REFORMY (132) Financial Markets Evolution/3

11. červen 2015 | 07.00 |

Poznámka česky na konci článku.


ANALYSIS OF FINANCIAL MARKETS EVOLUTION BY UTILIZING THE THEORY OF COOPERATIVE GAMES III.

Jan Červenka, Ondřej Černík, Jiří Mihola, Radim Valenčík

3 Interpretation of the elementary model of supply and demand of investment opportunities and investment means as cooperative game

Each point in the closed interval 0, x2corresponds to certain split of revenues among economic entities. We determine that:

y(1)

0pt; line-height:115%;font-family:"Times New Roman","serif";mso-ansi-language:EN-GB"> is total revenue (hereinafter referred to as "salary") of the first entity (player)

y(2) is total revenue (hereinafter referred to as "salary") of the second entity (player)

If x moves in 0, x2, all achievable combinations of salaries will be on F(y(1), y(2)), where

0ʃxf(x).d(x), y(2)= xʃx2g(x 2 - x).d(x),

Graph No. 2: Salaries of players (total revenues of economic entities)

Source: Own creation

Where:

* Starting point if x=x1, je (y(1)1, y(2)1)

* Point (y(1)E, y(2)E) is the point of maximum sum of salaries.

* Point (y(1)2, y(2)2) are salaries of players after the compensation under conditions specified in Chapter 2.

Graph No. 2 demonstrates that both players received a higher salary, which corresponds to what is shown in Graph No. 1. If both players achieved in point (y(1)2, y(2)2) maximum salary with prices of investment means moving in Graph No. 1 in a closed interval y1, y2, individual rationality would be enough to consider point (y(1)2, y(2)2) an intuitively acceptable solution of the relevant cooperative game. However, this may not be the case, see Graph No. 3:

Graph No. 3: Salaries of players in case of changing prices of investment means

Source: Own creation

The figure on the left shows an increased salary of the first player and the second player with the price of investment means yi. The next picture shows a possible change if the price of investment means changed from yi to yE. The salary of the first player would decrease compared with the salary of the second player. That is quite an important moment. It shows that the condition of individual rationality may not be enough to find a definite solution. It is therefore suitable to transform the encountered problem in the form of Nash (S, d) bargaining problem.

Graph No. 4: Salaries of players in case of changing prices of investment means

Source: Own creation

Here:

S is a set of possible splits of salaries if maxy(1) and maxy(2) are smaller than y(1)E and y(1)E

S´ is a set of possible splits of salaries if maxy(1) and maxy(2) are bigger than y(1)E and y(1)E

If we indicate inversion function to function y=f(x) as x=f-1(y) and similarly y=g-1(x) inversion function to function y=g(x), then the values of Pareto improvement are (as functions of y, which represents here the price of investment means):

Δy(1)=x1ʃxif(x).d(x)- yi(xi-x1), kde x=f-1(y)

Δy(2)=yi(xi-x1)- x1ʃxig(x 2 - x).d(x), kde x=g-1(y)

Respective functions of Pareto improvements are in Graph No. 3. The full curve shows a case where maxy(1) and maxy(2) are smaller than y(1)E and y(1)E, the dashed curve shows a case where maxy(1) and maxy(2) are bigger than y(1)E and y(1)E. In the first case (provided that the price of investment means is constant and all investment opportunities with revenues higher than this price are used) the solution of the respective cooperative game is unequivocally determined by the condition of individual rationality. In the second case, trouble (our term) during the utilization of investment opportunities, far from it and there is a number of possible approaches to the solution.

Poznámka v češtině:

Příspěvek byl přednesen na 7. mezinárodní konferenci VŠFS na téma "FINANČNÍ TRHY V PROCESU GLOBALIZACE SVĚTOVÉ EKONOMIKY". V češtině i s poznámkami byl zveřejněn 19.-27.4.2015 na blogu:

http://radimvalencik.pise.cz

První pokračování viz:

http://radimvalencik.pise.cz/2244-reformy-78-evoluce-financnich-trhu-1.html

(Pokračování příspěvku v angličtině)

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