Walras's law

Last updated

Walras's law is a principle in general equilibrium theory asserting that budget constraints imply that the values of excess demand (or, conversely, excess market supplies) must sum to zero regardless of whether the prices are general equilibrium prices. That is:

Contents

where is the price of good j and and are the demand and supply respectively of good j.

Walras's law is named after the economist Léon Walras [1] of the University of Lausanne who formulated the concept in his Elements of Pure Economics of 1874. [2] Although the concept was expressed earlier but in a less mathematically rigorous fashion by John Stuart Mill in his Essays on Some Unsettled Questions of Political Economy (1844), [3] Walras noted the mathematically equivalent proposition that when considering any particular market, if all other markets in an economy are in equilibrium, then that specific market must also be in equilibrium. The term "Walras's law" was coined by Oskar Lange [4] to distinguish it from Say's law. Some economic theorists [5] also use the term to refer to the weaker proposition that the total value of excess demands cannot exceed the total value of excess supplies.

Definitions

Walras's law

Walras's law is a consequence of finite budgets. If a consumer spends more on good A then they must spend and therefore demand less of good B, reducing B's price. The sum of the values of excess demands across all markets must equal zero, whether or not the economy is in a general equilibrium. This implies that if positive excess demand exists in one market, negative excess demand must exist in some other market. Thus, if all markets but one are in equilibrium, then that last market must also be in equilibrium.

This last implication is often applied in formal general equilibrium models. In particular, to characterize general equilibrium in a model with m agents and n commodities, a modeler may impose market clearing for n – 1 commodities and "drop the n-th market-clearing condition." In this case, the modeler should include the budget constraints of all m agents (with equality). Imposing the budget constraints for all m agents ensures that Walras's law holds, rendering the n-th market-clearing condition redundant. In other words, suppose there are 100 markets, and someone saw that 99 are in equilibrium [note 1] , they would know the remaining market must also be in equilibrium without having to look.

In the former example, suppose that the only commodities in the economy are cherries and apples, and that no other markets exist. This is an exchange economy with no money, so cherries are traded for apples and vice versa. If excess demand for cherries is zero, then by Walras's law, excess demand for apples is also zero. If there is excess demand for cherries, then there will be a surplus (excess supply, or negative excess demand) for apples; and the market value of the excess demand for cherries will equal the market value of the excess supply of apples.

Walras's law is ensured if every agent's budget constraint holds with equality. An agent's budget constraint is an equation stating that the total market value of the agent's planned expenditures, including saving for future consumption, must be less than or equal to the total market value of the agent's expected revenue, including sales of financial assets such as bonds or money. When an agent's budget constraint holds with equality, the agent neither plans to acquire goods for free (e.g., by stealing), nor does the agent plan to give away any goods for free. If every agent's budget constraint holds with equality, then the total market value of all agents' planned outlays for all commodities (including saving, which represents future purchases) must equal the total market value of all agents' planned sales of all commodities and assets. It follows that the market value of total excess demand in the economy must be zero, which is the statement of Walras's law. Walras's law implies that if there are n markets and n – 1 of these are in equilibrium, then the last market must also be in equilibrium, a property which is essential in the proof of the existence of equilibrium.

Formal statement

Consider an exchange economy with agents and divisible goods.

For every agent , let be their initial endowment vector and their Marshallian demand function (demand vector as a function of prices and income).

Given a price vector , the income of consumer is . Hence, their demand vector is .

The excess demand function is the vector function:

Walras's law can be stated succinctly as:

This can be proven using the definition of excess demand:

The Marshallian demand is a bundle that maximizes the agent's utility, given the budget constraint. The budget constraint here is:

for each

Hence, all terms in the sum are 0 so the sum itself is 0. [6] :317–318

Implications

Labor market

Neoclassical macroeconomic reasoning concludes that because of Walras's law, if all markets for goods are in equilibrium, the market for labor must also be in equilibrium. Thus, by neoclassical reasoning, Walras's law contradicts the Keynesian conclusion that negative excess demand and consequently, involuntary unemployment, may exist in the labor market, even when all markets for goods are in equilibrium. The Keynesian rebuttal[ dubious discuss ] is that this neoclassical perspective ignores financial markets, which may experience excess demand (such as a "liquidity trap")[ clarification needed ] that permits an excess supply of labor and consequently, temporary involuntary unemployment, even if markets for goods are in equilibrium.[ dubious discuss ][ citation needed ]

See also

Related Research Articles

<span class="mw-page-title-main">Supply and demand</span> Economic model of price determination in a market

In microeconomics, supply and demand is an economic model of price determination in a market. It postulates that, holding all else equal, the unit price for a particular good or other traded item in a perfectly competitive market, will vary until it settles at the market-clearing price, where the quantity demanded equals the quantity supplied such that an economic equilibrium is achieved for price and quantity transacted. The concept of supply and demand forms the theoretical basis of modern economics.

In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium. General equilibrium theory contrasts with the theory of partial equilibrium, which analyzes a specific part of an economy while its other factors are held constant. In general equilibrium, constant influences are considered to be noneconomic, or in other words, considered to be beyond the scope of economic analysis. The noneconomic influences may change given changes in the economic factors however, and therefore the prediction accuracy of an equilibrium model may depend on the independence of the economic factors from noneconomic ones.

<span class="mw-page-title-main">Léon Walras</span> French mathematical economist (1834–1910)

Marie-Esprit-Léon Walras was a French mathematical economist and Georgist. He formulated the marginal theory of value and pioneered the development of general equilibrium theory. Walras is best known for his book Éléments d'économie politique pure, a work that has contributed greatly to the mathematization of economics through the concept of general equilibrium. The definition of the role of the entrepreneur found in it was also taken up and amplified by Joseph Schumpeter.

<span class="mw-page-title-main">Budget constraint</span> All available goods and services a customer can purchase with respect to their income

In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within their given income. Consumer theory uses the concepts of a budget constraint and a preference map as tools to examine the parameters of consumer choices. Both concepts have a ready graphical representation in the two-good case. The consumer can only purchase as much as their income will allow, hence they are constrained by their budget. The equation of a budget constraint is where is the price of good X, and is the price of good Y, and m is income.

In economics, aggregate demand (AD) or domestic final demand (DFD) is the total demand for final goods and services in an economy at a given time. It is often called effective demand, though at other times this term is distinguished. This is the demand for the gross domestic product of a country. It specifies the amount of goods and services that will be purchased at all possible price levels. Consumer spending, investment, corporate and government expenditure, and net exports make up the aggregate demand.

Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my utility?" It is a type of optimal decision problem. It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending (income), the prices of the goods and their preferences.

In microeconomics, a consumer's Marshallian demand function is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the standard demand function. It is a solution to the utility maximization problem of how the consumer can maximize their utility for given income and prices. A synonymous term is uncompensated demand function, because when the price rises the consumer is not compensated with higher nominal income for the fall in their real income, unlike in the Hicksian demand function. Thus the change in quantity demanded is a combination of a substitution effect and a wealth effect. Although Marshallian demand is in the context of partial equilibrium theory, it is sometimes called Walrasian demand as used in general equilibrium theory.

In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.

There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal. The requirements for perfect competition are these:

  1. There are no externalities and each actor has perfect information.
  2. Firms and consumers take prices as given.
<span class="mw-page-title-main">Walrasian auction</span>

A Walrasian auction, introduced by Léon Walras, is a type of simultaneous auction where each agent calculates its demand for the good at every possible price and submits this to an auctioneer. The price is then set so that the total demand across all agents equals the total amount of the good. Thus, a Walrasian auction perfectly matches the supply and the demand.

A Lindahl tax is a form of taxation conceived by Erik Lindahl in which individuals pay for public goods according to their marginal benefits. In other words, they pay according to the amount of satisfaction or utility they derive from the consumption of an additional unit of the public good. Lindahl taxation is designed to maximize efficiency for each individual and provide the optimal level of a public good.

<span class="mw-page-title-main">Local nonsatiation</span> Consumer preferences property

In microeconomics, the property of local nonsatiation (LNS) of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is strictly preferred to it.

The Sonnenschein–Mantel–Debreu theorem is an important result in general equilibrium economics, proved by Gérard Debreu, Rolf Mantel, and Hugo F. Sonnenschein in the 1970s. It states that the excess demand curve for an exchange economy populated with utility-maximizing rational agents can take the shape of any function that is continuous, has homogeneity degree zero, and is in accordance with Walras's law. This implies that the excess demand function does not take a well-behaved form even if each agent has a well-behaved utility function. Market processes will not necessarily reach a unique and stable equilibrium point.

Competitive equilibrium is a concept of economic equilibrium, introduced by Kenneth Arrow and Gérard Debreu in 1951, appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal standard by which other market structures are evaluated.

In microeconomics, excess demand, also known as shortage, is a phenomenon where the demand for goods and services exceeds that which the firms can produce.

In economics and consumer theory, a linear utility function is a function of the form:

Fisher market is an economic model attributed to Irving Fisher. It has the following ingredients:

In theoretical economics, an abstract economy is a model that generalizes both the standard model of an exchange economy in microeconomics, and the standard model of a game in game theory. An equilibrium in an abstract economy generalizes both a Walrasian equilibrium in microeconomics, and a Nash equilibrium in game-theory.

Market equilibrium computation is a computational problem in the intersection of economics and computer science. The input to this problem is a market, consisting of a set of resources and a set of agents. There are various kinds of markets, such as Fisher market and Arrow–Debreu market, with divisible or indivisible resources. The required output is a competitive equilibrium, consisting of a price-vector, and an allocation, such that each agent gets the best bundle possible given the budget, and the market clears.

In theoretical economics, an Arrow–Debreu exchange market is a special case of the Arrow–Debreu model in which there is no production - there is only an exchange of already-existing goods. An Arrow–Debreu exchange market has the following ingredients:

References

  1. Barron, John M.; Ewing, Bradley T.; Lynch, Gerald J. (2006), Understanding macroeconomic theory, Taylor & Francis, p. 1, ISBN   978-0-415-70195-2
  2. "Walras' Law". Investopedia . Retrieved March 17, 2015.
  3. Ariyasajjakorn, Danupon (2007), Trade, foreign direct investment, technological change, and structural change in labor usage, ProQuest, p. 55, ISBN   978-0-549-30654-2
  4. Lange, O. 1942. Say's law: A restatement and criticism. In Lange, O., F. McIntyre, and T. O. Yntema, eds., Studies in Mathematical Economics and Econometrics, in Memory of Henry Schultz, pages 49–68. University of Chicago Press, Chicago.
  5. Florenzano, M. 1987. On an extension of the Gale–Nikaido–Debreu lemma. Economics Letters 25(1):51–53.
  6. Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN   0-393-95735-7.
  1. Or whatever value of N-1 out of N total markets.