We propose an algorithm for executing transactions in object oriented databases. The object oriented database model generalizes the classical model of database concurrency control by permitting accesses to class and instance objects, by permitting arbitrary operations on objects as opposed to traditional read and write operations, and by allowing for the nested execution of transactions on objects. In this paper, we first develop a uniform methodology for treating both classes and instances. We then develop a two phase locking protocol with a new relationship between locks called ordered sharing for an object oriented database. Ordered sharing does not restrict the execution of conflicting operations. Finally, we extend the protocol to allow for nesting. The resulting protocol permits more concurrency that other known locking-based protocols.