Cost Functions
Having described, in general terms, the concept of a production function, one conceptual tool remains to be developed in order to analyze production by firms in a general equilibrium economy. If production functions (as the definitive structures of Neoclassical firms) emphatically determine the quantities of output that a firm can produce with given combinations of labor and capital, then the cost functions convey the objective factor market constraints that firms face in selecting how much labor and capital they can afford to rent from housholds in order to produce goods and services. In effect, the cost function expresses the budget constraint faced by firms, relative to their abilities to finance the rental of factors.
To reiterate the basic points developed in the previous sections, firms do not own factors of production. The land, labor, capital, technological information, and all other particulars of production all individually belong to households, and households will only rent them out, for a given finite period, in exchange for the consumption possibilities that arise from having their factors productively employed. That is to say, firms can only produce goods and services because households demand objects of consumption, and their demand for goods and services must be balanced by a willingness to commit the land, labor, capital, and technology necessary for production to a collective endeavor to produce them. In a general equilibrium economy, the various decisions of households regarding consumption of final goods and services and renting out of production factors are fully integrated such that these decisions are instantaneous. Again, this conclusion arises from an assumption of perfect information for both firms (on the most efficient possible means of producing goods and services using given production factors) and for households (in linking the separate utility maximizing decisions on what they want to consume and what production factors they will need to commit in order to secure its production). For the moment, I am not going to critique the steep information requirements evident here. It suffices to say that the firms, within this outline of general equilibrium thinking, face a relatively simple task while households face two extremely demanding decisions that will pose important consequences on the workings of the entire economic system.
The further point here is that the necessity of perfect information is satisfied, in a wholly decentralized way, through the functioning of markets. When individual households come together for market exchange, we will assume that they instantaneously obtain accurate information on how much land, labor, capital, and technology they will have to rent out to firms and how many final goods and services they can expect to receive in exchange. This raises the question of how it might be possible that market activity produces and circulates this information. When Leon Walras first developed his ideas about a general equilibrium economy in the 1870s, he treated all market activity in terms of auctioning behavior, and he personified its decentralized direction through the personage of an invisible auctioneer.
When individual households, thus, theoretically enter the combined, simultaneous (and continuous) market context for exchange of final goods and services and for exchange of production factors, they express (implicitly in reality, through their consumption and occupational choices) how much of each good and service they want and how much of each of the factors of production they own they are willing to commit in exchange for all of the goods and services they want to consume. In the end, once all households have expressed how much of every good and service they want and how much of each factor they are willing to supply, each decentralized market, through the fictional direction of the auctioneer, realizes an equilibrium of quantity supplied and quantity demanded. General equilibrium theory assumes that this process is continuous and the simultaneous expressions of In a broader sense, the total quantities of land, labor, capital, and technology supplied by households to firms enable firms to produce the exact quantities of final goods and services that households demand through final good and service markets.
We assume this equilibration of all factor and final commodity markets would occur even if the economy operated strictly through non-monetary barter exchange between households. In this sense, there is no need to have some monetary equivalent representing the purchasing power received by households in exchange for renting out their production factors to firms. On the other hand, the inclusion of such a monetary equivalent constitutes a simplifying assumption, enabling us to apply a superfluous money-name to each unit of final goods and services and to each unit of production factors, without otherwise transforming the basic market exchange processes, amounting to a barter exchange between households, that underlies the inscription of exchange objects with a monetary quantity. I want to stress the superfluous character of this inscription. For both the Walrasian/Paretian and Austrian sub-traditions in Neoclassical economics, money is neuter. It does not constitute a good, capable of delivering utilitarian satisfaction to a household in and of itself. The satisfaction derived from money manifests itself in the capacity of money to finance purchases of goods and services that will enhance the utility of households through consumption. Thus, for the purposes of the theory of the firm we are developing here, money is a value-free descriptor.
Having issued these further elaborations on general equilibrium thinking, especially with regard to the role of money, I can present a basic definition of the cost function faced by a firm, as:
TC1 = Pll1 + Pkk1 + Pnn1
Where TC expresses the total cost for the production of good 1 as a sum of the costs of renting labor(l), capital(k), and land(n) from households. Each of these cost components is simply a product arising from the multiplication of the total quantity of each factor (e.g. l1 for the total quantity of labor employed in producing good 1) by its objective monetary factor market price (e.g. Pl for the monetary wage rate per unit of labor). Again, the reason wht the factor market prices faced by the firm can be regarded as objective arises from my general equilibrium assumptions. If each firm is simply one small unit of production among an industry with many firms and among many industries all drawing from common pools of land, labor, and capital, then there is no way that any given firm can exercise power in factor markets to determine how much they will pay per unit of land, labor, or capital. Within the theory of the firm developed in this document, firms are always and in every conceivable context price takers, incapable of affecting the prices that they are charged for inputs or the prices that they receive for outputs.
I will assume, in this respect, that the kinds of technologies necessary to produce goods and services are in some way subsumed within capital, perhaps as some element of human capital. Again, this simplifies that factor market problem faced by the firm. I will also assume that in most cases firms do not demand special, quantitatively limited factors of production from households that command some particular, elevated rental price. Lastly, in order to put myself on the same page as my two-factor production function in the previous section, make the (somewhat controversial) simplifying assumption that land is available to the firm at zero cost. In a certain respect, the last two assumptions here are related. Land, in general, involves quantatively limited resources, even if the absolute limits to such resources as petroleum or even water do not always make themselves totally apparent. Rental rates on land/natural resources can be a fairly complicated subject. I plan to discuss it but not in this document - it goes far beyond my limited purposes. Beyond this, theorists in the Walrasian/Paretian tradition tend to proceed with ease in making the simplifying assumption that firms can get land at zero cost, limiting firms to the decision of how much labor and capital they will need to efficiently generate certain quantities of output, subject to implicit limitations on the quantity of land being used in relation to their production functions. With all this in mind, for our purposes, the cost function we will use can be stated as:
TC1 = Pll1 + Pkk1
This cost function, thus, contains two independent variables and a dependent variable expressed as the sum of two products. The right hand arguments of the function express how much labor and capital a firm can purchase for any given level of total cost at given factor market prices. Thus, the function encapsulates a range of combinations of labor and capital for every level of total cost and conveys the possibility of substitution between factors in order to maintain a given cost level at given monetary factor market prices. Graphically, in two-dimensional lk space, the cost function can be expressed as a linear constraint and a locus of all possible combinations of labor and capital attainable at each given level of total costs. These linear constraints, shown below in figure 5, will henceforth be denoted as isocost curves, meaning "same-cost."
Figure 5: Isocost Curves in lk space at three Total Cost (TC) Levels
Elaborating, at each total cost level, the linear isocost curves subsume a continuous range of substitutions between quantities of labor and capital. Figure 5 illustrates the maximum levels for labor and capital as l and k intercept values (lmax and kmax), respectively, for three distinct levels of total cost, where TCC < TCB < TCA. As such, if the firm selects a production process exclusively utilizing labor (i.e. a selection represented as an intercept with the l-axis) at total cost level TCA, then, manipulating the cost function algebraically, we conclude:
TCA/Pl - (Pk/Pl)kA = lA (1)
TCA/Pl = lA = lmax(TCA) (2) Because kA = 0, at the l-intercept.
Thus, graphically, the isocost curves depict a continuous set of points between the k and l intercepts (and inclusive of the intercepts) where a combination of labor and capital, at given factor market prices completely expends a given total cost level. Any movement along a particular isocost curve, such as the movement between points γ and τ at total cost TCA in figure 6, involves no change in the total cost incurred by the firm.
Figure 6: Change in Combination of Labor (l) and Capital (k) rented at a fixed Total Cost (TC) Level A
Conversely, we can pattern any increase in total costs incurred by a firm through a shift from one isocost curve to a higher isocost curve, denoting a higher level of total costs. This is illustrated through an increase in the amount of labor rented by a firm from level β to α, holding capital (k*) constant, in figure 7.
Figure 7: Change in Total Costs (TC) from Change in Labor Rented at Fixed Wage Rates (Pl)
At this point, we need to develop a connection between the concept of the cost function elaborated in this section and the production function elaborated in the previous section. The latter function provides determinate quantities of output for every combination of labor and capital rented by a firm. By contrast, the cost function tells us nothing about output quantities. It simply tells the firm how much it will cost to rent specific quantities of labor and capital, which will, subsequently be combined to produce output quantities specified by a production function. Graphically illustrated in xlk space, the isocost curves of a cost function appear as series of vertical planes, anchored at intercept quantities on the l and k axes, as illustrated by figure 8.
Figure 8: A Cost Function in xlk space at three Total Cost (TC) Levels
Elaborating, for every output level x0 and any given set of factor market prices, there exists a continuous series of isocost curves. These isocost curves are vertical and extend to infinity along the x axis. In this regard, cost functions and the isocost curves that they generate exist independently of x output levels.
Closing out this section with a summary of the assumptions I have developed here:
1. Firms in a general equilibrium economy are factor market price takers. Given their size relative to factor markets, they are incapble of independently selecting wage rates for the labor they rent from households, interest rates for the capital/technology they rent from households, or rental rates for land/natural resources they rent from households. Therefore, we will treat factor market prices as continuously objective.
2. Households supply factors of production in exchange for consumption possibilities. It is conceviable to amend this assumption in reference to intertemporal consumption portfolios, with positive rates of interest, but this redefinition would just readjust consumption possibilities over a broader timeframe in order to expand future consumption possibilities relative to present-time consumption. In either case, the sole purpose for households of supplying land, labor, and capital to firms is to consume articles that can derive some measure of utility.
3. Factor market prices, expressed in monetary terms, denote a set of real consumption possibilities. The point here is to definitively argue that the point of supplying land, labor and capital is not to receive monetary compensation but to receive an equivalent mass of consumption goods and services paid from the receiving of a certain quantity of the monetary equivalent. Money is neuter.
4. Land, as a factor of production, is considered fixed and, for purposes of cost analysis, has a zero rate of return. Again, this constitutes a controversial simplification on real economic transactions for the firm. It is employed to enable us to perform a simple, two-factor maximization in the following section. However, I intend to perform a critique of this simplification in another context.
5. Labor and Capital are continuously substitutable for the firm for any level of total costs, through the mathematical instrument of the cost function. This assumption is sufficient to denote the particular linear details expressed in our analysis of the cost function. Insofar as the isocost curves, defined by each, continuous level of total costs, are continuous between lk intercepts, the firm is capable of arriving at discrete substitutions between fixed quantities of abstract labor and capital.
Having described, in general terms, the concept of a production function, one conceptual tool remains to be developed in order to analyze production by firms in a general equilibrium economy. If production functions (as the definitive structures of Neoclassical firms) emphatically determine the quantities of output that a firm can produce with given combinations of labor and capital, then the cost functions convey the objective factor market constraints that firms face in selecting how much labor and capital they can afford to rent from housholds in order to produce goods and services. In effect, the cost function expresses the budget constraint faced by firms, relative to their abilities to finance the rental of factors.
To reiterate the basic points developed in the previous sections, firms do not own factors of production. The land, labor, capital, technological information, and all other particulars of production all individually belong to households, and households will only rent them out, for a given finite period, in exchange for the consumption possibilities that arise from having their factors productively employed. That is to say, firms can only produce goods and services because households demand objects of consumption, and their demand for goods and services must be balanced by a willingness to commit the land, labor, capital, and technology necessary for production to a collective endeavor to produce them. In a general equilibrium economy, the various decisions of households regarding consumption of final goods and services and renting out of production factors are fully integrated such that these decisions are instantaneous. Again, this conclusion arises from an assumption of perfect information for both firms (on the most efficient possible means of producing goods and services using given production factors) and for households (in linking the separate utility maximizing decisions on what they want to consume and what production factors they will need to commit in order to secure its production). For the moment, I am not going to critique the steep information requirements evident here. It suffices to say that the firms, within this outline of general equilibrium thinking, face a relatively simple task while households face two extremely demanding decisions that will pose important consequences on the workings of the entire economic system.
The further point here is that the necessity of perfect information is satisfied, in a wholly decentralized way, through the functioning of markets. When individual households come together for market exchange, we will assume that they instantaneously obtain accurate information on how much land, labor, capital, and technology they will have to rent out to firms and how many final goods and services they can expect to receive in exchange. This raises the question of how it might be possible that market activity produces and circulates this information. When Leon Walras first developed his ideas about a general equilibrium economy in the 1870s, he treated all market activity in terms of auctioning behavior, and he personified its decentralized direction through the personage of an invisible auctioneer.
When individual households, thus, theoretically enter the combined, simultaneous (and continuous) market context for exchange of final goods and services and for exchange of production factors, they express (implicitly in reality, through their consumption and occupational choices) how much of each good and service they want and how much of each of the factors of production they own they are willing to commit in exchange for all of the goods and services they want to consume. In the end, once all households have expressed how much of every good and service they want and how much of each factor they are willing to supply, each decentralized market, through the fictional direction of the auctioneer, realizes an equilibrium of quantity supplied and quantity demanded. General equilibrium theory assumes that this process is continuous and the simultaneous expressions of In a broader sense, the total quantities of land, labor, capital, and technology supplied by households to firms enable firms to produce the exact quantities of final goods and services that households demand through final good and service markets.
We assume this equilibration of all factor and final commodity markets would occur even if the economy operated strictly through non-monetary barter exchange between households. In this sense, there is no need to have some monetary equivalent representing the purchasing power received by households in exchange for renting out their production factors to firms. On the other hand, the inclusion of such a monetary equivalent constitutes a simplifying assumption, enabling us to apply a superfluous money-name to each unit of final goods and services and to each unit of production factors, without otherwise transforming the basic market exchange processes, amounting to a barter exchange between households, that underlies the inscription of exchange objects with a monetary quantity. I want to stress the superfluous character of this inscription. For both the Walrasian/Paretian and Austrian sub-traditions in Neoclassical economics, money is neuter. It does not constitute a good, capable of delivering utilitarian satisfaction to a household in and of itself. The satisfaction derived from money manifests itself in the capacity of money to finance purchases of goods and services that will enhance the utility of households through consumption. Thus, for the purposes of the theory of the firm we are developing here, money is a value-free descriptor.
Having issued these further elaborations on general equilibrium thinking, especially with regard to the role of money, I can present a basic definition of the cost function faced by a firm, as:
TC1 = Pll1 + Pkk1 + Pnn1
Where TC expresses the total cost for the production of good 1 as a sum of the costs of renting labor(l), capital(k), and land(n) from households. Each of these cost components is simply a product arising from the multiplication of the total quantity of each factor (e.g. l1 for the total quantity of labor employed in producing good 1) by its objective monetary factor market price (e.g. Pl for the monetary wage rate per unit of labor). Again, the reason wht the factor market prices faced by the firm can be regarded as objective arises from my general equilibrium assumptions. If each firm is simply one small unit of production among an industry with many firms and among many industries all drawing from common pools of land, labor, and capital, then there is no way that any given firm can exercise power in factor markets to determine how much they will pay per unit of land, labor, or capital. Within the theory of the firm developed in this document, firms are always and in every conceivable context price takers, incapable of affecting the prices that they are charged for inputs or the prices that they receive for outputs.
I will assume, in this respect, that the kinds of technologies necessary to produce goods and services are in some way subsumed within capital, perhaps as some element of human capital. Again, this simplifies that factor market problem faced by the firm. I will also assume that in most cases firms do not demand special, quantitatively limited factors of production from households that command some particular, elevated rental price. Lastly, in order to put myself on the same page as my two-factor production function in the previous section, make the (somewhat controversial) simplifying assumption that land is available to the firm at zero cost. In a certain respect, the last two assumptions here are related. Land, in general, involves quantatively limited resources, even if the absolute limits to such resources as petroleum or even water do not always make themselves totally apparent. Rental rates on land/natural resources can be a fairly complicated subject. I plan to discuss it but not in this document - it goes far beyond my limited purposes. Beyond this, theorists in the Walrasian/Paretian tradition tend to proceed with ease in making the simplifying assumption that firms can get land at zero cost, limiting firms to the decision of how much labor and capital they will need to efficiently generate certain quantities of output, subject to implicit limitations on the quantity of land being used in relation to their production functions. With all this in mind, for our purposes, the cost function we will use can be stated as:
TC1 = Pll1 + Pkk1
This cost function, thus, contains two independent variables and a dependent variable expressed as the sum of two products. The right hand arguments of the function express how much labor and capital a firm can purchase for any given level of total cost at given factor market prices. Thus, the function encapsulates a range of combinations of labor and capital for every level of total cost and conveys the possibility of substitution between factors in order to maintain a given cost level at given monetary factor market prices. Graphically, in two-dimensional lk space, the cost function can be expressed as a linear constraint and a locus of all possible combinations of labor and capital attainable at each given level of total costs. These linear constraints, shown below in figure 5, will henceforth be denoted as isocost curves, meaning "same-cost."
Elaborating, at each total cost level, the linear isocost curves subsume a continuous range of substitutions between quantities of labor and capital. Figure 5 illustrates the maximum levels for labor and capital as l and k intercept values (lmax and kmax), respectively, for three distinct levels of total cost, where TCC < TCB < TCA. As such, if the firm selects a production process exclusively utilizing labor (i.e. a selection represented as an intercept with the l-axis) at total cost level TCA, then, manipulating the cost function algebraically, we conclude:
TCA/Pl - (Pk/Pl)kA = lA (1)
TCA/Pl = lA = lmax(TCA) (2) Because kA = 0, at the l-intercept.
Thus, graphically, the isocost curves depict a continuous set of points between the k and l intercepts (and inclusive of the intercepts) where a combination of labor and capital, at given factor market prices completely expends a given total cost level. Any movement along a particular isocost curve, such as the movement between points γ and τ at total cost TCA in figure 6, involves no change in the total cost incurred by the firm.
Conversely, we can pattern any increase in total costs incurred by a firm through a shift from one isocost curve to a higher isocost curve, denoting a higher level of total costs. This is illustrated through an increase in the amount of labor rented by a firm from level β to α, holding capital (k*) constant, in figure 7.
At this point, we need to develop a connection between the concept of the cost function elaborated in this section and the production function elaborated in the previous section. The latter function provides determinate quantities of output for every combination of labor and capital rented by a firm. By contrast, the cost function tells us nothing about output quantities. It simply tells the firm how much it will cost to rent specific quantities of labor and capital, which will, subsequently be combined to produce output quantities specified by a production function. Graphically illustrated in xlk space, the isocost curves of a cost function appear as series of vertical planes, anchored at intercept quantities on the l and k axes, as illustrated by figure 8.
Elaborating, for every output level x0 and any given set of factor market prices, there exists a continuous series of isocost curves. These isocost curves are vertical and extend to infinity along the x axis. In this regard, cost functions and the isocost curves that they generate exist independently of x output levels.
Closing out this section with a summary of the assumptions I have developed here:
1. Firms in a general equilibrium economy are factor market price takers. Given their size relative to factor markets, they are incapble of independently selecting wage rates for the labor they rent from households, interest rates for the capital/technology they rent from households, or rental rates for land/natural resources they rent from households. Therefore, we will treat factor market prices as continuously objective.
2. Households supply factors of production in exchange for consumption possibilities. It is conceviable to amend this assumption in reference to intertemporal consumption portfolios, with positive rates of interest, but this redefinition would just readjust consumption possibilities over a broader timeframe in order to expand future consumption possibilities relative to present-time consumption. In either case, the sole purpose for households of supplying land, labor, and capital to firms is to consume articles that can derive some measure of utility.
3. Factor market prices, expressed in monetary terms, denote a set of real consumption possibilities. The point here is to definitively argue that the point of supplying land, labor and capital is not to receive monetary compensation but to receive an equivalent mass of consumption goods and services paid from the receiving of a certain quantity of the monetary equivalent. Money is neuter.
4. Land, as a factor of production, is considered fixed and, for purposes of cost analysis, has a zero rate of return. Again, this constitutes a controversial simplification on real economic transactions for the firm. It is employed to enable us to perform a simple, two-factor maximization in the following section. However, I intend to perform a critique of this simplification in another context.
5. Labor and Capital are continuously substitutable for the firm for any level of total costs, through the mathematical instrument of the cost function. This assumption is sufficient to denote the particular linear details expressed in our analysis of the cost function. Insofar as the isocost curves, defined by each, continuous level of total costs, are continuous between lk intercepts, the firm is capable of arriving at discrete substitutions between fixed quantities of abstract labor and capital.
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