DUALITY IN MONOPOLY
Ilko Vrankić, Mira Krpan
University of Zagreb, Faculty of Economics and Business
Trg J. F. Kennedy 6, 10000 Zagreb, Croatia
{ivrankic, mkrpan}@efzg.hr
Abstract: This paper follows our theoretical research on duality in microeconomic theory and
applies the duality principles, which rest on the price taking behaviour of economic subjects, on the
monopolistic behaviour. The standard approach of deriving the profit function for the monopolist
from the production function and defined pseudo production function is accompanied by the
alternative approach in which the starting point is the pseudo cost function. Finally, starting from the
derived profit function the pseudo production function is recovered and a version of Hotelling’s
lemma is given. All results are illustrated with a numerical example.
Keywords: duality, pseudo production function, pseudo cost function, profit function, Hotelling’s
lemma.
1 INTRODUCTION
Rationality of economic agents in microeconomic theory is described by the optimization
problems in which convex sets play an important role. The possibility of characterizing them
in two ways naturally gives rise to duality in microeconomic theory [3,4]. Duality in
microeconomic theory includes derivation and recovering of the alternative representations of
the consumer preferences and the production technology. Since duality in this case rests on
the price taking behaviour, the question is how can the principles of duality be applied in the
case of a monopolistic firm where the single producer has influence on the price which he
charges for the product [1,2,5,7]? In this paper we show how the profit function for the
monopolist for the given demand function can be obtained by introducing the pseudo cost
function [6] and solving the profit maximization problem. The standard approach in which
the production function and defined pseudo production function is the starting point is also
derived. Finally, starting from the derived profit function we show how the pseudo
production function can be recovered. Although Hotelling's lemma, a famous result from the
microeconomic theory, defined below, cannot be directly applied in obtaining the
monopolist's supply function for the given demand function, a version of Hotelling's lemma
is illustrated.
2 FROM THE PSEUDO PRODUCTION FUNCTION TO THE PROFIT
FUNCTION
The emphasis is on a monopolist whose objective is to maximize profit. The monopolist
faces the following negatively sloped inverse demand function [5]
p0  wDx0  ,
(1)
where p 0 is the price of the monopolist’s product and w  0 represents the influence on
demand of “other variables”, for example income, and D is a function of x0 for which
D ' ( x0 )  0 . The monopolist’s technology is described by the production function x0  F (x) ,
where x is the vector of inputs used in the production of the monopolist’s product x 0 . Since
the total revenue function is described by p0 x0  wD  F (x) F (x)  wF (x) D  F (x) , the profit
maximization model for the monopolist in which the quantities of inputs are the choice
variables reduces to
maks wF (x) DF (x)  wx ,
(2)
x
where w is the vector of input prices. If we form the pseudo production function [5]
F * (x)  F (x) DF (x) 
p o x0
w
(3)
as deflated revenue function and interpret the parameter w in (2) as the price of the pseudo
product, then the known results from duality theory in microeconomics could be applied. It is
assumed that sufficient regularity conditions are satisfied so that the maximum exists [1,5,7].
The first order necessary conditions for the profit maximization problem reduce to
w
F * (x)
 wi , i
xi
(4)
Therefore, the monopolist will hire the levels of inputs for which the marginal revenue of the
F * (x)
corresponding input, w
, is equal to its marginal cost, wi . An interior solution is
xi
assumed. It is assumed also that the input market is perfectly competitive so that the
monopolist takes input prices as given.
Solving the first order necessary conditions leads us to the input demand functions. By
substituting the derived input demand functions in the production function, for the given
demand function, we obtain the monopolist's supply function. The profit function is obtained
by substituting the derived functions in the goal function in (2). To illustrate this procedure,
the production function
1
1
x0  f ( x1 , x 2 )  x14 x 24
(5)
is chosen [6]. Let us start from the linear inverse market demand function,
p0  w(a  bx0 ) ,
(6)
for which the pseudo production function is defined as follows
1
1
1
1
1
1
1
1
F * (x)  (a  bx14 x24 ) x14 x24  ax14 x24  bx12 x22 .
(7)
The profit maximization model reduces to
1
4
1
1
4
2
1
2
1
1
2
2
maks w(ax x  bx x )  w1 x1  w2 x2 .
x1 . x2
(8)
First order necessary conditions are expressed by the following system of equations
w
1 1
F * (x)
a 3 1 b 1 1
w 3 1
 w( x1 4 x24  x1 2 x22 )  x1 4 x24 (a  2bx14 x24 )  w1
x1
4
2
4
1 1
F * (x)
a 14  34 b 12  12
w 14  34
w
 w( x1 x2  x1 x2 )  x1 x2 (a  2bx14 x24 )  w2
x2
4
2
4
,
(9)
x2 w1
. We can express x2 as a function of x1 ,

x1 w2
Dividing both expressions in (9) gets us to
1
1
1
w
w4
w  3 w4 1
x2  1 x1 , whose substitution in one of (9) leads us to x1 4 11 x14 (a  2bx12 11 )  w1 . By
w2
4
4
4
w2
solving
it
we
the
get
monopolist’s
demand
w2
function
for
the
first
input,
1
x1* 
a 2 w2 w24
1
2
1
1
2
1
1
2
2
. By inserting it in x2 
4 w (2 w w  bw) 2
2
w1
x1 , we get the monopolist’s demand
w2
1
4
1
2
a ww
function for the second input, x2* 
1
2
2
1
2
1
.
1
2
2
4 w (2 w w  bw)
The
supply
function
for
the
2
aw
monopolist, for the given demand function, is then x0* 
1
2
1
1
2
2
. By inserting the
2(2 w w  bw)
derived supply and demand functions in the goal function of a producer, the profit function is
obtained,
* 
a 2 w2
1
2
1
1
2
2

(10)
4(2w w  bw)
3 FROM THE PSEUDO COST FUNCTION TO THE PROFIT FUNCTION
Another method for obtaining the monopolist's profit function for the given demand function
is to start from the monopolist’s cost function, c( x0 , w) , which is an alternative way of
describing technology [9]. The decision variable in this model of the monopolist’s profit
maximization problem is the quantity of production,
maks p0 x0  c( x0 , w)  maks wD( x0 ) x0  c( x0 , w) .
x0
x0
(11)
Taking into account the definition of the pseudo production function, F * (x)  D( x0 ) x0  y ,
the quantity of production can be expressed as the function of pseudo production, x0  g ( y) .
In this case the profit maximization problem reduces to
maks wy  cw, g ( y )
y
 maks wy  c* ( y, w ),
(12)
y
where c* ( y, w) can be called the pseudo cost function. The first order necessary condition
for the profit maximization problem gives the following equation which needs to be solved to
get us to the pseudo production function,
c* ( y, w)
.
(13)
y
Therefore, profit maximization for the given demand function in this model is characterized
by the equality between the price of the pseudo product and the marginal pseudo cost. To
illustrate how the profit function for the monopolist can be derived by starting from the
pseudo cost function, we start from the chosen production function in (5). The cost function
is derived from the model of cost minimization for the given level of production1,
w
c( x0 , w )  min wx
x
(14)
1 1
4 4
1 2
subject to x x  x0.
From the theory of production it is known that the economic efficiency is characterized by
f ( x1 , x 2 )
x1
the equality between the marginal rate of technical substitution MRTS 
and the
f ( x1 , x 2 )
x 2
w1
input
price
ratio
[8].
In
our
case
it
is
w2
x
w
MRTS  2  1 , from which the long-run expansion path is derived, which describes the
x1 w2
wx
optimal combinations of inputs at each output level as output expands, x2  1 1 . Substituting
w2
it in the constraint, we get the conditional input demand functions, which give the cost
minimizing input levels for the given output level,
1


1
1
x1 ( w1 , w2 , x0 )  w1 2 w22 x 02
and
1
x 2 ( w1 , w2 , x0 )  w12 w2 2 x02 , whose substitution in the goal function of (14) gives us the cost
function
1
1
c( w1 , w2 , x0 )  2 w12 w22 ( x0 ) 2 .
(15)
Our goal is to express it in terms of the pseudo production function and to obtain the pseudo
cost function. For the given inverse demand function in (6) the pseudo production function is
defined as y  x0 (a  bx0 )  ax0  bx02 . Therefore, the quantity of production is related to the
pseudo production function by the following quadratic equation bx0   ax0  y  0 from
2
which it follows x0 
a  a 2  4by
2b
and the pseudocost function as a function of
pseudoproduction function collapses to
1
1
c* ( y, w1 , w2 )  2w12 w22 (
a  a 2  4by 2
) .
2b
Therefore, the following optimization model needs to be solved
1
More on derivation and the properties of the cost function can be found in [8].
(16)
1
1
maks wy  2w12 w22 (
y
a  a 2  4by 2
) .
2b
(17)
The first order necessary condition is
1
2
1
1
2
2
1
2
1
1
2
2
2w w a
2w w
c


 w.
2
y
b
b a  4by
(18)
Solving the equation for y gives the pseudoproduction function,
1
1
a 2 4ww12 w22  bw2
y
,
2
4  1 1

2 2
 2w1 w2  bw 


(19)
whose substitution into the goal function leads to the same profit function as before,
1
1
1 1
a 2 4ww12 w22  bw2
a 2 w2
a 2 w2
2 2
 w

2
w
w


1
2
2
1 1
1 1
4  1 1
2

2
2
2
2
2 2
4(2w1 w2  bw)
4(2 w1 w2  bw)
 2w1 w2  bw 


*
(20)
4 HOTELLING’S LEMMA
According to Hotelling’s lemma [1,2,5,7,8], the derivative of the profit function for a
perfectly competitive, price-taking firm with respect to the product price of a firm is equal to
the firm’s quantity supplied. Since the monopolist is not the price taker, the question is how
can be Hotelling’s lemma applied in this case. Starting from the monopolist’s profit function
and looking at the parameter w as the price of the pseudo product, we can apply Hotelling’s
lemma and get the pseudo production function [6],
1
1
1
1
 * a 2 2 w(2w12 w22  bw)  bw2 a 2 4ww12 w22  bw2
y


.
1 1
1 1
w
4
4
2
2
(2 w12 w22  bw)
(2 w12 w22  bw)
(21)
The supply function of a monopolist can be obtained from the pseudo production function for
 * p0* x0* w(a  bx0* ) x0*


 ax0*  b( x0* ) 2 or
the
given
demand
function,
w
w
w
*

b( x0* ) 2  ax0* 
 0 . So we got the quadratic equation which needs to be solved to get the
w
supply function, x0* , for the given demand function
a  a 2  4b
x0* 
2b
1 1
2 2
1
2
1 1
2 2
1
2
4ww w  bw2
a a a b

(2w w  bw)2
w 

2b
*
2
2
aw
1
2
1
1
2
2
2(2w w  bw)
.
(22)
5 FROM THE PROFIT FUNCTION TO THE PSEUDO PRODUCTION
FUNCTION
Starting from the profit function for the monopolist, how can we go back and recover the
pseudo production function? Since the profit function gives the maximum profit for every
combination of input prices, w1 and w2 , and the parameter w, its definition brings us to the
following inequality
 (w, w1 , w2 )  wF (x) DF (x)  w1 x1  w2 x2
w1 , w2 , w .
(23)
Definition of the pseudo production function enables us to rewrite the previous inequality as
 (w, w1 , w2 )  wF * (x)  w1 x1  w2 x2
w1 , w2 , w ,
(24)
from which it follows that the pseudo production function is the result of the following
optimization problem
F * (x)  max y : wy  w1 x1  w2 x2   ( w, w1 , w2 )
.
(25)
By normalizing the price of the pseudo product and dividing all the input prices by w ,
w
w
W1  1 ,W2  2 , the previous optimization problem reduces to
w
w
F * (x)  max y : y  W1 x1  W2 x2   (1,W1 ,W2 ) , W1 ,W2
and
(26)
F * (x)  min W1 x1  W2 x2   (1,W1 ,W2 ) .
(27)
W1 ,W2
Its solution is the monopolist's pseudo production function. Below we illustrate how to
recover the pseudo production function from the profit function. Starting from our derived
profit
function
in
(10),
the
normalized
profit
function2
is
2
a
and the pseudo production function can be obtained as the
 (1,W1 ,W2 ) 
1
1
2
2
4(2W1 W2  b)
solution to the following optimization problem
F * (x)  min W1 x1  W2 x2 
W1 ,W2
a2
1
2
.
1
2
2
(28)
4(2W1 W  b)
The system of equations that expresses the first order necessary conditions follows,
1
1
1
1

f
a2
 x1  (2W12W22  b) 2W1 2W22  0
W1
4
f
a
 x2  (2W1 W  b) 2W1 W2  0
w2
4
2
2
The profit function
 (1,
1
2
1
2
2
1
2

(29)
1
2
w1 w2
, ) is actually the conjugate function to the pseudoproduction function
w w
F * (x) if F * (x) is concave [5 ]. More on the conjugacy approach to duality theory can be found in [7].
1
Multiplying
1
2
quantities
of
inputs
gives
x1 x2 
1
a2
(2W12W22  b)  4
16
which
implies
1
2
2
1
1
a  14  14
a  14  14 b
2
2
2W1 W  b  x1 x1 and W1 W2  x1 x1  . The product of the corresponding input
2
4
2
1 1
1 1
a
b
a 1 1 b 1 1
with its price reduces to W1 x1  x14 x24  x12 x22 and W2 x2  x14 x24  x12 x22 . Inserting the
4
2
4
2
given results in the goal function gives
1 1
1 1
a 1 1
b 1 1 a 1 1
F * (x)  2 x14 x24  2 x12 x22  x14 x24  ax14 x24  bx12 x22 ,
4
2
2
(30)
which is the pseudo production function we started from.
6 CONCLUSION
Since duality in microeconomics rests on price taking behaviour, the main idea was to apply
the known duality principles to the monopolistic case. In deriving the profit function for the
monopolist the standard approach is shown which includes starting from the production
function and the defined pseudo production function. We give another approach by starting
from the pseudo cost function. Finally, starting from the derived profit function the pseudo
production function is recovered. A version of Hotelling’s lemma, important from the
empirical standpoint, is given. Application of the duality theory in monopoly to the real data
is left for the future research.
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