A Pure Neoclassical Theory of the Firm VI: Conclusion (Microeconomics)

Factor Markets and the Product Exhaustion Theorem

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.

    Figure 26: Labor Market Equilibrium
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. 

A Pure Neoclassical Theory of the Firm V (Microeconomics)

Changes in Factor Prices

I finished the previous section arguing, among other things, that, within a general equilibrium economy, changes in household preferences, increasing or decreasing quantities of output demanded at all prices, solicit immediate adjustments by firms in output quantities produced, accompanied by changes in financial allocations from households, such that firms will produce the exact quantities demanded by households.  Provided firms conform to our assumptions regarding the homotheticity of production functions and constant returns to scale/constant average total costs, such changes may generate no changes in output prices, if transformations of household preferences do not simultaneously affect the relative prices of labor and capital.  In this section, I will consider the possibility that a transformation of household preferences or changes in production technologies will impact relative factor prices.  In order to reinforce the conclusions made here, I will develop a quantitative example, utilizing a Cobb-Douglas type production function to evaluate the effect of a change in factor prices. 
               Reiterating one of our larger assumptions about general equilibrium economies, firms not only operate as price takers in output markets among large numbers of other firms, but approach factor markets as price takers because they draw from labor and capital markets in competition with all other firms in their industry and with all other firms in all other industries.  In this light, the relative intensities of labor and capital employment across all industries must impact, in part, the relative compensation rates for the production factors.  If, at some moment in time, households demand relative large quantities of labor-intensive goods and services, then a sudden shift in preferences toward goods and services utilizing capital more intensively in their production should be expected to raise the relative price of capital across all markets.  Given the integrated nature of a general equilibrium economy, such a change would impact the equilibrium levels of output in every market. 
                Playing this scenario out graphically, figure 20 illustrates the effects of a shift in aggregate demand for production factors, raising the relative price of capital.

Figure 20: Shift from relatively capital intensive to relatively labor intensive expansion path resulting from an increase in relative price of capital.
Elaborating on figure 20, the increase in the relative price of capital is illustrated by the pivoting of isocost IC11 downward into IC11', with a common maximum quantity of labor and a reduced maximum quantity of capital for the same level of total cost TC11 (thus, TC11 = TC11').  The increase in capital costs shifts the firm from isoquant curve I1(x11) to I0(x10), where x11 > x10, without altering its total costs.  In this manner, the average total cost per unit of output for the firm increases. 
            I attempt to show two separate effects of the transformation of factor market prices here.  First, holding output levels constant, an increase in the relative price of capital generates a factor substitution effect evident in the movement along isoquant curve I1 from point A to C, a point of tangency with isocost curve IC12(TC12), where TC12 > TC11.  Isocost IC12 is parallel to IC11', denoting a common set of relative prices defining the slope of the curve (-Pl/Pk).  It, thus, lies on an expansion path EP2 constituted as a locus of all profit maximizing/cost minimizing factor combinations at the new set of relative factor prices.  At point C, the firm selects lC labor (lC > lA) and kC capital (kC < kA).  This factor combination reflects the fact that, as the relative price of capital increases, the firm has an incentive to substitute labor for capital in order to arrive at a lower level of total costs relative to its factor combination at point A.  Thus, holding the quantity of output constant, the firm maximizes profit/minimizes costs by renting more labor and less capital.
            The factor substitution effect becomes evident when we ask how much of each factor of production would the firm rent at the new set of relative prices if it were to continue to produce the same quantity of output.  By contrast, the scale effect of a change in factor prices measures the extent to which, given a substitution between factors, the firm may be compelled to reduce its employment of both factors if it continues to produce at the same initial level of total costs.  As such, the scale effect measures the degree to which a change in factor prices will affect output levels.  This effect is illustrated by the movement along expansion path EP2 from point C to point B, resulting in the decrease in output of  (x11 -x10).  At output level x10, the firm selects lB labor (lB < lC) and kB capital (kB < kC).  Thus, accounting for the inability of the firm to maintain its previous level of output under the elevated capital prices, the firm reduces its employment of both labor and capital in order to maximize profit/minimize costs under the same initial total cost level. 
              Proceeding to the market level, we must account for the fact that the change in fact prices has reduced output from all representative firms in the industry (because, given our conclusion in the last section derived from the general equilibrium assumption of perfect information, all firms operate with identical production functions).  Moreover, firms continue to operate with constant average total costs per unit of output except that the level of average total costs increases relative to the initial level.  Hence, we get an upward shift in the market level supply function, shown in figure 21.

    Figure 21: Effect of higher relative price of capital on market equilibrium.
Arguing out the connection between figures 19 and 20, it might make sense to interpret X11' as the aggregation of x10s in figure 19, implying that the effect of an increase in the relative price of capital, barring any other change in household preferences, is a decrease in the quantity of commodity 1 supplied and demanded by households in competitive equilibrium.  The higher output price P11' likewise reflects the elevated price of capital as a component to the cost functions of all representative firms within the industry. 
         Having laid out this graphical example of an increase in the relative price of capital, certain outcomes remain unclear beyond the specification of factor substitution and scale effects.  For example, it is certain that the change in relative prices will place the firm on expansion path EP2 and that, given the increase in the relative price of capital, such a shift between expansion paths will be accompanied by an increase in average total costs, leading to shift of the market supply function.  On the other hand, the exact level of output and total cost level at which the firm will actually produce remains unknown.  That is to say, the specification of the above effects proceeds as if the firm will be constrained by its level of household financing to produce at the same initial level of total costs.  It may be the case that households demand higher or lower quantities of output than x10 from the firm.  It may be the case that households seek quantities of output higher than x11.  This is ultimately a question that must be answered by households through their continuous negotiation of output quantities and factor supply levels, and determinations of the relative prices that will bring everything into equilibrium across all markets (in Walras' French terminology  tâtonnement, implying jostling/groping along between competing agents toward a resolution).   The one thing that we know for a fact is that the previous level of output is now more costly for households to obtain from firms and that, subsequently, if households seek to consume a constant level of output from this industry, then, ceteris paribus (all other things being equal), they will need to supply larger levels of both labor and capital, reducing the relative prices of both factors, in order to achieve the same output.  I will approach this dimension in the next section discussing factor market supply decisions directly.            
             Proceeding beyond our graphical example, I want to develop a quantitative example here that lays out both the methodology heretofore presented on profit maximization/cost minimization in relation to the household decisions on output quantities demanded and responds to speculative changes in factor prices.  Thus, let us say that we have a firm operating in a single, perfectly competitive industry, say coffee roasting, in an economy that includes a large number of different industries competing with one another for relatively scarce quantities of labor and capital.  The firm operates with the following production function:
xcoffee = 4 (k^.6)(l^.4)
This production function is a Cobb-Douglas form.  Such production functions take the generalized form:
x1 = b (k^α)(l^β)
Where b is an adjustment factor for total factor productivity, and α and β are output elasticities, measuring the effect of a change in output in relation to a change in quantities of capital and labor respectively.  This construction is particular useful within the larger framework of general equilibrium analysis because, assuming each of the output elasticities take on a value between 0 and 1, any isolated change in one of the factors, holding the other factor constant, will manifest diminishing marginal productivity for the adjusted factor.  Conversely, if the sum of the two output elasticities is equal to 1, then any proportionate change of both production factors will result in a proportional increase in output (i.e. constant returns to scale/constant average total costs).  Cobb Douglas production functions are, thus, homothetic. 
               In regard to factor markets, the firm faces prices of $20 per hour of capital usage and $10 per labor hour.  Therefore, we can express the firm's cost function accordingly:
TCcoffee = $20k + $10l
Following our methodolgy to determine profit maximizing/cost minimizing combinations of capital and labor by taking total differentials, the production function yield the following first order partial derivatives for capital k and labor l:

δxcoffee/δk = 2.4[(l^.4)/(k^.4)] (Marginal Product of Capital)

δxcoffee/δl = 1.6[(k^.6)/(l^.6)] (Marginal Product of Labor)
We can define the total differential as:
δxcoffee = (δxcoffee/δk)Δk + (δxcoffee/δl)Δl
Plugging in our first order partial derivatives/marginal products and setting the change in output equal to 0 (again, because we are attempting to define a relationship between the marginal products for each factor of production at fixed levels of output), we get:
0 = 2.4[(l^.4)/(k^.4)]Δk + 1.6[(k^.6)/(l^.6)]Δl
Algebraically manipulating this equation, we get:
Δk/Δl = -(2/3)(k/l) = MRTSk
Where the marginal rate of technical substitution of capital (MRTSk) expresses the slope of the isoquant surfaces for each given fixed level of output.  Again, the term is negative, reflecting the negative slope of the isoquant surfaces and, thus, the necessity to substitute more of one factor of production as we reduce the other in order to maintain constant levels of output. It is, therefore, the first piece in the determination of a profit maximizing/cost minimizing combination of labor and capital in the production of coffee.  In order to arrive at this combination, we need to further account for the firm's cost function and the effect of factor market prices on the firm's decision to rent labor and capital.  Following the same methodology, we need to define the total differential for the cost function.  The cost function yields the following first order partial derivatives:
δTCcoffee/δk = $20
δTCcoffee/δl = $10
Simply speaking, the effect of any unit change for either factor of production on total cost will be equal to its market price per unit.  Utilizing, again, the first order partial derivatives from the cost function in order to define the total differential and setting the change in total costs equal to 0, we get:
0 = $20Δk + $10Δl
Algebraically manipulating the equation, we derive the following:
Δk/Δl = -(1/2) = -Pl/Pk
Expressing the slope of isocost surfaces as a ratio of factor prices. 
         From here, we need to relate the marginal rate of technical substitution of capital (slope of the isoquant) to the ratio of factor prices (slope of the isocosts) in order to define a relationship at which the firm will enjoy a profit maximum/cost minimum (where the two slopes are equal).  Setting the two rates of change above equal, we arrive at:
Δk/Δl = -(1/2) = -(2/3)(k/l)
Algebraically manipulating, we arrive at the following profit maximizing/cost minimizing relation between capital and labor:
k = (3/4)l
Thus, the firm maximizes its profits and minimizes its costs along an expansion path defined by this constant relation between capital and labor for all quantities of output.  Graphically, this can be illustrated as a ray from the origin with a slope of 3/4. 
               This concludes our arithmetic analysis of the production conditions faced by the firm.  As I have argued so far, our assumption of perfect information assures us that every firm in the coffee roasting industry produces coffee using this production function because it must be the most efficient possible way to produce coffee.  Therefore, our analysis of the production relations of this one firm mirrors the production relations for every firm in the market.  If all firms realize that, in order to maximize profits, they must rent 3 units of capital for every 4 units of labor they rent, then the sole question that remains for firms concerns where they are along their expansion paths.  This is question answered not by the firms themselves but, jointly, by the households that demand coffee and by the households that supply the finances to firms that produce coffee.  We can begin this stage of the analysis from either side (the consumption side or the financing side) because, ultimately, in a general equilibrium economy, both sides must be harmonized.  For purposes of argument, I am going to begin with the financiers.  Let us say that, given perfect knowledge of how much coffee households want to consume and perfect knowledge of how much each coffee roasting firm would be able to contribute to the market while maximizing profits, these households supply the firm with $900 to rent labor and capital for some given period of production.  Incorporating this information into our cost function, along with our profit maximizing/cost minimizing condition, we get:
$900 = $20[(3/4)l] + $10l
Solving separately for capital k and labor l, we arrive at the following combination:
l = 36
k = 27
Plugging these quantities into our production function, we get:
xcoffee = 4 (27^.6)(36^.4)
Yielding the (rounded off) total output:
xcoffee = 121
We can illustrate this result graphically through figure 22.

    Figure 22: Profit maximizing factor combination at $900 total cost and 121 units of output.
Let us say that there are 3000 identical firms in the market for coffee roasting.  If each produces 121 units/pounds of coffee for this production period, then the market will consume 363,000 pounds of coffee.  Moreover, each of these firms faces an identical average total cost for producing each pound of coffee at all quantities along its expansion path:
ATCcoffee = TCcoffee/x1 = $900/121 = $7.44
With this information in mind, given our assumption that price competition between firms will force all firms to charge their average total costs per unit of output, we can illustrate the market outcome graphically through figure 23.

    Figure 23: Market equilibrium for coffee.
Where I have penciled in an arbitrary non-linear downward sloping demand schedule to speculate on the nature of household consumption demand for roasted coffee under assumptions about diminishing marginal utility in coffee consumption that seem, at least in my mind (as an enthusiastic caffeine addict), to be fairly reasonable. 
            We can also arrive at this outcome by proceeding in the opposite direction (and, in a sense, given the knowledge possessed by household financiers, we already have!).  If across the economy consuming households demand, for any given period, 363,000 pounds of roasted coffee, then, given 3,000 identical firms producing an identical/homogeneous product, each firm must produce 121 pounds of roasted coffee.  Plugging this quantity and our profit maximizing condition into our production function, we get:
121 = 4 [(3/4l)^.6](l^.4)
Solving this equation separately for capital and labor, we arrive at the same factor combination that we find by proceeding in the other direction (i.e. 36 units of labor and 27 units of capital).  Finally, plugging these quantities into our cost function, we would arrive at a total cost of $900, the financing for which households would need to supply to firms. 
               Now, to evaluate the effect of a change in factor prices, let us say that something happens in another market, say the market for meat products.  This market has been affected by a rapid change in technologies that allows firms to produce meat products with much smaller quantities of labor (i.e. automation of production lines).  Such a change has led firms to substitute larger quantities of capital for labor and, as a consequence, a relative abundance of labor now exists within labor markets, driving the price for labor down, ceteris paribus.  For purposes of our example, let us say the price of labor goes down to $7.50 per labor hour from $10.
                Beginning our analysis of the effects of this price change on production levels and total costs for the firms in this industry, we need to first assess the effect of a reduction in labor costs on the cost function of our representative firm.  In contrast to my initial graphical example, where capital costs were increasing, the firm in this case is enjoying lower labor costs and, therefore, a reduction in its average total costs at all quantities of output.  With this in mind, I will, first, ask how much labor and capital our representative firm will rent under the new set of relative factor prices and how much output this new combination will produce under the assumption that the firm receives the same level of financing from households.  To do this, I will need to repeat my total differential analysis with respect to the cost function.  I do not need to repeat the procedure for the production function because the technological conditions that govern coffee production have not changed.  Thus, the firm now faces the cost function:
TCcoffee' = $20k + $7.50l
Differentiating, I get the following first order partial derivatives:
δTCcoffee'/δk = $20
δTCcoffee'/δl = $7.50
The new total differential for the cost function is:
δTCcoffee' = $20Δk + $7.50Δl
Setting the change in total cost to 0 and algebraically manipulating this equation, we get the following ratio of factor prices:
Δk/Δl = -.375 = -Pl'/Pk
Reintroducing the marginal rate of technical substitution of capital (MRTSk) derived from the production function and equalizing our two rates of change, we get:
Δk/Δl = -.375 = -(2/3)(k/l)
Algebraically manipulating, we arrive at the new profit maximizing/cost minimizing condition for the firm, defining the slope of its new expansion path:
k = (.5625)l
Now, working under the assumption that the firm receives the same quantity of financing from households, we get:
$900 = $20[(.5625)l] + $7.50l
Solving for capital k and labor l, we get:
l = 48
k = 27
Inserting these quantities into our production function, we get:
xcoffee' = 4 (27^.6)(48^.4)
Solving for Xcoffee', our (rounded off) new output quantity:
xcoffee' = 136
Finally, dividing our fixed level of total costs $900 by our new output quantity of 136, we get the following average total cost:
ATCcoffee' = TCcoffee'/x1' = $900/136= $6.62
This solution represents the net effect of a change in the price of labor under the assumption that the firm will continue to receive a constant level of financing from households to undertake production.  In order to complete my analysis and enable myself to disaggregate the factor substitution and scale effects of the price change, I now have to ask, conversely, how much labor and capital the firm would rent and what level of total costs it would incur if households demanded a constant quantity of output from each firm of 121 pounds of coffee.  Thus, starting this time from the production function, I set the quantity produced by the firm at 121 pounds, incorporating our new profit maximizing condition:
121 = 4 ([(.5625)l]^.6)(l^.4)
Solving for capital k and labor l, we get (rounding off):
l = 43
k = 24
Inserting this combination into our cost function, we get:
TCcoffee'' = $802.50
And, finally, dividing the new total cost by the fixed output quantity, we get the average total cost:
ATCcoffee'' = TCcoffee''/x1 = $802.50/136= $6.63
Confirming (more or less, after rounding calculations off!) that our two calculations (the first holding total costs constant, the second holding output quantities constant) are both on the same expansion path and that, at every quantity along this path, the firm is minimizing costs/maximizing profits subject to the new set of factor prices that it faces.  Figure 24 attempts to capture this change in factor prices graphically.    

    Figure 24: Effects of change in price of labor on profit maximization.
Where the dashed isocost curve IC11' represents the financing/cost constraint of the firm when we hold the quantity produced fixed at 121 pounds of coffee with reduced labor costs and the parallel solid isocost curve IC12 represents its financing/cost constraint when we hold the financing level constant at $900 with reduced labor costs.  Before evaluating the effect of this price change on the market as a whole, I want to measure the factor substitution and scale effects based on the calculations presented above.
               First, holding output levels constant at 121 pounds of coffee and asking the effect of a change in labor prices on the amount of labor and capital rented by the firm, we see that the firm substitutes an additional 7 units of labor (43 - 36 = 7) for 3 units of capital (24 - 27 = -3).  This is the factor substitution effect of the change in the price of labor.  If we proceed from this amended combination of labor and capital at new factor prices (i.e. l = 43, k = 24) and allow the financial/cost constraint faced by the firm to increase to its original level of $900, the firm adds another 5 units of labor (48 - 43 = 5) and 3 units of capital (27 - 24 = 3).  This is the scale effect of the price change for labor.  The net effect of the price change on labor rented by the firm is an increase of 12 units (addition of 7 units measured by the factor substitution effect plus additional 5 units measured by the scale effect).  The net effect of the price change on capital rented by the firm is zero change (loss of 3 units measured by the factor substitution effect plus addition of 3 units by the scale effect).
                The market level effects of a decrease in the price of labor may be more uncertain.  Let me explain why.  The one thing that we know here is that the average total costs faced by firms will decrease to (around) $6.62 per pound of coffee.  This is what consuming households will pay per pound of coffee.  We do not, however, know how much coffee households will want to consume at this price because I have not specified the outcomes of multifarious, diverse individual household utility functions, aggregating to define the market demand function.  We can assume predictively that households will experience various degrees of diminishing marginal utility as they continue to consume larger quantities of coffee.  In this manner, the market demand function will slope downward, but we do not know the extent to which market demand will exhibit elasticity with respect to a change in the price of coffee beyond a general assumption that demand will not be perfectly elastic (i.e. a horizontal demand function).  In any case, within the larger structure of a general equilibrium economy, we will hold to the conclusion that the market level demand for coffee will, in some manner, be harmonized with the market supply of money capital/financing to coffee roasting firms.  This means that, whatever consuming households are willing to consume will be communicated to households supplying money capital to firms, such that the total cost constraint of firms along their expansion paths will enable the 3,000 identical firms roasting coffee in the market to produce exactly what households are willing to consume.  Figure 25 illustrates such an outcome without assigning a number to the speculative equilibrium quantity.     

    Figure 25: Market equilibrium generated by decrease in labor costs.

A Pure Neoclassical Theory of the Firm IV (Microeconomics)

Expansion Pathways

The purpose of this section is to engage further with the question of scale.  Having defined profit maximization/cost minimization in relation to production at a defined quantity of output, I need to address the scale of production, if only because it implicates the broader structure of a general equilibrium economy.  Refreshing our larger set of assumptions on general equilibrium thinking, we are dealing with a fully integrated structure of markets, with many industries, many firms in each industry, and many, many households, as owners of factors of production and consumers of final goods and services.  I have argued that within this structure, firms are captive agents to the decisions of households in both factor markets and final good and service markets.  One dimension of that captivity concerns the scale of output demanded from firms by households. 
              I have attempted to suggest in earlier sections of this document that limitations on the scale of production by firms are implicated at every stage in the the firm's operation, from the assembling of financing (money capital) to the assembling of machinery, raw materials, and labor to the pricing of outputs (again, firms are always price takers - they do not price goods, households do through the functioning of markets).  Assuming relative scarcity in the availability of financing, firms cannot finance the rental of production factors from households without limitations.  Households will only supply money capital to firms for outputs that will be liquidated at objectively given output prices (noting the fact that the assembly of financing in a general equilibrium theoretic structure is simultaneous to the determination of output prices and factor prices!).  With strictly limited finances, firms are limited in the combinations of labor and capital that they are able to employ in production.  As price takers, finally, firms cannot alter their prices in final commodity markets to liquidate excess outputs from operating at a scale in excess of what the market can accomodate. 
              From the previous section, we know that, given an objective set of factor prices (i.e. a cost function) and access to the appropriate production techniques to assemble the factors in order to produce final commodities (i.e. a production function), a profit maximizing/cost minimizing combination of production factors can be identified for any given level of output demand by households.  Here, I am going to graphically formalize this insight in xlk-space, reduced for simplicity in figure 12 to lk-space. 

     Figure 12: Expansion Path under constant capital-to-labor ratio of 1.
In figure 12, a firm, characterized by the production function represented in isoquant curves I1, I2, and I3, is shown with three separate total cost constraints (TC11, TC12, and TC13), reflected in three isocost curves (IC11, IC12, and IC13).  For each total cost level, the firm has an objective profit maximizing/cost minimizing combination of labor and capital to rent from households.  These profit maximizing/cost minimizing combinations are illustrated as points A, B, and C, where isoquant curves intersect with isocost curves at unique points of tangency.  The output quantities, defined by the production function, are such that xA <xB < xC, so that the movement between these points of tangency follows a pathway of expanded output (or, moving in the opposite direction, of contracting output).  If I draw a line connecting these points, I can, therefore, define a relationship between factor combinations and output quantities as the firm expands or contracts its outputs.  I will call this relationship the firm's expansion path.  In figure 12, this is illustrated by the line EP1.  A firm's expansion path is a locus of all profit maximizing/cost minimizing combinations of labor and capital for each level of output. 
           Expansion path EP1 in figure 12 displays some important characteristics concerning the particular form of production function characterizing the firm.  Notably, I have drawn it as a ray extending from the lk origin with a constant slope in lk-space defined by the capital-to-labor ratio.  In other words, the ratio at which the firm rents capital and labor in order to maximize profits and minimize costs is constant, given each particular set of relative prices for labor and capital reflected in the firm's cost function.  That is to say, if the relative prices for production factors were to change (reflected, again, by a change in the slopes of the cost function/isocost curves), then the slope of the expansion path extending from the origin would change, but it would continue to be represented by a ray with a constant slope.  If the relative price of capital increased, for example, then we would expect the firm to substitute more labor for capital at all output scales, generating a less steep expansion path.  As such, the production function illustrated in figure 12 could be characterized mathematically as homothetic, implying that any multiplication of the production factors by a common positive constant will not affect the marginal rates of technical substitution derived from the production function. 
             Homothetic production functions make the problem of analyzing the scale of production relatively simple because we do not need to inquire into how changes in scale affect the capital or labor intensity of production.  Factor intensities are only affected by changes to relative prices in factor markets. This pattern may not be consistent with actual production processes in real economies, but, as always, the purpose of this document is to outline a pure theoretic model and to advance conditions that can be very readily analyzed without excessive complexities. Figure 13, below, however, is suggestive (without elaborating on a function form) of circumstances characterized by a nonlinear expansion path, where the production process characterizing a firm displays an initially high intensity of capital usage (a high initial k/l ratio) eventually giving way to increasingly high intensity of labor usage (a steadily diminishing k/l ratio). Thus, the capital-to-labor ratio (k/l) is a function of the level of output, or, succinctly, the changing marginal productivities of both production factors as we increase the scale of output.

    Figure 13: Nonlinear expansion path with increasing labor intensity of production.
Offering a cursory interpretation of the process suggested in figure 13, it is probable that, beyond a critical level of output, the marginal product of capital diminishes rapidly over the range under consideration relative to the rental rate for capital within factor markets and relative to the marginal product of labor and wage rate.
              This analysis implicates function forms that are inconsistent with our assumptions about a general equilibrium economy. Further, it raises concerns about the returns to scale. To reiterate a critical assumption, firms in a general equilibrium economy will be assumed to operate with production functions chracterized by constant returns to scale. In this regard, homothetic production functions are quite important because they are characterized by invariance of the marginal rate of technical substitution as we scale production factors. We need production functions that embody another mathematical property as well, however. That is, they must be homogeneous of degree 1 (linear homogeneity). This means that if we scale the production factors by a given positive constant, then we will scale outputs by exactly the same proportion. Mathematically:
F[α(l1), α(k1)] = α^k[F(l1, k1)] Where k = 1.
This is precisely the definition of constant returns to scale that I offered earlier.  Graphically speaking, constant returns implies that the firm's expansion path, in both two-dimensional/two-factor space and output-fixed factor ratio space, will be linear, as displayed below in figures 14A and B, respectively. 

                                             A.                                                                       B.
Figures 14A and B: Expansion Path with Constant Returns in Two-Dimensional/Two-Factor Space (A) and Output/Fixed Factor Ratio Space (B).
In this manner, every change in scale along expansion path EP* involves a proportional increase in the capital and labor employed by the firm sufficient to maintain the capital/labor ratio (k*/l*).  In each case of an expansion from some initial scale represented at the point D, output increases by the same proportion.  Thus, if we multiply each of the factors in the initial factor combination by some positive constant a, then output will increase by exactly a.  Expressing this outcome in two-dimensional/two-factor space in figure 14A, we obtain a series of parallel isoquant level curves separated by distances proportional to the change in employment of factors.  Altering the representation of expansion path EP* slightly to express its shape in output/fixed factor ratio space in figure 14B, EP* appears as a ray drawn at 45 degrees from the output/multi-factor origin, reflecting the fact that every change in scale of factor utilization maintaining the fixed factor ratio (k*/l*) by the firm must produce a proportional change in outputs.                      
           Constant returns to scale/linear homogeneity has important consequences for larger functioning of a general equilibrium economy that I will elaborate in the next section and for the distribution of revenues between factors of production that I will elaborate in the succeeding section. For now, I will simply summarize the insights of this section by noting that the sorts of production functions that characterize firms within our larger analysis articulate linear expansion paths on which the distances between isoquant level curves are proportional to the change in scale of production factors in each given profit maximizing/cost minimizing factor combination.

Analyzing Final Commodity Markets from the Insular Window of the Firm

So far, I have attempted to elaborate on the theoretic structure within which Walrasian/Paretian (and Austrian) firms act in order to maximize profits and minimize costs from the production of final commodities (i.e. goods and services).  Within these theories, the actions of the firm are strictly mathematical.  Firms perform a constrained maximization in conformity with the boundaries set, on the one hand, by factor market pricing and, on the other hand, by the quantitative demands of households in final commodity markets.  This section attempts to flesh out the final commodity market constraint experienced by the firm, both in terms of the static determination of output quantities and in regard to market dynamics, most emphatically the transformation of household preferences for final goods and services.
             To reiterate the broader outlines of a general equilibrium economy within which we have situated the firm, we are talking about an economy with large numbers of industries (all drawing from common pools of homogeneous production factors), large numbers of firms in each industry, and very large numbers of households, consuming final commodities and renting out factors of production.  Such a structure ensures that there will be rigorous competition between firms, depriving firms of any ability to set prices.  All firms in a general equilibrium economy are price takers - markets set prices.  The questions for this section concern how markets set prices and how firms respond when markets change.  To the extent that we are talking about an integrated market system here, the reasoning involves an overlap between the foundational analyses of microeconomics and the systemic elaborations of macroeconomic theory, an observation that reveals a great deal about the mindset of Neoclassical theory with regard to aggregates.  Fundamentally, for both the microeconomic theory of the firm and macroeconomic theorizations in subjects like growth theory, the most important foundational unit is the household.  This is where we have to start. 
             I will elaborate a more detailed theory of household demand for goods and services elsewhere, but for now it suffices to say that demand is structured by the maximization of a household utility functions subject to income constraints.  We can pattern household utility functions mathematically in the form:
um = f(x1, x2, ..., xn)
For the m-th household defining the utility it obtains from consuming a market bundle of n different commodities.  Similarly, the income (y) constraint for the household can be patterned in the form:
ym = p1x1 + p2x2 + ... + pnxn
Where the income of the m-th household, derived from its renting out of production factors to firms, equates to a stream of n consumption goods and services at objective final commodity market prices. 
            The first order partial derivatives for such household utility functions will be assumed to be universally positive, with negative second order partial derivatives, implying diminishing marginal utility from consumption of larger quantities of any particular commodity.  As a consequence of diminishing marginal utility, individual household demand curves for individual commodities will be, downward sloping in terms of price.  As households consume more of a given commodity, holding quantities of all other commodities constant, the additional utility derived from consuming the last unit of the commodity will be less than the additional utility received from the previous unit.  Thus, the price the household will be willing to pay in order to consume progressively larger quantities of a good or service will decline as quantities consumed increase.  Aggregating across all households, market demand functions, defining the quantities of a particular good or service demanded by households, will, by assumption, decrease in relation to price.  Graphically, market demand curves will, thus, slope downward in price-quantity space, demonstrating decreasing quantity demanded as prices increase, as illustrated in figure 15.

    Figure 15: Market demand curve for commodity 1.
Elaborating briefly, figure 14 argues that, aggregating across all households, market demand is a function of the own price of a commodity (P1), all other commodity prices (P-1), the incomes of households derived from renting out of production factors (ym, where each ym constitutes an element within the vector Y of household incomes), and the household preferences of all households defined from individual household utility functions (um, where each um constitutes an element within the matrix U of household utility functions).  In simple terms, market demand is a function of a combination of market variables (relative prices) and irreducible individual household variables (divergent individual household preferences for individual commodities and income levels, likewise reflecting the divergent preferences of individual households).         
             These insights on final commodity demand might be relevant to the firm if it was to engage in bargaining with households over the price of final commodities and the quantities the firm would be willing to supply.  However, such behavior extends too much liberty to firms with respect to their interactions with household consumers.  Firms do not negotiate their prices with households - they accept the prices that the market determines.  The larger point that I have attempted so far to establish is that profit maximization/cost minimization by firms is not an entrepreneurial problem, demanding novelty and the virtue of risk-taking in an uncertain environment, but strictly a mathematical problem, requiring a decent scientific calculator!  Assuming both firms and households enjoy perfect information, both regarding technological capacities in the production process and the utilitarian characteristics of particular articles of consumption to discrete consumption population dynamics, there is no way that firms will not succeed at the task of maximizing profit. 
             Notwithstanding its apparent mathematical simplicity, the larger question for us concerns the quantity of output that will be produced, given the inverse relationship of quantity demanded and final commodity price.  This is because, in accordance with our assumptions and the previous analyses of expansion paths, the production function defining the firm is characterized by constant returns to scale and homotheticity.  These two characteristics convey themselves to the larger conclusion that the firm experiences constant average total costs per unit of output, an argument that I will prove in a succeeding section.  For now, I want to advance that, as a consequence of constant average total costs, market supply curves in a general equilibrium economy display universal perfect elasticity with respect to changes in consumer preferences and incomes.  Further, I want to advance a proposition that I will more rigorously address subsequently that, in perfectly competitive equilibria, firms with constant returns to scale/constant average total costs will charge their average total costs to consumers as the market price.  Graphically, the market supply curves drawn in accordance with this theory are perfectly horizontal at the level of average total cost.  Figure 16 attempts to illustrate the relationship between market demand and market supply in light of this proposition. 

    Figure 16: Equilibrium quantities in the market for commodity 1 at three prices. 
Elaborating, the market supply function, aggregating output quantities provided by all firms within the market for commodity 1, is a function of quantity produced (by virtue of the individual underlying production functions/technologies employed by the firms) and of the prices of factor inputs, labor and capital (in turn a function of household preferences in the supply of production factors).  In effect, the market supply schedule simply incorporates the arguments in the profit maximizing/cost minimizing procedures undertaken by each individual firm within the market.
               Holding to the assumption that firms face constant returns to scale/constant average total costs, the existence of three distinct supply functions for three different levels of output must reflect particular changes in the arguments of the supply function.  That is to say, it has to reflect either a change in relative prices in factor markets (and, hence, a change in the capital-to-labor ratio adopted in profit maximizing combinations) or an overall change in production technologies (raising or lowering costs per unit of output for all scales of production as a result of changes to the production functions of individual firms).  In all cases, however, I will maintain the proposition that firms face constant returns to scale/constant average total costs.
               Elaborating briefly on the argument that firms charge their average total costs in competitive equilibrium, the point is that, by assumption, firms are price takers, too small relative to the larger market to set their own prices.  In this manner, competitive pressures must act on all firms to drive prices down to some minimum price at which all firms would be willing to produce quantities of output demanded by households.  In practice, this might convey itself to the conclusion that costs determine the output price that will be charged within individual markets and that, therefore, firms collectively set prices across each market.  Such a conclusion constitutes a deviation from general equilibrium reasoning.  Again, firms are captive agents, continuously acting at the behest of households.  In this manner, a larger explanation of why firms charge their average total costs in competitive equilibrium as a price minimum will have to connect the simultaneous utility maximizing decisions of households in final commodity and factor markets to show why this is the only possible outcome and why it is an outcome wholly orchestrated through the integrated nature of market processes rather than through the conscious strategies of firms.  I will return to this question when I discuss factor markets.     
               At this point, I want to assert the counterintuitive nature of figure 16 in relation to alternative conceptions of the firm in Neoclassical theory.  In particular, the market supply functions that I will draw in reference to Marshallian firms will be characterized by a positive relationship between price and quantity, denoting increasing marginal costs per unit of output as a firm increases output in the short run.  Likewise, Marshallian firms, in the mold defined arithmetically and graphically by the American economist Jacob Viner, incorporate a long run supply function, that may slope upward (decreasing economies of scale), downward (increasing economies of scale), or feature constant long run average total costs (zero slope/long run perfect elasticity of supply with respect to changes in market demand).  Such features are relevant in considering the adjustment mechanisms performed by firms that have an actual, entrepreneurial role to play in short run market pricing and short run determination of output quantities in response to fluctuations in market demand.  In these terms, it makes sense to discuss upward sloping short run supply curves for Marshallian firms, as I plan to explain, because such firms encounter particular short run rigidities due to fixed factors that will raise marginal costs even if long run average total costs are constant. 
               I have sought here, by contrast, to assert that a general equilibrium economy is both continuous and timeless.  The negotiation of output commodities demanded and factor inputs supplied by households is a simultaneous and continuous process, seamless in its functioning with neither the necessity nor the freedom for firms to perform short run price and quantitative adjustments.  In such a theoretic context, where neither a short run nor a long run exists, per se, the assumptions defining the larger theory command firms to operate in accordance with a given logic, and these assumptions include constant returns to scale/constant average total costs as a foundation in the functioning of a general equilibrium economy.  Hence, market supply curves at any moment in time are perfectly elastic at the level of average total costs for all units of output. 
                Arguing in this manner, we can now piece together the logic of general equilibrium from the final commodity market side of the firm.  Households collectively determine that they want particular quantities of output for all commodities in the economy.  They simultaneously determine how much of each factor of production they will willingly rent out to firms in exchange for these commodities.  Market mechanisms across all markets, through the guise of an all encompassing auctioneer, set relative prices in order to bring all final commodity and factor markets into equilibrium, so that households can produce exactly the quantity of final commodities that they want to consume. 
                Interpreting figure 16 in this light, output quantities are set for each market price, by market demand functions characterized, in the aggregate across households, by diminishing marginal utility.  I will endeavor to argue in the next section that households undertake a separate but simultaneous process of utility maximization in factor markets.  These processes jointly determine the parameters for firms to resolve their profit maximization/cost minimization problems.  In particular, if households are willing to supply sufficiently large quantities of labor and capital, under relative prices for labor and capital that set the average total cost per unit of output at P1B, then households will exactly demand X1B from all firms operating within the market.  The problem for individual firms, moreover, reduces itself to the selection of an appropriate combination of labor and capital in order to produce their individual x1B shares of total output for the market, based on the share of financing (money capital) allocated by households to each firm for the production of this commodity.  Figure 17 attempts to depict such a solution for an individual firm with a particularly capital intensive expansion path in which the firm rents k1B capital and l1B labor for a total cost of TC1B to produce its x1B output share.

    Figure 17: Profit maximization for an individual firm producing x1B output.  
Holding firmly to our assumptions of constant costs and homotheticity in regard to production functions, any of the production levels along expansion path EP1 for the representative firm in figure 17 will have the same average total cost per unit of output.  Thus, any shift in output demand arising from changes in household preferences that does not simultaneously result in a change in relative prices for production factors will force the firm to select new factor combinations, based on its level of financing from households renting out money capital, along expansion path EP1.  If quantities of commodity 1 demanded by households increase, together with the willingness of households to rent out larger quantities of money capital to firms producing commodity 1, then our representative firm may begin to produce at level x1E.  Conversely, a decrease in quantities demanded, together with a decrease in the willingness of households to finance commodity 1 production, might bring the firm to producing at level x1D.  Conceptually, in reference to the larger market outcome, we can portray such shifts in demand in the manner illustrated in figure 18.

Figure 18: Equilibrium quantities for commodity 1 for three demand schedules with perfectly elastic supply.
Again, the point here is that, barring any change in the relative prices of factors of production, firms within any given industry, producing a particular commodity, will be perfectly responsive to changes in demand for the commodity by households because such changes will simultaneously be reflected in changes to the availability of financing to firms.  The interconnected nature of a general equilibrium economy ensures that any change in household demand for final goods and services must be accompanied by reinforcing changes to factor markets (demand for more goods and services must be accompanied by increased supply of factors).  In the event that such changes do not permanently change relative factor prices, profit maximization by representative firms, like the one in figure 16, will be restricted to factor combinations along the same expansion path, in which average total costs per unit of output are constant.  Constant average total costs generate perfectly elastic market supply schedules, like S1 in figure 18, reflecting the capacity of firms within the industry to vary production levels without encountering overall increases in per unit costs as output increases.
              Following from our general equilibrium assumption on the access of firms and households to perfect information on production technologies, the representative firm in figure 16 must be employing technologies that are most efficient in relation to the use of production factors to produce commodity 1.  That is to say, no other ways of combining factors of production to produce a given level of commodity 1 output exist such that the firm can reduce its employment and rental costs for one factor of production without having to increase its employment and costs for the other production factor to produce the same level of output.  If each firm within the industry, moreover, is assumed to be engaged in price competition with every other firm in the industry, then all firms must be using identical least cost technologies.  Thus, the production function used by the firm above, the cost function it faces, and the expansion path which the production function and cost function jointly articulate must be identical for every other firm within the industry. 
              For the sake of argument and as a means of drawing parallels to the analysis that I will advance in discussing Marshallian firms, I want to deviate from this conclusion to argue that some firms might enjoy some cost advantage that they are able to isolate from other firms.  What effect might such a cost advantage be expected to have on competition between firms and on market equilibria?  It is my contention here that the presence of such a cost advantage will have no effect on the competitiveness of firms possessing access to it and will not change the equilibrium price charged for the commodity in the market.  Here is the reason why.
             Let us say that in a given market, say the market for bread, all firms have access to universally available technologies that enable each firm to produce bread for the lowest possible cost per loaf of bread.  Households pay a price for bread that is exactly equal to this lowest possible cost, because no bakery is capable of charging anything above the lowest possible cost without being competed out of the market.  Suddenly, a new technology is developed that can produce bread at a lower cost in terms of both labor and capital expended to produce all levels of output.  However, this technology is so new that not every bakery is able to access it.  Moreover, somebody owns the capacity to grant access to the new technology.  This somebody is a household that owns the technology and, maybe through a patent right, has the ability to extract rents from bakeries that it licenses to use the technology.  The household has no incentive to see certain bakeries profit excessively from the use of its technology (any excess revenues earned by a bakery will just enable it to pay other households more than the market will command for their labor and capital), so it will charge bakeries that want to use the technology exactly the difference between the average total cost per loaf of bread using the older, universally available lowest cost technology and the average total cost per loaf using the household's new technology.  In this manner, innovative bakeries, using the new technology, will be forced to charge exactly the same price per loaf of bread as every other bakery using the older technology.  I attempt to illustrate this situation in figure 19.

Figure 19: Market equilibrium under competition with certain firms using a costly technology to reduce labor and capital costs per unit of output.
Elaborating on figure 18 in reference to our bakery example, the price per loaf of bread is P1B, which equals the average total cost per loaf of bread (TC1B/X1B) expended by bakeries in the market.  However, certain firms can get access to new baking technology, restricted by the household that invented the technology.  In terms of labor and capital costs per loaf, these bakeries are producing bread for TC1B'/X1B'.  On the other hand, they face the additional cost PT for gaining access to the new technology.  This cost, that must be paid to the household that invented the new way of baking bread, will equal the difference between TC1B/X1B and TC1B'/X1B'.  Because these bakeries have to pay a rent out to the inventor of their technology, they face total costs that are exactly equal to those using the older, less efficient technology.  Thus, they should be perfectly indifferent in selecting between the older, less efficient technology and the newer, lower cost technology.  In other words, their cost advantage disappears entirely, so they have no incentive to produce larger quantities of bread than those that would be produced if every firm was still using the older technology.  The equilibrium output here will be X1B.
               In passing from the example illustrated by figure 19, I want to point out something that I will comment at length about later in this document.  If all firms within the industry had access to the new technology, under constant household preferences, they would produce X1F output.  In other words, the development of a new, more efficient technology means that households can enjoy larger quantities of consumption, but the ability to restrict the use of the technology to the advantage of its inventor (who receives rent from firms as a condition of use) means that the technology will have no impact of output levels.  This outcome has consequences that can be measured through Neoclassical welfare analysis.