Capacity constraints and voluntary output cutback in dynamic Cournot competition

This paper considers Cournot competition in which duopolists play a quantity-setting game with sticky prices and capacity constraints, A feedback (subgame-perfect) equilibrium is derived and analyzed, With sufficient differentiability, the paper shows the existence and the uniqueness of symmetric feedback equilibrium. We show that there arises a range of constraints defined by two threshold values in which the firms' outputs are below the constraint level in the steady state. Yet the outputs depend nontrivially on the constraint. For outside observers, the constraint does not seem to be binding. But the constraint is binding in that a marginal change in the level of capacity constraint affects firms' steady-state outputs. The result implies that under this constraint the firms cut back their steady-state outputs voluntarily more than stipulated by the constraint. We also show that the procedure to construct a constrained solution from an unconstrained solution by truncation may not always be appropriate. This paper's analysis utilizes the concept of connectable points associated with the solutions to the auxiliary equation of the Hamilton-Jacobi-Bellman equation. Since the concept of connectable points and their characterization does not directly rely upon the linear-quadratic structure of this game, the method can be modified and applied to other differential games with restrictions on the control space(s).