Acta Universitatis Danubius. Œconomica, Vol 10, No 6 (2014)

A Study of Cobb-Douglas

Production Function with Differential Geometry



Alin Cristian Ioan1



Abstract: In this paper we shall made an analysis of Cobb-Douglas production function from the differential point of view. We shall obtain some interesting results about the nature of the points of the surface, the total curvature, the conditions when a production function is minimal and finally we give the equations of the geodesics on the surface i.e. the curves of minimal length between two points.

Keywords: production functions; metric; curvature; geodesic; Cobb-Douglas

JEL Classification: E23



1. Introduction

In the theory of production functions, all computations and phenomenons are studied for a constant level of production. In order to detect many aspects of them, a complete analysis can be made only at the entire surface.

We therefore 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. Because in a production process, depending on the nature of applied technology, but also its specificity, not any amount of resources are possible, we restrict the production area to a subset DpSP called domain of production.

It is called a Cobb-Douglas production function an application:

Q:DpR+, (K,L)Q(K,L)=cKLR+ (K,L)Dp, ,R*+, c0

The production function is C-differentiable and homogenous of degree +.



2. The Differential Geometry of Cobb-Douglas Surface

The graph representation of a production function is a surface.

Let note in what follows:

(1) p= , q= , r= , s= , t=

We have after simple calculations:

(2) p= , q= , r= , s= , t=

The bordered Hessian:

(3) Hf= =

therefore, because:

(4) B1= = 0, B2= = 0

we obtain that Q is quasiconcave, that is for any aR, Q-1([a,)) is convex in R2.

For a constant value of one parameter we obtain a curve on the surface, that is Q=Q(K,L0) or Q=Q(K0,L) are both curves on the production surface. They are obtained from the intersection of the plane L=L0 or K=K0 with the surface Q=Q(K,L).

In the study of the surfaces, two quadratic forms are very useful.

The first fundamental quadratic form of the surface is:

(5) g=g11dL2+2g12dLdK+g22dK2

where: g11=1+p2, g12=pq, g22=1+q2.

In our case:

(6) g11= , g12= , g22=

The area element is:

(7) d= dKdL= dKdL=

and the surface area A when (K,L)R (a region in the plane K-O-L) is A= .

The second fundamental form of the surface is:

(8) h=h11dL2+2 h12dLdK+ h22dK2

where: h11= , h12= , h22= .

In our case:

(9) h11= , h12= ,

h22=

Considering the quantity =h11h22-h122 we have that:

(10) =

  • If >0 in each point of the surface, we will say that it is elliptical. Such surfaces are the hyperboloid with two sheets, the elliptical paraboloid and the ellipsoid.

  • If 0 in each point of the surface, we will say that it is hyperbolic. Such surfaces are the hyperboloid with one sheet and the hyperbolic paraboloid.

  • If =0 in each point of the surface, we will say that it is parabolic. Such surfaces are the cone surfaces and the cylinder surfaces.

From (10) we find that:

  • : the production surface is elliptical;

  • : the production surface is parabolic;

  • : the production surface is hyperbolic

The curvature of a curve is, from an elementary point of view, the degree of deviation of the curve relative to a straight line. Considering a surface S and an arbitrary curve through a point P of the surface who has the tangent vector v in P, let the plane determined by the vector v and the normal N in P at S. The intersection of with S is a curve Cn named normal section of S. Its curvature is called normal curvature.

If we have a direction m= in the tangent plane of the surface in an arbitrary point P we have that the normal curvature is given by:

(11) k(m)=

The extreme values k1 and k2 of the function k(m) are called the principal curvatures of the surface in that point. They satisfy also the equation:

(12) (g11g22-g122)k2-(g11h22-2g12h12+g22h11)k+(h11h22-h122)=0

The values of m, who give the extremes, call principal directions in that point.

They also satisfy the equation:

(13) (g11s-g12r)m2+(g11t-g22r)m+(g12t-g22s)=0

The curve =m (where m is one of the principal directions) is called line of curvature on the surface. On such a curve we have the maximum or minimum variation of the value of Q in a neighborhood of P.

The quantity K=k1k2 is named the total curvature in the considered point and H= is named the mean curvature of the surface in that point.

We have therefore:

(14) K= = =

(15) H= =

A surface with K=constant call surface with constant total curvature and if H=0 call minimal surface. In our case we can see that K=0 if and only if: +=1.

If we consider now in the tangent plane at the surface in a point P a direction m, if h11m2+2 h12m+h22=0 we will say that m is an asymptotic direction, and the equation: gives the asymptotic curves of the surface in the point P.

In our case, the asymptotic directions are:

(16) m1= , m2=

If +=1 then both asymptotic directions are equal.

With notations x1=L, x2=K, let define now the Christoffel symbols of first order:

(17) ij,k=

and of second order:

(18) =gi1jk,1+gi2jk,2

where g11= G, g12=- F, g22= E are the components of the inverse matrix of .

We have now:

(19) 11,1= , 11,2= , 12,1= , 12,2= ,

22,1= , 22,2=

(20) =g1111,1+g1211,2= ,

=g2111,1+g2211,2= ,

=g1112,1+g1212,2= ,

=g2112,1+g2212,2= ,

=g1122,1+g1222,2= ,

=g2122,1+g2222,2=

From the upper we find that:

(21) 11,1= , 11,2= ,

12,1= , 12,2= ,

22,1= , 22,2=

(22) = ,

= ,

= ,

=