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The Walrasian model considers equilibrium in the markets. Walrasian equilibrium. Consumer Demand Equations

Consider the mathematical modeling of the market according to Walras. The initial concepts of the Walrasian model are:

· disaggregation of market participants: individual consumers and individual producers are considered;

The perfection of competition

general balance.

The latter concept means considering the equilibrium for all goods at once, and not for individual goods. Consequently, the concept of general equilibrium (i.e., equilibrium for all goods) is introduced into the Walrasian model.

We will assume that two types of goods are sold and bought on the market: finished goods that are a product of production (final consumption goods) and production resources (primary factors of production). Therefore, we will consider the "extended" space of goods , where is the number of types of all goods. The components of the vector are both outputs and inputs (primary factors). To distinguish between them, the costs are given a negative sign (therefore, we write , and not ). If there is a vector of net output, then all its components corresponding to costs will be equal to zero; if there is a vector of only primary factors, then all its components corresponding to final products will be equal to zero.

Indices (types) of goods, as before, will be denoted by the letter , consumer indices - by letter and manufacturers' indexes - by letter . Denote by the vector of commodity prices.

Entering the market, each consumer or producer becomes both a buyer of one and a seller of other goods. Consumer, i.e. a market participant "not directly engaged in production" can sell the primary factors at his disposal and buys the goods of producers. Manufacturer, i.e. a market participant "directly engaged in production" sells its finished product and buys primary inputs from consumers.

Therefore, every consumer i as a market participant is characterized by three parameters: the initial stock of goods, the income function and the vector function of demand for production products

Each manufacturer j characterized by two parameters: the vector function of the sentence finished products and the cost demand vector function . However, in the Walrasian model, a somewhat generalized characterization of the manufacturer is used - with the help of one set interpreted as the set of his (optimal) production plans. In the input-output language, this set can be defined as follows: where is the production function. Obviously, .

Therefore, the mathematical model of the market is understood as a set of elements:

(4.3.1)

where is the price space of goods, N is the set of all market participants ( N contains elements).

Without qualitative losses, instead of (4.3.1), as a market model, we can consider the set

The nature of the population elements (4.3.1) here is somewhat different from that which was characterized by an isolated consideration of the consumer and production sectors.

First, the vector contains the prices of both final consumption goods and inputs. Next, we will proceed from the volatility of prices. Moreover, prices change not at the request of individual market participants, but solely under the influence of aggregate demand and aggregate supply. Therefore, one of the key questions is: are there prices that suit both consumers and producers?

Based on technical considerations, we will assume that the price space P includes zero space, i.e. we will assume the existence of zero prices.

Secondly, as mentioned above, each market participant acts in two persons: as a buyer and as a seller. Obviously, the number of sellers and buyers for different goods will be different. Therefore, the numbers and should not be associated with the number of sellers and buyers.

Thirdly, the income of each consumer is assumed to consist of two components: 1) the proceeds from the sale of the initial stock of goods belonging to him, 2) the income received from his participation in the profits of the production sector (let us denote, for example, through the acquisition of securities and other types of investment and labor activity.Thus, we assume that

/ (4.3.2)

In the Walrasian model, it is assumed that the entire income of the manufacturing sector is completely distributed among consumers:

where , and the scalar product on the right, taking into account the structure of the vectors , is interpreted as the profit of the entire production sector. Note that the summation of vectors is carried out component by component.

Fourth, the supply and demand functions are assumed to be vectorial and multivalued. For example, for a function, the first property means that where is a scalar function of demand for the -th product (see (2.5.3)). The second property means that the function for each p maps more than one vector and the set of such vectors, i.e. This takes place, for example, when in relation (2.5.2), which determines demand, the maximum is reached not only at one point.

In the Walrasian model, the concepts of aggregate demand and supply are formalized as follows.

Definition 4.1. Aggregate (market) demand function

(4.3.3)

Aggregate (market) supply function is called a multivalued function

(4.3.4)

With this definition, the meaning of aggregate demand fully corresponds to the method of formation of market demand based on solutions to optimization problems individual consumers. Specifically, it is the sum of individual consumer demand functions. The definition of the aggregate supply function requires additional explanation. To this end, we introduce the notation:

By definition, any element of a set Y can be represented as a vector , where Since there are many optimal producer plans j, then the components of the vector are the optimal volumes of output and costs, and all of them constitute a solution to the same optimization problem. Thus, part of the components of vector , as well as vectors , reflects the supply of finished products, and part - the demand for primary factors. Therefore, a vector cannot be called unambiguously a sentence. At the same time, the vector can be interpreted as an aggregate supply, since the part of the components of the vector corresponding to the demand is "compensated" by the vector b.

Let us show that for any p and , i.e. the domain of change of the aggregate functions is the same space as for the individual functions. Consider first two consumers. For any set is formed by shifting the set in the direction of the vector x by the length of this vector (Fig. 4.4).

Rice. 4.4 The sum of a vector and a set

Consider three consumers. For any set formed by shifting the set in the direction of the vector x the length of this vector. That's why

Continuing these considerations, we get

In the same way, the inclusion is established Since and therefore , then the set is formed by the displacement of the set Y in the direction of the vector b the length of this vector. So

Having formalized the concepts of aggregate supply and demand functions, the market model (4.3.1) can be represented as a set of the form

(4.3.5)

Any vector is called aggregate demand (corresponding to the price vector p); any vector - by the total supply (corresponding to the price vector p). These vectors are the (optimal) responses of the aggregate buyer and aggregate seller to the established price vector in the market. If at the same time, then there is a shortage of goods on the market, and at , their surpluses appear. Such prices cannot be considered satisfactory, since in one case the interests of buyers are infringed, and in the other - of sellers. Obviously, the best option for the economy is equality. This ideal case is not always the case in practice. Therefore, it is advisable to somehow weaken it. In the Walrasian model, the most humane version of the generalization of the concept of economic equilibrium is allowed from the point of view of the interests of consumers.

Definition 4.2. The set of vectors is called competitive equilibrium in the market (4.3.5) if

(4.3.6)

(4.3.8)

In this case is called the equilibrium price vector.

By definition of the aggregate supply and demand functions, it follows from the inclusions (4.3.6)

Where

where

those. aggregate demand and supply are formed as the total values ​​of individual consumer demands and individual proposals of producers. Therefore, in expanded form, the equilibrium conditions (4.3.6)-(4.3.8) can be rewritten as:

(4.3.9)

(4.3.10)

(4.3.11)

(4.3.12)

Consider the economic content of the conditions that determine the competitive equilibrium in the market (4.3.5). Condition (4.3.6) shows that each consumer and each producer responds to prices in the best possible way. This is clearly seen from relations (4.3.9) and (4.3.10) . Condition (4.3.7) ensures that the aggregate supply is not less than the aggregate demand. Condition (4.3.8) requires that, in value terms, aggregate demand equals aggregate supply. Condition (4.3.8) is automatically satisfied if strict equality takes place in (4.3.7). In this case, the equilibrium will be given by the relations:

those. the need for condition (4.3.8) disappears.

Rice. 4.5 Oversupply

Suppose that for some product in (4.3.7) there is a strict inequality: . Then, in terms of value, we obtain the inequality not corresponding to the condition (4.3.8). Value called surplus. According to the law of supply, if there is a surplus, the price of a commodity must be reduced. But this will lead to a change in the equilibrium price. Can we find a way out of this contradiction? Obviously,

Consequently, to restore condition (4.3.8) it is necessary to eliminate the excess. Taking into account the sign, this is only possible for But then

and

those. product k generally excluded from circulation on the market.

The justification of fairness (4.3.8) by the fact that the goods supplied in excess of the existing demand receives a zero price makes economic sense, but does not lend itself to adequate formalization. Indeed, for a fixed number the inequality

incompatible with equality

Thus, the formal way out of the situation under consideration is to consider the price of the reproducible goods equal to zero. Purely theoretically, this technique is consistent, since it does not lead to contradictions in the future.

At the same time, it should be recognized that there is no economically meaningful explanation for the existence of a product with a zero price. Declaring such a product free seems untenable. Strictly speaking, there are no free goods in the economy, any by-product can find a use, i.e. has a non-zero price. We cannot agree with the modification of the law of supply and demand, well known to economists, about the existence of reproducible goods with a zero price, since in the case of overproduction, the requested part of this product is sold at a non-zero price. For the economy, the existence of a surplus is just as bad as the existence of a deficit. All this speaks in favor of the expediency of defining equilibrium in the form (4.3.13) .

The market model according to Walras is built. As you can see, the central place in it is occupied by the concept of competitive equilibrium. The attractiveness of equilibrium as a state of the market (and the economy as a whole) lies in the possibility of selling all goods produced and in satisfying the demand of all consumers. The process of formation of market prices can be conditionally compared with the work of some algorithm consisting of four blocks (Fig. 4.6).

Rice. 4.6 Equilibrium price formation scheme

In the first block, a price vector is formed. Vector Information p enters the blocks and , in which the sets and , respectively, are formed, the content of which, in turn, is transferred to the block . Pairwise comparison of elements is carried out in the block . If there exists a pair or pairs for which the condition (or conditions (4.3.7) , (4.3.8)) is satisfied, then the process ends. Otherwise prices p are rejected, which is signaled to the block where new prices are formed. The procedure continues until an equilibrium price vector is found.

An affirmative answer to this question is related to the resolution of two important problems:

1. establishing the fact of the existence of competitive equilibrium in the Walras model;

2. development of a computational procedure (method) converging to the equilibrium price for the formation of market prices.

The existence of an equilibrium in the Walrasian model has not been established. The reason lies in the level of formalism of this model - it is very abstract. By concretizing the definitions of its constituent elements and clarifying their functional properties, one can obtain various modifications of the Walras model. The most famous of them is called the Arrow-Debré model, after the names of its creators.

The problem of developing numerical methods for calculating equilibrium prices is related to the establishment of necessary and sufficient signs of equilibrium. It is necessary that they be constructive, i.e. generated a convergent iterative procedure, such as the web-like model (see Fig. 4.2).

This model is an attempt to present all the equations describing the general equilibrium in the economy, in order to compare the number of these equations with the number of variables they include. If the number of equations is equal to the number of variables, then general equilibrium is possible.

Imagine an economy with the following characteristics: in any market of this economy there is perfect competition ( big number buyers and sellers, full awareness, no costs for entry and exit from the market, each consumer and firm act independently of the others); it is also assumed that there is no external effects and public goods.

There is t types of consumer goods, each of which is produced under conditions of perfect competition by many independent firms. Each firm maximizes its profit.

The farm has P types of resources that are owned by consumers and are provided by the latter firms at certain prices. Each consumer can own any number of types of resources and does not necessarily offer for sale the entire amount of the available resource. Consumers distribute the received income among different consumer goods, maximizing their utility functions.

Let a fixed amount of each resource be needed to produce a unit of each good. Thus, there is a matrix of size pht, separate element ats, which shows the amount of resource j, necessary for the production of a good /:

Thus, in total in the economy there is P resource markets and t markets for consumer goods. In every market, there are two variables - price and quantity. In the market for a particular good, this is P, and Q t , and in the market of a separate resource -pj and qj. In total it turns out 2 P + 2t unknown.

Let us now determine the number of equations describing the economic system. There are four groups of equations that describe Various types functional dependencies in the economy: 1) equations for the demand for consumer goods, 2) equations for the supply of resources, 3) equations for equilibrium in the industry, 4) equations for the demand for resources. The first two groups describe the equilibrium of consumers, the second two define the equilibrium of producers.

1. Consumer demand equations

The individual consumer's demand for each good is defined as a function of the prices of all consumer goods i prices of all resources

Since the demand of each consumer depends on these variables, it can be said that the market demand is defined as the sum of the individual demands. Therefore, to write down the market demand function for a good, you need to write down the following equality:

where qi- the volume of production of the good;

- the total demand of all consumers in the market

good I.

Because we have t markets for goods, we have exactly t such demand equations.

2. Resource supply equations

Because consumers must also choose the amount of supply of resources they possess, one must write down their supply functions. The individual supply of a resource also depends on the prices of consumer goods. (P, P t) and prices of all resources (p h p „). It is these two series of values ​​that make it possible to estimate the benefits from the sale of resources. Since the individual supply of each consumer is defined in a similar way, we can represent the market supply function of an individual resource as a function of all prices on the farm and write the following equation:

where q, - sales volume in the resource market j;

Resource suggestion function j all household consumers.

Since the economy has P resource markets, we have exactly P such offer functions.

Note that one price vector defines volumes

demand and supply at once in all markets of goods and resources, since the choice of an individual consumer consists in the simultaneous determination of his demand and supply in all markets of the economy at given prices.

In addition, in this vector of prices, it is the ratio of prices of various goods and resources that is important, and not their absolute value. A proportional change in all prices will not cause a change in supply and demand in all markets. For example, if both the prices of goods and the prices of resources increase by exactly 2 times, no consumer will have an incentive to change his behavior.

3. Equilibrium equations in the industry

According to the logic already used, we would now have to write down the supply functions in the market for each good on the basis of the supply function of an individual firm. But we cannot do this due to the assumption of fixed coefficients. After all, fixed coefficients mean no economies of scale and no diminishing marginal productivity. The supply function of any good in this situation must have infinite elasticity, and the size of the firm turns out to be indeterminate.

In this situation, we can ignore the supply functions as such and write down another condition for the equilibrium of an individual producer in a particular market - the equality of profit to zero. Since there is perfect competition in all markets, the general equilibrium will be reached if the profitability of the production of all goods is the same and equal to zero. Or, what is the same, the average cost will be equal to the price of the good. Thus, we have

those. the price of a good i is broken down into the cost of acquiring resources to produce a unit of the good. Since every good must be produced under similar conditions, we have t such equations. Here, too, only the ratio of prices is essential: their proportional change does not violate equality (67.3).

4. Demand equations for resources

In determining the demand for resources, we are faced with the same problem as when considering the equation of equilibrium in the industry. Since the production coefficients are constant, the resource demand functions will have infinite elasticity. But as in the previous case, we can cheat and write down the general equilibrium condition - the demand for each resource will be presented in such an amount that is necessary to produce an equilibrium set of goods according to the existing production coefficients. Formally, this is also a resource demand function, in which not the prices of goods and resources are written as arguments, but the already selected quantities of goods produced. Therefore, we can write

where qi- the volume of production of the good i.

Since this equality must hold for all resources, we also have P such equations.

Since we are analyzing relative prices and abstracting from their absolute values, in order to measure prices, we need to choose one good that will serve as a unit of account. The price of this good is taken equal to one and therefore is not unknown. So the number of unknowns is 2p + 2t - 1.

Now we can sum up. In total, our system has 2 P + 2t equations and 2p + 2t- 1 unknown. As you can see, there are fewer unknowns than equations, and this shows that one of the equations is redundant. If it can be excluded from the system by proving its dependence on the rest, then the general equilibrium is possible.

One equation can be eliminated based on the following consideration. In general equilibrium, all the income received by consumers from the sale of resources is spent in the markets for consumer goods. This means that the total cost of resources should be equal to total cost good. Therefore, in conditions of general equilibrium, knowing the prices and quantities in all markets for resources and goods, except for the market for the good chosen as the unit of account, we can calculate the volume of demand in this market in a residual way. As a result of this, one of the demand equations turns out to be dependent on all other equations in the system, and it can be excluded. Remains 2 P + 2t- 1 independent equations.

Thus, the number of equations turns out to be equal to the number of unknowns, and this means the possibility of achieving a general equilibrium in the economy.

The necessity of equality of the number of unknowns to the number of equations to achieve general equilibrium in the economy does not mean the sufficiency of this condition. First, if the functions are non-linear, then the system of equations may have several solutions. This means that there are several equilibrium points (supply and demand curves in individual markets may intersect more than once). Secondly, as a result of solving this system of equations, we can get negative prices and quantities for individual goods, which will not make economic sense, and the general equilibrium with such absurd prices and quantities will be impossible.

The first rigorous proof of the existence of general equilibrium was carried out in the 1930s. German mathematician and statistician A. Wald. Subsequently, this proof was improved in the 1950s. K. Arrow and J. Debre. As a result, it was shown that there is a unique general equilibrium state with non-negative prices and quantities, if two conditions are met: 1) there is a constant or decreasing returns to scale of production; 2) for any good there is one or more other goods that are with him in relation to substitution.

To prove the possibility of achieving a general equilibrium, it is necessary to determine the mechanism for achieving equilibrium prices and volumes in each market. Walras himself used the theory of groping to prove the achievement of equilibrium, which is as follows.

First, it is necessary to answer the question of whether the system will move towards equilibrium prices and volumes. This is proved "by contradiction": if one imagines that at first some arbitrary price vector is realized, which does not correspond to the equilibrium one, this will mean a surplus in some markets and a shortage in others. This condition will lead to higher prices in markets where there is a shortage, and lower prices in those markets where there is a surplus. The change in prices will continue until the equilibrium vector of prices is "groped".

The first economist to build a mathematical model using a system of equations to prove the possibility of the existence of a general equilibrium was the Swiss economist Léon Walras (1834-1910). He suggested that the national economy consists of consumers using n interrelated goods, the production of which is carried out using m different factors of production. Under conditions:

Given the utility functions of each consumer and his budget,

Equality of the budget of the consumer of the value of his factors of production,

If the volume of its factors of production is fixed (the absolute inelasticity of their supply), it is possible to construct the demand function of the i-th consumer for j-th benefit:

M i is the budget of the i-th consumer,

P j , r t - prices of goods and factors, respectively, j = 1,2,..n, t=1,2,...m,

F S i , t is the given volume of the t-th factor belonging to the i-th consumer.

For the sake of simplicity, let us assume that each firm produces only one kind of good. With a given technology and known prices for goods and factors of production, a profit maximizing firm forms a supply function for a good and a demand function for factors. The sum of the offers of all firms producing the same good forms the industry supply:

The total demand of these firms for factors is the industry demand for each of the factors:

On the basis of functions (6)-(8), a microeconomic general equilibrium model is constructed, consisting of three groups of equations:

1. equilibrium conditions in the goods market:

2. equilibrium conditions in the markets for factors of production:

3. budgetary restrictions of firms in the market of perfect competition in the form of equality of total revenue to total costs:

The system of equations (9)-(11) contains 2n+m unknowns and the same number of equations. But only 2n+m-1 equations are independent. This is due to the budget constraint of consumers, due to which the total excess demand of any consumer is equal to zero.

Assume that there are only 2 markets for goods and 1 market for factors. The budget constraint (equation) of the th consumer has the form:

This equality says that the expenses of the th consumer (left side) should be equal to his income from the sale of his goods and factors of production (right side).

In parentheses - excess demand of the th consumer in each of the markets, i.e. the equality of the total excess demand to zero for any consumer is only another form of representation of his budget constraint. Let us sum up the budget equations of all participants in market transactions:



It follows from equality (13) that if the system of prices P 1 , P 2 , r ensures equilibrium in any two markets, then the equilibrium will also be in the third one. This conclusion, which is true for any number of markets, is called Walras' law.

In accordance with Walrasian law the system of equations (9)-(11) contains 2n+m-1 independent equations. At the time of Walras, there was no mathematical apparatus for solving it. Walras took the path of grouping equations, and considered the movement towards equilibrium as a gradual process - a “groping search” for the correct proportions of exchange, especially at the stage of a preliminary contract.

In order for the system to have a solution, one more independent equation must be added, or the number of unknowns must be reduced by 1. The first option - macroeconomic - introduces an additional equation for the equilibrium of supply and demand in the money market. The second - the microeconomic price of the chosen good is taken as 1, and the system of relative prices is sufficient to explain microeconomic phenomena.

General equilibrium under conditions of pure exchange with limited resources and goods provides a solution to the economic problem - the placement of a limited number of goods among consumers. One of better ways such an arrangement is the box (box) of Francis Edgeworth (English economist, 1845-1926), in 1891. Wrote Mathematical Psychology.

(nxm) elements, while V. Leontiev has a square matrix (sxn) of elements. In addition, the Leontief model structured gross output (it is subdivided into intermediate and final product), shows the sources of value added production, available, a balance sheet section with the characteristics of the use of elements of the final product and a redistribution section of income. But still the most essential idea of ​​the cost-output model was already contained in the conclusions of Walras.

Comparing the number of unknowns (2m + 2n - 1) and the number of equations, we find that they do not coincide and the system is not in equilibrium in the economic sense. Walras comes out of the situation by eliminating one equation from the model. The result is the equality of the number of equations and the number of unknowns. The system of equations gets the economic meaning of equilibrium. This, of course, does not mean that it will necessarily be solved, because it is still necessary to prove the existence, uniqueness, and positivity of the solution. From a practical point of view, it is also hardly possible to solve a system of equations that would include tens of millions of product items with specific indicators of the costs incurred for their production. Therefore, the meaning of the Walrasian model is precisely theoretical - it shows the market ideally.

To understand what an ideal market-type macroeconomic equilibrium is, Walras and Pareto contributed the most weighty ideas. The Walrasian model served as the starting point for new horizons of equilibrium analysis.

Quite meaningful is the modified Walrasian model, understood as a synthesis of models of behavior of firms, consumers and general economic equilibrium. The new model assumes the following conditions

Conclusions from the Walras model

The Walrasian model is a simplified, conditional picture of the national economy. It does not consider how equilibrium is established in development, dynamics. It does not take into account many factors that operate in practice, for example, psychological motives, expectations. The model considers established markets, well-established infrastructure that meets the needs of the market.

SECTION 2. Walras model

Section 2 Walras Model 221

Section 2 Walras Model 223

Section 2 Walras Model 225

Thus, in the case when the demand curve has a negative and the supply curve has a positive slope, the models of Walras and Marshall lead to the same stable equilibrium. However, do supply and demand curves always look like this? Recall Fig. 6a from section 2 of lecture 1, which depicts the so-called "curving" labor supply curve. In its upper part, this curve has a negative slope. The supply curves in the foreign exchange market can also be characterized by a negative slope (this issue will be considered in the next issues of our publication). Let us now consider a market with a negatively sloping supply curve to see if the Walrasian and Marshall models lead us to the same conclusions about the conditions for the stability of the equilibrium in this case.

Thus, the models of Walras and Marshall lead, at least from a theoretical point of view, to different conditions balance stability. The reason for these differences is the different initial ideas about the functioning of the market mechanism that underlie the models we are considering. Is it possible to say that the Walrasian model correctly describes the action of the market mechanism, and the Marshall model - incorrectly (or vice versa) Probably not. Indeed, the process of establishing equilibrium in the short run is better described using the Walrasian model, when, for example, excess demand leads to a rise in price to the equilibrium value.

In the theory created by him and C. Arrow in the 1950s, a set of conditions was described that guarantee the existence of general equilibrium under assumptions that are less stringent than in the Walras model (the Arrow-Debré theorem). But his research was not limited to the question of the existence of equilibrium. He analyzed both its normative characteristics and uniqueness. K. Arrow and J. Debret formulated the conditions under which the action of the price mechanism, combined with the desires of consumers, leads to the efficient use of resources. Developing the theory of general equilibrium, Debre created new methods of economic analysis, which are now used by many economists.

In purely theoretical terms, the new classic is a modified version of the Walrasian model. Modification, oh

In the Walrasian model, all information is contained in prices, moreover, in equilibrium prices. Deviations in the system from equilibrium can only be the result of various kinds of imperfections (from incomplete information to price inflexibility and delays in people's response to external disturbances)4.

Let us now consider how the statement "about the role of dual estimates of y" in the framework of the Walras-Wald model differs from similar, but incorrect statements in the framework of the optimal planning model (3.P).

Built on the basis of this theoretical concept, the Walrasian model is a general economic equilibrium model, a kind of one-shot snapshot of the national economy in its purest form. As for the state of equilibrium, according to Walras, it presupposes the presence of three conditions

In short, he developed an innovative model using a primitive mathematical apparatus. Cournot (ournot), a better mathematician than Walras, and a man who had a great influence on him, shied away from solving this problem because of its extreme difficulty. Despite the fact that the Walrasian model was constantly revised and improved, its general concept remains unchanged to this day. In his History of Economic Analysis (1954) Schumpeter (Shumpeter) wrote Walras was in my opinion the greatest of all economists.

The result obtained by von Neumann makes it possible to realize the importance and aspect of equilibrium, which was not revealed in the Walras model, namely equilibrium - this is the maximum output in monetary terms - MIII and the minimum income of factors. This conclusion is, and in another language, Smith's statement about the equality of value and output and the amount of income in the economy.

Lange and Lerner proposed a model for a decentralized economy that consists of state-owned enterprises, consumers, and a governing body, the Central Planning Committee. The latter, in fact, BJ plays the role of an auctioneer from the Walrasian model, calculates the optimal prices, primarily the prices of production factors, for some speculative economy, and sets them to economic subjects. Managers of state-owned enterprises make their own decisions, focusing on the parameters of fixed prices. In doing so, they are guided by two rights

Obviously, the Walrasian model cannot be used to explain those mechanisms that are perceived as a violation of macroeconomic proportions. According to the new classics, it is impossible to derive macroeconomic disequilibrium from microeconomic equilibrium.

The Keynesians rightly noted that the new classics p k judged within the framework of the Walrasian model, which excludes any false non-price signals, which does not agree well with the real behavior of people. If we admit that even in conditions of price flexibility people react not only to price signals, then the overall picture of economics differs significantly both from the Walrasian one and from the one drawn by the new classics.

For the first time, the theoretical model of general economic equilibrium in a classical market was developed by the Swiss economist L. Walras (1834 - 1910) as a theory of general competitive equilibrium. The proposed model is macroeconomic in form, but in content it is based on microeconomic indicators that characterize the behavior of individual producers and consumers of goods in the markets. The Walrasian model is based on the use of equilibrium prices, which ensure the equality of supply and demand for each product. The rational element of the model of L. Walras is the formulation of an extremal problem for the national economy as a whole and the approach to prices as an integral element of finding a general optimum (equilibrium).

In a market economy, prices determine the volume of output, and output largely determines prices. The prices of consumer goods and services depend on the prices of resources. And the prices of resources - from the prices of economic goods for which there is effective demand. The relationship in the economy turns out to be closed, resembling a circle, out of which you can only solve a certain system of equations.

L. Walras, analyzing the equilibrium system of the economy, tried to describe the general economic equilibrium using a system of equations. He showed that the number of equations is equal to the number of unknowns. This means, firstly, the fundamental possibility of solving a system of equations, that is, achieving a general equilibrium; secondly, from the mathematical point of view, the uniqueness of such a solution. By substituting concrete values ​​of prices into the production function obtained as a result of the solution, on the basis of its mathematical analysis, it is possible to obtain the quantities of exchanged economic goods.

The system of equations obtained by L. Walras is called the system of general equilibrium equations. An analysis of the solution of the system of equations led L. Walras to the correct conclusion that the general equilibrium system is stable and, being taken out of this state, tends to it again through the mechanism of relative prices.

It should be recognized that the model of L. Walras in a certain respect idealized economic reality. It provided that consumers know their supply and demand functions, technical coefficients, restrictions, and many other data included in the model. In addition, the general equilibrium model proceeds from perfect competition, which assumes the ideal mobility of all resources, the full awareness of the participants, absolutizes the state of equilibrium, while in reality, disproportions, failures, imbalances, and shortcomings are much more common. The market economy is static, does not always take into account scientific and technological progress, uncertainty factors, institutional development conditions.

L. Walras first developed a model, and then went from it to real economic reality, and not vice versa. However, this model can be modified, complicated and simplified by including new variables. The latter can be set both endogenously and exogenously, reflecting both economic processes and phenomena, and the evolution of the institutional conditions for the functioning of a market economy.

The theory of general economic equilibrium (according to Walras) is based on the following ideas and provisions:

- every market economy strives for equilibrium in the form of a trend, which is hallmark many natural economic processes;

- there is a principle of interdependence of the main elements of a market economy, which ensures the unity of the system and affects the implementation of the desire for equilibrium;

- the starting point of the equilibrium analysis is the analysis of the exchange of products between producers and consumers, when the exchange is carried out on the basis of mutual benefit and equivalence.

Equilibrium in the economy is not reduced to the market equilibrium of individual commodity producers. But equilibrium can only be achieved through the market mechanism, through exchange. Price is the main instrument in this mechanism. Alignment (establishment of equilibrium) of supply and demand for goods (according to Walras) occurs through "groping", the search for mutually acceptable prices, acting as equilibrium prices.

It is important to emphasize that L. Walras described the path for theoretical economic science, which, as J. Schumpeter rightly noted, is still being followed today.

The head of the Lausanne school (mathematical school) Pareto (1848 - 1923) believed that economic theory should study the mechanism that establishes a balance between the needs of people and the limited means of satisfying them, for which it is necessary wide application mathematical method of analysis. He sought to theoretically substantiate the model of interdependence of all economic factors, including price. V. Pareto tried to improve the theory of general economic equilibrium of L. Walras.

Unlike the latter, he considered a number of equilibrium states in time, and also allowed for the variation of the coefficients of the production function depending on the size of output. V. Pareto considered it fundamentally possible to provide a model describing the main relationships in the economy with statistical material, although he did not develop methods for its aggregation. V. Pareto sought to cleanse the theory of equilibrium from psychologism and eliminate the hedonistic explanation (hedonism is an ethical position that affirms pleasure as the highest good and reduces to it the whole variety of moral requirements) of the motives of economic behavior, which is so characteristic of the theory of marginal utility. For this, they introduced new tool studies of economic behavior - indifference curves, borrowed from F. Edgeworth.

The Italian economist V. Pareto formulated the principle of optimality, which states that the maximum welfare, or total utility, is achieved when the desire for the well-being of individuals does not lead to a decrease in the standard of living of any member of society. In his opinion, this principle can be implemented in conditions of unlimited competition.

In economic theory, there have been many attempts to improve the original model of L. Walras, aimed primarily at overcoming its mechanistic nature and extreme simplification. One of the most developed is the Hicks-Allen model, in which the system describing the state of general equilibrium contains three groups of equations. The first reflects the process of achieving maximum utility for each consumer while limiting the amount of his income; in the second - obtaining the maximum profit for each entrepreneur while limiting the nature and size of his products; in the third, equations are given that describe the conditions for the equality of supply and demand for the entire range of goods under consideration and the formation of profit as the difference between the sale and purchase prices. Thus, the general equilibrium system is based on the achievement of consistency between the partial profit optima for all entrepreneurs and the partial utility optima for all consumers.

The neoclassical concept of general economic equilibrium (L. Walras, A. Marshall, J. B. Clark, V. Pareto, etc.) is a model of the functioning of a private capitalist economy in conditions of ideal freedom of competition, absolute price elasticity, complete rationality in the behavior of firms seeking maximize profits at minimum production costs, and the absence of abrupt dynamic changes associated with technical progress and government intervention. This theory is called microeconomic - although we are talking about general equilibrium, since it comes to the analysis of the latter from the point of view of the behavior of individual economic units. Its theoretical foundation is formed by theories of marginal utility, marginal productivity, and the theory of imputation based on it. The essence of this concept boils down to the fact that in the conditions of market forces, prices are set at a level where, on the one hand, they express the existing preferences of consumers and the relative utility of goods, and on the other hand, they reflect the minimum production costs. This minimum cost is reached when the combination of factors of production is such that their marginal products are proportional to their prices. If the prices of factors of production (labor and capital) change freely in accordance with changes in their relative supply, then on the national economic scale, according to the model, not only the minimum costs will be achieved, but also the most complete and effective use available resources. Thus, the focus of the general economic equilibrium model is on price and the market. This model was used to adapt the mechanism of free competition to the market element.