Optimal plans and timing under additive transformations to rewards

The nature and role of additive transformations to rewards are elucidated for a general class of deterministic, nonautonomous, optimal control problems with many state and control variables. Conditions relating to the optimal choice of initial and terminal times and initial and terminal values of state variables are identified such that additive transformations affect optimal plans. General comparative static results are derived and the framework is extended to cover two common classes of stochastic control problems. Three applications are presented: the canonical adjustment cost model of a firm, a stochastic extension of an irreversible pollution accumulation problem with regime-switching, and an extension of a lifecycle model of retirement in which an agent’s retirement wealth evolves stochastically.