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.   

A Pure Neoclassical Theory of the Firm VIII: Critique II (Microeconomics)

The Questions of Temporality and Spatiality

Proceeding beyond the limited abstract terms by which we have heretofore leveled a critique of Walrasian/Paretian conceptions of the firm in a general equilibrium economy, we need to approach a critical ontological problem concerning the dimensionality of economic processes.  Is it fair to characterize a general equilibrium economy as either timeless or spaceless? 
           In order to answer this question, we need to, first, clarify exactly what we have advanced as a portrait of general equilibrium.  However we specify the particular mechanisms of price adjustment in a general equilibrium economy, we have assumed that the conditions exist to ensure that such an economy exists in permanent and continuous equilibrium.  That is to say, any changes in factor supplies, production technologies, household preferences, etc. must induce an instantaneous and smooth readjustment of relative prices.  The instantaneous character of the process of tâtonnement  (i.e. bargaining between households over prices across all markets) implies that the process is, itself, continuous.  Households never emerge from price negotiations with other households.  Tâtonnement goes on and on and on forever.  In this sense, the identification of any particular, momentary equilibrium outcome constitutes a singular static outcome in the never ending adjustment and readjustment of prices to guarantee that all markets clear at every moment in time.  Therefore, we do not need to inquire into the temporal structure of market price determination.  Every moment in time is an instance of tâtonnement. 
           At the outset, I want to acknowledge that the theory mirrors real economic processes here in certain ways.  If we acknowledge that real households engage in recurring and temporally overlapping evaluations of final commodity and factor prices in relation to their own preferences and that these evaluations collectively determine market pricing outcomes (i.e. forcing firms to periodically readjust prices in relation to shifts in final consumer demand or factor supply), then we would have to conclude that, in the aggregate, new offers and counter-offers in commodity and factor pricing are repetitively being issued across all markets in ways that suggest that a continuous, non-stop auction metaphor might not be far from the truth.  In this sense, general equilibrium theory merely simplifies the overarching complexities of discontinuous readjustment processes between subsets of households and firms by positing that the entire process can be treated as if the readjustment process was actually continuous for the entire integrated system.  Again, as I suggested in the previous section of this critique, such a result can be entirely unsatisfying to economists proceeding from a decidely more empiricist/positivist epistemological standpoint, for whom the analysis of real decisions by households must take temporal variations (e.g. seasonal shifts in demand for certain goods and services, long term factor supply contracting, etc.) into account. 
            The particular focus of this larger document on firms and the economics of production raises more pointed concerns with regard to time, introducing problems that extend beyond the continuous nature of  tâtonnement to muddy the larger connection of Walrasian/Paretian commodity producing entities to firms in real economies.  As argued, Walrasian/Paretian firms exist as contingent assemblages of production factors in which individual households continue to supply their production factors only to the extent that doing so achieves utility maximizing outcomes.  Unlike real firms, Walrasian/Paretian firms lack tangible property in infrastructure and fixed capital.  If, in these terms, we can asssume that infrastructure (e.g. roads, rail networks, buildings/facilities) and fixed capital (durable machinery) exist, then we must simultaneously divorce the ownership of such physical components from the purely contingent and transitory existence of firms.  We would, thus, need to envision of world of fixed capital and infrastructure in place, where firms rent fixed capital, rent labor, purchase circulating raw materials, and rent the money capital to pay for all of the other factors, subject to reimbursement in accordance with the marginal revenue products for each of the production factors (i.e. conforming to product exhaustion).  This imagery effectively separates the notion of capital accumulation from the existence of firms - firms do not accumulate capital, households do by virtue of their deferred consumption of income. 
               Proceeding from this imagery of spatially displaced/free-floating firms (an imagery of which I will subsequently pursue a spatially-oriented criticism!), we have a theoretic basis against which to approach the temporal framework of production.  Insofar as firms do not accumulate capital but only rent capital from households, there is no basis for considering temporal frameworks within which firms shift between diverse scales of production by accumulating new capital.  Firms do not need time to invest in new fixed capital or build new plant/infrastructure because firms do not undertake such investments.  If accumulated masses of fixed capital exist to be rented from households by firms and rates of interest are adequate to reimburse households for use of their fixed capital by firms, then firms can always achieve given scales of output, supported by household demand for final commodities, at any given moment in time.  If an existing production facility or quantity of fixed capital (machinery) is inadequate to produce quantities of output desired by consumers, then a firm can always move to larger production facilities or rent new fixed capital to produce desired output quantities, increasing the total expenditure on factors of production but simultaneously increasing total revenues by an equivalent quantity.
             Rounding out the implications arising from our assumptions about firms in general equilibrium economies, if firms are continuously dealing with homothetic production functions, with constant returns to scale and constant average total costs per unit of output, then any change in scale should be effected smoothly, with compositions of labor and capital corresponding with profit maximizing marginal rates of technical substitution.  In this manner, firms never have to depart from their expansion paths as they either increase or decrease their scales of production.  Appropriate infrastructures and quantities of fixed capital must always exist in place for firms to rent in order to produce quantities of output that maximize profits/minimize costs.  This continuous availability of infrastructures and fixed capital for firms enables us to argue that there is no differentiation between a short run and a long run in the Walrasian/Paretian theory of the firm.  By contrast, we will find that such a differentiation exists for firms within Marshallian theory simply because Marshallian firms have to invest in fixed capital and infrastructures to alter their scales of operation, and such investments are costly in terms of time - new manufacturing plants or other facilities cannot be built overnight!
             Thus, our assumptions regarding the transitory existence of firms and the homotheticity of production functions lead us to a world in which households continuously invest in new fixed capital and new infrastructures, available for rent by firms in accordance with their respective scalar requirements for fixed capital and infrastructure.  Such investment processes (suitably adjusted to account for depreciation and technological obsolescence) can be patterned through a variety of different economic growth models (e.g. the Solow model), conforming to all of our assumptions about general equilibrium economies.  Growth models of this kind perform the function of a dynamical bridge from the static mechanism of tâtonnement to the multi-scalar shifting of macroeconomic production possibilities and sectoral responses to changing household preferences over time.  And this temporal consideration is critical.  General equilibrium economies change because households invest savings into new capital and new infrastructure.  In the absence of such investments (and in the absence of either capital depreciation, technological change, or changes in household preferences) a general equilibrium economy would operate as an infinitely reproducing, never ending repetition of identical market outcomes. 
           At this point, I want to reassert that, approaching from a strictly rationalist standpoint, general equilibrium theorizations are not identical to the real economies that they attempt to mirror through theory.  Rather, they simply posit abstract models in an attempt to generate predictions on the workings of market systems as if the system functioned in accordance with their abstract models.  In these terms, there is absolute nothing wrong with the timeless character of static equilibria in a general equilibrium system as long as models of this kind continue to produce relevant insights into the functioning of real market systems, and I am not going to contest that, in certain respects, they do.  On the other hand, we have to confront the peculiar ontological characteristics of the market systems theorized by Walrasian/Paretian economics. 
           Through macroeconomic growth theory, the logic of capital accumulation attains a sort of abstract existence separated from particular production processes or aggregate demand for outputs of final commodities, per se.  In some respect, household investment in capital must be wholly speculative.  On the other hand, within a general equilibrium system, such speculation has to be supported by demands for new capital and infrastructure by firms.  Any income saved must be capital invested to produce larger quantities goods and services that will be instantaneously demanded by households as increased consumption possibilities.  Alternately, if we assume some level of depreciation, there must be some basic level of investment in new capital and infrastructure to replace depreciating implements in order to preserve a constant level of consumption possibilities over time.  But here we run into the problem that investment in capital and infrastructure take time to come on line as productive assets. 
            As such, our Walrasian/Paretian theory of economic growth (introduced to harmonize our assumptions on changes in scale by firms with the static/timeless nature of tâtonnement) runs squarely into a quintessentially Keynesian concern: expectations about the future under conditions of uncertainty.  Why would utility maximizing households undertake investments in fixed capital and infrastructure if these investments enjoyed an inherently uncertain rate of return?  This question transcends the unique boundaries of Walrasian/Paretian general equilibrium theorization to become a universal concern in economic discourse.  Moreover, theorization of subjective probability distributions on future income possibilities from investments notwithstanding, there is no meaningful way to answer this question.   
           To summarize where we have ventured to this point in considering the role of time/timelessness in Walrasian/Paretian general equilibrium theorizations, we have argued that the process of tâtonnement is continuous and, thus, timeless/static.  At any moment in time, a general equilibrium system must generate a vector of relative prices that will bring all markets within the system into equilibrium.  If such a system is not, however, to be a truly stationary system (i.e. a permanently unchanging economy), then there must be some mechanisms by which changes in household preferences for final commodities and/or changes in production technologies can circulate throughout the system.  Insofar as our theory of the firm precludes the possibility that firms occupy a role as anything other than a means for efficient production of commodities, households must be the agents enacting such shifts in production, including overall increases in consumption possibilities over time.  Thus, the role of households in achieving smooth transitions in production over time leads us to the problem of capital accumulation and speculative investment with uncertain future returns.  It almost certainly, likewise, necessitates that speculative investments in capital and infrastructures by households may result in the accumulation of fixed capital and infrastructures that, at the time of their completion, remain totally idle, awaiting conditions in which they can be inserted into production processes at rates of return satisfactory to their household owners (rates that may never arise!).  That is to say, the household owners of physical capital invest to accumulate capital for the sake of accumulation, without any guarantee that their investment will yield a return, in order to ensure that the larger system will have adequate reserves of capital to smoothly transition to higher quantities of output or to maintain existing levels of output under depreciation.  For a theory that assumes that households are pure, self-centered utility maximizing agents, the idea that households should invest speculatively to secure the needs of the larger system seems just a little quixotic!  I will revisit this idea in this critique when I directly confront capital as a factor of production. 
          The second dimension of interest in this section is space/spatiality.  Like time, spatiality implicates itself in every facet of commodity production and market exchange.  If commodity production has a time and, hence, a temporal relationship (e.g. a unique and unalterable sequencing) to processes of exchange and household consumption, it must simultaneously have a place and a spatial relationship (e.g. colocation) to exchange and consumption.  As I have argued in this section, the notion that firms should be able to continuously readjust their scales of production in response to changes in demand requires that reserves of fixed capital and infrastructure be readily available in place.  This necessarily implies the existence of spatial distribution/dispersion of fixed capital and infrastructure, defining a spatial relationship between the various assets utilized by a firm at a given moment in time (i.e. the spatiality of firms renting multiple production facilities at different locations).  Moreover, contemporary innovations in online marketing notwithstanding, exchange processes must have both temporal and spatial dimensions that can be identified and related temporally and spatially to other processes.  If I purchase a textbook online with my laptop in a coffee shop in Massachusetts from a textbook supplier in Oregon and receive the textbook through the U.S. Postal Service four days later, then we have a more complicated temporal and spatial distribution to the process than would have obtained had I purchased the same textbook from a bookstore down the street.  However, as a matter of ontological necessity, space and time are universally implicated in all economic processes, no less in this case than in any other. 
         We know from our analysis above that Walrasian/Paretian general equilibrium theory deals with time by denying its influence on theoretic structures.  Rather, the theory imposes on market exchange and commodity production a peculiar timelessness, manifest in the capacity of firms to perfectly adjust to changes in commodity demand, factor supply, or technology instantaneously.  This imposition has particular consequences, most emphatically evident in the necessity that households engage in continuous speculative investments in fixed capital and infrastructure at uncertain future rates of return.  I will argue now that Walrasian/Paretian theory does not deny (or contort) spatiality as much as it ignores spatiality as a problem outright.  At no point in our analysis of the firm, for example, did I elaborate on the problem of transportation costs under circumstances where sites of commodity production are physically/geographically distinct from points of exchange and/or consumption.  Such considerations might serve to illuminate important decisions made in the operation of real firms.  On the other hand, I will argue that any considerations on the location of production facilities and/or transportation costs holds the potential to undermine key assumptions on competition in the functioning of a general equilibrium economy. 
            Specifically, the relative proximity of competing firms to sites of market exchange must necessarily impose differential transportation costs, generating centripetal pressures in the location of production facilities relative to key markets.  Such a conclusion assumes, importantly, that market exchange articulates a particular, centralized geography.  It, further, assumes that the average total cost of transportation services consumed by firms remains constant per unit distance as the distance traveled by commodities from production facilities to sites of market exchange increases.  If these conditions hold, then, given the absolute finitude of land-space, firms must compete for spaces closer to relevant market sites.  In turn, firms closer to markets must enjoy lower average total costs, taking transportation costs into account, than firms farther away.  In a system predicated on the principle of perfect competition, firms located at greater distances from given markets must, therefore, be competed away.  In the limit as firms select locations at infinitesimally small distances from market sites, limitations on the availability of space must undermine competition, thereby enabling firms to exercise market power in determining commodity prices. 
             Thankfully, there is a way out of this quandry.  Insofar as land-space exists as a factor of production owned and rented out by households and the marginal utility of holding land-space for divergent uses in proximity to market sites diminishes with distance from such places, it must be the case that households will charge higher rents closer to markets.  Thus, firms that choose to minimize their transportation costs by locating closer to a relevant market site will encounter higher land rents eliminating their cost advantages in relation to firms that locate at greater distances from markets.  As such, the centripetal pull of transportation cost minimization must be counterbalanced by a centrifugal push of differential land rent minimization, ensuring that firms will be indifferent in making location decisions.  We can formally pattern such a relationship between land rent and transportation cost for an economic space delineated by a even circular plane with a market site at the center by means of a von Thünen model, named after an early Nineteenth century German pioneer in geographical economic analysis.  This model is elaborated in figure 27.

   Figure 27: Von Thünen-type model for land rent/transportation cost relationship 
Elaborating in reference to the general equilibrium theorization of firms that we have developed so far, this model posits that, for some arbitrarily large economic space with a single basin of attraction for market activity, land rents should decline monotonically as we approach the economy's periphery.  By contrast, transportation costs should be expected to rise monotonically over the same interval.  Assuming that net revenues for all firms are adequate to compensate labor and capital in accordance with their marginal productivities irrespective of locational decisions by each firm, the model defines net revenues to the exclusion of factor payments for land.  Thus, we arrive at product exhaustion via the additional step of accounting for land rent and transportation costs.  That is to say, firms will always arrive at product exhaustion/zero profit.  Our land rent model simply explains the ratio of transportation costs to land rent that each firm will encounter as it selects a spatial site for production such that each firm will be perfectly indifferent in selecting locations because the balance of land rent and transportation expenditures will exactly exhaust net revenues after labor and capital have been compensated.  As such, land rent is derived as a residual after we have accounted for the cost of transportation from each distance within our monocentric economy. 
       Consequently, if we assume that all markets operate with such a spatial pull-push mechanism, then we can ignore the influence of spatiality as a determinant of market pricing and a potential source undermining perfect competition.  Any advantageous locational decision made by a firm in relation to its targeted markets will be eliminated by the countervailing imposition of land costs, bringing us right back to product exhaustion.  On the other hand, it is one thing to assume, within a theoretic model, that geography/spatiality will exert no influence on relative prices.  It is another thing to actually articulate the arguments that might lead to such a result and lay bare the assumptions against which such arguments might seem to mirror economic reality.  That is to say, if economists want to conclude that space does not matter, then we should always provide a persuasive explanation to justify our conclusion. 
              

A Pure Neoclassical Theory of the Firm VII: Critique I (Microeconomics)

Critique of a Pure Walrasian/Paretian Theory of the Firm
A Caricature of a Caricature
The theoretic matter presented in this document should have conveyed an overwhelming impression of disconnection from the everyday economic reality of firms.  Succinctly, what we have considered here has been the theoretic construction of an abstract economy, within which certain tendencies of economic behavior (i.e. in the production and distribution/exchange of relatively scarce goods, services, and factors of production) have been accentuated to the detriment of other behavioral patterns.  In this manner, the final consumption demands of households constitute the functional rationale for the workings of the entire system and the profit-driven capitalist motivations of entrepreneurs fall entirely out of the structure of the theory.  Such a theory enables us to pose the production and distribution of goods and services as a collective action problem concerning coordination of many, many households contributing labor and capital to the collective endeavors in commodity production that we label firms.  It emphatically seeks to tell us that a market economy is a vast collective endeavor resting on a base of independent households of sovereign, utility maximizing agents. 
          In fact, the theoretic/arithmetic processual counterbalance to utility maximization by households, profit maximization/cost minimization by firms, conveys a sterile arithmetic formality.  Profit maximization simply constitutes an intermediary step, sandwiched between two household utility maximization procedures (i.e. in factor market supply and final commodity demand), explaining why fully integrated free market economies produce exactly what households desire to consume at the least possible cost for their production, in terms of factor resources spent and foregone opportunities for increased consumption.  That is to say, the firm, as an agent, typifies efficient production arising through the decentralized, market-oriented cooperation between households.  Beyond this, the firm does not need to possess an independent existence/reality, and so it does not. 
          Such abstractions from the real of everyday economic processes, fulfill the larger epistemological goals of Walrasian/Paretian theory, as I understand them.  Emphatically, they reduce the real of economics to the sovereign choices of households in regard to factor supply and final commodity demand.  In the process, they reduce the firm to an intermediary formality.  Within the larger theory, we might as well write the firm out entirely, insofar as the firm is simply a way for collections of households to produce things they want that they would otherwise be unable to produce without the help of other households.  In this sense, the theory seeks to rationally condense the truth about economic reality by theoretically carving away the appearance that firms possess an existence independent of households.  In turn, by reducing economics to the essential truths of household utility maximization, Walrasian/Paretian theory shapes the way that real economic agents view market systems in order to reinforce a particular body of normative economics, defining the appropriateness of certain social policies addressing or otherwise impacting the functioning of markets. 
          With this in mind, I am prepared to argue that my particular presentation of a theory of the firm in a general equilibrium economy is unique for some of the assumptions that I make in order to enforce a particular understanding of Walrasian/Paretian theory.  From the moment that Léon Walras published his Éléments d’économie politique pure, ou la théorie de la richesse sociale (Elements of Pure Economics, or the Theory of Social Wealth) in 1873, there have been diverse expositions of general equilibrium thinking, most far more sophisticated and compelling in their accounts on the nature of equilibrium than the theory I have presented here.  In particular, the version of general equilibrium theory developed in the 1950s by American economist Kenneth Arrow and French economist Gérard Debreu has been extremely influential in defining the theoretic conditions for the existence, optimality, and stability of equilibrium.  The issues raised by Arrow-Debreu equilibrium extend far beyond the limited terms of my treatment of firms in a general equilibrium economy.  Rather, my limited intentions in this document concern the situating of firms in relation to utility maximizing household agents.  In this respect, I hope that I adequately make the point, through my overly emphatic argumentation, that the existence of firms is a formality within an economy that operates strictly to facilitate the circulation of goods and services between households to achieve a more favorable distribution of articles of consumption.  For this reason, general equilibrium theorists in the (Neo-)Walrasian/Paretian tradition appear to struggle, at least in part, to find meaningful ways to introduce firms and production into their general equilibrium theorizations without simultaneously transforming/truncating the centrality of exchange and of utility maximization by households. 
            In my view, the main sources of originality in my approach to firms arises from my emphasis of the interconnected themes of timelessness (continuous negotiation of equilibrium and instantaneous responses to changes in household preferences by firms) and perfect elasticities of final commodity supply and factor demand with respect to changes in relative prices by firms.  In regard to my particular effort to decisively subordinate the firm to the overall collective direction of an equilibrium economy by households, I consider these two outcomes to arise necessarily from the theoretic demands of general equilibrium.  Moreover, they decisively differentiate firms within the framework of a general equilibrium economy from Marshallian firms, operating under a wide range of Neoclassical theoretic assumptions but otherwise unconstrained by the particular arthmetic and logical demands of a fully integrated general equilibrium economy.  In this respect, if Walrasian/Paretian general equilibrium theorizations, in general, constitute a caricature of the way real economies operate, then the theorization presented here might constitute a caricature of a caricature, intended to accentuate the tendencies that I find latent within Walrasian/Paretian theory with respect to firms and production.

Epistemology and the Nature of Theoretic Abstraction

Having thus far advanced the argument that theories of general equilibrium are irreducible to the economic reality of firms, this critique sets forth a problem that demands some explanation.  Specifically, what purpose does theory serve if it does not produce a rigorously faithful description of the material reality of the world?  The issues involved here are sufficiently broad to justify book length explanations on the role of theory as an adjunct to the broader subject of epistemology, the theory of knowledge/knowing.  Thankfully, I am not going to be advancing such an expansive discursus here!  However, I do want to advance a particular set of epistemological propositions, intended to clarify the particular approach that will structure our larger study of microeconomics.
           To start out, I want to define a critical differentiation between two paradigms of epistemological conventions on the possibility of knowledge: realism and performativity.  This dichotomy parallels, in a broader cultural context, the distinction between modernism and post-modernism.  Moreover, realism subsumes a range of divergent approaches to the relationship between theorization and empirical analysis.  The key issue concerns the capacity to arrive at objective truths about material processes within the world. 
          Realist approaches to theory and analysis, in some sense, accept that it is innately possible for theorists and analysts of material processes (e.g. market exchange) to derive truths that can explain these processes and situate such processes objectively within the world.  Moreover, such truths must be inter-subjectively valid (i.e. perspectives do not matter - a truth is a truth from anyone's perspective).  Truths penetrate the universalistic character of reality.  In this regard, theoretic and analytic processes are quintessentially quests to find the truth about material existence. 
           Some realists approach the objects of their theorization/analysis with an assumption that they can realize the truth by subjecting their objects to observation by means of the senses, suitably augmented by technological/diagnostic media.  They amass information from sight, sound, smell, taste, and touch, and assemble these observations to derive hypotheses about the facts/phenomena to which their senses have alluded.  Then they set about testing these hypotheses with the aim of disproving what their senses have told them through rigorous investigation of new sets of observations.  If they fail to disprove their hypotheses, then they codify these hypotheses as theories concerning the objects of their scientific investigations.  We can label this subset of realists empiricists or positivists. 
           Other realists regard the empircists and their scientific method as an ill founded methodology to derive objective truths about the world.  These theorists embody a skepticism in regard to the capacity of the scientific method to generate sufficiently exhaustive sets of factual information against which hypotheses can be tested, on the one hand, and doubt that the scientific method can adequately isolate the causal mechanisms for the phenomena theorists and analysts observe and hypothesize, on the other hand.  In these terms, they argue that the scientific method both lacks generality/applicability to wider ranges of phenomena beyond the limited boundaries of individual experiments, and occludes valid, essential cause-and-effect relationships by saturating experimental contexts with inessential phenomena.  In order to avoid these divergent problems, such critics of empiricism argue that we can only arrive at the truth by reasoning introspectively/intuitively, applying formal logical and unbiased arithmetic rules, to carve the truth of particular phenomena away from the material complexities and backgrounds of inessential relationships within which these phenomena are situated.  Such thinkers may not trust their eyes or ears to deliver to them the truth, but they believe fervently in the sovereign capacity of the human mind to illuminate the overwhelming complexities characterizing the mysteries of the universe.  We can label these realists rationalists.
(Two of my theoretic mentors have a more detailed and informative account on these realist epistemologies, substantially in accord with my own. Read Resnick, Stephen A. and Richard D. Wolff (1987), Knowledge and Class: A Marxian Critique of Political Economy, 30-33.  Chicago: University of Chicago Press.)
            By contrast, the spectrum of approaches to theorization/analysis that I label performative discounts the capacity of theorists and analysts to realize objective/inter-subjective truths about material existence.  For various reasons, performative epistemologies acknowledge that neither the scientific method nor rational introspective inquiry is capable producing objective accounts on material reality.  In contrast to expositors of realist epistemological foundations, performative theorists consider theory and analysis to be processes contained within the broader structure of material existence.  In this manner, theorists actively participate, by means of their theorizations and analyses, in the way that the objects of their theories are both understood and lived.  By reshaping the way some particular material process is understood, theorists reshape the practices of agents engaged within the process.  Thus, theory does not merely describe material reality; it actively seeks to perform material reality and, perhaps, transform it by transforming the way it is understood. 
            To draw a comparison with realist epistemologies, performative approaches may vary widely in their understandings of the role of the theorist/analyst relative to the objects of theory/analysis.  Some performative theorists retain a strong connection to the belief that objective truths exist and that the purpose of theory and analysis is to uncover them.  For example, various theorists within the loose community identified with Actor Network Theory (ANT), some of whom I plan to acknowledge further within this critique, define their epistemological approach as performative and acknowledge that theory constitutes a moment of agency contained within the broader assemblages of human and non-human agency that form actor networks.  The epistemological frameworks from which ANT theorists proceed is, therefore, founded on a theory of material being/ontology that can be characterized as relentlessly constructivist (material processes are the outcomes of piecewise assemblages/constructions of diverse, spatially and temporally divergent moments of agency).  However, these theorists retain a highly skeptical belief in the objectivity of science, holding implicitly or explicitly that the purpose of theory is to derive objective truths about material existence even if such truths never actually crystalize into matters of fact (Latour, Bruno (2005), Reassembling the Social, 87-120.  New York: Oxford University Press.).  In this sense, the idea of performativity adopted by these theorists rightly can be characterized as straddling a boundary between empiricist realism and the performative embeddedness of the theoretic process as a participant within the construction of material reality.
           Like ANT, the epistemological framework embodied in my approach to theory is founded on a connection to ontology - all performative epistemologies acknowledge that theory/analysis cannot be separated from its object in a way that renders theory/analysis neutral, impartial, and wholly objective.  Moreover, in certain ways, the constructivist network metaphors that characterize ANT might be used as a starting point to describe my understanding of ontology.  However, I regard material reality as an irreducibly complex (overdetermined) totality of processes seamlessly connected spatially and temporally.  This irreducible complexity/overdetermination ensures that theory/analysis can never produce objective accounts that unravel the complexities through which particular material processes are constituted/constructed.  The most that theory and analysis can do is to develop coherent and discursively persuasive explanations of material phenomena constructed by piecing together arbitrary subsets of causal linkages between material processes. 
          In rejecting the possibility of objectivity, I attribute a manifestly partisan role to theory/analysis.  That is to say, theory/analysis reconstructs the reality of its objects by selecting and piecing together explanatory/causal components in particular ways that conform to the theorist's subjectively-defined agenda.  I regard this underlying subjectivity of theory/analysis to be no less true for the natural and physical sciences than it is for the social sciences.  Thus, a performative analytical account in, say, climate science might accumulate substantial quantities of empirical data conforming to a particular hypothesis on ecological transformations to congeal an argument that seeks to persuasively reshape its audience's understanding of climate change.  The relative truthfulness of such an account (i.e. the degree to which it exhaustively accounts for contextual variations in the phenomenon it describes/interprets) is, perhaps, less important here than its capacity to be convincing to its audience even as it simultaneously acknowledges its incompleteness and its incapacity to tell the absolute, objective truth of climate change.  In this view, the specifically scientific character of the account derives not from its objectivity but from its willingness to acknowledge the contested nature of truth, against which it is simply one perspective whose relative value must be adjudged in relation to its consequential impact. 
          Two aspects of the framework are critical in differentiating it from alternative performative approaches on epistemology.  First, it envisions the process of theory as a contained within the struggle between competing ideas in every conceivable disciplinary context where theories/analyses are deployed.  Second, it emphatically compels the theorist/analyst to self-consciously acknowledge his/her own subjectivity and support for a particular contested/contestable body of ideas, notwithstanding the incapacity of the theorist/analyst to appeal to objectivity as the rationale for supporting such ideas.  Conversely, in rejecting the objectivity of his/her own accounts, the theorist/analyst can rightly counter that all competing accounts likewise lack the potential to be objective. 
           An ANT theorist would certainly dispute this overdeterminist rejection of objectivity.  ANT and overdeterminist conceptions of performativity diverge in regard to the potentiality for theories/analyses to realize objectivity.  In broader terms, such differences shape the understanding of both performative perspectives on the sources of ontological uncertainty in deploying theoretic accounts.  That is to say, accounts consistent with each approach would be apt to evaluate the effects of the theoretic process in different ways, and these differences constitute the particular terms in which the theoretic process might be characterized as performing the reality of its theoretic objects.  The ontological presumptions manifest by ANT approaches prioritize the capacity of agents to mediate the range of agencies influencing their own actions.  In these terms, the theorist, as an agent involved in the performance of his/her theoretic objects, may fail to produce the ends predicted by the theory simply because his/her theoretic objects maintain the capacity to resist the particular influences of the theoretic process.  Thus, for ANT, the theorist may strive to make his/her theoretic objects perform in accordance with his/her objective theoretic predictions, but theory operates within a larger context of mediation.  By contrast, in rejecting objectivity as a possibility for the theorist, the approach to performativity that I label overdeterminist counters that the relationship between the theorist and his/her theoretic objects is continuously shaped by an infinitely varied range of other processes, ensuring that theorist can never comprehend the particular ways in which the theoretic process shapes reality.  Given this incapacity of the theorist to unravel the complexities of his/her relationship to theoretic objects, the most that theory can do is produce empirically and/or rationally consistent and persuasive arguments conforming to the theorist's theoretic agenda in the hope that his/her intervention into material reality will produce the desired ends. 
          Having laid out a range of divergent realist and performative epistemological approaches, my larger goal in this section is to situate Walrasian/Paretian theoretic conceptions in relation to their underlying epistemological presumptions.  In this regard, there is always a danger inherent in attributing a particular epistemological approach when a given body of theory never explicitly spells out its relationship to objectivity.  Moreover, approaching from my own overdeterminist conception of performative epistemology, there is a further danger in attributing a partisan agenda on a given theory where none is explicitly enunciated.  With these dangers in mind, the business of evaluating theory and performing criticism is, in my view, an inherently partisan exercise, even to the extent that we are crossing the intractable boundaries of epistemological paradigms.  From my perspective, a theorist engaging in criticism must strive to achieve a thorough comprehension of the theories he/she is critiquing, not to evaluate whether or not these theories present an objectively true account of material reality, but to ask what particular material consequences might arise if the theories are taken seriously and acted upon.  Invariably, even to the extent that we acknowledge that a body of theory is conceived as an effort to unravel the objective truth in some aspect of material reality, a recognition that every theory is consequential to material reality demands that we inquire into the potential sources of given perspectives. 
           In the spirit of taking Walrasian/Paretian theories of the general equilibrium seriously and inquiring into their effects on the functioning of markets and determinations of economic policy, I want to advance a set of propositions.  First, any derivation of the characteristics of firms, the roles performed by firms, and the structural/arithmetic features of profit maximization/cost minimization must recognize that these processes derive from a larger theoretic context in which the role of households in achieving utility maximization is conceived as dominant.  In this sense, before Walrasian/Paretian theorist ever go about the business of collecting data or performing statistical analysis to validate the objectivity of their presumptions about the role of firms in particular market contexts, they have already introspectively truncated the reality that they are analyzing to determine the relative importance of particular economic facts.  Walrasian/Paretian theories emanate from theoretic processes that can be characterized as rationalist. 
           Second, as a body of rationalist theory, Walrasian/Paretian general equilibrium theory seeks validation of assumptions and predictions on the functioning of market processes, formed a priori (i.e. before any empirical evaluation of real markets), in order to determine whether markets function as if they operated according to the predictions of general equilibrium theorizing.  Moreover, the invalidation of particular assumptions (e.g. homotheticity of production functions) may not invalidate the larger structure of the theory to the extent that important predictions derived from the theory (e.g. stable positive cross price elasticities between particular complementary commodities, negative impacts of labor-saving technological innovations on labor compensation rates) continue to hold.  In this sense, we might differentiate between those assumptions that promote the internal logical consistency of the theory and those conclusions, emanating from theoretic assumptions, that enable the theory to make realistic predictions on the functioning of actual markets.  It may be that the assumptions of the theory do not conform to the realities of actual firms and actual markets, but if the theory continues to make accurate predictions regarding the functioning of sectoral economies or entire market systems, then it continues to enjoy some degree of validity, against the larger validity criteria of rationalist theories.  
            Third, in conformity with its larger commitment to epistemological realism (i.e. its implicit assumption that the purpose of the theoretic process is to unravel the objective truth in its theoretic objects), Walrasian/Paretian theory has no explicit partisan agenda.  On the contrary, it might be possible to extract partisan agendas from the theorizations of individual theorists within the larger tradition.  For example, we might conclude that Léon Walras' particular partisan position against the private ownership of land, as a factor of production, shaped the particular way in which general equilibrium theory has approached land, inasmuch as general equilibrium theory has continuously sought to divorce land, theoretically if not practically, from the larger mechanics in the determination of equilibrium pricing (i.e. tâtonnement)(see Walras, Léon (1896). Études d’économie sociale: Théorie de la répartition de la richesse sociale.  Paris: F. Pichon, Imprimeur-Éditeur. Located at the archive "gallica.bnf.fr" of the Bibliothèque nationale de France, at: http://gallica.bnf.fr/ark:/12148/bpt6k111751z).  Likewise, the quasi-collectivist, cooperative utilitarian imperative in general equilibrium economics on which I will subsequently elaborate comments in this critique appears to emerge, most succinctly, from the partisan motivations of later Walrasian/Paretian theorists like Oskar Lange (who, after merging Paretian analytic methods with socialistic planning principles and theoretically debating the nature of state planning against Austrian rigorous free market supporters, went on to serve advise the post-World War II communist government in his native Poland on central planning of consumption).  Whatever Lange's exact motivations were, it stands to reason that his particular emphasis on the relationship between utilitarian consumption preferences over entrepreneurial imperatives contributed to the larger focus of general equilibrium theory. 
            On some level, we have to accept that such theorists were engaged on a conscious search for objective truth and that this motivation constitutes the overriding imperative, in their minds, driving the development of their theories.  On the other hand, we have to conclude that the specific manner in which individual theorists approached the theoretic process was shaped by prior conceptions, leading each to pursue particular abstract methodologies in order to clarify the myriad sources of empirical data revealing the workings of real economic systems.  Such abstractions must, thus, be recognized as moments in which the subjective operation of individual perspectives, shaped in turn by myriad prior life experiences and influences, determine the development of theory. 
            Concluding, it is my contention that the particular abstractions developed within Walrasian/Paretian theorizations of the firm reflect particular partisan agendas, even to the extent that such agendas were oblivious to individual Walrasian/Paretian theorists, who conceived their work in realist terms.  These agendas, over time, promoted an imagery of free market systems that prioritized individual choice by consuming households in relation to traditionally capitalist imperatives of profit maximization.  Hence, the autonomous role of the firm disappears in Walrasian/Paretian theorizations, replaced by sterile, arithmetic profit maximization/cost minimization processes, sandwiched between autonomous household utility maximization processes. 
             Abstraction, as a methodological tactic, seeks to clarify impossibly complex contexts to extract the truth underlying market exchange and production.  However, to the degree that abstraction constitutes a wholly rational, logical tool exercised by individuals with overdetermined perspectives, it can never objectively resolve the complexities contained with market exchange and define the unique linkages among final commodity markets and between these and factor markets.  My intention is, thus, to definitively oppose the capacity of Walrasian/Paretian theory to present objective theories on market exchange.  Having dispensed with the objectivity of these theories, however, I seek to resuscitate those elements of the tradition that, indisputably offer important insights into market dynamics, especially in this era of globalization.