Selected mathematical-economics theoretical models of the static equilibrium of the demand-supply type

Sustainable development programs are often clearly linked to the problem of equilibrium of demand and supply. In this context, the study of supply and demand equilibrium models is an important economic aspect related to sustainable development. The article presents general solutions and the number of solutions for some models of static equilibrium regarding the value of demand and supply. The general method for solving linear static equilibrium models is presented. It has been noticed that if in the non-linear models of the equilibrium of two goods the corresponding coefficients are the same, then the equilibrium values are the same. The number of solutions for some non-linear models of static equilibrium was found, provided that the values of demand and supply were given by means of real polynomials of any natural degree. In addition, it was noted that if some types of nonlinear models can be reduced to linear models, then the equilibrium values of these models are the same at different equilibrium prices, when the number of price indices in the tested models is the same, and the corresponding coefficients in these models are the same. The final part of the article presents the use of the E.Waring's hypothesis and lower estimation in the Waring problem in order to determine the number of equilibrium conditions in certain specific static equilibrium models. The use of the Waring hypothesis gives a good theoretical basis for solving some of the demand-supply equilibrium models in non-negative integers numbers. Such solutions are particularly valuable from the economic point of view. The article does not cover the often trivial cases in which the supply and demand equilibrium models had an infinite number of solutions.