Until now this webportal has somwhat neglected the neoclassical production theory. The present column makes up for this omission by describing the micro-economic theory of production functions. Essential concepts such as the complementarity and the substitution of production factors are explained. The isoquant is introduced, as well as the occurrence of various types of scaling effects. The problems of macro-economic production functions, such as in the model of Solow, are addressed. Although the theory in this column remains at the introductory level, she is useful and even indispensable for future columns about the technical development in the production1.
When in an enterprise a quantity Q of some good or service is produced (output), then production factors are required (input). The production factors are quite diverse, and vary from labour, raw materials, expedients and semi-manufactured articles to the machinery and the buildings. Suppose that N of such factors exist, with their corresponding quantities given by the set (q1, q2, ... , qN). Together the quantities form an n-dimensional vector q. Now the production function is defined as the function Q = f(q), which describes the relation between the inputs and outputs2.
The production function is a typical find of the neoclassical paradigm, which tries to describe the economy at the micro level of the enterprises and the households. The function is an abstract representation of a certain production process or enterprise. That is to say, she symbolizes a certain production technique. The variables refer to the material situation, and ignore the monetary value of the production factors and of the products. The choice for a production function is made by the entrepreneur himself. In the current theory the organizational, managerial and administrative regulations are not explicitely mentioned in the formulation of the production function3. She does admit, that the environment of the entrepreneur affects Q. Since the approach is neutral with respect to the system, Leninist economists do not make ideological objections against the use of production functions.
It is commonly assumed, that f(q) describes a technically efficient production. The quantities qi (with n=1, ..., N) are completely employed. For although in f(q) the value is absent, the neoclassical paradigm does require that the costs are minimized. Moreover, since she expects that the products are sold at their heighest prices, she defines the problem of the entrepreneur as4
(1) maximize for the q-space f(q) × p − Σn-1N qn × pn
In the formula 1, p is the product price, and pn is the factor price of the input n. It is obvious that the production is merey viable, as long as the difference between the benefits and the costs is positive. In the situation with pure competition the entrepreneur can not influence the prices, so that he becomes a price taker, who adjusts the quantities. The choice for a certain volume Q of production is dictated by the behaviour of f(q). For the sake of convenience the economists like to use in their models the so-called homogeneous production functions of degree γ, that satisfy the relation
(2) λγ × Q = λγ × f(q) = f(λ × q)
In the formula 2, λ is a real number, which allows to scale the production volume in upward or downward direction. This is called the level variation. Insertion of the formula 2 in the formula 1 gives5
(3) maximize for the q-space q1 × { q1γ-1 × f(q / q1) − (q / q1) · (p / p) }
In the formula 3 the last term represents the mathematical inner product of the vectors q/q1 and p/p. The formula 3 is written in such a manner, that q1 is a scaling factor, whereas the ratio qn/q1 represents the technique.
Figure 1: various scaling effects
The formula 3 does not imply, that q1 must be maximized. That depends on the value of the term Ψ(q1, q/q1) = q1γ-1 × f(q / q1) − (q / q1) · (p / p). A special case occurs for γ=1, because then Ψ depends only on the technique, but not on q1. Note that in this situation one will have Ψ=0 6. The case γ=1 has apparently the hallmark, that the maximization does not depend on the scale of the production process ( constant returns to scale). This type of production functions with γ=1 is called linear-homogeneous.
On the other hand there are positive scaling effects (increasing returns to scale), when one has γ>1 (more-than-linear). For, then a given technique q/q1 has a positive Ψ for an increasing q1, due to the rising term q1γ-1 × f(q / q1). Such a situation furthers the formation of monopolies, so that the perfect competition will be undermined. The case γ<1 (less-than-linear) has negative scaling effects. The formula 3 shows that in this case an increasing scale of production will finally lead to a negative Ψ and thus to losses for the enterprise. Here the entrepreneur must try to find the optimal volume of production Q = Qopt. The figure 1 displays in a graphical manner the behaviour of the production volume as a function of the scaling factor λ, for the three regimes of γ.
The basic model of the neoclassical paradigm assumes a perfect competition, and therefore chooses a linear-homogeneous f(q). Unfortunately, this is not very realistic. Besides, the figure 1 already suggests, that none of the homogeneous production functions is universally applicable (not even with γ<1 or γ>1). In general, the reality is a mix of those cases. Notably for a small volume of production an increased scale is beneficial due to the falling costs per unit of product, because some production factors are not divisible. Consider especially the "fixed" costs, such as the administration, which is necessary even for Q=0. On the other hand, the services often exhibit a neutral scaling. The fixed costs are small.
Reversely, a very large scale can lead to a rigid and expanding bureaucracy, which stifles the efficiency. Incidentally, the neoclassical paradigm can not explain this latter phenomenon. In its perspective the conflicts of interest within the organization are ignored. There are no conflicts of target. Furthermore, due to exhaustion all kinds of mining and reclamation on a large scale will often suffer from decreasing surpluses. Anyway, this shows that in the true production function the scaling effects can change, depending on the volume of production. In reality an increased scale will sometimes even be accompanied by the selection of a completely different production function.
When the behaviour of the production functions is studied, the isoquants are a useful aid. They are defined by the equation Q0 = f(q), where Q0 represents a constant value. A fascinating question is whether the function allows to produce Q0 with various vectors q, for instance with q1 and q2. When this is impossible, then the proportions qn/qm are apparently fixed in advance. Production factors with this behaviour are called complementary. The factor intensities qn/qm are fixed, and thus also the production coefficients qn/Q0. The isoquant is simply a point in the q space. These are called limitational production functions or Leontief production functions7.
The neoclassical paradigm is not interested in complementary production functions. For, it describes the behaviour of the entrepreneurs by means of the formula 3. The entrepreneur searches for the production functions with the lowest prices pn. In the neoclassical perspective he is always sufficiently innovative to develop new production techniques, with more profitable factor intensities. In other words, the factors are substitutive, and in the neoclassical paradigm they can even be substituted completely (with the exception, that for instance the quantity of the factor labour L and of certain essential capital goods K can never become totally zero). Now the isoquant is a curve in the q space. The figure 2 gives an illustration.
When one moves along the isoquant, then a continuous substitution of production factors occurs. The marginal substitution rate of the two production factors n and m is defined as MSVnm = -dqm / dqn. The minus sign makes MSV positive. Thus along the isoquant MSV satisfies8
(4) MSVnm = (∂Q / ∂qn) / (∂Q / ∂qm)
A term of the form ∂Q / ∂qn is called the marginal productivity of the production factor n.
Until now the development of Q in time has been ignored. However, this column wants to study the dynamics of the technical progress. New techniques will raise the productivity of the production factors. Thus the time must be included in the production function, which then becomes Q(t) = F(q, t). In order to limit the complexity of the theory, henceforth merely two production factors will be considered, namely the labour L and a capital good K. The famous economist J.R. Hicks has proposed to model the technical progress by means of production functions of the form
(5) Q(t) = A(t) × F(K, L)
The factor A(t) is a measure of the level of the technology at the time t, and is called the total factor productivity. The technical progress, which is expressed by the formula 5, is called Hicks neutral. Note that although the quantities of the production factors K and L can change with time, nevertheless their functional behaviour F(K, L) remains conserved. This probably explains the term "neutral" for this form of progress. One can define A(0) = 1 without any loss of generality. Since progress implies an increasing productivity, one must have A(t) > 1 for t>0.
The technical development under Hicks-neutral conditions can be illustrated by means of the isoquant Q(t) = Q(0) = Q0. For, in a situation with a constant Q0 and a rising productivity the quantities of K and L can decrease. On the isoquant the increase of A(t) is compensated by the fall of K and L. When the function F(K, L) is linear-homogeneous, like the neoclassical paradigm assumes, then one has Q0 = F(K × A(t), L × A(t)). When at t=0 the starting point is (K, L) on the isoquant with capital intensity k=K/L, then at a later time t>0 that point will move towards the origin without changing k. The figure 3 illustrates this time behaviour of isoquants under Hicks-neutral conditions9.
The famous economist R.F. Harrod was not particularly satisfied by the formula 5 for the technical development, because he foresees problem in the empirical determination of the quantity K of the capital good. He believes that it is easier to compare situations of equal K, and to study exclusively the improvement of the labour productivity Q/L. This leads to production functions of the form
(6) Q(t) = F(K, A(t) × L)
The technical progress, which is described by the formula 6, is called Harrod neutral. Also here the functional behaviour remains conserved, provided that the calculations use F(K, Λ(t)) and Λ(t) = A(t) × L. Suppose for convenience again that A(0) = 1, then one has Λ(0) = L. Λ increases for t>0, not because the working-hours are increased, but because the productivity of the quantity L of labour rises10. Also under Harrod-neutral condition the technical development can be illustrated by means of the isoquant Q(t) = Q(0) = Q0. The increase of A(t) must be compensated on the isoquant by a proportional decrease of L. When at t=0 the starting point is (K, L) on the isoquant with capital coefficient κ=K/Q, then at a later time t>0 that point will move to the K-axis without changing κ. The figure 4 illustrates this time behaviour of isoquants under Harrod-neutral conditions11.
Finally this micro-economic argument must be completed with a much-used mathematical formula for the production function. The names of the North-Americans Cobb and Douglas are attached to this function, because they have made her popular. She is12
(7) Q = A × Kβ × Lα
Since this function is applied mainly to static situations, the variable t is omitted. The constants α and β are positive. As always, K and L are complete substitutes for each other. The formula 7 is very convenient for practical applications. For instance, the formula 7 can be rewritten as K = (Q / A)1/β / Lα/β, so that apparently the isoquants behave as hyperbolas in the (L, K) plane.
The reader can check easily that the Cobb-Douglas functions is homogeneous of degree α+β. Therefore in the popular linear-homogeneous case β equals 1−α. And the marginal productivities are ∂Q / ∂K = β × Q/K and ∂Q / ∂L = α × Q/L. When α and β are less than 1, which is a common assumption, then the marginal productivity of a factor is apparently less than the average productivity (namely, Q/K and Q/L). It follows immediately that the marginal productivity of a factor decreases, according as he is more abundantly present. According to the formula 4 another interesting property of the Cobb-Douglas function is, that the marginal substitution rate MSVLK equals k × α/β. MSVLK remains unaltered along a line of constant capital intensity.
Cobb and Douglas became famous, because they have applied the function of the formula 7 at the macro level. The nation-state is presented as an enormous enterprise, as it were. It is obvious, that there are large practical problems attached to this approach13. For, it is an impossible task to aggregate all production factors of the state into a single production function. They are too numerous. Perhaps the loyal reader will object, that at least fot the case of complementary production factors the theory of Sraffa can be applied. But even those simple formulas are only applied in situations with separate economic sectors, and not at the level of the separate enterprises.
However, the studies at the sectoral level must necessarily already use aggregation. The separate capital goods are collected in a limited number of categories. The aggregation of heterogeneous goods is only possible by means of their monetary value. In the Cobb-Douglas function this happens in a rigorous manner, because only a single capital factor K remains. The same approach is used in the well-known growth model of the economist R.M. Solow. However, problems surface when values are used in the production functions. For, the production functions represent a certain technique. As soon as values are included, then the social variables such as the wage level and the interest rate will also affect the result.
At the micro level, for the separate enterprise, the production prices and the factor prices are given, at least as long as the market is ruled by perfect competition. But this assumption is no longer valid at the aggregated macro level, because there the prices are mutually coupled. For instance, as soon as production factors are substituted, the prices will change, as well as the resulting aggregated values. Thus the total product Q can change in value, although perhaps the material itself remains the same. This results in apparently strange phenomena, certainly for the neoclassical economists, such as the reswitching of previously rejected production techniques.
The difference between the micro and macro level can also be illustrated well by means of the relation between the marginal productivities and the factor prices. The formula 1 shows that at the micro level the entrepreneur tries to make his yield as large as possible, and to minimize his costs. His method is shown in a graphical manner in the figure 2. The production costs are TC = K × pK + L × pL. This is the green straight line in the (L, K) plane, and is called the budget line. In the same plane the isoquants are drawn, which belong to his production function f(K, L).
Now the entrepreneur tries to produce at the highest possible isoquant, that is within his reach. That reach is limited by the sum TC, which he owns. The optimum of the entrepreneur is the point of the budget line, that just touches the isoquant. A higher isoquant is beyond his range. In the optimum the slope of the budget line (which is -pL/pK) is exactly equal to the slope of the isoquant (which is MSVLK, and thus satisfies the formula 4). This leads to the formula14
(8) (∂Q / ∂L) / pL = 1/π = (∂Q / ∂K) / pK
In the formula 8 π is a constant, whose value can be determined by a simple argument. Namely, in most circumstances the marginal productivity ∂Q / ∂L will be a positive but nonetheless falling function of L 15. An entrepreneur continues to hire workers until the value of their marginal productivity has fallen to the wage level. That endpoint satisfies pL = ∂(p × Q) / ∂L = p × ∂Q / ∂L. This is precisely the formula 8, with π=p. A similar argument can be made for the factor capital.
Does the relation pn = p × ∂Q / ∂qn also hold at the macro level? Even nowadays some introductory textbooks state that this is true16. But it is wrong, because at the macro level the product prices p and the factor prices pn are themselves variables, and they depend on Q and qn. The micro-economical argument can not be copied in macro-economics.
In conclusion, the reader is once more reminded that this critique on macro-economical applications also holds for the growth model of Solow. For, that model also aggregrates the physical products into monetary sums, namely the factor capital K(t) and the nett product N(t). The values of all those monetary sums will vary, according as different production techniques are used. In fact the production function of Solow does not represent a technique any more17.