Acta Universitatis Danubius. Œconomica, Vol 10, No 6 (2014)
Generalized CobbDouglas
Function for Two Inputs and Linear Elasticity
Catalin Angelo Ioan^{1}, Gina Ioan^{2}
Abstract: The article deals with a production function of two factors with constant scale return where the elasticity of one of the factors is a function of first degree. After the examination of parameters conditions according to the axioms of the production functions, there are computed the main indicators. Also, the combination of factors is determined in order to maximize the total output under a given cost.
Keywords: production function; CobbDouglas
JEL Classification: E23
1. General Aspects of the Production Functions
In any economic activity, obtaining a result of it implies, by default, there is a certain number of resources, supposedly indivisible needed for the proper functioning of the production process.
We therefore 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.
Because in a production process, depending on the nature of applied technology, but also its specificity, not any amount of resources is possible, we restrict the production area to a subset D_{p}SP called domain of production.
In a context of the existence of the domain of production, we put the question of determining its output depending on the level of inputs of D_{p}.
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.
FP1. The domain of production is convex;
FP2. Q(0,0)=0;
FP3. The production function is of class C^{2} on D_{p} that is it admits partial derivatives of order 2 and they are continuous;
FP4. The production function is monotonically increasing in each variable;
FP5. 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. The Generalized CobbDouglas Function for Two Inputs
Consider now a production function Q:D_{p}R_{+}, (K,L)Q(K,L)R_{+} (K,L)D_{p} with constant return to scale, where =()>0.
Considering the function q such that: Q(K,L)=L we have: = = = =().
From here we find that: . Let F be a primitive function of . From we obtain: where C  constant strictly positive.
In particular, for ()= we have:
and the production function becomes (after an obvious renoting of C):
=
or other:
If ()= then:
4. The Generalized CobbDouglas Function for Two Inputs and Linear Elasticity
Consider now the production function: Q(K,L)= , K,L0, a,b,C0.
Because the function is elementary follows that it is of class C^{} on the definition domain.
We now have:
,
Considering 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.
In the present case:
= , =
It is obvious that 0. For 0 it is necessary and sufficient that: K,L0. With the substitution the statement is equivalent to 0.
Because the discriminant follows that if 0 then . Therefore: 0, a0. We get: = . But is equivalent to a(0,1).
From the above, for a(0,1), b0, the function is quasiconcave on D_{p}= R^{2}.
Also, relative to the monotonically increasing in each variable, we have: 0 and because: = 0 we get 0.
As an example, for C=1, a=0.2 and b=1 the graph is:
Figure 1
5 Main Indicators of the Generalized CobbDouglas Function for two Inputs and Linear Elasticity
We can compute, after section 2, the main indicators for the production function defined above. We have therefore:

The marginal productivity:
= = , = =

The average productivity:
= , =

The partial marginal substitution rate:
RMS(K,L)= , RMS(L,K)=

The elasticity of output:
= = = , = = =

The elasticity of the marginal rate of technical substitution:
= =
6. The Problem of Determining the Maximum of Production in terms of Given Total Cost
Let now the following problem:
where CT is the total cost of the production which is suppose to be a given constant.
From the KarushKuhnTucker conditions we have the necessary and sufficient conditions (taking into account that the restriction is affine):
From section 5 we get that the system becomes:
or: