## Vedenyapin A.D., Mitasov S.A. The Problem of Finding the Optimal Number of Countries in a Limited Area

Vedenyapin Aleksandr Dmitrievich

Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Mathematical Analysis and Function Theory, Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Mitasov Sergey Aleksandrovich

Master Student, Department of Mathematical Analysis and Function Theory,
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. We assumed that the world consists of two continents. We consider the long term – any period of time during which there is a change of state borders, the division of states. All exchange rates are of 1 : 1. Prices of goods are the same everywhere, both at domestic markets and foreign, since all countries in the world are in the WTO. For this reason, prices in various currencies take the same value.
The process of new states formation in one of the continents is under the assumption that every state is required to structure occupied by the emission of the national currency in the following way – part of the gross domestic product of the state each year is given to another continent (“owner” of structures) in exchange for the opportunity to print their own currency in order to redistribute the remaining annual product within the state between its population and for the implementation of the function of money – a medium of exchange.
In such circumstances, for each state there are special coefficients which with the growth of the area of the state reflect the growth of annual production and the decrease in the annual volume of the product.
Search for the optimal number of states in the continent can take place both in the interest of another continent and in the interests of people of the unfortunate continent. For this purpose two respective functions are built

G(S1,S2,...,Sn)=∑SiKi(Si)Pi(Si)

F(S1,S2,...,Sn)=∑SiKi(Si)(1-Pi(Si))

when S1 +S2 +...+Sn = L = const, n ≠ const.

Accordingly, their highs are searched by the maximizations of benefits principle.
Also a system of equations is introduced, which reflects necessary state for each country – when the annual volume of the given product is equal to the annual volume of remaining product.
At the end of this paper we present a theorem, that under certain conditions the most profitable for the population of the “unfortunate” continent is the union into the single state.

Key words: maximization of benefits, circulation medium, effect of production scale, Lagrange's theorem on finite increments, mathematical model.