Model of Optimal Consumption with Possibility of Taking Loans at Random Moments of Time

In this paper, we consider the classical problem of maximizing discounted utility, provided that the moment of the next purchase and receipt of a loan is random (Poisson). The purpose of the study is to take into account the uncertain waiting period for receipt of a credit in consumption decision-making. The model is formulated as the problem of optimal stochastic control. The consumer at random moments buys the product at a non-random price and at the same random moments can take and return indefinite loans. For loans, the agent continuously pays interest. He constantly receives dividends in the form of external receipt of money into the account and can accumulate non-interest non-cash money. The optimality conditions are obtained using the Lagrange multiplier method. Sufficient optimality conditions reduce to partial differential equations with variable and unknown delay. They can only be solved by using a combinations of analytic expansions with respect to a small parameter. A special difficulty is the regularization ("softening") of the conditions of complementary slackness. As a result, functions were obtained that determine the optimal control of consumption purchases and the size of the loan. One can see how the consumption expenditures change as the end of the planning period approaches. First, consumption depends on money and debt not separately, but on their difference – own means of the consumer. Secondly, far from the planning horizon, consumption is small and grows as the final point in time approaches. This model can be used as part of the description of the consumer agent in dynamic stochastic general equilibrium models.