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EFZG WORKIN G PAPER SERIES
EFZG SERIJA ČLANAKA U NAS TAJANJU
ISSN 1849-6857
UDK 33:65
No. 15-04
Vedran Kojić
Solving the utility maximization
problem with CES and CobbDouglas utility function via
mathematical inequalities
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Solving the utility maximization
problem with CES and Cobb-Douglas
utility function via mathematical
inequalities
Vedran Kojić*
[email protected]
Faculty of Economics and Business
University of Zagreb
Trg J. F. Kennnedyja 6
10000 Zagreb, Croatia
The views expressed in this working paper are those of the author(s) and not necessarily represent those of the Faculty of
Economics and Business – Zagreb. The paper has not undergone formal review or approval. The paper is published to
bring forth comments on research in progress before it appears in final form in an academic journal or elsewhere.
* The author is grateful to professor Zrinka Lukač, PhD, for fruitful comments.
Copyright July 2015 by Vedran Kojić
All rights reserved.
Sections of text may be quoted provided that full credit is given to the source.
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Abstract
This paper presents a new, non-calculus approach to solving the utility maximization problem with
CES utility function, as well as with Cobb-Douglas utility function in case of n≥2 commodities.
Instead of using the Lagrange multiplier method or some other method based on differential calculus,
these two maximization problems are solved by using Jensen's inequlity and weighted arithmeticgeometric mean (weighted AM-GM) inequality. In comparison with calculus methods, this approach
does not require checking the first and the second order conditions.
Key words:
Utility maximization problem, CES and Cobb-Douglas utility function,
mathematical inequalities, without calculus
JEL Classification
C69, D11
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1. INTRODUCTION
In the last thirty years mathematical inequalities have been applied to various economic problems. In
praticular, mathematical inequalities such as arithmetic-geometric mean inequality and CauchyBuniakowsky-Schwarz inequality, have benn used to solve many optimization problems in the field of
inventory theory (see [2], [3], [8]). A good review of papers that deal with applications of
mathematical inequalities to inventory theory models can be found in [3]. The most significant
contribution of these papers is reflected in the fact that some important optimization problems in
economics which cannot be trivially solved by using methods based on calculus are solved in a way
much easier to understand, thus providing better insight into the nature of the problem. In this paper
we consider two standard and very important microeconomic problems: the utility maximization
problem with CES utility function and the utility maximization problem with Cobb-Douglas utility
function. These two problems are usually solved by using differential calculus. However, standard
microeconomic textbooks show solution only in the case of n=2 commodities (see for instance [5] and
[7]). Therefore, the aim of this paper is to show how to solve these problems in an easier manner via
mathematical inequalities for arbitrary n≥2. In this paper we use the following mathematical
inequalities.
Theorem 1. (Jensen's inequality) Let n be a positive integer. If f is a convex function on [a, b], then
for any choice of t1,…,tn  [0, 1] such that
n
t
i 1
i
 1 , and for all x1,…,xn  [a, b] the inequality
f  t1x1   tn xn   t1 f  x1   tn f  xn 
(1)
holds. The equality in (1) holds if and only if x1=…=xn. Note: if f is concave function, then in (1)
reverse inequality holds.
Theorem 2. (Weighted AM-GM inequality) Let n be a positive integer. If x1,…,xn and t1,…,tn are
n
positive numbers such that
t
i 1
i
 1 , then
x1t1 xntn  t1x1  tn xn .
The equality in (2) holds if and only if x1=…=xn.
Proofs of the previous theorems can be found, for instance, in [1] or [4].
(2)
2. CASE n=2
The consumer utility maximization problem can be stated as
max u  x1 ,..., xn 
(3)
x1 ,..., xn 0
n
s. t.  pi xi  y
(4)
i 1
where n is a positive integer and represents the number of commodities that consumer buys, xi≥0 is
the quantity of commodity i, pi>0 is the price per unit of commodity i, y>0 is a consumer’s fixed
money income and u is a strictly increasing and strictly quasiconcave utility function. If u is the CES
utility function, then in case of n=2 commodities the problem (3)-(4) becomes the utility
maximization problem with CES utility function (5)-(6):
1
max u CES  x1 , x2   A 1 x1   2 x2  
(5)
s. t. p1x1  p2 x2  y
(6)
x1 , x2 0
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where   ,1 \ 0 , coefficients 0  1 , 2  1 describe consumer preferences and A>0 is the scale
of production. Similarly, if u is the Cobb-Douglas utility function, then in case of n=2 problem (3)-(4)
becomes the utility maximization problem with Cobb-Douglas utility function (7)-(8):
max u C D  x1 , x2   Ax11 x22
(7)
x1 , x2 0
s. t. p1x1  p2 x2  y .
Theorem 3. The maximum utility in problem (5)-(6) is equal to
(8)
1 
1
1
 1 1

CES
umax
 Ay  11  p11    21  p21  




*,CES
*,CES
and it is achieved for the unique global maximizer  x1 , x2  where
(9)
y
xk*,CES 
(10)
, k  1, 2.
1
 pi  k   1
pi   

i 1
 pk  i 
Proof. We proceed as in [6]. From (6) we get
y p1
x2 
 x1 .
(11)
p2 p2
Substituting (11) in (5), we transform the problem (5)-(6) into an unconstrained maximization problem
2
1
where u
CES
 

CES
 y p1 x1  


(12)
max u  x1   A 1 x1   2  
  ,
x1 0

p2
p2  



1

, t2 
is a function of one variable. Let   0, t1 
. Note that 0<t1, t2<1, t1+t2=1.
1 
1 
Note that function u
CES
from (12) can equivalently be written as
1
u
CES
 

 2  y p1  


x

A
1


t

x

1


t

x
 1   1 1 1   2 
 
  p2 p2 1  


1
1
 1    
 
1


1

 
       2    y p1   
A  t1  1 x1       x1   .

 

 
      p2 p2   


(13)
1
Let us consider functions f , g : 0,    IR, f  x   x , g  x   x . It is trivial to show that if


  ,0 , then f is convex and g is decreasing. If   0,1 , then f is concave and g is increasing. If
we combine these facts with Theorem 1, then from (13) it follows
1
 1

1
  y
CES

p 



2
u  x1   1     A  t11 x1  t2     1 x1   .

    p2 p2  


(14)
1
    p1
x1  0 for all x1. This is possible if and
The equality in (14) holds if and only if t11 x1  t2  2 
   p2
1
only if
1
1
t    p
  p
t11  t2  2  1  1   2  1 .
t2  1  p2
   p2
1

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Since t1+t2=1 and
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t1 1
 , from (15) we obtain
t2 

1
     p 1 
  2   1  ,
 1   p2 
(16)

1
t1 
1
 
1  2 
 1 
1
1 
 p1 
 
 p2 

1 
  2 1   p1 1 
   

p
, t2   1  1 2   .
  1   p 1 
1  2   1 
 1   p2 
(17)
If we substitute (16) and (17) into (14), we obtain the maximum level of utility in case of CES utility
CES
function u max , that is
CES
umax
1

 11 1

1 
 1
 Ay  1 p1   2 p2 




1 

.
(18)
We still need to obtain the unique level of commodity quantities x1 and x2 for which the global
maximum is obtained. By applying Theorem 1 to (11), we obtain that the maximum utility level (18)
is achieved if and only if
1
    y p 
1 x1   2    1 x1  .
    p2 p2 
1
(19)


*,CES
*,CES
Now, from (11), (16), (17) and (19), the unique global maximizer x1 , x2
for the problem (5)-
(6) can be easily obtained as
y
xk*,CES 
1
, k  1, 2.
(20)
 pi  k 
 
i 1
 k i 
The results (18) and (20) have the same form as in [5]. Q.E.D.
 1
2
p  p
i
Theorem 4. The maximum utility in problem (7)-(8) is equal to
A11 2 2 y1  2
C D
,
umax

 
1   2  1 2 p11 p22
(21)
and it is achieved for the unique global maximizer  x1*,C D , x2*,C D  where
k y
, k  1, 2.
pk 1   2 
Proof. By substituting (11) in (7), we obtain the unconstrained maximization problem
xk*,C  D 
2
 y p 
max u  x1   Ax1   1 x1  .
x1 0
 p2 p2 
Let us transform (23) in the following way:
C D
(22)
1
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u
C D
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 x1   2
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A
 1  2
p1 p2
A
Since
1  2
  1
 x1 1 2





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15-04
1  2
2


  y  p1 x1 1  2 


2
1
2

  1
1  2
1
2



 2 p1 x1  1 2 1
1 y  1 p1 x1 1  2
2




1  2 
2
1
p11 p2 2 
1
2
1
(24)
1  2
2
  2 p1 x1   1 y  1 p1 x1  
1
1  2




2



.
1
2

 1 , by applying Theorem 2 to (24), we obtain
1   2 1   2
u
C D

1
2
2
 2 p1 x1 
1 y 
1 p1 x1 

p1 p2  1   2
1   2
1   2

1  2
1  2
1  2
y
A
1  2  1  2 .
p1 p2
1   2 
 x1   A
1  2 
2
1
1
2
(25)
Thus, by Theorem 2, the maximum utility level is equal to
A11 2 2 y1  2
C D
(26)
umax

 
1   2  1 2 p11 p22
and it is achieved if and only if
(27)
 2 p1x1  1 y  1 p1x1 .
From (27) we get the unique optimal quantities of commodities 1 and 2, i.e. the unique global
maximizer  x1*,C D , x2*,C D  , where
x1*,C  D 
1 y
2 y
.
, x2*,C D 
p1 1   2 
p2 1   2 
(28)
Q.E.D.
One of the common assumption in economics is that utility function needs to be quasiconcave. In
order for the Cobb-Douglas function to meet this condition, coefficients 1 , 2  0 have to satisfy the
condition 1   2  1 . If 1   2  1 , that is if the Cobb-Douglas function exhibit constant returns to
scale, then (26) and (28) can be simplified as
1
2
   
y
 y
u
 Ay  1   2  , x1*,C  D  1 , x2*,C  D  2 .
(29)
p1
p2
 p1   p2 
Results (9), (10) and (29) are already known in contemporary microeconomic literature. However, to
the best of my knowledge, this is the first time that these results are derived by using mathematical
inequalities.
In the next section, the results are generalized for the case of n>2 commodities. To the best of my
knowledge, the results obtained for the general case are unknown in contemporary economic and
mathematical literature.
C D
max
3. GENERAL CASE n>2
We consider the following utility maximization problem in case of CES utility function for n>2:
max u CES  x1 ,..., xn   A 1x1 
x1 ,..., xn 0
1
  n xn  
s. t. p1x1   pn xn  y .
Theorem 5. The maximum level of utility in problem (30)-(31) is equal to
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(30)
(31)
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1 
1

 n

CES
umax
 Ay   i1  pi 1 
 i 1



and it is achieved for the unique global maximizer  x1*,CES , , xn*,CES  , where
y
xk*,CES 
(32)
, k  1, 2,..., n.
1
(33)
 pi  k 
 
i 1
 k i 
Proof. We prove the theorem by using mathematical induction over the number of commodities n≥2.
(i)
The claim of Theorem 5 holds for n=2, as shown in Theorem 3.
(ii)
Assume that the claim of Theorem 5 holds for k=2, 3, …, n.
(iii)
Let us prove that the claim of Theorem 5 holds for k=n+1.
Consider the following utility maximization problem
 1
n
p  p
i
1
max u CES  x1 ,..., xn1   A 1 x1 
 n1xn1  
x1 ,..., xn1 0
s. t. p1x1 
(34)
 pn1xn1  y .
(35)
From (35) we obtain
xn1 
y
p
 1 x1 
pn1 pn1
pn
xn .
pn1

(36)
Substituting (36) into (35) we obtain

xn1   A  1 x1 


u CES  x1 ,
where xn  y  p1x1 
1
 
 xn
 
pn

 n xn   n1 

xn  
 pn1 pn1  
(37)
 pn1xn1 . By the same reasoning as when obtaining (12) and (18), we apply

 xn

p
 n xn  from (37) and thus obtain
the claim (i) to terms  n xn and  n1 
 pn1 pn1 

u CES  x1 ,

, xn1   A  1 x1 




  n1 xn1  x n  


 A 1 x1 
1
1 
n

  n1 xn1   n x n


 1
pn

1
1 
n 1
1 

 1

pn1 


1




(38)
1

,
1 
1

 1 

where  n    n1  pn 1   n11 pn11  . Note that maximizing the expression on the left hand side of




(38) is equivalent to the following problem:
max
x1 ,..., xn1 , x n 0
u
CES
 x ,..., x
1
n1





, x n  A 1 x1 
  n1 xn1   n x n
s.t. p1x1  ...  pn1xn1  1 xn  y .
By applying claim (ii) to (39)-(40), from (38) we get
 n1
 Ay   
 i 1

Now, the equality in (38) is achieved if and only if
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
 1
pi



(39)
(40)

1
 n1 1 

u CES  x1 ,..., xn , xn1   Ay   i1  pi 1   1n  1 1 
 i 1



1
1 
i

1

1 

.
1 

(41)
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xn
xn 
1
 1
p
 
pn  pn1  n1  n 
 pn  n1 
Furthermore, equality in (41) is achieved if and only if
y
xl 
 p 
pl  i l 

i 1
 pl i 
where l=1,2,…,n-1. For l=n we have
n 1
1
 1
y
xn 
 pi  n 

i 1
i 

Combining (42) and (44) we get
n 1
 p  1 
1
 1
i
 1  l 
 1

 pl  n 
 p  
pn  n  n1 
 pn1  n 
1
 1
n 1
y
1
 1
i
n 1
 pi l 

 l i
 p  p
1

1   1
n
n
1
 1
l

p
 pi 

 i
 p 
1

1
1 
n
1
1 
n 1
 pn1
,
1
(43)
 1
pn1

 1
pn
(42)

 1

y
, xn1 
.
1
 1
y

i 1

 y
i 1
xn 
xn
, xn1 
1
 1
1
1 
n 1

 1
.
(44)
pn1
.
(45)
 pi n 
 p 
pi  i n1 


i 1
i 1
 n i
 pn1 i 
Since (41), (43) and (45) prove the claim (iii), Theorem 5 is completely proved. Q.E.D.
Let us now consider the utility maximization problem in case of Cobb-Douglas utility function for
n>2:
max u C D  x1 ,..., xn   Ax11 xnn ,
(46)
n 1
p  p
n 1
i
x1 ,..., xn 0
s. t. p1x1   pn xn  y .
Theorem 6. The maximum utility in problem (46)-(47) is equal to
(47)
n
 i

 i1
i
n
 y 
 i 
C D


umax  A n
  ,

  
i 1  pi 
 i 
 i 1 
and it is achieved for the unique global maximizer  x1*,C D , , xn*,C D  , where
xk*,C  D 
k y
n
pk  i
, k  1, 2,..., n.
(48)
(49)
i 1
n
Note: If

i 1
i
 1 , then (48) and (49) can be written in a simplified form as
i
 
u
 Ay  i  ,
(50)
i 1  pi 
 y
xk*,C  D  k , k  1, 2,..., n.
(51)
pk
Proof. By mathematical induction over n≥2. The idea is the same as in the proof of Theorem 5, where
the claim (i) holds as shown in Theorem 4. Q.E.D.
C D
max
n
Remark 1. If

i 1
i
n
 1 , then Cobb-Douglas utility function is a limit of CES utility function as ρ
approaches to zero, i.e.
Page 9 of 11
EFZG
WORKING
PAPER
SERIES
15-04
1
n
 n

lim A   i xi   A xii .
 0
 i 1

i 1
(52)
Proof. Let
1
 n

L  lim   i xi  .
 0
 i 1

By taking logarithm of (53) and by using the L’Hospital rule, we get
n
n
 n


ln   i xi 

x
ln
x
 i ln xi


i
i
i
n
   0  L' H lim i 1
i 1
ln L  lim  i 1


ln
xii .

   0 n
n
 0

0

 
i 1
x


i 1

i i
i 1
(53)
(54)
i
n
Thus, L   xii which proves Remark 1. Q.E.D.
i 1
n
Remark 2. If

i 1
i
 1 , then (50) is a limit of (32) as ρ approaches to zero, i.e.
 n 11 1 
lim Ay   i pi 
 i 1

 0


Proof. It is sufficient to prove that
1 
i
 
 Ay  i  .
i 1  pi 

n
1 

1

i
n
 n

 
L  lim   i1  pi 1     i  .

 0 
i 1  pi 
 i 1

By taking logarithm and by using L’Hospital rule, we get
1

n
 n 11 1 
 
1
1 
 1
ln    i pi 
 i pi
ln  i 

2
 i 1

1     pi 
   0  L' H lim i 1
ln L  lim 


1

 0

 0   0  n 1   1 
1
   i pi  
2
 1
 i 1
 1   
n
 
 i ln  i  n
i
i

n
pi 
 i 
 i 
i 1


  ln    ln    .
n
i 1
i 1  pi 
 pi 

i 1
(55)
(56)
(57)
i
i
n
 
Thus, L    i  . This proves Remark 2. Q.E.D.
i 1  pi 
n
Remark 3. If

i 1
i
 1 , then (51) is a limit of (33) as ρ approaches to zero, i.e.
y
lim
 0
 p 
pi  i k 

i 1
 pk i 
n
1
 1
for all k=1,2,…,n.
Proof.
Page 10 of 11

k y
pk
,
(58)
EFZG
WORKING
PAPER
SERIES
y
lim
 0
15-04
 p 
pi  i k 

i 1
 pk i 
n
1
 1

y
 y
 k ,
p 
pk
pi  k  i

p

i 1
i
k
n
(59)
for all k=1,2,…,n.
4. REFERENCES
[1] R. Bulajich Manfrino, J. A. Gomez Ortega and R. Valdez Delgado, Inequalities: A Mathematical
Olympiad Approach, Basel: Birkhauser Verlag, 2009.
[2] L. E. Cardenas-Barron, An easy method to derive EOQ and EPQ inventory models with
backorders, Computer and Mathematics with Applications, 59, 2 (2010), 948-952.
[3] L. E. Cardenas-Barron, The derivation of EOQ/EPQ inventory models with two backorders costs
using analytic geometry and algebra, Applied Mathematical Modeling, 35, 5, (2011), 2394-2407.
[4] P. K. Hung, Secrets in Inequalities, GIL Publishing House, 2007.
[5] G. A. Jehle and P. J. Reny, Advanced Microeconomic Theory, FT Prentice Hall, 2011.
[6] V. Kojić, A non-calculus approach to solving the utility maximization problem using the CobbDouglas and CES utility function, Croatian Operational Research Review, 6, 1 (2015), 269-277.
[7] A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic Theory, New York: Oxford
University Press, 1995.
[8] J.-T. Teng, A simple method to compute economic order quantities, European Journal of
Operational Research, 198, (2009), 351-353.
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