Acta Universitatis Danubius. Œconomica, Vol 13, No 2 (2017)

Some Aspects of Production

Functions Differential Geometry

Cătălin Angelo Ioan¹

Abstract: The article deals with some aspects of differential production functions with examples for Cobb-Douglas function in two or three variables. There are studied in each case, the conditions of the parameters in order that the sectional curvature be constant.

Keywords: production function; metric; Riemann; sectional curvature

JEL Classification: C02

1. Introduction

Let H be a hypersurface in Rⁿ⁺¹ of equation:

xⁿ⁺¹=f(x₁,...,x_n), (x₁,...,x_n)DRⁿ, D – open

We will assume in what follows that fC²(D).

A parametric representation of hypersurfaces is:

H:

We will note sometimes: x= Rⁿ.

Considering the Jacobian matrix J_f= we will assume that all hypersurfaces points are regular i.e. rank =n.

We define the tangent hyperplane to the hypersurface at a point x₀ and noted :

which is the locus of the tangents to the curves on H passing through x₀. Any element of is called the tangent vector to the hypersurface.

The normal at the hypersurface is the straight line orthogonal in x₀ on H and has equation:

We define (Eisenhart, 1926) the metric tensor of the hypersurface as having components of matrix g=(g_ij) where:

, i,j=

The tensor g, called the first fundamental form of the hypersurfaces, allows to define the length of a vector v as , the angle of two non-null vectors being:

where g(X,Y)= , X_i and Y^j being the components of the vectors X,Y .

Taking into account the parametric expression of hypersurfaces, we have:

= = , i,j=

where is the Kronecker's symbol.

We therefore fundamental matrix of the first forms of hypersurfaces:

g=

Consider also the inverse of the metric tensor: g^-1=(g^ij) where g^ij are the inverse matrix elements g^-1: where g= 0.

We define now Christoffel's symbols of the first kind:

, i,j,k=

and of the second kind:

= , i,j,k=

For considered hypersurface we have so:

, i,j,k=

= , i,j,k=

Let us define also for two vectors tangent to the hypersurface at a point x₀= D the covariant derivative of Y= relative to X= where Xⁱ,Yⁱ:DR, = , i= :

=

So we have for the considered hypersurface:

= =

The covariant derivative is the generalization of the concept of directional derivative, in the sense that if X is, locally, tangent to a curve x_i=x_i(t), i= and Y is the restriction of a field of vectors along the curve, then represents “the rate of change” of Y in the movement on the curve. In other words, in a flat space, the components of are given by the derivatives in the direction of the tangent to the curve.

A curve x_i=c_i(t), i= , t(a,b), ab, a,bR is called geodesic if, considering the vector X= occurs: =0. The equation of a geodesic is: (Ianuș, 1983)

, i=

Considering 2-space  generated by two vectors tangent to the hypersurface at the point x₀, then all geodesics passing through x₀ and have as tangent vectors to the curve vectors of , they generate a surface S passing through x₀ and having  as tangent plane. Considering the normal to S in x₀, this, together with an arbitrary tangent vector from  generates a normal plane. It intersects the surface S after a curve called normal section. Considering their curvature, that is their “deviation” from a straight line, there are obtain several curvatures whose extreme (minimum and maximum) form the so-called principal curvatures. The product of the two principal curvatures is called Gaussian curvature of the surface S. In the case of hypersurfaces, corresponding Gaussian curvature to the plane  determined by two vectors X,Y is called the sectional curvature corresponding to the vectors X,Y in x₀ noted with k(x₀,) where =<X,Y> (the subspace generated by X and Y).

We define now the Riemann curvature tensor: R(X,Y)Z= and the Riemann-Christoffel curvature tensor: R(X,Y,Z,V)=g(R(X,Y)V,Z) X,Y,Z,V .

We will note:

, ,

,

Taking into account that: we have for the considered hypersurface:

=

=

The significance of the Riemann curvature tensor is that X,Y :

R(X,Y,X,Y)=k(x₀,)

A hypersurface is said to have constant curvature if the sectional curvature is the same independent of the point x₀ and the 2-space  determined by two arbitrary vectors of . It shows that if hypersurface has constant curvature k then:

R(X,Y,Z,V)= X,Y,Z,V

Schur's theorem states that if the hypersurfaces dimension is larger or equal to 3 and the sectional curvature depends only on the point and not tangent vectors (isotropy property) than the sectional curvature is constant.

If Riemann curvature tensor is null then it can build a coordinate system in which metric tensor components are constant. If the metric tensor is constant then the Riemann curvature tensor is trivial null.

Considering Riemann curvature tensor, it defines the Ricci tensor as: S(X,Y) with and S_ij= and also the scalar curvature S= .

For considered hypersurface:

S_ij= =

S= = =

A hypersurface is called Einstein hypersurface if S(X,Y)=g(X,Y) X,Y x₀H.

If a hypersurface has constant curvature, then it is Einstein, and if it is Einstein and has dimension 3 then it has constant curvature.

We now define the curvature tensor of Weyl:

C(X,Y)Z=R(X,Y)Z-XL(Y,Z)+YL(X,Z)+g(X,Z)lY-g(Y,Z)lX X,Y,Z

where L(X,Y)= and

and also: C(X,Y,Z,V)=g(C(X,Y)V,Z) from where:

On a hypersurface of dimension 3 the Weyl tensor is null.

A hypersurface is said to be applied conformal to another space (e.g. the flat Euclidean space) if there is an application that preserves angles between any two tangent vectors. If hypersurface is conformal to an Euclidian space we will say that it is conformally flat.

Weyl theorem states that any hypersurface of dimension 2 (i.e. surface in the usual sense) is conformally flat, one of dimension 3 is conformally flat if and only if the tensor Riemann vanishes identically and if the dimension is greater than 3, then the necessary and sufficient condition to be conformally flat is that the Weyl tensor vanishes identically.

2. The Production Function

We define on Rⁿ – the production space for n resources: SP=(x₁,...,x_n)x₁,...,x_n0 where x=(x₁,...,x_n)SP 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 will restrict the production area to a subset DPSP called domain of production.

It is called production function an application Q:DPR₊, (x₁,...,x_n)Q(x₁,...,x_n)R₊ (x₁,...,x_n)DP.

The production function must satisfy a number of axioms:

• The domain of production is convex;

• Q(0,...,0)=0;

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

• The production function is monotonically increasing in each variable;

• The production function is quasi-concave that is: Q(x+(1-)y)min(Q(x),Q(y)) [0,1] x,yDP.

One of the most used production function in microeconomics or macroeconomics analysis is the Cobb-Douglas function:

Q: R₊, , A0, k₁,...,k_n0

3. The Differential Geometry of Cobb-Douglas Function in 2 Variables

In what follows we will consider the Cobb-Douglas function: Q: R₊, (where for simplification we took A=1, ,0.

The equation of the surface is therefore: u= .

The parametric representation of this surface is:

H:

First, we have:

, , , ,

The tangent plane to the surface at a point is:

:

that is:

:

We can see that the tangent plane pass through origin if and only if +=1 that is the function is homogenous of degree 1.

The unitary normal at the surface is:

N(_{ },_{ },_{ })

The metric tensor has the components:

_{ }, _{ }, _{ }

and the inverse:

_{ },_{ },_{ }

Christoffel's symbols of the first kind are:

_{ },

_{ },_{ }, _{ },

_{ }, _{ }

Christoffel's symbols of the second kind are:

_{ }, _{ },

_{ }, _{ },

_{ }, _{ }

Riemann-Christoffel curvature tensor is:

The components of Ricci tensor are:

_{ }, _{ },

and the scalar curvature:

S=_{ }

Finally, the sectional curvature is:

where K(1,2) corresponds to the plane determined by the vectors: = and = .

We can see that the sectional curvature vanishes if and only if +=1 that is the function is homogenous of degree 1. In this case we have that Riemann-Christoffel curvature tensor vanishes and obviously Ricci tensor and the scalar of curvature.

4. The Differential Geometry of Cobb-Douglas Function in 3 Variables

In what follows we will consider the Cobb-Douglas function: Q: R₊, (where for simplification we took A=1, ,,0.

The equation of the surface is therefore: u= .

The parametric representation of this hypersurface is:

H:

First, we have:

, , ,

, , ,

, ,

The tangent hyperplane to the hypersurface at a point is:

: that is:

:

We can see that the tangent hyperplane pass through origin if and only if ++=1 that is the function is homogenous of degree 1.

The unitary normal at the hypersurface is:

N(_{ },_{ },_{ },_{ })

Computing as in the upper we find thate the sectional curvatures are:

where K(1,2) corresponds to the plane determined by the vectors: = and = , K(1,3) corresponds to the plane determined by the vectors: = and = and K(2,3) corresponds to the plane determined by the vectors: = and = .

We can see that the upper sectional curvatures vanishes if and only if +=1, +=1 and +=1 from where we find that === that is the function becomes: .

5. References

Chiang, A.C. (1984). Fundamental Methods of Mathematical Economics. McGraw-Hill Inc.

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

Eisenhart, L.P. (1926). Riemannian Geometry. Princeton University Press.

Harrison M. & Waldron, P. (2011). Mathematics for Economics and Finance. Routledge.

Ianuș, S. (1983). Differential Geometry with Applications in the Relativity Theory. Academiei Publishing.

Ioan, C.A. & Ioan, G. (2011). n-Microeconomics. Galati: Zigotto Publishing.

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

Refbacks

• There are currently no refbacks.