About an optimum model of market...

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,

APPLIED ECONOMICS, BUSINESS and DEVELOPMENT

ISSN: 1790-5109 47 ISBN: 978-960-474-184-7

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:


ISSN: 1790-5109 50 ISBN: 978-960-474-184-7

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


ISSN: 1790-5109 51 ISBN: 978-960-474-184-7

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:


ISSN: 1790-5109 52 ISBN: 978-960-474-184-7

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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ISSN: 1790-5109 53 ISBN: 978-960-474-184-7

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