This is a job for a handyman, and its such a small job I would look around your home for a few more "small" jobs, wonky electrical outlets etc.
You "might" need a new fixture, or it might be possible to replace just the socket, or if the metal contacts in the socket are dirty or slightly bent away from making good contact with the bulb just to clean them slightly bend them back. Cheap socket syndrome.
Make friends with someone who can do this kind of stuff, or get a good reference for a handyman and use him for all the small repairs.
Generally you only need a real electrician for serious repairs and installing new wiring, not making repairs to existing stuff.
Handyman typically half to third cost of Electrician or plumber, but about 10% of the skill and training.