Why is there no unique input price and quantity of input hired in the bilateral monopoly case? What factors will play a role in determining that price and quantity?
A bilateral monopoly is a market structure consisting of both monopoly (a single seller) and monopsony (a single buyer). Bilateral monopoly situations are typically analyzed using the theory of Nash bargaining games, and market price and output will be determined by forces like bargaining power of both buyer and seller. A bilateral monopoly model is often used in situations where the switching costs of both sides are prohibitively high.
An example of a bilateral monopoly would be when a labor union (a monopolist in the supply of labor) faces a single large employer in a factory town (a monopsonist). A peculiar one exists in the market for nuclear-powered aircraft carriers in the United States, where the buyer (the U.S. Navy) is the only one demanding the product, and there is only one seller (Huntington Ingulls Industries) by stipulation of the regulations promulgated by the buyer's parent organization (the U.S. Department of Defense, which has thus far not licensed any other firm to manufacture, overhaul, or decommission nuclear-powered aircraft carriers).