Analysis and comparison of the two interchangeable goods market model and the two complementary goods market model

Trukhan E.V.

Belorussian State University

Analysis and comparison of the two interchangeable goods market model and the two complementary goods market model

In the current article the mathematical first order models of two interchangeable and two complementary goods were investigated. The models were built on the basis of Kalitin B.S. conception [5] in the form of the linear ordinary differential equations system regarding the price vector. The system takes into account the interactions on the market of buyers and sellers, the effect of the government interference and the competition effect on the market. The primary model’s parameter is the price vector. It reflects the model participants’ interaction and their influence to the market situation.

The interchangeable goods model has the following form [1-3]

(1)

The complementary goods model represented in the form of the following differential equations system

(2)

where in the models (1) and (2) the are the positive model’s parameters describing the strength of economic forces of buyers, sellers, competition and government.

In the described models the following designations are used:

p_j(t) – the price of j-th good in the point of time t;

– the equilibrium price of j-th good;

q_j(p) – the sales volume of j-th good,;

– the equilibrium sales volume of j-th good;

– the lower liminal value of j-th good price;

– the upper liminal value of j-th good price;

– the surplus of the sellers price [2];

– the surplus of the buyers price;

, - the lower and the upper liminal values of the j-th good sales volume.

The price obeys to the condition

Each sales volume function q_j(p), supposed to obey to the following conditions:

where according to the common designations [4] the values

e_j e_ji

represent accordingly the elasticity of the j-th good demand regarding the j-th good price and the cross elasticity of the j-th good demand regarding the i-th good price in the point .

According to the Raus-Gurvic criterion the economic equilibrium asymptotic stability conditions have the following form:

1) 2) (3)

where S_j=, ,

The appropriate conditions for the complementary goods model are following:

(4)

For the described markets the optimal government taxation policy problems are formulated. This policy should take into account at least such conditions: 1) preservation of the existing market relations stability (the conditions (3), (4)); 2) replenishment of the state budget in the best way. The both economic assumptions were considered in the following linear programming problems for the interchangeable goods market and complementary goods market accordingly:

+ ® max,

(5)

+ ® max,

(6)

The optimal plans for the problems (5) and (6) were built by the geometrical method on the (r₁,r₂) variables plane.

The described problems depend from many parameters. Therefore all the possible decisions regarding the different parameters values were examined. The optimal values of the variables r₁ and r₂ brought to the following tables including the decisions regarding the one of the symmetric goods. The interchangeable goods problem decisions are in the tables 1-5, the complementary goods problem decisions are accordingly in the tables 6-8.

Table1 (the inelastic demand regarding the price: )

1-E>0

under

1-E£0

under

1-E=0

Table 2 (the unitary elasticity of the demand regarding the price exists)

, under

under

e₁=1, e₂=1

Table 3 (the unitary and elastic demands regarding the price: e₁=1, e₂>1)

>0, <0

=0,

<0, <0

<0,

<0, <0

Table 4 (the elastic demand regarding the price: e₁>1, e₂>1)

1-E>0, G₁>0

G₁³0,

1-E<0, L>0

G₁<0, G₂<0,

1-E<0, L>0

in case

under

G₁<0, G₂<0

1-E=0,

Table 5 (the elastic and inelastic demands regarding the price: e₁>1, 0<e₂<1)

G₂>0

G₂<0, L<0

In all the preceding and following tables current designations are in use:

(7)

Table 6 (the inelastic demand regarding the price: )

1-E>0

, under

1-E£0

, , under

1-E=0

Table 7 (the unitary elasticity of the demand regarding the price exists)

e₁=1, e₂=1

e₁=1,

e₂>1

Таблица 8 (the elastic demand regarding the price: e₁>1, e₂>1)

1-E0

1-E<0

The comparative analysis of the decisions is realized and the following economic interpretation of the received results is given.

In the case of the inelastic demand and under the condition 1-E<0 (i.e. the value of the cross elasticity is sufficiently small) the optimal taxation rate for the interchangeable goods under the appropriate competition coefficients constraint can be higher then the complementary goods taxation rate. The competition coefficients constraint reveals that the tax is greater for the having the higher competition coefficient agent.

The research of the inelastic case under the equal coefficients displayed that under the condition of equal income of both market agents the taxation rate can be higher for the complementary goods market. The agent with the higher income level will have the higher taxation rate on the interchangeable goods market. In the case of the lower income level the taxation rate is higher for the complementary goods market. Thus it’s possible to assume that for the current case the complementary goods market is more stable regarding the taxation rate increasing under the appropriate competition coefficients ratio conditions.

The case of the inelastic demand as under the condition 1-E=0 (the cross elasticity and elasticity are equitable) so under the condition 1-E<0 (the cross elasticity is greater then the elasticity) displays the same conclusions like the previous case but with modified competition coefficient constraint. The case of the homogeneous goods (the goods with the equal market characteristics) under the same conditions reveals the possibility of taxation rate increasing on the market of the interchangeable goods to a greater extent under the cross elasticity rate constraint. These results help to make such conclusion: the interchangeable goods market with a high cross elasticity rate is less stable regarding the taxation rate increasing then the complementary goods market.

The unitary elasticity case investigation revealed that in this case the optimal taxation rate can be greater on the complementary goods market.

In the case of the elastic demand under the condition 1-E>0 (the cross elasticity is low) the taxation rate can be higher on the interchangeable goods market under the necessary competition coefficient constraint for the appropriate market agent: the top constraint for the high cross elasticity, the lower constraint for the low cross elasticity. In the same case under the condition 1-E<0 the taxation rate increasing possible on the interchangeable goods market under the cross elasticity constraint.

From the acquired results it’s evident that the high cross elasticity provides the destabilizing market effect under the taxation rate increasing. The competition coefficient can provide as stabilizing so destabilizing effect. In the case of the inelastic demand the competition coefficient constraint is built regarding the other agent competition coefficient rate involving some another parameters of the opponent agent’s position including the cross elasticity parameter. In the case of elastic demand this constraint doesn’t take into consideration the competition coefficient rate of the next market agent (it includes only the cross elasticity coefficient).

Bibliography

1. Калитин Б.С. // Вестн. БГУ. Сер. 1. 2003. № 3. С. 67-72.

2. Калитин Б.С.. // Экономика. Управление. Право. 2004. № 2. С. 3-6.

3. Калитин Б.С. // Вестн. БГУ. Сер. 1. 2005. № 2. С. 80-84..

4. Dolan A.J. The market: microeconomic model // A.J. Dolan and D. Lindsey, 1992.

5. Kalitin B. S. The mathimatical methods of economic: Minsk: BSU, 2004.