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Theoretical and Applied Economics
No. 2 / 2020 (623), Summer

Mathematical model used in substantiating optimal contract

Constantin ANGHELACHE
Bucharest University of Economic Studies, Romania
“Artifex” University of Bucharest, Romania
Dumitru MARIN
Bucharest University of Economic Studies, Romania
Dana Luiza GRIGORESCU
Bucharest University of Economic Studies, Romania

Abstract. The theory of optimal contracts refers to the market state, in which the bidders, the participants in the execution of transactions, have a certain number of certain information higher or lower. Normally, anyone who concludes a contract, regardless of its nature and we refer to the business environment, seeks to place himself on an optimal solution, i.e. one that according to the Latin principle “aurea mediocritas” gives him a chance to average, but protects him the realization of a contract subject to many risks. Normally, the conclusion of any contract is based on an interest, which starting from the principle of the free market, based on the ratio between supply and demand, may end up in the situation of concluding one or another of the contracts.

In the literature the problem of optimal contracts is not so new, only that there is less information and materials published by specialists on this topic. In principle, in the capital market, those operating in the capital market must consider the possibility of concluding a contract on optimal terms.

Nowadays, when we are in the Big Data era, in which databases are enormous, it is essential that companies or agencies that mediate business know very clearly the information that underlies the transaction to be concluded. As always, there is a clear enough difference between the level of information held by one or another of the customers.

In this article we started the theoretical problem in very synthetic terms, because it is known and we tried to substantiate a model that could be the basis for renting optimal contracts. It should be noted that it started from the utility function, in the sense of von Neumann-Morgenstern, as well as the Lagrange function and last but not least from the Kuhn-Tucker multiplier, often used in microeconomic analyzes based on consistent models.

In this article we started from the estimation of the multiplier between two equations to obtain an optimal result. The optimal contract for an agent, best placed, is a solution given by a system of two equations that lead to Pareto optimality, or if you will to Pareto efficiency. On the other hand, the optimal contract in the situation of asymmetric information for the interested agent is the optimal Pareto or Pareto-optimal. In this situation, in this article we took a numerical case, from whose analysis it is clear how the study should be based in the perspective of substantiating the decisions to conclude an optimal contract.

Finally, the mathematical model is concretely formulated, imposing some participation restrictions or compatibility restrictions that must always be taken into account by the one who enters the market and wants to conclude an optimal contract. It follows that the model can be simplified, in order to remove some restrictions or to establish that some of them can be satisfied by using the Kuhn-Tucker multiplier. Thus, if a second agent or another market participant signs the contract, he assumes some risks. There are a lot of solutions in the market study, however, and then it must be borne in mind that these contracts are satisfactory if we take into account the conditions of the mathematical model we talked about.

In this article we started from the theoretical elements, analyzing them in the mathematicaleconomic sense, in order to reach the relationship that the conditions for concluding an optimal contract imply.

Keywords: optimal contract, models, functions, multiplier, agents, capital market.

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