Edgeworth theorem

The Edgeworth theorem (Edgeworth contraction) is a theorem by Francis Ysidro Edgeworth (born February 8, 1845 in Edgeworthstown, County Longford , Ireland , † February 13, 1926 in Oxford , Oxfordshire, England), the spectrum possible from free Describes the exchange of resulting allocations . According to the theorem, the determination of the result depends on the number of individuals acting in the market . As the number of market participants increases , the possible outcomes converge towards a certain allocation.

The theorem

"Contract without competition is indeterminate, contract with perfect competition is perfectly determinate, contract with more or less perfect competition is less or more indeterminate."

"The exchange without competition is indefinite, the exchange in perfect competition is exactly determined and the exchange in more or less perfect competition is more or less indefinite."

- Francis Ysidro Edgeworth : Mathematical Psychics

Origin of the theorem

Francis Ysidro Edgeworth described the theorem, which was later named after him, in his book "Mathematical Psychis" from 1881. He received the impetus for this from the book by William Stanley Jevons "Theory of Political Economy" and an anonymous criticism of the same in the "Saturday Review" (1872). In "Theory of Political Economy" Jevon explored the use of basic mathematics and psychology to analyze the exchange of goods in perfect markets. He referred to the assumption of the acting individuals as price takers and neglected the number of actors in the market. This was the void in Jevon's analysis that Edgeworth intended to fill. His aim was to explicitly include the number of market participants in the equilibrium analysis of markets. He managed to show to what extent competition or the number of market participants through exchange processes leads to an exchange allocation that is identical to the allocation of the market equilibrium with price-taking individuals.

Theoretical basics

With the help of the Edgeworth box , Edgeworth analyzes the exchange behavior of market participants and the resulting allocations . Edgeworth does not assume a perfect market . However, he assumes that transaction costs can be neglected and that the exchange takes place under normal competitive conditions . He found out that with a small number of market participants a spectrum of possible exchange results arises and the allocation would therefore be indefinite. An increase in the number of participating individuals reduces the spectrum due to internal competition until the result adjusts to a certain allocation with a sufficiently large number of market participants. This allocation is comparable to the equilibrium allocation on the market assuming price-taking market participants. If there are several individuals operating in the market, they can make agreements in order to strengthen their negotiating positions and thus increase their individual benefit . In the course of the negotiations, however, the coalitions formed in this way can also be broken up again by outsiders by attracting at least one of the coalition partners with better conditions.

Graphic 1: Edgeworth-Box
U _i : indifference curves (utility levels)
Point (x ^A ₁ , x ^A ₂ ), (x ^B ₁ , x ^B ₂ ): initial configuration

The Edgeworth Diagram

→ Main article : Edgeworth box

The Edgeworth diagram is used to analyze the exchange of two goods between two individuals (A, B). It allows the respective equipment and and preferences in the form of Indifferenzkurven U _A U _B in a diagram, consisting of two mirror-inverted composite amounts amounts diagrams to represent. Hence the name " Edgeworth-Box ". While the width of the box measures the total amount of the good , the height represents the total amount of the good . The consumption possibilities of individual A are measured from the bottom left and the consumption possibilities of individual B from the top right. Now every point inside the Edgeworth box represents a possible equipment of the respective individuals (diagram 1). ${\displaystyle x^{A}=(x_{1}^{A},x_{2}^{A})}$ ${\displaystyle x^{B}=(x_{1}^{B},x_{2}^{B})}$ ${\displaystyle x_{1}=(x_{1}^{A}+x_{1}^{B})}$ ${\displaystyle x_{2}=(x_{2}^{A}+x_{2}^{B})}$

Graphic 2: Exchange of two individuals
U _i : indifference curves (utility levels) contract curve

Exchange of two individuals

With the Edgeworth box as a basis, the trade or exchange of individuals (A, B) can be modeled well (Figure 2). Exchanges are made on a voluntary basis, i.e. nobody exchanges if a new bundle of goods does not provide them at least as well as the equipment in their possession. Every individual has certain preferences in the form of indifference curves. The further the indifference curve is from the respective origin, the higher the utility level U _i that it represents. An exchange with mutual consent only occurs if this exchange represents a Pareto improvement . A Pareto improvement does not make any of the individuals worse and at least one does better. At a point where a Pareto-improving exchange is no longer possible and consequently no more voluntary exchange takes place, there is a Pareto-efficient allocation. From the tangential condition it follows that there are many Pareto-efficient allocations within the Edgeworth box. The contract curve is created from this set of Pareto-efficient points . When exchanging two individuals, all points on the contract curve are possible final allocations. Thus, the final allocation that ultimately results is indeterminate and depends primarily on the respective negotiating skills and negotiating power of both individuals.

Graph 3:
2 types of contract curve

Note: In Edgeworth's original notation, the and represent the amounts exchanged and not the amounts consumed. For this reason, the contract curve is only located within the area enclosed by the indifference curves (Figure 3). ${\displaystyle (x_{1}^{A},x_{1}^{B})}$ ${\displaystyle (x_{2}^{A},x_{2}^{B})}$

Bargaining power

→ Main article : bargaining power

Bargaining power describes the relative strength of the bargaining position between people or organizations during a reconciliation of interests. The level of bargaining power depends on the individual initial equipment and the number of competing market participants. The higher the initial level and the lower the number of competing market participants, the higher the level of bargaining power. As soon as one of the negotiating partners has achieved monopoly power, he can choose any point on the contract curve as the swap allocation. In our case, B ₁ could thus choose the point . ${\displaystyle C}$

Exchange between more than two individuals

Here the number of market participants increases symmetrically, the individuals A ₂ and B _{2 are} added. The preferences of individual A ₂ are identical to those of individual A ₁ ; the same applies to individuals B ₁ and B ₂ . The weighted average of two separate bundles of the respective pairs (A ₁ , A ₂ ), (B ₁ , B ₂ ) would raise each individual to a higher level of utility. This is true according to Edgeworth's utilitarian distribution approach, because the distribution in equal parts among individuals with identical preferences leads to maximum benefit for them.

Figure 4: Increase in utility level
U _i : indifference curves (utility levels)
contract curve
price vector
U _A ': new utility level

With only two individuals it would have been possible for B ₁ to convince A _{1 to} swap at point by using his potentially high bargaining power . Thus, B ₁ could have received the entire benefit resulting from the exchange. A _1, on the other hand, would have remained at its original level of utility. When exchanging between more than two individual pairs, the four individuals A _1,2 and B _1,2 are initially in the market. Assuming B ₁ would try to exclude B ₂ and continue to trade with A ₁ at the point , both A _{i could} better position themselves by splitting their endowments due to convex indifference curves . The exchanged equipment of A ₁ plus the initial equipment of A ₂ increases the benefit level of both. Thus, the utility level of both A increases _i by the simple addition of A ₂ at point (Figure 4), even if A ₁ to the worst possible conditions at point exchanged. ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle P}$ ${\displaystyle C}$

Mathematical formulation

${\displaystyle A_{1,2}}$ have each ${\displaystyle x^{A}=(x_{1}^{A},x_{2}^{A})}$

${\displaystyle A_{1,2}}$ have each ${\displaystyle x^{B}=(x_{1}^{B},x_{2}^{B})}$

${\displaystyle (x_{1},x_{2})}$ let the exchanged sets at point C.

The result of the exchange at point C: at point P with each at point C with the quantities remain ${\displaystyle A_{i}}$ ${\displaystyle (x_{1}^{A}-{\frac {x_{1}}{2}},{\frac {x_{2}}{2}})}$ ${\displaystyle B_{1}}$ ${\displaystyle (x_{1},x_{2}^{B}-x_{2})}$ ${\displaystyle B_{2}}$ ${\displaystyle (0,x_{2}^{B})}$

Chart 5: Reduced contract curve (C * to C '*) due to a symmetrical increase in market participants

B ₂ cannot prevent this trade, but has incentives to make A _i a more advantageous offer. One of the A _i could accept this offer, which in turn would force B ₁ to reconsider its offer. Due to this internal competition , it is no longer possible to convince the A _{i to} swap at the point . The new worst possible dip for A _i is on the contract curve at a point that offers both A _i at least as high a level of utility as the point . As a result of increased competition, the B _i have lost part of their original bargaining power. The best exchange point on the contract curve that the B _{i can} still reach is the point . By dividing it equally between A ₁ and A ₂ , no other point between and can be reached. That implies as the new limit (constraint) of the contract curve. Conversely, the same approach can be applied to the point from the other side with A _i as the holder of the bargaining power . The contract curve and thus the amount of possible exchange results has been reduced to the curve segment from to (Figure 5). ${\displaystyle C}$ ${\displaystyle P}$ ${\displaystyle C^{*}}$ ${\displaystyle P}$ ${\displaystyle C^{*}}$ ${\displaystyle C^{*}}$ ${\displaystyle C'}$ ${\displaystyle C'^{*}}$ ${\displaystyle C^{*}}$

Exchange in perfect competition

Figure 6: Symmetrical increase in the number of individuals enables higher levels of benefit to be achieved

A similar analysis can be carried out with each additional pair (A, B). When considering the situation with three pairs each (A _i , B _i ) again swaps (A ₁ , A ₂ ) with (B ₁ , B ₂ ) at the point . If the exchanged equipment of (A ₁ , A ₂ ) plus the initial equipment of A _{3 is} divided equally between the A _i , the point shifts to , i.e. a higher indifference curve (Figure 6). All A _i thus achieve a higher benefit. With every additional pair (A, B), the point approaches the contract curve through the same division between the A _i . As a result, the benefit level of the A _{i increases} . ${\displaystyle C^{*}}$ ${\displaystyle P}$ ${\displaystyle P_{2}}$ ${\displaystyle P}$

Mathematical formulation

With is the "achievable" stretch ratio of about given : ${\displaystyle A_{i},B_{i};i=1,...,N}$ ${\displaystyle {\overline {EP_{i}}}}$ ${\displaystyle {\overline {EC_{i-1}^{*}}}}$ ${\displaystyle C_{0}^{*}=C^{'}}$

${\displaystyle {\frac {\overline {EP_{i}}}{\overline {EC_{i-1}^{*}}}}={\frac {(N-1)}{N}}}$

${\displaystyle \lim _{n\to \infty }{\frac {(N-1)}{N}}\to 1}$

With a sufficiently large point and point are congruent. This congruence is achieved at the point (Figure 7). ${\displaystyle N}$ ${\displaystyle C^{*}}$ ${\displaystyle P_{i}}$ ${\displaystyle G}$

Figure 7: Equilibrium in perfect competition,
price vector = price vector of the perfect market with individuals as price takers

As the limit theorem shows tends the aspect ratio from the origin to and the distance ratio from the origin to having to 1. At sufficiently large , the points covered and at the point (Chart 7). By definition, there are corresponding indifference curves U _A , U _B , which are tangent at this point . In perfect competition with a sufficiently high number of market participants, the contract curve is reduced until the exact allocation is determined. This allocation is comparable to the equilibrium in the perfect market assuming that all market participants are price takers . However, agreements between the actors are allowed. The straight line running from the origin through the point thus corresponds to the price vector from the above-mentioned market equilibrium, which represents the price ratio of x ₁ and x ₂ . ${\displaystyle P_{i}}$ ${\displaystyle C_{i-1}^{*}}$ ${\displaystyle C=C_{0}}$ ${\displaystyle N}$ ${\displaystyle P}$ ${\displaystyle C^{*}}$ ${\displaystyle G}$ ${\displaystyle G}$

implication

There are essentially two relevant implications. The first is that the end result of small group exchanges is indeterminate and depends largely on what Edgeworth calls "non-economic" factors. An example of this is the bargaining power listed above . The second implication justifies the assumption of market participants as price takers due to the equality of the determined result in sufficiently large groups and the market equilibrium . This also applies under the permitted coalition formation.

criticism

Edgeworth's assertion that his theorem also applies to general cases, when there is an unequal number of market participants and different preferences of the individuals must be viewed critically . Formal approaches to prove this can be found in the works of Debreu and Scarf (1963) and Robert Aumann (1964). However, it becomes clear that the results can only be achieved under more stringent assumptions. For example, Aumann's proof is based on the existence of an infinite number of players in the market. There was also a debate between Edgeworth and Marshall about a method developed by Marshall to escape the vagueness of the final allocation when exchanging two individuals. The equilibrium allocation is at the point at which the marginal utility of the exchanged goods coincide. In other words, the point at which none of the actors can further increase their benefit by exchanging a good. However, constant, marginal benefits of the individuals are assumed in order to achieve a definite balance. In addition, the assumed free exchange of information as well as the neglected transaction costs are criticized, since they, like other assumptions, such as the perfect market , are viewed as unrealistic.

literature

• Avinash K. Dixit, Susan Skeath (Eds.): Games of Strategy. WW Norton & Company, 2004, ISBN 0-393-92499-8 .
• Javaid R. Khwaja: Toward a General Theory of Exchange: Strategic Decisions and Complexity. iUniverse, 2013, ISBN 978-1-4759-9739-2 .
• Francis Ysidro Edgeworth: Mathematical Psychics. London 1881
• Francis Ysidro Edgeworth: New and Old Methods of Ethics. James Parker and CO, Oxford / London 1877
• Francis Ysidro Edgeworth, Peter K. Newman (Eds.): Mathematical Psychics and Further Papers on Political Economy. Oxford University Press, Oxford 2009, ISBN 978-0-19-828712-4 .
• John Creedy: Edgeworth and the Development of Neoclassical Economics. Basil Blackwell, Oxford 1986, ISBN 0-631-14923-6 .
• Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics. Pearson, Munich 2009, ISBN 978-3-8273-7282-6 .
• Hal R. Varian: Fundamentals of Microeconomics. Oldenbourg Verlag, Munich 2009, ISBN 978-3-486-70453-2 .
• Gerard Debreu, Herbert Scarf: A Limit Theorem on the Core of an Economy. In: International Economic Review. Vol. 4, No. 3, Sep 1963, pp. 235-246.
• John Creedy: The Development of the Theory of Exchange. In: History of Economic Reviews. Summer Edition, No. 28, pp. 1-45.
• Robert J. Aumann: Markets with a Continuum of Trades. In: Econometrica. Vol. 32, No. 1/2, Jan-Apr 1964, pp. 39-50.

Web links

• Determinacy Restored (June 17, 2015)
• Edgeworth's Conjecture (June 17, 2015)

Individual evidence

1. ^ Francis Ysidro Edgeworth: Mathematical Psychics. London 1881, p. 20.
2. ^ John Creedy: Edgeworth and the Development of Neoclassical Economics. Basil Blackwell, Oxford 1986, pp. 44ff.
3. ↑ Hal R. Varian: Fundamentals of Microeconomics. Oldenbourg Verlag, Munich 2009, p. 648ff.
4. ^ Francis Ysidro Edgeworth: Mathematical Psychics. London 1881, pp. 20ff.
5. ^ John Creedy: Edgeworth and the Development of Neoclassical Economics. Basil Blackwell, Oxford 1986, pp. 55ff.
6. ^ Avinash K. Dixit, Susan Skeath (Ed.): Games of Strategy. 2nd Edition. WW Norton & Company, 2004, pp. 571f.
7. ↑ Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics. Pearson, Munich 2009, p. 702.
8. ^ Francis Ysidro Edgeworth: New and Old Methods of Ethics. James Parker and CO, Oxford / London 1877, p. 43.
9. ^ John Creedy: Edgeworth and the Development of Neoclassical Economics. Basil Blackwell, Oxford 1986, pp. 63ff.
10. ^ Francis Ysidro Edgeworth: Mathematical Psychics. London 1881, p. 34ff.
11. ^ John Creedy: Edgeworth and the Development of Neoclassical Economics. Basil Blackwell, Oxford 1986, p. 69.
12. ^ John Creedy: Edgeworth and the Development of Neoclassical Economics. Basil Blackwell, Oxford 1986, pp. 68ff.
13. ^ Francis Ysidro Edgeworth: Mathematical Psychics. London 1881, p. 30ff.
14. ^ Gerard Debreu, Herbert Scarf: A Limit Theorem on the Core of an Economy. In: International Economic Review. Vol. 4, No. 3, Sep 1963, pp. 235-246.
15. ^ Robert J. Aumann: Markets with a Continuum of Trades. In: Econometrica. Vol. 32, No. 1/2, Jan-Apr 1964, pp. 39-50.
16. ↑ Javaid R. Khwaja: Toward a General Theory of Exchange: Strategic Decisions and Complexity. iUniverse, 2013, p. 62ff.