[fest]

Another thing higher-ups like is having several options presented:

a) I can do this in 4-6 weeks, but we can also hire a consultant who can probably do this in 6-8 weeks and it is going to cost roughly this much (in this particular example I had intimate knowledge of the system and was proficient with the framework/tools).

b) Look, we don't have anyone on the team particularly suited for this project. We can either train ourselves, but the project will take about two months, or we can hire an expert who can probably do that in a month, and this is going to cost roughly this much.

Of course, be very conservative with your estimates and think hard, if someone from outside of the company can do the project faster (green-field project with unknown tech) or slower (upgrading internal system, for which the original developer is still around, but busy earning money for the company)