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# Labor, Marginal Product of

THE MARGINAL PRODUCT OF LABOR AND THE DISTRIBUTION OF INCOME

THE LAW OF VARIABLE PROPORTIONS AND THE LAW OF DIMINISHING PRODUCT

HISTORICAL DEVELOPMENT OF THE CONCEPT

SOME CRITICISMS

BIBLIOGRAPHY

Consider a firm producing a homogeneous output y employing labor l and other n factors of production: x 1, x 2, , xn. The change in output resulting from the addition of one extra unit of labor, with the other inputs being held constant, is called the marginal, or physical, product of labor (MPL). Formally, assuming that the technology of the firm is described by the production function y = f(l, x1,2, , xn ), MPL is expressed as follows:

Broadly speaking, the concept of MPL describes the ratio of change in output stemming from a small, or marginal, increase in the use of labor. Moreover, modern neoclassical production theory assumes that labor (and the other inputs it is combined with) can be employed at a continuous level (a realistic assumption labor is measured in labor time), and that the production function is continuous, with continuous first derivatives. The marginal product of labor can then be rigorously expressed by the first partial derivative of f with respect to l :

Although the same name is generally given to the two definitions reported aboveand what follows will conform to this general useit would be more appropriate to call the former (the discrete case) the marginal product of labor, while the latter (the continuous case) would be more appropriately defined as the marginal productivity of labor. These definitions may be easily extended to the case of joint production, that is, when a firm, for a given combination of inputs, produces more than one output (the so-called multiproduct firm).

## THE MARGINAL PRODUCT OF LABOR AND THE DISTRIBUTION OF INCOME

The concept of MPL has a central place in marginal productivity theory, since at the very core of this approach is the understanding that in a competitive market the remuneration of each productive factor should be equal to its marginal contribution to production. A direct implication is that as the relative prices of factors change, the proportion in which they are employed will change as well, in order to reestablish equality between marginal product and input prices. This is known as the principle of variation. With respect to labor (p and w are, respectively, the unit prices of output and labor), profit maximization requires that, at equilibrium, the real wage should be equal to This can also be written as (the second order condition for a maximum is guaranteed by the assumption that MPL is, at least to a certain level of l on, a decreasing function of l ). The equation represents the value of the marginal product of labor (VMPL) and its schedule below the maximum of the value of average product of labor represents the short-run firms demand curve for labor. If the firm operates in a noncompetitive market for its product, profit maximization requires that MPL × MR = w, where MR is the marginal revenue generated by an extra unit of output (in perfect competition MR = p ). MPL × MR is called the marginal revenue product, and in this case its schedule (below the maximum of VAPL) represents the short-run firms demand curve for labor (a deeper analysis of these matters may be found in any standard microeconomic text).

## THE LAW OF VARIABLE PROPORTIONS AND THE LAW OF DIMINISHING PRODUCT

Consider, for the sake of simplicity, a canonical short-run production function: y = f(K 0 , l) = F (l), where l (as before) is labor, y is the output of the firm, and K 0 is a fixed given amount of another input that is conventionally defined as capital (or better services of a capital good). Neoclassical production theory admits various possibilities concerning the variation of MPL as l changes. According to the law of variable proportions, as the firm increases the use of labor, holding K constant at a given level (K 0), the output will first increase at an increasing rate up to a point, say l = l*, and thereafter increase at a decreasing rate, depicting a S -shaped production function. This law expresses the idea that, for given input prices, there is an optimal employment ratio of K and l in correspondence of each quantity of output produced, but since capital is an indivisible input the firm cannot use less capital than K 0. Consequently, the first increments of l will cause y to increase at an increasing rate, since the addition of further units of labor leads to a more favorable proportion between l and K 0. However, successive additions of labor will lead, sooner or later, to a point (E in Fig. 1) where the constraint of the constant factor will operate in the opposite manner. From this point on, further units of labor lead to a less favorable proportion between l and K 0, so that each additional unit of labor will add less output than its predecessors.

Another widely used assumption about MPL is the so-called law of diminishing product, which states that as l increases, output will always increase less than proportionately, implying a concave production function (the explanation is the same as for the one given above from point E on). Although this second law avoids the indivisibility

assumption (which is not necessary in the neoclassical productivity theory, and may lead to some theoretical contradictions), it rests on an idea that is sometimes difficult to justify: that MPL will always decrease whatever the given amount of K may be.

Both laws admit the possibility that a level of l may be reached at which MPL = 0, after which further increments of l will cause the MPL to become negative. One could say that at this point the given capital stock will treat further additions of labor as a disservice.

## HISTORICAL DEVELOPMENT OF THE CONCEPT

The first formulation of the law of variable proportions can be found in a work by the French economist Anne-Robert-Jacques Turgot (17271781), who showed that when advances (seeds) on a given piece of land are increased, the product will first rise more than proportionately, but that after a certain point the diminishing product will prevail, until the fertility of the soil being exhausted an addition to the advances will add nothing whatever to the produce (quoted in Sraffa 1925, pp. 327328). The role of MPL in the explanation of labors reward may be found in various forerunners of marginal revolution, and in the works of the earlier neoclassical economistsparticularly William Stanly Jevons (18351882) and Carl Menger (18401921). Only in the 1890s, however, with the contributions of John Bates Clark, Alfred Marshall, Knut Wicksell, and Philip H. Wicksteed, was a coherent theory of distribution fully established. Marshall, also through numerical examples, proved that profit maximization requires that real wages equal MLP. Clark claimed that, according to the marginal theory, production and distribution become faces of the same process of determination of value (undoubtedly one of the main goals of neoclassical theory), and that each social class gets what it contributes, under natural law, to the general output of industry (Clark 1891, p. 313; see also Clark 1899, p. v).

A few years later, however, Wicksteed showed that the marginal theory of distribution, in order to be consistent, must assume a homogeneous production function of degree 1 (i.e., constant returns to scale). Only under this assumption, in fact, would the sum of factors rewards, determined on the basis of their marginal products, be equal to the total output produced (this is the product exhaustion theorem ). The assumptions about production processes and the laws governing income distribution thus became strictly interconnected, requiring the marginal theory of distribution, if it is to be consistent, to postulate a particular configuration of productive processes. Wicksell thus suggested using the following production function (later called Cobb-Douglas): y = lαKβ, with α > 0, β > 0, α + β = 1. This function, and the law of diminishing product of labor (and of capital) it incorporates, is still the most commonly used in neoclassical micro- and macroeconomic theory. It should be noted that the assumption of constant returns to scale represents a serious limitation, since it rules out the important phenomenon of increasing returns to scalethat is, it rules out the case for: α + β > 1 (see Debreu 1959, p. 41, for a similar statement with regard to general economic equilibrium analysis).

## SOME CRITICISMS

Soon after its formulation, the neoclassical theory of wages raised many grumbles, as Dennis Robertson put it. One criticism concerned the distributive justice implied by what Clark defined as natural law. It was pointed out that only the lastor better, the marginalworker would be paid according to his or her contribution to production (equal to the marginal product), while all the other workers would be paid less than that, because under the assumption of diminishing product of labor, their marginal product is higher than that of the last worker employed. The marginal theory of distribution would therefore imply labor exploitation.

Two points should be noted here. First, it makes sense to talk of a first worker, a second worker, and so on, only in a logical sense, since labor l is a homogeneous factor of production, and all that can really be discussed is the marginal product of the homogeneous factor l (Clark 1891, p. 308). Second, the exploitation (of both labor and other inputs) is denied by the assumption (fully compatible with other hypothesis about MPL) of constant returns to scale, which guarantees product exhaustion.

Contrary to Clarks claim, however, even if labor is considered a homogeneous factor, empirical evidence suggests that discrimination on labor markets may lead to wage differentials for equally productive worker groups. Beginning with the seminal work of Gary Becker in 1957, neoclassical theory has explained discrimination mainly as the outcome of personal prejudices of some groups (particularly employers) against others (women, racial minorities, etc.). Suppose that an employer (in a competitive product market) has a prejudice against a group, say women (although females and males have, by assumption, the same marginal productivity). On the basis of this prejudice, the employer will undervalue womens contribution to production, with the consequence that different equilibrium wages for the two groups will be fixed. The monetary wages of females and males are represented by wf and wm, respectively, so that labor market equilibrium conditions are given by for males, and by for females, where d > 0 is the discrimi nation coefficient that measures the employers prejudice. Since MPL is the same for both men and women, it is easy to realize that women will be paid less than their VMPL (i.e., wf = wm d ). According to this model, discrimination is possible only in the short run, however, since competition among employers and workers will eventually equalize the wages of the two groups.

Alternative non-neoclassical models explain the persistence of discrimination by focusing on the segmented labor market approach, in a general noncompetitive theoretical framework. According to this approach, if the labor market is split into submarkets (think of the dual labor-market case, with a primary and a secondary segment), and if there are barriers to employee mobility, discrimination may take the form of occupational segregation, since workers who are discriminated against will be crowded into the only accessible labor market characterized by a lower equilibrium wage. In this case, however, unlike the neoclassical taste model presented above, all workers are paid according to their actual VMPL and wage differentials are the result of different labor supply conditions in the two submarkets, given the presence of barriers to employee mobility (see Smith 2003, ch. 6 and Ehrenberg-Smith 2003, ch. 12, for an exhaustive overview of discrimination theories).

Other criticisms concentrate on the impossibility of disentangling the specific product of the various factors of production, even at the margin of their application (Robertson 1931, p. 224). This argument rejects the very possibility of defining and measuring the marginal product of a factor (and particularly of labor). Among the various authors who have raised objections of this kind is J. A. Hobson, who did a good job synthesizing the various implications of the argument. In his view, if a productive process is characterized by cooperating factors employed in a well-defined proportion, an increase or a reduction of one of these factors (e.g., labor) will imply a reorganization of the entire process, thus modifying the average and marginal product of all the other fixed factors. If this is the case, it is not possible to attribute the corresponding change in output to one variable factor alone. Marshall, following Francis Y. Edgeworth, claimed that Hobson mistakenly assumed a discrete violent check in the supply of one input, rather than referring to continuous variations at the margin. In this case, the changes in the marginal product of the fixed factors would be negligible. In other words, Hobsons critics maintain that he was wrong because he was referring to a broad variation, while rigorously marginal productivity theory is concerned with the infinitesimal variation. Here, the relevance of the distinction between the marginal product and the marginal productivity of labor becomes apparent.

Hobson, however, was clearly discussing quite a different problem, which can be explained in the following way: If inputs used in a productive process must be employed in a well-defined fixed proportion, the principle of variation is not effective, and consequently the marginal product of labor (or of any other input) is a meaningless concept. An increase in one unit of labor will probably yield no additional output, while the withdrawal of one unit of labor will cause a notable diminution of both total product and the contribution of other inputs. In other words, if capital has to be considered as a specified physical endowment of goods, it could be combined only with a specified amount of labor (unless it is a sort of butter capital or jelly capital, as John B. Clark put it, that can employ a widely variable quantity of labor). This position is still supported by many opponents to neoclassical theory, and it was partially supported by some of the older neoclassical economists, such as Léon Walras and Vilfredo Pareto.

There is a less radical way to consider Hobsons criticism. Although an extra unit of labor may cause an increase in total output, this can only occur if the productive process is reorganized by way of a change in technique of production, such as an increase in the division of labor. But this means that one is not working on the same production function, but rather jumping from one function to another one. In this case, there would be little sense in talking about the marginal product of labor in a static framework. The idea of a change in the technological configuration as the employment of labor is changed is a characteristic feature of modern theory of economic growth and particularly of the literature on increasing returns (a line of thought of this kind can be traced back to Adam Smith, passing through Allyn A. Young up to the contributions in the late 1990s of James Buchanan and Yong J. Yoon with their notion of generalized increasing returns).

Hobsons criticism anticipated most of the elements later discussed in the famous Cambridge Capital Controversies that raged from the mid-1950s to the mid-1970s between neoclassical economists in Cambridge, Massachusetts, and post-Keynesian economists in Cambridge, United Kingdom. Although the controversy was mainly concerned with the nature and the unit of measure of capital, it nevertheless had serious implications for the consistency of the whole marginal theory of distribution. In fact, the main theoretical result of the debate was that if one considers a production process that employs (given the assumption of constant returns to scale) at least one other factor of production in addition to labor (i.e., capital), then, in general, it will be impossible to derive distributive shares from the specification of production function. Furthermore, the critics pointed out that, in general, other strange phenomenasuch as capital inversion or the re-switching of techniquesmay occur, which would undermine the principle of variation.

In conclusion, the only case consistent with the marginal productivity theory is that of a one-commodity economy, in which capital (in line with Clarks definition) is indistinguishable from consumption goods, and in which the only specific input is represented by labor. But in this case, as the critics have pointed out, both Ricardian and Marxian theories of value and distribution hold.

SEE ALSO Becker, Gary; Cambridge Capital Controversy; Crowding Hypothesis; Economics, Labor; Economics, Marxian; Economics, Neoclassical; Economics, Neo-Ricardian; Economics, Post Keynesian; Employment; Exploitation; Income Distribution; Labor Demand; Labor Force Participation; Labor Market; Labor Market Segmentation; Labor Supply; Marginalism; Marshall, Alfred; Returns to Scale; Returns, Diminishing; Smith, Adam; Wages

## BIBLIOGRAPHY

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Giorgio Colacchio