In our previous paper, we derive the demand function in the form p=du(x)/dx and we apply the Weber-Fechner's law to the utility function and we obtain the demand function in the familiar form p=A/x. We compare our derivation of the demand function with the standard one. The differences are i) different functional form of the utility function, ii) different objective function to aximize, iii) different treatment of the budget condition. We also study how much quantity of goods we should distribute to N persons in n kinds of goods. By adding each person's demand function, we obtain the total demand function. By the market equilibrium, we obtain the only unique solution of how much quantity of goods we should distribute to each person. The quantity of goods is distributed according to each person's preference.Weber-Fechner's law, utility function, demand function, Pareto optimum

Kazuyasu Shigemoto

English

Research Papers in Economics

Weber-Fechner’s Law, Demand Function and Related Topics
Kazuyasu Shigemoto 1
Abstract
In our previous paper, we derive the demand function in the form p =
du(x)
dx
and we
apply the Weber-Fechner’s law to the utility function and we obtain the demand function
in the familiar form p =
A
x
. We compare our derivation of the demand function with the
standard one. The diﬀerences are i) diﬀerent functional form of the utility function, ii)
diﬀerent objective function to maximize, iii) diﬀerent treatment of the budget condition.
We also study how much quantity of goods we should distribute to N persons in n kinds
of goods. By adding each person’s demand function, we obtain the total demand function.
By the market equilibrium, we obtain the only unique solution of how much quantity of
goods we should distribute to each person. The quantity of goods is distributed according
to each person’s preference.
1 Introduction
Recently the new trend of the economics appears. In 2002, Kahnemann won the Nobel
prize because of the contribution of applying the psychology to the economics[1][2]. His
Nobel prize typically represents this new trend of the economics, trying to understand various
phenomena in the economics more deeply from the psychological point of view.
The important law of the economics also comes from the knowledge of the behavioral
psychology. In this context, we consider the economics as the application of the behavioral
psychology. In micro economics, the utility function is one of the most important things,
and the functional form of this utility function must be determined from the knowledge of
the behavioral psychology. But the functional form of the utility function is still in the
qualitative level as people use this or that functional form without any scientiﬁc reason. The
most important law in the behavioral psychology is the Weber-Fechner’s law[3][4][5], which
1Department of Economics, Tezukayama University, Nara 631, Japan
E-mail address: shigemot@tezukayama-u.ac.jp
1represents the relation between the magnitude of physical stimulus and the magnitude of
psychological sense in human being. Then, in our previous paper, we try to understand the
functional form of the demand function from the point of view of the behavioral psychology
and we apply this Weber-Fechner’s law to the utility function, which gives the explicit
functional form for the utility function and the demand function[6] [7].
In this paper, we compare our derivation of the demand function with the standard one
and we clarify the diﬀerences. We also study the problem of how much quantity of goods
we should distribute to each person in various kinds of goods.
2 Demand Function from the Weber-Fechner’s Law:
Review of Our Previous Result
We ﬁrst review our previous result of deriving the demand function[6][7]. We consider n
kinds of goods. We denote the utility function as u(x1,x2,···,xn) where xi is the quantity
of the i-th goods. We assume that this utility function is of separable type
u(x1,x2,···,xn) = u1(x1) + u2(x2) + ··· + un(xn), (1)
in the zero-th approximation, which is the quite natural assumption as each goods separately
contributes to the total utility in the zero-th approximation[8]. Then this utility function of
the i-th goods ui is the function only of the quantity of the i-th goods xi. Then the Gossen’s
law becomes in the form
du1(x1)/dx1
p1
=
du2(x2)/dx2
p2
= ··· =
dun(xn)/dxn
pn
= k, (2)
where pi is the price of the i-th goods and k becomes constant, independent of the kind of
goods. By rescaling the price, we can choose k = 1. Then we can rewrite Eq.(2) in the form
pi =
dui(xi)
dxi
, (i = 1,2,···,n), (3)
and the demand function pi = Di(xi) is expressed in the form
pi = Di(xi) =
dui(xi)
dxi
, (i = 1,2,···,n), (4)
2by using the utility function. Then we use the Weber-Fechner’s law[3][4] for the utility
function of the form
ui(xi) =



Ai log

xi/x
(0)
i

if xi ≥ x
(0)
i
0 xi < x
(0)
i
, (5)
where x
(0)
i is the threshold of the quantity of goods. Hereafter, we consider only the case
that the quantity of goods is greater than the threshold. Then we have the familiar form of
the demand function in the form
pi =
Ai
xi
. (6)
Next, we consider the principle to determine the demand function from the point of view
of the consumer’s surplus. The demand function pi = Di(xi) is given by maximizing the
consumer’s surplus
(consumer
0s surplus) =
Z xi
0
Di(x
0
i)dx
0
i − pixi, (7)
for the i-th goods. Substituting the expression Eq.(4) into Eq.(7), we have
(consumer
0s surplus) =
Z xi
0
dui(x0
i)
dx0
i
dx
0
i − pixi = ui(xi) − pixi, (8)
where we use ui(0) = 0. Then we deﬁne the ”consumer’s proﬁt” ρi(xi) as the utility minus
expenditure, that is, ui(xi) − pixi and we have
(consumer
0s surplus) = (”consumer
0s proﬁt”) = ρi(xi) = ui(xi) − pixi. (9)
Then we can rewrite that the principle to determine the demand function is to maximize
the ”consumer’s proﬁt” ui(xi) − pixi.
3 Comparison of the Derivation of the Demand Func-
tion
3.1 Standard derivation of the demand function
We ﬁrst review the standard method to derive the demand function[8][9]. For simplicity,
we assume that there is one person with two kinds of goods and each quantity of goods is
3given by x1 and x2. We denote each price of goods as p1 and p2 and the budget amount as
m. We maximize the utility function of the Cobb-Douglas type
u(x1,x2) = Ax
α
1x
β
2, (10)
with the budget constraint m = p1x1 + p2x2. This problem is equivalent to maximize the
following objective function
f(x1,x2,λ) = Ax
α
1x
β
2 − λ(p1x1 + p2x2 − m). (11)
Maximizing this objective function, we have the following equations
λp1 = Aαx
α−1
1 x
β
2, λp2 = Aβx
α
1x
β−1
2 , m = p1x1 + p2x2. (12)
Then the demand quantities of goods x1, x2 and the value of the multiplier are determined
and we have the demand function of the form
p1 =
α
(α + β)
m
x1
, p2 =
β
(α + β)
m
x2
, λ =
Aααββ(α + β)1−α−β
m1−α−βpα
1p
β
2
. (13)
In this way, the expenditure of each goods pixi (i = 1,2) is determined in such a way as the
whole budget amount is relatively distribute into two parts according to the preference of
goods, that is, p1x1 =
α
(α + β)
m, p2x2 =
β
(α + β)
m.
3.2 Modiﬁed derivation of the demand function
Instead of the Cobb-Douglas type utility function, we use the separate type and Weber-
Fechner’s type utility function
u(x1,x2) = A1 log(x1/x
(0)
1 ) + A2 log(x2/x
(0)
2 ). (14)
Assuming x
(0)
1 = x
(0)
2 , Ai (i = 1,2) represents the person’s preference of the i-th goods. If
A1 > A2, the person prefer the ﬁrst goods to the second goods. Maximizing this utility
function with the budget constraint m = p1x1 + p2x2, the demand quantities of goods x1,
x2 and the value of the multiplier are determined and we have the demand function of the
form
p1 =
A1
(A1 + A2)
m
x1
, p2 =
A2
(A1 + A2)
m
x2
, λ =
(A1 + A2)
m
. (15)
Making the correspondence α ↔ A1, β ↔ A2, this demand function is in the same functional
form as that in the standard derivation.
43.3 Our derivation of the demand function
According to our principle, we maximize the total ”consumer’s proﬁt” for each goods.
The total ”consumer’s proﬁt” ρtotal(x1,x2), which is the sum of the ”consumer’s proﬁt” for
each goods ρ1(x1) and ρ2(x2), is given by
ρ
total(x1,x2) = A1 log(x1/x
(0)
1 ) + A2 log(x2/x
(0)
2 ) − p1x1 − p2x2. (16)
Then the demand quantities of goods x1, x2 are determined and we have the demand function
becomes of the form
p1 =
A1
x1
, p2 =
A2
x2
. (17)
In our derivation, each demand function is determined independently, which is the natural
consequence because we behave according to some demand function even if there exists only
one goods. The expenditure of each goods pixi (i = 1,2) is proportional to the person’s
preference of goods Ai (i = 1,2), but the expenditure is independent of the budget amount.
Then if p1x1+p2x2 = A1+A2 ≤ m, we purchase both types of goods with the above quantity
x1, x2, but we do not use up all budget in general. While if p1x1 +p2x2 = A1 +A2 > m, we
do not purchase any type of goods.
3.4 Comparison of various derivation of the demand function
The utility function of the Cobb-Douglas type is not natural as it is not the separable
type, then the modiﬁed derivation will be more natural than the standard one.
Further, the demand function of the ﬁrst goods is determined in a relative way with
that of the second goods in the standard derivation. But the demand function must be
determined even if there exist only one kind of goods. In this sense, our derivation will be
more natural than the standard one.
Furthermore, our derivation gives the similar result compared with the standard and the
modiﬁed derivations, but the treatment of the budget condition is quite diﬀerent so that
the economical consequence becomes quite diﬀerent. In the standard derivation and in the
modiﬁed derivation, we use up all budget for the given prices p1, p2 and the given budget
m. While in our derivation, we purchase if and only if the expenditure is less or equal to the
budget amount. In this sense, our derivation will give the more reasonable consequence in
the actual human behavior.
54 Problem of Distributing the Goods to Each Person
Next we consider the problem to distribute the quantity of goods to each persons. We
consider that there are N persons and n kinds of goods. We denote the distributed quantity
of the i-th goods for the j-th person as x
(j)
i (i = 1,2,···,n), (j = 1,2,···,N) and we take the
j-th person’s utility function of the i-th goods in the form u
(j)
i (x
(j)
i ) = A
(j)
i log(x
(j)
i /x
(j)(0)
i ).
We take the price pi of i-th goods to be common for all persons because of the law of one
price. Then we maximize j-th person’s ”consumer’s proﬁt” of the form
ρ
(j)(x
(j)
1 ,x
(j)
2 ,···,x
(j)
n ) =
n X
i=1
A
(j)
i log(x
(j)
i /x
(j)(0)
i ) −
n X
i=1
pix
(j)
i , (j = 1,2,···,N), (18)
which gives
pi =
A
(j)
i
x
(j)
i
, (i = 1,2,···,n), (j = 1,2,···,N). (19)
Then the total demand quantity of the i-th goods Xi is given by Xi =
N X
j=1
x
(j)
i . Using
x
(j)
i =
A
(j)
i
pi
, we have the total demand function of the i-th goods pi = Dtotal
i (Xi) in the
form
pi = D
total
i (Xi) =
Atotal
i
Xi
, (20)
where A
total
i =
N X
j=1
A
(j)
i . Combining this total demand function with the total supply function
pi = Stotal
i (Xi), we have the equilibrium quantity X∗
i and the equilibrium price p
∗
i =
Atotal
i
X∗
i
for the i-th goods. Using x
∗(j)
i =
A
(j)
i
p∗
i
, the j-th person’s distribution of the quantity of the
i-th goods is given by
x
∗(j)
i =
A
(j)
i
Atotal
i
X
∗
i , (i = 1,2,···,n), (j = 1,2,···,N). (21)
6In this way, according to each person’s preference of the i-th goods, the quantity of that
goods is distributed, that is, if A
(j)
i > A
(k)
i , we have x
∗(j)
i > x
∗(k)
i . In the above, we assume
that all persons have enough budget amount, that is,
n X
i=1
p
∗
ix
∗(j)
i ≤ m
(j), (j = 1,2,···,N).
While if there exist the M poor persons who have not enough budget in the form
n X
i=1
p
∗
ix
∗(j)
i > m
(j), (j = 1,2,···,M), the above theoretical equilibrium is not realized. Then
the optimum distribution is not realized. But if the sum of all person’s budget amount
is larger than the sum of all person’s expenditure, that is,
N X
j=1
n X
i=1
p
∗
ix
∗(j)
i ≤
N X
j=1
m
(j), we can
realize the optimum state by taking the tax from the rich persons and giving the ﬁnancial
aid to the poor persons in such a way as all persons can purchase all kinds of goods, though
such policy cause all persons to loose the incentive to work.
5 Summary and Discussion
In our previous paper, we derive the demand function by maximizing the ”consumer’s
proﬁt” deﬁned by the utility minus expenditure and we obtain the demand function in the
form p =
du(x)
dx
. We apply the Weber-Fechner’s law to this utility function, which gives the
demand function in the familiar form p =
A
x
.
In this paper, we compare our derivation of the demand function with that of the standard
one. In our derivation, the demand function is derived i) by using the utility function which
satisﬁes the Weber-Fechner’s law, ii) by maximizing the ”consumer’s proﬁt”, iii) by making
the choice that we purchase if and only if the expenditure is less or equal to the budget
amount. While in the standard derivation, the demand function is derived i) by using the
utility function of the Cobb-Douglas type, ii) by maximizing the utility function with the
budget constraint, iii) by use up all budget.
We also study the problem to distribute the quantity of goods to each person. We
ﬁnd that we can distribute the quantity of goods according to each person’s preference by
maximizing each person’s ”consumer’s proﬁt”. According to our method, the distributed
quantity of goods is uniquely determined.
The standard way of distributing the quantity of goods to each person is realized by the
Pareto optimum state. The key point of the Pareto optimum state is that i) the Pareto
7optimum state is realized by increasing each person’s utility, ii) the quantity of goods is ex-
changed between person to person but not in the market, so that the total quantity of each
kind of goods is ﬁxed as there is no additional supply ﬂow through the market, iii) there are
inﬁnitely many Pareto optimum states even if we start from the same Pareto non-optimum
state. While, the key point of our method is that i) our optimum state is realized by maximiz-
ing each person’s ”consumer’s proﬁt”, ii) our optimum state is realized through the market
equilibrium, iii) our optimum state is uniquely determined as the market equilibrium state
is uniquely determined. Nowadays, we obtain almost all kinds of goods through the market.
Then our optimum mechanism is more realistic than the Pareto optimum mechanism.
References
[1] D. Kahnemann, P. Slovic and A. Tversky (Eds.), “Judgment Under Uncertainty:
Heuristics and Biases”, (Cambridge Univ. Press, Cambridge, 1982).
[2] H. Takayasu (Ed.), “Empirical Science of Financial Fluctuations: The advent of econo-
physics ”, (Springer Verlag, Tokyo, 2002).
[3] E.H. Weber, “De pulsu, resorptione, audita et tactu. -Annotationes anatomicae et
physiologicae-”, 1834 (Trs. by H.E. Ross, Academic Press, New York, 1978)D
[4] G.T. Fechner, “Elemente der Psychophysik I u. II”, (Breitkopf u. Hartel, Leipzig,
1907).
[5] S.S. Stevens, “On the psychophysical law”, Psychological Review, 64 (1957), 153-181
; “To honor Fechner and repeal his law, Science, 133 (1961), 80-133.
[6] K. Shigemoto, “Weber-Fechner’s law and Cobb-Douglas Function ”, Tezukayama Eco-
nomic Papers, 6 (1997), 69-73.
[7] K. Shigemoto, “Weber-Fechner’s law and Demand Function ”, Tezukayama Academic
Review, 9 (2002), 41-46.
[8] P.R. Layard and A.A. Walters, “Microeconomic Theory ”, (McGraw-Hill, 1978).
[9] H.R. Varian, “Intermediate Microeconomics ”, 5th edition, (W.W. Norton & Company,
1999); “Microeconomic Analysis ”, 2nd edition, (W.W. Norton & Company, 1978).
8

(Eds.), “Judgment Under Uncertainty: Heuristics and Biases”,

De pulsu, resorptione, audita et tactu. -Annotationes anatomicae et physiologicae-”, 1834 (Trs. by H.E.

Elemente der Psychophysik I u. II”, (Breitkopf u.

Empirical Science of Financial Fluctuations: The advent of econophysics ”,

Intermediate Microeconomics ”, 5th edition, (W.W.

Microeconomic Theory ”,

On the psychophysical law”,

Weber-Fechner’s law and Cobb-Douglas Function ”,

Weber-Fechner’s law and Demand Function ”,

Weber-Fechner's Law, Demand Function and Related Topics

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Weber-Fechner's Law, Demand Function and Related Topics

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