Model of optimal producer’s behavior in the presence of random moments of receiving loans and investment

This paper presents the approach to solving optimal control problems that appear in economic models using the Lagrange’s multipliers method. This method is not as widely used as it might be, taking into accounts its benefits and convenience. The power of this method for intertemporal general equilibrium allows building complex structural models of the whole economy and work with the system of differential and finite equations instead of integral and functional ones as in dynamic programming. Not only deterministic, but also stochastic economic models might be solved using this method, although the theory of Lagrange’s method for stochastic optimal control models is less developed than the dynamic programming approach. In this paper, we develop their approach and specify the particular mathematical constructions underlying the mathematical formulation of the stochastic control problem. The presented paper demonstrates the Lagrange’s method on the example of the problem of a firm that makes decisions regarding investment, production and payment of dividends to the owners of the firm. The stochastic component in this model is represented by the random process of moments of time when making transactions is possible. The agent’s problem on a finite horizon differs from the infinite horizon problem by the presence of the boundary layer where the analysis might significantly change compared to the analysis of the solution within the planning horizon.