. In principle, consider a general function: y = ¦ (x1, x2,..., xm) Now, a function of this type is called homogeneous of degree r if by multiplying allarguments by a constant scalar l , we increase the value of the function by a r, i.e. l ry = ¦ (l x1, l x2,...., l xm) If r = 1, we call this a linearly homogenous function. Now, if we interpret this function to be a production function, then the implications are obvious. If r = 1, then l r = l , so increasing inputs by factor l will increase output by the same factor l . This, of course, is the very definition of constant returns to scale. If r > 1, then l r > l , which implies that when we increase inputs by scalar l , output will increase by more than proportionally. This is the definition of increasing returns to scale. Finally, if 0 < r < 1, then l r < l , which implies that increasing inputs by a scalar l will lead output to increase by less than proportionally. This is the definition of decreasing returns to scale. We can gain a clearer understanding of returns to scale by examining diagramatically via isoquants as in Figure 3.2 - thus we must assume a two-input, one-output production function, Y = ¦ (L, K). Consider a factor configuration denoted by (K*, L*), which yields output Y =Y*, as noted by point e in Figure 3.2. If we increase both capital and labor by the factor l , then notice that we obtain a new factor configuration (l K*, l L*) which yields output Y = lrY*, as noted by point e¢ in Figure 3.2. This "increasing scale" is represented by a movement along the ray from the origin with slope L*/K*.. Figure 3.2 - Returns to Scale in Isoquants Consequently, whether we have increasing, constant or decreasing returns to scale depends upon r, the degree of homogeneity. It is simply a matter of labelling our isoquants appropriately. If r = 1, then l rY* = l Y* = ¦ (l K*, l L*), thus we have constant returns. If r > 1, then l rY* > l Y*, so we have increasing returns. If r < 1, then l rY* < l Y*, so we have decreasing returns to scale

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