Еconomic equilibrium


Экономическая теория и математическое моделирование

Partial equilibrium is an equilibrium in the one market of goods, services, factors of production (analyzes only a part of the market, ceteris paribus). Example: supply and demand model is a partial equilibrium model...



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Partial equilibrium is an equilibrium in the one market of goods, services, factors of production (analyzes only a part of the market, ceteris paribus). Example: supply and demand model is a partial equilibrium model (prices of all substitutes and complements, and income levels of consumers are constant).

General equilibrium explains the equilibrium of supply, demand and prices in the whole economy (macroeconomic equilibrium) with several or many interacting markets. General economic equilibrium is a structural optimum of the economic system. The society aspires to this optimum, but never reaches it entirely, because of the constant change of the optimum / ideal equilibrium (Самуэльсон П. Экономика. - М.: Инфра-М, 2005. с. 159). General economic equilibrium is a condition of economy, when all markets are simultaneously in equilibrium state and every agent maximizes its target function (e.g. reaches its aim) – И.В.Гальперин. Микроэкономика.

In frames of neoclassical school Léon Walras was the first, who in his book Elements of Pure Economics (1874) attempted to model prices for a whole economy. He proved that general equilibrium is comparable with the economic system, where conditions of perfect competition exist in every market (that’s why economists call this model as a competitive equilibrium model). It means that if all the buyers and sellers are price-takers, one can find a price system in which all markets are simultaneously in equilibrium, and each actor of these markets maximizes his objective function in given constraints.

In the Walrasian model general equilibrium is a result of solution of system of equations, where unknown variables are prices of all goods and factors of production and their quantity, bought and sold by every consumer and producer. Equations reflect maximizing behavior of consumers and producers. Some of them (behavioral equations) show a function of supply and demand of all customers in all markets; other equations are equations market clearance conditions, i.e. equations of their equilibrium. Such system of equations can be solved if the number of independent equations equals the number of unknown variables in the system. Walras said it.

However, “number of independent equations = number of unknown variables” is only the necessary but not sufficient condition for solution of the equations of general equilibrium. Walras and his followers could not prove the existence of general equilibrium. In the real world, non-perfect competitive markets dominate and production processes are characterized by indivisibility, scientists still can not to state if general exists or not. Nevertheless, the theory of general equilibrium is a very important branch of microeconomics, since the system of perfect competitive markets provides an efficient allocation of resources in the economy.

Introduction of Walras’ model of general economic equilibrium using the function of excess demand (ED), not the function of supply and demand, which usually determines the market equilibrium, although there is a direct connection between two approaches. Demand for some goods is a function of prices of all other goods, income and number of consumers. For a given amount of income and consumer demand function for i-commodity is a function of all prices for m-commodities. It can be represented as

QDi = Di(P1, ┘, Pi, ┘, Pm), i = 1, 2, ..., т.

In a perfectly competitive market, supply of a product (let’s say i-commodity) is also a function of all prices for all m-commodities.1 It can be represented as

QSi = Si(P1, ┘, Pi, ┘, Pm), i= 1, 2, ..., m

Than the ED function for i-commodity can be represented as a function of the diminution between demand and supply function. Let’s denote the excess demand for i-commodity as Ei. Than

Ei = Di(P1, ..., Pi, ..., Pm) - Si(P1, ..., Pi, ..., Pm)

The excess demand curve can be constructed by subtracting the horizontal supply curve of the demand curve. Let’s analyze the figure, where the functions of supply and demand are Di and Si. When price is P* volumes of supply and demand are equal and, therefore, ED=0. At any higher price the value of supply exceeds the value of demand, so that excess demand is negative. Finally, if price is P1, when the value of supply equals zero, the excess demand equals its entire value. Obviously, on linear functions of supply and demand the linear function will be the function of excess demand. Thus, while the functions of supply and demand is

QD = А - аР и QS = B - bР,

The function of excess demand will be

EQ = (А - В) - (а + b)Р

And reversed function will be

EP = A' - a'Q,

Where А' = (А - В)(а + b); а' = 1/(а + b).

The function of excess demand has some specificities, which make its use more convenient than the usual demand and supply functions, while analyzing  competitive equilibrium. The definition and function of excess demand allows to consider supply as a negative excess demand, and demand - as a positive surplus.

Conditions of equilibrium:

  1.  Excess demand must equal zero

EDi(P1, ..., Pm) = 0

  1.  SPiEDi(P1, ┘, Pm) = 0

This is the Walras’ Law (If all markets except one (i.e. m-1 markets) are in equilibrium, then the remaining (m-1)th market is also in equilibrium. This means that the number of independent equations in the system is m - 1.

  1.  It’s impossible to solve the system consisting of m -1 independent equations with respect to m variables. However, the number of m variables can be reduced by 1, by selecting one commodity as the unit of account (French “numйraire”) and dividing all prices by P1. Then EDi(P1, ..., Pm) = 0 will be

EDi(1, P2/P1, P3/P1, ┘, Pm/P1) = 0. 

Thus, we have a system consisting of m - 1 equations of the form of EDi(1, P2/P1, P3/P1, ┘, Pm/P1) = 0, which admits the only solution regarding (m – 1)th price.

General competitive equilibrium and Pareto efficiency/optimality.

Pareto efficiency is a state of the system, in which the value of each particular criteria, which described the state of the system, can not be improved without deterioration of other elements.

The definition of Pareto efficiency consists of three components. In other words one can state necessary conditions of Pareto optimal state of economy.

  1.  efficiency in the allocation of benefits between consumers (efficiency in exchange) – if it’s impossible to redistribute the benefits so, that welfare of at least one consumer has increased without decreasing the welfare of others.


  1.  efficiency in production - if it’s impossible to increase production of one or more products without reducing the production of others.


  1.  efficiency in the structure of production - If it is impossible to increase the welfare of at least one individual without reducing the welfare of others, by changing the structure (combination) of produced goods.


All three conditions are implemented in conditions of the perfect competition, not only for two consumers or two companies, but also for a large number of them.

The fact that a general competitive equilibrium and Pareto optimality assume the fulfillment of the same conditions means that there is a close relationship between them, which is summarized in two basic theorems of the theory of social welfare.

The first theorem of the theory of social welfare states that in a condition of general equilibrium the allocation of economic resources is Pareto-optimal.

The second fundamental theorem of the theory of social welfare: upon conditions that all indifference curves and isoquants are dome shaped, respective to the origin of coordinates, for any Pareto-efficient allocation of resources, there is a price system, which provides a general economic equilibrium.

Graph: http://microeconomica.economicus.ru/img/ru/galperin/galperin_w16_3.gif

Maximum utility of B is achieved in point Eb. As a result, deficit occurs in the market of X good; and there is excess in the market of Y good. Thus, the dome shape of indifference curves is necessary for any Pareto-efficient allocation of goods for possibility to find the price system, providing a general competitive equilibrium.

Goods differ in whether they are excludable and whether they are rival. A good is excludable if it is possible to prevent someone from using it. A good is rival if one person’s enjoyment of the good prevents other people from enjoying the same unit of the good. Markets work best for private goods, which are both excludable and rival. Markets do not work as well for other types of goods. 

Public goods are neither excludable nor rival. Examples of public goods include fireworks displays, national defense, system of justice, street light and the creation of fundamental knowledge. Because people are not charged for their use of the public good, they have an incentive to free ride when the good is provided privately. Therefore, governments provide public goods (by means of private and legal entities’ taxes), making their decision about the quantity based on cost-benefit analysis.

Common-pool resources (CPR), also called common property resource is a type of good consisting of a natural or human-made resource system (e.g. an irrigation system or fishing grounds), whose size or characteristics makes it costly, but not impossible, to exclude potential beneficiaries from obtaining benefits from its use. Unlike pure public goods, common pool resources face problems of congestion or overuse, because they are subtractable. A common-pool resource typically consists of a core resource (e.g. water or fish), which defines the stock variable, while providing a limited quantity of extractable fringe units, which defines the flow variable. While the core resource is to be protected or entertained in order to allow for its continuous exploitation, the fringe units can be harvested or consumed. Examples: irrigation systems, fishing grounds, pastures, forests, water, the atmosphere. Because their core resources are vulnerable, common-pool resources are generally subject to the problems of congestion, overuse, pollution, and potential destruction unless harvesting or use limits are devised and enforced. Common-pool resources may be owned by national, regional or local governments as public goods, by communal groups as common property resources, or by private individuals or corporations as private goods. When they are owned by no one, they are used as open access resources. Having observed a number of common pool resources throughout the world, Elinor Ostrom noticed that a number of them are governed by common property regimes — arrangements different from private property or state administration — based on self-management by a local community. Her observations contradict claims that common-pool resources should be privatized or else face destruction in the long run due to collective action problems leading to the overuse of the core resource (see: Tragedy of the commons).

Public economics (or economics of the public sector) is the study of government policy through the lens of economic efficiency and equity. At its most basic level, public economics provides a framework for thinking about whether or not the government should participate in economics markets and to what extent its role should be. In order to do so, microeconomic theory is utilized to assess whether the private market is likely to provide efficient outcomes in the absence of governmental interference. Inherently, this study involves the analysis of government taxation and expenditures. This subject encompasses a host of topics including market failures, externalities, and the creation and implementation of government policy. Public economics builds on the theory of welfare economics and is ultimately used as a tool to improve social welfare. Broad methods and topics include: analysis and design of public policy; public-finance theory and its application; distributional effects of taxation and government expenditures; analysis of market failure and government failure; optimal taxation; the theory of public goods and so on.

The condition of equilibrium in the sphere of public goods production:


It means that the amount of private good, Y, which must be given for the production of an additional unit of public good, X, must be equal to the sum of all the good Y, which consumers want to give up (without changing their welfare or utility) for the consumption of one additional unit of public good , X. In other words, the production of public goods is Pareto efficient if the sum of individual marginal rates of substitution of private social benefits equals the marginal rate of product transformation. For more than two, the number of customers MRSAXY + MRSBXY = MRPTXY can be generalized as


See the graph for the equilibrium: http://microeconomica.economicus.ru/index1.php?file=17_3

The need for government regulation appears not only because of imperfection of particular market (non-uniqueness of equilibrium, its instability, incomplete accounting of costs and benefits), but also because of the need to solve macroeconomic problems (fight with inflation, aspiration to the full employment, the combination of the principles of economic efficiency and social justice and others). Such regulation may have an aim to stabilize equilibriums or its shift, forth coming to equilibrium or distancing from it. It can be implemented by the direct control over the level of prices and volumes of the market (establishment of neccessary government price or market quotas) by using financial instruments (taxes and subsidies), some other methods.

In economics, the term economic efficiency refers to the use of resources so as to maximize the production of goods and services. An economic system is said to be more efficient than another (in relative terms) if it can provide more goods and services for society without using more resources. In absolute terms, a situation can be called economically efficient if:

  •  No one can be made better off without making someone else worse off (commonly referred to as Pareto efficiency).
  •  No additional output can be obtained without increasing the amount of inputs.
  •  Production proceeds at the lowest possible per-unit cost.

One of alternative criteria of economic efficiency is a social welfare function. In economics, a social welfare function is a real-valued function that ranks conceivable social states (alternative complete descriptions of the society) from lowest to highest.

There is a problem of financing of public goods production. If the private firm appoints a price for the good for covering its expenses, the government imposes taxes. Besides, the issue of determining the optimal amount of public goods, which is Pareto efficient, emerges. Determination of the volume of production of public goods requires collective action. They are require to identify public preferences for solving free-rider problem. In particular, the identification of social preferences in the public sector is conducted by means of a mechanism of social choice. In conditions of political democracy this mechanism is voting.

So, the most important issues in the state provision of public goods are: funding for the production of public goods; determination of the optimal production volume; Identifying public preferences; solution of free rider problem.

Free-rider problem is a problem of consumption of public goods without the appropriate fee for it. In the consumption of public goods involved many people. In this case everyone gets incentive not to pay for a provided benefit; everyone hopes to "travel without a ticket." Imagine that everyone believes his ability to pay for the benefit is small compared to the total cost of the public good, and his decision to pay or not pay will not affect the collective decision. It makes no sense to pay. In addition, one hopes that his good will be made in any case, if many people will not paid. If all the consumers of the public good supports this argument, none of them will pay, and the public good will be not produced.

Public finance is the study of the role of the government in the economy.

The sphere of public finance is considered to be threefold: governmental effects on (1) efficient allocation of resources, (2) distribution of income, and (3) macroeconomic stabilization.

Economists classify government expenditures into three main types: government consumption (government purchases of goods and services for current use), government investment (government purchases of goods and services intended to create future benefits), transfer payments (government expenditures that are not purchases of goods and services, and instead just represent transfers of money).

Government operations are those activities involved in the running of a state or a functional equivalent of a state (for example, tribes, secessionist movements or revolutionary movements) for the purpose of producing value for the citizens.

Income distribution is some forms of government expenditure are specifically intended to transfer income from some groups to others.

Governmental expenditures include pensions, public education, health care, public financing of electoral campaigns and so on.

Governments receive their revenues for the sake of financing their expenditures from several sources: government revenue (non-tax revenue (revenue from government-owned corporations, sovereign wealth funds, Seigniorage) and taxes), governments borrowing, printing of money and inflation, privatization.

In economics, an externality, or transaction spillover, is a cost or benefit that is not transmitted through prices and is incurred by a party who did not agree to the action causing the cost or benefit. The cost of an externality is a negative externality, or external cost, while the benefit of an externality is a positive externality.

Externalities may emerge among consumers, among producers and during interaction between consumers and producers. An example of a negative externality may be dumping waste into the river, used for fishing and swimming. An example of positive externalities can be threes in the park positively influencing for the utility of environment and people.

During determining of the volume of production, consumption, sales or purchases of market participants pursue their private interests only and do not take into account externalities, both negative and positive. Therefore, they produce too much goods which production is accompanied by negative externalities, and goods, which production is accompanied by positive externalities, by contrast, is too small. As a result, Pareto-efficiency of the structure of production  is not achieved.

Market failure is a concept within economic theory describing when the allocation of goods and services by a free market is not efficient. That is, there exists another conceivable outcome where a market participant may be made better-off without making someone else worse-off. So, the outcome is not Pareto optimal. Market failures can be viewed as scenarios where individuals' pursuit of pure self-interest leads to results that are not efficient – that can be improved upon from the societal point-of-view. The term was used for the first time in Francis M. Bator’s article “The Anatomy of Market Failure” (1958), however the concept has been traced back to the Victorian philosopher Henry Sidgwick.

Usually researchers discern these categories of market failures: information asymmetries, non-competitive markets, externalities, public goods, monopolies, natural monopolies, bounded rationality.

The existence of a market failure is often used as a justification for government intervention in a particular market. Some types of government policy interventions, such as taxes, subsidies, bailouts, wage and price controls, and regulations, including attempts to correct market failure, may also lead to an inefficient allocation of resources, sometimes called government failure. Thus, there is sometimes a choice between imperfect outcomes, i.e. imperfect market outcomes with or without government interventions. But either way, if a market failure exists the outcome is not Pareto efficient. Mainstream neoclassical and Keynesian economists believe that it may be possible for a government to improve the inefficient market outcome, while several heterodox schools of thought disagree with this.

For example, see (table): http://fictionbook.ru/static/bookimages/04/01/24/04012425.bin.dir/h/i_014.jpg

Until the middle of the XX century the most famous and popular ways of addressing the external effects were: 1) internalization of externalities and 2) determination of the taxes.

Internalization of external effects can be achieved by the merger of enterprises which produce different goods. In this case, the private marginal cost of one enterprise is identical to the social marginal cost of other one. Reducing its production the joint venture will reduce external costs and internalize other parts, including these expenditures in joint-venture’s private expenditures. Probable inefficiency of scale limits the borders of internalization.

English economist Arthur Pigou suggested to implement the tax for every additional unit of production made by organization generating negative externalities. Pigou considered that governments are able to diminish the gap between private and public goods by support or restriction  investments in different spheres. Subsidies and taxes are the most important ways of support and restrictions.

In practice, it’s difficult to determine the external expenditures for setting the Pigouvian tax rate, especially because these expenditures can differ significantly in different companies. Besides, the external effect will be felt differently in densely populated and sparsely populated areas.

The traditional approach to solve the problem of external effects (internalization and the Pigouvian tax) remained dominant until I960, when the American economist Ronald Coase wrote an article "The problem of social cost." He showed that the problem of externalities has reciprocal nature.

R. Coase: "The question is usually understood as “A damages B, and there is a need to decide how to restrict actions of A”. But this is false. We have a problem of mutually obliging nature. We damage A, while avoid the damage for B. Realistic problem to decide is whether to allow A to damage B or should B allowed to damage A? The problem is how to avoid the most serious injury"

If rights of property are not established, negative externalities appeares during the competition between different variants of using of resources. Coase conclusion subsequently called as the Coase theorem. Externalities can be internalized through the attaching of property rights on objects (generators of these externalities) and the exchange of those rights if it is not associated with high transaction costs. If these rights are well defined and marketable, the market mechanism can bring the parties to the effective agreement. If the firm has a legal right to pollute the environment, those who suffer from the pollution may purchase the right on the minimum of pollutions from the company. If inhabitants of surrounding areas have right for a clean environment, the firm can buy from them a permission for its pollution.

Theorem of Coase (example of negative externality emerged because of plants’ contamination of water also used for fishing): http://microeconomica.economicus.ru/img/ru/galperin/galperin_w17_4.gif

The solution suggested by Coase is particularly attractive for those economists who are inclined to underestimate the importance of government intervention in the economy in general and to the solution of the problems caused by external effects, in particular. But Coase theorem has several weaknesses.

First, the Coase theorem requires the cost of the negotiations not to be as high as to become almost irresistible barrier for achieving an effective agreement.

Secondly, the Coase decision can be made if the resource owners can identify the sources of the damage and legally prevent this damage. Even if the right to clean air will be legally stated, it is unclear how it will be possible to identify those, whose activity causes the ozone hole and acid rain, and in what extent these negative externalities should be applied to different economic entities.

The solution suggested by Coase is more applicable, when the limited number of participants are involved and the sources of negative externalities are easily determined.

Public choice theory is the use of modern economic tools to study problems that traditionally are in the province of political science. From the perspective of political science, it is the subset of positive political theory that models voters, politicians, and bureaucrats as mainly self-interested.

In particular, it studies such agents and their interactions in the social system either as such or under alternative constitutional rules. These can be represented in a number of ways, including standard constrained utility maximization, game theory, or decision theory. Public choice analysis has roots in positive analysis ("what is") but is often used for normative purposes ("what ought to be"), to identify a problem or suggest how a system could be improved by changes in constitutional rules, the subject of constitutional economics.

Public choice theory is intimately related to social choice theory, which uses mathematical tools to study voting and voters. Much early work had aspects of both, and both use the tools of economics and game theory. Since voter behavior influences the behavior of public officials, public choice theory often uses results from social choice theory.

The modern literature in Public Choice began with Duncan Black, who in 1948 identified the underlying concepts of what would become median voter theory. He also wrote The Theory of Committees and Elections in 1958. Gordon Tullock refers to him as the "father of public choice theory".

Public choice theory is often used to explain how political decision-making results in outcomes that conflict with the preferences of the general public. For example, many advocacy groups (lobbyists) are not desire of the overall democracy.

One way to organize the subject matter studied by public choice theorists is to begin with the foundations of the state itself. According to this procedure, the most fundamental subject is the origin of government. Although some work has been done on anarchy, autocracy, revolution, and even war, the bulk of the study in this area has concerned the fundamental problem of collectively choosing constitutional rules.

Another major sub-field is the study of bureaucracy. The usual model depicts the top bureaucrats as being chosen by the chief executive and legislature, depending on whether the democratic system is presidential or parliamentary. The typical image of a bureau chief is a person on a fixed salary who is concerned with pleasing those who appointed him. The latter have the power to hire and fire him more or less at will. The bulk of the bureaucrats, however, are civil servants whose jobs and pay are protected by a civil service system against major changes by their appointed bureau chiefs. This image is often compared with that of a business owner whose profit varies with the success of production and sales, who aims to maximize profit, and who can in an ideal system hire and fire employees at will.

A field that is closely related to public choice is "rent-seeking". Rent-seeking is broader than Public Choice in that it applies to autocracies as well as democracies and, therefore, is not directly concerned with collective decision-making. However, the obvious pressures it exerts on legislators, executives, bureaucrats, and even judges are factors that public choice theory must account for in its analysis of collective decision-making rules and institutions. Moreover, the members of a collective who are planning a government would be wise to take prospective rent-seeking into account.

Public Choice Theory has been developed largely in the context of democratic political systems of the variety that exist in Europe and North America. There’s only one book (by Muzaffar Ali Isani), which characterizes rules and institutions in LDCs.

There are also critical views on democracy. One of the basic claims that results from public choice theory is that good government policies in a democracy are an underprovided public good, because of the rational ignorance of the voters. Each voter is faced with a tiny probability that his vote will change the result of the elections, while gathering the relevant information necessary for a well-informed voting decision requires substantial time and effort. Therefore, the rational decision for each voter is to be generally ignorant of politics and perhaps even abstain from voting. Rational choice theorists claim that this explains the gross ignorance of most citizens in modern democracies as well as low voter turnout. Rational abstention creates the so-called "Paradox of voting" in which a strict cost-benefit analysis implies that no one should vote.

There is a need to identify and accord individual preferences for identifying a social welfare function. Voting (based on egalitarian principle “1 man – 1 vote”) is one of the simplest and logical way of doing it.

It’s obvious that “1 man – 1 vote” principle doesn’t reflect differences in intensities of individual preferences.

The voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive), even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.

Director’s Law (by American economist Aaron Director) was an important consequence of this theorem. Director’s Law holds that governments of democratic countries conduct redistribution policies in favor of the middle class.

Kenneth Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox (1951) states that, when voters have three or more distinct alternatives (options), no voting system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specific set of criteria. These criteria are called unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. K. Arrow: it’s impossible to create an algorithm for collective decision-making that meets all these requirements.

More on political mechanisms of aggregation of preferences for public goods: http://microeconomica.economicus.ru/index.php?file=16_4


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