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TH T H E E E EV VO OL LU UT TI IO ON NA AR R Y Y C CO OM MP PU UT TA AT TI IO ON N O ON N T TH H E E MA M AR RK KE ET T. . T TH HE E C C OM O MP PE ET TI IT TI IO ON N A AN N E EQ Q UI U IL LI IB BR RI I UM U M I IN N

SA S AL LE E C CO OS ST TS S A AN ND D QU Q UA AN NT TI I TI T IE ES S

Keywords: market, sale costs, optimization, equilibrium

Abstract

The theory of games usually concentrates attention in the market dynamical analysis on finding an the equilibrium and on the optimization behavior of sellers. It should be mentioned that for the application of the developed results. It is very important to know that the equilibrium doesn’t only exists but also can be reached by market participants.

By this reason the above approach was criticized by some researchers. (For example the Austrian market process theory (Mises[21], Hayek[12] and Kirzner[18]) and the orthodox neoclassical theory (Marshall, Robinson, Chamberlin and Walras)). The problem of the equilibrium stability has been discussed in our papers earlier [7 and 8].

We have proposed to consider the problem of the attainability of the market equilibrium as an asymptotic stability task with some specially constructed iterations procedures. The result of its analysis was the theorem of the sufficient condition of the competition at the market for enterpreneurs (33). We try to renew the mentioned approach in this paper. It will be found the conditions of the local stability of the equilibrium in cases, when compete two and more products.

Definitions:

1. n homogeneous products compete at the market.

2. The commercial demand of the market is S.

3. a_iis the quality of the product i.

4. The entrepreneur, which produces the product i, use the marketing strategyxi ≥0. It generalizes all sell costs of the product i, somehow:

advertisement, discounts, wrapping, etc. The product i will desappear from the market, whenxi =0 .

5. The entrepreneur, who produces the product j, receives a profit g_i from the sale of every piece.

6. a_ig_i ≥a_i+1g_i+1,∀i,i=1÷n−1

7. The sales V_jand the net profit P_jof the product j are:

1

, 1

j j n j

i i i

V S a x j n

=

a x

= = ÷



₁ ^{, 1}

j j

j j j j j n j

i i i

P V g x Sg a x x j n

=a x

= − = − = ÷



If the entrepreneur maximize the net profit from the sales of the product i, we receives:

1 1

1 1 0

j j j j j

n n

j i i i i i i

P Sa g a x x

₌

ax

₌

ax

 

∂ =  −  − =

∂    

^{or j *j ni1 i i*}

(

^{j j ni1 i i*}

)

j j

a x a x Sa g a x Sa g

=

=



− =



⁽¹⁾

The current information of the competitors’ strategies is unknown in reality.

However, every competitor receives a data about the marketing strategies of his rivals. For this reason, the market strategy for the product i depends from the previous marketing strategies of all competitors. In this case the market strategy

( , )

x i m

is at the sale session m:

^{j j}

^{( )}

^{ni 1 i i}

^{( 1)} (

^{j j in1 i i}

^{( 1)} )

j j

a x m

a x m Sa g a x m

Sa g

=

=

=  − −  −

(2)

The model of the competition for five products can be illustrated on the pictures below. There is shown the division of the market between the competitors.

The behaviour of 5 competitors

0%

20%

40%

60%

80%

100%

t

part of market

Fig. 1.

The equilibrium for 5 com petitors

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

t

part of the market

Fig. 2.

The pictures above describe the behavior of the five competitors with the same parameters a_i andg_i in both cases. Nevertheless we see at the Fig. 1. unsteady process. In the second case (Fig. 2.) the competitors move to the point of the nontrivial equilibrium. The distinction of the first case from the second one consists in the different positions of the competitors of the market. Thus the completion of the products sometimes doesn’t reach the equilibrium of the market strategies.

Now we are trying to examine the conditions of it in the general case. For the analysis of the stability around of the equilibrium is convenient to use the common application of the all marketing strategies

1 ( )

n

m i i i

A a x m

=



= (3) The region of values of the common application of all marketing strategies is always positive (Am >0), because in other case the competition not exist. Then the iteration procedure (2) transforms to the form below:

₁

1

₁

m m n m

A A n A

S α

− −

 

=  − 

 

(2*)

Resolving equation

1

A A n

n

A

S α

∗

=

∗

  −

∗

 

 

we have got two states of the

equilibriums of the market strategies:

A₀^* =0, and ₁

1

n

A S n α

∗

= −

(4)

where ⁿ ₁

1

n j

j j

α = 

₌

a g

is the computational index of the market. It characterizes the dispersion of the market parameters of the product i (the quality a_i and the profitg_i) from the average level of it. From the definition we have:

n n

n a g

n g

a

n ≤α ≤

1 1

and when ^{1 1} ≈1

n ng a

g

a , that n ^≈



_iⁿ₌ aigi

α

1 (5)

The competition around the equilibrium can be described by the function:

₁ 1 ² ₁

) 2

( ₋

− − −

= m n m

m y

n S y

y

α

(6)

where

y

_m

= A

_m

− A

^∗. The condition of the stability around the equilibrium is,

when

lim

_m

0

m

y

→∞

=

. The trivial equilibrium A₀^* =0 is not stable, because from the linear of equation (2*) in the vicinity of it one can write an equality

| A

_m

| | = A

_m−1

| n

. The behavior of two competitors can be illustrated at the picture below:

The competition of 2 sellers at the trivial equilibrium

market strategy 1

market strategy 2

Fig. 3.

Analyzing the equation (6) we can write the necessary and sufficient local asymptotical stability condition of the equilibriumA₁^* in a form of inequality2−n <1. This means that it will be stable if and only if n=2^{. In} this case one has an inequality y_m <|y_m−1| in the sufficiently small vicinity of equilibrium. But in our case for two competitors linear approximation has form

m 0

y = . For this reason we have to start nonlinear analysis of the behavior of

the two competitors. From the inequality y_m <|y_m−1| follows 1 ₁

| _ny_m | 1 S

α

₋ < ^. Thus for n=2the region of the equilibrium stability has form:

2

| | S y ^<α ^. Remember (3) and (6) we have y_m−1= A_m−1−A m^*, ∈N^{. Thus}

2 2

1

α α

S

A_m− − S < or S m N

A_m < ∀ ∈

< ₋ 2 , 0

2

1 α

This means that exists the vicinity, which guarantee to reach the equilibrium of the market strategies for both competitors. Continuing to examine the equilibrium in the case n>2. We have a different behaviory_mfor its different initial values and the quantity of competitors:

The behaviour of y(0) for three com pe titors around the equilibrium

y(m -1)

y(m)

Fig. 4.

Behaviour of the four competitors around the equilibrium

-4 -3 -2 -1 0 1 2

-4 -3 -2 -1 0 1 2

y(m-1)

y(m)

Fig. 5.

The drawings above illustrate the behavior of y_mfor three (Fig. 4.) and four(Fig.

5.) competitors, when the initial signification y₀ gives the possibility to reach the equilibrium.

The behaviour of five com petitors around the e quilibrium

y ( m - 1)

Fig. 6.

The pattern above illustrates the behavior of y_mfor five competitors. Thus, we can see that possibility to reach the equilibrium depends from the initial position

y0and the number of competitors n.

Consequently, the local stability around the equilibrium exists, when the inequity is responsible:

N n S y

y n y

m n

m

m <  − − ₋ < ∈

−

, 1 )

2 (

1 ₁

1

α

.

Thus we receive the sufficient condition of the local stability around the equilibrium:

n m N S

n y S

n m

n

− ∈

<

− <

3 , 1

α

α . (7)

To comparison (7) with the definition of the functiony_m, we have received, that the region of the function exist only in the case, when n≤3. Becausey_m = A_m−A* we receive two inequities, which describes the sufficient condition of the equilibrium stability for n≤3:

N n m

m S x S a

n j j

n i n

− ∈

<

<



₌ ^{( ) 2 ( 1),}

2

1 α

α

From (4) we have received the inequality for the state A₁^*, when n≤3.

n i

i n n

n

n S g a n n

S S

α α

α α

) 1 ) (

) 1 1 (

)( 1 ( 2

2 < − − − < −

The right part of the inequality is always true, when

n i i

g n

a α

−1

> . Hence we have the condition, when the three products can to compete around the nontrivial equilibriumA₁^*:

^{i i 4 ,}

{

^{1, 2, 3}

}

n

a g i

>α ∈ (8) Conclusion 1:

The product i can to compete at the market if the equation above is true for its market characteristics.

The equation (6) can be used only for the analyses of behavior of 2 and 3 competitors.

Examining (1) further we propose that the marketing strategies of the competitors have used information about the marketing strategies of his rivals.

Hence we can find the market strategy x_i^*in the state of the equilibrium:

Sa g a x a x i n

x a

n

i j j

j j n

i j

j i j

i i i

i = −

 

= ÷

≠

≠ =

= ) , 1

1 (

, 1

* ,

1

*

* (1*)

* 2 1,

arg max ( ) 1 ( ) , 1

i ( )

n

i i j j i j j

x

P i Sa g a x i n

a i ^{= ≠}

= −



= ÷

In this case the market strategyx_i(m)at the sale session m have been another.

As in the model above the current information of the competitors’ strategies is unknown. To suppose that i competitor have used the previous marketing strategies of his rivals. In this case the market strategy x_i(m) is:

N m n i m x a m

x a g

a Sa m

x _{i i} ⁿ_{j j i j j} ⁿ_{j j i j j}

i

i^{( )}= ^{1 (} −



_=1,≠ ⁽ −¹⁾⁾



_=1,≠ ⁽ −^1), =¹÷ ^, ∈

Sales of the product i is unprofitable, if P_i(m)<0. Hence we can receive the borders of the area of the competition:

N m n i

m x m

x a m

x a

m x g Sa

n j i j

j j j

i i

i i

i − ≥ = ÷ ∈

− +

−¹⁾



_{= ≠} ^{( 1) ( ) 0, 1 ,}

(

) (

, 1

Hence ⁿ a x m Sa_jg_j i n n N

i j

j j j − ≤ ∀ = ÷ ∈

≤



_{= ≠} ^{( 1) , 1 ,}

0 1, and

N n n i

i g Sa m

x_i ≤ _{i i} ∀ = ÷ ∈

≤ ( ) , , 1 ,

0 .

Thus, the behavior of the all producers of the homogeneous products i, where N

m n

i=1÷ & ∈ can be described by the new iteration procedure:

1, 1,

1,

1,

1( ( 1)) ( 1),

( ) ( 1) , 1 ,

0, ( 1) , 1 ,

n n

i i j j i j j j j i j j

i n

i j j i j j i i

n

j j i i

j j i

Sa g a x m a x m

a

x m when a x m Sa g i n m N

when a x m Sa g i n m N

= ≠ = ≠

= ≠

= ≠

 − − −



= − ≤ = ÷ ∈

 − > = ÷ ∈



 





(9)

The set ^{Xn =}

{

^{xj: 0≤x mj( )≤Sgj, j= ÷1 n m, ∈N}

}

is invariant relatively to the iteration procedure (9).

At the first step we examine the situation, when two competitors (the products 1 and 2) appear at the market. The behavior of both rivals can be described by next equations:

(

^{( ) ( )}

)

) 1 1

( _{1 1 2 2 2 2}

1

1 Sa g a x m a x m

m a

x + = − , if Sa₁g₁ ≥a₂x₂(m)

(

^{( ) ( )}

)

) 1 1

( _{2 2 1 1 1 1}

2

2 Sa g a x m a x m

m a

x + = − , if Sa₂g₂ ≥a₁x₁(m)

The competition begins, when^xi(^m)>0,ⁱ=

{ }

1,2,^m∈^N. These conditions are

right, when the inequality m N

a g S a m

x ≤ ∈

≤ ( ) ,

0

1 2 2

1 is true. Hence we have

received the inequality above:

2 2 2

2 2

2 1

1 ( 1) ( 1)

0≤ Sag a x m− −a x m− ≤Sa g

It is right, when the inequalitya₁g₁−4a₂g₂ ≤0 is true. Thus we receive the sufficient condition for the case, when two competitors always reach the equilibrium of their market strategies in the set X². This was proved in [11].

1 4

2 2

1

1 ≤

≤ a g g

a (10)

The equilibrium of the market strategies for the two competitors can be found from the system of equations:







−

=

−

=

1 1 1 1 2 2 2

2

2 2 2 2 1 1 1

1

x a x a x Sa x

a

x a x a x Sa x

a

Thus we have found two states of equilibrium for each of two competitors:

( )

^0;0

0 =

B and

















+

= +

2 1 1 2

1 1 2 2

2 2

1 1 2 2 1 1

2 ) 1

(

; ) 1

(

1

g a g a

g a g a Sg g

a g Sg a

B

From the definition (6) we receive that 1. 1

1

1 1 2 2

1 2 2

+ a g <

g a

g a g a

By this reason: _{1, 1}

1 Sg

x _B < and _{1, 2}

1 Sg

x _B < . ThusB₀ ∈X²,B₁∈X². These results can be illustrated by the drawing below:

The non-trivial equilibrium for 2 market strategies

market strategy 1

market strategy 2

Fig. 7.

The picture shows us the behavior of the two competitors, when (10) is true. If both competitors use the strategies from the set X²their strategies reach the equilibrium in B₁after a few sessions m at any case .

The new procedure permits to define the region of local asymptotical stability.

The necessary and sufficient condition of the local equilibrium stability in vicinity of the state B₁depends on the eigenvalues of Yacoby matrix for the iteration procedure (9). This eigenvalues should be situated inside of the circle of the unit radius with the centre in 0. For the case n=2we have received:

1

1 2 2 1 2 2 1

1 ≤

∂

∂

∂

− ∂

∂

∂

∂

∂

x f x

f x f x

f 0

2 2 1

1 =

∂

= ∂

∂

∂

x f x

f .

Since the definition (6),

2 1

2 2 1 1

2 2

2 ga g a g a x

f = −

∂

∂ and

2 2

1 1 2 2

1 2

2a g g a g a x

f = −

∂

∂ , we have

the next inequity:

) 1 2 ( ) 2 ( ) 1 ( ) 1 ( 4

)) 2 ( ) 2 ( ) 1 ( ) 1 (

( ²

− ≤ g a g a

g a g a

That we have found the necessary and sufficient condition of the local stability in the stateB₁:

(1) (1)

1 3 2 2

(2) (2) a g a g

≤ ≤ + (11)

These results can be illustrated by the pictures below:

The consumer 1 disloges the 2nd competitor.

x(1)

x(2)

Fig. 8.

The 2 competitors at the market

x(1)

x(2)

Fig. 9.

The pictures above illustrate the influence of the origin positions of the competitors at the market with their relation of the market characteristics responded inequality (1) (1)

4 3 2 2

(2) (2) a g a g

≤ ≤ + . The Fig. 8. shows us, what can happen the second competitor chooses his market strategies very far from equilibrium. In the second case, we have concluded that there exists the bounded attractor if the initial position of two competitors.

Continue the investigation of the equilibrium in a general case. The equilibrium of the market strategies for all homogeneous products on the market S means, thatx m_i( + =1) x m_i( ),∀i m i, , = ÷1 n m, ∈N. To resolve the system of the equations (1), we have received two solutions:

n i

x

B₀^* : _i^* =0, =1÷ , and i n

g a

g a n g a a

n x S B

n k

k k n k

i i k k

i

i = ÷

− −

= −





=

=

1 , 1 ) (

1 1

) 1 : (

1 2 1

*

*

1 (4*)

The trivial equilibrium B₀^* =0 isn’t stable, because it situated out of the region of the valuesxi >0.

If we use the summary competition index α_nof the market, we receive the new form ofB₁^*:

2

*

1 )

1 (

n i i n

i i

g a n

a n x S

α α − ⁻

= −

The profit in the nontrivial equilibriumB₁^*:x_i^*,i=1÷nis:

2 2

2

1

* 1)

( 1 )

1 (

i i n n

i n

k i i

i i i

i a g

Sg n

g a

g a Sg n

P = − − = − −



=

α α

If we summarize a_ix_i^*at the equilibriumB₁^*by all i,i=1÷n, we receive:

* 1 1

1 2

1 2

* ( 1)

1 ) ) 1 ( )(

1 ) (

( 1 ) 1

( S n A

g n a

n n S g a n n

x S a

n n

i i i n

n n

i

i i n n

i n

i

i = − − − = − − −



= − =



₌



₌

=

α α

α α α

Thus we prove, that both (2 and 9) iteration procedures have the same nontrivial equilibrium. Hence we can use the results above and the procedure y_mto the further analysis of the competition. The both iteration procedures show us the definitive role of the computational index of the market n ₁

1

n j

j j

α = 

₌

a g

^{. It}

reflects the dispersion of the market parameters of the products (the quality a_i and the profitg_i). As is seen above, the correlation of the quality a_i and the profitg_iof product i with this index determinate its part of the market. This gives the possibility for the entrepreneur to make a decision of entering the market and to avoid the risk to lose money with the production of noncompetitive commodities.

II. The equilibrium stability of the competition in quantities.

The concept of competitive equilibrium underlies the modern economic theory. According to known " to theorems of well-being ", competitive equilibrium is an optimum condition of market economy. Consequence of its deviation is decrease in economic efficiency. The description of conditions of the perfect competition is not constructive, because:

• it does not allow to define for the concrete market, whether these conditions are satisfied;

• how much can deviate the prices from equilibrium by Walras.

Above we have studied a competition of several firms in sale costs at the one- product market, which is close to A.Cournout’s model. The question of stability

of equilibrium competition in quantities is studied based on generalization of imperfect competition. We expand a model, having made following assumptions:

1. In the market compete the firms making similar (homogeneous) products with qualitya_i, where0< ≤ ∀ =a_i 1, i i, 1,..,n. There is

,1

j ≤ ≤j n, at which a_j =1( (the standard of quality);

2. Each firm has an opportunity to make production equal to the maximal size of demandS₀. During a production cycle (trade session), the firm idelivers the certain volume of production on the marketq_i. It cannot changed during session. Each firm takes the quantity set by its competitors as a given, evaluates its residual demand, and then behaves as a monopoly;

3. Function of demand depends formQ=ϕ(S₀, , )γ p ^{, where} γ ^is elasticity of demand by the price p. All the products during session are on sale under one pricep. The market price is set at a level such that demand equals the total quantity produced by all firms. The price is established depending on total quantity of the goods, which has been supplied during the session on the market by all firms;

4. At an output during session on the market firm ispends sale costsx_i. It generalizes a marketing strategy of the iproduct, somehow:

advertisement, discounts, wrapping, etc. The sell costs and quality of the product has determined the part of market, which ifirm can control during a session. During the sale session, the firms compete in sell costs, and choose their sell costs simultaneously. The product idisappears from the market, ifxi ≤0;

5. Each firm has a cost function. Normally the cost functions are treated as common knowledge. The cost functions are the same among firms. The cost function of firm iconsists of marginal costs c_i ≥0, if i< j^, thatc_i ≤c_j, (proportional to volume of made production), and sell costs

xi;

6. The firms are economically rational and act strategically, usually seeking to maximize profit given their competitors' decisions. There is strategic behavior by all firms;

7. Firms do not cooperate.

The firm, which produces the product j, receives a profit g_j = −p c_j ^{from the} sale of every piece of product. In

[ ]

¹ we accounted the net profit P_jfrom the product j. If the firm iduring sale session maximizes the net profit from the sales of the product i. Examining (2) further we propose that the marketing strategies of the competitors have used information about the marketing strategies of his rivals. In

[ ]

¹ we found the market strategy x_i^*in the state of the equilibrium (3).

Hence, we shall create the general model of the market division. The share of the market of the firm will be defined as:

1

1

1 1

n i i

i k k n

i i j j

j

a x n

q q Q

a x βa g

=

=

 − 

= =  − 

 

 

^{, where 1}

n i i

Q=



₌ q . (9) According to (1)-(3) profit of the firm for session is:

1 2

( ) ( ) ( ) 1

i i ( )

i i

P p Q p g p n

α

g p

β

 − 

=  − 

 

Each firm aspires to receive the maximal profit. Assuming that they iterative

change quantity of the products one may use the

formula

1

1 ( 1)

m m i

i i

i

q q m

q

ϕ

⁻

− ∂

= + ∆ −

∂ .We can show that the maximum of function of profit is reached in an internal point of a set of definition. The Fig. 1 illustrates this.

Let us study a situation for n=2. under assumption

0 0

( , , )

^p

Q = ϕ S γ p = S e

^−γ

From

[ ]

¹ we have the profit functions for every firm:

2 3

0 2

1 1 2 2

( ) ( , , ) ( )

( ( ) ( ))

i i

i

a g p

P p S p

a g p a g p ϕ γ

= +

Each competitor wishes to maximize own profit by changing pricep. If we take a derivative by p of the above and equate it to zero, we receive system of the

equations:

3 3 2 2 3

1 1 2 2 1 1 2 2 1 2

2 2 4

1 1 2 2 1 1 2 2 1 1 2 2

3 ( ) 2( )( )

0, 1,2

( ) ( ) ( )

j j j j

p p p

i g g g a g a g a g a g a a g

P e e e j

p a g a g a g a g ag a g

γ γ γ γ

− ′ − + − + + −

 

∂ = =− + = =

∂  +  + +

1 1 2 2 1 2

(3 − γ g

_i

)( a g + a g ) − 2( a + a g )

_i

= 0, j = 1, 2

(10)

Then the set of points of an extremum is defined by equality

g

₂

− g

₁

= 0

and a g_{1 1} + a g_{2 2} = 0

The later leads to equality g₁=g₂ =0. If

c

₁

≠ c

₂, it is not true. From

2 1 0

g −g = and (6) we receive

1 1 1 2 2 1 2 1

(3 − γ g )( a g + a g ) − 2( a + a g ) = 0

We have found coordinates of a condition of equilibrium on plane

{ }

g1^,g2 ):

− γ g ˆ

₁

+ = − 1 0, γ g ˆ

₂

+ = 1 0

Hence, at an output on the market with the prices of both firms reach an extremum with the prices:

1 1 2 2

1 1

ˆ ,ˆ

p c p c

γ γ

= + = +

To assume, that both firms try to maximize their profit, changing at m at occurrence on the market sizeg m( )by recurrent procedure:

[ ]

[ ]

1 1 1 1 1 2 2 1 1 2

2 2 2 1 1 2 2 2 1 2

( ) ( 1) (3 ( 1))( ( 1) ( 1)) ( 1)2( )

( ) ( 1) (3 ( 1))( ( 1) ( 1)) ( 1)2( )

g m g m g m a g m a g m g m a a

g m g m g m a g m a g m g m a a

γ γ

 = − + ∆ − − − + − − − +

 

= − +∆ − − − + − − − +



The Fig. 2 illustrate this behaviour.

Fig. 10. A profit function

0.1 0.2 0.3 0.4 0.5 ⁱ

−0.15

−0.1

−0.05 0.05 0.1

Profit

g

Fig. 11. Behaviour of two competitors with the different margin costs and different initial conditions

We can show, that the extremum

1 1 2 2

1 1

ˆ , ˆ

p c p c

γ γ

= + = +

found above is a unique stable state of equilibrium, because the Jacoby matrix described above iterative procedure has negative own values.

{ } { }

1 2 2

1 1 2

2 2 2

1 2 1 2 1 2 1 2 1 2 1 2

1 2

3 2

2 3 ,

et ( 3 )(3 ) 4 3 3 6 3( ) 0,

Sp 4 4 0

a a a

J a a a

D J a a a a a a a a a a a a

J a a

− −

 

=   − −  

= + + − = + + = + >

= − − <

The analysis above has shown us, that strategy of optimization of quantity conducts to the different prices for each firm. It also means that the optimum size of the supply for each of firms will be distinguished.

1

0 ^ci , 1, 2

Qi =S e^{−γ −} i= Conclusion 2:

The supply of firm does not depend on quality of the products if it adheres to optimum marketing strategy.

Then we can present profit function of ifirm as

2 2

( ) 1 1 ⁱ

i i

i i

n q

P q Qg p

g Q

α β

 − 

=  −  =

  (11) For further analysis, we have substituted the function of demand as

0

1 Q S

γp

= + (12) At non-significant numbers ofγ this function is indistinguishable from (12).

Besides it, unlike the standard linear approximation of function of demand, this function keeps property of "camber". It is extremely important for the analysis of

0.65 0.7 0.75 0.8 0.85 0.9 0.95 ¹

0.6 0.7 0.8 0.9

2

g

profit function. From here we receive, that 1 ⁰ S 1 p

γ

Q

 

=  − 

 .Hence the profit function receives a form:

2 2 2

0 0

2

1 1

( ) ⁱ 1 ^{i i}

i i i

q S S q q

P q c c

Q

γ

Q

γ

Q Q

γ

     

=   − − = −  + 

 

 

 

As function of profit can accept any values, we shall find its regular maximum.

0 2

2 1

2 2

i i i i i

i i

P S q q q q

q

γ

Q Q Q

γ

c Q

    

∂ =  − −  +  − 

∂     

From here we shall receive system of the linear equations from which it is possible to find the price of the equilibrium:

0 2( _i) (1 _i)(2 _i) 0

S Q q c Q q

Q − − +γ − = or ⁰

0

2 2

i i

i

S Q c Q

q Q

S Q c Q γ

γ

= − −

− − (13) Optimizing behaviour of the five firms, we receive a following pattern. Thus, the quota of the weakest competitors gradually decreases. The Fig. 3 illustrates, that after a while there is stabilization of the total offer and, as consequence, the prices and shows us, that the quota of the weakest competitors gradually decreases. Summarizes both parts (10), we receive the equation from which is possible to find an equilibrium condition of the market:

0

1 0 1

0 0

1 1

2 2 2

n i n

i i

i i

S Q c Q

n S

S Q c Q S Q c Q

γ

γ γ

= =

− − = − =

− − − −

 

From here it is possible to find market equilibrium in quantities:

1

0 0

1 2 1

, , 1,

2 (1 ) 2

n i

i

n i i n

S γc Q S

=

= − ∀ =



− + or price:

1 1

1 2 1 1

, 1, 1,

1 2 2 1 2

n n i

i i

i i

p n c

n i n

p c p c

γ γ

γ γ γ γ

= =

+ = − − = − =

+ − + −

 

Conclusion 3:

Necessary condition of competitiveness of the product at market is performance of an inequality 1

ci

<γ , i.e. the margin cost of product should not exceed value the return of elasticity of demand by price.

If all firms have identical margin costs it is possible to find their quotas. They are equal q₁= = ∀ =q_i q_n, i i, 1,n^{. Then:}

0

2 1

(2 1)(1 ) Q nq S n

n cγ

= = −

− +

The division of the market under the above conditions is characterized by Fig. 4

Division of market

0 10 20 30 40 50

1 3 5 7 9 11 13 15 17 19 21 23 25 m

quotas

q5 q4 q3 q2 q1

Fig. 12. Division of the market between five market between five firms at different entry conditions

Market division

0 10 20 30 40 50 60

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 m

quotas

Series5 Series4 Series3 Series2 Series1

Fig. 13. Division of the market between five market between five equal competitors and same initial positions

Conclusion 4.

The greater number of contestant firms with the identical margin cost in the market leads to increase in the offer and reduction of price.

The price in state of equilibrium, exceeding cost price, since an inequality (11) is always true, whenn>1.

1 0 1 (2 1)(1 )

1 ( 1)

2( 1)

S n c

p c

Q n

γ

γ γ

  − +

=  − = − >

  − (14) Let us consider a case of two firms (duopoly), whenc₁ <c₂. Profit of the manufacturer for session:

2 2

0 2

( ) ^{i i} (1 )

i i

S q q

P q c

Q Q

γ γ

= − + ^{, where} Q= +q1 q2

Its maximum is reached when:

0 0

2

2

2 1 2

2 2 2 ( ) (1 )(2 ) 0

i i i i i i

i i i i

i

P S q q q q q S

c Q q c Q q

q Q Q Q Q Q Q

γ

γ γ γ

      

∂ =  − −  +  − =  − − + − =

∂       

or 2 ⁰

( _i) (1 _i)(2 _i) 0

S Q q c Q q

Q − − +γ − =

Hence, we receive the system of the equations defining behaviour of competitors:

0

2 1 2

0

1 2 1

2 (1 )( ) 0

2 (1 )( ) 0

S q c Q q

Q

S q c Q q

Q

γ γ

 − + + =



 − + + =



(15)

Fig. 5 illustrates us, that both firms, maximising profit, after several outputs on the market reach equilibrium.

market division

0 1 2 3 4 5 6 7

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 m

quota Series2

q1

Fig. 14. Division of the market in duopoly with different margin costs and initial positions

market division

0 1 2 3 4 5 6 7 8

1 3 5 7 9 11 13 15 17 19

m

quotas

Series2 Series1

Fig. 15. Behaviour of two firms with the same margin costs and diferent initial positions

Let's designate b₁= +1 γc₁and b₂ = +1 γc₂. From (12) it is received, accordingly:

1 2 1

2

0 0 1

1

2 2

b b

q Q Q

S S Qb

Q b

= =

− −

and ₁ ^{2 2 2}

0 0 2

2

1 1

2 2

b b

q Q Q

S S b Q

Q b

+ +

= =

− −

(16)

2 2 1

1 2

0 2 0 1

2 2

b b

q q Q Q

S b Q S bQ

 

+ = =  + 

− −

  ^or

2 2

0 2 2 1 1 1 2 2 1

2 ( S b q − b q ) − bb q ( − q ) 0 =

It is a hyperbole on a plane, passing through the beginning of coordinates. The second equation q₁+q₂ =2S₀ is straight line. For these reasons, the solution must have not more, then two roots.

Thus, еhey correspond to two states of the market equilibrium:

^{1,2 0}

(

^{1 2 12 1 2 22}

)

1 2

2 3

Q S b b b b b b

= b b + ± − + (17) From (13) and (14) we receive two states of equilibrium: E₁ =( ,q q% %_{1 2}) andE₂ =( ,q q_{1 2}). In a stateE₁=( ,q q% %_{1 2}) the quotas of the both firms are:

2 2

1 2 1 2 1 1 2 2

0

1 2 2

1 2 1 2 1 1 2 2

( )

2

3 2

b b b b b b b b q S

b b b b b b b b

− + − +

= − + − +

% ,

2 2

1 2 1 2 1 1 2 2

0

2 2 2

1 2 2 1 1 1 2 2

( )

2

3 2

b b b b b b b b q S

b b b b b b b b

− + − +

= − + − +

%

Similarly, we can recalculate quotas for the state E₂ =( ,q q_{1 2}):

2 2

1 2 1 2 1 1 2 2

0

1 2 2

1 2 1 2 1 1 2 2

( )

2

3 2

b b b b b b b b q S

b b b b b b b b

− + − +

= − + − + ^,

(

^{1 2 12 1 2 22}

)

²

0

2 2 2

1 2 2 1 1 1 2 2

2

3 3

b b b b b b

q S

b b b b b b b b

+ + − +

= − + − +

From here we shall find the prices for both states of equilibrium E₁ =( ,q q% %_{1 2}) and E₂ =( ,q q_{1 2}).