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.
