On the Solution of a Deterministic and Stochastic Household Problem with a Finite Planning Horizon

The article uses the example of an optimization problem of a household that makes a de cision on the volumes of consumption and investment to show what difficulties arise in deterministic and stochastic formulations on a finite time interval. In order to make the problem solvable on a finite time interval, a special terminal condition on the agent's equity capital is added, gene ralizing the standard versions of such conditions.

The article considers two settings. The first setting is a deterministic case, assuming that the household knows the trajectories of all exogenous variables over the entire time interval under consideration. An analytical solution to this problem is found and it is shown that by choosing the parameter of the terminal constraint in the problem on a finite time interval, it is always possible to obtain a consumption trajectory from the solution of a similar problem set for an infinite planning horizon. If the coefficient of the terminal condition is chosen so that the optimal consumption trajectory continues the previous value, then with a certain combination of initial conditions, the household's problem can either be solvable only up to a certain planning horizon, or be completely unsolvable.

The second statement is a stochastic case, when the household knows only the distribution law of exogenous variables. In this case, it is not possible to provide a complete analytical solution, but a sequential algorithm is proposed that allows one to obtain a step-by-step description of the calculation of such a solution. The study of the properties of the constructed model al lows one to show how different the work with stochastic optimization problems for the analysis of deviations from a certain selected trajectory of states (balanced growth) in response to the implementation of other states (shocks) is from the problem of analyzing specific realized trajectories of the agent's variables.