A lot of government agencies, as well as groups within large companies, want to keep their budgets (or increase them), so there isn't an incentive to be cost effective -- as you mentioned.

But, as another person commented, the overhead around government projects (documentation, reports, meetings, etc) will bloat things beyond a simple time/materials development contract.

The FBO has a site listing the contacts, but there are actually some pretty interesting projects that pop up to be bid upon. Only downside for someone new to the process (or a small shop) is that the big boys (SAIC, etc) have this part pretty streamlined as well as sometimes having in's with the agency requesting the proposal.

Edit: The site -- https://www.fbo.gov/