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

**A
Study of Allen Production **

**Function
with Differential Geometry **

**Alin
Cristian Ioan**^{1}

**Abstract.
**In
this paper we shall made an analysis of Allen production function
from the differential point of view. We shall obtain some interesting
results about the nature of the points of the surface and the total
curvature.

**Keywords:**
production functions; metric; curvature; Allen

**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 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 **the**
**set
of resources**.
Because 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
an **Allen
production function**
an application:

Q:*D*_{p}**R**_{+},
(K,L)Q(K,L)=
**R**_{+}
(K,L)*D*_{p},
a,b,c,d**R**^{*}_{+},
c0

The production
function is C^{}-differentiable
and homogenous of degree 1.

**2.
The Differential Geometry of Allen 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}=
=

we obtain that
in order to Q be quasiconcave
(*that
is for any a*****R***,
Q*^{-1}*([a,****))
is convex*)
we must have
0.

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 K-O-L) 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) =0

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 parabolic.

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: k(m)=
.
The extreme values k_{1}
and k_{2}
of the function k(m) are the principal curvatures of the surface in
that point. The quantity K=k_{1}k_{2}
is named the total curvature in the considered point. We have K=0
therefore the surface has null constant total curvature.

**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.
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J. (1967). On Estimation of the CES Production Function.
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T.C. & Hildebrand, G.H. (1965). Manufacturing
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Univ. Press, Ithaca.

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

Revankar,
N.S. (1971). A Class of Variable Elasticity of Substitution
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Sato,
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1 Nicolae Oncescu College, Braila, Address: 1-3 ŞOS. Brăilei, City: Ianca, Brăila County, Tel.: +40239-668 494, Corresponding author: alincristianioan@yahoo.com.

**AUDŒ,
Vol. 11, no. 1, pp. 115-119**

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