We offer simple and intuitive proofs of the Euler equation and the maximum principle based on Gossen's Second Law, one of the best known results in economics.Euler equation, Gossen's second law, Marginal utility, Maximum principle.

Guang-zhen Sun

English

Research Papers in Economics

Exploiting Gossen’s second law: a simple proof of the Euler equation 
and the maximum principle 
 
Guang-Zhen Sun
∗
Department of Economics, Monash University, Clayton, Vic.3800, 
Australia.  
(Email: guang-zhen.sun@buseco.monash.edu.au).  
 
 
Abstract: We offer simple and intuitive proofs of the Euler equation and 
the maximum principle based on Gossen’s Second Law, one of the best 
known results in economics.  
Keywords: Euler equation, Gossen’s second law, Marginal utility, 
Maximum principle.  
JEL: C61, D91. 
                                                 
∗ Tel: 61-3-99052409; fax: 61-3-99055476. E-mail address: guang-zhen.sun@buseco.monash.edu.au.  
  11. Introduction  
Gossen’s Second Law, literally stating that the marginal utility of one extra 
dollar (MUD) spent on each consumption good is the same for all the 
consumption goods as required by budget-constrained utility maximization, 
is unquestionably one of the best known results to students in economics. 
On the other hand, in the inter-temporal decision context, Euler’s equation 
proves to be a far more powerful tool, from which one can readily obtain 
Gossen’s second law in its inter-temporal version wherein the same goods 
(or services) at different times are formally viewed as different goods 
defined by the date and hence MUD remains the same across time. This 
short article aims to show that one can indeed reverse the reasoning, 
making use of Gossen’s second law to prove the Euler equation without 
resorting to the calculus of variations. (The proof of the Euler equation 
using the calculus of variations is found in almost any textbook in 
mathematical economics, see e.g., Lancaster 1987, pp.377-9 and Léonard 
and Van Long 1995, pp.170-1). Furthermore, by similar argument, the 
maximum principle can also be established. Our approach has an obvious 
advantage: it is essentially based on one basic result found in any textbook 
on intermediate microeconomics, suggesting that, in addition to the familiar 
exercises of establishing theorems/propositions in economics by using 
mathematical reasoning, economic intuition may sometimes help establish 
theorems in mathematics as well. Serving as a nice example, Gossen’s 
  2second law enables one to gain far more insights into optimization 
problems than conventionally presumed.   
 
2. Intuition 
Consider 
∫ ∈
=
T
t
T t C t x
dt t t x t x F Max V
0
0
1 ) ), ( ), ( (
) , ( ) (
&    where   (1) 
2 C F ∈
where T could be infinity. Let  ) ( ) ( t x t y & ≡ . One may interpret problem (1) as 
utility optimization in that the time-dependent “utility” F derives from the 
amount of goods  ,  , and the amount of   ,  , for any  . 
Suppose, for convenience, that   is measured in dollars. Note the utility 
interpretation of problem (1) holds regardless of whether or not F increases 
with   or  . For the sake of illustration, we may view the decision 
horizon of problem (1) as a period from year   to year T. To simplify 
notation, the solution to problem (1) is denoted as   still in the rest of 
this section. 
t X ) (t x t Y ) (t y ] , [ 0 T t t ∈
) (t x
) (t x ) (t x &
0 t
) (t x
  Hypothetically adding one extra dollar at any point of time   in the 
X-Type commodity space 
* t
), ( { t x ]} , [ 0 T t t ∈  leads to the same change in V, by 
Gossen’s second law. But such a change in year   automatically causes a 
change in   for any  , but has no effect on   for any  . That 
is: 
* t
) (t x * t t > ) (t x * t t <
  3    ,    
⎩
⎨
⎧ +
→
) (
1 $ ) (
) (
t x
t x
t x
when
when
*
*
t t
t t
<
≥
This is illustrated in Figure 1. 
 
     Figure 1 inserted here. 
 
What about  ? Intuitively speaking, the change rate in  , viz.  , can 
be understood as the difference in   between two successive years. For 
any year before  , there is no change in   and hence   remains 
unchanged. For any year after  , all   increase by 1, thus   also 
remains unchanged. For the particular year  , the amount of   suddenly 
jumps by one unit, resulting in an increase in the growth rate,   , by one. 
Thus,  
) (t x & ) (t x ) (t x &
) (t x
* t ) (t x ) (t x &
* t ) (t x ) (t x &
* t * t X
) (t x &
         
⎩
⎨
⎧ +
→
) (
1 ) (
) (
t x
t x
t x
&
&
&
for
otherwise
t t * =
,       
As a consequence, the change in V caused by a hypothetical increase in   
by one is 
) (t x
) ( ) ( t x
F
ds
s x
F
T
t & ∂
∂
+
∂
∂
∫ , which, by Gossen’s second law, must be the 
same for any   and differentiation of which wrt time   consequently 
equals zero. Hence,  the Euler equation, 
] , [ 0 T t t ∈ t
0
) (
)
) (
( =
∂
∂
−
∂
∂
t x
F
t x
F
dt
d
&
. 
 
 
  43. A new proof of the Euler equation 
We now present a rigorous treatment of the above economic intuition. As 
above, the optimal solution to (1) is denoted as  ), ( { t x ]} , [ 0 T t t ∈ . Arbitrarily 
choose   and increase   by an arbitrarily small number  ) , ( * 0 T t t ∈ *) (t x δ  but 
no change is deliberately made about   for all  ) (t x & ] , [ 0 T t t ∈ . Thus, after such 
a marginal change is made about   at  ,   also increases by  ) (t x * t ) (t x δ  for 
any  ,  (refer to Figure 1) and remains unchanged for any  . Note 
the hypothetically adjusted  
* t t ≥ * t t <
), *, ; ( ~ { δ t t x ]} , [ 0 T t t ∈ ), ( { t x ≡ for   and  * t t <
, ) ( δ + t x  for   is not continuous at point   and` hence  *} t t ≥ * t
) , ( ) *, ; ( ~
0
1 T t C t t x ∉ δ  for any   . 0 > δ  Or, more precisely,  ) *, ; ( ~ δ t t x  is right-
continuous but not left-continuous at  * t t =  for any   . 0 > δ  Letting  0 → δ , 
the component of the effect of change in   through changing 
on V is thus 
*) (t x
), ( { t x ]} , [ 0 T t t ∈
     ∫ ∂
∂
T
t
dt
t x
F
* ) (
        ( 2 )  
Clearly,  .  , hence  .  Consider an 
arbitrarily small interval 
* t t ≠ ∀ ) ( ) *, ; ( ~ t x t t x & & = δ 0 / ) *, ; ( ~ = δ δ d t t x d&
*] , * [ t t ε −  where ε  is a small positive number. Let 
) *, , ; ( δ ε t t x   δ
ε
ε
⋅
− −
+ ≡
) * (
) (
t t
t x  for any  *] , * [ t t t ε − ∈ . 
Then, ε δ δ ε / ) ( ) *, , ; ( + = t x t t x & & , hence  ε δ δ ε / 1 / ) *, , ; ( = d t t x d& ,  *) , * ( t t t ε − ∈ ∀ . 
Thus,  δ δ ε d t t x d / ) *, , ; ( &  becomes a delta function as  0 → ε . But 
δ δ ε d t t x d / ) *, , ; ( &  is the same as  at point  . Hence,  δ δ d t t x d / ) *, ; ( ~ & * t t =
  5δ δ d t t x d / ) *, ; ( ~ &  is also a delta function, for any value ofδ , including, in 
particular, zero; that is,  0 / ) *, ; ( ~
= δ δ δ d t t x d&  is a delta function. Note 
) (t x
F
& ∂
∂  is 
continuous for any  , due to  and  . Also notice that  ) , ( 0 T t t ∈
2 C F ∈
1 ) ( C t x ∈
) ( ) *, ; ( ~
0 t x t t x = = δ δ  and  ) ( ) *, ; ( ~
0 t x t t x & & = = δ δ  for any  ) , ( 0 T t t ∈ . Thus  0 ) *, ; ( ~ = ∂
∂
δ δ t t x
F
&  
is a continuous function. By the substitution property of the delta function 
(see,  e.g., Tuckwell 1988, p.51), the effect of change in  through 
changing   on V equals,  
*) (t x
*) (t x &
∫ = = ⋅
∂
∂
dt
d
t t x d
t t x
F
}
) *, ; ( ~
) *, ; ( ~ { 0 0 δ δ δ
δ
δ
&
& *) (t x
F
& ∂
∂
=    (3) 
Thus, it follows from (2) and (3) that the effect on V of an infinitesimal 
change in  , denoted as  *) (t x ) ( / * δ o t x V ∂ ∂  for notational convenience, is  
    ) ( / * δ o t x V ∂ ∂  =∫ ∂
∂
T
t
dt
t x
F
* ) ( *) (t x
F
& ∂
∂
+      (4) 
Gossen’s second law requires the above must be the same for any value of 
 Differentiation of the RHS of (4) wrt   thus equals zero, i.e.,  . * t * t
*) (t x
F
∂
∂
−  
* )
) (
( t t t x
F
dt
d
= ∂
∂
+
&
  ,  . The Euler equation is established. Note our 
proof applies to multi-dimensional Euler equations as well, for which the 
analysis is essentially the same, yet entailing more cumbersome notations.     
0 = ) , ( * 0 T t t ∈ ∀
 
4. A new proof of the maximum principle 
  6We consider the control problem with a fixed value of the state variable at 
the terminal-point (e.g., Léonard and Van Long 1995, Chapter 4), 
      where     ∫ =
T
t
t u dt t t u t x F Max V
0
) ), ( ), ( (
) (
1 C F ∈
  s.t.,  ,  ) ), ( ), ( ( ) ( t t u t x f t x = & 0 0) ( x t x =  and  T x T x = ) (          (5) 
The solution to problem (5) is denoted as   and the corresponding state 
values over   as  . Clearly, for any arbitrary differentiable 
function 
) ( ˆ t u
] , [ 0 T t t ∈ ) ( ˆ t x
) (t π ,  
     
  s.t.,  , 
) (t u
Max V = ∫ − +
T
t
dt t x t t u t x f t t t u t x F
0
)]} ( ˆ ) ), ( ), ( ˆ ( )[ ( ) ), ( ), ( ˆ ( { & π
) ), ( ), ( ˆ ( ) ( ˆ t t u t x f t x = &
0 0) ( ˆ x t x =  and  T x T x = ) ( ˆ    (6) 
As analysed above, Gossen’s second law requires that, to adopt the notation 
introduced in equation (4),  ) ( / δ o t u V ∂ ∂  be constant. But,  ) ( / δ o t u V ∂ ∂  
= ds
s u
f
s
s u
F
T
t∫ ∂
∂
⋅ +
∂
∂
}
) (
) (
) (
{ π , of which differentiation wrt   thus equals zero; 
that is,  
t
   
) (
) (
) ( t u
f
t
t u
F
∂
∂
⋅ +
∂
∂
π = 0      ( 7 )  
We signify the solution of (6) as   for the sake of 
notational neatness.
) ), ( ˆ ), ( ˆ ( ) ( t t x t x u t u & =
1 Problem (5) can thus be equivalently formulated as 
                                                 
1  A careful reader might be concerned with the possibility that   may also depend on the values of 
 and   for some  , even on the path  . But that is not a 
problem, for apparently our argument below applies to such a general case.   
) (t u
) ' ( ˆ t x ) ' ( ˆ t x & t t ≠ ' ]} , [ )), ( ˆ ), ( ˆ {( 0 T t s s x s x ∈ &
  7    (8) 
) (t x
Max V = ∫ − +
T
t
dt t x t t t x t x u t x f t t t x t x u t x F
0
)]} ( ) ), ), ( ), ( ( ), ( ( )[ ( ) ), ( ), ( ( ), ( ( { & & & π
The solution to (8) is still signified as   hereafter to simplify notation. A  ) (t x
similar argument to that in the preceding subsection on  0 ) *, ; ( ~
= δ δ
δ
t t x
d
d &  
being a delta function at  , where  ) ), ( ˆ ), ( ˆ ( ) ( t t x t x u t u & = * t
), *, ; ( ~ { δ t t x ]} , [ 0 T t t ∈ ), ( { t x ≡ for  * t t <  and  , ) ( δ + t x  for   for any  *} t t ≥ 0 ≥ δ ,  
yields,  ) ( / δ o t x V ∂ ∂  = ) ( }
) (
) (
) (
{ t ds
s x
f
s
s x
F
T
t
π π −
∂
∂
⋅ +
∂
∂
∫  
ds
s x
u
s u
f
s
s u
F
T
t∫ ∂
∂
⋅
∂
∂
⋅ +
∂
∂
+
) (
}
) (
) (
) (
{ π  
) (
]
) (
) (
) (
[
t x
u
t u
f
t
t u
F
& ∂
∂
∂
∂
⋅ +
∂
∂
+ π . In the light of (7),  
) ( / δ o t x V ∂ ∂   ) ( }
) (
) (
) (
{ t ds
s x
f
s
s x
F
T
t
π π −
∂
∂
⋅ +
∂
∂
=∫ , which  by Gossen’s second law 
must be the same for any t, and of which differentiation wrt t therefore 
equals zero. Hence, 
) (
) (
) (
) (
t x
f
t
t x
F
t
∂
∂
⋅ −
∂
∂
− = π π & ) ( t t u t x t x = & . Of course,   
holds. QED. 
) ), ( ), ( ( f
 
 
 
 
  8References 
Tuckwell, H. C., 1988. Elementary Applications of Probability Theory 
(Chapman and Hall, New York).   
 
Lancaster, K., 1987. Mathematical Economics (Dover, New York). 
 
Léonard, D., Van Long, N., 1995. Optimal Control Theory and Static 
Optimization in Economics (Cambridge University Press, New York). 
  9  10 
Figure 1: Marginal change in  *) (t x . 
 
 
 
 
 
 
 
          
 
t*  t  0 t  
x(t) 
$1  $1 
$1 

Exploiting Gossen's Second Law: A Simple Proof of the Euler Equation and the Maximum Principle

http://www.buseco.monash.edu.au/eco/research/papers/2005/1405gossens2ndlaw.pdf

Exploiting Gossen's Second Law: A Simple Proof of the Euler Equation and the Maximum Principle

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics