We consider the Debreu private ownership economy in which all consumption plans belong to a proper linear subspace of the commodity-price space ℝ^l. This geometric property of consumption sets means that there is a dependency between quantities of some commodities in all consumption plans. Competitive mechanism makes producers adjust their plans of action to the same dependency. It results in the mild evolution of the production sector to offer production plans which are also contained in the given subspace of ℝ^l. Modified production system and the initial consumption system can form an economy in equilibrium. The aim of this paper is to model gentle changes of producers’ activity that give equilibrium in the Debreu economy with consumption system reduced to a proper subspace of ℝ^l without considering additional costs.

The paper considers a private ownership economy in which economic agents could realize their aims at given prices, Walras Law is satisfied but agents’ optimal plans of action do not lead to an equilibrium in the economy. It means that the market clearing condition is not satisfied for agents’ optimal plans of action. In this context, the paper puts forward three specific adjustment processes resulting in equilibrium in a transformation of the initial economy. Specifically, it is shown, by the use of strict mathematical reasoning, that if there is no equilibrium in a private ownership economy at given prices, then, under some natural economic assumptions, after a mild evolution of the production sector, equilibrium at unchanged prices can be achieved.

We examine new mechanisms that introduce environmentally friendly eco-changes involving the elimination of noxious commodities and take into account the structure of demand without a detrimental effect to agents' position. In the era of the fourth industrial revolution, these mechanisms allow eliminating unnecessary services or goods that are being replaced by modern technologies. We define optimal mechanisms under the criterion of distance minimization, when a small number of detrimental commodities is excluded from production processes as well as when producers are change-averse. The results have the form of theorems with rigorous proofs.

Arrow and Debreu proved that, if in the mathematical model of the economy that they defined some conditions are satisfied, then there is a price system under which there is Walras equilibrium in this economy. We examine a procedure, defined on the basis of a continuous mapping, of transformation of production sets of the Arrow-Debreu economies in which at least one of the assumptions concerning the production sets is not satisfied. Finally, we get a class of ArrowDebreu economies in which, after modification, there is Walras equilibrium. The research methods are based on the analysis of the linear mappings in real vector space of finite dimension, and the results have the form of mathematical theorems.