As we know, Theorems in differential topology and algebraic topology facilitated the development of many crucial concepts in economics, namely the Nash equilibrium—a solution concept in game theory established by John F. Nash, Jr.—which was ultimately proved by the Brouwer fixed-point theorem in topology (Dilkina et al. 42- 43). Many other economic theories, such as the microeconomic general equilibrium theory, largely depend on topological theorems. Moreover, analyzing different topological networks of economic systems can provide mathematical insight into how the society is financially functioning. For instance, by producing empirical models of labor markets which connect individuals’ employment situations, economists found that the differences in the topology of such networks greatly influence the inequalities in wages as well as the duration and correlation of their unemployment rates (Klinedinst et al. 11). but I ask if there any other applications in economic.
Given the importance of the economy in sustaining life. what you see?
Adiya K. Hussein
There are also some works on a topological analysis of social choice (where essentially Arrow's condition of independence of irrelevant alternatives is replaced by continuity, which, of course, necessitates an appropriate framework). Main contributors are Graciela Chichilnisky, Geoffrey Heal and Nick Baigent.
Dear Adiya K. Hussein
If you are interested in the application of mathematics on economics, please ready two of my papers:
(1) International trade theory and exotic algebra
https://www.researchgate.net/publication/280646264_International_trade_theory_and_exotic_algebra
(2) Subtropical Convex Geometry as Ricardian Theory of International Trade
https://www.researchgate.net/publication/236020268_Subtropical_Convex_Geometry_as_the_Ricardian_Theory_of_International_Trade
(1) is formerly published as
Yoshinori Shiozawa (2015) International trade theory and exotic algebra. Evolutionary and Institutional Economics Review. 12(1): 177–212.
The Ricardian trade economy without input trade has a mathematical structure that is named subtropical algebra after Tropical Algebra. The Ricardo-Sraffa trade economy (i.e. trade economy with input trade) has a much more difficult structure and its algebraic structure is not yet known. Please see
(3) The New Theory of International Values: An Overview
https://www.researchgate.net/publication/304717720_The_New_Theory_of_International_Values_An_Overview
(3) is formerly published as Chapter 1 in Shiozawa et al. (ed.) A new costruction of Ricardian theory of international values: Analytical and historical approach. Singapore: Springer, 2017.
Dear Yoshinori Shiozawa
Thank you very much for this valuable offer and for your valuable researchs.
Dear Adiya K. Hussein ,
Your question is an interesting one, and I think topology can be introduced into almost all branches of sciences. Economic is essential in our life, related to production, data distribution, goods consumption, and services.
So many factors as networking and graphs should be considered in many mathematical models to describe economic situations as transportations and strategies of game theory, and hence topology is strongly presented to facilitate the possible solution and to make decisions.
An interesting statistical quantity is the Jaccard distance. It is used to compare the similarity and diversity of sample sets.
One can show it can be considered as a metric over the set elements.
The following was my answer in a previous RG question :
Using the definition of Jaccard distance J(A,B)=1 - |A∩B|/|A∪B|,
we know that |A∩B| ≤ | A∪B|, i.e, |A∩B|/|A∪B| ≤1, we have
(1) J(A,B)=1- |A∩B|/|A∪B| ≥ 0.
(2) J(A,A)=1- |A∩A|/|A∪A| = 1 - 1 = 0.
(3) J(A,B)=1- |A∩B|/|A∪B| = 1-|B∩A|/|B∪A| = J(B,A).
(4) Is J(A,B) + J(B,C) ≥ J(A,C) ?
To prove part (4), one needs to consider all possible cases:
Case (0) One, two or the three sets A, B or C is φ.
(1) A∩B=φ,A∩C=φ and B∩C=φ
A, B and C are disjoint.
(2) A⊂B⊂C ( concentric sets)
(3) B ⊂ A∩C
(4) A∩B∩C ≠ φ
(*) Check all other similar cases.
Proof:
Case (0) is trivial.
Case(1) J(A,B)=J(B,C)=J(A,C)=1
and inequality (1) satisfied.
Case (2) For A⊂B⊂C
Let |A|=a, |B|=b, |C| = c, a < b < c, we have
J(A,B)=1- |A∩B|/|A∪B| = 1- |A|/|B|
J(B,C) = 1-|B∩C|/|B∪C| =1 - |B|/|C|
J(A,C)=1- |A|/|C|
we compute,
J(A,B) + J(B,C) - J(A,C) =
(1-(a/b))+(1-(b/c))-(1-(a/c)) = ((a-b)(b-c)/(bc) > 0
therefore inequality (4) holds.
You can tackle all other cases in a similar way, consequently,
J can be considered as a metric over the data space.
Best wishes
Dear Issam Kaddoura
Thanks Dr. for this interesting and valuable explanation.
Dear K. Hussein A.,
We have also topology in “qualitative input-output analysis” (QIOA) when we examine which productive j sector is connected (following a given criteria) with other productive sectors i (i's adjacency), the i sectors being themselves connected with other productive sectors k (i's adjacency of the adjacency), etc., up to infinite. See for example:
- de Mesnard L. 2001. “On Boolean topological methods of structural analysis”, in Input-Output Analysis: Frontiers and Extensions, in honor to Ronald E. Miller, Michael Lahr and Erik Dietzenbacher Eds., Palgrave: 269-80.
- de Mesnard L. 1995. “A Note on Qualitative Input-Output Analysis”, Economic Systems Research, 7, 4: 439-45.
Yours sincerely
Dear Prof. Louis De Mesnard
Thank you very much for your nice answer and your valuable researches.
Article A foundation of location theory: Consumer preferences and demand
Existing economic problems associated with a particular theorem may be "reframed" using new topological approaches:
http://math.uchicago.edu/~may/REU2015/REUPapers/Chia.pdf
Providing an alternative route to understand a problem. Framing the problem in a more suitable way may in turn lead to new insights.
Article The Mathematical Logic of Creative Economics
. Dear Shian-Loong Bernard Lew
Thank you very much for your nice answer and valuable researches.
Hi. As far as I remember, the renewal of the general equilibrium theory in the 1950's and 1960's were related to new may to establish its existence and study their conditions within a topological framework. It resulted in the Nobel prize awarded to Gérard Debreu.
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