The overall cost of home printing really depends on your location. In the USA it is possible to get really big discounts on printers, paper, and inksets. If you can afford to wait for the sales, and buy in bulk when they are on you could reduce the cost of home printing to match that of a good pro lab, for print sizes from 10×8 /A4 to 19×13/A3+. For small prints in bulk a lab will usually always be cheapest.
In the rest of the world it seems the potential discounts on home printing supplies, and printers, are much smaller. Usually the per print costa from a good print lab can be substantially less than for printing at home. I only have a relativly basic multifunction A4 Canon Pixma MG5150, I use Canon PT101 Pro Platinum paper and OEM inks. The quality is even so very good. The cost of an A4 print is around £2.50 for paper/ink. This is OK where I only need a single print doing. The pro lab I use though is much cheaper, and the quality is excellent. A 16×12 or A3 print on Fuji Crystal Archive C Type paper is only £1.15 including VAT. P&P is only £4.50 and is usually next day, if I have five prints made they work out cheaper per print than A4 does printed at home.
If my printer could print the larger size the cost would be around £4.50 for the roughly doubled area. That cost doesn't include anything to cover the cost of the printer. At £1 a print it would take around 400 prints to amortize a Pro 100 or 650 to cover the Pro 1, against single prints from the lab at a toatal of £5.65 each. If I did one print a week then it would take between eight and twelve years, depending on the printer, to amortize the cost of the printers. If I needed more than one print a week the lab just looks even better from a cost point of view, ordering once a week shouldn't be a problem.
Home printing is not usually the most cost effective way of printing, and with a good lab the quality will be high too. Really the only thing that home printing offers is the satsifaction of doing everything yourself. Even then there is a big difference in digital printing, where all the skill is in producing a great file to feed to the automated digital printing device, be it on your desk or in a lab, and working in the old, all manual, wet darkroom.
Alan