A **blockchain** is a growing list of records, called *blocks*, that are linked together using cryptography.^{[1]}^{[2]}^{[3]}^{[4]} Each block contains a cryptographic hash of the previous block, a timestamp, and transaction data (generally represented as a Merkle tree). The timestamp proves that the transaction data existed when the block was published in order to get into its hash. Blocks contain the hash of the previous block, forming a chain, with each additional block reinforcing the ones before it. Therefore, blockchains are resistant to modification of their data because once recorded, the data in any given block cannot be altered retroactively without altering all subsequent blocks. Continue reading “Blockchain”

### Statistics

**Statistics** is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.^{[1]}^{[2]}^{[3]} In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as “all people living in a country” or “every atom composing a crystal”. Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.^{[4]} Continue reading “Statistics”

### Drawdown (economics)

The **drawdown** is the measure of the decline from a historical peak in some variable (typically the cumulative profit or total open equity of a financial trading strategy).^{[1]}

Somewhat more formally, if {\textstyle X(t),\;t\geq 0} is a stochastic process with {\textstyle X(0)=0}, the drawdown at time {\displaystyle T}, denoted {\textstyle D(T)}, is defined as: Continue reading “Drawdown (economics)”

### Arrow–Debreu model

In mathematical economics, the **Arrow–Debreu model** suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.^{[1]}

The model is central to the theory of general (economic) equilibrium and it is often used as a general reference for other microeconomic models. It is named after Kenneth Arrow, Gérard Debreu,^{[2]} and sometimes also Lionel W. McKenzie for his independent proof of equilibrium existence in 1954^{[3]} as well as his later improvements in 1959.^{[4]}^{[5]}

The A-D model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium (or Walrasian equilibrium) of an economy. In general, there may be many equilibria; however, with extra assumptions on consumer preferences, namely that their utility functions be strongly concave and twice continuously differentiable, a unique equilibrium exists. With weaker conditions, uniqueness can fail, according to the Sonnenschein–Mantel–Debreu theorem.

## Convex sets and fixed points

In 1954, McKenzie and the pair Arrow and Debreu independently proved the existence of general equilibria by invoking the Kakutani fixed-point theorem on the fixed points of a continuous function from a compact, convex set into itself. In the Arrow–Debreu approach, convexity is essential, because such fixed-point theorems are inapplicable to non-convex sets. For example, the rotation of the unit circle by 90 degrees lacks fixed points, although this rotation is a continuous transformation of a compact set into itself; although compact, the unit circle is non-convex. In contrast, the same rotation applied to the convex hull of the unit circle leaves the point *(0,0)* fixed. Notice that the Kakutani theorem does not assert that there exists exactly one fixed point. Reflecting the unit disk across the y-axis leaves a vertical segment fixed, so that this reflection has an infinite number of fixed points.

### Non-convexity in large economies

The assumption of convexity precluded many applications, which were discussed in the *Journal of Political Economy* from 1959 to 1961 by Francis M. Bator, M. J. Farrell, Tjalling Koopmans, and Thomas J. Rothenberg.^{[6]} Ross M. Starr (1969) proved the existence of economic equilibria when some consumer preferences need not be convex.^{[6]} In his paper, Starr proved that a “convexified” economy has general equilibria that are closely approximated by “quasi-equilbria” of the original economy; Starr’s proof used the Shapley–Folkman theorem.^{[7]}

## Economics of uncertainty: insurance and finance

Compared to earlier models, the Arrow–Debreu model radically generalized the notion of a commodity, differentiating commodities by time and place of delivery. So, for example, “apples in New York in September” and “apples in Chicago in June” are regarded as distinct commodities. The Arrow–Debreu model applies to economies with maximally complete markets, in which there exists a market for every time period and forward prices for every commodity at all time periods and in all places.^{[citation needed]}

The Arrow–Debreu model specifies the conditions of perfectly competitive markets.

In financial economics the term “Arrow–Debreu” is most commonly used with reference to an Arrow–Debreu security. A canonical Arrow–Debreu security is a security that pays one unit of numeraire if a particular state of the world is reached and zero otherwise (the price of such a security being a so-called “state price”). As such, any derivatives contract whose settlement value is a function on an underlying whose value is uncertain at contract date can be decomposed as linear combination of Arrow–Debreu securities.

Since the work of Breeden and Lizenberger in 1978,^{[8]} a large number of researchers have used options to extract Arrow–Debreu prices for a variety of applications in financial economics.^{[9]}

## See also

- Model (economics)
- Incomplete markets
- Fisher market – a simpler market model, in which the total quantity of each product is given, and each buyer comes only with a monetary budget.

## References

**^**Arrow, K. J.; Debreu, G. (1954). “Existence of an equilibrium for a competitive economy”.*Econometrica*.**22**(3): 265–290. doi:10.2307/1907353. JSTOR 1907353.**^**EconomyProfessor.com Archived 2010-01-31 at the Wayback Machine, Retrieved 2010-05-23**^**McKenzie, Lionel W. (1954). “On Equilibrium in Graham’s Model of World Trade and Other Competitive Systems”.*Econometrica*.**22**(2): 147–161. doi:10.2307/1907539. JSTOR 1907539.**^**McKenzie, Lionel W. (1959). “On the Existence of General Equilibrium for a Competitive Economy”.*Econometrica*.**27**(1): 54–71. doi:10.2307/1907777. JSTOR 1907777.**^**For an exposition of the proof, see Takayama, Akira (1985).*Mathematical Economics*(2nd ed.). London: Cambridge University Press. pp. 265–274. ISBN 978-0-521-31498-5.- ^ Jump up to:
^{a}^{b}Starr, Ross M. (1969), “Quasi–equilibria in markets with non–convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37)”,*Econometrica*,**37**(1): 25–38, CiteSeerX 10.1.1.297.8498, doi:10.2307/1909201, JSTOR 1909201. **^**Starr, Ross M. (2008). “Shapley–Folkman theorem”. In Durlauf, Steven N.; Blume, Lawrence E. (eds.).*The New Palgrave Dictionary of Economics*.**4**(Second ed.). Palgrave Macmillan. pp. 317–318. doi:10.1057/9780230226203.1518. ISBN 978-0-333-78676-5.**^**Breeden, Douglas T.; Litzenberger, Robert H. (1978). “Prices of State-Contingent Claims Implicit in Option Prices”.*Journal of Business*.**51**(4): 621–651. doi:10.1086/296025. JSTOR 2352653.**^**Almeida, Caio; Vicente, José (2008). “Are interest rate options important for the assessment of interest risk?” (PDF).*Working Papers Series n. 179, Central Bank of Brazil*.

### Euro short-term rate (€STR) (Ofer Abarbanel online library)

**Euro short-term rate** (**€STR**) is a reference rate for the currency euro. The €STR is calculated by the European Central Bank (ECB) and is based on the money market statistical reporting of the Eurosystem. The working group on euro risk-free rates has recommended €STR as a replacement for the EMMI Euro Overnight Index Average (EONIA) as the Euro risk-free rate for all products and contracts. Continue reading “Euro short-term rate (€STR) (Ofer Abarbanel online library)”

### Commercial mortgage (Ofer Abarbanel online library)

A **commercial mortgage** is a mortgage loan secured by commercial property, such as an office building, shopping center, industrial warehouse, or apartment complex. The proceeds from a commercial mortgage are typically used to acquire, refinance, or redevelop commercial property. Continue reading “Commercial mortgage (Ofer Abarbanel online library)”

### Crossing of cheques (Ofer Abarbanel online library)

A **crossed cheque** is a cheque that has been marked specifying an instruction on the way it is to be redeemed. Continue reading “Crossing of cheques (Ofer Abarbanel online library)”

### BankTrack (Ofer Abarbanel online library)

**BankTrack** is a non-governmental organization working in the field of private sector banks and sustainability. BankTrack uses direct action, lobbying, and research to achieve its goals. BankTrack focuses primarily on the work of private sector banks and their involvement in projects that are a risk to the environment, society or human rights. Continue reading “BankTrack (Ofer Abarbanel online library)”

### Braco’s Banking House (Ofer Abarbanel online library)

**Braco’s Banking House** is a three-storey town house in Elgin, Moray in Scotland. The home and business place of banker William Duff of Braco from 1703 to 1722, the house has borne his name ever since. It was designated a Category A listed building in 1970. Continue reading “Braco’s Banking House (Ofer Abarbanel online library)”

### Banking in India (Ofer Abarbanel online library)

Modern **banking in India** originated in the last decade of the 18th century. Among the first banks were the Bank of Hindustan, which was established in 1770 and liquidated in 1829–32; and the General Bank of India, established in 1786 but failed in 1791.^{[1][2][3][4]} Continue reading “Banking in India (Ofer Abarbanel online library)”