A Pure Neoclassical Theory of the Firm XI: Critique V (Microeconomics)

The Ambiguity of Capital: A Brief Introduction to an Insoluble Theoretic Problem

If heterogeneities between diverse manifestations of labor services, even in a single production process, complicate the notion of homogenizing labor in ways that might make it amenable to marginal productivity calculations as we add successive quantities of labor to a production process, then the idea of homogenizing capital to calculate its marginal productivity is complicated by the absence of any clear conception of what capital is, per se.  In the same way that labor has held a ubiquitous place in the history of economic theory as divergent theoretic traditions have attributed a different significance to direct productive human activity, multifarious opposing conceptions of the meaning and significance of capital reside across opposing traditions.
          For the Classicals, including Marx, capital had been synonymous with the ownership of particular assets that conferred on capital the capacity to organize the employment and utilization of labor in the production of goods and services.  In these terms, the capacity of capital to command a rate of return beyond the cost of its own reproduction (i.e. surplus, in a Marxian sense) defines the class structure called capitalism.  By contrast, for the Neoclassicals, again, capital represents a purely technical argument in a theory stressing productive efficiency across ranges of substitutable factor combinations.  While it, thus, makes sense to discuss capital as a household asset capable of being rented out to firms as a factor of production, the notion of representing the collective body of households owning capital as a class with collective interests and a collective political disposition and power extends beyond the theoretic frontier of the Neoclassical tradition.  To the extent that we can discuss capitalism within Neoclassical theory, the term amorphously encapsulates free market exchange, individual consumer choice, and the freedom of entrepreneurs to engage in innately uncertain investments without any discernible connection between such a system and theories of production and exchange except as an implicit background for the latter.  As we shall see, moreover, such a background is not even entirely evident for Walrasian/Paretian theory, insofar as general equilibrium theorizations operate with an implicitly decentralized, cooperative, non-entrepreneurial, quasi-non-market structure.
       If Neoclassical theory, in general, operates with an unclear conception of capitalism, then it confounds the problem by having unclear conceptions of capital, as a factor of production, in the heart of its theorizations of technological efficiency in production.  Such a problem is succinctly evident for Walrasian/Paretian theory, which embarked on several years of theoretic conflict against a collection of left-leaning Marshallian/Keynesian and Neo-Ricardian/Marxian theorists over the meaning of capital and the significance of compensation rates for the owners of capital (i.e. the so-called “Cambridge controversy” of the 1950s and 60s).  To the extent that these debates ended with no conclusive consensus in academic economics over its subject, it would, at least, be worthwhile to posit, as a point of criticism, that the theoretical ground on which Walrasian/Paretian general equilibrium economics is situated remains perilously unstable!  In this respect, we might benefit from a brief diversion from Walrasian/Paretian approaches to capital in order to develop an understanding of alternative approaches within the Neoclassical tradition.  Most emphatically, Austrian theory can be credited with the development of the most theoretically rigorous conception of capital in Neoclassical economics.
       As a discursive introduction, let us say that we are operating in a rudimentary economy in which individuals undertake the production of goods and services independently.  In order to produce the goods and services that they want to consume or to exchange with others, each individual has full access to a primordial common allotment of natural resources that can, for our purposes, be assumed infinitely expansive relative to the  capacity of individuals to withdraw its materials.  No tools or machinery exists to assist in production, but, importantly, the knowledge necessary to build such machinery is universally available.  However, if an individual chooses to produce a tool or machine to aid in production, the tool will be completely consumed in a single production process.  Therefore, in deciding what to produce, each individual is faced with a set of productive technological choices.  Should they utilize a technology maximizing direct human labor to achieve the desired output or should they take time out to build some tools that might allow them to produce more of the good or service they desire?
         Critically, the issue for Austrian production theory is about time.  It takes time to produce output enhancing tools and machinery that might otherwise be spent producing the thing that you want to consume with universally available, minimalistic technologies that will generate minimal outputs.  Individuals who choose to produce goods and services utilizing more “roundabout” techniques will enjoy larger quantities of such outputs and will, thus, create more wealth in objects of consumption.  For Austrian theory, capital is constituted in the utilization of production techniques that are more roundabout, and, thus, capitalism, as a system of technologically-driven production, consists of a pervasive drive to make production more roundabout.  As such, Austrian theory attributes the greater wealth of capitalist economies to the willingness of individual entrepreneurs, engaged in production, to defer their desire to consume in order to take the time necessary to develop the technologies that will expand the quantity of outputs available under more rudimentary, minimalistic technologies.
            If we take this theoretic definition of capital a step forward toward the problem of compensation, the relationship between time and output quantities remains key.  Roundaboutness in production always involves a deferral of desired consumption possibilities.  This deferral constitutes a real material cost, measurable through time.  In order to induce an individual to engage in the production of tools/machinery that will expand output beyond output levels that he would be able to consume with existing technologies right now, he must receive a quantity of outputs at some indeterminate time in the future that will compensate him for the inconvenience of having to defer consumption.  This conception of a future reward on deferred contemporaneous consumption defines the Austrian conception of the rate of interest on capital investments, where the compensatory payment for capital is defined as a rate of interest.  In accordance with the broader understanding of capital investment in Austrian theory, the reward for deferring consumption must increase as a function of time.  On the other hand, if we assume that individual utility functions discount future consumption possibilities relative to contemporaneous consumption (i.e. individuals would rather eat their cake today than eat the same cake tomorrow), then, as production processes become more and more roundabout, the interest rate would have to increase to account for the fact that more prolonged deferrals of consumption command a steadily declining level of marginal utility for individual entrepreneurs/investors as a function of time.
                 The thing that unifies capital as a concept and renders it susceptible to homogenization in the interest of analyzing its marginal productivity is its temporal relationship to production under abstractly rudimentary technological conditions.  In the same sense that the time taken to produce a fishing pole represents a capital investment in relation to the practice of catching fish with ones bare hands, the time taken to learn accepted teaching practices for younger/primary aged students in particular subjects represents a capital investment relative to the practice of amateur home-schooling by untrained parents.  The return to each of these capital investments, conceived as a rate of interest, compensates different manifestations of the same species of production factor (i.e. mechanical capital v. human capital).  Acknowledging the genealogical commonality of these two investments as capital, each presents us with a problem in anthropological economics - can we universally conceive of an abstractly rudimentary labor process converted by a capital investment to a state of greater roundaboutness enabling the enhanced productivity of labor?  
         The fishing example might work exceptionally well in demonstrating an Austrian conception of capital and, in particular, for measuring the comparative productivity enhancement from increased roundaboutness.  Maybe we can say the same thing about the training of primary school teachers in relation to home schooling, if only by means of a qualitative comparison on the range of knowledge and relative pedagogical efficiency of trained educators.  On the other hand, how do we make meaningful comparisons between the industrial production of structural steel in basic oxygen furnaces relative to some rudimentary practice of steel production (e.g. puddling of wrought iron in a primitive clay furnace)?  In regard to human capital investments, can we really make a serious comparison between the legal training of jurists under the bar or the medical training of brain surgeons relative to some imaginary rudimentary juridical practices or brain surgery by amateurs?!  At some point, in a wide range of production processes for goods and services, we have to concede that we lack a convincing rudimentary baseline against which to evaluate the gains from increasing roundaboutness because the most rudimentary practices in question happen to be extremely capital intensive.  In this regard, we need to amend our initial definition of capital to recognize that capital represents a continuum within which we can identify successive degrees of roundaboutness and, further, comprehend production processes that demand significant levels of roundaboutness to enable production to take place at all.  In fact, to the extent that we approach our image of firms in a general equilibrium economy from a standpoint in which all profit maximizing factor combinations are points of tangency between an isoquant and isocost function with positive quantities of both labor and capital, we have to conclude that some level of roundaboutness in production is implicit within our basic understanding of a Walrasian/Paretian firm.  
             Proceeding henceforth from a basic Austrian definition of capital as temporal roundaboutness in production, on the further conclusion that all roundaboutness is relative (and need not reflect a connection to some primordial rudimentary labor-intensive process), we need to further inquire into the determination of compensatory rates for capital.  Acknowledging that such compensation rates, especially for human capital, may embody a degree of ambiguity, enforced by intense degrees of complementarity between divergent production factors (e.g. basic homogenized labor services and human capital, manifest as occupation-specific training), the resulting compensation rate received by a given household supplying capital to the firm may exist as composite of multiple interest rates (for diverse. heterogeneous manifestations of capital) and wage rates (for labor services distinct from the accumulation of human capital by individuals).  As with labor services, the homogenization of capital is complicated by innate heterogeneities.  In fact, the more that labor services are simplified through reorganization and automation of labor processes, the more we will have to contend with heterogeneous composites of labor services and capital, insofar as reductions of basic skill levels in labor processes effectively demand inputs of technical learning on the structuration of production processes manifest as human capital.  All of this leads us to the basic conclusion that capital is heterogeneous and commands divergent rates of interest as compensation, dependent upon the relationship of particular forms of capital to particular production processes. 
              If capital, thus, exists as a heterogeneous set of factors, incorporating divergent temporal displacements in the production of goods and services collectively resulting in some degree of roundaboutness, then, as with labor services, the firm encounters a vector of capital factors, each manifesting a certain, defined elasticity of substitution with other all other factors varying between 1 (perfect substitutability) and -1 (perfect complementarity).  In this regard, the same comments and criticisms applied to the problem of heterogeneity in labor services also applies to heterogeneity in capital.  Our simplistic, two-factor production model has to give way to a production model with multiple vectors in which certain labor services are intimately tied to particular forms of capital, especially human capital, and, given a particular vector of compensatory rates for diverse forms of labor services and capital, firms may achieve profit maximization with factor combinations that leave certain types of labor services and capital wholly unused.  And, again, such outcomes may result in product exhaustion without offering a continuously differentiable production function through which we would want to calculate the marginal productivities of each form of capital. As such, compensatory rates for capital would certainly be determined by imputation from output market pricing, conventional to Austrian theory (which, in any case, rejects the whole project of mathematically deriving marginal productivities as a basis for factor compensation). 
  In certain respects, Walrasian/Paretian theory has taken the lead of Austrian theory in interpreting capital as deferred consumption, commanding a rate of interest through future expanded consumption possibilities.  As such, we may view Austrian theory as the implicit backstory/prequel underlying the Walrasian/Paretian tale of an integrated general equilibrium economy governed by tâtonnement.  On the other hand, significant inconsistencies arise when we introduce this conception of capital into our theory of the firm.  Most notably, introducing capital, in an Austrian sense, as a factor of production defined by a temporal relationship between entrepreneurs and particular processes of production, becomes innately problematic in a body of theory noteworthy for its timeless character.  The Walrasian/Paretian theory of the firm presented in this document requires that firms be continuously and instantaneously capable of adjusting scales of production to account for changes in output demand.  Again, such a requirement demands that capital be readily available in place if firms need to readjust to larger scales of output - there is no time in a pure general equilibrium economy to invest in producing the capital needed to expand scales of output if it is not already available to firms that need to expand.  Implicitly, for the Austrian theory of capital to be consonant with Walrasian/Paretian general equilibrium theory, (entrepreneurial) households have to incidentally invest in capital assets, necessary to expand output levels, without any certainty that the capital assets being produced will ever actually be rented by firms at a positive rate of return/interest.  As argued earlier, such uncertainties may not be a problem for either Austrian or Marshallian (and Keynesian) theories of capital investment, where future outcomes in market economies exhibit inexplicable elements of risk, but they have no place in a Walrasian general equilibrium system.
           Approaching the incongruities between Austrian and Walrasian/Paretian conceptions of capital from a slightly different focus, the idea of roundaboutness, critical to Austrian theories of capital and interest, reflects differences between relatively labor intensive technologies and relatively capital intensive ones, where the theory manifests an implicit productivity bias for the latter by definition.  That is to say, as implied by Austrian theory, entrepreneurs should only invest in more roundabout production technologies if they are expected to yield positive returns, evident as increased outputs for the same expenditure of labor.  It is possible that this deduction overstates the Austrian emphasis on roundaboutness relative to profit maximization to some degree, however.  I have no doubt that an Austrian theorist would concede that an entrepreneur seeking to maximize profits in an economy with relatively abundant supplies of unemployed laborers and a consequentluy depressed compensation rate for labor services should select a relatively labor intensive technology.  The point for Austrian theory, in this sense, is not that a theoretic preference for roundaboutness as a formula for increasing the material wealth of an economy should override the contextual advantages of utilizing relatively abundant factor resources.
            At this point, I want to once again address the particular balancing act implied by the accumulation and investment of capital.  In neither a Walrasian/Paretian nor an Austrian sense is capital equivalent to savings/retained incomes or wealth.  The difference between savings and capital resides in the active character of the latter - capital is actively invested in production to explicitly expand output by means of roundaboutness.  Savings is simply a stock of retained income or wealth, manifest in diverse material forms that lies fallow, awaiting consumption or investment.  In a fully integrated general equilibrium economy, utility maximizing households will only retain income that could otherwise be consumed to enhance contemporaneous utility if and only if there is some future reward that will justify deferring their consumption.  The problem is complicated further by the notion that households have a time preference/temporal dimension to their utility functions through which they discount future consumption possibilities.  In this manner, all household savings must be invested as capital - Keynesian considerations on liquidity preferences in the maintenance of positive, non-interest earning cash balances by households have no place in Walrasian/Paretian or Austrian understandings on the relationship between savings and investment.  
          On the other hand, the possibility of a future return on retained incomes demands knowledge on the technical possibilities of future enhanced material returns.  In order to fully act rationally, households must have knowledge of a set of technological and/or market conditions that do not presently exist in order to be satisfied that deferred consumption today will result in more consumption or more satisfying consumption tomorrow or next month or five years hence.  Such information cannot possibly exist.  To borrow a page from post-Keynesian macroeconomic theory, the future outcomes of capital investments are characterized by fundamental uncertainty.  Considered in this manner, every retention of incomes by households in forms that might subsequently be employed as capital reflects a subjectively formed belief, on the part of the household, that by deferring consumption today, they will earned more or better consumption tomorrow.  Capital formation always demands a digression from strictly rational behavior through which saving households come to believe in the possibility that they will be better off not consuming all of their incomes in the present.  
           Transplanting this imagery of imperfectly rational households into our portrait of the firm in a general equilibrium economy, our linking of savings to capital investment remains the (entrepreneurial) backstory to the static, technologically-determined profit maximizing behavior of the firm (absent an entrepreneur).  The peculiarity of the story resides, however, in the fact that it has to be told across multiple periods of time as if everything was occurring simultaneously.  The accumulation of capital must begin at some moment in the past at which its future rate of return was merely an expectation in the minds of the households that deferred consumption in order to accumulate it.  The present equilibration of marginal productivity from utilization of a given quantity of a particular capital asset (within a vector of capital assets) to a particular rate of interest must imply that the past expectations for augmented future incomes of saving households were either satisfied or those expectations were updated to reflect unexpected changes in economic conditions.  In either case, the entire act through which savings was converted into usable capital and usable capital was injected into a production process reflects a slew of uncertainties and expectations that cannot be redeemed by any recourse to the strictly rational behavior of the household agents.         
           Finally, I need to reiterate the point that the Austrian backstory to Walrasian/Paretian general equilibrium is unabashedly entrepreneurial in nature, while entrepreneurship is thoroughly vacated from the Walrasian/Paretian firm, which remains a simplistic, technologically determined profit maximizer.  The palpable lack of an entrepreneur in Walrasian/Paretian theory is, in many ways, the most important point of contrast with Austrian theory and the principle reason why it would be disingenuous to argue, without addenda, that the theory of the firm presented here is a self-consciously Austrian theory rather than a theory that seeks to borrow judiciously from certain Austrian themes and influences.  To the extent that Austrian theory readily engages in a conception of free market activity, capital formation, and investment in productive roundaboutness in a world of acknowledged uncertainties, entrepreneurial risk, and, to utilize the terminology of Austrian theorist Joseph Schumpeter, creative destruction, a thoroughly Austrian economy might actually be described as capitalist, by the terms with which it labels capitalism.  By contrast, the mechanistic character of firms and implicitly cooperative nature of tâtonnement between utility maximizing household agents renders any attribution of the term capitalism to a general equilibrium vision of market economics inherently murky and ambiguous.
         Having concluded that the Walrasian/Paretian conception of capital is, at best, a caricature of Austrian theory, I want to further elaborate two additional points on compensation to the owners of capital.  First, we have argued so far that firms seek to maximize their profits by renting production factors up to the point at which the marginal revenue product derived from renting the last unit of each production factor equals the factor market price for this last unit, under circumstances where, due to competition, the firm accepts both output and factor market prices as given.  This result is given in our consideration of the product exhaustion theorem, and it applies not only to our two-factor case but also to cases in which the firm is utilizing n >2 distinct factors of production.  As such, any multiplication of production factors to account for greater heterogeneities must still require the firm to equalize its marginal revenue products to the factor market price for each factor of production in its labor and capital vectors.  Alternatively, if we cannot strictly derive marginal revenue products through differentiation of the production function, then the output price-imputed value derived from the last unit of each factor rented must similarly be equated to the factor market price for this unit.  If this must be the case and we can otherwise clearly determine how much output the firm receives from the last unit of each capital factor it rents, then what is the conceptual significance of the factor market compensation rate that it evaluates in order to realize profit maximization?  That is to say, how does this factor market price relate to our quasi-Austrian theory of capital, where the compensation rate for capital is a rate of interest for deferred consumption?  
      In answering this question, we are interested in numerous details about the particular factor under consideration.  What is its durability (i.e. can the unit be injected in multiple iterations of the production process or is it completely consumed in one iteration)?  Can it be readily reproduced beyond the exhaustion of its productive life?  Is the factor usable in a wide range of production processes or is it restricted to an extremely finite range of processes?  How many units of the factor are available to firms (i.e. is the supply available to firms restricted in some way that will tend to enhance the market power of household agents in possession of the factor)?  Beyond these concerns, we are, again, evidently concerned with the expectations that households have regarding compensation?  What rate of interest would be sufficient to compensate the household for investing in the production of capital?  
      I want to give a more detailed explanation on these questions when I explicitly elaborate on household agents in a Walrasian/Paretian general equilibrium economy, but, for now, the simple answer that I want to advance is that the compensatory rate that households must demand from firms will reflect their expectations with respect to both the total quantity of compensation they expect to receive for their capital investment and their expectations regarding how long their capital asset will be usable and, hence, productive for renting firms.  In this sense, households in possession of capital assets approach firms in factor market contexts with subjectively formed expectations regarding how much they should be willing to receive for their asset today if their asset will be available at the same factor market price for t expected periods into the future, sufficient to realize the equality:
tΣi=1(pkiki) = I(1+r)^t
where pki = Market price of a capital asset k at time i
ki = Quantity of capital asset k rented at time i
I = Total investment in production of capital asset k
r = Rate of interest expected by household from investment in k
t = Total periods in expected rental life of k
Without giving too much away from subsequent critiques of household utility maximization, no objectively verifiable means exists to enable household agents to determine whether they will be capable of securing a sufficient return over t periods to realize the total future expected value of a capital investment over t periods. Moreover, borrowing again from the Austrian theoretic account from which the Walrasian/Paretian conception of capital owes its inspiration, there is no way to determine a priori the number of iterations t through which a capital asset will continue to earn interest for its owner. The determination of this number is an outcome of the diversity of production processes and firms to which the capital asset may be rented, the degree to which the productivity of the capital asset depreciates over time and the extent to which the owner is faced with competition from investors in newer, more productive substitute capital assets, and the extent to which technologies change, rendering this particular capital assets obsolescent. Here, again, we encounter the Austrian theme of creative destruction, manifest in the prolonged or swift reduction in the relative productivity of a capital asset in relation to newer technologies. All such considerations must, in some way, reconfigure the production functions of firms and, as such, reshape the context within which Walrasian/Paretian firms achieve profit maximization.  
Lastly, a critical point of departure in terminology between the Classical and Neoclassical traditions regarding capital remains to be discussed. Namely, theorists in the Classical tradition, at least, to some degree, until the development of Marxian theory, regard the rate of return to capital as a rate of profit. The reasons why such a return might accrue for the owners of capital are entirely unclear before Marx. Notably, for Ricardo, profit appears, for lack of a specific explanation, as a natural rate of return for investment of capital, distinguished from land and labor. As such, Ricardo's most detailed attempt at an explanation for the origins of profits on capital concern agricultural investments tied to naturally occurring variations of the productivity of land. For Marx, profit is expressly attributable to the exploitation of labor power/capital capable of varying its rate of value productivity in relation to its socially determined reproductive expenses. Thus, in order for profit to exist, direct human labor must generate a surplus value in excess of the expenses accrued by firms to hire/rent labor power.  
In the Neoclassical tradition, the definition of compensatory payments to the household owners of capital changes starkly, in ways that hold significant ethical consequences. As shown, the Austrians regard the rate of return to capital as interest, accruing both as a function of deferred consumption and enhanced roundaboutness of the production process. In this manner, the ethical consequences of capital accumulation are transformed to accommodate both the frugality of investors (in relation to prodigal segments of the population in an economy for whom poverty and an absence of discretionary income to employ as savings is a character flaw!) and the technological savvy of entrepreneurs. The conceptual significance and ethical consequences of profit, by contrast, relate emphatically to variations in market conditions, generating static divergences in the prices that firms may receive for their goods and services. Such an explanation of profit readily applies to Austrian theory and, as we shall see, to the Marshallian conception of the firm, where short term variations on market conditions and the capacity to vary certain factors while others remain fixed are pertinent. To the extent that these approaches are mutually sensitive to the effects of short term variations in market conditions on the revenues realized by firms, the conception of firms as profit maximizing entities exudes something more than a bland, mathematically-determinate significance. Moreover, in reducing profit to a short run market phenomenon (in contrast to long run returns to scarce or otherwise restricted resources defined as rent), profits lose a substantial degree of ethical significance. Can entrepreneurs really be blamed today for profiting from short term appreciations in market demand for goods and services that might be competed away tomorrow?  
To the extent that Walrasian/Paretian general equilibrium theory avoids the developmental consequences of capital accumulation by taking the quantities of capital available to firms in a given temporal context as given, we can entirely avoid the ethical consequences of our respective terminologies for the compensatory returns of capital. Rather, insofar as Walrasian/Paretian firms operate in a timeless/spaceless environment where any short term variation in demand for goods and services will instantaneously be resolved to realize a continuous zero-profit condition across all markets, the theory of the firm presented in this document dispenses with the possibility of profit entirely. Thus, we are left with our Austrian inspired conception of interest as the default alternative in assessing the virtues of supplying capital to production.
         To conclude this section, the conception of capital contained within our Walrasian/Paretian theory of the firm remains, in certain respects, unclear.  As with labor, we have to acknowledge a substantial degree of heterogeneity in capital, susceptible to patterning through vectors of discrete capital factors incorporating tools, equipment, and machinery but also human capital, manifest as experiential or education/training-driven knowledge.  The common thread connecting these factors remains the capacity of each to enhance the productivity of a given production process by making the process more temporally roundabout.  On the other hand, capital introduces complementarities and complications that we will continue to sort out across subsequent theoretic approaches to the firm.  
          

A Pure Neoclassical Theory of the Firm X: Critique IV (Microeconomics)

The Homogenization of Labor Services and Determination of Labor Compensation

Moving forward from consideration of the Walrasian/Paretian production function to evaluate Walrasian/Paretian treatments of the factors of production, we need to emphatically question the notion that we are dealing with homogeneous units that can be readily aggregated to generate determinate quantities of output from given factor combinations.  In this manner, we are after the ready quantification of inputs to production in forms that can be easily combined and amenable to calculations to yield marginal contributions to the output of the firm.  Bearing in mind that we are, once again, dealing with a rigorously rationalist body of theory that constructs ideal, abstract models and validates them in reference to their explanatory power in relation to real markets and real production processes, it would not be fair to argue that abstract homogenization of labor, land, and capital presents an unrealistic portrait of real production inputs.  Reality is not at stake in Walrasian theory - the capacity to explain reality through an abstract model in a persuasive manner is.  In these terms, we need to examine the particular ways in which the Walrasian/Paretian theory of the firm homogenizes the production factors and illuminate the particular consequences of such theorizations.
        Beginning with labor, homogeneous representations of labor services are ubiquitous within multifarious shades of economic theory.  The English Classicals approached labor as a common denominator in the production of value.  For Adam Smith, labor appears as a generalization of human activity, subject to development through the technical organization of production processes and acquisition of skill and dexterity through experience.  As such, task divisions of labor at the minute level in individual production processes (e.g. Smith's pin factory) had the effect of sharpening the particular skills of laborers at particular tasks at the expense of diffusive, complex experiential knowledge of a larger production process.  This developmental understanding of labor appealed to a particular imagery of labor through which abstractly unskilled laborers could be transformed to enhance individual and collective productivity, without simultaneously developing an integral understanding by workers of the broader range of tasks involved in commodity production.  In this sense, Smith's comprehension of labor incorporated a transition from abstract universality to dynamical particularity, characteristic of an age in which the social utilization of human labor was rapidly changing, from small scale, localized artisan production of goods to technologically transformed manufacturing of goods under a task division of labor, and gradually and unevenly to introduction of machinery and artificial sources of power that could entirely replace the skills and energy of human laborers.  In the end, however, for Smith, labor commands a universalized dual significance as both the transformative process through which natural endowments could be transformed into objects of consumption and the source of value underlying such objects.
            Almost half a century later, David Ricardo sharpened Smith's understanding of labor by distinguishing between the notion of labor time expended in the production of goods as the source of marketable value in most commodities and the notion of labor services as a marketable commodity commanding a wage.  Later in the Nineteenth century, Karl Marx offered the most rigorous formalization of Ricardo's insights, homogenizing the understanding of labor time embodied in the production of goods and services as a socially necessary average calculation, conveyed and standardized through the operation of markets and the force of competition, and formalizing the concept of marketable labor capacity as labor power, as a fundamental ingredient to production under capitalism.  As such, the Marxian counter-homogenization of abstract social labor time and labor power represent the theoretic apogee of the Classical tradition beginning with Smith.
           These Classical formulations of the homogeneity of labor differ in important ways from the understanding of labor that emerges within the Neoclassical tradition, either for Walrasian theory proper or for Austrian and Marshallian variations.  Preeminently, developmental concerns with labor disappear, subsumed within the unseen technological contours of the production function.  In particular ways, the Neoclassical tradition reintegrates the role of labor in value creation with the question of compensatory rates, simultaneously undermining certain aspects of labor considered exceptional for labor by the Classicals in relation to the other factors of production.  Emphatically, labor time loses its primacy as the source of value in exchange advanced by the Ricardians and the Marxists.  Moreover, the Neoclassicals implicitly depreciate the Classical emphasis on income distribution between classes.  If the Classicals, from Smith to Marx, are preeminently concerned with the distribution of income between classes defined overwhelmingly in accordance with ownership of divergent productive assets, then the Neoclassicals, from the early Marginalists into the Twentieth century, are preeminently concerned with questions of technical efficiency in the employment of substitutable factor combinations and how such considerations, seen through the prism of relative factor scarcity and market competition, reflect upon the ethics of compensation.  In these respects, marginal productivity factor pricing theory, as anticipated or otherwise implied by Walras in his early elucidations of general equilibrium theory and refined into a full-fledged theoretic doctrine by the American J.B. Clark, is above all as much an ethical proposition about wage rate determination as it is a technical condition of profit maximization/cost minimization.  My considerations on labor homogeneity and compensation here, are, thus, bound up with marginal productivity factor pricing although my account does not expressly offer a critique of the latter.
          The homogenization of all factors of production is predicated on the need to calculate the marginal product from adding additional increments of each factor.  Such calculations can only make sense if each additional unit of each factor is virtually identical to every other unit of the same factor already employed or capable of being employed.  Considered in this manner, we need to establish both the technical boundaries of the production process and the manner in which labor services are technologically integrated with other factors in order to generate output.  These pieces of information can inform the theoretic homogenization of labor services to the extent that they posit the level to which concrete variations in real labor activity can be erased in the effort to define a given homogeneous factor input.
           Fundamentally, we need to differentiate between two distinct senses of homogenization with regard to labor services.  On the one hand, there is the real simplification of labor processes, related to the concept of scientific management and the reorganization of industrial tasks through subdivision and, at least in part, automation.  Conceptions of homogeneity in labor services along these lines emphasize the real interchangeable character of human labor, enabling a radical reduction in basic necessary skill levels to perform industrial tasks.  Arguably, such a real simplification of labor services goes far beyond what Walrasian/Paretian theory has in mind.  Rather, in my view, we are pursuing an abstract theoretic homogenization of labor services, through which even the most complicated tasks in a complex production process can be conceptually related to more basic tasks, perhaps, in certain circumstances, by reinterpreting a particular form of labor services as a composite of labor and (human) capital.  In this manner, the boundaries between labor and capital may become blurred, but such ambiguities are likely inevitable when we attempt to make calculative generalizations concerning something as heterogeneous as the performance of labor.
           If we pursue the abstract homogenization of labor services to its logical extreme, then we would reduce labor services to a single factor of production, contained within a vast array of production processes in varying proportions in combination with a single homogeneous substance called capital.  Such a conception of labor services manifests itself within certain aggregate production functions, conceived on a macroeconomic level.  Such theoretic devices, again, operate beyond the range of inquiry contained by this document - we are interested in developing an understanding of production functions and factors within the limited theoretic framework of the firm, abstracted, in part, from the dynamics of the wider general equilibrium economy within which it is contained.  Briefly, however, labor at this extreme level of abstraction detaches from the particular technological contours of individual production functions/individual firms to represent an abstract substance commanding a rate of return related, somehow, to its marginal productivity across all production processes and all commodity markets.  In short, it represents an attempt by Neoclassical macroeconomic theory to come to terms with the general rate of return to labor as a productive asset and, thus, to explicate short run or long run patterns of change and/or stagnation in real wages relative to technological factors across an entire macroeconomy.  Without elaborating further, such theorizations expand the explanatory power of theoretic tools within the Neoclassical tradition far beyond their logical limits!
         Returning to the bounded level of abstraction contained within this document, it is at least conceivable that we could homogenize labor services adequately to establish a particular relationship between a production process utilizing labor services in combination with particular forms of capital and the compensatory rate offered to labor for performance within the production process.  Through the prism of Walrasian/Paretian theory, such a relationship would involve the equalization of the marginal revenue product of labor, derived from the individual firm's production function, with the market wage rate for labor services.  The relationship is, in turn, grounded in the process of tâtonnement.  That is to say, households on the commodity consumption side of the firm's production problem establish the prices that they are willing to pay for given quantities of the firm's output commodity, defined on the market level at which all firms are subject to rigorous price competition.  On the factor supply side, households determine how much of each factor of production they are willing to supply to the firm for each rental/wage rate, again defined on the market level at which the firm must likewise accept the position of price taker in competition with many, many other firms. 
           Relative factor prices enable the firm to determine, through the objective technological contours of the production function, what combination of factors will achieve profit maximization/cost minimization.  At a factor combination characterized by constant returns to scale/constant costs, the sole remaining question concerns how much of each factor will be rented by the firm to harmonize output market demand at operative costs per unit of output (where the marginal cost for the last unit of output must equal the average cost across all units of output) with factor market supply at operative compensation rates (where each factor will be compensated in accordance with its marginal productivity), consistent with the firm's financial constraint (i.e. its capacity to rent money capital at given rates of interest to compensate the factors).  Again, the results of this balancing act is, in turn, technologically (pre-)determined for the firm by its production function.  Under the constraints of perfect competition/perfect information, Walrasian/Paretian firms can only react to the market outcomes they face.
          With this process in mind, is it really necessary to abstractly reduce labor services to a single factor of production if, on the one hand, technological subdivisions of the production process can be readily defined within the contours of the production function and, on the other hand, such subdivisions are mirrored by the presence of readily separable factor markets with divergent compensation rates?  If we accepted a certain level of heterogeneity in the definition of production factors and in the equilibration of marginal revenue products with factor market compensation rates, then we might obtain a more accurate and nuanced portrait of the production process.  This approach is implied in the suggestion by Gérard Debreu, one of the theoretic fathers of contemporary Neo-Walrasian general equilibrium theory, that we need to recognize a much greater degree of heterogeneity between commodity markets based on divergent, contingent conditions of exchange (see Debreu (1959), The Theory of Value: An Axiomatic Analysis of General Equilibrium, 28-32.  New Haven, CT: Yale University Press, at: http://digamo.free.fr/debreu59.pdf).
       The point of Debreu's argument, in this respect, is that a range of heterogeneous conditions structure individual contexts within which commodities are exchanged and that this range of heterogeneities actually constitutes boundaries between commodity markets.  As such, for every individual moment in which a fully integrated economic system arrives at a general equilibrium, the set of commodity markets that equilibrate is different from the set of commodity markets that equilibrates at some other moment.  If a continuity exists in the realization of general equilibrium, such a continuity is expressly manifest in the process of tâtonnement without which we would lack any meaningful explanation for market clearing where the commodities offered in trade are, by contextual structuration, continuously changing.  The timeless character of a general equilibrium economy would be taken to its extreme at which every continuity in objects of exchange would disappear except as an object of memory/simile in the minds of consumers.    
       Debreu's definitive conditioning of the meaning of commodities offers us a relevant means for sidestepping the problem of theoretic homogenization for labor services.  If the labor services supplied by each individual can be readily distinguished from the labor services supplied by every other individual, or, conversely, if the labor services demanded by firms at each stage of a production process can be readily distinguished from those demanded at all other stages, then we would have to admit of a multiplicity of labor markets.  Each labor market would, in turn, realize an equilibrium in which the marginal revenue productivity derived from each individual form of labor services would determine the compensatory rate for each respective form of labor services.  That is to say, different markets might exist for the labor services offered by every individual and the marginal productivity of each would be calculated in reference to a set of temporal units of labor services offered by the individual where all other conditions for the utilization of labor services (e.g. quantities/qualities/varieties of capital employed) are held equal.
        If Debreu's redefinition of the commodity offers us a way out of abstract homogenization, then we need to develop the implications of pursuing such an alternative.  Notably, accepting Debreu's radical heterogeneity of labor services, we can no longer talk about the simple two-factor production function, characteristic of most Walrasian/Paretian theorizations of general equilibrium.  Rather, every production function would have to embody multiple vectors of production factors in which each individual offering labor services would enter as a distinct factor to the production process, each with its own marginal productivity.  Such radical heterogeneity may exceed the practical limits of our theory of the firm, but it remains as an asymptotic imagery of what Walrasian/Paretian economics might be able to pattern theoretically if it concedes that real firms must negotiate the real heterogeneities evident in the employment of diverse individual laborers, including divergent internalized skill sets and market conditions at the time of employment.  Theoretically, all such conditions would have to be subsumed within either the technological contours of the production function or defined within the cost function by virtue of market processes/tâtonnement.  
        Developing the implications of radical heterogeneity further, two salient and linked concerns become evident.  First, if every discrete manifestation of labor services constitutes its own market, then how can we evaluate the influence of competition between households/individuals in mitigating compensatory rates charged to firms?  Second, if we expand a production function to include an indeterminately large vector of individual labor service factors, then how large would such a vector have to be and how would we deal with combinations in which certain factors remained unused?  At the intersection of these two concerns exists the problem of substitutability as a practical matter for the firm in selecting between relatively homogeneous labor services.  
         The first of our concerns here can be reduced to the determination of market boundaries.  Again, theoretically speaking, we can opt for Debreu's radical heterogeneity of commodities/factors, but accepting such radical heterogeneity would imply that every supplier enjoys a strict monopoly in the provision of their services.  As such, in order to evaluate the effects of competition between suppliers of labor services, we would have to acknowledge that a broad range of substitutes exists for every particular manifestation of labor services demanded by a firm.  Every discrete labor service factor, therefore, would have to exhibit some degree of substitutability in relation to every other factor, expressed as an elasticity, where unitary elasticity would reflect perfect substitutes.  If the average elasticity of substitution between labor services suppliers approached unity across all suppliers accessible to a given firm for a given production process, we might conclude that the firm faced something approaching a single market for labor services, characterized by perfect competition (with perfectly homogeneous labor services).  Considered in this manner, the question of boundaries between markets is intimately linked, for the firm, to the contours of its production function in relation to each individual labor services factor, and, again, as technological compatibilities reduce the necessary degree of heterogeneity between individual labor service factors, the boundaries constituted by discrete, individual skill sets erode and collapse, constituting singular labor markets for exchange of relatively homogeneous labor services.  Recognizing the potential for near-perfect substitution of labor services between discrete individuals, in most cases, as an outcome of technological compatibility across a range of possible substitutes, the question of boundaries between markets for labor services reduces, in part, to a technological matter framed by the firm's production function. 
        If we concede the point that technologically-induced compatibilities in the substitution of labor services between individuals can break down individual differences between skill sets, then we must simultaneously concede that legitimate and insurmountable differences between individual skill sets are liable to remain, more prominently in certain production processes than in others, and, consequently, such differences must continue to structure labor market boundaries.  In real economies, many of these market boundaries appear subsumed within the institutional boundaries of the firm as internal labor markets,  As individuals employed within firms accumulate greater quantities of workplace-specific experience in the performance of a given occupation, their development of an occupationally-specific skill set differentiates them from potential new hires in ways that are apt to be reflected by the firm's production function through enhanced productivity at the temporal margin.  Thus, at least asymptotically, more experienced individuals would command higher rates of compensation than new hires for reasons of skill-based differentials in marginal productivity.  Such differences must appear, within structure of the firm's production function, as differences between two or more distinct forms of labor services, each with its own discrete market.    
           More generally, it would stand to reason that the set of individuals competing for positions in the labor market for home healthcare assistants would differ from the set competing for positions as paralegal aids for reasons not limited to the particular forms of training required to participate in these fields.  If we can reasonably conclude that investment in particular training processes, as a form of human capital, differentiates individual labor market participants, then we should also conclude that the nature of labor services performed in particular fields differentiates participants prior to the accumulation of human capital.  Individual physical attributes and motivational proclivities for particular forms of work can both determine the scale of particular, discrete labor markets, per se, and the extent to which individuals will invest in occupationally-specific forms of training as human capital, in the limit based on the expectations individuals have that they will enjoy a positive return in relation to other individuals by investing time to accumulate occupation-specific skills.  
           Finally, if we were to consider the scale of discrete labor markets faced by real firms for particular production processes, we have to evaluate the basic ontological dimensions of temporality and spatiality encountered as such firms undertook production decisions.  Notably, what are the temporal conditions of the production process for which labor services are required?  How long will employees engage in production before final commodities are generated for exchange?  What subsequent conditions might govern the possibility for contractual renewal across multiple iterations of a particular production process?  Such questions must condition the temporal length of any labor contract, however structured or implicit.  Conversely, how geographically expansive will be the search for suppliers of particular labor services?  Is there an expectation that suppliers with a particular skill and/or training/experience set can be found within areas proximate to the site of production or should firms engage in wider searches?  All of these conditions must structure the scale and skill/qualitative dimensions of particular markets for labor services, in turn determining the degree to which labor services of a particular, detailed form can be considered homogeneous in relation to relatively close substitutes.  As each market manifests a greater degree of homogeneity between individual suppliers, we would expect that competition between suppliers would propel real wage rates to approach some mutually acknowledged compensatory minimum, however defined, if only because the presence of readily available substitutes would tend to dampen the expectations individuals have for compensation.
             From the perspective of the firm, framed by the technological contours of its production function, insurmountable differences between the labor services offered by particular individuals or sets of individuals would, again, have to be patterned as differences between discrete factors of production, contained by discrete markets for labor services and included as discrete mathematical arguments in the firm's production function.  Differences between individual employees with substantial quantities of on-the-job experience and new hires would constitute the terms by which the firm would compare divergent factors of production to achieve profit maximization/cost minimization.  As suggested, such differences might, invariably, be patterned as differential accumulations of human capital, and, as such, particular employment decisions by the firm might involve combinations of "pure" labor services and accumulated training/experience as human capital.  In these terms, it is possible that a firm, undertaking a given production process, might enjoy multiple, discrete profit maximizing/cost minimizing factor combinations along a continuum defined by its production function.  Moreover, it is possible that certain forms of labor services, defined within the firm's production function, might remain entirely unused within its selection of a profit maximizing/cost minimizing combination of production factors.  Finally, respecting the possibility that production decisions by a range of firms within a given market context with multiple, technologically-driven profit maximizing/cost minimizing combinations might have some impact on demand for particular labor services, it is at least conceivable that demand side decisions by firms in discrete factor markets will shape the process of equilibrium price determination.  We need to evaluate each of these conclusions.
             To begin, the basic presence of heterogeneities between labor services implies that firms face a vector of alternative labor services factors, where each factor must vary slightly before we even begin to account for accumulations of human capital by individuals.  Theoretically, considered across all production processes within an integrated general equilibrium economy, such vectors might be defined to include all labor services factors offered for rent within the economy (i.e. the labor services of all individuals within the employable labor force).  In practical terms, such a situation is unrealistic - the employment decisions of firms, patterned as vectors of available labor services factors, are limited to those individuals who expressly offer their labor services or might otherwise be recruited by a firm because of their physical/mental attributes and/or their particular accumulations of generalized or occupationally-specific human capital.  As such, the labor services vector faced by each firm is limited in its scale and scope by a range of possible considerations.
            This limitation of the range of available labor services factors simplifies the employment decisions of the firm to at least some degree, but it can never reduce the problem to one of selecting the appropriate quantity of homogeneous labor services in relation to some quantity of homogeneous capital.  In order to achieve profit maximization, the firm is still constrained to select between diverse combinations of relatively heterogeneous labor services and heterogeneous capital, including training/skill acquisition by potential employees as human capital.  That is to say, a firm engaged in some production process, say dining services, must select between available labor services offered by some individuals with little experience and no training at relatively low compensation rates (reflecting diminished expectations on marginal productivity), other individuals with multiple years of experience preparing foods and accumulation of formal training at relatively high compensation rates, and individuals with various quantities of experience and training in between.  If each of these possibilities corresponds with a different factor combination on the set defined by its production function, then the firm must decide which combination will maximize its profits (assuming, for our purposes, that such a decision takes the production function as an objective, determinate formula for the production of outputs, even if our previous critique of the production function casts doubt on the firm's ability to do so!).
          With regard to the firm's labor services vector and its set of technologically available factor combinations from its production function, two general outcomes arise for profit maximization/cost minimization: an internal solution exists in which all labor services factors in its vector are utilized in positive quantities by the firm, and a corner solution in which some factors are used in positive quantities while others are wholly unused.  The former case, which becomes less likely as the labor services vector expands and the elasticities of substitution between discrete labor-human capital combinations approach unity, might at least afford the possibility that marginal productivity calculations for each factor obtain strictly through differentiation of a production function rigorously corresponding to Walrasian/Paretian assumptions.  Conversely, if the firm's production function incorporates a labor services vector in which certain factors are utilized in zero quantities, then it becomes impossible to gauge the marginal productivity of any other factor by means of differentiation.  Rather, we would be compelled to maximize profits through linear optimization in which the factor compensation rates are determined through price imputation.
         As suggested in the previous section on the production function, there is nothing innately wrong with imputed factor pricing so long as we recognize that, if we are operating within a general equilibrium framework, then both factor and output market prices are subject to simultaneous determination through tâtonnement.  On the other hand, we should further concede that, even to the extent that we realize product exhaustion, price imputation divorces factor pricing from technological productivity, manifest strictly as a mathematical outcome of differentiation (i.e. obtaining a marginal product calculation by differentiating a production function characterized by continuous substitutability and constant returns to scale).  As such, we cannot obtain the marriage of technological efficiency and ethical distributive justice implied in J.B. Clark's formalization of marginal productivity factor pricing theory.  The latter simply cannot stand if we have to diverge from strict reliance on differentiation of simple production functions with homogeneous factors.
          Concluding this section, I want to emphasize that the abstract theoretic homogenization of labor services, prescribed by Walrasian/Paretian production theory as a means to account for direct human labor within the firm's production function, is entirely possible and, perhaps, necessary if we want to achieve all of the useful insights of general equilibrium theory regarding the efficiency and distributive justice of integrated, unregulated market/cooperative economies relative to rent and utilization of production factors.  This reduction of labor services to a homogeneous factor is not equivalent to the real simplification of labor processes in industrial production of goods and services, however the pervasive breakdown of differences between threshold occupational skill sets in different production processes may shape our capacity to envision labor services as a homogeneous factor.  Moreover, the capacity to view labor services, as a basic factor, in combination with accumulated skills, experience, and training as human capital, further facilitates our ability to reduce labor to a homogeneous common denominator entering into to labor process.
         Conversely, if we proceed in the opposite direction by recognizing radical heterogeneities in individual labor services, then we introduce a more nuanced, if not "realistic," imagery of employment decisions involved in profit maximization by the firm.  Such employment decisions concern the selection between ranges of discrete labor/human capital combinations with (relatively) independently determined compensation rates and expectations on productivity at the temporal margin (i.e. as we increase the number of labor hours utilized from a given, relatively heterogeneous factor, holding all other variable factors constant).  On the other hand, the incorporation of heterogeneity between labor services available to the firm complicates the mathematical process of profit maximization and, at least to some extent, undermines the ethical distributive claims associated with marginal productivity factor pricing.
         In the end, I want to emphasize that the importance in choosing between strict, abstract homogenization of production factors and the introduction of radical heterogeneity between discrete factors pertains to the overall discursive/argumentative/rhetorical aims of the theory.  We invariably return to the question of what the theory means to tell us about the reality that it is theorizing.  In these terms, incorporation of relative heterogeneity may simply constitute a means through which Walrasian/Paretian general equilibrium theorizations can connect a little more persuasively with the radical heterogeneities of real economies.

A Pure Neoclassical Theory of the Firm IX: Critique III (Microeconomics)

Production Functions as Ideal Representations in Production Analysis

In this elaboration, I have taken utmost caution to specify what constitutes a firm within the particular parameters of Walrasian/Paretian general equilibrium economics, noting that such a construct is not analogous to the firm within other bodies of economic thought (e.g. Marshallian Neoclassical approaches, Marxian theory) or to firms operating within "real" economies.  The firm within Walrasian/Paretian theory emphatically represents the behavior of firms within the real world in negotiating market exchange within a fully integrated economic system, reflected in the notion of general equilibrium.  The conceptual fulcrum in this representation is the production function, which prescribes an optimal combination of land, labor, and capital units for every set of relative prices for production factors (defined within the firm's cost function) in order to produce desired output quantities at the lowest possible cost.  The production function describes what a firm is not by indicating what a firm has (in Walrasian/Paretian theory, again, firms own nothing - they rent production factors from households) but by demonstrating what a firm does.  Firms constitute the spatio-temporal context within which utility maximizing households combine the factors of production at their disposal in order to produce the things that they want to consume or, alternately, the things that they want to exchange for desired articles of consumption.  By mathematically characterizing the relationship between given quantities of production factors and determinate quantities of output, the production function expresses the contingent reality of production as a pure and simple technical problem. 
            If, to some extent, production functions are ubiquitous to all Neoclassical economic approaches, general equilibrium economics imposes certain relevant constraints on the form of the production function, intended to realize the larger analytical goals of the theory.  Specifically, Walrasian/Paretian production functions must be characterized by linear homogeneity and homotheticity, ensuring both that optimal combinations of production factors are invariant with respect to the scale of production (constant returns/constant costs) and that compensation to households for rental of production factors in accordance with the marginal productivity for each factors must result in product exhaustion.  Moreover, in a general analytical sense, the mathematical arguments of a Walrasian/Paretian production function must represent the most efficient technical means known for combining factors to derive outputs.  That is to say, the production function axiomatically embodies state of the art technology and, within a perfectly competitive market framework, such technology must be available to all firms.  Continuous substitutability between factors ensures that discrete changes in the relative prices of production factors can be addressed through discrete variations in factor utilization.  As such, for any given production process, a single production function must exist and contain the most efficient existing means to combine factors of production to produce all quantities of output at the lowest possible cost.  Any firm without access to such a production function would, indisputably, be competed out of the market. 
            In approaching a critique of the Walrasian/Paretian production function, I want to separate the issues introduced in this section from criticism of the abstract and homogeneous nature of production factors to which I want to devote other sections.  Summarizing the characteristics that I think are pertinent here, Walrasian/Paretian production functions express determinate quantities of output in relation to specified quantities of production factors and irrespective of myriad exogenous conditions not directly related to the production factors.  Any given set of relative prices for production factors will generate a unique optimal combination of land, labor, and capital, at which output and profits will be maximized and costs minimized.  For each set of relative factor prices, changes in the scale of output will occur with fixed proportions of the production factors, rendering scalar differences between firms invariant with respect to the calculated average production costs per unit of output (i.e. constant costs).  Finally, the assumption of perfect competition mandates that all firms must enjoy perfect information with regard to production technologies, ensuring that all firms within a particular market for a particular good or service must be employing the same production function.  We will approach a critique of the Walrasian/Paretian production function in reference to these characteristics. 
             First, what does it mean to say that a Walrasian/Paretian production function constitutes a determinate formula in the combination of production factors to obtain output quantities?  As a minimum, it must mean that, under a predictable set of circumstances, given quantities of land, labor, and capital must, on average, produce specified quantities of output as revealed by the mathematical arguments of the function.  As such, over repeated iterations of a production process over an extended period of time, the formula represented by the production function must deliver relatively predictable output quantities in relation to factor inputs with, at most, a narrow range of inexplicable temporary divergences from the predicted output quantities.  If these descriptions make it sound as if we were veering into a statistical/econometric problem, that is because the tools of econometric analysis can illuminate the terms of a critique against the determinate nature of production functions, relating actual production outcomes to potential estimated values in a statistical regression.  The point here is not to argue that an interpretation of the theoretically determinate character of a production function is too rigorous to survive empirical scrutiny.  It is, rather, to open up the derivation of production functions to statistical methods, through which corrections for systematic deviations can reveal underlying exogenous sources of variation. 
              Econometric estimation of production functions enjoys a long history in applied economics.  Emphatically, the Cobb-Douglas relationship had its origins in the work of University of Chicago economist Paul Douglas, attempting to estimate a relationship between diminishing marginal productivity and the distribution of revenues to labor and capital within larger economic aggregates, effectively validating J.B. Clark's theories on marginal productivity factor pricing.  In these terms, Douglas' work took the critical features of the Walrasian/Paretian production function (i.e. linear homogeneity, constant returns to scale) as its points of departure without simultaneously respecting the microfoundational assumptions of general equilibrium thinking (i.e. extrapolation to define an aggregate production function in order to validate theoretic concepts strictly applicable to individual production agents) (on the econometric development of the Cobb-Douglas production function and its early uses, see Jeff E. Biddle, "The Introduction of the Cobb-Douglas Regression and Its Early Uses by Agricultural Economists," (dated: October 2010; downloaded: 28 July 2015), at: http://econ.msu.edu/faculty/biddle/docs/October%20revision--Conference%20Paper.pdf). 
              Econometric estimation of production functions, truly consistent with Walrasian/Paretian general equilibrium analysis, should be restricted to data from individual firms, relating quantities of factor inputs to produced outputs over a temporal range.  That is to say, we are dealing with time series analysis or, at the most, panel data across multiple firms or multiple production units of a given firm with very similar production processes.  In this regard, I am not concerned with the sort of aggregate production analysis that characterizes certain Neoclassical macroeconomic theorizations. Considered graphically in three-dimensional, two-factor/one-commodity space, data for individual production processes might appear as a scatterplot of data points in the strictly positive octant, under the assumption that any positive quantities of commodity x requires strictly positive combinations of labor l and capital k (temporarily excluding quantities of land as a statistical argument).  Figure 28 seeks to represent such a scatterplot, where points A through D represent distinct scales of output x produced by distinct combinations of labor and capital.

Figure 28: Scatterplot of Output Quantities and Factor Combinations with Selected Output and Factor Combinations A to D in Two Factor/Single Output Space.
The point of regressing output quantities against factor combinations in such circumstances to statistically identify a function that might enable us to draw isoquant level curves is to say something definitive about the nature of substitutability between factors at each scale of output and to define expansion paths through successive scales of output at fixed factor ratios.  In other words, it involves the utilization of an intertemporal data set, possibly augmented/complicated by inter-spatial/multi-unit data, to define a determinate functional relationship expressing two distinct directions of variation negotiated by firms - between diverse factor combinations at a given fixed scale of output and across scales of output at fixed factor ratios. 
              Problems immediately exist in the selection of variables for such estimations.  That is to say, the very notion that we could isolate a set of data points as in figure 27 assumes that we can find suitable variables representing each production factor (considered in an abstract, homogeneous sense) and quantities of output.  Assuming that we can devise suitable variables representing the quantitative utilization of production factors as regressors, we would still need to devise a suitable dependent variable quantitatively measuring productivity/output.  Ideally, we are seeking to regress a flow measure of output (i.e. the volume of a particular commodity produced by a firm from a given production process over successive periods) against flow measures of factor inputs.  Given deficiencies in the availability of detailed output data from individual firms on given production processes, especially for firms involved in multiple, heterogeneous production processes at given facilities, it is unlikely that real assemblages of data would exactly correspond to the ideal theoretic assumptions on function forms.  Rather, actual estimations might incorporate information on value added from multiple production processes against stock estimations of labor and capital variables. 
             Again, proceeding from an assumption that we can adequately homogenize the production factors in order to relate abstract input quantities of land, labor, and capital to some dependent variable reflecting output quantities, either as a volume or a value-based variable, problems arise when systematic and/or stochastic unobserved sources of variation skew input-output relationships, raising standard errors from individual coefficient estimates.  Various econometric methodologies exist to correct for the presence of such variation (e.g. instrumental variables, fixed effects regressions).  The problem essentially consists of statistically endogenizing otherwise unobserved exogenous relationships through single or multiple stages of estimation.  For example, variations in managerial styles or other observable qualitative differences between production processes over time or across units of a multi-unit firm might account for variations in output quantities not accountable to factor productivity.  It is conceivable that some recourse to dichotomous/dummy variables might correct for the presence of output variation resulting from differences in managerial styles, logistical/inventory maintenance, or other patterns exogenous to the theoretic parameters of general equilibrium analysis.  Resolution of ambiguities in statistical estimations, in this respect, may demand detailed analysis of intangible, non-quantitative variations between production processes in order to comprehend how particular, persistent differences in the estimation of factor coefficients might arise.
              Beyond this, as a matter of time series analysis, we would need to ensure the stationarity of production data (i.e. that the temporal frame itself does not exert some influence on the estimation of the coefficient values).  In particular, assuming exogenously given technological parameters can be held constant over extended periods of time and that technologies allow for continuous, smooth, and, within a general equilibrium structure, instantaneous substitution between factors of production in response to changes in factor prices, variations in market conditions at given points in time should exert no influence over the estimation of the technological parameters conveyed within the production function.  Any correction in the utilization of labor and capital to account for increase in the relative prices for the production factors should be reflected in the marginal rates of technical substitution emanating from the production function.  As such, production functions estimated under Walrasian/Paretian general equilibrium assumptions should be wholly stationary, barring significant transformations in technology over the period estimated.  In the same manner, analyses of technological change, incorporated statistically into the estimation process, should enable researchers to render a time series estimation over an extended period stationary by accounting for technologically driven sources of temporal variation in factor productivities in the same way that hedonic pricing models operate to render time series analysis of consumer product markets stationary.
            Not seeking to understate the problems inherent in addressing these issues, a rigorous effort to analyze particular firms and particular production processes might succeed in developing models sensitive enough to account for divergent sources of endogenous variation, across multi-unit firms and over extended periods of time.  The larger problem is that, in order to truly estimate a production function rigorously consistent with Walrasian/Paretian theory, we would need to explicitly apply the assumptions of general equilibrium analysis and, thus, ensure, among other things, that the functional forms and coefficient values demonstrate linear homogeneity/constant returns to scale and continuous substitutability of production factors at each output level.  Testing such assumptions on real world production processes through variously constrained statistical regression procedures might result in inflated standard errors in the estimation of coefficient values if such processes typically operate under technologies inconsistent with Walrasian assumptions.  As such, it might be worth inquiring into the theoretic importance of such assumptions if we are strictly interested in defining a quantitative relationship between output quantities and quantities of input factors.
           As a preliminary consideration, what does the hypothetical relationship between divergent factor combinations look like at given output scales?  I posit "hypothetical" here insofar as any particular data point, representing a particular, ossified moment in time, can only convey the relationship between output and factor inputs for a single factor combination at the particular set of factor prices that prompted the selection of the combination in question.  Without a broader array of data points, no estimation can definitively convey the shape of an isoquant level curve for the same output scale with an alternate set of factor prices.  On the other hand, real data points, reflecting the actual production decisions of real firms, would certainly reflect a broader array of variables, determining the mix of factors selected in production and the output quantities demanded at multiple points in time.  Over time, technologies change and factor prices adjust for multifarious reasons sensitive to short run variations (e.g. financial market fluctuations affecting interest rates on savings/capital).  Absent data reflecting real factor substitution under divergent factor market compensation rates, holding all other variables at a given moment in time constant, we cannot estimate the actual shape of isoquant level curves to a production function.
            To illustrate the nature of the problem here, let us take a hypothetical firm producing a given commodity x1 with production data for a given set of production periods, denoted in two-dimensional labor/capital space in figure 29 with two sets of cost curves reflecting two different sets of factor market prices under a single level of total costs (TC').  If the output profile for such a firm over the larger period in question is essentially flat, then, given shifting factor market compensation rates (i.e. from Pl'/Pk' to Pl''/Pk''), we might assume that the two clusters of scatter plots reside on or around a single isoquant level curve, denoted in figure 29 as isoquant I1(x1), defined by a production function displaying all of the necessary features in Walrasian/Paretian general equilibrium theory.  I have drawn this isoquant as dashed line to emphasize that, in constrast to our factor market prices, the isoquant curve and its associated production function exist only hypothetically (even to our hypothetical firm!) as an object of Walrasian/Paretian theory.  Relatively higher labor costs (Pl'/Pk') command factor combinations that are relatively capital intensive and relatively higher capital costs (Pl''/Pk'') command combinations that are relatively labor intensive to maintain a common scale of ouput and total costs.  The ability of the firm to perform discrete substitutions of labor for capital and vice versa along its isoquant surfaces would enable it to realize, more or less accurately, profit maximizing/cost minimizing scales of output given each set of factor market compensation rates.  Presumably, if we had such data points of the sort displayed below, manipulated through a constrained regression method, we could arrive at an estimated production function that might trace smooth, continuous level curves for each output scale of the kind speculatively drawn. 

Figure 29: Scatterplot of Factor Combinations for a Firm Producing Commodity X1 at a Given Output Scale under Two Sets of Factor Market Prices 
The problem here is that, given a relatively static scale of output, we have information about the firm selecting a broadly labor intensive technology when capital costs are relatively high and a broadly capital intensive technology when labor costs are relatively high.  What we do not have is any information on intervening factor combinations located between the two clusters of data points along our hypothetical isoquant surface.  What happens for factor market compensation rates between those represented by our two cost curves?  In particular, is it possible that a clustering of data points for two broadly divergent factor intensities of production might reflect two discrete production technologies between which no intervening factor combinations can yield positive quantities of output?  Emphatically, such a situation is entirely conceivable.  A dichotomous choice between two sets of proportional factor combinations at at least one scale of output would have to be interpreted as a case of perfect substitutability in which the firm would be compelled to choose one or the other combination of factors, given a range of possible factor prices.  Such a situation would violate particular Walrasian/Paretian assumptions about profit maximization/cost minimization, at least insofar as we accept that firms should be able to select unique factor combinations that will objectively maximize profit subject to the firm's financing constraint and each prevailing set of factor market prices. 
         So what happens if we dispense with continuous substitutability between factor combinations as a feature of the Walrasian/Paretian production function?  Succinctly, if we are dealing with a discrete and finite number of technologically mandated factor combinations, firms would still be profit maximizing/cost minimizing production agents insofar they would continue to select the combination of production factors maximizing output relative to the firm's financing constraint.  However, the profit maximizing combination would cease to equalize the ratio of factor market compensation rates to the marginal rate of technical substitution, measuring the curvature of isoquant surfaces.  Instead of intersecting the production function at a point of tangency, the firm's isocost curves would either intersect at the vertices, defined by particular factor combinations, or converge with the isoquant surface for prolonged ranges, defining perfect substitutability and indifference between two discrete factor combinations at a particular factor price ratio. 
         To illustrate, let us examine our previous hypothetical case of a firm with two discrete clusterings of data points, one around a relatively capital intensive locus and the other around a relatively labor intensive locus.  Dispensing with our assumption that labor and capital should be continuously substitutable along a smooth isoquant surface, we can draw piecewise unit isoquants with vertices at each locus denoting the presence of two discrete technologies incorporating fixed proportions of each factor.  This situation is illustrated in figure 30, with three isocost curves denoting different factor market compensation rates under a common financial constraint. 

Figure 30: Production Process with Two Discrete Technologies Defined Along Divergent Expansion Paths with Fixed Factor Proportions

Elaborating, the firm with the production function represented in figure 30 should choose its available capital intensive production technology, with capital-labor ratio (k/l)', if the factor price ratio exceeds a threshold value (Pl/Pk)*.  It should select its available labor intensive technology, with capital-labor ratio (k/l)'', if the factor price ratio is less than (Pl/Pk)*.  At the threshold factor price ratio (Pl/Pk)*, the firm will be indifferent between its available technologies.  In this manner, the isocost curve with slope -(Pl/Pk)* is coterminous with the slope of the dashed isoquant surface I2 connecting factor combinations along the labor and capital intensive expansion paths.  Because both technologies will maximize profits/minimize costs at this threshold price ratio, the firm may select either technology.  If it chooses the labor intensive technology, then any subsequent increase in the market compensation rate for labor will result in selection of the capital intensive technology.  The opposite would be true of the capital intensive technology if it was initially selected and the compensation rate for capital increases.  The steeper isocost curve with slope -(Pl/Pk)' reaches its highest point of convergence with an isoquant surface at the vertex defined by the factor ratio (k/l)'.  The flatter isocost curve with slope -(Pl/Pk)'' reaches its highest point of convergence with an isoquant surface at the vertex defined by factor ratio (k/l)''.  At either of these corner solutions, any subsequent increases in the compensation rate of the less intensively used factor will not result in any substitution away from this factor - no alternative technologies exists that would allow the firm to continue to substitute more of the less costly factor for less of the more costly one. 

             Reiterating, the rigorous limitations on factor substitutability introduced in this example present a violation of general equilibrium assumptions regarding factor selection and compensation relative to the marginal productivity of each factor.  At each vertex, the factor combination constitutes a profit maximum for a range of factor price ratios.  As such, it stands to reason that the marginal products of each factor, as components in the marginal rate of technical substitution, can, at best, only constitute a limiting condition on the selection of each combination.  That is to say, if the marginal rate of technical substitution, measuring the slope of the isoquant, exactly equals the factor price ratio, then the firm will be indifferent between the two perfectly substitutable technological combinations.  If any other factor price ratio obtains, then the marginal rate of technical substitution can tell us nothing about the choice of factors.  This is only, strictly, a problem if we are attempting to relate factor compensation to marginal productivity, but such a principle remains at the heart of Walrasian/Paretian theory!

             Pragmatic-minded microeconomic analysts have developed the tools to deal with a finite set of efficient production technologies with fixed factor combinations and constant returns to scale.  In particular, given detailed production engineering analysis to establish a feasible set of all efficient factor combinations characterized by constant returns to scale, linear optimization techniques can establish the conditions under which one or more production technologies can be utilized to efficiently maximize revenues given a particular set of output market prices under specified technological constraints.  Such optimization methodologies, in turn, establish conditions for product exhaustion through which unknown factor market prices can be imputed from known output market price coefficients for specified technological processes in order to satisfy a zero profit condition.  On the other hand, such a methodology obviates the grounding of Walrasian/Paretian production theory in continuously differentiable production functions, from which we can extract marginal rates of technical substitutions derived from the marginal products of each factor.  Rather, the reliance of linear optimization on Austrian-inspired price imputation, at best, mimics Walrasian tâtonnement, if we assume that factor and output market price determination are actually, in fact, simultaneous and mutually constitutive, and, at worst, represents a unidirectional dictation of "higher-order" factor market prices from "lower-order" output market pricing (or vice versa, in a more Ricardian Classical vein).  Either way, the production process and the marginal productivity of individual factors becomes palpably disconnected from the determination of factor market compensation rates.  Even to the extent that we arrive at product exhaustion, we lose the potential normative consequences attendant to marginal productivity factor pricing except by virtue of interpretation.  Acknowledging, thus, that production processes with finite sets of efficient factor combinations can successfully be optimized to yield product exhaustion/a zero-profit condition, continuously differentiable production functions, with continuous substitutability of factors along smooth isoquant level curves, may not be strictly necessary to achieve the critical insights of Walrasian/Paretian general equilibrium economics.  

           Having dispensed entirely with the need for continuous substitutability of factors at given scales of output, what can we say about the need for constant returns to scale?  Again, the relevance of constant returns to scale involves the relationship between marginal productivity factor pricing and product exhaustion.  From our exposition of the Walrasian/Paretian theory of the firm, we know the implication that, under constant returns to scale, the average products of each factor of production are equal to their marginal products.  Consequently, if the firm compensates each unit of each factor in accordance with the marginal revenue product, measured as the marginal product times the output price, then it exactly achieves product exhaustion.  For this reason, if we assume that Walrasian/Paretian firms operate with production functions characterized by linear homogeneity/constant returns to scale, then the problem of scale can be reduced to harmonizing levels of financing under a given set of relative factor prices to determine the appropriate scale of outputs for each profit maximizing combination of factor inputs.  We need not inquire into the scalar profile/topography of Walrasian/Paretian production functions in multi-factor/output space because, under every expansion path for every factor combination along the surface of each isoquant level curve characterized by continuous substitutability, constant returns to scale will obtain - our expansion paths will always exist as 45 degree lines in multi-factor/output space.  

              As such, to the extent that Walrasian/Paretian firms instantaneously adjust to every change in household consumer demand and household factor supply, we remain justified in drawing flat, perfectly elastic output market supply curves and factor market demand curves -  adjustment of output supplies and factor demands that do not result in a shift between production technologies/expansion paths to account for changes in relative factor market prices or consumer preferences (i.e. the variables controlled by households) will not impact the cost of production per unit of output.  Moreover, even to the extent that we dispense with the notion of continuous substitutability/smooth isoquant surfaces, if a discrete set of production technologies, each with constant returns, exists, then, under imputed factor pricing/linear optimization, firms can achieve product exhaustion and, at the market level, output supply and factor demand will continue to demonstrate perfect elasticity with respect to changes in household consumer demand and factor supply, respectively.  In either case, increases or decreases in output that do not force firms to shift between technologies will never result in either increasing or decreasing costs per unit of output for firms or markets, as a whole.  Finally, insofar as any change in technologies employed by particular industries, within the broader structure of a general equilibrium economy may result in changes to the mix of outputs supplied and factor inputs demanded, the universality of constant returns with every technology/optimal factor combination means that any technological changes will simply place firms on new expansion paths at which they will achieve product exhaustion through a new distribution of factor payments under marginal productivity factor pricing.  At the market level, this may result in upward or downward shifts in output supply and factor demand functions, but these functions will remain perfectly elastic with respect to changes in household consumer demand and factor supply respectively.  

           With all this in mind, constant returns constitutes a linchpin to general equilibrium economics, as a whole, and to marginal productivity factor pricing, in particular.  If we dispense with constant returns to scale as a characteristic of Walrasian/Paretian production functions, then any adjustment of consumer demand or factor supply may generate non-linear changes in output quantities, complicating the larger process of tâtonnement.  Certain outputs may experience increasing per unit costs, while others may generate cost savings, unrealized at lower production quantities, all of which would have to enter into the broader determination of equilibrating price vectors.  With regard to the determination of output prices for various levels of output, we would be unable to confidently draw perfectly elastic output market supply and factor market demand curves because firms might face increasing costs as they increase factors proportionately along their expansion paths.  Certain output markets, for example, might experience increasing costs while others see costs diminish with increases in the scale of industry output, assuming again, as a matter of general equilibrium theory, that all firms enjoy perfect information on available technological methods to minimize cost/maximize profit.  In sum, the calculation of output and factor market pricing under general equilibrium would become exceptionally complicated if we were unable to assume constant returns to scale as a general condition for all firms in all industries.   

           On the other hand, as a distributional matter, marginal productivity factor pricing at fixed factor combinations will not achieve product exhaustion if constant returns do not obtain.  Emphatically, if a firm experiences decreasing returns to scale at its profit maximizing fixed factor ratio on its expansion path, then, if it compensates each factor at its marginal revenue productivity, it will obtain a surplus over and above total factor compensation.  Conversely, firms operating with production functions on a range of increasing returns will encounter a deficit in revenues from paying each of the factors according to their marginal revenue products.  As such, dispensing with the assumption of constant returns and the concomitant linearly homogeneous production function, at best, complicates the process of locating a market clearing price vector and, at worst, forces Walrasian/Paretian theory to dispense with marginal productivity factor pricing, product exhaustion, and the zero profit condition, at least as a technical characteristic emerging from the function form of production functions.  

          In this regard, is it realistic to assume that real firms across all industries within real market economies operate with production functions characterized by linear homogeneity/constant returns to scale?  Or, again approaching the problem metaphorically from an econometric standpoint, would it be possible to gloss over the presence of increasing or decreasing returns in the scalar profile of a real firm by means of statistical manipulation, raising standard errors in order to artificially establish a constant return expansion path through multi-factor/output space?  Let us say that we have a firm performing a single production process at which, over an extended period of time, it has faced fairly static relative factor prices.  Given the set of factor market compensation rates that it faces and its set of available technologies, it selects a relatively capital intensive technology.  Figure 31 portrays a collection of data points of factor combinations for selected periods, with three solid isocost lines for given levels of the firm's financing constraint.  Absent data on possible factor combinations under other factor price ratios and acknowledging our previous critique of continuous substitutability, we omit any hypothetical isoquant level curves.

Figure 31: Linear Expansion Path with Data Points in Two-Dimensional/Two-Factor Space       

Acknowledging that expansion path EP* is linear in two-dimensional/two-factor space, defining a fixed factor ratio for the firm under relatively constant factor compensation rates, we have to come to terms with variations in output level from the various data points.  In these regards, we are interested in whether the firm encounters constant returns to scale, per Walrasian/Paretian assumptions, or whether it faces some variation on increasing or decreasing returns to scale.  Along these lines, figure 32 is presented as a plausible representation of output variations within our hypothetical set of data points, with two possible alternative expansion paths that might be generated by econometric estimations under divergent estimation processes: EPcr drawn as a homogeneous function at a 45 degree angle from the origin with constant returns to scale, and EPdr drawn as a non-linear/non-homogeneous function with global decreasing returns to scale.  

Figure 32: Homogeneous Constant Return and Non-Homogeneous Decreasing Return Expansion Paths with Data Points in Output/Fixed-Factor Ratio Space

As an initial point of elaboration, the intention of figure 32 is, again, to extrapolate the curvature/topography of the expansion path that would, in two-dimensional/two-factor space, otherwise be indicated by the intervals between isoquant level surfaces.  In the case of the constant returns, every multiplicative scaling of the production factors will generate a proportional scaling of output and, thus, the expansion path must be a 45 degree ray from the origin for all combinations with our fixed-factor ratio, expansion path EPcr.  Expansion path EPdr, by contrast, follows a likely asymptotic trajectory in which initially high marginal returns to the factor combination perpetually decline with increases in scale.  I posit some arbitrary data point A, with factor combination (aki,ali) as a convergent scale of output between linear and non-linear estimations of the two expansion paths. As I have presented this collection of hypothetical data points and possible linear and non-linear estimations, my purpose has been to argue that there may be good reasons to insist that, if we are after a valid specification of the relationship between factor combinations and output quantities as we increase the scale of production, then we may not want to accept preconditions that mandate constant returns either for reasons of arithmetic simplicity or analytical utility.       

         As with the previous examples in this section, figures 31 and 32 command as much scientific validity as scribble on a classroom chalkboard.  My purposes in pursuing an econometric metaphor in advancing a critique of the Walrasian/Paretian production function has, again, been wholly rhetorical. The point of such a hypothetical drawing of estimated regression lines from fabricated but, conceivably, plausible data points is to argue that the notion of constant returns to scale, however essential to Walrasian/Paretian general equilibrium theorizations, may be difficult to substantiate in data from real firms.  If we can accept that real production data incorporates multifarious sources of variation that may elude the capacities of the best econometric specialists to model within a regression procedure, then it is conceivable that production data might be more effectively modeled by non-linear estimations in which we would have to dispense entirely with the assumption of constant returns to scale.  If this is the case, however, then, as argued, we encounter (not insurmountable) obstacles to the viability of general equilibrium theorization per se. 

          Concluding these reflections on the problem of constant returns to scale as a defining feature of the Walrasian/Paretian production, I have two critical comments that I want to elaborate.  First, it might be the case that the specification of a production function characterized by linear homogeneity/constant returns is not as critical to Walrasian/Paretian theorizations of the firm as the theory itself seems to suggest.  Second, maybe my hypothetical/rhetorical econometric evaluation cannot adequately account for the actual functional form operative for real firms.  

          In regard to the first of these critiques, the mathematical tractability of general equilibrium analysis is at stake in the linear homogeneity of production functions.  If we divest of constant returns, then we still might have an edifice of logical principles anchored in the notion of tâtonnement, with an explicable concept of a zero-profit condition arising from competition, but the mathematical architecture that brings all of the pieces together into a consistent production-side model, sandwiched between household utility maximization problems, would have evaporated.  For certain other approaches in Neoclassical theory, this would never be a concern.  In particular, the Austrians, for whom the entire project of general equilibrium analysis is both superfluous and vulnerable to logical and mathematical criticism, output market competition alone is necessary and sufficient to constitute a zero-profit condition for firms, and imputed factor market prices can mathematically generate product exhaustion without any recourse to a production function with all of the Walrasian/Paretian conditionalities.  In this light, linear homogeneity appears as an analytical prejudice unique to Walrasian/Paretian theory, with its need to rationally anchor its imagery of a fully integrated, decentralized market economy on functions that can consistently yield the desired outcomes of the theory: unique, stable equilibria and mechanisms through which changes in underlying household preferences will instantaneously be conveyed throughout a smoothly readjusting system.  For Walrasian, Neo-Walrasian, and Paretian theorists, the idea that we could dispense with linearly homogeneous production functions would, thus, be anathema.  It is only when we diverge from the theoretic straitjacket of general equilibrium thinking that we realize the unnecessarily restrictive character of its assumptions.

          With regard to the metaphorical allusion to econometric estimation employed in this section, we have to separate two distinct critical arguments.  On the one hand, no econometric or statistical analysis can ever account for possible outcomes not represented within a data set.  That is to say, it might be the case that real firms, within real market contexts, operate with production processes that could be patterned through rigorous linear homogeneity and continuous factor substitutability, but no data set will ever fully account for either of these conditions because the breadth of data necessary to objectively validate them are unlikely to exist.  The best that we can do is test certain hypotheses about parameters based on Walrasian/Paretian theoretic assumptions regarding constant returns and factor substitutability, asking simply how plausible it is that Walrasian/Paretian assumptions prevail in the operations of a given production process.  

         On the other hand, maybe the idea of econometrically estimating production parameters from data sets of production processes misinterprets the significance and meaning of the Walrasian/Paretian production function.  In this regard, criticisms have already been advanced by theorists on the margins of the Neoclassical tradition like Joan Robinson that we need to differentiate between ex ante and ex post specifications of production functions.  If production functions are conceived as the implicit underlying operative mechanism determining output quantities from specific factor combinations, then, of course, we should always be capable of estimating production functions econometrically from production data that will strictly and accurately explain how firms take particular factor inputs and generate commodity outputs.  If, on the other hand, a production function is simply a conventional, heuristic device to organize planning by firms to frame expectations of future production processes around technically derived formulae commanding a certain threshold of confidence on the accuracy of estimated outputs, then the project of econometrically estimating production functions for individual production processes may, at least in some degree, be ill-founded.  In the very least, we would have to dispense with the notion that a production function could ever be viewed as a determinate mechanism.  Such a reinterpretation of the production function would, however, extract us from the rationally constricted, determinate world of Walrasian/Paretian thinking and plant us in the more experiential, empirically-grounded, and probabilistic terrain of Marshallian theory, perhaps invested with a generous set of Keynesian assumptions on the inherent uncertainty of future outcomes.  Ex ante production functions are guideposts for firms and investors lacking any place in a theoretic environment that lacks the space for uncertainty.