Mean Field Production Output And Water Conservation Control With Sticky Prices

In recent years,for the sake of resource conservation and environmental protection,our country has established water right trading system.Water consumption index can be sold in the water right trading market to encourage firms to save water spontaneously.Hence when firms determine their production output strategies,they should also take the profit or loss from water right trading into consideration.On the other hand,the production price in the market is sticky in usual,which is to say the production price is resistant to change.The optimal control problem of production output under price stickiness assumption is always a research hotspot.When there is more than one firm in the market producing the same production,the profit function of a firm is tightly coupled with others’ by the production price and water right price.For firms in the market,how to choose their production output and water conservation strategies is a complex stochastic differential game problem.When the amount of firms is huge,the problem is extremely hard to solve.In this thesis,we studied the optimal control problem of production output and water conservation by the theory of mean field games.Specifically speaking,we first regarded the limiting optimal control problem of infinitely many firms as an approximation of the case of huge amounts of firms.We obtained the optimal production output and water conservation strategy of the representative firm in the limit case.Then we transformed the consistence condition into a fixed-point problem.We proved the existence and uniqueness of the fixed point of our preseted mapping.At last,inspired by the mean field Nash equilibrium derived above,we constructed a reasonable and explicit production output and water conservation control strategy for each firm.We proved the aforementioned strategy combination is a approximate Nash equilibrium.