# 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 () or on the conditions of their
indicators (, , , , , ). Another approach can be done in terms of purely geometric
properties (, ), but also through the generalization of existing functions and then putting in
obviousness their common characteristics (, ).
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.