Acta Universitatis Danubius. Œconomica, Vol 11, No 1 (2015)

The Complete Theory of Cobb-Douglas Production Function



Cătălin Angelo Ioan1, Gina Ioan2



Abstract. The paper treats various aspects concerning the Cobb-Douglas production function. On the one hand were highlighted conditions for the existence of the Cobb-Douglas function. Also were calculated the main indicators of it and short and long-term costs. It has also been studied the dependence of long-term 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; Cobb-Douglas; 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 Rn the production space for n fixed resources as SP=(x1,...,xn)xi0, i= where xSP, x=(x1,...,xn) 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 DpSP – called the domain of production.

We will call a production function an application:

Q:DpR+, (x1,...,xn)Q(x1,...,xn)R+ (x1,...,xn)Dp

which satisfies the following axioms:

A1. Q(0,...,0)=0;

A2. The production function is of class C2 on Dp that is it admits partial derivatives of order 2 and they are continuous on Dp;

A3. The production function is monotonically increasing in each variable, that is: 0, i= ;

A4. The production function is quasi-concave (see Appendix).

Considering a production function Q:DpR+ and R+ - fixed, the set of inputs which generate the production called isoquant. An isoquant is therefore characterized by: {(x1,...,xn)DpQ(x1,...,xn)= } or, in other words, it is the inverse image .

We will say that a production function Q:DpR+ is constant return to scale if Q(x1,...,xn)=Q(x1,...,xn), with increasing return to scale if Q(x1,...,xn)>Q(x1,...,xn) and decreasing return to scale if Q(x1,...,xn)Q(x1,...,xn) (1,) (x1,...,xn)Dp.



2. The Cobb-Douglas Production Function

The Cobb-Douglas function has the following expression:

Q:D -{0}R+, (x1,...,xn)Q(x1,...,xn)= R+ (x1,...,xn)D, A , 1,...,nR*

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 quasi-concave. Also, if the function is quasi-concave we have that 0.

But from the axiom A3 we must have that 0 that is i0. After these considerations we have that if i0, i= the Cobb-Douglas function is strictly quasi-concave.

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(x1,...,xn)= = = = - the Cobb-Douglas production function.

Considering now again the Cobb-Douglas production: Q(x1,...,xn)= let search the dependence of the parameters 1,...,n.

We have: = = 0 xi1, i= . From this relation we have that at an increasing of a parameter i the production Q will increase also.

In particular, for the Cobb-Douglas function related to capital K and labor L: Q=AKL we have that the main indicators are:

  • =

  • =

  • =

  • =

  • RMS(K,L)=RMS(K)=

  • RMS(L,K)= RMS(L)=

  • =

  • =

  • =KL=-1



3. The Costs of the Cobb-Douglas Production Function

Considering now the problem of minimizing costs for a given production Q0, where the prices of inputs are pi, 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. xk, from the value pk to