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 R² – the production space for two resources: K – capital and L - labor as SP=(K,L)K,L0 where xSP, 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 Cobb-Douglas production function an application:

Q:D_pR₊, (K,L)Q(K,L)=cK^L^R₊ (K,L)D_p, ,R^*₊, c0

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) H_f= =

therefore, because:

(4) ^B₁= = 0, ^B₂= = 0

we obtain that Q is quasiconcave, that is for any aR, 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 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 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) =gⁱ¹jk,1+gⁱ²jk,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) =g¹¹11,1+g¹²11,2= ,

=g²¹11,1+g²²11,2= ,

=g¹¹12,1+g¹²12,2= ,

=g²¹12,1+g²²12,2= ,

=g¹¹22,1+g¹²22,2= ,

=g²¹22,1+g²²22,2=

From the upper we find that:

(21) 11,1= , 11,2= ,

12,1= , 12,2= ,

22,1= , 22,2=

(22) = ,

= ,

= ,

= ,

= ,

=

A geodesic is in common language the shortest curve between two points. It is useful when we try to determine the shortest way to go from a production at other in a minimum time. The equation of a geodesic is:

(23)

that is:

(24)

(25)

or, with the quantities determined:

(24)

(25)

The equations of geodesics are: L=L(s), K=K(s) where s is the element of arc on the curves.

3. References

Arrow, K.J., Chenery, H.B., Minhas, B.S. & Solow, R.M. (1961). Capital Labour Substitution and Economic Efficiency. Review of Econ and Statistics, 63, pp. 225-250.

Cobb, C.W. & Douglas, P.H. (1928). A Theory of Production. American Economic Review, 18, pp. 139–165.

Ioan, C.A. (2007). Applications of the space differential geometry at the study of production functions. Euroeconomica, 18, pp. 30-38.

Ioan, C.A. (2004). Applications of geometry at the study of production functions. The Annals of Danubius University, Fascicle I, Economics, pp. 27-39.

Kadiyala, K.R. (1972). Production Functions and Elasticity of Substitution. Southern Economic Journal, 38(3), pp. 281-284.

Kmenta, J. (1967). On Estimation of the CES Production Function. International Economic Review, 8(2), pp. 180-189.

Liu, T.C. & Hildebrand, G.H. (1965). Manufacturing Production Functions in the United States, 1957. Ithaca: Cornell Univ. Press.

Mishra, S.K. (2007). A Brief History of Production Functions. Shillong, India: North-Eastern Hill University.

Revankar, N.S. (1971). A Class of Variable Elasticity of Substitution Production Functions. Econometrica, 39(1), pp. 61-71.

Sato, R. (1974). On the Class of Separable Non-Homothetic CES Functions. Economic Studies Quarterly, 15, pp. 42-55.

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