Adv Syst Sci Appl 2023; 02:184–194 Published online at https://ijassa.ipu.ru. Nonuniqueness of Equilibrium in Closed Market Model Alexander Kotyukov1*, Natal’ya Pavlova1,2,3,4 1V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia 2 Moscow Institute of Physics and Technology, Moscow, Russia 3 RUDN University, Moscow, Russia 4 Derzhavin Tambov State University, Tambov, Russia Abstract: In this paper, we consider a closed market model. In this model the supply and the demand functions are restored by their price elasticities. We obtain sufficient conditions on nonuniqueness of equilibrium in this model. For several special cases of closed market models we get a criteria of equilibrium uniqueness. The obtained results are illustrated with the example of the market with two goods. Keywords: supply, demand, equilibrium, coincidence point, covering map 1. INTRODUCTION Equilibrium is one of the main concepts of modern economics. It is a state of the market in which all the produced goods are sold and every market participant does not want to change their position. Consider an economic system which consists of producers and consumers. The producers manufacture goods which are sold to the consumers. We assume that the consumers work for the producers and spend all their salaries to purchase the goods and do not make savings. The total amount of the goods produced is called a supply. The total amount of the goods sold is called a demand. If the supply exceeds the demand, the producers suffer profit loss due to the goods unsold. This makes them cut salaries of the consumers. If the consumers have less money, they purchase less goods which, in turn, decrease the profit of the producers even more. It is easy to see that this process leads to a catastrophic economical situation. On the other hand, if the demand exceeds the supply, some consumers cannot purchase the goods they need since these goods are not available for purchase. This leads to an unfavorable situation in the region, e.g. a massive hunger, a pandemic, lack of building resources etc. Therefore, we are interested in maintaining the market state in which the total amount of the goods produced is equal to the total amount of the goods needed, i.e., if the supply equals to the demand. Such market state is called an equilibrium. The prices on the goods on such market are called equilibrium prices. Equilibrium as an economical concept appeared in the second half of 18th century in the works of Stewart [14] and Smith [13]. The first mathematical notion of the concept was introduced by Walras [15] in 1874. At that time there were no developed mathematical theory ∗Corresponding author: amkotyukov@mail.ru NONUNIQUENESS OF EQUILIBRIUM IN CLOSED MARKET MODEL 185 to obtain the substantial results on equilibrium existence. Such results were first obtained by Arrow and Debreu [1] in 1956. In this work, we obtain sufficient conditions on equilibrium nonuniqueness in a closed market model. Here the demand and the supply functions are restored by their price elasticities. The elasticity shows how one economic variable changes with a change of another. An equilibrium in this model is considered as the coincidence point of the supply and the demand functions. To obtain the mentioned conditions we use the results in the theory of covering maps and coincidence points [2]– [5]. Let us formalize the problem. 2. PROBLEM STATEMENT Denote Rn + = {p = (p1, ..., pn) ∈ Rn : pi > 0 ∀i = 1, n}. Let the market consist of n goods. The prices on these goods are described by a vector p ∈ Rn + which satisfies the inequalities: c1i ≤ pi ≤ c2i ∀i = 1, n. Here c1 = (c11, ..., c1n), c2 = (c21, ..., c2n) ∈ Rn + are natural constraints on the prices p (denote P = [c11; c21]× ...× [c1n; c2n]). Let the demand function D : Rn + → Rn +, D(p) = (D1(p), ..., Dn(p)), and the supply function S : Rn + → Rn +, S(p) = (S1(p), ..., Sn(p)), be given. We assume that for some known prices p∗ ∈ P we know the demand D∗ ∈ Rn +, D∗ = (D∗ 1, ..., D ∗ n), and the supply S∗ ∈ Rn +, S ∗ = (S∗ 1 , ..., S ∗ n), i.e., D∗ = D(p∗), S∗ = S(p∗). Next, suppose that we have a matrix E = (Eij)i,j=1,n where Eij ∈ R is the jth good price elasticity of the ith good demand. This quantity shows how the demand on the ith good changes with the change of the price on the jth good. Similarly, we have a matrix Ẽ = (Ẽij)i,j=1,n where Ẽij ∈ R is the jth good price elasticity of the ith good supply. This quantity shows how the demand on the ith good changes with the change of the price on the jth good. Definition 2.1. A closed market model is a following set of parameters: σ = (c1, c2, p ∗, D∗, S∗, E , Ẽ). This set uniquely defines the demand function D by: Di(p1, ..., pn) = D∗ i n∏ j=1 (p∗j) −Eijp Eij j , i = 1, n; (2.1) and the supply function S by: Si(p1, ..., pn) = S∗ i n∏ j=1 (p∗j) −Ẽijp Ẽij j , i = 1, n. (2.2) Formulas (2.1), (2.2) are the solutions to the following systems of equations: Eij = ∂Di ∂pj pj Di , i, j = 1, n; (2.3) Ẽij = ∂Si ∂pj pj Si , i, j = 1, n (2.4) Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 186 A. KOTYUKOV, N. PAVLOVA with initial conditions Di(p ∗) = D∗ i , Si(p ∗) = S∗ i , i = 1, n. (2.5) Formulas (2.3), (2.4) are the definitions of ith good price elasticity of jth good demand and supply respectively. Definition 2.2. A vector p0 ∈ P is called an equilibrium prices vector (an equilibrium) in the model σ if D(p0) = S(p0). (2.6) 3. ON THE UNIQUENESS AND NONUNIQUENESS OF EQUILIBRIUM IN A CLOSED MARKET MODEL Sufficient conditions on equilibrium existence were obtained in [3]. Substitute (2.1), (2.2) into (2.6) and obtain the following system of linear equations: D∗ i n∏ j=1 (p∗j) −Eijp Eij j = S∗ i n∏ j=1 (p∗j) −Ẽijp Ẽij j , i = 1, n. By taking a logarithm of the left-hand and right-hand sides of the last equation we get: n∑ j=1 ( Eij − Ẽij ) ln pj = lnS∗ i − lnD∗ i + n∑ j=1 (Eij − Ẽij) ln p ∗ j , i = 1, n. (3.7) Note that this system is linear by ln pj, j = 1, n. Theorem 3.1. Let the parameters of the model σ satisfy the condition Eij = Ẽij ∀i, j = 1, n. (3.8) Then ∀p ∈ P is an equilibrium in the model σ iff S∗ i = D∗ i ∀i = 1, n. Proof Indeed, let the parameters of the model σ satisfy (3.8). Hence, System (3.7) is equivalent to the following system: lnS∗ i − lnD∗ i = 0, i = 1, n. (3.9) It is obvious that any vector p ∈ P is an equilibrium in the model σ iff S∗ i = D∗ i ∀i = 1, n. Corollary 3.1. Let the parameters of the model σ satisfy (3.8). Then the condition: ∃i = 1, n : S∗ i ̸= D∗ i is a criteria of equilibrium absence in the model σ. Remark 3.1. The condition (3.8) means that the consumers and the producers react to the price changes in the same way. Example 3.1. Let us describe several examples of the market models satisfying the conditions of Lemma 3.1. 1. Consider a market model of electric energy inside the given country. There is only one good – electric energy, and only one producer – a number of state energetic companies which possess the natural monopoly on this good. In the short term the price elasticity of demand equals zero [9]. This is conditioned by the fact that the companies cannot rearrange their technological processes and change the volume of production rapidly. In this model the price elasticity of demand also equals zero in the short term [7]. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) NONUNIQUENESS OF EQUILIBRIUM IN CLOSED MARKET MODEL 187 2. The models of raw material market have zero price elastitcities of both supply and demand [6]. For consumers the cost of raw material has small influence to the goods’ prices. For producers raw material are inelastic for several reasons, such as the load capacity, production factors interchangeability or raw material availability [11]. Now we introduce the following notation: A = (aij)i,j=1,n, aij = Eij − Ẽij; ω = (ω1, ..., ωn), ωi = lnS∗ i − lnD∗ i + n∑ j=1 (Eij − Ẽij) ln p ∗ j . Theorem 3.2. Let the parameters of the model σ satisfy the condition detA ̸= 0. Then there exists a unique equilibrium prices vector p0i = exp(A−1ω)i , i = 1, n (3.10) iff the parameters of the model satisfy the following condition: max i=1,n 2 ln c2i − ln c1i ∣∣∣∣(A−1ω)i − ln c1i + ln c2i 2 ∣∣∣∣ ≤ 1 (3.11) where (A−1ω)i is the ith component of the vector A−1ω. Proof In conditions of Lemma the system (3.7) is consistent by Rouché-Capelli theorem [12, Theorem 2.14] since detA ̸= 0. Moreover, by Theorem [12, Theorem 2.15] its solution is unique. Firstly we prove the necessity. Let p0 defined by (3.10) be an equilibrium in the model σ. Then p0 satisfies (3.7). Moreover, p0 ∈ P , i.e.: ln c1i ≤ (A−1ω)i, (A −1ω)i ≤ ln c2i, i = 1, n, (3.12) where (A−1ω)i is the ith component of the vector A−1ω. Subtract (ln c1i + ln c2i)/2 from the inequalities (3.12): ln c1i − ln c1i + ln c2i 2 ≤ (A−1ω)i − ln c1i + ln c2i 2 , (A−1ω)i − ln c1i + ln c2i 2 ≤ ln c2i − ln c1i + ln c2i 2 , i = 1, n. Thus, − ln c2i − ln c1i 2 ≤ (A−1ω)i − ln c1i + ln c2i 2 ≤ ln c2i − ln c1i 2 , i = 1, n. Hence, we have: ∣∣∣∣(A−1ω)i − ln c1i + ln c2i 2 ∣∣∣∣ ≤ ln c2i − ln c1i 2 , i = 1, n. By dividing this inequality by (ln c2i − ln c1i)/2 > 0 we obtain: 2 ln c2i − ln c1i ∣∣∣∣(A−1ω)i − ln c1i + ln c2i 2 ∣∣∣∣ ≤ 1, i = 1, n. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 188 A. KOTYUKOV, N. PAVLOVA Therefore, if p0 defined by (3.10) is an equilibrium in the model σ, the following condition holds: max i=1,n 2 ln c2i − ln c1i ∣∣∣∣(A−1ω)i − ln c1i + ln c2i 2 ∣∣∣∣ ≤ 1. Now we prove the sufficiency. Let the condition (3.11) be satisfied. Since detA ̸= 0, the matrix A is invertible and, hence, the solution to the system (3.7) is unique and has the form: p = exp (A−1ω). From the condition (3.11) we obtain that p ∈ P . Therefore, there exists a unique equilibrium p0 in the model σ. Corollary 3.2. Let the parameters of the model σ satisfy Lemma 3.2 and the condition S∗ i = D∗ i ∀i = 1, n. Then p0 = p∗. Now we consider the case rangA = rang(A|ω) = k < n. Theorem 3.3. In the model σ there exist an infinite number of equilibrium prices vectors iff the parameters of the model σ satisfy the following conditions: 1) rangA = rang(A|ω) = k < n; 2) max i=1,n 2 ln c2i − ln c1i ∣∣∣∣∣ n−k∑ j=1 Cjxji − ln c1i + ln c2i 2 ∣∣∣∣∣ < 1, (3.13) where xji ∈ R is the ith component of the vector Xj ∈ X , i = 1, n and X = {X1, ..., Xn−k} is the fundamental system of solutions to the system (3.7). Proof First let an infinite number of equilibrium prices vectors exist in the model σ. Since any equilibrium p ∈ P is a solution to the system (3.7), we have rangA = rang(A|ω) = k (since the system (3.7) is consistent) and k < n (since system (3.7) has more than one solution). Let X = {X1, ..., Xn−k}, Xi ∈ Rn (3.14) be the fundamental system of solutions for the system (3.7). Then any equilibrium p (which is the solution to the system (3.7)) can be written in the following form: ln pi = n−k∑ j=1 CjXj (3.15) where Cj ∈ R, j = 1, n− k, are some constants. From the inclusion p ∈ P we get ln c1i ≤ n−k∑ j=1 CjXj, n−k∑ j=1 CjXj ≤ c2i, i = 1, n. By repeating the steps conducted in the proof of Lemma 3.2 we obtain that if the vector p is an equilibrium in the model σ, then max i=1,n 2 ln c2i − ln c1i ∣∣∣∣∣ n−k∑ j=1 Cjxji − ln c1i + ln c2i 2 ∣∣∣∣∣ ≤ 1 Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) NONUNIQUENESS OF EQUILIBRIUM IN CLOSED MARKET MODEL 189 where xji is the ith component of the vector Xj , i = 1, n. From this we easily obtain what we want. Now let conditions 1) and 2) of the Theorem be satisfied. Then by Rouché-Capelli theorem this system is consistent, but its solution is not unique since rangA ̸= n. Using (3.14) we get that any solution p ∈ Rn + to the system (3.7) can be written in the form (3.15). Moreover, from condition 2) inverting the conclusions made in the first part of the proof we obtain that p ∈ P . Hence, any solution to the system (3.7) satisfying condition 2) is an equilibrium in the model σ. To demonstrate these results consider the following example. Example 3.2. Let n = 2, p∗ = (1; 1), c1 = (1; e), c2 = (1/(4e); 1). In Table 3.1 we can see the existence and uniqueness and non-uniqueness for different parameters D∗, S∗, ω. Table 3.1. Existence, uniqueness and non-uniqueness of the solution in the case n = 2, p∗ = (1; 1), c1 = (1; e), c2 = (1/(4e); 1). S∗, D∗, ω∗ (1; 1) (1; 1) (0; 0) (2; 2) (1; 1) (ln 2; ln 2) (1; 1) (2; 2) (− ln 2;− ln 2) (2; 2) (1; 1 2 ) (ln 2; 2 ln 2) A ( 0 0 0 0 ) Theorem 3.1 ∀p∈P Corollary 3.1 ∅ Corollary 3.1 ∅ Corollary 3.1 ∅( 1 0 0 1 ) Corollary 3.2 (1;1) Theorem 3.2 (2;2) Theorem 3.2 (1/2;1/2) Theorem 3.2 (2;4)( 0 1 1 0 ) Corollary 3.2 (1;1) Theorem 3.2 (2;2) Theorem 3.2 (1/2;1/2) Theorem 3.2 (4;2)( 1/2 1/2 1/2 1/2 ) Theorem 3.3 (p1, 1 p1 ), p1∈[1;e] Theorem 3.3 (p1; 4 p1 ),p1∈[1;e] Theorem 3.3 (p1; 1 4p1 ),p1∈[1;e] Corollary 3.1 ∅( 1 0 1/2 1/2 ) Corollary 3.2 (1;1) Theorem 3.2 (2;2) Theorem 3.2 (1/2;1/2) Theorem 3.2 (2;8) Now we consider the case n = 2. The system (3.7) is equivalent to the following system: a11 ln p1 + a12 ln p2 = ω1, a21 ln p1 + a22 ln p2 = ω2, (3.16) which is considered under the conditions ln c11 ≤ ln p1 ≤ ln c21, ln c12 ≤ ln p2 ≤ ln c22. (3.17) Theorem 3.4. Let n = 2. Then: 1. if detA ̸= 0 and ln c11 ≤ a22ω1 − a12ω2 a11a22 − a12a21 ≤ ln c21, ln c12 ≤ a11ω2 − a21ω1 a11a22 − a12a21 ≤ ln c22, then there exists a unique equilibrium( a22ω1 − a12ω2 a11a22 − a12a21 , a11ω2 − a21ω1 a11a22 − a12a21 ) in the model σ; Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 190 A. KOTYUKOV, N. PAVLOVA 2. if rangA = rang(A|ω) = 1 and ai2 = 0 ∀i = {1, 2}, then there exists an infinite number of equilibrium prices vectors in the model σ which belong to the set: ln p1 = ω1 a11 = ω2 a21 , c12 ≤ p2 ≤ c22. 3. if rangA = rang(A|ω) = 1 and ai1 = 0 ∀i = {1, 2}, then there exists an infinite number of equilibrium prices vectors in the model σ which belong to the set: ln p2 = ω1 a12 = ω2 a22 , c11 ≤ p1 ≤ c21. 4. if rangA = rang(A|ω) = 1, a11a12 > 0 and ω1 = a11 ln c11 + a12 ln c12, then there exists a unique equilibrium (c11, c12) in the model σ; 5. if rangA = rang(A|ω) = 1, a11, a12 > 0 and ω1 = a11 ln c21 + a12 ln c22, then there exists a unique equilibrium (c21, c22) in the model σ; 6. if rangA = rang(A|ω) = 1, a11a12 < 0 and ω1 = a11 ln c21 + a12 ln c12, then there exists a unique equilibrium (c21, c12) in the model σ; 7. if rangA = rang(A|ω) = 1, a11a12 < 0 and ω1 = a11 ln c11 + a12 ln c12, then there exists a unique equilibrium (c11, c12) in the model σ; 8. if rangA = rang(A|ω) = 1 and either: • a11a12 > 0 and a11 ln c11 + a12 ln c12 < ω1 < a11 ln c21 + a12 ln c22; or • a11a12 < 0 and a11 ln c21 + a12 ln c12 < ω1 < a11 ln c11 + a12 ln c22; then there exists an infinite number of equilibrium prices vectors in the model σ defined by the formula: ln p2 = ω1 − a11 ln p1 a12 , ln c11 ≤ p1 ≤ ln c21. Proof Consider all the cases consequently. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) NONUNIQUENESS OF EQUILIBRIUM IN CLOSED MARKET MODEL 191 1. Let detA ̸= 0. Then by Rouché-Capelli theorem the system (3.16) is consistent and by Theorem [12, Theorem 2.15] its solution is unique. Let us find this solution using, e.g., Cramer’s rule: A = ∣∣∣∣a11 a12 a21 a22 ∣∣∣∣ = a11a22 − a12a21, A1 = ∣∣∣∣ω1 a12 ω2 a22 ∣∣∣∣ = a22ω1 − a12ω2, A2 = ∣∣∣∣a11 ω1 a21 ω2 ∣∣∣∣ = a11ω2 − a21ω1. Then the solution to this system has the form: ln p1 = A1/A = a22ω1 − a12ω2 a11a22 − a12a21 , ln p2 = A2/A = a11ω2 − a21ω1 a11a22 − a12a21 . This solution must satisfy (3.17). Therefore, if ln c11 ≤ a22ω1 − a12ω2 a11a22 − a12a21 ≤ ln c21, ln c12 ≤ a11ω2 − a21ω1 a11a22 − a12a21 ≤ ln c22, then there exists a unique equilibrium in the model σ. Otherwise an equilibrium in this model does not exist. Now let rangA = rang(A|ω) = 1. 2. In the case ai2 = 0, ∀i ∈ {1, 2} the system (3.16) is equivalent to the following system: a11 ln p1 = ω1, from which it follow that all equilibrium prices vectors has the form: ln p1 = ω1 a11 = ω2 a21 , c12 ≤ p2 ≤ c22. 3. In the case ai1 = 0, ∀i ∈ {1, 2} all equilibrium prices vectors belong to the set: {(p1, p2) ∈ P | ln p2 = ω1 a12 = ω2 a22 , c11 ≤ p1 ≤ c21}. Hereinafter we suppose that aij ̸= 0, i, j = 1, 2. Then all the solutions to the system (3.16) belong to the line: ln p2 = ω1 − a11 ln p1 a12 . (3.18) Now we find the conditions under which an equilibrium is unique even in the case when the system (3.16) has an infinite number of solutions. Note that the angle of the line (3.18) equals −a11/a12. We consider several cases. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 192 A. KOTYUKOV, N. PAVLOVA 4. Let a11a12 > 0. Then an equilibrium in the model σ is unique in two cases: (a) if: ln c12 = −a11 ln c11 a12 + ω1 a12 ; (3.19) (b) if: ln c22 = −a11 ln c21 a12 + ω1 a12 ; (3.20) In the case 4a from (3.19) we obtain that if ω1 = a11 ln c11 + a12 ln c12, then in the model σ there exists a unique equilibrium p0 = (c11, c12). 5. In the case 4b from (3.20) we obtain that if: ω1 = a11 ln c21 + a12 ln c22, then in the model σ there exists a unique equilibrium p0 = (c21, c22). In the case when a11 ln c11 + a12 ln c12 < ω1 < a11 ln c21 + a12 ln c22, (3.21) in the model σ there exist an infinite number of equilibrium prices vectors belonging to the set: ln p2 = ω1 − a11 ln p1 a12 , ln c11 ≤ p1 ≤ ln c21. 6. Let a11a12 < 0. This case is considered similarly to the previous case. Here an equilibrium in the model σ is unique in two cases: (a) if: ln c22 = −a11 ln c11 a12 + ω1 a12 ; (3.22) (b) if: ln c21 = −a11 ln c12 a12 + ω1 a12 ; (3.23) In the case 6a from (3.22) we obtain that if: ω1 = a11 ln c11 + a12 ln c22, then in the model σ there exists a unique equilibrium p0 = (c11, c22). 7. In the case 6b from (3.23) we obtain that if: ω1 = a11 ln c12 + a12 ln c21, then in the model σ there exists a unique equilibrium p0 = (c12, c21). In the case when: a11 ln c12 + a12 ln c21 < ω1 < a11 ln c11 + a12 ln c22, (3.24) in the model σ there exist an infinite number of equilibrium prices vectors belonging to the set: ln p2 = ω1 − a11 ln p1 a12 , ln c11 ≤ p1 ≤ ln c21. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) NONUNIQUENESS OF EQUILIBRIUM IN CLOSED MARKET MODEL 193 8. The last statement is obtained from inequalities (3.21) and (3.24) with the corresponding conditions on the parameters a11, a12. Example 3.3. Consider a market of electric energy in the Russian Federation in 2022. In the table below we can see the data collected from Federal State Statistics Service [8]: Table 3.2. The production and consumption of electric energy in the Russian Federation in 2021–2022 2021 2022 Production, trillion kW*hr 1.100 1.167 Consumption, trillion kW*hr 0.965 1.110 Prices, rubles per kW*hr 4.23 4.78 In terms of the model σ we have: p∗ = 4.78, S∗ = 1.167, D∗ = 1.11. We put c1 = 2.69, c2 = 4.81 as the lowest and the highest mean prices on electric energy in the Russian Federation in 10 years, respectively. Using the economic definition of elasticity we obtain that: E = 1.15, Ẽ = 0.46. Now we check the conditions of Theorem 3.2. Here A = 0.69, ω = 1.1295. From (3.11) we obtain that 2 ln 8.5− ln 2.63 ∣∣∣∣1.12950.69 − ln 8.5 + ln 2.63 2 ∣∣∣∣ = 0.1422 < 1. Therefore, the conditions of Theorem 3.2 are satisfied. Hence, there exists a unique equilibrium in the model σ. ACKNOWLEDGEMENTS Theorem 3.2 is due to N. Pavlova who was supported by the Russian Science Foundation (Project No. 23-11-20020, https://rscf.ru/project/23-11-20020/). Theorems 3.3 and 3.4 are due to A. Kotyukov who was supported by the Russian Science Foundation (Project No. 20-11-20131, https://rscf.ru/project/20-11-20131/). REFERENCES 1. Arrow, K., & Debreu, G. (1954). Existence of an equilibrium for a competitive economy, Econometrica, 22(3). 2. Arutyunov, A. (2014). Coincidence points of two maps, Funct. Anal. Its. Appl., 48, 72– 75. 3. Arutyunov, A., Kotyukov, A. & Pavlova, N. (2021). Equilibrium in Market Models with Known Elasticities, Advances in Systems Science and Applications, 24(4), 130–144. 4. Arutyunov, A., & Zhukovskiy. S. (2010). Existence and properties of inverse mappings, Proc. Steklov Inst. Math., 271, 12–22. 5. Arutyunov, A., Zhukovskiy, S., & Pavlova, N. (2013). Equilibrium price as a coincidence point of two mappings, Comput. Math. Math. Phys., 53(2), 158–169. 6. Bond, M.E., (1987). An Econometric Study of Primary Commodity Exports from Developing Country Regions to the World, Staff Papers (International Monetary Fund), 34(2), 191–227. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 194 A. KOTYUKOV, N. PAVLOVA 7. Burke, P.J., & Abayasekara, A. (2018). The price elasticity of electricity demand in the United States: A three-dimensional analysis, The Energy Journal, 39(2), 123–146. 8. Federal State Statistics Service. (2023, May). Proizvodstvo osnovnih vidov produkcii v natural’nom virajenii [The production of basic commodities in natural equivalent (annual data)]. [in Russian]. [Online]. Available https://gks.ru/free doc/new site/ business/prom/natura/god.htm 9. Knaut, A. & Paulus, S. (2016) Hourly price elasticity pattern of electricity demand in the German day-ahead market, EWI Working Paper, 16/07. 10. Kotyukov, A., & Pavlova, N. (2021). Equilibrium in Market Models, Proc of. 14th International Conference Management of Large-Scale System Development (MLSD) (Moscow, Russia), 1–5. 11. Radetzki, M. (2010, August). Primary Commodities: Historical Perspectives and Prospects. [Online]. Available https://www.imf.org/external/np/seminars/eng/2010/ afrfin/pdf/Radetzki2.pdf 12. Shafarevich, I. & Remizov, A. (2012). Linear Algebra and Geometry. Luxemburg: Springer Science & Business Media. 13. Smith, A. (1776). An Inquiry into the Nature and Cause of the Wealth of Nations. 2 Vols. London, UK: W. Strahan and T. Cadell. 14. Stewart, J. (1767). An Inquiry into the Principles of Political Economy, Vol. 1. London, UK: A. Millar and T. Cadell, Strand. 15. Walras, L. (1874). Elements d’Economie Politique Pure [Elements of Pure Political Economy]. Lausanne, France: Revue de Théologie et de Philosophie et Compterendu des Principales Publications Scientifiques, [in French]. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) https://gks.ru/free_doc/new_site/business/prom/natura/god.htm https://gks.ru/free_doc/new_site/business/prom/natura/god.htm https://www.imf.org/external/np/seminars/eng/2010/afrfin/pdf/Radetzki2.pdf https://www.imf.org/external/np/seminars/eng/2010/afrfin/pdf/Radetzki2.pdf Introduction Problem Statement On the uniqueness and nonuniqueness of equilibrium in a closed market model