economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...

, a complete market (aka Arrow-Debreu market or complete system of markets) is a market with two conditions: # Negligible

transaction costs In economics and related disciplines, a transaction cost is a cost in making any economic trade when participating in a market. Oliver E. Williamson defines transaction costs as the costs of running an economic system of companies, and unlike produ ...

and therefore also

perfect information In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market pr ...

, # there is a price for every asset in every possible state of the world In such a market, the complete set of possible bets on future states of the world can be constructed with existing

asset In financial accountancy, financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value ...

s without

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...

. Here, goods are state-contingent; that is, a good includes the time and state of the world in which it is consumed. For instance, an umbrella tomorrow if it rains is a distinct good from an umbrella tomorrow if it is clear. The study of complete markets is central to state-preference theory. The theory can be traced to the work of

Kenneth Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972. In economics ...

(1964),

Gérard Debreu Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize ...

(1959), Arrow & Debreu (1954) and

Lionel McKenzie Lionel Wilfred McKenzie (January 26, 1919 – October 12, 2010) was an American economist. He was the Wilson Professor Emeritus of Economics at the University of Rochester. He was born in Montezuma, Georgia. He completed undergraduate studies at ...

(1954). Arrow and Debreu were awarded the

Nobel Memorial Prize in Economics The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...

(Arrow in 1972, Debreu in 1983), largely for their work in developing the theory of complete markets and applying it to the problem of

general equilibrium 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 ov ...

States of the world

A state of the world is a complete specification of the values of all relevant variables over the relevant time horizon. A state-contingent claim, or state claim, is a contract whose future payoffs depend on future states of the world. For example, suppose you can bet on the outcome of a coin toss. If you guess the outcome correctly, you will win one dollar, and otherwise you will lose one dollar. A bet on heads is a state claim, with payoff of one dollar if heads is the outcome, and payoff of negative one dollar if tails is the outcome. "Heads" and "tails" are the states of the world in this example. A state-contingent claim can be represented as a payoff vector with one element for each state of the world, e.g. (payoff if heads, payoff if tails). So a bet on heads can be represented as ($1, −$1) and a bet on tails can be represented as (−$1, $1). Notice that by placing one bet on heads and one bet on tails, you have a state-contingent claim of ($0, $0); that is, the payoff is the same regardless of which state of the world occurs. The bet on a coin toss is a simplistic example but illustrates widely applicable concepts, especially in

finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...

. If markets are complete, it is possible to arrange a portfolio with any conceivable payoff vector. That is, the state claims available for purchase, represented as payoff vectors,

span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan ester ...

the payoff space. A ''pure security'' or simple

contingent claim In finance, a contingent claim is a derivative whose future payoff depends on the value of another “underlying” asset,Dale F. Gray, Robert C. Merton and Zvi Bodie. (2007). Contingent Claims Approach to Measuring and Managing Sovereign Credit Ri ...

is a state claim that pays off in only one state. Any state-contingent claim can be regarded as a collection of pure securities. A system of markets is complete if and only if the number of attainable pure securities equals the number of possible states. Formally, a market is complete with respect to a

trading strategy In finance, a trading strategy is a fixed plan that is designed to achieve a profitable return by going long or short in markets. The main reasons that a properly researched trading strategy helps are its verifiability, quantifiability, consistency ...

s

, if there exists a

self-financing trading strategy In financial mathematics, a self-financing portfolio is a portfolio having the feature that, if there is no exogenous infusion or withdrawal of money, the purchase of a new asset must be financed by the sale of an old one. Mathematical definition ...

s_0

such that at any time

t

, the returns of the two strategies,

s

and

s_0

are equal. This is equivalent to stating that for a complete market, all cash flows for a trading strategy can be replicated using a similar synthetic trading strategy. Because a trading strategy can be simplified into a set of simple

contingent claims In finance, a contingent claim is a derivative whose future payoff depends on the value of another “ underlying” asset,Dale F. Gray, Robert C. Merton and Zvi Bodie. (2007). Contingent Claims Approach to Measuring and Managing Sovereign Credit R ...

(strategies paying 1 in one state and 0 in every other state), a complete market can be generalized as the ability to replicate cash flows of all simple contingent claims.

Dynamically-complete market

In order for a market to be complete, it must be possible to ''instantaneously'' enter into any position regarding any future state of the market. In contrast, a market is called dynamically complete if it is possible to construct a self-financing trading strategy that will have the same cash-flow. In other words, a complete market allows you to place all of your bet at once, while a dynamically complete market may require that you execute subsequent trades after making your initial investment. The requirement that the strategy be self-financing means that subsequent trades must be cash-flow neutral (you cannot contribute or withdraw any additional funds). Any complete market is also dynamically complete.

References

{{Reflist

States of the world

Dynamically-complete market

See also

References

Further reading