This research discusses the Internet service provider (ISP) bandwidth allocation and pricing problems for a duopoly bandwidth market with two competitive ISPs. According to the contracts between Internet subscribers and ISPs, Internet subscribers can enjoy their services up to their contracted bandwidth limits. However, in reality, many subscribers may experience the facts that their on-line requests are denied or their connection speeds are far below their contracted speed limits. One of the reasons is that ISPs accept too many subscribers as their subscribers. To avoid this problem, ISPs can set limits for their subscribers to enhance their service qualities. This paper develops constrained nonlinear programming to deal with this problem for two competitive ISPs. The condition for reaching the equilibrium between the two competitive firms is derived. The market equilibrium price and bandwidth resource allocations are derived as closed form solutions.