Super contact and related optimality conditions

Under transactions costs, and generally any sort of friction, economic agents, acting dynamically in a stochastic environment, may find it optimal, in some region of the state space, to take no action at all. Action is triggered when the state of the economic system reaches the boundary of the ‘region of no action’. In this note, first-order conditions for the choice of the region of no action, and for the type of action to be taken, are established, assuming that the uncontrolled state variable follows a diffusion process. The generic name for these conditions is ‘smooth pasting’ or ‘high contact’ conditions; they require that marginal utility should take the same value before and after the action has been taken, In this note, we show that, in some cases, these conditions involve the first derivatives of the value function of the dynamic program, while, in other cases, they involve the second derivatives and require a higher form of tangency which Dumas (1988) called ‘super contact’.