Walrasian general equilibrium model. Equilibrium theory L
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NON-GOVERNMENTAL EDUCATIONAL INSTITUTION
HIGHER PROFESSIONAL EDUCATION
"SIBERIAN INSTITUTE OF BUSINESS AND INFORMATION TECHNOLOGY"
Rabstract
by discipline « Main directions of economic thought"
on the topic: "Leon Walras' General Equilibrium Model"
Completed by a student
1st year Kel Maria Vyacheslavovna
Group: EV-114(2)
Introduction
1. The concept and essence of general economic equilibrium
2. The theory of general economic equilibrium by L. Walras
Conclusion
References
Introduction
In the very general view equilibrium in the economy is the balance and proportionality of its main parameters, in other words, a situation when participants economic activity there are no incentives to change the current situation.
In relation to the market, equilibrium is the correspondence between the production of goods and the effective demand for them.
At the macro level, a distinction is made between partial and general equilibrium. Partial equilibrium is the quantitative correspondence of two interrelated macroeconomic parameters or individual aspects of the economy. This is, for example, the balance of production and consumption, budget income and expenditure, supply and demand, etc.
In contrast to partial, general economic equilibrium means the correspondence and coordinated development of all spheres of the economic system.
General equilibrium analysis can be very useful for assessing the overall functioning of the economy, for understanding special economic problems and for policy formation, which makes the research topic relevant.
Thus, the object of study is general economic equilibrium.
The problem of general economic equilibrium was studied various schools and scientists such as Quesnay, Keynes, Marx, Pareto, Friedman, Samuelson, etc. However, the very concept of general economic equilibrium was developed by L. Walras. The scientist formulated the basic conditions for the structural correspondence of demand and supply of goods, and quantitatively described the relationship between the key economic parameters of production and exchange within the framework of a simple auction scheme.
The purpose of the work is to study the concept of general economic equilibrium.
To achieve this goal, it is necessary to complete the following tasks: define the concept and study the essence of general economic equilibrium; consider the theory of general economic equilibrium of L. Walras; explore the interaction of markets as a factor of equilibrium; analyze the joint equilibrium in the markets of goods, money and capital.
The main task of economists is to find out which imbalances are important for analysis this issue, and which ones can be neglected. In any case, understanding the moment of general equilibrium is mandatory for analyzing and assessing the economy, which is of practical importance in economic forecasting and planning.
1. The concept and essence of general economic equilibrium
economic equilibrium capital
Macroeconomic equilibrium is the central problem of the national economy and a key category of economic theory and economic policy. It characterizes the balance and proportionality of economic processes: production and consumption, supply and demand, production costs and results, material and financial flows. Equilibrium reflects the choice that suits everyone in society.
At the microeconomics level, the problem of equilibrium is considered in relation to a separate market - partial equilibrium, i.e. equilibrium in a particular market of goods and services, factors of production.
However, in real life The economy of each country is a collection of markets for individual goods, intertwined with a complex system of relationships. This is explained by the fact that all producers are also consumers, and all goods are either directly or indirectly related to each other as components of the total commodity mass in the form of interchangeable or complementary goods.
General equilibrium is the equilibrium state of the entire market system, which means the establishment of equality of supply and demand in all interconnected markets. Kamaev V.D. Textbook on the basics of economic theory. - M., 2007. - P. 62.
A more precise definition of general economic equilibrium is given in the economic dictionary - “an equilibrium state of the economy that develops as a result of the balancing interaction of demand for goods, services, resources, their supply in the markets and the price system formed under the influence of supply and demand.” Raizberg B.A., Lozovsky L.Sh., Starodubtseva E.B. Modern economic dictionary. - M., 2006. - P. 96.
General equilibrium, unlike partial equilibrium, is achieved much more difficult and less often. In the market for final goods and services, equilibrium will mean that producers maximize income and consumers receive maximum utility from the products they purchase. Equilibrium in the factor of production market shows that all production resources entering it have found their buyer, and the marginal income of resource owners, which forms demand, is equal to the marginal product of each resource, which forms supply. Equilibrium in the money market characterizes the situation when the number of expected cash equal to the amount of money that the population and entrepreneurs want to have.
Under conditions of free competition, the set of prices for goods corresponds to a state of general equilibrium if the following three conditions are satisfied:
1) all consumers maximize their utility under given budget constraints;
2) all firms maximize their profits with a given technology;
3) for each product, supply is equal to demand.
“The general equilibrium model includes two types of markets - goods and factors of production - in the general circulation. General equilibrium will be achieved when both types of markets - goods and factors - are simultaneously in a state of equilibrium.” Sidorovich A.V. Economic theory. - M., 2008. - P. 51.
There is a distinction between ideal and real equilibrium.
The ideal (theoretically desired) balance is achieved in the economic behavior of individuals with the full optimal realization of their interests in all structural elements, sectors, and spheres of the national economy.
Achieving such balance presupposes compliance with the following conditions of reproduction:
All individuals must find consumer goods in the market;
All entrepreneurs must find factors of production in the market;
All of last year's product must be sold.
Ideal equilibrium is based on the premises of perfect competition and the absence of externalities - side effects.
In the real economy, various violations of these requirements are observed. Cyclical and structural crises, inflation take the economy out of balance. At the same time, even in the conditions of these imbalances, the economic system can be brought into dynamic equilibrium, which will reflect market realities with all their contradictions.
“Real macroeconomic equilibrium is the equilibrium established in the economic system under conditions of imperfect competition and external factors influencing the market.”
To understand the specifics of the current economic situation and the implementation of economic policy, it is important to establish whether the economic equilibrium is stable or unstable. If, in response to an external impulse that disrupts the equilibrium, the system itself, under the influence of internal forces, returns to an equilibrium state, then such an equilibrium is called stable, but if it does not restore itself, then it is unstable. Therefore, along with determining the conditions for establishing general economic equilibrium, it is necessary to examine whether it will be stable or not. Storchevoy M.A. Fundamentals of Economics / ed. P.A. Vatnik. - St. Petersburg, 2009. - P. 68.
Achieving general economic equilibrium does not mean that now every participant in the market economy is satisfied with his position; equilibrium simply states that by changing the volume and structure of purchases or sales, no one will be able to improve their well-being in the current conditions.
General economic equilibrium is not a typical state of a market economy, since plans of sovereign entities developed independently of each other can only by chance turn out to be mutually agreed upon. Due to the constantly changing needs of the population and production technology, the economy often finds itself in a state of transition from one equilibrium state to another. Therefore, in reality, both individual markets and the national economy as a whole often find themselves in a disequilibrium state than in an equilibrium state. However, the behavior of economic entities in a market economy directs it towards equilibrium: until the plans of participants in market transactions are consistent, they will adjust the economic situation through changes in supply and demand. Grebennikov P.I., Leussky A.I., Tarasevich L.S. Macroeconomics. - M., 2006. - P. 45.
Thus, general economic equilibrium is characterized by the coincidence of the plans of all buyers regarding purchase volumes with the plans of sellers regarding sales volumes. The discrepancy between these plans causes imbalance in the national economy.
To determine the state of general economic equilibrium means to find out under what conditions all participants in a market economy will be able to realize their intended goals. Therefore, economic equilibrium corresponds not only to a certain volume and structure of the supply of goods, but also to the satisfaction of each participant in market transactions with the implementation of his plans.
General economic equilibrium means that by changing the volume and structure of purchases or sales, no one will be able to improve their well-being in the current conditions.
2. The theory of general economic equilibriumL. Walras
IN economic science There are many models of macroeconomic equilibrium that reflect the views of different directions of economic thought on this problem:
F. Quesnay model of simple reproduction using the example of economics France XVIII centuries;
K. Marx schemes of simple and expanded capitalist social reproduction;
L. Walras model of general economic equilibrium under the law of free competition;
V. Leontiev “input-output” model;
J. Keynes model of short-term economic equilibrium.
Let us consider in more detail the model of general economic equilibrium of L. Walras.
According to some researchers in the field of the history of economic thought, L. Walras (1834-1910) is the greatest economist of the nineteenth century. He earned such recognition for developing a system of general market equilibrium, which was called the closed model of economic equilibrium, set out in his main work “Elements of Pure Political Economy” (1874).
General equilibrium presupposes the establishment of equilibrium in exchange and production. Equilibrium in exchange means that the effective (actual) demand for productive services (products) is equal to the effective supply of productive services (products). Equilibrium in production means that the price of each product is equal to the costs of its production, which includes normal profit as a reward for capital.
Such a state of equilibrium in production and exchange is an ideal case, not a real one. It never happens that the selling price of a product is absolutely exactly equal to the costs of producing this good, just as there is no exact correspondence between effective demand and effective supply. But such a state can be called normal in the sense that an economy operating in conditions of absolutely free competition tends to it. In such a situation, if the price of a product exceeds the cost of its production, entrepreneurs receive excess profits and begin to expand production. If the price of a product is lower than the cost of its production, entrepreneurs suffer losses and begin to reduce output. As a result, the prices of final goods change and a general equilibrium is established. Agapova I. History of economic thought. - M., 2008. - P. 126
For the sake of simplicity, let us consider a model of a barter economy in which there is no production.
There are n goods in this economy, and n-e good acts as a unit of account, or money. The price of each good is expressed in this unit of account.
Let Pi/Pn be the price of the i-th good divided by the price of the n-th good (relative price).
Suppose that Pn =1, then the price of the i-th good will be equal to Pi.
At the beginning of the exchange, each economic entity has a certain reserve (allotment) of various goods, including money. The total utility of this stock depends on the marginal utility of each good at the individual's disposal. The individual's goal is to maximize his utility. He can achieve this by exchanging goods belonging to him with less marginal utility for goods belonging to other individuals and representing greater utility to him. Naturally, the marginal utility of each good is weighed taking into account its relative price (Gossen’s second law), as well as the relative prices of other goods. Consequently, the demand for the i-th good, as well as the supply of this good, are functions of the relative prices of all goods:
Di = Di(P1,..., Pn-1);
Si = Si(P1,..., Pn-1).
General economic equilibrium means that supply and demand in each market are equal, that is, the amount of a good put up for sale is equal to the amount of the good that buyers want to purchase. The equality of these quantities is ensured due to the relative price of the good. The equilibrium price is established in the Walrasian model during the so-called “groping” process.
There is a special person in the market - the auctioneer - who observes the progress of affairs in the economy and shouts out the relative prices of goods. Then the participants in the exchange tell the auctioneer how much of a particular good they would like to sell or buy at given prices. If demand is not equal to supply (there is excess demand (Di > Si) or excess supply (Di< Si), аукционщик назначает новые цены. Причем здесь действует следующее правило: если был избыток спроса, - цена повышается, если избыток предложения, - цена понижается. Обмен состоится только тогда, когда набор относительных цен, объявленный аукционщиком, окажется равновесным.
Mathematically, to find this set consisting of n-1 prices, it is necessary to solve the n-1 equation (the price of the n-th good - money - is given):
Di(P1,..., Pn-1) = Si(P1,..., Pn-1); i = 1,.. n-1.
The number of equations here is equal to the number of unknowns, and therefore this system will have a unique solution, i.e., an equilibrium set of relative prices exists and it is unique. From here we can derive the so-called Walras law:
which states that the value of aggregate demand is equal to the value of aggregate supply. Galperin V. M. Macroeconomics. - St. Petersburg, 2005. - P. 124 In other words, the sum of excess demand and supply in all markets should always be equal to zero. Therefore, if n-1 is in equilibrium (i.e., there is neither excess demand nor excess supply for any of them), then nth market must also be in balance.
Thus, Walras's law does not at all imply that the economy is always in equilibrium, that is, there is no excess demand or supply in all markets. It’s just that at the level of the entire national economy, all these surpluses “cancel out each other” in value terms.
As can be seen from the general equilibrium model, money plays in it the passive role of a counting unit (measure of value), in which the value of other goods is expressed. Here it is necessary to distinguish between relative and absolute prices. Relative price is the price of one good relative to the price of another good. The absolute price is the price of money (Pn), or the general price level. Business entities are only interested in relative prices. The absolute price depends on the amount of money in circulation. A change in the money supply leads to a proportional change in the absolute price level. Thus, if the quantity of money triples, then absolute prices must triple: the price of each good triples, and relative prices remain unchanged. As a consequence, a change in the money supply does not entail a change in real values (demanded and offered quantities of goods). Tarasevich L.S., Grebennikov P.I., Leussky A.I. Macroeconomics. - M., 2006. - P. 47.
In addition, due to the actions of the auctioneer, in the Walrasian general equilibrium model, purchases and sales turn out to be absolutely synchronized in time. Therefore, business entities have no incentive to use money as a medium of exchange and a store of value. Thus, using the model of L. Walras it is impossible to explain the existence of money in a market economy.
Walras's model, although logically complete, is too abstract in nature, since it excludes many important elements of real economic life.
In addition to the lack of accumulation, some of the oversimplifications include:
The static nature of the model (assumes the invariability of the stock and range of products, as well as the invariance of production methods and consumer preferences),
The assumption of the existence of perfect competition and ideal awareness of production subjects.
In other words, the problems of economic growth, innovation, changes in consumer tastes, and economic cycles remained outside the scope of the Walras model. Walras's merit lies more in posing the problem than in solving it. It gave impetus to economic thought to search for models of dynamic equilibrium and economic growth. We find the development of Walras's ideas in the works of the American economist V. Leontiev, whose algebraic theory of analysis of the input-output model in the forties of the twentieth century made it possible to numerically solve large systems of equations, called “balance equations”. However, the first economist who studied the issues of dynamic development within the framework of neoclassical theory was J. Schumpeter.
However, Leon Walras's model became the basis for the entire theory of economic equilibrium in the neoclassical school. Even those who would later criticize neoclassical theory used models based on the model of L. Walras, making the necessary changes to it.
“Complete economic equilibrium is the structural optimum of the economic system, to which society strives, but never fully achieves it due to the constant change in the optimum itself, the ideal of proportionality.” Samuelson P. Economics. - M., 2005. - P. 159.
The fundamental possibility of achieving general equilibrium under conditions of perfect competition in mathematical form was first proven by L. Walras. Expressing the OER model by a system of equations, he proved that in an economic system consisting of n interconnected markets, nth market There will always be equilibrium if equilibrium is achieved in the (n-1)th market. It should be noted that L. Walras's model has been subject to critical analysis by many authors.
Conclusion
In this course work We examined the main theoretical and practical aspects associated with general economic equilibrium. Based on the study, the following conclusions can be drawn.
The most important economic problem is the problem of general economic equilibrium - a choice in which the way in which limited productive resources (labor, land, capital) are used to create various goods and their distribution among various members of society are balanced.
General economic equilibrium, as defined by Walras, “is a state in which the effective supply and effective demand for productive services are equalized in the market for services, the effective supply and effective demand for products are equalized in the market for products, and, finally, the selling price is equal to the cost of production, expressed in productive services."
Typically, equilibrium is achieved by either limiting needs (in the market they always appear in the form of effective demand) or increasing and optimizing the use of resources.
As you know, the economy is in constant motion, continuous development: cycle phases, market conditions, incomes change, and shifts in demand occur. All this suggests that the equilibrium state can only conditionally be considered static.
General economic equilibrium is the balance of the entire economy of the country, a system of interconnected and mutually agreed upon proportions in all spheres, industries, in all markets, among all participants in economic activity, ensuring the normal development of the national economy.
Using the analysis of the Walras model, we determined that in a state of market equilibrium, aggregate demand is equal to aggregate supply.
Literature
1. Agapova I. History of economic thought. - M., 2008;
2. Galperin V. M. Macroeconomics. - St. Petersburg, 2005;
3. Grebennikov P.I., Leussky A.I., Tarasevich L.S. Macroeconomics. - M., 2006;
4. Kamaev V.D. Textbook on the basics of economic theory. - M., 2007;
5. Raizberg B.A., Lozovsky L.Sh., Starodubtseva E.B. Modern economic dictionary. - M., 2006;
6. Samuelson P. Economics. - M., 2005;
7. Sidorovich A.V. Economic theory. - M., 2008;
8. Storchevoy M.A. Fundamentals of Economics / ed. P.A. Vatnik. - St. Petersburg, 2009;
9. Tarasevich L.S., Grebennikov P.I., Leussky A.I. Macroeconomics. - M., 2006.
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In economics, there are many models of macroeconomic equilibrium, reflecting the views of different directions of economic thought on this problem:
- - F. Quesnay model of simple reproduction using the example of the French economy of the 18th century;
- - K. Marx schemes of simple and expanded capitalist social reproduction;
- - L. Walras model of general economic equilibrium under the law of free competition;
- - V. Leontiev “input-output” model;
- - J. Keynes model of short-term economic equilibrium.
Let us consider in more detail the model of general economic equilibrium of L. Walras.
According to some researchers in the field of the history of economic thought, L. Walras (1834-1910) is the greatest economist of the nineteenth century. He earned such recognition for developing a system of general market equilibrium, which was called the closed model of economic equilibrium, set out in his main work “Elements of Pure Political Economy” (1874).
General equilibrium presupposes the establishment of equilibrium in exchange and production. Equilibrium in exchange means that the effective (actual) demand for productive services (products) is equal to the effective supply of productive services (products). Equilibrium in production means that the price of each product is equal to the costs of its production, which includes normal profit as a reward for capital.
Such a state of equilibrium in production and exchange is an ideal case, not a real one. It never happens that the selling price of a product is absolutely exactly equal to the costs of producing this good, just as there is no exact correspondence between effective demand and effective supply. But such a state can be called normal in the sense that an economy operating in conditions of absolutely free competition tends to it. In such a situation, if the price of a product exceeds the cost of its production, entrepreneurs receive excess profits and begin to expand production. If the price of a product is lower than the cost of its production, entrepreneurs suffer losses and begin to reduce output. As a result, the prices of final goods change and a general equilibrium is established. Agapova I. History of economic thought. - M., 2008. - P. 126
For the sake of simplicity, let us consider a model of a barter economy in which there is no production.
In this economy there are n goods, and the nth good acts as a unit of account, or money. The price of each good is expressed in this unit of account.
Let Pi/Pn be the price of the i-th good divided by the price of the n-th good (relative price).
Suppose that Pn =1, then the price of the i-th good will be equal to Pi.
At the beginning of the exchange, each economic entity has a certain reserve (allotment) of various goods, including money. The total utility of this stock depends on the marginal utility of each good at the individual's disposal. The individual's goal is to maximize his utility. He can achieve this by exchanging goods belonging to him with less marginal utility for goods belonging to other individuals and representing greater utility to him. Naturally, the marginal utility of each good is weighed taking into account its relative price (Gossen’s second law), as well as the relative prices of other goods. Consequently, the demand for the i-th good, as well as the supply of this good, are functions of the relative prices of all goods:
Di = Di(P1,..., Pn-1);
Si = Si(P1,..., Pn-1).
General economic equilibrium means that supply and demand in each market are equal, that is, the amount of a good put up for sale is equal to the amount of the good that buyers want to purchase. The equality of these quantities is ensured due to the relative price of the good. The equilibrium price is established in the Walrasian model during the so-called “groping” process.
There is a special person in the market - the auctioneer - who observes the progress of affairs in the economy and shouts out the relative prices of goods. Then the participants in the exchange tell the auctioneer how much of a particular good they would like to sell or buy at given prices. If demand is not equal to supply (there is excess demand (Di > Si) or excess supply (Di< Si), аукционщик назначает новые цены. Причем здесь действует следующее правило: если был избыток спроса, - цена повышается, если избыток предложения, - цена понижается. Обмен состоится только тогда, когда набор относительных цен, объявленный аукционщиком, окажется равновесным.
Mathematically, to find this set consisting of n-1 prices, it is necessary to solve the n-1 equation (the price of the n-th good - money - is given):
Di(P1,..., Pn-1) = Si(P1,..., Pn-1); i = 1,.. n-1.
The number of equations here is equal to the number of unknowns, and therefore this system will have a unique solution, i.e., an equilibrium set of relative prices exists and it is unique. From here we can derive the so-called Walras law:
which states that the value of aggregate demand is equal to the value of aggregate supply. Galperin V. M. Macroeconomics. - St. Petersburg, 2005. - P. 124 In other words, the sum of excess demand and supply in all markets should always be equal to zero. Therefore, if n-1 is in equilibrium (that is, there is neither excess demand nor excess supply in any of them), then the nth market must also be in equilibrium.
Thus, Walras's law does not at all imply that the economy is always in equilibrium, that is, there is no excess demand or supply in all markets. It’s just that at the level of the entire national economy, all these surpluses “cancel out each other” in value terms.
As can be seen from the general equilibrium model, money plays in it the passive role of a counting unit (measure of value), in which the value of other goods is expressed. Here it is necessary to distinguish between relative and absolute prices. Relative price is the price of one good relative to the price of another good. The absolute price is the price of money (Pn), or the general price level. Business entities are only interested in relative prices. The absolute price depends on the amount of money in circulation. A change in the money supply leads to a proportional change in the absolute price level. Thus, if the quantity of money triples, then absolute prices must triple: the price of each good triples, and relative prices remain unchanged. As a consequence, a change in the money supply does not entail a change in real values (demanded and offered quantities of goods). Tarasevich L.S., Grebennikov P.I., Leussky A.I. Macroeconomics. - M., 2006. - P. 47.
In addition, due to the actions of the auctioneer, in the Walrasian general equilibrium model, purchases and sales turn out to be absolutely synchronized in time. Therefore, business entities have no incentive to use money as a medium of exchange and a store of value. Thus, using the model of L. Walras it is impossible to explain the existence of money in a market economy.
Walras's model, although logically complete, is too abstract in nature, since it excludes many important elements of real economic life.
In addition to the lack of accumulation, some of the oversimplifications include:
- - static nature of the model (assumes the invariability of the stock and range of products, as well as the invariance of production methods and consumer preferences),
- - the assumption of the existence of perfect competition and ideal awareness of production subjects.
In other words, the problems of economic growth, innovation, changes in consumer tastes, and economic cycles remained outside the scope of the Walras model. Walras's merit lies more in posing the problem than in solving it. It gave impetus to economic thought to search for models of dynamic equilibrium and economic growth. We find the development of Walras's ideas in the works of the American economist V. Leontiev, whose algebraic theory of analysis of the input-output model in the forties of the twentieth century made it possible to numerically solve large systems of equations, called “balance equations”. However, the first economist who studied the issues of dynamic development within the framework of neoclassical theory was J. Schumpeter.
However, Leon Walras's model became the basis for the entire theory of economic equilibrium in the neoclassical school. Even those who would later criticize neoclassical theory used models based on the model of L. Walras, making the necessary changes to it.
“Complete economic equilibrium is the structural optimum of the economic system, to which society strives, but never fully achieves it due to the constant change in the optimum itself, the ideal of proportionality.” Samuelson P. Economics. - M., 2005. - P. 159.
The fundamental possibility of achieving general equilibrium under conditions of perfect competition in mathematical form was first proven by L. Walras. Expressing the OER model as a system of equations, he proved that in an economic system consisting of n interconnected markets, there will always be equilibrium in the nth market if equilibrium is achieved in the (n-1)th market. It should be noted that L. Walras's model has been subject to critical analysis by many authors.
The first economist to construct a general equilibrium model was JI. Walras. The national economy, according to Walras, consists of I households consuming n varieties of goods, for the production of which m different factors of production are used.
Households' preferences for goods and factors of production are given by their utility functions. The consumer's budget is formed as a result of the sale of factors of production belonging to him. Market supply and demand curves are formed as a result of the addition of individual functions.
Based on the derived utility functions, budget constraints, market demand and supply, Walras presented a general equilibrium model consisting of three groups of equations that show:
1) equilibrium conditions in the goods market: , where Q j is the quantity j-th good (j = 1, n) consumed by all households;
2) equilibrium conditions in factor markets: , where F t is the amount of the t-factor of production ( t= 1, t), available to all households;
3) budgetary restrictions of firms in conditions of perfect competition in the form of equality of total revenue to total costs:
P j = g w:val="EN-US"/>
The system of equations contains (2 n + m- 1) independent equations. If the incomes of consumers are known, then by substituting the real values of pennies into the equations, we obtain the amount of goods and services exchanged. JI. Walras, solving a system of equations, made two important conclusions:
1) in the absence of general economic equilibrium, the sum of surpluses in some markets is equal to the sum of deficits in others;
2) if a certain price system ensures equilibrium in any three markets, then equilibrium will be observed in the fourth market. This conclusion was called Walras' law.
Let's consider the Walras model using a specific example.
Example 9.2
Suppose that one product is produced - crackers, and only flour and sugar are consumed to produce them. We denote the demand for crackers by Q, and take the price of crackers equal to one. Technological coefficients are given in the table.
The volumes of supply of flour and sugar are given by the formulas
q 1 = 2+r 1 ; q 2 = 6+2 r 2 .
Based on the data available in the problem statement, we write:
a) equilibrium equation for the cracker production industry: 1 = 0.25r 1 + 0.5r 2 ;
b) demand equation for flour and sugar: q 1 = 0.25 Q, q 2 =0.5Q. Let us solve a system of five equations, assuming that the volumes
products and resources are expressed in thousands of tons. As a result, we find that in a state of general equilibrium, the industry produces 16 thousand tons of crackers, while 4 thousand tons of flour and 8 thousand tons of sugar are consumed.
Let's consider mathematical modeling of the market according to Walras. The original concepts of the Walras model are:
· disaggregation of market participants: individual consumers and individual producers are considered;
· perfection of competition;
general balance.
The latter concept means considering the equilibrium for all goods at once, rather than for individual goods. Consequently, the Walras model introduces the concept of general equilibrium (i.e., equilibrium for all goods).
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 costs (primary factors). To distinguish them, costs are given a negative sign (therefore we write , 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 - letter and manufacturer indices - letter . Let us denote the vector of prices of goods.
Entering the market, each consumer or producer becomes simultaneously a buyer of some goods and a seller of other goods. Consumer, i.e. a market participant “not directly involved 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 products and buys primary factors 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 proposal finished products and the vector function of cost demand. However, in the Walras model, a somewhat generalized characteristic of the manufacturer is used - using one set interpreted as the set of his (optimal) production plans. In input-output language, this set can be defined as follows: where is the production function. Obviously, .
Therefore, under mathematical model market understand a set of elements:
(4.3.1)
where is the space of prices of goods, N- 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 elements of the aggregate (4.3.1) here is somewhat different from that which was characterized when considering the consumer and production sectors in isolation.
First, the vector 
contains prices of both final consumption goods and inputs. Next, we will proceed from price variability. Moreover, prices do not change 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 the 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 will be different for different products. Therefore, numbers 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) proceeds from the sale of his initial stock of goods, 2) income received from his participation in the profits of the production sector (denoted, for example, by purchasing securities and other types of investment and labor activity. Thus we assume that
/ (4.3.2)
In the Walrasian model, it is assumed that all income from the production 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 componentwise.
Fourth, the supply and demand functions are assumed to be vector and multivalued. For example, for a function, the first property means that 
where is the scalar demand function for the ith product (see (2.5.3)). The second property means that a function to each p matches more than one vector 
and the set of such vectors, i.e. 
This occurs, for example, when in relation (2.5.2), which determines demand, the maximum is achieved not only at one point.
In the Walras model, the concepts of aggregate supply and demand are formalized as follows.
Definition 4.1. Function of aggregate (market) demand
(4.3.3)
Function of aggregate (market) supply called a multivalued function
(4.3.4)
With this definition, the meaning of aggregate demand fully corresponds to the method of forming market demand based on solutions to the optimization problems of individual consumers. Specifically, it is the sum of individual consumer demand functions. The definition of the aggregate supply function requires additional explanation. For this purpose, we introduce the following notation:
By definition, any element of the set Y can be represented by the vector , where Since there are many optimal plans of the manufacturer j, then the components of the vector are the optimal volumes of output and costs, and they all constitute a solution to the same optimization problem. Thus, part of the vector components, like vectors, reflects the supply of finished products, and part - the demand for primary factors. Therefore, a vector cannot be uniquely called a sentence. At the same time, the vector can be interpreted as an aggregate supply, since the part of the vector components corresponding to demand is “compensated” by the vector b.
Let us show that for any p and , i.e. the area of change of aggregate functions is the same space as for individual functions. Let's first consider two consumers. For any, the 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 Sum of vector and set
Let's consider three consumers. Plenty for anyone 
is formed by shifting the set 
in the direction of the vector x by the length of this vector. Therefore
Continuing these arguments, we get
The inclusion is established in exactly the same way: Since and because , the set is formed by shifting the set Y in the direction of the vector b by the length of this vector. That's why 
Having formalized the concepts of functions of aggregate demand and supply, the market model (4.3.1) can be represented by a set of the form
(4.3.5)
Any vector is called aggregate demand (corresponding to the price vector p); any vector - aggregate supply (corresponding to the price vector p). These vectors are the (optimal) reactions of the aggregate buyer and aggregate seller to the price vector established in the market. If at the same time, then there is a shortage of goods on the market, and if, their surplus appears. 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 economics is equality. This ideal case does not always occur in practice. Therefore, it is advisable to somehow weaken it. Walras's model allows for the most humane version of the generalization of the concept of economic equilibrium from the point of view of consumer interests.
Definition 4.2. A set of vectors is called competitive equilibrium on the market (4.3.5) if
(4.3.6)
(4.3.8)
In this case it is called the equilibrium price vector.
By definition of the functions of aggregate supply and demand, from inclusions (4.3.6) it follows
Where 
Where 
those. aggregate supply and demand are formed as the total values of individual demands of consumers and individual offers 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)
Let us consider the economic content of the conditions that determine 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 way. This is clearly seen from relations (4.3.9) and (4.3.10). Condition (4.3.7) ensures that aggregate supply is not less than 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 holds 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
Let us assume that for some product in (4.3.7) there is a strict inequality: . Then in cost terms we obtain the inequality 
not meeting condition (4.3.8). Magnitude 
called surplus. According to the law of supply, if there is a surplus, the price of a product must be reduced. But this will lead to a change in the equilibrium price. Let's find a way out of this contradiction? Obviously,
Consequently, to restore condition (4.3.8) it is necessary to eliminate the surplus. Taking into account the sign, this is only possible if But then
And 
those. product k generally excluded from circulation on the market.
The justification of fairness (4.3.8) by the fact that a product supplied in excess of existing demand receives a zero price is economically meaningful, but cannot be adequately formalized. 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 overproduced product to be equal to zero. Purely theoretically, this technique is sound, since it does not lead to further contradictions.
At the same time, it should be recognized that there is no economically meaningful explanation for the existence of a zero-price product. Declaring such a product free seems untenable. Strictly speaking, there are no free goods in the economy; any by-product can be used, i.e. has a non-zero price. We also cannot agree with the modification of the law of supply and demand, well known to economists, about the existence of overproduced goods with a zero price, since in the case of overproduction the demanded 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).
A market model according to Walras has been built. As can be seen, 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 produced goods and satisfying the demand of all consumers. The process of market price formation can be roughly compared to the work of a certain algorithm consisting of four blocks (Fig. 4.6).
Rice. 4.6 Scheme for the formation of equilibrium prices
In the first block, a price vector is formed. Vector information p enters the blocks and , in which the sets and are formed, respectively, the contents of which, in turn, are transferred to the block . The block performs pairwise comparison of elements 
. If there is a pair or pairs for which the condition is satisfied (or conditions (4.3.7), (4.3.8)), then the process ends. Otherwise prices p are rejected, which sends a signal 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 associated with 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) for the formation of market prices that converges to the equilibrium price.
The existence of equilibrium in the Walras model has not been established. The reason lies in the level of formalism of this model - it is very abstract. By specifying the definitions of its constituent elements and clarifying their functional properties, it is possible to obtain various modifications of the Walras model. The most famous of them is called the Arrow-Debreu model, after the names of its creators.
The problem of developing numerical methods for calculating equilibrium prices is associated with establishing necessary and sufficient signs of equilibrium. It is necessary that they be constructive, i.e. generated a convergent iterative procedure, such as, for example, the web-shaped model (see Fig. 4.2).
Walrasian equilibrium
Marshall equilibrium
The equilibrium price is formed for the following reasons:
The demand price coincides with the supply price in the case of equilibrium volumes of supply and demand.
16. General concept of elasticity. Elasticity formula
Elasticity- the degree of reaction of one quantity to a change in another.
Elasticity coefficient- assessment of elasticity in percentage terms; the ratio of the percentage change in one quantity to the percentage change in another.
Demand elasticity value– elasticity can be measured using coefficient. elast.
Elasticity =((Q1 – Q2) / (Q1+Q2)) / ((P2 – P1) / (P1 + P2)). P1 – price before the change, P2 – price after the change, Q1 – quantity demanded before the change, Q2 – quantity demanded after the change.
Price elastic demand (P): changes in demand for a product depend on changes in its price. Three types of price elasticity:
1) Elastic demand (with a slight drop in price, sales volume rises). E>1.
2). Unit elasticity (% price change is equal to % change in sales volume). E = 1.
3).Inelastic demand (when the price changes, there is no significant change in sales.).E<1
Factors of elastic demand.
1. Indispensability - if there is a substitute, therefore demand will be more elastic).
2. The importance of goods for the consumer. - (inelasticity is the demand for essential goods).
3. Share of income and expenses. (Goods are elastic if a significant share of funds is spent on them).
4. Time frame – elasticity increases over the long term and becomes less elastic over short periods of time.
Change in total revenue with the same changes in the price of the product. with different elast. demand:
1. With elastic demand, a fall in price causes such an increase in sales volume, which leads to an increase in total revenue.
2. When demand has a unitary elasticity, the increase in sales volume with a decrease in prices is such that total revenue remains unchanged.
3. With inelastic demand, a fall in price leads to a small increase in sales, and the volume of total revenue decreases.
Elast. demand by income- as the ratio of the percentage change in the volume of demand for a product to the percentage change in income (I): E D =Q\Q: I\I. Since the consumer changes the demand for various goods differently when income changes, the indicator can have different positive and negative values. If the consumer increases the volume of purchases with an increase in income, then elast. in terms of income is positive (E I >0; we are talking about a standard normal product, for example, an additional pair of shoes). If demand growth outstrips income growth high elasticity. by income (E I >1; durable goods: cars, computer - people spend on their savings or take out a loan). If, as income increases, demand decreases E I<0 - аномальные или низкокачественные товары (дешевые сорта колбасы).
P 
cross elasticity.
Eсross=Q A \Q A: P B \P B . Q A - volume of demand for product A, P B - price of product B
Elasticity of supply- the sensitivity of supply volume to changes in market prices, or - the degree of change in the quantity of goods and services offered for sale in response to changes in market prices.
Factors: time factor, raw material prices, salary level, etc.
Price elasticity coefficient of supply (Es) = Qa\Q: P\P
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