We derive the dynamic Nash equilibrium path of the toll rate of the bridge which connects the mainland and the island. The game is played between the bridge authority and the firm in the island.
The firm in the island is assumed to be owned by individuals of the island, that is, the behavior of the firm is given as the optimal behavior of the island residents. The island exports their product to the mainland and import goods from the mainland. The island and the mainland trade goods of their products by using the bridge. The time path of the export and the import, i.e., the island's consumptions of the imported goods and the domestic goods, are determined so as to maximize the discounted sum of the future utilities of the individuals of the island, given the time path of the toll rates determined by the bridge authourity. At the same time, the optimal time path of the investment of the flrm in the island is determined.
The bridge authority determines the optimal time path of the toll rate of the bridge so as to maximize the discounted sum of the future utilities of the island residents. The constraint of the authority is that the construction cost of the bridge should be refund by the toll revenue of the bridg in finite periods. Also, the authority takes the time path of the investment of the island as given.
We derive that when the utility function of the island is strictly concave CES type, the equilibrium path of the toll rate is decreasing in time.