Bilateral monopoly:

A bilateral monopoly is said to exist when one producer has an output monopoly and there is only one buyer for the product. The following table will help you to understand the concepts of perfect competition, monopoly and bilateral monopoly.  

Since there is only one demander and one seller, the price and quantity will be determined by negotiation. However, we can always find the upper and lower limits of price and quantities. We will consider the single seller as all powerful and the single buyer as all powerful, alternatively. The situation is described by the following figure.

DD is the demand curve and MR is the marginal revenue curve. MC_S is the marginal cost curve of the single seller.  If the monopolist seller is all powerful, she will make the buyer behave as if there were many buyers. She will equate MC_S to MR and will produce X_S and will charge price Ps.

However, if the single buyers is the all powerful, she can make the monopolist behave like a perfect competitor. Thus, the MC_S would be the supply curve of the monopolist. Corresponding to this supply curve, we can construct the MC_B curve, which shows the marginal cost of buying an additional unit. MC_B exceeds the price because in order to purchase an additional unit, the buyer must pay a price and that higher price must pertain to all the units purchased.

The buyer equates her marginal cost of buying an additional unit with the marginal value of an additional unit (as given by the demand curve) and purchases X_B amount. Since the seller is behaving like a competitor with supply curve MCs, she will sell the X_B units at per unit price P_B.  The actual solution for the bilateral monopoly problem is indeterminate, depending on the bargaining power of the seller and the buyer.