'Undercutting' the going rate because you're new can work well though it is a contentious issue for many, and you risk those students telling all their friends that rate and if they then contact you in 6 or 12 months' time to have their work proofread, they will expect the same price.
I know some people who have struggled to raise their prices after starting low so can't say I would recommend it. Selling yourself 'cheap' can also backfire because it can make people question whether you're good enough.
I read this blog just the other day, and it may be worth you having a look -- it's not specifically related to proofreading but the principles are the same.http://www.leavingworkbehind.com/freela ... ing-rates/
If you are new to the work, charging on a par with others by the 1,000 words will make you seem competitive, even if they don't realise it's taking you twice as long. Yes, that means your hourly wage will be lower, but you'll be able to speed up and increase the hourly wage without having to increase the cost to the client.
One final thought -- if the head of a huge multi-million pound business came to you asking for your proofreading services, what would you charge? And if that person was your sister's friend, would you charge less?! Personally, I'd steer clear of mates rates, and to coin a Kwik Fit phrase, just go with great rates all the time. You have your price, stick with it.