Acta Universitatis Danubius. Œconomica, Vol 10, No 4 (2014)
A Rational Production Function
Cătălin Angelo Ioan^{1}, Alin Cristian Ioan^{2}
Abstract: The article deals with a rational production function of two factors with constant scale return. It was determined the compatibility conditions with the axioms of production function resulting inequality of a single variable.
Keywords: production function; marginal productivity; average productivity
JEL Classification: C80
1. Introduction
In what follows we shall presume there is a certain number of resources, supposedly indivisible needed for the proper functioning of the production process.
We define on R^{2} – the production space for two resources: K – capital and L  labor as SP=(K,L)K,L0 where xSP, x=(K,L) is an ordered set of resources and we restrict the production area to a subset D_{p}SP called domain of production.
It is called production function an application Q:D_{p}R_{+}, (K,L)Q(K,L)R_{+} (K,L)D_{p}.
For an efficient and complex mathematical analysis of a production function, we impose a number of axioms both its definition and its scope.

The domain of production is convex;

Q(0,0)=0 (if it is defined on (0,0));

The production function is of class C^{2} on D_{p} that is it admits partial derivatives of order 2 and they are continuous;

The production function is monotonically increasing in each variable;

The production function is quasiconcave that is: Q(x+(1)y)min(Q(x),Q(y)) [0,1] x,yR_{p}.
From a geometric point of view, a quasiconcave function having the property of being above the lowest value 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}.
2. The Main Indicators of Production Functions
Consider now a production function: Q:D_{p}R_{+}, (K,L)Q(K,L)R_{+} (K,L)D_{p}.
We call marginal productivity relative to an input x_{i}: = and represents the trend of variation of production to the variation of x_{i}.
We call average productivity relative to an input x_{i}: = the value of production at a consumption of a unit of factor x_{i}.
We call partial marginal substitution rate of factors i and j the opposite change in the amount of factor j as a substitute for the amount of change in the factor i in the case of a constant level of production and we have: RMS(i,j)= .
We call elasticity of output with respect to an input x_{i}: = = and represents the relative variation of production to the relative variation of the factor x_{i}.
Considering now a production function Q:D_{p}R_{+} with constant return to scale that is Q(K,L)= Q(K,L), let note = . It is called the elasticity of the marginal rate of technical substitution = .
3. A Rational Production Function
Consider now a production function Q:D_{p}R^{2}R_{+}, (K,L)Q(K,L)R_{+} (K,L)D_{p} with constant return to scale:
K,L0
where P and R are homogenous polynomials in K and L, deg P=n, deg R=n1, n2.
Because the function is elementary follows that it is of class C^{} on the definition domain.
Let note also: .
In what follows we put the question of determining the conditions so that the axioms 4 and 5 to be verified.
We now have:
,
Because of homogeneity, we have:
that is:
Note now: = , = , = , = . We have:
After many computations, we obtain:
=
=
=
=
The conditions that: 0, 0 become:
Considering now the bordered Hessian matrix:
=
and the minors:
= = =
= = =
it is known that if <0, >0 the function is quasiconcave. Conversely, if the function is quasiconcave then: 0, 0. Therefore, a sufficient condition for the validity of axiom 5 is:
From these two sets of conditions we obtain finally:
or, more simple (taking inot account that Q,K,L0):
Theorem 1
A function Q:D_{p}R^{2}R_{+}, (K,L)Q(K,L)R_{+} (K,L)D_{p} with constant return to scale:
K,L0
is a production function if:
where: = , = , = , = .
Because P and R are homogenous, we have:
,
with the obviously notations: , .
If we have:
=
=
Analogously: = , =
We obtain therefore that the conditions of the above theorem become:
If we note for simplify: we finally have:
Because