This final analytical section of this document addresses a theme that I consider a linchpin to the larger structure of Walrasian/Paretian and Austrian Neoclassical theory. Namely, it concerns the relation of factor compensation to the total revenues generated by firms. Proceeding in this manner, within a general equilibrium framework, we know that factor markets and final commodity markets are inseparably intertwined. The determination of prices in both sets of markets is simultaneous and continuous, and any changes on either side are liable to be reflected by changes in prices elsewhere. Thus, if at one moment an economy is characterized by substantial consumption of relatively labor intensive commodities, then a change to consumption of larger quantities of relatively capital intensive commodities will impact factor markets, raising the price of capital relative to labor.
On the other hand, we have so far made certain critical assumptions about firms operating within our general equilibrium economic framework. Most importantly, we have imposed the condition that production functions are homothetic and demonstrate constant returns to scale. In this section, we are going to see the significance of making such an assumption, and in the forthcoming critique of the larger theory, I will endeavor to show the arbitrary nature of the assumption. For now, the central theme of this section concerns the relevance of a postulate called the product exhaustion theorem, describing the interaction of firms with factor markets.
To start, without elaborating to any great degree in this document, the supply sides in factor markets reflect decisions by households on the rental of factors of production to firms in exchange for determinate quantities of final commodity consumption. Again, the very nature of the problem here links factor markets to markets for final goods and services. In particular, households balance the disutility of supplying factors of production against the utility of consuming commodities. That is to say, we assume that it is unpleasant to households to have to supply labor hours to firms. Likewise, it is unpleasant to defer consumption of money on purchasing goods and services in order to make capital available to firms in order to invest in increasing the productivity of the production process. It is, finally, unpleasant to part, even temporally, with the enjoyment of particular, absolutely scarce assets (e.g. land, natural resources, technological knowledge/formulae) in order to enable firms to derive a particular market advantage from the use of such assets. The pain inflicted on households with the rental of these factors of production ensures that they will not be given to firms without cost. Rather, the basis for the supply side of factor markets arises from the relative disutility to households in having to part with their factors of production for a finite span of time for which households must be compensated.
All factor payments to households generally constitute a rental payment by firms, insofar as firms own nothing that does not exist for free use in the public sphere/commons. In reference to the assumptions that I have so far made in this document, the commons represents a fairly substantial space. In my consideration of cost functions ("A Pure Neoclassical Theory of the Firm II"), I allowed land to be a freely available factor of production in order to simplify our subsequent analysis of profit maximization/cost minimization. From here, I could conceivably reintroduce land as a factor of production for which firms must compensate the household owners, but it would tend to complicate our analysis in ways that I would rather avoid. As such, I will continue to place land in a special category, along with technological innovations that can be isolated from broader use, outside of the variable factors labor and capital. In this regard, we might assume that all firms must pay some basic, unaccounted, sunk cost to compensate the household owners of land, but any firm enjoying a price advantage from particular forms of land must pay an additional rent that will, conceptually, erase any advantage that they receive for the particular assets they are using. In any case, payments for land will not occupy us in our larger analysis here.
In regard to labor and capital, particular divergent principles determine factor supply and compensation rates. The determination of these compensation rates will not be the central focus in this discussion. Rather, I intend to simply point out, that factor supply decisions by households balance the relative disutility of supplying labor hours and defering consumption against the wage rate for labor and the interest rate payable to households for capital, respectively. The more important concept here resides on the demand side of factor markets. Specifically, what interests us are the relationships between the marginal products of labor and capital, the market compensation rates for labor and capital, and the total revenue earned by the firm for producing final goods and services.
In the simplest terms possible, we have so far argued, by assumption, that price competition between firms in output markets would drive the market price down to the average total cost of output. In this regard, the total revenue TR1 received by the firm, defined as the level of output sold times the output market price, will equal the firm's total cost TC1 of producing the output, define by the firm's cost function. According to our definition of the cost function, this condition implies that total revenues must be exhausted in paying for the two factors of production, labor and capital. We will get back to this result and its conceptual importance, but first we have to arithmetically specify the relationship between factor prices and the distribution of output market revenues between the factors. Once again, defining total cost as:
TC1 = Pkk1 + Pll1
And defining total revenue as:
TR1 = P1x1 = P1f(l1,k1)
Which is simply the output price P1 times output quantity x1. We can define profit, or net revenue as:
NR1 = TR1 -TC1
Or:
NR1 = P1f(l1,k1) - (Pkk1 + Pll1)
Which simply argues that the firm's profits/revenues net of total costs are the difference between its total revenues and total costs. Taking the first order partial derivatives for this equation in terms of the factors of production (that is, allowing the factors to vary in accordance with the capacity of the firm to select different combinations of labor and capital), we get:
δNR1/δk1 = P1MPk - Pk
δNR1/δl1 = P1MPl - Pl
Noting that the firm's profits are maximized where the changes in net revenue arising from changes in the factors of production are equal to zero, we need to set each of these partial derivatives equal to zero. The idea here is to isolate a set of conditions in the firm's decision to rent factors of production where it can no longer change/increase its net revenue by varying the labor and capital it rents from households. Doing so and rearranging each algebraically, we get:
Pk= P1MPk
Pl = P1MPl
Which argues that the firm should rent factors of production up to the point at which the factor market price for each of the production factors equals the output market value of its marginal product. We can define this latter term, scaling each marginal product by the output price, as the marginal revenue products of labor and capital. Noting, once again, our conclusion that, in general equilibrium, output prices will equal the average total cost of production to firms and, thus, the condition:
TC1 = TR1
Or:
Pkk1 + Pll1 = P1x1
We can substitute the marginal revenue products for the factor market prices to obtain:
P1MPkk1 + P1MPll1 = P1x1
Dividing both sides by the output price P1, we get:
MPkk1 + MPll1 = x1
Implying that, subject again to our assumption that firms operate with production functions characterized by linear homogeneity (homotheticity and constant returns to scale), the marginal products for each of the factors of production are equal to their respective average products. Thus, if each factor of production is paid its marginal product, in accordance with our maximization condition relative to factor rental decisions by the firm, the firm will exactly exhaust its total product in compensating households for the factors of production rented out to the firm. This conclusion constitutes the product exhaustion theorem.
Briefly, in reference to the larger history of this theorem, product exhaustion makes use of a finding within pure mathematical theory, Euler's theorem. This theorem states that for any function
f(xi) homogeneous of a degree n, such that:
f(tx1,...,txi) = t^nf(xi)
Then, letting t = 1, the following relationship obtains:
(δy/δx1)x1 + ...+ (δy/δxi)xi = nf(xi)
Translating this theorem into economic analysis, if a production function is homogeneous of a degree n = 1, then the outputs generated by the function will be proportional to the sum of marginal products for each factor of production times the quantity of the production factor for all scales of output. In other words, an optimal combination of production factors, at which no factor of production can be altered to increase net revenues, will be invariant in relation to the scale of production and, if under such a combination the factors of production are paid their marginal products, factor compensation will exactly exhaust the total product.
Quantitively, I can further prove the outcome of this theoretic analysis with reference to the coffee example employed in the preceding section. If we take our initial production function:
xcoffee = 4 (k^.6)(l^.4)
Supplemented by our initial factor market prices of $20 per unit of capital and $10 per unit of labor, and the initial output market price of $7.44 per unit of coffee, then I can solve the following equation for capital k:
$10 = $7.44 [1.6 (k^.6)/(l^.6)]
Derived from the equation of the factor market price of capital and the marginal revenue product of capital. This gives me the profit maximizing/cost minimizing relationship:
.75l = k
The same profit maximizing/cost minimizing condition that I obtain by going through total differential analysis. (Try it. You'll see that it actually works!)
Graphically, we can combine our argument that firms are price takers that rent factors of production up to the point where the price for a factor of production equals its marginal revenue product with the supply side assertion that households seek to balance the diminishing marginal utility of final commodity consumption against the rising marginal disutility of supplying factors, we can portray a labor market equilibrium through figure 26.
Where supply SL is a function of the price of labor PL, the price of all other factors of production and all final commodities P-L, the utility functions for each of m households that include labor time as an argument and, by assumption, demonstrates increasing marginal disutility, and non-labor sources of income y-l for each of m households. The demand side of the labor market, by contrast, is illustrated by a perfectly elastic demand schedule determined by the price of labor PL, the price of other factors of production P-L, and the technological requirements for labor from each of n firms across all industries. Again, the perfectly elastic nature of this schedule reflects both the fact that firms are price takers and that firms, within a general equilibrium economy, continuously operate along their expansion paths at profit maximizing/cost minimizing combinations of labor and capital. As a result, market output supply schedules are perfectly elastic with respect to changes in quantity demanded and market factor demand schedules are perfectly elastic with respect to changes in the quantities of labor and capital households are willing to supply.
Concluding this section, we need to evaluate the conceptual significance of the product exhaustion theorem. Specifically, we face two definitive conclusions, in regard to firms operating within a general equilibrium economy:
1. Firms pay households renting out factors of production exactly what each unit of each factor adds in productivity to the production process.
2. Firms exhaust all of their revenues paying households for the use of their factors of production.
I will consider each of these conclusions in order.
First, the conclusion that firms must pay each of the factors of production their (marginal) productivity derives, more specifically, from the profit maxmizing condition above, that firms should continue renting factors of production until the last unit of each production factor returns, as a share of marginal revenue from the last unit of output sold, the price of renting the last unit of the factor. This is a condition that applies for all types of production function, with increasing, decreasing, and constant returns to scale. Logically, if a firm continues to earn marginal revenues in excess of the marginal costs it incurs from renting additional units of production factors under diminishing marginal productivity, then it should continue to rent more production factors until its marginal revenues equal its marginal costs. If the firm stops, then it will forego the additional revenues that it could earn by renting more production factors. This process simply describes profit maximization.
This condition is not, however, equivalent to the conclusion that a firm will achieve product exhaustion if it pays the factors their marginal revenue products. If, for example, a firm operates within some range of a production function characterized by decreasing returns to scale, then, if it pays the household owners of production factors their marginal revenue products, it will actually pay factor compensation rates below their average contributions to total revenue for at least one of the factors of production (i.e. the firm will earn total revenues in excess of total costs). Conversely, if a firm operates within ranges of a production function characterized by increaseing returns to scale and pays the factors of production their marginal revenue products, it will actually pay at least one of the production factors a rate of compensation higher than its average contribution to the firm's total revenues (i.e. the firm will be compelled to pay out more in factor compensation than it receives from the sale of its outputs). In accordance with our assumptions about a general equilibrium economy, neither of these situations occur with the sorts of firms that we are considering.
Again, I intend to offer a criticism of the constraints that we need to impose on an economy operating within the larger scheme of general equilibrium thinking, but, insofar as we are dealing with firms with production functions characterized by homotheticity and constant returns to scale/linear homogeneity, product exhaustion will continuously be the rule to the extent that firms pay the household owners of their factors their marginal revenue products. That is to say, moreover, at no time will firms not compensate the household owners of production factors their marginal products. In this sense, the invocation of the production exhaustion theorem conveys a special significance within general equilibrium thinking, connecting factor markets to markets for final goods and services.
Conceptually, the product exhaustion theorem implies that firms exist as pure technological intermediaries and temporal placeholders for transactions that ultimately occur between households operating through markets. That is to say, certain households aggregate their diverse factors of production through the intermediary device of a firm in order to produce commodities that they would otherwise be incapable of producing themselves, on their own. At the completion of the production process, these households receive a mass of final commodities, equivalent to the marginal products of the factors they contributed to the production process, and exchange them for final commodities produced by other households through the intermediary device of other firms. Again, if monetary equivalents creep into this larger series of exchanges, such equivalents occlude what is otherwise a process through which households perform diverse sets of utility maximizing choices for factor and final commodity markets to decide how much of each final commodity is desired and how much of each factor of production will be provided. In the end, each household will receive exactly what it contributed to the larger process and engage in exchanges to receive the exact bundle of final goods and services that it desires. In this sense, with product exhaustion, we have come full circle, back to the assertion early on in this document that firms do not enjoy any tangible reality beyond that of the production function. For Walrasian/Paretian theory, the household emphatically represents the real of economic theory, while firms exist as a phenomenal signature of the need for households to assemble their factors of production to produce the goods and services they demand.
There might be an ethical corollary regarding the justice of paying households the marginal revenue products from use of their production factors. That is to say, if firms pay households the marginal revenue products of their factors, then the households will receive in return exactly the same value that they contributed to the production process. In a certain sense, I want to acknowledge the relevance of this ethical corollary while, further, promising to subject it to a critique in relation to alternative theories of factor compensation (e.g. Marxian approaches developed in reference to social reproduction). Furthermore, historically, this argument was not lost on the expositors of Neoclassical economic theory. John Bates Clark, a late Nineteenth century American economist who largely developed the first theory of marginal productivity factor pricing/compensation, explicitly had this corollary in mind. I would argue that his larger purpose in developing marginal productivity factor pricing was to offer it as an ethical counterargument against the efforts of the burgeoning American labor movement to raise wages, seeking to state, emphatically, that unorganized industrial workers received exactly what they deserved to get (i.e. their marginal products) from American labor markets.
On the other hand, evaluated from the framework of Walrasian/Paretian general equilibrium thinking, ethical arguments concerning the appropriate levels of compensation to households, in markets for labor or any other factor markets, appear somewhat misplaced. As I will argue in my critical comments on the Walrasian/Paretian theory of the firm, general equilibrium thinking conveys the imagery of a virtual-cooperatively organized market system, operating through the mutual consent and mutual benefit of participating households. In these terms, distributive justice is procedurally built into the larger organization of the system, and any household that regards either factor market outcomes or the virtual-cooperative (democratic) process of tâtonnement as unjust can simply opt out. That is to say, Walrasian/Paretian general equilibrium thought takes for granted that market systems will operate under procedures that are fair to all parties, regardless of the distributive outcomes that they generate. As a minimum, markets will generate outcomes that will be at least as good for all participating households as those that the households would have encountered if they decided not to participate.
Moreover, a defense of marginal productivity factor pricing as an ethical principle underlying capitalist labor markets (i.e. labor markets supporting, at least within Marxian theory, exploitation of human labor) presumes something very important that I have consistently argued within this document to be absent from Walrasian/Paretian firms. Namely, the firm as we have studied it here not only lacks any substantial, permanent existence but also lacks the critically important feature of an entrepreneur. The Walrasian/Paretian firm is a transitory, cooperative endeavor of households, whose existence ceases when participating households cease to derive any utilitarian benefit from its operation. No entrepreneurial figure exists to assemble the factors of production and take the risk of producing commodities that may or may not sell within final commodity markets - these are features of entirely different theories of the firm. Thus, we do not need an ethical argument against exploitation by an entrepreneurial employer here because our firms lack employers. With this in mind, I am going to defer criticisms regarding J.B. Clark's ethical defense of marginal productivity factor pricing until my consideration of Marshallian firms.
My final comments here follow logically from what I have argued above. Notably, if firms exhaust all of their revenues paying households for the use of their production factors, then, it follows, that firms generate zero profits/net revenues. Mathematically, this result is obvious. For the equilibrium quantities of output production, total revenues generated by the firm must equal total costs if competition between firms ensures that all firms in all industries continuously charge their average total costs per unit of output. This outcome can be further defended, however, in regard to the larger conception of a general equilibrium economy and the structural role assigned to firms. If firms are simply a transitory technological medium (defined by the production function) for bringing households together to produce commodities, then, once households are paid their marginal contributions to the process, then why we would expect any residual to exist? If such a residual were to arise, then who would be able to exercise a claim on it?
In functional terms, we might argue that profits do not exist because profits do not serve any needs within the structure of the economy. The household owners of labor receive exactly what they contributed to the project of the firm as wages. The household owners of money capital/financing receive exactly what they contributed as interest. The household owners of land/natural resources receive exactly what they contributed as rent. All of these payments are subject to marginal productivity factor pricing, ensuring that all households are paid exactly their marginal contribution to production. There is no one left to exercise a claim on profits even if they were to arise. Competition between firms might be an adequate condition to ensure that there will be no profits arising from production, but, in a larger sense, there is no functional actor that can claim some distribution called profit in relation to the marginal productivity of their factors.
Therefore, at the conclusion of our theoretic story of the pure Neoclassical Walrasian/Paretian (and, at least partially, Austrian) firm, we discover that firms in a general equilibrium economy generate zero profit both because competition between firms in every market drives prices down to average total costs, at which firms achieve product exhaustion, and because there is no structural agent that can exercise any claim to a payment in excess of the payments to the household owners of production factors.
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