Optimal Control Problems with State Specific Jumps in the State Equation

An important class of control problems in economics are those in which the state equation switches (jumps) whenever the state variable crosses a threshold. An example is a life-cycle problem in which a household faces higher rates on borrowing than on lending, and therefore the interest rate on the household's asset balance switches discretely each time the asset balance switches signs [6]. The existing method for solving such a problem is notoriously difficult to compute because the first-order conditions include a continuum of complementary slackness conditions. In this paper we provide an easy solution method that utilizes the standard Maximum Principle for unconstrained optimization problems. No inequality constraints (and therefore no complementary slackness conditions) are required.