Capital Accumulation with Constraint: A Mean Field Type Control Framework

To succeed in a competitive business environment, optimal capital investment plays a significant role. A firm cannot ignore the penalty associated with carrying excessive or insufficient production capacity. We provide a model of the optimal rate of capital investment under uncertainty incorporating a penalty to study the key impact. The penalty is modeled as a squared deviation between the expected and the desired levels. The payoff functional thus incorporates a nonlinear function of the expected capital level. This control problem is of the mean field type. We obtain a closed form solution by a direct method. As expected for mean field type control problems, the optimal feedback depends not only on the current states, but also on the initial conditions. We perform numerical studies to quantitatively address how risk control in capital level deviation affects the optimal investment policy.