Acta Universitatis Danubius. Œconomica, Vol 10, No 6 (2014)
A Study of CobbDouglas
Production Function with Differential Geometry
Alin Cristian Ioan^{1}
Abstract: In this paper we shall made an analysis of CobbDouglas 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; CobbDouglas
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 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 are possible, we restrict the production area to a subset D_{p}SP called domain of production.
It is called a CobbDouglas production function an application:
Q:D_{p}R_{+}, (K,L)Q(K,L)=cK^{}L^{}R_{+} (K,L)D_{p}, ,R^{*}_{+}, c0
The production function is C^{}differentiable and homogenous of degree +.
2. The Differential Geometry of CobbDouglas 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) H_{f}= =
therefore, because:
(4) ^{B}_{1}= = 0, ^{B}_{2}= = 0
we obtain that Q is quasiconcave, that is for any aR, Q^{1}([a,)) is convex in R^{2}.
For a constant value of one parameter we obtain a curve on the surface, that is Q=Q(K,L_{0}) or Q=Q(K_{0},L) are both curves on the production surface. They are obtained from the intersection of the plane L=L_{0} or K=K_{0} 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=g_{11}dL^{2}+2g_{12}dLdK+g_{22}dK^{2}
where: g_{11}=1+p^{2}, g_{12}=pq, g_{22}=1+q^{2}.
In our case:
(6) g_{11}= , g_{12}= , g_{22}=
The area element is:
(7) d= dKdL= dKdL=
and the surface area A when (K,L)R (a region in the plane KOL) is A= .
The second fundamental form of the surface is:
(8) h=h_{11}dL^{2}+2 h_{12}dLdK+ h_{22}dK^{2}
where: h_{11}= , h_{12}= , h_{22}= .
In our case:
(9) h_{11}= , h_{12}= ,
h_{22}=
Considering the quantity =h_{11}h_{22}h_{12}^{2} 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 C_{n} 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 k_{1} and k_{2} of the function k(m) are called the principal curvatures of the surface in that point. They satisfy also the equation:
(12) (g_{11}g_{22}g_{12}^{2})k^{2}(g_{11}h_{22}2g_{12}h_{12}+g_{22}h_{11})k+(h_{11}h_{22}h_{12}^{2})=0
The values of m, who give the extremes, call principal directions in that point.
They also satisfy the equation:
(13) (g_{11}sg_{12}r)m^{2}+(g_{11}tg_{22}r)m+(g_{12}tg_{22}s)=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=k_{1}k_{2} 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 h_{11}m^{2}+2 h_{12}m+h_{22}=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) m_{1}= , m_{2}=
If +=1 then both asymptotic directions are equal.
With notations x^{1}=L, x^{2}=K, let define now the Christoffel symbols of first order:
(17) ij,k=
and of second order:
(18) =g^{i1}jk,1+g^{i2}jk,2
where g^{11}= G, g^{12}= F, g^{22}= 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) =g^{11}11,1+g^{12}11,2= ,
=g^{21}11,1+g^{22}11,2= ,
=g^{11}12,1+g^{12}12,2= ,
=g^{21}12,1+g^{22}12,2= ,
=g^{11}22,1+g^{12}22,2= ,
=g^{21}22,1+g^{22}22,2=
From the upper we find that:
(21) 11,1= , 11,2= ,
12,1= , 12,2= ,
22,1= , 22,2=
(22) = ,
= ,
= ,
=