Adv Syst Sci Appl 2021; 04:130–144 Published online at https://ijassa.ipu.ru. Equilibrium in Market Models with Known Elasticities Aram Arutyunov1, Alexander Kotyukov2*, Natalia Pavlova1,2,3 1 RUDN University, Moscow, Russia 2 V.A. Trapeznikov Institute of Control Sciences of RAS, Moscow, Russia 3 Moscow Institute of Physics and Technology, Moscow, Russia Abstract: Closed and open market models, in which the supply and demand functions are restored by their price elasticities, are studied. For the closed market model criteria on the existence of equilibrium is obtained as the corollary of existence theorems for the solutions to systems of linear equations and inequalities. The results of the covering maps theory, namely existence theorems of a coincidence point, are applied to obtain sufficient conditions on the existence of an equilibrium in open market model and to develop search algorithm of an equilibrium in this model. Numerical experiments illustrating the obtained results are conducted. Keywords: demand, supply, elasticity, market model, economical equilibrium 1. INTRODUCTION The concept of equilibrium plays an important role in the research of market models. Basically it is a situation on the market in which none of the participants is interested in the change of their state. Let us explain the meaning of equilibrium. Assume that the market consists of the producers and the consumers. The producers tend to maximize their own income by manufacturing some goods. The consumers spend their budget to purchase these goods satisfying their needs. The total volume of goods produced is called the supply. The total volume of goods needed is called the demand. It is obvious that insufficient amount of goods on the market can lead to unfavorable consequences such as hunger, epidemic or even death depending on the market described. Therefore, there must be enough goods on the market to satisfy the needs of the consumers. On the other hand, if some goods produced is not sold to the consumers, the producers suffer an obvious income loss. Hence, they must produce enough goods to satisfy the needs of the consumers, but not more than this required amount. From these considerations we obtain that the amount of goods produced must be equal to the amount of goods needed. In other words, the value of the supply must be equal to the value of the demand. This situation on the market is called an equilibrium. The concept of equilibrium allows us to determine the best prices for the producers and the consumers to set in the modeled region. Indeed, with these prices (which are called equilibrium prices) producers obtain maximal profit since they sell all the goods produced, and consumers obtain the goods needed in the full amount. However, the mathematical approach to this concept remained poor due to the state of the progress of mathematics at the time. ∗Corresponding author: amkotyukov@mail.ru EQUILIBRIUM IN MARKET MODELS WITH KNOWN ELASTICITIES 131 The advancement in the theory of covering mappings and coincidence points (see, e.g., [1], [2]) makes the deep research into the equilibrium theory possible. This mathematical field allows us to make a research on the equilibrium even in the nonlinear market models. For example, in [5]– [7] theorems on the existence of coincidence points for mappings of metric spaces are applied to obtain sufficient conditions on the existence of an equilibrium in the market models. In [3], [4], [7] the mappings of supply and demand are determined as solutions of extremal problems. The supply function can also be obtained as the realization of some production model (see, e.g., [5], [13]). Here we restore the supply and demand function using the concept of elasticitiy. Elasticities are the values that connect supply and demand volumes with parameters obtained from the statistical data (e.g., prices, income, transport costs etc.). That allows us to determine the supply and demand functions that describe real market of some region. Using equilibrium theory, we can improve the economical situation in the region. In this paper, we investigate the existence of equilibrium in open and closed market models. Here, we used price elasticity coefficients of goods presented on the market to obtain the explicit forms of the supply and demand functions. For these functions, we obtained a criteria for the existence of equilibrium in the closed market model. Also for the open market model we obtained sufficient conditions using corollaries of coincidence points theorems. We also developed a search algorithm of an equilibrium for the open market model and conducted a numerical experiment. This algorithm is based on pattern search method and illustrate the effectiveness of the obtained sufficient conditions. This method is chosen due to its simplicity and visibility. The obtained results show that equilibrium is found in the case when the model satisfies the sufficient conditions with moderate precision. 2. MARKET MODELS This section provides sample equations, figures, and tables. 2.1. Closed Market Model Consider a market of n goods with uniform prices set: p = (p1, . . . , pn), pi ∈ [c1i, c2i], where vectors c1, c2 set the natural restrictions on these prices. Suppose that we have n× n matrices E = (Eij)i,j=1,n, Ẽ = (Ẽij)i,j=1,n (Eij, Ẽij ∈ R for all i, j = 1, n, vectors c̄1 = (c11, ..., c1n), c̄2 = (c21, ..., c2n) ∈ Rn + with 0 < c1i < c2i for all i = 1, n, a vector p̄∗ = (p∗1, ..., p ∗ n) ∈ Rn + such that c1i ≤ p∗i ≤ c2i for all i = 1, n, and vectors D̄∗ = (D∗1, ..., D ∗ n), S̄∗ = (S∗1 , ..., S ∗ n) ∈ Rn +. We call σc = (E , Ẽ , D̄∗, S̄∗, p̄∗, c̄1, c̄2) (2.1) a closed market model (the goods presented on the market are manufactured by the producers only). The set of all σc is denoted by Σc. The parameters of the model have the following sense. Elements of E are the price elasticities of demand: Eij = ∂Di ∂pj pj Di , i, j = 1, n. (2.2) Here pi is the price of the ith good, Di = Di(p1, ..., pn) is a total demand on the ith good, Eij is the elasticity of demand on the ith good for the price of the jth good, i, j = 1, n. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 132 A. ARUTYUNOV, A. KOTYUKOV, N. PAVLOVA Elements of Ẽ are the price elasticities of supply: Ẽij = ∂Si ∂pj pj Si , i, j = 1, n. (2.3) Here Si = (Si(p1, ..., pn) is a total supply of the ith good, Ẽij is the elasticity of supply on the ith good for the price of the jth good, i, j = 1, n. Components of the vectors c̄1 and c̄2 generate natural conditions on the prices of the goods: c1i ≤ pi ≤ c2i, i = 1, n. Component of the vector D̄∗ = D̄(p∗1, ..., p ∗ n) is the demand on the corresponding good and component of the vector S̄∗ = S̄(p∗1, ..., p ∗ n) is the supply of the corresponding good for the prices set p̄∗ = (p∗1, ..., p ∗ n). Parameters (E , Ẽ , D̄∗, S̄∗, p̄∗, c̄1, c̄2) uniquely define the function of demand D : [c11; c21]× ...× [c1n; c2n]→ Rn +, D(p1, ..., pn) = (D1(p1, ..., pn), ..., Dn(p1, ..., pn)) (2.4) and the function of supply S : [c11; c21]× ...× [c1n; c2n]→ Rn +, S(p1, ..., pn) = (S1(p1, ..., pn), ..., Sn(p1, ..., pn)) (2.5) Solving the system of partial differential equations (2.2) we obtain the explicit form of the demand function: Di(p1, ..., pn) = D∗i n∏ j=1 (p∗j) −Eijp Eij j . (2.6) Similarly, solving the system of partial differential equations (2.3) we get the explicit form of the supply function: Si(p1, ..., pn) = S∗i n∏ j=1 (p∗j) −Ẽijp Ẽij j . (2.7) Definition 2.1: A vector p̄ ∈ [c11, c21]× ...× [c1n, c2n] is called an equilibrium price vector (an equilibrium), in the model σc if D(p̄) = S(p̄). 2.2. Open Market Model Suppose that we have n× n matrices E = (Eij)i,j=1,n, Ẽ = (Ẽij)i,j=1,n (Eij, Ẽij ∈ R for all i, j = 1, n, vectors c̄1 = (c11, ..., c1n), c̄2 = (c21, ..., c2n) ∈ Rn + with 0 < c1i < c2i for all i = 1, n, a vector p̄∗ = (p∗1, ..., p ∗ n) ∈ Rn + such that c1i ≤ p∗i ≤ c2i for all i = 1, n, and vectors D̄∗ = (D∗1, ..., D ∗ n), S̄∗ = (S∗1 , ..., S ∗ n) ∈ Rn +. Moreover, we suppose that a vector ā = (a1, ..., an) ∈ Rn +, such that there exists at least one number i = 1, n : ai > 0, is given. We call σo = (E , Ẽ , D̄∗, S̄∗, p̄∗, c̄1, c̄2, ā) (2.8) an open market model (at least one kind of the goods is imported externally). The set of all σo is denoted by Σo. The parameters E , Ẽ , D̄∗, S̄∗, p̄∗, c̄1, c̄2 of the model (2.8) have the same sense as the ones for the model (2.1). The component of vector āwith the number i = 1, n equals to the volume of imported ith good. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) EQUILIBRIUM IN MARKET MODELS WITH KNOWN ELASTICITIES 133 The parameters (E , Ẽ , D̄∗, S̄∗, p̄∗, c̄1, c̄2, ā) uniquely define the demand function (2.4) and supply function (2.5). These functions are defined by (2.6) and (2.7) respectively. Definition 2.2: A vector p̄ ∈ [c11, c21]× ...× [c1n, c2n] is called an equilibrium price vector (an equilibrium), in the model σo if D(p̄) = S(p̄) + ā. 3. EQUILIBRIUM IN MARKET MODELS The following theorems (criteria on the existense of an equilibrium) are extenstions of the results obtained in [8] for the closed market model. Theorem 3.1: In closed market model σC ∈ ΣC there exists an equilibrium vector iff for any vectors λ̄ = (λ1, . . . , λn) ∈ Rn + and x̄ = (x1, . . . , xn) ∈ Rn the following conditions hold: n∑ i,j=1 λiaijxj + n∑ i=1 λibi + n∑ i=1 λixi = 0, n∑ i=1 λi ln c1i c2i ≤ 0 with aij = Ẽij − Eij, i, j = 1, n, bi = ln D∗i S∗i + n∑ k=1 (Ẽik − Eik) ln p∗k, i = 1, n. Proof follows from Alexandrov theorem and Fan Ky theorem on the consistency of linear inequality system (see, e.g., [9]) and the fact that for closed market model σc ∈ Σc the condition on the existence of an equilibrium is equivalent to the condition on the existence of the solution to the following system of equations (which is linear by ln pj, j = 1, n): n∑ j=1 (Ẽij − Eij) ln pj = ln D∗i S∗i + n∑ j=1 (Ẽij − Eij) ln p∗j , i = 1, n, satisfying inequalities: c1j ≤ pj ≤ c2j ∀j = 1, n. The following Theorem is another criteria on the existence of equilibrium. Its conditions (unlike Theorem 3.1) can be easily verified. To do that, one must simply calculate the finite number of determinants. Theorem 3.2: For closed market model σC ∈ ΣC there exists an equilibrium vector iff there exist such natural numbers ik ≤ n, jl ≤ n, k, l = 1, r with r = rang ( Ẽ − E ) and square matrix A = {aikjl}, aikjl = Ẽikjl − Eikjl with rang r such that the following conditions hold: Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 134 A. ARUTYUNOV, A. KOTYUKOV, N. PAVLOVA detA 6= 0, det  A bi1 ... bir aij1 · · · aijr bi  = 0 ∀i = 1, n, det  A bi1 ... bir δij1 · · · δijr ln c1i  ≤ 0, det  A bi1 ... bir δij1 · · · δijr ln c2i  ≥ 0 ∀i = 1, n with bi = ln D∗i S∗i + n∑ k=1 (Ẽik − Eik) ln p∗k, i = 1, n. and δij is Kronecker delta. Proof The vector p̄ = (p1, ..., pn) is an equilibrium prices vector in model σc iff it is the solution to the following system of equations and inequalities n∑ j=1 (Ẽij − Eij) ln pj = ln D∗i S∗i + n∑ j=1 (Ẽij − Eij) ln p∗j , pi ≥ c1i, pi ≤ c2i, i = 1, n. (3.9) The system (3.9) is consistent iff the following system is consistent n∑ j=1 aijxi = bi, xi ≥ C1i, xi ≤ C2i, i = 1, n. (3.10) Here C1i = ln c1i, i = 1, n;C2i = ln c2i, i = 1, n. Applying Theorem 1.3 from [10] to the system of linear equations and inequalities (3.10) we obtain the conditions of the Theorem. 3.1. Equilibrium in Open Market Models Let us introduce the following notation: ᾱ(σ) = [ max i=1,n (( S∗i n∏ j=1 (p∗j) −Ẽij min { c |Ẽij | 1j , c −|Ẽij | 2j })−1 × n∑ k=1 c2k − c1k 2 c2k|Ẽ−1ki | )]−1 , β̄(σ) = max i=1,n (( D∗i n∏ j=1 (p∗j) −Eij max { c |Eij | 2j , c −|Eij | 1j }) × n∑ k=1 ck2 − c1k 2c1k |Eik| ) , Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) EQUILIBRIUM IN MARKET MODELS WITH KNOWN ELASTICITIES 135 γ̄(σ) = max i=1,n |Si(c̃) + ai −Di(c̃)| , where E−1ij is the element of the inverse matrix E−1 to Ẽ , c̃ = c1+c2 2 . Theorem 3.3: Let the parameters of the open market model σo ∈ Σo satisfy the conditions: • β̄(σ) < ᾱ(σ); • γ̄(σ) < ᾱ(σ)− β̄(σ). Then there exists an equilibrium prices vector p̄ ∈ Rn + with c̄1 ≤ p̄ ≤ c̄2. To prove this Theorem, we need the following definitions and results from the theory of covering maps. Let us formulate them. Let (X, ρX) and (Y, ρY ) be the metric spaces and Ψ and Φ be the maps from X to Y . By BX(x, r) denote a closed ball with the center at point x and radius r in the space X . Analogously we define BY (x, r). Definition 3.1: (see [11]). A map Ψ : X → Y is called α-covering if Ψ(BX(x, r)) ⊇ BY (Ψ(x), αr)∀x ∈ X, ∀r > 0. Definition 3.2: (see [11]). A map Ψ is called metrically κ-regular if ∀x0 ∈ X, y ∈ Y ∃ ∈ X : Ψ(x) = y and ρX(x, x0) ≤ κρY (y,Ψ(x0)). Proposition 3.1: A map Ψ is α-covering iff Ψ is 1/α-regular. Note that the maps Ψ and Φ are, obviously, surjective. It is easy to show that from the properties of metric regularity we can obtain the following Proposition. Proposition 3.2: Let σo ∈ Σo, S,D : Rn + → Rn be such that S is α-covering and D is Lipschitz continuous with Lipschitz constant β, ∃p∗, p∗∗ ∈ Rn + : S(p∗) = D(p∗), S(p∗∗) = D(p∗). If S(p∗) = S(p∗∗), then p∗ = p∗∗. Proof Indeed, if S is α-covering, then S is 1/α-regular and ρX(p∗, p∗∗) ≤ 1 α ρY (S(p∗), S(p∗∗)) = 0, which leads to p∗ = p∗∗. Theorem 3.4: (see Theorem 1 from [11]) Let the space X be complete, x0 ∈ X,α > 0, R > 0. Let the map Ψ : X → Y be closed and α-covering on BX(x0, R). Then for any nonnegative β < α and any map Φ : BX(x0, R)→ Y satisfying Lipschitz condition with the constant β and such that ρY (Ψ(x0),Φ(x0)) ≤ (α− β)R, there exists a coincidence point ξ ∈ X for the maps Ψ,Φ, i.e., Ψ(ξ) = Φ(ξ), such that ρY (x0, ξ) ≤ ρY (Ψ(x0),Φ(x0)) α− β . Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 136 A. ARUTYUNOV, A. KOTYUKOV, N. PAVLOVA Proof Proof of Theorem 3.3. Introduce the following notation. By cov(S|M) denote the supremum of all α > 0 such that S is α-covering on M . By lip(D|M) denote the infimum of all β ≥ 0 such that D satisfies Lipschitz condition with a constant β. Then lip(D|M) = sup p∈intM ∥∥∥∥∂D∂p (p) ∥∥∥∥ . In spaces Rn +,Rn define the norms by ‖x‖1 = 2 max j=1,n |xj| c2j − c1j ∀x = (x1, ..., xn) ∈ Rn, ‖y‖2 = max j=1,n |yj|∀y = (y1, ..., yn) ∈ Rn. Let X = Rn +, Y = Rn. Put M = BX(c̃, 1). It is obvious that M = [c11, c21]× ...× [c1n, c2n]. Consider the metric spaces (X, ρY ) and (Y, ρY ) with the metrics ρX , ρY defined by the norms ‖ · ‖1, ‖ · ‖2 correspondingly. It is obvious that Rn + is not complete, but we only need that BX(c̃, 1) is complete. Let us estimate lip(D|M). To do that, estimate ∥∥∥∂D ∂p (p) ∥∥∥ first. From (2.2) it follows that∥∥∥∥∂D∂p (p) ∥∥∥∥ = D∗iEik pk n∏ j=1 (p∗j) −Eijp Eij j Therefore, ∥∥∥∥dDdp ∥∥∥∥ = max ‖x‖=1 ∥∥∥∥dDdp x ∥∥∥∥ = max ‖x‖=1 max i=1,n n∑ k=1 ∣∣∣∣∂Di ∂pk xk ∣∣∣∣ ≤ ≤ max ‖x‖=1 max i=1,n n∑ k=1 D∗i |Eik| pk |xk| n∏ j=1 (p∗j) −Eijp Eij j ≤ ≤ max i=1,n ( n∏ j=1 (p∗j) −Eijp Eij j D∗i ) n∑ k=1 c2k − c1k 2pk |Eik| ≤ ≤ max i=1,n D∗i ( n∏ j=1 (p∗j) −Eij max { c Eij 2j , c Eij 1j }) × × n∑ k=1 ck2 − c1k 2c1k |Eik| = β̄(σ). Now we estimate cov(S|M). According to Proposition 3.1, if the map S is α-covering, the inverse map S−1 is 1/α-Lipschitz continuous. We obtain the estimate using this proposition. Firstly we find (∂S/∂p)−1. By (2.3) we have ∂Si ∂pk (p) = ẼikS ∗ i pk n∏ j=1 (p∗j) −Ẽijp Ẽij j Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) EQUILIBRIUM IN MARKET MODELS WITH KNOWN ELASTICITIES 137 Therefore, det ∂S(p) ∂p = n∏ i=1 ( S∗i p −1 i n∏ j=1 (p∗j) −Ẽijp Ẽij j ) det Ẽ . By Sik, Ẽik denote a cofactor to element ∂S/∂pk, Ẽik of ∂S/∂p, Ẽ correspondingly. Thus: Sik =  n∏ m=1 m 6=k p−1m  n∏ l=1 l 6=i ( S∗l n∏ j=1 (p∗j) −Ẽljp Ẽlj j ) Ẽik. Hence, the element of inverse matrix (∂S/∂p)−1:( ∂S(p) ∂p )−1 ki = Ẽ−1ki p−1k S∗i n∏ j=1 (p∗j) −Ẽijp Ẽij j , where Ẽ−1ki is the element of inverse matrix to Ẽ . Now we estimate the Lipschitz constant of (∂D/∂p)−1:∥∥∥∥∥ ( ∂S(p) ∂p )−1∥∥∥∥∥ = max ‖x‖1=1 max i=1,n n∑ k=1 ∥∥∥∥∥ ( ∂Si(p) ∂pk )−1∥∥∥∥∥ ≤ ≤ max ‖x‖1=1 max i=1,n n∑ k=1 pk ∣∣∣Ẽ−1ki xk ∣∣∣ S∗i n∏ j=1 (p∗j) −Ẽij min { c Ẽij 1j , c Ẽij 2j } ≤ ≤ max i=1,n ( S∗i n∏ j=1 (p∗j) −Ẽij min { c Ẽij 1j , c Ẽij 2j })−1 × × n∑ k=1 ∣∣∣Ẽ−1ki ∣∣∣ c2k − c1k 2 c2k = 1 ᾱ(σ) . By the conditions of Theorem and inequalities cov(S|M) ≥ ᾱ(σ), lip(D|M) ≤ β̄(σ) there exist positive numbers α, β such that β̄(σ) < β < α < ᾱ(σ), S is α-covering on M, D is β-Lipschitz continuous on M. Since ρY (S(c̃+ a,D(c̃)) = γ̄(σ), from Condition 2 of the Theorem it follows that ρY (S(c̃+ a,D(c̃)) < α− β. Therefore, there exists a vector p̄ ∈ X such that D(p̄) = S(p̄) + a and ρY (p̄, c̃) ≤ ρY (S(c̃) + a,D(c̃)) α− β . From the last inequality it follows that p ∈ intM , since M = BX(c̃, 1) and ρy(S(c̃) + a,D(c̃)) < α− β. Hence, c1j < pj < c2j, j = 1, n. Theorem 3.5: Let the following conditions be satisfied for the model σo ∈ Σo: Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 138 A. ARUTYUNOV, A. KOTYUKOV, N. PAVLOVA • ᾱ(σ) > 2β̄(σ); • ∃p̃ = (p̃1, . . . , p̃n) ∈ Rn +: S(p̃1, . . . , p̃n) = D(p∗1, . . . , p ∗ n), where (p∗1, . . . , p ∗ n) is an equilibrium in model σo. Then in the model σo there exists an equilibrium prices vector not equal (p∗1, . . . , p ∗ n). Proof follows from Theorem 1 in [12]. The results similar to Theorems 3.3–3.5 for dynamic market models may be obtained, applying the results of [13]– [16] besides existence theorems for coincidence points. In such models the supply and demand functions depend not only on prices on the goods, but on price change rates. In turn, the question of the existence for an equilibrium can be considered as the question of existence for the solution of a system of differential equations. 4. ITERATION PROCESS Define the norms in the spaces Rn + and Rn: ‖x‖X = 2 max j=1,n |xj| c2j − c1j ∀x = (x1, ..., xn) ∈ Rn, ‖y‖Y = max j=1,n |yj| ∀y = (y1, ..., yn) ∈ Rn, and consider metric spaces (Rn +, ρX), (Rn, ρY ) with the metrics ρX and ρY generated by the norms ‖ · ‖X and ‖ · ‖Y correspondingly. Fix an arbitrary δ > 0. From Definition (3.1) we obtain that ρY (S(x′), D(x)) ≤ δρY (S(x), D(x)), (4.11) ρX(x′, x) ≤ α−1ρY (S(x), D(x)). (4.12) Based on these inequalities we can construct the following iteration process. Let S,D : X → Y , S be α-covering and D be Lipschitz continuous with the constant β. Fix an arbitrary p0 ∈ X and a sequence of nonnegative numbers {δi}: β + α lim i→∞ δi < α. (4.13) Then there exists a sequence {pi}, i = 0, 1, ... such that ρY (S(pi+1), D(pi)) ≤ δiρY (S(pi), D(pi)), (4.14) ρX(pi+1, pi) ≤ α−1ρY (S(pi), D(pi)). (4.15) Theorem 4.1 (A. Arutyunov, see [1]): Let the space X be complete, S : X → Y be α-covering, gphS = {(x, y) ∈ X × Y : y = S(x)} be closed and D : X → Y satisfy Lipschitz condition with a constant β < α and, moreover, β + α lim i→∞ δi < α. (4.16) Then ∀p0 ∈ X there exists a sequence {pi} satisfying (4.14), (4.15) ∀i, and any such sequence converges to some coincidence point ξ = ξ(x0) with ρX(ξ, p0) ≤ α−1 ( 1 + ∞∑ j=1 i∏ j=1 ( δi + β α )) ρY (S(p0), D(p0)). Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) EQUILIBRIUM IN MARKET MODELS WITH KNOWN ELASTICITIES 139 We construct search algorithm on the base of iteration process (4.14), (4.15) and Theorem 4.1. The search of a new point on every step of this iteration process is based on direct search method. Namely, we search the new point of the iteration process in the form: pi+1 = pi + h = (p1i + h1, ..., pni + hn), (4.17) where h = (h1, ..., hn) is the step that needs to be defined on every iteration. The search radius, i.e., the maximal value of every coordinate of the step, can be found using (4.15). Indeed, from (4.15) we obtain that: max i=1,n |hi| ci1 − ci2 6 α−1 2 max i=1,n |Si(p)−Di(p)|, where S(p) = (S1(p), ..., Sn(p)), D(p) = (D1(p), ..., Dn(p)). Then: |hi| ci1 − ci2 6 α−1 2 max i=1,n |Si(p)−Di(p)|. Hence we obtain the estimate for the search radius: |hi| 6 ci1 − ci2 2α max i=1,n |Si(p)−Di(p)|. (4.18) Next we must define the elements of the sequence {δi}. From (4.16) it follows that: lim i→∞ δi < 1− β α . (4.19) Here we put: δi = δ = 1− β α − ε, (4.20) where ε > 0 is an arbitrary small positive number (process error). Constructed sequence obviously satisfies (4.19). Now we describe the step of the iteration process. Let pi be the point obtained on ith iteration. Define the search radius by considering an equality in (4.18) and put p̃i+1 = (p1i + h1, p2i, ..., pni), i.e., take the initial point with ¡¡incremented¿¿ first coordinate. For this point we check the condition (4.14). If it is satisfied, we put pi+1 = p̃i+1 but we continue the search process by decreasing h1. If we find another point satisfying (4.14), we compare it with the previous one by the value at ρ2(S(·), D(pi)) and take the one with the minimal value. Then we decrease h1 and continue the search while h1 > ε. After we complete the search by the first coordinate in positive direction we put p̃i+1 = (p1i − h1, p2i, ..., pni), and conduct this search for this point. Once we finish this search (i.e. h1 < ε), we ¡¡increment¿¿ the second coordinate and conduct this search for it etc. When we complete the search for the last coordinate we finish the iteration and obtain the point pi+1. Numerical experiments show that on the first step of the iteration process δi chosen above can violate this process. Condition (4.19) allows us to redefine δi to satisfy (4.14), from which we obtain that max p∈Uh(pi) ρY (S(p), D(pi)) ≤ δiρY (S(pi), D(pi)), Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 140 A. ARUTYUNOV, A. KOTYUKOV, N. PAVLOVA where Uh(pi) = {x ∈ Rn + : |pij − xj| ≤ hj, j = 1, n}. From the last inequality we get the lower estimate for δi: δi ≥ (ρY (S(pk), D(pk)))−1 max p∈Uh(pk) ρY (S(p), D(pk)). It is possible to define δi as shown above on each iteration but this can significantly decrease the iteration process. 5. CONCLUSION The open and closed market models are studied. The existence of an equilibrium is investigated. For the closed market model we obtained necessary and sufficient conditions for the existence of an equilibrium price vector. To do that, we used corollaries of consistence theorems for the systems of linear equations and inequalities. These conditions can be easily verified numerically. For the open market model we obtained sufficient existence conditions. To do that, we used corollaries of the theorems on coincidence points for covering and Lipschitz continuous mappings. These conditions can be also verified numerically. Also we obtained the conditions for the uniqueness and nonuniqueness of equilibrium in the open and closed market models. They are easily obtained as corollaries from coincidence points theorems. All these results can be applied to investigate the power of the set of equilibrium price vectors and to find them numerically. ACKNOWLEDGEMENTS This paper was written with the financial support of RFBR (grant no. 19-01-00080, 20- 01-00610). Theorem 3.3 is obtained by the first author under financial support of Russian Science Foundation (Project No. 20-11-20131). REFERENCES 1. Arutyunov, A.V. (2014) Coincidence points of two maps, Funct. Anal. Its. Appl., 48, 72–75. 2. Arutyunov, A.V., Zhukovskiy, S.E. 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N um er ic al re su lts (n = 1) . a E Ẽ c 1 c 2 D ∗ S ∗ p ∗ E xa ct so lu tio n A pp ro xi m at e so lu tio n D iff er en ce 12 ,0 49 67 25 7 0, 49 84 97 05 0, 31 75 12 80 3, 42 51 30 70 3, 90 65 44 50 52 ,3 29 66 27 0 39 ,7 80 66 12 0 3, 66 58 37 60 3, 53 06 37 18 3, 53 07 15 53 0, 00 00 78 35 23 ,9 46 10 55 9 0, 37 99 69 09 0, 87 86 58 00 1, 08 46 55 00 2, 26 08 07 60 62 ,5 57 06 23 0 42 ,4 19 23 65 0 1, 67 27 31 30 1, 10 99 18 76 1, 11 00 59 81 0, 00 01 41 05 75 ,3 90 41 16 9 0, 61 29 43 72 0, 94 26 46 90 2, 00 81 69 40 2, 23 83 04 10 82 ,0 56 52 56 0 7, 06 13 41 90 2, 12 32 36 75 2, 14 25 03 95 2, 14 24 84 99 0, 00 00 18 96 3, 46 47 92 08 0, 84 91 81 28 0, 98 56 63 00 1, 17 15 36 60 1, 86 12 72 60 14 ,1 80 57 21 0 10 ,3 13 52 89 0 1, 51 64 04 60 1, 21 93 94 22 1, 21 96 33 61 0, 00 02 39 39 12 ,0 49 67 25 4 0, 49 84 97 05 1, 31 75 12 80 3, 42 51 30 70 3, 90 65 44 50 52 ,3 29 66 27 0 39 ,7 80 66 12 0 3, 66 58 37 60 3, 73 46 44 90 3, 73 46 49 99 0, 00 00 05 09 45 ,7 59 60 18 8 0, 21 80 86 95 1, 36 43 36 50 1, 68 36 60 00 3, 10 19 71 30 83 ,7 68 17 07 0 36 ,6 72 68 59 0 2, 39 28 15 65 2, 49 14 02 47 2, 49 13 66 71 0, 00 00 35 76 57 ,9 20 58 65 8 0, 29 80 26 99 1, 39 38 11 70 5, 47 84 52 30 6, 84 55 39 20 98 ,1 63 66 61 0 37 ,7 83 17 68 0 6, 16 19 95 75 6, 76 01 78 29 6, 76 01 31 04 0, 00 00 47 25 52 ,0 17 25 02 3 1, 78 50 43 26 0, 74 30 21 60 1, 31 06 19 30 1, 41 11 85 60 83 ,4 28 42 96 0 27 ,7 29 30 28 0 4, 00 61 36 80 1, 32 13 04 17 1, 32 13 83 81 0, 00 00 19 28 45 ,8 26 52 17 4 1, 48 40 95 65 0, 97 46 88 80 3, 82 62 40 60 4, 18 60 33 00 61 ,0 37 24 68 0 15 ,9 49 39 75 0 2, 94 41 30 05 4, 04 54 58 93 4, 04 54 39 65 0, 00 00 40 24 25 ,4 09 51 76 1 1, 11 90 65 93 0, 25 28 51 10 2, 84 52 80 20 3, 04 29 79 90 97 ,1 41 77 20 0 74 ,4 14 36 64 0 4, 27 68 34 30 3, 03 15 88 98 3, 03 15 48 74 0, 00 00 11 91 24 ,6 69 58 65 5 1, 35 44 84 68 0, 81 66 33 60 3, 81 63 91 70 4, 73 72 76 90 80 ,8 88 50 35 0 61 ,8 26 36 12 0 7, 17 07 06 85 4, 66 85 27 86 4, 66 85 15 95 0, 00 03 75 21 8, 24 24 59 60 1, 74 15 68 54 1, 94 87 13 80 1, 51 48 66 20 1, 92 54 00 30 10 ,9 46 97 02 0 2, 62 02 12 60 6, 90 77 53 25 1, 70 97 24 80 1, 70 97 35 03 0, 00 00 03 48 59 ,3 89 03 65 3 1, 97 22 58 93 1, 92 99 66 70 6, 59 08 41 10 7, 22 46 65 40 76 ,1 39 83 21 0 20 ,3 01 44 89 0 5, 10 37 47 60 7, 12 53 61 38 7, 12 53 57 90 0, 00 03 58 18 25 ,9 81 30 49 0 1, 06 15 89 55 1, 96 27 97 30 4, 74 29 31 00 5, 46 45 64 20 45 ,8 81 08 57 0 20 ,3 65 32 23 0 3, 88 47 47 45 5, 41 41 73 42 5, 41 38 15 24 0, 00 02 70 41 4, 17 41 79 59 1, 18 72 77 26 1, 12 76 44 60 3, 24 92 54 90 4, 52 02 40 00 86 ,8 19 87 70 0 82 ,2 83 27 37 0 0, 75 43 47 15 3, 74 61 59 23 3, 74 64 29 64 0, 00 06 71 30 4, 86 89 45 08 0, 45 99 18 70 -0 ,6 65 85 27 0 7, 67 71 90 70 8, 79 74 88 20 9, 24 30 15 45 4, 38 45 89 11 8, 23 73 39 45 8, 24 94 31 89 8, 24 94 31 35 0, 00 00 00 54 8, 49 60 13 14 1, 80 79 54 21 -1 ,4 85 82 48 0 1, 25 34 82 80 1, 37 77 86 50 8, 12 00 03 01 0, 12 08 65 98 1, 31 56 34 65 1, 35 90 74 73 1, 35 90 74 84 0, 00 00 00 11 0, 35 32 40 02 2, 73 43 24 05 -2 ,7 00 23 78 0 0, 27 42 14 80 8, 02 55 73 70 0, 60 35 63 16 4, 57 84 65 68 4, 14 98 94 25 6, 26 31 73 87 6, 26 31 73 74 0, 00 00 00 13 4, 29 33 12 68 3, 94 83 31 92 -3 ,4 95 45 33 0 6, 40 13 22 30 7, 26 92 56 00 9, 98 52 49 81 4, 81 04 50 34 6, 83 52 89 15 6, 72 75 93 15 6, 72 75 93 30 0, 00 00 00 15 Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) EQUILIBRIUM IN MARKET MODELS WITH KNOWN ELASTICITIES 143 Ta bl e 5. 2. N um er ic al re su lts (n = 2) . a E Ẽ c 1 c 2 D ∗ S ∗ p ∗ E xa ct so lu tio n A pp ro xi m at e so lu tio n D iff er en ce (1 0 − 4 ) 5, 44 92 46 34 0. 18 90 0. 88 20 0. 78 92 0. 20 51 6, 15 66 38 00 11 ,8 27 80 96 0 8, 12 07 83 80 0, 97 49 50 70 8, 99 22 23 80 7, 45 51 09 45 7, 45 50 90 20 0, 00 00 19 25 1, 50 26 96 78 0. 69 93 0. 96 98 0. 03 57 0. 86 06 3, 73 11 10 20 13 ,4 09 70 39 0 7, 41 03 93 50 4, 25 43 88 20 8, 57 04 07 05 6, 24 36 31 97 6, 24 36 31 97 6, 08 64 05 69 2. 27 65 1. 37 60 0. 81 56 0. 24 02 0, 77 74 84 50 0, 93 91 76 60 6, 40 65 20 17 6, 25 21 56 23 0, 85 83 30 55 0, 93 91 72 82 0, 93 89 80 26 0, 00 02 71 11 5, 29 11 89 33 4. 52 51 7. 37 93 0. 03 05 0. 97 65 5, 38 25 48 20 5, 45 60 87 70 3, 79 43 71 35 8, 70 64 45 23 5, 41 93 17 95 5, 45 60 70 10 5, 45 58 79 26 7, 84 59 66 91 6. 13 38 8. 32 64 0. 91 73 0. 31 66 7, 42 23 43 60 7, 50 77 17 10 8, 47 62 63 15 6, 92 24 53 70 7, 46 50 30 35 7, 50 77 10 74 7, 50 77 01 23 0, 00 00 14 74 2, 77 77 15 60 9. 25 62 6. 30 13 0. 66 89 0. 69 81 2, 18 74 84 60 2, 34 25 98 40 2, 37 61 39 70 5, 24 03 01 45 2, 26 50 41 50 2, 34 25 80 83 2, 34 25 69 57 0, 93 75 43 08 6. 22 04 4. 73 21 0. 98 47 0. 16 37 7, 34 77 19 70 7, 45 62 99 40 7, 56 45 49 29 9, 52 37 66 09 7, 40 20 09 55 7, 45 62 72 36 7, 45 61 14 10 0, 00 01 98 82 3, 03 70 09 92 8. 44 00 5. 62 22 0. 31 53 0. 70 35 4, 77 91 20 10 4, 87 65 29 30 5, 34 61 82 34 9, 61 66 58 66 4, 82 78 24 70 4, 87 64 83 42 4, 87 63 63 07 32 ,6 44 72 26 0 89 .9 77 7 22 .7 80 0 9. 67 36 6. 45 68 87 ,4 42 05 80 0 87 ,4 83 79 64 0 36 ,5 58 47 46 8 81 ,6 56 30 91 9 87 ,4 62 92 72 0 87 ,4 83 79 04 5 87 ,4 83 67 33 1 0, 00 02 81 63 1, 97 42 33 17 43 .3 59 5 23 .2 55 9 6. 97 41 6. 63 99 4, 94 66 53 80 5, 03 38 54 10 51 ,9 18 10 77 8 80 ,3 05 65 52 0 4, 99 02 53 95 5, 03 38 53 05 5, 03 35 96 94 6, 62 85 81 47 83 .4 10 5 37 .7 34 4 5. 96 30 0. 33 56 94 ,7 63 27 61 0 94 ,8 75 13 01 0 40 ,5 95 97 35 9 81 ,5 09 84 57 6 94 ,8 19 20 31 0 94 ,8 75 07 15 0 94 ,8 75 04 71 1 0, 00 00 70 92 75 ,8 26 74 31 4 40 .7 93 5 73 .9 58 4 0. 98 47 8. 39 60 44 ,1 84 89 15 0 44 ,3 11 03 04 0 99 ,7 88 15 74 4 93 ,8 29 97 67 7 44 ,2 47 96 09 5 44 ,3 11 00 34 0 44 ,3 10 93 68 1 0, 93 75 43 08 6. 22 04 4. 73 21 0. 98 47 0. 16 37 7, 34 77 19 70 7, 45 62 99 40 7, 56 45 49 29 9, 52 37 66 09 7, 40 20 09 55 7, 45 62 72 36 7, 45 60 67 59 0, 00 02 61 15 3, 03 70 09 92 8. 44 00 5. 62 22 0. 31 53 0. 70 35 4, 77 91 20 10 4, 87 65 29 30 5, 34 61 82 34 9, 61 66 58 66 4, 82 78 24 70 4, 87 64 83 42 4, 87 63 21 34 2, 60 77 08 48 4. 01 64 9. 47 26 0. 31 68 0. 42 43 7, 55 57 16 60 7, 56 82 48 90 6, 04 56 77 02 9, 75 81 69 92 7, 56 19 82 75 7, 56 82 45 33 7, 56 82 12 05 0, 00 00 66 78 5, 41 95 22 09 3. 87 16 1. 42 76 0. 79 69 0. 63 16 5, 03 93 81 00 5, 06 07 73 20 5, 25 34 99 15 8, 17 91 71 63 5, 05 00 77 10 5, 06 07 65 88 5, 06 07 07 98 7, 58 08 78 11 2. 81 64 1. 45 88 0. 71 91 0. 22 19 3, 12 70 26 60 3, 32 65 84 70 9, 14 18 50 23 9, 46 44 01 26 3, 22 68 05 65 3, 32 65 67 08 3, 32 65 84 67 0, 00 00 61 68 0, 60 92 50 99 4. 16 22 6. 04 16 0. 29 73 0. 68 45 9, 16 04 05 30 9, 23 57 00 10 7, 33 79 82 84 8, 85 95 19 59 9, 19 80 52 70 9, 23 56 87 40 9, 23 56 28 28 0, 37 63 45 16 0. 96 67 0. 57 79 -0 .5 49 2 -0 .1 57 6 5, 03 02 07 50 5, 30 80 02 40 9, 06 97 82 03 8, 77 30 20 14 5, 16 91 04 95 5, 09 26 56 85 5, 08 56 82 69 0, 00 69 74 16 0, 37 63 45 16 0. 45 16 0. 22 16 -0 .8 89 3 -0 .6 16 6 5, 03 02 07 50 5, 30 80 02 40 9, 06 97 82 03 8, 77 30 20 14 5, 16 91 04 95 5, 20 89 78 41 5, 30 17 36 41 0, 37 63 45 16 0. 96 67 0. 57 79 -0 .5 49 2 -0 .1 57 6 5, 03 02 07 50 5, 30 80 02 40 9, 06 97 82 03 8, 77 30 20 14 5, 16 91 04 95 5, 08 54 88 70 5, 08 56 82 69 0, 00 01 93 99 0, 37 63 45 16 0. 45 16 0. 22 16 -0 .8 89 3 -0 .6 16 6 5, 03 02 07 50 5, 30 80 02 40 9, 06 97 82 03 8, 77 30 20 14 5, 16 91 04 95 5, 30 18 90 90 5, 30 17 36 41 5, 34 46 57 76 0. 08 06 0. 91 13 -0 .2 61 3 -0 .7 45 8 5, 89 70 07 00 6, 01 06 39 50 9, 77 38 45 69 4, 51 11 80 83 5, 95 38 23 25 5, 98 00 72 40 5, 98 01 63 42 0, 00 00 91 02 5, 34 46 57 76 0. 41 83 0. 21 31 -0 .7 01 8 -0 .2 88 8 5, 89 70 07 00 6, 01 06 39 50 9, 77 38 45 69 4, 51 11 80 83 5, 95 38 23 25 6, 01 06 39 50 6, 01 06 39 48 4, 43 20 44 23 0. 32 71 0. 38 22 -0 .9 38 8 -0 .0 42 8 7, 05 48 84 90 7, 14 02 09 00 5, 94 09 92 36 1, 49 78 93 01 7, 09 75 46 95 7, 07 48 50 70 7, 07 48 52 27 0, 00 00 01 57 4, 43 20 44 23 0. 05 78 0. 65 83 -0 .0 55 6 -0 .6 05 9 7, 05 48 84 90 7, 14 02 09 00 5, 94 09 92 36 1, 49 78 93 01 7, 09 75 46 95 7, 09 22 56 90 7, 09 22 73 83 Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 144 A. ARUTYUNOV, A. KOTYUKOV, N. PAVLOVA Ta bl e 5. 3. N um er ic al re su lts (n = 3) . a E Ẽ c 1 c 2 D ∗ S ∗ p ∗ E xa ct so lu tio n A pp ro xi m at e so lu tio n D iff er en ce 0, 00 06 71 64 -0 ,5 23 1 0, 21 73 0, 96 89 0, 55 78 -0 ,7 80 4 0, 54 95 0, 00 09 08 00 0, 00 15 87 80 0, 00 08 13 57 0, 00 05 03 46 0, 00 12 47 90 0, 00 12 83 20 0, 00 14 04 21 0, 00 01 21 01 0, 00 01 12 86 0, 72 10 -0 ,1 72 6 0, 98 37 0, 69 00 -0 ,3 33 0 0, 28 00 0, 00 05 04 00 0, 00 09 08 60 0, 00 08 67 91 0, 00 08 75 20 0, 00 07 06 30 0, 00 09 08 60 0, 00 07 69 05 0, 00 02 74 98 0, 00 00 0, 62 94 -0 ,9 42 3 -0 ,6 84 8 -0 ,4 30 6 0, 26 99 0, 00 02 05 00 0, 00 09 78 10 0, 00 06 58 43 0, 00 05 11 56 0, 00 05 91 55 0, 00 08 88 50 0, 00 08 35 23 0, 00 01 07 50 0, 22 74 0, 85 85 -0 ,3 71 0 0, 07 15 0, 84 18 0, 60 96 0, 00 00 22 80 0, 00 05 24 10 0, 00 09 35 67 0, 00 03 90 17 0, 00 02 73 45 0, 00 01 60 20 0, 00 04 59 87 0, 00 02 99 67 0, 00 09 57 15 -0 ,7 54 7 -0 ,5 59 8 -0 ,2 74 8 0, 60 20 0, 77 08 0, 96 75 0, 00 00 48 70 0, 00 04 08 10 0, 00 09 23 20 0, 00 08 66 16 0, 00 02 28 40 0, 00 01 35 30 0, 00 01 44 92 0, 00 05 39 50 0, 00 00 -0 ,2 46 3 0, 04 60 0, 70 82 0, 90 49 -0 ,7 07 5 0, 00 00 62 60 0, 00 07 87 40 0, 00 07 71 32 0, 00 01 36 70 0, 00 04 25 00 0, 00 06 20 40 0, 00 07 49 18 0, 00 00 39 63 -0 ,5 02 6 0, 43 13 0, 76 71 0, 57 29 -0 ,9 93 4 0, 38 43 0, 00 00 20 70 0, 00 00 45 80 0, 00 00 58 73 0, 00 00 05 03 0, 00 00 33 25 0, 00 00 45 80 0, 00 00 45 80 0, 00 00 00 00 0, 00 00 71 97 0, 12 15 0, 12 76 -0 ,0 76 8 -0 ,8 32 8 0, 62 71 0, 88 35 0, 00 00 15 50 0, 00 01 11 90 0, 00 00 58 27 0, 00 00 87 90 0, 00 00 63 70 0, 00 00 15 60 0, 00 00 15 50 0, 00 00 60 47 0, 00 00 -0 ,8 04 4 -0 ,7 78 7 -0 ,1 94 1 -0 ,7 38 3 0, 10 19 0, 00 00 79 00 0, 00 01 28 10 0, 00 00 35 24 0, 00 00 83 54 0, 00 01 03 55 0, 00 00 86 80 0, 00 00 79 00 0, 00 00 84 32 0, 62 10 0, 83 08 0, 77 80 -0 ,2 94 9 -0 ,1 87 4 -0 ,2 29 7 0, 00 08 44 50 0, 00 10 34 60 0, 00 00 93 84 0, 00 03 80 78 0, 00 09 39 55 0, 00 10 34 20 0, 00 10 34 60 0, 00 00 00 40 0, 00 02 08 63 0, 12 16 0, 01 03 0, 61 53 0, 19 70 -0 ,2 77 6 0, 96 37 0, 00 02 54 50 0, 00 05 62 60 0, 00 03 01 63 0, 00 01 79 09 0, 00 04 08 55 0, 00 03 18 30 0, 00 02 81 44 0, 00 03 21 73 0, 00 00 -0 ,8 86 7 0, 35 45 -0 ,1 95 0 0, 59 71 0, 48 74 0, 00 00 27 30 0, 00 04 11 80 0, 00 06 81 58 0, 00 01 49 39 0, 00 02 19 55 0, 00 04 58 20 0, 00 03 98 54 0, 00 00 58 37 -0 ,5 24 3 -0 ,1 60 8 -0 ,0 33 4 -0 ,0 84 5 0, 58 85 -0 ,9 71 8 0, 00 00 52 80 0, 00 01 32 00 0, 00 00 74 52 0, 00 00 55 78 0, 00 00 92 40 0, 00 01 32 00 0, 00 01 29 00 0, 00 00 03 00 0, 00 00 15 50 -0 ,3 62 9 -0 ,4 25 6 -0 ,9 42 7 0, 20 73 -0 ,5 53 4 0, 94 00 0, 00 00 94 80 0, 00 01 68 60 0, 00 00 99 64 0, 00 00 37 64 0, 00 01 31 70 0, 00 00 94 90 0, 00 00 94 95 0, 00 00 95 41 0, 00 00 0, 08 58 0, 49 31 0, 50 91 -0 ,8 26 4 -0 ,7 45 4 0, 00 00 81 70 0, 00 01 73 70 0, 00 00 57 52 0, 00 00 41 15 0, 00 01 27 70 0, 00 01 17 00 0, 00 01 29 71 0, 00 00 35 87 0, 83 21 -0 ,8 92 6 0, 15 00 0, 78 44 -0 ,2 13 1 -0 ,8 08 9 0, 00 00 85 00 0, 00 01 24 10 0, 00 00 42 21 0, 00 00 81 18 0, 00 01 04 55 0, 00 00 85 00 0, 00 00 85 00 0, 00 00 00 00 0, 00 00 85 47 0, 81 50 0, 73 89 -0 ,3 04 9 0, 60 97 0, 68 85 -0 ,2 39 1 0, 00 00 31 30 0, 00 00 31 70 0, 00 00 88 74 0, 00 00 10 68 0, 00 00 31 50 0, 00 00 31 30 0, 00 00 31 30 0, 00 00 56 85 0, 00 00 -0 ,7 91 3 -0 ,3 91 7 -0 ,2 20 6 -0 ,8 45 8 -0 ,7 43 1 0, 00 00 52 80 0, 00 01 27 90 0, 00 00 78 93 0, 00 00 46 32 0, 00 00 90 35 0, 00 00 99 20 0, 00 01 27 90 0, 00 00 90 64 -0 ,4 21 7 -0 ,6 80 1 0, 23 54 -0 ,7 93 6 0, 21 52 0, 69 38 0, 00 00 33 70 0, 00 00 44 70 0, 00 00 57 39 0, 00 00 80 24 0, 00 00 39 20 0, 00 00 44 70 0, 00 00 44 67 0, 00 00 00 03 0, 00 00 90 87 0, 74 60 -0 ,4 54 8 -0 ,6 92 4 0, 80 62 0, 03 82 0, 27 16 0, 00 00 51 40 0, 00 00 84 90 0, 00 00 30 60 0, 00 00 26 84 0, 00 00 68 15 0, 00 00 51 40 0, 00 00 51 40 0, 00 00 63 11 0, 00 00 0, 07 96 -0 ,2 79 9 0, 96 50 -0 ,5 81 7 0, 35 58 0, 00 00 00 50 0, 00 00 56 90 0, 00 00 28 20 0, 00 00 53 24 0, 00 00 28 70 0, 00 01 12 30 0, 00 00 22 13 0, 00 00 67 69 -0 ,6 24 0 -0 ,6 59 5 0, 22 90 -0 ,5 79 1 -0 ,9 20 8 -0 ,1 69 8 0, 00 00 27 90 0, 00 01 08 20 0, 00 00 74 69 0, 00 00 91 47 0, 00 00 68 05 0, 00 01 08 20 0, 00 01 08 20 0, 00 00 00 00 0, 00 00 59 05 0, 94 22 -0 ,0 28 3 -0 ,2 84 5 0, 20 06 -0 ,7 92 1 -0 ,6 68 9 0, 00 00 22 30 0, 00 00 28 60 0, 00 00 12 85 0, 00 00 18 99 0, 00 00 25 45 0, 00 00 28 60 0, 00 00 28 60 0, 00 00 28 81 0, 00 00 -0 ,6 35 1 0, 17 13 -0 ,9 94 1 0, 06 97 -0 ,6 77 9 0, 00 00 45 70 0, 00 00 83 70 0, 00 00 19 68 0, 00 00 34 36 0, 00 00 64 70 0, 00 00 54 00 0, 00 00 81 22 0, 00 00 77 05 0, 91 51 0, 78 77 0, 13 82 0, 91 13 -0 ,4 71 5 0, 57 01 0, 00 00 39 90 0, 00 01 34 70 0, 00 00 48 45 0, 00 00 22 84 0, 00 00 87 30 0, 00 01 34 70 0, 00 01 34 50 0, 00 00 00 20 0, 00 00 32 30 -0 ,0 20 8 0, 77 08 0, 18 09 -0 ,7 56 5 -0 ,2 71 6 -0 ,4 08 5 0, 00 00 76 80 0, 00 01 34 10 0, 00 00 09 04 0, 00 00 59 78 0, 00 01 05 45 0, 00 01 34 10 0, 00 01 34 10 0, 00 00 87 44 0, 00 00 -0 ,8 70 8 0, 86 00 -0 ,4 45 5 0, 64 63 0, 54 18 0, 00 00 51 60 0, 00 01 23 80 0, 00 00 58 10 0, 00 00 48 79 0, 00 00 87 70 0, 00 01 51 60 0, 00 01 10 73 Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) Introduction Market Models Closed Market Model Open Market Model Equilibrium in Market Models Equilibrium in Open Market Models Iteration process Conclusion