In terms of cost-effectiveness, it also depends whether you'd use the car more, doesn't it?
While we now know this doesn't quite fit with the OP's situation, if you live in a normal UK town then your local supermarket might be a 10-20 minute walk away. When you're at home on maternity leave it is easy to find yourself going to the shop every day, especially if you want to find a reason to get out and about every day (and don't live somewhere nice and rural with lovely walks to go on). A 10-20 minute walk, to me, is nicer than going in the car to the shop and yes, as some people have said, the baby might easily fall asleep in the pram. And then you have the pram to carry the shopping back, which is undoubtedly harder with a baby in a sling.
All of which waffle is to say that if you end up using the car a lot more then it might not always be cost effective. You're possibly making a massive saving anyway not paying for your journey to work, whether it's petrol or the train, but if you end up using the car every day I imagine that would be more expensive than a cheap pushchair in the long-term?
Also, I can't help thinking everyone has a different idea what a "pram" is. I know quite a few people who have managed without a lay-flat pram for the first few months as the baby is so tiny and did enjoy the sling more. But over a few months old, when they could go in a "pushchair", I think it would start to get harder. Ergo, no pram - fine, no pushchair - but more problematic...