Acta Universitatis Danubius. Œconomica, Vol 12, No 1 (2016)
On Certain Conditions for Generating Production Functions  I
Catalin Angelo Ioan^{1}, Gina Ioan^{2}
Abstract: The article is the first in a series that will treat underlying conditions to generate a production function. The importance of production functions is fundamental to analyze and forecast the various indicators that highlights different aspects of the production process. How often forgets that these functions start from some premises, the article comes just meeting these challenges, analyzing different initial conditions. On the other hand, where possible, we have shown the concrete way of determining the parameters of the function.
Keywords: production function; productivity; marginal rate of substitution
JEL Classification: D43
1 Introduction
Theory of production functions is vitally important in microeconomic analysis.
The need of economic phenomena mathematization, not only from a desire to give legitimacy to scientific economic theory but rather, to draw conclusions and prediction of enterprise activity required a careful analysis of them.
Wellthought literature profile, but especially practical applications encountered in all kinds of handouts, printed or online, we drew a number of issues that sometimes are neglected (probably considered insignificant) or omitted with true intent.
The first issue found by us is that of verification of sufficient conditions (not always necessary, but depending on the actual nature of the problem) as a function to be truly of production.
Another aspect which seems essential is the practical applicability. One question that could be asked of any student from any part of the Earth, is: “Departing from a series of discrete data, how you will generate the output and, especially, what kind of production function will choose?”
By own researches, I realized that maybe over 90% of production functions presented in teaching applications are of CobbDouglas type (requiring, however, the constancy of elasticity), the remainder being more or less created artificial (often they even unverified existing conditions).
It might object here that the learning exercises aims to increase math skills with these functions. The problem is not this, but what follow...
I rarely saw concrete applications, showing clearly how to practically apply these functions. Without this approach, the theory remains dry, with beautiful graphics (as an aside, all graphs of production looks pretty much the same, what will result in the following) and without practical application.
Following these minimum considerations, we will try in the following pages to generate major production functions based on practical conditions (the approach being not new, meeting in original papers), but systematized and then explaining in each case how can apply them practically.
2 General Notions
In what follows, we assume that resources are infinitely divisible, which implies the use of specific tools of mathematical analysis to analyze specific phenomena.
We thus define on R^{n} the space of production for n fixed resources as:
SP=(x_{1},...,x_{n})x_{i}0, i=
where xSP, x=(x_{1},...,x_{n}) is an ordered set of resources (inputs).
Because within a production process, depending on the nature of applied technology, but also its specificity, not any amount of resources possible, we will restrict the production area to a subset D_{p}SP called production domain.
It is now called production function (output) an application:
Q:D_{p}R_{+}, (x_{1},...,x_{n})Q(x_{1},...,x_{n})R_{+} (x_{1},...,x_{n})D_{p}
For an effective and complex mathematical analysis of a production function we will require a number of axioms (not all essential) both its scope and its definition.
A1.
The production domain D_{p}
is convex i.e. x=(x_{1},...,x_{n}),
y=(y_{1},...,y_{n})D_{p}
[0,1]
follows
(1)x+y=((1)x_{1}+y_{1},...,(1)x_{n}+y_{n})D_{p}.
Axiom A1 only mean that in the process of changing of the inputs from a level x to y, the linear shift is achieved through a series of successive steps which keeps them in the field of production, so by default the possibility of using the production function chosen. The condition could relax here, requiring domain to be, for example, connected by arches, that to be a continuous path between any two nuple inputs.
A2. Q(0,0,...,0)=0
The axiom reflects a common sense assumption namely that in the absence of any input can not get any output.
A3. The production function is continuous.
Continuity, in purely mathematical sense, represents that for any fixed point of the production domain D_{p} and any string of inputs (y_{k})_{k}_{}_{1}, y_{k}= which converges to (or otherwise i= ) the production converges to .
More simply, the continuity of the production function means that for two sets of resources (x_{1},...,x_{n}) and (y_{1},...,y_{n})D_{p} close enough, result outputs Q(x_{1},...,x_{n}) and Q(y_{1},...,y_{n}) close enough. In other words, a very small change of inputs lead to a reasonable production obtained.
An axiom, not necessarily required, but particularly useful for obtaining significant results (using differential calculus) is:
A4. The production function is of class C^{2}(D_{p}) i.e. admits 2nd order continous partial derivatives.
The condition of belonging to the class C^{2} may seem, at first glance, restrictive, but is not really. All basic functions (constant, power, exponential, logarithmic, trigonometric functions as those obtained from them by arithmetic operations of addition, subtraction, multiplication, division, power lifting, composing or reversal) are of C^{} class (implicitly of class C^{2}) on the definition domain i.e. have their partial derivatives of any order and these are continues. As a function of class C^{k}, k0 is continuous implies that axiom A3, given that accept A4, is a simple consequence of the latter, so it can be removed.
What is actually at least C^{1} class differentiability? If for a continuous function means, at an immediately approach (without much mathematical rigor) that its graph is not „broken” on the definition domain, the derivativability of class C^{1} means that it does not have „corners” or „folds”, the graph being smooth. In addition, for example in a corner point (for functions of one variable – different left and right derivatives) we can not make predictions, the behavior at left/right not anticipates the behavior at right/left.
A5. The production function is monotonically increasing in each variable.
A5 axiom states that in “ceteris paribus” hypotesis, i= if x_{i}y_{i} then> 0, k= , ki such that , D_{p}. If the function Q is at least C^{1}(D_{p}) the character of monotonically increasing becomes 0, i= . In terms of a “classic” production function with two variables: K – capital and L  labor, we have: 0, 0.
Also from the axiom A5 result, as an immediate consequence, that if x_{1}y_{1},...,x_{n}y_{n} then: Q(x_{1},x_{2},...,x_{n})Q(y_{1},x_{2},...,x_{n})Q(y_{1},y_{2},...,x_{n})...Q(y_{1},y_{2},...,y_{n}). It is obvious that the relationship occurs only if the nature of the inequalities between components is the same for all of them.
A condition often referred to in the definition of the production function is:â
A6. The production function is quasiconcave.
The quasiconcavity of a function means:
Q(x+(1)y)min(Q(x),Q(y)) [0,1] x,yR_{p}
Geometrically speaking, a quasiconcave function has property to be above the lowest values recorded at the end of a certain segment. The property is equivalent to the convexity of the set Q^{1}[a,) aR, where Q^{1}[a,)={xR_{p}Q(x)a}.
What does the quasiconcavity so? Convexity of the set Q^{1}[a,) lies in that if Q(x)a, Q(y)a then Q ((1)x+y)a. This specifies, in conjunction with the axiom A1, that the transition from one set of inputs x to y is at a production level equal to or greater than a specified lower limit. Neither this condition would not necessarily be required, existing situations (for example, the transition to a market economy of the former communist states) the refurbishment (thus changing the structure of inputs) was made with temporary dip in the level of production. But as economic analysis, most often refers (unfortunately) to the processes that are somewhat stabilized, we will retain this condition.
Considering so a production function Q:D_{p}R_{+}, (x_{1},...,x_{n})Q(x_{1},...,x_{n})R_{+} (x_{1},...,x_{n})D_{p} let the bordered Hessian matrix:
H^{B}(f)=
and  the boarded principal diagonal determinants formed with the first (k+1) rows and columns of the matrix H^{B}(f). We have the following theorem:
Theorem If Q is a quasiconcave function then 0, k= . If 0 then Q is quasiconcave function.
Notes from the theorem that if at least one determinant is null we have not ensured the existence of quasiconcavity.
For classical production functions Q=Q(K,L) the sufficient condition for quasiconcavity becomes: 0 therefore: 0.
Recall, near the end of this introduction, that a function is called homogeneous if rR such that: Q(x_{1},...,x_{n})=^{r}Q(x_{1},...,x_{n}) R*. r is called the degree of homogeneity of the function.
We say that a production function Q:D_{p}R_{+} is with constant return to scale if Q(x_{1},...,x_{n})=Q(x_{1},...,x_{n}) (so homogeneous of first degree), with increasing return to scale if Q(x_{1},...,x_{n})>Q(x_{1},...,x_{n}) and with decreasing return to scale if Q(x_{1},...,x_{n})<Q(x_{1},...,x_{n}) (1,) (x_{1},...,x_{n})D_{p}. The fact that a return to production is at constant scale means that the production has the same multiplication factor with those of the two factors. Similarly, the return of increasing (decreasing) scale production is multiplied by a factor higher (lower) than that of inputs.
We will note below for functions Q=Q(K,L): = .
In what follows we will analyze production functions of the form: Q=Q(K,L)
3 Main Indicators of a Production Function
Let a production function:
Q:D_{p}R_{+}, (x_{1},...,x_{n})Q(x_{1},...,x_{n})R_{+} (x_{1},...,x_{n})D_{p}
We will call the marginal productivity relative to a production factor x_{i}: = and represents the trend of variation of production at the variation of the factor x_{i}. In particular, for a production function of the form: Q=Q(K,L) we have _{K}=  called the marginal productivity of capital and _{L}=  called the marginal productivity of labor.
If the output is given by discrete values, we define: meaning the mean variation of the production on the interval of length .
We call also the average productivity relative to a production factor x_{i}: = and represents the value of production at the consumption of a unit of factor x_{i}.
In particular, for a production function of the form: Q=Q(K,L) we have: w_{K}=  called the productivity of capital, and w_{L}=  the productivity of labor.
From [4], we have that in the general case of the variation of all inputs, for k_{1} units of input 1,...,k_{n} units of input n, and Q(0,...,0)=0:
Q(k_{1},...,k_{n})=
In particular, for Q=Q(K,L) we have: Q(K,L)= .
Again, from [7], considering the factors i and j with ij, we define the restriction of production area: P_{ij}=(x_{1},...,x_{n})x_{k}=a_{k}=const, k= , ki,j, x_{i},x_{j}D_{p} relative to the two factors when the others have fixed values and D_{ij}=(x_{i},x_{j})(x_{1},...,x_{n})P_{ij}  the domain of production relative to factors i and j.
Defining Q_{ij}:D_{ij}R_{+}  the restriction of the production function to the factors i and j, i.e.: Q_{ij}(x_{i},x_{j})=Q(a_{1},...,a_{i1},x_{i},a_{i+1},...,a_{j1},x_{j},a_{j+1},...,a_{n}) we obtain that Q_{ij} define a surface in R^{3} for every pair of factors (i,j).
We call partial marginal rate of technical substitution of the factors i and j, relative to D_{ij} (caeteris paribus), the opposite change in the amount of factor j to substitute a variation of the quantity of factor i in the situation of conservation production level and note: RMS(i,j, )= = in an arbitrary point = . We define also ([7]) the global marginal rate of substitution between the ith factor and the others as: RMS(i, )= . The global marginal rate of technical substitution is the minimum (in the meaning of norm) of changes in consumption of factors so that the total production remain unchanged.
In particular, for a production function of the form: Q=Q(K,L) we have:
RMS(K,L)= , RMS(L,K)=
It is called elasticity of production in relation to a production factor x_{i}: = =  the relative variation of production at the relative variation of factor x_{i}. In particular, for a production function of the form: Q=Q(K,L) we have _{K}= =  called the elasticity of production in relation to the capital and _{L}= =  the elasticity factor of production in relation to the labor.
If the production function is homogenous of degree r, after Euler’s relation: we obtain that .
4 Conditions of Marginal Productivity
4.1. =constant=, constant
In this case, we have: Q(K,L)= = = K+Lg(K,L). Because we have that that is g=g(L). Therefore: Q(K,L)=K+f(L). Now =f’(L)0fconstant.
The conditions from the axioms become:

Q(0,0)=0f(0)=0

f – continuous

fC^{2}(D_{p})

=0

= f’(L)0

=^{2}f”(L)0f”(L)0
After these considerations we obtain that 0 and f is a monotonically increasing, strictly concave differentiable function of class at least two and vanishing in 0.
If now Q is homogenous, we have: rR: Q(K,L)=^{r}Q(K,L) that is: K+f(L)=^{r}(K+f(L)).
If r1K+f(L)=^{r}K+^{r}f(L) .Because K and L are independent variables follows K=constant therefore contradiction. We have r=1 that is: f(L)=f(L), f being linear: f(L)=L. We obtained: Q=K+L – the linear production function. Let note in this case that Q is quasiconcave even though f”(L)=0 for f(L)=L.
For the linear production function, the determination of the parameters is very simple (using Least Square Method).
Let (K_{i},L_{i},Q_{i})_{i=1,...,n} values of the capital, labor and production at the moments 1 to n. The minimum condition of the expression: E= (relative to and ) becomes:
therefore:
4.2. =constant=, constant
Like previous, we obtain (permuting K with L): Q(K,L)=L+f(K) with f satisfying the same conditions like above. The determination of the parameters is as above.
4.3. =constant=, =constant=
Q(K,L)= = =K+L – the linear production function. The determination of the parameters is as above.
4.4. =^{}=
Q(K,L)= = = .
But from where: = therefore: Q(K,L)= =