The Journal of Accounting and Management, Vol 5, No 2 (2015)
The Complete Theory of Generalized CES Production Function
Cătălin Angelo IOAN1, Gina IOAN2
Abstract: The paper treats various aspects concerning the generalized CES production function. On the one hand were highlighted conditions for the existence of the generalized CES function. Also were calculated the main indicators of it and short and long-term costs. It has also been studied the dependence of long-term cost of the parameters of the production function. The determination of profit was made both for perfect competition market and maximizes its conditions. Also we have studied the effects of Hicks and Slutsky and the production efficiency problem.
Keywords: production function; generalized CES; Hicks; Slutsky
JEL Classification: C02; C65
1 Introduction
To conduct any economic activity is absolutely indispensable the existence of inputs, in other words of any number of resources required for a good deployment of the production process. We will assume that all resources are indefinitely divisible.
We
define on Rn
the production space for n fixed resources as SP=(x1,...,xn)xi0,
i=
where xSP,
x=(x1,...,xn)
is
an ordered set of resources and, because inside a production process,
depending on the nature of applied technology, not any amount of
resources is possible, we will restrict production space to a convex
subset DpSP
– called the domain of production.
We will call a production function an application:
Q:DpR+, (x1,...,xn)Q(x1,...,xn)R+ (x1,...,xn)Dp
which satisfies the following axioms:
A1. Q(0,...,0)=0;
A2. The production function is of class C2 on Dp that is it admits partial derivatives of order 2 and they are continuous on Dp;
A3.
The production function is monotonically increasing in each variable,
that is:
0,
i=
;
A4. The production function is quasi-concave, that is: Q(x+(1-)y)min(Q(x),Q(y)) [0,1] x,yDp
Considering
a production function Q:DpR+
and
R+
- fixed, the set of inputs
which generate the production
called isoquant. An isoquant is therefore characterized
by: {(x1,...,xn)DpQ(x1,...,xn)=
}
or, in other words, it is the inverse image
.
We will say that a production function Q:DpR+ is constant return to scale if Q(x1,...,xn)=Q(x1,...,xn), with increasing return to scale if Q(x1,...,xn)>Q(x1,...,xn) and decreasing return to scale if Q(x1,...,xn)Q(x1,...,xn) (1,) (x1,...,xn)Dp.
2 The generalized CES production function
The generalized CES function has the following expression:
Q:Dp
-{0}R+,
(x1,...,xn)Q(x1,...,xn)=
R+
(x1,...,xn)Dp,
,1...,n0,
(-,0)(0,1), 0
For = we have the classical CES production function.
Computing the partial derivatives of first and second order, we get:
i=
ij=
i=
Let the bordered Hessian matrix:
We
find (not so easy):
=
,
s=
.
Because
(-1)s
=
if
0,
k=
it follows that the function is strictly quasi-concave. Also, if the
function is quasi-concave we have that
0.
But from the definition this condition is equivalent with
definition’s conditions.
We
have now:
=
=
and the homogeneity degree: r=
.
The main indicators are:
=
,
i=
- the
marginal productivity relative to the production factor xi;
=
=
,
i=
- the
average productivity relative the production factor xi;
RMS(i,j)=
,
i,j=
- the partial
marginal rate of technical substitution of the factors i and j;
RMS(i)=
,
i=
- the
global marginal rate of substitution between the i-th factor and the
others;
=
,
i=
- the
elasticity of production in relation to the production factor xi;
ij=
,
i,j=
,
ij
- the relative variation of marginal rate of technical substitution
relative to factors i and j at the relative variation of the factor
endowment ratio with factor i relative to factor j.
Reciprocally,
if for a homogenous production function of degree r: ij=
,
i,j=
,
ij,
1
we have that: ij=
=-1.
But,
in terms of
we obtain:
ij=
=-1,
i,j=
,
ij
in=
=-1,
i=
From
the first relations, multiplying by
and summing with ji:
Replacing in (2) we find that:
,
i=
After
multiplying (3) with
it follows from (1):
,
i,j=
,
ij
,
i=
Let
note:
,
i=
where R*.
We have that:
,
i,j=
Because
from (6) we have:
it follows from (4) and (5):
We have now the first differential of Xi:
=
therefore:
from where:
.
Taking into account the definition of Xi
we finally find that:
Noting
g=
we obtained from (4), (5):
But
g=
implies that:
,
,
,
from where:
therefore
=0
that is A=constant.
After
these considerations:
and summing for i=
:
Denoting
=
we finally have:
therefore q is a CES production function.
Considering
now again the generalized CES production: Q(x1,...,xn)=
let search the dependence of the parameters 1,...,n,,.
We have:
=
=
=
=
=
=
From
these relations we have that at an increasing of a parameter j
the production Q will increase also if 0
and decreases if 0.
Because 0
it follows that Q will increase if
increases. On the restriction of production’s domain at
D’p=(x1,...,xn)xi0,
i=
,
1
we have that Q will decrease at an increasing of
and on the restriction of production’s domain at
D”p=(x1,...,xn)xi0,
i=
,
1
we have that Q will increase at an increasing of .
In
particular, for the generalized CES function related to capital K and
labor L: Q=
we have that the main
indicators are:
=
,
=
=
,
=
RMS(K,L)=
RMS(K)=
,
RMS(L,K)= RMS(L)=
=
,
=
=
If = and K=, L=1- we obtain the main indicators of the classical CES production function:
=
,
=
=
,
=
RMS(K,L)=
RMS(K)=
,
RMS(L,K)= RMS(L)=
=
,
=
=
3 The costs of the generalized CES production function
Considering
now the problem of minimizing costs for a given production Q0,
where the prices of inputs are pi,
i=
,
we have:
From
the obvious relations:
we obtain:
and from the second equation:
.
Noting
r=
we finally obtain:
,
i=
The total cost is:
TC=
=
At
a price change of one factor, i.e. xk,
from the value pk
to
we have:
=
where the relative variation of the total cost is:
Let us now consider the behavior of the total cost of production function at a parameters variation. We have:
Therefore,
if 0
we have that
0
and if 0:
0.
If
we consider now for a given output Q0,
the inputs x1,...,xn
such that:
let
.
We have STCk=
=
representing the short-term total cost when factors
remain constant (^
means that the term is missing).
We put now the question of determining the envelope of the family of hypersurfaces:
=
Conditions to be met are:
After
the elimination of parameters
we have either the locus of singular points of hypersurfaces (which
is not the case for the present issue) or envelope sought.
We have therefore:
From
the second equation, we have for
:
.
Multiplying with i
and summing for all
:
therefore:
.
Replacing
we have now:
.
From
the first equation: TC=
.
We obtained so that the envelope of the family of hypersurfaces of the short-term total cost when all inputs are constant except one is just the long-term cost obtained from nonlinear optimization problem with respect to the minimizing of the cost for a given production.
Calculating the costs derived from the (long-term or short-term) total cost now, we have:
ATC=
=
(average
long-term total cost)
MTC=
=
=
(marginal
long-term total cost)
ASTCk=
=
(average
short-term total cost)
MCk=
=
(marginal
short-term total cost)
VTCk=
(variable
short-term total cost)
AVTCk=
=
(average
variable short-term total cost)
FTCk=
(fixed
short-term total cost)
AFTCk=
=
(average
fixed short-term total cost)
Finally we have:
-
the coefficient of elasticity of long-term total cost with respect to
the price factor k
-
the coefficient of elasticity of long-term total cost with respect to
the production Q0
=
- the coefficient of elasticity of average long-term total cost with
respect to the price factor k
=
- the coefficient of elasticity of marginal long-term total cost with
respect to the price factor k
In
particular, for the generalized CES function related to capital K and
labor L: Q=
we have:
=
=
TC=
On
the short-term, we have for constancy of K: STCL=
and
ATC=
MTC=
ASTCL=
MCL=
VTCL=
AVTCL=
FTCL=
AFTCL=
,
,
,
=
=
,
=
=
.
In particular, for = and K=, L=1- we obtain for the classical CES production function:
=
=
TC=
STCL=
ATC=
MTC=
ASTCL=
MCL=
VTCL=
AVTCL=
FTCL=
AFTCL=
,
,
,
=
=
,
=
=
.
4 The profit
Now consider a sale price of output Q0: p(Q0). The profit is therefore:
(Q0)= p(Q0)Q0-TC(Q0)
It is known that in a market with perfect competition, the price is given and equals marginal cost. The profit on long-term becomes:
(Q0)=p(Q0)Q0-TC(Q0)=MTC(Q0)Q0-TC(Q0)=
=
In
particular, for the generalized CES function related to capital K and
labor L: Q=
we have: (Q0)=
and for the classical CES function (=
and K=,
L=1-):
(Q0)=0.
On
short-term, when factors
remain constant, we have:
(Q0)=p(Q0)Q0-STCk(Q0)=MTC(Q0)Q0-STCk(Q0)=
therefore:
(Q0)=
For
Q=
we have that if K=constant:
(Q0)=
and if =
and K=,
L=1-
for classical CES: (Q0)=
The
condition of profit maximization for an arbitrarily price p,
depending on the factors of production, is:
=
from where
,
i=
or otherwise:
from where:
.
But Q=
implies:
therefore:
.
Because Q is quasi-concave the solution of the characteristic system is the unique point of maximum.
The
maximum profit is:
=
.
For
Q=
we have that:
,
,
=
.
If now = and K=, L=1- for classical CES:
,
,
=
.
5 The Hicks and Slutsky effects for the generalized CES production function
Now
consider the production function Q(x1,...,xn)=
and factor prices
.
The non-linear programming problem relative to maximize production at
a given total cost (CT0)
is:
Because the objective function is quasi-concave and also the restriction (being affine) and the partial derivatives are all positive we find that the Karush-Kuhn-Tucker conditions are also sufficient. Therefore, we have:
From the first equations we obtain:
therefore:
Substituting the first n-1 relations into the last we finally find that:
,
k=
and the appropriate production: Q0(x1,...,xn)=
.
Suppose
now that some of the prices of factors of production (possibly after
renumbering, we may assume that they are: x1,...,xs)
is modified to values
,
the rest remain constant.
From the above, it results:
We will apply in the following, the method of Hicks. To an input price change, let consider that the production remains unchanged, leading thus to a change of the total cost. We therefore have:
from where:
With the new total cost, the optimal amounts of inputs become:
The Hicks substitution effect which preserves the production is therefore:
The difference caused by the old cost instead the new total cost one is therefore:
For
Q=
we have that, for s=1 (that is for a capital price change):
We shall apply now the Slutsky method for our analysis.
At the modify of the price of the factors x1,...,xs, the total cost for the same optimal combination of factors is:
therefore:
The appropriate production is:
The Slutsky substitution effect which not preserves the production is therefore:
and the difference caused by the old production instead the new production one is therefore:
For
Q=
we have that, for s=1 (that is for a capital price change):
6 Production efficiency of generalized CES production function
Let
now two generalized CES production functions for two goods
,
and a number of n inputs F1,...,Fn
available in quantities
.
Production functions of
or
are:
,
appropriate
to the consumption of xk
units of factor Fk,
k=
.
We
have seen that:
,
,
i=
.
The production contract curve satisfies:
,
i=
Dividing
for ij:
and for i=1:
,
j=
.
Finally, for x1=
we have the equation of production contract curve:
If we consider now the input prices: p1,...,pn we have that for the production contract curve: x1=g1(),...,xn=gn(), R:
and:
,
j=
.
For
=1
we then obtain: p1=1,
,
j=
.
If
the initial allocation of factors of production was
we have that
therefore:
from where it follows
.
For
this value we find now the final allocation:
7 The concrete determination of the generalized CES production function
Considering
an affine function: f:RnR,
f(x1,...,xn)=1x1+...+nxn+n+1
and a set of m>n+1 data:
,
k=
the problem of determining i,
i=
using the least square method is to minimize the expression:
that is to solve the system:
Considering the matrix:
and
the cofactor of the (i,j)-element in
we will obtain:
Considering
now a production function Q(x1,...,xn)=
we put the problem of concrete determination of the parameters A, ,
,
i,
i=
.
Let
therefore a set of m>n+1 data:
,
k=
.
Considering
the equation of Q in the form:
we will take for the beginning A=1 (because A will enter in the
structure of i)
and we have:
.
Considering
fixed
and ,
we will modify the data set to the new one:
,
k=
.
From
above, we will obtain the values of i,
i=
.
For an accurate determination of Q we will vary the values of
and
till we will find the maximum value of the correlation coefficient.
8 Conclusions
The above analysis reveals several aspects. On the one hand were highlighted conditions for the existence of the generalized CES function. Also were calculated the main indicators of it and short and long-term costs. It has also been studied the dependence of long-term cost of the parameters of the production function. The determination of profit was made both for perfect competition market and maximize its conditions. Also we have studied the effects of Hicks and Slutsky and the production efficiency problem.
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1 Danubius University of Galati, Department of Finance and Business Administration, catalin_angelo_ioan@univ-danubius.ro.
2 Danubius University of Galati, Department of Finance and Business Administration, ginaioan@univ-danubius.ro.
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