This note introduces a new mathematical tool, nonstandard analysis, for the analysis of an important class of problems in mathematical economics-the relation between bargaining and the competitive price system.

limit theorem A theorem stating that the core of an economy shrinks to the set of Walrasian equilibria as the number of agents increases to infinity. That This glossary of economics is a list of definitions containing terms and concepts used in economics, its sub-disciplines, and related fields. competition law Also called an antitrust law or anti-monopoly law. Any law that promotes or seeks to maintain market competition by regulating anti-competitive conduct by companies. economic equilibrium A situation in which economic forces such as supply and demand are balanced and in which, in the absence of external influences, the values of economic variables do not change. For example, in the standard textbook model of perfect competition, equilibrium occurs at the point at which quantity demanded and quantity supplied are equal. Market equilibrium in this case is a condition in which a market price is established through competition such that the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers. This price is often called the competitive price or market clearing price and will tend not to change unless demand or supply changes, and the quantity is called the "competitive quantity" or market clearing quantity. However, the concept of equilibrium in economics also applies to imperfectly competitive markets, where it takes the form of a Nash equilibrium. Edgeworth's limit theorem A theorem stating that the core of an economy shrinks to the set of Walrasian equilibria as the number of agents increases to infinity. That is, among all possible outcomes which may result from free market exchange or barter between groups of people, while the precise location of the final settlement (the ultimate division of goods) between the parties is not uniquely determined, as the number of traders increases, the set of all possible final settlements converges to the set of Walrasian equilibria. full employment output (Y*) How accelerator effect A positive effect on private fixed investment because of the growth of the market economy. Rising GDP usually implies that profit expectations and business confidence rise, encouraging businesses to build more factories and other buildings and to install more machinery. capitalism An economic system based on the private ownership of the means of production and their operation for profit. Central characteristics of capitalism include capital accumulation, competitive markets, price systems, private property, property rights recognition, voluntary exchange, and wage labor. competition law Also called an antitrust law or anti-monopoly law. Any law that promotes or seeks to maintain market competition by regulating anti-competitive conduct by companies. competitive market A market in which many sellers compete against each other to attract customers. Each seller has an incentive to sell at the lowest price possible to attract customers, so prices tend to be driven so low that the sellers can just barely make a profit. crowding out A phenomenon that occurs when increased government involvement in a sector of a market economy substantially affects the remainder of the market, either on the supply or demand side of the market. economic calculation problem (ECP) A criticism of using economic planning as a substitute for market-based allocation of the factors of production. It is argued that economy planning necessarily leads to an irrational and inefficient allocation of resources. economic equilibrium A situation in which economic forces such as supply and demand are balanced and in which, in the absence of external influences, the values of economic variables do not change. For example, in the standard textbook model of perfect competition, equilibrium occurs at the point at which quantity demanded and quantity supplied are equal. Market equilibrium in this case is a condition in which a market price is established through competition such that the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers. This price is often called the competitive price or market clearing price and will tend not to change unless demand or supply changes, and the quantity is called the "competitive quantity" or market clearing quantity. However, the concept of equilibrium in economics also applies to imperfectly competitive markets, where it takes the form of a Nash equilibrium. economic growth An increase in the inflation-adjusted market value of the goods and services produced by an economy over time. It is conventionally measured as the percent rate of increase in real gross domestic product, or real GDP. Edgeworth's limit theorem A theorem stating that the core of an economy shrinks to the set of Walrasian equilibria as the number of agents increases to infinity. That is, among all possible outcomes which may result from free market exchange or barter between groups of people, while the precise location of the final settlement (the ultimate division of goods) between the parties is not uniquely determined, as the number of traders increases, the set of all possible final settlements converges to the set of Walrasian equilibria. full employment output (Y*) How much output is produced in the economy when full employment exists in the labor market. household The sector of the economy which purchases goods from the product market and sells labor, land, and entrepreneurship ability to the factor market in the circular flow market. market economy An economy in which almost all economic activity happens in markets, with little or no interference by the government; often referred to as a laissez-faire ("leave alone") economic system. mixed economy An economic system blending elements of a market economy with elements of a planned economy, free markets with state interventionism, or private enterprise with public enterprise.

The main result of this paper is in some sense (weaker assumptions, other proof methods are necessary) an equivalence theorem of Aumann's type: Under assumptions , the set of all private value allocations in the economy E coincides with the set of competitive allocations. E belongs to an (exactly defined) class of exchange economies. The point of the proof of the equivalence is, to derive from E a nonatomic coalitional game (the market game) and to use its properties, especially the core, and then saddle point theorems, since a Slater condition can be shown to be fullfilled. As a closing remark it is said, that the assumptions for the utility functions of the traders can be weakened: there is a version of the equivalence theorem, where the utility functions are strictly increasing not on the entire commodity space but in its interior and vanish on its boundary.

We consider exchange economies with a continuum of agents and differential information about finitely many states of nature. It was proved in Einy, Moreno and Shitovitz (2001) that if we allow for free disposal in the market clearing (feasibility) constraints then an irreducible economy has a competitive (or Walrasian expectations) equilibrium, and moreover, the set of competitive equilibrium allocations coincides with the private core. However when feasibility is defined with free disposal, competitive equilibrium allocations may not be incentive compatible and contracts may not be enforceable (see e.g. Glycopantis, Muir and Yannelis (2002)). This is the main motivation for considering equilibrium solutions with exact feasibility. We first prove that the results in Einy et al. (2001) are still valid without free-disposal. Then we define an incentive compatibility property motivated by the issue of contracts’ execution and we prove that every Pareto optimal exact feasible allocation is incentive compatible, implying that contracts of a competitive or core allocations are enforceable.

The paper considers the problem of allocating private goods in a society whose members act in the sole pursuit of self-interest but recognize the value of adopting equitable allocation rules. We define the Equitable Bargaining set as a solution concept that combines the concern for equitability with the need to prevent possible deviations from agents seeking better opportunities. Our definition grounds on a two-step veto mechanism as in Mas-Colell (1989), but it considers only objections and counter-objections based on equitable agreements between agents. We study conditions under which our notion of bargaining set provides insights on other solution concepts. We show that the Equitable Bargaining set coincides with that of Mas-Colell when the underlying economy is atomless, but not in general. Then we provide two sets of conditions for economies with market imperfections that apply to finite economies and to mixed market economies. In the first case our conditions imply that the Equitable Bargaining set is a subset of the core, and so it converges to the set of competitive allocations if the economy is replicated. In the second case, we show that all allocations in the Equitable Bargaining set are competitive, extending the Walras-bargaining equivalence of Mas-Colell (1989) to the framework of mixed markets. All the conditions we use follow from well-established assumptions from the literature in finite and mixed market economies.

The paper shows that the aspiration core of a TU-game coincides with the set of competitive wages arising in a labor market economy in which time is indivisible, but workers and firms can sign contingent labor contracts and trade in employment lotteries. The set of firms that are active in the market is endogenously determined at equilibrium and it coincides with the generating collection of the corresponding aspiration core allocation.