Acta Universitatis Danubius. Œconomica, Vol 11, No 1 (2015)
The Complete Theory of Cobb-Douglas Production Function
Cătălin Angelo Ioan1, Gina Ioan2
Abstract. The paper treats various aspects concerning the Cobb-Douglas production function. On the one hand were highlighted conditions for the existence of the Cobb-Douglas 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; Cobb-Douglas; Hicks; Slutsky
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 (see Appendix).
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 Cobb-Douglas Production Function
The Cobb-Douglas function has the following expression:
Q:D
-{0}R+,
(x1,...,xn)Q(x1,...,xn)=
R+
(x1,...,xn)D,
A
,
1,...,nR*
Computing the partial derivatives of first and second order, we get:
i=
ij=
i=
Let the bordered Hessian matrix:
We
find (not so easy):
=
,
k=
.
Because
(-1)k
=
,
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 axiom A3 we must have that
0
that is i0.
After these considerations we have that if i0,
i=
the Cobb-Douglas function is strictly quasi-concave.
We
have now:
=
=
and r=
.
The main indicators are:
=
=
,
i=
=
=
,
i=
RMS(i,j)=
,
i,j=
RMS(i)=
,
i=
=
,
i=
ij=-1,
i,j=
Reciprocally,
if for a homogenous production function of degree r:
=
,
i=
we have that:
,
i=
and
.
But now, we have:
=
(where ^ means that the variable is missing).
We
have now:
.
For ji
we obtain now:
therefore:
.
Integrating with respect to
:
therefore:
.
Analogously, by recurrence:
with A=constant with respect to
.
But:
.
After these considerations it follows that if it is homogenous of
degree r, r must be
.
Finally:
implies that: Q(x1,...,xn)=
=
=
=
- the Cobb-Douglas production function.
Considering
now again the Cobb-Douglas production: Q(x1,...,xn)=
let search the dependence of the parameters 1,...,n.
We
have:
=
=
0
xi1,
i=
.
From this relation we have that at an increasing of a parameter i
the production Q will increase also.
In particular, for the Cobb-Douglas function related to capital K and labor L: Q=AKL we have that the main indicators are:
=
=
=
=
RMS(K,L)=RMS(K)=
RMS(L,K)=
RMS(L)=
=
=
=KL=-1
3. The Costs of the Cobb-Douglas 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:
=
,
k=
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:
0
.
If we note:
0
and M=
0
we obtain that:
0
.
Considering
now the function f:(0,)R,
f(x)=
we have
0
therefore f is strictly increasing. Because
and
the equation:
has a unique solution
called cost threshold with respect to the k-th parameter. After these
considerations we have that for k
the total cost will increase at an increasing of k
and after it will decrease.
The
situation may seem paradoxical that at the growth of the elasticity
of one input, total cost increases. Fortunately, due to the sharp
rise of f, the values of
are very small so it does not significantly affect processes.
Like an example, considering the production function Q(K,L)=KL, ,0 we have that the behavior of 0 related to is (for Q=5):
Figure 1
The long-term total cost for =0.7 and variable is shown in figure 2:
Figure 2
where the maximum value is reached for =0.025.
If
we consider for a given output Q0,
the inputs x1,...,xn
such that:
let
where ^ means that the term is missing.
We
have STCk=
=
representing the short-term total cost when factors
remain constant.
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:
Noting:
=
it follows:
from
where:
.
Finally:
=
and replacing:
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)
The
extreme of the function ASTCk(Q0)=
are given by:
=
=0
from where:
when
k1
and the minimum value is:
=
.
If
k1
it follows that
0
therefore the average short-term total cost will decrease.
Finally we have:
-
the coefficient of elasticity of long-term total cost with respect to
the price factor i
-
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 i
-
the coefficient of elasticity of marginal long-term total cost with
respect to the price factor i
In particular, for the Cobb-Douglas function related to capital K and labor L: Q=AKL we have:
=
=
TC=
On
the short-term, we have for constancy of K: STCL=
and
ATC=
MTC=
ASTCL=
MCL=
VTCL=
AVTCL=
FTCL=
AFTCL=
The
extreme of the function: ASTCL(Q0)
is given by:
when 1
and the minimum value is:
.
,
,
,
,
,
,
.
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 Cobb-Douglas function related to capital K and
labor L: Q=AKL
we have: (Q0)=
.
On
short-term, when factors
remain constant, we have:
(Q0)=p(Q0)Q0-STCk(Q0)=MTC(Q0)Q0-STCk(Q0)=
therefore:
(Q0)=
=
Like
a conclusion, the company will make a profit in the short-term if,
under constancy factors
,
the amount of factor xk
will be higher than
if k1
and less than
if k1.
If k=1
then the firm will incur losses.
For
Q=AKL
we have that if K=constant, the company will make a profit in the
short-term in the case 1
if L
and in the case 1
if L
.
If =1
the firm will incur losses.
The
condition of profit maximization for an arbitrarily price p,
depending on the factors of production, is:
=
from where
,
i=
or otherwise:
and finally:
.
Because Q is quasi-concave the solution of the characteristic system
is the unique point of maximum. How Q=
we obtain that the appropriate production is:
=
therefore, if r1:
and the factors:
,
i=
.
The
maximum profit is:
=
.
If
r=1 the necessary condition for profit maximization is:
or p must be: p=
therefore the amount of factors are not independent, that is, for a
fixed factor, let say xs:
and:
,
i=
,
is.
The profit is:
=0
for any amount of xs.
For Q=AKL we have that, if +1:
,
,
,
and
if +=1
the necessary condition for profit maximization is:
,
,
,
.
At
a variable price p(Q) we have now: (Q)=p(Q)Q-CT(Q)
therefore the necessary condition for profit maximization is
=0
therefore:
.
Substituting
the expression of MTC we obtain:
where we noted:
=
.
But this differential equation gives us:
,
CR+
and the profit (Q)=p(Q)Q-CT(Q)=C.
Therefore, for the maximum of the profit (Q)=C
we must have the price
,
CR+.
Because:
we have that if r1,
the price will decrease with production and if r1
for
the price will decrease and for
the price will increase. If r=1 we have that
and the price will decrease with production.
For
Q=AKL
we have
and for the profit (Q)=C
we must have the price
.
If +1,
the price will decrease with production and if +1
for
the price will decrease and for
the price will increase. If +=1
we have that
and the price will decrease with production.
5. The Hicks and Slutsky Effects for the Cobb-Douglas 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:
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 it 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:
Let now calculate the new prices influence to the effects of substitution and of new cost in the Hicks effect.
We have:
After
these relations, it follows that the effect of substitution at the
increase of the price xk,
k=
is reduced, while the effect of new cost is reduced if
or it increase if
.
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:
Let us now calculate the influence of the new prices on the effects of substitution and new cost in Slutsky effect.
We have:
Therefore
the effect of substitution at a price increase of the factor xk,
k=
is reduced, while the effect due to production decrease in case
0
or it increase if
0.
6. Production Efficiency of Cobb-Douglas Production Function
Let
now two Cobb-Douglas production functions for two goods
,
and a number of n inputs F1,...,Fn
available in quantities
.
The 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 xi=
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:
=
where r1=
.
For
this value we find now the final allocation:
If
now the two production functions are:
,
we have that for limited quantities of capital
and labor
the equation of production contract curve is:
and
the final allocation for an initial one:
:
where
.
7. The Concrete Determination of the Cobb-Douglas 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 logarithm of Q, we have:
therefore we will modify the data set to the new one:
,
k=
.
From above:
where
.
For
the Cobb-Douglas Q=Q(K,L)=AKL
we have therefore, for the set:
:
,
,
8. Conclusions
The above analysis reveals several aspects. On the one hand were highlighted conditions for the existence of the Cobb-Douglas function. Also were calculated the main indicators of its 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.
9. Appendix
A.1. Mathematical concepts
A function Q:DRnR, D – convex set, is quasi-concave if:
Q(x+(1-)y)min(Q(x),Q(y)) [0,1] x,yD
and is strictly quasi-concave if:
Q(x+(1-)y)min(Q(x),Q(y)) (0,1) x,yD
A function Q:DRnR, D – convex set, is quasi-convex if:
Q(x+(1-)y)max(Q(x),Q(y)) [0,1] x,yD
and is strictly quasi-convex if:
Q(x+(1-)y)max(Q(x),Q(y)) (0,1) x,yD
Geometrically speaking, a quasi-concave function has the property to be above the lowest values recorded at the ends of some segment. This property is equivalent with the convexity of the set Q-1[a,)={xDQ(x)a} aR.
Note also that if f and g are arbitrary functions:
f – quasi-concave (quasi-convex) implies that -f is quasi-convex (quasi-concave);
f – strictly quasi-concave (quasi-convex) implies that f is quasi-concave (quasi-convex);
f - quasi-concave (quasi-convex) implies that f is quasi-concave (quasi-convex) for any 0;
f,g - quasi-concave (quasi-convex) imply that min(f,g) (max(f,g)) is quasi-concave (quasi-convex) for any ,0;
f - quasi-concave (quasi-convex) and g:RR is increasing imply that gf:DR is quasi-concave (quasi-convex);
fC1(D)
is (strictly) quasi-concave
if
and only if: f(x)f(y)
x,yD;
fC1(D)
is (strictly) quasi-convex
if
and only if: f(x)f(y)
x,yD;
A monotonically function f:DRR is quasi-concave and quasi-convex;
Any affine function is quasi-concave and quasi-convex.
Considering now the bordered hessian matrix:
=
and the bordered principal diagonal determinants:
=
,
k=
we have the following theorems:
Theorem
If
the function f:D
R,
D – convex, fC2(D)
is quasi-concave then
0,
k=
.
Theorem
In order that the function f:D
R,
D – convex, fC2(D)
be quasi-concave is sufficient that
0,
k=
.
A.2. The main indicators of production functions
Let a production function:
Q:DpR+, (x1,...,xn)Q(x1,...,xn)R+ (x1,...,xn)Dp
We
will call the marginal productivity relative to a production factor
xi:
=
representing
the trend of variation of production at the variation of the factor
xi.
In particular, for a production function of the form: Q=Q(K,L) we
have K=
- called
the marginal productivity of capital and L=
- called
the marginal productivity of labor.
We
call the average productivity relative to a production factor xi:
=
representing the value
of production at the consumption of a unit of factor xi.
In particular, for a production function of the form: Q=Q(K,L) we
have: wK=
- called
the productivity of capital, and wL=
- the productivity
of labor.
Considering
the factors i and j with ij,
we define the restriction of production area:
Pij=(x1,...,xn)xk=ak=const,
k=
,
ki,j,
xi,xjDp
relative
to the two factors when the others have fixed values. Also, let:
Dij=(xi,xj)(x1,...,xn)Pij
- the
domain of production relative to factors i and j.
We define: Qij:DijR+ - the restriction of the production function to the factors i and j, i.e.: Qij(xi,xj)=Q(a1,...,ai-1,xi,ai+1,...,aj-1,xj,aj+1,...,an). The functions Qij define a surface in R3 for every pair of factors (i,j).
We will call partial marginal rate of technical substitution of the factors i and j, relative to Dij (caeteris paribus), the opposite change in the amount of factor j to substitute a variation of the quantity of factor i in the situation of conservation production level.
We
will note: RMS(i,j)=
and we have, since
Qij(xi,xj)=Q0=constant:
RMS(i,j)=
.
Obviously
.
We
also define the global marginal rate of substitution between the i-th
factor and the others: RMS(i)=
.
The global
marginal rate of technical substitution is the minimum (in the
meaning of norm) of changes in consumption of factors so that the
total production remain unchanged.
In
particular, for a production function of the form: Q=Q(K,L) we have:
RMS(K,L)=
RMS(K)=
,
RMS(L,K)=RMS(L)=
.
It
is called elasticity of production in relation to a production factor
xi:
=
=
- the relative
variation of production at the relative variation of factor xi.
In
particular, for a production function of the form: Q=Q(K,L) we have
K=
- called
the elasticity of production in relation to the capital and L=
- the
elasticity factor of production in relation to the labor.
Let
note now for arbitrary factors xi,
xj:
ij=
,
i,j=
,
ij
and we call the factor endowment ratio with the factor i relative to
factor j.
It
is called the elasticity of marginal rate of technical substitution
for a production function relative to inputs i and j: ij=
,
i,j=
,
ij
and represents 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.
We
have therefore: ij=
=
.
Considering
now a production function Q:DpR+,
(x1,...,xn)Q(x1,...,xn)R+
(x1,...,xn)Dp,
homogenous of degree r, let
note for an arbitrary factor (for example xn):
i=
,
i=
.
Of
course: ij=
.
We obviously have:
Q(x1,...,xn)=
=
Considering
the restriction of the production function at Dp
:
=
we can write:
Q(x1,...,xn)=
With the new function introduced, the above indicators are:
=
,
i=
=
=
,
i=
=
RMS(i,j)=
,
i,j=
RMS(i,n)=
,
i=
RMS(i)=
,
i=
RMS(n)=
=
,
i=
=
ij=
in=
For
a production function of
the form: Q=Q(K,L),
,
:
=
=
=
=
RMS(K,L)=RMS(K)=
=
=
=KL=
A.3. Necessary and sufficient conditions for nonlinear optimization
Considering now the non-linear programming problem:
where
f,gC2(Dp)
and a solution
=
the Karush-Kuhn-Tucker conditions occur: iR+,
i=
so that:
where
F
is the gradient of F defined by: F=
and =1
for the case of maximizing and =-1
in the case of minimizing.
If
f,gi,
i=
are of class C2,
from [1] follows, for the maximizing case, the sufficiency of
Karush-Kuhn-Tucker conditions takes place in the broader framework of
quasi-concavity of functions f and g and, moreover, if for a solution
=
one of the conditions occurs:
k=
such that
0;
k=
such that
0
and
0;
0;
f is concave.
For
the problem:
replacing
f with –f and taking into account that
=
follows that Karush-Kuhn-Tucker conditions becomes:
,
iR+,
i=
and sufficiency reduces to one of the cases:
k=
such that
0;
k=
such that
0
and
0;
0;
f is convex.
In
the particular case of the problem of minimizing the total cost (TC)
relative to a production function Q=Q(x1,...,xn)
and pi,
i=
- the prices of inputs:
the Karush-Kuhn-Tucker conditions are:
.
Because =0
implies pk=0
– which is absurd in economic terms, results: 0
then:
or, with another expression:
.
Because the objective function
is affine, Q is quasi-concave and, in addition
=pk0
follows, from the foregoing, that these conditions are sufficient.
A.4. Production efficiency
Let
us consider in the following two goods
,
and a number of n inputs F1,...,Fn
available in quantities
,
and the production functions of
or
as follows:
,
appropriate
to the consumption of xk
units of factor Fk,
k=
.
We will assume that the production functions are of class C2
inside space production SP.
We
will build the Edgeworth’s box consisting in a n-dimensional
parallelepiped:
the quantities of
being relative to
and those of
relative to
on the parallelepiped sides. Let consider an initial allocation of
inputs for
and
:
,
where
ai+bi=
,
i=
.
The productions appropriate to the initial allocation are:
,
relative to O and F, respectively. Because bi=
-ai,
i=
we have:
=
.
The production function of
is therefore:
and means the production of
relative to the origin of axes. We have now:
,
,
i,j=
therefore
is also quasi-concave but with negative partial derivatives of order
1. Considering the isoproduction hypersurfaces, it follows that
(relative to O) those of
is convex, while that of
is concave.
Let
- the production zone of
superior to
and
- the production zone of
superior to
.
Suppose
now that int
(int
means the interior of the set, i.e. those points for which there is a
n-dimensional cube centered in them sufficiently small side and
included in the given set).
Let
now a point
int
and also let the straight line that passes through the origin and C.
Let note
the intersection with the isoproduction hypersurface
and
the intersection with the isoproduction hypersurface
.
We have now
and
.
Because
is convex we obtain that:
and
- concave implies that
or
.
After these inequalities follows that the production of each good can
increase, so the initial allocation is not optimal.
We call Pareto’s efficiency the situation where new production can not improve without affecting the other’s production. From the foregoing, it follows that the Pareto’s efficiency is obtained if the isoproduction hypersurfaces are tangent.
The
condition of tangency for Q=
and Q=
=
is reduced to the determination of those points
where
,
i=
,
R
i.e. those points where hypersurfaces intersect and have the same
tangent hyperplane (directors
parameters are proportional).
Taking into account that
=
we have that:
,
i=
,
=-R.
In
marginal notation, we have:
,
i=
,
R.
For
two inputs (K and L) the above relations are equivalent with:
.
On the other hand:
=
- marginal rate of technical substitution of capital for
and
=
- marginal rate of technical substitution of capital for
.
The upper equality becomes:
=
.
All of the points where the allocation is Pareto’s efficient generates the production contract curve.
Contract curve represents all combinations of goods for which no party can maximize its production without diminishing the other’s production. On the other hand, any point on the curve represents a possible allocation contracts. The problem is this: if one good will be produced in order to reach the maximum level, what will do the other?
Considering
now the prices of n inputs as p1,...,pn
the total cost is: TC=
and it maximize the production if it is tangent to the isoproduction
hypersurface. But each good want to be produced in maximum quantity
therefore:
,
or, in other words, the cost hyperplane will be tangent to both isoproduction hypersurfaces, that is it will coincide with the common tangent hyperplane.
Considering the production contract curve of the form:
x1=g1(),...,xn=gn(), R
follows:
from where:
pk=
,
0,
k=
We
note that prices are determined up to a multiplicative factor, which
does not affect the result of the problem and can therefore consider
=1.
If the initial allocation of factors of production was
,
the total cost of production of
is
=
.
The new amounts of factors (which also satisfy the same total cost)
involves:
.
Replacing the values of pk
into this equation:
hence
we will find R.
Substituting in the appropriate expressions, will result pk
and xk,
k=
.
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1 Associate Professor, PhD, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd, Galati, Romania, Tel.: +40372 361 102, Fax: +40372 361 290, Corresponding author: catalin_angelo_ioan@univ-danubius.ro.
2 Assistant Professor, PhD in progress, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd, Galati, Romania, Tel.: +40372 361 102, Fax: +40372 361 290, E-mail: gina_ioan@univ-danubius.ro.
AUDŒ, Vol. 11, no. 1, pp. 74-114
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