Duopoly Game Models of Water Supply in the Presence of Market Power
Lu, Haishu, Journal of Digital Information Management
Water resource trading is a water market policy which can reduce unnecessary and exorbitant waste of water resource by using economic means and it has features of both water quantity assurance and cost-effective in the control of water consumption [1-2]. It is a more effective means by using the invisible hand of market to control water resource utilization. To some extent, although economists mainly concerned with fair and efficient problems, they have committed themselves to realizing water resource trading in the process of the international political agenda [3-4]. Now in the process of the implementation of water market policy, economists must deliberate on the effective allocation of water supply in the real world. Therefore, water resource trading attracts the universal concern of all countries in the world. In order to implement the system of the tradable water supply, we must first solve a key problem on the allocations of water supply from both a theoretical and a practical level .
According to the existing literatures, we find that many authors made a lot of efforts to study the allocation of emission right and water supply by using homogeneous oligopoly game model. Heller  started to pay great attention to the initial emission permits allocation problem and carried on the discussion in this aspect. By using a homogenous oligopoly game model and auction theory, Sunnevag  studied the allocation of permits under two auction mechanisms. Recently, by using homogeneous oligopoly game theory, Foellmi and Meister  established a model of water supply and product market competition. On the basis of the results of the model, they analyzed and compared the relevant welfare gains and showed that production efficiency and retail prices may differ depending on the initial cost differential, the application of regulations and the distribution of bargaining power. Using a theoretical model, they showed that at higher initial production cost differentials, welfare is higher under competitive conditions, even in a lower-bound benchmark case without any regulation. Considering market power, Ansin and Houba  established a water market model as multi-market Cournot competition with a river structure. In their model, suppliers are connected through water balances, which impose resource constraints, and they are connected to heterogeneous water users via a water delivery infrastructure. Furthermore, they gave conditions for the existence of water market equilibrium and assessed the effects of market power on water extraction, delivery, and water prices.
Inspired and motivated by the above results in this research field, in this paper, we establish two duopoly game models of product quantity and the water supply for two firms to produce differentiated product under the settings of complete information and incomplete information, respectively and obtain the corresponding allocation results of product quantity and the water supply under the condition that the market power exists, and next, we compare the allocation results obtained under the setting of incomplete information on marginal cost with that obtained under the setting of complete information on marginal cost.
2. The basic model
In this paper, we consider the allocation model of the water supply of two firms producing two differentiated good. Let [p.sub.1] = A - [Bq.sub.1] - [Dq.sub.2] and [p.sub.2] = A - [Bq.sub.2] - [Dq.sub.1] be the linear inverse demand curves facing firm 1 and firm 2 respectively, where q. denotes the output of firm i (i = 1, 2); [p.sub.1] and [p.sub.2] denote the price of two different products produced by firm 1 and firm 2 respectively. Parameter A measures the market size or the reservation price, which is assumed to be equal across varieties for the sake of simplicity. As for parameters B and D, we assume that 0 [less than or equal to] D [less than or equal to] B. Notice that parameter D captures the degree of substitutability between the two different goods produced by two firms. …