Acta Universitatis Danubius. Œconomica, Vol 10, No 4 (2014)

A Rational Production Function

Cătălin Angelo Ioan1, Alin Cristian Ioan2

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 R2the 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 DpSP called domain of production.

It is called production function an application Q:DpR+, (K,L)Q(K,L)R+ (K,L)Dp.

For an efficient and complex mathematical analysis of a production function, we impose a number of axioms both its definition and its scope.

  1. The domain of production is convex;

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

  3. The production function is of class C2 on Dp that is it admits partial derivatives of order 2 and they are continuous;

  4. The production function is monotonically increasing in each variable;

  5. The production function is quasiconcave that is: Q(x+(1-)y)min(Q(x),Q(y)) [0,1] x,yRp.

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,)= {xRpQ(x)a}.

2. The Main Indicators of Production Functions

Consider now a production function: Q:DpR+, (K,L)Q(K,L)R+ (K,L)Dp.

We call marginal productivity relative to an input xi: = and represents the trend of variation of production to the variation of xi.

We call average productivity relative to an input xi: = the value of production at a consumption of a unit of factor xi.

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 xi: = = and represents the relative variation of production to the relative variation of the factor xi.

Considering now a production function Q:DpR+ 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:DpR2R+, (K,L)Q(K,L)R+ (K,L)Dp with constant return to scale:


where P and R are homogenous polynomials in K and L, deg P=n, deg R=n-1, 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:DpR2R+, (K,L)Q(K,L)R+ (K,L)Dp with constant return to scale:


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: