The main purpose of this paper is to formalize the modelling process, analysis and mathematical definition of corruption when entering into a contract between principal agent and producers. The formulation of the problem and the definition of concepts for the general case are considered. For definiteness, all calculations and formulas are given for the case of three producers, one principal agent and one intermediary. Economic analysis of corruption allowed building a mathematical model of interaction between agents. Financial resources distribution problem in a contract with a corrupted intermediary is considered.Then proposed conditions for corruption emergence and its possible consequences. Optimal non-corruption schemes of financial resources distribution in a contract are formed, when principal agent's choice is limited first only by asymmetrical information and then also by external influences.Numerical examples suggesting optimal corruption-free agents' behaviour are presented.
Nowadays problem of corruption gets a lot of attention in both scientific and political spheres. The reason for such interest is due to harmful effects it has on society and economy. Corruption undermines the functioning of the host state and lowers the efficiency of production (Rose-Ackerman, 1999). It also hampers economic development and growth, especially since it has negative impact on investment rates (Wei and Wu, 2001). Corruption is most commonly defined as the misuse or the abuse of public office for private gain (World Bank, 1997). Recent studies are focusing more and more on specific anti-corruption programmes based on countries' experiences. A numerous research is coming from developing countries describing current economy situation and proposing anti-corruption measures.
The president of Russian Federation has set a very important task for Russia to enter into the top twenty countries in the «Ease of doing business» ranking provided by the World Bank. A high ease of doing business ranking means the regulatory environment is more conducive to the starting and operation of a local firm. Empirical research funded by the World Bank to justify their work shows that the effect of improving these regulations on economic growth is strong (World Bank, 2013).
National anti-corruption committee ran by Kirill Kabanov is conducting a research for Kremlin on corruption strategies used in Russia and other countries. As a result of this analysis, the head of the National anti-corruption committee will propose to include additional legal mechanism for fighting against corruption. National Anti-Corruption Strategy was introduced in Russia in 2010, but there are some questions raised about its efficiency. Due to the fact that currently in Russia there is no thorough analysis of corruption practices and no thorough analysis of anti-corruption mechanisms, running anti-corruption plan does not prove to be effective.
The purpose of the conducted research, the results of which are presented in this paper, is to formalise the modelling process, analysis and mathematical definition of corruption when entering into a contract and to find optimal strategies for a corruption-free interaction between principal agent and producers.
, where principal agent 0 F enters into a contract with producers i F , in order to produce i q amount of goods for i s amount of money (here and below ) iI . During the life of the contract the prodcuers i F might face decrease (or increase) in the production levels , depending on the amount of funds delivered by the principal agent.
The principal agent 0 F can contact an intermediary F in order to get information about the change in the level of production .
In case there is an increase in production level f , the producer can spend less money than the amount of funds provided by the principal agent 0 F , and thus gain additional profit. The producer i F wants to conceal this information, and hence he/she offers a bribe i z to the intermediary F . In this model we consider a case of a corrupted intermediary. If there is a decrease in the production level d , then the provided funds are not enough to sustain production level, and hence the producer wants the official report to contain true information, and thus, he/she is required to offer bribe i z to the intermediary F , as the intermediary threatens to hide this information. If the intermediary F is not aware of the above, he/she does not tend to conceal any information. In order to stop intermediary F and the producer i F from negotiating between themselves, the principal agent 0 F can make a legal payment S to the intemediary F . In this case the intermediary presents true information in the report R .
Then the principal agent makes a decision about funds distribution i s S = between the producers i F based on the intermediary’s F report R .
Consider a model with three producers. The producers 1 F , 2 F and 3 F together account for the following amount of goods
where 1 2 3 ,, q q q -the amount of goods produced by the producers 1 F , 2 F and 3 F respectively,
-is the change in the level of production, which is a private information held by the producers. The payoff of the producers is defined by the formula:
where , = 1, 2,3 i si -is the size of financial funding provided by the principal agent 0 F . The amount of funding spent by i F on the production of goods is described by the function
. The side payment 0 i z is offered by the producer i F to the intermediary F . The payoff function of the intermediary F is as follows:
where 0 S —is the financial funding provided by the principal agent,
—are the costs for the intermediary to conceal side payments done by the producers i F .
F examines and analyses the information about the producers 1 F , 2 F and 3
F and either finds out about the change in the production level ( { , }, = 1, 2,3
, or he/she does not find out anything (
. In case the intermediary
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