MATHEMATICAL MODELING MULTI-PRODUCT DISPERSED MARKET. NUMERICAL METHODS FOR SOLVING PROBLEMS Текст научной статьи по специальности «Математика»

ECONOMIC SCIENCES

MATHEMATICAL MODELING MULTI-PRODUCT DISPERSED MARKET. NUMERICAL

METHODS FOR SOLVING PROBLEMS

Kovalenko A.

The Federal Research Center "Informatics and Control Management" RAN

Build. 44, Vavilova Str. Moscow Samara National Research University named of Academician S.P. Korolev

Zlotov A.

The Federal Research Center "Informatics and Control Management" RAN

Build. 44, Vavilova Str. Moscow DOI: 10.24412/3453-9875-2021-77-3-3-13

Abstract

Mathematical models are constructed that develop Walras economic models, both centralized and decentralized spatially dispersed systems with the interaction of subjects of perfect and imperfect competition. The subjects of the economic system are a household that buys various types of goods and produces and sells various types of labor; multi-food enterprises buying various types of commodity resources and labor resources; resellers, buyers of goods in local markets at a low price and selling in another local market at a higher price and transport network of households to enterprises. The paper describes methods for finding the state of equilibrium and practical applications.

Keywords: Arrow-Debreu model, imperfect and perfect competition, network problems, theory of hydraulic systems, search for equilibrium states

Introduction

The predecessors of the models under consideration.

The problem of describing the economic system and the existence of its equilibrium was outlined in the work of L. Walras [1]. In it, L. Walras wrote that "... this theory is mathematical, the presentation and proof of the existence of equilibrium must be mathematical." The author of this mathematical model was the Nobel Prize winner Kenneth Joseph Arrow, together with Gerard Debreu. The model was of purely theoretical interest, the interactions of the subjects were based only on perfect competition. In this model, there was no spatial component of the economy, and, accordingly, its main carrier is a reseller.

The work of the Nobel laureate Jean Tyrol [2] is devoted to the models of markets of imperfect competition (analysis of market power and its regulation). However, in the constructed models there is no spatial division of markets.

Objective:

a) develop models linking the dispersed markets of various types of goods and various types of labor of spatially located households and enterprises;

b) take into account the developed model's various types of interaction between market entities.

c) develop numerical methods for finding the equilibrium state on the analyzed market structures.

d) show that the model allows performing structural units of the economic system, such as clusters, districts, regions, countries, etc.

Novelty

a) introduction of market entities into models - resellers who distribute products between local markets.

b) introduction of carriers of labor into the model, which carries out the movement of various types of labor along with the transport network from households to enterprises.

d) description of the functioning of households using utility functions, these households consume resources for their existence - various types of goods, receiving goods produce various types of labor.

e) when searching for an equilibrium state, the problems of market subjects are used in extreme settings.

f) organizing various types of interactions of subjects in the commodity markets, markets of perfect and imperfect competition are being built.

g) development of numerical methods for analyzing the developed models. nnn

h) numerical methods for finding the equilibrium state of the considered models are based on vector optimization methods [4] - [6].

Precursors of numerical methods for finding the equilibrium state of the considered models.

The use of numerical methods for the analysis of economic systems was introduced by Vasily Leontiev. He applied these methods to input-output balance models. Numerical methods for analyzing multi-product dispersed markets are discussed in the article [8].

Reasons leading to a spatially dispersed description of the market.

The behavior of the market of perfect competition for a homogeneous product (one product market) is described by the curves of consumer demand and supply of goods manufacturers. Market equilibrium is achieved at the intersection of these curves.

Definition.

The market will be called concentrated (local) if the behavior of all its subjects (consumers and producers) can be described by one demand curve and one supply curve.

A concentrated market implicitly assumes that there are two subjects in the market (generally speaking, aggregated ones): a producer and a consumer. These subjects are located where the exchange takes place, or their curves are reduced to the conditions of this place. In economic theory, these curves are constructed as the response functions of the extreme problems of optimal control of the subjects of exchange.

The real economy also has the following properties:

S uneven geographical and physical properties of territories,

S scattered distribution of various types of natural resources,

S scattered and limited places for a comfortable life of people,

S the tendency of people to certain types of labor and, accordingly, to the production of certain types of products,

S the need to consume products that are produced far from places of residence.

This is not a complete list of the reasons that lead the market to spatial dispersal, and, accordingly, to the emergence of entities that play the role of resellers, buying products in one local market and selling them in another. Generally speaking, there are no non-dispersed markets. The history of human development shows that resellers are no less important than manufacturers and consumers, they are one of the driving forces of knowledge and development of new territories.

The mathematical formulation of the model of a single-product dispersed market of perfect competition is described in [9]. In it, consumers, producers, resellers are described, respectively, by the curves of demand, supply, trade, and transport curves, the model is the problem of flow distribution of the Theory of hydraulic networks (TGN) [5], the methods of solving which are known.

The development of this model for the multi-product case with perfect and imperfect competition is given in [10]. An important point of market exchange is the nature of the interaction of market entities. If in the local market none of the subjects is a leader (i.e., does not affect the price), then this local market is a market of perfect competition. If all local markets are perfect competitors, then the economic system will be the perfect competitor. If at least one local market has a leader, then the economic system will also be imperfect competition. Each node can have its leader from the set of subjects of this node. One of its subjects becomes the leader (which does not affect the price). In this case, the dispersed market becomes a market of imperfect competition.

Equilibrium in a market of perfect competition is spontaneous. According to Scottish economist Adam Smith, the balance is established by the "invisible hand of the market" - the "BALANCE OF DEMAND AND SUPPLY", which can be considered the leader in this

market. Note that the complete analysis of the causes of the wealth of nations, which was carried out by Smith. The resources that feed the economy are different both in type, volume, and spatial arrangement.

This work is devoted to the next stage in the development of distributed market models. In the economic system, the influence of the subjects depends on the degree of remoteness, leadership within the local market, mutual arrangement in the structure of the system's connections. A large number of local markets and the possibility of variation in them by subjects holding a strategic variable provide a huge number of options for market structures. To analyze these structures, we propose numerical optimization methods, numerical methods of Hydraulic network theory (HNT).

1. Description of the economic system.

The economic system model consists of:

♦ ♦ many households whose livelihood is based

on:

- the consumption of products purchased from various markets.

- households receive budgetary funds for buying goods through the sale of their labor,

- as well as through shares from enterprises, shares from dealers who buy and sell goods.

♦ ♦ many businesses consuming:

- household labor (money-labor exchange is carried out in dispersed labor markets);

- goods - resources produced by other enterprises (commodity-money exchange in dispersed commodity markets);

- resources of the territory in which they are located;

- the profit of enterprises is distributed to wages, payments for acquired resources, payment for shares of households-owners of enterprises.

♦ ♦ many resellers who:

- buy goods from enterprises;

- resell among themselves, and ultimately, transport goods through an appropriate transport system;

- sell goods to households;

♦ ♦ transport system,

through which carriers of labor (representatives of households) carry their labor from places of residence to places of consumption (enterprises), in return receiving payment for labor.

To analyze such systems, highly aggregated mac-roeconomic models are currently widely used. In this article, we use the development of microeconomic models [11].

2. Mathematical models of subjects of the economic system

2.1. Models of the type (goods ^ labor) of households based on utility functions. Household task.

2.1.1.Construction of a mathematical model of the functioning of the household.

In accordance with the modern understanding of economic theory, under the household we mean the totality of persons who jointly extract and consume the necessary items (benefits) of existence. A household

can consist of one person living independently. Activities aimed at obtaining livelihoods will be called labor. The use, the use of the obtained objects of existence, we will call consumption. The household is the lowest, indivisible subject of the economic system.

Let W there be a finite set of types of goods of the economic system,

w be the type of goods weW. Let L be a finite set of types of labor of the economic system, i - type of labor. Mathematical models of households in the microeco-nomic analysis are widely known [11]. The models are built based on the theory of binary relations, specified in the space of consumption vectors, on the one hand, and in the space of labor vectors. These relations make it possible to construct the so-called ordinal utility functions, the maximization of which under budget constraints gives the consumer's task. A very productive approach, but the question is, where does the budget come from? You need to earn it, that is. due to labor. Modern microeconomics uses the natural division of objects (since further analysis is associated with the exchange, we will use the word "commodity") of consumption into separate types. And for each type, there is a corresponding unit of measurement that allows for a quantitative comparison of vectors of goods within the same type. Comparison of vectors of goods carries a constructed binary ratio and, accordingly, a utility function.

The situation is more complicated with labor models. Labor is an activity aimed, directly or indirectly, at the acquisition of consumer goods. The division of labor into separate types is natural. By the type of labor, we mean homogeneous activities, indistinguishable from each other. Examples of such types of work are the activities of locksmiths, the activities of turners, carpenters, etc. Note that these activities can also, in turn, be subtyped. We will assume that such a partition is finite. A characteristic feature of the type of labor is the ability to measure it. The most natural (but not obligatory) unit of change in physical types of labor in the production of a particular product is the amount of energy expended, or time spent. We will assume that each type of labor is measured in conventional units. Two species can be considered the same if they can produce the same product and the difference can only be in costs.

We denote by xw the vector of consumer goods, xl the vector of labor performed, then each component of the vector x = (xw, xl) has its unit of measurement. If XW is the space of types of consumer goods, and the space XL of types of labor, then the direct product X=XWxXL of these spaces gives a complete space that describes the state of the economic system.

We will assume that for any household in space X it is possible to define a binary relation of the strict order « ^ » and the order of equivalence « ~ », which establishes the preference on it. On this basis, the utility function u = u(x) = u(xw, xl) can be constructed.

Let's designate HH as a set of households in the economic system. Let ieHH. We assume that for any x, y eX household i:

or x ^ >y (we read "x is preferable to y for household i ),

or y ^ x (we read "y is preferable to x for household i),

or x ~i y (we read "x is equivalent to y" for household i).

Based on the order introduced in this way, the utility function ui = ui(x) = ui (xw, xl) can be obtained. Usually, for households, the vectors xweXW are a boon, xleXL an anti-boon.

In the future, since each i has its vector of goods and its vector of types of labor, we will use XW and XL.

2.1.2.Models like ((wages for labor + wages for savings) consumer goods) ^ labor)) based on utility functions. The functioning of the household in the structure of the economic system.

To describe the structure of connections between objects of a spatially dispersed economic system, we will use the notation of graph theory. These structures will be fully described below, the designations used here correspond to them, and do not interfere with the understanding of the described models.

Each household of the ieHH node uses its total income to purchase consumer goods and supplies its labor to labor markets. The total income consists of:

1) income Di1 from the sale of their types of labor;

2) income Dt2 from owning shares of enterprises;

3) income D3 from the ownership of resellers' shares.

Income on shares is investments in businesses and dealers in previous periods.

We will assume that each household ieHH, as a result of its activities, maximizes its utility. The maximum utility of the household will be

3. ut = ut (x) =ut ((xwv)

'vV (i)

;(xlv ) ) ^ max

(,) ®HH (i)

Where ®HH (i) is the vector of variables (parameters) for which optimization is

carried out, in particular

®HH(i) =| (xwv or(xlv(i)

V+ (i) - a set of arcs along which goods come from

commodity markets,

V[ (i) - the set of arcs along which labor enters the labor markets.

For v e V+ (i) the price of the goods arriving along

with the arc v from the node hi (v), is equal to Phi(v), so the cost of purchasing the goods will look like

Z Phi

. So the budget constraint will look like

X Phl(v) xwv < H i + H2+ H 3

v e V + (■)

Income from the sale of labor The first part of the sale of labor is calculated according to the dependence

H 1 = ZPlh

h2(v)

■ xl

veVi (i)

Where V[ (i) is the set of arcs going from node i to nodes of labor markets h2(v), v e V[ (i); p2(v) - the price of labor in these nodes. Income from shares of enterprises.

Let us denote Ji by the set of enterprises, the shareholder of which is the household i, j c pr Take an

enterprise j e J , the profit of that enterprise ^ . We

J 1 j

denote by Ij - the set of households - shareholders of enterprise j, among which this profit is distributed. Let's introduce the coefficients ak- the share of profit received by the household k from the enterprise j. These coefficients must have the properties

Zak = 1, a * 0 k e ij.

keI j

Then the second part of the household budget will be equal to H 2= Za'jftj

jeJ,

Income on shares from resellers. Let w - some product, weW, u - reseller of this product u eVW, denote

VV = y v - the set of all resellers. Let us denote by JV

wew

the set of resellers, the shareholder of which is the household i. Let's take a reseller jveJV, a reseller's profit % . Let us designate JVjv the set of households -the reseller's shareholders jv, among which this profit is

distributed. Let's introduce the coefficients ^ - the share of the profit received by the household k from the reseller jv. These coefficients must have the properties

Zpv=1, P * 0 k e

IK,.

Then the third part of the household budget will be equal to h ]=

jveJVj

Household task.

For node i, the household problem HH(i) has the

form

Ui = u(x) =u<((xwv{iy (xlv)veV_w) ^ Max

(2.1)

under budget constraints

» xw»

Z Ph1») XW» - ZPlh2» ■ xlv + Z jj + ZP»

(2.2)

»V (i)

JeJ, JveJV,

Since we believe that households are directly connected by arcs with the nodes of local markets of the economic system, flows along the arcs are indicated qv, the beginning of the arc is indicated h1(v), the end of the arc is indicated h2(v), then the household problem HH(i) will have the form

u, = u(x) =u,((q»Xev+ (0; (q»(,)) ^

Max

(qv \ev+ (,);(q» W (,)

(2.3)

Z pi(v) q»- ZPL(V) ■ q» + Z jj + Zp>3v

veVL (i)

jeJ,

j»eJV,

»eV ,,(,)

For arc v e v+(i) the node h1(v) belongs to the market of the product w, for which h1(v) e Ew. For arc v e Vm, the node h1(v) belongs to the labor market l, for which the node is h2(v) e E1.

2.2. Models of the type (goods + resources of the territory + labor ^ goods) enterprises based on production functions. Enterprise task.

Let us denote by PR the set of enterprises of the economic system that produce goods from the set W. It is assumed that each enterprise can produce several types of goods. Let ie PR, PR(i) be the enterprise of the i-th node. We denote Vw (j) by the set of arcs along

which the enterprise PR(i) sends the produced goods to the corresponding markets. For any arc v e Vw (1) start

e h1(v) = 1. The end of the arc j = h2(v) belongs to the

market set Rw of the product w, we W. The arc v defines the type of goods sent from the enterprise.

We denote by the v£ (i) set of arcs along which

goods are supplied to the enterprise PR(i) as consumed resources. For any arc v e V(1) h2(v) = 1. The beginning of the arc j=h1(v) of the market Rw of the product weW. The arc v defines the type of goods received by the enterprise. A detailed description of the product markets will be given below.

Various types of labor are involved in the production of goods by the PR(i) enterprise. Let's designate V/ (i) the set of arcs along which labor is supplied to

this enterprise. Let veV*(1), j=h1(v). Among the

many types of labor, there is a type leL such that j e e,

, where jeEi is the set of nodes of a finite directed graph G =E ,Vl,H;) describing the structure of connections

between local labor markets of the form l. The details of the structure of labor markets will be described below.

Let v <=vw (j), gv be the volume of the product sent along with the arc v to the node j=h2(v). This node belongs to one and only one set of nodes Ew, weW. Similarly, if v ev^ (j), then qv is the volume of the product

received as a consumed resource along with the arc v from the node j=h 1(v). This node belongs to one and only one set of node Ew, weW.

The vector of own resources of the territory belonging to the enterprise will be denoted by r,, (r. < r)

, where r is the maximum volume of the enterprise's resources, < is the sign of inequality in space, the dimensions of the vector n. We write the product release model as

(qv W Coe F (q W j),(qv V(i), r))

Where F is the set of the attainability of the volumes of manufactured products for the given values of

the arguments ((qv)vev+ (0,(qv)vij),r) . The vector of

own resources rt can be interpreted as resources extracted on the territory and distributed for consumption

(generally speaking, in a processed form) at other enterprises and households of the economic system. The Objective of the enterprise PR (i)

% = ZPh2(vqv-( ZpkKv)q + ZpkKv)q+{pr,ty^max

veVx(i) veV} (i) veVf (i) pr(1)

with restrictions

fav X.

e F (q )vEV, (iy(qv )vi (iy r ))

F = P 1p

h2(v)qv

P

hl(v)qv

Z P

h\(u)H hl(u)

~{Prv, rv)

rv - the vector of the maximum volumes of these resources,

Pr - the vector of prices for these resources, the components of the price vector are external variables;

After transformation we get

1v>veVw (i) ri <i ri

OpR(i) - a list of variables for which the maximum is taken. So in the markets in which the enterprise is a monopolist, this list, among others, may include the variables

Ph2M, v eVw (i), PH(v), v e V+ (i), PH(v), v e V+ (i).

2.3. Model of the type (commodity + labor-commodity) of the functioning of a reseller of the market

Let w be the type of product, w eW to this type of product corresponds to the dispersed market Rw , which

G = (E V H \

has a structure given by the graph w ^ w, w, w'.

Consider an arc v eVw. To begin with, we will assume

P > P

that h2(v'> h1(v'>, or (p -p > 0). In this case, the

reseller RES(v) corresponding to the arc v buys the goods (and accordingly becomes its owner) in volume qv > 0 at the node market h\(v) at the price p ,

transports, and sells at the node market h2(v) at the price P . The reseller's profit will be as follows:

(2-4) Fv = (Ph2(v) Phl(v))qv

\

Z PhKu) q*K„) +( Prv > rv)

\ueV; J

The task of the reseller RES (v) will take the form

f \

Fv = (Ph2(v) Phl(v)) qv

Z Phl(u) VhKu) +( Prv , rv)

^ max,

®RES(v)

0< qv < f((qhm\ev + ;rv)

0 < r < rv ,

v v v v v '

where f{{qhKu-,) v+\r) - is the production function that limits the volume of transportation;

®REsiv) - a list of parameters for which the maximum is taken.

Let p < p . In this case, the reseller buys the

goods (becomes its owner) in volume qv at the node h2(v) by the price p }, transports it to the node k\(v)

and sells by the price P . Since the flow moves from the end of the arc to its beginning, then q < o. The reseller's profit will look like

(

Fv = Phl(v) ( qv )

Ph2(v)(-qv) + Z Phi(u) qhiu) +(P rv, rv)

\

the first term is the revenue from the sale in the node h2(v), the second is the cost, which contains: S purchase of goods in a node h\(y), S purchase of resources in markets hl(u), u e V+ where V+ - a set of arcs along which resources flow,

S the cost of resources that are the property of

the reseller, r - the desired vector of the volumes of rey v

sources used in the transport of purchased products,

the first term is the revenue from the entire operation, the second is the cost. After some simple transformations

f A

Fv = —{Ph2(v) — Ph1(v))}(—qv )— Z Ph1(u) qh1(u) + {P rv, rJ)

\ ueV.v j

The reseller's extreme task will take the form

Fv = (Ph2(v) Phl(v)) Iv

Z Phl(u) qhl(u)

+ ( P

rv , rv)

\

^ max,

®RES(v)

; rv ),

0 < - qv < fv i(qh1(u))ue , 0 < r < rv.

v v v v v

Combining both cases, we get the problem of the reseller RES (v)

A

Fv ~ (Ph2(v) Ph1(v)) qv

\qv\ < fv ((.qhi(u))UeV:; r I

0 < r < rv,

v v v v v ?

sign(yv ) = sign(ph2(v) —phl(v) )

where:

S q | -the volume of traffic along the arc v limited by the value of the production function fv, the sign qv coincides with the sign (Ph2(v) —Phl(v)) and determines

the direction of the flow along the arc;

S purchase of resources in the markets h1(u), u e V+ where V+ - a set of arcs along which resources flow,

ZPhl(u)qhl(u) +(P rv, rv)

^ max,

®UFK (v)

(2.5)

S rv - required vector of the volumes of resources, which is used in the transport of purchased products,

S Prv - the cost of the reseller's resources, - the vector of prices for resources are external variables;

S rv is the vector of the maximum volumes of these resources;

S ®res(v) is a list of parameters of a reseller of

arc v, along which maximization is carried out, in this list, depending on the structure of the local market, may

V ueV;

include variable p ,, p

h2(v) , Ph1(v) ,

, components of the vec- corresponds to a variable

the volume of goods

tor rv.

Comment. The model assumes that the connection of enterprises, resellers, and households d1rect1y w1th the local market 1s carried out w1thout 1ntermed1ar1es (resellers).

3. Spatially dispersed commodity markets

3.1. Description of the structure of the model.

As above, W - many types of goods of the economic system. Each type of w eW market corresponds to its own single-product spatially dispersed market Rw [10]. We define the market structure as a directed graph

G = (E ,V , H ), where

w \ W W wf

S Ew - a set of nodes where goods are exchanged;

S Vw - a set of arcs, a reseller of goods corresponds to an arc;

S Hw - mappings for arcs H (v) = (hl(v), h2(v)), h\(v) - node, beginning of arc v; h2(v) - node, end of arc v.

To each j e ew we assign a variable p, that denotes the exchange price of a commodity. Each v ev

transported by the reseller, q > o if the direction of the flow coincides with the direction of the arc, q < o otherwise. The variable q is determined by the reseller's model of this arc.

3.2. Boundary conditions, conditions of the product balance of the single-product market of goods.

We split the set of nodes Ew of the product w into three disjoint parts. Ew = £„'u£»2u£„3, where Ewlr-,Ew2 =0, Ew1nEw3=0, Ew2 <^Ew3 =0.

Z — free variable

P — constant, P = P'i

i e El

n*

Z — constant, z = B i P — free variable z, = B* (P )

i e EW i e E,3,

(3.1)

(3.2) (3.3)

For Zi = '( i ) elasticity is not equal to zero and not equal to infinity. Under equilibrium conditions, the equality

( Zq»- Z q»)+( Zq»- Z q»)+(— Z q»)+(— Z q»)=:

yeV + (i) yeV - (i) »eV+N (i) yeVm (i) yeVRES(i) »eVHH (i)

(3.4)

S The first bracket denotes the sum of flows that dealers bring into the node, minus the sum of flows that dealers take out of the node. V+ (i) - the set of arcs entering the node i; V ~ (i) - the set of arcs leaving the node i; S the second parenthesis is the sum of the flows of goods from the nodes h\(v), v e V*p(i) brought in for sale to the node i, minus the sum of the flows of goods that are taken out by the enterprises h2(v), v e VEN(i) from the node i. These enterprises use

the exported flows as resources of their production;

S the third parenthesis is the sum of flows that resellers take out of the hub and use as resources for the

transport of goods. Arcs v eVRES(i) of non-standard definition, the beginning of the arc hl(v) is node i, the end of the arc h2(v) is, in turn, also an arc from the set of RES;

S the fourth parenthesis is the sum of flows that households take out for consumption. The beginning of the arc v e VHH(i) is node i, that is. h1(v) = 1 ;

The expression on the right in (3.4) is the foreign trade balance.

3.3. For nodes i e E1W in a state of equilibrium, the

foreign trade balance is a calculated value and is calculated by the expression

( Z q»— Z q» )+( Z q»— Z q» )+(— Z q» )+(— Z q» )=,l e +W

i» /

yeV + (i) yeV— (i)

(3.5)

»eV+ (i) »eV+N(i)

»eVRES (0

»eVHH (0

3.3. The balance of cash flows at the nodes of equivalent value is transferred. The calculation of the

the markets. equivalent cost of exchange of goods is derived from

In the nodes, a commodity-money exchange is the relation (3.4) carried out, along with the movement of goods, their

( Z qyPt - Z qyP<)+( Z qP - Z )+(- Z qJ,i)+(- Z qvP)=z P

»eV + ( i)

»eV — (i)

»eV+N (i)

or

»eVRES(f)

»eVHH(i)

( Z q»P + Z q»P )—( Z q»P + Z q»P + Z q»P + Z q»P ) = z P

»eV + ( i)

»eV,+p(i)

»eV — (i)

»eV+N (i)

»eVRES (i)

»eVHH (i)

The direction of movement of cash flows that participate in the exchange is opposite to the direction of

»eV + (i)

»eV+N (i)

»eV — (i)

the flow of commodities. Note, that if the system is closed, that is z. = 0 , then

( Z qyPt + Z qyPt )—( Z qyPt + Z qvP> + Z qP + Z q»P< )=o

»eV+N (i) yeVREs(i) »eVH^i)

households, and consumers of labor - enterprises. To describe the structure of labor markets, as in the description of product markets, we will use the concept of a graph. Each type of labor leL has its own graph Gi . In it the nodes are local labor markets, places of direct communication of markets with enterprises, and places of transfers when moving along the transport network.

4. Mathematical model of the labor market in the economic and geographical system of the world economy

4.1. Descriptive model.

The basis for constructing a model of a spatially dispersed labor market is the model of the movement of carriers of types of labor between sources of labor,

q

y

yeV+N (i)

Resellers also use labor markets, so non-standard arcs are added to the graph, the beginning of which is in the node - the labor market, the end - the arc corresponding to the reseller. We will assume that stopping points are in close proximity to consumers and suppliers of labor. This approach does not detract from the connection with reality. Several different arcs of the graph of different types of labor can correspond to the same section of the transport network.

4.2. Mathematical model of the labor market. As above, L is a finite set of types of labor of the economic-geographical system. Let leL, the dispersed labor market l will be denoted by Rl . It corresponds to a finite directed graph g = (E ,Vl, H;), describing the

structure of connections between local labor markets of the type l. El - stopping points and transfer points, where carriers of labor can change the route of movement. Let i e Et, V + (i), V~ (i) - the sets of arcs, respectively, entering and leaving from node i to incident nodes from El, through v*h(i), VEN(i), VRES(i) is denotes the sets of arcs, respectively, entering households (set HH), and leaving in the direction of enterprises (set of EN) and in the direction of resellers (set of RES). Let us denote by Pi the price of labor at node i for v e V, qv - the value of the flow of labor of the type

l, moving along the arc v.

Price relationship of subjects of labor markets The following 2 cases are possible: First case. P(h2(v) > P(h1(v)), the flow moves from the node h1(v) to h2(v), therefore q > 0. Let us denote by 0v q) - the cost of transport of carriers of

labor per unit of flow with a total flow qv along the arc v. In a state of equilibrium will be performed P(h2(v) = P(h1(v)) + 0v (qv).

Second case. P(h2(v)) < P(hl(v)) the labor flow moves from node h2(v) to node h1(v). In this case q„ < 0, 0V(-qv) - the cost of transport of carriers of labor per unit flow for the total flow qv along the arc v. In a state of equilibrium will be performed P(h\(v) = P(h2(v)) + 0v (—qv )

Combining both cases, we obtain that for any qv

P(h2(v)) — P(h1(v)) = sign(qv 00 (\qv\) ,

or,

0 (\ qv\) = sign(P(h2(v)) — P(h\(v))) (P(h2(v)) — P(h\(v))) Let us denote by q — the inverse function for qv, then = 0v-l| P(h2(v)) — P(h1(v))|or

qv = sign(P(h2(v)) — P(h\(v))) 0;P(h2(v))— P(h\(v))\.

i e E]

, i e E2 i e E3

(4.2)

(4.3)

(4.4)

qv = sign(Av )0V-\AV) (4.1)

Border conditions.

Let E= E/uE^uE?3, where E/nE2 =0, E/nE3=0, EfnEi3

zi — free variable

P — constant, p = P

Z — constant, z = B'i P — free variable

z> = B'i (P ) This is at the labor market. For i e e3 the foreign trade balance zi is related to

the price Pi by the function B*(P). For this function,

elasticity is not equal to zero and not equal to infinity. Relations (4.2) correspond to the case when the modeled system cannot affect the prices of the systems of nodes i e E], relations (4.3) correspond to the case

when the volume of consumption (supply) by the nodes i e Ej is constant and does not depend on the equilibrium price at the node. Nodal labor balance.

Let i e E \ E1 (that is i e E,2 U E,3),

v e (V + (i) U V- (i))u (U V+h(i) U VEN(j))U VRES(i) , for each v e (V + (i) U V- (i))U (U V+h(i) U V£n(i))U VRES(i)

in

state

of

equilibrium

Denoting a= (P(h2(v))—P(h1(v)))the previous dependence, we can write

F = ( Z^v(Pi,P—i)— Z^v(Pi,P—i)) + ( Z^v(Pi,P—i)— Z^v(Pi,P—i))

Z qv— Z qv + Z qv— Z qv— Z qv=B (P),

veV + (i) veV- (i) veV++ (i) veV^(') veVms(i)

For ieE1

Z qv— Z qv +Z qv— Z qv— Z qv = z< (4.5)

veV* (i) veV- (i) veV++(i) veVE,r(i) veVms(i)

(4.5) is the formula for calculating the export-import balance.

5. Nodal interactions of the economic-geographical system of the world economy, perfect in the equilibrium model.

5.1. Food balance in the form of an extreme problem.

The "invisible hand" of the local food market Consider an arbitrary type of product w eW. We do not take the nodes i eEl, since for them the variable

Pi is a constant, there is no restriction on the variable zi, it can take any values per (3.5).

Let us fix for all nodes of commodity markets j e E \ {i} and all labor markets j e e! the value of the

price Pj . The variables Pj, j e (Ew \{i}) U E will be denoted p.. If there is perfect competition at node i, then the variable Pi is independent, the variables P , its functions, and (3.4) can be written:

veV* (i)

veV— (i)

Where qv = ^ (p, p.) - the function is the response

of the flow to the variable P at constant values of the

product, the value of P-i for:

- V + (i) resellers of sellers to node i for the price p.

veVSp(i) veVnp(i) (5.1)

- V (i) resellers of node i for the price P ;

- v eV*N (i) enterprises h1(v)- sellers for the price

p;

v eVEN(i) enterprises h2(v) of buyers for the

price P ;

a

- v eVRES(i) resellers h2(v) of buyers for the price

- v eVHH (i) households h2(v) on the price; h2(v) The left-hand side of equality (5.1) will be called

unbalanced at node i at price Pi and denoted n,. (p,P_t)

, then (5.1) takes the form

N (Pi, p) = 0

(5.2)

Further, equations (5.2) will be called central for node i.

Definition of IE1. We will say that node i is in a state of equilibrium at a price p* at fixed p , if it is

satisfied n,. (P*, P_t ) = 0. The value p* will be called the

equilibrium market price of node i with fixed values p.

Definition of IE2. We will say that an economic system is in equilibrium if all product and labor markets are in equilibrium. Thus from tasks:

- household (2.3);

- enterprises (2.4);

- tasks of a reseller (2.5), as well as conditions:

- at the boundaries of communication with other economic systems (boundary conditions) (3.1) - (3.3);

- product balance (3.4), (3.5)

we obtain an equivalent problem consisting of the boundary conditions

(3.1) - (3.3) and the system of nodal equations (5.2). Let - p j j e e 1 u E 2 u E 3 - the solution of the

J 5 J w w w

system of nodal equations (5.2) and boundary conditions (4.1) - (4.4), then for any p,, j e E 1 u E 2 u E 3

" J ~ j w w w

Pi, is a solution of one equation

N (p, p—i ) = 0 (5.3)

of one unknown variable Pi. Problem (5.3) of finding an equilibrium state at a node can be written as an extremal problem:

If np,p_,) = 0 then in (5.4) NB'(p',p ') = 0, taking into account that NB (P ) > 0, we obtain that Pi the optimal solution (5.4).

In the opposite direction, if in problem (5.4) Pi the optimal solution is and NB (p,, p-i ) = 0, then N, {pi, p-i ) = 0, that is (5.3) holds.

Using Adam Smith's term, problem (5.4) will be called the "invisible hand of the market' problem for node ieEw2u Ew3. If at some node ieEw2uEw3 the value of Pi turns out to be such that NB (p, p—i ) > 0, then the "invisible hand of the market' makes the transition of the node to such a price p' at which

NB (p, p—i )it will be minimal, that is strive for a state of balance. The equilibrium in the entire network will correspond to such prices p* ieE„2uEw3, at which,

NB (p* , p_*) = 0, ieEw2uEw3 , that is all nodes will be

in equilibrium. In the (hydraulic network theories) TGN, this approach is one of the network nodal linking approaches.

Algorithm of "invisible hand of commodity mar-

We organize variation of the variable Pi to minimize the functional (5.4):

For each value of Pi, perform:

1. For each v eV+ (1) and every value of Pi, we look for a response q = 7 (p, p.) based on the solution

of the reseller problem (2.5);

2. For each v e V- (1) and every value Pi , we look

for a response q = 7 (p, p.) based on the solution of

the reseller problem (2.5);

3. For each v eVEN (i) enterprise-seller

j = hl(v) e EN we search a response q = 7(p, p.) to price p based on the solution of problem (2.4) of node j with fixed p.;

4. For each v eVnp (1) buyer-enterprise

j = h2(v) e nP we search response q = 7(p,p.) to

price Pi based on the solution of problem (2.4) for node j with fixed p.;

5. For each v eVRES(i) reseller - buyer u = h2(v) e VV = y V , we look for a response

weW

q = (p, p ) to the price Pi based on the solution of problem (2.4) of node j at fixed values p.;

6. For each v e VHH (i) household j = h2(v) e HH we look for a response q =7 (p, p .) to the price Pi

based on the solution of the problem (2.3);

7. Looking for the response of the export-import balance z, = B*i (p, p) of external subjects of node i.

8. Using the ones obtained in paragraphs 1.-6. values qv, as well as the value zi from node i, based on dependence (3.4), we obtain the value of the "market hand' functional NB (p, p-i) Using the value of Pi and

NB (p, p-i) rejecting non-optimal solutions.

Remark 5.1. It is easy to see that the described algorithm minimizes a function of one variable with the calculation of only the values of the function; therefore, the algorithm can use the one-dimensional optimization algorithms of the 0-th order.

Remark 5.2. The transition to a state of equilibrium in one node can unbalance other nodes. In [11], theorems are given, in which it is proved, that if the local one-product market of each node tends to a state of equilibrium, then the entire system converges to a state of equilibrium. However, as the convergence estimates show, the transition of the entire system to the equilibrium state can be a rather lengthy process.

Remark 5.3. The described system is a game with a Nash equilibrium state. The "1nv1s1ble hands of the markets'' of the nodes ieEw2uEw3 are the subjects of this game. Functional (5.4) describe the criteria for minimizing each subject of the game and the mutual influence between them.

5.2. Description of labor balance conditions in the form of an extreme problem. "The invisible hand of the local labor market" Let for the type of labor leL, 1 e E2 ^E]. We do not take the nodes 1 e e1 , since for them the variable Pi

-w UEw

is a constant, there are no restrictions on the variable zi, it can take any values in per (4.5).

We will assume that for all nodes of commodity markets and labor markets j e E \ {/}, the value of the

price Pj is fixed. The variables Pj, j e e \{/}, will be

denoted in what follows P-j. In conditions of perfect

competition, all market entities i are price recipients, therefore, for all v e v it is true (4.1)

Where a„ = (P(h2(v)) — P(h1(v))),or

qv = sgn(Ph2(v) —Ph1(v)) 0V' 1(Ph2(v) —Ph1(v))

For v e V + (i) h2(v) = i, for v e V" (i) h1(v) = i, therefore (4.3) can be written

Z sgniP -Phi(v)) d;\Pt-PhiW)-Z sgn(Ph2(v) -Pi ) 0:1(Ph2(v) -p )+

veV+ (i) veV~ (i)

+ Z(«v)- Zk)- Zk)=4(P)

veVHH(i) veVEN(i) veVRES(i)

For v e V* qv is the response function to the variable Pj for fixed P-j household tasks HH(j) for j = hl(v). For v e VEN(0 qv is the response function to the variable Pi for fixed P-j tasks of enterprises HH(j) for j = hl(v). For v e VßN(o qv, it is the response function to the variable Pi for fixed -i, respectively, re-

(5.4)

is the response function to the variable Pi for fixed respectively, resellers' tasks RES(u) for j = h2(v) (recall that resellers in the notation system correspond to graph arcs, therefore the end of the arc v refers to the arc u).

Thus (5.4) can be written

sellers' tasks EN(j) for j = h2(v). For

v eV,

RES(i)

qv, it

Z sgn( P -P„m)0~ '(Pi -Pl(v))- Z sgn( Ph2(v) -Pi ) O;1 (Ph2(v) -P ) +

veV+ (i) veV~ (i)

+ Z(qv)- Z(qv)- Z(qv)-B(P)=0

(5.6)

veV+ (i)

veVm(i)

veVnPK (i)

The left-hand side of equality (5.6) is called the unbalance at the node i and at the price Pi and fixed

N (P P )

prices P-i and we denote it i i i , then in the state of the node equilibrium

N (P, P_t) = 0 (5.7)

By designating NBi(P,P,.) = (N,.(P,P,.))2 , the problem of finding the equilibrium state in (5.7) can be written as

P* = argmin NBt(P,p) (5.8)

p if P =const

Using Adam Smith's term, we will call problem (5.8) proclaiming the "invisible hand of the market' of node i eEfuEi3. If for all ieE^uE3 (that is, for product markets too) the values P* are such that

NB(P*,P,*) = 0, then the resulting values give the

equilibrium state of perfect competition for the economic system as a whole.

6. Numerical methods for the analysis of nodal interactions of imperfect competition in the general equilibrium model of a spatially dispersed economic system

The transition from perfect competition to imperfect competition consists in changing the leadership of the "hand of the market" to the leadership of one of the market entities. Depending on this, we get one or another structure of the local market, and, accordingly, the structure of the entire system. The search for the equilibrium state of the entire system, as above, consists in the node-by-node linking of all the nodal problems.

Let us present models of nodal problems for some structures of imperfect competition and algorithms for finding an equilibrium state in them.

6.1. Description of the algorithm the leadership of the manufacturer - the seller in the commodity market. Nodal monopoly

Suppose that at node i the producer j = h1(u), u e Vnp + (i) takes the position of the leader. We get a monopoly of producer j at this node i. Under the monopoly of producer j at node i, market power is transferred to him. He has the ability to choose the value of the strategic variable Pi , varying it in order to maximize his profit. At the same time, we note that, as before, the product balance condition must be met in the node.

Let us transform the previous mathematical model of perfect competition as follows. As an example, consider the market structure shown in the following figure 1.

Fig. 1 Scheme of connections of the market of node i with incident nodes In this figure, solid lines are arcs of connections with enterprises, sellers and consumers; dotted line -arcs, corresponding to resellers; dash-dotted line - arcs of connections with households; the bold arc is the connection between the monopolist j and the market i. The leader who owns the choice of the strategic variable Pi

is enterprise j. Since i = h2(u) the leader's task will take the form.

X = P,qu + XPmvflv -( XPhKv)qv + Xpmq +(Prj, J) ^ max (6J)

veV, ( j)\{u} "" '

with limitation

q,q)6F'((qvW<jrVw,ri)) rj < j rj

We divide the sought variables

®UF (j) = (Pi , qu, (qv )v6Vw (j)W (qv W U)>(qv W U), rj) ' over which the maximum is taken, into 3 parts: p , q and ((q ) ■ (q ) (q ) ■ r))

KK^vJv6Vj (j)\{u)'^vJv6V+ (jy^^v6V* (j)' j"

Per this, we construct a three-level maximization scheme 7:

Level 1. organization of enumeration by variable

Pi .

Level 2. for each fixed p we solve: 2a) for each k = hl(v), v 6 V*P(i) \ {u} task EN(k), we obtain the volume of sales qv by enterprises - perfect competitors;

®ur( j)

v6lW (j) v6"L (j )

2b) for each k = h2(v), v 6VEN (i) task EN(k),, we

obtain the volume qv of resource consumption from the market i;

2c) for each v 6v + (i) task of resellers RES(v),

we obtain the volume of imports qv;

2d) for each v 6v~ (i) task of resellers RES(v),

we obtain the volume of export qv;

2e) for each v 6 VHH (i) , k = h2 (v), according to the household problem hh(k), we find the volume of consumption qv from the market i. 2f) calculate zi (P); Level 3.

3a) Based on the product balance (3.4), we find the volume of production that the monopolist should produce:

qu

=-( Xqv- X qv) - ( X qv- X qv) - (- X qv) - (- X qv)+

veV + (i) veV - (i)

vsVnP(i)

v^VnFK (i')

b) for a given Pi and obtained qu, we solve problem (6.1), as a result we obtain n ■ In this case, the maximum is taken over the variables

(qv )

veV,- (JW

(qv )

vV (J)

(q, )

' r

veV+ (J)' J

As above, we do not specify how the variation with respect to the variable Pi is carried out; for this, a variation of any method of one-dimensional optimization of the 0-th order can be applied. Also, we do not specify the methods for solving problems of the second level. This can be done using any of the conditional optimization methods. Since each of these problems can take a solution from the previous iteration as an initial approximation, the method of returning directions is interesting, which modifies the method of possible directions [16].

6.2. Description of the algorithm for the leadership of the household in the node. Nodal monopsony.

In order to obtain a monopsony model at a node, the leadership of the enterprise should be replaced with the leadership of the household. Problem (6.1) will be replaced by problem (2.2).

6.3. Algorithm of the reseller's leadership incident to the local market. Nodal monopoly of a reseller.

Consider the case when the reseller u 6 V+ (i), i = h2(u), is the leader in the market of node i. (The case u 6 V~(i) is similar).

Algorithm of "reseller leadership.

As above, we organize the variation of the variable Pi to maximize the functional of the reseller problem (2.5) with u 6 V+ (i):

For each value of Pi we do:

1. for each v 6 V+ (i)\ u and the value of Pi, we look for a response qv = ^ (p, p.) based on the solution of the reseller problem (2.5);

2. for each v e V- (i) and the value of Pi we look for a response qv = ^ (p, p. ) based on the solution of

the reseller problem (2.5);

3. for each v eV+N (i) enterprise-seller

J = h1(v) e EN, we seek a response q = ^ (p, p. ) to the price p based on the solution of problem (2.4) of node j with fixed p. ;

4. for each v eVEN (i) enterprise - buyer J = h2(v) eEN, we look for a response q = ^(p,p.) to the price p based on the solution of problem (2.4) of node j with fixed p. ;

5. for each v eVRES (i) reseller - buyer u = h2(v) e VV = y V , we look for a response to the

weW

price p based on the solution of problem (2.4) of node j at fixed values p ;

6. for each v e VHH (i) household j = h2(v) e HH, we are looking for a response q =^v(p,p.) to the pricep based on the solution of problem (2.3);

7. looking for a response - (export-import balance) z = B*i p, p. ) of external subjects of node i.

8. using the values of qv obtained in items 1.- 6., as well as the value of zi of item 7, based on dependence (3.4), we obtain the value of the functional of the reseller problem (2.5) at u e V + (i).

9. Using the value Pi and value of the functional of the reseller problem (2.5) at u e V + (i), we reject obviously non-optimal solutions.

Remarks 5.1. and 5.2 are also validfor the above algorithm.

veV,+p(i)\{U}

Conclusion

As noted in the review [12], general equilibrium models are widely used in applied economic research of spatially dispersed systems because they allow one to quantify the relationship between various subsystems of the economic system, as well as the impact of various factors. But as noted in [13] that "despite numerous attempts, it was not possible to find any general and natural conditions that ensure the uniqueness and stability of equilibrium." The proposed models develop and refine the Arrow - Debreu models, and make it possible to answer the questions posed in [13]. In a real economy, both the utility function and the production function used to describe it can be linear. Changes in prices at nodes can lead to jumps in extreme problems describing the behavior of market entities, which in turn leads to an imbalance in the entire system. To obtain answers to the fundamental questions posed in [13], it is advisable to use functions of the CES type.

For models of a dispersed market for a homogeneous product of perfect competition (this is a spatially dispersed layer of the general equilibrium model), convergence to an equilibrium state is proved in [14]. But the proof uses the special properties of the supply and demand functions. The solution of the analogs of the TGS did not raise doubts about the uniqueness of the solution, but in the TGS, when solving problems to describe the flow movement, quadratic functions are used. Problems of instability and non-uniqueness arise when the flow regime (laminar, turbulent, etc.) is taken into account. There is no such thing as a laminar -> turbulent water hammer transition - the perturbations are too small. It's just that chaos in the flow grows abruptly instead of order (laminar regime).

This work contains a toolkit for finding an equilibrium state for cases when there may be a monopolist in any local market. As a result, we get a huge number of different structures, ranging from perfect competition, cascades of monopolies [15] to centralized management.

As noted above, an analog of the search for the equilibrium state of dispersed markets is the algorithm of the point-by-point linkage of the theory of hydraulic networks. The experience of using this algorithm has a long history. Solving specific problems and numerical experiments in this area have shown that they can be used in simulation decision-making systems. Numerical methods for finding an equilibrium state become much easier if, when solving, it is allowed for enterprises to use cost functions I. 7) and supply functions

7 = nt(Pt); for resellers to use cost functions iv(y) and

trade and transport curves yv = (Ph2(v) - Pmm).

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