about an optimum model of market...
TRANSCRIPT
About an optimum model of market equilibrium
CODRUłA CORNELIA DURA, ILIE MITRAN, IMOLA DRIGA
The Economics Department, The Mathematical Department
University of Petroşani,
University Street, No.20, Petroşani
ROMANIA
[email protected], [email protected], [email protected]
Abstract: - The determination of the equilibrium price within the context of the economic analysis of the supply – demand relation, represents in fact, a particular case of solving the minmax equation from
uncooperative games theory. Thus, considering two decision makers and a zero-sum game, we shall have the
following problem: y yx x
maxmin f (x, y) min max f (x, y)= . This papers deals with some optimal properties of the
equilibrium price; these properties, which are presented using an original approach, lead us to important economic interpretations. The originality of the paper consists of the modality used in order to determine the
equilibrium price starting from the elasticity of supply and demand functions. In case the analytical expressions of the supply and demand functions remain unknown, we specify an original proceeding in order
to determine the equation of price dynamics by using a method which is appropriate for higher order linear differential equations.
Key-Words: equilibrium price, supply-demand relation, the elasticity of the supply function, the elasticity of the demand function, the equation of price dynamics, the dynamic index of prices.
1 Considerations upon supply-demand
functions and upon the equilibrium
price An important feature of the models of behavior for producers and consumers is that prices are assumed to be known [10].
One of the characteristics of the models of market equilibrium is that we have to determine the
price for the equality achieved (at a certain period of time or at different periods of time) between the consumer’s demand and the producer’s supply.
If prevailing market price is denoted by p, the supply and demand functions (the current variable p)
are marked by the usual C and O, respectively and they meet the following requirements:
1) C,O :[0, ) R,∞ → differentiable functions
2) C (p) 0,O (p) 0, p 0′ ′< > ∀ ≥
We use p * to mark the typical equilibrium price
which is in fact the solution of the following equation:
C(p) O(p)= (1)
Remark 1. Based on the conditions 2), it is
obvious that the demand function is monotonously decreasing, while the supply function is monotonously increasing. The equilibrium price
corresponds to the intersection of the two curves of
the graph (fig.1).
Fig.1 The equilibrium price
2 The determination of the
equilibrium price and economic
interpretation The actual determination of the equilibrium price p* is generally a difficult problem and it requires knowledge of analytical expressions of the functions
C and O.
2.1 The direct approach
However, when the equation C (p) = O (p) is a first,
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second and third degree algebraic equation, or when it is a bi-square or a mutual equation, the solution p*
can be determined precisely. Otherwise p* can only be approximated.
In addition to the usual techniques of approximation, p* can also be determined through the linearization of equation (1.) (Practically,
developing both the demand function and the supply function into Taylor and Mc-Laurin series at a
convenient chosen point). For example, developing the two members of the equation (1) into Mc-Laurin series, we are led to the
following equation:
C(0) pC (0) O(0) pO (0)′ ′+ = + (2)
hence the immediate solution p*:
O(0) C(0)p*
O (0) C (0)
−= −
′ ′− (3)
and the equilibrium volume of transactions:
C (0)O(0) C(0)O (0)C(p*) O(p*)
C (0) O (0)
′ ′−= =
′ ′−
If the two members of the equation (1) are developed into Taylor series at a certain point (but
fixed point) p , there follows the equation:
C(p) (p p)C (p) O(p) (p p)O (p)′ ′+ − = + −
and therefore:
O(p) C(p)p* p
O (p) C (p)
−= − +
′ ′− (4)
Remark 2. If the supply and demand functions are differentials of higher order, of course, the
developments of the two members of the equation (1) into Mc-Laurin series and Taylor series,
respectively can be extend to several terms and we shall be required to solve some algebraic equations of higher order .
Remark 3. In particular cases: C(p) ap b,O(p) cp d,a 0,c 0= − + = − > >
where the equilibrium price is given by the
following relation:
b dp*
a c
+=
+ (5)
and the supply and demand function for the equilibrium price is:
bc adC(p*) O(p*)
a c
−= =
+ (6)
a graphical representation looks like this:
Fig.2 The particular cases of equilibrium price
2.2 Determining the equilibrium point
starting from the elasticity of supply -
demand functions We shall take into consideration the following
elements:
• The minimum prices for which the supply –
demand functions are defined are marked with
1,1p and 2,1p respectively;
• The elasticity of the supply function is 1e and
the elasticity of the demand function is 2ie .
Both 1e and 2ie are assumed to be the n degree
polynomial in relation to price p, namely: 2 n
1 0 1 2 ne a a p a p a p= + + + +⋯ (7) 2 m
1 0 1 2 me b b p b p b p= + + + +⋯ (8)
We shall run through the following steps:
1. The elasticity for the supply-demand functions can be determined as we can see in the analytical expressions (7) and (8), respectively provided we
know the prices 1,1 1,2 1,np ;p ;...;p corresponding to
each moment 1,2,...,n and also the prices
2,1 2,2 2,np ;p ;...;p corresponding to each moment
1,2,...,m taken from statistical data. By using the
least squares method, we can calculate the values of
coefficients 0 1 na ,a ,...,a and 0 1 nb ,b ,...,b ; these
values represents the solutions of the following algebraic systems:
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n n n n0 1 n
0 1,i 1 1,i n 1,i 1,i
i 1 i 1 i 1 i 1
n n n n1 2 n 1
0 1,i 1 1,i n 1,i 1,i 1,i
i 1 i 1 i 1 i 1
nn
0 1,i
i 1
a p a p ... a p O(p )
a p a p ... a p p O(p )
.........................................................................
a p
= = = =
+
= = = =
=
+ + + =
+ + + =
∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
n n nn 1 2n n
1 1,i n 1,i 1,i 1,i
i 1 i 1 i 1
a p ... a p p O(p )+
= = =
+ + + =∑ ∑ ∑ ∑
(9)
m m m m0 1 m
0 2,i 1 2,i m 2,i 2,ii 1 i 1 i 1 i 1
m m m m1 2 m 1
0 2,i 1 2,i m 2,i 2,i 2,ii 1 i 1 i 1 i 1
m m m mm m 1 2 m m
0 2,i 1 2,i m 2,i 2,i 2,ii 1 i 1 i 1 i 1
b p b p ... b p C(p )
b p b p ... b p p C(p )
b p b p ... b p p C(p )
= = = =
+
= = = =
+ ⋅
= = = =
+ + + =
+ + + =
+ + + =
∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
(10)
2. Because O O(p),C C(p),= = taking into
consideration the elasticity concept we are led to the following finite differential equations [9]:
1
O pe
O p
∆ ∆= , 2
C pe
C p
∆ ∆= (11)
which yields the following first degree differential equations[9]:
1
dO dpe
O p= , 2
dC dpe
C p= (12)
By integrating these differential equations and taking into account the equalities (7) and (8), we
shall get:
( ) ( )1,1 1,1
1,1
p* p*
*
1 1,1
p p
p* 2 n
0 1 2 n
p
dO dpe O p O p
O p
a a p a p a pdp
p
= ⇒ − =
+ + + +=
∫ ∫
∫⋯
(13)
( ) ( )2,1 2,1
2,1
p* p*
*
2 1,1
p p
p* 2 n
0 1 2 n
p
dC dpe C p C p
C p
b b p b p b pdp
p
= ⇒ − =
+ + + +=
∫ ∫
∫⋯
(14)
Accordingly, we shall have:
( ) ( )i i** n1,1*
1,1 0 i
i 11,1
p ppO(p ) O(p ) a ln a
p i=
−= + +∑ (15)
( ) ( )i i** m2,1*
2,1 0 i
i 12,1
p ppC(p ) C(p ) b ln b
p i=
−= + +∑ (16)
3.Taking into consideration that *p represents an
equilibrium point, it is obvious that C(p*) O(p*)= ,
and consequently, *p represents the solution of the
equation (17). Generally, it is extremely difficult to solve the
above mentioned equation and the equilibrium price
*p can be approximately determined by using
specific approximate solving methods of the algebraic equations (successive approximation method, Newton method etc.).
( ) ( ) ( )
( ) ( ) ( )
i i** n1,1
1,1 0 i
i 11,1
i i** m2,1
2,1 0 i
i 12,1
p ppO p a ln a
p i
p ppC p b ln b
p i
=
=
−+ + =
−= + +
∑
∑
(17)
Particular cases
1) If n m= and we mark 0 1,1 2,1p p p= = , then the
previous equation becomes:
( ) ( ) ( )( )
nii i* *
0 0
i 1
ni i i
0 0 0 0 0 0
i 1
a ba b ln p p
i
a b(a b ) ln p p C(p ) O(p )
i
=
=
−− + =
−= − + + −
∑
∑(18)
2) If n m= , 0 1,1 2,1p p p= = and the elasticity
functions are linear (i.e. 1 0 1e a a p= + ,
2 0 1e b b p= + ), the equilibrium price *p represents
the solution of the following equation:
( ) ( )* *
0 0 1 1a b ln p a b p A− + − = (19)
where:
( ) ( )0 0 0 1 1 0 0 0A a b ln p a b p C(p ) O(p )= − + − + − (20)
3) If the logarithmic function can be written in a
linear form, for the point p 1= , after an immediate
calculation, we shall get:
( ) ( )( ) ( )
*
' '
O p C pp p
O p C p
−= −
− (21)
This result is concordant with the equality (4).
Economic Interpretation
It is obvious that the equilibrium point *p represents the solution of the following problem:
ppmax O(p) min C(p)= (22)
From equalities
O(p) O(p) (p p)O (p)′= + − (23)
C(p) C(p) (p p)C (p)′= + − (24)
we can make, immediately, the following deduction:
C(p) O(p)
C(p) O(p) (p p)(C (p) O (p))
− =
′ ′= − + − − (25)
Therefore, the area between the graphic
representations (fig.3) of the curves O O(p)= ,
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C C(p)= and the lines np 0,p p= = (marked with
nA ) can be determined using the formula:
np
n
0
nn
A (C(p) O(p))dp
pp (C(p) O(p) ( p)(C (p) O (p)))
2
= − =
′ ′= − + − −
∫ (26)
Taking into consideration the requirements *
n nn n
lim p p , lim A 0= = and the equality
* C(p) O(p)p p
C (p) O (p)
−= −
′ ′− (27)
after some calculation we can get the following relation:
C(p) O(p) p(C (p) O (p))′ ′− = − (28)
In case *p p= (namely the development of both
the demand function and the supply function into Taylor series is made exactly in the equilibrium
point), from (6) equality we can obtain:
* *C (p ) O (p )′ ′= (29)
From economic point of view, this equality
shows that the marginal values of the demand and of the supply functions have equal absolute values in
the equilibrium point *p .
Fig.3
3 Determining the equation of price
dynamics We shall start from the model known as the "cobweb
model" due to graphical images generated by the supply and demand functions.
3.1 The case when analytical expressions of
supply-demand functions are known There are practical situations in which the demand is affected by the proposed price at the time it was placed, while the supply is influenced by the
prevailing market price from a previous period of time.
For example, in the case of agricultural products, between the intent to provide and the supply itself
there is a gap of time (of almost half a year). Therefore, we shall note pt, and pt-1, respectively the prices at time t (when the demand was made)
and time t-1 (the period of time of the previous offer).
The condition of equilibrium is, in this case, as follows:
t t 1C(p ) O(p )−= (30)
and it will lead to a recurrence relationship between pt and pt-1 (called "the recurring price equation”).
The recurring price equation can be determined most comfortable through the linearization of the two members of the equation (30) (developing Mc-
Laurin and Taylor series and retaining only the first two terms of the development).
Thus, when developing Mc-Laurin series out of the two members of the equation (30) we get:
t t 1C(0) p C (0) O(0) p O (0)−′ ′+ = + ,
from where:
t t 1
O (0)p p O(0) C(0)
C (0)−
′= + −
′ (31)
If we note: O (0)
A ,B O(0) C(0)C (0)
′= = −
′
t tx p p*= − (i.e. xt measures the deviation from the
prevailing price at time t and the equilibrium price p* given by (3)), then the recurrent relationship (31)
becomes:
t t 1p Ap B−= + (32)
which, after an immediate calculation results in: t
t 0p A (p p*) p *= − + (33)
Practically, the equality (33) reflects the price
dynamics (which is why it is called "dynamic pricing equation”).
During a situation of equilibrium t t 1p p p *−= = ,
from equation (32) we get:
t t 1x p* A(x p*) B−+ = + + , which results in:
t t 1x Ax −= (34)
As a consequence, t
t 1
xA
x −
= , thus the size of
O (0)A
C (0)
′=
′represents the dynamic index of the
current price deviation from the equilibrium price.
Remark 4. If the demand and supply functions are linear and the following analytical expressions are known:
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t t t
t t 1 t 1
C C(p ) ap b,a 0
O O(p ) Cp d,c 0− −
= = − + >
= = + >
then, the dynamic index of current price deviation from the equilibrium price A can be calculated
directly as a ratio of the sensitivity of the supply towards the price S0 and the sensitivity of the
demand towards the price Sc: 0 cA S / S=
Moreover the sensitivity is determined as a derivative of the supply and the demand functions in
relation to the current price pt:
t t0 c
t t
dO dCS , S
dp dp= = (35)
Therefore, in addition to the following equalities *
t t
*
t 1 t 1
x p pA
x p p− −
−= =
− (36)
There is another method that determines the dynamic index:
1 1
1 1
dO dCA /
dp dp= (37)
Obviously, this equality has the value A a / c= −
in case the supply and demand are linear, but it can
also be used in other circumstances in which the analytical expressions of these functions are more
general. Based on the economic interpretation of size A,
the equality (33) may yield the following conclusions: 1. the dynamic index has a subunit module (i.e.
A <1). In this case, tt
limp p*= , so that we have a
situation of equilibrium (fig.4).
Fig.4
2. the dynamic index has an improper module
(i.e. A >1). In this case, the series (pt)t is divergent,
and practically we have a situation of hyperinflation
and imbalance (fig.5);
Fig.5
3. the dynamic index has the value A = 1 or the
value A = -1. In this case t 0t
limp p= , and thus we
are faced with the development of the same two-
state equilibrium values (alternative), p0 and p1 (fig. 6).
Fig.6 Practically, the influence of the dynamic index is shown in figure 4, figure 5 and figure 6.
3.2. The case when there are no known
analytical expressions of the supply and
demand functions When there are no known analytical expressions of
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the demand and supply functions but we are aware of the interdependence between the base price pt and
previous prices pt-1, pt-2,…,pt-k,, the determination of the equation of dynamic pricing is done by going
through several stages that involve relatively simple calculations. The interdependence between the basic price and previous prices can be established from
statistical data and using well- known approximation methods (interpolation methods, i.e., approximation
by polynomials, the method of the least squares etc.). There are two cases:
Case I. The interdependence between the basic
price and previous prices is t t 1 t kf (p ,p ,...,p ) 0− − = , f
being a known function. This is commonly known as the homogeneous case. Prices take the following form:
t t 1 t k
t t 1 t kp pr ,p pr ,...,p pr ,p, r 0− −− −= = = > (38)
The functional given interdependence
t t 1 t kf (p ,p ,...,p ) 0− − = turns into an equivalent one of
the following form t t 1 t kF(r , r ,..., r ) 0− − = , called a
characteristic equation. Let us note r1, r2, …,rk+1 the real and non-zero
solutions of this last equation. The price at time t has following form:
* t t t t
t 1 1 2 2 3 3 k kp c r c r c r ... c r= + + + + (39)
where c1, c2, …,ck are real constants to be
determined from the initial conditions, that is to say that at the initial moment and at k-1 previous moments, the prices are known.
Case II. The interdependence between the basic price and previous prices is
t t 1 t kf (p ,p ,...,p ) g(t)− − = (40)
where f and g are known functions and g is different from the required function (the homogeneous case).
The price at time t is denoted tp and it has the
following form: * 0
t t tp p p= + (41)
where *
tp is the price given by the homogeneous
equation t t 1 t kf (p ,p ,...,p ) 0− − = (which is determined
by previous methodology) and 0
tp is a particular
solution of the equation t t 1 t kf (p ,p ,...,p ) g(t)− − = ; the
form of 0
tp being given by shape of the right
member. More precisely if g (t) is polynomial 0
tp
will be polynomial as well; if g (t) is exponential
then 0
tp will be exponential, too, etc.
Particular cases
1. The recurrence equation of prices has the following form:
n n 1 n 2p ap bp− −= + , n 2,n N≥ ∈ (42)
where prices corresponding to moments
t 0, t 1= = are 0 1p , p and they are assumed to be
known; likewise a and b are two real constants arbitrary chosen.
Under these circumstances, we shall make the
substitution n
np pr ,n 2, n N= > ∈ , and we are led to
the following quadratic equation:
2r ar b 0− − = (43)
If we mark with 1 2r , r the solutions of the above
equation: 2
1,2
a a 4br
2
± += , we can mark out the
following situations:
a) 1 2r r≠
In this case, the general term of series (42) is
given by the following equality:
n n
n 1 1 2 2p c r c r= + , where constants 1c and 2c are the
solutions of the following system:
1 2 1
1 1 2 2 2
c c p
r c r c p
+ =
+ = (44)
By solving system (44), after performing some relatively simple calculations we shall get:
1 1 2 0
1 2
2 0 1 1
1 2
1c (p r p )
r r
1c (p r p )
r r
= − − = − −
(45)
Therefore, we have: n n t t
1 2 1 2 2 1n 1 0
1 2 1 2
r r r r r rp p p
r r r r
− −= +
− − (46)
b) 1 2r r= (i.e. 2a 4b 0+ = )
After performing some analogous calculations
we shall get:
n 1
n 1 1 0 0p r n(p p ) p−= − + (47)
Remark 5.
It turns out that in case 0 1a b 1, p p 1= = = = , the
series defined by the recurrence relation (1) represents, in fact, the Fibonacci series. In this case,
we have: n n
n
1 5 1 5 1 5 1 5p
2 10 2 2 10 2
+ −= + + −
(48)
2. The recurrence equation of prices has the
following form:
n 1 n n 2 1 1p p p , p 0,p 1+ += + = = (49)
Under these circumstances, we can demonstrate by
mathematical induction that:
APPLIED ECONOMICS, BUSINESS and DEVELOPMENT
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n
2p sin(n 1)
33
π= − (50)
It is obvious that, in this case, there is no
equilibrium point because price series n n(p ) has two
limit points: 1 2
* *p 0,p 1= = .
4 Conclusions • The paper marks out the following challenging
elements: o The determination of the equilibrium price
taking into consideration different situations: the demand – supply functions
are effectively known; the elasticity of the demand and of the supply functions is also known; the dependence between prices at
different moments is assumed to be known;
o Calculations made in order to establish the dynamic pricing equation in various situations;
o The economic interpretations of the results obtained.
• The results presented within the paper can be
easily developed if we take taxes into account (because the tax is perceived by the producer as an additional cost and, consequently, the
producer would try to recover this amount through prices). From practical point of view,
the equilibrium model established under the circumstances of taxation, is based upon the demand-supply equality in relation with
different prices;
• The equilibrium interest (usually marked by *i )
represents a basic element for the market equilibrium models. There are various
possibilities for determining *i ; nevertheless
the most rigorous (but not necessary the most comfortable as far as calculations are
concerned) is the method based on the determination of the loan supply elasticity
(which implies the saving through deposits) and the determination of the loan demand elasticity.
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APPLIED ECONOMICS, BUSINESS and DEVELOPMENT
ISSN: 1790-5109 53 ISBN: 978-960-474-184-7