I don't think there is a completely straightforward answer as it depends on a lot of factors:
- How much you print
- How often you print
- What size you print
- What quality you print
- What ink you buy
- What paper you use
- How long the printer lasts
- What you choose to print
For example, if you print a lot and often in large size and use discount-ink in high quality, chances are that - over time - it's cheaper than doing the same thing online.
However, it may be slightly lesser quality - which is only important if the difference matters to you.
If you print only occasionally, smaller sized prints, use top-of-the-line ink and choose to print drafts and stuff you know you aren't going to use anyway but just want to see, there is a chance it may be noticeably more expensive.
There are many stories that these printers - when not used often - have the nozzles caked with dried ink, and ink becomes less usable, for example. Having a printer also may alter behavior - since it's there, one might choose to print stuff that you wouldn't get printed online.
If you print often with a home printer, the acquisition-price of the printer itself is spread out over many prints, thus lowering the per-print total cost.
The other pat is availability of funds: you buy a printer you pay now - without a single print in your hand. Using print-services, you pay only what what you print.
It would be possible to plug all these factors into a spreadsheet and map out the per-print price based on volumes of prints over time.
But now that you already have a printer, it is perhaps less critical to know in such detail.
As a very wide generalization, I'd say that it's tradeoff: buying a printer and printing is - at first - more expensive, but you get the convenience and flexibility of printing whatever, whenever.
Using only print-services, you are perhaps more selective, possibly have higher quality, don't have to worry about maintenance of equipment and only have to pay-as-you-go. As such it's comparable to the math that goes into choosing a price-plan for your cell-phone.
My apologies if I've confused the subject more than clarifying it