I think the opposite happens most often. There are plenty of times when a person, such as the head of a department, has responsability but not authority. He or she might be held responsible for making something happen with coworkers or subordiates, but not the acutal authority to enforce any directives or incentives. This is one of the major stressors that middle managers face.
Because authority comes in the form of titles, ranks, and positions, authority can be delegated to one who is appointed. Responsibility, however, is a personal choice: One can choose to be responsible or not. It is an individual quality rather than a concrete "name." If one is assigned an authoritative post, that choice is essentially eliminated.
Hence, a supervisor can bestow titles and authority, but it remains up to the named person to live up to the responsibility inferred by positions of trust. Both words can be used as nouns, but they each have related but different implications. One can be responsible without being an authority, just as one can be an authority without being responsible.
In a business (or any organization, really), you have certain responsibilities that are part of your job. You can give a staff person the authority to do something or make a decision that is part of your job, but you remain responsible for the outcome.
In an organizational set up delegation of authority from superior to his subordinate is not only possible but essential. Once the authority to take a decision is delegated to the subordinate, the boss free from performing those tasks himself, and can devote his or her time and energies to tasks that subordinate cannot perform successfull for one reason or other. This way work of the organization gets done more effectively and efficiently.
However when it comes to delegating responsibility things are somewhat different. When authority is delegated to the subordinate for performing some specific work, the subordinate must also assume responsibility for successful execution of the work done. But does this mean that subordinate in no longer responsible for the results of the subordinates. The answer is an emphatic "No".
If the responsibility is also delegated then the boss can wash off his hand from all that his subordinates, and as managers get most of their results through the efforts of their subordinates, it would amount to them having no or limited responsibility. There will be no one to take responsibility for the organization as a whole and steer it in the right direction.