# The determination of the main indicators of a production function

 Author Catalin Angelo Ioan Position Associate Professor, PhD, Danubius University of Galati, Faculty of Economic Sciences, Romania Pages 626-635
European Integration - Realities and Perspectives 2012
626
The Determination of the Main Indicators of a Production Function
Using the Bernoulli Equations
Catalin Angelo Ioan
1
Abstract: Production functions are an essential tool for analysis of production processes. The indicators of
marginal production, marginal rates of substitution, elasticities of production and the marginal elasticity of
technical s ubstitution characterized, fr om different p oint of view, the behavior of the production under the
action of factors of labor or capital. This paper presents a new way of determining using the first-or der
differential equation of Bernoulli type, giving also a useful tool for the creation of new production functions.
Keywords: production function; marginal rate of substitution; elasticity
JEL Classification: D01
1Introduction
In the analysis of production processes, fundamentally important are the production functions, which
are accompanied by a series of indicators that provide useful information i n t he economic analysis.
The production functions approach can be based on practical needs ([10]) or on the conditions of their
indicators ([1], [3], [9], [11], [12], [13]). Another approach can be done in terms of purely geometric
properties ([4], [5]), but also through the generalization of existing functions and then putting in
obviousness their common characteristics ([6], [8]).
In this paper, we will broach the problem of determining the main indicators from a different point of
view. Even if they come from different economical considerations, we will construct a first-order
differential equation of Bernoulli type, whose coefficients will allow the immediate determination of
those indicators.
At first glance, the question arises: how to use such an approach? The problem has an immediate
response, although not very visible. The differential equation, by assigning different expressions to the
composing functions, can be a true generator of production functions!
Let therefore a production function Q:(0,)
2
R
+
, (K,L)Q(K,L) homogeneous of first degree.
Let note, also χ=
L
K
- the technical endowment of labor. We have:
Q(K,L)=
=
1,
L
K
LQL,
L
K
LQ
=Lq(χ)
where q(χ)=Q(χ,1).
The partial first-order derivations of Q, are obtained easily:
1
Associate Professor, PhD, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd,
Galati, Romania, tel: +40372 361 102, fax: +40372 361 290, Corresponding author: catalin_angelo_ioan@univ-danubius.ro.