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.
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.
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.
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.
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.
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.
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.
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.
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.
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