The idea that I am attempting to convey here is that the firm in mainstream Neoclassical economic theory is
not necessarily the firm in the real world. It is an object of theory. But some Neoclassical theories are more real than others.
What I mean by this is that there isn't a single, unitary Neoclassical tradition. Rather, Neoclassical economics, like every other theoretic tradition in the history of economics, is an amalgam of different theories, some of which contrast very starkly.
The Walrasian and Paretian traditions in
Neoclassical economics, which effectively constitute a single theoretic approach originating with the general equilibrium theory of the French theorist Leon Walras, is highly mathematical and makes substantial use of both linear algebraic methods and differential calculus to develop an abstract approach to consumer and firm behavior.
The idea of the circular flow, represented in class and used heavily in macroeconomic classes, is a product of Walrasian theory, representing the idea of a general equilibrium economy in a simple diagram. The pure theory of
the firm that I am developing here is based on the assumptions of a general
equilibrium economy evident in the circular flow. Walrasian theory is
not, however, the only version of Neoclassical economic theory. Marshallian
theory, based on the theories of the English theorist Alfred Marshall, represents an alternative approach to the theory of the firm. Marshall, who had been a trained mathematician, presented a theory that attempted to be less
mathematical and much more empirical, specifically accounting for the fact that the
operation of firms operated within specific time frames (i.e. a long run and a short
run). In differentiating between a long run and a short run, Marshall and his theoretic descendants (including not only the Chicago School economists
(Friedman, Stigler, Coase, etc.) but also Marshall's eventual replacement as
the chair of economics at Cambridge, J.M. Keynes) follow a distinction that was recognized by most of the Classical economists (e.g. Ricardo, Mill, Marx, etc.).
It is, moreover, relevant that no microeconomics textbook worth its salt would not theorize the distinction between the operation of firms in the long run and the short run.
In fact, I would make the point that the theory of the firm that has come down through the development of Neoclassical theory to be learned by most contemporary economics undergraduate students constitutes a patchwork of Marshallian
and Paretian/Walrasian insights.
With that in mind, I am going to suggest that
a pure Walrasian perspective on the firm has no conception of time frame. By this I mean
that it has no long run, and the short run holds no distinction between fixed and variable factors of production. For Walrasian general equilibrium theory, economic activity is continuous and the imposition of a temporal framework is
arbitrary. The decisions of households on factor supply and output demand
are simultaneous, and firms, which exist merely as contingent assemblies of factors of production, just provide a mechanism for households to achieve the
production of the goods and services they need. Household decisions on factor supply
and output demand occur on a continuous basis - there is no way to meaningfully break up economic activity into a long and short run, and there are no firms,
as real entities that exist outside of theory, to make decisions about long and short run operations. There are only production functions that can tell
us the most efficient way to put factors of production together to produce
commodities.
The fact that Walrasian theory has no
conception of the long run reflects the further implication of the Austrian tradition of Neoclassical theory that there is no durable social institution of the firm. Austrian theory, as the most libertarian tradition of Neoclassical economics, holds the
conviction that all economic activity must occur under the conditions of total moral freedom for the individual, as a utility maximizer. Any social institutions that constrain individual freedom are, thus, harmful to the individual's ability to maximize their utility and, hence, individuals will
only
take part in collective, social processes if such processes contribute to the individual's utility and do not constrain the individual's liberty. By
recourse to this reasoning, no free individual utility maximizer would submit to participate in the firm, supplying his or her factors of the production, if
they were not capable of augmenting their utility, and if they were constrained in any way by their participation in the firm. By implication, the firm,
as a social institution, is just an assembly of free individuals, who supply factors of production, with the expectation of augmenting their utility by cooperating to produce commodities. If they ceased to enjoy they expectation that
they
could augment their utility, the firm would cease to exist and the factors of production would revert to their individual owners.
What I characterize as a pure Neoclassical
theory of the firm is, thus, a combination of the Walrasian/Paretian and Austrain conceptions of the firm, leaning heavily on the former for mathematical/analytical tools, and on the latter for assumptions on the social durability of the firm as an institution and its relationship to utility maximizing individuals. I will contrast
this
conception of the firm with the Marshallian conception and, more generally, Marshallian-inspired conceptions. The reason why I
want to do so is, as I suggested, performative in nature. Specifically, I want to
put into relief a particular interpretation of the differentiation of the long and short run, as a practical insight of the Marshallian approach that, in my view,
calls the entire project of the Neoclassical tradition, including both supply and demand-side analysis, into question.
Firms as Production Functions
In Walrasian theory, there are two agents in an economic
system: households and firms. Households
demand final goods and services and supply factors of production in order to
produce goods and services for consumption.
Firms rent factors of production and produce goods and services. This conception of the relationship between
households and firms is indicative of a one-sided dynamic. Firms are not entrepreneurial, they do not
produce goods and services under conditions of risk in order to appeal to the
demands of consumers, and they do not invest in uncertain expansions of scale
to speculate on changes to market conditions.
Rather, households determine what and how many goods and services they
mean to consume and, likewise, how many of each factor of production they
intend to supply to firms in order to produce.
Firms merely respond to the demands of consumers. This is a distinctly different conception of
firms from that contained in Marshallian theory, and one profoundly at variance
with the existence of firms in the real world.
We can go
even further in drawing a distinction between the Walrasian theoretic firm and
real firms by enquiring into the nature of property relations within the
firm.
Real firms can be classified as
sole proprietorships, partnerships, or corporations based on the legal
relations of ownership over assets possessed by the firm.
In proprietorships and partnerships, the
owners of firms include the assets of the firm within their personal property,
together with the owners’ residential and personal financial resources.
In corporations, ownership of the firm’s
assets are vested in the corporation as an artificial person, created under
specific legal provisions under conditions of limited liability to the
corporation’s shareholders.
In all such
cases, the firm clearly holds property.
This is not the case for Walrasian
theory.
The Walrasian firm assembles
factors of production without owning any of these factors.
The land/space within which the firm produces
commodities is rented from households, who obtain a rate of compensation in the
form of rent.
The machinery used by the
firm is also rented from households.
In
this circumstance, it may not be the case that households directly own
machinery, but they own the money capital necessary to finance the production
and purchase of machinery, for which they must be compensated with a rate of
interest.
The labor undertaken by
workers for the firm is most clearly not owned by the firm – their labor is
rented by the firm from households in exchange for a wage.
Even the technology/information required by
the firm is, in some sense, rented from households, who receive compensation
for special forms of labor services.
The point here is that Walrasian
theory advances a specific interpretation of real firms, emphasizing the
contingent nature of collective action by individual households in the context
of commodity production.
In point of
fact, the assets of a corporation like General Motors are held by shareholders,
who can be seen as the household owners of land, capital, and information and
entrepreneurial services.
If General
Motors files for chapter 11 bankruptcy this year, this property, to the extent
that it can be recovered as a definite quantity of wealth through liquidation,
must revert to its owners (i.e. to the household financiers of General Motors,
whose money capital has been rented on the contingent assumption that they
would enjoy a positive rate of return for deferring consumption of their
income).
As abstract as this may then
seem on its face, there is something to the Walrasian perspective.
The approach, together with the partly allied
perspective of the Austrian tradition, is performative in shaping the way
individuals view the real existence of firms to argue that firms have no
existence without factors of production, but these
factors are never alienated
in perpetuity from the households that supply them.
If the firm ceases to exist, the rented
factors of production just go back to their respective owners.
The approach seeks to emphasize, critically,
that
all economic processes turn on the utility maximizing decisions of
households, of which
firms are captive agents.
If the firm is not reducible to the
factors of production it possesses but does not own, it is, in some sense,
reducible to the production process it undertakes.
Even if it does not own the production
technologies necessary to put factors of production together into a determinate
recipe for final commodities, it does constitute the context within which
production does take place.
This makes
the idea of a firm identical to the idea of a production process and, more
theoretically, with the concept of a production function.
A production function is the determinate
mathematical recipe that the firm follows in order to produce commodities.
Without getting into a rigorous criticism of
the concept of a production function (yet), my intention is to elaborate on how
the firm uses its production function to find the most efficient technologies
to produce commodities for household consumption.
General Equilibrium and the Role of Firms: The Assumptions
The pure theory of firms is based on the logic of a general
equilibrium economy. Such an economy, as
suggested, operates as a continuous series of simultaneous household utility
maximization problems. Households decide
how many and what kind of commodities they want, and, as a consequence, how
much and what kind of production factors in their ownership they will provide
in order to produce such commodities.
Certain households provide only labor services and specialize in
particular production processes, while others contribute labor and capital
(deferred consumption income) to a range of production processes. In the larger structure of the economy, the
utility maximizing decisions of households converge through two sets of
exchange processes that distribute factors of production to firms based on the
quantity of final goods and services demanded by households through the
mechanism of pricing. That is, if
households want more cars, then the price of cars will increase, reflecting the increase in demand for cars and allowing firms
producing cars to demand more labor and capital at higher rates of composition
relative to other industries. Once the
appropriate quantity of labor and capital has been distributed to car
production, the rate of compensation of labor and capital in the industry
reverts to the average rate in other industries. In this sense, market pricing in final good
and factor markets, driven wholly by the choices of households, determines the
distribution of factors to firms and composition of final goods and services
produced.
Again,
firms, as contexts for production with no substantial existence, simply follow
the direction of households with regard to both these sets of markets.
This makes the analysis of what firms do
relatively simple – they maximize their production functions to produce as efficiently
as possible relative to both input and output prices.
To build a simple account of what firms do,
we need a set of assumptions, however.
First, in any given market context,
both on the input (factor market) and output (final commodity) side, firms operate in competition with large
numbers of other firms.
If we further
make the assumption that the
labor, land, and capital that firms borrow from
households are perfectly homogeneous, we can further argue that a firm
specialized in producing one type of commodity competes for factors of
production with firms producing entirely different types of commodities.
This sort of competitive environment ensures
that firms will be
price takers.
They will be unable to affect input market prices, because any attempt
to lower their compensation of production factors will drive factors off to
other firms.
For the same reason, they
will be unable to affect output market prices, because
outputs are perfectly
homogeneous and any attempt to raise prices above the output market price will
drive away consumers who can purchase virtually identical commodities from
other firms at the market price.
The
firm, therefore, treats input and output market prices as objective
constraints.
The nature of competition is
buttressed by the fact that
access to productive technological information is perfect
for all firms.
No firm in any given
production process enjoys any technological advantage relative to other
firms producing the same commodity.
Further, any cost advantage from
access to particular resources is completely compensated to households in the
form of rent.
That is to say, particular
firms might enjoy access to good input resources needed to produce commodities,
but in order to get such access, they must compensate the households who own
these resources with rental payments.
By
the same sort of reasoning, in the off chance that particular households owned
technological information on more efficient production processes to which they
could control access, they would have a source of extracting rental payments
from firms as a condition of using the technology.
In either case, there is no way that firms
can gain access to any source of material advantage against other firms without
having to compensate households for the source of their advantage.
Finally, as
the most basic assumption about firms in a general equilibrium economy,
all
firms are profit-maximizing agents.
This condition constitutes a requisite of perfectly rational behavior by firms as commodity producing agents. Functionally, this means that the firm selects an output level relative
to the price of final commodities households are willing to pay and selects a
cost minimizing combination of production factors subject to this output/total
cost level.
At this combination, the
additional
return that the firm gets for the last unit of the commodity produced (i.e. its
marginal revenue) will exactly equal the additional cost of producing the last
unit (i.e. its marginal cost). If
the firm produced an output level where marginal revenue exceeded marginal
cost, it would forego additional revenue that could be generated from an
increase in production and sales to consumers.
This foregone revenue would constitute an opportunity cost of the
production process that could be minimized by increasing production.
If it produced at an output level where
marginal cost exceeded marginal revenue, then the last unit produced would
generate less revenue than it cost the firm to produce this unit.
This would constitute an excess cost that could
be minimized by reducing production quantities.
Thus, profit maximization for the firm takes place at the level where
the opportunity costs of foregone production at existing prices and excess
production costs above market prices have both been eliminated: where marginal
cost equals marginal revenue.
To summarize, in part, and amplify the range of assumptions on a general equilibrium economy embodied within this pure Neoclassical theory of the firm, we can argue:
1. A general equilibrium economy operates continuously. I can avoid any reference to a timeframe here because at every moment the "dynamic" assumptions associated with a general equilibrium economy are always operating to ensure that the economy continuously realizes a condition in which firms produce exactly the quantity of goods and services that households demand such that every market clears.
2. A general equilibrium economy has many industries and many firms in each industry. This assumption is basic if we are to have perfect competition in both output and input markets, but perfect competition requires the following conditions, as well.
3. Factors of production (land, labor, and capital) are universally owned by households and, except in rare cases of special, quantitatively limited factors that command economic rents from firms, each factor is widely distributed among large numbers of households so that no individual households can exercise market power.
4. Factors of production are perfectly homogeneous and capable of being substituted at will between any production process. Assumptions 3 and 4 ensure that factors of production must be perfectly competitive across all firms and all industries, because all firms draw from the same markets for land, labor, and capital in which the individual households supplying the factors cannot exert market power to demand higher rates of compensation.
5. All firms engaged in any particular production process have perfect access to technological information on the process. If information is perfectly distributed in these manner, then every firm must be capable of arriving at the most efficient combination of factors of production required to produce its goods and services at the lowest possible cost.
6. All firms act as perfectly rational profit maximizing agents. Again, this means that, given the choice, firms will select the most efficient available technology to combine the factors of production it has rented from household in order to produce the goods and services demanded by households at the lowest possible cost for the last unit of output produced (marginal cost). By doing so, it simultaneously maximizes the additional revenue that it receives for the last unit of output produced (marginal revenue).
7. Firms instantaneously respond to every change in household preferences for goods and services by substituting factors of production between production processes. This assumption underlies the "dynamics" of a general equilibrium economy. I can treat a general equilibrium economy as continuous and timeless (assumption 1) because firms are assumed capable of instantaneously responding to every change in markets. The point, again, is that we are not discussing real firms here, but firms existing as a construct under a pure theoretic vision of how economies operate. If firms are capable of continuously substituting factors of production between production processes, then we do not need to ask questions about the long run effects of adjusting to changes in final commodity market demand - we continuously operate within a short run economy in which every factor of production is assumed to be
variable.
8. The vehicles driving the substitution of factors of production between production processes, firms, and industries are relative prices between final goods and services, which, in turn, reflect the priorities placed on each good and service by households. This final assumption on the general equilibrium economy reinforces the larger message of Walrasian/Paretian and Austrian theoretic approaches, that the underlying rationale of unregulated free market activity exists in the utility maximizing decisions of households, determining what mix of goods and services needs to be produced in order to satisfy household desires. Thus, as households demand more bread relative to pasta, the relative price of bread in terms of pasta should be expected to rise. This increase will cause firms producing bread to demand larger amounts of land, labor, and capital to produce more bread. As larger quantities of bread come into market circulation, the price of bread in relation to pasta will decline until it arrives at a level at which both markets clear. Assumption 7 asserts that this "dynamic" shifting of factors of production between production processes occurs instantaneously.
The remainder of this document takes the rigorous, abstract theoretic assumptions listed in this section as a point of departure for how firms operate. My intention for the remainder of this document is, thus, to separate an individual firm from the larger theoretic structure of a general equilibrium economy in order to analyze its particular operations. This analysis is predicated on the workings of the firm's production function.
The Production Function
I argue above that, in some sense, Walrasian/Paretian firms can be reduced to the production functions that mathematically characterize what it is that they do. In this section, I want to more rigorously characterize these production function in a way that will enable me to draw important insights about how firms determine how much is to be produced and what mix of factors of production will minimize costs/maximize profits.
The idea of a production function is both remarkably simple
and extraordinarily abstract in substance.
It represents a mathematical recipe through which a firm is able
to combine diverse combinations of abstract factors of production (land,
labor, and capital) in order to produce determinate quantities of particular
outputs. On its face, this proposition
makes a lot of sense. It should be
possible to construct reliable mathematical relationships between quantities of
inputs to a production process and the quantities of outputs emerging from the
process. Thus, we could assemble
technically refined recipes for baking cakes or for manufacturing automotive
engine blocks, including not only the quantities of flour and baking soda
required to bake twenty two-layer cakes and the quantities and qualities of
low-carbon steel required for twenty engine blocks but also the depreciation on specific types
of machinery required to produce each and the quantities of energy (both
electricity and heat) expended in the production of each. The task involved in the construction of
such a concrete mathematical recipe might fall somewhere between the realms of
expertise of a production engineer and a cost accountant. I have no doubt that most real firms,
especially large, corporate manufacturing firms, undertake detailed production
analyses of this nature in order to determine how best to maximize output and
minimize costs. On the other hand, it
goes far beyond the level of abstraction conveyed in basic Neoclassical
economic production theory.
For our
purposes, the particular specificities involved in real production processes by
real firms can be reduced to a set of quantitative relations between three
simplified
production factors (land, labor, and capital).
Without engaging in a critique of these production factors for now, I
can apply some simple definitions to each.
Land encompasses all natural resources, including the actual land
space in which I production process takes place.
Labor includes all forms of basic human physical and mental
exertion in a production process
not requiring an extensive investment
in education and training.
Capital,
as a catch-all, includes every other input to a production process by which
labor and land are made by more productive of outputs in relation to time.
In this manner, an investment to make labor
more productive by increasing the skill of workers in a production process
through education is a form of capital (i.e.
human capital).
Likewise, the incorporation of a machine to
replace certain forms of labor and transform the labor undertaken by other
workers, increasing the total quantity of physical output produced per worker,
is a different form of capital (i.e. labor-saving mechanical capital).
Both forms of capital share a basic
characteristic in
increasing the overall productivity of a production
process within a given unit of production time by investing a certain quantity
of time in advance.
In defining
capital, the economists of the Austrian tradition of Neoclassical theory, thus,
argued that we should understand capital as the capacity to make a production
process more productive by making it temporally more “roundabout.”
Our ability
to define a Neoclassical production function relies on our willingness to
accept the
factors of production as simplified abstract representations of
the complex, concrete inputs to real production processes,
as if
land, labor, and capital could stand in for long lists of inputs in ways that
could still yield meaningful insights into the nature of production processes.
In the interest of developing an
understanding of the Neoclassical theory of production, I will not presently
dispute this necessary assumption.
Further, I am going to assert a second, simplifying assumption.
For purposes of our production analysis, I
am going to treat
land as a continuously fixed factor of production.
Thus, the land/natural resources a firm has available for use
in its production process can neither be increased or decreased in any way that
will affect the additional, marginal costs or the additional, marginal
productivity of the firm.
As such,
land, as a factor of production, can simply be left out of our analysis of
production.
This assumption adds a
useful degree of simplicity to the mathematics involved here, reducing the
production problem of the firm to two independent variables (labor (
l)
and capital (
k), as
continuously variable factors of production) and one dependent variable (output (
x)).
Mathematically, we can express this as:
x1 = f(l1, k1)
Where the sub-scripts on each of the variables in this equation identify the relationship of specific quantities of labor and capital to some particular commodity 1. Geometrically, such a production function might be
patterned in three-dimensional (xlk) space, where specific combinations of labor and
capital generate determinate quantities of output, mapped out on a three
dimensional Cartesian grid, in the form below:
Figure 1: A Production Function in xlk space
Thus, in three-dimensional
xlk-space, a sheet, representing production possibilities for positive quantities of
l and
k arises from the
x origin. The level curves (
I1, I2, I3) illustrate particular output quantity levels that I will henceforth identify as isoquant curves. Each of these curves represents a locus of combinations of labor and capital at which the quantity of output produced is equal. Thus, a movement along each curve substitutes some of one factor of production for more of the other without changing the quantity of output produced. Figure 2 attempts to illustrate this idea in two-dimensional lk space, in the same way that a geographical representation of contour lines on a mountain might be represented in two-dimensions.
Figure 2: Isoquant Curves in lk space.
Elaborating, the isoquants for this production function are smooth, negatively sloped curves, convex in relation to the
xlk origin. They may be described as a continuous
family of curves, with each curve representing a particular level of output and a particular locus of labor and capital combinations generating each level of output. To the extent that each curve represents a unique level of output, it follows that the
isoquants never intersect or cross each other - such an outcome would imply that, at points of intersection, a given combination of labor and capital generates
multiple distinct quantities of output, logically undermining the explanatory value of the production function.
Geomtrically, the smoothness of the isoquants in figure 2 implies that labor and capital can be continuously substituted along each curve to obtain a given quantity of output with different combinations of the production factors. This characteristic happens to be useful to the analysis that I plan to undertake here, but isoquants do not necessarily need to take on this particular curvature. Rather, they could take on forms represented in figures 3A and 3B, denoting production functions for commodities characterized by production processes with perfect substitution between factors of production and perfect complementarity of factors, respectively. I will explain why these cases take on their particular form in the next section, analyzing production by Walrasian firms, but, for now, it suffices to state that these involve particular conditions evident in very specific types of production functions diverging from the general case of continuous substitutability between factors of production that I am developing here.
Figures 3A and 3B: Isoquants for Production Functions with Perfect Substitutability (A) and Perfect Complementarity (B) between Production Factors
Elaborating further with respect to figure 2, the points
A and
B are both located on isoquant curve
I14. Thus, both of these points represent combinations of labor and capital generating an equal quantity of output. However, point A represents a production process utilizing a greater quantity of capital (k1A > k1B) and a lesser quantity of labor (l1A < l1B) than the production process represented by point B. A movement from point A to point B, therefore, involves a substitution by the firm of labor for capital without any change in output. I will derive the slope of the isoquants in the context of analyzing production in the next section.
Fleshing out the particularities of this production function a little farther, I am going to make a small number of important assumptions. First, the production function above, and any production function identified in this document, is
continuously defined in positive
xlk space. That is, for all
positive quantities of
labor and capital, a determinate positive quantity of output
x exists. The basic idea here is that production always requires positive quantities of both labor and capital. With this in mind, the boundaries of the function, where either
l or
k have
zero quantities, have zero quantities of output - if a firm has large quantities of labor but no capital (or vice versa), then it can produce nothing.
While I am still considering the geometry of the production function in
xlk space, it is worth asking the question:
does the production function have an absolute maximum value? My answer to this question is
maybe, but the question itself is relatively unimportant given my particular explanation of production functions. The absolute maximum of the production function represents a combination of labor and capital,
under a fixed quantity of land, where output reaches its highest level. That is to say, additional quantities of labor and capital beyond the maximum will yield no additional output and, conceivably, will result in a decline in total output, as the actual space of production becomes overcrowded with variable factors of production (i.e. labor and capital). Such an interpretation of the production function is relevant if only because it requires me to re-integrate the fixed factor of production land back into an explanation of production processes in order to account for the fact that the variable factors of production must approach some absolute constraint beyond which more labor and capital cannot increase output. As such,
production functions may not be infinitely increasing in xlk space. This this condition will be especially relevant in dealing with production functions for Marshallian firms in the next document. However, for our purposes, I will assume that the production functions in this document will be increasing for the particular ranges of variable factor quantities considered here. In practical terms, we will be dealing with the production function where the
isoquant level curves are continuously convex and negatively sloped relative to the xlk-origin, as reflected in figure 2.
Delving into the differential calculus of the production function, differentiating the function yields
positive first order partial derivatives at least for the relevant range of the function under consideration here, implying that the function is continuously upward sloping in both the
l and
k directions (i.e. up to the absolute maximum of the function). Any increase in either labor
or capital will generate a positive change in output at any level off the boundaries of the production function. Expressed in slightly different terms, the
marginal products of labor and capital, defined as the additional product generated by increasing labor or capital by one unit, are positive for the ranges of labor and capital under consideration. I can express these marginal products as:
δx1/δk1 = fk(k1, l1) = MPk (Marginal
Product of Capital)
δx1/δl1 = fl(k1, l1) = MPl (Marginal Product of Labor)
A final assumption concerns the question of
returns to scale.
Returns to scale define the relationship between a change in the quantity of all variable production factors hired and the effect of such a change on output levels. Specifically, if we increase both variable factors of production by a given multiplicative factor of
α, what effect will such a change in the scale of production have on output. There are three possibilities here, all of which will characterize certain ranges of a production function up to the function's absolute maximum:
F[α(l1), α(k1)] > αF(l1, k1) Increasing Returns to Scale
F[α(l1), α(k1)] < αF(l1, k1) Decreasing Returns to Scale
F[α(l1), α(k1)] = αF(l1, k1) Constant Returns to Scale
These conditions arise from the particular technical characteristics of the production processes represented through the mechanism of the production function. It is possible that a production function will be characterized exclusively by increasing, decreasing, or constant returns to scale, but it is also possible that production functions include separate ranges characterized by increasing, constant, and decreasing returns to scale. Again, the production functions for Marshallian firms in the next document will take on this form. The particular significance of changing returns to scale concerns the particular cost structures that such a condition will mandate. Thus, such production functions have certain ranges in which a doubling of the labor and capital utilized by the firm will more than double output (increasing returns to scale). In other ranges of the production function, a doubling of labor and capital will less than double output (decreasing returns to scale). In still other ranges or levels, a doubling of labor and capital will exactly double output. Graphically, returns to scale are reflected in the slope of the production function along rays emanating from the xlk origin, denoting the extent to which output increases as a particular combination of labor and capital is increased by multiplicative factors. This is shown in figure 4.
Figure 4: Isoquants for a Production Function with Changing Returns to Scale
Elaborating, along the ray A(l=k), unit increases of production scale from a base level (k0 and l0) generate increases in output by changing factors. Thus, doubling the initial scale of inputs (from k0,l0 to 2k0, 2l0) generates 2.5 times more output than under the initial scale of production. By contrast, tripling the initial scale generates 3 times more output than under the initial scale, and quadrupling the initial scale yields 3.6 times more output than initially. Interpreting this outcome with regard to the slope of the production function, it might help to compare the outcome in figure 4 to the hypothesis that unit increases in the scale of factors will achieve constant returns to scale. Under this hypothesis, we would expect that doubling labor and capital from the initial scale would yield 20 units of output, tripling the factors would yield 30 units of output, and quadrupling would yield 40 units of output (that is, each unit increase in scale increases output by 10 units). In actuality, the slope of the production function is initially steeper than the slope of a ray denoting constant returns to scale. It cross such a ray at 30 units. Beyond this point, the slope of the production function is less steep than the ray denoting constant returns to scale. Thus, the changing slope of the production function implies that we must have an initial range of increasing returns to scale, a point of transition with constant returns to scale, and a concluding range of decreasing returns to scale.
For our purposes, I will assume that firms within the context of a general equilibrium economy described in the previous section will continuously operate under constant returns to scale, implying that no opportunities for cost reductions per unit output from a change in the scale of operations exist for any firms. Firms, as perfectly rational profit maximizing agents, continuously select a scale of operations characterized by constant returns to scale at which no opportunity to reduce costs per unit output by increasing or decreasing their scale of operation remains unexploited. As a simplifying matter, I will, therefore, assume that the production functions henceforth analyzed in this document will be characterized by constant returns to scale over their entire range.
At this point, I want to quickly summarize the assumptions that I have made in this section:
1. The types of production functions developed in this document are continuously defined in positive xlk space, and take on zero output values on the boundaries where either labor or capital take on a value of zero.
2. The production functions may have an absolute maximum value (implying that they are not infinitely increasing in xlk with increases in production factors), but we will only be interested in ranges in which the production function is increasing in value with increases in labor and capital.
3. On the ranges that interest us, level curves/isoquant curves will be smooth, negatively sloped, and convex in relation to the xlk origin, implying that labor and capital are continuously substitutable at defined levels of output.
4. On the ranges that interest us, the first order l and k partial derivatives of the production, defining the marginal products of labor and capital, are continuously positive.
5. Production functions may be characterized, for their entire range, by increasing, decreasing, or constant returns to scale, they may have separate ranges of increasing, constant, and decreasing returns to scale.
6. Firms in a general equilibrium economy will operate with production functions characterized by constant returns to scale over their entire range, implying that no opportunities exist for the firm to reduce costs per unit output by altering their scale of operation.
