Acta Universitatis Danubius. Œconomica, Vol 11, No 1 (2015)
The Complete Theory of CobbDouglas Production Function
Cătălin Angelo Ioan^{1},^{ }Gina Ioan^{2}
Abstract. The paper treats various aspects concerning the CobbDouglas production function. On the one hand were highlighted conditions for the existence of the CobbDouglas function. Also were calculated the main indicators of it and short and longterm costs. It has also been studied the dependence of longterm cost of the parameters of the production function. The determination of profit was made both for perfect competition market and maximizes its conditions. Also we have studied the effects of Hicks and Slutsky and the production efficiency problem.
Keywords: production function; CobbDouglas; Hicks; Slutsky
1. Introduction
To conduct any economic activity is absolutely indispensable the existence of inputs, in other words of any number of resources required for a good deployment of the production process. We will assume that all resources are indefinitely divisible.
We define on R^{n} the production space for n fixed resources as SP=(x_{1},...,x_{n})x_{i}0, i= where xSP, x=(x_{1},...,x_{n}) is an ordered set of resources and, because inside a production process, depending on the nature of applied technology, not any amount of resources is possible, we will restrict production space to a convex subset D_{p}SP – called the domain of production.
We will call a production function an application:
Q:D_{p}R_{+}, (x_{1},...,x_{n})Q(x_{1},...,x_{n})R_{+} (x_{1},...,x_{n})D_{p}
which satisfies the following axioms:
A1. Q(0,...,0)=0;
A2. The production function is of class C^{2} on D_{p} that is it admits partial derivatives of order 2 and they are continuous on D_{p};
A3. The production function is monotonically increasing in each variable, that is: 0, i= ;
A4. The production function is quasiconcave (see Appendix).
Considering a production function Q:D_{p}R_{+} and R_{+}  fixed, the set of inputs which generate the production called isoquant. An isoquant is therefore characterized by: {(x_{1},...,x_{n})D_{p}Q(x_{1},...,x_{n})= } or, in other words, it is the inverse image .
We will say that a production function Q:D_{p}R_{+} is constant return to scale if Q(x_{1},...,x_{n})=Q(x_{1},...,x_{n}), with increasing return to scale if Q(x_{1},...,x_{n})>Q(x_{1},...,x_{n}) and decreasing return to scale if Q(x_{1},...,x_{n})Q(x_{1},...,x_{n}) (1,) (x_{1},...,x_{n})D_{p}.
2. The CobbDouglas Production Function
The CobbDouglas function has the following expression:
Q:D {0}R_{+}, (x_{1},...,x_{n})Q(x_{1},...,x_{n})= R_{+} (x_{1},...,x_{n})D, A , _{1},...,_{n}R^{*}
Computing the partial derivatives of first and second order, we get:
i=
ij=
i=
Let the bordered Hessian matrix:
We find (not so easy): = , k= .
Because (1)^{k} = , if 0, k= it follows that the function is strictly quasiconcave. Also, if the function is quasiconcave we have that 0.
But from the axiom A3 we must have that 0 that is _{i}0. After these considerations we have that if _{i}0, i= the CobbDouglas function is strictly quasiconcave.
We have now: = = and r= .
The main indicators are:

= = , i=

= = , i=

RMS(i,j)= , i,j=

RMS(i)= , i=

= , i=

_{ij}=1, i,j=
Reciprocally, if for a homogenous production function of degree r: = , i= we have that: , i= and .
But now, we have:
= (where ^ means that the variable is missing).
We have now: . For ji we obtain now: therefore: . Integrating with respect to :
therefore: . Analogously, by recurrence: with A=constant with respect to . But: . After these considerations it follows that if it is homogenous of degree r, r must be . Finally: implies that: Q(x_{1},...,x_{n})= = = =  the CobbDouglas production function.
Considering now again the CobbDouglas production: Q(x_{1},...,x_{n})= let search the dependence of the parameters _{1},...,_{n}.
We have: = = 0 x_{i}1, i= . From this relation we have that at an increasing of a parameter _{i} the production Q will increase also.
In particular, for the CobbDouglas function related to capital K and labor L: Q=AK^{}L^{} we have that the main indicators are:

=

=

=

=

RMS(K,L)=RMS(K)=

RMS(L,K)= RMS(L)=

=

=

=_{KL}=1
3. The Costs of the CobbDouglas Production Function
Considering now the problem of minimizing costs for a given production Q_{0}, where the prices of inputs are p_{i}, i= , we have:
From the obvious relations: we obtain: and from the second equation: . Noting r= we finally obtain:
= , k=
The total cost is:
TC= = .
At a price change of one factor, i.e. x_{k}, from the value p_{k} to