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One of the fundamental elements in database management systems is the locking mechanism. Locking is vital for maintaining data integrity and consistency, especially when multiple transactions operate concurrently. In its essence, a database locking scheme is a method that governs how transactions will lock data in the database to prevent conflicting operations that could lead to errors or data corruption.

A lock is typically a marker that restricts access to the data item, ensuring that other transactions cannot modify it while it's being used by one process. Different types of locks, such as shared (S) locks for reading data and exclusive (X) locks for writing data, can be applied depending on the operation. With Multiple Granularity Locking (MGL), introduced in the exercise, locking can be applied at various levels – from an entire database down to a single record. MGL enhances efficiency by minimizing the number of lock and unlock operations.

Transaction concurrency refers to the ability of a database system to allow multiple transactions to access the database at the same time. The key to successful concurrency is to enable these transactions to proceed without waiting for each other, while still ensuring the correct sequencing of database operations. This is critical for users to access the system at the same time without interfering with each other's operations.

When concurrency is properly managed, the database system can achieve high levels of performance and responsiveness. However, without proper locking mechanisms like MGL, transactions could interfere with one another leading to issues such as 'lost updates', 'temporary inconsistencies', and 'phantoms'. In the context of Multiple Granularity Locking, the hierarchical structure allows transactions to obtain the appropriate level of lock, enabling better management of concurrency.

Database efficiency is often a reflection of how well the system handles and processes multiple transactions concurrently while maintaining data integrity and minimizing resource costs. Fine-grained locking control, as enabled by MGL, plays a significant role in enhancing efficiency. The ability to lock data at multiple levels of granularity means that a transaction doesn't have to lock more data than necessary.

For instance, when a transaction only needs to access a single record, it wouldn't be sensible to lock the entire table, as it would impede other transactions from accessing other records. With MGL, it is possible to be precise with lock levels, which reduces wait times and increases the throughput of transactions. Consequently, this controlled locking approach results in a more efficient database system, as the resources are utilized judaciously, and more work is completed in less time.